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THE PHASE RULE AND ITS APPLICATIONS

by

ALEX. FINDLAY, M.A., PH.D., D.SC.

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THE PHASE RULE AND ITS APPLICATIONS

BY

ALEX. FINDLAY, M.A., PH.D., D.SC.

Lecturer on Physical Chemistry, University of Birmingham

With One Hundred and Thirty-Four Figures in the Text

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AND ADVICE



{vii}

PREFACE TO THE SECOND EDITION.

During the two years which have elapsed since the first edition of this
book appeared, the study of chemical equilibria has been prosecuted with
considerable activity, and valuable additions have been made to our
knowledge in several departments of this subject. In view of the scope of
the present work, it has been, of course, impossible to incorporate all
that has been done; but several new sections have been inserted, notably
those on the study of basic salts; the interpretation of cooling curves,
and the determination of the composition of solid phases without analysis;
the equilibria between iron, carbon monoxide, and carbon dioxide, which are
of importance in connection with the processes occurring in the blast
furnace; and the Phase Rule study of the ammonia-soda process. I have also
incorporated a short section on the reciprocal salt-pair barium
carbonate--potassium sulphate, which had been written for the German
edition of this book by the late Professor W. Meyerhoffer. The section on
the iron-carbon alloys, which in the first edition was somewhat
unsatisfactory, has been rewritten.

A. F.

_September, 1906._



{viii}

PREFACE

Although we are indebted to the late Professor Willard Gibbs for the first
enunciation of the Phase Rule, it was not till 1887 that its practical
applicability to the study of Chemical Equilibria was made apparent. In
that year Roozeboom disclosed the great generalization, which for upwards
of ten years had remained hidden and unknown save to a very few, by
stripping from it the garb of abstract Mathematics in which it had been
clothed by its first discoverer. The Phase Rule was thus made generally
accessible; and its adoption by Roozeboom as the basis of classification of
the different cases of chemical equilibrium then known established its
value, not only as a means of co-ordinating the large number of isolated
cases of equilibrium and of giving a deeper insight into the relationships
existing between the different systems, but also as a guide in the
investigation of unknown systems.

While the revelation of the principle embedded in the Phase Rule is
primarily due to Roozeboom, it should not be forgotten that, some years
previously, van't Hoff, in ignorance of the work of Willard Gibbs, had
enunciated his "law of the incompatibility of condensed systems," which in
some respects coincides with the Phase Rule; and it is only owing to the
more general applicability of the latter that the very {ix} important
generalization of van't Hoff has been somewhat lost sight of.

The exposition of the Phase Rule and its applications given in the
following pages has been made entirely non-mathematical, the desire having
been to explain as clearly as possible the principles underlying the Phase
Rule, and to illustrate their application to the classification and
investigation of equilibria, by means of a number of cases actually
studied. While it has been sought to make the treatment sufficiently
elementary to be understood by the student just commencing the study of
chemical equilibria, an attempt has been made to advance his knowledge to
such a stage as to enable him to study with profit the larger works on the
subject, and to follow with intelligence the course of investigation in
this department of Physical Chemistry. It is also hoped that the volume may
be of use, not only to the student of Physical Chemistry, or of the other
branches of that science, but also to the student of Metallurgy and of
Geology, for whom an acquaintance with at least the principles of the Phase
Rule is becoming increasingly important.

In writing the following account of the Phase Rule, it is scarcely
necessary to say that I have been greatly indebted to the larger works on
Chemical Equilibria by Ostwald ("Lehrbuch"), Roozeboom ("Die Heterogenen
Gleichgewichte"), and Bancroft ("The Phase Rule"); and in the case of the
first-named, to the inspiration also of personal teaching. My indebtedness
to these and other authors I have indicated in the following pages.

In conclusion, I would express my thanks to Sir William Ramsay, whose
guidance and counsel have been constantly {x} at my disposal; and to my
colleagues, Dr. T. Slater Price and Dr. A. McKenzie, for their friendly
criticism and advice. To Messrs. J. N. Friend, M.Sc., and W. E. S. Turner,
B.Sc., I am also indebted for their assistance in reading the proof-sheets.

A. F.

_November, 1903._




{xi}

CONTENTS

                                                                       PAGE

  CHAPTER I

  INTRODUCTION                                                            1

  General, I. Homogeneous and heterogeneous equilibrium,
  5. Real and apparent equilibrium, 5.

  CHAPTER II

  THE PHASE RULE                                                          7

  Phases, 8. Components, 10. Degree of freedom. Variability
  of a system, 14. The Phase Rule, 16. Classification of systems
  according to the Phase Rule, 17. Deduction of the Phase
  Rule, 18.

  CHAPTER III

  TYPICAL SYSTEMS OF ONE COMPONENT                                       21

  A. _Water._ Equilibrium between liquid and vapour. Vaporization
  curve, 21. Upper limit of vaporization curve, 23.
  Sublimation curve of ice, 24. Equilibrium between ice and
  water. Curve of fusion, 25. Equilibrium between ice, water,
  and vapour. The triple point, 27. Bivariant systems of water,
  29. Supercooled water. Metastable state, 30. Other systems
  of the substance water, 32. B. _Sulphur_, 33. Polymorphism, 33.
  Sulphur, 34. Triple point--Rhombic and monoclinic sulphur
  and vapour. Transition point, 34. Condensed systems, 36.
  Suspended transformation, 37. Transition curve--Rhombic
  and monoclinic sulphur, 37. Triple point--Monoclinic sulphur,
  liquid, and vapour. Melting point of monoclinic sulphur, 38.
  Triple point--Rhombic and monoclinic sulphur and liquid, 38.
  Triple point--Rhombic sulphur, liquid, and vapour. Metastable
  triple point, 38. Fusion curve of rhombic sulphur, 39.
  Bivariant systems, 39. C. _Tin_, 41. Transition point, 41.
  {xii}
  Enantiotropy and monotropy, 44. D. _Phosphorus_, 46. Enantiotropy
  combined with monotropy, 51. E. _Liquid Crystals_, 51.
  Phenomena observed, 51. Nature of liquid crystals, 52. Equilibrium
  relations in the case of liquid crystals, 53.

  CHAPTER IV

  GENERAL SUMMARY                                                        55

  Triple point, 55. Theorems of van't Hoff and of Le Chatelier,
  57. Changes at the triple point, 58. Triple point solid--solid--vapour,
  62. Sublimation and vaporization curves,
  63. Fusion curve--Transition curve, 66. Suspended transformation.
  Metastable equilibria, 69. Velocity of transformation,
  70. Law of successive reactions, 73.

  CHAPTER V

  SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION                   76

  Different systems of two components, 77. PHENOMENA OF
  DISSOCIATION. Bivariant systems, 79. Univariant systems,
  80. Ammonia compounds of metal chlorides, 82. Salts with
  water of crystallization, 85. Efflorescence, 86. Indefiniteness
  of the vapour pressure of a hydrate, 87. Suspended transformation,
  89. Range of existence of hydrates, 90. Constancy
  of vapour pressure and the formation of compounds, 90.
  Measurement of the vapour pressure of hydrates, 91.

  CHAPTER VI

  SOLUTIONS                                                              92

  Definition, 92. SOLUTIONS OF GASES IN LIQUIDS, 93.
  SOLUTIONS OF LIQUIDS IN LIQUIDS, 95. Partial or limited
  miscibility, 96. Phenol and water, 97. Methylethylketone
  and water, 100. Triethylamine and water, 101. General form
  of concentration-temperature curve, 101. Pressure-concentration
  diagram, 102. Complete miscibility, 104. Pressure-concentration
  diagram, 104.

  CHAPTER VII

  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
  VOLATILE                                                              106

  General, 106. The saturated solution, 108. Form of the
  solubility curve, 108. A. ANHYDROUS SALT AND WATER.
  {xiii}
  The solubility curve, 111. Suspended transformation and
  supersaturation, 113. Solubility curve at higher temperatures,
  114. (1) _Complete miscibility of the fused components._ Ice as
  solid phase, 116. Cryohydrates, 117. Changes at the quadruple
  point, 119. Freezing mixtures, 120. (2) _Partial miscibility of
  the fused components._ Supersaturation, 124. Pressure-temperature
  diagram, 126. Vapour pressure of solid--solution--vapour,
  126. Other univariant systems, 127. Bivariant systems, 129.
  Deliquescence, 130. Separation of salt on evaporation, 130.
  General summary, 131.

  CHAPTER VIII

  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
  VOLATILE                                                              133

  B. HYDRATED SALT AND WATER, (1) _The compounds
  formed do not have a definite melting point._ Concentration-temperature
  diagram, 133. Sodium sulphate and water, 134.
  Suspended transformation, 137. Dehydration by means of
  anhydrous sodium sulphate, 138. Pressure-temperature diagram,
  138. (2) _The compounds formed have a definite melting point._
  Solubility curve of calcium chloride hexahydrate, 145.
  Pressure-temperature diagram, 149. The indifferent point, 150.
  The hydrates of ferric chloride, 151. Suspended transformation,
  155. Evaporation of solutions at constant temperature, 155.
  Inevaporable solutions, 157. Illustration, 158.

  CHAPTER IX

  EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS                            161

  General, 161. Iodine and chlorine, 161. Concentration-temperature
  diagram, 162. Pressure-temperature diagram, 165.
  Bivariant systems, 167. Sulphur dioxide and water, 169.
  Pressure-temperature diagram, 170. Bivariant systems, 173.

  CHAPTER X

  SOLID SOLUTIONS. MIXED CRYSTALS                                       175

  General, 175. Solution of gases in solids, 176. Palladium
  and hydrogen, 178. Solutions of solids in solids. Mixed
  crystals, 180. Formation of mixed crystals of isomorphous
  substances, 182. I. The two components can form an unbroken
  series of mixed crystals. (_a_) _The freezing points of all mixtures
  lie between the freezing points of the pure components._ Examples,
  183. Melting-point curve, 183. (_b_) _The freezing-point curve passes
  through a maximum._ Example, 186. (_c_) _The freezing-point
  curve passes through a minimum._ Example, 188. Fractional
  {xiv}
  crystallization of mixed crystals, 188. II. The two components
  do not form a continuous series of mixed crystals. (_a_) _The
  freezing-point curve exhibits a transition point_, 190. Example,
  190. (_b_) _The freezing-point curve exhibits a eutectic point_, 191.
  Examples, 192. Changes in mixed crystals with the temperature,
  192.

  CHAPTER XI

  EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES                                195

  Temperature-concentration diagram, 196. Transformation
  of the unstable into the stable form, 201. Examples, 203.
  _Benzaldoximes_, 203. _Acetaldehyde and paraldehyde_, 204.

  CHAPTER XII

  SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF
  TWO COMPONENTS                                                        207

  Summary of the different systems of two components, 208.
  (1) _Organic compounds_, 212. (2) _Optically active substances_,
  213. Examples, 216. Transformations, 217. (3) _Alloys_, 220.
  Iron--carbon alloys, 223. Determination of the composition of
  compounds without analysis, 228. Formation of minerals, 232.

  CHAPTER XIII

  SYSTEMS OF THREE COMPONENTS                                           234

  General, 234. Graphic representation, 235.

  CHAPTER XIV

  SOLUTIONS OF LIQUIDS IN LIQUIDS                                       240

  1. _The three components form only one pair of partially
  miscible liquids_, 240. Retrograde solubility, 245. The influence
  of temperature, 247. 2. _The three components can form two
  pairs of partially miscible liquids_, 249. 3. _The three components
  form three pairs of partially miscible liquids_, 251.

  CHAPTER XV

  PRESENCE OF SOLID PHASES                                              253

  A. The ternary eutectic point, 253. Formation of compounds,
  255. B. Equilibria at higher temperatures. Formation
  of double salts, 258. Transition point, 258. Vapour pressure.
  {xv}
  Quintuple point, 261. Solubility curves at the transition point,
  264. Decomposition of the double salt by water, 267. Transition
  interval, 270. Summary, 271.

  CHAPTER XVI

  ISOTHERMAL CURVES AND THE SPACE MODEL                                 272

  Non-formation of double salts, 272. Formation of double
  salt, 273. Transition interval, 277. Isothermal evaporation,
  278. Crystallization of double salt from solutions containing
  excess of one component, 280. Formation of mixed crystals,
  281. Application to the characterization of racemates, 282.
  _Representation in space._ Space model for carnallite, 284.
  Summary and numerical data, 287. Ferric chloride--hydrogen
  chloride--water, 290. Ternary systems, 291. The isothermal
  curves, 294. Basic Salts, 296. Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O, 298.
  Basic mercury salts, 301. Indirect determination of the composition
  of the solid phase, 302.

  CHAPTER XVII

  ABSENCE OF LIQUID PHASE                                               305

  Iron, carbon monoxide, carbon dioxide, 305.

  CHAPTER XVIII

  SYSTEMS OF FOUR COMPONENTS                                            312

  Reciprocal salt-pairs. Choice of components, 313. Transition
  point, 314. Formation of double salts, 315. Transition
  interval, 315. Graphic representation, 316. Example, 317.
  Ammonia-soda process, 320. Preparation of barium nitrite, 327.
  Barium carbonate and potassium sulphate, 328.

  APPENDIX

  EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT                    331

  I. The dilatometric method, 331. II. Measurement of
  the vapour pressure, 334. III. Solubility measurements, 335.
  IV. Thermometric method, 337. V. Optical method, 338.
  VI. Electrical methods, 338.

  NAME INDEX                                                            341

  SUBJECT INDEX                                                         345

       *       *       *       *       *


{1}

THE PHASE RULE

CHAPTER I

INTRODUCTION

General.--Before proceeding to the more systematic treatment of the Phase
Rule, it may, perhaps, be not amiss to give first a brief forecast of the
nature of the subject we are about to study, in order that we may gain some
idea of what the Phase Rule is, of the kind of problem which it enables us
to solve, and of the scope of its application.

It has long been known that if water is placed in a closed, exhausted
space, vapour is given off and a certain pressure is created in the
enclosing vessel. Thus, when water is placed in the Torricellian vacuum of
the barometer, the mercury is depressed, and the amount of depression
increases as the temperature is raised. But, although the pressure of the
vapour increases as the temperature rises, its value at any given
temperature is constant, no matter whether the amount of water present or
the volume of the vapour is great or small; if the pressure on the vapour
is altered while the temperature is maintained constant, either the water
or the vapour will ultimately disappear; the former by evaporation, the
latter by condensation. At any given temperature within certain limits,
therefore, water and vapour can exist permanently in contact with one
another--or, as it is said, be in equilibrium with one another--only when
the pressure has a certain definite value. The same law of constancy of
vapour pressure at a given {2} temperature, quite irrespective of the
volumes of liquid and vapour,[1] holds good also in the case of alcohol,
ether, benzene, and other pure liquids. It is, therefore, not unnatural to
ask the question, Does it hold good for all liquids? Is it valid, for
example, in the case of solutions?

We can find the answer to these questions by studying the behaviour of a
solution--say, a solution of common salt in water--when placed in the
Torricellian vacuum. In this case, also, it is observed that the pressure
of the vapour increases as the temperature is raised, but the pressure is
no longer independent of the volume; as the volume increases, the pressure
slowly diminishes. If, however, solid salt is present in contact with the
solution, then the pressure again becomes constant at constant temperature,
even when the volume of the vapour is altered. As we see, therefore,
solutions do not behave in the same way as pure liquids.

Moreover, on lowering the temperature of water, a point is reached at which
ice begins to separate out; and if heat be now added to the system or
withdrawn from it, no change will take place in the temperature or vapour
pressure of the latter until either the ice or the water has
disappeared.[2] Ice, water, and vapour, therefore, can be in equilibrium
with one another only at one definite temperature and one definite
pressure.

In the case of a solution of common salt, however, we may have ice in
contact with the solution at different temperatures and pressures. Further,
it is possible to have a solution in equilibrium not only with anhydrous
salt (NaCl), but also with the hydrated salt (NaCl, 2H_{2}O), as well as
with ice, and the question, therefore, arises: Is it possible to state in a
general manner the conditions under which such different systems can exist
in equilibrium; or to obtain some insight {3} into the relations which
exist between pure liquids and solutions? As we shall learn, the Phase Rule
enables us to give an answer to this question.

The preceding examples belong to the class of so-called "physical"
equilibria, or equilibria depending on changes in the physical state. More
than a hundred years ago, however, it was shown by Wenzel and Berthollet
that "chemical" equilibria can also exist; that chemical reactions do not
always take place completely in one direction as indicated by the usual
chemical equation, but that before the reacting substances are all used up
the reaction ceases, and there is a condition of equilibrium between the
reacting substances and the products of reaction. As an example of this,
there may be taken the process of lime-burning, which depends on the fact
that when calcium carbonate is heated, carbon dioxide is given off and
quicklime is produced. If the carbonate is heated in a closed vessel it
will be found, however, not to undergo entire decomposition. When the
pressure of the carbon dioxide reaches a certain value (which is found to
depend on the temperature), decomposition ceases, and calcium carbonate
exists side by side with calcium oxide and carbon dioxide. Moreover, at any
given temperature the pressure is constant and independent of the amount of
carbonate or oxide present, or of the volume of the gas; _nor does the
addition of either of the products of dissociation, carbon dioxide or
calcium oxide, cause any change in the equilibrium_. Here, then, we see
that, although there are three different substances present, and although
the equilibrium is no longer due to physical, but to chemical change, it
nevertheless obeys the same law as the vapour pressure of a pure volatile
liquid, such as water.

It might be supposed, now, that this behaviour would be shown by other
dissociating substances, _e.g._ ammonium chloride. When this substance is
heated it dissociates into ammonia and hydrogen chloride, and at any given
temperature the pressure of these gases is constant,[3] and is independent
of the amounts of solid and gas present. So far, therefore, ammonium
chloride behaves like calcium carbonate. If, however, one of the {4}
products of dissociation be added to the system, it is found that the
pressure is no longer constant at a given temperature, but varies with the
amount of gas, ammonia or hydrogen chloride, which is added. In the case of
certain dissociating substances, therefore, addition of one of the products
of dissociation alters the equilibrium, while in other cases it does not.
With the help of the Phase Rule, however, a general interpretation of this
difference of behaviour can be given--an interpretation which can be
applied not only to the two cases cited, but to all cases of dissociation.

Again, it is well known that sulphur exists in two different crystalline
forms, octahedral and prismatic, each of which melts at a different
temperature. The problem here is, therefore, more complicated than in the
case of ice, for there is now a possibility not only of one solid form, but
of two different forms of the same substance existing in contact with
liquid. What are the conditions under which these two forms can exist in
contact with liquid, either singly or together, and under what conditions
can the two solid forms exist together without the presence of liquid
sulphur? To these questions an answer can also be given with the help of
the Phase Rule.

These cases are, however, comparatively simple; but when we come, for
instance, to study the conditions under which solutions are formed, and
especially when we inquire into the solubility relations of salts capable
of forming, perhaps, a series of crystalline hydrates; and when we seek to
determine the conditions under which these different forms can exist in
contact with the solution, the problem becomes more complicated, and the
necessity of some general guide to the elucidation of the behaviour of
these different systems becomes more urgent.

It is, now, to the study of such physical and chemical equilibria as those
above-mentioned that the Phase Rule finds application; to the study, also,
of the conditions regulating, for example, the formation of alloys from
mixtures of the fused metals, or of the various salts of the Stassfurt
deposits; the behaviour of iron and carbon in the formation of steel and
the {5} separation of different minerals from a fused rock-mass.[4] With
the help of the Phase Rule we can group together into classes the large
number of different isolated cases of systems in equilibrium; with its aid
we are able to state, in a general manner at least, the conditions under
which a system can be in equilibrium, and by its means we can gain some
insight into the relations existing between different kinds of systems.

Homogeneous and Heterogeneous Equilibrium.--Before passing to the
consideration of this generalization, it will be well to first make mention
of certain restrictions which must be placed on its treatment, and also of
the limitations to which it is subject. If a system is uniform throughout
its whole extent, and possesses in every part identical physical properties
and chemical composition, it is called _homogeneous_. Such is, for example,
a solution of sodium chloride in water. An equilibrium occurring in such a
homogeneous system (such as the equilibrium occurring in the formation of
an ester in alcoholic solution) is called _homogeneous equilibrium_. If,
however, the system consists of parts which have different physical
properties, perhaps also different chemical properties, and which are
marked off and separated from one another by bounding surfaces, the system
is said to be _heterogeneous_. Such a system is formed by ice, water, and
vapour, in which the three portions, each in itself homogeneous, can be
mechanically separated from one another. When equilibrium exists between
different, physically distinct parts, it is known as _heterogeneous
equilibrium_. It is, now, with heterogeneous equilibria, with the
conditions under which a heterogeneous system can exist, that we shall deal
here.

Further, we shall not take into account changes of equilibrium due to the
action of electrical, magnetic, or capillary forces, or of gravity; but
shall discuss only those which are due to changes of pressure, temperature,
and volume (or concentration).

Real and Apparent Equilibrium.--In discussing equilibria, also, a
distinction must be drawn between real and {6} apparent equilibria. In the
former case there is a state of rest which undergoes continuous change with
change of the conditions (_e.g._ change of temperature or of pressure), and
for which the chief criterion is that _the same condition of equilibrium is
reached from whichever side it is approached_. Thus in the case of a
solution, if the temperature is maintained constant, the same concentration
will be obtained, no matter whether we start with an unsaturated solution
to which we add more solid, or with a supersaturated solution from which we
allow solid to crystallize out; or, in the case of water in contact with
vapour, the same vapour pressure will be obtained, no matter whether we
heat the water up to the given temperature or cool it down from a higher
temperature. In this case, water and vapour are in _real_ equilibrium. On
the other hand, water in contact with hydrogen and oxygen at the ordinary
temperature is a case only of _apparent_ equilibrium; on changing the
pressure and temperature continuously within certain limits there is no
continuous change observed in the relative amounts of the two gases. On
heating beyond these limits there is a sudden and not a continuous change,
and the system no longer regains its former condition on being cooled to
the ordinary temperature. In all such cases the system may be regarded as
undergoing change and as tending towards a state of true or real
equilibrium, but with such slowness that no change is observed.

Although the case of water in contact with hydrogen and oxygen is an
extreme one, it must be borne in mind that the condition of true
equilibrium may not be reached instantaneously or even with measurable
velocity, and in all cases it is necessary to be on one's guard against
mistaking apparent (or false) for real (or true) equilibrium. The
importance of this will be fully illustrated in the sequel.

       *       *       *       *       *


{7}

CHAPTER II

THE PHASE RULE

Although the fact that chemical reactions do not take place completely in
one direction, but proceed only to a certain point and there make a halt,
was known in the last quarter of the eighteenth century (Wenzel, 1777;
Berthollet, 1799); and although the opening and subsequent decades of the
following century brought many further examples of such equilibria to our
knowledge, it was not until the last quarter of the nineteenth century that
a theorem, general in its application and with foundations weakened by no
hypothetical assumptions as to the nature or constitution of matter, was
put forward by Willard Gibbs;[5] a generalization which serves at once as a
golden rule by which the condition of equilibrium of a system can be
tested, and as a guide to the similarities and dissimilarities existing in
different systems.

Before that time, certainly, attempts had been made to bring the different
known cases of equilibria--chemical and physical--under general laws. From
the very first, both Wenzel[6] and Berthollet[7] recognized the influence
exercised by the _mass_ of the substances on the equilibrium of the system.
It was reserved, however, for Guldberg and Waage, by their more general
statement and mathematical treatment of the Law of Mass Action,[8] to
inaugurate the period of quantitative study of equilibria. The law which
these investigators enunciated {8} served satisfactorily to summarize the
conditions of equilibrium in many cases both of homogeneous and, with the
help of certain assumptions and additions, of heterogeneous equilibrium. By
reason, however, of the fact that it was developed on the basis of the
kinetic and molecular theories, and involved, therefore, certain
hypothetical assumptions as to the nature and condition of the substances
taking part in the equilibrium, the law of mass action failed, as it
necessarily must, when applied to those systems in which neither the number
of different molecular aggregates nor the degree of their molecular
complexity was known.

Ten years after the law of mass action was propounded by Guldberg and
Waage, Willard Gibbs,[9] Professor of Physics in Yale University, showed
how, in a perfectly general manner, free from all hypothetical assumptions
as to the molecular condition of the participating substances, all cases of
equilibrium could be surveyed and grouped into classes, and how
similarities in the behaviour of apparently different kinds of systems, and
differences in apparently similar systems, could be explained.

As the basis of his theory of equilibria, Gibbs adopted the laws of
thermodynamics,[10] a method of treatment which had first been employed by
Horstmann.[11] In deducing the law of equilibrium, Gibbs regarded a system
as possessing only three independently variable factors[12]--temperature,
pressure, and the concentration of the components of the system--and he
enunciated the general theorem now usually known as the _Phase Rule_, by
which he defined the conditions of equilibrium as a relationship between
the number of what are called the phases and the components of the system.

Phases.--Before proceeding farther we shall first consider what exactly is
meant by the terms _phase_ and _component_. We have already seen (p. 5)
that a heterogeneous system is made {9} up of different portions, each in
itself homogeneous, but marked off in space and separated from the other
portions by bounding surfaces. These homogeneous, physically distinct and
mechanically separable portions are called _phases_. Thus ice, water, and
vapour, are three phases of the same chemical substance--water. A phase,
however, whilst it must be physically and chemically homogeneous, need not
necessarily be chemically simple. Thus, a gaseous mixture or a solution may
form a phase; but a heterogeneous mixture of solid substances constitutes
as many phases as there are substances present. Thus when calcium carbonate
dissociates under the influence of heat, calcium oxide and carbon dioxide
are formed. There are then _two_ solid phases present, viz. calcium
carbonate and oxide, and one gas phase, carbon dioxide.

The _number of phases_ which can exist side by side may vary greatly in
different systems. In all cases, however, there can be but one gas or
vapour phase on the account of the fact that all gases are miscible with
one another in all proportions. In the case of liquid and solid phases the
number is indefinite, since the above property does not apply to them. The
number of phases which can be formed by any given substance or group of
substances also differs greatly, and in general increases with the number
of participating substances. Even in the case of a single substance,
however, the number may be considerable; in the case of sulphur, for
example, at least eight different solid phases are known (_v._ Chap. III.).

It is of importance to bear in mind that equilibrium is _independent of the
amounts_ of the phases present.[13] Thus it is a familiar fact that the
pressure of a vapour in contact with a {10} liquid (_i.e._ the pressure of
the saturated vapour) is unaffected by the amounts, whether relative or
absolute, of the liquid and vapour; also the amount of a substance
dissolved by a liquid is independent of the amount of solid in contact with
the solution. It is true that deviations from this general law occur when
the amount of liquid or the size of the solid particles is reduced beyond a
certain point,[14] owing to the influence of surface energy; but we have
already (p. 5) excluded such cases from consideration.

Components.--Although the conception of phases is one which is readily
understood, somewhat greater difficulty is experienced when we come to
consider what is meant by the term _component_; for the components of a
system are not synonymous with the chemical elements or compounds present,
_i.e._ with the _constituents_ of the system, although both elements and
compounds may be components. By the latter term there are meant only those
constituents the concentration of which can undergo _independent_ variation
in the different phases, and it is only with these that we are concerned
here.[15]

To understand the meaning of this term we shall consider briefly some cases
with which the reader will be familiar, and at the outset it must be
emphasized that the Phase Rule is concerned merely with those constituents
which take part in the state of real equilibrium (p. 5); for it is only to
the final state, not to the processes by which that state is reached, that
the Phase Rule applies.

Consider now the case of the system water--vapour or ice--water--vapour.
The number of constituents taking part in the equilibrium here is only one,
viz. the chemical substance, water. Hydrogen and oxygen, the constituents
of water, are not to be regarded as components, because, in the first
place, they are {11} not present in the system in a state of real
equilibrium (p. 6); in the second place, they are combined in definite
proportions to form water, and their amounts, therefore, cannot be varied
independently. A variation in the amount of hydrogen necessitates a
definite variation in the amount of oxygen.

In the case, already referred to, in which hydrogen and oxygen are present
along with water at the ordinary temperature, we are not dealing with a
condition of true equilibrium. If, however, the temperature is raised to a
certain point, a state of true equilibrium between hydrogen, oxygen, and
water-vapour will be possible. In this case hydrogen and oxygen will be
components, because now they do take part in the equilibrium; also, they
need no longer be present in definite proportions, but excess of one or the
other may be added. Of course, if the restriction be arbitrarily made that
the free hydrogen and oxygen shall be present always and only in the
proportions in which they are combined to form water, there will be, as
before, only one component, water. From this, then, we see that a change in
the conditions of the experiment (in the present case a rise of
temperature) may necessitate a change in the number of the components.

It is, however, only in the case of systems of more than one component that
any difficulty will be found; for only in this case will a choice of
components be possible. Take, for instance, the dissociation of calcium
carbonate into calcium oxide and carbon dioxide. At each temperature, as we
have seen, there is a definite state of equilibrium. When equilibrium has
been established, there are three different substances present--calcium
carbonate, calcium oxide, and carbon dioxide; and these are the
constituents of the system between which equilibrium exists. Now, although
these constituents take part in the equilibrium, they are not all to be
regarded as components, for they are not mutually independent. On the
contrary, the different phases are related to one another, and if two of
these are taken, the composition of the third is defined by the equation

  CaCO_{3} = CaO + CO_{2}

{12} Now, in deciding the number of components in any given system, not
only must the constituents chosen be capable of independent variation, but
a further restriction is imposed, and we obtain the following rule: _As the
components of a system there are to be chosen the_ smallest number _of
independently variable constituents by means of which the composition of
each phase participating in the state of equilibrium can be expressed in
the form of a chemical equation._

Applying this rule to the case under consideration, we see that of the
three constituents present when the system is in a state of equilibrium,
only two, as already stated, are independently variable. It will further be
seen that in order to express the composition of each phase present, two of
these constituents are necessary. The system is, therefore, one of _two
components_, or a system of the second order.

When, now, we proceed to the actual choice of components, it is evident
that any two of the constituents can be selected. Thus, if we choose as
components CaCO_{3} and CaO, the composition of each phase can be expressed
by the following equations:--

  CaCO_{3} = CaCO_{3} + 0CaO
       CaO = CaO + 0CaCO_{3}
    CO_{2} = CaCO_{3} - CaO

As we see, then, both zero and negative quantities of the components have
been introduced; and similar expressions would be obtained if CaCO_{3} and
CO_{2} were chosen as components. The matter can, however, be simplified
and the use of negative quantities avoided if CaO and CO_{2} are chosen;
and it is, therefore, customary to select these as the components.

While it is possible in the case of systems of the second order to choose
the two components in such a way that the composition of each phase can be
expressed by positive quantities of these, such a choice is not always
possible when dealing with systems of a higher order (containing three or
four components).

From the example which has just been discussed, it might {13} appear as if
the choice of the components was rather arbitrary. On examining the point,
however, it will be seen that the arbitrariness affects only the _nature_,
not the _number_, of the components; a choice could be made with respect to
which, not to how many, constituents were to be regarded as components. As
we shall see presently, however, it is only the number, not the nature of
the components that is of importance.

After the discussion of the conditions which the substances chosen as
components must satisfy, another method may be given by which the number of
components present in a system can be determined. Suppose a system
consisting of several phases in equilibrium, and the composition of each
phase determined by analysis. If each phase present, regarded as a whole,
has the same composition, the system contains only one component, or is of
the first order. If two phases must be mixed in suitable quantities in
order that the composition of a third phase may be obtained, the system is
one of two components or of the second order; and if three phases are
necessary to give the composition of a fourth coexisting phase, the system
is one of three components, or of the third order.[16]

Although the examples to be considered in the sequel will afford sufficient
illustration of the application of the rules given above, one case may
perhaps be discussed to show the application of the method just given for
determining the number of components.

Consider the system consisting of Glauber's salt in equilibrium with
solution and vapour. If these three phases are analyzed, the composition of
the solid will be expressed by Na_{2}SO_{4}, 10H_{2}O; that of the solution
by Na_{2}SO_{4} + _x_H_{2}O, while the vapour phase will be H_{2}O. The
system evidently cannot be a one-component system, for the phases have not
all the same composition. By varying the amounts of two phases, however
(_e.g._ Na_{2}SO_{4}, 10H_{2}O and H_{2}O), the composition of the third
phase--the solution--can be obtained. The system is, therefore, one of _two
components_.

But sodium sulphate can also exist in the anhydrous form and as the hydrate
Na_{2}SO_{4}, 7H_{2}O. In these cases there may {14} be chosen as
components Na_{2}SO_{4} and H_{2}O, and Na_{2}SO_{4}, 7H_{2}O and H_{2}O
respectively. In both cases, therefore, there are two components. But the
two systems (Na_{2}SO_{4}, 10H_{2}O--H_{2}O, and Na_{2}SO_{4},
7H_{2}O--H_{2}O) can be regarded as special cases of the system
Na_{2}SO_{4}--H_{2}O, and these two components will apply to all systems
made up of sodium sulphate and water, no matter whether the solid phase is
anhydrous salt or one of the hydrates. In all three cases, of course, the
_number_ of components is the same; but by choosing Na_{2}SO_{4} and H_{2}O
as components, the possible occurrence of negative quantities of components
in expressing the composition of the phases is avoided; and, further, these
components apply over a much larger range of experimental conditions.
Again, therefore, we see that, although the number of the components of a
system is definite, a certain amount of liberty is allowed in the choice of
the substances; and we also see that the choice will be influenced by the
conditions of experiment.

Summing up, now, we may say--

(1) The components are to be chosen from among the constituents which are
present when the system is in a state of true equilibrium, and which take
part in that equilibrium.

(2) As components are to be chosen the _smallest number_ of such
constituents necessary to express the composition of each phase
participating in the equilibrium, zero and negative quantities of the
components being permissible.

(3) In any given system the _number_ of the components is definite, but may
alter with alteration of the conditions of experiment. A certain freedom of
choice, however, is allowed in the (qualitative, not quantitative)
selection of the components, the choice being influenced by considerations
of simplicity, suitability, or generality of application.[17]

Degree of Freedom. Variability of a System.--It is well known that in
dealing with a certain mass of gas or vapour, _e.g._ water vapour, if only
one of the independently variable factors--temperature, pressure, and
concentration (or volume)--is fixed, the state of the gas or vapour is
undefined; while occupying the same volume (the concentration, therefore,
remaining {15} unchanged), the temperature and the pressure may be altered;
at a given temperature, a gas can exist under different pressures and
occupy different volumes, and under any given pressure the temperature and
volume may vary. If, however, two of the factors are arbitrarily fixed,
then the third factor can only have a certain definite value; at any given
values of temperature and pressure a given mass of gas can occupy only a
definite volume.

Suppose, however, that the system consists of water in contact with vapour.
The condition of the system then becomes perfectly defined on arbitrarily
giving one of the variables a certain value. If the temperature is fixed,
the pressure under which water and water vapour can coexist is also
determined; and conversely, if a definite pressure is chosen, the
temperature is also defined. Water and vapour can coexist under a given
pressure only at a definite temperature.

Finally, let the water and vapour be cooled down until ice begins to
separate out. So soon as the third phase, ice, appears, the state of the
system as regards temperature and pressure of the vapour is perfectly
defined, and none of the variables can be arbitrarily changed without
causing the disappearance of one of the phases, ice, water, or vapour.

We see, therefore, that in the case of some systems two, in other cases,
only one of the independent variables (temperature, pressure,
concentration) can be altered without destroying the nature of the system;
while in other systems, again, these variables have all fixed and definite
values. We shall therefore define the number of degrees of freedom[18] of a
system as the _number of the variable factors, temperature, pressure, and
concentration of the components, which must be arbitrarily fixed in order
that the condition of the system may be perfectly defined_. From what has
been said, therefore, we shall describe a gas or vapour as having two
degrees of freedom; the system water--vapour as having only one; and the
system ice--water--vapour as having no degrees of freedom. We may also
speak of the {16} _variability_ or _variance_ of a system, and describe a
system as being invariant, univariant, bivariant, multivariant,[19]
according as the number of degrees of freedom is nought, one, two, or more
than two.

A knowledge of its variability is, therefore, of essential importance in
studying the condition and behaviour of a system, and it is the great merit
of the Phase Rule that _the state of a system is defined entirely by the
relation existing between the number of the components and the phases
present_, no account being taken of the molecular complexity of the
participating substances, nor any assumption made with regard to the
constitution of matter. It is, further, as we see, quite immaterial whether
we are dealing with "physical" or "chemical" equilibrium; in principle,
indeed, no distinction need be drawn between the two classes, although it
is nevertheless often convenient to make use of the terms, in spite of a
certain amount of indefiniteness which attaches to them--an indefiniteness,
indeed, which attaches equally to the terms "physical" and "chemical"
process.[20]

The Phase Rule.--The Phase Rule of Gibbs, which defines the condition of
equilibrium by the relation between the number of coexisting phases and the
components, may be stated as follows: A system consisting of n components
can exist in _n_ + 2 phases only when the temperature, pressure, and
concentration have fixed and definite values; if there are _n_ components
in _n_ + 1 phases, equilibrium can exist while one of the factors varies,
and if there are only _n_ phases, two of the varying factors may be
arbitrarily fixed. This rule, the application of which, it is hoped, will
become clear in the sequel, may be very concisely and conveniently
summarized in the form of the equation--

  P + F = C + 2, or F = C + 2 - P

where P denotes the number of the phases, F the degrees of freedom, and C
the number of components. From the second form of the equation it can be
readily seen that the greater the number of the phases, the fewer are the
degrees of freedom. With increase in the number of the phases, therefore,
the {17} condition of the system becomes more and more defined, or less and
less variable.

Classification of Systems according to the Phase Rule.--We have already
learned in the introductory chapter that systems which are apparently quite
different in character may behave in a very similar manner. Thus it was
stated that the laws which govern the equilibrium between water and its
vapour are quite analogous to those which are obeyed by the dissociation of
calcium carbonate into carbon dioxide and calcium oxide; in each case a
certain temperature is associated with a definite pressure, no matter what
the relative or absolute amounts of the respective substances are. And
other examples were given of systems which were apparently similar in
character, but which nevertheless behaved in a different manner. The
relations between the various systems, however, become perfectly clear and
intelligible in the light of the Phase Rule. In the case first mentioned,
that of water in equilibrium with its vapour, we have one
component--water--present in two phases, _i.e._ in two physically distinct
forms, viz. liquid and vapour. According to the Phase Rule, therefore,
since C = 1, and P = 2, the degree of freedom F is equal to 1 + 2 - 2 = 1;
the system possesses one degree of freedom, as has already been stated. But
in the case of the second system mentioned above there are two components,
viz. calcium oxide and carbon dioxide (p. 12), and three phases, viz. two
solid phases, CaO and CaCO_{3}, and the gaseous phase, CO_{2}. The number
of degrees of freedom of the system, therefore, is 2 + 2 - 3 = 1; this
system, therefore, also possesses one degree of freedom. We can now
understand why these two systems behave in a similar manner; both are
univariant, or possess only one degree of freedom. We shall therefore
expect a similar behaviour in the case of all univariant systems, no matter
how dissimilar the systems may outwardly appear. Similarly, all bivariant
systems will exhibit analogous behaviour; and generally, systems possessing
the same degree of freedom will show a like behaviour. In accordance with
the Phase Rule, therefore, we may classify the different systems which may
be found into invariant, univariant, bivariant, multivariant, {18}
according to the relation which obtains between the number of the
components and the number of coexisting phases; and we shall expect that in
each case the members of any particular group will exhibit a uniform
behaviour. By this means we are enabled to obtain an insight into the
general behaviour of any system, so soon as we have determined the number
of the components and the number of the coexisting phases.

The adoption of the Phase Rule for the purposes of classification has been
of great importance in studying changes in the equilibrium existing between
different substances; for not only does it render possible the grouping
together of a large number of isolated phenomena, but the guidance it
affords has led to the discovery of new substances, has given the clue to
the conditions under which these substances can exist, and has led to the
recognition of otherwise unobserved resemblances existing between different
systems.

Deduction of the Phase Rule.--In the preceding pages we have restricted
ourselves to the statement of the Phase Rule, without giving any indication
of how it has been deduced. At the close of this chapter, therefore, the
mathematical deduction of the generalization will be given, but in brief
outline only, the reader being referred to works on Thermodynamics for a
fuller treatment of the subject.[21]

All forms of energy can be resolved into two factors, the _capacity_ factor
and the _intensity_ factor; but for the production of equilibrium, only the
intensity factor is of importance. Thus, if two bodies having the same
temperature are brought in contact with each other, they will be in
equilibrium as regards heat energy, no matter what may be the amounts of
heat (capacity factor) contained in either, because the intensity
factor--the temperature--is the same. But if the temperature of the two
bodies is different, _i.e._ if the intensity factor of heat energy is
different, the two bodies will no longer be in equilibrium; but heat will
pass from the hotter to the colder until both have the same temperature.

As with heat energy, so with chemical energy. If we have a substance
existing in two different states, or in two different {19} phases of a
system, equilibrium can occur only when the intensity factor of chemical
energy is the same. This intensity factor may be called the _chemical
potential_; and we can therefore say that a system will be in equilibrium
when the chemical potential of each component is the same in all the phases
in which the component occurs. Thus, for example, ice, water, and vapour
have, at the triple point, the same chemical potential.

The potential of a component in any phase depends not only on the
composition of the phase, but also on the temperature and the pressure (or
volume). If, therefore, we have a system of C components existing in P
phases, then, in order to fix the composition of unit mass of each phase,
it is necessary to know the masses of (C - 1) components in each of the
phases. As regards the composition, therefore, each phase possesses (C - 1)
variables. Since there are P phases, it follows that, as regards
composition, the whole system possesses P(C - 1) variables. Besides these
there are, however, two other variables, viz. temperature and pressure, so
that altogether a system of C components in P phases possesses P(C - 1) + 2
variables.

In order to define the state of the system completely, it will be necessary
to have as many equations as there are variables. If, therefore, there are
fewer equations than there are variables, then, according to the deficiency
in the number of the equations, one or more of the variables will have an
undefined value; and values must be assigned to these variables before the
system is entirely defined. The number of these undefined values gives us
the variability or the degree of freedom of the system.

The equations by which the system is to be defined are obtained from the
relationship between the potential of a component and the composition of
the phase, the temperature and the pressure. Further, as has already been
stated, equilibrium occurs when the potential of each component is the same
in the different phases in which it is present. If, therefore, we choose as
standard one of the phases in which all the components occur, then in any
other phase in equilibrium with {20} it, the potential of each component
must be the same as in the standard phase. For each phase in equilibrium
with the standard phase, therefore, there will be a definite equation of
state for each component in the phase; so that, if there are P phases, we
obtain for each component (P - 1) equations; and for C components,
therefore, we obtain C(P - 1) equations.

But we have seen above that there are P(C - 1) + 2 variables, and as we
have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2
- P variables undefined. That is to say, the degree of freedom (F) of a
system consisting of C components in P phases is--

  F = C + 2 - P

       *       *       *       *       *


{21}

CHAPTER III

TYPICAL SYSTEMS OF ONE COMPONENT

A. _Water._

For the sake of rendering the Phase Rule more readily intelligible, and at
the same time also for the purpose of obtaining examples by which we may
illustrate the general behaviour of systems, we shall in this chapter
examine in detail the behaviour of several well-known systems consisting of
only one component.

The most familiar examples of equilibria in a one-component system are
those furnished by the three phases of water, viz. ice, water, water
vapour. The system consists of one component, because all three phases have
the same chemical composition, represented by the formula H_{2}O. As the
criterion of equilibrium we shall choose a definite pressure, and shall
study the variation of the pressure with the temperature; and for the
purpose of representing the relationships which we obtain we shall employ a
temperature-pressure diagram, in which the temperatures are measured as
abscissæ and the pressures as ordinates. In such a diagram invariant
systems will be represented by points; univariant systems by lines, and
bivariant systems by areas.

Equilibrium between Liquid and Vapour. Vaporization Curve.--Consider in the
first place the conditions for the coexistence of liquid and vapour.
According to the Phase Rule (p. 16), a system consisting of one component
in two phases has one degree of freedom, or is univariant. We should
therefore expect that it will be possible for liquid water to coexist with
water vapour at different values of temperature and {22} pressure, but that
if we arbitrarily fix one of the variable factors, pressure, temperature,
or volume (in the case of a given mass of substance), the state of the
system will then be defined. If we fix, say, the temperature, then the
pressure will have a definite value; or if we adopt a certain pressure, the
liquid and vapour can coexist only at a certain definite temperature. Each
temperature, therefore, will correspond to a definite pressure; and if in
our diagram we join by a continuous line all the points indicating the
values of the pressure corresponding to the different temperatures, we
shall obtain a curve (Fig. 1) representing the variation of the pressure
with the temperature. This is the curve of vapour pressure, or the
_vaporization curve_ of water.

[Illustration: FIG. 1.]

Now, the results of experiment are quite in agreement with the requirements
of the Phase Rule, and at any given temperature the system water--vapour
can exist in equilibrium only under a definite pressure.

The vapour pressure of water at different temperatures has been subjected
to careful measurement by Magnus,[22] Regnault,[23] Ramsay and Young,[24]
Juhlin,[25] Thiesen and Scheel,[26] and others. In the following table the
values of the vapour pressure from -10° to +100° are those calculated from
the measurements of Regnault, corrected by the measurements of Wiebe and
Thiesen and Scheel;[27] those from 120° to 270° were determined {23} by
Ramsay and Young, while the values of the critical pressure and temperature
are those determined by Battelli.[28]

  VAPOUR PRESSURE OF WATER.

  -------------+-----------------+--------------+--------------------
               |                 |              |
  Temperature. | Pressure in cm. | Temperature. |   Pressure in cm.
               |    mercury.     |              |      mercury.
  -------------+-----------------+--------------+--------------------
               |                 |              |
      -10°     |     0.213       |     120°     |   148.4
        0°     |     0.458[29]   |     130°     |   201.9
      +20°     |     1.752       |     150°     |   356.8
       40°     |     5.516       |     200°     |  1162.5
       60°     |    14.932       |     250°     |  2973.4
       80°     |    35.54        |     270°     |  4110.1
      100°     |    76.00        |     364.3°   | 14790.4 (194.6 atm.)
               |                 | (critical    | (critical pressure).
               |                 | temperature) |
  -------------+-----------------+--------------+--------------------

The pressure is, of course, independent of the relative or absolute volumes
of the liquid and vapour; on increasing the volume at constant temperature,
a certain amount of the liquid will pass into vapour, and the pressure will
regain its former value. If, however, the pressure be permanently
maintained at a value different from that corresponding to the temperature
employed, then either all the liquid will pass into vapour, or all the
vapour will pass into liquid, and we shall have either vapour alone or
liquid alone.

Upper Limit of Vaporization Curve.--On continuing to add heat to water
contained in a closed vessel, the pressure of the vapour will gradually
increase. Since with increase of pressure the density of the vapour must
increase, and since with rise of temperature the density of the liquid must
decrease, a point will be reached at which the density of liquid and vapour
become identical; the system ceases to be heterogeneous, and passes into
one homogeneous phase. The temperature at which this occurs is called the
_critical temperature_. To this temperature there will, of course,
correspond a certain definite pressure, called the _critical pressure_. The
curve representing the {24} equilibrium between liquid and vapour must,
therefore, end abruptly at the critical point. At temperatures above this
point no pressure, however great, can cause the formation of the liquid
phase; at temperatures above the critical point the vapour becomes a gas.
In the case of water, the critical temperature is 364.3°, and the critical
pressure 194.6 atm.; at the point representing these conditions the
vapour-pressure curve of water must cease.

Sublimation Curve of Ice.--Vapour is given off not only by liquid water,
but also by solid water, or ice. That this is so is familiar to every one
through the fact that ice or snow, even at temperatures below the melting
point, gradually disappears in the form of vapour. Even at temperatures
considerably lower than 0°, the vapour pressure of ice, although small, is
quite appreciable; and it is possible, therefore, to have ice and vapour
coexisting in equilibrium. When we inquire into the conditions under which
such a system can exist, we see again that we are dealing with a univariant
system--one component existing in two phases--and that, therefore, just as
in the case of the system water and vapour, there will be for each
temperature a certain definite pressure of the vapour, and this pressure
will be independent of the relative or absolute amounts of the solid or
vapour present, and will depend solely on the temperature. Further, just as
in the case of the vapour pressure of water, the condition of equilibrium
between ice and water vapour will be represented by a line or curve showing
the change of pressure with the temperature. Such a curve, representing the
conditions of equilibrium between a solid and its vapour, is called a
_sublimation curve_. At temperatures represented by any point on this
curve, the solid (ice) will sublime or pass into vapour without previously
fusing. Since ice melts at 0° (_vide infra_), the sublimation curve must
end at that temperature.

The following are the values of the vapour pressure of ice between 0° and
-50°.[30]

{25}

  VAPOUR PRESSURE OF ICE.

  ---------------------------------------------------------------
  Temperature. | Pressure in mm. | Temperature. | Pressure in mm.
               |     mercury.    |              |     mercury.
  -------------+-----------------+--------------+----------------
     -50°      |     0.050       |    -8°       |     2.379
     -40°      |     0.121       |    -6°       |     2.821
     -30°      |     0.312       |    -4°       |     3.334
     -20°      |     0.806       |    -2°       |     3.925
     -15°      |     1.279       |     0°       |     4.602
     -10°      |     1.999       |              |
  ----------------------------------------------------------------

Equilibrium between Ice and Water. Curve of Fusion.--There is still another
univariant system of the one component water, the existence of which, at
definite values of temperature and pressure, the Phase Rule allows us to
predict. This is the system solid--liquid. Ice on being heated to a certain
temperature melts and passes into the liquid state; and since this system
solid--liquid is univariant, there will be for each temperature a certain
definite pressure at which ice and water can coexist or be in equilibrium,
independently of the amounts of the two phases present. Since now the
temperature at which the solid phase is in equilibrium with the liquid
phase is known as the melting point or point of fusion of the solid, the
curve representing the temperatures and pressures at which the solid and
liquid are in equilibrium will represent the change of the melting point
with the pressure. Such a curve is called the _curve of fusion_, or the
melting-point curve.

It was not until the middle of the nineteenth century that this connection
between the pressure and the melting point, or the change of the melting
point with the pressure, was observed. The first to recognize the existence
of such a relationship was James Thomson,[31] who in 1849 showed that from
theoretical considerations such a relationship must exist, and predicted
that in the case of ice the melting point would be lowered by pressure.
This prediction was fully confirmed by his brother, W. Thomson[32] (Lord
Kelvin), who found that under a pressure {26} of 8.1 atm. the melting point
of ice was -0.059°; under a pressure of 16.8 atm. the melting point was
-0.129°.

The experiments which were first made in this connection were more of a
qualitative nature, but in recent years careful measurements of the
influence of pressure on the melting point of ice have been made more
especially by Tammann,[33] and the results obtained by him are given in the
following table and represented graphically in Fig. 2.

  FUSION PRESSURE OF ICE.

  ---------------------------------------------------------------------
                  | Pressure in kilogms. per | Change of melting point for
  Temperature.    |  sq. cm.[34]             | an increase of pressure of
                  |                          | 1 kilogm. per sq. cm.
  ---------------------------------------------------------------------
      -0°         |        1                 |
      -2.5°       |      336                 |       0.0074°
      -5°         |      615                 |       0.0090°
      -7.5°       |      890                 |       0.0091°
      -10.0°      |     1155                 |       0.0094°
      -12.5°      |     1410                 |       0.0100°
      -15.0°      |     1625                 |       0.0116°
      -17.5°      |     1835                 |       0.0119°
      -20.0°      |     2042                 |       0.0121°
      -22.1°      |     2200                 |       0.0133°
  ---------------------------------------------------------------------

From the numbers in the table and from the figure we see that as the
pressure is increased the melting point of ice is lowered; but we also
observe that a very large change of pressure is required in order to
produce a very small change in the melting point. The curve, therefore, is
very steep. Increase of pressure by one atmosphere lowers the melting point
by only 0.0076°,[35] or an increase of pressure of 135 atm. is required to
produce a lowering of the melting point of 1°. We see further that the
fusion curve bends slightly as the pressure is increased, which signifies
that the variation of {27} the melting point with the pressure changes; at
-15°, when the pressure is 1625 kilogm. per sq. cm., increase of pressure
by 1 kilogm. per sq. cm. lowers the melting point by 0.012°. This curvature
of the fusion curve we shall later (Chap. IV.) see to be an almost
universal phenomenon.

[Illustration: FIG. 2.]

[Illustration: FIG. 3.]

Equilibrium between Ice, Water, and Vapour. The Triple Point.--On examining
the vapour-pressure curves of ice and water (Fig. 3), we see that at a
temperature of about 0° and under a pressure of about 4.6 mm. mercury, the
two curves cut. At this point liquid water and solid ice are each in
equilibrium with vapour at the same pressure. Since this is so, they must,
of course, be in equilibrium {28} with one another, as experiment also
shows. At this point, therefore, ice, water, and vapour can be in
equilibrium, and as there are three phases present, the point is called a
_triple point_.[36]

The triple point, however, does not lie exactly at 0° C., for this
temperature is defined as the melting point of ice under atmospheric
pressure. At the triple point, however, the pressure is equal to the vapour
pressure of ice and water, and this pressure, as we see from the tables on
pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less than in the
previous case. Now, we have just seen that a change of pressure of 1 atm.
corresponds to a change of the melting point of 0.0076°; the melting point
of ice, therefore, when under the pressure of its own vapour, will be very
nearly +0.0076°, and the pressure of the vapour will be very slightly
greater than 4.579 mm., which is the pressure at 0° (p. 21). The difference
is, however, slight, and may be neglected here. At the temperature, then,
of +0.0076°, and under a pressure of 4.6 mm. of mercury, ice, water, and
vapour will be in equilibrium; the point in our diagram representing this
particular temperature and pressure is, therefore, the triple point of the
system ice--water--vapour.

Since at the triple point we have three phases of one component, the system
at this point is invariant--it possesses no degrees of freedom. If the
temperature is changed, the system will undergo alteration in such a way
that one of the phases will disappear, and a univariant system will result;
if heat be added, ice will melt, and we shall have left water and vapour;
if heat be abstracted, water will freeze, and we shall have left ice and
vapour; if, when the temperature is altered, the pressure is kept constant,
then we shall ultimately obtain only one phase (see Chap. IV.).

The triple point is not only the point of intersection of the vaporization
and sublimation curves, but it is also the end-point of the fusion curve.
The fusion curve, as we have seen, is the curve of equilibrium between ice
and water; and since at the triple point ice and water are each in
equilibrium with {29} vapour of the same pressure, they must, of course,
also be in equilibrium with one another.

[Illustration: FIG. 4.]

Bivariant Systems of Water.--If we examine Fig. 4, we see that the curves
OA, OB, OC, which represent diagrammatically the conditions under which
water and vapour, ice and vapour, and water and ice are in equilibrium,
form the boundaries of three "fields," or areas, I., II., III. These areas,
now, represent the conditions for the existence of the single phases,
solid, liquid, and vapour respectively. At temperatures and pressures
represented by any point in the field I., solid only can exist as a stable
phase. Since we have here one component in only one phase, the system is
bivariant, and at any given temperature, therefore, ice can exist under a
series of pressures; and under any given pressure, at a series of
temperatures, these pressures and temperatures being limited only by the
curves OB, OC. Similarly also with the areas II. and III.

We see, further, that the different areas are the regions of stability of
the phase common to the two curves by which the area is enclosed.[37] Thus,
the phase common to the two systems {30} represented by BO (ice and
vapour), and OA (water and vapour) is the vapour phase; and the area BOA is
therefore the area of the vapour phase. Similarly, BOC is the area of the
ice phase, and COA the area of the water phase.

Supercooled Water. Metastable State.--When heated under the ordinary
atmospheric pressure, ice melts when the temperature reaches 0°, and it has
so far not been found possible to raise the temperature of ice above this
point without liquefaction taking place. On the other hand, it has long
been known that water can be cooled below zero without solidification
occurring. This was first discovered in 1724 by Fahrenheit,[38] who found
that water could be exposed to a temperature of -9.4° without solidifying;
so soon, however, as a small particle of ice was brought in contact with
the water, crystallization commenced. Superfused or supercooled
water--_i.e._ water cooled below 0°--is unstable only in respect of the
solid phase; so long as the presence of the solid phase is carefully
avoided, the water can be kept for any length of time without solidifying,
and the system supercooled water and vapour behaves in every way like a
stable system. A system, now, which in itself is stable, and which becomes
instable only in contact with a particular phase, is said to be
_metastable_, and the region throughout which this condition exists is
called the metastable region. Supercooled water, therefore, is in a
metastable condition. If the supercooling be carried below a certain
temperature, solidification takes place spontaneously without the addition
of the solid phase; the system then ceases to be metastable, and becomes
_instable_.

Not only has water been cooled to temperatures considerably below the
melting point of ice, but the vapour pressure of the supercooled water has
been measured. It is of interest and importance, now, to see what
relationship exists between the vapour pressure of ice and that of
supercooled water at the same temperature. This relationship is clearly
shown by the numbers in the following table,[39] and is represented in Fig.
3, {31} p. 27., and diagrammatically in Fig. 4, the vapour pressures of
supercooled water being represented by the curve OA', which is the unbroken
continuation of AO.

  VAPOUR PRESSURE OF ICE AND OF SUPERCOOLED WATER.

  ---------------------------------------------------------------------
                 |       Pressure in mm. mercury.
                 ------------------------------------------------------
  Temperature.   |               |               |
                 |     Water.    |     Ice.      |    Difference.
  ---------------------------------------------------------------------
         0°      |     4.618     |     4.602     |    0.016[40]
        -2°      |     3.995     |     3.925     |    0.070
        -4°      |     3.450     |     3.334     |    0.116
        -8°      |     2.558     |     2.379     |    0.179
       -10°      |     2.197     |     1.999     |    0.198
       -15°      |     1.492     |     1.279     |    0.213
       -20°      |     1.005     |     0.806     |    0.199
  ---------------------------------------------------------------------

At all temperatures below 0° (more correctly +0.0076°), at which
temperature water and ice have the same vapour pressure, the vapour
pressure of supercooled water is _greater_ than that of ice at the same
temperature.

From the relative positions of the curves OB and OA (Fig. 4) we see that at
all temperatures above 0°, the (metastable) sublimation curve of ice, if it
could be obtained, would be higher than the vaporization curve of water.
This shows, therefore, that at 0° a "break" must occur in the curve of
states, and that in the neighbourhood of this break the curve above that
point must ascend less rapidly than the curve below the break. Since,
however, the differences in the vapour pressures of supercooled water and
of ice are very small, the change in the direction of the vapour-pressure
curve on passing from ice to water was at first not observed, and Regnault
regarded the sublimation curve as passing continuously into {32} the
vaporization curve. The existence of a break was, however, shown by James
Thomson[41] and by Kirchhoff[42] to be demanded by thermo-dynamical
considerations, and the prediction of theory was afterwards realized
experimentally by Ramsay and Young in their determinations of the vapour
pressure of water and ice, as well as in the case of other substances.[43]

From what has just been said, we can readily understand why ice and water
cannot exist in equilibrium below 0°. For, suppose we have ice and water in
the same closed space, but not in contact with one another, then since the
vapour pressure of the supercooled water is higher than that of ice, the
vapour of the former must be supersaturated in contact with the latter;
vapour must, therefore, condense on the ice; and in this way there will be
a slow distillation from the water to the ice, until at last all the water
will have disappeared, and only ice and vapour remain.[44]

Other Systems of the Substance Water.--We have thus far discussed only
those systems which are constituted by the three phases--ice, water, and
water vapour. It has, however, been recently found that at a low
temperature and under a high pressure ordinary ice can pass into two other
crystalline varieties, called by Tammann[45] ice II. and ice III., ordinary
ice being ice I. According to the Phase Rule, now, since each of these
solid forms constitutes a separate phase (p. 9), it will be possible to
have the following (and more) systems of water, in addition to those
already studied, viz. water, ice I., ice II.; water, ice I., ice III.;
water, ice II., ice III., forming invariant systems and existing in
equilibrium only at a definite triple point; further, water, ice II.;
water, ice III.; ice I., ice II.; ice I., ice III.; ice II., ice III.,
forming univariant systems, existing, therefore, at definite corresponding
values of {33} temperature and pressure; and lastly, the bivariant systems,
ice II. and ice III. Several of these systems have been investigated by
Tammann. The triple point for water, ice I., ice III., lies at -22°, and a
pressure of 2200 kilogms. per sq. cm. (2130 atm.), as indicated in Fig. 2,
p. 27.[46] In contrast with the behaviour of ordinary ice, the temperature
of equilibrium in the case of water--ice II., and water--ice III., is
_raised_ by increase of pressure.

B. _Sulphur._

Polymorphism.--Reference has just been made to the fact that ice can exist
not only in the ordinary form, but in at least two other crystalline
varieties. This phenomenon, the existence of a substance in two or more
different crystalline forms, is called _polymorphism_. Polymorphism was
first observed by Mitscherlich[47] in the case of sodium phosphate, and
later in the case of sulphur. To these two cases others were soon added, at
first of inorganic, and later of organic substances, so that polymorphism
is now recognized as of very frequent occurrence indeed.[48] These various
forms of a substance differ not only in crystalline shape, but also in
melting point, specific gravity, and other physical properties. In the
liquid state, however, the differences do not exist.

According to our definition of phases (p. 9), each of these polymorphic
forms constitutes a separate phase of the particular substance. As is
readily apparent, the number of possible systems formed of one component
may be considerably increased when that component is capable of existing in
different crystalline forms. We have, therefore, to inquire what are the
conditions under which different polymorphic forms can coexist, either
alone or in presence of the liquid and vapour phase. For the purpose of
illustrating the general behaviour of such systems, we shall study the
systems formed by the different crystalline forms of sulphur, tin, and
benzophenone.

{34}

Sulphur exists in two well-known crystalline forms--rhombic, or octahedral,
and monoclinic, or prismatic sulphur. Of these, the former melts at 114.5°;
the latter at 120°.[49] Further, at the ordinary temperature, rhombic
sulphur can exist unchanged, whereas, on being heated to temperatures
somewhat below the melting point, it passes into the prismatic variety. On
the other hand, at temperatures above 96°, prismatic sulphur can remain
unchanged, whereas at the ordinary temperature it passes slowly into the
rhombic form.

If, now, we examine the case of sulphur with the help of the Phase Rule, we
see that the following systems are theoretically possible:--

  I. _Bivariant Systems: One component in one phase._
    (_a_) Rhombic sulphur.
    (_b_) Monoclinic sulphur.
    (_c_) Sulphur vapour.
    (_d_) Liquid sulphur.

  II. _Univariant Systems: One component in two phases._
    (_a_) Rhombic sulphur and vapour.
    (_b_) Monoclinic sulphur and vapour.
    (_c_) Rhombic sulphur and liquid.
    (_d_) Monoclinic sulphur and liquid.
    (_e_) Rhombic and monoclinic sulphur.
    (_f_) Liquid and vapour.

  III. _Invariant Systems: One component in three phases._
    (_a_) Rhombic and monoclinic sulphur and vapour.
    (_b_) Rhombic sulphur, liquid and vapour.
    (_c_) Monoclinic sulphur, liquid and vapour.
    (_d_) Rhombic and monoclinic sulphur and liquid.

[Illustration: FIG. 5.]

Triple Point--Rhombic and Monoclinic Sulphur and Vapour. Transition
Point.--In the case of ice, water and vapour, we saw that at the triple
point the vapour pressures of ice and water are equal; below this point,
ice is stable; above this point, water is stable. We saw, further, that
below 0° the vapour pressure of the stable system is lower than that of the
metastable, and therefore that at the triple point there is a break in the
vapour pressure curve of such a kind that above {35} the triple point the
vapour-pressure curve ascends more slowly than below it. Now, although the
vapour pressure of solid sulphur has not been determined, we can
nevertheless consider that it does possess a certain, even if very small,
vapour pressure,[50] and that at the temperature at which the vapour
pressures of rhombic and monoclinic sulphur become equal, we can have these
two solid forms existing in equilibrium with the vapour. Below that point
only one form, that with the lower vapour pressure, will be stable; above
that point only the other form will be stable. On passing through the
triple point, therefore, there will be a change of the one form into the
other. This point is represented in our diagram (Fig. 5) by the point O,
the two curves AO and OB representing diagrammatically the vapour pressures
of rhombic and monoclinic sulphur respectively. If the vapour phase is
absent and the system maintained under a constant pressure, _e.g._ {36}
atmospheric pressure, there will also be a definite temperature at which
the two solid forms are in equilibrium, and on passing through which
complete and reversible transformation of one form into the other occurs.
This temperature, which refers to equilibrium in absence of the vapour
phase, is known as the _transition temperature_ or _inversion temperature_.

Were we dependent on measurements of pressure and temperature, the
determination of the transition point might be a matter of great
difficulty. When we consider, however, that the other physical properties
of the solid phases, _e.g._ the density, undergo an abrupt change on
passing through the transition point, owing to the transformation of one
form into the other, then any method by which this abrupt change in the
physical properties can be detected may be employed for determining the
transition point. A considerable number of such methods have been devised,
and a description of the most important of these is given in the Appendix.

In the case of sulphur, the transition point of rhombic into monoclinic
sulphur was found by Reicher[51] to lie at 95.5°. Below this temperature
the octahedral, above it the monoclinic, is the stable form.

Condensed Systems.--We have already seen that in the change of the melting
point of water with the pressure, a very great increase of the latter was
necessary in order to produce a comparatively small change in the
temperature of equilibrium. This is a characteristic of all systems from
which the vapour phase is absent, and which are composed only of solid and
liquid phases. Such systems are called _condensed systems_,[52] and in
determining the temperature of equilibrium of such systems, practically the
same point will be obtained whether the measurements are carried out under
atmospheric pressure or under the pressure of the vapour of the solid or
liquid phases. The transition point, therefore, as determined in open
vessels at atmospheric pressure, will differ only by a very slight amount
from the triple point, or point at which the two solid or liquid phases are
in equilibrium under the pressure of their vapour. {37} The determination
of the transition point is thereby greatly simplified.

Suspended Transformation.--In many respects the transition point of two
solid phases is analogous to the melting point of a solid, or point at
which the solid passes into a liquid. In both cases the change of phase is
associated with a definite temperature and pressure in such a way that
below the point the one phase, above the point the other phase, is stable.
The transition point, however, differs in so far from a point of fusion,
that while it is possible to supercool a liquid, no definite case is known
where the solid has been heated above the triple point without passing into
the liquid state. Transformation, therefore, is suspended only on one side
of the melting point. In the case of two solid phases, however, the
transition point can be overstepped in both directions, so that each phase
can be obtained in the metastable condition. In the case of supercooled
water, further, we saw that the introduction of the stable, solid phase
caused the speedy transformation of the metastable to the stable condition
of equilibrium; but in the case of two solid phases the change from the
metastable to the stable modification may occur with great slowness, even
in presence of the stable form. This tardiness with which the stable
condition of equilibrium is reached greatly increases in many cases the
difficulty of accurately determining the transition point. The phenomena of
suspended transformation will, however, receive a fuller discussion later
(p. 68).

Transition Curve--Rhombic and Monoclinic Sulphur.--Just as we found the
melting point of ice to vary with the pressure, so also do we find that
change of pressure causes an alteration in the transition point. In the
case of the transition point of rhombic into monoclinic sulphur, increase
of pressure by 1 atm. raises the transition point by 0.04°-0.05°.[53] The
transition curve, or curve representing the change of the transition point
with pressure, will therefore slope to the right away from the pressure
axis. This is curve OC (Fig. 5).

{38}

Triple Point--Monoclinic Sulphur, Liquid, and Vapour. Melting Point of
Monoclinic Sulphur.--Above 95.5°, monoclinic sulphur is, as we have seen,
the stable form. On being heated to 120°, under atmospheric pressure, it
melts. This temperature is, therefore, the point of equilibrium between
monoclinic sulphur and liquid sulphur under atmospheric pressure. Since we
are dealing with a condensed system, this temperature may be regarded as
very nearly that at which the solid and liquid are in equilibrium with
their vapour, _i.e._ the triple point, solid (monoclinic)--liquid--vapour.
This point is represented in the diagram by B.

Triple Point--Rhombic and Monoclinic Sulphur and Liquid.--In contrast with
that of ice, the fusion point of monoclinic sulphur is _raised_ by increase
of pressure, and the fusion curve, therefore, slopes to the right. The
transition curve of rhombic and monoclinic sulphur, as we have seen, also
slopes to the right, and more so than the fusion curve of monoclinic
sulphur. There will, therefore, be a certain pressure and temperature at
which the two curves will cut. This point lies at 151°, and a pressure of
1320 kilogm. per sq. cm., or about 1288 atm.[54] It, therefore, forms
another triple point, the existence of which had been predicted by
Roozeboom,[55] at which rhombic and monoclinic sulphur are in equilibrium
with liquid sulphur. It is represented in our diagram by the point C.
_Beyond this point monoclinic sulphur ceases to exist in a stable
condition._ At temperatures and pressures above this triple point, rhombic
sulphur will be the stable modification, and this fact is of mineralogical
interest, because it explains the occurrence in nature of well-formed
rhombic crystals. Under ordinary conditions, prismatic sulphur separates
out on cooling fused sulphur, but at temperatures above 151° and under
pressures greater than 1288 atm., the rhombic form would be produced.[56]

Triple Point--Rhombic Sulphur, Liquid, and Vapour. Metastable Triple
Point.--On account of the slowness with {39} which transformation of one
form into the other takes place on passing the transition point, it has
been found possible to heat rhombic sulphur up to its melting point
(114.5°). At this temperature, not only is rhombic sulphur in a metastable
condition, but the liquid is also metastable, its vapour pressure being
greater than that of solid monoclinic sulphur. This point is represented in
our diagram by the point b.

From the relative positions of the metastable melting point of rhombic
sulphur and the stable melting point of monoclinic sulphur at 120°, we see
that, of the two forms, the metastable form has the lower melting point.
This, of course, is valid only for the relative stability in the
neighbourhood of the melting point; for we have already learned that at
lower temperatures rhombic sulphur is the stable, monoclinic sulphur the
metastable (or unstable) form.

Fusion Curve of Rhombic Sulphur.--Like any other melting point, that of
rhombic sulphur will be displaced by increase of pressure; increase of
pressure raises the melting point, and we can therefore obtain a metastable
fusion curve representing the conditions under which rhombic sulphur is in
equilibrium with liquid sulphur. This metastable fusion curve must pass
through the triple point for rhombic sulphur--monoclinic sulphur--liquid
sulphur, and on passing this point it becomes a stable fusion curve. The
continuation of this curve, therefore, above 151° forms the stable fusion
curve of rhombic sulphur (curve CD).

These curves have been investigated at high pressures by Tammann, and the
results are represented according to scale in Fig. 6,[57] _a_ being the
curve for monoclinic sulphur and liquid; _b_, that for rhombic sulphur and
liquid; and _c_, that for rhombic and monoclinic sulphur.

Bivariant Systems.--Just as in the case of the diagram of states of water,
the areas in Fig. 5 represent the conditions for the stable existence of
the single phases: rhombic sulphur in the area to the left of AOCD;
monoclinic sulphur in the area OBC; liquid sulphur in the area EBCD;
sulphur vapour below the curves AOBE. As can be seen from the diagram, {40}
the existence of monoclinic sulphur is limited on all sides, its area being
bounded by the curves OB, OC, BC. At any point outside this area,
monoclinic sulphur can exist only in a metastable condition.

[Illustration: FIG. 6.]

Other crystalline forms of sulphur have been obtained,[58] so that the
existence of other systems of the one-component sulphur besides those
already described is possible. Reference will be made to these later
(p. 51).

{41}

C. _Tin._

Another substance capable of existing in more than one crystalline form, is
the metal tin, and although the general behaviour, so far as studied, is
analogous to that of sulphur, a short account of the two varieties of tin
may be given here, not only on account of their metallurgical interest, but
also on account of the importance which the phenomena possess for the
employment of this metal in everyday life.

After a winter of extreme severity in Russia (1867-1868), the somewhat
unpleasant discovery was made that a number of blocks of tin, which had
been stored in the Customs House at St. Petersburg, had undergone
disintegration and crumbled to a grey powder.[59] That tin undergoes change
on exposure to extreme cold was known, however, before that time, even as
far back as the time of Aristotle, who spoke of the tin as "melting."[60]
Ludicrous as that term may now appear, Aristotle nevertheless unconsciously
employed a strikingly accurate analogy, for the conditions under which
ordinary white tin passes into the grey modification are, in many ways,
quite analogous to those under which a substance passes from the solid to
the liquid state. The knowledge of this was, however, beyond the wisdom of
the Greek philosopher.

For many years there existed considerable confusion both as to the
conditions under which the transformation of white tin into its allotropic
modification occurs, and to the reason of the change. Under the guidance of
the Phase Rule, however, the confusion which obtained has been cleared
away, and the "mysterious" behaviour of tin brought into accord with other
phenomena of transformation.[61]

Transition Point.--Just as in the case of sulphur, so also in the case of
tin, there is a transition point above which the {42} one form, ordinary
white tin, and below which the other form, grey tin, is the stable variety.
In the case of this metal, the transition point was found by Cohen and van
Eyk, who employed both the dilatometric and the electrical methods
(Appendix) to be 20°. Below this temperature, grey tin is the stable form.
But, as we have seen in the case of sulphur, the change of the metastable
into the stable solid phase occurs with considerable slowness, and this
behaviour is found also in the case of tin. Were it not so, we should not
be able to use this metal for the many purposes to which it is applied in
everyday life; for, with the exception of a comparatively small number of
days in the year, the temperature of our climate is below 20°, and _white
tin is, therefore, at the ordinary temperature, in a metastable condition_.
The change, however, into the stable form at the ordinary temperature,
although slow, nevertheless takes place, as is shown by the partial or
entire conversion of articles of tin which have lain buried for several
hundreds of years.

On lowering the temperature, the velocity with which the transformation of
the tin occurs is increased, and Cohen and van Eyk found that the
temperature of maximum velocity is about -50°. Contact with the stable form
will, of course, facilitate the transformation.

The change of white tin into grey takes place also with increased velocity
in presence of a solution of tin ammonium chloride (pink salt), which is
able to dissolve small quantities of tin. In presence of such a solution
also, it was found that the temperature at which the velocity of
transformation was greatest was raised to 0°. At this temperature, white
tin in contact with a solution of tin ammonium chloride, and the grey
modification, undergoes transformation to an appreciable extent in the
course of a few days.

Fig. 7 is a photograph of a piece of white tin undergoing transformation
into the grey variety.[62] The bright surface of the tin becomes covered
with a number of warty masses, formed of the less dense grey form, and the
number and size of these continue to grow until the whole of the white tin
has passed {43} into a grey powder. On account of the appearance which is
here seen, this transformation of tin has been called by Cohen the "tin
plague."

