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% Project Gutenberg's Eight Lectures on Theoretical Physics, by Max Planck%
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% This eBook is for the use of anyone anywhere at no cost and with        %
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% Title: Eight Lectures on Theoretical Physics                            %
%        Delivered at Columbia University in 1909                         %
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% Author: Max Planck                                                      %
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% Translator: A. P. Wills                                                 %
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% Release Date: February 29, 2012 [EBook #39017]                          %
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% Language: English                                                       %
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% *** START OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES ***            %
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Project Gutenberg's Eight Lectures on Theoretical Physics, by Max Planck

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: Eight Lectures on Theoretical Physics
       Delivered at Columbia University in 1909

Author: Max Planck

Translator: A. P. Wills

Release Date: February 29, 2012 [EBook #39017]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK EIGHT LECTURES ***


Produced by Brenda Lewis, Keith Edkins and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/Canadian Libraries)
\end{verbatim}
\pagestyle{empty}
\newpage

%-----File: 001.png---\redacted\--------
\begin{center}
{\small \so{COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK}}

\vspace{0.5\baselineskip}

{\footnotesize PUBLICATION NUMBER THREE\\
OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH\\
ESTABLISHED DECEMBER 17\textsc{th}, 1904}

\rule{4in}{0.5pt}

\vspace{-2.5ex}

\rule{4in}{0.5pt}

\vspace{\baselineskip}

{\LARGE \textbf{EIGHT LECTURES}\\[1ex]
\textbf{ON THEORETICAL PHYSICS}}

\vspace{\baselineskip}

{\footnotesize DELIVERED AT COLUMBIA UNIVERSITY\\
IN 1909}

\vspace{\baselineskip}

{\tiny BY}

{\small MAX PLANCK}

\vspace{0.5\baselineskip}

{\tiny PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN\\[-1.5ex]
LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1909}

\vspace{\baselineskip}

{\scriptsize TRANSLATED BY}

{\footnotesize A. P. WILLS}

{\tiny PROFESSOR OF MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY}

\vspace{3\baselineskip}

\pngcent{illo001.png}{534}

\vspace{3\baselineskip}

{\small NEW YORK\\
COLUMBIA UNIVERSITY PRESS\\
1915}
\end{center}

{\scriptsize \noindent \textsc{Transcriber's Note:} \emph{A few typographical errors have been corrected -
these are noted at the end of the text.}}

%-----File: 002.png---\redacted\--------
\newpage
\begin{center}
\textsc{Translated and Published by Arrangement with\\
S.~Hirzel, Leipzig, owner of the original copyright\\
Copyright 1915 by Columbia University Press}

\vspace{6in}

\textsf{\footnotesize PRESS OF\\
THE NEW ERA PRINTING COMPANY\\
LANCASTER, PA.}

{\small 1915}
\end{center}

%-----File: 003.png---\redacted\--------
\newpage
\begin{spacing}{0.9}{\small On the seventeenth day of December, nineteen hundred and four, Edward Dean
Adams, of New York, established in Columbia University ``The Ernest Kempton
Adams Fund for Physical Research'' as a memorial to his son, Ernest Kempton
Adams, who received the degrees of Electrical Engineering in~1897 and Master of
Arts in~1898, and who devoted his life to scientific research. The income of this
fund is, by the terms of the deed of gift, to be devoted to the maintenance of a
research fellowship and to the publication and distribution of the results of scientific
research on the part of the fellow. A generous interpretation of the terms of the
deed on the part of Mr.~Adams and of the Trustees of the University has made it
possible to issue these lectures as a publication of the Ernest Kempton Adams Fund.}\end{spacing}

\begin{center}

\vspace{-\baselineskip}\rule{4in}{0.5pt}

\vspace{-2.5ex}

\rule{4in}{0.5pt}\vspace{\baselineskip}

\textbf{Publications of the\\
Ernest Kempton Adams Fund for Physical Research}
\end{center}