[Illustration: FIG. 7.]

{44}

Enantiotropy and Monotropy.--In the case of sulphur and tin, we have met
with two substances existing in polymorphic forms, and we have also learned
that these forms exhibit a definite transition point at which their
relative stability is reversed. Each form, therefore, possesses a definite
range of stable existence, and is capable of undergoing transformation into
the other, at temperatures above or below that of the transition point.

Another class of dimorphous substances is, however, met with as, for
instance, in the case of the well-known compounds iodine monochloride and
benzophenone. Each crystalline form has its own melting point, the
dimorphous forms of iodine monochloride melting at 13.9° and 27.2°,[63] and
those of benzophenone at 26° and 48°.[64] This class of substance differs
from that which we have already studied (_e.g._ sulphur and tin), in that
at all temperatures up to the melting point, only one of the forms is
stable, the other being metastable. There is, therefore, no transition
point, and transformation of the crystalline forms can be observed _only in
one direction_. These two classes of phenomena are distinguished by the
names _enantiotropy_ and _monotropy_; enantiotropic substances being such
that the change of one form into the other is a reversible process (_e.g._
rhombic sulphur into monoclinic, and monoclinic sulphur into rhombic), and
monotropic substances, those in which the transformation of the crystalline
forms is irreversible.

[Illustration: FIG. 8.]

[Illustration: FIG. 9.]

These differences in the behaviour can be explained very well in many cases
by supposing that in the case of enantiotropic substances the transition
point lies below the melting point, while in the case of monotropic
substances, it lies above the melting point.[65] These conditions would be
represented by the Figs. 8 and 9.

In these two figures, O_{3} is the transition point, O_{1} and O_{2} the
melting points of the metastable and stable forms {45} respectively. From
Fig. 9 we see that the crystalline form I. at all temperatures up to its
melting point is metastable with respect to the form II. In such cases the
transition point could be reached only at higher pressures.

Although, as already stated, this explanation suffices for many cases, it
does not prove that in all cases of monotropy the transition point is above
the melting point of the two forms. It is also quite possible that the
transition point may lie below the melting points;[66] in this case we have
what is known as _pseudomonotropy_. It is possible that graphite and
diamond,[67] perhaps also the two forms of phosphorus, stand in the
relation of pseudomonotropy (_v._ p. 49).

The disposition of the curves in Figs. 8 and 9 also explains the phenomenon
sometimes met with, especially in organic chemistry, that the substance
first melts, then solidifies, and remelts at a higher temperature. On again
determining the melting point after re-solidification, only the higher
melting point is obtained.

The explanation of such a behaviour is, that if the determination of the
melting point is carried out rapidly, the point O_{1}, the melting point of
the metastable solid form, may be realized. At this temperature, however,
the liquid is metastable with respect to the stable solid form, and if the
temperature is {46} not allowed to rise above the melting point of the
latter, the liquid may solidify. The stable solid modification thus
obtained will melt only at a higher temperature.

D. _Phosphorus._

An interesting case of a monotropic dimorphous substance is found in
phosphorus, which occurs in two crystalline forms; white phosphorus
belonging to the regular system, and red phosphorus belonging to the
hexagonal system. From determinations of the vapour pressures of liquid
white phosphorus, and of solid red phosphorus,[68] it was found that the
vapour pressure of red phosphorus was considerably lower than that of
liquid white phosphorus at the same temperature, the values obtained being
given in the following table.

  VAPOUR PRESSURES OF WHITE AND RED PHOSPHORUS.

  -------------------------------------------------------------------------
    Vapour pressure of liquid white phosphorus.    | Vapour pressure of red
                                                   |     phosphorus.
  -------------------------------------------------+-----------------------
  Temperature.| Pressure | Temperature.|  Pressure | Temperature.| Pressure
              |  in cm.  |             |   in atm. |             |  in atm.
  ------------+----------+-------------+-----------+-------------+---------
    165°      |  12      |    360°     |    3.2    |    360°     |   0.1
    180°      |  20.4    |    440°     |    7.5    |    440°     |   1.75
    200°      |  26.6    |    494°     |   18.0    |    487°     |   6.8
    219°      |  35.9    |    503°     |   21.9    |    510°     |  10.8
    230°      |  51.4    |    511°     |   26.2    |    531°     |  16.0
    290°      |  76.0    |     --      |     --    |    550°     |  31.0
    --        |   --     |     --      |     --    |    577°     |  56.0
  -------------------------------------------------------------------------

These values are also represented graphically in Fig. 10.

[Illustration: FIG. 10.]

At all temperatures above about 260°, transformation of the white into the
red modification takes place with appreciable velocity, and this velocity
increases as the temperature is raised. Even at lower temperatures, _e.g._
at the ordinary temperature, the velocity of transformation is increased
under the influence {47} of light,[69] or by the presence of certain
substances, _e.g._ iodine,[70] just as the velocity of transformation of
white tin into the grey modification was increased by the presence of a
solution of tin ammonium chloride (p. 40). At the ordinary temperature,
therefore, white phosphorus must be considered as the less stable
(metastable) form, for although it can exist in contact with red phosphorus
for a long period, its vapour pressure, as we have seen, is greater than
that of the red modification, and also, its solubility in different
solvents is greater[71] than that of the red modification; as we shall find
later, the solubility of the metastable form is always greater than that of
the stable.

The relationships which are met with in the case of phosphorus can be best
represented by the diagram, Fig. 11.[72]

In this figure, BO_{1} represents the conditions of equilibrium of the
univariant system red phosphorus and vapour, which ends at O_{1}, the
melting point of red phosphorus. By heating in capillary tubes of hard
glass, Chapman[73] found that red phosphorus melts at the melting point of
potassium iodide, _i.e._ about 630°,[74] but the pressure at this
temperature is unknown.

At O_{1}, then, we have the triple point, red phosphorus, liquid, and
vapour, and starting from it, we should have the {48} vaporization curve of
liquid phosphorus, O_{1}A, and the fusion curve of red phosphorus, O_{1}F.
Although these have not been determined, the latter curve must, from
theoretical considerations (_v._ p. 58), slope slightly to the right;
_i.e._ increase of pressure raises the melting point of red phosphorus.

[Illustration: FIG. 11.]

When white phosphorus is heated to 44°, it melts. At this point, therefore,
we shall have another triple point, white phosphorus--liquid--vapour; the
pressure at this point has been calculated to be 3 mm.[75] This point is
the intersection of three curves, viz. sublimation curve, vaporization
curve, and the fusion curve of white phosphorus. The fusion curve, O_{2}E,
has been determined by Tammann[76] and by G. A. Hulett,[77] and it was
found that increase of pressure by 1 atm. raises the melting point by
0.029°. The sublimation curve of white phosphorus has not yet been
determined.

As can be seen from the table of vapour pressures (p. 46), the vapour
pressure of white phosphorus has been determined up to 500°; at
temperatures above this, however, the velocity with which transformation
into red phosphorus takes place is so great as to render the determination
of the vapour pressure {49} at higher temperatures impossible. Since,
however, the difference between white phosphorus and red phosphorus
disappears in the liquid state, the vapour pressure curve of white
phosphorus must pass through the point O_{1}, the melting point of red
phosphorus, and must be continuous with the curve O_{1}A, the vapour
pressure curve of liquid phosphorus (_vide infra_). Since, as Fig. 10
shows, the vapour pressure curve of white phosphorus ascends very rapidly
at higher temperatures, the "break" between BO_{1} and O_{1}A must be very
slight.

As compared with monotropic substances like benzophenone, phosphorus
exhibits the peculiarity that transformation of the metastable into the
stable modification takes place with great slowness; and further, the time
required for the production of equilibrium between red phosphorus and
phosphorus vapour is great compared with that required for establishing the
same equilibrium in the case of white phosphorus. This behaviour can be
best explained by the assumption that change in the molecular complexity
(polymerization) occurs in the conversion of white into red phosphorus, and
when red phosphorus passes into vapour (depolymerization).[78]

This is borne out by the fact that measurements of the vapour density of
phosphorus vapour at temperatures of 500° and more, show it to have the
molecular weight represented by P_{4},[79] and the same molecular weight
has been found for phosphorus in solution.[80] On the other hand, it has
recently been shown by R. Schenck,[81] that the molecular weight of red
phosphorus is at least P_{8}, and very possibly higher.

In the case of phosphorus, therefore, it is more than possible that we are
dealing, not simply with two polymorphic {50} forms of the same substance,
but with polymeric forms, and that there is no transition point at
temperatures above the absolute zero, unless we assume the molecular
complexity of the two forms to become the same. The curve for red
phosphorus would therefore lie below that of white phosphorus, for the
vapour pressure of the polymeric form, if produced from the simpler form
with evolution of heat, must be lower than that of the latter. A transition
point would, of course, become possible if the sign of the heat effect in
the transformation of the one modification into the other should change.
If, further, the liquid which is produced by the fusion of red phosphorus
at 630° under high pressure also exists in a polymeric form, greater than
P_{4}, then the metastable vaporization curve of white phosphorus would not
pass through the melting point of red phosphorus, as was assumed above.[82]

We have already seen in the case of water (p. 31) that the vapour pressure
of supercooled water is greater than that of ice, and that therefore it is
possible, theoretically at least, by a process of distillation, to transfer
the water from one end of a closed tube to the other, and to there condense
it as ice. On account of the very small difference between the vapour
pressure of supercooled water and ice, this distillation process has not
been experimentally realized. In the case of phosphorus, however, where the
difference in the vapour pressures is comparatively great, it has been
found possible to distil white phosphorus from one part of a closed tube to
another, and to there condense it as red phosphorus; and since the vapour
pressure of red phosphorus at 350° is less than the vapour pressure of
white phosphorus at 200°, it is possible to carry out the distillation from
a _colder_ part of the tube to a _hotter_, by having white phosphorus at
the former and red phosphorus at the latter. Such a process of distillation
has been carried out by Troost and Hautefeuille between 324° and 350°.[83]

Relationships similar to those found in the case of phosphorus are also met
with in the case of cyanogen and {51} paracyanogen, which have been studied
by Chappuis,[84] Troost and Hautefeuille,[85] and Dewar,[86] and also in
the case of other organic substances.

Enantiotropy combined with Monotropy.--Not only can polymorphic substances
exhibit enantiotropy or monotropy, but, if the substance is capable of
existing in more than two crystalline forms, both relationships may be
found, so that some of the forms may be enantiotropic to one another, while
the other forms exhibit only monotropy. This behaviour is seen in the case
of sulphur, which can exist in as many as eight different crystalline
varieties. Of these only monoclinic and rhombic sulphur exhibit the
relationship of enantiotropy, _i.e._ they possess a definite transition
point, while the other forms are all metastable with respect to rhombic and
monoclinic sulphur, and remain so up to the melting point; that is to say,
they are monotropic modifications.[87]

E. _Liquid Crystals._

Phenomena observed.--In 1888 it was discovered by Reinitzer[88] that the
two substances, cholesteryl acetate and cholesteryl benzoate, possess the
peculiar property of melting sharply at a definite temperature to milky
liquids; and that the latter, on being further heated, suddenly become
clear, also at a definite temperature. Other substances, more especially
_p_-azoxyanisole and _p_-azoxyphenetole, were, later, found to possess the
same property of having apparently a double melting point.[89] On cooling
the clear liquids, the reverse series of changes occurred.

The turbid liquids which were thus obtained were found to possess not only
the usual properties of liquids (such as the {52} property of flowing and
of assuming a perfectly spherical shape when suspended in a liquid of the
same density), but also those properties which had hitherto been observed
only in the case of solid crystalline substances, viz. the property of
double refraction and of giving interference colours when examined by
polarized light; the turbid liquids are _anisotropic_. To such liquids, the
optical properties of which were discovered by O. Lehmann,[90] the name
_liquid crystals_, or crystalline liquids, was given.

Nature of Liquid Crystals.--During the past ten years the question as to
the nature of liquid crystals has been discussed by a number of
investigators, several of whom have contended strongly against the idea of
the term "liquid" being applied to the crystalline condition; and various
attempts have been made to prove that the turbid liquids are in reality
heterogeneous and are to be classed along with emulsions.[91] This view was
no doubt largely suggested by the fact that the anisotropic liquids were
turbid, whereas the "solid" crystals were clear. Lehmann found, however,
that, when examined under the microscope, the "simple" liquid crystals were
also clear,[92] the apparent turbidity being due to the aggregation of a
number of differently oriented crystals, in the same way as a piece of
marble does not appear transparent although composed of transparent
crystals.[93]

Further, no proof of the heterogeneity of liquid crystals has yet been
obtained, but rather all chemical and physical investigations indicate that
they are homogeneous.[94] No separation {53} of a solid substance from the
milky, anisotropic liquids has been effected; the anisotropic liquid is in
some cases less viscous than the isotropic liquid formed at a higher
temperature; and the temperature of liquefaction is constant, and is
affected by pressure and admixture with foreign substances exactly as in
the case of a pure substance.[95]

[Illustration: FIG. 12.]

Equilibrium Relations in the Case of Liquid Crystals.--Since, now, we have
seen that we are dealing here with substances in two crystalline forms
(which we may call the solid and liquid[96] crystalline form), which
possess a definite transition point, at which, transformation of the one
form into the other occurs in both directions, we can represent the
conditions of equilibrium by a diagram in all respects similar to that
employed in the case of other enantiotropic substances, _e.g._ sulphur
(p. 35).

{54}

In Fig. 12 there is given a diagrammatic representation of the
relationships found in the case of _p_-azoxyanisole.[97]

Although the vapour pressure of the substance in the solid, or liquid
state, has not been determined, it will be understood from what we have
already learned, that the curves AO, OB, BC, representing the vapour
pressure of solid crystals, liquid crystals, isotropic liquid, must have
the relative positions shown in the diagram. Point O, the transition point
of the solid into the liquid crystals, lies at 118.27°, and the change of
the transition point with the pressure is +0.032° pro 1 atm. The transition
curve OE slopes, therefore, slightly to the right. The point B, the melting
point of the liquid crystals, lies at 135.85°, and the melting point is
raised 0.0485° pro 1 atm. The curve BD, therefore, also slopes to the
right, and more so than the transition curve. In this respect azoxyanisole
is different from sulphur.

The areas bounded by the curves represent the conditions for the stable
existence of the four single phases, solid crystals, liquid crystals,
isotropic liquid and vapour.

The most important substances hitherto found to form liquid crystals
are[98]:--

  ----------------------------------+------------+--------
                                    |            |
          Substance.                | Transition | Melting
                                    |   point.   | point.
  ----------------------------------+------------+--------
                                    |            |
  Cholesteryl benzoate              |   145.5°   |  178.5°
  Azoxyanisole                      |   118.3°   |  135.9°
  Azoxyphenetole                    |   134.5°   |  168.1°
  Condensation product from         |            |
   benzaldehyde and benzidine       |   234°     |  260°
  Azine of _p_-oxyethylbenzaldehyde |   172°     |  196°
  Condensation product from         |            |
   _p_-tolylaldehyde and benzidine  |   231°     |   --
  _p_-Methoxycinnamic acid          |   169°     |  185°
  ----------------------------------+------------+--------

       *       *       *       *       *


{55}

CHAPTER IV

GENERAL SUMMARY

In the preceding pages we have learned how the principles of the Phase Rule
can be applied to the elucidation of various systems consisting of one
component. In the present chapter it is proposed to give a short summary of
the relationships we have met with, and also to discuss more generally how
the Phase Rule applies to other one-component systems. On account of the
fact that beginners are sometimes inclined to expect too much of the Phase
Rule; to expect, for example, that it will inform them as to the exact
behaviour of a substance, it may here be emphasized that the Phase Rule is
a general rule; it informs us only as to the general conditions of
equilibrium, and leaves the determination of the definite, numerical data
to experiment.

Triple Point.--We have already (p. 28) defined a triple point in a
one-component system, as being that pressure and temperature at which three
phases coexist in equilibrium; it represents, therefore, an invariant
system (p. 16). At the triple point also, three curves cut, viz. the curves
representing the conditions of equilibrium of the three univariant systems
formed by the combination of the three phases in pairs. The most common
triple point of a one-component system is, of course, the triple point,
solid, liquid, vapour (S-L-V), but other triple points[99] are also
possible when, as in the case of {56} sulphur or benzophenone, polymorphic
forms occur. Whether or not all the triple points can be experimentally
realized will, of course, depend on circumstances. We shall, in the first
place, consider only the triple point S-L-V.

As to the general arrangement of the three univariant curves around the
triple point, the following rules may be given. (1) The prolongation of
each of the curves beyond the triple point must lie between the other two
curves. (2) The middle position at one and the same temperature in the
neighbourhood of the triple point is taken by that curve (or its metastable
prolongation) which represents the two phases of most widely differing
specific volume.[100] That is to say, if a line of constant temperature is
drawn immediately above or below the triple point so as to cut the three
curves--two stable curves and the metastable prolongation of the third--the
position of the curves at that temperature will be such that the middle
position is occupied by that curve (or its metastable prolongation) which
represents the two phases of most widely differing specific volume.

Now, although these rules admit of a considerable variety of possible
arrangements of curves around the triple point,[101] only two of these have
been experimentally obtained in the case of the triple point
solid--liquid--vapour. At present, therefore, we shall consider only these
two cases (Figs. 13 and 14).

[Illustration: FIG. 13.]

[Illustration: FIG. 14.]

An examination of these two figures shows that they satisfy the rules laid
down. Each of the curves on being prolonged passes between the other two
curves. In the case of substances of the first type (Fig. 13), the specific
volume of the solid is greater than that of the liquid (the substance
contracts on fusion); the difference of specific volume will, therefore, be
greatest between liquid and vapour. The curve, therefore, for liquid and
vapour (or its prolongation) must lie between the other two curves; this is
seen from the figure to be the case. Similarly, the rule is satisfied by
the arrangement of curves in Fig. 14, where the difference of specific
volumes is {57} greatest between the solid and vapour. In this case the
curve S-V occupies the intermediate position.

As we see, the two figures differ from one another only in that the fusion
curve OC in one case slopes to the right away from the pressure axis, thus
indicating that the melting point is raised by increase of pressure; in the
other case, to the left, indicating a lowering of the melting point with
the pressure. These conditions are found exemplified in the case of sulphur
and ice (pp. 29 and 35). We see further from the two figures, that O in
Fig. 13 gives the highest temperature at which the solid can exist, for the
curve for solid--liquid slopes back to regions of lower temperature; in
Fig. 14, O gives the lowest temperature at which the liquid phase can exist
as stable phase.[102]

Theorems of van't Hoff and of Le Chatelier.--So far we have studied only
the conditions under which various systems exist in equilibrium; and we now
pass to a consideration of the changes which take place in a system when
the external conditions of temperature and pressure are altered. For all
such changes there exist two theorems, based on the laws of thermodynamics,
by means of which the alterations in a system can be qualitatively
predicted.[103] The first of these, usually {58} known as van't Hoff's _law
of movable equilibrium_,[104] states: When the temperature of a system in
equilibrium is raised, that reaction takes place which is accompanied by
absorption of heat; and, conversely, when the temperature is lowered, that
reaction occurs which is accompanied by an evolution of heat.

The second of the two theorems refers to the effect of change of pressure,
and states:[105] When the pressure on a system in equilibrium is increased,
that reaction takes place which is accompanied by a diminution of volume;
and when the pressure is diminished, a reaction ensues which is accompanied
by an increase of volume.

The demonstration of the universal applicability of these two theorems is
due chiefly to Le Chatelier, who showed that they may be regarded as
consequences of the general law of action and reaction. For this reason
they are generally regarded as special cases of the more general law, known
as the _theorem of Le Chatelier_, which may be stated in the words of
Ostwald, as follows:[106] _If a system in equilibrium is subjected to a
constraint by which the equilibrium is shifted, a reaction takes place
which opposes the constraint, _i.e._ one by which its effect is partially
destroyed._

This theorem of Le Chatelier is of very great importance, for it applies to
all systems and changes of the condition of equilibrium, whether physical
or chemical; to vaporization and fusion; to solution and chemical action.
In all cases, whenever changes in the external condition of a system in
equilibrium are produced, processes also occur within the system which tend
to counteract the effect of the external changes.

_Changes at the Triple Point._--If now we apply this theorem to equilibria
at the triple point S-L-V, and ask what changes will occur in such a system
when the external conditions of pressure and temperature are altered, the
general answer to the question will be: So long as the three phases are
present, no {59} change in the temperature or pressure of the system can
occur, but _only changes in the relative amounts of the phases_; that is to
say, the effect on the system of change in the external conditions is
opposed by the reactions or changes which take place within the system
(according to the theorems of van't Hoff and Le Chatelier). We now proceed
to discuss what these changes are, and shall consider first the effect of
alteration of the temperature at constant volume and constant pressure, and
then the effect of alteration of the pressure both when the temperature
remains constant and when it varies.

When the volume is kept constant, the effect of the addition of heat to a
system at the triple point S-L-V differs somewhat according as there is an
increase or diminution of volume when the solid passes into the liquid
state. In the former and most general case (Fig. 14), addition of heat will
cause a certain amount of the solid phase to melt, whereby the heat which
is added becomes latent; the temperature of the system therefore does not
rise. Since, however, the melting of the solid is accompanied by an
increase of volume, whereby an increase of pressure would result, a certain
portion of the vapour must condense to liquid, in order that the pressure
may remain constant. The total effect of addition of heat, therefore, is to
cause both solid and vapour to pass into liquid, _i.e._ there occurs the
change S + V --> L. It will, therefore, depend on the relative quantities
of solid and vapour, which will disappear first. If the solid disappears
first, then we shall pass to the system L-V; if vapour disappears first, we
shall obtain the system S-L. Withdrawal of heat causes the reverse change,
L --> S + V; at all temperatures below the triple point the liquid is
unstable or metastable (p. 30).

When fusion is accompanied by a diminution of volume (_e.g._ ice, Fig. 13),
then, since the melting of the solid phase would decrease the total volume,
_i.e._ would lower the pressure, a certain quantity of the solid must also
pass into vapour in order that the pressure may be maintained constant. On
addition of heat, therefore, there occurs the reaction S --> L + V;
withdrawal of heat causes the reverse change L + V --> S. Above the
temperature of the triple point the {60} solid cannot exist; below the
triple point both systems, S-L and S-V, can exist, and it will therefore
depend on the relative amounts of liquid and vapour which of these two
systems is obtained on withdrawing heat from the system at constant volume.

The same changes in the phases occur when heat is added or withdrawn at
constant pressure, so long as the three phases are present. Continued
addition of heat, however, at constant pressure will ultimately cause the
formation of the bivariant system vapour alone; continued withdrawal of
heat will ultimately cause the formation of solid alone. This will be
readily understood from Fig. 15. The dotted line D'OD is a line of constant
pressure; on adding heat, the system passes along the line OD into the
region of vapour; on heat being withdrawn, the system passes along OD' into
the area of solid.

[Illustration: FIG. 15.]

Similar changes are produced when the volume of the system is altered.
Alteration of volume may take place either while transference of heat to or
from the system is cut off (adiabatic change), or while such transference
may occur (isothermal change). In the latter case, the temperature of the
system will remain constant; in the former case, since at the triple point
the pressure must be constant so long as the three phases are present,
increase of volume must be compensated by the evaporation of liquid. This,
however, would cause the temperature to fall (since communication of heat
from the outside is supposed to be cut off), and a portion of the liquid
must therefore freeze. In this way the latent heat of evaporation is
counterbalanced by the latent heat of fusion. As the result of increase of
volume, therefore, the process occurs L --> S + V. Diminution of volume,
without transference of heat, will bring about the opposite change, S + V
--> L. In the former case there is ultimately obtained the univariant
system S-V; in the latter case there will be {61} obtained either S-L or
L-V according as the vapour or solid phase disappears first.

This argument holds good for both types of triple point shown in Figs. 13
and 14 (p. 57). A glance at these figures will show that increase of volume
(diminution of pressure) will lead ultimately to the system S-V, for at
pressures lower than that of the triple point, the liquid phase cannot
exist. Decrease of volume (increase of pressure), on the other hand, will
lead either to the system S-L or L-V, because these systems can exist at
pressures higher than that of the triple point. If the vapour phase
disappears and we pass to the curve S-L, continued diminution of volume
will be accompanied by a fall in temperature in the case of systems of the
first type (Fig. 13), and by a rise in temperature in the case of systems
of the second type (Fig. 14).

[Illustration: FIG. 16.]

[Illustration: FIG. 17.]

Lastly, if the temperature is maintained constant, _i.e._ if heat can pass
into or out of the system, then on changing the volume the same changes in
the phases will take place as described above until one of the phases has
disappeared. Continued increase of volume (decrease of pressure) will then
cause the disappearance of a second phase, the system passing along the
dotted line OE' (Figs. 16, 17), so that ultimately there remains only the
vapour phase. Conversely, diminution of volume (increase of pressure) will
ultimately lead either to solid (Fig. 16) or to liquid alone (Fig. 17), the
system passing along the dotted line OE. {62}

In discussing the alterations which may take place at the triple point with
change of temperature and pressure, we have considered only the triple
point S-L-V. The same reasoning, however, applies, _mutatis mutandis_, to
all other triple points, so that if the specific volumes of the phases are
known, and the sign of the heat effects which accompany the transformation
of one phase into the other, it is possible to predict (by means of the
theorem of Le Chatelier) the changes which will be produced in the system
by alteration of the pressure and temperature.

In all cases of transformation at the triple point, it should be noted that
all _three phases are involved in the change_,[107] and not two only; the
fact that in the case, say, of the transformation from solid to liquid, or
liquid to solid, at the melting point with change of temperature, only
these two phases appear to be affected, is due to there generally being a
large excess of the vapour phase present and to the prior disappearance
therefore of the solid or liquid phase.

In the case of triple points at which two solid phases are in equilibrium
with liquid, other arrangements of the curves around the triple point are
found. It is, however, unnecessary to give a general treatment of these
here, since the principles which have been applied to the triple point
S-L-V can also be applied to the other triple points.[108]

Triple Point Solid--Solid--Vapour.--The triple point solid--solid--vapour
is one which is of considerable importance. Examples of such a triple point
have already been given in sulphur and tin, and a list of other substances
capable of yielding two solid phases is given below. The triple point S-S-V
is not precisely the same as the transition point, but is very nearly so.
The transition point is the temperature at which the relative stability of
the two solid phases undergoes change, when the vapour phase is absent and
the pressure is 1 atm.; whereas at the triple point the pressure is that of
the system itself. The transition point, therefore, bears the same relation
to the triple point S-S-V as the melting point to the triple point S-L-V.

{63}

In the following table is given a list of the most important polymorphous
substances, and the temperatures of the transition point.[109]

  ------------------------------------+-------------
                                      |
              Substance.              |  Transition
                                      | temperature.
  ------------------------------------+-------------
                                      |
  Ammonium nitrate--                  |
   [beta]-rhombic --> [alpha]-rhombic |     35°
   [alpha]-rhombic --> rhombohedral   |     83°
   Rhombohedral --> regular           |    125°
  Mercuric iodide                     |    126°
  Potassium nitrate                   |    129°
  Silver iodide                       |    145°
  Silver nitrate                      |    160°
  Sulphur                             |     95.5°
  Tetrabrommethane                    |     46.8°
  Thallium nitrate--                  |
   Rhombic --> rhombohedral           |     80°
   Rhombohedral --> regular           |    142.5°
  Thallium picrate                    |     46°
  Tin                                 |     20°
  ------------------------------------+-------------

Sublimation and Vaporization Curves.--We have already seen, in the case of
ice and liquid water, that the vapour pressure increases as the temperature
rises, the increase of pressure per degree being greater the higher the
temperature. The sublimation and vaporization curves, therefore, are not
straight lines, but are bent, the convex side of the curve being towards
the temperature axis in the ordinary _pt_-diagram.

In the case of sulphur and of tin, we assumed vapour to be given off by the
solid substance, although the pressure of the vapour has not hitherto been
measured. The assumption, however, is entirely justified, not only on
theoretical grounds, but also because the existence of a vapour pressure
has been observed in the case of many solid substances at temperatures much
below the melting point,[110] and in some cases, _e.g._ camphor,[111] the
vapour pressure is considerable.

{64}

As the result of a large number of determinations, it has been found that
all vapour pressure curves have the same general form alluded to above.
Attempts have also been made to obtain a general expression for the
quantitative changes in the vapour pressure with change of temperature, but
without success. Nevertheless, the _qualitative_ changes, or the general
direction of the curves, can be predicted by means of the theorem of Le
Chatelier.

As we have already learned (p. 16), the Phase Rule takes no account of the
molecular complexity of the substances participating in an equilibrium. A
dissociating substance, therefore, in contact with its vaporous products of
dissociation (_e.g._ ammonium chloride in contact with ammonia and hydrogen
chloride), will likewise constitute a univariant system of one component,
provided the composition of the vapour phase as a whole is the same as that
of the solid or liquid phase (p. 13). For all such substances, therefore,
the conditions of equilibrium will be represented by a curve of the same
general form as the vapour pressure curve of a non-dissociating
substance.[112] The same behaviour is also found in the case of substances
which polymerize on passing into the solid or liquid state (_e.g._ red
phosphorus). Where such changes in the molecular state occur, however, the
time required for equilibrium to be established is, as a rule, greater than
when the molecular state is the same in both phases.

From an examination of Figs. 13 and 14, it will be easy to predict the
effect of change of pressure and temperature on the univariant systems S-V
or L-V. If the volume is kept constant, addition of heat will cause an
increase of pressure, the system S-V moving along the curve AO until at the
triple point the liquid phase is formed, and the system L-V moving along
the curve OB; so long as two phases are present, the condition of the
system must be represented by these two curves. Conversely, withdrawal of
heat will cause condensation of vapour, and therefore diminution of
pressure; the system will therefore move along the vaporization or
sublimation curve to lower temperatures and pressures, so long as the
system remains univariant.

{65}

If transference of heat to or from the system is prevented, increase of
volume (diminution of pressure) will cause the system L-V to pass along the
curve BO; liquid will pass into vapour and the temperature will fall.[113]
At O solid may appear, and the temperature of the system will then remain
constant until the liquid phase has disappeared (p. 57); the system will
then follow the curve OA until the solid phase disappears, and we are
ultimately left with vapour. On the other hand, diminution of volume
(increase of pressure) will cause condensation of vapour, and the system
S-V will pass along the curve AO to higher temperatures and pressures; at O
the solid will melt, and the system will ultimately pass to the curve OB or
to OC (p. 57).

Addition or withdrawal of heat at constant pressure, and increase or
diminution of the pressure at constant temperature, will cause the system
to pass along lines parallel to the temperature and the pressure axis
respectively; the working out of these changes may be left to the reader,
guided by what has been said on pp. 60 and 61.

The sublimation curve of all substances, so far as yet found, has its upper
limit at the melting point (triple point), although the possibility of the
existence of a superheated solid is not excluded. The lower limit is,
theoretically at least, at the absolute zero, provided no new phase, _e.g._
a different crystalline modification, is formed. If the sublimation
pressure of a substance is greater than the atmospheric pressure at any
temperature below the point of fusion, then the substance will _sublime
without melting_ when heated in an open vessel; and fusion will be possible
only at a pressure higher than the atmospheric. This is found, for example,
in the case of red phosphorus (p. 47). If, however, the sublimation
pressure of a substance at its triple point S-L-V is less than one
atmosphere, then the substance will melt when heated in an open vessel.

In the case of the vaporization curve, the upper limit lies at the critical
point where the liquid ceases to exist;[114] the {66} lower limit is
determined by the range of the metastable state of the supercooled liquid.

The interpolation and extrapolation of vapour-pressure curves is rendered
very easy by means of a relationship which Ramsay and Young[115] found to
exist between the vapour-pressure curves of different substances. It was
observed that in the case of closely related substances, the ratio of the
absolute temperatures corresponding to equal vapour pressures is constant,
_i.e._ T_{1}/T'_{1} = T_{2}/T'_{2}. When the two substances are not closely
related, it was found that the relationship could be expressed by the
equation T_{1}/T'_{1} = T_{2}/T'_{2} + _c_(_t_' - _t_) where _c_ is a
constant having a small positive or negative value, and _t_' and _t_ are
the temperatures at which one of the substances has the two values of the
vapour pressure in question. By means of this equation, if the
vapour-pressure curve of one substance is known, the vapour-pressure curve
of any other substance can be calculated from the values at any two
temperatures of the vapour pressure of that substance.

Fusion Curve--Transition Curve.--The fusion curve represents the conditions
of equilibrium between the solid and liquid phase; it shows the change of
the melting point of a substance with change of pressure.

As shown in Figs. 13 and 14, the fusion curve is inclined either towards
the pressure axis or away from it; that is, increase of pressure can either
lower or raise the melting point. It is easy to predict in a qualitative
manner the different effect of pressure on the melting point in the two
cases mentioned, if we consider the matter in the light of the theorem of
Le Chatelier (p. 58). Water, on passing into ice, expands; therefore, if
the pressure on the system ice--water be increased, a reaction will take
place which is accompanied by a diminution in volume, _i.e._ the ice will
melt. Consequently, a lower temperature will be required in order to
counteract the effect of increase of pressure; or, in other words, the
melting point will {67} be lowered by pressure.[116] In the second case,
the passage of the liquid to the solid state is accompanied by a diminution
of volume; the effect of increase of pressure will therefore be the reverse
of that in the previous case.

If the value of the heat of fusion and the alteration of volume
accompanying the change of state are known, it is possible to calculate
_quantitatively_ the effect of pressure.[117]

We have already seen (p. 25) that the effect of pressure on the melting
point of a substance was predicted as the result of theoretical
considerations, and was first proved experimentally in the case of ice.
Soon after, Bunsen[118] showed that the melting point of other substances
is also affected by pressure; and in more recent years, ample experimental
proof of the change of the melting point with the pressure has been
obtained. The change of the melting point is, however, small; as a rule,
increase of pressure by 1 atm. changes the melting point by about 0.03°,
but in the case of water the change is much less (0.0076°), and in the case
of camphor much more (0.13°). In other words, if we take the mean case, an
increase of pressure of more than 30 atm. is required to produce a change
in the melting point of 1°.

Investigations which were made of the influence of pressure on the
melting-point, showed that up to pressures of several hundred atmospheres
the fusion curve is a straight line.[119] Tammann[120] has, however, found
that on increasing the pressure the fusion curve no longer remains
straight, but bends towards the pressure axis, so that, on sufficiently
increasing the pressure, a maximum temperature might at length be reached.
This maximum has, so far, however, not been attained, although the melting
point curves of various substances have been studied up to pressures of
4500 atm. This is to be accounted for partly {68} by the fact that the
probable maximum temperature in the case of most substances lies at very
great pressures, and also by the fact that other solid phases make their
appearance, as, for example, in the case of ice (p. 32).

As to the upper limit of the fusion curve, the view has been expressed[121]
that just as in the case of liquid and vapour, so also in the case of solid
and liquid, there exists a critical point at which the solid and the liquid
phase become identical. Experimental evidence, however, does not appear to
favour this view.[122]

The _transition point_, like the melting point, is also influenced by the
pressure, and in this case also it is found that pressure may either raise
or lower the transition point, so that the transition curve may be inclined
either away from or towards the pressure axis. The direction of the
transition curve can also be predicted if the change of volume accompanying
the passage of one form into the other is known. In the case of sulphur, we
saw that the transition point is raised by increase of pressure; in the
case of the transition of rhombohedral into [alpha]-rhombic form of
ammonium nitrate, however, the transition point is lowered by pressure, as
shown by the following table.[123]

  -------------+----------
               |
  Temperature. | Pressure.
  -------------+----------
               |
     85.85°    |   1 atm.
     84.38°    | 100  "
     83.03°    | 200  "
     82.29°    | 250  "
  -------------+----------

So far as investigations have been carried out, it appears that in most
cases the transition curve is practically a straight line.

It has, however, been found in the case of Glauber's salt, that with
increase of pressure the transition curve passes through a point of maximum
temperature, and exhibits, therefore, a form similar to that assumed by
Tammann for the fusion curve.[124]

{69}

Suspended Transformation. Metastable Equilibria.--Hitherto we have
considered only systems in stable equilibrium. We have, however, already
seen, in the case of water, that on cooling the liquid down to the triple
point, solidification did not necessarily take place, although the
conditions were such as to allow of its formation. Similarly, we saw that
rhombic sulphur can be heated above the transition point, and monoclinic
sulphur can be obtained at temperatures below the transition point,
although in both cases transformation into a more stable form is possible;
the system becomes metastable.

The same reluctance to form a new phase is observed also in the phenomena
of superheating of liquids, and in the "hanging" of mercury in barometers,
in which case the vapour phase is not formed. In general, then, we may say
that _a new phase will not necessarily be formed immediately the system
passes into such a condition that the existence of that phase is possible_;
but rather, instead of the system undergoing transformation so as to pass
into the most stable condition under the existing pressure and temperature,
this transformation will be "suspended" or delayed, and the system will
become metastable. Only in the case of the formation of the liquid from the
solid phase, in a one-component system, has this reluctance to form a new
phase not been observed.

_To ensure the formation of the new phase, it is necessary to have that
phase present._ The presence of the solid phase will prevent the
supercooling of the liquid; and the presence of the vapour phase will
prevent the superheating of the liquid. However, even in the presence of
the more stable phase, transformation of the metastable phase occurs with
very varying velocity; in some cases so quickly as to appear almost
instantaneous; while in other cases, the change takes place so slowly as to
require hundreds of years for its achievement. It is this slow rate of
transformation that renders the existence of metastable forms possible,
when in contact with the more stable phase. Thus, for example, although
calcite is the most stable form of calcium carbonate at the ordinary
temperature,[125] the less stable {70} modification, aragonite,
nevertheless exists under the ordinary conditions in an apparently very
stable state.

As to the amount of the new phase required to bring about the
transformation of the metastable phase, quantitative measurements have been
carried out only in the case of the initiation of crystallization in a
supercooled liquid.[126] As the result of these investigations, it was
found that, in the case of superfused salol, the very small amount of 1 ×
10^{-7} gm. of the solid phase was sufficient to induce crystallization.
Crystallization of a supercooled liquid, however, can be initiated only by
a "nucleus" of the same substance in the solid state, or, as has also been
found, by a nucleus of an isomorphous solid phase; it is not brought about
by the presence of any chance solid.

Velocity of Transformation.--Attention has already been drawn to the
sluggishness with which reciprocal transformation of the polymorphic forms
of a substance may occur. In the case of tin, for example, it was found
that the white modification, although apparently possessing permanence, is
in reality in a metastable state, under the ordinary conditions of
temperature and pressure. This great degree of stability is due to the
tardiness with which transformation into the grey form occurs.

What was found in the case of tin, is met with also in the case of all
transformations in the solid state, but the velocity of the change is less
in some cases than in others, and appears to decrease with increase of the
valency of the element.[127] To this fact van't Hoff attributes the great
permanence of many really unstable (or metastable) carbon compounds.

Reference has been made to the fact that the velocity of transformation can
be accelerated by various means. One of the most important of these is the
employment of a liquid which has a solvent action on the solid phases. Just
as we have seen that at any given temperature the less stable form has the
higher vapour pressure, but that at the transition point the vapour
pressure of both forms becomes identical, so also it can be proved
theoretically, and be shown experimentally, that {71} at a given
temperature the solubility of the less stable form is greater than that of
the more stable, but that at the transition point the solubility of the two
forms becomes identical.[128]

If, then, the two solid phases are brought into contact with a solvent, the
less stable phase will dissolve more abundantly than the more stable; the
solution will therefore become supersaturated with respect to the latter,
which will be deposited. A gradual change of the less stable form,
therefore, takes place through the medium of the solvent. In this way the
more rapid conversion of white tin into grey in presence of a solution of
tin ammonium chloride (p. 42) is to be explained. Although, as a rule,
solvents accelerate the transformation of one solid phase into the other,
they may also have a retarding influence on the velocity of transformation,
as was found by Reinders in the case of mercuric iodide.[129]

The velocity of inversion, also, is variously affected by different
solvents, and in some cases, at least, it appears to be slower the more
viscous the solvent;[130] indeed, Kastle and Reed state that yellow
crystals of mercuric iodide, which, ordinarily, change with considerable
velocity into the red modification, have been preserved for more than a
year under vaseline.

Temperature, also, has a very considerable influence on the velocity of
transformation. The higher the temperature, and the farther it is removed
from the equilibrium point (transition point), the greater is the velocity
of change. Above the transition point, these two factors act in the same
direction, and the velocity of transformation will therefore go on
increasing indefinitely the higher the temperature is raised. Below the
transition point, however, the two factors act in opposite directions, and
the more the temperature is lowered, the more is the effect of removal from
the equilibrium point counteracted. A point will therefore be reached at
which the velocity is a maximum. Reduction of the temperature {72} below
this point causes a rapid falling off in the velocity of change. The point
of maximum velocity, however, is not definite, but may be altered by
various causes. Thus, Cohen found that in the case of tin, the point of
maximum velocity was altered if the metal had already undergone
transformation; and also by the presence of different liquids.[131]

Lastly, the presence of small quantities of different substances--catalytic
agents or catalyzers--has a great influence on the velocity of
transformation. Thus, _e.g._, the conversion of white to red phosphorus is
accelerated by the presence of iodine (p. 47).

Greater attention, however, has been paid to the study of the velocity of
crystallization of a supercooled liquid, the first experiments in this
direction having been made by Gernez[132] on the velocity of
crystallization of phosphorus and sulphur. Since that time, the velocity of
crystallization of other supercooled liquids has been investigated; such as
acetic acid and phenol by Moore;[133] supercooled water by Tumlirz;[134]
and a number of organic substances by Tammann,[135] Friedländer and
Tammann,[136] and by Bogojawlenski.[137]

In measuring the velocity of crystallization, the supercooled liquids were
contained in narrow glass tubes, and the time required for the
crystallization to advance along a certain length of the tube was
determined, the velocity being expressed in millimetres per minute. The
results which have so far been obtained may be summarized as follows. For
any given degree of supercooling of a substance, the velocity of
crystallization is constant. As the degree of supercooling increases, the
velocity of crystallization also increases, until a certain point is
reached at which the velocity is a maximum, which has a definite
characteristic value for each substance. This maximum velocity remains
constant over a certain range of {73} temperature; thereafter, the velocity
diminishes fairly rapidly, and, with sufficient supercooling, may become
zero. The liquid then passes into a glassy mass, which will remain
(practically) permanent even in contact with the crystalline solid.

In ordinary glass we have a familiar example of a liquid which has been
cooled to a temperature at which crystallization takes place with very
great slowness. If, however, glass is heated, a temperature is reached,
much below the melting point of the glass, at which crystallization occurs
with appreciable velocity, and we observe the phenomenon of
devitrification.[138]

When the velocity of crystallization is studied at temperatures above the
maximum point, it is found that the velocity is diminished by the addition
of foreign substances; and in many cases, indeed, it has been found that
the diminution is the same for equimolecular quantities of different
substances. It would hence appear possible to utilize this behaviour as a
method for determining molecular weights.[139] The rule is, however, by no
means a universal one. Thus it has been found by F. Dreyer,[140] in
studying the velocity of crystallization of formanilide, that the
diminution in the velocity produced by equivalent amounts of different
substances is not the same, but that the foreign substances exercise a
specific influence. Further, von Pickardt's rule does not hold when the
foreign substance forms mixed crystals (Chap. X.) with the crystallizing
substance.[141]

Law of Successive Reactions.--When sulphur vapour is cooled at the ordinary
temperature, it first of all condenses to drops of liquid, which solidify
in an amorphous form, and only after some time undergo crystallization; or,
when phosphorus vapour is condensed, white phosphorus is first formed, and
not the more stable form--red phosphorus. It has also been observed that
even at the ordinary temperature (therefore much below the transition
point) sulphur may crystallize out from solution in benzene, alcohol,
carbon disulphide, and other {74} solvents, in the prismatic form, the less
stable prismatic crystals then undergoing transformation into the rhombic
form;[142] a similar behaviour has also been observed in the transformation
of the monotropic crystalline forms of sulphur.[143]

Many other examples might be given. In organic chemistry, for instance, it
is often found that when a substance is thrown out of solution, it is first
deposited as a liquid, which passes later into the more stable crystalline
form. In analysis, also, rapid precipitation from concentrated solution
often causes the separation of a less stable and more soluble amorphous
form.

On account of the great frequency with which the prior formation of the
less stable form occurs, Ostwald[144] has put forward the _law of
successive reactions_, which states that when a system passes from a less
stable condition it does not pass directly into the most stable of the
possible states; but into the next more stable, and so step by step into
the most stable. This law explains the formation of the metastable forms of
monotropic substances, which would otherwise not be obtainable. Although it
is not always possible to observe the formation of the least stable form,
it should be remembered that that may quite conceivably be due to the great
velocity of transformation of the less stable into the more stable form.
From what we have learned about the velocity of transformation of
metastable phases, we can understand that rapid cooling to a low
temperature will tend to preserve the less stable form; and, on account of
the influence of temperature in increasing the velocity of change, it can
be seen that the formation of the less stable form will be more difficult
to observe in superheated than in supercooled systems. The factors,
however, which affect the readiness with which {75} the less stable
modification is produced, appear to be rather various.[145]

Although a number of at least apparent exceptions to Ostwald's law have
been found, it may nevertheless be accepted as a very useful generalization
which sums up very frequently observed phenomena.

       *       *       *       *       *


{76}

CHAPTER V

SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION

In the preceding pages we have studied the behaviour of systems consisting
of only one component, or systems in which all the phases, whether solid,
liquid, or vapour, had the same chemical composition (p. 13). In some
cases, as, for example, in the case of phosphorus and sulphur, the
component was an elementary substance; in other cases, however, _e.g._
water, the component was a compound. The systems which we now proceed to
study are characterized by the fact that the different phases have no
longer all the same chemical composition, and cannot, therefore, according
to definition, be considered as one-component systems.

In most cases, little or no difficulty will be experienced in deciding as
to the _number_ of the components, if the rules given on pp. 12 and 13 are
borne in mind. If the composition of all the phases, each regarded as a
whole, is the same, the system is to be regarded as of the first order, or
a one-component system; if the composition of the different phases varies,
the system must contain more than one component. If, in order to _express_
the composition of all the phases present when the system is in
equilibrium, two of the constituents participating in the equilibrium are
necessary and sufficient, the system is one of two components. Which two of
the possible substances are to be regarded as components will, however, be
to a certain extent a matter of arbitrary choice.

The principles affecting the choice of components will best be learned by a
study of the examples to be discussed in the sequel. {77}

Different Systems of Two Components.--Applying the Phase Rule

  P + F = C + 2

to systems of two components, we see that in order that the system may be
invariant, there must be four phases in equilibrium together; two
components in three phases constitute a univariant, two components in two
phases a bivariant system. In the case of systems of one component, the
highest degree of variability found was two (one component in one phase);
but, as is evident from the formula, there is a higher degree of freedom
possible in the case of two-component systems. Two components existing in
only one phase constitute a tervariant system, or a system with three
degrees of freedom. In addition to the pressure and temperature, therefore,
a third variable factor must be chosen, and as such there is taken the
_concentration of the components_. In systems of two components, therefore,
not only may there be change of pressure and temperature, as in the case of
one-component systems, but the concentration of the components in the
different phases may also alter; a variation which did not require to be
considered in the case of one-component systems.

[Illustration: FIG. 18.]

Since a two-component system may undergo three possible {78} independent
variations, we should require for the graphic representation of all the
possible conditions of equilibrium a system of three co-ordinates in space,
three axes being chosen, say, at right angles to one another, and
representing the three variables--pressure, temperature, and concentration
of components (Fig. 18). A curve (_e.g._ AB) in the plane containing the
pressure and temperature axes would then represent the change of pressure
with the temperature, the concentration remaining unaltered (_pt_-diagram);
one in the plane containing the pressure and concentration axes (_e.g._ AF
or DF), the change of pressure with the concentration, the temperature
remaining constant (_pc_-diagram), while in the plane containing the
concentration and the temperature axes, the simultaneous change of these
two factors at constant pressure would be represented (_tc_-diagram). If
the points on these three curves are joined together, a surface, ABDE, will
be formed, and any line on that surface (_e.g._ FG, or GH, or GI) would
represent the simultaneous variation of the three factors--pressure,
temperature, concentration. Although we shall at a later point make some
use of these solid figures, we shall for the present employ the more
readily intelligible plane diagram.

The number of different systems which can be formed from two components, as
well as the number of the different phenomena which can there be observed,
is much greater than in the case of one component. In the case of no two
substances, however, have all the possible relationships been studied; so
that for the purpose of gaining an insight into the very varied behaviour
of two-component systems, a number of different examples will be discussed,
each of which will serve to give a picture of some of the relationships.

Although the strict classification of the different systems according to
the Phase Rule would be based on the variability of the systems, the study
of the many different phenomena, and the correlation of the comparatively
large number of different systems, will probably be rendered easiest by
grouping these different phenomena into classes, each of these classes
being studied with the help of one or more typical examples. The order of
treatment adopted here is, of course, quite arbitrary; {79} but has been
selected from considerations of simplicity and clearness.

PHENOMENA OF DISSOCIATION.

Bivariant Systems.--As the first examples of the equilibria between a
substance and its products of dissociation, we shall consider very briefly
those cases in which there is one solid phase in equilibrium with vapour.
Reference has already been made to such systems in the case of ammonium
chloride. On being heated, ammonium chloride dissociates into ammonia and
hydrogen chloride. Since, however, in that case the vapour phase has the
same total composition as the solid phase, viz. NH_{3} + HCl = NH_{4}Cl,
the system consists of only one component existing in two phases; it is
therefore univariant, and to each temperature there will correspond a
definite vapour pressure (dissociation pressure).[146]

If, however, excess of one of the products of dissociation be added, the
system becomes one of two components.

In the first place, analysis of each of the two phases yields as the
composition of each, solid: NH_{4}Cl (= NH_{3} + HCl); vapour: _m_NH_{3} +
_n_HCl. Obviously the smallest number of substances by which the
composition of the two phases can be expressed is two; that is, the number
of components is two. What, then, are the components? The choice lies
between NH_{3} + HCl, NH_{4}Cl + NH_{3}, and NH_{4}Cl + HCl; for the three
substances, ammonium chloride, ammonia, hydrogen chloride, are the only
ones taking part in the equilibrium of the system.

Of these three pairs of components, we should obviously choose as the most
simple NH_{3} and HCl, for we can then represent the composition of the two
phases as the _sum_ of the two components. If one of the other two possible
pairs of components be chosen, we should have to introduce negative
quantities of one of the components, in order to represent the composition
of the vapour phase. Although it must be allowed that the introduction of
negative quantities of a component in such cases is quite permissible,
still it will be {80} better to adopt the simpler and more direct choice,
whereby the composition of each of the phases is represented as a sum of
two components in varying proportions (p. 12).

If, therefore, we have a solid substance, such as ammonium chloride, which
dissociates on volatilization, and if the products of dissociation are
added in varying amounts to the system, we shall have, in the sense of the
Phase Rule, a _two-component system existing in two phases_. Such a system
will possess two degrees of freedom. At any given temperature, not only the
pressure, but also the composition, of the vapour-phase, _i.e._ the
concentration of the components, can vary. Only after one of these
independent variables, pressure or composition, has been arbitrarily fixed
does the system become univariant, and exhibit a definite, constant
pressure at a given temperature.

Now, although the Phase Rule informs us that at a given temperature change
of composition of the vapour phase will be accompanied by change of
pressure, it does not cast any light on the relation between these two
variables. This relationship, however, can be calculated theoretically by
means of the Law of Mass Action.[147] From this we learn that in the case
of a substance which dissociates into equivalent quantities of two gases,
the product of the partial pressures of the gases is constant at a given
temperature.

This has been proved experimentally in the case of ammonium hydrosulphide,
ammonium cyanide, phosphonium bromide, and other substances.[148]

Univariant Systems.--In order that a system of two components shall possess
only one degree of freedom, three phases must be present. Of such systems,
there are seven possible, viz. S-S-S, S-S-L, S-S-V, L-L-L, S-L-L, L-L-V,
S-L-V; S denoting solid, L liquid, and V vapour. In the present chapter we
shall consider only the systems S-S-V, _i.e._ those systems in which there
are two solid phases and a vapour phase present.

{81}

As an example of this, we may first consider the well-known case of the
dissociation of calcium carbonate. This substance on being heated
dissociates into calcium oxide, or quick-lime, and carbon dioxide, as shown
by the equation CaCO_{3} <--> CaO + CO_{2}. In accordance with our
definition (p. 9), we have here two solid phases, the carbonate and the
quick-lime, and one vapour phase; the system is therefore univariant. To
each temperature, therefore, there will correspond a certain, definite
maximum pressure of carbon dioxide (dissociation pressure), and this will
follow the same law as the vapour pressure of a pure liquid (p. 21). More
particularly, it will be independent of the relative or absolute amounts of
the two solid phases, and of the volume of the vapour phase. If the
temperature is maintained constant, increase of volume will cause the
dissociation of a further amount of the carbonate until the pressure again
reaches its maximum value corresponding to the given temperature.
Diminution of volume, on the other hand, will bring about the combination
of a certain quantity of the carbon dioxide with the calcium oxide until
the pressure again reaches its original value.

The dissociation pressure of calcium carbonate was first studied by
Debray,[149] but more exact measurements have been made by Le
Chatelier,[150] who found the following corresponding values of temperature
and pressure:--

  -------------+-------------------------
               |
  Temperature. | Pressure in cm. mercury.
  -------------+-------------------------
               |
      547°     |             2.7
      610°     |             4.6
      625°     |             5.6
      740°     |            25.5
      745°     |            28.9
      810°     |            67.8
      812°     |            76.3
      865°     |           133.3
  -------------+-------------------------

From this table we see that it is only at a temperature of about 812° that
the pressure of the carbon dioxide becomes equal to atmospheric pressure.
In a vessel open to {82} the air, therefore, the complete decomposition of
the calcium carbonate would not take place below this temperature by the
mere heating of the carbonate. If, however, the carbon dioxide is removed
as quickly as it is formed, say by a current of air, then the entire
decomposition can be made to take place at a much lower temperature. For
the dissociation equilibrium of the carbonate depends only on the partial
pressure of the carbon dioxide, and if this is kept small, then the
decomposition can proceed, even at a temperature below that at which the
pressure of the carbon dioxide is less than atmospheric pressure.

Ammonia Compounds of Metal Chlorides.--Ammonia possesses the property of
combining with various substances, chiefly the halides of metals, to form
compounds which again yield up the ammonia on being heated. Thus, for
example, on passing ammonia over silver chloride, absorption of the gas
takes place with formation of the substances AgCl,3NH_{3} and
2AgCl,3NH_{3}, according to the conditions of the experiment. These were
the first known substances belonging to this class, and were employed by
Faraday in his experiments on the liquefaction of ammonia. Similar
compounds have also been obtained by the action of ammonia on silver
bromide, iodide, cyanide, and nitrate; and with the halogen compounds of
calcium, zinc, and magnesium, as well as with other salts. The behaviour of
the ammonia compounds of silver chloride is typical for the compounds of
this class, and may be briefly considered here.

It was found by Isambert[151] that at temperatures below 15°, silver
chloride combined with ammonia to form the compound AgCl,3NH_{3}, while at
temperatures above 20° the compound 2AgCl,3NH_{3} was produced. On heating
these substances, ammonia was evolved, and the pressure of this gas was
found in the case of both compounds to be constant at a given temperature,
but was greater in the case of the former than in the case of the latter
substance; the pressure, further, was independent of the amount decomposed.
The behaviour of these two substances is, therefore, exactly analogous to
that shown by calcium carbonate, and the explanation is also similar.

{83}

Regarded from the point of view of the Phase Rule, we see that we are here
dealing with two components, AgCl and NH_{3}. On being heated, the
compounds decompose according to the equations:--

  2(AgCl,3NH_{3}) <--> 2AgCl,3NH_{3} + 3NH_{3}.
    2AgCl,3NH_{3} <--> 2AgCl + 3NH_{3}.

There are, therefore, three phases, viz. AgCl,3NH_{3}; 2AgCl,3NH_{3}, and
NH_{3}, in the one case; and 2AgCl,3NH_{3}; AgCl, and NH_{3} in the other.
These two systems are therefore univariant, and to each temperature there
must correspond a definite pressure of dissociation, quite irrespective of
the amounts of the phases present. Similarly, if, at constant temperature,
the volume is increased (or if the ammonia which is evolved is pumped off),
the pressure will remain constant so long as two solid phases, AgCl,3NH_{3}
and 2AgCl,3NH_{3}, are present, _i.e._ until the compound richer in ammonia
is completely decomposed, when there will be a sudden fall in the pressure
to the value corresponding to the system 2AgCl,3NH_{3}--AgCl--NH_{3}. The
pressure will again remain constant at constant temperature, until all the
ammonia has been pumped off, when there will again be a sudden fall in the
pressure to that of the system formed by solid silver chloride in contact
with its vapour.

The reverse changes take place when the pressure of the ammonia is
gradually increased. If the volume is continuously diminished, the pressure
will first increase until it has reached a certain value; the compound
2AgCl,3NH_{3} can then be formed, and the pressure will now remain constant
until all the silver chloride has disappeared. The pressure will again
rise, until it has reached the value at which the compound AgCl,3NH_{3} can
be formed, when it will again remain constant until the complete
disappearance of the lower compound. _There is no gradual change of
pressure_ on passing from one system to another; but the changes are
abrupt, as is demanded by the Phase Rule, and as experiment has
conclusively proved.[152]

The dissociation pressures of the two compounds of silver {84} chloride and
ammonia, as determined by Isambert,[153] are given in the following
table:--

  -------------------------+-------------------------
                           |
          AgCl,3NH_{3}.    |        2AgCl,3NH_{3}.
  -------------+-----------+--------------+----------
               |           |              |
  Temperature. | Pressure. | Temperature. | Pressure.
  -------------+-----------+--------------+----------
               |           |              |
         0°    |  29.3 cm. |     20.0°    |   9.3 cm.
      10.6°    |  50.5 "   |     31.0°    |  12.5 "
      17.5°    |  65.5 "   |     47.0°    |  26.8 "
      24.0°    |  93.7 "   |     58.5°    |  52.8 "
      28.0°    | 135.5 "   |     69.0°    |  78.6 "
      34.2°    | 171.3 "   |     71.5°    |  94.6 "
      48.5°    | 241.4 "   |     77.5°    | 119.8 "
      51.5°    | 413.2 "   |     83.5°    | 159.3 "
      54.0°    | 464.1 "   |     86.1°    | 181.3 "
               |           |     88.5°    | 201.3 "
  -------------+-----------+--------------+----------

The conditions for the formation of these two compounds, by passing ammonia
over silver chloride, to which reference has already been made, will be
readily understood from the above tables. In the case of the triammonia
mono-chloride, the dissociation pressure becomes equal to atmospheric
pressure at a temperature of about 20°; above this temperature, therefore,
it cannot be formed by the action of ammonia at atmospheric pressure on
silver chloride. The triammonia dichloride can, however, be formed, for its
dissociation pressure at this temperature amounts to only 9 cm., and
becomes equal to the atmospheric pressure only at a temperature of about
68°; and this temperature, therefore, constitutes the limit above which no
combination can take place between silver chloride and ammonia under
atmospheric pressure.

Attention may be here drawn to the fact, to which reference will also be
made later, that _two_ solid phases are necessary in order that the
dissociation pressure at a given temperature shall be definite; _and for
the exact definition of this pressure it is necessary to know, not merely
what is the substance undergoing dissociation, but also what is the solid
product of dissociation formed_. For the definition of the equilibrium, the
latter is as important as the former. We shall presently find proof of this
in the case {85} of an analogous class of phenomena, viz. the dissociation
of salt hydrates.

Salts with Water of Crystallization.--In the case of the dehydration of
crystalline salts containing water of crystallization, we meet with
phenomena which are in all respects similar to those just studied. A salt
hydrate on being heated dissociates into a lower hydrate (or anhydrous
salt) and water vapour. Since we are dealing with two components--salt and
water[154]--in three phases, viz. hydrate _a_, hydrate _b_ (or anhydrous
salt), and vapour, the system is univariant, and to each temperature there
will correspond a certain, definite vapour pressure (the dissociation
pressure), which will be independent of the relative or absolute amounts of
the phases, _i.e._ of the amount of hydrate which has already undergone
dissociation or dehydration.

[Illustration: FIG. 19.]

The constancy of the dissociation pressure had been proved experimentally
by several investigators[155] a number of years before the theoretical
basis for its necessity had been given. In the case of salts capable of
forming more than one hydrate, we should obtain a series of dissociation
curves (_pt_-curves), as in the case of the different hydrates of copper
sulphate. In Fig. 19 there are represented diagrammatically the
vapour-pressure curves of the following univariant systems of copper
sulphate and water:--

  Curve OA: CuSO_{4},5H_{2}O <--> CuSO_{4},3H_{2}O + 2H_{2}O.
  Curve OB: CuSO_{4},3H_{2}O <--> CuSO_{4},H_{2}O + 2H_{2}O.
  Curve OC: CuSO_{4},H_{2}O   <--> CuSO_{4} + H_{2}O.

Let us now follow the changes which take place on {86} increasing the
pressure of the aqueous vapour in contact with anhydrous copper sulphate,
the temperature being meanwhile maintained constant. If, starting from the
point D, we slowly add water vapour to the system, the pressure will
gradually rise, without formation of hydrate taking place; for at pressures
below the curve OC only the anhydrous salt can exist. At E, however, the
hydrate CuSO_{4},H_{2}O will be formed, and as there are now three phases
present, viz. CuSO_{4}, CuSO_{4},H_{2}O, and vapour, the system becomes
_univariant_; and since the temperature is constant, the pressure must also
be constant. Continued addition of vapour will result merely in an increase
in the amount of the hydrate, and a decrease in the amount of the anhydrous
salt. When the latter has entirely disappeared, _i.e._ has passed into
hydrated salt, the system again becomes _bivariant_, and passes along the
line EF; the pressure gradually increases, therefore, until at F the
hydrate 3H_{2}O is formed, and the system again becomes univariant; the
three phases present are CuSO_{4},H_{2}O, CuSO_{4},3H_{2}O, vapour. The
pressure will remain constant, therefore, until the hydrate 1H_{2}O has
disappeared, when it will again increase till G is reached; here the
hydrate 5H_{2}O is formed, and the pressure once more remains constant
until the complete disappearance of the hydrate 3H_{2}O has taken place.

Conversely, on dehydrating CuSO_{4},5H_{2}O at constant temperature, we
should find that the pressure would maintain the value corresponding to the
dissociation pressure of the system
CuSO_{4},5H_{2}O--CuSO_{4},3H_{2}O--vapour, until all the hydrate 5H_{2}O
had disappeared; further removal of water would then cause the pressure to
fall _abruptly_ to the pressure of the system
CuSO_{4},3H_{2}O--CuSO_{4},H_{2}O--vapour, at which value it would again
remain constant until the tri-hydrate had passed into the monohydrate, when
a further sudden diminution of the pressure would occur. This behaviour is
represented diagrammatically in Fig. 20, the values of the pressure being
those at 50°.

Efflorescence.--From Fig. 19 we are enabled to predict the conditions under
which a given hydrated salt will effloresce when exposed to the air. We
have just learned that copper {87} sulphate pentahydrate, for example, will
not be formed unless the pressure of the aqueous vapour reaches a certain
value; and that conversely, if the vapour pressure falls below the
dissociation pressure of the pentahydrate, this salt will undergo
dehydration. From this, then, it is evident that a crystalline salt hydrate
will effloresce when exposed to the air, if the partial pressure of the
water vapour in the air is lower than the dissociation pressure of the
hydrate. At the ordinary temperature the dissociation pressure of copper
sulphate is less than the pressure of water vapour in the air, and
therefore copper sulphate does not effloresce. In the case of sodium
sulphate decahydrate, however, the dissociation pressure is greater than
the normal vapour pressure in a room, and this salt therefore effloresces.

[Illustration: FIG. 20.]

Indefiniteness of the Vapour Pressure of a Hydrate.--Reference has already
been made (p. 84), in the case of the ammonia compounds of the metal
chlorides, to the importance of the solid product of dissociation for the
definition of the dissociation pressure. Similarly also in the case of a
hydrated salt. A salt hydrate in contact with vapour constitutes only a
bivariant system, and can exist therefore at different values of
temperature and pressure of vapour, as is seen from the diagram, Fig. 19.
Anhydrous copper sulphate can exist in contact with water vapour at all
values of temperature and pressure lying in the field below the curve OC;
and the hydrate CuSO_{4},H_{2}O can exist in contact with vapour at all
values of temperature and pressure in the field BOC. Similarly, each of the
other hydrates can exist in contact with vapour at different values of
temperature and pressure.

From the Phase Rule, however, we learn that, in order that at a given
temperature the pressure of a two-component system {88} may be constant,
there must be three phases present. Strictly, therefore, we can speak only
of the vapour pressure of a _system_; and since, in the cases under
discussion, the hydrates dissociate into a solid and a vapour, any
statement as to the vapour pressure of a hydrate has a definite meaning
_only when the second solid phase produced by the dissociation is given_.
The everyday custom of speaking of the vapour pressure of a hydrated salt
acquires a meaning only through the assumption, tacitly made, that the
second solid phase, or the solid produced by the dehydration of the
hydrate, is the _next lower_ hydrate, where more hydrates than one exist.
That a hydrate always dissociates in such a way that the next lower hydrate
is formed is, however, by no means certain; indeed, cases have been met
with where apparently the anhydrous salt, and not the lower hydrate (the
existence of which was possible), was produced by the dissociation of the
higher hydrate.[156]

That a salt hydrate can exhibit different vapour pressures according to the
solid product of dissociation, can not only be proved theoretically, but it
has also been shown experimentally to be a fact. Thus CaCl_{2},6H_{2}O can
dissociate into water vapour and either of two lower hydrates, each
containing four molecules of water of crystallization, and designated
respectively as CaCl_{2},4H_{2}O[alpha], and CaCl_{2},4H_{2}O[beta].
Roozeboom[157] has shown that the vapour pressure which is obtained differs
according to which of these two hydrates is formed, as can be seen from the
following figures:--

  -------------+----------------------------------------------------------
               |                  Pressure of System.
  Temperature. +-----------------------------+----------------------------
               | CaCl_{2},6H_{2}O; CaCl_{2}, | CaCl_{2},6H_{2}O; CaCl_{2},
               | 4H_{2}O[alpha]; vapour.     | 4H_{2}O[beta]; vapour.
  -------------+-----------------------------+----------------------------
       -15°    |         0.027 cm.           |         0.022 cm.
         0     |         0.092  "            |         0.076  "
       +10     |         0.192  "            |         0.162  "
        20     |         0.378  "            |         0.315  "
        25     |         0.508  "            |         0.432  "
        29.2   |            --               |         0.567  "
        29.8   |         0.680  "            |            --
  -------------+-----------------------------+---------------------------

{89}

By reason of the non-recognition of the importance of the solid
dissociation product for the definition of the dissociation pressure of a
salt hydrate, many of the older determinations lose much of their value.

Suspended Transformation.--Just as in systems of one component we found
that a new phase was not necessarily formed when the conditions for its
existence were established, so also we find that even when the vapour
pressure is lowered below the dissociation pressure of a system,
dissociation does not necessarily occur. This is well known in the case of
Glauber's salt, first observed by Faraday. Undamaged crystals of
Na_{2}SO_{4},10H_{2}O could be kept unchanged in the open air, although the
vapour pressure of the system Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
is greater than the ordinary pressure of aqueous vapour in the air. That is
to say, the possibility of the formation of the new phase Na_{2}SO_{4} was
given; nevertheless this new phase did not appear, and the system therefore
became metastable, or unstable with respect to the anhydrous salt. When,
however, a trace of the new phase--the anhydrous salt--was brought in
contact with the hydrate, transformation occurred; the hydrate effloresced.

The possibility of suspended transformation or the non-formation of the new
phases must also be granted in the case where the vapour pressure is raised
above that corresponding to the system hydrate--anhydrous salt (or lower
hydrate)--vapour; in this case the formation of the higher hydrate becomes
a possibility, but not a certainty. Although there is no example of this
known in the case of hydrated salts, the suspension of the transformation
has been observed in the case of the compounds of ammonia with the metal
chlorides (p. 82). Horstmann,[158] for example, found that the pressure of
ammonia in contact with 2AgCl,3NH_{3} could be raised to a value higher
than the dissociation pressure of AgCl,3NH_{3} without this compound being
formed. We see, therefore, that even when the existence of the higher
compound in contact with the lower became possible, the higher compound was
not immediately formed.

Range of Existence of Hydrates.--In Fig. 19 the vapour {90} pressure curves
of the different hydrates of copper sulphate are represented as maintaining
their relative positions throughout the whole range of temperatures. But
this is not necessarily the case. It is possible that at some temperature
the vapour pressure curve of a lower hydrate may cut that of a higher
hydrate. At temperatures above the point of intersection, the lower hydrate
would have a higher vapour pressure than the higher hydrate, and would
therefore be metastable with respect to the latter. The range of stable
existence of the lower hydrate would therefore end at the point of
intersection. This appears to be the case with the two hydrates of sodium
sulphate, to which reference will be made later.[159]

Constancy of Vapour Pressure and the Formation of Compounds.--We have seen
in the case of the salt hydrates that the continued addition of the vapour
phase to the system caused an increase in the pressure until at a definite
value of the pressure a hydrate is formed; the pressure then becomes
constant, and remains so, until one of the solid phases has disappeared.
Conversely, on withdrawing the vapour phase, the pressure remained constant
so long as any of the dissociating compound was present, independently of
the degree of the decomposition (p. 86). This behaviour, now, has been
employed for the purpose of determining whether or not definite chemical
compounds are formed. Should compounds be formed between the vapour phase
and the solid, then, on continued addition or withdrawal of the vapour
phase, it will be found that the vapour pressure remains constant for a
certain time, and will then suddenly assume a new value, at which it will
again remain constant. By this method, Ramsay[160] found that no definite
hydrates were formed in the case of ferric and aluminium oxides, but that
two are formed in the case of lead oxide, viz. 2PbO,H_{2}O and 3PbO,H_{2}O.

The method has also been applied to the investigation of the so-called
palladium hydride,[161] and the results obtained appear to show that no
compound is formed. Reference will, however, be made to this case later
(Chap. X.).

{91}

Measurement of the Vapour Pressure of Hydrates.--For the purpose of
measuring the small pressures exerted by the vapour of salt hydrates, use
is very generally made of a differential manometer called the
_Bremer-Frowein tensimeter_.[162]

This apparatus has the form shown in Fig. 21. It consists of a U-tube, the
limbs of which are bent close together, and placed in front of a millimetre
scale. The bend of the tube is filled with oil or other suitable liquid,
_e.g._ bromonaphthalene. If it is desired to measure the dissociation
pressure of, say, a salt hydrate, concentrated sulphuric acid is placed in
the flask _e_, and a quantity of the hydrate, well dried and powdered,[163]
in the bulb d. The necks of the bulbs _d_ and _e_ are then sealed off.
Since, as we have learned, suspended transformation may occur, it is
advisable to first partially dehydrate the salt, in order to ensure the
presence of the second solid product of dissociation; the value of the
dissociation pressure being independent of the degree of dissociation of
the hydrate (p. 86). The small bulbs _d_ and _e_ having been filled, the
apparatus is placed on its side, so as to allow the liquid to run from the
bend of the tube into the bulbs _a_ and _b_; it is then exhausted through
_f_ by means of a mercury pump, and sealed off. The apparatus is now placed
in a perpendicular position in a thermostat, and kept at constant
temperature until equilibrium is established. Since the vapour pressure on
the side containing the sulphuric acid may be regarded as zero, the
difference in level of the two surfaces of liquid in the U-tube gives
directly the dissociation pressure of the hydrate in terms of the
particular liquid employed; if the density of the latter is known, the
pressure can then be calculated to cm. of mercury.

[Illustration: FIG. 21.]

       *       *       *       *       *


{92}

CHAPTER VI

SOLUTIONS

Definition.--In all the cases which have been considered in the preceding
pages, the different phases--with the exception of the vapour
phase--consisted of a single substance of definite composition, or were
definite chemical individuals.[164] But this invariability of the
composition is by no means imposed by the Phase Rule; on the contrary, we
shall find in the examples which we now proceed to study, that the
participation of phases of variable composition in the equilibrium of a
system is in no way excluded. To such phases of variable composition there
is applied the term _solution_. A solution, therefore, is to be defined as
_a homogeneous mixture, the composition of which can undergo continuous
variation within certain limits_; the limits, namely, of its
existence.[165]

From this definition we see that the term solution is not restricted to any
particular physical state of substances, but includes within its range not
only the liquid, but also the gaseous and solid states. We may therefore
have solutions of gases in liquids, and of gases in solids; of liquids in
liquids or in solids; of solids in liquids, or of solids in solids.
Solutions of gases in gases are, of course, also possible; since, however,
gas solutions never give rise to more than one phase, their {93} treatment
does not come within the scope of the Phase Rule, which deals with
heterogeneous equilibria.

It should also be emphasized that the definition of solution given above,
neither creates nor recognizes any distinction between solvent and
dissolved substance (solute); and, indeed, a too persistent use of these
terms and the attempt to permanently label the one or other of two
components as the solvent or the solute, can only obscure the true
relationships and aggravate the difficulty of their interpretation. In all
cases it should be remembered that we are dealing with equilibria between
two components (we confine our attention in the first instance to such),
the solution being constituted of these components in variable and varying
amounts. The change from the case where the one component is in great
excess (ordinarily called the solvent) to that in which the other component
predominates, may be quite gradual, so that it is difficult or impossible
to say at what point the one component ceases to be the solvent and becomes
the solute. The adoption of this standpoint need not, however, preclude one
from employing the conventional terms solvent and solute in ordinary
language, especially when reference is made only to some particular
condition of equilibrium of the system, when the concentration of the two
components in the solution is widely different.

SOLUTIONS OF GASES IN LIQUIDS.

As the first class of solutions to which we shall turn our attention, there
may be chosen the solutions of gases in liquids, or the equilibria between
a liquid and a gas. These equilibria really constitute a part of the
equilibria to be studied more fully in Chapter VIII.; but since the
two-phase systems formed by the solutions of gases in liquids are among the
best-known of the two-component systems, a short section may be here
allotted to their treatment.

When a gas is passed into a liquid, absorption takes place to a greater or
less extent, and a point is at length reached when the liquid absorbs no
more of the gas; a condition of equilibrium is attained, and the liquid is
said to be saturated {94} with the gas. In the light of the Phase Rule,
now, such a system is bivariant (two components in two phases); and two of
the variable factors, pressure, temperature, and concentration of the
components, must therefore be chosen in order that the condition of the
system may be defined. If the concentration and the temperature are fixed,
then the pressure is also defined; or under given conditions of temperature
and pressure, the concentration of the gas in the solution must have a
definite value. If, however, the temperature alone is fixed, the
concentration and the pressure can alter; a fact so well known that it does
not require to be further insisted on.

As to the way in which the solubility of a gas in a liquid varies with the
pressure, the Phase Rule of course does not state; but guidance on this
point is again yielded by the theorem of van't Hoff and Le Chatelier. Since
the absorption of a gas is in all cases accompanied by a diminution of the
total volume, this process must take place with increase of pressure. This,
indeed, is stated in a quantitative manner in the law of Henry, according
to which the amount of a gas absorbed is proportional to the pressure. But
this law must be modified in the case of gases which are very readily
absorbed; the _direction of change_ of concentration with the pressure
will, however, still be in accordance with the theorem of Le Chatelier.

If, on the other hand, the pressure is fixed, then the concentration will
vary with the temperature; and since the absorption of gases is in all
cases accompanied by the evolution of heat, the solubility is found, in
accordance with the theorem of Le Chatelier, to diminish with rise of
temperature.

In considering the changes of pressure accompanying changes of
concentration and temperature, a distinction must be drawn between the
total pressure and the partial pressure of the dissolved gas, in cases
where the solvent is volatile. In these cases, the law of Henry applies not
to the total pressure of the vapour, but only to the partial pressure of
the dissolved gas. {95}

SOLUTIONS OF LIQUIDS IN LIQUIDS.

When mercury and water are brought together, the two liquids remain side by
side without mixing. Strictly speaking, mercury undoubtedly dissolves to a
certain extent in the water, and water no doubt dissolves, although to a
less extent, in the mercury; the amount of substance passing into solution
is, however, so minute, that it may, for all practical purposes, be left
out of account, so long as the temperature does not rise much above the
ordinary.[166] On the other hand, if alcohol and water be brought together,
complete miscibility takes place, and one homogeneous solution is obtained.
Whether water be added in increasing quantities to pure alcohol, or pure
alcohol be added in increasing amount to water, at no point, at no degree
of concentration, is a system obtained containing more than one liquid
phase. At the ordinary temperature, water and alcohol can form only two
phases, liquid and vapour. If, however, water be added to ether, or if
ether be added to water, solution will not occur to an indefinite extent;
but a point will be reached when the water or the ether will no longer
dissolve more of the other component, and a further addition of water on
the one hand, or ether on the other, will cause the formation of two liquid
layers, one containing excess of water, the other excess of ether. We
shall, therefore, expect to find all grades of miscibility, from almost
perfect immiscibility to perfect miscibility, or miscibility in all
proportions. In cases of perfect immiscibility, the components do not
affect one another, and the system therefore remains unchanged. Such cases
do not call for treatment here. We have to concern ourselves here only with
the second and third cases, viz. with cases of complete and of partial
miscibility. There is no essential difference between the two classes, for,
as we shall see, {96} the one passes into the other with change of
temperature. The formal separation into two groups is based on the
miscibility relations at ordinary temperatures.

Partial or Limited Miscibility.--In accordance with the Phase Rule, a pure
liquid in contact with its vapour constitutes a univariant system. If,
however, a small quantity of a second substance is added, which is capable
of dissolving in the first, a bivariant system will be obtained; for there
are now two components and, as before, only two phases--the homogeneous
liquid solution and the vapour. At constant temperature, therefore, both
the composition of the solution and the pressure of the vapour can undergo
change; or, if the composition of the solution remains unchanged, the
pressure and the temperature can alter. If the second (liquid) component is
added in increasing amount, the liquid will at first remain homogeneous,
and its composition and pressure will undergo a continuous change; when,
however, the concentration has reached a definite value, solution no longer
takes place; two liquid phases are produced. Since there are now three
phases present, two liquids and vapour, the system is univariant; at a
given temperature, therefore, the concentration of the components in the
two liquid phases, as well as the vapour pressure, must have definite
values. Addition of one of the components, therefore, cannot alter the
concentrations or the pressure, but can only cause a change in the relative
amounts of the phases.

The two liquid phases can be regarded, the one as a solution of the
component I. in component II., the other as a solution of component II. in
component I. If the pressure is maintained constant, then to each
temperature there will correspond a definite concentration of the
components in the two liquid phases; and addition of excess of one will
merely alter the relative amounts of the two solutions. As the temperature
changes, the composition of the two solutions will change, and there will
therefore be obtained two solubility curves, one showing the solubility of
component I. in component II., the other showing the solubility of
component II. in component I. Since heat may be either evolved or absorbed
when one liquid dissolves in another, the solubility may diminish or
increase {97} with rise of temperature. The two solutions which at a given
temperature correspond to one another are known as _conjugate solutions_.

The solubility relations of partially miscible liquids have been studied by
Guthrie,[167] and more especially by Alexejeff[168] and by Rothmund.[169] A
considerable variety of curves have been obtained, and we shall therefore
discuss only a few of the different cases which may be taken as typical of
the rest.

Phenol and Water.--When phenol is added to water at the ordinary
temperature, solution takes place, and a homogeneous liquid is produced.
When, however, the concentration of the phenol in the solution has risen to
about 8 per cent., phenol ceases to be dissolved; and a further addition of
it causes the formation of a second liquid phase, which consists of excess
of phenol and a small quantity of water. In ordinary language it may be
called a solution of water in phenol. If now the temperature is raised,
this second liquid phase will disappear, and a further amount of phenol
must be added in order to produce a separation of the liquid into two
layers. In this way, by increasing the amount of phenol and noting the
temperature at which the two layers disappear, the so-called solubility
curve of phenol in water can be obtained. By noting the change of the
solubility with the temperature in this manner, it is found that at all
temperatures below 68.4°, the addition of more than a certain amount of
phenol causes the formation of two layers; at temperatures above this,
however, two layers cannot be formed, no matter how much phenol is added.
At temperatures above 68.4°, therefore, water and phenol are miscible in
all proportions.

On the other hand, if water is added to phenol at the ordinary temperature,
a liquid is produced which consists chiefly of phenol, and on increasing
the amount of water beyond a certain point, two layers are formed. On
raising the temperature these two layers disappear, and a homogeneous
solution is again obtained. The phenomena are exactly analogous to those
already described. Since, now, in the second {98} case the concentration of
the phenol in the solution gradually decreases, while in the former case it
gradually increases, a point must at length be reached at which the
composition of the two solutions becomes the same. On mixing the two
solutions, therefore, one homogeneous liquid will be obtained. But the
point at which two phases become identical is called a critical point, so
that, in accordance with this definition, the temperature at which the two
solutions of phenol and water become identical may be called the _critical
solution temperature_, and the concentration at this point may be called
the _critical concentration_.

[Illustration: FIG. 22.]

From what has been said above, it will be seen that at any temperature
below the critical solution temperature, two conjugate solutions containing
water and phenol in different concentration can exist together, one
containing excess of water, the other excess of phenol. The following table
gives the composition of the two layers, and the values are represented
graphically in Fig. 22.[170]

  PHENOL AND WATER.

  C_{1} is the percentage amount of phenol in the first layer.
  C_{2}         "          "          "           second layer.
  -------------+--------+--------
  Temperature. |  C_{1}.|  C_{2}.
  -------------+--------+--------
      20°      |   8.5  |  72.2
      30°      |   8.7  |  69.9
      40°      |   9.7  |  66.8
      50°      |  12.0  |  62.7
      55°      |  14.2  |  60.0
      60°      |  17.5  |  56.2
      65°      |  22.7  |  49.7
      68.4°    |  36.1  |  36.1
  -------------+--------+--------

{99}

The critical solution temperature for phenol and water is 68.4°, the
critical concentration 36.1 per cent. of phenol. At all temperatures above
68.4°, only homogeneous solutions of phenol and water can be obtained;
water and phenol are then miscible in all proportions.

At the critical solution point the system exists in only two phases--liquid
and vapour. It ought, therefore, to possess two degrees of freedom. The
restriction is, however, imposed that the composition of the two liquid
phases, coexisting at a point infinitely near to the critical point,
becomes the same, and this disposes of one of the degrees of freedom. The
system is therefore univariant; and at a given temperature the pressure
will have a definite value. Conversely, if the pressure is fixed (as is the
case when the system is under the pressure of its own vapour), then the
temperature will also be fixed; that is, the critical solution temperature
has a definite value depending only on the substances. If the vapour phase
is omitted, the temperature will alter with the pressure; in this case,
however, as in the case of other condensed systems, the effect of pressure
is slight.

From Fig. 22 it is easy to predict the effect of bringing together water
and phenol in any given quantities at any temperature. Start with a
solution of phenol and water having the composition represented by the
point _x_. If to this solution phenol is added at constant temperature, it
will dissolve, and the composition of the solution will gradually change,
as shown by the dotted line _xy_. When, however, the concentration has
reached the value represented by the point _y_, two liquid layers will be
formed, the one solution having the composition represented by _y_, the
other that represented by _y'_. The system is now univariant, and on
further addition of phenol, the composition of the two liquid phases will
remain unchanged, but their relative amounts will alter. The phase richer
in phenol will increase in amount; that richer in water will decrease, and
ultimately disappear, and there will remain the solution _y'_. Continued
addition of phenol will then lead to the point _x'_, there being now only
one liquid phase present.

Since the critical solution point represents the highest temperature at
which two liquid phases consisting of phenol and {100} water can exist
together, these two substances can be brought together in any amount
whatever at temperatures higher than 68.4°, without the formation of two
layers. It will therefore be possible to pass from a system represented by
_x_ to one represented by _x'_, without at any time two liquid phases
appearing. Starting with _x_, the temperature is first raised above the
critical solution temperature; phenol is then added until the concentration
reaches the point _x__{2}. On allowing the temperature to fall, the system
will then pass into the condition represented by _x'_.

[Illustration: FIG. 23.]

Methylethylketone and Water.--In the case just described, the solubility of
each component in the other increased continuously with the temperature.
There are, however, cases where a maximum or minimum of solubility is
found, _e.g._ methylethylketone and water. The curve which represents the
equilibria between these two substances is given in Fig. 23, the
concentration values being contained in the following table:[171]--

  METHYLETHYLKETONE AND WATER.

  --------------+-----------------+-----------------
  Temperature.  | C_{1} per cent. |  C_{2} per cent.
  --------------+-----------------+-----------------
       -10°     |      34.5       |     89.7
       +10°     |      26.1       |     90.0
        30°     |      21.9       |     89.9
        50°     |      17.5       |     89.0
        70°     |      16.2       |     85.7
        90°     |      16.1       |     84.8
       110°     |      17.7       |     80.0
       130°     |      21.8       |     71.9
       140°     |      26.0       |     64.0
       151.8°   |      44.2       |     44.2
  --------------+-----------------+-----------------

{101}

These numbers and Fig. 23 show clearly the occurrence of a minimum in the
solubility of the ketone in water, and also a minimum (at about 10°) in the
solubility of water in methylethylketone. Minima of solubility have also
been found in other cases.

[Illustration: FIG. 24.]

Triethylamine and Water.--Although in most of the cases studied the
solubility of one liquid in another increases with rise of temperature,
this is not so in all cases. Thus, at temperatures below 18°, triethylamine
and water mix together in all proportions; but, on raising the temperature,
the homogeneous solution becomes turbid and separates into two layers. In
this case, therefore, the critical solution temperature is found in the
direction of lower temperature, not in the direction of higher.[172] This
behaviour is clearly shown by the graphic representation in Fig. 24, and
also by the numbers in the following table:--

  TRIETHYLAMINE AND WATER.

  -------------+-----------------+----------------
  Temperature. | C_{1} per cent. |  C_{2} per cent.
  -------------+-----------------+----------------
       70°     |      1.6        |     --
       50°     |      2.9        |     --
       30°     |      5.6        |     96
       25°     |      7.3        |     95.5
       20°     |     15.5        |     73
      ±18.5°   |    ±30          |    ±30
  -------------+-----------------+----------------

General Form of Concentration-Temperature Curve.--From the preceding
figures it will be seen that the general {102} form of the solubility curve
is somewhat parabolic in shape; in the case of triethylamine and water, the
closed end of the curve is very flat. Since for all liquids there is a
point (critical point) at which the liquid and gaseous states become
identical, and since all gases are miscible in all proportions, it follows
that there must be some temperature at which the liquids become perfectly
miscible. In the case of triethylamine and water, which has just been
considered, there must therefore be an upper critical solution temperature,
so that the complete solubility relations would be represented by a closed
curve of an ellipsoidal aspect. An example of such a curve is furnished by
nicotine and water. At temperatures below 60° and above 210°, nicotine and
water mix in all proportions.[173] Although it is possible that this is the
general form of the curve for all pairs of liquids, there are as yet
insufficient data to prove it.

With regard to the closed end of the curve it may be said that it is
continuous; the critical solution point is not the intersection of two
curves, for such a break in the continuity of the curve could occur only if
there were some discontinuity in one of the phases. No such discontinuity
exists. The curve is, therefore, not to be considered as two solubility
curves cutting at a point; it is a curve of equilibrium between two
components, and so long as the phases undergo continuous change, the curve
representing the equilibrium must also be continuous. As has already been
emphasized, a distinction between solvent and solute is merely conventional
(p. 93).

Pressure-Concentration Diagram.--In considering the pressure-concentration
diagram of a system of two liquid components, a distinction must be drawn
between the total pressure of the system and the partial pressures of the
components. On studying the total pressure of a system, it is found that
two cases can be obtained.[174]

So long as there is only one liquid phase, the system is bivariant. The
pressure therefore can change with the concentration and the temperature.
If the temperature is maintained {103} constant, the pressure will vary
only with the concentration, and this variation can therefore be
represented by a curve. If, however, two liquid phases are formed, the
system becomes univariant: and if one of the variables, say the
temperature, is arbitrarily fixed, the system no longer possesses any
degree of freedom. _When two liquid phases are formed, therefore, the
concentrations and the vapour pressure have definite values, which are
maintained so long as the two liquid phases are present_; the temperature
being supposed constant.

In Fig. 25 is given a diagrammatic representation of the two kinds of
pressure-concentration curves which have so far been obtained. In the one
case, the vapour pressure of the invariant system (at constant temperature)
lies higher than the vapour pressure of either of the pure components; a
phenomenon which is very generally found in the case of partially miscible
liquids, _e.g._ ether and water.[175] Accordingly, by the addition of water
to ether, or of ether to water, there is an increase in the _total_ vapour
pressure of the system.

[Illustration: FIG. 25.]

With regard to the second type, the vapour pressure of the systems with two
liquid phases lies between that of the two single components. An example of
this is found in sulphur dioxide and water.[176] On adding sulphur dioxide
to water there is an increase of the total vapour pressure; but on adding
water to liquid sulphur dioxide, the total vapour pressure is diminished.

The case that the vapour pressure of the system with two {104} liquid
phases is _less_ than that of each of the components is not possible.

With regard to the _partial pressure_ of the components, the behaviour is
more uniform. The partial pressure of one component is in all cases lowered
by the addition of the other component, the diminution being approximately
proportional to the amount added. If two liquid phases are present, the
partial pressure of the components, as well as the total pressure, is
constant, and is the same for both phases. That is to say, in the case of
the two liquids, saturated solution of water in ether, and of ether in
water, the partial pressure of the ether in the vapour in contact with the
one solution is the same as that in the vapour over the other
solution.[177]

Complete Miscibility.--Although the phenomena of complete miscibility are
here treated under a separate heading, it must not be thought that there is
any essential difference between those cases where the liquids exhibit
limited miscibility and those in which only one homogeneous solution is
formed. As has been already pointed out, the solubility relations alter
with the temperature; and liquids which at one temperature can dissolve in
one another only to a limited extent, are found at some other temperature
to possess the property of complete miscibility. Conversely, we may expect
that liquids which at one temperature, say at the ordinary temperature, are
miscible in all proportions, will be found at some other temperature to be
only partially miscible. Thus, for example, it was found by Guthrie that
ethyl alcohol and carbon disulphide, which are miscible in all proportions
at the ordinary temperature, possess only limited miscibility at
temperatures below -14.4°.[178] Nevertheless, it is doubtful if the
critical solution temperature is in all cases experimentally realizable.

Pressure-Concentration Diagram.--Since, in the cases of complete
miscibility of two liquid components, there are never more than two phases
present, the system must always be bivariant; and two of the variables
pressure, temperature or concentration of the components, must be
arbitrarily chosen {105} before the system becomes defined. For this reason
the Phase Rule affords only a slight guidance in the study of such
equilibria; and we shall therefore not enter in detail into the behaviour
of these homogeneous mixtures. All that the Phase Rule can tell us in
connection with these solutions, is that at constant temperature the vapour
pressure of the solution varies with the composition of the liquid phase;
and if the composition of the liquid phase remains unchanged, the pressure
also must remain unchanged. This constancy of composition is exhibited not
only by pure liquids, but also by liquid solutions in all cases where the
vapour pressure of the solution reaches a maximum or minimum value. This is
the case, for example, with mixtures of constant boiling point.[179]

       *       *       *       *       *


{106}

CHAPTER VII

SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE

General.--When a solid is brought into contact with a liquid in which it
can dissolve, a certain amount of it passes into solution; and the process
continues until the concentration reaches a definite value independent of
the amount of solid present. A condition of equilibrium is established
between the solid and the solution; the solution becomes _saturated_. Since
the number of components is two, and the number of phases three, viz.
solid, liquid solution, vapour, the system is univariant. If, therefore,
one of the factors, pressure, temperature, or concentration of the
components (in the solution[180]), is arbitrarily fixed, the state of the
system becomes perfectly defined. Thus, at any given temperature, the
vapour pressure of the system and the concentration of the components have
a definite value. If the temperature is altered, the vapour pressure and
also, in general, the concentration will undergo change. Likewise, if the
pressure varies, while the system is isolated so that no heat can pass
between it and its surroundings, the concentration and the temperature must
also undergo variation until they attain values corresponding to the
particular pressure.

That the temperature has an influence, sometimes a very considerable
influence, on the amount of substance passing into solution, is
sufficiently well known; the effect of pressure, although less apparent, is
no less certain. If at any given temperature the volume of the vapour phase
is diminished, {107} vapour will condense to liquid, in order that the
pressure may remain constant, and so much of the solid will pass into
solution that the concentration may remain unchanged; for, so long as the
three phases are present, the state of the system cannot alter. If,
however, one of the phases, _e.g._ the vapour phase, disappears, the system
becomes bivariant; at any given temperature, therefore, there may be
different values of concentration and pressure.

The direction in which change of concentration will occur with change of
pressure can be predicted by means of the theorem of Le Chatelier, if it is
known whether solution is accompanied by increase or diminution of the
total volume. If diminution of the total volume of the system occurs on
solution, increase of pressure will increase the solubility; in the reverse
case, increase of pressure will diminish the solubility.

This conclusion has also been verified by experiment, as is shown by the
following figures.[181]

  ---------------------------------------------------------------
                    |Change of       |  Solubility (at 18°) (grams salt
                    |volume by       |  in 1 gram of solution).
                    |dissolving 1 gm.|---------------------------
       Salt.        |of salt in the  |          |
                    |saturated       | Pressure | Pressure
                    |solution.       | = 1 atm. | = 500 atm.
  ------------------+----------------+----------+----------------
  Sodium chloride   |     -0.07      |  0.264   |   0.270
  Ammonium chloride |     +0.10      |  0.272   |   0.258
  Alum              |     -0.067     |  0.115   |   0.142
                    |                |          |(_p_ = 400 atm.)
  -------------------------------------------------------------

As can be seen, a large increase of the pressure brings about a no more
than appreciable alteration of the solubility; a result which is due, as in
the case of the alteration of the fusion point with the pressure, to the
small change in volume accompanying solution or increase of pressure. For
all practical purposes, therefore, the solubility as determined under
atmospheric pressure may be taken as equal to the true {108} solubility,
that is, the solubility when the system is under the pressure of its own
vapour.

The Saturated Solution.--From what has been said above, it will be seen
that the condition of saturation of a solution can be defined only with
respect to a certain solid phase; if no solid is present, the system is
undefined, for it then consists of only two phases, and is therefore
bivariant. Under such circumstances not only can there be at one given
temperature solutions of different concentration, all containing less of
one of the components than when that component is present in the solid
form, but there can also exist solutions containing more of that component
than corresponds to the equilibrium when the solid is present. In the
former case the solutions are _unsaturated_, in the latter case they are
_supersaturated with respect to a certain solid phase_; in themselves, the
solutions are stable, and are neither unsaturated nor supersaturated.
Further, if the solid substance can exist in different allotropic
modifications, the particular form of the substance which is in equilibrium
with the solution must be known, in order that the statement of the
solubility may be definite; for each form has its own solubility, and, as
we shall see presently, the less stable form has the greater solubility
(cf. p. 47). In all determinations of the solubility, therefore, not only
must the concentration of the components in the solution be determined, but
equal importance should be attached to the characterisation of the solid
phase present.

In this connection, also, one other point may be emphasised. For the
production of the equilibrium between a solid and a liquid, time is
necessary, and this time not only varies with the state of division of the
solid and the efficiency of the stirring, but is also dependent on the
nature of the substance.[182] Considerable care must therefore be taken
that sufficient time is allowed for equilibrium to be established. Such
care is more especially needful when changes may occur in the solid phase,
and neglect of it has greatly diminished the value of many of the older
determinations of solubility.

Form of the Solubility Curve.--The solubility curve--that {109} is, the
curve representing the change of concentration of the components in the
solution with the temperature--differs markedly from the curve of vapour
pressure (p. 63), in that it possesses no general form, but may vary in the
most diverse manner. Not only may the curve have an almost straight and
horizontal course, or slope or curve upwards at varying angles; but it may
even slope downwards, corresponding to a decrease in the solubility with
rise of temperature; may exhibit maxima or minima of solubility, or may, as
in the case of some hydrated salts, pass through a point of maximum
temperature. In the latter case the salt may possess two values of
solubility at the same temperature. We shall consider these cases in the
following chapter.

[Illustration: FIG. 26.]

The great variety of form shown by solubility curves is at once apparent
from Fig. 26, in which the solubility curves of various substances (not,
however, drawn to scale) are reproduced.[183]

Varied as is the form of the solubility curve, its _direction_,
nevertheless, can be predicted by means of the theorem of van't Hoff and Le
Chatelier; for in accordance with that theorem (p. 57) increase of
solubility with the temperature must occur in those cases where the process
of solution is accompanied by an _absorption_ of heat; and a decrease in
the solubility with rise of temperature will be found in cases where
solution occurs with _evolution_ of heat. Where there is no heat effect
accompanying solution, {110} change of temperature will be without
influence on the solubility; and if the sign of the heat of solution
changes, the direction of the solubility curve must also change, _i.e._
must show a maximum or minimum point. This has in all cases been verified
by experiment.[184]

In applying the theorem of Le Chatelier to the course of the solubility
curve, it should be noted that by heat of solution there is meant, not the
heat effect produced on dissolving the salt in a large amount of solvent
(which is the usual signification of the expression), but the heat which is
absorbed or evolved when the salt is dissolved in the almost saturated
solution (the so-called last heat of solution). Not only does the heat
effect in the two cases have a different value, but it may even have a
different sign. A striking example of this is afforded by cupric chloride,
as the following figures show:[185]--

  -----------------------------------------------------------
  Number of gram-molecules of        |
  CuCl_{2}, 2H_{2}O dissolved in 198 |       Heat effect.
  gram-molecules of water.           |
  -----------------------------------+-----------------------
              1                      |          +37 K
              2.02                   |          +66 "
              4.15                   |         +105 "
              7.07                   |         +117 "
              9.95                   |         +117 "
             11                      |          +91 "
             18.8                    |          -10 "
             19.6                    |          -31 "
             24.75                   |         -198 "
  ------------------------------------------------------------

In the above table the positive sign indicates evolution of heat, the
negative sign, absorption of heat; and the values of the heat effect are
expressed in centuple calories. Judging from the heat effect produced on
dissolving cupric chloride in a large bulk of water, we should predict that
the solubility of that salt would diminish with rise of temperature; as a
matter of fact, it increases. This is in accordance with the fact that
{111} the last heat of solution is _negative_ (as expressed above), _i.e._
solution of the salt in the almost saturated solution is accompanied by
absorption of heat. We are led to expect this from the fact that the heat
of solution changes sign from positive to negative as the concentration
increases; experiment also showed it to be the case.

Despite its many forms, it should be particularly noted that the solubility
curve of any substance is _continuous_, so long as the solid phase, or
solid substance in contact with the solution, remains unchanged. If any
"break" or discontinuous change in the direction of the curve occurs, it is
a sign that the _solid phase has undergone alteration_. Conversely, if it
is known that a change takes place in the solid phase, a break in the
solubility curve can be predicted. We shall presently meet with examples of
this.[186]

A.--ANHYDROUS SALT AND WATER.

The Solubility Curve.--In studying the equilibria in those systems of two
components in which the liquid phase is a solution or phase of varying
composition, we shall in the present chapter limit the discussion to those
cases where no compounds are formed, but where the components crystallise
out in the pure state. Since some of the best-known examples of such
systems are yielded by the solutions of anhydrous salts in water, we shall
first of all briefly consider some of the results which have been obtained
with them.

For the most part the solubility curves have been studied only at
temperatures lying between 0° and 100°, the solid phase in contact with the
solution being the anhydrous salt. For the representation of these
equilibria, the concentration-temperature {112} diagram is employed, the
concentration being expressed as the number of grams of the salt dissolved
in 100 grams of water, or as the number of gram-molecules of salt in 100
gram-molecules of water. The curves thus obtained exhibit the different
forms to which reference has already been made. So long as the salt remains
unchanged the curve will be continuous, but if the salt alters its form,
then the solubility curve will show a break.

[Illustration: FIG. 27.]

Now, we have already seen in Chapter III. that certain substances are
capable of existing in various crystalline forms, and these forms are so
related to one another that at a given temperature the relative stability
of each pair of polymorphic forms undergoes change. Since each crystalline
variety of a substance must have its own solubility, there must be a break
in the solubility curve at the temperature of transition of the two
enantiotropic forms. At this point the two solubility curves must cut, for
since the two forms are in equilibrium with respect to their vapour, they
must also be in equilibrium with respect to their solutions. From the table
on p. 63 it is seen that potassium nitrate, ammonium nitrate, silver
nitrate, thallium nitrate, thallium picrate, are capable of existing in two
or more different enantiotropic crystalline forms, the range of stability
of these forms being limited by definite temperatures (transition
temperature). Since the transition point is not altered by a solvent
(provided the latter is not absorbed by the solid phase), we should find on
studying the solubility of these substances in water that the solubility
curve would exhibit a change in direction at the temperature of transition.
As a matter of fact this has been verified, more especially in the case of
ammonium nitrate[187] {113} and thallium picrate.[188] The following table
contains the values of the solubility of ammonium nitrate obtained by
Müller and Kaufmann, the solubility being expressed in gram-molecules
NH_{4}NO_{3} in 100 gram-molecules of water. In Fig. 27 these results are
represented graphically. The equilibrium point was approached both from the
side of unsaturation and of supersaturation, and the condition of
equilibrium was controlled by determinations of the density of the
solution.

  SOLUBILITY OF AMMONIUM NITRATE.

  ------------------------------------------------------------
   Temperature. | Solubility. | Temperature. | Solubility.
  --------------+-------------+--------------+----------------
      12.2°     |    34.50    |    32.7°     |    57.90
      20.2°     |    43.30    |    34.0°     |    58.89
      25.05°    |    48.19    |    35.0°     |    59.80
      28.0°     |    51.86    |    36.0°     |    61.00
      30.0°     |    54.40    |    37.5°     |    62.90
      30.2°     |    54.61    |    38.0°     |    63.60
      31.9°     |    57.20    |    39.0°     |    65.09
      32.1°     |    57.60    |    40.0°     |    66.80
  ------------------------------------------------------------

From the graphic representation of the solubility given in Fig. 27, there
is seen to be a distinct change in the direction of the curve at a
temperature of 32°; and this break in the curve corresponds to the
transition of the [beta]-rhombic into the [alpha]-rhombic form of ammonium
nitrate (p. 63).

Suspended Transformation and Supersaturation.--As has already been learned,
the transformation of the one crystalline form into the other does not
necessarily take place immediately the transition point has been passed;
and it has therefore been found possible in a number of cases to follow the
solubility curve of a given crystalline form beyond the point at which it
ceases to be the most stable modification. Now, it will be readily seen
from Fig. 27 that if the two solubility curves be prolonged beyond the
point of intersection, the solubility of the less stable form is greater
than that of the more stable. A solution, therefore, which is saturated
with respect to the less stable form, _i.e._ which is in equilibrium with
that form, is _supersaturated with respect to the more stable
modification_. If, {114} therefore, a small quantity of the more stable
form is introduced into the solution, the latter must deposit such an
amount of the more stable form that the concentration of the solution
corresponds to the solubility of the stable form at the particular
temperature. Since, however, the solution is now _unsaturated_ with respect
to the less stable variety, the latter, if present, must pass into
solution; and the two processes, deposition of the stable and solution of
the metastable form, must go on until the latter form has entirely
disappeared and a saturated solution of the stable form is obtained. There
will thus be a conversion, through the medium of the solvent, of the less
stable into the more stable modification. This behaviour is of practical
importance in the determination of transition points (_v._ Appendix).

From the above discussion it will be seen how important is the statement of
the solid phase for the definition of saturation and supersaturation.[189]

Solubility Curve at Higher Temperatures.--On passing to the consideration
of the solubility curves at higher temperatures, two chief cases must be
distinguished.

    (1) The two components in the fused state can mix in all proportions.

    (2) The two components in the fused state cannot mix in all
    proportions.

1. _Complete Miscibility of the Fused Components._

[Illustration: FIG. 28.]

The best example of this which has been studied, so far as anhydrous salts
and water are concerned, is that of silver nitrate and water. The
solubility of this salt at temperatures {115} above 100° has been studied
chiefly by Etard[190] and by Tilden and Shenstone.[191] The values obtained
by Etard are given in the following table, and represented graphically in
Fig. 28.

  SOLUBILITY OF SILVER NITRATE.

  ---------------------------------------------------
      Temperature.    |Parts of dry salt in 100 parts
                      |   of solution.
  --------------------+------------------------------
          -7°         |         46.2
          -1°         |         52.1
          +5°         |         56.3
          10°         |         61.2
          20°         |         67.8
          40.5°       |         76.8
          73°         |         84.0
         135°         |         92.8
         182°         |         96.9
  ---------------------------------------------------

In this figure the composition of the solution is expressed in parts of
silver nitrate in 100 parts by weight of the solution, so that 100 per
cent. represents pure silver nitrate. As can be seen, the solubility
increases with the temperature. At a temperature of about 160° there should
be a break in the curve due to change of crystalline form (p. 63). Such a
change in the direction of the solubility curve, however, does not in any
way alter the essential nature of the relationships discussed here, and may
for the present be left out of account. On following the solubility curve
of silver nitrate to higher temperatures, therefore, the concentration of
silver nitrate in the solution gradually increases, until at last, at a
temperature of 208°,[192] the melting point of pure silver nitrate is
reached, and the concentration of the water has become zero. The curve
throughout its whole extent represents the equilibrium between silver
nitrate, solution, and vapour. Conversely, starting with pure silver
nitrate in contact with the fused salt, addition of water will lower the
melting point, _i.e._ will lower the temperature at which the solid salt
can exist in contact with the liquid; {116} and the depression will be all
the greater the larger the amount of water added. As the concentration of
the water in the liquid phase is increased, therefore, the system will pass
back along the curve from higher to lower temperatures, and from greater to
smaller concentrations of silver nitrate in the liquid phase. The curve in
Fig. 28 may, therefore, be regarded either as the solubility curve of
silver nitrate in water, or as the freezing point curve for silver nitrate
in contact with a solution consisting of that salt and water.

As the temperature of the saturated solution falls, silver nitrate is
deposited, and on lowering the temperature sufficiently a point will at
last be reached at which ice also begins to separate out. Since there are
now four phases co-existing, viz. silver nitrate, ice, solution, vapour,
the system is invariant, and the point is a _quadruple point_. This
quadruple point, therefore, forms the lower limit of the solubility curve
of silver nitrate. Below this point the solution becomes metastable.

Ice as Solid Phase.--Ice melts or is in equilibrium with water at a
temperature of 0°. The melting point, will, however, be lowered by the
solution of silver nitrate in the water; and the greater the concentration
of the salt in the solution the greater will be the depression of the
temperature of equilibrium. On continuing the addition of silver nitrate, a
point will at length be reached at which the salt is no longer dissolved,
but remains in the solid form along with the ice. We again obtain,
therefore, the invariant system ice--salt--solution--vapour. The
temperature at which this invariant system can exist has been found by
Middelberg[193] to be -7.3°, the solution at this point containing 47.1 per
cent. of silver nitrate.

The same general behaviour will be found in the case of all other systems
of two components belonging to this class; that is, in the case of systems
from which the components crystallise out in the pure state, and in which
the fused components are miscible in all proportions. In all such cases,
therefore, the solubility curves (curves of equilibrium) can be represented
diagrammatically as in Fig. 29. In this figure OA represents the solubility
curve of the salt, and OB the freezing {117} point curve of ice. O is the
quadruple point at which the invariant system exists, and may be regarded
as the point of intersection of the solubility curve with the
freezing-point curve. Since this point is fixed, the condition of the
system as regards temperature, vapour pressure, and concentration of the
components (or composition of the solution), is perfectly definite. From
the way, also, in which the condition is attained, it is evident that the
quadruple point is the lowest temperature that can be obtained with
mixtures of the two components in presence of vapour. It is known as the
_cryohydric point_, or, generally, the _eutectic point_.[194]

[Illustration: FIG. 29.]

Cryohydrates.[195]--On cooling a solution of common salt in water to a
temperature of -3°, Guthrie observed that the hydrate NaCl,2H_{2}O
separated out. This salt continued to be deposited until at a temperature
of -22° opaque crystals made their appearance, and the liquid passed into
the solid state without change of temperature. A similar behaviour was
found by Guthrie in the case of a large number of other salts, a
temperature below that of the melting point of ice being reached at which
on continued withdrawal of heat, the solution solidified at a constant
temperature. When the system had attained this minimum temperature, it was
found that the composition of the solid and the liquid phases was the same,
and remained unchanged throughout the period of solidification. This is
shown by the following figures, which give the composition of different
samples of the solid phase deposited from the solution at constant
temperature.[196]

{118}

  ---------------------------------
  No. | Temperature of  |   NaCl.
      | solidification. | Per cent.
  ----|-----------------|----------
   1  |  -21° to -22°   |   23.72
   2  |      -22°       |   23.66
   3  |      -22°       |   23.73
   4  |      -23°       |   23.82
   5  |      -23°       |   23.34
   6  |      -23°       |   23.35
  ---------------------------------
             Mean           23.6
  ---------------------------------

Conversely, a mixture of ice and salt containing 23.6 per cent. of sodium
chloride will melt at a definite and constant temperature, and exhibit,
therefore, a behaviour supposed to be characteristic of a pure chemical
compound. This, then, combined with the fact that the solid which was
deposited was crystalline, and that the same constant temperature was
attained, no matter with what proportions of water and salt one started,
led Guthrie to the belief that the solids which thus separated at constant
temperature were definite chemical compounds, to which he gave the general
name _cryohydrate_. A large number of such cryohydrates were prepared and
analysed by Guthrie, and a few of these are given in the following table,
together with the temperature of the cryohydric point:[197]--

  CRYOHYDRATES.

  ------------------------------------------------------------------
         Salt.       | Cryohydric point. | Percentage of anhydrous
                     |                   | salt in the cryohydrate.
  ------------------------------------------------------------------
  Sodium bromide     |        -24°       |          41.33
  Sodium chloride    |        -22°       |          23.60
  Potassium iodide   |        -22°       |          52.07
  Sodium nitrate     |        -17.5°     |          40.80
  Ammonium sulphate  |        -17°       |          41.70
  Ammonium chloride  |        -15°       |          19.27
  Sodium iodide      |        -15°       |          59.45
  Potassium bromide  |        -13°       |          32.15
  Potassium chloride |        -11.4°     |          20.03
  Magnesium sulphate |         -5°       |          21.86
  Potassium nitrate  |         -2.6°     |          11.20
  Sodium sulphate    |         -0.7°     |           4.55
  ------------------------------------------------------------------

{119}

The chemical individuality of these cryohydrates was, however, called in
question by Pfaundler,[198] and disproved by Offer,[199] who showed that in
spite of the constancy of the melting point, the cryohydrates had the
properties, not of definite chemical compounds, but of mixtures; the
arguments given being that the heat of solution and the specific volume are
the same for the cryohydrate as for a mixture of ice and salt of the same
composition; and it was further shown that the cryohydrate had not a
definite crystalline form, but separated out as an opaque mass containing
the two components in close juxtaposition. The heterogeneous nature of
cryohydrates can also be shown by a microscopical examination.

At the cryohydric point, therefore, we are not dealing with a single solid
phase, but with two solid phases, ice and salt; the cryohydric point,
therefore, as already stated, is a quadruple point and represents an
invariant system.

Although on cooling a solution to the cryohydric point, separation of ice
may occur, it will not necessarily take place; the system may become
metastable. Similarly, separation of salt may not take place immediately
the cryohydric point is reached. It will, therefore, be possible to follow
the curves BO and AO beyond the quadruple point,[200] which is thereby
clearly seen to be the point of intersection of the solubility curve of the
salt and the freezing-point curve of ice. At this point, also, the curves
of the univariant systems ice--salt--vapour and ice--salt--solution
intersect.

Changes at the Quadruple Point.--Since the invariant system
ice--salt--solution--vapour can exist only at a definite temperature,
addition or withdrawal of heat must cause the disappearance of one of the
phases, whereby the system will become univariant. So long as all four
phases are present the temperature, pressure, and concentration of the
components in the solution must remain constant. When, therefore, heat is
added to or withdrawn from the system, mutually compensatory changes will
take place within the system whereby the {120} condition of the latter is
preserved. These changes can in all cases be foreseen with the help of the
theorem of van't Hoff and Le Chatelier; and, after what was said in Chap.
IV., need only be briefly referred to here. In the first place, addition of
heat will cause ice to melt, and the concentration of the solution will be
thereby altered; salt must therefore dissolve until the original
concentration is reached, and the heat of fusion of ice will be
counteracted by the heat of solution of the salt. Changes of volume of the
solid and liquid phases must also be taken into account; an alteration in
the volume of these phases being compensated by condensation or
evaporation. All four phases will therefore be involved in the change, and
the final state of the system will be dependent on the amounts of the
different phases present; the ultimate result of addition or withdrawal of
heat or of change of pressure at the quadruple point will be one of the
four univariant systems: ice--solution--vapour; salt--solution--vapour;
ice--salt--vapour; ice--salt--solution. If the vapour phase disappear,
there will be left the univariant system ice--salt--solution, and the
temperature at which this system can exist will alter with the pressure.
Since in this case the influence of pressure is comparatively slight, the
temperature of the quadruple point will differ only slightly from that of
the cryohydric point as determined under atmospheric pressure.

Freezing Mixtures.--Not only will the composition of a univariant system
undergo change when the temperature is varied, but, conversely, if the
_composition_ of the system is caused to change, corresponding changes of
temperature must ensue. Thus, if ice is added to the univariant system
salt--solution--vapour, the ice must melt and the temperature fall; and if
sufficient ice is added, the temperature of the cryohydric point must be at
length reached, for it is only at this temperature that the four phases
ice--salt--solution--vapour can coexist. Or, on the other hand, if salt is
added to the system ice--solution--vapour, the concentration of the
solution will increase, ice must melt, and the temperature must thereby
fall; and this process also will go on until the cryohydric point is
reached. In both cases ice melts and there is a change in the {121}
composition of the solution; in the former case, salt will be
deposited[201] because the solubility diminishes as the temperature falls;
in the latter, salt will pass into solution. This process may be
accompanied either by an evolution or, more generally, by absorption of
heat; in the former case the effect of the addition of ice will be
partially counteracted; in the latter case it will be augmented.

These principles are made use of in the preparation of _freezing mixtures_.
The lowest temperature which can be reached by means of these (under
atmospheric pressure) is the cryohydric point. This temperature-minimum is,
however, not always attained in the preparation of a freezing mixture, and
that for various reasons. The chief of these are radiation and the heat
absorbed in cooling the solution produced. The lower the temperature falls,
the more rapid does the radiation become; and the rate at which the
temperature sinks decreases as the amount of solution increases. Both these
factors counteract the effect of the latent heat of fusion and the heat of
solution, so that a point is reached (which may lie considerably above the
cryohydric point) at which the two opposing influences balance. The
absorption of heat by the solution can be diminished by allowing the
solution to drain off as fast as it is produced; and the effect of
radiation can be partially annulled by increasing the rate of cooling. This
can be done by the more intimate mixing of the components. Since, under
atmospheric pressure, the temperature of the cryohydric point is constant,
the cryohydrates are very valuable for the production of baths of constant
low temperature.

2. _Partial Miscibility of the Fused Components._

On passing to the study of the second class of systems of two components
belonging to this group, namely, those in which the fused components are
not miscible in all proportions, we find that the relationships are not
quite so simple as {122} in the case of silver nitrate and water. In the
latter case, only one liquid phase was possible; in the cases now to be
studied, two liquid phases can be formed, and there is a marked
discontinuity in the solubility curve on passing from the cryohydric point
to the melting point of the second (non-volatile) component.

Paratoluidine dissolves in water, and the solubility increases as the
temperature rises.[202] At 44.2°, however, paratoluidine in contact with
water melts, and two liquid phases are formed, viz. a solution of water in
fused paratoluidine and a solution of fused paratoluidine in water. We
have, therefore, the phenomenon of _melting under the solvent_. This
melting point will, of course, be lower than the melting point of the pure
substance, because the solid is now in contact with a solution, and, as we
have already seen, addition of a foreign substance lowers the melting
point. Such cases of melting under the solvent are by no means rare, and a
review of the relationships met with may, therefore, be undertaken here. As
an example, there may be chosen the equilibrium between succinic nitrile,
C_{2}H_{4}(CN)_{2} and water, which has been fully studied by
Schreinemakers.[203]

[Illustration: FIG. 30.]

If to the system ice--water at 0° succinic nitrile is added, the
temperature will fall; and continued addition of the nitrile will lead at
last to the cryohydric point _b_ (Fig. 30), at which solid nitrile, ice,
solution, and vapour can coexist. The temperature of the cryohydric point
is -1.2°, and the composition of the solution is 1.29 mol. of nitrile in
100 mol. of solution. From _a_ to _b_ the solid phase in contact with the
solution is ice. {123} If the temperature be now raised so as to cause the
disappearance of the ice, and the addition of nitrile be continued, the
concentration of the nitrile in the solution will increase as represented
by the curve _bc_. At the point _c_ (18.5°), when the concentration of the
nitrile in the solution has increased to 2.5 molecules per cent., the
nitrile melts and two liquid phases are formed; the concentration of the
nitrile in these two phases is given by the points _c_ and _c'_. As there
are now four phases present, viz. solid nitrile, solution of fused nitrile
in water, solution of water in fused nitrile, and vapour, the system is
_invariant_. Since at this point the concentration, temperature, and
pressure are completely defined, addition or withdrawal of heat can only
cause a change in the relative amounts of the phases, _but no variation of
the concentrations_ of the respective phases. As a matter of fact,
continued addition of nitrile and addition of heat will cause an increase
in the amount of the liquid phase containing excess of nitrile (_i.e._ the
solution of water in fused nitrile), whereas the other liquid phase, the
solution of fused nitrile in water, will gradually disappear. When it has
completely disappeared, the system will be represented by the point _c'_,
where the molecular concentration of nitrile is now 75 per cent., and again
becomes univariant, the three phases being solid nitrile, liquid phase
containing excess of nitrile, and vapour; and as the amount of the water is
diminished the temperature of equilibrium rises, until at 54° the melting
point of the pure nitrile is reached.

Return now to the point c. At this point there exists the invariant system
solid nitrile, two liquid phases, vapour. If heat be added, the solid
nitrile will disappear, and there will be left the univariant system,
consisting of two liquid phases and vapour.[204] Such a system will exhibit
relationships similar to those already studied in the previous chapter. As
the temperature rises, the mutual solubility of the two fused components
becomes greater, until at _d_ (55.5°) the critical solution temperature is
reached, and the fused components become miscible in all proportions.

At all temperatures and concentrations lying to the right {124} of the
curve _abcdc'e_ there can be only one liquid phase; in the field _cdc'_
there are two liquid phases.

From the figure it will be easy to see what will be the result of bringing
together succinic nitrile and water at different temperatures and in
different amounts. Since _b_ is the lowest temperature at which liquid can
exist in stable equilibrium with solid, ice and succinic nitrile can be
mixed in any proportions at temperatures below _b_ without undergoing
change. Between _b_ and _c_ succinic nitrile will be dissolved until the
concentration reaches the value on the curve _bc_, corresponding to the
given temperature. On adding the nitrile to water at temperatures between
_c_ and _d_, it will dissolve until a concentration lying on the curve _cd_
is attained; at this point two liquid phases will be formed, and further
addition of nitrile will cause the one liquid phase (that containing excess
of nitrile) to increase, while the other liquid phase will decrease, until
it finally disappears and there is only one liquid phase left, that
containing excess of nitrile. This can dissolve further quantities of the
nitrile, and the concentration will increase until the curve _c'e_ is
reached, when the concentration will remain unchanged, and addition of
solid will merely increase the amount of the solid phase.

If a solution represented by any point in the field lying below the curve
_bcd_ is heated to a temperature above _d_, the critical solution
temperature, then the concentration of the nitrile can be increased to any
desired amount without at any time two liquid phases making their
appearance; the system can then be cooled down to a temperature represented
by any point between the curves _dc'e_. In this way it is possible to pass
continuously from a solution containing excess of one component to
solutions containing excess of the other, as represented by the dotted line
_xxxx_ (_v._ p. 100). At no point is there formation of two liquid phases.

Supersaturation.--Just as suspended transformation is rarely met with in
the passage from the solid to the liquid state, so also it is found in the
case of the melting of substances under the solvent that suspended fusion
does not occur; but that when the temperature of the invariant point is
reached at which, therefore, the formation of two liquid layers is
possible, {125} these two liquid layers, as a matter of fact, make their
appearance. Suspended transformation can, however, take place from the side
of the liquid phase, just as water or other liquid can be cooled below the
normal freezing point without solidification occurring. The question,
therefore, arises as to the relative solubilities of the solid and the
supercooled liquid at the same temperature.

[Illustration: FIG. 31.]

The answer to this question can at once be given from what we have already
learned (p. 113), if we recollect that at temperatures below the point of
fusion under the solvent, the solid form, at temperatures above that point,
the liquid form, is the more stable; at this temperature, therefore, the
relative stability of the solid and liquid forms changes. Since, as we have
already seen, the less stable form has the greater solubility, it follows
that the supercooled liquid, being the less stable form, must have the
greater solubility. This was first proved experimentally by Alexejeff[205]
in the case of benzoic acid and water, the solubility curves for which are
given in Fig. 31. As can be seen from the figure, the prolongation of the
curve for liquid--liquid, which represents the solubility of the
supercooled liquid benzoic acid, lies above that for the solubility of the
{126} solid benzoic acid in water; the solution saturated with respect to
the supercooled liquid is therefore supersaturated with respect to the
solid form. A similar behaviour has been found in the case of other
substances.[206]

Pressure-Temperature Diagram.--Having considered the changes which occur in
the concentration of the components in a solution with the temperature, we
may conclude the discussion of the equilibrium between a salt and water by
studying the variation of the vapour pressure.

Since in systems of two components the two phases, solution and vapour,
constitute a bivariant system, the vapour pressure is undefined, and may
have different values at the same temperature, depending on the
concentration. In order that there may be for each temperature a definite
corresponding pressure of the vapour, a third phase must be present. This
condition is satisfied by the system solid--liquid (solution)--vapour; that
is, by the saturated solution (p. 108). In the case of a saturated
solution, therefore, the pressure of the vapour at any given temperature is
constant.

Vapour Pressure of Solid--Solution--Vapour.--It has long been known that
the addition of a non-volatile solid to a liquid in which it is soluble
lowers the vapour pressure of the solvent; and the diminution of the
pressure is approximately proportional to the amount of substance dissolved
(Law of Babo). The vapour-pressure curve, therefore, of a solution of a
salt in water must lie below that for pure water. Further, in the case of a
pure liquid, the vaporization curve is a function only of the temperature
(p. 63), whereas, in the case of a solution, the pressure varies both with
the temperature and the _concentration_. These two factors, however, act in
opposite directions; for although the vapour pressure in all cases
increases as the temperature rises, increase of concentration, as we have
seen, lowers the vapour pressure. Again, since the concentration itself
varies with the temperature, two cases have to be considered, viz. where
the concentration increases with rise of {127} temperature, and where the
concentration diminishes with rise of temperature.

The relations which are found here will be best understood with the help of
Fig. 32.[207] In this figure, OB represents the sublimation curve of ice,
and BC the vaporization curve of water; the curve for the solution must lie
below this, and must cut the sublimation curve of ice at some temperature
below the melting point. The point of intersection A is the cryohydric
point. If the solubility increases with rise of temperature, the increase
of the vapour pressure due to the latter will be partially annulled. Since
at first the effect of increase of temperature more than counteracts the
depressing action of increase of concentration, the vapour pressure will
increase on raising the temperature above the cryohydric point. If the
elevation of temperature is continued, however, to the melting point of the
salt, the effect of increasing concentration makes itself more and more
felt, so that the vapour-pressure curve of the solution falls more and more
below that of the pure liquid, and the pressure will ultimately become
equal to that of the pure salt; that is to say, practically equal to zero.
The curve will therefore be of the general form AMF shown in Fig. 32. If
the solubility should diminish with rise of temperature, the two factors,
temperature and concentration, will act in the same direction, and the
vapour-pressure curve will rise relatively more rapid than that of the pure
liquid; since, however, the pure salt is ultimately obtained, the
vapour-pressure curve must in this case also finally approach the value
zero.

[Illustration: FIG. 32.]

Other Univariant Systems.--Besides the univariant system {128}
salt--solution--vapour already considered, three others are possible, viz.
ice--solution--vapour, ice--salt--solution, and ice--salt--vapour.

The fusion point of a substance is lowered, as we have seen, by the
addition of a foreign substance, and the depression is all the greater the
larger the quantity of substance added. The vapour pressure of the water,
also, is lowered by the solution in it of other substances, so that the
vapour pressure of the system ice--solution--vapour must decrease as the
temperature falls from the fusion point of ice to the cryohydric point.
This curve is represented by BA (Fig. 32), and is coincident with the
sublimation curve of ice.

This, at first sight, strange fact will be readily understood when we
consider that since ice and solution are together in equilibrium with the
same vapour, they must have the same vapour pressure. For suppose at any
given temperature equilibrium to have been established in the system
ice--solution--vapour, removal of the ice will not alter this equilibrium.
Suppose, now, the ice and the solution placed under a bell-jar so that they
have a common vapour, but are not themselves in contact; then, if they do
not have the same vapour pressure, distillation must take place and the
solution will become more dilute or more concentrated. Since, at the
completion of this process, the ice and solution are now in equilibrium
when they are not in contact, they must also be in equilibrium when they
are in contact (p. 32). But if distillation has taken place the
concentration of the solution must have altered, so that the ice will now
be in equilibrium with a solution of a different concentration from before.
But according to the Phase Rule ice cannot at one and the same temperature
be in equilibrium with two solutions of different concentration, for the
system ice--solution--vapour is univariant, and at any given temperature,
therefore, not only the pressure but also the _concentration of the
components in the solution must be constant_. Distillation could not,
therefore, take place from the ice to the solution or _vice versâ_; that is
to say, the solution and the ice must have the same vapour pressure--the
sublimation pressure of ice. The reason of the coincidence is the
non-volatility of the salt: had {129} the salt a measurable vapour pressure
itself, the sublimation curve of ice and the curve for
ice--solution--vapour would no longer fall together.

The curve AO represents the pressures of the system ice--salt--vapour. This
curve will also be coincident with the sublimation curve of ice, on account
of the non-volatility of the salt.

The equilibria of the fourth univariant system ice--salt--solution are
represented by AE. Since this is a condensed system, the effect of a small
change of temperature will be to cause a large change of pressure, as in
the case of the fusion point of a pure substance. The direction of this
curve will depend on whether there is an increase or diminution of volume
on solidification; but the effect in any given case can be predicted with
the help of the theorem of Le Chatelier.

Since the cryohydric point is a quadruple point in a two-component system,
it represents an invariant system. The condition of the system is,
therefore, completely defined; the four phases, ice, salt, solution,
vapour, can co-exist only when the temperature, pressure, and concentration
of the solution have constant and definite values. Addition or withdrawal
of heat, therefore, can cause no alteration of the condition of the system
except a variation of the relative amounts of the phases. Addition
of heat at constant volume will ultimately lead to the system
salt--solution--vapour or the system ice--solution--vapour, according as
ice or salt disappears first. This is readily apparent from the diagram
(Fig. 32), for the systems ice--salt--solution and ice--salt--vapour can
exist only at temperatures below the cryohydric point (provided the curve
for ice--salt--solution slopes towards the pressure axis).

Bivariant Systems.--Besides the univariant systems already discussed,
various bivariant systems are possible, the conditions for the existence of
which are represented by the different areas of Fig. 32. They are as
follows:--

  _Area._             _System._

  OAMF     Salt--vapour.
  CBAMF    Solution--vapour; salt--solution.
  EABD     Salt--solution; ice--solution.
  EAO      Ice--salt.

{130}

Deliquescence.--As is evident from Fig. 32, salt can exist in contact with
water vapour at pressures under those represented by OAMF. If, however, the
pressure of the vapour is increased until it reaches a value lying on this
curve at temperatures above the cryohydric point, solution will be formed;
for the curve AMF represents the equilibria between salt--solution--vapour.
From this, therefore, it is clear that if the pressure of the aqueous
vapour in the atmosphere is greater than that of the saturated solution of
a salt, that salt will, on being placed in the air, form a solution; it
will _deliquesce_.

Separation of Salt on Evaporation.--With the help of Fig. 32 it is possible
to state in a general manner whether or not salt will be deposited when a
solution is evaporated under a constant pressure.[208]

The curve AMF (Fig. 32) is the vapour-pressure curve of the saturated
solutions of the salt, _i.e._ it represents, as we have seen, the maximum
vapour pressure at which salt can exist in contact with solution and
vapour. The dotted line _aa_ represents atmospheric pressure. If, now, an
unsaturated solution, the composition of which is represented by the point
_x_, is heated in an open vessel, the temperature will rise, and the vapour
pressure of the solution will increase. The system will, therefore, pass
along a line represented diagrammatically by _xx'_. At the point _x'_ the
vapour pressure of the system becomes equal to 1 atm.; and as the vessel is
open to the air, the pressure cannot further rise; the solution boils. If
the heating is continued, water passes off, the concentration increases,
and the boiling point rises. The system will therefore pass along the line
_x'm_, until at the point _m_ solid salt separates out (provided
supersaturation is excluded). The system is now univariant, and continued
heating will no longer cause an alteration of the concentration; as water
passes off, solid salt will be deposited, and the solution will evaporate
to dryness.

If, however, the atmospheric pressure is represented not by _aa_ but by
_bb_, then, as Fig. 32 shows, the maximum vapour {131} pressure of the
system salt--solution--vapour never reaches the pressure of 1 atm. Further,
since the curve _bb_ lies in the area of the bivariant system
solution--vapour there can at no point be a separation of the solid form;
for the system solid--solution--vapour can exist only along the curve AMF.

On evaporating the solution of a salt in an open vessel, therefore, salt
can be deposited only if at some temperature the pressure of the saturated
solution is equal to the atmospheric pressure. This is found to be the case
with most salts. In the case of aqueous solutions of sodium and potassium
hydroxide, however, the vapour pressure of the saturated solution never
reaches the value of 1 atm., and on evaporating these solutions, therefore,
in an open vessel, there is no separation of the solid. Only a homogeneous
fused mass is obtained. If, however, the evaporation be carried out under a
pressure which is lower than the maximum pressure of the saturated
solution, separation of the solid substance will be possible.

General Summary.--The systems which have been discussed in the present
chapter contained water as one of their components, and an anhydrous salt
as the other. It will, however, be clear that the relationships which were
found in the case of these will be found also in other cases where it is a
question of the equilibria between two components, which crystallize out in
the pure state, and only one of which possesses a measurable vapour
pressure. A similar behaviour will, for example, be found in the case of
many pairs of organic substances; and in all cases the equilibria will be
represented by a diagram of the general appearance of Fig. 29 or Fig. 30.
That is to say: Starting from the fusion point of component I., the system
will pass, by progressive addition of component II., to regions of lower
temperature, until at last the cryohydric or eutectic point is reached. On
further addition of component II., the system will pass to regions of
higher temperature, the solid phase now being component II. If the fused
components are miscible with one another in all proportions a continuous
curve will be obtained leading up to the point of fusion of component II.
Slight changes of direction, it is true, due to changes in the crystalline
form, may be found along this curve, {132} but throughout its whole course
there will be but one liquid phase. If, on the other hand, the fused
components are not miscible in all proportions, then the second curve will
exhibit a marked discontinuity, and two liquid phases will make their
appearance.

       *       *       *       *       *


{133}

CHAPTER VIII

SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE

B.--HYDRATED SALT AND WATER.

In the preceding chapter we discussed the behaviour of systems formed of
two components, only one of which was volatile, in those cases where the
two components separated from solution in the pure state. In the present
chapter we shall consider those systems in which combination between the
components can occur with the formation of definite compounds; such as are
found in the case of crystalline salt hydrates. Since a not inconsiderable
amount of study has been devoted to the systems formed by hydrated salts
and water, systems which are of great chemical interest and importance, the
behaviour of these will first call for discussion in some detail, and it
will be found later that the relationships which exist in such systems
appear also in a large number of other two-component systems.

The systems belonging to this group may be divided into two classes
according as the compounds formed possess a definite melting point, _i.e._
form a liquid phase of the same composition, or do not do so. We shall
consider the latter first.

1. _The Compounds formed do not have a Definite Melting Point._

Concentration-Temperature Diagram.--In the case of salts which can form
crystalline hydrates, the temperature-concentration diagram, representing
the equilibria of the {134} different possible systems, must necessarily be
somewhat more complicated than where no such combination of the components
occurs. For, as has already been pointed out, each substance has its own
solubility curve; and there will therefore be as many solubility curves as
there are solid phases possible, _the curve for each particular solid phase
being continuous so long as it remains unchanged in contact with the
solution_. As an example of the relationships met with in such cases, we
shall first of all consider the systems formed of sodium sulphate and
water.

[Illustration: FIG. 33.]

Sodium Sulphate and Water.--At the ordinary temperatures, sodium sulphate
crystallises from water with ten molecules of water of crystallisation,
forming Glauber's salt. On determining the solubility of this salt in
water, it is found that the solubility increases as the temperature rises,
the values of the solubility, represented graphically by the curve AC (Fig.
33), being given in the following table.[209] The numbers denote grams of
sodium sulphate, calculated as anhydrous salt, dissolved by 100 grams of
water.

  SOLUBILITY OF Na_{2}SO_{4},10H_{2}O.

  --------------------------
  Temperature. | Solubility.
  --------------------------
       0°      |     5.02
      10°      |     9.00
      15°      |    13.20
      18°      |    16.80
      20°      |    19.40
      25°      |    28.00
      30°      |    40.00
      33°      |    50.76
      34°      |    55.00
  --------------------------

{135}

On continuing the investigation at higher temperatures, it was found that
the solubility no longer increased, but _decreased with rise of
temperature_. At the same time, it was observed that the solid phase was
now different from that in contact with the solution at temperatures below
33°; for whereas in the latter case the solid phase was sodium sulphate
decahydrate, at temperatures above 33° the solid phase was the anhydrous
salt. The course of the solubility curve of anhydrous sodium sulphate is
shown by BD, and the values of the solubility are given in the following
table:--[210]

  SOLUBILITY OF ANHYDROUS SODIUM SULPHATE.

  --------------------------
  Temperature. | Solubility.
  --------------------------
    18°        |    53.25
    20°        |    52.76
    25°        |    51.53
    30°        |    50.37
    33°        |    49.71
    34°        |    49.53
    36°        |    49.27
    40.15°     |    48.78
    50.40°     |    46.82
  --------------------------

As is evident from the figure, the solubility curve which is obtained when
anhydrous sodium sulphate is present as the solid phase, cuts the curve
representing the solubility of the decahydrate, at a temperature of about
33°.

If a solution of sodium sulphate which has been saturated at a temperature
of about 34° be cooled down to a temperature below 17°, while care is taken
that the solution is protected against access of particles of Glauber's
salt, crystals of a second hydrate of sodium sulphate, having the
composition Na_{2}SO_{4},7H_{2}O, separate out. On determining the
composition of the solutions in equilibrium with this hydrate at different
temperatures, the following values were obtained, these values being
represented by the curve FE (Fig. 33):--

{136}

  SOLUBILITY OF Na_{2}SO_{4},7H_{2}O.

  --------------------------
  Temperature. | Solubility.
  --------------------------
       0°      |    19.62
      10°      |    30.49
      15°      |    37.43
      18°      |    41.63
      20°      |    44.73
      25°      |    52.94
      26°      |    54.97
  --------------------------

Since, as has already been stated, each solid substance has its own
solubility curve, there are three separate curves to be considered in the
case of sodium sulphate and water. Where two curves cut, the solution must
be saturated with respect to two solid phases; at the point B, therefore,
the point of intersection of the solubility curve of anhydrous sodium
sulphate with that of the decahydrate, the solution must be saturated with
respect to these two solid substances. But a system of two components
existing in four phases, anhydrous salt--hydrated salt--solution--vapour,
is invariant; and this invariability will remain even if only three phases
are present, provided that one of the factors, pressure, temperature, or
concentration of components retains a constant value. This is the case when
solubilities are determined in open vessels; the pressure is then equal to
atmospheric pressure. Under these circumstances, then, the system,
anhydrous sodium sulphate--decahydrate--solution, will possess no degree of
freedom, and can exist, therefore, only at one definite temperature and
when the solution has a certain definite composition. The temperature of
this point is 32.482° on a mercury thermometer, or 32.379° on the hydrogen
thermometer.[211]

{137}

Suspended Transformation.--Although it is possible for the anhydrous salt
to make its appearance at the temperature of the quadruple point, it will
not necessarily do so; and it is therefore possible to follow the
solubility curve of sodium sulphate decahydrate to a higher temperature.
Since, however, the solubility of the decahydrate at temperatures above the
quadruple point is greater than that of the anhydrous salt, the solution
which is _saturated_ with respect to the former will be _supersaturated_
with respect to the latter. On bringing a small quantity of the anhydrous
salt in contact with the solution, therefore, anhydrous salt will be
deposited; and all the hydrated salt present will ultimately undergo
conversion into the anhydrous salt, through the medium of the solution. In
this case, as in all cases, the solid phase, which is the most stable at
the temperature of the experiment, has at that temperature the least
solubility.

Similarly, the solubility curve of anhydrous sodium sulphate has been
followed to temperatures below 32.5°. Below this temperature, however, the
solubility of this salt is greater than that of the decahydrate, and the
saturated solution of the anhydrous salt will therefore be supersaturated
for the decahydrate, and will deposit this salt if a "nucleus" is added to
the solution. From this we see that at temperatures above 32.5° the
anhydrous salt is the stable form, while the decahydrate is unstable (or
metastable); at temperatures below 32.5° the decahydrate is stable. This
temperature, therefore, is the _transition temperature_ for decahydrate and
anhydrous salt.

From Fig. 33 we see further that the solubility curve of the anhydrous salt
(which at all temperatures below 32.5° is metastable) is cut by the
solubility curve of the heptahydrate; and this point of intersection (at a
temperature of 24.2°) must be the _transition point_ for heptahydrate and
anhydrous salt. Since at all temperatures the solubility of the
heptahydrate is greater than that of the decahydrate, the former hydrate
must be metastable with respect to the latter; so that throughout its whole
course the solubility curve of the heptahydrate {138} represents only
metastable equilibria. Sodium sulphate, therefore, forms only one stable
hydrate, the decahydrate.

The solubility relations of sodium sulphate illustrate very clearly the
importance of the solid phase for the definition of saturation and
supersaturation. Since the solubility curve of the anhydrous salt has been
followed backwards to a temperature of about 18°, it is readily seen, from
Fig. 33, that at a temperature of, say, 20° three different _saturated_
solutions of sodium sulphate are possible, according as the anhydrous salt,
the heptahydrate or the decahydrate, is present as the solid phase. Two of
these solutions, however, would be metastable and _supersaturated with
respect to the decahydrate_.

Further, the behaviour of sodium sulphate and water furnishes a very good
example of the fact that a "break" in the solubility curve occurs when, and
only when, the solid phase undergoes change. So long as the decahydrate,
for example, remained unaltered in contact with the solution, the
solubility curve was continuous; but when the anhydrous salt appeared in
the solid phase, a distinct change in the direction of the solubility curve
was observed.

Dehydration by Means of Anhydrous Sodium Sulphate.--The change in the
relative stability of sodium sulphate decahydrate and anhydrous salt in
presence of water at a temperature of 32.5° explains why the latter salt
cannot be employed for dehydration purposes at temperatures above the
transition point. The dehydrating action of the anhydrous salt depends on
the formation of the decahydrate; but since at temperatures above 33° the
latter is unstable, and cannot be formed in presence of the anhydrous salt,
this salt cannot, of course, effect a dehydration above that temperature.

Pressure-Temperature Diagram.--The consideration of the
pressure-temperature relations of the two components, sodium sulphate and
water, must include not only the vapour pressure of the saturated
solutions, but also that of the crystalline hydrates. The vapour pressures
of salt hydrates have already been treated in a general manner (Chap. V.),
so that it is only necessary here to point out the connection between the
two classes of systems. {139}

In most cases the vapour pressure of a salt hydrate, _i.e._ the vapour
pressure of the system hydrate--anhydrous salt (or lower hydrate)--vapour,
is at all temperatures lower than that of the system anhydrous salt (or
lower hydrate)--solution--vapour. This, however, is not a necessity; and
cases are known where the vapour pressure of the former system is, under
certain circumstances, equal to or higher than that of the latter. An
example of this is found in sodium sulphate decahydrate.

On heating Na_{2}SO_{4},10H_{2}O, a point is reached at which the
dissociation pressure into anhydrous salt and water vapour becomes equal to
the vapour pressure of the saturated solution of the anhydrous salt, as is
apparent from the following measurements;[212] the differences in pressure
being expressed in millimetres of a particular oil.

  Temperature:      29.0°   30.83°   31.79°   32.09°   32.35°   32.6°
  Difference of
  pressure:         23.8    10.8      5.6      3.6      1.6      0

At 32.6°, therefore, the vapour pressures of the two systems

  Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
  Na_{2}SO_{4}--solution--vapour

are equal; at this temperature the four phases, Na_{2}SO_{4},10H_{2}O;
Na_{2}SO_{4}; solution; vapour, can coexist. From this it is evident that
when sodium sulphate decahydrate is heated to 32.6°, the two new phases
anhydrous salt and solution will be formed (suspended transformation being
supposed excluded), and the hydrate will appear to undergo _partial
fusion_; and during the process of "melting" the vapour pressure and
temperature will remain constant.[213] This is, however, not a true but a
so-called _incongruent_ melting point; for the composition of the liquid
phase is not the same as that of the solid. As has already been pointed out
(p. 137), we are dealing here with the _transition point_ of the
decahydrate and anhydrous salt, _i.e._ with the reaction
Na_{2}SO_{4},10H_{2}O <--> Na_{2}SO_{4} + 10H_{2}O.

Since at the point of partial fusion of the decahydrate four {140} phases
can coexist, the point is a quadruple point in a two-component system, and
the system at this point is therefore invariant. The temperature of this
point is therefore perfectly definite, and on this account the proposal has
been made to adopt this as a fixed point in thermometry.[214] The
temperature is, of course, practically the same as that at which the two
solubility curves intersect (p. 112). If, however, the vapour phase
disappears, the system becomes univariant, and the equilibrium temperature
undergoes change with change of pressure. The transition curve has been
determined by Tammann,[215] and shown to pass through a point of maximum
temperature.

[Illustration: FIG. 34.]

The vapour pressure of the different systems of sodium sulphate and water
can best be studied with the help of the diagram in Fig. 34.[216] The curve
ABCD represents the vapour-pressure curve of the saturated solution of
anhydrous sodium sulphate. GC is the pressure curve of decahydrate +
anhydrous salt, which, as we have seen, cuts the curve ABCD at the
transition temperature, 32.6°. Since at this point the solution is
saturated with respect to both the anhydrous salt and the decahydrate, the
vapour-pressure curve of the saturated solution of the latter must also
pass through the point C.[217] As at temperatures below this point the
solubility of the decahydrate is less than that of the anhydrous salt, the
vapour pressure of the solution will, in accordance with Babo's law
(p. 126), be higher than that of the solution of the anhydrous salt; which
was also found experimentally to be the case (curve HC).

{141}

In connection with the vapour pressure of the saturated solutions of the
anhydrous salt and the decahydrate, attention must be drawn to a
conspicuous deviation from what was found to hold in the case of
one-component systems in which a vapour phase was present (p. 31). There,
it was seen that the vapour pressure of the more stable system was always
_lower_ than that of the less stable; in the present case, however, we find
that this is no longer so. We have already learned that at temperatures
below 32.5° the system decahydrate--solution--vapour is more stable than
the system anhydrous salt--solution--vapour; but the vapour pressure of the
latter system is, as has just been stated, lower than that of the former.
At temperatures above the transition point the vapour pressure of the
saturated solution of the decahydrate will be lower than that of the
saturated solution of the anhydrous salt.

This behaviour depends on the fact that the less stable form is the more
soluble, and that the diminution of the vapour pressure increases with the
amount of salt dissolved.

With regard to sodium sulphate heptahydrate the same considerations will
hold as in the case of the decahydrate. Since at 24° the four phases
heptahydrate, anhydrous salt, solution, vapour can coexist, the
vapour-pressure curves of the systems hydrate--anhydrous salt--vapour
(curve EB) and hydrate--solution--vapour (curve FB) must cut the pressure
curve of the saturated solution of the anhydrous salt at the above
temperature, as represented in Fig. 34 by the point B. This constitutes,
therefore, a second quadruple point, which is, however, metastable.

From the diagram it is also evident that the dissociation pressure of the
heptahydrate is higher than that of the decahydrate, although it contains
less water of crystallization. The system heptahydrate--anhydrous
salt--vapour must be metastable with respect to the system
decahydrate--anhydrous salt--vapour, and will pass into the latter.[218]
Whether or not there is a temperature at which the vapour-pressure curves
of the two systems intersect, and below which the heptahydrate becomes the
more stable form, is not known.

{142}

In the case of sodium sulphate there is only one stable hydrate. Other
salts are known which exhibit a similar behaviour; and we shall therefore
expect that the solubility relationships will be represented by a diagram
similar to that for sodium sulphate. A considerable number of such cases
have, indeed, been found,[219] and in some cases there is more than one
metastable hydrate. This is found, for example, in the case of nickel
iodate,[220] the solubility curves for which are given in Fig. 35. As can
be seen from the figure, suspended transformation occurs, the solubility
curves having in some cases been followed to a considerable distance beyond
the transition point. One of the most brilliant examples, however, of
suspended transformation in the case of salt hydrates, and the sluggish
transition from the less stable to the more stable form, is found in the
case of the hydrates of calcium chromate.[221]

[Illustration: FIG. 35.]

In the preceding cases, the dissociation-pressure curve of the hydrated
salt cuts the vapour-pressure curve of the saturated {143} solution of the
anhydrous salt. It can, however, happen that the dissociation-pressure
curve of one hydrate cuts the solubility curve, not of the anhydrous salt,
but of a lower hydrate; in this case there will be more than one stable
hydrate, each having a stable solubility curve; and these curves will
intersect at the temperature of the transition point. Various examples of
this behaviour are known, and we choose for illustration the solubility
relationships of barium acetate and its hydrates[222] (Fig. 36).

[Illustration: FIG. 36.]

At temperatures above 0°, barium acetate can form two stable hydrates, a
trihydrate and a monohydrate. The solubility of the trihydrate increases
very rapidly with rise of temperature, and has been determined up to 26.1°.
At temperatures above 24.7°, however, the trihydrate is metastable with
respect to the monohydrate; for at this temperature the solubility curve of
the latter hydrate cuts that of the former. This is, therefore, the
transition temperature for the trihydrate and monohydrate. The solubility
curve of the monohydrate succeeds that of the trihydrate, and exhibits a
conspicuous point of minimum solubility at about 30°. Below 24.7° the {144}
monohydrate is the less stable hydrate, but its solubility has been
determined to a temperature of 22°. At 41° the solubility curve of the
monohydrate intersects that of the anhydrous salt, and this is therefore
the transition temperature for the monohydrate and anhydrous salt. Above
this temperature the anhydrous salt is the stable solid phase. Its
solubility curve also passes through a minimum.

The diagram of solubilities of barium acetate not only illustrates the way
in which the solubility curves of the different stable hydrates of a salt
succeed one another, but it has also an interest and importance from
another point of view. In Fig. 36 there is also shown a faintly drawn curve
which is continuous throughout its whole course. This curve represents the
solubility of barium acetate as determined by Krasnicki.[223] Since,
however, three different solid phases can exist under the conditions of
experiment, it is evident, from what has already been stated (p. 111), that
the different equilibria between barium acetate and water could not be
represented by one _continuous_ curve.

Another point which these experiments illustrate and which it is of the
highest importance to bear in mind is, that in making determinations of the
solubility of salts which are capable of forming hydrates, it is not only
necessary to determine the composition of the solution, but _it is of equal
importance to determine the composition of the solid phase in contact with
it_. In view of the fact, also, that the solution equilibrium is in many
cases established with comparative slowness, it is necessary to confirm the
point of equilibrium, either by approaching it from higher as well as from
lower temperatures, or by actually determining the rate with which the
condition of equilibrium is attained. This can be accomplished by actual
weighing of the dissolved salt or by determinations of the density of the
solution, as well as by other methods.

{145}

2. _The Compounds formed have a Definite Melting Point._

In the cases which have just been considered we saw that the salt hydrates
on being heated did not undergo complete fusion, but that a solid was
deposited consisting of a lower hydrate or of the anhydrous salt. It has,
however, been long known that certain crystalline salt hydrates (_e.g._
sodium thiosulphate, Na_{2}S_{2}O_{3},5H_{2}O, sodium acetate,
NaC_{2}H_{3}O_{2},3H_{2}O) melt completely in their water of
crystallization, and yield a liquid of the _same composition_ as the
crystalline salt. In the case of sodium thiosulphate pentahydrate the
temperature of liquefaction is 56°; in the case of sodium acetate
trihydrate, 58°. These two salts, therefore, have a definite melting point.
For the purpose of studying the behaviour of such salt hydrates, we shall
choose not the cases which have just been mentioned, but two others which
have been more fully studied, viz. the hydrates of calcium chloride and of
ferric chloride.