\advert{Number One. \textbf{Fields of Force.} By \textsc{Vilhelm Friman Koren Bjerknes}, Professor of Physics
in the University of Stockholm. A course of lectures delivered at Columbia University,
1905--6.}
{Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on
application of hydrodynamics to meteorology. 160~pp.}\vspace{-1.5\baselineskip}

\advert{Number Two. \textbf{The Theory of Electrons and its Application to the Phenomena of Light and
Radiant Heat.} By \textsc{H.~A. Lorentz}, Professor of Physics in the University of Leyden.
A course of lectures delivered at Columbia University, 1906--7. With added notes.
332~pp. Edition exhausted. Published in another edition by Teubner.}{}\vspace{-2.5\baselineskip}

\advert{Number Three. \textbf{Eight Lectures on Theoretical Physics.} By \textsc{Max Planck}, Professor of
Theoretical Physics in the University of Berlin. A course of lectures delivered at
Columbia University in 1909, translated by \textsc{A.~P. Wills}, Professor of Mathematical
Physics in Columbia University.}
{Introduction: Reversibility and irreversibility.  Thermodynamic equilibrium in dilute solutions.
Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory.
Statistical theory. Principle of least work. Principle of relativity. 130~pp.}\vspace{-1.5\baselineskip}

\advert{Number Four. \textbf{Graphical Methods.} By \textsc{C. Runge}, Professor of Applied Mathematics in the
University of Göttingen. A course of lectures delivered at Columbia University,
1909--10.}
{Graphical calculation. The graphical representation of functions of one or more independent variables.
The graphical methods of the differential and integral calculus. 148~pp.}\vspace{-1.5\baselineskip}

\advert{Number Five. \textbf{Four Lectures on Mathematics.} By \textsc{J. Hadamard}, Member of the Institute,
Professor in the Collége de~France and in the École Polytechnique. A course of lectures
delivered at Columbia University in~1911.}
{Linear partial differential equations and boundary conditions. Contemporary researches in differential
and integral equations. Analysis situs. Elementary solutions of partial differential equations
and Green's functions. 53~pp.}\vspace{-1.5\baselineskip}

\advert{Number Six. \textbf{Researches in Physical Optics, Part~I,} with especial reference to the radiation
of electrons. By \textsc{R.~W. Wood}, Adams Research Fellow, 1913, Professor of Experimental
Physics in the Johns Hopkins University. 134~pp. With 10~plates. Edition exhausted.}{}\vspace{-2.5\baselineskip}

\advert{Number Seven. \textbf{Neuere Probleme der theoretischen Physik.} By \textsc{W.~Wien}, Professor of
Physics in the University of Würzburg. A course of six lectures delivered at Columbia
University in~1913.}
{Introduction: Derivation of the radiation equation. Specific heat theory of Debye. Newer radiation
theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein
fluctuations. Theory of Röntgen rays. Method of determining wave length. Photo-electric effect and
emission of light by canal ray particles. 76~pp.}\vspace{-1\baselineskip}

\begin{spacing}{0.9}{\small These publications are distributed under the Adams Fund to many libraries
and to a limited number of individuals, but may also be bought at cost from the
Columbia University Press.}\end{spacing}

%-----File: 004.png---\redacted\--------
% [Blank Page]
%-----File: 005.png---\redacted\--------

\newpage
\Section{}{\emph{PREFACE TO ORIGINAL EDITION.}}

The present book has for its object the presentation of the
lectures which I delivered as foreign lecturer at Columbia University
in the spring of the present year under the title: ``The
Present System of Theoretical Physics.'' The points of view
which influenced me in the selection and treatment of the
material are given at the beginning of the first lecture. Essentially,
they represent the extension of a theoretical physical
scheme, the fundamental elements of which I developed in an
address at Leyden entitled: ``The Unity of the Physical Concept
of the Universe.'' Therefore I regard it as advantageous to
consider again some of the topics of that lecture. The presentation
will not and can not, of course, claim to cover exhaustively
in all directions the principles of theoretical physics.