Solubility Curve of Calcium Chloride Hexahydrate.[224]--Although calcium
chloride forms several hydrates, each of which possesses its own
solubility, it is nevertheless the solubility curve of the hexahydrate
which will chiefly interest us at present, and we shall therefore first
discuss that curve by itself.

[Illustration: FIG. 37.]

The solubility of this salt has been determined from the cryohydric point,
which lies at about -55°, up to the melting point of the salt.[225] The
solubility increases with rise of temperature, as is shown by the figures
in the following table, and by the (diagrammatic) curve AB in Fig. 37. In
the table, the numbers under the heading "solubility" denote the number of
grams of CaCl_{2} dissolved in 100 grams {146} of water; those under the
heading "composition," the number of gram-molecules of water in the
solution to one gram-molecule of CaCl_{2}.

  SOLUBILITY OF CALCIUM CHLORIDE HEXAHYDRATE.

  -----------------------------------------
  Temperature. | Solubility. | Composition.
  -----------------------------------------
      -55°     |     42.5    |    14.5
      -25°     |     50.0    |    12.3
      -10°     |     55.0    |    11.2
        0°     |     59.5    |    10.37
       10°     |     65.0    |     9.49
       20°     |     74.5    |     8.28
       25°     |     82.0    |     7.52
       28.5°   |     90.5    |     6.81
       29.5°   |     95.5    |     6.46
       30.2°   |    102.7    |     6.00
       29.6°   |    109.0    |     5.70
       29.2°   |    112.8    |     5.41
  -----------------------------------------

So far as the first portion of the curve is concerned, it resembles the
most general type of solubility curve. In the present case the solubility
is so great and increases so rapidly with rise of temperature, that a point
is reached at which the water of crystallization of the salt is sufficient
for its complete solution. This temperature is 30.2°; and since the
composition of the solution is the same as that of the solid salt, viz. 1
mol. of CaCl_{2} to 6 mols. of water, this temperature must be the melting
point of the hexahydrate. At this point the hydrate will fuse or the
solution will solidify without change of temperature and without change of
composition. Such a melting point is called a _congruent_ melting point.

But the solubility curve of calcium chloride hexahydrate differs markedly
from the other solubility curves hitherto considered in that it possesses a
_retroflex portion_, represented in the figure by BC. As is evident from
the figure, therefore, calcium chloride hexahydrate exhibits the peculiar
and, as it was at first thought, impossible behaviour that it can be in
equilibrium at one and the same temperature with two different solutions,
one of which contains more, the other less, water than the solid hydrate;
for it must be remembered that {147} throughout the whole course of the
curve ABC the solid phase present in equilibrium with the solution is the
hexahydrate.

Such a behaviour, however, on the part of calcium chloride hexahydrate will
appear less strange if one reflects that the melting point of the hydrate
will, like the melting point of other substances, be lowered by the
addition of a second substance. If, therefore, water is added to the
hydrate at its melting point, the temperature at which the solid hydrate
will be in equilibrium with the liquid phase (solution) will be lowered; or
if, on the other hand, anhydrous calcium chloride is added to the hydrate
at its melting point (or what is the same thing, if water is removed from
the solution), the temperature at which the hydrate will be in equilibrium
with the liquid will also be lowered; _i.e._ the hydrate will melt at a
lower temperature. In the former case we have the hydrate in equilibrium
with a solution containing more water, in the latter case with a solution
containing less water than is contained in the hydrate itself.

It has already been stated (p. 109) that the solubility curve (in general,
the equilibrium curve) is continuous so long as the solid phase remains
unchanged; and we shall therefore expect that the curve ABC will be
continuous. Formerly, however, it was considered by some that the curve was
not continuous, but that the melting point is the point of intersection of
two curves, a solubility curve and a fusion curve. Although the earlier
solubility determinations were insufficient to decide this point
conclusively, more recent investigation has proved beyond doubt that the
curve is continuous and exhibits no break.[226]

{148}

Although in taking up the discussion of the equilibria between calcium
chloride and water, it was desired especially to call attention to the form
of the solubility curve in the case of salt hydrates possessing a definite
melting point, nevertheless, for the sake of completeness, brief mention
may be made of the other systems which these two components can form.

[Illustration: FIG. 38.]

Besides the hexahydrate, the solubility curve of which has already been
described, calcium chloride can also crystallize in two different forms,
each of which contains four molecules {149} of water of crystallization;
these are distinguished as [alpha]-tetrahydrate, and [beta]-tetrahydrate.
Two other hydrates are also known, viz. a dihydrate and a monohydrate. The
solubility curves of these different hydrates are given in Fig. 38.

On following the solubility curve of the hexahydrate from the ordinary
temperature upwards, it is seen that at a temperature of 29.8° represented
by the point H, it cuts the solubility curve of the [alpha]-tetrahydrate.
This point is therefore a quadruple point at which the four phases
hexahydrate, [alpha]-tetrahydrate, solution, and vapour can coexist. It is
also the transition point for these two hydrates. Since, at temperatures
above 29.8°, the [alpha]-tetrahydrate is the stable form, it is evident
from the data given before (p. 146), as also from Fig. 38, that the portion
of the solubility curve of the hexahydrate lying above this temperature
represents _metastable_ equilibria. The realization of the metastable
melting point of the hexahydrate is, therefore, due to suspended
transformation. At the transition point, 29.8°, the solubility of the
hexahydrate and [alpha]-tetrahydrate is 100.6 parts of CaCl_{2} in 100
parts of water.

The retroflex portion of the solubility curve of the hexahydrate extends to
only 1° below the melting point of the hydrate. At 29.2° crystals of a new
hydrate, [beta]-tetrahydrate, separate out, and the solution, which now
contains 112.8 parts of CaCl_{2} to 100 parts of water, is saturated with
respect to the two hydrates. Throughout its whole extent the solubility
curve EDF of the [beta]-tetrahydrate represents metastable equilibria. The
upper limit of the solubility curve of [beta]-tetrahydrate is reached at
38.4° (F), the point of intersection with the curve for the dihydrate.

Above 29.8° the stable hydrate is the [alpha]-tetrahydrate; and its
solubility curve extends to 45.3° (K), at which temperature it cuts the
solubility curve of the dihydrate. The curve of the latter hydrate extends
to 175.5° (L), and is then succeeded by the curve for the monohydrate. The
solubility curve of the anhydrous salt does not begin until a temperature
of about 260°. The whole diagram, therefore, shows a succession of stable
hydrates, a metastable hydrate, a metastable melting point and retroflex
solubility curve. {150}

Pressure-Temperature Diagram.--The complete study of the equilibria between
the two components calcium chloride and water would require the discussion
of the vapour pressure of the different systems, and its variation with the
temperature. For our present purpose, however, such a discussion would not
be of great value, and will therefore be omitted here; in general, the same
relationships would be found as in the case of sodium sulphate (p. 138),
except that the rounded portion of the solubility curve of the hexahydrate
would be represented by a similar rounded portion in the pressure
curve.[227] As in the case of sodium sulphate, the transition points of the
different hydrates would be indicated by breaks in the curve of pressures.
Finally, mention may again be made of the difference of the pressure of
dissociation of the hexahydrate according as it becomes dehydrated to the
[alpha]- or the [beta]-tetrahydrate (p. 88).

The Indifferent Point.--We have already seen that at 30.2° calcium chloride
hexahydrate melts congruently, and that, provided the pressure is
maintained constant, addition or withdrawal of heat will cause the complete
liquefaction or solidification, without the temperature of the system
undergoing change. This behaviour, therefore, is similar to, but is not
quite the same as the fusion of a simple substance such as ice; and the
difference is due to the fact that in the case of the hexahydrate the
emission of vapour by the liquid phase causes an alteration in the
composition of the latter, owing to the non-volatility of the calcium
chloride; whereas in the case of ice this is, of course, not so.

Consider, however, for the present that the vapour phase is absent, and
that we are dealing with the two-phase system solid--solution. Then, since
there are two components, the system is bivariant. For any given value of
the pressure, therefore, we should expect that the system could exist at
different temperatures; which, indeed, is the case. It has, however,
already been noted that when the composition of the liquid phase becomes
the same as that of the solid, the system then behaves as a _univariant_
system; for, at a given pressure, the system solid--solution can exist only
at _one_ temperature, change of temperature producing complete
transformation in {151} one or other direction. _The variability of the
system has therefore been diminished._

This behaviour will perhaps be more clearly understood when one reflects
that since the composition of the two phases is the same, the system may be
regarded as being formed of _one component_, just as the system NH_{4}Cl
<--> NH_{3} + HCl was regarded as being composed of one component when the
vapour had the same total composition as the solid (p. 13). One component
in two phases, however, constitutes a univariant system, and we can
therefore see that calcium chloride hexahydrate in contact with solution of
the same composition will constitute a univariant system. The temperature
of equilibrium will, however, vary with the pressure;[228] if the latter is
constant, the temperature will also be constant.

A point such as has just been referred to, which represents the special
behaviour of a system of two (or more) components, in which the composition
of two phases becomes identical, is known as an _indifferent point_,[229]
and it has been shown[230] that at a given pressure the temperature in the
indifferent point is the _maximum_ or _minimum_ temperature possible at the
particular pressure[231] (cf. critical solution temperature). At such a
point a system loses one degree of freedom, or behaves like a system of the
next lower order.

The Hydrates of Ferric Chloride.--A better illustration of the formation of
compounds possessing a definite melting point, and of the existence of
retroflex solubility curves, is afforded by the hydrates of ferric
chloride, which not only possess definite points of fusion, but these
melting points are stable. A very brief description of the relations met
with will suffice.[232]

{152}

Ferric chloride can form no less than four stable hydrates, viz.
Fe_{2}Cl_{6},12H_{2}O, Fe_{2}Cl_{6},7H_{2}O, Fe_{2}Cl_{6},5H_{2}O, and
Fe_{2}Cl_{6},4H_{2}O, and each of these hydrates possesses a definite,
stable melting point. On analogy with the behaviour of calcium chloride,
therefore, we shall expect that the solubility curves of these different
hydrates will exhibit a series of _temperature maxima_; the points of
maximum temperature representing systems in which the composition of the
solid and liquid phases is the same. A graphical representation of the
solubility relations is given in Fig. 39, and the composition of the
different saturated solutions which can be formed is given in the following
tables, the composition being expressed in molecules of Fe_{2}Cl_{6} to 100
molecules of water. The figures printed in thick type refer to transition
and melting points.

[Illustration: FIG. 39.]

{153}

  COMPOSITION OF THE SATURATED SOLUTIONS OF FERRIC CHLORIDE AND ITS
      HYDRATES.

  (_The name placed at the head of each table is the solid phase._)

  ICE.
  ---------------------------
  Temperature. | Composition.
  ---------------------------
     ±-55°     |    ±2.75
      -40°     |     2.37
      -27.5°   |     1.90
      -20.5°   |     1.64
      -10°     |     1.00
        0°     |     0
  ---------------------------

  Fe_{2}Cl_{6},12H_{2}O.
  ---------------------------
  Temperature. | Composition.
  ---------------------------
     -55°      |    ±2.75
     -41°      |     2.81
     -27°      |     2.98
       0°      |     4.13
      10°      |     4.54
      20°      |     5.10
      30°      |     5.93
      35°      |     6.78
      36.5°    |     7.93
      37°      |     8.33
      36°      |     9.29
      33°      |    10.45
      30°      |    11.20
      27·4°    |    12.15
      20°      |    12.83
      10°      |    13.20
       8°      |    13.70
  ---------------------------

  Fe_{2}Cl_{6},7H_{2}O.
  ---------------------------
  Temperature. | Composition.
  ---------------------------
      20°      |    11.35
      27·4°    |    12.15
      32°      |    13.55
      32.5°    |    14.29
      30°      |    15.12
      25°      |    15.54
  ---------------------------

  Fe_{2}Cl_{6},5H_{2}O.
  ---------------------------
  Temperature. | Composition.
  ---------------------------
      20°      |    11.35
      12°      |    12.87
      20°      |    13.95
      27°      |    14.85
      30°      |    15.12
      35°      |    15.64
      50°      |    17.50
      55°      |    19.15
      56°      |    20.00
      55°      |    20.32
  ---------------------------

  Fe_{2}Cl_{6},4H_{2}O
  ---------------------------
  Temperature. | Composition.
  ---------------------------
      20°      |    11.35
      50°      |    19.96
      55°      |    20.32
      60°      |    20.70
      69°      |    21.53
      72.5°    |    23.35
      73.5°    |    25.00
      72.5°    |    26.15
      70°      |    27.90
      66°      |    29.20
  ---------------------------

  Fe_{2}Cl_{6} (ANHYDROUS).
  ---------------------------
  Temperature. | Composition.
  ---------------------------
      20°      |    11.35
      66°      |    29.20
      70°      |    29.42
      75°      |    28.92
      80°      |    29.20
     100°      |    29.75
  ---------------------------

The lowest portion of the curve, AB, represents the equilibria between ice
and solutions containing ferric chloride. It represents, in other words,
the lowering of the fusion point of ice by addition of ferric chloride. At
the point B (-55°), the cryohydric point (p. 117) is reached, at which the
solution is in equilibrium with ice and ferric chloride dodecahydrate. As
{154} has already been shown, such a point represents an invariant system;
and the liquid phase will, therefore, solidify to a mixture of ice and
hydrate without change of temperature. If heat is added, ice will melt and
the system will pass to the curve BCDN, which is the solubility curve of
the dodecahydrate. At C (37°), the point of maximum temperature, the
hydrate melts completely. The retroflex portion of this curve can be
followed backwards to a temperature of 8°, but below 27.4° (D), the
solutions are supersaturated with respect to the heptahydrate; point D is
the eutectic point for dodecahydrate and heptahydrate. The curve DEF is the
solubility curve of the heptahydrate, E being the melting point, 32.5°. On
further increasing the quantity of ferric chloride, the temperature of
equilibrium is lowered until at F (30°) another eutectic point is reached,
at which the heptahydrate and pentahydrate can co-exist with solution. Then
follow the solubility curves for the pentahydrate, the tetrahydrate, and
the anhydrous salt; G (56°) is the melting point of the former hydrate, J
(73.5°) the melting point of the latter. H and K, the points at which the
curves intersect, represent eutectic points; the temperature of the former
is 55°, that of the latter 66°. The dotted portions of the curves represent
metastable equilibria.

As is seen from the diagram, a remarkable series of solubility curves is
obtained, each passing through a point of maximum temperature, the whole
series of curves forming an undulating "festoon." To the right of the
series of curves the diagram represents unsaturated solutions; to the left,
supersaturated.

If an unsaturated solution, the composition of which is represented by a
point in the field to the right of the solubility curves, is cooled down,
the result obtained will differ according as the composition of the
solution is the same as that of a cryohydric point, or of a melting point,
or has an intermediate value. Thus, if a solution represented by _x__{1} is
cooled down, the composition will remain unchanged as indicated by the
horizontal dotted line, until the point D is reached. At this point,
dodecahydrate and heptahydrate will separate out, and the liquid will
ultimately solidify completely to a mixture or "conglomerate" of these two
hydrates; the temperature of {155} the system remaining constant until
complete solidification has taken place. If, on the other hand, a solution
of the composition _x__{3} is cooled down, ferric chloride dodecahydrate
will be formed when the temperature has fallen to that represented by C,
and the solution will completely solidify, without alteration of
temperature, with formation of this hydrate. In both these cases,
therefore, a point is reached at which complete solidification occurs
without change of temperature.

Somewhat different, however, is the result when the solution has an
intermediate composition, as represented by _x__{2} or _x__{4}. In the
former case the dodecahydrate will first of all separate out, but on
further withdrawal of heat the temperature will fall, the solution will
become relatively richer in ferric chloride, owing to separation of the
hydrate, and ultimately the eutectic point D will be reached, at which
complete solidification will occur. Similarly with the second solution.
Ferric chloride dodecahydrate will first be formed, and the temperature
will gradually fall, the composition of the solution following the curve CB
until the cryohydric point B is reached, when the whole will solidify to a
conglomerate of ice and dodecahydrate.

Suspended Transformation.--Not only can the upper branch of the solubility
curve of the dodecahydrate be followed backwards to a temperature of 8°, or
about 19° below the temperature of transition to the heptahydrate; but
suspended transformation has also been observed in the case of the
heptahydrate and the pentahydrate. To such an extent is this the case that
the solubility curve of the latter hydrate has been followed downwards to
its point of intersection with the curve for the dodecahydrate. This point
of intersection, represented in Fig. 39 by M, lies at a temperature of
about 15°; and at this temperature, therefore, it is possible for the two
solid phases dodecahydrate and pentahydrate to coexist, so that M is a
eutectic point for the dodecahydrate and the pentahydrate. It is, however,
a metastable eutectic point, for it lies in the region of supersaturation
with respect to the heptahydrate; and it can be realized only because of
the fact that the latter hydrate is not readily formed.

Evaporation of Solutions at Constant Temperature.--On {156} evaporating
dilute solutions of ferric chloride at constant temperature, a remarkable
series of changes is observed, which, however, will be understood with the
help of Fig. 40. Suppose an unsaturated solution, the composition of which
is represented by the point _x__{1}, is evaporated at a temperature of
about 17° - 18°. As water passes off, the composition of the solution will
follow the dotted line of constant temperature, until at the point where it
cuts the curve BC the solid hydrate Fe_{2}Cl_{6},12H_{2}O separates out. As
water continues to be removed, the hydrate must be deposited (in order that
the solution shall remain saturated), until finally the solution dries up
to the hydrate. As dehydration proceeds, the heptahydrate can be formed,
and the dodecahydrate will finally pass into the heptahydrate; and this, in
turn, into the pentahydrate.

[Illustration: FIG. 40.]

But the heptahydrate is not always formed by the dehydration of the
dodecahydrate, and the behaviour on evaporation is therefore somewhat
perplexing at first sight. After the solution has dried to the
dodecahydrate, as explained above, further removal of water causes
liquefaction, and the system is now represented by the point of
intersection at _a_; at this point the solid hydrate is in equilibrium with
a solution containing relatively more ferric chloride. If, therefore,
evaporation is continued, the solid hydrate must _pass into solution_ in
order that the composition of the latter may remain unchanged, so that
ultimately a liquid will again be obtained. A very slight further
dehydration will bring the solution into the state represented by _b_, at
which the pentahydrate is formed, and the solution will at last disappear
and leave this hydrate alone.

Without the information to be obtained from the curves in Figs. 39 and 40,
the phenomena which would be observed on carrying out the evaporation at a
temperature of about 31 - 32° {157} would be still more bewildering. The
composition of the different solutions formed will be represented by the
perpendicular line _x__{2}12345. Evaporation will first cause the
separation of the dodecahydrate, and then total disappearance of the liquid
phase. Then liquefaction will occur, and the system will now be represented
by the point 2, in which condition it will remain until the solid hydrate
has disappeared. Following this there will be deposition of the
heptahydrate (point 3), with subsequent disappearance of the liquid phase.
Further dehydration will again cause liquefaction, when the concentration
of the solution will be represented by the point 4; the heptahydrate will
ultimately disappear, and then will ensue the deposition of the
pentahydrate, and complete solidification will result. On evaporating a
solution, therefore, of the composition _x__{2}, the following series of
phenomena will be observed: solidification to dodecahydrate; liquefaction;
solidification to heptahydrate; liquefaction; solidification to
pentahydrate.[233]

Although ferric chloride and water form the largest and best-studied series
of hydrates possessing definite melting points, examples of similar
hydrates are not few in number; and more careful investigation is
constantly adding to the list.[234] In all these cases the solubility curve
will show a point of maximum temperature, at which the hydrate melts, and
will end, above and below, in a cryohydric point. Conversely, if such a
curve is found in a system of two components, we can argue that a definite
compound of the components possessing a definite melting point is formed.

Inevaporable Solutions.--If a saturated solution in contact with two
hydrates, or with a hydrate and anhydrous salt is heated, the temperature
and composition of the solution will, of course, remain unchanged so long
as the two solid phases are present, for such a system is invariant. In
addition to this, however, the _quantity_ of the solution will also remain
unchanged, the water which evaporates being supplied by the higher hydrate.
The same phenomenon is also observed in the case of cryohydric points when
ice is a solid phase; so long as the latter is present, evaporation will be
accompanied {158} by fusion of the ice, and the quantity of solution will
remain constant. Such solutions are called _inevaporable_.[235]

[Illustration: FIG. 41.]

Illustration.--In order to illustrate the application of the principles of
the Phase Rule to the study of systems formed by a volatile and a
non-volatile component, a brief description may be given of the behaviour
of sulphur dioxide and potassium iodide, which has formed the subject of a
recent investigation. After it had been found[236] that liquid sulphur
dioxide has the property of dissolving potassium iodide, and that the
solutions thus obtained present certain peculiarities of behaviour, the
question arose as to whether or not compounds are formed between the
sulphur dioxide and the potassium iodide, and if so, what these compounds
are. To find an answer to this question, Walden and Centnerszwer[237] made
a complete investigation of the solubility curves (equilibrium curves) of
these two components, the investigation extending from the freezing point
to the critical point of sulphur dioxide. For convenience of reference, the
results which they obtained are represented diagrammatically in Fig. 41.
The freezing point (A) of pure sulphur dioxide was found to be -72.7°.
Addition of potassium iodide lowered the freezing point, but the maximum
depression obtained was very small, and was reached when the concentration
of the potassium iodide in the solution was only 0.336 mols. per cent.
Beyond this point, an increase in the concentration of the iodide was
accompanied by an elevation of the freezing point, the change of the
freezing point with the concentration being represented by the curve BC.
The solid {159} which separated from the solutions represented by BC was a
bright _yellow_ crystalline substance. At the point C (-23.4°) a
temperature-maximum was reached; and as the concentration of the potassium
iodide was continuously increased, the temperature of equilibrium first
fell and then slowly rose, until at +0.26° (E) a second temperature-maximum
was registered. On passing the point D, the solid which was deposited from
the solution was a _red_ crystalline substance. On withdrawing sulphur
dioxide from the system, the solution became turbid, and the temperature
remained constant. The investigation was not pursued farther at this point,
the attention being then directed to the equilibria at higher temperatures.

When a solution of potassium iodide in liquid sulphur dioxide containing
1.49 per cent. of potassium iodide was heated, solid (potassium iodide) was
deposited at a temperature of 96.4°. Solutions containing more than about 3
per cent. of the iodide separated, on being heated, into two layers, and
the temperature at which the liquid became heterogeneous fell as the
concentration was increased; a temperature-minimum being obtained with
solutions containing 12 per cent. of potassium iodide. On the other hand,
solutions containing 30.9 per cent. of the iodide, on being heated,
deposited potassium iodide; while a solution containing 24.5 per cent. of
the salt first separated into two layers at 89.3°, and then, on cooling,
solid was deposited and one of the liquid layers disappeared.

Such are, in brief, the results of experiment; their interpretation in the
light of the Phase Rule is the following:--

The curve AB is the freezing-point curve of solid sulphur dioxide in
contact with solutions of potassium iodide. BCD is the solubility curve of
the yellow crystalline solid which is deposited from the solutions. C, the
temperature-maximum, is the melting point of this _yellow_ solid, and the
composition of the latter must be the same as that of the solution at this
point (p. 145), which was found to be that represented by the formula
KI,14SO_{2}. B is therefore the eutectic point, at which solid sulphur
dioxide and the compound KI,14SO_{2} can exist together in equilibrium with
solution and vapour. The curve DE is the solubility curve of the _red_
crystalline solid, and the {160} point E, at which the composition of
solution and solid is the same, is the melting point of the solid. The
composition of this substance was found to be KI,4SO_{2}.[238] D is,
therefore, the eutectic point at which the compounds KI,14SO_{2} and
KI,4SO_{2} can coexist in equilibrium with solution and vapour. The curve
DE does not exhibit a retroflex portion; on the contrary, on attempting to
obtain more concentrated solutions in equilibrium with the compound
KI,4SO_{2}, a new solid phase (probably potassium iodide) was formed. Since
at this point there are four phases in equilibrium, viz. the compound
KI,4SO_{2}, potassium iodide, solution, and vapour, the system is
invariant. E is, therefore, the _transition point_ for KI,4SO_{2} and KI.

Passing to higher temperatures, FG is the solubility curve of potassium
iodide in sulphur dioxide; at G two liquid phases are formed, and the
system therefore becomes invariant (cf. p. 121). The curve GHK is the
solubility curve for two partially miscible liquids; and since complete
miscibility occurs on _lowering_ the temperature, the curve is similar to
that obtained with triethylamine and water (p. 101). K is also an invariant
point at which potassium iodide is in equilibrium with two liquid phases
and vapour.

The complete investigation of the equilibria between sulphur dioxide and
potassium iodide, therefore, shows that these two components form the
compounds KI,14SO_{2} and KI,4SO_{2}; and that when solutions having a
concentration between those represented by the points G and K are heated,
separation into two layers occurs. The temperatures and concentrations of
the different characteristic points are as follows:--

  -------------------------------------------------------------
                              |                | Composition of
              Point.          |  Temperature.  |  the solution
                              |                |  per cent. KI.
  -------------------------------------------------------------
  A (m.p. of SO_{2})          |        -72.7°  |       --
  B (eutectic point)          |        --      |      0.86
  C (m.p. of KI,14SO_{2})     |        -23.4°  |     17.63
  E (m.p. of KI,4SO_{2})      |         +0.26° |     39.33
  G (KI + two liquid phases)  | (about) 88°    |     24.0
  H (critical solution point) |         77.3°  |     12
  K (KI + two liquid phases)  | (about) 88°    |      2.7
  -------------------------------------------------------------

       *       *       *       *       *


{161}

CHAPTER IX

EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS

General.--In the two preceding chapters certain restrictions were imposed
on the discussion of the equilibria between two components; but in the
present chapter the restriction that only one of the components is volatile
will be allowed to fall, and the general behaviour of two volatile[239]
components, each of which is capable of forming a liquid solution with the
other, will be studied. As we shall see, however, the removal of the
previous restriction produces no alteration in the general aspect of the
equilibrium curves for concentration and temperature, but changes to some
extent the appearance of the pressure-temperature diagram. The latter would
become still more complicated if account were taken not only of the total
pressure but also of the partial pressures of the two components in the
vapour phase; this complication, however, will not be introduced in the
present discussion.[240] In this chapter we shall consider the systems
formed by the two components iodine and chlorine, and sulphur dioxide and
water.

Iodine and Chlorine.--The different systems furnished by iodine and
chlorine, rendered classical by the studies of Stortenbeker,[241] form a
very complete example of equilibria in a two-component system. We shall
first of all consider the {162} relations between concentration and
temperature, with the help of the accompanying diagram, Fig. 42.

[Illustration: FIG. 42.]

Concentration-Temperature Diagram.--In this diagram the temperatures are
taken as the abscissæ, and the composition of the solution, expressed in
atoms of chlorine to one atom of iodine,[242] is represented by the
ordinates. In the diagram, A represents the melting point of pure iodine,
114°. If chlorine is added to the system, a solution of chlorine in liquid
iodine is obtained, and the temperature at which solid iodine is in
equilibrium with the liquid solution will be all the lower the greater the
concentration of the chlorine. We therefore obtain the curve ABF, which
represents the composition of the solution {163} with which solid iodine is
in equilibrium at different temperatures. This curve can be followed down
to 0°, but at temperatures below 7.9° (B) it represents metastable
equilibria. At B iodine monochloride can be formed, and if present the
system becomes invariant; B is therefore a quadruple point at which the
four phases, iodine, iodine monochloride, solution, and vapour, can
coexist. Continued withdrawal of heat at this point will therefore lead to
the complete solidification of the solution to a mixture or conglomerate of
iodine and iodine monochloride, while the temperature remains constant
during the process. B is the eutectic point for iodine and iodine
monochloride.

Just as we found in the case of aqueous salt solutions that at temperatures
above the cryohydric or eutectic point, two different solutions could
exist, one in equilibrium with ice, the other in equilibrium with the salt
(or salt hydrate), so in the case of iodine and chlorine there can be two
solutions above the eutectic point B, one containing a lower proportion of
chlorine in equilibrium with iodine, the other containing a higher
proportion of chlorine in equilibrium with iodine monochloride. The
composition of the latter solution is represented by the curve BCD. As the
concentration of chlorine is increased, the temperature at which there is
equilibrium between iodine monochloride and solution rises until a point is
reached at which the composition of the solution is the same as that of the
solid. At this point (C), iodine monochloride melts. Addition of one of the
components will lower the temperature of fusion, and a continuous
curve,[243] exhibiting a retroflex portion as in the case of
CaCl_{2},6H_{2}O, will be obtained. At temperatures below its melting
point, therefore, iodine monochloride can be in equilibrium with two
different solutions.

The upper portion of this curve, CD, can be followed downwards to a
temperature of 22.7°. At this temperature iodine trichloride can separate
out, and a second quadruple {164} point (D) is obtained. This is the
eutectic point for iodine monochloride and iodine trichloride.

By addition of heat and increase in the amount of chlorine, the iodine
monochloride disappears, and the system passes along the curve DE, which
represents the composition of the solutions in equilibrium with solid
iodine trichloride. The concentration of chlorine in the solution increases
as the temperature is raised, until at the point E, where the solution has
the same composition as the solid, the maximum temperature is reached; the
iodine trichloride melts. On increasing still further the concentration of
chlorine in the solution, the temperature of equilibrium falls, and a
continuous curve, similar to that for the monochloride, is obtained. The
upper branch of this curve has been followed down to a temperature of 30°,
the solution at this point containing 99.6 per cent. of chlorine.[244] The
very rounded form of the curve is due to the trichloride being largely
dissociated in the liquid state.

One curve still remains to be considered. As has already been mentioned,
iodine monochloride can exist in two crystalline forms, only one of which,
however, is stable at temperatures below the melting point; the two forms
are _monotropic_ (p. 44). The stable form which melts at 27.2°, is called
the [alpha]-form, while the less stable variety, melting at 13.9°, is known
as the [beta]-form. If, now, the presence of [alpha]-ICl is excluded, it is
possible to obtain the [beta]-form, and to study the conditions of
equilibrium between it and solutions of iodine and chlorine, from the
eutectic point F to the melting point G. As the [beta]-ICl becomes less
stable in presence of excess of chlorine, it has not been possible to study
the retroflex portion of the curve represented by the dotted continuation
of FG.

The following table gives some of the numerical data from which Fig. 42 was
constructed.[245]

{165}

  IODINE AND CHLORINE.

  I. _Invariant systems._

  -------------------------------------------------------------------------
          |          |                  Phases present.
   Temper-| Pressure.+--------------------+-----------------+--------------
   ature. |          |      Solid.        |    Liquid.      |   Vapour.
  --------+----------+--------------------+-----------------+--------------
   7.9°   | 11 mm.   | I_{2},[alpha]-ICl  | I[wavy]Cl_{0.66}| I + Cl_{0.92}
   0.9°   |   --     | I_{2},[beta]-ICl   | I[wavy]Cl_{0.72}|     --
   22.7°  | 42 mm.   | [alpha]-ICl,ICl_{3}| I[wavy]Cl_{1.19}| I + Cl_{1.75}
   [-102° | <1 atm.  | ICl_{3},Cl_{2}     | I[wavy]Cl_{m}   | I + Cl_{n}]
  --------+----------+--------------------+-----------------+--------------

  II. _Melting points._

  A. Iodine,[246] 114.15° (pressure 89.8 mm.).
  C. [alpha]-Iodine monochloride, 27.2° (pressure 37 mm.).
  E. Iodine trichloride, 101° (pressure 16 atm.).
  G. [beta]-Iodine monochloride, 13.9°.

Since the vapour pressure at the melting point of iodine trichloride
amounts to 16 atm., the experiments must of course be carried out
in closed vessels. At 63.7° the vapour pressure of the system
trichloride--solution--vapour is equal to 1 atm.

Pressure-Temperature Diagram.--In this diagram there are represented the
values of the vapour pressure of the saturated solutions of chlorine and
iodine. To give a complete picture of the relations between pressure,
temperature, and concentration, a solid model would be required, with three
axes at right angles to one another along which could be measured the
values of pressure, temperature, and concentration of the components in the
solution. Instead of this, however, there may be employed the accompanying
projection figure[247] (Fig. 43), the lower portion of which shows the
projection of the equilibrium curve on the surface containing the
concentration and temperature axes, while the upper portion is the
projection on the plane containing the pressure and temperature axes. The
lower portion is therefore a concentration-temperature diagram; {166} the
upper portion, a pressure-temperature diagram. The corresponding points of
the two diagrams are joined by dotted lines.

[Illustration: FIG. 43.]

Corresponding to the point C, the melting point of pure iodine, there is
the point C_{1}, which represents the vapour pressure of iodine at its
melting point. At this point three curves cut: 1, the sublimation curve of
iodine; 2, the vaporization curve of fused iodine; 3, C_{1}B_{1}, the
vapour-pressure curve of the saturated solutions in equilibrium with solid
iodine. Starting, therefore, with the system solid iodine--liquid iodine,
addition of chlorine will cause the temperature of equilibrium to fall
continuously, while the vapour pressure will first increase, pass through a
maximum and then fall continuously {167} until the eutectic point, B
(B_{1}), is reached.[248] At this point the system is invariant, and the
pressure will therefore remain constant until all the iodine has
disappeared. As the concentration of the chlorine increases in the manner
represented by the curve B_f_H, the pressure of the vapour also increases
as represented by the curve B_{1}_f__{1}H_{1}. At H_{1}, the eutectic point
for iodine monochloride and iodine trichloride, the pressure again remains
constant until all the monochloride has disappeared. As the concentration
of the solution passes along the curve HF, the pressure of the vapour
increases as represented by the curve H_{1}F_{1}; F_{1} represents the
pressure of the vapour at the melting point of iodine trichloride. If the
concentration of the chlorine in the solution is continuously increased
from this point, the vapour pressure first increases and then decreases,
until the eutectic point for iodine trichloride and solid chlorine is
reached (D_{1}). Curves Cl_{2} solid and Cl_{2} liquid represent the
sublimation and vaporization curves of chlorine, the melting point of
chlorine being -102°.

Although complete measurements of the vapour pressure of the different
systems of pure iodine to pure chlorine have not been made, the
experimental data are nevertheless sufficient to allow of the general form
of the curves being indicated with certainty.

Bivariant Systems.--To these, only a brief reference need be made. Since
there are two components, two phases will form a bivariant system. The
fields in which these systems can exist are shown in Fig. 43 and Fig. 44,
which is a more diagrammatic representation of a portion of Fig. 43.

    I. Iodine--vapour.
   II. Solution--vapour.
  III. Iodine trichloride--vapour.
  IV. Iodine monochloride--vapour.

[Illustration: FIG. 44.]

The conditions for the existence of these systems will probably be best
understood from Fig. 44. Since the curve B'A' {168} represents the
pressures under which the system iodine--solution--vapour can exist,
increase of volume (diminution of pressure) will cause the volatilization
of the solution, and the system iodine--vapour will remain. If, therefore,
we start with a system represented by _a_, diminution of pressure at
constant temperature will lead to the condition represented by _x_. On the
other hand, increase of pressure at _a_ will lead to the condensation of a
portion of the vapour phase. Since, now, the concentration of chlorine in
the vapour is greater than in the solution, condensation of vapour would
increase the concentration of chlorine in the solution; a certain amount of
iodine must therefore pass into solution in order that the composition of
the latter shall remain unchanged.[249] If, therefore, the volume of vapour
be sufficiently great, continued diminution of volume will ultimately lead
to the disappearance of all the iodine, and there will remain only solution
and vapour (field II.). As the diminution of volume is continued, the
vapour pressure and the concentration of the chlorine in the solution will
increase, until when the pressure has reached the value _b_, iodine
monochloride can separate out. The system, therefore, again becomes
univariant, and at constant temperature the pressure and composition of the
phases must remain unchanged. Diminution of volume will therefore not
effect an increase of pressure, but a condensation of the vapour; and since
this is richer in chlorine than the {169} solution, solid iodine
monochloride must separate out in order that the concentration of the
solution remain unchanged.[250] As the result, therefore, we obtain the
bivariant system iodine monochloride--vapour.

A detailed discussion of the effect of a continued increase of pressure
will not be necessary. From what has already been said and with the help of
Fig. 44, it will readily be understood that this will lead successively to
the univariant system (_c_), iodine monochloride--solution--vapour; the
bivariant system solution--vapour (field II.); the univariant system (_d_),
iodine trichloride--solution--vapour; and the bivariant system _x'_, iodine
trichloride--vapour. If the temperature of the experiment is above the
melting point of the monochloride, then the systems in which this compound
occurs will not be formed.

Sulphur Dioxide and Water.--In the case just studied we have seen that the
components can combine to form definite compounds possessing stable melting
points. The curves of equilibrium, therefore, resemble in their general
aspect those of calcium chloride and water, or of ferric chloride and
water. In the case of sulphur dioxide and water, however, the melting point
of the compound formed cannot be realized, because transition to another
system occurs; retroflex concentration-temperature curves are therefore not
found here, but the curves exhibit breaks or sudden changes in direction at
the transition points, as in the case of the systems formed by sodium
sulphate and water. The case of sulphur dioxide and water is also of
interest from the fact that two liquid phases can be formed.

The phases which occur are--Solid: ice, sulphur dioxide hydrate,
SO_{2},7H_{2}O. Liquid: two solutions, the one containing excess of sulphur
dioxide, the other excess of water, and represented by the symbols SO_{2}
[wavy] _x_H_{2}O (solution I.), and H_{2}O [wavy] _y_SO_{2} (solution II.).
Vapour: a mixture of sulphur dioxide and water vapour in varying
proportions. Since there are two components, sulphur dioxide and water, the
number of {170} possible systems is considerable. Only the following,
however, have been studied:--

  I. _Invariant Systems: Four co-existing phases._
    (_a_) Ice, hydrate, solution, vapour.
    (_b_) Hydrate, solution I., solution II., vapour.

  II. _Univariant Systems: Three co-existing phases._
    (_a_) Hydrate, solution I., vapour.
    (_b_) Hydrate, solution II., vapour.
    (_c_) Solution I., solution II., vapour.
    (_d_) Hydrate, solution I., solution II.
    (_e_) Hydrate, ice, vapour.
    (_f_) Ice, solution II., vapour.
    (_g_) Ice, hydrate, solution II.

  III. _Bivariant Systems: Two co-existing phases._
    (_a_) Hydrate, solution I.
    (_b_) Hydrate, solution II.
    (_c_) Hydrate, vapour.
    (_d_) Hydrate, ice.
    (_e_) Solution I., solution II.
    (_f_) Solution I., vapour.
    (_g_) Solution I., ice.
    (_h_) Solution II., vapour.
    (_i_) Solution II., ice.
    (_j_) Ice, vapour.

[Illustration: FIG. 45.]

Pressure-Temperature Diagram.[251]--If sulphur dioxide is passed into water
at 0°, a solution will be formed and the temperature at which ice can
exist in equilibrium with this solution will fall more and more as the
concentration of the sulphur dioxide increases. At -2.6°, however, a
cryohydric point is reached at which solid hydrate separates out,
and the system becomes invariant. The curve AB (Fig. 45) therefore
represents the pressure of the system ice--solution II.--vapour, and B
represents the temperature and pressure at which the invariant system
ice--hydrate--solution II.--vapour can exist. At this point the temperature
is -2.6°, and the pressure 21.1 cm. If heat is withdrawn from this
system, the solution will ultimately {171} solidify to a mixture of
ice and hydrate, and there will be obtained the univariant system
ice--hydrate--vapour. The vapour pressure of this system has been
determined down to a temperature of -9.5°, at which temperature the
pressure amounts to 15 cm. The pressures for this system are represented by
the curve BC. If at the point B the volume is diminished, the pressure must
remain constant, but the relative amounts of the different phases will
undergo change. If suitable quantities of these are present, diminution of
volume will ultimately lead to the total condensation of the vapour phase,
and there will remain the univariant system ice--hydrate--solution. The
temperature of equilibrium of this system will alter with the pressure,
but, as in the case of the melting point of a simple substance, great
differences of pressure will cause only comparatively small changes in the
temperature of equilibrium. The change of the cryohydric point with the
pressure is represented by the line BE; the actual values have not been
determined, but the curve must slope towards the pressure axis because
fusion is accompanied by diminution of volume, as in the case of pure ice.
{172}

A fourth univariant system can be formed at B. This is the system
hydrate--solution II.--vapour. The conditions for the existence of
this system are represented by the curve BF, which may therefore be
regarded as the vapour-pressure curve of the saturated solution of
sulphur dioxide heptahydrate in water. Unlike the curve for iodine
trichloride--solution--vapour, this curve cannot be followed to the melting
point of the hydrate. Before this point is reached, a second liquid phase
appears, and an invariant system consisting of hydrate--solution
I.--solution II.--vapour is formed. We have here, therefore, the phenomenon
of melting under the solution as in the case of succinic nitrile and water
(p. 122). This point is represented in the diagram by F; the temperature at
this point is 12.1°, and the pressure 177.3 cm. The range of stable
existence of the hydrate is therefore from -2.6° to 12.1°; nevertheless,
the curve FB has been followed down to a temperature of -6°, at which point
ice formed spontaneously.

So long as the four phases hydrate, two liquid phases, and vapour are
present, the condition of the system is perfectly defined. By altering the
conditions, however, one of the phases can be made to disappear, and a
univariant system will then be obtained. Thus, if the vapour phase is made
to disappear, the univariant system solution I.--solution II.--hydrate,
will be left, and the temperature at which this system is in equilibrium
will vary with the pressure. This is represented by the curve FI; under a
pressure of 225 atm. the temperature of equilibrium is 17.1°. Increase of
pressure, therefore, raises the temperature at which the three phases can
coexist.

Again, addition of heat to the invariant system at F will cause the
disappearance of the solid phase, and there will be formed the univariant
system solution I.--solution II.--vapour. In the case of this system the
vapour pressure increases as the temperature rises, as represented by the
curve FG. Such a system is analogous to the case of ether and water, or
other two partially miscible liquids (p. 103). As the temperature changes,
the composition of the two liquid phases will undergo change; but this
system has not been studied fully.

The fourth curve, which ends at the quadruple point F, is {173} that
representing the vapour pressure of the system hydrate--solution I.--vapour
(FH). This curve has been followed to a temperature of 0°, the pressure at
this point being 113 cm. The metastable prolongation of GF has also been
determined. Although, theoretically, this curve must lie below FH, it was
found that the difference in the pressure for the two curves was within the
error of experiment.

Bivariant Systems.--The different bivariant systems, consisting of two
phases, which can exist within the range of temperature and pressure
included in Fig. 45, were given on p. 170. The conditions under which these
systems can exist are represented by the areas in the diagram, and the
fields of the different bivariant systems are indicated by letters,
corresponding to the letters on p. 170. Just as in the case of
one-component systems (p. 29), we found that the field lying between any
two curves gave the conditions of existence of that phase which was common
to the two curves, so also in the case of two-component systems, a
bivariant two-phase system occurs in the field enclosed[252] by the two
curves to which the two phases are common. As can be seen, the same
bivariant system can occur in more than one field.

As is evident from Fig. 45, three different bivariant systems are capable
of existing in the area HFI; which of these will be obtained will depend on
the relative masses of the different phases in the univariant or invariant
system. Thus, starting with a system represented by a point on the curve
HF, diminution of volume at constant temperature will cause the
condensation of a portion of the vapour, which is rich in sulphur dioxide;
since this would increase the concentration of sulphur dioxide in the
solution, it must be counteracted by the passage of a portion of the
hydrate (which is relatively poor in sulphur dioxide) into the solution.
If, therefore, the amount of hydrate present is relatively very small, the
final result of the compression will be the production of the system _f_,
solution I.--vapour. On the other hand, if the vapour is present in
relatively small amount, it will be the first phase to disappear, {174} and
the bivariant system _a_, hydrate--solution I., will be obtained. Finally,
if we start with the invariant system at F, compression will cause the
condensation of vapour, while the composition of the two solutions will
remain unchanged. When all the vapour has disappeared, the univariant
system hydrate--solution I.--solution II. will be left. If, now, the
pressure is still further increased, while the temperature is kept below
12°, more and more hydrate must be formed at the expense of the two liquid
phases (because 12° is the lower limit for the coexistence of the two
liquid phases), and if the amount of the solution I. (containing excess of
sulphur dioxide) is relatively small, it will disappear before solution
II., and there will be obtained the bivariant system hydrate--solution II.
(bivariant system _b_).

In a similar manner, account can be taken of the formation of the other
bivariant systems.

A behaviour similar to that of sulphur dioxide and water is shown by
chlorine and water and by bromine and water, although these have not been
so fully studied.[253] In the case of hydrogen bromide and water, and of
hydrogen chloride and water, a hydrate, viz. HBr,2H_{2}O and HCl,2H_{2}O,
is formed which possesses a definite melting point, as in the case of
iodine trichloride. In these cases, therefore, a retroflex curve is
obtained. Further, just as in the case of the chlorides of iodine the upper
branch of the retroflex curve ended in a eutectic point, so also in the
case of the hydrate HBr,2H_{2}O the upper branch of the curve ends in a
eutectic point at which the system dihydrate--monohydrate--solution--vapour
can exist. Before the melting point of the monohydrate is reached, two
liquid phases are formed, as in the case of sulphur dioxide and water.

       *       *       *       *       *


{175}

CHAPTER X

SOLID SOLUTIONS. MIXED CRYSTALS

General.--With the conception of gaseous and liquid solutions, every one is
familiar. Gases can mix in all proportions to form homogeneous solutions.
Gases can dissolve in or be "absorbed" by liquids; and solids, also, when
brought in contact with liquids, "pass into solution" and yield a
homogeneous liquid phase. On the other hand, the conception of a _solid
solution_ is one which in many cases is found more difficult to appreciate;
and the existence and behaviour of solid solutions, in spite of their not
uncommon occurrence and importance, are in general comparatively little
known.

The reason of this is to be found, to some extent, no doubt, in the fact
that the term "solid solution" was introduced at a comparatively recent
date,[254] but it is probably also due in some measure to a somewhat hazy
comprehension of the definition of the term "solution" itself. As has
already been said (p. 92), a solution is a homogeneous phase, the
composition of which can vary continuously within certain limits; the
definition involves, therefore, no condition as to the physical state of
the substances. Accordingly, solid solutions are homogeneous solid phases,
the composition of which can undergo continuous variation within certain
limits. Just as we saw that the range of variation of composition is more
limited in the case of liquids than in the case of gases, so also we find
that the limits of miscibility are in general still more restricted in the
case of solids. Examples of complete miscibility are, however, not unknown
even in the case of solid substances.

Solid solutions have long been known, although, of course, {176} they were
not defined as such. Thus, the phenomena of "occlusion" of gases by metals
and other substances (occlusion of hydrogen by palladium; occlusion of
hydrogen by iron) are due to the formation of solid solutions. The same is
probably also true of the phenomena of "adsorption," as in the removal of
organic colouring matter by charcoal, although, in this case, surface
tension no doubt plays a considerable part.[255]

As examples of the solution of gases in solids there may be cited (in
addition to the phenomena of occlusion already mentioned), the hydrated
silicates and the zeolites. During dehydration these crystalline substances
remain clear and transparent, and the pressure of the water vapour which
they emit varies with the degree of hydration or the concentration of water
in the mineral.[256] As examples of the solution of solids in solids we
have the cementation of iron by charcoal, the formation of glass, and the
crystallization together of isomorphous substances.

Although we have here spoken of the glasses as "solid solutions," it should
be mentioned that the term "solid" is used in its popular sense. Strictly
speaking, the glasses are to be regarded as supercooled liquids (see also
p. 53, footnote).

In discussing the equilibria in systems containing a solid solution, it is
of essential importance to remember that a solid solution constitutes only
_one_ phase, a phase of varying composition, as in the case of liquid
solutions.

Solution of Gases in Solids.--Comparatively little work has been done in
this connection, the investigations being limited chiefly to the phenomena
of occlusion or adsorption of gases by charcoal.[257] We shall, therefore,
indicate only briefly {177} and in a general manner, the behaviour which
the Phase Rule enables us to foresee.[258]

In dealing with the systems formed by the two phases gas--solid, three
chief cases call for mention:--

I. _The gas is not absorbed by the solid, but when the pressure reaches a
certain value, combination of the two components can result._

[Illustration: Fig. 46.]

The graphic representation of such a system is shown in Fig. 46, the
ordinates being the pressures of the gas, and the abscissæ the
concentrations of the gaseous component in the solid phase. Since there is
no formation of a solid solution, the concentration of gas in the solid
phase remains zero until the pressure has increased to the point A. At this
point combination can take place. There will now be three phases present,
viz. solid component, compound, and vapour. The system is therefore
univariant, and if the temperature is maintained constant, the vapour
pressure will be constant, irrespective of the amount of compound formed,
_i.e._ irrespective of the relative amounts of gas and solid. This is
indicated by the line AB. When the solid component has entirely
disappeared, the system ceases to be univariant, and if no absorption
occurs, the pressure will increase again, as shown by BC. If a second
compound can be formed, then a second _pc_-line will be obtained, similar
to the preceding. To this group belong the salt hydrates (Chap. VII.).

II. _The gas may be absorbed and may also form a compound._

If absorption of gas occurs with formation of a solid solution, then, as
the system consists of two phases, solution--vapour, it is bivariant. At
constant temperature, therefore, the pressure will still vary with the
concentration of the gaseous component in the solid phase. This is
represented by the curve AB in Fig. 47. When, however, the pressure has
reached a certain value, combination can take place; and since there are
now three phases present, the system is {178} univariant, and at constant
temperature the pressure is constant, as shown by the line BC.

III. _Absorption of gas occurs, but at a certain concentration the solid
solution can separate into two immiscible solid solutions._

We have seen, in Chapter VI., that two liquids can form two immiscible
solutions, and the same has also been found true of solid solutions, as we
shall presently learn more fully. If, now, two immiscible solutions are
formed, then the system will become univariant, and at constant temperature
the _pc_-curve will be a straight line, as in the case of the formation of
a compound (cf. p. 86). The behaviour of this system will, therefore, also
be represented diagrammatically by Fig. 47.

[Illustration: FIG. 47.]

_Palladium and Hydrogen._--The phenomenon of the absorption of hydrogen by
palladium, to which Graham gave the name "occlusion," is one that has
claimed the attention of several investigators. Although Graham was not of
opinion that a compound is formed, but rather that the gas undergoes very
great condensation, acts as a quasi-metal (to which he gave the name
hydrogenium), and forms a homogeneous alloy with the palladium, later
investigations, especially those of Troost and Hautefeuille,[259] pointed
to the formation of a definite chemical compound, having the formula
Pd_{2}H. This conclusion has, however, not been confirmed by subsequent
investigation.[260]

Roozeboom and Hoitsema[261] sought to arrive at a final decision as to the
nature of the phenomenon by an investigation of the equilibrium between
hydrogen and palladium on the basis of the Phase Rule classification given
above. If a compound is formed, diminution of volume would cause no
increase of pressure, but only an increase in the amount of the compound.

As this is the only case of gas absorption which has been {179} accurately
studied from this point of view, a brief account of the results obtained
will be given here, although these are not so clear and free from ambiguity
as one would desire.

The scientists just mentioned investigated the variation of the pressure of
hydrogen with the amount absorbed by the metal at different temperatures,
and a few of their results, typical of all, are represented graphically in
Fig. 48; the curves indicating the variation of the gas pressure with the
concentration of the hydrogen in the palladium at the temperatures 120°,
170°, and 200°. As can be seen, the curve consists of three parts, an
ascending portion which passes gradually and continuously into an almost
horizontal but slightly ascending middle part, which in turn passes without
break into a second rapidly ascending curve. This, as Fig. 48 indicates, is
the general form of the curve; but the length of the middle portion varies
with the temperature, being shorter at higher than at lower temperatures.

[Illustration: FIG. 48.]

What is the interpretation to be put on these curves? With regard to the
two end portions, these represent bivariant, two-phase systems, consisting
of a solid solution and gas. They correspond, therefore, to curve AB in
Fig. 47. If the middle portion were horizontal, it would indicate either
the formation of a compound or of two immiscible solid solutions. If a
compound Pd_{2}H were formed, then the middle portion would at all
temperatures end at the same value of the concentration, viz. that
corresponding to 0.5 atoms of hydrogen to 1 atom of palladium. As the
figure shows, however, this is not the case; the higher the temperature,
the lower is the concentration at which the middle passes into the terminal
portion of the curve. {180} Such a behaviour would, however, agree with the
assumption of the formation of two solid solutions, the "miscibility" of
which increases with the temperature, as in the case of the liquid
solutions of phenol and water (p. 97). Nevertheless, although the
assumption of the formation of two solid solutions is more satisfactory
than that of the formation of a compound, it does not entirely explain the
facts. If two solid solutions are formed, the pressure curve should be
horizontal, but this is not the case; and the deviation from the horizontal
does not appear to be due to impurities either in the gas or in the metal,
but is apparently a peculiarity of the system. Further, the gradual instead
of abrupt passage of the three portions of the curve into one another
remains unexplained. Hoitsema has expressed the opinion that the occlusion
of hydrogen by palladium is a process of continuous absorption, the
peculiar form of the curve--the flat middle portion--being possibly due to
a condensation of the gas, even at temperatures far above the critical
temperature of liquid hydrogen.

While, therefore, the occlusion of hydrogen by palladium still presents
some unexplained phenomena, the behaviour found by Hoitsema would appear to
disprove conclusively the formation of a definite chemical compound.[262]

SOLUTION OF SOLIDS IN SOLIDS. MIXED CRYSTALS.

The introduction by van't Hoff of the term "solid solution" resulted from
the discovery of a number of deviations from the Raoult-van't Hoff law for
the depression of the freezing point by dissolved substances. In all cases,
the depression was too small; in some instances, indeed, the freezing point
may be raised. To explain these irregularities, van't Hoff assumed that the
dissolved substance crystallized out along with the solid solvent; and he
showed how this would account for the {181} deviations from the law of the
depression of the freezing point, which had been developed on the
assumption that only the pure solvent crystallized out from the
solution.[263]

The "mixed crystals" which were thus obtained, and which van't Hoff called
dilute solid solutions, showed great resemblance in their behaviour to
ordinary liquid solutions, and obeyed the laws applicable to these. These
laws, however, can no longer be applied in the case of the concentrated
solid solutions formed by the crystallization together of isomorphous
substances, and known as isomorphous mixtures. Indeed, it has been
contended[264] that these isomorphous mixtures should not be considered as
solid solutions at all, although no sharp line of demarcation can be drawn
between the two classes. The differences, however, in the behaviour of the
two groups are of a quantitative rather than a qualitative nature; and
since we are concerned at present only with the qualitative behaviour, we
shall make no distinction between the crystalline solid solutions and the
isomorphous mixtures, but shall study the behaviour of the two classes
under the head of "mixed crystals."

Mixed crystals can be formed either by sublimation[265] or from a liquid
phase; and in the latter case the mixed crystals can be deposited either
from solution in a common solvent or from a mixture of the fused
components. In this method of formation, which alone will be discussed in
the present chapter, we are dealing with the fusion curves of two
substances, where, however, the liquid solution is in equilibrium not with
one of the pure components, but with a solid solution or mixed crystal. The
simple scheme (Fig. 29, p. 117) which was obtained in the case of two
components which crystallize out in the pure state, is no longer sufficient
in the case of the formation of mixed crystals. With the help of the Phase
Rule, however, the different possible systems can be classified; and
examples of the different cases predicted by the Phase Rule have also been
obtained by experiment.

{182}

We shall now consider briefly the formation of mixed crystals by
isomorphous substances; the consideration of the formation of mixed
crystals of isodimorphous substances will, on account of the complexity of
the relationships, not be undertaken here.[266]

_Formation of Mixed Crystals of Isomorphous Substances._

For the purpose of representing the relationships found here we shall
employ a temperature-concentration diagram,[267] in which the ordinates
represent the temperature and the abscissæ the concentration of the
components. Since there are two solutions, the liquid and the solid, and
since the concentration of the components in these two phases is not, in
general, the same, two curves will be required for each system, one
relating to the liquid phase, the other relating to the solid. The
temperature at which solid begins to be deposited from the liquid solution
will be called the _freezing point_ of the mixture, and the temperature at
which the solid solution just begins to liquefy will be called the _melting
point_ of the solid solution. The temperature-concentration curve for the
liquid phase will therefore be the freezing-point curve; that for the solid
solution, the melting-point curve. The latter will be represented by a
dotted line.[268]

{183}

I.--THE TWO COMPONENTS CAN FORM AN UNBROKEN SERIES OF MIXED CRYSTALS.

Since, as has already been pointed out (p. 176), a mixed crystal (solid
solution) constitutes only one phase, it is evident that if the two
components are miscible with one another in all proportions in the solid
state, there can never be more than one solid phase present, viz. the solid
solution or mixed crystal. If the components are completely miscible in the
solid state, they will also be completely miscible in the liquid state, and
there can therefore be only one liquid phase. The system can at no point
become invariant, because there can never be more than three phases
present. When, therefore, the two components form a continuous series of
mixed crystals, the equilibrium curve must also be continuous. Of these
systems three types are found.

[Illustration: FIG. 49.]

(_a_) _The freezing points of all mixtures lie between the freezing points
of the pure components_ (Curve I., Fig. 49).

Examples.--This type of curve is represented by the mixed crystals of
naphthalene and [beta]-naphthol.[269] The addition of [beta]-naphthol to
naphthalene raises the freezing point of the latter, and the rise is
directly proportional to the amount of naphthol added. The freezing point
curve is therefore a straight line joining the melting points of the two
components. This behaviour, however, is rather exceptional, the
freezing-point curve lying generally above, sometimes also below, the
straight line joining the melting points of the pure components. Thus the
freezing-point curve of mixtures of [alpha]-monochlorocinnamic aldehyde and
[alpha]-monobromocinnamic aldehyde[270] lies above the {184} straight line
joining the melting points of the pure components (31.22° and 69.56°), as
is evident from the following table:--

  ----------------------------------------------------------------------
  Molecules of bromo-        |                 |
  cinnamic aldehyde in       | Freezing point. | Deviation from straight
  100 mols. of mixture.      |                 | line.
  ----------------------------------------------------------------------
          0.00               |   31.22°        |       --
         10.48               |   37.28°        |      2.04°
         21.91               |   43.12°        |      3.50°
         30.07               |   46.80°        |      4.05°
         45.04               |   52.94°        |      4.45°
         62.16               |   58.82°        |      3.77°
         82.98               |   65.07°        |      2.03°
         93.50               |   67.91°        |      0.84°
        100.00               |   69.56°        |       --
  ----------------------------------------------------------------------

Melting-point Curve.--This curve, like the freezing-point curve, must also
be continuous, and the melting points of the different solid solutions will
lie between the melting points of the pure components. This is represented
by the dotted line in Fig. 49, I. The relative position of the two curves,
which can be deduced with the help of thermodynamics and also by
experimental determination, is found in all cases to be in accordance with
the following rule: At any given temperature, _the concentration of that
component by the addition of which the freezing point is depressed, is
greater in the liquid than in the solid phase_; or, conversely, _the
concentration of that component by the addition of which the freezing point
is raised, is greater in the solid than in the liquid phase_. An
illustration of this rule is afforded by the two substances chloro- and
bromo-cinnamic aldehyde already mentioned. As can be seen from the above
table, the addition of chlorocinnamic aldehyde lowers the melting point of
the bromo-compound. In accordance with the rule, therefore, the
concentration of the chloro-compound in the liquid phase must be greater
than in the solid phase; and this was found experimentally. At a
temperature of 49.44°, the liquid contained 58.52 per cent., the solid only
52.57 per cent. of the chlorocinnamic aldehyde.

From this it will also be clear that on cooling a fused mixture of two
substances capable of forming mixed crystals, {185} the temperature of
solidification will not remain constant during the separation of the solid;
nor, on the other hand, will the temperature of liquefaction of the solid
solution be constant. Thus, for example, if a liquid solution of two
components, A and B, having the composition represented by the point _x_
(Fig. 50), is allowed to cool, the system will pass along the line _xx'_.
At the temperature of the point _a_, mixed crystals will be deposited, the
composition of which will be that represented by b. As the temperature
continues to fall, more and more solid will be deposited; and since the
solid phase is relatively rich in the component B, the liquid will become
relatively poorer in this. The composition of the liquid solution will
therefore pass along the curve _ad_, the composition of the solid solution
at the same time passing along the curve _bc_; at the point _c_ the liquid
will solidify completely.[271]

[Illustration: FIG. 50.]

Conversely, if mixed crystals of the composition and at the temperature
_x'_ are heated, liquefaction will begin at the temperature _c_, yielding a
liquid of the composition d. On continuing to add heat, the temperature of
the mass will rise, more of the solid will melt, and the composition of the
two phases will change as represented by the curves _da_ and _cb_. When the
temperature has risen to _a_, complete liquefaction will have occurred. The
process of solidification or of liquefaction is therefore extended over a
temperature interval _ac_.

Even when the freezing-point curve is a straight line joining {186} the
melting points of the pure components, the melting-point curve will not
necessarily coincide with the freezing-point curve, although it may
approach very near to it; complete coincidence can take place only when the
melting points of the two components are identical. An example of this will
be given later (Chap. XII.).

(_b_) _The freezing-point curve passes through a maximum_ (Curve II., Fig.
49).

[Illustration: FIG. 51.]

This curve exhibits the greatest degree of contrast to the freezing-point
curve which is obtained when the pure components crystallize out. For,
since the curve passes through a maximum, it is evident that the freezing
point of each of the components must be _raised_ by the addition of the
other component.

Example.--Very few cases belonging to this type are known. The best example
is found in the freezing-point curve of mixtures of _d_- and
_l_-carvoxime[272] (C_{10}H_{14}N.OH). The freezing points and melting
points of the different mixtures of _d_- and _l_-carvoxime are given in the
following table, and represented graphically in Fig. 51:--

  ---------------+----------------+-----------------+-----------------
  Per cent. of   | Per cent. of   | Freezing point. | Melting point.
  _d_-carvoxime. | _l_-carvoxime. |                 |
  ---------------+----------------+-----------------+-----------------
     100         |      0         |      72.0°      |     72.0°
      99         |      1         |      72.4°      |      --
      98         |      2         |      73.0°      |      --
      95         |      5         |      75.4°      |     73.0°
      90         |     10         |      79.0°      |     75.0°
      80         |     20         |      84.6°      |     80.0°
      70         |     30         |      88.2°      |     85.0°
      60         |     40         |      90.4°      |      --
      50         |     50         |      91.4°      |     91.4°
      25         |     75         |      86.4°      |     82.0°
       8         |     92         |      77.4°      |      --
       1         |     99         |      72.4°      |      --
       0         |    100         |      72.0°      |     72.0°
  ---------------+----------------+-----------------+-----------------

{187}

In this figure, the melting-point curve, _i.e._ the
temperature-concentration curve for the mixed crystals, is represented by
the lower curve. Since the addition of the lævo-form to the dextro-form
raises the melting point of the latter, the concentration of the lævo-form
(on the right-hand branch of the curve) must, in accordance with the rule
given, be greater in the solid phase than in the liquid. Similarly, since
addition of the dextro-form raises the melting point of the lævo-form, the
solid phase (on the left-hand branch of the curve) must be richer in
dextro- than in lævo-carvoxime. At the maximum point, the melting-point and
freezing-point curves touch; at this point, therefore, the composition of
the solid and liquid phases must be identical. It is evident, therefore,
that at the maximum point the liquid will solidify, or the solid will
liquefy completely without change of temperature; and, accordingly, mixed
crystals of the composition represented by the maximum point will exhibit a
definite melting point, and will in this respect behave like a simple
substance.

(_c_) _The freezing-point curve passes through a minimum_ (Curve III., Fig.
49).

In this case, as in the case of those systems where the pure components are
deposited, a minimum freezing point is obtained. In the latter case,
however, there are two freezing-point curves which intersect at a eutectic
point; in the case where mixed crystals are formed there is only one
continuous curve. On one side of the minimum point the liquid phase
contains relatively more, on the other side relatively less, of the one
component than does the solid phase; while at the minimum point the
composition of the two phases is the same. At this point, therefore,
complete solidification or complete liquefaction will occur without change
of temperature, and the mixed crystals will accordingly exhibit a definite
melting point.

[Illustration: FIG. 52.]

{188}

Example.--As an example of this there may be taken the mixed crystals of
mercuric bromide and iodide.[273] Mercuric bromide melts at 236.5°, and
mercuric iodide at 255.4°. The mixed crystal of definite constant melting
point (minimum point) contains 59 mols. per cent. of mercuric bromide, the
melting point being 216.1°.

The numerical data are contained in the following table, and represented
graphically in Fig. 52:--

  -----------------------------------------------------
  Mols. per cent. of |                 |
  HgBr_{2}.          | Freezing point. | Melting point.
  -----------------------------------------------------
        100          |     236.5°      |      236°
         90          |     228.8°      |      226°
         80          |     222.2°      |      219°
         70          |     217.8°      |      217°
         65          |     216.6°      |      216°
         60          |     216.1°      |      215.5°
         55          |     216.3°      |      216°
         50          |     217.3°      |      216°
         40          |     221.1°      |      218°
         30          |     227.8°      |      223°
         20          |     236.2°      |      231°
         10          |     245.5°      |      242°
          0          |     255.4°      |      254°
  -----------------------------------------------------

[Illustration: FIG. 53.]

Fractional Crystallization of Mixed Crystals.--With the help of the
diagrams already given it will be possible to predict what will be the
result of the fractional crystallization of a fused mixture of two
substances which can form mixed crystals. Suppose, for example, a fused
mixture of the composition _x_ (Fig. 53) is cooled down; then, as we have
already seen, when the temperature has fallen to _a_, mixed crystals of
composition, _b_, are deposited. If the temperature is allowed to fall
{189} to _x'_, and the solid then separated from the liquid, the mixed
crystals so obtained will have the composition represented by e. If, now,
the mixed crystals _e_ are completely fused and the fused mass allowed to
cool, separation of solid will occur when the temperature has fallen to the
point _f_. The mixed crystals which are deposited have now the composition
represented by _g_, i.e. _they are richer in B than the original mixed
crystals_. By repeating this process, the composition of the successive
crops of mixed crystals which are obtained approximates more and more to
that of the pure component B, while, on the other hand, the composition of
the liquid phase produced tends to that of pure A. By a systematic and
methodical repetition of the process of fractional crystallization,
therefore, a _practically_ complete separation of the components can be
effected; a perfect separation is theoretically impossible.

From this it will be readily understood that in the case of substances the
freezing point of which passes through a maximum, fractional
crystallization will ultimately lead to mixed crystals having the
composition of the maximum point, while the liquid phase will more and more
assume the composition of either pure A or pure B, according as the initial
composition was on the A side or the B side of the maximum point. In those
cases, however, where the curves exhibit a minimum, the solid phase which
separates out will ultimately be one of the pure components, while a liquid
phase will finally be obtained which has the composition of the minimum
point.

II.--THE TWO COMPONENTS DO NOT FORM A CONTINUOUS SERIES OF MIXED CRYSTALS.

This case corresponds to that of the partial miscibility of liquids. The
solid component A can "dissolve" the component B until the concentration of
the latter in the mixed crystal has reached a certain value. Addition of a
further amount of B will not alter the composition of the mixed crystal,
but there will be formed a second solid phase consisting {190} of a
solution of A in B. At this point the four phases, mixed crystals
containing excess of A, mixed crystals containing excess of B, liquid
solution, vapour, can coexist; this will therefore be an invariant point.
The temperature-concentration curves will therefore no longer be
continuous, but will exhibit a break or discontinuity at the point at which
the invariant system is formed.

(_a_) _The freezing-point curve exhibits a transition point_ (Curve I.,
Fig. 54).

As is evident from the figure, addition of B raises the melting point of A,
and, in accordance with the rule previously given, the concentration of B
in the mixed crystals will be greater than in the solution. This is
represented in the figure by the dotted curve AD. On the other hand,
addition of A lowers the melting point of B, and the two curves BC and BE
are obtained for the liquid and solid phases respectively. At the
temperature of the line CDE the liquid solution of the composition
represented by C is in equilibrium with the two different mixed crystals
represented by D and E. At this temperature, therefore, the _tc_-curve for
the solid phase exhibits a discontinuity; and, since the solid phase
undergoes change at this point, the freezing-point curve must show a break
(p. 111).

[Illustration: FIG. 54.]

Example.--Curves of the form given in Fig. 54 I. have been found
experimentally in the case of silver nitrate and sodium nitrate.[274] The
following table contains the numerical data, which are also represented
graphically in Fig. 55:--

{191}

  -----------------------------------------------------
  Molecules NaNO_{3} | Freezing point. | Melting point.
      per cent.      |                 |
  -----------------------------------------------------
          0          |      208.6°     |     208.6°
          8          |      211.4°     |     210°
         15.06       |      215°       |     212°
         19.46       |      217.2°     |     214.8°
         21.9        |      222°       |     215°
         26          |      228.4°     |     216.5°
         29.7        |      234.8°     |     217.5°
         36.2        |      244.4°     |     217.5°
         47.3        |      259.4°     |     237.6°
         58.9        |      272°       |     257°
         72          |      284°       |     274°
        100          |      308°       |     308°
  -----------------------------------------------------

The temperature of the transition point is 217.5°; at this point the liquid
contains 19.5, and the two conjugate solid solutions 26 and 38 molecules of
sodium nitrate per cent. respectively.

[Illustration: FIG. 55.]

[Illustration: FIG. 56.]

(_b_) _The freezing-point curve exhibits a eutectic point_ (Curve II., Fig.
54). {192}

In this case the freezing point of each of the components is lowered by the
addition of the other, until at last a point is reached at which the liquid
solution solidifies to a mixture or conglomerate of two mixed crystals.

Examples.--Curves belonging to this class have been obtained in the case of
potassium and thallium nitrates[275] and of naphthalene and monochloracetic
acid.[276] The data for the latter are given in the following table and
represented in Fig. 56:--

  -------------------------------------------------------------------------
               |         Liquid solution.     |        Solid solution.
               ------------------------------------------------------------
  Temperature. |              |               |              |
               |  Per cent.   |   Per cent.   |  Per cent.   |   Per cent.
               | naphthalene. |     acid.     | naphthalene. |     acid.
  -------------------------------------------------------------------------
      62°      |     --       |     100       |     --       |     100
      60°      |      4.0     |      96.0     |      1.7     |      98.3
      55°      |     21.0     |      79.0     |      2.1     |      97.9
      53.5°    |     29.4     |      70.0     |     --       |      --
      55°      |     31.3     |      68.7     |     59.6     |      40.4
      60°      |     42.4     |      57.6     |     80.3     |      19.7
      65°      |     53.3     |      46.7     |     89.2     |      10.8
      70°      |     69.7     |       2.3     |     95.4     |       4.6
      75°      |     84.4     |      15.6     |     96.6     |       3.4
      79.9°    |    100       |      --       |    100       |      --
  -------------------------------------------------------------------------

At the eutectic point the liquid solution is in equilibrium with two
different mixed crystals the composition of which is represented by D and E
respectively. If, therefore, a fused mixture containing the two components
A and B in the proportions represented by C is cooled down, it will, when
the temperature has reached the point C, solidify completely to a
_conglomerate_ of mixed crystals, D and E.

[Illustration: FIG. 57.]

[Illustration: FIG. 58.]

Changes in Mixed Crystals with the Temperature.--In the case of the
different types of systems represented in Fig. 49, a homogeneous liquid
solution of the two components will exist at temperatures above the
freezing-point curve, a homogeneous mixed crystal at temperatures below the
melting-point curve, while at any point between the freezing-point and
melting-point {193} curves the mixture will separate into a solid phase and
a liquid phase. In the case, however, of the two types shown in Fig. 54 the
relationships are somewhat more complicated. As before, the area above the
freezing-point curve gives the conditions under which homogeneous liquid
solutions can exist; but below the melting-point curve two different mixed
crystals can coexist. This will be best understood from Figs. 57 and 58. D
and E represent, as we have seen, the composition of two mixed crystals
which are in equilibrium with the liquid solution at the temperature of the
point C. These two mixed crystals represent, in the one case, a saturated
solution of B in A (point D), and the other a saturated solution of A in B
(point E). Just as we saw that the mutual solubility of two liquids varied
with the temperature, so also in the case of two solids; as the temperature
alters, the solubility of the two solid components in one another will
change. This alteration is indicated diagrammatically in Figs. 57 and 58 by
the dotted curve similar to the solubility curves for two mutually soluble
liquids (p. 101).

Suppose, now, that a mixed crystal of the composition _x_ is cooled down,
it will remain unchanged until, when the temperature has fallen to _t'_,
the homogeneous mixed crystal breaks up into a conglomerate of two mixed
crystals the composition of {194} which is represented by _x'_ and _x"_
respectively. From this, then, it can be seen that in the case of
substances which form two solid solutions, the mixed crystals which are
desposited from the liquid fused mass need not remain unchanged in the
solid state, but may at some lower temperature lose their homogeneity. This
fact is of considerable importance for the formation of alloys.[277]

A good example of this will soon be met with in the case of the iron and
carbon alloys. The alloys of copper and tin also furnish examples of the
great changes which may take place in the alloy between the temperature at
which it separates out from the fused mass and the ordinary temperature.
Thus, for example, one of the alloys of copper and tin which separates out
from the liquid as a solid solution breaks up, on cooling, into the
compound Cu_{3}Sn and liquid:[278] a striking example of a solid substance
partially liquefying on being cooled.

       *       *       *       *       *


{195}

CHAPTER XI

EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES

It has long been known that certain substances, _e.g._ acetoacetic ester,
are capable when in solution or in the fused state, of reacting as if they
possessed two different constitutions; and in order to explain this
behaviour the view was advanced (by Laar) that in such cases a hydrogen
atom oscillated between two positions in the molecule, being at one time
attached to oxygen, at another time to carbon, as represented by the
formula--

  CH_{3}.C--CH.CO_{2}C_{2}H_{5}
         .  ^
         .  |
         O<-H

When the hydrogen is in one position, the substance will act as an
hydroxy-compound; with hydrogen in the other position, as a ketone.
Substances possessing this double function are called _tautomeric_.

Doubt, however, arose as to the validity of the above explanation, and this
doubt was confirmed by the isolation of the two isomerides in the solid
state, and also by the fact that the velocity of change of the one
isomeride into the other could in some cases be quantitatively measured.
These and other observations then led to the view, in harmony with the laws
of chemical dynamics, that tautomeric substances in the dissolved or fused
state represent a _mixture_ of two isomeric forms, and that equilibrium is
established not by _intra_- but by _inter_-molecular change, as expressed
by the equation--

  CH_{3}.CO.CH_{2}.CO_{2}C_{2}H_{5} <--> CH_{3}.C(OH):CH.CO_{2}C_{2}H_{5}

{196} In the solid state, the one or other of the isomerides represents the
stable form; but in the liquid state (solution or fusion) the stable
condition is an equilibrium between the two forms.

A similar behaviour is also found in the case of other isomeric substances
where the isomerism is due to difference of structure, _i.e._ structure
isomerism (_e.g._ in the case of the oximes

  C_{6}H_{5}.C.H      C_{6}H_{5}.C.H
             ||   and            ||  ),
             N.OH             HO.N

or to difference in configuration, _i.e._ stereoisomerism (_e.g._ optically
active substances), or to polymerism (_e.g._ acetaldehyde and paraldehyde).
In all such cases, although the different solid forms correspond to a
single definite constitution, in the liquid state a condition of
equilibrium between the two modifications is established. As a general name
for these different classes of substances, the term "dynamic isomerides"
has been introduced; and the different kinds of isomerism are classed
together under the title "dynamic isomerism."[279]

By reason of the importance of these phenomena in the study more especially
of Organic Chemistry, a brief account of the equilibrium relations
exhibited by systems composed of dynamic isomerides may be given here.[280]

In studying the fusion and solidification of those substances which exhibit
the relationships of dynamic isomerism, the phenomena observed will vary
somewhat according as the reversible transformation of the one form into
the other takes place with measurable velocity at temperatures in the
neighbourhood of the melting points, or only at some higher temperature. If
the transformation is very rapid, the system will behave like a
one-component system, but if the isomeric change is comparatively slow, the
behaviour will be that of a two-component system.

Temperature-Concentration Diagram.--The relationships which are met with
here will be most readily understood with {197} the help of Fig. 59.
Suppose, in the first instance, that isomeric transformation does not take
place at the temperature of the melting point, then the freezing point
curve will have the simple form ACB; the formation of compounds being for
the present excluded. This is the simplest type of curve, and gives the
composition of the solutions in equilibrium with the one modification
([alpha] modification) at different temperatures (curve AC); and of the
solutions in equilibrium with the other modification ([beta] modification)
at different temperatures (curve BC). C is the eutectic point at which the
two solid isomerides can exist side by side in contact with the solution.

[Illustration: FIG. 59.]

Now, suppose that isomeric transformation takes place with measurable
velocity. If the pure [alpha]-modification is heated to a temperature _t'_
above its melting point, and the liquid maintained at that temperature
until equilibrium has been established, a certain amount of the [beta]-form
will be present in the liquid, the composition of which will be represented
by the point _x'_. The same condition of equilibrium will also be reached
by starting with pure [beta]. Similarly, if the temperature of the liquid
is maintained at the temperature _t"_, equilibrium will be reached, we
shall suppose, when the solution has the composition _x"_. The curve DE,
therefore, which passes through all the different values of _x_
corresponding to different values of _t_, will represent the change of
equilibrium with the temperature. It will slope to the right (as in the
figure) if the transformation of [alpha] into [beta] is accompanied by
absorption of heat; to the left if the transformation is accompanied by
evolution of heat, in accordance with van't Hoff's Law of movable
equilibrium. If transformation occurs without heat effect, the equilibrium
will be independent of the {198} temperature, and the equilibrium curve DE
will therefore be perpendicular and parallel to the temperature axis.

We must now find the meaning of the point D. Suppose the pure [alpha]- or
pure [beta]-form heated to the temperature _t'_, and the temperature
maintained constant until the liquid has the composition _x'_ corresponding
to the equilibrium at that temperature. If the temperature is now allowed
to fall sufficiently slowly so that the condition of equilibrium is
continually readjusted as the temperature changes, the composition of the
solution will gradually alter as represented by the curve _x'_D. Since D is
on the freezing point curve of pure [alpha], this form will be deposited on
cooling; and since D is also on the equilibrium curve of the liquid, D is
the only point at which solid can exist in stable equilibrium with the
liquid phase. (The vapour phase may be omitted from consideration, as we
shall suppose the experiments carried out in open vessels.) All systems
consisting of the two hylotropic[281] isomeric substances [alpha] and
[beta] will, therefore, ultimately freeze at the point D, which is called
the "natural" freezing point[282] of the system; provided, of course, that
sufficient time is allowed for equilibrium to be established. From this it
is apparent that _the stable modification at temperatures in the
neighbourhood of the melting point is that which is in equilibrium with the
liquid phase at the natural freezing point_.

From what has been said, it will be easy to predict what will be the
behaviour of the system under different conditions. If pure [alpha] is
heated, a temperature will be reached at which it will melt, but this
melting point will be sharp only if the velocity of isomeric transformation
is comparatively slow; _i.e._ slow in comparison with the determination of
the melting point. If the substance be maintained in the fused condition
for some time, a certain amount of the [beta] modification will be formed,
and on lowering the temperature the pure [alpha] form will be deposited,
not at the temperature of the melting point, but at some lower temperature
depending on the concentration of the [beta] modification in the liquid
phase. If isomeric transformation {199} takes place slowly in comparison
with the rate at which deposition of the solid occurs, the liquid will
become increasingly rich in the [beta] modification, and the freezing point
will, therefore, sink continuously. At the eutectic point, however, the
[beta] modification will also be deposited, and the temperature will remain
constant until all has become solid. If, on the other hand, the velocity of
transformation is sufficiently rapid, then as quickly as the [alpha]
modification is deposited, the equilibrium between the two isomeric forms
in the liquid phase will continuously readjust itself, and the end-point of
solidification will be the natural freezing point.

Similarly, starting with the pure [beta] modification, the freezing point
after fusion will gradually fall owing to the formation of the [alpha]
modification; and the composition of the liquid phase will pass along the
curve BC. If, now, the rate of cooling is not too great, or if the velocity
of isomeric transformation is sufficiently rapid, complete solidification
will not occur at the eutectic point; for at this temperature solid and
liquid are not in stable equilibrium with one another. On the contrary, a
further quantity of the [beta] modification will undergo isomeric change,
the liquid phase will become richer in the [alpha] form, and the freezing
point will _rise_; the solid phase in contact with the liquid being now the
[alpha] modification. The freezing point will continue to rise until the
point D is reached, at which complete solidification will take place
without further change of temperature.

The diagram also allows us to predict what will be the result of rapidly
cooling a fused mixture of the two isomerides. Suppose that either the
[alpha] or the [beta] modification has been maintained in the fused state
at the temperature _t'_ sufficiently long for equilibrium to be
established. The composition of the liquid phase will be represented by
_x'_. If the liquid is now _rapidly_ cooled, the composition will remain
unchanged as represented by the dotted line _x'_G. At the temperature of
the point G solid [alpha] modification will be deposited. If the cooling is
not carried below the point G, so as to cause complete solidification, the
freezing point will be found to rise with time, owing to the conversion of
some of the [beta] form into the [alpha] form {200} in the liquid phase;
and this will continue until the composition of the liquid has reached the
point D. From what has just been said, it can also be seen that if the
freezing point curves can be obtained by actual determination of the
freezing points of different synthetic mixtures of the two isomerides, it
will be possible to determine the condition of equilibrium in the fused
state at any given temperature without having recourse to analysis. All
that is necessary is to rapidly cool the fused mass, after equilibrium has
been established, and find the freezing point at which solid is deposited;
that is, find the point at which the line of constant temperature cuts the
freezing point curve. The composition corresponding to this temperature
gives the composition of the equilibrium mixture at the given temperature.

It will be evident, from what has gone before, that the degree of
completeness with which the different curves can be realised will depend on
the velocity with which isomeric change takes place, and on the rapidity
with which the determinations of the freezing point can be carried out. As
the two extremes we have, on the one hand, practically instantaneous
transformation, and on the other, practically infinite slowness of
transformation. In the former case, only one melting and freezing point
will be found, viz. the natural freezing point; in the latter case, the two
isomerides will behave as two perfectly independent components, and the
equilibrium curve DE will not be realised.

The diagram which is obtained when isomeric transformation does not occur
within measurable time at the temperature of the melting point is somewhat
different from that already given in Fig. 59. In this case, the two
freezing point curves AC and BC (Fig. 60) can be readily realized, as no
isomeric change occurs in the liquid phase. Suppose, however, that at a
higher temperature, _t'_, reversible isomeric transformation can take
place, the composition of the liquid phase will alter until at the point
_x'_ a condition of equilibrium is reached; and the composition of the
liquid at higher temperatures will be represented by the curve _x'_F. Below
the temperature _t'_ the position of the equilibrium curve is hypothetical;
but as the temperature {201} falls the velocity of transformation
diminishes, and at last becomes _practically_ zero. The equilibrium curve
can therefore be regarded as dividing into two branches _x'_G and _x'_H. At
temperatures between G and _t'_ the [alpha] modification can undergo
isomeric change leading to a point on the curve G_x'_; and the [beta]
modification can undergo change leading to a point on the curve H_x'_. The
same condition of equilibrium is therefore not reached from each side, and
we are therefore dealing not with true but with false equilibrium (p. 5).
Below the temperatures G and H, isomeric transformation does not occur in
measurable time. We shall not, however, enter into a detailed discussion of
the equilibria in such systems, more especially as they are not systems in
true equilibrium, and as the temperature at which true equilibrium can be
established with appreciable velocity alters under the influence of
catalytic agents.[283] Examples of such systems will no doubt be found in
the case of optically active substances, where both isomerides are
apparently quite stable at the melting point. In the case of such
substances, also, the action of catalytic agents in producing isomeric
transformation (racemisation) is well known.

[Illustration: FIG. 60.]

Transformation of the Unstable into the Stable Form.--As has already been
stated, the stable modification in the neighbourhood of the melting point
is that one which is in equilibrium with the liquid phase at the natural
freezing point. In the case of polymorphic substances, we have seen (p. 39)
that that form which is stable in the neighbourhood of the melting point
melts at the higher temperature. That was a {202} consequence of the fact
that the two polymorphic forms on melting gave identical liquid phases. In
the present case, however, the above rule does not apply, for the simple
reason that the liquid phase obtained by the fusion of the one modification
is not identical with that obtained by the fusion of the other. In the case
of isomeric substances, therefore, the form of lower melting point _may_ be
the more stable; and where this behaviour is found it is a sign that the
two forms are isomeric (or polymeric) and not polymorphic.[284] An example
of this is found in the case of the isomeric benzaldoximes (p. 203).

Since in Fig. 59 the [alpha] modification has been represented as the
stable form, the transformation of the [beta] into the [alpha] form will be
possible at all temperatures down to the transition point. At temperatures
below the eutectic point, transformation will occur without formation of a
liquid phase; but at temperatures above the eutectic point liquefaction can
take place. This will be more readily understood by drawing a line of
constant temperature, HK, at some point between C and B. Then if the [beta]
modification is maintained for a sufficiently long time at that
temperature, a certain amount of the [alpha] modification will be formed;
and when the composition of the mixture has reached the point H, fusion
will occur. If the temperature is maintained constant, isomeric
transformation will continue to take place in the liquid phase until the
equilibrium point for that temperature is reached. If this temperature is
higher than the natural melting point, the mixture will remain liquid all
the time; but if it is below the natural melting point, then the [alpha]
modification will be deposited when the system reaches the condition
represented by the point on the curve AC corresponding to the particular
temperature. As isomeric transformation continues, the freezing point of
the system will rise until it reaches the natural freezing point D.
Similarly, if the [alpha] modification is maintained at a temperature above
that of the point D, liquefaction will ultimately occur, and the system
will again reach the final state represented by D.[285]

{203}

Examples.--_Benzaldoximes._ The relationships which have just been
discussed from the theoretical point of view will be rendered clearer by a
brief description of cases which have been experimentally investigated. The
first we shall consider is that of the two isomeric benzaldoximes:[286]--

  C_{6}H_{5}.C.H            C_{6}H_{5}.C.H
             ||                        ||
          HO.N                         N.OH

  Benzantialdoxime          Benzsynaldoxime
  ([alpha]-modification).    ([beta]-modification).

Fig. 61 gives a graphic representation of the results obtained.

The melting point of the [alpha] modification is 34-35°; the melting point
of the unstable [beta]-modification being 130°. The freezing curves AC and
BC were obtained by determining the freezing points of different mixtures
of known composition, and the numbers so obtained are given in the
following table.

{204}

  ----------------------------------------------------
  Grams of the [alpha] modification |
    in 100 gm. of mixture.          | Freezing point.
  ----------------------------------+-----------------
               26.2                 |     101°
               49.2                 |      79°
               73.7                 |      46°
               91.7                 |      26.2°
               95.0                 |      28.6°
               96.0                 |      30.0°
  ----------------------------------------------------

[Illustration: FIG. 61.]

The eutectic point C was found to lie at 25-26°, and the natural freezing
point D was found to be 27.7°. The equilibrium curve DE was determined by
heating the liquid mixtures at different temperatures until equilibrium was
attained, and then rapidly cooling the liquid. In all cases the freezing
point was practically that of the point D. From this it is seen that the
equilibrium curve must be a straight line parallel to the temperature axis;
and, therefore, isomeric transformation in the case of the two
benzaldoximes is not accompanied by any heat effect (p. 197). This
behaviour has also been found in the case of acetaldoxime.[287]

The isomeric benzaldoximes are also of interest from the fact that the
stable modification has the _lower_ melting point (_v._ p. 202).

_Acetaldehyde and Paraldehyde._--As a second example of the equilibria
between two isomerides, we shall take the two isomeric (polymeric) forms of
acetaldehyde, which have recently been exhaustively studied.[288]

{205}

In the case of these two substances the reaction

  3CH_{3}.CHO <--> (CH_{3}.CHO)_{3}

takes place at the ordinary temperature with very great slowness. For this
reason it is possible to determine the freezing point curves of
acetaldehyde and paraldehyde. The three chief points on these curves,
represented graphically in Fig. 62, are:--

  m.p. of acetaldehyde - 118.45°
  m.p. of paraldehyde  +  12.55°
  eutectic point       - 119.9°

[Illustration: FIG. 62.]

In order to determine the position of the natural melting point, it was
necessary, on account of the slowness of transformation, to employ a
catalytic agent in order to increase the velocity with which the
equilibrium was established. A drop of concentrated sulphuric acid served
the purpose. In presence of a trace of this substance, isomeric
transformation very speedily occurs, and leads to the condition of
equilibrium. Starting in the one case with fused paraldehyde, and in the
other case with acetaldehyde, the same freezing point, viz. 6.75°, was
obtained, the solid phase being paraldehyde. This temperature, 6.75°, is
therefore the natural freezing point, and paraldehyde, the solid in
equilibrium with the liquid phase at this point, is the stable form.

With regard to the change of equilibrium with the temperature, it was found
that whereas the liquid phase contained 11.7 molecules per cent. of
acetaldehyde at the natural freezing point, the liquid at the temperature
of 41.6° contains 46.6 molecules per cent. of acetaldehyde. As the
temperature {206} rises, therefore, there is increased formation of
acetaldehyde, or a decreasing amount of polymerisation. This is in harmony
with the fact that the polymerisation of acetaldehyde is accompanied by
evolution of heat.

While speaking of these isomerides, it may be mentioned that at the
temperature 41.6° the equilibrium mixture has a vapour pressure equal to
the atmospheric pressure. At this temperature, therefore, the equilibrium
mixture (obtained quickly with the help of a trace of sulphuric acid)
boils.[289]

       *       *       *       *       *


{207}

CHAPTER XII

SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF TWO
COMPONENTS

In this concluding chapter on two-component systems, it is proposed to
indicate briefly how the Phase Rule has been applied to the elucidation of
a number of problems connected with the equilibria between two components,
and how it has been employed for the interpretation of the data obtained by
experiment. It is hoped that the practical value of the Phase Rule may
thereby become more apparent, and its application to other cases be
rendered easier.

The interest and importance of investigations into the conditions of
equilibrium between two substances, lie in the determination not only of
the conditions for the stable existence of the participating substances,
but also of whether or not chemical action takes place between these two
components; and if combination occurs, in the determination of the nature
of the compounds formed and the range of their existence. In all such
investigations, the Phase Rule becomes of conspicuous value on account of
the fact that its principles afford, as it were, a touchstone by which the
character of the system can be determined, and that from the form of the
equilibrium curves obtained, conclusions can be drawn as to the nature of
the interaction between the two substances. In order to exemplify the
application of the principles of the Phase Rule more fully than has already
been done, illustrations will be drawn from investigations on the
interaction of organic compounds; on the equilibria between optically
active compounds; and on alloys. {208}

Summary of the Different Systems of Two Components.--Before passing to the
consideration of the application of the Phase Rule to the investigation of
particular problems, it will be well to collect together the different
types of equilibrium curves with which we are already acquainted; to
compare them with one another, in order that we may then employ these
characteristic curves for the interpretation of the curves obtained as the
result of experiment.

In investigating the equilibria between two components, three chief classes
of curves will be obtained according as--

I. No combination takes place between the two components.

II. The components can form definite compounds.

III. The components separate out in the form of mixed crystals.

The different types of curves which are obtained in these three cases are
represented in Figs. 63, 64, 65. These different diagrams represent the
whole series of equilibria, from the melting point of the one component (A)
to that of the other component (B). The curves represent, in all cases, the
composition of the solution, or phase of variable composition; the
temperature being measured along one axis, and the composition along the
other.

We shall now recapitulate very briefly the characteristics of the different
curves.

[Illustration: FIG. 63.]

If no compound is formed between the two components, {209} the general form
of the equilibrium curve will be that of curve I. or II., Fig. 63. Type I.
is the simplest form of curve found, and consists, as the diagram shows, of
only two branches, AC and BC, meeting at the point C, _which lies below the
melting point of either component_. The solid phase which is in equilibrium
with the solutions AC is pure A; that in equilibrium with BC, pure B. C is
the eutectic point. Although at the eutectic point the solution solidifies
entirely without change of temperature, the solid which is deposited is not
a homogeneous solid phase, but a mixture, or conglomerate of the two
components. _The eutectic point, therefore, represents the melting or
freezing point, not of a compound, but of a mixture_ (p. 119).

Curve II., Fig. 63, is obtained when two liquid phases are formed. C is an
eutectic point, D and F are transition points at which there can co-exist
the four phases--solid, two liquid phases, vapour. DEF represents the
change in the composition of the two liquid phases with rise of
temperature; the curve might also have the reversed form with the critical
solution point below the transition points D and F.

[Illustration: FIG. 64.]

In the second class of systems (Fig. 64), that in which combination between
the components occurs, there are again two types according as the compound
formed has a definite melting point (_i.e._ can exist in equilibrium with a
solution of the same composition), or undergoes only partial fusion; that
is, exhibits a transition point.

If a compound possessing a definite melting point is formed, the
equilibrium curve will have the general form shown by curve I., Fig. 64. A,
B, and D are the melting points of pure A, pure B, and of the compound
A_{x}B_{y} respectively. AC {210} is the freezing point curve of A in
presence of B; BE that of B in presence of A; and DC and DE the freezing
point curves of the compound in presence of a solution containing excess of
one of the components. C and E are eutectic points at which mixtures of A
and A_{x}B_{y}, or B and A_{x}B_{y} can co-exist in contact with solution.
The curve CDE may be large or small, and the melting point of the compound,
D, may lie above or below that of each of the components, or may have an
intermediate position. If more than one compound can be formed, a series of
curves similar to CDE will be obtained (_cf._ p. 152).

On the other hand, if the compound undergoes transition to another solid
phase at a temperature below its melting point, a curve of the form II.,
Fig. 64, will be found. This corresponds to the case where a compound can
exist only in contact with solutions containing excess of one of the
components. The metastable continuation of the equilibrium curve for the
compound is indicated by the dotted line, the summit of which would be the
melting point of the compound. Before this temperature is reached, however,
the solid compound ceases to be able to exist in contact with solution, and
transition to a different solid phase occurs at the point E (_cf._ p. 134).
This point, therefore, represents the limit of the existence of the
compound AB. If a series of compounds can be formed none of which possess a
definite melting point, then a series of curves will be obtained which do
not exhibit a temperature-maximum, and there will be only one eutectic
point. The limits of existence of each compound will be marked by a break
in the curve (_cf._ p. 143).

[Illustration: FIG. 65.]

Turning, lastly, to the third class of systems, in which formation of mixed
crystals can occur, five different types of curves can be obtained, as
shown in Fig. 65. With regard to the first three types, curves I., II., and
III., {211} these differ entirely from those of the previous classes, in
that they are continuous; they exhibit no eutectic point, and no transition
point. Curve II. bears some resemblance to the melting-point curve of a
compound (_e.g._ CDE, Fig. 64, I.), but differs markedly from it in not
ending in eutectic points.

Further, in the case of the formation of a compound, the composition of the
solid phase remains unchanged throughout the whole curve between the
eutectic points; whereas, when mixed crystals are produced, the composition
of the solid phase varies with the composition of the liquid solution. On
passing through the maximum, the relative proportions of A and B in the
solid and the liquid phase undergo change; on the one side of the maximum,
the solid phase contains relatively more A, and on the other side of the
maximum, relatively more B than the liquid phase. Lastly, when mixed
crystals are formed, the temperature at which complete solidification
occurs changes as the composition of the solution changes, whereas in the
case of the formation of compounds, the temperature of complete
solidification for all solutions is a eutectic point.

The third type of curve, Fig. 65, can be distinguished in a similar manner
from the ordinary eutectic curve, Fig. 63, I., to which it bears a certain
resemblance. Whereas in the case of the latter, the eutectic point is the
temperature of complete solidification of all solutions, the point of
minimum temperature in the case of the formation of mixed crystals, is the
solidification point only of solutions having one particular composition;
that, namely, of the minimum point. For all other solutions, the
temperature of complete solidification is different. Whereas, also, in the
case of the simple eutectic curve, the solid which separates out from the
solutions represented by either curve remains the same throughout the whole
extent of that curve, the composition of the mixed crystal varies with
variation of the composition of the liquid phase, and the relative
proportions of the two components in the solid and the liquid phase are
reversed on passing through the minimum.[290]

In a similar manner, type IV., Fig. 65, can be distinguished from type II.,
Fig. 64, by the fact that it does not exhibit a {212} eutectic point, and
that the composition of the solid phase undergoes continuous variation with
variation of the liquid phase on either side of the transition point.
Lastly, type V., which does exhibit a eutectic point, differs from the
eutectic curve of Fig. 63, in that the eutectic point does not constitute
the point of complete solidification for all solutions, and that the
composition of the solid phase varies with the composition of the liquid
phase.

Such, then, are the chief general types of equilibrium curves for
two-components; they are the pattern curves with which other curves,
experimentally determined, can be compared; and from the comparison it will
be possible to draw conclusions as to the nature of the equilibria between
the two components under investigation.

1. _Organic Compounds._

[Illustration: FIG. 66.]

The principles of the Phase Rule have been applied to the investigation of
the equilibria between organic compounds, and Figs. 66-69 reproduce some of
the results which have been obtained.[291]

{213}

Fig. 66, the freezing point curve (curve of equilibrium) for
_o_-nitrophenol and _p_-toluidine, shows a curve of the simplest type[292]
(type I., Fig. 63), in which two branches meet at an eutectic point. The
solid phase in equilibrium with solutions represented by the left-hand
branch of the curve was _o_-nitrophenol (m.p. 44.1°); that in equilibrium
with the solutions represented by the right-hand branch, was _p_-toluidine
(m.p. 43.3°). At the eutectic point (15.6°), these two solid phases could
co-exist with the liquid phase. This equilibrium curve, therefore, shows
that _o_-nitrophenol and _p_-toluidine do not combine with one another.

In connection with this curve, attention may be called to the interesting
fact that although the solid produced by cooling the liquid phase at the
eutectic point has a composition approximating to that of a compound of
equimolecular proportions of the phenol and toluidine, and a constant
melting point, it is nevertheless a _mixture_. Although, as a rule, the
constituents of the eutectic mixture are not present in simple molecular
proportions, there is no reason why they should not be so; and it is
therefore necessary to beware of assuming the formation of compounds in
such cases.[293]

Fig. 67, on the other hand, indicates with perfect certainty the formation
of a compound between phenol and [alpha]-naphthylamine.[294] (_Cf._ curve
I., Fig. 64.)

Phenol freezes at 40.4°, but the addition of [alpha]-naphthylamine lowers
the freezing point as represented by the curve AC. At C (16.0°) the
compound C_{6}H_{5}OH,C_{10}H_{7}NH_{2} is formed, and the system becomes
invariant. On increasing the amount of the amine, the temperature of
equilibrium rises, the solid phase now being the compound. At D, the curve
passes through a maximum (28.8°), at which the solid and liquid phases have
the same composition. This is the melting point of the compound. Further
addition of the amine lowers the temperature of equilibrium, until at E
solid [alpha]-naphthylamine separates out, and a second eutectic point
(24.0°) is obtained. BE is the {214} freezing-point curve of
[alpha]-naphthylamine in presence of phenol, the freezing point of the pure
amine being 48.3°.

On account of the great sluggishness with which the compound of phenol and
[alpha]-naphthylamine crystallizes, it was found possible to follow the
freezing point curves of phenol and the amine to temperatures considerably
below the eutectic points, as shown by the curves CF and EG.

[Illustration: FIG. 67.]

Phenol can also combine with _p_-toluidine in equimolecular proportions;
and this compound is of interest, from the fact that it exists in two
crystalline forms melting at 28.5° and 30°. Each of these forms now must
have its own equilibrium curve, and it was found that the intermediate
portion of the freezing point curve was duplicated, as shown in Fig.
68.[295]

{215}

[Illustration: FIG. 68.]

[Illustration: FIG. 69.]

{216}

Lastly, a curve is given, Fig. 69,[296] which corresponds with curve II.,
Fig. 64. Picric acid and benzene can form a compound, which, however, can
exist only in contact with solutions _containing excess of benzene_. When
the temperature is raised, a point (K) is reached at which the compound
melts with separation of solid picric acid. The point, K, is, therefore, a
_transition point_; analysis, however, showed that the composition of the
solution at this point is very nearly that of the compound
C_{6}H_{2}(NO_{2})_{3}OH,C_{6}H_{6}, so that the melting point of the
compound can almost be reached. The fusion of the compound of benzene and
picric acid with separation of the latter is analogous to the (partial)
fusion of Glauber's salt with separation of anhydrous sodium sulphate.

2. _Optically Active Substances._

The question as to whether a resolvable inactive body is a mixture of the
two oppositely active constituents (a _dl_-mixture), or a racemic compound,
is one which has given rise to considerable discussion during the past
decade; and several investigators have endeavoured to establish general
rules by which the question could be decided. In the case of inactive
liquids it is a matter of great difficulty to arrive at a certain
conclusion as to whether one is dealing with a mixture or a compound, for
in this case the usual physical methods give but a dubious answer; and
although the existence of a racemate in the liquid state (in the case of
conine) has been asserted,[297] most chemists incline to the belief that
such a thing is improbable.

Even in the case of crystalline substances, where the differences between
the various forms is greater, it was not always easy to discriminate
between the _dl_-mixture and the racemic compound. The occurrence of
hemihedral faces was considered by Pasteur to be a sufficient criterion for
an optically active substance. It has, however, been found that hemihedry
in crystals, although a frequent accompaniment of {217} optical activity,
is by no means a necessary or constant expression of this property. Other
rules, also, which were given, although in some cases reliable, were in
other cases insufficient; and all were in so far unsatisfactory that they
lacked a theoretical basis.

With the help of the Phase Rule, however, it is possible from a study of
the solubility or fusion curves of the optically active and inactive
substances, to decide the nature of the inactive substance, at least under
certain conditions. On account of the interest and importance which these
compounds possess, a brief description of the application of the Phase Rule
to the study of such substances will be given here;[298] the two optical
antipodes being regarded as the two components.

In the present chapter we shall consider only the fusion curves, the
solubility curves being discussed in the next section on three-component
systems. The rules which are hereby obtained, have reference only to the
nature of the inactive substance in the neighbourhood of the melting
points.

I. _The inactive substance is a _dl_-mixture._

In this case the fusion curves will have the simple form shown in type I,
Fig. 63. A and B are the melting points of the two optical isomerides, and
C the eutectic point at which the inactive mixture consisting of equal
amounts of d- and l-form melts. Owing to the similar effect of the one form
on the freezing point of the other, the figure is symmetrical. No example
of this simple case has been investigated.

II. _The two components form a racemic compound._

In this case there will be three melting point curves as in Fig. 64, type
I. In this case also the figure must be symmetrical.

Examples.--As examples of this, may be taken dimethyl tartrate and mandelic
acid, the freezing point curves of which are given in Figs. 70 and 71.[299]
As can be seen, the curve for the racemic tartrate occupies a large part of
the diagram, {218} while that for racemic mandelic acid is much smaller. In
the case of dimethyldiacetyl tartrate, this middle portion is still less.

[Illustration: FIG. 70.]

[Illustration: FIG. 71.]

[Illustration: FIG. 72.]

Active dimethyl tartrate melts at 43.3°; racemic dimethyl tartrate at
89.4°. Active mandelic acid melts at 132.8°; the racemic acid at 118.0°. In
the one case, therefore, the racemic compound has a higher, in the other a
lower melting point than the active forms. {219}

In the case of partially racemic compounds (_i.e._ the compound of a
racemate with an optically active substance) the type of curve will be the
same, but the figure will no longer be symmetrical. Such a curve has been
found in the case of the l-menthyl esters of d- and l-mandelic acid (Fig.
72).[300] The freezing point of l-menthyl d-mandelate is 97.2°, of
l-menthyl l-mandelate 77.6°, and of l-menthyl r-mandelate 83.7.° It will be
observed that the summit of the curve for the partially racemic mandelate
is very flat, indicating that the compound is largely dissociated into its
components at the temperature of fusion.

III. _The inactive substance is a pseudo-racemic mixed crystal._

In cases where the active components can form mixed crystals, the
freezing-point curve will exhibit one of the forms given in Fig. 65. The
inactive mixed crystal containing 50 per cent. of the dextro and laevo
compound, is known as a pseudo-racemic mixed crystal.[301] So far, only
curves of the types I. and II. have been obtained.

Examples.--The two active camphor oximes are of interest from the fact that
they form a continuous series of mixed crystals, _all of which have the
same melting point_. The curve which is obtained in this case is,
therefore, a straight line joining the melting points of the pure active
components; the melting point of the active isomerides and of the whole
series of mixed crystals being 118.8°.

[Illustration: FIG. 73.]

In the case of the carvoximes mixed crystals are also formed, but the
equilibrium curve in this case exhibits a maximum (Fig. 73). At this
maximum point the composition of the solid and of the liquid solution is
the same. Since the curve must be symmetrical, this maximum point must
occur in the case of the solution containing 50 per cent. {220} of each
component, which will therefore be inactive. Further, this inactive mixed
crystal will melt and solidify at the same temperature, and behave,
therefore, like a chemical compound (p. 187). The melting point of the
active compounds is 72°; that of the inactive pseudo-racemic mixed crystal
is 91.4°·

Transformations.--As has already been remarked, the conclusions which can
be drawn from the fusion curves regarding the nature of the inactive
substances formed hold only for temperatures in the neighbourhood of the
melting points. At temperatures below the melting point transformation may
occur; _e.g._ a racemate may break up into a _dl_-mixture, or a
pseudo-racemic mixed crystal may form a racemic compound. We shall at a
later point meet with examples of a racemic compound changing into a
_dl_-mixture at a definite transition point; and the pseudo-racemic mixed
crystal of camphoroxime is an example of the second transformation.
Although at temperatures in the neighbourhood of the melting point the two
active camphoroximes form only mixed crystals but no compound, a racemic
compound is formed at temperatures below 103°. At this temperature the
inactive pseudo-racemic mixed crystal changes into a racemic compound; and
in the case of the other mixed crystals transformation to racemate and
(excess of) active component also occurs, although at a lower temperature
than in the case of the inactive mixed crystal. Although this behaviour is
one of considerable importance, this brief reference to it must suffice
here.[302]

3. _Alloys._

One of the most important classes of substances in the study of which the
Phase Rule has been of very considerable importance, is that formed by the
mixtures or compounds of metals with one another known as alloys. Although
in the investigation of the nature of these bodies various methods are
employed, one of the most important is the determination of the character
of the freezing-point curve; for from the form of this, valuable
information can, as we have already learned, be {221} obtained regarding
the nature of the solid substances which separate out from the molten
mixture.

Although it is impossible here to discuss fully the experimental results
and the oftentimes very complicated relationships which the study of the
alloys has brought to light, a brief reference to these bodies will be
advisable on account both of the scientific interest and of the industrial
importance attaching to them.[303]

We have already seen that there are three chief types of freezing-point
curves in systems of two components, viz. those obtained when (1) the pure
components crystallize out from the molten mass; (2) the components form
one or more compounds; (3) the components form mixed crystals. In the case
of the metals, representatives of these three classes are also found.

1. _The components separate out in the pure state._

In this case the freezing-point curve is of the simple type, Fig. 63, I.
Such curves have been obtained in the case of a number of pairs of metals,
_e.g._ zinc--cadmium, zinc--aluminium, copper--silver (Heycock and
Neville), tin--zinc, bismuth--lead (Gautier), and in other cases. From
molten mixtures represented by one branch of the freezing-point curve one
of the metals will be deposited; while from mixtures represented by the
other branch, the other metal will separate out. At the eutectic point the
molten mass will solidify to a _heterogeneous mixture_ of the two metals,
forming what is known as the _eutectic alloy_. Such an alloy, therefore,
will melt at a definite temperature lower than the melting point of either
of the pure metals.

{222}

In the following table are given the temperature and the composition of the
liquid at the eutectic point, for three pairs of metals:--

  -------------------------------------------------------------------
                  | Temperature. |      Composition of liquid.
  -------------------------------------------------------------------
  Zinc--cadmium   |    264.5°    | 73.5 atoms per cent. of cadmium.
  Zinc--aluminium |    380.5°    | 11     "      "         aluminium.
  Copper--silver  |    778°      | 40     "      "         copper.
  -------------------------------------------------------------------

The melting points of the pure metals are, zinc, 419°; cadmium, 322°;
silver, 960°; copper, 1081°; aluminium, 650°.

2. _The two metals can form one or more compounds._

In this case there will be obtained not only the freezing-point curves of
the pure metals, but each compound formed will have its own freezing-point
curve, exhibiting a point of maximum temperature, and ending on either side
in an eutectic point. The simplest curve of this type will be obtained when
only one compound is formed, as is the case with mercury and thallium.[304]
This curve is represented in Fig. 74, where the summit of the intermediate
curve corresponds with a composition TlHg_{2}. Similar curves are also
given by nickel and tin, by aluminium and silver, and by other metals, the
formation of definite compounds between these pairs of metals being thereby
indicated.[305]

[Illustration: FIG. 74.]

{223}

A curve belonging to the same type, but more complicated, is obtained with
gold and aluminium;[306] in this case, several compounds are formed, some
of which have a definite melting point, while others exhibit only a
transition point. The chief compound is AuAl_{2}, which has practically the
same melting point as pure gold.

3. _The two metals form mixed crystals (solid solutions)._

The simplest case in which the metals crystallize out together is found in
silver and gold.[307] The freezing-point curve in this case is an almost
straight line joining the freezing points of the pure metals (_cf._ curve
I., Fig. 65, p. 210). These two metals, therefore, can form an unbroken
series of mixed crystals.

In some cases, however, the two metals do not form an unbroken series of
mixed crystals. In the case of zinc and silver,[308] for example, the
addition of silver _raises_ the freezing point of the mixture, until a
transition point is reached. This corresponds with curve IV., Fig. 65.
Silver and copper, and gold and copper, on the other hand, do not form
unbroken series of mixed crystals, but the freezing-point curve exhibits an
eutectic point, as in curve V., Fig. 65.

Not only may there be these three different types of curves, but there may
also be combinations of these. Thus the two metals may not only form
compounds, but one of the metals may not separate out in the pure state at
all, but form mixed crystals. In this case the freezing point may rise (as
in the case of silver and zinc), and one of the eutectic points will be
absent.

Iron-Carbon Alloys.--Of all the different binary alloys, probably the most
important are those formed by iron and carbon: alloys consisting not of two
metals, but of a metal and a non-metal. On account of the importance of
these alloys, an attempt will be made to describe in brief some of the most
important relationships met with.

Before proceeding to discuss the applications of the Phase Rule to the
study of the iron-carbon alloys, however, the main {224} facts with which
we have to deal may be stated very briefly. With regard to the metal
itself, it is known to exist in three different allotropic modifications,
called [alpha]-, [beta]-, and [gamma]-ferrite respectively. Like the two
modifications of sulphur and of tin, these different forms exhibit
transition points at which the relative stability of the forms changes.
Thus the transition point for [alpha]- and [beta]-ferrite is about 780°;
and below this temperature the [alpha]- form, above it the [beta]- form is
stable. For [beta]- and [gamma]-ferrite, the transition point is about
870°, the [gamma]- form being the stable modification above this
temperature.

The different modifications of iron also possess different properties.
Thus, [alpha]-ferrite is magnetic, but does not possess the power of
dissolving carbon; [beta]-ferrite is non-magnetic, and likewise does not
dissolve carbon; [gamma]-ferrite is also non-magnetic, but possesses the
power of dissolving carbon, and of thus giving rise to solid solutions of
carbon in iron.

Various alloys of iron and carbon, also, have to be distinguished. First of
all, there is _hard steel_, which contains varying amounts of carbon up to
2 per cent. Microscopic examination shows that these mixtures are all
homogeneous; and they are therefore to be regarded as solid solutions of
carbon in iron ([gamma]-ferrite). To these solutions the name _martensite_
has been given. _Pearlite_ contains about 0.8 per cent. of carbon, and, on
microscopic examination, is found to be a heterogeneous mixture. If heated
above 670°, pearlite becomes homogeneous, and forms martensite. Lastly,
there is a definite compound of iron and carbon, iron carbide or
_cementite_, having the formula Fe_{3}C.

A short description may now be given of the application of the Phase Rule
to the two-component system iron--carbon; and of the diagram showing how
the different systems are related, and with the help of which the behaviour
of the different mixtures under given conditions can be predicted.
Although, with regard to the main features of this diagram, the different
areas to be mapped and the position of the frontier lines, there is general
agreement; a final decision has not yet been reached with regard to the
interpretation to be put on all the curves.

[Illustration: FIG. 75.]

The chief relationships met with in the case of the {225} iron-carbon
alloys are represented graphically in Fig. 75.[309] The curve AC is the
freezing-point curve for iron,[310] BC the unknown freezing-point curve for
graphite. C is an eutectic point. Suppose, now, that we start with a
mixture of iron and carbon, represented by the point _x_. On lowering the
temperature, a point, _y_, will be reached at which solid begins to
separate out. This solid phase, however, is not pure iron, but a solid
solution of carbon in iron, having the composition represented by _y'_ (cf.
p. 185). As the temperature continues to fall, the {226} composition of the
liquid phase changes in the direction of _y_C, while the composition of the
solid which separates out changes in the direction _y'_D; and, finally,
when the composition of the molten mass is that of the point C (4.3 per
cent. of carbon), the whole mass solidifies to a heterogeneous mixture of
two solid solutions, one of which is represented by D (containing 2 per
cent. of carbon), while the other will consist practically of pure
graphite, and is not shown in the figure. The temperature of the eutectic
point is 1130°.

Even below the solidification point, however, changes can take place. As
has been said, the solid phase which finally separates out from the molten
mass is a solid solution represented by the point D; and the curve DE
represents the change in the composition of this solid solution with the
temperature. As indicated in the figure, DE forms a part of a curve
representing the mutual solubility of graphite in iron and iron in
graphite; the latter solutions, however, not being shown, as they would lie
far outside the diagram. As the temperature falls below 1130°, more and
more graphite separates out, until at E, when the temperature is 1000°, the
solid solution contains only 1.8 per cent. of carbon. At this temperature
cementite also begins to be formed, so that as the temperature continues to
fall, separation of cementite (represented by the line E'F') occurs, and
the composition of the solid solution undergoes alteration, as represented
by the curve EF. Below the temperature of the point F (670°) the martensite
becomes heterogeneous, and forms pearlite.

From the above description, therefore, it follows that if we start with a
molten mixture of iron and carbon, the composition of which is represented
by any point between D and C (from 2 to 4.3 per cent. of carbon), we shall
obtain, on cooling the mass, first of all solid solutions, the composition
of which will be represented by points on the line AD; that then, after the
mass has completely solidified at 1130°, further cooling will lead to a
separation of graphite and a change in the composition of the martensite
(from 2 to 1.8 per cent. of carbon). On cooling below 1000°, however, the
martensite and graphite will give rise to cementite and solid solutions
{227} containing less carbon than before, until, at temperatures below
670°, we are left with a mixture of pearlite and cementite.

We have already said that iron consists in three allotropic modifications,
the regions of stability of which are separated by definite transition
points. The transition point for [alpha]- and [beta]-ferrite (780°) is
represented in Fig. 75 by the point H; and the transition point for [beta]-
and [gamma]-ferrite (870°) by the point I. Since neither the [alpha]- nor
the [beta]-ferrite dissolves carbon, the transition point will be
unaffected by addition of carbon, and we therefore obtain the horizontal
transition curve HG. In the case of the [beta]- and [gamma]-ferrite,
however, the latter dissolves carbon, and the transition point is
consequently affected by the amount of carbon present. This is shown by the
line IG.

If a martensite containing less carbon than that represented by the point G
is cooled down from a temperature of, say, 900°, then when the temperature
has fallen to that, represented by a point on the curve IG, [beta]-ferrite
will separate out, and, as the temperature falls, the composition of the
solid solution will alter as represented by IG. On passing below the
temperature of HG, the [beta]-ferrite will be converted into
[alpha]-ferrite, and, as the temperature falls, the latter will separate
out more and more, while the composition of the solid solution alters in
the direction GF. On passing to still lower temperatures, the solid
solution at F (0.8 per cent. of carbon) breaks up into pearlite. If the
percentage of carbon in the original solid solution was between that
represented by the points G and F, then, on cooling down, no
[beta]-ferrite, but only [alpha]-ferrite would separate out.

We see, therefore, that when martensite is allowed to cool _slowly_, it
yields a heterogeneous mixture either of ferrite and pearlite (when the
original mixture contained up to 0.8 per cent. of carbon), or pearlite and
cementite (when the original mixture contained between 0.8 and 2 per cent.
of carbon). These heterogeneous mixtures constitute soft steels, or, when
the carbon content is low, wrought iron.

The case, however, is different if the solid solution of carbon in iron is
_rapidly_ cooled (quenched) from a temperature above the curve IGFE to a
temperature below this {228} curve. In this case, the rapid cooling does
not allow time for the various changes which have been described to take
place; so that the homogeneous solid solution, on being rapidly cooled,
remains homogeneous. In this way hard steel is obtained. By varying the
rapidity of cooling, as is done in the tempering of steel, varying degrees
of hardness can be obtained.

The interpretation of the curves given above is that due essentially to
Roozeboom, who concluded from the experimental data that at temperatures
below 1000° the stable systems are martensite and cementite, or ferrite and
cementite, graphite being labile. It has, however, been pointed out, more
especially by E. Heyn,[311] that this is not in harmony with the facts of
metallurgy, which show that graphite is undoubtedly formed on slow cooling,
and more especially when small quantities of silicon are present in the
iron.[312] While, therefore, the relationships represented by Fig. 75 are
obtained under certain conditions (especially when manganese is present),
Heyn considers that all the curves in that figure, except ACB, represent
_metastable_ systems--systems, therefore, akin to supercooled liquids.
Rapid cooling will favour the production of the metastable systems
containing cementite, and therefore give rise to relationships represented
by Fig. 75; whereas slow cooling will lead to the stable system ferrite and
graphite. Presence of silicon tends to prevent, presence of manganese tends
to assist, the production of the metastable systems.

Although this view put forward by Heyn has not been conclusively proved, it
must be said that there is much evidence in its favour. Further
investigation is, however, required before a final decision as to the
interpretation of the curves can be reached.

Determination of the Composition of Compounds, without Analysis.--Since the
equilibrium between a solid and a liquid phase depends not only on the
composition of the liquid (solution) but also on that of the solid, it is
necessary {229} to determine the composition of the latter. In some cases
this is easily effected by separating the solid from the liquid phase and
analyzing it. In other cases, however, this method is inapplicable, or is
accompanied by difficulties, due either to the fact that the solid phase
undergoes decomposition (_e.g._ when it contains a volatile constituent),
or to the difficulty of completely separating the mother liquor; as, for
example, in the case of alloys. In all such cases, therefore, recourse must
be had to other methods.

In the first place, synthetic methods may be employed.[313] In this case we
start with a solution of the two components, to which a third substance is
added, which, however, does not enter into the solid phase.[314] We will
assume that the initial solution contains _x_ gm. of A and _y_ gm. of B to
1 gm. of C. After the solution has been cooled down to such a temperature
that solid substance separates out, a portion of the liquid phase is
removed with a pipette and analyzed. If, now, the composition of the
solution is such that there are _x'_ gm. of A and _y'_ gm. of B to 1 gm. of
C., then the composition of the solid phase is _x_ - _x'_ gm. of A and _y_
- _y'_ gm. of B. When _x_ = _x'_, the solid phase is pure B; when _y_ =
_y'_, the solid phase is pure A.

We have assumed here that there is only one solid phase present, containing
A and B. To make sure that the solid phase is not a solid solution in which
A and B are present in the same ratio as in the liquid solution, a second
determination of the composition must be made, with different initial and
end concentrations. If the solid phase is a solid solution, the composition
will now be found different from that found previously.

The composition of the solid phase can, however, be determined in another
manner, viz. by studying the fusion curve and the curve of cooling. From
the form of the fusion curve alone, it is possible to decide whether the
two components {230} form a compound or not; and if the compounds which may
be formed have a definite melting point, the position of the latter gives
at once the composition of the compounds (cf. p. 231).

This method, however, cannot be applied when the compounds undergo
decomposition before the melting point is reached. In such cases, however,
the form of the cooling curve enables one to decide the composition of the
solid phase.[315] If a solution is allowed to cool slowly, and the
temperature noted at definite times, the graphic representation of the rate
of cooling will give a continuous curve; _e.g._ _ab_ in Fig. 76. So soon,
however, as a solid phase begins to be formed, the rate of cooling alters
abruptly, and the cooling curve then exhibits a break, or change in
direction (point _b_). When the eutectic point is reached, the temperature
remains constant, until all the liquid has solidified. This is represented
by the line _cd_. When complete solidification has occurred, the fall of
temperature again becomes uniform (_de_).

[Illustration: FIG. 76.]

[Illustration: FIG. 77.]

[Illustration: FIG. 78.]

The length of time during which the temperature remains constant at the
point _c_, depends, of course, on the eutectic solution. If, therefore, we
take equal amounts of solution having a different initial composition, the
period of constant temperature in the cooling curve will evidently be
greatest in the case of the solution having the composition of the eutectic
point; and the period will become less and less as we increase the amount
of one of the components. The relationship between initial composition of
solution and the duration of constant temperature at the eutectic point is
represented by the curve _a'c'b'_ (Fig. 77). When a compound possessing a
definite melting point is formed, it behaves as a pure substance. If,
therefore, the initial composition of the {231} solution is the same as
that of the compound, no eutectic solution will be obtained; and therefore
no line of constant temperature, such as _cd_ (Fig. 76). In such a case, if
we represent graphically the relation between the initial composition of
the solution and the duration of constant temperature, a diagram is
obtained such as shown in Fig. 78. The two maxima on the time-composition
curve represent eutectic points, and the minima, _a'_, _b'_, _e'_, pure
substances. The position of _e'_ gives the composition of the compound.
When a series of compounds is formed, then for each compound a minimum is
found on the time-composition curve.

[Illustration: FIG. 79.]

If the compound formed has no definite melting point, the diagram obtained
is like that shown in Fig. 79. If we start with a solution, the composition
of which is represented by a point between _d_ and _b_, then, on cooling,
_b_ will separate out first, and the temperature will fall until the point
_d_ is reached. The temperature then remains constant until the component
_b_, which has separated out, is converted into the compound. After this
the temperature again falls, until it again remains constant at the
eutectic point c. In the case of the first halt, the period of constant
temperature is greatest when the initial composition of the solution is the
same as that of the compound; and it becomes shorter and shorter with {232}
increase in the amount of either component. In this way we obtain the
time-composition curve _b'e"d'_, of which the maximum point _e"_ gives the
composition of the compound.

On the other hand, the period of constant temperature for the eutectic
point _c_ is greatest in the case of solutions having the same initial
_composition_ as that corresponding with the eutectic point; and it
decreases the more the initial composition approaches that of the pure
component _a_ or the component e. In this way we obtain the
time-composition curve _a'c'e'_. Here also the point _e'_ represents the
composition of the compound. We see, therefore, that from the graphic
representation of the freezing-point curve, and from the duration of the
temperature-arrests on the cooling curve, for solutions of different
initial composition, it is possible, without having recourse to analysis,
to decide what solid phases are formed, and what is their composition.

Formation of Minerals.--Important and interesting as is the application of
the Phase Rule to the study of alloys, its application to the study of the
conditions regulating the formation of minerals is no less so; and although
we do not propose to consider different cases in detail here, still
attention must be drawn to certain points connected with this interesting
subject.

In the first place, it will be evident from what has already been said,
that that mineral which first crystallizes out from a molten magma is not
necessarily the one with the highest melting point. The _composition_ of
the fused mass must be taken into account. When the system consists of two
components which do not form a compound, one or other of these will
separate out in a pure state, according as the composition of the molten
mass lies on one or other side of the eutectic composition; and the
separation of the one component will continue until the composition of the
eutectic point is reached. Further cooling will then lead to the
simultaneous separation of the two components.

If, however, the two components form a stable compound (_e.g._ orthoclase,
from a fused mixture of silica and potassium aluminate), then the
freezing-point curve will resemble that {233} shown in Fig. 64; _i.e._
there will be a middle curve possessing a dystectic point, and ending on
either side at a eutectic point. This curve would represent the conditions
under which orthoclase is in equilibrium with the molten magma. If the
initial composition of the magma is represented by a point between the two
eutectic points, orthoclase will separate first. The composition of the
magma will thereby change, and the mass will finally solidify to a mixture
of orthoclase and silica, or orthoclase and potassium aluminate, according
to the initial composition.

What has just been said holds, however, only for stable equilibria, and it
must not be forgotten that complications can arise owing to suspended
transformation (when, for example, the magma is rapidly cooled) and the
production of metastable equilibria. These conditions occur very frequently
in nature.

The study of the formation of minerals from the point of view of the Phase
Rule is still in its initial stages, but the results which have already
been obtained give promise of a rich harvest in the future.[316]

       *       *       *       *       *


{234}

CHAPTER XIII

SYSTEMS OF THREE COMPONENTS

General.--It has already been made evident that an increase in the number
of the components from one to two gives rise to a considerable increase in
the possible number of systems, and introduces not a few complications into
the equilibrium relations of these. No less is this the case when the
number of components increases from two to three; and although examples of
all the possible types of systems of three components have not been
investigated, nor, indeed, any one type fully, nevertheless, among the
systems which have been studied experimentally, cases occur which not only
possess a high scientific interest, but are also of great industrial
importance. On account not only of the number, but more especially of the
complexity of the systems constituted of three components, no attempt will
be made to give a full account, or, indeed, even a survey of all the cases
which have been subjected to a more or less complete experimental
investigation; on the contrary, only a few of the more important classes
will be selected, and the most important points in connection with the
behaviour of these described.

On applying the Phase Rule

  P + F = C + 2

to the systems of three components, we see that in order that the system
shall be invariant, no fewer than five phases must be present together, and
an invariant system will therefore exist at a _quintuple_ point. Since the
number of liquid phases can never exceed the number of the components, and
since there can be only one vapour phase, it is evident that in this case,
{235} as in others, there must always be at least one solid phase present
at the quintuple point. As the number of phases diminishes, the variability
of the system can increase from one to four, so that in the last case the
condition of the system will not be completely defined until not only the
temperature and the total pressure of the system, but also the
concentrations of two of the components have been fixed. Or, instead of the
concentrations, the partial pressures of the components may also be taken
as independent variables.

Graphic Representation.--Hitherto the concentrations of the components have
been represented by means of rectangular co-ordinates, although the
numerical relationships have been expressed in two different ways. In the
one case, the concentration of the one component was expressed in terms of
a fixed amount of the other component. Thus, the solubility of a salt was
expressed by the number of grams of salt dissolved by 100 grams of water or
other solvent; and the numbers so obtained were measured along one of the
co-ordinates. The second co-ordinate was then employed to indicate the
change of another independent variable, _e.g._ temperature. In the other
case, the combined weights of the two components A and B were put equal to
unity, and the concentration of the one expressed as a fraction of the
whole amount. This method allows of the representation of the complete
series of concentrations, from pure A to pure B, and was employed, for
example, in the graphic representation of the freezing point curves.

Even in the case of three components rectangular co-ordinates can also be
employed, and, indeed, are the most convenient in those cases where the
behaviour of two of the components to one another is very different from
their behaviour to the third component; as, for example, in the case of two
salts and water. In these cases, the composition of the system can be
represented by measuring the amounts of each of the two components in a
given weight of the third, along two co-ordinates at right angles to one
another; and the change of the system with the temperature can then be
represented by a third axis at right angles to the first two. In those
cases, {236} however, where the three components behave in much the same
manner towards one another, the rectangular co-ordinates are not at all
suitable, and instead of these a _triangular diagram_ is employed. Various
methods have been proposed for the graphic representation of systems of
three components by means of a triangle, but only two of these have been
employed to any considerable extent; and a short description of these two
methods will therefore suffice.[317]

[Illustration: FIG. 80.]

In the method proposed by Gibbs an equilateral triangle of unit height is
used (Fig 80).[318] The quantities of the different components are
expressed as fractional parts of the whole, and the sum of their
concentrations is therefore equal to unity, and can be represented by the
height of the triangle. The corners {237} of the triangle represent the
pure substances A, B, and C respectively. A point on one of the sides of
the triangle will give the composition of a mixture in which only two
components are present, while a point within the triangle will represent
the composition of a ternary mixture. Since every point within the triangle
has the property that the sum of the perpendiculars from that point on the
sides of the triangle is equal to unity (the height of the triangle), it is
evident that the composition of a ternary mixture can be represented by
fixing a point within the triangle such that the lengths of the
_perpendiculars_ from the point to the sides of the triangle are equal
respectively to the fractional amounts of the three components present; the
fractional amount of A, B, or C being represented by the perpendicular
distance from the side of the triangle _opposite_ the corners A, B, and C
respectively.

The location of this point is simplified by dividing the normals from each
of the corners on the opposite side into ten or one hundred parts, and
drawing through these divisions lines at right angles to the normal and
parallel to the side of the triangle. A network of rhombohedra is thus
obtained, and the position of any point can be read off in practically the
same manner as in the case of rectangular co-ordinates. Thus the point P in
Fig. 80 represents a ternary mixture of the composition A = 0.5, B = 0.3, C
= 0.2; the perpendiculars P_a_, P_b_, and P_c_ being equal respectively to
0.5, 0.2, and 0.3 of the height of the triangle.

Another method of representation, due to Roozeboom, consists in employing
an equilateral triangle, the length of whose _side_ is made equal to unity,
or one hundred; the sum of the fractional or percentage amounts of the
three components being represented therefore by a side of the triangle. In
this case the composition of a ternary mixture is obtained by determining,
not the _perpendicular_ distance of a point P from the three sides of the
triangle, but the distance in a direction _parallel_ to the sides of the
triangle (Fig. 81). Conversely, in order to represent a mixture consisting
of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one
side of the triangle, say AB, is first of all divided into ten or one {238}
hundred parts; a portion, B_x_ = _a_, is then measured off, and represents
the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off
and represents the fractional amount of B, while the remainder, _xx'_ =
_c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel
to the sides of the triangle, and the point of intersection, P, represents
the composition of the ternary mixture of given composition; for, as is
evident from the figure, the distance of the point P from the three sides
of the triangle, when measured in directions _parallel_ to the sides, is
equal to _a_, _b_, and _c_ respectively. From the division marks on the
side AB, it is seen that the point P in this figure also represents a
mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives
exactly the same result as the previous method. The employment of a
right-angled isosceles triangle has also been suggested,[319] but is not in
general use.

[Illustration: FIG. 81.]

In employing the triangular diagram, it will be of use to note a property
of the equilateral triangle. A line drawn from one corner of the triangle
to the opposite side, represents the composition of all mixtures in which
the _relative_ amounts of two of the components remain unchanged. Thus, as
Fig. 82 shows, if the component C is added to a mixture x, in which A and B
are present in the proportions of _a_ : _b_, a mixture _x'_, which is
thereby obtained, also contains A and B in the ratio _a_ : b. For the two
triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and
KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_
and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ :
E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore,
the ratio of A to B is the same.

[Illustration: FIG. 82.]

[Illustration: FIG. 83.]

If it is desired to represent at the same time the change of another
independent variable, _e.g._ temperature, this can be done by measuring the
latter along axes drawn perpendicular to the corners of the triangle. In
this way a right prism (Fig. 83) is obtained, and each section of this cut
parallel to the base represents therefore an _isothermal surface_.

       *       *       *       *       *


{240}

CHAPTER XIV

SOLUTIONS OF LIQUIDS IN LIQUIDS

We have already seen (p. 95) that when two liquids are brought together,
they may mix in all proportions and form one homogeneous liquid phase; or,
only partial miscibility may occur, and two phases be formed consisting of
two mutually saturated solutions. In the latter case, the concentration of
the components in either phase and also the vapour pressure of the system
had, at a given temperature, perfectly definite values. In the case of
three liquid components, a similar behaviour may be found, although
complete miscibility of three components with the formation of only one
liquid phase is of much rarer occurrence than in the case of two
components. When only partial miscibility occurs, various cases are met
with according as the three components form one, two, or three pairs of
partially miscible liquids. Further, when two of the components are only
partially miscible, the addition of the third may cause either an increase
or a diminution in the mutual solubility of these. An increase in the
mutual solubility is generally found when the third component dissolves
readily in each of the other two; but when the third component dissolves
only sparingly in the other two, its addition diminishes the mutual
solubility of the latter.

We shall consider here only a few examples illustrating the three chief
cases which can occur, viz. (1) A and B, and also B and C are miscible in
all proportions, while A and C are only partially miscible. (2) A and B are
miscible in all proportions, but A and C and B and C are only partially
miscible. (3) A and B, B and C, and A and C are only partially miscible. A,
B, and C here represent the three components.

1.--_The three components form only one pair of partially miscible
liquids._ {241}

An example of this is found in the three substances: chloroform, water, and
acetic acid.[320] Chloroform and acetic acid, and water and acetic acid,
are miscible with one another in all proportions, but chloroform and water
are only partially miscible with one another. If, therefore, chloroform is
shaken with a larger quantity of water than it can dissolve, two layers
will be formed consisting one of a saturated solution of water in
chloroform, the other of a saturated solution of chloroform in water. The
composition of these two solutions at a temperature of about 18°, will be
represented by the points _a_ and _b_ in Fig. 84; _a_ representing a
solution of the composition: chloroform, 99 per cent.; water, 1 per cent.;
and _b_ a solution of the composition: chloroform, 0.8 per cent.; water,
99.2 per cent. When acetic acid is added, it distributes itself between the
two liquid layers, and two conjugate _ternary_ solutions, consisting of
chloroform, water, and acetic acid are thereby produced which are in
equilibrium with one another, and the composition of which will be
represented by two points inside the triangle. In this way a series of
pairs of ternary solutions will be obtained by the addition of acetic acid
to the mixture of chloroform and water. By this addition, also, not only do
the two liquid phases become increasingly rich in acetic acid, but the
mutual solubility of the chloroform and water increases; so that the layer
_a_ becomes relatively richer in water, and layer _b_ relatively richer in
chloroform. This is seen from the following table, which gives the
percentage composition of different conjugate ternary solutions at 18°.

  -------------------------------------------------------------------------
             Heavier layer.           |            Lighter layer.
  -------------------------------------------------------------------------
  Chloroform. | Water. | Acetic acid. | Chloroform. | Water. | Acetic acid.
  -------------------------------------------------------------------------
     99.01    |  0.99  |     0        |     0.84    | 99.16  |     0
     91.85    |  1.38  |     6.77     |     1.21    | 73.69  |    25.10
     80.00    |  2.28  |    17.72     |     7.30    | 48.58  |    44.12
     70.13    |  4.12  |    25.75     |    15.11    | 34.71  |    50.18
     67.15    |  5.20  |    27.65     |    18.33    | 31.11  |    50.56
     59.99    |  7.93  |    32.08     |    25.20    | 25.39  |    49.41
     55.81    |  9.58  |    34.61     |    28.85    | 23.28  |    47.87
  -------------------------------------------------------------------------

{242}

By the continued addition of acetic acid, the composition of the successive
conjugate solutions in equilibrium with one another becomes, as the table
shows, more nearly the same, and a point is at length reached at which the
two solutions become identical. This will therefore be a _critical point_
(p. 98). Increased addition of acetic acid beyond this point will lead to a
single homogeneous solution.

These relationships are represented graphically by the curve _a_K_b_, Fig.
84. The points on the branch _a_K represent the composition of the
solutions relatively rich in chloroform (heavier layer), those on the curve
_b_K the composition of solutions relatively rich in water (lighter layer);
and the points on these two branches representing conjugate solutions are
joined together by "tie-lines." Thus, the points _a'b'_ represent conjugate
solutions, and the line _a'b'_ is a tie-line.

[Illustration: FIG. 84.]

Since, now, acetic acid when added to a heterogeneous mixture of chloroform
and water does not enter in equal amounts into the two layers, but in
amounts depending on its coefficient of distribution between chloroform and
water,[321] the {243} tie-lines will not be parallel to AB, but will be
inclined at an angle. As the solutions become more nearly the same, the
tie-lines diminish in length, and at last, when the conjugate solutions
become identical, shrink to a point. For the reason that the tie-lines are,
in general, not parallel to the side of the triangle, the critical point at
which the tie-line vanishes will not be at the summit of the curve, but
somewhere below this, as represented by the point K.

The curve _a_K_b_, further, forms the boundary between the heterogeneous
and homogeneous systems. A mixture of chloroform, water, and acetic acid
represented by any point outside the curve _a_K_b_, will form only one
homogeneous phase; while any mixture represented by a point within the
curve, will separate into two layers having the composition represented by
the ends of the tie-line passing through that point. Thus, a mixture of the
total composition _x_, will separate into two layers having the composition
_a'_ and _b'_ respectively.

Since three components existing in three phases (two liquid and a vapour
phase) constitute a bivariant system, the final result, _i.e._ the
composition of the two layers and the total vapour pressure, will not
depend merely on the temperature, as in the case of two-component systems
(p. 102), but also on the composition of the mixture with which we start.
At constant temperature, however, all mixtures, the composition of which is
represented by a point on one and the same tie-line, will separate into the
same two liquid phases, although the relative _amounts_ of the two phases
will vary. If we omit the vapour phase, the condition of the system will
depend on the pressure as well as on the temperature and composition of the
initial mixture. By keeping the pressure constant, _e.g._ at atmospheric
pressure (by working with open vessels), the system again becomes
bivariant. We see, therefore, that the position of the curve _a_K_b_, or,
in other words, the composition of the different conjugate ternary
solutions, will vary with the temperature, and only with the temperature,
if we assume either constancy of pressure or the presence of the vapour
phase. Since at the critical point the condition is imposed that the two
liquid phases become identical, one degree of freedom is thereby {244}
lost, and therefore only one degree of freedom remains. The critical point,
therefore, depends on the temperature, and only on the temperature; always
on the assumption, of course, that the pressure is constant, or that a
vapour phase is present. Fig. 84, therefore, represents an isothermal
(p. 239).

It is of importance to note that the composition of the different ternary
solutions obtained by the addition of acetic acid to a heterogeneous
mixture of chloroform and water, will depend not only on the amount of
acetic acid added, but also on the relative amounts of chloroform and water
at the commencement. Suppose, for example, that we start with chloroform
and water in the proportions represented by the point _c'_ (Fig. 84). On
mixing these, two liquid layers having the composition _a_ and _b_
respectively will be formed. Since by the addition of acetic acid the
relative amounts of these two substances in the system as a whole cannot
undergo alteration, the total composition of the different ternary systems
which will be obtained must be represented by a point on the line C_c'_
(p. 238). Thus, for example, by the addition of acetic acid a system may be
obtained, the total composition of which is represented by the point _c"_.
Such a system, however, will separate into two conjugate ternary solutions,
the composition of which will be represented by the ends of the tie-line
passing through the point _c"_. So long as the total composition of the
system lies below the point S, _i.e._ the point of intersection of the line
C_c'_ with the boundary curve, two liquid layers will be formed; while all
systems having a total composition represented by a point on the line
C_c'_, above S, will form only one homogeneous solution.

From the figure, also, it is evident that as the amount of acetic acid is
increased, the relative amounts of the two liquid layers formed differ more
and more until at S a limiting position is reached, when the amount of the
one liquid layer dwindles to nought, and only one solution remains.

The same reasoning can be carried through for different initial amounts of
chloroform and water, but it would be fruitless to discuss all the
different systems which can be obtained. The reason for the preceding
discussion was to show that {245} although the addition of acetic acid to a
mixture of chloroform and water will, in all cases, lead ultimately to a
limiting system, beyond which homogeneity occurs, that point is not
necessarily the critical point. On the contrary, in order that addition of
acetic acid shall lead to the critical mixture, it is necessary to start
with a binary mixture of chloroform and water in the proportions
represented by the point _c'_. In this case, addition of acetic acid will
give rise to a series of conjugate ternary solutions, the composition of
which will gradually approach to one another, and at last become identical.

From the foregoing it will be evident that the amount of acetic acid
required to produce a homogenous solution, will depend on the relative
amounts of chloroform and water from which we start, and can be ascertained
by joining the corner C with the point on the line AB representing the
total composition of the initial binary system. The point where this line
intersects the boundary curve _a_K_b_ will indicate the minimum amount of
acetic acid which, under these particular conditions, is necessary to give
one homogeneous solution.

Retrograde Solubility.--As a consequence of the fact that acetic acid
distributes itself unequally between chloroform and water, and the critical
point K, therefore, does not lie at the summit of the curve, it is possible
to start with a homogeneous solution in which the percentage amount of
acetic acid is greater than at the critical point, and to pass from this
first to a heterogenous and then again to a homogenous system merely by
altering the relative amounts of chloroform and water. This phenomenon, to
which the term _retrograde solubility_ is applied, will be observed not
only in the case of chloroform, water, and acetic acid, but in all other
systems in which the critical point lies below the highest point of the
boundary curve for heterogeneous systems. This will be seen from the
diagram, Fig. 85. Starting with the homogeneous system represented by _x_,
in which, therefore, the concentration of C is greater than in the critical
mixture (K), if the relative amounts of A and B are altered in the
direction _xx'_, while the amount of C is maintained constant, the system
will become heterogeneous when the composition reaches the point _y_, and
will remain {246} heterogeneous with changing composition until the point
_y'_ is passed, when it will again become homogeneous. If the relative
concentration of C is increased above that represented by the line SS, this
phenomenon will, of course, no longer be observed.

[Illustration: FIG. 85.]

Relationships similar to those described for chloroform, water, and acetic
acid are also found in the case of a number of other trios, _e.g._ ether,
water, and alcohol; chloroform, water, and alcohol.[322] They have also
been observed in the case of a considerable number of molten metals.[323]
Thus, molten lead and silver, as well as molten zinc and silver, mix in all
proportions; but molten lead and zinc are only partially miscible with one
another. When melted together, therefore, the last two metals will separate
into two liquid layers, one rich in lead, the other rich in zinc. If silver
is now added, and the temperature maintained above the freezing point of
the mixture, the silver passes for the most part, in accordance with the
law of distribution, into the upper layer, which is rich in zinc; silver
being more soluble in molten zinc than in molten lead. This is clearly
shown by the following figures:--[324]

{247}

  --------------------------------------------------
       Heavier alloy.     |     Lighter alloy.
  --------------------------------------------------
    Percentage amount of  |  Percentage amount of
  Silver. | Lead. | Zinc. | Silver. | Lead. | Zinc.
  --------------------------------------------------
    1.25  | 96.69 |  2.06 |  38.91  |  3.12 | 57.97
    1.71  | 96.43 |  1.86 |  45.01  |  3.37 | 51.62
    5.55  | 93.16 |  1.29 |  54.93  |  4.21 | 40.86
  --------------------------------------------------

The numbers in the same horizontal row give the composition of the
conjugate alloys, and it is evident that the upper layer consists almost
entirely of silver and zinc. On allowing the mixture to cool slightly, the
upper layer solidifies first, and can be separated from the still molten
lead layer. It is on this behaviour of silver towards a mixture of molten
lead and zinc that the Parkes's method for the desilverization of lead
depends.[325] If aluminium is also added, a still larger proportion of
silver passes into the lighter layer, and the desilverization of the lead
is more complete.[326]

[Illustration: FIG. 86.]

[Illustration: FIG. 87.]

The Influence of Temperature.--As has already been said, a ternary system
existing in three phases possesses two degrees of freedom; and the state of
the system is therefore dependent not only on the relative concentration of
the components, but also on the temperature. As the temperature changes,
therefore, the boundary curve of the heterogeneous system will also alter;
and in order to represent this alteration we shall make use of the right
prism, in which the temperature is measured upwards. In this way the
boundary curve passes into a boundary surface (called a dineric surface),
as shown in Fig. 86. In this figure the curve _akb_ is the isothermal for
the ternary system; the curve _a_K_b_ shows the change in the _binary_
system AB with the temperature, with {248} a critical point at K. This
curve has the same meaning as those given in Chapter VI. The curve _k_K is
a critical curve joining together the critical points of the different
isothermals. In such a case as is shown in Fig. 86, there does not exist
any real critical temperature for the ternary system, for as the
temperature is raised, the amount of C in the "critical" solution becomes
less and less, and at K only two components, A and B, are present. In the
case, however, represented in Fig. 87, a real ternary critical point is
found. In this figure _ak'b_ is an isothermal, _ak"_ is the curve for the
binary system, and K is the ternary critical point. All points outside the
helmet-shaped boundary surface represent homogeneous ternary solutions,
while all points within the surface belong to heterogeneous systems. Above
the temperature of the point K, the three components are miscible in all
proportions. An example of a ternary system yielding such a boundary
surface is that consisting of phenol, water, and acetone.[327] In this case
the critical temperature K is 92°, and the composition at this ternary
critical point is--

  Water   59 per cent.
  Acetone 12    "
  Phenol  29    "

[Illustration: FIG. 88.]

The difference between the two classes of systems just mentioned, is seen
very clearly by a glance at the Figs. 88 and 89, which show the projection
of the isothermals on the base of the prism. In Fig. 88, the projections
yield paraboloid curves, the two branches of which are cut by one side of
the triangle; and the critical point is represented by a point on {249}
this side. In the second case (Fig. 89), however, the projections of the
isothermals form ellipsoidal curves surrounding the supreme critical point,
which now lies _inside the triangle_. At lower temperatures, these
isothermal boundary curves are cut by a side of the triangle; at the
critical temperature, _k"_, of the binary system AB, the boundary curve
_touches_ the side AB, while at still higher temperatures the boundary
curve comes to lie entirely within the triangle. At any given temperature,
therefore, between the critical point of the binary system (_k"_), and the
supreme critical point of the ternary system (K), each pair of the three
components are miscible with one another in all proportions; for the region
of heterogeneous systems is now bounded by a closed curve lying entirely
within the triangle. Outside this curve only homogeneous systems are found.
Binary mixtures, therefore, represented by any point on one of the sides of
the triangle must be homogeneous, for they all lie outside the boundary
curve for heterogeneous states.

[Illustration: FIG. 89.]

2. _The three components can form two pairs of partially miscible liquids._

In the case of the three components water, alcohol, and succinic nitrile,
water and alcohol are miscible in all proportions, but not so water and
succinic nitrile, or alcohol and succinic nitrile.

[Illustration: FIG. 90.]

[Illustration: FIG. 91.]

As we have already seen (p. 122), water and succinic nitrile can form two
liquid layers between the temperatures 18.5° and 55.5°; while alcohol and
nitrile can form two liquid layers between 13° and 31°. If, then, between
these two temperature limits, alcohol is added to a heterogeneous mixture
of water and nitrile, or water is added to a mixture of alcohol and
nitrile, two heterogeneous ternary systems will be formed, {250} and two
boundary curves will be obtained in the triangular diagram, as shown in
Fig. 90.[328] On changing the temperature, the boundary curves will also
undergo alteration, in a manner similar to that just discussed. As the
temperature falls, the two curves will spread out more and more into the
centre of the triangle, and might at last meet one another; while at still
lower temperatures we may imagine the curves still further expanding so
that the two heterogeneous regions flow into one another and form a _band_
on the triangular diagram (Fig. 91). This, certainly, has not been realized
in the case of the three components mentioned, because at a temperature
higher than that at which the two heterogeneous regions could fuse
together, solid separates out.

[Illustration: FIG. 92.]

The gradual expansion of a paraboloid into a band-like area of
heterogeneous ternary systems, has, however, been observed in the case of
water, phenol, and aniline.[329] In Fig. 92 are shown three isothermals,
viz. those for 148°, 95°, and 50°. At 148°, water and aniline form two
layers having the composition--

  Water,   83.5 per cent. }     { water,   20 per cent.
                          } and {
  Aniline, 16.5    "      }     { aniline, 80    "

{251}

and the critical point _k'_ has the composition--

  Water, 65; phenol, 13.2; aniline, 21.8 per cent.

At 95°, the composition of the two binary solutions is--

  Water,  93 per cent. }     { water     8 per cent.
                       } and {
  Aniline, 7    "      }     { aniline, 92    "

while the point _k"_ has the composition

  Water, 69.9; phenol, 26.6; aniline, 3.5 per cent.

At 50°, the region of heterogeneous states now forms a band, and the two
layers formed by water and aniline have the composition--

  Water,  96.5 per cent. }     { water,    5.5 per cent.
                         } and {
  Aniline, 3.5    "      }     { aniline, 94.5    "

while the two layers formed by water and phenol have the composition--

  Water,  89 per cent.}     { water,  38 per cent.
                      } and {
  Phenol, 11    "     }     { phenol, 62    "

All mixtures of water, phenol, and aniline, therefore, the composition of
which is represented by any point within the band _abcd_, will form two
ternary solutions; while if the composition is represented by a point
outside the band, only one homogeneous solution will be produced.

3. _The three components form three pairs of partially miscible liquids._

[Illustration: FIG. 93.]

The third chief case which can occur is that no two of the components are
completely miscible with one another. In this case, therefore, we shall
obtain three paraboloid boundary curves, as shown in Fig. 93. If, now, we
imagine these three curves to expand in towards the centre of the triangle,
as might happen, for example, by lowering the temperature, a point will
{252} be reached at which the curves partly overlap, and we shall get the
appearance shown in Fig. 94.