\begin{flushright}
   \textsc{The Author.\hspace*{1em}}
\end{flushright}

\textsc{Berlin}, 1909
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% [Blank Page]
%-----File: 007.png---\redacted\--------

\vspace{10ex}

\Section{}{\emph{TRANSLATOR'S PREFACE.}}

At the request of the Adams Fund Advisory Committee, and
with the consent of the author, the following translation of Professor
Planck's Columbia Lectures was undertaken. It is hoped
that the translation will be of service to many of those interested
in the development of theoretical physics who, in spite of
the inevitable loss, prefer a translated text in English to an
original text in German. Since the time of the publication of
the original text, some of the subjects treated, particularly that
of heat radiation, have received much attention, with the result
that some of the points of view taken at that time have undergone
considerable modifications. The author considers it desirable,
however, to have the translation conform to the original
text, since the nature and extent of these modifications can
best be appreciated by reference to the recent literature relating
to the matters in question.
\begin{flushright}
    \textsc{A. P. Wills.}\hspace*{1em}
\end{flushright}
%-----File: 008.png---\redacted\--------
% [Blank Page]
%-----File: 009.png---\redacted\--------

\newpage

\begin{center}{\Large CONTENTS.}\end{center}

\medskip\begin{center}\textsc{First Lecture.}\end{center}

\vspace{-3ex}\hfill{\tiny PAGE}

Introduction. Reversibility and Irreversibility \dotfill   \pageref{Lect1}

\medskip\begin{center}\textsc{Second Lecture.}\end{center}

Thermodynamic States of Equilibrium in Dilute Solutions \dotfill \pageref{Lect2}

\medskip\begin{center}\textsc{Third Lecture.}\end{center}

Atomic Theory of Matter \dotfill  \pageref{Lect3}

\medskip\begin{center}\textsc{Fourth Lecture.}\end{center}

Equation of State for a Monatomic Gas \dotfill   \pageref{Lect4}

\medskip\begin{center}\textsc{Fifth Lecture.}\end{center}

Heat Radiation. Electrodynamic Theory \dotfill   \pageref{Lect5}

\medskip\begin{center}\textsc{Sixth Lecture.}\end{center}

Heat Radiation. Statistical Theory \dotfill  \pageref{Lect6}

\medskip\begin{center}\textsc{Seventh Lecture.}\end{center}

General Dynamics. Principle of Least Action \dotfill    \pageref{Lect7}

\medskip\begin{center}\textsc{Eighth Lecture.}\end{center}

General Dynamics. Principle of Relativity \dotfill   \pageref{Lect8}

%-----File: 010.png---\redacted\--------
% [Blank Page]
%-----File: 011.png---\redacted\--------

\Chapter{First Lecture.}
{Introduction: Reversibility and Irreversibility.}\label{Lect1}
\pagestyle{plain}


\First{Colleagues, ladies and gentlemen:} The cordial invitation, which
the President of Columbia University extended to me to
deliver at this prominent center of American science some
lectures in the domain of theoretical physics, has inspired in
me a sense of the high honor and distinction thus conferred
upon me and, in no less degree, a consciousness of the
special obligations which, through its acceptance, would be
imposed upon me. If I am to count upon meeting in some
measure your just expectations, I can succeed only through
directing your attention to the branches of my science with
which I myself have been specially and deeply concerned, thus
exposing myself to the danger that my report in certain respects
shall thereby have somewhat too subjective a coloring.

From those points of view which appear to me the most
striking, it is my desire to depict for you in these lectures the
present status of the system of theoretical physics. I do not
say: the present status of theoretical physics; for to cover this
far broader subject, even approximately, the number of lecture
hours at my disposal would by no means suffice. Time limitations
forbid the extensive consideration of the details of this great
field of learning; but it will be quite possible to develop for you, in
bold outline, a representation of the system as a whole, that is, to
give a sketch of the fundamental laws which rule in the physics
of today, of the most important hypotheses employed, and of
the great ideas which have recently forced themselves into the
subject. I will often gladly endeavor to go into details, but not
in the sense of a thorough treatment of the subject, and only with
the object of making the general laws more clear, through appropriate
%-----File: 012.png---\redacted\--------
specially chosen examples. I shall select these examples
from the most varied branches of physics.

If we wish to obtain a correct understanding of the achievements
of theoretical physics, we must guard in equal measure
against the mistake of overestimating these achievements, and
on the other hand, against the corresponding mistake of underestimating
them. That the second mistake is actually often
made, is shown by the circumstance that quite recently voices
have been loudly raised maintaining the bankruptcy and,
débâcle of the whole of natural science. But I think such
assertions may easily be refuted by reference to the simple fact
that with each decade the number and the significance of the
means increase, whereby mankind learns directly through the
aid of theoretical physics to make nature useful for its own
purposes. The technology of today would be impossible without
the aid of theoretical physics. The development of the whole
of electro-technics from galvanoplasty to wireless telegraphy
is a striking proof of this, not to mention aerial navigation. On
the other hand, the mistake of overestimating the achievements
of theoretical physics appears to me to be much more dangerous,
and this danger is particularly threatened by those who have
penetrated comparatively little into the heart of the subject.
They maintain that some time, through a proper improvement
of our science, it will be possible, not only to represent completely
through physical formulae the inner constitution of the
atoms, but also the laws of mental life. I think that there is
nothing in the world entitling us to the one or the other of
these expectations. On the other hand, I believe that there is
much which directly opposes them. Let us endeavor then to
follow the middle course and not to deviate appreciably toward
the one side or the other.

When we seek for a solid immovable foundation which is able
to carry the whole structure of theoretical physics, we meet
with the questions: What lies at the bottom of physics? What
is the material with which it operates? Fortunately, there is
%-----File: 013.png---\redacted\--------
a complete answer to this question. The material with which
theoretical physics operates is measurements, and mathematics
is the chief tool with which this material is worked. All physical
ideas depend upon measurements, more or less exactly carried
out, and each physical definition, each physical law, possesses
a more definite significance the nearer it can be brought into
accord with the results of measurements. Now measurements
are made with the aid of the senses; before all with that of sight,
with hearing and with feeling. Thus far, one can say that the
origin and the foundation of all physical research are seated in
our sense perceptions. Through sense perceptions only do we
experience anything of nature; they are the highest court of
appeal in questions under dispute. This view is completely
confirmed by a glance at the historical development of physical
science. Physics grows upon the ground of sensations. The
first physical ideas derived were from the individual perceptions
of man, and, accordingly, physics was subdivided into: physics
of the eye (optics), physics of the ear (acoustics), and physics of
heat sensation (theory of heat). It may well be said that so
far as there was a domain of sense, so far extended originally
the domain of physics. Therefore it appears that in the beginning
the division of physics was based upon the peculiarities
of man. It possessed, in short, an anthropomorphic character.
This appears also, in that physical research, when not occupied
with special sense perceptions, is concerned with practical life,
and particularly with the practical needs of men. Thus, the
art of geodesy led to geometry, the study of machinery to mechanics,
and the conclusion lies near that physics in the last
analysis had only to do with the sense perceptions and needs
of mankind.

In accordance with this view, the sense perceptions are the
essential elements of the world; to construct an object as opposed
to sense perceptions is more or less an arbitrary matter of will.
In fact, when I speak of a tree, I really mean only a complex of
sense perceptions: I can see it, I can hear the rustling of its
%-----File: 014.png---\redacted\--------
branches, I can smell its fragrance, I experience pain if I knock
my head against it, but disregarding all of these sensations,
there remains nothing to be made the object of a measurement,
wherewith, therefore, natural science can occupy itself. This is
certainly true. In accordance with this view, the problem of
physics consists only in the relating of sense perceptions, in accordance
with experience, to fixed laws; or, as one may express
it, in the greatest possible economic accommodation of our ideas
to our sensations, an operation which we undertake solely
because it is of use to us in the general battle of existence.