The points _a_, _b_, and _c_ represent the points where the three curves
cut, and the triangle _abc_ is a region where the curves overlap. From this
diagram we can see that any mixture having a composition represented by a
point in one of the clear spaces at the corners of the larger triangle,
will form a homogeneous solution; if the composition corresponds to any
point lying in one of the quadrilateral regions _x__{1}, _x__{2} or
_x__{3}, two ternary solutions will be formed; while, if the composition is
represented by any point in the inner triangle, separation into three
layers will occur.

[Illustration: FIG. 94.]

Since in the clear regions at the corners of the triangle we have three
components in two phases, liquid and vapour, the systems have three degrees
of freedom. At constant temperature, therefore, the condition of the system
is not defined until the concentrations of two of the components are fixed.
A system belonging to one of the quadrilateral spaces has, as we have seen,
two degrees of freedom; besides the temperature, one concentration must be
fixed. Lastly, a system the composition of which falls within the inner
triangle _abc_, will form three layers, and will therefore possess only one
degree of freedom. If the temperature is fixed, the composition of the
three layers is also determined, viz. that of the points _a_, _b_, and _c_
respectively; and a change in the composition of the original mixture can
lead only to a difference in the relative amounts of the three layers, not
to a difference in their composition.

An example of a system which can form three liquid phases is found in
water, ether, and succinic nitrile.[330]

       *       *       *       *       *


{253}

CHAPTER XV

PRESENCE OF SOLID PHASES

A. The Ternary Eutectic Point.--In passing to the consideration of those
ternary systems in which one or more solid phases can exist together with
one liquid phase, we shall first discuss not the solubility curves, as in
the case of two-component systems, but the simpler relationships met with
at the freezing point. That is, we shall first of all examine the freezing
point curves of ternary systems.

[Illustration: FIG. 95.]

Since it is necessary to take into account not only the changing
composition of the liquid phase, but also the variation of the temperature,
we shall employ the right prism for the graphic representation of the
systems, as shown in Fig. 95. A, B, and C in this figure, therefore, denote
the melting points of the pure components. If we start with the component A
at its melting point, and add B, which is capable of dissolving in liquid
A, the freezing point of A will be lowered; and, similarly, the freezing
point of B by addition of A. In this way we get the freezing point curve
A_k__{1}B for the binary system; _k__{1}; being an eutectic point. This
curve will of course lie in the plane formed by one face of the prism. In a
similar manner we obtain the freezing point curves A_k__{2}C and B_k__{3}C.
These curves give the composition of the binary liquid phases in
equilibrium {254} with one of the pure components, or at the eutectic
points, with a mixture of two solid components. If, now, to the system
represented say by the point _k__{1}, a small quantity of the third
component, C, is added, the temperature at which the two solid phases A and
B can exist in equilibrium with the liquid phase is lowered; and this
depression of the eutectic point is all the greater the larger the addition
of C. In this way we obtain the curve _k__{1}K, which slopes inwards and
downwards, and indicates the varying composition of the ternary liquid
phase with which a mixture of solid A and B are in equilibrium. Similarly,
the curves _k__{2}K and _k__{3}K are the corresponding eutectic curves for
A and C, and B and C in equilibrium with ternary solutions. At the point K,
the three solid components are in equilibrium with the liquid phase; and
this point, therefore, represents _the lowest temperature attainable with
the three components given_. Each of the ternary eutectic curves, as they
may be called, is produced by the intersection of two surfaces, while at
the ternary eutectic point, three surfaces, viz. A_k__{1}K_k__{2},
B_k__{1}K_k__{3}, and C_k__{1}K_k__{3} intersect. Any point on one of these
surfaces represents a ternary solution in equilibrium with only one
component in the solid state; the lines or curves of intersection of these
represent equilibria with two solid phases, while at the point K, the
ternary eutectic point, there are three solid phases in equilibrium with a
liquid and a vapour phase. The surfaces just mentioned represent bivariant
systems. One component in the solid state can exist in equilibrium with a
ternary liquid phase under varying conditions of temperature and
concentration of the components in the solution; and before the state of
the system is defined, these two variables, temperature and composition of
the liquid phase, must be fixed. On the other hand, the curves formed by
the intersection of these planes represent univariant systems; at a given
temperature two solid phases can exist in equilibrium with a ternary
solution, only when the latter has a definite composition. Lastly, the
ternary eutectic point, K, represents an invariant system; three solid
phases can exist in equilibrium with a ternary solution, only when the
latter has one fixed composition and when the temperature has a definite
value. This eutectic point, therefore, {255} has a perfectly definite
position, depending only on the nature of the three components.

Instead of employing the prism, the change in the composition of the
ternary solutions can also be indicated by means of the _projections_ of
the curves _k__{1}K, _k__{2}K, and _k__{3}K on the base of the prism, the
particular temperature being written beside the different eutectic points
and curves. This is shown in Fig. 96.

[Illustration: FIG. 96.]

The numbers which are given in this diagram refer to the eutectic points
for the system bismuth--lead--tin, the data for which are as
follows:--[331]

  --------------------------------------------------------------------
  Melting point of | Percentage composition of | Temperature of binary
     pure  metal.  |  binary eutectic mixture. |    eutectic point.
  --------------------------------------------------------------------
                   |      Bi    Pb    Sn       |
    Bismuth, 268°  |      55    45    --       |      Bi--Pb, 127°
    Lead, 325°     |      58    --    42       |      Bi--Sn, 133°
    Tin, 232°      |      --    37    63       |      Pb--Sn, 182°
  --------------------------------------------------------------------

  --------------------------------------------------
  Percentage composition of | Temperature of ternary
  ternary eutectic mixture. |    eutectic point.
  --------------------------------------------------
       Bi    Pb    Sn       |
       52    32    16       |          96°
  --------------------------------------------------

Formation of Compounds.--In the case just discussed, the components
crystallized out from solution in the pure state. If, however, combination
can take place between two of the components, the relationships will be
somewhat different; the curves which are obtained in such a case being
represented in Fig. 97. From the figure, we see that the two components B
{256} and C form a compound, and the freezing point curve of the binary
system has therefore the form shown in Fig. 64 (p. 209). Further, there are
two _ternary_ eutectic points, K_{1} and K_{2}, the solid phases present
being A, B, and compound, and A, C, and compound respectively.

[Illustration: FIG. 97.]

The particular point, now, to which it is desired to draw attention is
this. Suppose the ternary eutectic curves projected on a plane parallel to
the face of the prism containing B and C, _i.e._ suppose the concentrations
of the two components B and C, between which interaction can occur,
expressed in terms of a constant amount of the third component A,[332]
curves will then be obtained which are in every respect analogous to the
freezing point curves of binary systems. Thus, suppose the eutectic curves
_k__{1}K and _k__{2}K in Fig. 95 projected on the face BC of the prism,
then evidently a curve will be obtained consisting of two branches
meeting in an eutectic point. On the other hand, the projection of the
ternary eutectic curves in Fig. 97 on the face BC of the prism, will
give a curve consisting of three portions, as shown by the outline
_k__{1}K_{1}K_{2}_k__{2} in Fig. 97.

Various examples of this have been studied, and the following table
contains some of the data for the system ethylene bromide (A), picric acid
(B), and [beta]-naphthol (C), obtained by Bruni.[333]

{257}

  -------------------------------------------------------------------------
                     | Temperature |            Solid phases present.
  -------------------------------------------------------------------------
  Point _k__{1}      |    9.41°    | Ethylene bromide, picric acid.
  Curve _k__{1}K_{1} |     --      |         "               "
  Point K_{1}        |    9.32°    | Ethylene bromide, picric acid, and
                     |             |  [beta]-naphthol picrate.
  Curve K_{1}D'K_{2} |     --      | Ethylene bromide,
                     |             |               [beta]-naphthol picrate.
  Point D'           |    9.75°    |     "       "         "        "
  Point K_{2}        |    8.89°    |     "       "     [beta]-naphthol,
                     |             |                           and picrate.
  Curve K_{2}_k__{2} |     --      |     "       "     [beta]-naphthol.
  Point _k__{2}      |    9.04°    |     "       "               "
  -------------------------------------------------------------------------

From what has been said, it will be apparent that if the ternary eutectic
curve of a three-component system (in which one of the components is
present in constant amount) is determined, it will be possible to state,
from the form of curve obtained, whether or not the two components present
in varying amount crystallize out pure or combine with one another to form
a compound. It may be left to the reader to work out the curves for the
other possible systems; but it will be apparent, that the projections of
the ternary eutectic curves in the manner given will yield a series of
curves alike in all points to the binary curves given in Figs. 63-65,
pp. 208-210.

Since, from the method of investigation, the temperatures of the eutectic
curves will depend on the melting point of the third component (A), it is
possible, by employing substances with widely differing melting points, to
investigate the interaction of the two components (_e.g._ two optical
antipodes) B and C over a range of temperature; and thus determine the
range of stability of the compound, if one is formed. Since, in some cases,
two substances which at one temperature form mixed crystals combine at
another temperature to form a definite compound, the relationships which
have just been described can be employed, and indeed, have been employed,
to determine the temperature at which this change occurs.[334] By means of
this method, Adriani found that below 103° _i_-camphoroxime exists as a
racemic compound, while above {258} that temperature it occurs as a racemic
mixed crystal[335] (_cf._ p. 219).

B. Equilibria at Higher Temperatures. Formation of Double Salts.--After
having studied the relationships which are found in the neighbourhood of
the freezing points of the components, we now pass to the discussion of the
equilibria which are met with at higher temperatures. In this connection we
shall confine the discussion entirely to the systems formed of two salts
and water, dealing more particularly with those cases in which the water is
present in relatively large amount and acts as solvent. Further, in
studying these systems, one restriction must be made, viz. that the single
salts are salts either of the same base or of the same acid; or are, in
other words, capable of yielding a common ion in solution. Such a
restriction is necessary, because otherwise the system would be one not of
three but of four components.[336]

Transition Point.--As is very well known, there exist a number of hydrated
salts which, on being heated, undergo apparent partial fusion; and in
Chapter V. the behaviour of such hydrates was more fully studied in the
light of the Phase Rule. Glauber's salt, or sodium sulphate decahydrate,
for example, on being heated to a temperature of about 32.5°, partially
liquefies, owing to the fact that the water of crystallization is split off
and anhydrous sodium sulphate formed, as shown by the equation--

  Na_{2}SO_{4},10H_{2}O = Na_{2}SO_{4} + 10H_{2}O

The temperature of 32.5°, it was learned, constituted a _transition point_
for the decahydrate and anhydrous salt plus water; decomposition of the
hydrated salt occurring above this temperature, combination of the
anhydrous salt and water below it.

Analogous phenomena are met with in systems constituted of two salts and
water in which the formation of double salts can take place. Thus, for
example, if _d_-sodium potassium {259} tartrate is heated to above 55°,
apparent partial fusion occurs, and the two single salts, _d_-sodium
tartrate and _d_-potassium tartrate, are deposited, the change which occurs
being represented by the equation--

  4NaKC_{4}O_{6}H_{4},4H_{2}O = 2Na_{2}C_{4}O_{6}H_{4},2H_{2}O
                              + 2K_{2}C_{4}O_{6}H_{4},½H_{2}O + 11H_{2}O

On the other hand, if sodium and potassium tartrates are mixed with water
in the proportions shown on the right side of the equation, the system will
remain partially liquid so long as the temperature is maintained above 55°
(in a closed vessel to prevent loss of water), but on allowing the
temperature to fall below this point, complete solidification will ensue,
owing to the formation of the hydrated double salt. Below 55°, therefore,
the hydrated double salt is the stable system, while above this temperature
the two single salts plus saturated solution are stable.[337]

A similar behaviour is found in the case of the double salt copper
dipotassium chloride (CuCl_{2},2KCl,2H_{2}O or CuK_{2}Cl_{4},2H_{2}O).[338]
When this salt is heated to 92°, partial liquefaction occurs, and the
original blue plate-shaped crystals give place to brown crystalline needles
and white cubes; while on allowing the temperature to fall, re-formation of
the blue double salt ensues. The temperature 92° is, therefore, a
transition point at which the reversible reaction--

  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O

takes place.

The decomposition of sodium potassium tartrate, or of copper dipotassium
chloride, differs in so far from that of Glauber's salt that _two_ new
solid phases are formed; and in the case of copper dipotassium chloride,
one of the decomposition products is itself a double salt.

In the two examples of double salt decomposition which have just been
mentioned, sufficient water was yielded to cause a partial liquefaction;
but other cases are known where this is not so. Thus, when copper calcium
acetate is heated to a {260} temperature of 75°, although decomposition of
the double salt into the two single salts occurs as represented by the
equation[339]--

  CuCa(C_{2}H_{3}O_{2})_{4},8H_{2}O = Cu(C_{2}H_{3}O_{2})_{2},H_{2}O
                                    + Ca(C_{2}H_{3}O_{2})_{2},H_{2}O
                                    + 6H_{2}O

the amount of water split off is insufficient to give the appearance of
partial fusion, and, therefore, only a change in the crystals is observed.

The preceding examples, in which decomposition of the double salt was
effected by a rise of temperature, were chosen for first consideration as
being more analogous to the case of Glauber's salt; but not a few examples
are known where the reverse change takes place, formation of the double
salt occurring _above_ the transition point, and decomposition into the
constituent salts below it. Instances of this behaviour are found in the
case of the formation of astracanite from sodium and magnesium sulphates,
and of sodium ammonium racemate from the two sodium ammonium tartrates, to
which reference will be made later. Between these various systems, however,
there is no essential difference; and whether decomposition or formation of
the double salt occurs at temperatures above the transition point, will of
course depend on the heat of change at that point. For, in accordance with
van't Hoff's law of movable equilibrium (p. 58), that change will take
place at the higher temperature which is accompanied by an absorption of
heat. If, therefore, the formation of the double salt from the single salts
is accompanied by an absorption of heat, the double salt will be formed
from the single salts on raising the temperature; but if the reverse is the
case, then the double salt on being heated will decompose into the
constituent salts.[340]

In those cases, now, which have so far been studied, the change at the
transition point is accompanied by a taking up or a splitting off of water;
and _in such cases the general rule can be given, that if the water of
crystallization of the two constituent {261} salts together is greater than
that of the double salt, the latter will be produced from the former on
raising the temperature_ (_e.g._ astracanite from sodium and magnesium
sulphates); _but if the double salt contains more water of crystallization
than the two single salts, increase of temperature will effect the
decomposition of the double salt_. When we seek for the connection between
this rule and the law of van't Hoff, it is found in the fact that the heat
effect involved in the hydration or dehydration of the salts is much
greater than that of the other changes which occur, and determines,
therefore, the sign of the total heat effect.[341]

Vapour Pressure. Quintuple Point.--In the case of Glauber's salt, we saw
that at a certain temperature the vapour pressure curve of the hydrated
salt cut that of the saturated solution of anhydrous sodium sulphate. That
point, it will be remembered, was a quadruple point at which the four
phases sodium sulphate decahydrate, anhydrous sodium sulphate, solution,
and vapour, could co-exist; and was also the point of intersection of the
curves for four univariant systems. In the case of the formation of double
salts, similar relationships are met with; and also certain differences,
due to the fact that we are now dealing with systems of three components.
Two cases will be chosen here for brief description, one in which
formation, the other in which decomposition of the double salt occurs with
rise of temperature.

On heating a mixture of sodium sulphate decahydrate and magnesium sulphate
heptahydrate, it is found that at 22° partial liquefaction occurs with
formation of astracanite. At this temperature, therefore, there can coexist
the five phases--

  Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O;
  solution; vapour.

This constitutes, therefore, a _quintuple point_; and since there are three
components present in five phases, the system is invariant. This point,
also, will be the point of intersection of curves for five univariant
systems, which, in this case, must each be composed of four phases. These
systems are--

{262}

    I. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
    Na_{2}Mg(SO_{4})_{2},4H_{2}O; vapour.

    II. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; solution; vapour.

    III. MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution; vapour.

    IV. Na_{2}SO_{4},10H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution;
    vapour.

    V. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
    Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution.

[Illustration: FIG. 98.]

On representing the vapour pressures of these different systems
graphically, a diagram is obtained such as is shown in Fig. 98,[342] the
curves being numbered in accordance with the above list. When the system I.
is heated, the vapour pressure increases until at the quintuple point the
liquid phase (solution) is formed, and it will then depend on the relative
amounts of the different phases whether on further heating there is formed
system III., IV., or V. If either of the first two is produced, we shall
obtain the vapour pressure of the solutions saturated with respect to both
double salt and one of the single salts; while if the vapour phase
disappears, there will be obtained the pressure of the condensed systems
formed of double salt, two single salts and solution. This curve,
therefore, indicates the _change of the transition point with pressure_;
and since in the ordinary determinations of the transition point in open
vessels, we are in reality dealing with condensed systems under the
pressure of 1 atm., it will be evident that the transition point does not
accurately coincide with the quintuple point (at which the system is under
the pressure of its own vapour). As in the case of other condensed systems,
however, pressure has only a slight influence on the temperature of the
transition point. Whether or not pressure raises or lowers the transition
point will depend on whether transformation is accompanied by an increase
or {263} diminution of volume (theorem of Le Chatelier, p. 58). In the case
of the formation of astracanite, expansion occurs, and the transition point
will therefore be raised by increase of pressure. Although measurements
have not been made in the case of this system, the existence of such a
curve has been experimentally verified in the case of copper and calcium
acetates and water (v. _infra_).[343]

[Illustration: FIG. 99.]

The vapour pressure diagram in the case of copper calcium acetate and water
(Fig. 99), is almost the reverse of that already discussed. In this case,
the double salt decomposes on heating, and the decomposition is accompanied
by a contraction. Curve I. is the vapour pressure curve for double salt,
two single salts (p. 260), and vapour; curves II. and III. give the vapour
pressures of solutions saturated with respect to double salt and one of the
single salts; curve IV. is the curve of pressures for the solutions
saturated with respect to the two single salts; while curve V. again
represents the change of the transition point with pressure. On examining
this diagram, it is seen that whereas {264} astracanite could exist both
above and below the quintuple point, copper calcium acetate can exist only
_below_ the quintuple point. This behaviour is found only in those cases in
which the double salt is decomposed by rise of temperature, and where the
decomposition is accompanied by a diminution of volume.[344]

As already mentioned, the decomposition of copper calcium acetate into the
single salts and saturated solution is accompanied by a contraction, and it
was therefore to be expected that increase of pressure would _lower_ the
transition point. This expectation of theory was confirmed by experiment,
for van't Hoff and Spring found that although the transition point under
atmospheric pressure is about 75°, decomposition of the double salt took
place even at the ordinary temperature when the pressure was increased to
6000 atm.[345]

Solubility Curves at the Transition Point.--At the transition point, as has
already been shown, the double salt and the two constituent salts can exist
in equilibrium with the same solution. The transition point, therefore,
must be the point of intersection of two solubility curves; the solubility
curve of the double salt and the solubility curve of the mixtures of the
two constituent salts. It should be noted here that we are not dealing with
the solubility curves of the single salts separately, for since the systems
are composed of three components, a single solid phase can, at a given
temperature, be in equilibrium with solutions of different composition, and
two solid phases in contact with solution (and vapour) are therefore
necessary to give an univariant system. The same applies, of course, to the
solubility of the double salt; for a double salt also constitutes a single
phase, and can therefore exist in equilibrium with solutions of varying
composition. If, however, we make the restriction (which we do for the
present) that the double salt is not decomposed by water, then the solution
will contain the constituent salts in the same relative proportions as they
are contained in the double salt, and the system may therefore be regarded
as one of _two_ components, viz. double salt and water. In this case one
solid phase is sufficient, with solution and {265} vapour, to give an
univariant system; and at a given temperature, therefore, the solubility
will have a perfectly definite value.

Since in almost all cases the solubility is determined in open vessels, we
shall in the following discussion consider that the vapour phase is absent,
and that the system is under a constant pressure, that of the atmosphere.
With this restriction, therefore, four phases will constitute an invariant
system, three phases an univariant, and two phases a bivariant system.

It has already been learned that in the case of sodium sulphate and water,
the solubility curve of the salt undergoes a sudden change in direction at
the transition point, and that this is accompanied by a change in the solid
phase in equilibrium with the solution. The same behaviour is also found in
the case of double salts. To illustrate this, we shall briefly discuss the
solubility relations of a few double salts, beginning with one of the
simplest cases, that of the formation of rubidium racemate from rubidium
_d_- and _l_-tartrates. The solubilities are represented diagrammatically
in Fig. 100, the numerical data being contained in the following table, in
which the solubility is expressed as the number of gram-molecules
Rb_{2}C_{4}H_{4}O_{6} in 100 gm.-molecules of water.[346]

  ---------------------------------------------------------------
  Temperature. | Solubility of tartrate | Solubility of racemate.
               |        mixture.        |
  ---------------------------------------------------------------
      25°      |         13.03          |          10.91
      35°      |          --            |          12.63
      40.4°    |          --            |          13.48
      40.7°    |         13.46          |           --
      54°      |         13.83          |           --
  ---------------------------------------------------------------

In Fig. 100 the curve AB represents the solubility of the racemate, while
A'BC represents the solubility of the mixed tartrates. Below the transition
point, therefore, the solubility of the racemate is less than that of the
mixed tartrates. The solution, saturated with respect to the latter, will
be supersaturated with respect to the racemate; and if a nucleus of this is
present, racemate will be deposited, and the mixed tartrates, if present in
equimolecular amounts, will ultimately {266} entirely disappear, and only
racemate will be left as solid phase. The solution will then have the
composition represented by a point on the curve AB. Conversely, above the
transition point, the saturated solution of the racemate would be
supersaturated with respect to the two tartrates, and transformation into
the latter would ensue. If, therefore, a solution of equimolecular
proportions of rubidium _d_- and _l_-tartrates is allowed to evaporate at a
temperature above 40°, a mixture of the two tartrates will be deposited;
while at temperatures below 40° the racemate will separate out.

[Illustration: FIG. 100.]

Similar relationships are met with in the case of sodium ammonium _d_- and
_l_-tartrate and sodium ammonium racemate; but in this case the racemate is
the stable form in contact with solution above the transition point
(27°).[347] Below the transition point, therefore, the solubility curve of
the mixed tartrates will lie below the solubility curve of the racemate.
Below the transition point, therefore, sodium ammonium racemate will break
up in contact with solution into a mixture of sodium ammonium _d_- and
_l_-tartrates. At a higher temperature, 35°, sodium ammonium racemate
undergoes decomposition into sodium racemate and ammonium racemate.[348]

The behaviour of sodium ammonium racemate is of interest from the fact that
it was the first racemic substance to be resolved into its optically active
forms by a process of crystallization. On neutralizing a solution of
racemic tartaric acid, half with soda and half with ammonia, and allowing
the solution to evaporate, Pasteur[349] obtained a mixture of sodium
ammonium {267} _d_- and _l_-tartrates. Since Pasteur was unaware of the
existence of a transition point, the success of his experiment was due to
the happy chance that he allowed the solution to evaporate at a temperature
below 27°; for had he employed a temperature above this, separation of the
racemate into the two enantiomorphous forms would not have occurred. For
this reason the attempt of Staedel to perform the same resolution met only
with failure.[350]

Decomposition of the Double Salt by Water.--In the two cases just
described, the solubility relationships at the transition point are of a
simpler character than in the case of most double salts. If, at a
temperature above the transition point, a mixture of rubidium _d_- and
_l_-tartrates in equimolecular proportions is brought in contact with water
a solution will be obtained, which is saturated with respect to both
enantiomorphous forms; and since the solubility of the two optical
antipodes is identical, and the effect of one on the solubility of the
other also the same, the solution will contain equimolecular amounts of the
_d_- and _l_-salt. If, now, the solution is cooled down in contact with the
solid salts to just below the transition point, it becomes supersaturated
with respect to the racemate, and this will be deposited. The solution
thereby becomes unsaturated with respect to the mixture of the active
salts, and these must therefore pass into solution. As the latter are
equally soluble, equal amounts of each will dissolve, and a further
quantity of the racemate will be deposited. These processes of solution and
deposition will continue until the single tartrates have completely
disappeared, and only racemate is left as solid phase. As a consequence of
the identical solubility of the two tartrates, therefore, no excess of
either form will be left on passing through the transition point. From this
it will be evident that the racemate can exist as single solid phase in
contact with its saturated solution at the transition point; or, in other
words, the racemate is not decomposed by water at the transition point. The
same behaviour will evidently be exhibited by sodium ammonium racemate at
27°, for the two enantiomorphous sodium ammonium tartrates have also
identical solubility.

{268}

Very different, however, is the behaviour of, say, astracanite, or of the
majority of double salts; for the solubility of the constituent salts is
now no longer the same. If, for example, excess of a mixture of sodium
sulphate and magnesium sulphate, in equimolecular proportions, is brought
in contact with water below the transition point (22°), more magnesium
sulphate than sodium sulphate will dissolve, the solubility of these two
salts in a common solution being given by the following figures, which
express number of molecules of the salt in 100 molecules of water.[351]

  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
  Na_{2}SO_{4},10H_{2}O AND MgSO_{4},7H_{2}O.

  ----------------------------------------
  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
  ----------------------------------------
      18.5°    |      2.16     |   4.57
      24.5°    |      3.43     |   4.68
  ----------------------------------------

At the transition point, then, it is evident that the solution contains
more magnesium sulphate than sodium sulphate: and this must still be the
case when astracanite, which contains sodium sulphate and magnesium
sulphate in equimolecular proportions, separates out. If, therefore, the
temperature is raised slightly above the transition point, magnesium
sulphate and sodium sulphate will pass into solution, the former, however,
in larger quantities than the latter, and astracanite will be deposited;
and this will go on until all the magnesium sulphate has disappeared, and a
mixture of astracanite and sodium sulphate decahydrate is left as solid
phases. Since there are now three phases present, the system is univariant
(by reason of the restriction previously made that the vapour phase is
absent), and at a given temperature the solution will have a definite
composition; as given in the following table:--

  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
  Na_{2}Mg(SO_{4})_{2},4H_{2}O AND Na_{2}SO_{4},10H_{2}O.

  ----------------------------------------
  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
  ----------------------------------------
      22°      |      2.95     |   4.70
      24.5°    |      3.45     |   3.62
  ----------------------------------------

{269}

From the above figures, therefore, it will be seen that at a temperature
just above the transition point a solution in contact with the two solid
phases, astracanite and Glauber's salt, contains a relatively smaller
amount of sodium sulphate than a pure solution of astracanite would; for in
this case there would be equal molecular amounts of Na_{2}SO_{4} and
MgSO_{4}. A solution which is saturated with respect to astracanite alone,
will contain more sodium sulphate than the solution saturated with respect
to astracanite plus Glauber's salt, and the latter will therefore be
deposited. From this, therefore, it is clear that if astracanite is brought
in contact with water at about the transition point, it will undergo
decomposition with separation of Glauber's salt (supersaturation being
excluded).

[Illustration: FIG. 101.]

This will perhaps be made clearer by considering Fig. 101. In this diagram
the ordinates represent the ratio of sodium sulphate to magnesium sulphate
in the solutions, and the abscissæ represent the temperatures. The line AB
represents solutions saturated with respect to a mixture of the single
salts (p. 268); BC refers to solutions in equilibrium with astracanite and
magnesium sulphate; while BX represents the composition of solutions in
contact with the solid phases astracanite and Glauber's salt. The values of
the solubility are contained in the following table, and in that on p. 268,
and are, as before, expressed in gm.-molecules of salt in 100 gm.-molecules
of water.[352]

{270}

  -------------------------------------------------------------------------
               | Astracanite                | Astracanite
  Temperature. |        + sodium sulphate.  |        + magnesium sulphate.
               |----------------------------|------------------------------
               | Na_{2}SO_{4}. |  MgSO_{4}. | Na_{2}SO_{4}. |  MgSO_{4}.
  -------------------------------------------------------------------------
      18.5°    |       --      |      --    |      3.41     |    4.27
      22°      |      2.95     |     4.70   |      2.85     |    4.63
      24.5°    |      3.45     |     3.62   |      2.68     |    4.76
      30°      |      4.58     |     2.91   |      2.30     |    5.31
      35°      |      4.30     |     2.76   |      1.73     |    5.88
  -------------------------------------------------------------------------

At the transition point the ratio of sodium sulphate to magnesium sulphate
is approximately 1 : 1.6. In the case of solutions saturated with respect
to both astracanite and Glauber's salt, the relative amount of sodium
sulphate increases as the temperature rises, while in the solutions
saturated for astracanite and magnesium sulphate, the ratio of sodium
sulphate to magnesium sulphate decreases.

If, now, we consider only the temperatures above the transition point, we
see from the figure that solutions represented by points above the line BX
contain relatively more sodium sulphate than solutions in contact with
astracanite and Glauber's salt; and solutions lying below the line BC
contain relatively more magnesium sulphate than solutions saturated with
this salt and astracanite. These solutions will therefore not be stable,
but will deposit in the one case, astracanite and Glauber's salt, and in
the other case, astracanite and magnesium sulphate, until a point on BX or
BC is reached. All solutions, however, lying to the right of CBX, will be
_unsaturated_ with respect to these two pairs of salts, and only the
solutions represented by the line XY (and which contain equimolecular
amounts of sodium and magnesium sulphates) will be saturated with respect
to the pure double salt.

Transition Interval.--Fig. 101 will also render intelligible a point of
great importance in connection with astracanite, and of double salts
generally. At temperatures between those represented by the points B and X,
the double salt when brought in contact with water will be decomposed with
separation of sodium sulphate. Above the temperature of the point {271} X,
however, the solution of the pure double salt is stable, because it can
still take up a little of either of the components. At temperatures, then,
above that at which the solution in contact with the double salt and the
less soluble single salt, contains the single salts in the ratio in which
they are present in the double salt, solution of the latter will take place
without decomposition. _The range of temperature between that at which
double salt can begin to be formed (the transition point) and that at which
it ceases to be decomposed by water is called the transition
interval._[353] If the two single salts have identical solubility at the
transition point, the transition interval diminishes to nought.

In those cases where the double salt is the stable form below the
transition point, the transition interval will extend downwards to a lower
temperature. Fig. 101 will then have the reverse form.

Summary.--With regard to double salts we have learned that their formation
from and their decomposition into the single salts, is connected with a
definite temperature, the _transition temperature_. At this transition
temperature two vapour pressure curves cut, viz. a curve of dehydration of
a mixture of the single salts and the solubility curve of the double salt;
or the dehydration curve of the double salt and the solubility curve of the
mixed single salts. The solubility curves, also, of these two systems
intersect at the transition point, but although the formation of the double
salt commences at the transition point, complete stability in contact with
water may not be attained till some temperature above (or below) that
point. _Only when the temperature is beyond the transition interval, will a
double salt dissolve in water without decomposition (_e.g._ the alums)._

       *       *       *       *       *


{272}

CHAPTER XVI

ISOTHERMAL CURVES AND THE SPACE MODEL

In the preceding chapter we considered the changes in the solubility of
double salts and of mixtures of their constituent salts with the
temperature; noting, more especially, the relationships between the two
systems at the transition point. It is now proposed to conclude the study
of the three-component systems by discussing very briefly the solubility
relations at constant temperature, or the isothermal solubility curves. In
this way fresh light will be thrown on the change in the solubility of one
component by the addition of another component, and also on the conditions
of formation and stable existence of double salts in solution. With the
help of these isothermal curves, also, the phenomena of crystallization at
constant temperature--phenomena which have not only a scientific interest
but also an important bearing on the industrial preparation of double
salts--will be more clearly understood.[354]

A brief description will also be given of the method of representing the
variation of the concentration of the two salts in the solution with the
temperature.

Non-formation of Double Salts.--In Fig. 102 are shown the solubility curves
of two salts, A and B, which at the given temperature do not form a double
salt.[355] The ordinates represent the amount of A, the abscissæ the amount
of B in a _constant amount_ of the third component, the solvent. The {273}
point A, therefore, represents the solubility of the salt A at the given
temperature; and similarly, point B represents the solubility of B. Since
we are dealing with a three-component system, one solid phase in contact
with solution will constitute a bivariant system (in the absence of the
vapour phase and under a constant pressure). At any given temperature,
therefore, the concentration of the solution in equilibrium with the solid
can undergo change. If, now, to a pure solution of A a small quantity of B
is added, the solubility of A will in general be altered; as a rule it is
diminished, but sometimes it is increased.[356] The curve AC represents the
varying composition of the solution in equilibrium with the solid component
A. Similarly, the curve BC represents the composition of the solutions in
contact with pure B as solid phase. At the point, C, where these two curves
intersect, there are two solid phases, viz. pure A and pure B, in
equilibrium with solution, and the system becomes invariant. At this point
the solution is saturated with respect to both A and B, and at a given
temperature must have a perfectly definite composition. To take an example,
if we suppose A to represent sodium sulphate decahydrate, and B, magnesium
sulphate heptahydrate, and the temperature to be 18.5° (_i.e._ below the
transition point), the point C would represent a solution containing 2.16
gm.-molecules Na_{2}SO_{4} and 4.57 gm.-molecules MgSO_{4} per 100
gm.-molecules of water (p. 268). The curve ACB is the boundary curve for
saturated solutions; solutions lying outside this curve are supersaturated,
those lying within the area ACBO, are unsaturated.

[Illustration: FIG. 102.]

[Illustration: FIG. 103.]

[Illustration: FIG. 104.]

Formation of Double Salt.--We have already learned in the preceding chapter
that if the temperature is outside[357] the {274} transition interval, it
is possible to prepare a pure saturated solution of the double salt. If,
now, we suppose the double salt to contain the two constituent salts in
equimolecular proportions, its saturated solution must be represented by a
point lying on the line which bisects the angle AOB; _e.g._ point D, Fig.
103. But a double salt constitutes only a single phase, and can exist,
therefore, in contact with solutions of varying concentration, as
represented by EDF.

Let us compare, now, the relations between the solubility curve for the
double salt, and those for the two constituent salts. We shall suppose that
the double salt is formed from the single salts when the temperature is
raised above a certain point (as in the formation of astracanite). At a
temperature below the transition point, as we have already seen, the
solubility of the double salt is greater than that of a mixture of the
single salts. The curve EDF, therefore, must lie above the point C, in the
region representing solutions supersaturated with respect to the single
salts (Fig. 104). Such a solution, however, would be metastable, and on
being brought in contact with the single salts would deposit these and
yield a solution represented by the point C. At this particular
temperature, therefore, the isothermal solubility curve will consist of
only two branches.

[Illustration: FIG. 105.]

Suppose, now, that the temperature is that of the transition point. At this
point, the double salt can exist together with the single salts in contact
with solution. The solubility curve {275} of the double salt must,
therefore, pass through the point C, as shown in Fig. 105.

From this figure, now, it is seen that a solution saturated with respect to
double salt alone (point D), is supersaturated with respect to the
component A. If, then, at the temperature of the transition point, excess
of the double salt is brought in contact with water,[358] and if
supersaturation is excluded, _the double salt will undergo decomposition
and the component A will be deposited_. The relative concentration of the
component B in the solution will, therefore, increase, and the composition
of the solution will be thereby altered in the direction DC. When the
solution has the composition of C, the single salt ceases to be deposited,
for at this point the solution is saturated for both double and single
salt; and the system becomes invariant.

This diagram explains very clearly the phenomenon of the decomposition of a
double salt at the transition point. As is evident, this decomposition will
occur when the solution which is saturated at the temperature of the
transition point, with respect to the two single salts (point C), does not
contain these salts in the same ratio in which they are present in the
double salt. If point C lay on the dotted line bisecting the right angle,
then the pure saturated solution of the double salt would not be
supersaturated with respect to either of the single salts, and the double
salt would, therefore, not be decomposed by water. As has already been
mentioned, this behaviour is found in the case of optically active
isomerides, the solubilities of which are identical.

At the transition point, therefore, the isothermal curve also consists of
two branches; but the point of intersection of the two branches now
represents a solution which is saturated not {276} only with respect to the
single salts, but also for the double salt in presence of the single salts.

We have just seen that by a change of temperature the two solubility
curves, that for the two single salts and that for the double salt, were
made to approach one another (_cf._ Figs. 104 and 105). In the previous
chapter, however, we found that on passing the transition point to the
region of stability for the double salt, the solution which is saturated
for a mixture of the two constituent salts, is supersaturated for the
double salt. In this case, therefore, point C must lie above the solubility
curve of the pure double salt (Fig. 106), and a solution of the composition
C, if brought in contact with double salt, will deposit the latter. If the
single salts were also present, then as the double salt separated out, the
single salts would pass into solution, because so long as the two single
salts are present, the composition of the solution must remain unaltered.
If one of the single salts disappear before the other, there will be left
double salt plus A or double salt plus B, according to which was in excess;
and the composition of the solution will be either that represented by D
(saturated for double salt plus A), or that of the point F (saturated for
double salt plus B).

[Illustration: FIG. 106.]

In connection with the isothermal represented in Fig. 106, it should be
noted that at this particular temperature a solution saturated with respect
to the pure double salt is no longer supersaturated for one of the single
salts (point D); so that at the temperature of this isothermal the double
salt is not decomposed by water. At this temperature, further, the boundary
curve consists of three branches AD, DF, and FB, which give the composition
of the solutions in equilibrium with pure A, double salt, and pure B
respectively; while the points D and F represent solutions saturated for
double salt plus A and double salt plus B.

On continuing to alter the temperature in the same direction {277} as
before, the relative shifting of the solubility curves becomes more marked,
as shown in Fig. 107. At the temperature of this isothermal, the solution
saturated for the double salt now lies in a region of distinct unsaturation
with respect to the single salts; and the double salt can now exist as
solid phase in contact with solutions containing both relatively more of A
(curve ED), and relatively more of B (curve DF), than is contained in the
double salt itself.

[Illustration: FIG. 107.]

Transition Interval.--From what has been said, and from an examination of
the isothermal diagrams, Figs. 104-107, it will be seen that by a variation
of the temperature we can pass from a condition where the double salt is
quite incapable of existing in contact with solution (supersaturation being
excluded), to a condition where the existence of the double salt in
presence of solution becomes possible; only in the presence, however, of
one of the single salts (_transition point_, Fig. 105). A further change of
temperature leads to a condition where the stable existence of the pure
double salt in contact with solution just becomes possible (Fig. 106); and
from this point onwards, pure saturated solutions of the double salt can be
obtained (Fig. 107). _At any temperature, therefore, between that
represented by Fig. 105, and that represented by Fig. 106, the double salt
undergoes partial decomposition, with deposition of one of the constituent
salts._ The temperature range between the transition point and the
temperature at which a stable saturated solution of the pure double salt
just begins to be possible, is known as the _transition interval_ (p. 270).
As the figures show, the transition interval is limited on the one side by
the transition temperature, and on the other by the temperature at which
the solution saturated for double salt and the less soluble of the single
salts, contains the component salts in the same ratio as they are present
in the double salt. The greater the difference in the solubility of the
single salts, the larger will be the transition interval. {278}

Isothermal Evaporation.--The isothermal solubility curves are of great
importance for obtaining an insight into the behaviour of a solution when
subjected to isothermal evaporation. To simplify the discussion of the
relationships found here, we shall still suppose that the double salt
contains the single salts in equimolecular proportions; and we shall, in
the first instance, suppose that the unsaturated solution with which we
commence, also contains the single salts in the same ratio. The composition
of the solution must, therefore, be represented by some point lying on the
line OD, the bisectrix of the right angle.

From what has been said, it is evident that when the formation of a double
salt can occur, three temperature intervals can be distinguished, viz. the
single-salt interval, the transition interval, and the double-salt
interval.[359] When the temperature lies in the first interval, evaporation
leads first of all to the crystallization of one of the single salts, and
then to the separation of both the single salts together. In the second
temperature interval, evaporation again leads, in the first place, to the
deposition of one of the single salts, and afterwards to the
crystallization of the double salt. In the third temperature interval, only
the double salt crystallizes out. This will become clearer from what
follows.

[Illustration: FIG. 108.]

[Illustration: FIG. 109.]

If an unsaturated solution of the two single salts in equimolecular
proportion (_e.g._ point _x_, Fig. 108) is evaporated at a temperature at
which the formation of double salt is impossible, the component A, the
solubility curve of which is {279} cut by the line OD, will first separate
out; the solution will thereby become richer in B. On continued
evaporation, more A will be deposited, and the composition of the solution
will change until it attains the composition represented by the point C,
when both A and B will be deposited, and the composition of the solution
will remain unchanged. The result of evaporation will therefore be a
mixture of the two components.

If the formation of double salt is possible, but if the temperature lies
within the transition interval, the relations will be represented by a
diagram like Fig. 109. Isothermal evaporation of the solution X will lead
to the deposition of the component A, and the composition of the solution
will alter in the direction DE; at the latter point the double salt will be
formed, and the composition of the solution will remain unchanged so long
as the two solid phases are present. As can be seen from the diagram,
however, the solution in E contains less of component A than is contained
in the double salt. Deposition of the double salt at E, therefore, would
lead to a relative decrease in the concentration of A in the solution, and
to counterbalance this, _the salt which separated out at the commencement
must redissolve_.

Since the salts were originally present in equimolecular proportions, the
final result of evaporation will be the pure double salt. If when the
solution has reached the point E the salt A which had separated out is
removed, double salt only will be left as solid phase. At a given
temperature, however, a single solid phase can exist in equilibrium with
solutions of different composition. If, therefore, isothermal evaporation
is continued after the removal of the salt A, double salt will be
deposited, and the composition of the solution will change in the direction
EF. At the point F the salt B will separate out, and on evaporation both
double salt and the salt B will be deposited. In the former case (when the
salt A disappears on evaporation) we are dealing with an _incongruently
saturated solution_; but in the latter case, where both solid phases
continue to be deposited, the solution is said to be _congruently
saturated_.[360]

A "congruently saturated solution" is one from which the {280} solid phases
are continuously deposited during isothermal evaporation to dryness,
whereas in the case of "incongruently saturated solutions," at least one of
the solid phases disappears during the process of evaporation.

[Illustration: FIG. 110.]

Lastly, if the temperature lies outside the transition interval, isothermal
evaporation of an unsaturated solution of the composition X (Fig. 110) will
lead to the deposition of pure double salt from beginning to end. If a
solution of the composition Y is evaporated, the component A will first be
deposited and the composition of the solution will alter in the direction
of E, at which point double salt will separate out. Since the solution at
this point contains relatively more of A than is present in the double
salt, both the double salt and the single salt A will be deposited on
continued evaporation, in order that the composition of the solution shall
remain unchanged. In the case of solution Z, first component B and
afterwards the double salt will be deposited. The result will, therefore,
be a mixture of double salt and the salt B (congruently saturated
solutions),

It may be stated here that the same relationships as have been explained
above for double salts are also found in the resolution of racemic
compounds by means of optically active substances (third method of
Pasteur). In this case the single salts are doubly active substances
(_e.g._ strychnine-_d_-tartrate and strychnine-_l_-tartrate), and the
double salt is a partially racemic compound.[361]

Crystallization of Double Salt from Solutions containing Excess of One
Component.--One more case of isothermal crystallization may be discussed.
It is well known that a double salt which is decomposed by pure water can
nevertheless be obtained pure by crystallization from a solution containing
excess of one of the single salts (_e.g._ in the case of carnallite). Since
the double salt is partially decomposed by water, the temperature of the
experiment must be within the transition {281} interval, and the relations
will, therefore, be represented by a diagram like Fig. 109. If, now,
instead of starting with an unsaturated solution containing the single
salts in equimolecular proportions, we commence with one in which excess of
one of the salts is present, as represented by the point Y, isothermal
evaporation will cause the composition to alter in the direction YD', the
relative amounts of the single salts remaining the same throughout. When
the composition of the solution reaches the point D', pure double salt will
be deposited. The separation of double salt will, however, cause a relative
decrease in the concentration of the salt A, and the composition of the
solution will, therefore, alter in the direction D'F. If the evaporation is
discontinued before the solution has attained the composition F, only
double salt will have separated out. Even within the transition interval,
therefore, pure double salt can be obtained by crystallization, provided
the original solution has a composition represented by a point lying
between the two lines OE and OF. Since, as already shown, the composition
of the solution alters on evaporation in the direction EF, it will be best
to employ a solution having a composition near to the line OE.

Formation of Mixed Crystals.--If the two single salts A and B do not
crystallize out pure from solution, but form an unbroken series of mixed
crystals, it is evident that an invariant system cannot be produced. The
solubility curve will therefore be continuous from A to B; the liquid
solutions of varying composition being in equilibrium with solid solutions
also of varying composition. If, however, the series of mixed crystals is
not continuous, there will be a break in the solubility curve at which two
solid solutions of different composition will be in equilibrium with liquid
solution. This, of course, will constitute an invariant system, and the
point will correspond to the point C in Fig. 108. A full discussion of
these systems would, however, lead us too far, and the above indication of
the behaviour must suffice.[362]

{282}

Application to the Characterization of Racemates.--The form of the
isothermal solubility curves is also of great value for determining whether
an inactive substance is a racemic compound or a conglomerate of equal
proportions of the optical antipodes.[363]

As has already been pointed out, the formation of racemic compounds from
the two enantiomorphous isomerides, is analogous to the formation of double
salts. The isothermal solubility curves, also, have a similar form. In the
case of the latter, indeed, the relationships are simplified by the fact
that the two enantiomorphous forms have identical solubility, and the
solubility curves are therefore symmetrical to the line bisecting the angle
of the co-ordinates. Further, with the exception of the partially racemic
compounds to be mentioned later, there is no transition interval.

In Fig. 111, are given diagrammatically two isothermal solubility curves
for optically active substances. From what has been said in the immediately
preceding pages, the figure ought really to explain itself. The upper
isothermal _acb_ represents the solubility relations when the formation of
a racemic compound is excluded, as, _e.g._ in the case of rubidium _d_- and
_l_-tartrates above the transition point (p. 265). The solution at the
point _c_ is, of course, inactive, and _is unaffected by addition of either
the _d_- or _l_- form_. The lower isothermal, on the other hand, would be
obtained at a temperature at which the racemic compound could be formed.
The curve _a'e_ is the solubility curve for the _l_- form; _b'f_, that for
the _d_- form; and _edf_, that for the racemic compound in presence of
solutions of varying concentration. The point _d_ corresponds to saturation
for the pure racemic compound.

[Illustration: FIG. 111.]

From these curves now, it will be evident that it will be possible, in any
given case, to decide whether or not an inactive body is a mixture or a
racemic compound. For this purpose, {283} two solubility determinations are
made, first with the inactive material alone (in excess), and then with the
inactive material plus excess of one of the optically active forms. If we
are dealing with a mixture, the two solutions thus obtained will be
identical; both will have the composition corresponding to the point _c_,
and will be inactive. If, however, the inactive material is a racemic
compound, then two different solutions will be obtained; namely, an
inactive solution corresponding to the point _d_ (Fig. 111), and an
_active_ solution corresponding either to _e_ or to _f_, according to which
enantiomorphous form was added.

_Partially racemic compounds._[364] In this case we are no longer dealing
with enantiomorphous forms, and the solubility of the two oppositely active
isomerides is no longer the same. The symmetry of the solubility curves
therefore disappears, and a figure is obtained which is identical in its
general form with that found in the case of ordinary double salts (Fig.
112). In this case there is a transition interval.

[Illustration: FIG. 112.]

The curves _acb_ belong to a temperature at which the partially racemic
compound cannot be formed; _a'dfb'_, to the temperature at which the
compound just begins to be stable in contact with water, and _a"ed'f'b"_
belongs to a temperature at which the partially racemic compound is quite
stable in contact with water. Suppose now solubility determinations, made
in the first case with the original material alone, and then with the
original body plus each of the two compounds, formed from the
enantiomorphous substances separately, then if the original body was a
mixture, identical solutions will be obtained in all three cases (point
_c_); if it was a partially racemic compound, three different solutions
(_e_, _d'_, and _f'_) will be obtained if the temperature was outside the
transition interval, and two solutions, _d_ and _f_, if the temperature
belonged to the transition interval.

{284}

_Representation in Space._

Space Model for Carnallite.--Interesting and important as the isothermal
solubility curves are, they are insufficient for the purpose of obtaining a
clear insight into the complete behaviour of the systems of two salts and
water. A short description will, therefore, be given here of the
representation in space of the solubility relations of potassium and
magnesium chlorides, and of the double salt which they form,
carnallite.[365]

[Illustration: FIG. 113.]

Fig. 113 is a diagrammatic sketch of the model for carnallite looked at
sideways from above. Along the X-axis is measured the concentration of
magnesium chloride in the {285} solution; along the Y-axis, the
concentration of potassium chloride; while along the T-axis is measured the
temperature. The three axes are at right angles to one another. The
XT-plane, therefore, contains the solubility curve of magnesium chloride;
the YT-plane, the solubility curve of potassium chloride, and in the space
between the two planes, there are represented the composition of solutions
containing both magnesium and potassium chlorides. Any _surface_ between
the two planes will represent the various solutions in equilibrium with
only one solid phase, and will therefore indicate the area or field of
existence of bivariant ternary systems. A _line_ or _curve_ formed by the
intersection of two surfaces will represent solutions in equilibrium with
two solid phases (viz. those belonging to the intersecting surfaces), and
will show the conditions for the existence of univariant systems. Lastly,
_points_ formed by the intersection of three surfaces will represent
invariant systems, in which a solution can exist in equilibrium with three
solid phases (viz. those belonging to the three surfaces).

We shall first consider the solubility relations of the single salts. The
complete equilibrium curve for magnesium chloride and water is represented
in Fig. 113 by the series of curves ABF_{1} G_{1} H_{1} J_{1} L_{1} N_{1}.
AB is the freezing-point curve of ice in contact with solutions containing
magnesium chloride, and B is the cryohydric point at which the solid phases
ice and MgCl_{2},12H_{2}O can co-exist with solution. BFG is the solubility
curve of magnesium chloride dodecahydrate. This curve shows a point of
maximum temperature at F_{1}, and a retroflex portion F_{1}G_{1}. The curve
is therefore of the form exhibited by calcium chloride hexahydrate, or the
hydrates of ferric chloride (Chapter VIII.). G_{1} is a transition point at
which the solid phase changes from dodecahydrate to octahydrate, the
solubility of which is represented by the curve G_{1}H_{1}. At H_{1} the
octahydrate gives place to the hexahydrate, which is the solid phase in
equilibrium with the solutions represented by the curve H_{1}J_{1}. J_{1}
and L_{1} are also transition points at which the solid phase undergoes
change, in the former case from hexahydrate to tetrahydrate; and in the
latter case, {286} from tetrahydrate to dihydrate. The complete curve of
equilibrium for magnesium chloride and water is, therefore, somewhat
complicated, and is a good example of the solubility curves obtained with
salts capable of forming several hydrates.

The solubility curve of potassium chloride is of the simplest form,
consisting only of the two branches AC, the freezing-point curve of ice,
and CO, the solubility curve of the salt. C is the cryohydric point. This
point and the two curves lie in the YT-plane.

On passing to the ternary systems, the composition of the solutions must be
represented by points or curves situated _between_ the two planes. We shall
now turn to the consideration of these. BD and CD are ternary eutectic
curves (p. 284). They give the composition of solutions in equilibrium with
ice and magnesium chloride dodecahydrate (BD), and with ice and potassium
chloride (CD). D is a _ternary cryohydric point_. If the temperature is
raised and the ice allowed to disappear, we shall pass to the solubility
curve for MgCl_{2},12H_{2}O + KCl (curve DE). At E carnallite is formed and
the potassium chloride disappears; EFG is then the solubility curve for
MgCl_{2},12H_{2}O + carnallite (KMgCl_{3},6H_{2}O). This curve also shows a
point of maximum temperature (F) and a retroflex portion. GH and HJ
represent the solubility curves of carnallite + MgCl_{2},8H_{2}O and
carnallite + MgCl_{2},6H_{2}O, G and H being transition points. JK is the
solubility curve for carnallite + MgCl_{2},4H_{2}O. At the point K we have
the _highest temperature at which carnallite can exist with magnesium
chloride in contact with solution_. Above this temperature decomposition
takes place and potassium chloride separates out.

If at the point E, at which the two single salts and the double salt are
present, excess of potassium chloride is added, the magnesium chloride will
all disappear owing to the formation of carnallite, and there will be left
carnallite and potassium chloride. The solubility curve for a mixture of
these two salts is represented by EMK; a simple curve exhibiting, however,
a temperature maximum at M. This maximum point corresponds with the fact
that dry carnallite melts at this temperature with separation of potassium
chloride. _At all temperatures {287} above this point, the formation of
double salt is impossible_. The retroflex portion of the curve represents
solutions in equilibrium with carnallite and potassium chloride, but in
which the ratio MgCl_{2} : KCl is greater than in the double salt.

Throughout its whole course, _the curve EMK represents solutions in which
the ratio of MgCl_{2} : KCl is greater than in the double salt_. As this is
a point of some importance, it will be well, perhaps, to make it clearer by
giving one of the isothermal curves, _e.g._ the curve for 10°, which is
represented diagrammatically in Fig. 114. E and F here represent solutions
saturated for carnallite plus magnesium chloride hydrate, and for
carnallite plus potassium chloride. As is evident, the point F lies above
the line representing equimolecular proportions of the salts (OD).

[Illustration: FIG. 114.]

Summary and Numerical Data.--We may now sum up the different systems which
can be formed, and give the numerical data from which the model is
constructed.[366]

  I. _Bivariant Systems._

  --------------------------------------
     Solid phase.   | Area of existence.
  --------------------------------------
  Ice               | ABDC
  KCl               | CDEMKLNO
  Carnallite        | EFGHJKM
  MgCl_{2},12H_{2}O | BF_{1}G_{1}GFED
  MgCl_{2},8H_{2}O  | G_{1}H_{1}HG
  MgCl_{2},6H_{2}O  | H_{1}I_{1}IH
  MgCl_{2},4H_{2}O  | I_{1}L_{1}LKI
  MgCl_{2},2H_{2}O  | L_{1}N_{1}NL
  --------------------------------------

II. _Univariant Systems._--The different univariant systems have already
been described. The course of the curves will be sufficiently indicated if
the temperature and composition of the solutions for the different
invariant systems are given.

{288}

  III.--_Invariant Systems--Binary and Ternary._

  -------------------------------------------------------------------------
         |                           |          | Composition of solution.
  Point. |      Solid Phases.        | Temper-  | Gram-molecules of salt
         |                           |  ature.  | per 1000 gram-mol. water.
  -------------------------------------------------------------------------
  A      | Ice                       |    0°    | --
         |                           |          |
  B      | Ice; MgCl_{2},12H_{2}O    |  -33.6°  | 49.2 MgCl_{2}
         |                           |          |
  C      | Ice; KCl                  |  -11.1°  | 59.4 KCl
         |                           |          |
  D      |{ Ice; MgCl_{2},12H_{2}O; }|  -34.3°  | 43 MgCl_{2}; 3 KCl
         |{ KCl                     }|          |
         |                           |          |
  E      |{ MgCl_{2},12H_{2}O; KCl; }|  -21°    | 66.1 MgCl_{2}; 4.9 KCl
         |{ carnallite              }|          |
         |                           |          |
  F_{1}  | MgCl_{2},12H_{2}O         |  -16.4°  | 83.33 MgCl_{2}
         |                           |          |
  F      |{ MgCl_{2},12H_{2}O;      }|  -16.6°  |{ Almost same as F_{1};
         |{ carnallite              }|          |{ contains small amount
         |                           |          |{ of KCl
         |                           |          |
  G_{1}  |{ MgCl_{2},12H_{2}O;      }|  -16.8°  | 87.5 MgCl_{2}
         |{ MgCl_{2},8H_{2}O        }|          |
         |                           |          |
  G      |{ MgCl_{2},12H_{2}O;      }|  -16.9°  |{ Almost same as G_{1},
         |{ MgCl_{2},8H_{2}O;       }|          |{ but contains small
         |{ carnallite              }|          |{ quantity of KCl
         |                           |          |
  H_{1}  |{ MgCl_{2},8H_{2}O;       }|   -3.4°  | 99 MgCl_{2}
         |{ MgCl_{2},6H_{2}O        }|          |
         |                           |          |
  H      |{ MgCl_{2},8H_{2}O;       }|ca. -3.4° |{ Almost same as H_{1},
         |{ MgCl_{2},6H_{2}O;       }|          |{ but contains small
         |{ carnallite              }|          |{ amount of KCl
         |                           |          |
  J_{1}  |{ MgCl_{2},6H_{2}O;       }|  116.67° | 161.8 MgCl_{2}
         |{ MgCl_{2},4H_{2}O        }|          |
         |                           |          |
  J      |{ MgCl_{2},6H_{2}O;       }|  115.7°  | 162 MgCl_{2}; 4 KCl
         |{ MgCl_{2},4H_{2}O;       }|          |
         |{ carnallite              }|          |
         |                           |          |
  K      |{ MgCl_{2},4H_{2}O; KCl;  }|  152.5°  | 200 MgCl_{2}; 24 KCl
         |{ carnallite              }|          |
         |                           |          |
  L_{1}  |{ MgCl_{2},4H_{2}O;       }|  181°    | 238.1 MgCl_{2}
         |{ MgCl_{2},2H_{2}O        }|          |
         |                           |          |
  L      |{ MgCl_{2},4H_{2}O;       }|  176°    | 240 MgCl_{2}; 41 KCl
         |{ MgCl_{2},2H_{2}O; KCl   }|          |
         |                           |          |
  M      | Carnallite; KCl           |  167.5°  | 166.7 MgCl_{2}; 41.7 KCl
         |                           |          |
  [N_{1} | MgCl_{2},2H_{2}O          |  186°    | ca. 241 MgCl_{2}]
         |                           |          |
  N      | MgCl_{2},2H_{2}O; KCl     |  186°    | 240 MgCl_{2}; 63 KCl
         |                           |          |
  [O     | KCl                       |  186°    | 195.6 KCl]
  -------------------------------------------------------------------------

With the help of the data in the preceding table and of the solid model it
will be possible to state in any given case what will be the behaviour of a
system composed of magnesium chloride, potassium chloride and water. One or
two different cases will be very briefly described; and the reader should
have no difficulty in working out the behaviour under other conditions with
the help of the model and the numerical data just given. {289}

In the first place it may be again noted that at a temperature above 167.5°
(point M) carnallite cannot exist. If, therefore, a solution of magnesium
and potassium chlorides is evaporated at a temperature above this point,
the result will be a mixture of potassium chloride and either magnesium
chloride tetrahydrate or magnesium chloride dihydrate, according as the
temperature is below or above 176°. The isothermal curve here consists of
only two branches.

Further, reference has already been made to the fact that all points of the
carnallite area correspond to solutions in equilibrium with carnallite, but
in which the ratio of MgCl_{2} to KCl is greater than in the double salt. A
solution which is saturated with respect to double salt alone will be
supersaturated with respect to potassium chloride. At all temperatures,
therefore, carnallite is decomposed by water with separation of potassium
chloride; hence all solutions obtained by adding excess of carnallite to
water will lie on the curve EM. _A pure saturated solution of carnallite
cannot be obtained._

If an unsaturated solution of the two salts in equimolecular amounts is
evaporated, potassium chloride will first be deposited, because the plane
bisecting the right angle formed by the X and Y axes cuts the area for that
salt. Deposition of potassium chloride will lead to a relative increase in
the concentration of magnesium chloride in the solution; and on continued
evaporation a point (on the curve EM) will be reached at which carnallite
will separate out. So long as the two solid phases are present, the
composition of the solution must remain unchanged. Since the separation of
carnallite causes a decrease in the relative concentration of the potassium
chloride in the solution, the portion of this salt which was deposited at
the commencement must _redissolve_, and carnallite will be left on
evaporating to dryness. (_Incongruently saturated solution._)

Although carnallite is decomposed by pure water, it will be possible to
crystallize it from a solution having a composition represented by any
point in the carnallite area. Since during the separation of the double
salt the relative amount of magnesium chloride increases, it is most
advantageous to {290} commence with a solution the composition of which is
represented by a point lying just above the curve EM (cf. p. 281).

From the above description of the behaviour of carnallite in solution, the
processes usually employed for obtaining potassium chloride will be readily
intelligible.[367]

Ferric Chloride--Hydrogen Chloride--Water.--In the case of another system
of three components which we shall now describe, the relationships are
considerably more complicated than in those already discussed. They deserve
discussion, however, on account of the fact that they exhibit a number of
new phenomena.

In the system formed by the three components, ferric chloride, hydrogen
chloride, and water, not only can various compounds of ferric chloride and
water (p. 152), and of hydrogen chloride and water be formed, each of which
possesses a definite melting point, but various ternary compounds are also
known. Thus we have the following solid phases:--

  2FeCl_{3},12H_{2}O   HCl,3H_{2}O   2FeCl_{3},2HCl,12H_{2}O
  2FeCl_{3},7H_{2}O    HCl,2H_{2}O   2FeCl_{3},2HCl,8H_{2}O
  2FeCl_{3},5H_{2}O    HCl,H_{2}O    2FeCl_{3},2HCl,4H_{2}O
  2FeCl_{3},4H_{2}O
  FeCl_{3}

From this it will be readily understood that the complete study of the
conditions of temperature and concentration under which solutions can
exist, either with one solid phase or with two or three solid phases, are
exceedingly complicated; and, as a matter of fact, only a few of the
possible equilibria have been investigated. We shall attempt here only a
brief description of the most important of these.[368]

If we again employ rectangular co-ordinates for the graphic {291}
representation of the results, we have the two planes XOT and YOT (Fig.
115): the concentration of ferric chloride being measured along the X-axis,
the concentration of hydrogen chloride along the Y-axis, and the
temperature along the T-axis. The curve ABCDEFGHJK is, therefore, the
solubility curve of ferric chloride in water (p. 152), and the curve
A'B'C'D'E'F' the solubility curve of hydrogen chloride and its hydrates. B'
and D' are the melting points of the hydrates HCl,3H_{2}O and HCl,2H_{2}O.
In the space between these two planes are represented those systems in
which all three components are present. As already stated, only a few of
the possible ternary systems have been investigated, and these are
represented in Fig. 116. The figure shows the model resting on the
XOT-plane, so that the lower edge represents the solubility curve of ferric
chloride, the concentration increasing from right to left. The
concentration of hydrogen chloride is measured upwards, and the temperature
forwards. The further end of the model represents the isothermal surface
for -30°. The surface of the model on the left does not correspond with the
plane YOT in Fig. 115, but with a parallel plane which cuts the
concentration axis for ferric chloride at a point representing 65
gm.-molecules FeCl_{3} in 100 gm.-molecules of water. The upper surface
corresponds with a plane parallel to the axis XOT, at a distance
corresponding with the concentration of 50 gm.-molecules HCl in 100
gm.-molecules of water.

[Illustration: FIG. 115.]

Ternary Systems.--We pass over the binary system FeCl_{3}--H_{2}O, which
has already been discussed (p. 152), and the similar system HCl--H_{2}O
(see Fig. 115), and turn to the discussion of some of the ternary systems
represented by {292} points on the surface of the model between the planes
XOT and YOT. As in the case of carnallite, a plane represents the
conditions of concentration of solution and temperature under which a
ternary solution can be in equilibrium with a _single_ solid phase
(bivariant systems), a line represents the conditions for the coexistence
of a solution with two solid phases (univariant systems), and a point the
conditions for equilibrium with three solid phases (invariant systems).

[Illustration: FIG. 116.]

In the case of a binary system, in which 2FeCl_{3},12H_{2}O is in
equilibrium with a solution of the same composition, addition of hydrogen
chloride must evidently lower the temperature at which equilibrium can
exist; and the same holds, of course, {293} for all other binary solutions
in equilibrium with this solid phase. In this way we obtain the surface I.,
which represents the temperatures and concentrations of solutions in which
2FeCl_{3},12H_{2}O can be in equilibrium with a ternary solution containing
ferric chloride, hydrogen chloride, and water. This surface is analogous to
the curved surface K_{1}K_{2}_k__{4}_k__{3} in Fig. 97 (p. 256). Similarly,
the surfaces II., III., IV., and V. represent the conditions for
equilibrium between the solid phases 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O;
2FeCl_{3},4H_{2}O; FeCl_{3} and ternary solutions respectively. The lines
CL, EM, GN, and IO on the model represent univariant systems in which a
ternary solution is in equilibrium with two solid phases, viz. with those
represented by the adjoining fields. These lines correspond with the
ternary eutectic curves _k__{3}K_{1} and _k__{4}K_{2} in Fig. 97. Besides
the surfaces already mentioned, there are still three others, VI., VII.,
and VIII., which also represent the conditions for equilibrium between one
solid phase and a ternary solution; but in these cases, the solid phase is
not a binary compound or an anhydrous salt, but a ternary compound
containing all three components. The solid phases which are in equilibrium
with the ternary solutions represented by the surfaces VI., VII., and
VIII., are 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
2FeCl_{3},2HCl,12H_{2}O respectively.

The model for FeCl_{3}--HCl--H_{2}O exhibits certain other peculiarities
not found in the case of MgCl_{2}--KCl--H_{2}O. On examining the model more
closely, it is found that the field of the ternary compound
2FeCl_{3},2HCl,8H_{2}O (VII.) resembles the surface of a sugar cone, and
has a projecting point, the end of which corresponds with a higher
temperature than does any other point of the surface. At the point of
maximum temperature the composition of the liquid phase is the same as that
of the solid. This point, therefore, represents the melting point of the
double salt of the above composition.

The curves representing univariant systems are of two kinds. In the one
case, the two solid phases present are both binary compounds; or one is a
binary compound and the other is one of the components. In the other case,
either one or both solid phases are ternary compounds. Curves belonging
{294} to the former class (so-called _border curves_) start from binary
eutectic points, and their course is always towards lower temperatures,
_e.g._ CL, EM, GN, IO. Curves belonging to the latter class (so-called
_medial curves_) would, in a triangular diagram, lie entirely within the
triangle. Such curves are YV, WV, VL, LM, MV, NS, ST, SO, OZ. These curves
do not always run from higher to lower temperatures, but may even exhibit a
point of maximum temperature. Such maxima are found, for example, at U
(Fig. 116), and also on the curves ST and LV.

Finally, whereas all the other ternary univariant curves run in valleys
between the adjoining surfaces, we find at the point X a similar appearance
to that found in the case of carnallite, as the univariant curve here rises
above the surrounding surface. The point X, therefore, does not correspond
with a eutectic point, but with a transition point. At this point the
ternary compound 2FeCl_{3},2HCl,12H_{2}O melts with separation of
2FeCl_{3},12H_{2}O, just as carnallite melts at 168° with separation of
potassium chloride.

The Isothermal Curves.--A deeper insight into the behaviour of the system
FeCl_{3}--HCl--H_{2}O is obtained from a study of the isothermal curves,
the complete series of which, so far as they have been studied, is given in
Fig. 117.[369] In this figure the lightly drawn curves represent isothermal
solubility curves, the particular temperature being printed beside the
curve.[370] The dark lines give the composition of the univariant systems
at different temperatures. The point of intersection of a dark with a light
curve gives the composition of the univariant solution at the temperature
represented by the light curve; and the point of intersection of two dark
lines gives the composition of the invariant solution in equilibrium with
three solid phases. The dotted lines represent metastable systems, and the
points P, Q, and R represent solutions of {295} the composition of the
ternary salts, 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
2FeCl_{3},2HCl,12H_{2}O.

[Illustration: FIG. 117.]

The farther end of the model (Fig. 116) corresponds, as already mentioned,
to the temperature -30°, so that the outline evidently represents the
isothermal curve for that temperature. Fig. 117 does not show this. We can,
however, follow the isothermal for -20°, which is the extreme curve on the
right in Fig. 117. Point A represents the solubility of 2FeCl_{3},12H_{2}O
in water. If hydrogen chloride is added, the concentration of ferric
chloride in the solution first decreases and then increases, until at point
34 the ternary double salt 2FeCl_{3},2HCl,12H_{2}O is formed. If the
addition of hydrogen chloride is continued, the ferric chloride disappears
ultimately, and only the ternary double salt remains. This salt can coexist
with solutions of the composition represented by the curve which passes
through the points 173, 174, 175. At the last-mentioned point, the ternary
salt with 8H_{2}O is formed. The composition of the solutions with which
this salt is in equilibrium at -20° is represented by the curve which
passes through a point of maximal concentration with respect to HCl, and
cuts the curve SN at the point 112, at which the solution is in equilibrium
with the two solid phases 2FeCl_{3},4H_{2}O and 2FeCl_{3},2HCl,8H_{2}O. The
succeeding portion of the isotherm represents the solubility curve at -20°
of 2FeCl_{3},4H_{2}O, which cuts the dark line OS at point 113, at which
the solution is in equilibrium with the two solid phases 2FeCl_{3},4H_{2}O
and 2FeCl_{3},2HCl,4H_{2}O. Thereafter comes the solubility curve of the
latter compound.

The other isothermal curves can be followed in a similar manner. If the
temperature is raised, the region of existence of the ternary double salts
becomes smaller and smaller, and at temperatures above 30° the ternary
salts with 12H_{2}O and 8H_{2}O are no longer capable of existing. If the
temperature is raised above 46°, only the binary compounds of ferric
chloride and water and the anhydrous salt can exist as solid phases.
The isothermal curve for 0° represents the solubility curve for
2FeCl_{3},12H_{2}O; 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O; and
2FeCl_{3},4H_{2}O. {296}

Finally, in the case of the system FeCl_{3}--HCl--H_{2}O, we find
_closed_ isothermal curves. Since, as already stated, the salt
2FeCl_{3},2HCl,8H_{2}O has a definite melting point, the temperature of
which is therefore higher than that at which this compound is in
equilibrium with solutions of other composition, it follows that the line
of intersection of an isothermal plane corresponding with a temperature
immediately below the melting point of the salt with the cone-shaped
surface of its region of existence, will form a closed curve. This is shown
by the isotherm for -4.5°, which surrounds the point Q, the melting point
of the ternary salt.

The following table gives some of the numerical data from which the curves
and the model have been constructed:--

  -------------------------------------------------------------------------
         |                             |          | Composition of the sol-
         |                             |          | ution in gm.-mols. salt
  Point. |         Solid phases.       | Temper-  | to 100 gm.-mols. water.
         |                             |  ature.  |------------------------
         |                             |          |   HCl   |  FeCl_{3}
  -------------------------------------------------------------------------
  A      | 2FeCl_{3},12H_{2}O          |  -20°    |   --    |   6.56
         |                             |          |         |
  C      |{ 2FeCl_{3},12H_{2}O;       }|   27.4°  |   --    |  24.30
         |{ 2FeCl_{3},7H_{2}O         }|          |         |
         |                             |          |         |
  E      |{ 2FeCl_{3},7H_{2}O;        }|   30°    |   --    |  30.24
         |{ 2FeCl_{3},5H_{2}O         }|          |         |
         |                             |          |         |
  G      |{ 2FeCl_{3},5H_{2}O;        }|   55°    |   --    |  40.64
         |{ 2FeCl_{3},4H_{2}O         }|          |         |
         |                             |          |         |
  J      | 2FeCl_{3},4H_{2}O; FeCl_{3} |   66°    |   --    |  58.40
         |                             |          |         |
         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
  L      |{ 2FeCl_{3},7H_{2}O;        }|   -7.5°  |  19.22  |  23.72
         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
         |                             |          |         |
         |{ 2FeCl_{3},7H_{2}O;        }|          |         |
  M      |{ 2FeCl_{3},5H_{2}O;        }|   -7.3°  |  23.08  |  28.55
         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
         |                             |          |         |
         |{ 2FeCl_{3},5H_{2}O;        }|          |         |
  N      |{ 2FeCl_{3},4H_{2}O;        }|  -16°    |  28.40  |  31.89
         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
         |                             |          |         |
         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
  S      |{ 2FeCl_{3},2HCl,8H_{2}O;   }|  -27.5°  |  32.33  |  34.21
         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
         |                             |          |         |
         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
  O      |{ FeCl_{3};                 }|   29°    |  33.71  |  49.84
         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
         |                             |          |         |
  U      |{ 2FeCl_{3},7H_{2}O;        }|   -4.5°  |  20.66  |  25.74
         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
         |                             |          |         |
         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
  V      |{ 2FeCl_{3},2HCl,12H_{2}O;  }|  -13°    |  22.40  |  18.00
         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
         |                             |          |         |
  X      |{ 2FeCl_{3},12H_{2}O;       }|  -12.5°  |  22.14  |  16.69
         |{ 2FeCl_{3},2HCl,12H_{2}O   }|          |         |
         |                             |          |         |
  Q      | 2FeCl_{3},2HCl,8H_{2}O      |   -3° (melting point)
  -------------------------------------------------------------------------

Basic Salts.--Another class of systems in the study of {297} which the
Phase Rule has performed exceptional service, is that of the basic salts.
In many cases it is impossible, by the ordinary methods of analysis, to
decide whether one is dealing with a definite chemical individual or with a
mixture. The question whether a solid phase is a chemical individual can,
however, be answered, in most cases, with the help of the principles which
we have already learnt. Let us consider, for example, the formation of
basic salts from bismuth nitrate, and water. In this case we can choose as
components Bi_{2}O_{3}, N_{2}O_{5}, and H_{2}O; since all the systems
consist of these in varying amounts. If we are dealing with a condition of
equilibrium at constant temperature between liquid and solid phases, three
cases can be distinguished,[371] viz.--

1. The solutions in different experiments have the same composition, but
the composition of the precipitate alters. In this case there must be two
solid phases.

2. The solutions in different experiments can have varying composition,
while the composition of the precipitate remains unchanged. In this case
only one solid phase exists, a definite compound.

3. The composition both of the solution and of the precipitate varies. In
this case the solid phase is a solid solution or a mixed crystal.

In order, therefore, to decide what is the nature of a precipitate produced
by the hydrolysis of a normal salt, it is only necessary to ascertain
whether and how the composition of the precipitate alters with alteration
in the composition of the solution. If the composition of the solution is
represented by abscissæ, and the composition of the precipitate by
ordinates, the form of the curves obtained would enable us to answer our
question; for vertical lines would indicate the presence of two solid
phases (1st case), horizontal lines the presence of only one solid phase
(2nd case), and slanting lines the presence of mixed crystals (3rd case).
This method of representation cannot, however, be carried out in most
cases. It is, however, {298} generally possible to find one pair or several
pairs of components, the _relative amounts_ of which in the solution or in
the precipitate undergo change when, and only when, the composition of the
solution or of the precipitate changes. Thus, in the case of bismuth,
nitrate, and water, we can represent the ratio of Bi_{2}O_{3} : N_{2}O_{5}
in the precipitate as ordinates, and N_{2}O_{5} : H_{2}O in the solution as
abscissæ. A horizontal line then indicates a single solid phase, and a
vertical line two solid phases. An example of this is given in Fig.
118.[372]

[Illustration: FIG. 118.]

Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O.--Although various systems have been
studied in which there is formation of basic salts,[373] we shall content
ourselves here with the description of some of the conditions for the
formation of basic salts of bismuth nitrate, and for their equilibrium in
contact with solutions.[374]

Three normal salts of bismuth oxide and nitric acid are known, viz.
Bi_{2}O_{3},3N_{2}O_{5},10H_{2}O(S_{10});
Bi_{2}O_{3},3N_{2}O_{5},4H_{2}O(S_{4}); and
Bi_{2}O_{3},3N_{2}O_{5},3H_{2}O(S_{3}). Besides these normal salts, there
are the following basic salts:--

{299}

  Bi_{2}O_{3},N_{2}O_{5},2H_{2}O   (represented by B_{1-1-2})
  Bi_{2}O_{3},N_{2}O_{5},H_{2}O    (     "      "  B_{1-1-1})
  6Bi_{2}O_{3},5N_{2}O_{5},9H_{2}O (     "      "  B_{6-5-9})
  2Bi_{2}O_{3},N_{2}O_{5},H_{2}O   (     "      "  B_{2-1-1})

Probably some others also exist. The problem now is to find the conditions
under which these different normal and basic salts can be in equilibrium
with solutions of varying concentration of the three components. Having
determined the equilibrium conditions for the different salts, it is then
possible to construct a model similar to that for MgCl_{2}--KCl--H_{2}O or
for FeCl_{3}--HCl--H_{2}O, from which it will be possible to determine the
limits of stability of the different salts, and to predict what will occur
when we bring the salts in contact with solutions of nitric acid of
different concentrations and at different temperatures.

For our present purpose it is sufficient to pick out only some of the
equilibria which have been studied, and which are represented in the model
(Fig. 119). In this case use has been made of the triangular method of
representation, so that the surface of the model lies within the prism.

[Illustration: FIG. 119.]

This model shows the three surfaces, A, B, and C, which represent the
conditions for the stable existence of the salts B_{1-1-1}, S_{10}, and
S_{3} in contact with solution at different {300} temperatures. The front
surface of the model represents the temperature 9°, and the farther end the
temperature 75.5°. The dotted curve represents the isotherm for 20°. The
prominences between the surfaces represent, of course, solutions which are
saturated in respect of two solid phases. Thus, for example, _pabc_
represents solutions in equilibrium with B_{1-1-1} and S_{10}; and the
ridge _qdc_, solutions in equilibrium with S_{10} and S_{3}. The point _b_,
which lies at 75.5°, is the point of maximum temperature for S_{10}. If the
temperature is raised above this point, S_{10} decomposes into the basic
salt B_{1-1-1} and solution. This point is therefore analogous to the point
M in the carnallite model, at which this salt decomposes into potassium
chloride and solution (p. 284); or to the point at which the salt
2FeCl_{3},2HCl,12H_{2}O decomposes into 2FeCl_{3},12H_{2}O and solution
(p. 294). The curve _pab_ has been followed to the temperature of 72°
(point _c_). The end of the model is incomplete, but it is probable that in
the neighbourhood of the point _c_ there exists a quintuple point at which
the basic salt B_{1-2-2} appears. In the neighbourhood of _e_ also there
probably exists another quintuple point at which S_{4} is formed. These
systems have, however, not been studied.

The following tables give some of the numerical data:--

  ISOTHERM FOR 20°.

  ----------------------------------------------------------------------
                               | Composition of the solution. Gram-mols.
                               | in 1000 gm.-mols. of water.
  Solid phase.                 |----------------------------------------
                               |    Bi_{2}O_{3}  |      N_{2}O_{5}
  -----------------------------|-----------------|----------------------
  B_{1-1-1}                    |      10.50      |       38.65
  --                           |      27.20      |       83.84
  B_{1-1-1}; S_{10}            |      30.15      |       97.97
  S_{10}                       |      29.70      |       96.57
  --                           |      19.65      |       98.76
  --                           |      10.51      |      162.58
  --                           |      33.51      |      355.87
  S_{10}; S_{3}                |      51.00      |      403.0
  S_{3}                        |      14.35      |      492.0
  --                           |       7.45      |      592.9
  ----------------------------------------------------------------------

SYSTEMS IN EQUILIBRIUM WITH B_{1-1-1} AND S_{10} (CURVE _pabc_).

  ------------------------------------------------------------
                     | Composition of the solution. Gram-mols.
                     |  in 1000 gm.-mols. of water.
  Temperature.       |----------------------------------------
                     |  Bi_{2}O_{3}  |  N_{2}O_{5}
  -------------------|---------------|------------------------
   9°                |    26.7       |     88.2
   20° (point _a_)   |    30.15      |     97.97
   30°               |    33.6       |    112.3
   50°               |    41.8       |    148.4
   65°               |    57.21      |    190.8
   75.5° (point _b_) |    87.9       |    288.4
   72° (point _c_)   |    96.0       |    327.0
  ------------------------------------------------------------

  SYSTEMS IN EQUILIBRIUM WITH S_{10} AND S_{3} (CURVE _qde_).

  --------------------------------------------------------
                | Composition of the solution. Gram-mols.
                |  in 1000 gm.-mols. of water.
  Temperature.  |-----------------------------------------
                |  Bi_{2}O_{3}  |  N_{2}O_{5}
  --------------|---------------|-------------------------
   11.5°        |    44.5       |    396
   20°          |    51.0       |    405.4
   50°          |    66.5       |    444.2
   65°          |    80.0       |    454.4
  --------------------------------------------------------

Basic Mercury Salts.--The Phase Rule has also been applied by A. J.
Cox[375] in an investigation of the basic salts of mercury, the result of
which has been to show that, of the salts mentioned in text-books, quite a
number are incorrectly stated to be chemical compounds or chemical
individuals (p. 92). The investigation, which was carried out essentially
in the manner described above, included the salts mentioned in the
following table; and of the basic salts said to be derived from them, only
those mentioned really exist. In the following table, the numbers in the
second column give the minimum values of the concentration of the acid,
expressed in equivalent normality, necessary for the existence of the {301}
corresponding salts in contact with solution at the temperature given in
the third column:--

  -------------------------------------------------------------
               Salt.             | Normality of  | Temperature.
                                 |     acid.     |
  -------------------------------------------------------------
  HgCrO_{4}                      |     1.41      |     50°
  3HgO.CrO_{3}                   | 2.6 × 10^{-4} |     50°
                                 |               |
  Hg(NO_{3})_{2}.H_{2}O          |    18.72      |     25°
  3HgO.N_{2}O_{5}                |     0.159     |     25°
                                 |               |
  HgSO_{4}                       |     6.87      |     25°
  3HgO.SO_{3}                    | 1.3 × 10^{-3} |     25°
                                 |               |
  HgF_{2}                        |     1.14      |     25°
                                 |               |
  HgNO_{3}.H_{2}O                |     2.95      |     25°
  5Hg_{2}O.3N_{2}O_{5}.2H_{2}O   | ca. 0.293     |     25°
  2Hg_{2}O.N_{2}O_{5}(?)         |     0.110     |     25°
  3Hg_{2}O.N_{2}O_{5}.2H_{2}O(?) | 1.7 × 10^{-3} |     25°
                                 |               |
  Hg_{2}SO_{4}                   | 4.2 × 10^{-3} |     25°
  2Hg_{2}O.SO_{3}.H_{2}O         | 5.6 × 10^{-4} |     25°
  -------------------------------------------------------------

Mercuric fluoride does not form any basic salt.

Since two succeeding members of a series can coexist only in contact with a
solution of definite concentration, we can prepare acid solutions of
definite concentration by bringing an excess of two such salts in contact
with water.

Indirect Determination of the Composition of the Solid Phase.--It has
already been shown (p. 228) how the composition of the solid phase in a
system of two components can be determined without analysis, and we shall
now describe how this can be done in a system of three components.[376]

We shall assume that we are dealing with the aqueous solution of two salts
which can give rise to a double salt, in which case we can represent the
solubility relations in a system of rectangular co-ordinates. In this case
we should obtain, as before (Fig. 120), the isotherm _adcb_, if we express
the {302} composition of the solution in gram-molecules of A or of B to 100
gram-molecules of water.

[Illustration: FIG. 120.]

Let us suppose, now, that the double salt is in equilibrium with the
solution at a definite temperature, and that the composition of the
solution is represented by the point e. The greater part of the solution is
now separated from the solid phase, and the latter, _together with the
adhering mother liquor_, is analyzed. The composition (expressed, as
before, in gram-molecules of A and B to 100 gram molecules of water) will
be represented by a point (_e.g._ _f_) on the line _e_S, where S represents
the composition of the double salt. That this is so will be evident when
one considers that the composition of the whole mass must lie between the
composition of the solution and that of the double salt, no matter what the
relative amounts of the solid phase and the mother liquor.

If, in a similar manner, we analyze a solution of a different composition
in equilibrium with the same double salt (not necessarily at the same
temperature as before), and also the mixture of solid phase and solution,
we shall obtain two other points, as, for example, _g_ and _h_, and the
line joining these must likewise pass through S. The method of finding the
{303} composition of an unknown double salt consists, therefore, in
finding, in the manner just described, the position of two lines such as
_ef_ and _gh_. The point of intersection of these lines then gives the
composition of the double salt.

If the double salt is anhydrous, the point S lies at infinity, and the
lines _ef_ and _gh_ are parallel to each other.

The same result is arrived at by means of the triangular method of
representation.[377] If we start with the three components in known
amounts, and represent the initial composition of the whole by a point in
the triangle, and then ascertain the final composition of the solution in
equilibrium with the solid phase at a definite temperature, the line
joining the points representing the initial and end concentration passes
through the point representing the composition of the solid phase. If two
determinations are made with solutions having different initial and final
concentrations in equilibrium with the same solid phase, then the point of
intersection of the two lines so obtained gives the composition of the
solid phase.

       *       *       *       *       *


{304}

CHAPTER XVII

ABSENCE OF A LIQUID PHASE

In the preceding chapters dealing with equilibria in three-component
systems, our attention was directed only to those cases in which liquid
solutions formed one or more phases. Mention must, however, be made of
certain systems which contain no liquid phase, and in which only solids and
gases are in equilibrium. Since, in all cases, there can be but one gas
phase, four solid phases will be necessary in order to form an invariant
system. When only three solid phases are present, the system is univariant;
and when only two solid phases coexist with gas, it is bivariant. If,
however, we make the restriction that the gas pressure is constant, we
diminish the variability by one.

On account of their great industrial importance, we shall describe briefly
some of the systems belonging to this class.

Iron, Carbon Monoxide, Carbon Dioxide.--Some of the most important systems
of three components in which equilibrium exists between solid and gas
phases are those formed by the three components--iron, carbon monoxide, and
carbon dioxide--and they are of importance especially for the study of the
processes occurring in the blast furnace.

If carbon monoxide is passed over reduced iron powder at a temperature of
about 600°, the iron is oxidized and the carbon monoxide reduced with
separation of carbon in accordance with the equation

  Fe + CO = FeO + C

This reaction is succeeded by the two reactions

  FeO + CO = Fe + CO_{2}
    CO_{2} + C = 2CO

{305}

[Illustration: FIG. 121.]

The former of these reactions is not complete, but leads to a definite
equilibrium. The result of the different reactions is therefore an
equilibrium between the three solid phases, carbon, iron, and ferrous
oxide, and the gas phase consisting of carbon monoxide and dioxide. We have
here four phases; and if the total pressure is maintained constant,
equilibrium can occur only at a definite temperature.

Since, under certain conditions, we can also have the reaction

  Fe_{3}O_{4} + CO = 3FeO + CO_{2}

{306} a second series of equilibria can be obtained of a character similar
to the former. These various equilibria have been investigated by Baur and
Glaessner,[378] and the following is a short account of the results of
their work.

Mixtures of the solid phases in equilibrium with carbon monoxide and
dioxide were heated in a porcelain tube at a definite temperature until
equilibrium was produced, and the gas was then pumped off and analyzed. The
results which were obtained are given in the following tables, and
represented graphically in Fig. 121.

  SOLID PHASES: Fe_{3}O_{4}; FeO.

  -----------------------------------------------------------------
      |             | Duration of    |              | Percentage of
  No. | Tube filled | the experiment | Temperature. |
      |    with     | in hours.      |              | CO_{2} | CO
  -----------------------------------------------------------------
    1 |    CO       |       14       |     600°     |  59.3  | 40.7
    2 |    CO       |       15       |     590°     |  54.7  | 45.3
    3 |    CO_{2}   |       16       |     590°     |  64.6  | 35.4
    4 |    CO       |       24       |     590°     |  58.4  | 41.6
    5 |    CO       |       22       |     730°     |  67.7  | 32.3
    6 |    CO_{2}   |       22       |     730°     |  86.1  | 31.9
    7 |    CO       |       22       |     750°     |  68.4  | 31.6
    8 |    CO_{2}   |       22       |     610°     |  64.9  | 35.1
    9 |    CO       |       23       |     420°     |  56.0  | 44.0
   10 |    CO       |       47       |     350°     |  65.6  | 34.4
   11 |    CO_{2}   |       46       |     350°     |  72.8  | 27.2
   12 |    CO       |       53       |     350°     |  64.0  | 36.0
   13 |    CO       |       18       |     570°     |  53.4  | 46.6
   14 |    CO       |       19       |     680°     |  60.5  | 39.5
   15 |    CO_{2}   |       24       |     540°     |  55.5  | 44.5
   16 |    CO       |       21       |     630°     |  57.5  | 42.5
   17 |    CO_{2}   |       17       |     690°     |  65.5  | 34.5
   18 |    CO_{2}   |       17       |     670°     |  67.0  | 33.0
   19 |    CO_{2}   |       24       |     410°     |  58.5  | 41.5
   20 |    CO       |       24       |     490°     |  51.7  | 48.8
   21 |    CO_{2}   |       23       |     590°     |  54.4  | 45.6
   22 |    CO_{2}   |        4       |     950°     |  77.0  | 23.0
   23 |    CO_{2}   |       15       |     850°     |  73.4  | 26.6
   24 |    CO       |        8       |     800°     |  71.2  | 28.8
   25 |    CO_{2}   |       24       |     540°     |  56.7  | 43.3
  -----------------------------------------------------------------

{307}

  SOLID PHASES: FeO; Fe.

  ------+-------------+-------------+--------------+---------------
        |             | Duration of |              | Percentage of
   No.  | Tube filled | experiment  | Temperature. |
        | with        | in hours.   |              | CO_{2} |  CO
  ------+-------------+-------------+--------------+--------+------
  I.    |    CO       |      15     |     800°     |  35.2  | 64.8
  II.   |    CO       |      18     |     530°     |  29.1  | 70.9
  III.  |    CO       |      13     |     880°     |  30.2  | 69.6
  IV.   |    CO_{2}   |      24     |     870°     |  32.3  | 67.7
  V.    |    CO       |      18     |     760°     |  36.9  | 63.1
  VI.   |    CO_{2}   |      16     |     820°     |  34.7  | 65.3
  VII.  |    CO_{2}   |      18     |     730°     |  41.1  | 58.9
  VIII. |    CO       |      18     |     630°     |  34.9  | 65.1
  IX.   |    CO_{2}   |      17     |     630°     |  61.6  | 58.4
  X.    |    CO       |      18     |     540°     |  25.0  | 75.0
  XI.   |    CO_{2}   |      25     |     540°     |  36.5  | 63.5
  ------+-------------+-------------+--------------+--------+------

As is evident from the above tables and from the curves in Fig. 121, the
curve of equilibrium in the case of the reaction

  Fe_{3}O_{4} + CO = 3FeO + CO_{2}

exhibits a maximum for the ratio CO : CO_{2}, at 490°, while, for the
reaction

  FeO + CO = Fe + CO_{2}

this ratio has a minimum value at 680°. From these curves can be derived
the conditions under which the different solid phases can exist in contact
with gas. Thus, for example, at a temperature of 690°, FeO and Fe_{3}O_{4}
can coexist with a mixture of 65.5 per cent. of CO_{2} and 34.5 per cent.
of CO. If the partial pressure of CO_{2} is increased, there occurs the
reaction

  3FeO + CO_{2} = Fe_{3}O_{4} + CO

and if carbon dioxide is added in sufficient amount, the ferrous oxide
finally disappears completely. If, on the other hand, the partial pressure
of CO is increased, there occurs the reaction

  Fe_{3}O_{4} + CO = 3FeO + CO_{2}

and all the ferric oxide can be made to disappear. We see, therefore, that
Fe_{3}O_{4} can only exist at temperatures and in {308} contact with
mixtures of carbon monoxide and dioxide, represented by the area which lies
below the under curve in Fig. 121. Similarly, the region of existence of
FeO is that represented by the area between the two curves; while metallic
iron can exist under the conditions of temperature and composition of gas
phase represented by the area above the upper curve in Fig. 121. If,
therefore, ferric oxide or metallic iron is heated for a sufficiently long
time at temperatures above 700° (to the right of the dotted line; _vide
infra_), complete transformation to ferrous oxide finally occurs.

In another series of equilibria which can be obtained, carbon is one of the
solid phases. In Fig. 121 the equilibria between carbon, carbon monoxide,
and carbon dioxide under pressures of one and of a quarter atmosphere, are
represented by dotted lines.[379]

If we consider only the dotted line on the right, representing the
equilibria under atmospheric pressure, we see that the points in which the
dotted line cuts the other two curves must represent systems in which
carbon monoxide and carbon dioxide are in equilibrium with FeO +
Fe_{3}O_{4} + C, on the one hand, and with Fe + FeO + C on the other. These
systems can only exist at one definite temperature, if we make the
restriction that the pressure is maintained constant (atmospheric
pressure). Starting, therefore, with the equilibrium FeO + Fe_{3}O_{4} + CO
+ CO_{2} at a temperature of about 670°, and then add carbon to the system,
the reaction

  C + CO_{2} = 2CO

will occur, because the concentration of CO_{2} is greater than what
corresponds with the system FeO + Fe_{3}O_{4} + C in equilibrium with
carbon monoxide and dioxide. In consequence of this reaction, the
equilibrium between FeO + Fe_{3}O_{4} and the gas phase is disturbed, and
the change in the composition of the gas phase is opposed by the reaction
Fe_{3}O_{4} + CO = 3FeO + CO_{2}, which continues until either all the
carbon {309} or all the ferric oxide is used up. If the ferric oxide first
disappears, the equilibrium corresponds with a point on the dotted line in
the middle area of Fig. 121, which represents equilibria between FeO + C as
solid phases, and a mixture of carbon monoxide and dioxide as gas phase. If
the temperature is higher than 685°, at which temperature the curve for
C--CO--CO_{2} cuts that for Fe--FeO--CO--CO_{2}; then, when all the ferric
oxide has disappeared, the concentration of CO_{2} is still too great for
the coexistence of FeO and C. Consequently, there occurs the reaction C +
CO_{2} = 2CO, and the composition of the gas phase alters until a point on
the upper curve is reached. A further increase in the concentration of CO
is opposed by the reaction FeO + CO = Fe + CO_{2}, and the pressure remains
constant until all the ferrous oxide is reduced and only iron and carbon
remain in equilibrium with gas. If the quantities of the substances have
been rightly chosen, we ultimately reach a point on the dotted curve in the
upper part of Fig. 121.

Fig. 121 shows us, also, what are the conditions under which the reduction
of ferric to ferrous oxide by carbon can occur. Let us suppose, for
example, that we start with a mixture of carbon monoxide and dioxide at
about 600° (the lowest point on the dotted line), and maintain the total
pressure constant and equal to one atmosphere. If the temperature is
increased, the concentration of the carbon dioxide will diminish, owing to
the reaction C + CO_{2} = 2CO, but the ferric oxide will undergo no change
until the temperature reaches 647°, the point of intersection of the dotted
curve with the curve for FeO and Fe_{3}O_{4}. At this point further
increase in the concentration of carbon monoxide is opposed by the
reduction of ferric oxide in accordance with the equation Fe_{3}O_{4} + CO
= 3FeO + CO_{2}. The pressure, therefore, remains constant until all the
ferric oxide has disappeared. If the temperature is still further raised,
we again obtain a univariant system, FeO + C, in equilibrium with gas
(univariant because the total pressure is constant); and if the temperature
is raised the composition of the gas must undergo change. This is effected
by the reaction C + CO_{2} = 2CO. When the {310} temperature rises to 685°,
at which the dotted curve cuts the curve for Fe--FeO, further change is
prevented by the reaction FeO + CO = Fe + CO_{2}. When all the ferrous
oxide is used up, we obtain the system Fe + C in equilibrium with gas. If
the temperature is now raised, the composition of the gas undergoes change,
as shown by the dotted line. The two temperatures, 647° and 685°, give,
evidently, the limits within which ferric or ferrous oxide can be reduced
directly by carbon.

It is further evident that at any temperature to the right of the dotted
line, carbon is unstable in presence of iron or its oxides; while at
temperatures lower than those represented by the dotted line, it is stable.
In the blast furnace, therefore, separation of carbon can occur only at
lower temperatures, and the carbon must disappear on raising the
temperature.

Finally, it may be remarked that the equilibrium curves show that ferrous
oxide is most easily reduced at 680°, since the concentration of the carbon
monoxide required at this temperature is a minimum. On the other hand,
ferric oxide is reduced with greatest difficulty at 490°, since at this
temperature the requisite concentration of carbon monoxide is a maximum.

Other equilibria between solid and gas phases are: Equilibrium between
iron, ferric oxide, water vapour, and hydrogen,[380] and the equilibria
between carbon, carbon monoxide, carbon dioxide, water vapour, and
hydrogen,[381] which is of importance for the manufacture of water gas.

       *       *       *       *       *


{311}

CHAPTER XVIII

SYSTEMS OF FOUR COMPONENTS

In the systems which have so far been studied, we have met with cases where
two or three components could enter into combination; but in no case did we
find double decomposition occurring. The reason of this is that in the
systems previously studied, in which double decomposition might have been
possible, namely in those systems in which two salts acted as components,
the restriction was imposed that either the basic or the acid constituent
of these salts must be the same; a restriction imposed, indeed, for the
very purpose of excluding double decomposition. Now, however, we shall
allow this restriction to fall, thereby extending the range of study.

Hitherto, in connection with four-component systems, the attention has been
directed solely to the study of aqueous solutions of salts, and more
especially of the salts which occur in sea-water, _i.e._ chiefly, the
sulphates and chlorides of magnesium, potassium, and sodium. The importance
of these investigations will be recognized when one recollects that by the
evaporation of sea-water there have been formed the enormous salt-beds at
Stassfurt, which constitute at present the chief source of the sulphates
and chlorides of magnesium and potassium. The investigations, therefore,
are not only of great geological interest as tending to elucidate the
conditions under which these salt-beds have been formed, but are of no less
importance for the industrial working of the deposits.

It is, however, not the intention to enter here into any detailed
description of the different systems which have so far been studied, and of
the sometimes very complex relationships {312} met with, but merely to
refer briefly to some points of more general import in connection with
these systems.[382]

Reciprocal Salt-Pairs. Choice of Components.--When two salts undergo double
decomposition, the interaction can be expressed by an equation such as

  NH_{4}Cl + NaNO_{3} = NaCl + NH_{4}NO_{3}

Since one pair of salts--NaCl + NH_{4}NO_{3}--is formed from the other
pair--NH_{4}Cl + NaNO_{3}--by double decomposition, the two pairs of salts
are known as _reciprocal salt-pairs_.[383] It is with systems in which the
component salts form reciprocal salt-pairs that we have to deal here.

It must be noted, however, that the four salts formed by two reciprocal
salt-pairs do not constitute a system of four, but only of _three_
components. This will be understood if it is recalled that only so many
constituents are taken as components as are necessary to _express_ the
composition of all the phases present (p. 12). It will be seen, now, that
the composition of each of the four salts which can be present together can
be expressed in terms of three of them. Thus, for example, in the case of
NH_{4}Cl, NaNO_{3}, NH_{4}NO_{3}, NaCl, we can express the composition of
NH_{4}Cl by NH_{4}NO_{3} + NaCl - NaNO_{3}; or of NaNO_{3} by NH_{4}NO_{3}
+ NaCl - NH_{4}Cl. In all these cases it will be seen that negative
quantities of one of the components must be employed; but that we have seen
to be quite permissible (p. 12). The number of components is, therefore,
three; but any three of the four salts can be chosen.

Since, then, two reciprocal salt-pairs constitute only three {313}
components or independently variable constituents, another component is
necessary in order to obtain a four-component system. As such, we shall
choose water.

Transition Point.--In the case of the formation of double salts from two
single salts, we saw that there was a point--the _quintuple point_--at
which five phases could coexist. This point we also saw to be a transition
point, on one side of which the double salt, on the other side the two
single salts in contact with solution, were found to be the stable system.
A similar behaviour is found in the case of reciprocal salt-pairs. The
four-component system, two reciprocal salt-pairs and water, can give rise
to an invariant system in which the six phases, four salts, solution,
vapour, can coexist; the temperature at which this is possible constitutes
a _sextuple point_. Now, this sextuple point is also a transition point, on
the one side of which the one salt-pair, on the other side the reciprocal
salt-pair, is stable in contact with solution.

The sextuple point is the point of intersection of the curves of six
univariant systems, viz. four solubility curves with three solid phases
each, a vapour-pressure curve for the system: two reciprocal
salt-pairs--vapour; and a transition curve for the condensed system: two
reciprocal salt-pairs--solution. If we omit the vapour phase and work under
atmospheric pressure (in open vessels), we find that the transition point
is the point of intersection of four solubility curves.

Just as in the case of three-component systems we saw that the presence of
one of the single salts along with the double salt was necessary in order
to give a univariant system, so in the four-component systems the presence
of a third salt is necessary as solid phase along with one of the
salt-pairs. In the case of the reciprocal salt-pairs mentioned above, the
transition point would be the point of intersection of the solubility
curves of the systems with the following groups of salts as solid phases:
Below the transition point: NH_{4}Cl + NaNO_{3} + NaCl; NH_{4}Cl + NaNO_{3}
+ NH_{4}NO_{3}; above the transition point: NaCl + NH_{4}NO_{3} + NaNO_{3};
NaCl + NH_{4}NO_{3} + NH_{4}Cl. From this we see that the two salts
NH_{4}Cl and NaNO_{3} would be able to exist together with solution below
the transition point, but not above it. This transition point has not been
determined. {314}

Formation of Double Salts.--In all cases of four-component systems so far
studied, the transition points have not been points at which one salt-pair
passed into its reciprocal, but at which a double salt was formed. Thus, at
4.4° Glauber's salt and potassium chloride form glaserite and sodium
chloride, according to the equation

  2Na_{2}SO_{4},10H_{2}O + 3KCl = K_{3}Na(SO_{4})_{2} + 3NaCl + 20H_{2}O

Above the transition point, therefore, there would be K_{3}Na(SO_{4})_{2},
NaCl and KCl; and it may be considered that at a higher temperature the
double salt would interact with the potassium chloride according to the
equation

  K_{3}Na(SO_{4})_{2} + KCl = 2K_{2}SO_{4} + NaCl

thus giving the reciprocal of the original salt-pair. This point has,
however, not been experimentally realized.[384]

Transition Interval.--A double salt, we learned (p. 277), when brought in
contact with water at the transition point undergoes partial decomposition
with separation of one of the constituent salts; and only after a certain
range of temperature (transition interval) has been passed, can a pure
saturated solution be obtained. A similar behaviour is also found in the
case of reciprocal salt-pairs. If one of the salt-pairs is brought in
contact with water at the transition point, interaction will occur and one
of the salts of the reciprocal salt-pair will be deposited; and this will
be the case throughout a certain range of temperature, after which it will
be possible to prepare a solution saturated only for the one salt-pair. In
the case of ammonium chloride and sodium nitrate the lower limit of the
transition interval is 5.5°, so that above this temperature and up to that
of the transition point (unknown), ammonium chloride and sodium nitrate in
contact with water would give rise to a third salt by double decomposition,
in this case to sodium chloride.[385]

{315}

Graphic Representation.--For the graphic representation of systems of four
components, four axes may be chosen intersecting at a point like the edges
of a regular octahedron (Fig. 122).[386] Along these different axes the
equivalent molecular amounts of the different salts are measured.

[Illustration: FIG. 122.]

[Illustration: FIG. 123.]

To represent a given system consisting of _x_B, _y_C, and _z_D in a given
amount of water (where B, C, and D represent equivalent molecular amounts
of the salts), measure off on OB and OC lengths equal to _x_ and _y_
respectively. The point of intersection _a_ (Fig. 122) represents a
solution containing _x_B and _y_C (_ab_ = _x_; _ac_ = _y_). From _a_ a line
_a_P is drawn parallel to OD and equal to _z_. P then represents the
solution of the above composition.

It is usual, however, not to employ the three-dimensional figure, but its
horizontal and vertical projections. Fig. 122, if projected on the base of
the octahedron, would yield a diagram such as is shown in Fig. 123. The
projection of the edges of the octahedron form two axes at right angles and
give rise to four quadrants similar to those employed for the
representation of ternary solutions (p. 273). Here, the point _a_
represents a ternary solution saturated with respect to B and C; and _a_P,
quaternary solutions in equilibrium with the same two salts as solid
phases. Such a diagram represents the conditions of equilibrium only for
one definite temperature, and corresponds, therefore, to the isothermal
diagrams for ternary systems (p. 273). In such a diagram, since the
temperature and {316} pressure are constant (vessels open to the air), a
surface will represent a solution in equilibrium with only one solid phase;
a line, a solution with two solid phases, and a point, one in equilibrium
with three solid phases.

[Illustration: FIG. 124.]

Example.--As an example of the complete isothermal diagram, there may be
given one representing the equilibria in the system composed of water and
the reciprocal salt-pair sodium sulphate--potassium chloride for the
temperature 0° (Fig. 124).[387] The amounts of the different salts are
measured along the four axes, and the composition of the solution is {317}
expressed in equivalent gram-molecules per 1000 gram-molecules of
water.[388]

The outline of this figure represents four ternary solutions in which the
component salts have a common acid or basic constituent; viz. sodium
chloride--sodium sulphate, sodium sulphate--potassium sulphate, potassium
sulphate--potassium chloride, potassium chloride--sodium chloride. These
four sets of curves are therefore similar to those discussed in the
previous chapter. In the case of sodium and potassium sulphate, a double
salt, _glaserite_ [K_{3}Na(SO_{4})_{2}] is formed. Whether glaserite is
really a definite compound or not is still a matter of doubt, since
isomorphic mixtures of Na_{2}SO_{4} and K_{2}SO_{4} have been obtained.
According to van't Hoff and Barscholl,[389] glaserite is an isomorphous
mixture; but Gossner[390] considers it to be a definite compound having the
formula K_{3}Na(SO_{4})_{2}. Points VIII. and IX. represent solutions
saturated with respect to glaserite and sodium sulphate, and glaserite and
potassium sulphate respectively.

The lines which pass inwards from these boundary curves represent solutions
containing three salts, but in contact with only two solid phases; and the
points where three lines meet, or where three fields meet, represent
solutions in equilibrium with three solid phases; with the phases, namely,
belonging to the three concurrent fields.

If it is desired to represent a solution containing the salts say in the
proportions, 51Na_{2}Cl_{2}, 9.5K_{2}Cl_{2}, 3.5K_{2}SO_{4}, the difficulty
is met with that two of the salts, sodium chloride and potassium sulphate,
lie on opposite axes. To overcome this difficulty the difference 51 - 3.5 =
47.5 is taken and measured off along the sodium chloride axis; and the
solution is therefore represented by the point 47.5Na_{2}Cl_{2},
9.5K_{2}Cl_{2}. In order, therefore, to find the amount of potassium
sulphate present {318} from such a diagram, it is necessary to know the
total number of salt molecules in the solution. When this is known, it is
only necessary to subtract from it the sum of the molecules of sodium and
potassium chloride, and the result is equal to twice the number of
potassium sulphate molecules. Thus, in the above example, the total number
of salt molecules is 64. The number of molecules of sodium and potassium
chloride is 57; 64 - 57 = 7, and therefore the number of potassium sulphate
molecules is 3.5.

Another method of representation employed is to indicate the amounts of
only two of the salts in a plane diagram, and to measure off the total
number of molecules along a vertical axis. In this way a solid model is
obtained.

The numerical data from which Fig. 124 was constructed are contained in the
following table, which gives the composition of the different solutions at
0°:--[391]

  ----------------------------------------
         |                               |
         |                               |
  Point. |       Solid phases.           |
         |                               |
  ----------------------------------------
      I. | NaCl                          |
         |                               |
     II. | KCl                           |
         |                               |
    III. | Na_{2}SO_{4},10H_{2}O         |
         |                               |
     IV. | K_{2}SO_{4}                   |
         |                               |
      V. | NaCl; KCl                     |
         |                               |
     VI. | NaCl; Na_{2}SO_{4},10H_{2}O   |
         |                               |
    VII. | KCl; K_{2}SO_{4}              |
         |                               |
   VIII. |{ Glaserite;                  }|
         |{ Na_{2}SO_{4},10H_{2}O       }|
         |                               |
     IX. |  Glaserite; K_{2}SO_{4}       |
         |                               |
      X. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
         |{ NaCl                        }|
         |                               |
     XI. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
         |{ glaserite                   }|
         |                               |
    XII. |  K_{2}SO_{4}; KCl; glaserite  |
  ----------------------------------------
  [Transcriber's note: table continued below...]
  -------------------------------------------------------------------------
   Composition of solution in gram-mols.                       | Total
   per 1000 gram-mols. water.                                  | number
  -------------------------------------------------------------| of salt
   Na_{2}Cl_{2}. | K_{2}Cl_{2}. | Na_{2}SO_{4}. | K_{2}SO_{4}. | molecules.
  -------------------------------------------------------------------------
        55       |      --      |      --       |      --      |   55
                 |              |               |              |
         --      |     34.5     |      --       |      --      |   34.5
                 |              |               |              |
         --      |      --      |      6        |      --      |    6
                 |              |               |              |
         --      |      --      |      --       |      9       |    9
                 |              |               |              |
        46.5     |     12.5     |      --       |      --      |   59
                 |              |               |              |
        47.5     |      --      |      8        |      --      |   55.5
                 |              |               |              |
         --      |     34.5     |      --       |      1       |   35.5
                 |              |               |              |
         --      |      --      |     10        |     10       |   20
                 |              |               |              |
                 |              |               |              |
         --      |      --      |      7.5      |     10       |   17.5
                 |              |               |              |
        51       |      9.5     |      --       |      3.5     |   64
                 |              |               |              |
                 |              |               |              |
        40.5     |     13       |      --       |      3.5     |   57
                 |              |               |              |
                 |              |               |              |
        18       |     23       |      --       |      3       |   44
  -------------------------------------------------------------------------

From the aspect of these diagrams the conditions under which the salts can
coexist can be read at a glance. Thus, {319} for example, Fig. 124 shows
that at 0° Glauber's salt and potassium chloride can exist together with
solution; namely, in contact with solutions having the composition X--XI.
This temperature must therefore be below the transition point of this
salt-pair (p. 314). On raising the temperature to 4.4°, it is found that
the curve VIII.--XI. moves so that the point XI. coincides with point X. At
this point, therefore, there will be _four_ concurrent fields, viz.
Glauber's salt, potassium chloride, glaserite, and sodium chloride. But
these four salts can coexist with solution only at the transition point; so
that 4.4° is the transition temperature of the salt-pair: Glauber's
salt--potassium chloride. At higher temperatures the line VIII.--XI. moves
still further to the left, so that the field for Glauber's salt becomes
entirely separated from the field for potassium chloride. This shows that
at temperatures above the transition point the salt-pair Glauber's
salt--potassium chloride cannot coexist in presence of solution.

[Illustration: FIG. 125.]

If it is only desired to indicate the mutual relationships of the different
components and the conditions for their coexistence (_paragenesis_), a
simpler diagram than Fig. 124 can be employed. Thus if the boundary curves
of Fig. 124 are so drawn that they cut one another at right angles, a
figure such as Fig. 125 is obtained, the Roman numerals here corresponding
with those in Fig. 124.

Ammonia-Soda Process.--One of the most important applications of the Phase
Rule to systems of four components with reciprocal salt-pairs has recently
been made by Fedotieff[392] in his investigations of the conditions for the
formation of sodium carbonate by the so-called ammonia-soda (Solvay) {320}
process.[393] This process consists, as is well known, in passing carbon
dioxide through a solution of common salt saturated with ammonia.

Whatever differences of detail there may be in the process as carried out
in different manufactories, the reaction which forms the basis of the
process is that represented by the equation

  NaCl + NH_{4}HCO_{3} = NaHCO_{3} + NH_{4}Cl

We are dealing here, therefore, with reciprocal salt-pairs, the behaviour
of which has just been discussed in the preceding pages. The present case
is, however, simpler than that of the salt-pair Na_{2}SO_{4}.10H_{2}O +
KCl, inasmuch as under the conditions of experiment neither hydrates nor
double salts are formed. Since the study of the reaction is rendered more
difficult on account of the fact that ammonium bicarbonate in solution,
when under atmospheric pressure, undergoes decomposition at temperatures
above 15°, this temperature was the one chosen for the detailed
investigation of the conditions of equilibrium. Since, further, it has been
shown by Bodländer[394] that the bicarbonates possess a definite solubility
only when the pressure of carbon dioxide in the solution has a definite
value, the measurements were carried out in solutions saturated with this
gas. This, however, does not constitute another component, because we have
made the restriction that the sum of the partial pressures of carbon
dioxide and water vapour is equal to 1 atmosphere. The concentration of the
carbon dioxide is, therefore, not independently variable (p. 10).

[Illustration: FIG. 126.]

In order to obtain the data necessary for a discussion of the conditions of
soda formation by the ammonia-soda process, solubility determinations with
the four salts, NaCl, NH_{4}Cl, NH_{4}HCO_{3}, and NaHCO_{3} were made,
first with the single salts and then {321} with the salts in pairs. The
results obtained are represented graphically in Fig. 126, which is an
isothermal diagram similar to that given by Fig. 124. The points I., II.,
III., IV., represent the composition of solutions in equilibrium with two
solid salts. We have, however, seen (p. 314) that the transition point,
when the experiment is carried out under constant pressure (atmospheric
pressure), is the point of intersection of four solubility curves, each of
which represents the composition of solutions in equilibrium with three
salts, viz. one of the reciprocal salt-pairs along with a third salt.
Since, now, it was found that the stable salt-pair at temperatures between
0° and 30° is sodium bicarbonate and ammonium chloride, determinations were
made of the composition of solutions in equilibrium with NaHCO_{3} +
NH_{4}Cl + NH_{4}HCO_{3} and with NaHCO_{3} + NH_{4}Cl + NaCl as solid
phases. Under the {322} conditions of experiment (temperature = 15°) sodium
chloride and ammonium bicarbonate cannot coexist in contact with solution.
These determinations gave the data necessary for the construction of the
complete isothermal diagram (Fig. 127). The most important of these data
are given in the following table (temperature, 15°):--

  -------------------------------------------------------------------------
         |                  | Composition of the solution in gram-molecules
         |                  | to 1000 gram-molecules
  Point. |  Solid phases.   | of water.
         |                  |----------------------------------------------
         |                  | NaHCO_{3} | NaCl | NH_{4}HCO_{3} | NH_{4}Cl
  -------------------------------------------------------------------------
   --    | NaHCO_{3}        |    1.08   |  --  |       --      |     --
   --    | NaCl             |     --    | 6.12 |       --      |     --
   --    | NH_{4}HCO_{3}    |     --    |  --  |      2.36     |     --
   --    | NH_{4}Cl         |     --    |  --  |       --      |    6.64
   I.    | NaHCO_{3}; NaCl  |    0.12   | 6.06 |       --      |     --
   II.   | NaCl; NH_{4}Cl   |     --    | 4.55 |       --      |    3.72
   III.  | NH_{4}Cl;        |     --    |  --  |      0.81     |    6.40
         | NH_{4}HCO_{3}    |           |      |               |
   IV.   | NaHCO_{3};       |    0.71   |  --  |      2.16     |     --
         | NH_{4}HCO_{3}    |           |      |               |
   P_{1} | NaHCO_{3};       |    0.93   | 0.51 |       --      |    6.28
         | NH_{4}HCO_{3};   |           |      |               |
         | NH_{4}Cl         |           |      |               |
   P_{2} | NaHCO_{3};       |    0.18   | 4.44 |       --      |    3.73
         | NaCl; NH_{4}Cl   |           |      |               |
  -------------------------------------------------------------------------

With reference to the solution represented by the point P_{1}, it may be
remarked that it is an incongruently saturated solution (p. 279). If sodium
chloride is added to this solution, the composition of the latter undergoes
change; and if a sufficient amount of the salt is added, the solution P_{2}
is obtained.

Turning now to the practical application of the data so obtained, consider
first what is the influence of concentration on the yield of soda. Since
the reaction consists essentially in a double decomposition between sodium
chloride and ammonium bicarbonate, then, after the deposition of the sodium
bicarbonate, we obtain a solution containing sodium chloride, ammonium
chloride, and sodium bicarbonate. In order to ascertain to what extent the
sodium chloride has been converted into solid sodium bicarbonate, it is
necessary to examine the composition of the solution which is obtained
{323} with definite amounts of sodium chloride and ammonium bicarbonate.

[Illustration: FIG. 127.]

Consider, in the first place, the solutions represented by the curve
P_{2}P_{1}. With the help of this curve we can state the conditions under
which a solution, saturated for ammonium chloride, is obtained, after
deposition of sodium bicarbonate. In the following table the composition of
the solutions is given which are obtained with different initial amounts of
sodium chloride and ammonium bicarbonate. The last two columns give the
percentage amount of the sodium used, which is deposited as solid sodium
bicarbonate (U_{Na}); and likewise the percentage amount of ammonium
bicarbonate which is usefully converted into sodium bicarbonate, that is to
say, the amount of the radical HCO_{3} deposited (U_{NH_{4}}):-- {324}

  ------+---------------------+
        |Initial composition  |
        |of the solutions:    |
        |grams of salt to 1000|
  Point.|grams of water.      |
        +------+--------------+
        | NaCl | NH_{4}HCO_{3}|
  ------+------+--------------+
  P_{2} | 479  |     295      |
  --    | 448  |     360      |
  --    | 417  |     431      |
  P_{1} | 397  |     496      |
  ------+------+--------------+
  [Transcriber's note: table continued below...]
  +----------------------------------+---------+----------
  |                                  |         |
  |Composition of solutions obtained:|         |
  |gram-equivalents per 1000 grams   |U_{Na}   |U_{NH_{4}}
  |of water.                         |per cent.|per cent.
  +----------+------+------+---------+         |
  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |         |
  +----------+------+------+---------+---------+----------
  |   0.18   | 8.17 | 4.62 |  3.73   |  43.4   |   95.1
  |   0.31   | 7.65 | 3.39 |  4.56   |  55.7   |   93.4
  |   0.51   | 7.13 | 2.19 |  5.45   |  69.2   |   90.5
  |   0.92   | 6.79 | 1.44 |  6.28   |  78.8   |   85.1
  +----------+------+------+---------+---------+----------

This table shows that the greater the excess of sodium chloride, the
greater is the percentage utilization of ammonia (Point P_{2}); and the
more the amount of sodium chloride decreases, the greater is the percentage
amount of sodium chloride converted into bicarbonate. In the latter case,
however, the percentage utilization of the ammonium bicarbonate decreases;
that is to say, less sodium bicarbonate is deposited, or more of it remains
in solution.

Consider, in the same manner, the relations for solutions represented by
the curve P_{2}IV, which gives the composition of solutions saturated with
respect to sodium bicarbonate and ammonium bicarbonate. In this case we
obtain the following results:--

  ------+---------------------+
        |Initial composition  |
        |of the solutions:    |
        |grams of salt to 1000|
  Point.|grams of water.      |
        +------+--------------+
        | NaCl | NH_{4}HCO_{3}|
  ------+------+--------------+
  P_{1} | 397  |     496      |
  --    | 351  |     446      |
  --    | 316  |     412      |
  --    | 294  |     389      |
  --    | 234  |     327      |
  ------+------+--------------+
  [Transcriber's note: table continued below...]
  +----------------------------------+------+----------
  |                                  |      |
  |Composition of solutions obtained:|      |
  |in gram-equivalents per 1000 grams|U_{Na}|U_{NH_{4}}
  |of water.                         |      |
  +----------+------+------+---------+      |
  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |      |
  +----------+------+------+---------+------+----------
  |   0.92   | 6.79 | 1.44 |  6.28   | 78.8 |   85.1
  |   0.99   | 6.00 | 1.34 |  5.65   | 77.7 |   82.5
  |   1.07   | 5.41 | 1.27 |  5.21   | 76.4 |   79.5
  |   1.12   | 5.03 | 1.23 |  4.92   | 75.5 |   75.1
  |   1.30   | 4.00 | 1.16 |  4.14   | 71.0 |   68.6
  +----------+------+------+---------+------+----------

As is evident from this table, diminution in the relative amount of sodium
chloride exercises only a slight influence {325} on the utilization of this
salt, but is accompanied by a rapid diminution of the effective
transformation of the ammonium bicarbonate. So far as the efficient
conversion of the sodium is concerned, we see that it reaches its maximum
at the point P_{1}, and that it decreases both with increase and with
decrease of the relative amount of sodium chloride employed; and faster,
indeed, in the former than in the latter case. On the other hand, the
effective transformation of the ammonium bicarbonate reaches its maximum at
the point P_{2}, and diminishes with increase in the relative amount of
ammonium bicarbonate employed. Since sodium chloride is, in comparison with
ammonia--even when this is regenerated--a cheap material, it is evidently
more advantageous to work with solutions which are relatively rich in
sodium chloride (solutions represented by the curve P_{1}P_{2}). This fact
has also been established empirically.

When, as is the case in industrial practice, we are dealing with solutions
which are saturated not for two salts but only for sodium bicarbonate, it
is evident that we have then to do with solutions the composition of which
is represented by points in the area P_{1}P_{2}I,IV. Since in the
commercial manufacture, the aim must be to obtain as complete a utilization
of the materials as possible, the solutions employed industrially must lie
in the neighbourhood of the curves P_{2}P_{1}IV, as is indicated by the
shaded portion in Fig. 127. The best results, from the manufacturer's
standpoint, will be obtained, as already stated, when the composition of
the solutions approaches that given by a point on the curve P_{2}P_{1}.
Considered from the chemical standpoint, the results of the experiments
lead to the conclusion that the Solvay process, _i.e._ passage of carbon
dioxide through a solution of sodium chloride saturated with ammonia, is
not so good as the newer method of Schlösing, which consists in bringing
together sodium chloride and ammonium bicarbonate with water.[395]

{326}

Preparation of Barium Nitrite.--Mention may also be made here of the
preparation of barium nitrite by double decomposition of barium chloride
and sodium nitrite.[396]

The reaction with which we are dealing here is represented by the equation

  BaCl_{2} + 2NaNO_{2} = 2NaCl + Ba(NO_{2})_{2}

It was found that at the ordinary temperature NaCl and Ba(NO_{2})_{2} form
the stable salt-pair. If, therefore, barium chloride and sodium nitrite are
brought together with an amount of water insufficient for complete
solution, transformation to the stable salt-pair occurs, and sodium
chloride and barium nitrite are deposited. When, however, a stable
salt-pair is in its transition interval (p. 315), a third salt--in this
case barium chloride--will be deposited, as we have already learned. On
bringing barium chloride and sodium nitrite together with water, therefore,
three solid phases are obtained, viz. BaCl_{2}, NaCl, Ba(NO_{2})_{2}. These
three phases, together with solution and vapour, constitute a univariant
system, so that at each temperature the composition of the solution must be
constant.

Witt and Ludwig found that the presence of solid barium chloride can be
prevented by adding an excess of sodium nitrite, as can be readily foreseen
from what has been said. Since the solution in presence of the three solid
phases must have a definite composition at a definite temperature, the
addition of sodium nitrite to the solution must have, as its consequence,
the solution of an equivalent amount of barium chloride, and the deposition
of an equivalent amount of sodium chloride and barium nitrite. By
sufficient addition of sodium nitrite, the complete disappearance of the
solid barium chloride can be effected, and there will remain only the
stable salt-pair sodium chloride and barium nitrite. As was pointed out by
Meyerhoffer, however, the disappearance of the barium chloride is effected,
not by a change in the {327} composition of the solution, but by the
necessity for the composition of the solution remaining constant.

[Illustration: FIG. 128.]

Barium Carbonate and Potassium Sulphate.--As has been found by
Meyerhoffer,[397] these two salts form the stable pair, not only at the
ordinary temperature, but also at the melting point. For the ordinary
temperatures this was proved in the following manner: A solution with the
solid phases K_{2}SO_{4} and K_{2}CO_{3}.2H_{2}O in excess can only coexist
in contact either with BaCO_{3} or with BaSO_{4}, since, evidently, in one
of the two groups the stable system must be present. Two solutions were
prepared, each with excess of K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O, {328} and
to one was added BaCO_{3} and to the other BaSO_{4}. After stirring for a
few days, the barium sulphate was completely transformed to BaCO_{3},
whereas the barium carbonate remained unchanged. Consequently, BaCO_{3} +
K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O is stable, and, therefore, so also is
BaCO_{3} + K_{2}SO_{4}. That BaCO_{3} + K_{2}SO_{4} is the stable pair also
at the melting point was proved by a special analytical method which allows
of the detection of K_{2}CO_{3} in a mixture of the four solid salts. This
analysis showed that a mixture of BaCO_{3} + K_{2}SO_{4}, after being fused
and allowed to solidify, contains only small amounts of K_{2}CO_{3}; and
this is due entirely to the fact that BaCO_{3} + K_{2}SO_{4} on fusion
deposits a little BaSO_{4}, thereby giving rise at the same time to the
separation of an equivalent amount of K_{2}CO_{3}.

The different solubilities are shown in Fig. 128. In this diagram the
solubility of the two barium salts has been neglected. A is the solubility
of K_{2}CO_{3}.2H_{2}O; addition of BaCO_{3} does not alter this. B is the
solubility of K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} + BaCO_{3}. A and B almost
coincide, since the potassium sulphate is very slightly soluble in the
concentrated solution of potassium carbonate. D gives the concentration of
the solution in equilibrium with K_{2}SO_{4} + BaSO_{4}. The most
interesting point is C. This solution is obtained by adding a small
quantity of water to BaCO_{3} + K_{2}SO_{4}, whereupon, being in the
transition interval, BaSO_{4} separates out and an equivalent amount of
K_{2}CO_{3} goes into solution. C is the end point of the curve CO, which
is called the Guldberg-Waage curve, because these investigators determined
several points on it.

In their experiments, Guldberg and Waage found the ratio K_{2}CO_{3} :
K_{2}SO_{4} in solution to be constant and equal to 4. This result is,
however, not exact, for the curve CO is not a straight line, as it should
be if the above ratio were constant; but it is concave to the abscissa
axis, and more so at lower than at higher temperatures.

The following table refers to the temperature of 25°. The Roman numbers in
the first column refer to the points in Fig. 128. The numbers in the column
[Sigma]_k__{2} give the amount, {329} in gram-molecules, of K_{2}CO_{3} +
K_{2}SO_{4} contained in 1000 gram-molecules of water:--

  SOLUBILITY DETERMINATIONS AT 25°.

  -----+-------------------------------------+-----------------------+
       |                                     | 100 gms. of the       |
       |                                     | solution contain,     |
   No. |           Solid phases.             | in grams,             |
       |                                     |           |           |
       |                                     |K_{2}CO_{3}|K_{2}SO_{4}|
  -----+-------------------------------------+-----------+-----------+
  I.   | K_{2}CO_{3}.2H_{2}O + BaCO_{3}      |    53.2   |     --    |
       |                                     |           |           |
  II.  |{ K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} }|    53.0   |    0.023  |
       |{ + BaCO_{3}                        }|           |           |
       |                                     |           |           |
  III.}| K_{2}SO_{4} + BaCO_{3}              |  { 28.5   |    0.886  |
  IV. }|                                     |  { 22.1   |    1.72   |
       |                                     |           |           |
  V.   | BaCO_{3} + K_{2}SO_{4} + BaSO_{4}   |    17.81  |    2.485  |
       |                                     |           |           |
  VI. }| K_{2}SO_{4} + BaSO_{4}              |  { 12.6   |    3.92   |
  VII.}|                                     |  {  5.85  |    6.76   |
       |                                     |           |           |
  VIII.| K_{2}SO_{4}                         |     --    |   10.76   |
       |                                     |           |           |
  IX. }| BaCO_{3} + BaSO_{4}                 |  {  7.35  |    0.602  |
  X.  }|                                     |  {  2.85  |    0.173  |
  -----+-------------------------------------+-----------+-----------+
  [Transcriber's note: table continued below...]
  -----+-----------------------+-----------------+-----------
       | 1000 moles            |                 |
       | of water contain,     |                 | K_{2}CO_{3}
   No. | in moles,             |[Sigma]_k__{2}   | -----------
       |           |           |                 | K_{2}SO_{4}
       |K_{2}CO_{3}|K_{2}SO_{4}|                 |
  -----+-----------+-----------+-----------------+-----------
  I.   |   147.9   |     --    |        --       |    --
       |           |           |                 |
  II.  |   147.8   |    0.051  |        --       |    --
       |           |           |                 |
       |           |           |                 |
  III.}|    52.58  |    1.296  |        --       |    --
  IV. }|    37.79  |    2.333  |        --       |    --
       |           |           |                 |
  V.   |    29.11  |    3.220  |       32.32     |   9.03
       |           |           |                 |
  VI. }|    19.66  |    4.853  |        --       |    --
  VII.}|     8.724 |    7.995  |        --       |    --
       |           |           |                 |
  VIII.|     --    |   12.47   |        --       |    --
       |           |           |                 |
  IX. }|    10.43  |    0.676  |       11.11     |  15.0
  X.  }|     3.828 |    0.184  |        4.0      |  21.0
  -----+-----------+-----------+-----------------+-----------

The Guldberg-Waage curve at 100° was also determined, and it was found that
the ratio K_{2}CO_{3}: K_{2}SO_{4} is also not constant, although the
variations are not so great as at 25°.

  GULDBERG-WAAGE CURVE AT 100°.

  ----------------------+-----------------------+-----------------+-------
                        |100 moles of water     |                 | K2CO3
       Solid phases.    |contain, in moles,     | [Sigma]_k__{2}  | -----
                        |           |           |                 | K2SO4
                        |K_{2}CO_{3}|K_{2}SO_{4}|                 |
  ----------------------+-----------+-----------+-----------------+-------
  BaCO_{3} + K_{2}SO_{4}|   23.9    |   12.65   |      35.65      |  1.82
         + BaSO_{4}     |           |           |                 |
  BaCO_{3} + BaSO_{4}   |    6.28   |    2.02   |       8.3       |  3.1
     "           "      |    3.17   |    0.851  |       4.025     |  3.7
  ----------------------+-----------+-----------+-----------------+-------

       *       *       *       *       *


{330}

APPENDIX

EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT

For the purpose of determining the transition temperature, a number of
methods have been employed, and the most important of these will be briefly
described here. In any given case it is sometimes possible to employ more
than one method, but all are not equally suitable, and the values of the
transition point obtained by the different methods are not always
identical. Indeed, a difference of several degrees in the value found may
quite well occur.[398] In each case, therefore, some care must be taken to
select the method most suitable for the purpose.

I. The Dilatometric Method.--Since, in the majority of cases,
transformation at the transition point is accompanied by an appreciable
change of volume, it is only necessary to ascertain the temperature at
which this change of volume occurs, in order to determine the transition
point. For this purpose the _dilatometer_ is employed, an apparatus which
consists of a bulb with capillary tube attached, and which constitutes a
sort of large thermometer (Fig. 129). Some of the substance to be examined
is passed into the bulb A through the tube B, which is then sealed off. The
rest of the bulb and a small portion of the capillary tube is then filled
with some liquid, which, of course, must be without chemical action on the
substance under investigation. A liquid, however, may be employed which
dissolves the substance, for, as we have seen (p. 70), the transformation
at the transition point is, as a rule, accelerated by the presence of a
solvent. On the other hand, the liquid must not dissolve in the substance
under examination, for the temperature of transformation would be thereby
altered.

{331}

In using the dilatometer, two methods of procedure may be followed.
According to the first method, the dilatometer containing the form stable
at lower temperatures is placed in a thermostat, maintained at a constant
temperature, until it has taken the temperature of the bath. The height of
the meniscus is then read on a millimetre scale attached to the capillary.
The temperature of the thermostat is then raised degree by degree, and the
height of the meniscus at each point ascertained. If, now, no change takes
place in the solid, the expansion will be practically uniform, or the rise
in the level of the meniscus per degree of temperature will be practically
the same at the different temperatures, as represented diagrammatically by
the line AB in Fig. 130. On passing through the transition point, however,
there will be a more or less sudden increase in the rise of the meniscus
per degree (line BC) if the specific volume of the form stable at higher
temperatures is greater than that of the original modification; thereafter,
the expansion will again be uniform (line CD). Similarly, on cooling,
contraction will at first be uniform and then at the transition point there
will be a relatively large diminution of volume.

[Illustration: FIG. 129.]

[Illustration: FIG. 130.]

If, now, transformation occurred immediately the transition point was
reached, the sudden expansion and contraction would take place at the same
temperature. It is, however, generally found that there is a lag, and that
with rising temperature the relatively large expansion does not take place
until a temperature somewhat higher than the transition point; and with
falling temperature the contraction occurs at a temperature somewhat below
the transition point. This is represented in Fig. 130 by the lines BC and
EF. The amount of lag will vary from case to case, and will {332} also
depend on the length of time during which the dilatometer is maintained at
constant temperature.

As an example, there may be given the results obtained in the determination
of the transition point at which sodium sulphate and magnesium sulphate
form astracanite (p. 268).[399] The dilatometer was charged with a mixture
of the two sulphates.

  --------------------------------------------------------
  Temperature. | Level of oil in capillary. | Rise per 1°.
  --------------------------------------------------------
     15.6°     |            134             |
     16.6°     |            141             |       7
     17.6°     |            148             |       7
     18.6°     |            154             |       6
     19.6°     |            161             |       7
     20.6°     |            168             |       7
     21.6°     |            241             |      73
     22.6°     |            243             |       2
     23.6°     |            251             |       8
     24.6°     |            259             |       8
  --------------------------------------------------------

The transition point, therefore, lies about 21.6° (p. 268).

The second method of manipulation depends on the fact that, while above or
below the transition point transformation of one form into the other can
take place, at the transition point the two forms undergo no change. The
bulb of the dilatometer is, therefore, charged with a mixture of the stable
and metastable forms and a suitable liquid, and is then immersed in a bath
at constant temperature. After the temperature of the bath has been
acquired, readings of the height of the meniscus are made from time to time
to ascertain whether expansion or contraction occurs. If expansion is
found, the temperature of the thermostat is altered until a temperature is
obtained at which a gradual contraction takes place. The transition point
must then lie between these two temperatures; and by repeating the
determinations it will be possible to reduce the difference between the
temperatures at which expansion and contraction take place to, say, 1°, and
to fix the temperature of the transition point, therefore, to within half a
degree. By this method the transition point, for example, of sulphur was
found to be 95.6° under a pressure of 4 atm.[400] The following are the
figures obtained by Reicher, who used a mixture {333} of 1 part of carbon
disulphide (solvent for sulphur) and 5 parts of turpentine as the measuring
liquid.

  TEMPERATURE 95.1°.

  -----------------------------------
  Time in minutes. | Level of liquid.
  -----------------------------------
          5        |      343.5
         30        |      340.5
         55        |      335.75
         65        |      333
  -----------------------------------

  TEMPERATURE 96.1°.

  -----------------------------------
  Time in minutes. | Level of liquid.
  -----------------------------------
          5        |      342.75
         30        |      354.75
         55        |      360.5
         60        |      361.5
  -----------------------------------

  TEMPERATURE 95.6°.

  -----------------------------------
  Time in minutes. | Level of liquid.
  -----------------------------------
          5        |      368.75
        100        |      368
        110        |      368.75
  -----------------------------------

At a temperature of 95.1° there is a contraction, _i.e._ monoclinic sulphur
passes into the rhombic, the specific volume of the former being greater
than that of the latter. At 96.1°, however, there is expansion, showing
that at this temperature rhombic sulphur passes into monoclinic; while at
95.6° there is neither expansion nor contraction. This is, therefore, the
transition temperature; and since the dilatometer was sealed up to prevent
evaporation of the liquid, the pressure within it was 4 atm.

II. Measurement of the Vapour Pressure.--In the preceding pages it has been
seen repeatedly that the vapour pressures of the two systems undergoing
reciprocal transformation become identical at the transition point (more
strictly, at the triple or {334} multiple point), and the latter can
therefore be determined by ascertaining the temperature at which this
identity of vapour pressure is established. The apparatus usually employed
for this purpose is the Bremer-Frowein tensimeter (p. 91).

Although this method has not as yet been applied to systems of one
component, it has been used to a considerable extent in the case of systems
containing water or other volatile component. An example of this has
already been given in Glauber's salt (p. 139).

III. Solubility Measurements.--The temperature of the transition point can
also be fixed by means of solubility measurements, for at that point the
solubility of the two systems becomes identical. Reference has already been
made to several cases in which this method was employed, _e.g._ ammonium
nitrate (p. 112), Glauber's salt (p. 134), astracanite and sodium and
magnesium sulphates (p. 268).

The determinations of the solubility can be carried out in various ways.
One of the simplest methods, which also gives sufficiently accurate results
when the temperature is not high or when the solvent is not very volatile,
can be carried out in the following manner. The solid substance is finely
powdered (in order to accelerate the process of solution), and placed in
sufficient quantity along with the solvent in a tube carefully closed by a
glass stopper; the latter is protected by a rubber cap, such as a rubber
finger-stall. The tube is then rotated in a thermostat, the temperature of
which does not vary more than one or two tenths of a degree, until
saturation is produced. The solution is withdrawn by means of a pipette to
which a small glass tube, filled with cotton wool to act as a filter, is
attached. The solution is then run into a weighing bottle, and weighed;
after which the amount of solid in solution is determined in a suitable
manner.

For more accurate determinations of the solubility, especially when the
solvent is appreciably volatile at the temperature of experiment, other
methods are preferable. In Fig. 131 is shown the apparatus employed by H.
Goldschmidt,[401] and used to a considerable extent in the laboratory of
van't Hoff. This consists essentially of three parts: _a_, a tube in which
the solvent and salt are placed; this is closed at the foot by an
india-rubber stopper. Through this stopper there passes the bent tube _cb_,
which connects the tube _a_ with the weighing-tube d. At _c_ there is a
plug of cotton wool. Tube _e_ is open to the air. The wider portion of the
tube _cb_, which passes through the rubber stopper in _a_, can be closed by
a plug {335} attached to a glass rod _ff_, which passes up through a hollow
Witt stirrer, _g_. After being fitted together, the whole apparatus is
immersed in the thermostat. After the solution has become saturated, the
stopper of the bent tube is raised by means of the rod _ff_ and a
suction-pump attached to the end of e. The solution is thereby drawn into
the weighing-tube _d_, the undissolved salt being retained by the plug at
c. The apparatus is then removed from the thermostat, tube _d_ detached and
immediately closed by a ground stopper. It is then carefully dried and
weighed.

[Illustration: FIG. 131.]

Another form of solubility vessel, due to Meyerhoffer and Saunders, is
shown in Fig. 132.[402] This consists of a single tube, and the stirring is
effected by means of a glass screw.

[Illustration: FIG. 132.]

The progress of the solution towards saturation can be very well tested by
determining the density of the solution from time to {336} time. This is
most conveniently carried out by means of the pipette shown in Fig.
133.[403] With this pipette the solution can not only be removed for
weighing, but the volume can be determined at the same time. It consists of
the wide tube _a_, to which the graduated capillary _b_, furnished with a
cap _c_, is attached. To the lower end of the pipette the tube _e_, with
plug of cotton wool, can be fixed. After the pipette has been filled by
sucking at the end of _b_, the stop-cock _d_ is closed and the cap _c_
placed on the capillary. The apparatus can then be weighed, and the volume
of the solution be ascertained by means of the graduations.

As has already been insisted, particular care must be paid to the
characterization of the solid in contact with the solution.

[Illustration: FIG. 133.]

IV. Thermometric Method.--If a substance is heated, its temperature will
gradually rise until the melting point is reached, and the temperature will
then remain constant until all the solid has passed into liquid. Similarly,
if a substance which can undergo transformation is heated, the temperature
will rise until the transition point is reached, and will then remain
constant until complete transformation has taken place.

This method, it will be remembered, was employed by Richards for the
determination of the transition point of sodium sulphate decahydrate
(p. 136). The following figures give the results obtained by Meyerhoffer in
the case of the transformation:--

  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O

the temperature being noted from minute to minute: 95°, 93°, 91.8°, 91.7°,
92°, 92.3°, 92.4°, 92.2°, 92.2°, 92°, 90.5°, 89°, and then a rapid fall in
the temperature. From this we see that the transition point is about 92.2°.
It is also evident that a slight supercooling took place (91.7°), owing to
a delay in the transformation, but that then the temperature rose to the
transition point. This is analogous to the supercooling of a liquid.

A similar halt in the temperature would be observed on passing from lower
to higher temperatures; but owing to a lag in the transformation, the same
temperature is not always obtained.

{337}

V. Optical Method.--The transition point can sometimes be determined by
noting the temperature at which some alteration in the appearance of the
substance occurs, such as a change of colour or of the crystalline form.
Thus mercuric iodide changes colour from red to yellow, and the blue
quadratic crystals of copper calcium acetate change, on passing the
transition point, into green rhombs of copper acetate and white needles of
calcium acetate (p. 260). Or again, changes in the double refraction of the
crystals may be also employed to ascertain the temperature of the
transition point. These changes are best observed by means of a microscope.

For the purpose of regulating the temperature of the substance a small
copper air-bath is employed.[404]

VI. Electrical Methods.--Electrical methods for the determination of the
transition point are of two kinds, based on measurements of conductivity or
of electromotive force. Both methods are restricted in their application,
but where applicable give very exact results.

The former method, which has been employed in several cases, need not be
described here. The second method, however, is of considerable interest and
importance, and calls for special reference.[405]

If two pieces, say, of zinc, connected together by a conducting wire, are
placed in a solution of a zinc salt, _e.g._ zinc sulphate, the potential of
the two electrodes will be the same, and no current will be produced in the
connecting wire. If, however, the zinc electrodes are immersed in two
solutions of _different_ concentration contained in separate vessels, but
placed in connection with one another by means of a bent tube filled with a
conducting solution, the potentials at the electrodes will no longer be the
same, and a current will now flow through the connecting wire. The
direction of this current _in the cell_ will be from the weaker to the more
concentrated solution.

The greater the difference in the concentration of the solutions with
respect to zinc, the greater will be the difference of the potential at the
two electrodes, or the greater will be the E.M.F. of the cell. When the
concentration of the two solutions becomes the same, the E.M.F. will become
zero, and no current will pass.

It will be understood now how this method can be made use of {338} for
determining the transition point of a salt, when we bear in mind that at
the transition point the solubility of the two forms becomes identical.
Thus, for example, the transition point of zinc sulphate heptahydrate into
hexahydrate could be determined in the following manner. Tube A (Fig. 134)
contains, say, a saturated solution of the heptahydrate along with some of
the solid salt; tube B, a saturated solution of the hexahydrate along with
the solid salt. The tube C is a connecting tube bent downwards so as to
prevent the mixing of the solutions by convection currents. ZZ are two zinc
electrodes immersed in the solution; the cell is placed in a thermostat and
the zinc electrodes connected with a galvanometer. Since, now, at
temperatures below the transition point the solubility of the hexahydrate
(the metastable form) is greater than that of the heptahydrate, a current
will be produced, flowing in the cell from heptahydrate to hexahydrate. As
the temperature is raised towards the transition point, the solubilities of
the two hydrates also approach, and the current produced will therefore
become weaker, because the E.M.F. of the cell becomes less; and when the
transition point is attained, the E.M.F. becomes zero, and the current
ceases. If the temperature is raised above this, the solubility of the
heptahydrate becomes greater than that of the hexahydrate, and a current
will again be produced, but in the opposite direction. By noting the
temperature, therefore, at which the current ceases, or the E.M.F. becomes
zero, the transition temperature can be ascertained.[406]

[Illustration: FIG. 134.]

In the case just described, the electrodes consisted of the same metal as
was contained in the salt. But in some cases, _e.g._ sodium sulphate,
electrodes of the metal contained in the salt cannot be employed.
Nevertheless, the above electrical method can be used {339} even in those
cases, if a suitable non-polarizable mercury electrode is employed.[407]

Although, as we saw, no current was produced when two pieces of zinc were
immersed in the same solution of zinc salt, a current will be obtained if
two different metals, or even two different modifications of the same
metal, are employed. Thus an E.M.F. will be established when electrodes of
grey and of white tin are immersed in the same solution of zinc salt, but
at the transition point this E.M.F. will become zero. By this method Cohen
determined the transition point of grey and white tin (p. 42).