All this appears extraordinarily simple and clear and, in accordance
with it, the fact may readily be explained that
this positivist view is quite widely spread in scientific circles
today. It permits, so far as it is limited to the standpoint here
depicted (not always done consistently by the exponents of
positivism), no hypothesis, no metaphysics; all is clear and
plain. I will go still further; this conception never leads to an
actual contradiction. I may even say, it can lead to no contradiction.
But, ladies and gentlemen, this view has never contributed
to any advance in physics. If physics is to advance, in
a certain sense its problem must be stated in quite the inverse
way, on account of the fact that this conception is inadequate
and at bottom possesses only a formal meaning.

The proof of the correctness of this assertion is to be found
directly from a consideration of the process of development
which theoretical physics has actually undergone, and which
one certainly cannot fail to designate as essential. Let us
compare the system of physics of today with the earlier and
more primitive system which I have depicted above. At the
first glance we encounter the most striking difference of all, that
in the present system, as well in the division of the various
physical domains as in all physical definitions, the historical
element plays a much smaller rôle than in the earlier system.
While originally, as I have shown above, the fundamental ideas
of physics were taken from the specific sense perceptions of man,
%-----File: 015.png---\redacted\--------
the latter are today in large measure excluded from physical
acoustics, optics, and the theory of heat. The physical definitions
of tone, color, and of temperature are today in no
wise derived from perception through the corresponding senses;
but tone and color are defined through a vibration number or
wave length, and the temperature through the volume change
of a thermometric substance, or through a temperature scale
based on the second law of thermodynamics; but heat sensation
is in no wise mentioned in connection with the temperature.
With the idea of force it has not been otherwise. Without
doubt, the word force originally meant bodily force, corresponding
to the circumstance that the oldest tools, the ax, hammer,
and mallet, were swung by man's hands, and that the first
machines, the lever, roller, and screw, were operated by men
or animals. This shows that the idea of force was originally
derived from the sense of force, or muscular sense, and was,
therefore, a specific sense perception. Consequently, I regard
it today as quite essential in a lecture on mechanics to refer, at
any rate in the introduction, to the original meaning of the force
idea. But in the modern exact definition of force the specific
notion of sense perception is eliminated, as in the case of color
sense, and we may say, quite in general, that in modern theoretical
physics the specific sense perceptions play a much smaller rôle
in all physical definitions than formerly. In fact, the crowding
into the background of the specific sense elements goes so far
that the branches of physics which were originally completely
and uniquely characterized by an arrangement in accordance
with definite sense perceptions have fallen apart, in consequence
of the loosening of the bonds between different and widely
separated branches, on account of the general advance towards
simplification and coordination. The best example of this is
furnished by the theory of heat. Earlier, heat formed a separate
and unified domain of physics, characterized through the
perceptions of heat sensation. Today one finds in well nigh all
physics textbooks dealing with heat a whole domain, that of
%-----File: 016.png---\redacted\--------
radiant heat, separated and treated under optics. The significance
of heat perception no longer suffices to bring together
the heterogeneous parts.

In short, we may say that the characteristic feature of the entire
previous development of theoretical physics is a definite elimination
from all physical ideas of the anthropomorphic elements, particularly
those of specific sense perceptions. On the other hand,
as we have seen above, if one reflects that the perceptions form
the point of departure in all physical research, and that it is impossible
to contemplate their absolute exclusion, because we cannot
close the source of all our knowledge, then this conscious
departure from the original conceptions must always appear
astonishing or even paradoxical. There is scarcely a fact in the
history of physics which today stands out so clearly as this.
Now, what are the great advantages to be gained through such
a real obliteration of personality? What is the result for the
sake of whose achievement are sacrificed the directness and
succinctness such as only the special sense perceptions vouchsafe
to physical ideas?