       *       *       *       *       *


{340}

NAME INDEX

          A
  Abegg, 52
  Adriani, 186, 217, 220
  Alexejeff, 97, 125
  Allan, 298
  Allen, L. E., 109
  Allen, R. W., 63
  Ampolla, 213
  Andreä, 109
  Aristotle, 41
  Armstrong, E. F., 313
  Armstrong, H. E., 196
  Arzruni, 33
  Aten, 147, 163
  Auerbach, 326

          B
  Babo, 126
  Bancroft, 102, 104, 161, 176, 196, 202, 229, 246, 260, 261, 272, 281, 302
  Barnes, 331, 339
  Barschall, 318
  Barus, 67
  Battelli, 23
  Baur, 233, 307
  Beckmann, 49
  Bell, 229
  Berthollet, 7
  Bodländer, 181, 247, 311, 321
  Bogojawlenski, 72
  Boudouard, 309, 311
  Braun, 107
  Brauns, 40, 51, 74
  Bredig, 52
  Bremer, 91
  Brodie, 34, 47
  Bruner, 126
  Bruni, 181, 182, 256, 257
  Bunsen, 67

          C
  Cady, 192
  Calvert, 130
  Cameron, 203
  Carnelley, 47
  Carpenter, 225
  Carveth, 204, 255
  Centnerszwer, 158
  Chapman, 47
  Chappuis, 51, 176
  Charpy, 255
  Churchill, 140
  Coehn, 52
  Cohen, 41, 72, 136, 139, 140
  Cooke, 331, 339
  Cox, 301

          D
  Dawson, 263
  Debray, 74, 81, 139
  Deville, 49, 74
  Dewar, 26, 51, 178
  Dietz, 157
  {341}
  Doelter, 233
  Donnan, 8, 18
  Dreyer, 73
  Duhem, 56, 151
  Dutoit, 204

          E
  Etard, 115, 135

          F
  Fahrenheit, 30
  Faraday, 82, 89
  Fath, 204
  Fedotieff, 315, 320
  Findlay, 111, 204, 206, 219
  Foote, 69
  Friedländer, 72
  Fritsche, 41
  Frowein, 91
  Füchtbauer, 75
  Fyffe, 143

          G
  Gattermann, 51, 52
  Gautier, 222, 223
  Gay-Lussac, 135
  Gernez, 72
  Gibbs, 7, 8, 151, 236
  Glaessner, 307
  Goldschmidt, E., 41
  Goldschmidt, H., 335
  Goldschmidt, V., 32
  Goossens, 26
  Gossner, 318
  Graham, 178
  Guertler, 73
  Guldberg, 7
  Guthrie, 97, 104, 117, 118, 119, 233

          H
  Haber, 311
  Hahn, 309, 311
  Hallock, 35
  Hammerl, 145
  Hautefeuille, 46, 49, 50, 51, 178
  Heller, 311
  Henry, 94
  Herold, 321
  Hertz, 49
  Heycock, 194, 221, 223
  Heyn, 225, 228
  Hickmans, 219
  Hiorns, 228
  Hissink, 115, 190
  Hoitsema, 14, 90, 177, 178, 298
  Hollmann, 204
  Holsboer, 110
  Horstmann, 8, 83, 89
  Hudson, 102
  Hulett, 10, 48, 52, 54, 67, 109

          I
  Isaac, 114
  Isambert, 80, 82, 84

          J
  Jaffé, 74, 114
  Joulin, 176
  Juhlin, 23, 24, 30
  von Jüptner, 225

          K
  Kastle, 71
  Kaufler, 49
  Kaufmann, 112
  Kayser, 176
  Keeling, 225
  Kelvin, 25
  Kenrick, 263, 297
  Kipping, 219
  Kirchhoff, 32
  Knorr, 203
  de Kock, 53, 182, 194
  Konowaloff, 102, 103, 104
  Krasnicki, 144
  Kremann, 147, 212
  Kuenen, 105
  Kultascheff, 233
  {342}
  Kuriloff, 216
  Kurnakoff, 221, 222, 230
  Küster, 72, 181, 183

          L
  Laar, 195
  Labenburg, 216
  Lattey, 101
  Le Chatelier, 58, 81, 233
  Lehfeldt, 338, 340
  Lehmann, 33, 52, 53
  Lidbury, 147
  Loewel, 134, 135
  Loewenherz, 134, 316
  Lowry, 196, 198
  Ludwig, 327
  Lumsden, 80, 109, 110
  Lussana, 68
  Luther, 22

          M
  Mack, 67
  Magnus, 22
  Mathews, 221
  Mellor, 80
  Meusser, 142
  Meyer, J., 71
  Meyer, V., 47
  Meyerhoffer, 158, 233, 259, 268, 271, 278, 279, 280, 284, 313, 315, 317,
      319, 327, 328, 336, 337
  Middelberg, 116
  Miers, 114
  Miller, 297
  Mitscherlich, 33, 49
  Mond, 178
  Moore, 72
  Moss, 66
  Müller, 112, 265
  Mylius, 109, 142, 157

          N
  Naumann, 49
  Neville, 194, 221, 223

          O
  Offer, 119
  Ostwald, 8, 10, 13, 16, 22, 44, 58, 68, 70, 74, 85, 88, 92, 102, 110,
      117, 125, 127, 130, 141, 198

          P
  Padoa, 73, 181
  Parsons, 298
  Pasteur, 266
  Paternò, 213
  Payen, 74
  Pedler, 47
  Pfaundler, 119
  Philip, 213, 214
  von Pickardt, 73
  Planck, 68
  Pope, 219
  Poynting, 68
  Preuner, 311
  Puschin, 222

          Q
  Quincke, 52

          R
  Rabe, 113
  Ramsay, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 90, 165, 178
  Raoult, 180
  Reed, 71
  Regnault, 22
  Reicher, 36, 37, 110, 260, 333
  Reinders, 71, 185, 188
  Reinitzer, 51, 52
  Richards, 136, 140
  Riddle, 47
  Riecke, 48, 55
  Roberts-Austen, 63, 194, 221, 223, 225
  Roloff, 117
  Roozeboom, 10, 38, 45, 47, 49, 50, 51, 54, 56, 57, 62, 63, 68, 88, 103,
      126, 145, 147, 150, 151, 157, 162, 170, 174, 178, 182, 196, 201, 211,
      217, 220, 225, 236, 238, 262, 264, 269, 272, 273, 281, 282, 290, 331
  {343}
  Rose, 223
  Rotarski, 52
  Rothmund, 97, 98, 100
  Rutten, 298

          S
  Saposchnikoff, 212
  Saunders, 313, 317, 319, 336, 337
  Saurel, 151
  Schaum, 49, 75
  Scheel, 22, 23, 30
  Schenck, 49, 52, 54, 311
  Schneider, 52
  Schönbeck, 75
  Schreinemakers, 122, 126, 246, 248, 250, 252, 290, 302
  Schrötter, 46
  Schukowsky, 52
  Schwarz, 331
  Seitz, 52
  Shenstone, 109, 115, 135
  Shepherd, 221, 255
  Shields, 178
  Skirrow, 130
  Spring, 63
  von Stackelberg, 107, 110
  Staedel, 267
  Stansfield, 194, 221
  Stokes, 236
  Stortenbeker, 44, 147, 161, 164, 281

          T
  Taber, 229
  Tammann, 26, 32, 33, 37, 38, 39, 48, 52, 65, 67, 68, 72, 73, 140, 151,
      176, 221, 230
  Thiesen, 22, 23, 30
  Thomson, J., 25, 28, 32
  Thomson, W., 25
  Tilden, 109, 115, 135
  Trevor, 16
  Troost, 46, 49, 50, 51
  Tumlirz, 72

          V
  Van Bemmelen, 180
  Van Deventer, 110, 139, 266, 267, 333
  Van Eyk, 41, 63, 192, 338
  Van't Hoff, 36, 38, 58, 70, 90, 92, 108, 127, 139, 140, 165, 175, 225,
      258, 260, 263, 265, 266, 267, 272, 284, 290, 313, 318, 333, 340
  Van Leeuwen, 259
  Van Wyk, 185
  Vogt, 5, 233

          W
  Waage, 7
  Wald, 92
  Walden, 158
  Walker, 80, 105, 122, 126, 143
  Wegscheider, 10, 49, 202
  Wells, 136
  Wenzel, 7
  Wiebe, 22
  Witt, 327
  Wright, 241, 246, 247
  von Wrochem, 109, 142

          Y
  Young, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 105, 165

          Z
  Zacharias, 180
  Zawidski, 63
  Zenghelis, 35
  Zimmermann, 311
  Zincke, 44
  Ziz, 141

       *       *       *       *       *


{344}

SUBJECT INDEX

          A
  Acetaldehyde and paraldehyde, 204
  Acetic acid, chloroform, water, 241
  Acetone, phenol, water, 248
  Adsorption, 176
  Alcohol, chloroform, water, 246
    ----, ether, water, 246
  Alloys, equilibrium curves of, 221
    ---- of copper and tin, liquefaction of, by cooling, 194
    ---- of iron and carbon, 223
    ---- of thallium and mercury, 222
    ----, ternary, 246
  Ammonia compounds of metal chlorides, 82
  Ammonia silver chlorides, 82
    ---- ---- ----, dissociation pressures of, 84
  Ammonia-soda process, 320
  Ammonium chloride, dissociation of, 3, 79
    ---- cyanide, dissociation of, 80
    ---- hydrosulphide, dissociation of, 80
    ---- nitrate, solubility of, 113
  Aniline, phenol, water, 250
  Astracanite, 260, 261, 268, 274

          B
  Babo, law of, 126
  Barium acetate, solubility of, 143
  Barium carbonate and potassium sulphate, 328
    ---- nitrite, preparation of, 327
  Basic salts, 296
  Benzaldoximes, 203
  Benzene and picric acid, 216
  Bismuth, effect of pressure on the melting point of, 67
    ----, lead, tin, 255
    ---- nitrates, basic, 298
  Bivariant systems, 16
  Bromocinnamic aldehyde and chlorocinnamic aldehyde, 183

          C
  Calcium carbonate, dissociation of, 3, 11, 81
    ---- chloride hexahydrate, solubility of, 146
    ---- ----, solubility of hydrates of, 148
    ---- ----, vapour-pressure of hydrates of, 88
  Camphor oximes, 219, 257
  Carnallite, 284
  Carvoximes, 186, 219
  Cementite, 224
  Chlorine and iodine, 161
  Chlorocinnamic aldehyde and bromocinnamic aldehyde, 183
  Chloroform, acetic acid, water, 241
    ----, alcohol, water, 246
  {345}
  Classification of systems, 17
  Component, 8, 10, 12
    ----, systems of one, 21, 55
  Components, choice of, 12, 13, 14, 76, 313
    ----, determination of number of, 13
    ----, systems of four, 312
    ----, ---- of three, 234
    ----, ---- of two, 76, 207
    ----, variation in number of, 11, 14
  Composition, determination of, without analysis, 228, 302
  Concentration-temperature curve for two liquids, 101
  Condensed systems, 36
  Constituent, 10
  Cooling curve, 230
  Copper calcium acetate, 260
    ---- chloride, heat of solution of, 110
    ---- dipotassium chloride, 259
    ---- sulphate, 85
  Critical concentration, 98, 242
    ---- pressure of water, 23
    ---- solution temperature, 98
    ---- temperature of water, 23
  Cryohydrates, 117, 118
  Cryohydric point, 117
    ---- ----, changes at the, 119
    ---- ---- for silver nitrate and ice, 116
  Crystals, liquid, 51
    ----, ----, equilibria of, 53
    ----, ----, list of, 54
    ----, ----, nature of, 52
    ----, mixed, 180
  Crystallization, velocity of, 72, 74
    ----, spontaneous, 114

          D
  Deliquescence, 130
  Devitrification, 73
  Diethylamine and water, solubility of, 101
  Dilatometer, determination of transition points by, 331
  Dineric surface, 247
  Dissociation equilibrium, effect of addition of dissociation products on,
      4
    ---- of ammonia compounds of metal chlorides, 82, 84
    ---- of ammonium chloride, 3, 79
    ---- ---- cyanide, 80
    ---- ---- hydrosulphide, 80
    ---- of calcium carbonate, 3, 81
    ---- of compounds, degree of, 147
    ---- of phosphonium bromide, 80
    ---- of salt hydrates, 85
    ----, phenomena of, 79
  Dissociation pressure, 81
  Distillation of supercooled liquid to solid, 32, 50
  Double salt interval, 278
    ---- salts, crystallization from solution, 280
    ---- ----, decomposition by water, 267
    ---- ----, formation of, 258, 273, 315

          E
  Efflorescence, 86
  Electrical methods of determining transition points, 338
  Enantiotropy, 44, 51
  Equilibria, Gibbs's theory of, 8
    ----, metastable, 69
  Equilibrium apparent (false), 5, 6
    ---- between ice and solution, 116
    ---- between ice and water, 25
    ---- between ice, water, vapour, 27
    ---- between water and vapour, 21
    ----, chemical, 3, 16
    ----, heterogeneous, 5
    ----, homogeneous, 5
    ----, independence of, on amounts of phases, 9
    ----, law of movable, 58
  {346}
    ----, physical, 3, 16
    ---- real (true), 5, 6
  Ether, alcohol, water, 246
    ----, succinic nitrile, water, 252
  Ethylene bromide, picric acid, [beta]-naphthol, 256
  Eutectic mixtures, 117, 191, 209, 255, 257
    ---- point, 117, 209, 213, 253

          F
  Ferric chloride, evaporation of solutions of, 155
    ---- ----, hydrates of, 151, 153
    ---- ----, hydrogen chloride and water, systems of, 290
  Ferrite, modifications of, 224
  Freedom, degree of, 14
  Freezing mixtures, 120
    ---- point, natural, 198
  Fusion curve, 66
    ---- ---- of ice, 25
    ---- of ice, influence of pressure on, 26
    ----, partial, 139

          G
  Glaserite, 315, 317
  Glasses, 176
  Glauber's salt, 13, 134
    ---- ----, transition curve of, 68, 140
  Graphic representation in space, 77, 284

          H
  Hydrates, range of existence of, 89
    ---- chloride and water, 174
  Hydrogen bromide and water, 174
  Hylotropic substances, 198

          I
  Ice I., 32
    ---- II., 32
    ---- III., 32
    ----, equilibrium between water and, 25
    ----, influence of pressure on melting point of, 25, 26
    ----, sublimation curve of, 24
    ----, vapour pressure of, 25, 31
  Indifferent point, 150
  Individual, chemical, 92
  Inversion temperature, 36
  Iodine and chlorine, 161
  Iron--carbon alloys, 223
    ----, carbon monoxide and carbon dioxide, 305
  Isomerides, dynamic, 195, 196
    ----, ----, equilibrium between, 195, 196
    ----, ----, equilibrium point of, 198
    ----, transformation of unstable into stable, 201
  Isomerism, dynamic, 196
  Isothermal evaporation, 278
    ---- solubility curves, 272

          L
  Lead, bismuth, tin, 255
    ----, desilverization of, 247
    ----, silver, zinc, 246
  Le Chatelier, theorem of, 57
  Lime, burning of, 3
  Liquidus curve, 182

          M
  Mandelic acid, 217
  Martensite, 224
  Mass action, law of, 7
  Melting point, influence of pressure on, 66
  {347}
    ---- ----, congruent, 146
    ---- ----, incongruent, 139
    ---- under the solvent, 122
  Menthyl mandelates, 219
  Mercuric bromide and iodide, 188
  Mercury salts, basic, 301
  Metastable equilibria, 69
    ---- region, 30
    ---- state, 30
  Methylethyl ketone and water, 100
  Minerals, formation of, 232
  Miscibility of liquids, complete, 95, 104, 114
    ---- ----, partial, 95, 96, 121
  Mixed crystals, 180, 281
    ---- ----, changes in, with temperature, 192
    ---- ----, examples of, 183, 186, 187, 190, 192, 219, 223
    ---- ----, formation of, 181, 182
    ---- ----, fractional crystallization of, 188
    ---- ----, freezing points of, 182
    ---- ----, melting points of, 182, 184
    ---- ----, pseudoracemic, 219
  Mixtures, isomorphous, 181
    ---- of constant boiling point, 105
    ---- of constant melting point, 117, 186, 187, 192, 209, 255, 257
  Monotropy, 44, 51
  Multivariant systems, 16

          N
  Naphthalene and monochloracetic acid, 192
    ---- and [beta]-naphthol, mixed crystals of, 183
  [beta]-Naphthol, ethylene bromide, picric acid, 256
  [alpha]-Naphthylamine and phenol, 213
  Nickel iodate, solubility of, 142
  _o_-Nitrophenol and _p_-toluidine, 213

          O
  Occlusion of gases, 176
  Optical method of determining transition points, 338
  Optically active substances, freezing-point curves of, 216
  Order of a system, 13
  Organic compounds, application of Phase Rule to, 212

          P
  Palladium and hydrogen, 90, 178
  Paragenesis, 320
  Paraldehyde and acetaldehyde, 204
  Partial pressures of two components, 102
  Pearlite, 224
  Phase, 8
    ---- Rule, 8, 16
    ---- ----, deduction of, 18
    ---- ----, scope of, 1
  Phases, formation of new, 69
    ----, number of, 9
  Phenol, acetone, water, 248
    ----, aniline, water, 250
    ---- and [alpha]-naphthylamine, 213
    ---- and _p_-toluidine, 214
    ---- and water, solubility of, 97
  Phosphonium bromide, dissociation of, 80
    ---- chloride, 65
  Phosphorus, 46
    ----, distillation of white to red, 50
    ----, melting point of red, 47
    ----, ---- ---- of white, 48
    ----, solubility of white and red, 47
    ----, vapour pressure of white and red, 46
  Picric acid and benzene, 216
    ---- ----, ethylene bromide, and [beta]-naphthol, 256
  Polymorphic forms, solubility of, 112
  {348}
    ---- substances, list of, 63
  Polymorphism, 33
  Potassium nitrate and thallium nitrate, 192
  Potential, chemical, 19
  Pressure-concentration diagram for two liquids, 102
  Pressure-temperature diagram for solutions, 126
  Pseudomonotropy, 45
  Pseudo-racemic mixed crystals, 21
  Pyridine and methyl iodide, 147
  Pyrometer, registering, 230

          Q
  Quadruple point, 116
  Quintuple point, 234, 261

          R
  Racemates, characterization of, 217, 282
  Reactions, law of successive, 73
  Reciprocal salt-pairs, 313
    ---- ----, transition point of, 314
  Rubidium tartrates, 265

          S
  Salt hydrates, 85
    ---- ----, indefiniteness of vapour pressure of, 87
    ---- ---- with definite melting point, 145
  Separation of salt on evaporation, 130
  Silicates, hydrated, 176
  Silver, lead, zinc, 246
  Silver nitrate, solubility of, 114
    ---- ---- and sodium nitrate, 190
  Single salt interval, 278
  Sodium ammonium tartrates, 266
    ---- nitrate and silver nitrate, 190
    ---- sulphate and water, equilibria between, 134
  Sodium sulphate and water, vapour pressures of, 138, 140
    ---- ----, anhydrous, dehydration by, 138
    ---- ----, solubility of, 135
    ---- ---- decahydrate, solubility of, 134
    ---- ---- ----, transition point of, 136, 139
    ---- ---- heptahydrate, solubility of, 136
    ---- ---- ----, transition point of, 137
  Solidus curve, 182
  Solubility curve at higher temperatures, 114
    ---- ----, form of, 108
    ---- ---- of anhydrous salts, 111
    ---- ----, retroflex, 146, 151, 162
    ---- curves, interpolation and extrapolation of, 111
    ---- ---- of three component systems, 264
    ----, determination of transition points by, 335
    ----, influence of pressure on, 107
    ----, ---- of subdivision on, 10
    ----, ---- of temperature on, 109
    ---- of metastable forms, 47, 112, 137
  Solubility of polymorphic forms, 112
    ---- of salt hydrates, 133, 145
    ---- of supercooled liquids, 125
    ----, retrograde, 245
  Solute, 93
  Solution, definition of, 92
    ----, heat of, 109, 110
    ----, saturated, 106, 108
    ----, supersaturated, 108
    ---- temperature, critical, 98
    ----, unsaturated, 108
  Solutions, bivariant systems, 129
    ----, congruently saturated, 279
    ---- conjugate, 97, 241
  {349}
    ----, incongruently saturated, 279, 289
    ----, inevaporable, 157
    ---- of gases in liquids, 93
    ---- ---- in solids, 176
    ---- of liquids in liquids (binary), 95
    ---- ---- ---- (ternary), 240
    ---- ----, influence of temperature on, 247
    ---- of solids in liquids, 106
    ---- ---- in solids, 180
    ----, solid, 175, 180
    ----, univariant systems, 127
  Space model for carnallite, 284
  Stability limit, 202
  Steel, formation of, 223
  Sublimation curve, 63
    ---- ---- of ice, 24
    ---- without fusion, 65
  Succinic nitrile and water, 122
    ---- ether, water, 252
  Sulphur, 33, 34
    ---- dioxide and water, 169
    ---- ---- and potassium iodide, 158
    ----, transition point of rhombic and monoclinic, 36
  Supersaturation, 113, 114, 124
    ----, limits of, 114
  Systems, condensed, 36
    ---- of one component, 21
    ---- of two components, 76, 77, 207

          T
  Tachydrite, influence of pressure on the transition point of, 263
  Tartrate, dimethyl, 217
    ----, sodium potassium, 259
  Tautomeric substances, 195
  Tensimeter, 91
  Thallium nitrate and potassium nitrate, 192
  Theorem of van't Hoff and Le Chatelier, 57
  Thermometric  determination of transition point, 337
  Tin, 41
    ----, lead, bismuth, 255
    ---- plague, 43
    ----, transition point of white and grey, 41
  _p_-Toluidine and _o_-nitrophenol, 213
    ---- and phenol, 214
  Transformation of optically active substances, 220
    ----, suspended, 37, 69, 89, 113, 137, 155
    ----, velocity of, 70
  Transition curve, 66
    ---- ---- of Glauber's salt, 68, 140
    ---- ---- of rhombic and monoclinic sulphur, 37
    ---- interval, 270, 277, 315
    ---- point, 34
    ---- ---- for double salts, 258
    ---- ----, influence of pressure on the, 68
    ---- points, as fixed points in thermometry, 140
    ---- ----, methods of determining, 331
    ---- ---- of polymorphic substances, 63
  Triangle, graphic representation by, 235
  Triethylamine and water, 101
  Triple point, 27, 55
    ---- ----, arrangement of curves round, 56
    ---- ----, changes at, 58
    ---- ----, ice, water, vapour, 27
    ---- ----, ice II., ice III., and water, 33
    ---- ----, metastable, 38
    ---- ----, monoclinic sulphur, liquid, vapour, 38
    ---- ----, monoclinic and rhombic sulphur, liquid, 38
    ---- ----, monoclinic and rhombic sulphur, vapour, 34
  {350}
    ---- ----, red phosphorus, liquid, vapour, 47
    ---- ----, rhombic sulphur, liquid, vapour, 38
    ---- ---- solid, solid, vapour, 62
    ---- ----, white phosphorus, liquid, vapour, 48

          U
  Univariant systems, 16

          V
  Van't Hoff, theorem of, 57
  Vaporization curve, 63
    ---- ----, interpolation and extrapolation of, 66
    ---- ---- of water, 21, 23
  Vapour pressure, constancy of, and formation of compounds, 90
    ---- ----, dependence of, on solid phase, 88
    ---- ----, influence of surface tension on, 2
    ---- ---- in three-component systems, 261
    ---- ----, measurement of, 91, 334
    ---- ---- of calcium chloride solutions, 150
    ---- ---- of ice, 25, 31
    ---- ---- of small drops, 10
    ---- ---- of sodium sulphate and water, 138
  Vapour pressure of solid, solution, vapour, 126
    ---- ---- of water, 21, 31
  Variability of a system, 14, 16
  Variance of a system, 16
  Volatile components, two, 161

          W
  Water, 21
    ----, acetic acid, chloroform, 241
    ----, acetone, phenol, 248
    ----, alcohol, ether, 246
    ----, ----, chloroform, 246
    ----, aniline, phenol, 250
    ----, bivariant systems of, 29
    ----, critical pressure of, 23
    ----, critical temperature of, 23
    ----, equilibrium between ice and, 25
    ----, ---- between vapour and, 21
    ----, ether, succinic nitrile, 252
    ----, supercooled, 30
    ----, ----, vapour pressure of, 31
    ----, vaporization curve of, 21
    ----, vapour pressure of, 23

          Z
  Zeolites, 176
  Zinc, lead, silver, 246
  ---- chloride in water, solubility of, 157

THE END

PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.

       *       *       *       *       *


NOTES

[1] Except when the volume of the liquid becomes exceedingly small, in
which case the surface tension exerts an influence on the vapour pressure.

[2] For reasons which will appear later (Chap. IV.), the volume of the
vapour is supposed to be large in comparison with that of the solid and
liquid.

[3] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.

[4] See, more especially, Vogt, _Die Silikatschmelzlösungen_. (Christiania,
1903, 1904.)

[5] _Trans. Connecticut Acad._, 1874-1878.

[6] Lehre von der chemischen Verwandtschaft der Körper, 1777.

[7] See Ostwald's _Klassiker_, No. 74.

[8] Etudes sur les affinités chimiques, 1867; Ostwald's _Klassiker_, No.
104.

[9] Died April, 1903.

[10] For a mathematical treatment of the Phase Rule the reader is referred
to the volume in this series on Thermodynamics, by F. G. Donnan.

[11] Liebig's _Annalen_, 1873, 170, 192; Ostwald, _Lehrbuch_, II. 2. 111.

[12] The action of gravity and other forces being excluded (see p. 5).

[13] It may seem as if this were a contradiction to what was said on p. 4
as to the effect of the addition of ammonia or hydrogen chloride to the
system constituted by solid ammonium chloride in contact with its products
of dissociation. There is, however, no contradiction, because in the case
of ammonium chloride the gaseous phase consists of ammonia and hydrogen
chloride in equal proportions, and in adding ammonia or hydrogen chloride
alone we are not adding the gaseous phase, but only a constituent of it.
Addition of ammonia and hydrogen chloride together in the proportions in
which they are combined to form ammonium chloride would cause no change in
the equilibrium.

[14] The vapour pressure of water in small drops is greater than that of
water in mass, and the solubility of a solid is greater when in a state of
fine subdivision than when in large pieces (_cf._ Hulett, _Zeitschr.
physikal. Chem._, 1901, 37. 385).

[15] See Ostwald, _Lehrbuch_, II. 2. 476, 934; Roozeboom, _Zeitschr.
physikal. Chem._, 1894, 15. 150; _Heterogene Gleichgewichte_, I. p. 16;
Wegscheider, _Zeitschr. physikal. Chem._, 1903, 43. 89.

[16] Ostwald, _Lehrbuch_, II. 2. 478.

[17] See also Hoitsema, _Zeitschr. physikal. Chem._ 1895, 17. 651.

[18] The term "degree of freedom" employed here must not be confused with
the same term used to denote the various movements of a gas molecule
according to the kinetic theory.

[19] Trevor, _Jour. Physical Chem._, 1902, 6. 136.

[20] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
Findlay, 2nd edit., p. 7. (Macmillan, 1904.)

[21] See the volume in this series on _Thermodynamics_ by F. G. Donnan.

[22] _Pogg. Annalen_, 1844, 61. 225.

[23] _Mémoires de l'Acad._, 26. 751.

[24] _Phil. Trans._ 1884, 175. 461; 1892, A, 183. 107.

[25] _Bihang Svenska Akad. Handl._ 1891, 17. I. 1.

[26] Abh_andl. physikal.-tech. Reichsanstalt_, 1900, 3. 71.

[27] Ostwald-Luther, _Physiko-chemische Messungen_, 2nd edit., p. 156.

[28] _Annales chim. et phys._, 1892 [6], 26. 425.

[29] The vapour pressure of water at 0° has recently been very accurately
determined by Thiesen and Scheel (_loc. cit._), and found to be 4.579 ±
0.001 mm. of mercury (at 0°), or equal to 0.006025 atm.

[30] Juhlin, _Bihang Svenska Akad. Handl._, 1891, 17. I. 58. See also
Ramsay and Young, _loc. cit._

[31] _Trans. Roy. Soc. Edin._, 1849, 16. 575.

[32] _Proc. Roy. Soc. Edin._, 1850, 2, 267.

[33] _Annalen der Physik_, 1899 [3], 68. 564; 1900 [4], 2. 1, 424. See
also Dewar, _Proc. Roy. Soc._, 1880, 30. 533.

[34] The pressure of 1 atmosphere is equal to 1.033 kilogm. per sq. cm.; or
the pressure of 1 kilogm. per sq. cm. is equal to 0.968 atm.

[35] Tammann, _loc. cit._, 1900, 2. 1, 424; cf. Goossens, _Arch. néerland_,
1886, 20. 449.

[36] J. Thomson, _Proc. Roy. Soc._, 1874, 22. 28.

[37] A field is "enclosed" by two curves when these cut at an angle less
than two right angles. It may be useful to remember that an invariant
system is represented by a _point_, a univariant system by a _line_, and a
bivariant system by an _area_.

[38] _Phil. Trans._, 1724, 39. 78.

[39] Juhlin, _loc. cit._, p. 61; cf. Ramsay and Young, _loc. cit._: Thiesen
and Scheel, _loc. cit._

[40] This small difference is due to experimental errors in the
determination of the vapour pressures; a differential method betrayed no
difference between the vapour pressure of ice and of water at 0°.

[41] _Phil. Mag._, 1874 [4], 47. 447; _Proc. Roy. Soc._, 1873, 22. 27.

[42] _Pogg. Annalen_, 1858, 103, 206.

[43] See _Phil. Trans._, 1884, 175, 461.

[44] This phenomenon of distillation from the supercooled liquid to the
solid has been very clearly observed in the case of furfuraldoxime (V.
Goldschmidt, _Zeitschr. f. Krystallographie_, 1897, 28. 169).

[45] _Annalen der Physik_, 1900 [4], 2. 1, 424.

[46] A similar triple point has been determined by Tammann in the case of
phenol (_Annalen der Physik_, 1902 [4], 9. 249).

[47] _Annales chim. et phys._, 1821, 19. 414.

[48] Lehmann, _Molekularphysik_, I. 153.; Arzruni, _Physikalische Chemie
der Krystalle_. (Graham-Otto, _Lehrbuch der Chemie_, I. 3.)

[49] Brodie, _Proc. Roy. Soc._, 1855, 7. 24.

[50] That solid sulphur does possess a certain vapour pressure has been
shown by Hallock, who observed the formation at the ordinary temperature of
copper sulphide in a tube containing copper and sulphur (_Amer. Jour.
Sci._, 1889 [3], 37. 405). See also Zenghelis, _Zeitschr. physikal.
Chem._, 1904, 50. 219.

[51] _Zeitschr. für Krystallographie_, 1884, 8. 593.

[52] Van't Hoff, _Studies on Chemical Dynamics_, p. 163.

[53] Reicher, _loc. cit._ See also Tammann, _Annalen der Physik_, 1899
[3], 68. 663.

[54] Tammann, _Annalen der Physik_, 1899 [3], 68. 633.

[55] Rec. Trav. _Chim. Pays-Bas_, 1887, 6. 314.

[56] Cf. van't Hoff, _Lectures on Physical Chemistry_, I., p. 27 (Arnold).

[57] _Annalen der Physik_, 1899 [3], 68. 663.

[58] Brauns, _Jahrbuch für Mineralogie_, 1899-1901, 13. Beilage, p. 39.

[59] Fritsche, _Ber._, 1869, 2. 112, 540.

[60] _De mirabilibus Auscultationibus_, Cap. 51 (_v._ Cohen, _Zeitschr.
physikal. Chem._, 1901, 36. 513).

[61] E. Cohen and C. van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 601;
Cohen, _ibid._, 1900, 33. 59; 35. 588; 1901, 36. 513; Cohen and E.
Goldschmidt, _ibid._, 1904, 50. 225.

[62] _Zeitschr. physikal. Chem._, 1900, 33, 58.

[63] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. Trav.
Chim. Pays-Bas_, 1888, 7. 152.

[64] Zincke, _Ber._, 1871, 4. 576.

[65] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 313.

[66] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.

[67] Roozeboom, _ibid._, p. 179.

[68] Schrötter, _Pogg. Annalen_, 1850, 81. 276; Troost and Hautefeuille,
_Annales de Chim. et Phys._ 1874 [5], 2. 153; _Ann. Scient. École Norm._
1868 [2], II. 266.

[69] Pedler, _Trans. Chem. Soc._, 1890, 57. 599.

[70] Brodie, _Trans. Chem. Soc._, 1853, 5, 289.

[71] This is a familiar fact in the case of the solubility in carbon
disulphide.

[72] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 170.

[73] _Trans. Chem. Soc._, 1899, 57. 734.

[74] Carnelley, _Trans. Chem. Soc._, 1876, 29. 489; 1878, 33. 275. V. Meyer
and Riddle, _Ber._, 1893, 26. 2443.

[75] Riecke, _Zeitschr. physikal. Chem._, 1890, 6. 411.

[76] _Annalen der Physik._, 1898 [3], 66. 492.

[77] _Zeitschr. physikal. Chem._, 1899, 28. 666.

[78] See Naumann, _Ber._, 1872, 4. 646; Troost and Hautefeuille, _Compt.
rend._, 1868, 66. 795; 1868, 67. 1345; Roozeboom, _Das Heterogene
Gleichgewicht_, I. pp. 62, 171.

[79] Mitscherlich, _Lieb. Annalen_, 1834, 12. 137; Deville and Troost,
_Compt. rend._, 1863, 56. 891.

[80] Beckmann, _Zeitschr. physikal. Chem._, 1890, 5. 79; Hertz, _ibid._, 6.
358.

[81] _Ber._, 1902, 35. 351. _Cf._ also, K. Schaum, _Annalen der Chem._,
1898, 300. 221; R. Wegscheider and Kaufler, _Sitzungsber. kaiserl. Akad.
Wissensch. in Wien_, 1901, 110, II. 606.

[82] See also Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.

[83] _Annales de Chim. et Phys._, 1874 [5], 2. 154.

[84] _Compt. rend._, 1887, 104. 1505.

[85] _Compt. rend._, 1868, 66. 795.

[86] _Phil. Mag._, 1884 [5], 18. 210. See also Roozeboom, _Das Heterogene
Gleichgewicht_, I. p. 177.

[87] Brauns, _Neues Jahrbuch für Mineralogie_, 1900, 13. Beilage-Band, p.
39; Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 181.

[88] _Monatshefte_, 1888, 9. 435.

[89] Gattermann, _Ber._, 1890, 53. 1738.

[90] _Zeitschr. physikal. Chem._, 1889, 4. 468; _Annalen der Physik_, 1900
[4], 2. 649.

[91] Quincke, _Annalen der Physik_, 1894 [3], 53. 613; Tammann, _Annalen
der Physik_, 1901 [4], 4. 524; 1902, 8. 103; Rotarski, _ibid._, 4. 528.

[92] _Annalen der Physik_, 1900 [4], 2. 649.

[93] _Annalen der Physik_, 1902 [4], 8. 911.

[94] See, more especially, O. Lehmann, _Annalen der Physik_, 1900 [4], 2.
649; Reinitzer, _Sitzungsber. kaiserl. Akad. zu Wien._, 1888, 94. (2), 719;
97. (1), 167; Gattermann, _loc. cit._; Schenck, _Zeitschr. physikal.
Chem._, 1897, 23. 703; 1898, 25. 337; 27. 170; 1899, 28. 280; Schenck and
Schneider, _ibid._, 1899, 29. 546; Abegg and Seitz, _ibid._, 1899, 29. 491;
Hulett, _ibid._, 1899, 28. 629; Coehn, _Zeitschr. Elektrochem._, 1904, 10.
856: Bredig and Schukowsky, _ibid._, 3419. For a full account of the
subject, the reader is referred to the work by Lehmann, _Flüssige
Kristalle_ (Engelmann, 1904), or the smaller monograph by Schenck,
_Kristallinische Flüssigkeiten und flüssige Kristalle_ (Engelmann, 1905).

[95] A. C. de Kock, _Zeitschr. physikal. Chem._, 1904, 48. 129.

[96] On account of the fact that all grades of rigidity have been realized
between the ordinary solid and the liquid state, in the case both of
crystalline and amorphous substances, it has been proposed to abandon the
terms "solid" and "liquid," and to class bodies as "crystalline" or
"amorphous," the passage from the one condition to the other being
discontinuous; crystalline bodies possess a certain regular orientation of
their molecules and a directive force, while in amorphous bodies these are
wanting (see Lehmann, _Annalen der Physik_, 1900 [4], 2. 696).

[97] Hulett, _loc. cit._

[98] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 144. See also
Schenck, _Kristallinische Flüssigkeiten und flüssige Kristalle_, p. 8
(Engelmann, 1904).

[99] The possible number of triple points in a one-component system is
given by the expression (_n_(_n_ - 1)(_n_ - 2))/1.2.3, where _n_ is the
number of phases (Riecke, _Zeitschr. physikal. Chem._, 1890, 6, 411). The
number of triple points, therefore, increases very rapidly as the number of
possible phases increases.

[100] Duhem, _Zeitschr. physikal. Chem._, 1891, 8. 371. _Cf._ Roozeboom,
_Das Heterogene Gleichgewicht_, p. 94 ff.

[101] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 99.

[102] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.

[103] These changes can be predicted quantitatively by means of the
thermodynamic equation, _dp_/_dt_ = Q/(T(_v_{2}_ - _v_{1}_)), provided the
specific volumes of the phases are known, and the heat effect which
accompanies the transformation of one phase into the other.

[104] _Studies on Chemical Dynamics_, translated by Ewan, p. 218.

[105] Le Chatelier, _Compt. rend._, 1884, 99. 786.

[106] See _Principles of Inorganic Chemistry_, translated by Findlay, 2nd
edit., p. 133. (Macmillan, 1904.)

[107] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.

[108] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 189.

[109] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 125. See also
Zawidski, _Zeitschr. physikal. Chem._, 1904, 47. 727; van Eyk, _ibid._,
1905, 51. 720.

[110] Roberts-Austen, _Proc. Roy. Soc._, 63. 454; Spring, _Zeitschr.
physikal. Chem._, 1894, 15. 65. See also p. 35.

[111] Ramsay and Young, _Phil. Trans._, 1884, 175. 461; Allen, _Trans.
Chem. Soc._, 1900, 77. 413.

[112] Ramsay and Young, _Phil. Trans._ 1886, 177. 87.

[113] This is exemplified in the well-known experiment with the cryophorus.

[114] Tammann has, however, found that the fusion curve (solid in contact
with liquid) of phosphonium chloride can be followed up to temperatures
above the critical point (_Arch. néer._, 1901 [2], 6. 244).

[115] _Phil. Mag._, 1886, 21. 33. See also S. A. Moss, _Physical Review_,
1903, 16. 356.

[116] This is found also in the case of bismuth. See Tammann, _Zeitschr.
anorgan. Chem._, 1904, 40. 54.

[117] See p. 57, footnote.

[118] _Pogg. Annalen_, 1850, 81. 562.

[119] Barus, _Amer. Jour. Sci._, 1892, 42. 125; Mack, _Compt. rend._, 1898,
127. 361; Hulett, _Zeitschr. physikal. Chem._, 1899, 38. 629.

[120] _Annalen der Physik_, 1899 [3], 68. 553, 629; 1900 [4], 1. 275; 2.
1; 3. 161. See also Tammann, _Kristallisieren und Schmelzen_ (Leipzig,
1903).

[121] Ostwald, _Lehrbuch_, II. 2. 373; Poynting, _Phil. Mag._, 1881 [5],
12. 2; Planck, _Wied. Annalen_, 1882, 15. 446.

[122] Bakhuis Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 91.

[123] Lussana, _Il nuovo Cimento_, 1895 [4], 1. 105.

[124] Tammann, _Zeitschr. physikal. Chem._, 1903, 46. 818.

[125] Foote, _Zeitschr. physikal. Chem._, 1900, 33. 740.

[126] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 289.

[127] Van't Hoff, _Arch, néer._, 1901, 6. 471.

[128] See, for example, the determinations of the solubility of rhombic and
monoclinic sulphur, by J. Meyer, _Zeitschr. anorg. Chem._, 1902, 33. 140.

[129] _Zeitschr. physikal. Chem._, 1899, 32. 506.

[130] Kastle and Reed, _Amer. Chem. Jour._, 1902, 27. 209.

[131] _Zeitschr. physikal. Chem._, 1900, 35. 581.

[132] _Compt. rend._, 1882, 95. 1278; 1884, 97. 1298, 1366, 1433.

[133] _Zeitschr. physikal. Chem._, 1893, 12. 545.

[134] _Sitzungsber. Wiener Akad._, 1894, 103. IIa. 226.

[135] _Zeitschr. physikal. Chem._, 23-29. See also Küster, _ibid._, 25-28.

[136] _Zeitschr. physikal. Chem._, 1897, 24. 152.

[137] _Ibid._, 1898, 27. 585.

[138] See W. Guertler, _Zeitschr. anorgan. Chem._, 1904, 40. 268; Tammann,
_Zeitschr. Elektrochem._, 1904, 10. 532.

[139] E. von Pickardt, _Zeitschr. physikal. Chem._, 1902, 42. 17.

[140] _Zeitschr. physikal. Chem._, 1904, 48. 467.

[141] M. Padoa, _Accad. Lincei, Atti_, 1904, 13. 329.

[142] Deville, _Compt. rend._, 1852, 34. 561; Payen, _ibid._, 1852, 34.
508; Debray, _ibid._, 1858, 46. 576. It has also been found by Jaffé
(_Zeitschr. physikal. Chem._, 1903, 43. 465) that when spontaneous
crystallization from solution occurs, the less stable form always separates
first when purification has been carried sufficiently far.

[143] Brauns, _Neues Jahrbuch für Mineralogie_, 1899, 13. (Beilage Band)
84.

[144] _Lehrbuch_, II. 2. 445. See also _Principles of Inorganic Chemistry_,
2nd edit., p. 210 ff.

[145] Schaum and Schönbeck, _Annalen der Physik_, 1902 [4], 8. 652. See
also Chr. Füchtbauer, _Zeitschr. physikal. Chem._, 1904, 48. 549.

[146] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.

[147] See volume in this series on _Chemical Dynamics_, by Dr. J. W.
Mellor.

[148] Isambert, _Compt. rend._, 1881, 92. 919; 1882, 94. 958; 1883, 96.
643. Walker and Lumsden, _Jour. Chem. Soc._, 1897, 71. 428.

[149] _Compt. rend._, 1867, 64. 603.

[150] _Compt. rend._, 1883, 102. 1243.

[151] _Compt. rend._, 1868, 66, 1259.

[152] Horstmann, _Ber._, 1876, 9. 749.

[153] _Loc. cit._

[154] For the reasons for choosing anhydrous salt and water instead of salt
hydrate and water as components, see p. 14.

[155] See Ostwald, _Lehrbuch_, II. 2. 527.

[156] Ostwald, _Lehrbuch_, II. 2. 538.

[157] _Zeitschr. physikal. Chem._, 1889, 4. 43.

[158] _Ber._, 1876, 9. 749.

[159] See, for example, van't Hoff, _Lectures on Theoretical and Physical
Chemistry_, I. p. 62 (Arnold).

[160] _Jour. Chem. Soc._, 1877, 32. 395.

[161] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.

[162] _Zeitschr. physikal. Chem._, 1887, 1. 5; 1895, 17. 52.

[163] It is important to powder the salt, since otherwise the dehydration
of the hydrate and the production of equilibrium occurs with comparatively
great tardiness.

[164] A chemical individual is a substance which persists as a phase of
constant composition when the conditions of temperature, pressure, and
composition of the other phases present, undergo continuous alteration
within certain limits--the limits of existence of the substance (Wald,
_Zeitschr. physikal. Chem._, 1897, 24. 648).

[165] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323; Ostwald,
_Lehrbuch_, I. 606.

[166] That mercury does dissolve in water can be argued from analogy, say,
with mercury and bromonaphthalene. At the ordinary temperature these two
liquids appear to be quite insoluble in one another, but at a temperature
of 280° the mercury dissolves in appreciable quantity; for on heating a
tube containing bromonaphthalene over mercury the latter sublimes _through_
the liquid bromonaphthalene and condenses on the upper surface of the tube.

[167] _Phil. Mag._, 1884, [5], 18. 22; 495.

[168] _Wied. Annalen_, 1886, 28. 305.

[169] _Zeitschr. physikal. Chem._, 1898, 26. 433.

[170] Rothmund, _loc. cit._

[171] Rothmund, _loc. cit._

[172] A similar behaviour is found in the case of diethylamine and water
(R. T. Lattey, _Phil. Mag._, 1905, [6], 10, 397).

[173] C. S. Hudson, _Zeitschr. physikal. Chem._, 1904, 47. 113.

[174] Konowaloff, _Wied. Annalen_, 1881, 14. 219. Ostwald, _Lehrbuch_, II.
2. 687. Bancroft, _Phase Rule_, p. 96.

[175] Konowaloff, _loc. cit._

[176] Roozeboom, _Zeitschr. physikal. Chem._, 1891, 8. 526; _Rec. Trav.
Chim. Pays-Bas_, 1884, 3. 38.

[177] Konowaloff, _loc. cit._ Cf. Bancroft, _Phase Rule_, p. 100.

[178] _Phil. Mag._, 1884 [5], 18. 503.

[179] See, for example, Walker, _Introduction to Physical Chemistry_, 3rd
edit., p. 86 (Macmillan, 1903). Consult also Young, _Fractional
Distillation_ (Macmillan, 1903), or Kuenen, _Verdampfung und Verflüssigung
von Gemischen_ (Barth, 1906), where the subject is fully treated.

[180] Since this is the only phase of variable composition present.

[181] E. von Stackelberg, _Zeitschr. physikal. Chem._, 1896, 20. 337. If
the change of volume which accompanies solution, and the heat effect are
known, the quantitative change of the solubility with the pressure can be
calculated (Braun, _Zeitschr. physikal. Chem._, 1887, 1. 259).

[182] Van't Hoff, _Arch. néerland._ 1901 [2], 6. 471.

[183] Tilden and Shenstone, _Phil. Trans._ 1884, 175. 23; Hulett and Allen,
_Jour. Amer. Chem. Soc._ 1902, 24. 667; Andreä, _Jour. prak. Chem._ 137.
474; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Mylius and v. Wrochem,
_Ber._ 1900, 33. 3689.

[184] E. von Stackelberg, _Zeitschr. physikal. Chem._ 1896, 20. 159; 1898,
26. 533; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Holsboer, _Zeitschr.
physikal. Chem._, 1902, 39. 691.

[185] Reicher and van Deventer, _Zeitschr. physikal. Chem._ 1890, 5. 559;
cf. Ostwald, _Lehrbuch_, II. 2. 803.

[186] It has been shown that the formula of Ramsay and Young (p. 66) can be
applied (with certain restrictions) to the interpolation and extrapolation
of the solubility curve of a substance provided two (or three) points on
the curve are known. In this case T, T_{1}, etc., refer to the temperatures
at which the two substances--one the solubility curve of which is known,
the other the solubility curve of which is to be calculated--have equal
solubilities, instead of, as in the previous case, equal vapour pressures.
(Findlay, _Proc. Roy. Soc._, 1902, 69. 471; _Zeitschr. physikal. Chem._,
1903, 42. 110.)

[187] W. Müller and P. Kaufmann, _Zeitschr. physikal. Chem._ 1903, 42. 497.

[188] W. O. Rabe, _Zeitschr. physikal. Chem._, 1901, 38. 175.

[189] With regard to the limits of supersaturation and the spontaneous
crystallization of the solute from supersaturated solutions, see Jaffé,
_Zeitschr. physikal. Chem._, 1903, 43. 565, and the very interesting paper
by Miers and Isaac, _Trans. Chem. Soc._, 1906, 89. 413.

[190] _Annales chim. phys._, 1894 [7], 2. 524.

[191] _Phil. Trans._, 1884, 175. 23.

[192] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 543.

[193] _Zeitschr. physikal. Chem._, 1903, 43. 313.

[194] Guthrie, _Phil. Mag._, 1875, [4], 49. 1; 1884, [5], 17. 462.

[195] See Roloff, _Zeitschr. physikal. Chem._, 1895, 17. 325; Guthrie,
_loc. cit._

[196] Guthrie, _Phil. Mag._, _loc. cit._ Cf. Ostwald, _Lehrbuch_, II. 2.
843.

[197] Guthrie, _Phil. Mag._, 1875 [4], 49. 269.

[198] _Ber._, 1877, 20. 2223.

[199] _Silz-Ber. Wien. Akad._, 1880, 81. II. 1058.

[200] Guthrie, _Phil. Mag._, 1875 [4], 49. 206.

[201] If in the neighbourhood of the cryohydric point solution should be
accompanied by an evolution of heat, then as the solubility would in that
case increase with fall of temperature, salt would pass into solution.

[202] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193.

[203] _Zeitschr. physikal. Chem._, 1897, 23. 418.

[204] Provided the solid nitrile is not present in too great excess.

[205] _Wied. Annalen_, 1886, 28. 328. Cf. Ostwald, _Lehrbuch_, II. 2. 872.

[206] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193. Schreinemakers,
_ibid._, 1897, 23. 417. Roozeboom, _Rec. trav. chim. Pays-Bays_, 1889, 8.
257. Bruner, _Zeitschr. physikal. Chem._, 1897, 23. 542.

[207] Van't Hoff, _Lectures on Theoretical Chemistry_, I. p. 42. Ostwald,
_Lehrbuch_, II. 2. 824.

[208] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
Findlay, 2nd edit., p. 453 (Macmillan, 1904); Skirrow and Calvert,
_Zeitschr. physikal. Chem._, 1901, 37. 217.

[209] _Vide_ Loewel, _Annales chim. phys._, 1857 [3], 49. 32. Cf.
Löwenherz, _Zeitschr. physikal. Chem._, 1895, 18. 82.

[210] Loewel, _loc. cit._ Gay-Lussac, _Annales chim. phys._, 1819, 11. 296.
For the solubility at higher temperatures, see Tilden and Shenstone, _Phil.
Trans._, 1884, 175. 23. Étard, _Annales chim. phys._, 1894 [7], 2. 548.

[211] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690; Richards and
Wells, _ibid._, 1903, 43. 465. This temperature is not quite the same as
that of the _quadruple point_ anhydrous salt--hydrated
salt--solution--vapour, because the latter is the temperature at which the
system is under the pressure of its own vapour. Since, however, the
influence of pressure on the solubility is very slight (p. 107), the
position of the two points will not be greatly different. The quadruple
point was found by Cohen (_Zeitschr. physikal. Chem._, 1894, 14. 90) to be
32.6° and 30.8 mm. of mercury.

[212] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
185. Cf. Cohen, _ibid._, 1894, 14. 88.

[213] Debray, _Compt. rend._, 1868, 66. 194.

[214] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690. A number of
other salt hydrates, having transition-points ranging from 20° to 78°,
which might be used for the same purpose, have been given by Richards and
Churchill, _ibid._, 1899, 28. 313.

[215] _Zeitschr. physikal. Chem._, 1903, 46. 818.

[216] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 67.

[217] Cohen, _Zeitschr. physikal. Chem._, 1894, 14. 90.

[218] Ziz, _Schweigger's Journal_, 1815, 15. 166. See Ostwald, _Lehrbuch_,
II. 2. 717.

[219] See, for example, the solubility determinations published in
_Wissenschaftliche Abhandl. der physikalisch-technischen Reichsanstalt_,
Vol. III., or in the _Berichte_, for the years 1897-1901.

[220] Meusser, _Ber._, 1901, 34. 2440.

[221] Mylius and von Wrochem, _Ber._, 1900, 33. 3693.

[222] Walker and Fyffe, _Jour. Chem. Soc._, 1903, 83. 180.

[223] _Monatshefte_, 1887, 8. 601.

[224] The equilibria between calcium chloride and water have been most
completely studied by Roozeboom (_Zeitschr. physikal. Chem._, 1889, 4. 31).

[225] Hammerl, _Sitzungsber. Wien. Akad._, 2^{te} Abteil, 1878, 78. 59.
Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.

[226] Lidbury, _Zeitschr. physikal. Chem._, 1902, 39. 453. The curvature at
the melting point is all the greater the more the compound is dissociated
into its components in the liquid state. If the compound is _completely
undissociated_, even in the vapour phase, the two branches of the curve
will _intersect_, (_e.g._ pyridine and methyl iodide; Aten, _Versl. Konink.
Akad. Wetensch. Amsterdam_, 1905, 13. 462). The smaller the degree of
dissociation, therefore, the sharper will be the bend. (See Stortenbeker,
_Zeitschr. physikal. Chem._, 1892, 10. 194.) From the extent of flattening
of the curve, it is also possible, with some degree of approximation, to
calculate the degree of dissociation of the substance in the fused state.
(See Roozeboom and Aten, _Zeitschr. physikal. Chem._, 1905, 53. 463;
Kremann, _Zeitschr. Elektrochem._, 1906, 12. 259.)

[227] See Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.

[228] Tammann, _Wied. Annalen_, 1899, 68. 577.

[229] Duhem, _Journ. Physical Chem._, 1898, 2. 31.

[230] Gibbs, _Trans. Conn. Acad._, 3. 155; Saurel, _Journ. Phys. Chem._,
1901, 5. 35.

[231] In the case of the fusion of a compound of two components with
formation of a liquid phase of the same composition, the temperature is a
maximum; in the case of liquid mixtures of constant boiling-point, the
temperature may be a minimum (p. 105).

[232] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477. The formula of
ferric chloride has been doubled, in order to avoid fractions in the
expression of the water of crystallization.

[233] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477.

[234] A similar series of hydrates is formed by zinc chloride and water
(Dietz and Mylius, _Zeitschr. anorg. Chem._, 1905, 44. 209).

[235] Meyerhoffer, _Ber._, 1897, 30. 1810.

[236] Walden, _Ber._, 1899, 32. 2863.

[237] _Zeitschr. physikal. Chem._, 1903, 42. 432.

[238] This composition was also confirmed by measurements of the vapour
pressure (cf. p. 90).

[239] Since all substances are no doubt volatile to a certain extent at
some temperature, it is to be understood here that the substances are
appreciably volatile at the temperature of the experiment.

[240] For a general discussion of the partial pressures in a system of two
components, see Bancroft, _Journ. Physical Chem._, 1899, 3. 1.

[241] _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. trav. chim.
Pays-Bas_, 1888, 7. 152.

[242] The composition of a solution is represented symbolically by placing
a double wavy line between the symbols of the components, and indicating
the number of atoms present in the ordinary manner: thus, I [wavy] Cl_{_x_}
represents a solution containing _x_ atoms of chlorine to one atom of
iodine (Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 450).

[243] Since iodine monochloride in the liquid state is only very slightly
dissociated, the bend at C is very sharp (see p. 147, footnote). See also
the investigation of the system pyridine and methyl iodide (Aten, _Versl.
Konink. Akad. Wetensch. Amsterdam_, 1905, 13. 462).

[244] This upper branch of the curve is not shown in the figure, as the
ordinate corresponding to 30° would be very great.

[245] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 22.

[246] Ramsay and Young, _Journ. Chem. Soc._, 1886, 49. 458.

[247] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 77 (Arnold).

[248] This is different from what we found in the case of non-volatile
solutes (p. 126). In the present case, the _partial pressure_ of the iodine
in the vapour will be lowered by addition of chlorine, but the _total
pressure_ is increased.

[249] The diminution of volume is supposed to be carried out at constant
temperature. The pressure and the composition of the phases must,
therefore, remain unchanged, and only the relative amounts of these can
undergo alteration.

[250] At point _b_ the ratio of chlorine to iodine in the solution is less
than in the monochloride, so that by the separation of this the excess of
chlorine yielded by the condensation of the vapour is removed.

[251] Roozeboom, _Rec. trav. chim. Pays-Bas_, 1884, 3. 29; 1885, 4. 65;
_Zeitschr. physikal. Chem._, 1888, 2. 450.

[252] Two curves "enclose" a field when they form with one another an angle
less than two right angles.

[253] Roozeboom, _Zeitschr. physikal. Chem._, _loc. cit._

[254] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323.

[255] Bancroft has proposed to restrict the term "occlusion" to the
formation of solid solutions, and to apply "adsorption" only to effects
which are primarily due to surface tension. Such a distinction, however,
would probably be very difficult to carry through, for although adsorption
may, in large measure, be due to surface tension, the behaviour of adsorbed
substances is similar to that of substances existing in solid solutions.

[256] Tammann, _Wied. Annalen_, 1897, 63. 16; _Zeitschr. physikal. Chem._,
1898, 27. 323.

[257] See, for example, Chappuis, _Wied. Annalen_, 1881, 12. 161; Joulin,
_Annal. chim. phys._, 1881, [5], 22. 398; Kayser, _Wied. Annalen_, 1881,
12. 526.

[258] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.

[259] _Annales chim. phys._, 1874, [5], 2. 279.

[260] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1; Dewar, _Phil.
Mag._, 1874, [4], 47, 324, 342; Mond, Ramsay and Shields, _Proc. Royal
Soc._, 1897, 62. 290.

[261] _Loc. cit._

[262] It is noteworthy that the form of curve obtained for hydrogen and
palladium bears a striking resemblance to that for the dehydration of
colloids containing absorbed water, _e.g._ silicic acid (_vide_ van
Bemmelen, _Zeitschr. anorg. Chem._, 1897-1900. Cf. Zacharias, _Zeitschr.
physikal. Chem._, 1902, 39. 480).

[263] _Zeitschr. physikal. Chem._, 1890, 5. 322.

[264] Küster, _Zeitschr. physikal. Chem._, 1895, 17. 367. Bodländer, _Neues
Jahrbuch f. Mineralogie_, 1898-99, Beilage Band, 12. 92.

[265] Bruni and Padoa, _Atti Accad. Lincei_, 1902 [5], 11. 1; 565.

[266] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 385; Bruni, _Rend.
Accad. Lincei_, 1898, 2. 138, 347. For a general account of "solid
solutions" the reader is referred to Bruni, "_Ueber feste Lösungen_"
(Ahrens'sche Sammlung), and to Bodländer, _loc. cit._ For the formation and
transformation of liquid mixed crystals, see A. C. de Kock, _Zeitschr.
physikal. Chem._, 1904, 48. 129.

[267] In discussing the various systems which may be obtained here,
Roozeboom (_loc. cit._) made use of the variation of the thermodynamic
potential (p. 29) with the concentration. In spite of the advantages which
such a treatment affords, the temperature-concentration diagram has been
adopted as being more readily understood and as more suitable for an
elementary discussion of the subject.

[268] These curves are also called the "liquidus" and the "solidus" curve
respectively.

[269] Küster, _Zeitschr. physikal. Chem._, 1895, 17. 360.

[270] Küster, _ibid._, 1891, 8. 589.

[271] It should be remarked that the behaviour described here will hold
strictly only when the solid mixed crystals undergo change sufficiently
rapidly to be always in equilibrium with the liquid. This, however, is not
always the case (see Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494;
van Wyk, _Zeitschr. anorg. Chem._, 1905, 48. 25), and complete
solidification will not in this case take place at the temperature
corresponding with the line _dc_ in Fig. 50, but only at a lower
temperature.

[272] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 469.

[273] Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494.

[274] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 542.

[275] Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 430.

[276] Cady, _Journ. Physical. Chem._, 1899, 3. 127.

[277] See Roberts-Austen and Stansfield, _Rapports du congrès international
de physique_, 1900, I. 363.

[278] Heycock and Neville, _Proc. Roy. Soc._, 1903, 71. 409. For the
partial liquefaction of mixed crystals on cooling, see also A. C. de Kock
(_Zeitschr. physikal. Chem._, 1904, 48. 129).

[279] Armstrong, _Watt's Dictionary of Chemistry_ (Morley and Muir), III.,
p. 88. See also Lowry, _Jour. Chem. Soc._, 1899, 75. 211.

[280] See Bancroft, _Journ. Physical Chem._, 1898, 2. 143; Roozeboom,
_Zeitschr. physikal. Chem._, 1899, 28. 288.

[281] Hylotropic substances are such as can undergo transformation into
other substances of the same composition (Ostwald, _Lehrbuch_, II. 2. 298).

[282] Also called Equilibrium Point (Lowry).

[283] For a discussion of these systems, see Roozeboom, _Zeitschr.
physikal. Chem._, _loc. cit_.

[284] See Bancroft, _loc. cit._, p. 147; Wegscheider, _Sitzungsber. Wiener
Akad._, 1902, 110. 908.

[285] Reference may be made here to the term "stability limit," introduced
by Knorr (_Annalen_, 1896, 293. 88) to indicate that temperature above
which liquefaction and isomeric change takes place. As employed by Knorr
and others, the term does not appear to have a very precise meaning, since
it is used to denote, not the temperature at which these changes can occur,
but the temperature at which the change is rapid (vide _Annalen_, 1896,
293. 91; 1899, 306. 334); and the introduction of an indefinite velocity of
change renders the temperature of the stability limit also somewhat
indefinite. The definiteness of the term is also not a little diminished by
the fact that the "limit" can be altered by means of catalytic agents.
Since, as we have seen, the stable modification can always undergo isomeric
change and liquefy at temperatures above the natural freezing point, but
not below that point; and, further, the less stable modification can
undergo isomeric transformation and liquefy at temperatures above the
eutectic point, but will not liquefy at temperatures below that; it seems
to the author that it would be more precise to identify these two
points--the natural freezing point and the eutectic point--which are not
altered by catalytic agents, with the "stability limits" of the stable and
unstable modification respectively. A perfectly definite meaning would
thereby be given to the term. In the case of those substances which do not
undergo appreciable isomeric change at the temperature of the melting
point, the stability limits would be the points G and H, Fig. 60.

[286] Cameron, _Journ. Physical Chem._, 1898, 2. 409.

[287] Carveth, _Journ. Phys. Chem._, 1898, 2. 159. See also Dutoit and
Fath, _Journ. chim. phys_., 1903, 1. 358; Findlay, _Trans. Chem. Soc._,
1904, 85. 403.

[288] Hollmann, _Zeitschr. physikal. Chem._, 1903, 43. 129.

[289] For other examples of the application of the Phase Rule to isomeric
substances, see _Journ. Physical Chem._, vols. 2. _et seq._; Findlay,
_Trans. Chem. Soc._, 1904, 85. 403.

[290] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 410.

[291] See also Saposchnikoff, _Zeitschr. physikal. Chem._, 49. 688;
Kremann, _Monatshefte_, 1904, 25. 1215, 1271, 1311.

[292] J. C. Philip, _Journ. Chem. Soc._, 1903, 83. 821.

[293] _Cf._ also Paterno and Ampolla, _Gazzetta chim. ital._, 1897, 27.
481.

[294] Philip, _loc. cit._, p. 826.

[295] Philip, _loc. cit._, p. 829. Compare curves for iodine monochloride,
Fig. 42, p. 162.

[296] Kuriloff, _Zeitschr. physikal. Chem._, 1897, 23. 676.

[297] Ladenburg, _Ber._, 1895, 28. 163; 1991.

[298] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 494; Adriani,
_ibid._, 1900, 33. 453.

[299] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 453.

[300] A. Findlay and Miss E. Hickmans.

[301] Kipping and Pope, _Journ. Chem. Soc._, 1897, 71. 993.

[302] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 512; Adriani,
_ibid._, 1900, 33. 473; 1901, 36. 168.

[303] In this connection reference should be made more especially to the
paper by Roberts-Austen and Stansfield, "Sur la constitution des alliages
métalliques," in the _Rapports du congrès international de physique_, 1900,
I. 363; J. A. Mathews, _Journ. of the Franklin Inst._, 1902; Gautier,
_Compt. rend._, 1896, 123. 109; Roberts-Austen, "Reports of the Alloys
Research Committee," in _Journ. Inst. Mechan. Engineers_, from 1891 to
1904; and the papers by Heycock and Neville, published in the _Journ. Chem.
Soc._, and the _Trans. Roy. Soc._ since 1897; also Neville, _Reports of the
British Association_, 1900, p. 131. Reference must also be made to the
important metallographic investigations by Tammann and his pupils, and of
Kurnakoff (_Zeitschr. anorgan. Chem._, vol. 40 and onwards), and also to
those of Shepherd, _Journ. Physical Chem._, 8. A bibliography of the alloys
is given in _Zeitschr. anorgan. Chem._, 1903, 35. 249.

[304] Kurnakoff and Puschin, _Zeitschr. anorgan. Chem._, 1902, 30. 104.

[305] Gautier, _Bull. Soc. d'Encouragement_, 1896 [5], 1. 1312.

[306] Heycock and Neville, _Phil. Trans._, 1900, 194. 201.

[307] Gautier, _loc. cit._ See also Roberts-Austen and Rose, _Proc. Roy.
Soc._, 1903, 71. 161.

[308] Heycock and Neville, _Journ. Chem. Soc._, 1897, 71. 414.

[309] See Roberts-Austen, _Introduction to Metallurgy_, 5th edit., p. 102;
Bakhuis Roozeboom, _Journ. Iron and Steel Inst._, 1900, II. 311; _Zeitschr.
physikal. Chem._, 1900, 34. 437; von Jüptner, _Siderology_, p. 223
(translation by C. Salter); van't Hoff, _Zinn, Gips, und Stahl_, p. 24, or
_Acht Vorträge über physikalische Chemie_, p. 37. Further, Roozeboom,
_Zeitschr. Elektrochem._, 1904, 10. 489; E. Heyn, _ibid._, p. 491;
Carpenter and Keeling, _Journ. Iron and Steel Inst._, 1904, 65. 224.

[310] The melting point of pure iron is given by Carpenter and Keeling
(_Journ. Iron and Steel Inst._, 1904, 65. 224) as 1505°.

[311] _Zeitschr. für Elektrochem._, 1904, 10. 491.

[312] See also Hiorns, _Journ. Soc. Chem. Ind._, 1906, 25. 50.

[313] Bancroft, _Jour. Physical Chem._, 1902, 6. 178; Bell and Taber,
_ibid._, 1906, 10. 120.

[314] The method to be followed when the third component enters into the
solid phase will be explained later.

[315] Tammann, _Zeitschr. anorg. Chem._, 1903, 37. 303; 1905, 45. 24.
Reference may be made here to the registering pyrometer of Kurnakoff,
_Zeitschr. anorg. Chem._, 1904, 42. 184.

[316] In this connection, see Doelter, _Physikalisch-chemisch Mineralogie_
(Barth, 1901); Meyerhoffer, _Zeitschr. f. Kristallographie_, 1902, 36. 593;
Guthrie, _Phil. Mag._, 1884 [5], 17. 479; Le Chatelier, _Compt. rend._,
1900, 130. 85; and especially E. Baur, _Zeitschr. physikal. Chem._, 1903,
42. 567; J. H. L. Vogt, _Zeitschr. Elektrochem._, 1903, 9. 852, and _Die
Silikatschmelzlösungen_, Parts I. and II. (Christiania, 1903, 1904). See
also N. V. Kultascheff, _Zeitschr. anorg. Chem._, 1903, 35. 187.

[317] G. G. Stokes, _Proc. Roy. Soc._, 1891, 49. 174; Gibbs, _Trans. Conn.
Acad._, 1876, 3. 176; Roozeboom, _Zeitschr. physikal. Chem._, 1894, 15.
147.

[318] This figure has been taken from Ostwald's _Lehrbuch_, II. 2. 984.

[319] Roozeboom, _Zeitschr. physikal. Chem._, 1893, 12. 369.

[320] C. R. A. Wright, _Proc. Roy. Soc._, 1891, 49. 174; 1892, 50. 375.

[321] The distribution coefficient will not remain constant because, apart
from other reasons, the mutual solubility of chloroform and water is
altered by the addition of the acid.

[322] Bancroft, _Physical Review_, 1895, 3. 21; Schreinemakers, _Zeitschr.
physikal. Chem._, 1897, 23. 652, and subsequent volumes.

[323] C. R. A. Wright, _Proc. Roy. Soc._, 1889-1893.

[324] C. R. A. Wright, _Proc. Roy. Soc._, 1892, 50. 390.

[325] Bodländer, _Berg- und Hüttenmänn. Ztg._, 1897, 56. 331.

[326] C. R. A. Wright, _Proc. Roy. Soc._, _loc. cit._

[327] Schreinemakers, _Zeitschr. physikal. Chem._, 1900, 33. 78.

[328] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 27. 95.

[329] Schreinemakers, _Zeitschr. physikal. Chem._, 1899, 29. 577.

[330] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 25. 543.

[331] Charpy, _Compt. rend._, 1898, 126. 1569. Compare the curves for the
system KNO_{3}--NaNO_{3}--LiNO_{3} (H. R. Carveth, _Journ. Physical Chem._,
1898, 2. 209). Also alloys of Pb--Sn--Bi (E. S. Shepherd, _Journ. Physical
Chem._, 1902, 6. 527).

[332] It should be remembered that in the triangular diagram a _line_
parallel to one of the sides indicates, at a given temperature, a constant
amount of the component represented by the opposite corner of the triangle;
and, hence, points in a _plane_, parallel to one face of a right prism,
will indicate for different temperatures, variation in the amounts of two
components, but constancy in the amount of the third.

[333] _Gazzetta chim. ital._, 1898, 28. II. 520.

[334] Bruni, _Gazzetta chim. ital._, 1898, 28. II. 508; 1900, 30. I. 35.

[335] _Zeitschr. physikal. Chem._, 1900, 36. 168.

[336] For a discussion of these systems, see van't Hoff, _Bildung und
Spaltung von Doppelsalzen_ (Leipzig, 1897).

[337] Van Leeuwen, _Zeitschr. physikal. Chem._, 1897, 23. 35.

[338] Meyerhoffer, _Zeitschr. physikal. Chem._, 1889, 3. 336; 1890, 5. 97.

[339] Reicher, _Zeitschr. physikal. Chem._, 1887, 1. 220.

[340] For other examples of the formation and decomposition of double salts
at a transition point, the reader is referred to the work by van't Hoff,
already cited, on the _Bildung und Spaltung von Doppelsalzen_; or to
Bancroft, _Phase Rule_, p. 180.

[341] Bancroft, _Phase Rule_, p. 183.

[342] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 514.

[343] The influence of pressure on the transition point in the case of
tachydrite has been determined by van't Hoff, Kenrick, and Dawson
(_Zeitschr. physikal. Chem._, 1901, 39. 27, 34; van't Hoff, _Zur Bildung
der ozeanischen Salzablagerungen_, I. p. 66--Brunswick, 1905). This salt is
formed from magnesium chloride and calcium chloride at 22°, in accordance
with the equation--

  2MgCl_{2}.6H_{2}O + CaCl_{2}.6H_{2}O = Mg_{2}CaCl_{6}.12H_{2}O + 6H_{2}O

Increase of pressure raises the transition point, because the formation of
tachydrite is accompanied by increase of volume; the elevation being 0.016°
for an increase of pressure of 1 atm. The number calculated from the
theoretical formula (p. 57) is 0.013° for 1 atm.

If one calculates the influence of the pressure of sea-water on the
temperature of formation of tachydrite (which is of interest on account of
the natural occurrence of this salt), it is found that a depth of water of
1500 metres, exerting a pressure of 180 atm., would alter the temperature
of formation of tachydrite by only 3°. The effect is, therefore,
comparatively unimportant.

[344] Roozeboom, _Zeitschr. physical. Chem._, 1887, 1. 227.

[345] _Zeitschr. physical. Chem._, 1887, 1. 227.

[346] Van't Hoff and Müller, _Ber._, 1898, 31. 2206.

[347] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
165.

[348] For a full discussion of the solubility relations of sodium ammonium
racemate, see van't Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 81.

[349] _Annales chim. phys._, 1848 [3], 24. 442.

[350] See Van't Hoff and van Deventer, _Zeitschr. phys. Chem._, 1887, 1.
165.

[351] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 121.

[352] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 518.

[353] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 109. On the
importance of the transition interval in the case of optically active
substances, see Meyerhoffer, _Ber._, 1904, 37. 2604.

[354] In connection with this chapter, see, more especially, van't Hoff,
_Bildung und Spaltung von Doppelsalzen_, p. 3, _ff._; Roozeboom, _Zeitschr.
physikal Chem._, 1892, 10. 158; Bancroft, _Phase Rule_, p. 201; 209.

[355] The same restriction must be made here as was imposed in the
preceding chapter, namely, that the two salts in solution give a common
ion.

[356] For example, addition of ammonium chloride to solutions of ferric
chloride (Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 149).

[357] It must, of course, be understood that the temperature is on that
side of the transition point on which the double salt is stable.

[358] Excess of the double salt must be taken, because otherwise an
unsaturated solution might be formed, and this would, of course, not
deposit any salt.

[359] Meyerhoffer, _Ber._, 1904, 37. 2605.

[360] Meyerhoffer, _Ber._, 1897, 30. 1809.

[361] Meyerhoffer, _Ber._, 1904, 37. 2604.

[362] Bancroft, _Phase Rule_, p. 203; Roozeboom, _Zeitschr. physikal.
Chem._, 1891, 8. 504, 531; Stortenbeker, _ibid._, 1895, 17. 643; 1897, 22.
60; 1900, 34. 108.

[363] Roozeboom, _Zeitschr. phys. Chem._, 1899, 28. 494; _Ber._, 1899, 32.
537.

[364] As, for instance, strychnine racemate, a compound of racemic acid
with the _optically active_ strychnine. This would be resolved into
strychnine _d_-tartrate and strychnine _l_-tartrate, which are not
enantiomorphous forms.

[365] Van't Hoff and Meyerhoffer, _Zeitschr. physikal Chem._, 1898, 27. 75;
1899, 30. 86. Fig. 113 is taken from the latter paper.

[366] Solid models constructed of plaster of Paris can be obtained from Max
Kaehler and Martini, Berlin.

[367] Instead of the present method of obtaining potassium chloride by
decomposing carnallite with water, advantage might be taken of the fact
that carnallite when heated to 168° undergoes decomposition with separation
of three-fourths of the potassium chloride (van't Hoff, _Acht Vorträge über
physikalische Chemie_, 1902, p. 32).

[368] Roozeboom and Schreinemakers, _Zeitschr. physikal. Chem._, 1894, 15.
588.

[369] These curves represent only portions of the isotherms, since the
systems in which a ternary solution is in equilibrium with solid hydrogen
chloride or a hydrate, have not been investigated.

[370] The numbers printed beside the points on the curves refer to the
number of the experiment in the original paper.

[371] Lash, Miller and Kenrick, _Journ. Physical. Chem._, 1903, 7. 259;
Allan, _Amer. Chem. Journ._, 1901, 25. 307.

[372] Allan, _Amer. Chem. Journ._, 1901, 25. 307.

[373] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 651; Allan, _loc.
cit._

[374] Rutten, _Zeitschr. anorgan. Chem._, 1902, 30. 342. Compare the system
BeO--SO_{3}--H_{2}O; Parsons, _Zeitschr. anorgan. Chem._, 1904, 42. 250.

[375] _Zeitschr. anorgan. Chem._, 1904, 40. 146.

[376] Schreinemakers, _Zeitschr. physikal. Chem._, 1893, 11. 76; Bancroft,
_Journ. Physical Chem._, 1902, 6. 179.

[377] _Zeitschr. anorgan. Chem._, 1904, 40. 148.

[378] _Zeitschr. physikal. Chem._, 1903, 43. 354.

[379] These equilibria were obtained by Boudouard, _Annales chim. phys._,
1901 [7], 24. 5. See also Hahn, _Zeitschr. physikal. Chem._, 1903, 42.
705; 44. 513.

[380] G. Preuner, _Zeitschr. physikal. Chem._, 1903, 47. 385.

[381] See Hahn, _Zeitschr. physikal. Chem._, 1903, 42. 705; 44. 513;
Boudouard, _Bull. Soc. chim._, [3], 25. 484; Bodländer, _Zeitschr. f.
Elektrochem._, 1902, 8. 833; R. Schenck and Zimmermann, _Ber._, 1903, 36.
1231, 3663; Schenck and Heller, _ibid._, 1905, 38. 2132; _Zeitschr. f.
Elektrochem._, 1903, 9. 691; Haber, _Thermodynamik technischer
Gasreaktionen_, p. 293 (Munich, 1903).

[382] A very useful summary of the investigations carried out by van't Hoff
and his pupils on the formation of the Stassfurt salt-beds is given by E.
F. Armstrong, in the _Reports of the British Association for 1901_, p. 262.
See also van't Hoff, _Zur Bildung der ozeanischen Salzablagerungen_
(Brunswick, 1905).

[383] See especially Meyerhoffer, _Silzungsber. Wien. Akad._, 1895, 104.
II. _b_, 840; Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899,
28. 453; 31. 370. The investigation of the equilibria between reciprocal
salt-pairs alone (three-component systems) is of great importance for the
artificial preparations of minerals, as also in analytical chemistry for
the proper understanding of the methods of conversion of insoluble systems
into soluble by fusion (see Meyerhoffer, _Zeitschr. physikal. Chem._, 1901,
38. 307).

[384] See Meyerhoffer, _Zeitschr. physikal. Chem._, 1899, 28. 459.

[385] Compare the reciprocal salt-pair NaCl--NH_{4}HCO_{3} (p. 321). In
this case the upper limit of the transition interval was found by
extrapolation of the solubility curve for NaHCO_{3} + NH_{4}Cl +
NH_{4}HCO_{3} and NaHCO_{3} + NH_{4}Cl + NaCl to be 32° (Fedotieff,
_Zeitschr. phys. Chem._, 1904, 49. 179).

[386] Löwenherz, _Zeitschr. physikal. Chem._, 1894, 13. 464.

[387] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.

[388] As the quantities of the salts are expressed in _equivalent_
gram-molecules, the molecule of sodium and potassium chloride must be
doubled in order to be equivalent to sodium sulphate and potassium
sulphate.

[389] _Sitz-Ber. der kgl. preuss. Akad. der Wiss._, 1903, p. 359. Van't
Hoff, _Zur Bildung der ozeanischen Salzablagerungen_, I. p. 34 (Brunswick,
1905).

[390] _Zeitschr. für Kristallographie_, 1904, 39. 155.

[391] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.

[392] _Zeitschr. physikal. Chem._, 1904, 49. 162.

[393] Another commercial process, in the study of which good service is
done by the Phase Rule, is the caustification of the alkali salts (G.
Bodländer, _Zeitschr. für Elektrochem._, 1905, 11. 186; J. Herold, _ibid._,
418).

[394] _Zeitschr. physikal. Chem._, 1900, 35. 32.

[395] Mention may also be made here of the equilibria between magnesium
carbonate and potassium carbonate, although these do not form a reciprocal
salt-pair (Auerbach, _Zeitschr. für Elektrochem._, 1904, 10. 161).

[396] O. N. Witt and K. Ludwig, _Ber._, 1903, 36. 4384; Meyerhoffer,
_ibid._, 1904, 37. 261, 1116.

[397] _Zeitschr. physikal. Chem._, 1905, 53. 513. Compare also, _ibid._,
1903, 38. 307.

[398] See Schwarz, _Beiträge zur Kenntnis der umkehrbaren Umwandlungen
polymorpher Korper_ (Göttingen, 1892); or, Roozeboom, _Heterogen.
Gleichgewicht_, I. p. 125. Also Barnes and Cooke, _Journ. Physical Chem._,
1902, 6. 172.

[399] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
173.

[400] Reicher, _Zeitschr. für Krystallographie_, 1884, 8. 593.

[401] _Zeitschr. physikal. Chem._, 1895, 17. 153.

[402] _Zeitschr. physikal. Chem._, 1899, 28. 464.

[403] Meyerhoffer and Saunders, _ibid._, p. 466.

[404] See Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 446.

[405] See in this connection the volume in this series on
_Electro-chemistry_, by Dr. R. A. Lehfeldt.

[406] Barnes and Cooke, _Journ. Physical Chem._, 1902, 6. 172.

[407] For a description and explanation of these, the reader should consult
the volume in this series by Dr. Lehfeldt on _Electro-chemistry_; and van't
Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 48 _ff._




       *       *       *       *       *




Changes made to the printed original.

Pages 30-31. "Fig. 3, p. 27.": 'p. 25." in original. So also page 33, "Fig.
2, p. 27".

Page 57. "pp. 29 and 35": 'pp. 25 and 38" in original.

Page 65. "p. 57.": 'p. 60" in original (twice).

Page 166. "there is the point C_{1}": C' in original.

Page 225. "C is an eutectic point": 'eutetic' in original.

Page 228. "Although this view put forward by Heyn": 'Athough' in original.

Page 232. "the period of constant temperature for the eutectic point c":
'the eutectic point e' in original.

Page 249. "two liquid layers between 13° and 31°": 'betwen' in original.

Page 257. Tables entries 4 and 7. "naphthol": 'napthol' in original.

Page 287. "from which the model is constructed": 'he model' in original.