The result is nothing more than the attainment of unity
and compactness in our system of theoretical physics, and, in
fact, the unity of the system, not only in relation to all of its
details, but also in relation to physicists of all places, all times,
all peoples, all cultures. Certainly, the system of theoretical
physics should be adequate, not only for the inhabitants of this
earth, but also for the inhabitants of other heavenly bodies.
Whether the inhabitants of Mars, in case such actually exist,
have eyes and ears like our own, we do not know,---it is quite
improbable; but that they, in so far as they possess the necessary
intelligence, recognize the law of gravitation and the principle of
energy, most physicists would hold as self evident: and anyone
to whom this is not evident had better not appeal to the physicists,
for it will always remain for him an unsolvable riddle that the
same physics is made in the United States as in Germany.

To sum up, we may say that the characteristic feature of the
%-----File: 017.png---\redacted\--------
actual development of the system of theoretical physics is an
ever extending emancipation from the anthropomorphic elements,
which has for its object the most complete separation possible
of the system of physics and the individual personality of the
physicist. One may call this the objectiveness of the system
of physics. In order to exclude the possibility of any misunderstanding,
I wish to emphasize particularly that we have here
to do, not with an absolute separation of physics from the
physicist---for a physics without the physicist is unthinkable,---but
with the elimination of the individuality of the particular
physicist and therefore with the production of a common system
of physics for all physicists.

\label{png17lab1}Now, how does this principle agree with the positivist conceptions
mentioned above? Separation of the system of physics
from the individual personality of the physicist? Opposed to
this principle, in accordance with those conceptions, each
particular physicist must have his special system of physics, in
case that complete elimination of all metaphysical elements is
effected; for physics occupies itself only with the facts discovered
through perceptions, and only the individual perceptions are
directly involved. That other living beings have sensations is,
strictly speaking, but a very probable, though arbitrary, conclusion
from analogy. The system of physics is therefore primarily an
individual matter and, if two physicists accept the same system,
it is a very happy circumstance in connection with their personal
relationship, but it is not essentially necessary. One can regard
this view-point however he will; in physics it is certainly quite
fruitless, and this is all that I care to maintain here. Certainly,
I might add, each great physical idea means a further advance
toward the emancipation from anthropomorphic ideas. This
was true in the passage from the Ptolemaic to the Copernican
cosmical system, just as it is true at the present time for the
apparently impending passage from the so-called classical mechanics
of mass points to the general dynamics originating in
the principle of relativity. In accordance with this, man and
%-----File: 018.png---\redacted\--------
the earth upon which he dwells are removed from the centre
of the world. It may be predicted that in this century the
idea of time will be divested of the absolute character with
which men have been accustomed to endow it (cf.\ the final
lecture). Certainly, the sacrifices demanded by every such
revolution in the intuitive point of view are enormous; consequently,
the resistance against such a change is very great. But
the development of science is not to be permanently halted
thereby; on the contrary, its strongest impetus is experienced
through precisely those forces which attain success in the struggle
against the old points of view, and to this extent such a
struggle is constantly necessary and useful.

Now, how far have we advanced today toward the unification
of our system of physics? The numerous independent domains
of the earlier physics now appear reduced to two; mechanics and
electrodynamics, or, as one may say: the physics of material
bodies and the physics of the ether. The former comprehends
acoustics, phenomena in material bodies, and chemical phenomena;
the latter, magnetism, optics, and radiant heat. But is
this division a fundamental one? Will it prove final? This
is a question of great consequence for the future development of
physics. For myself, I believe it must be answered in the
negative, and upon the following grounds: mechanics and electrodynamics
cannot be permanently sharply differentiated from
each other. Does the process of light emission, for example,
belong to mechanics or to electrodynamics? To which domain
shall be assigned the laws of motion of electrons? At first
glance, one may perhaps say: to electrodynamics, since with
the electrons ponderable matter does not play any rôle. But
let one direct his attention to the motion of free electrons in
metals. There he will find, in the study of the classical researches
of H.~A. Lorentz, for example, that the laws obeyed by
the electrons belong rather to the kinetic theory of gases than
to electrodynamics. In general, it appears to me that the
original differences between processes in the ether and processes
%-----File: 019.png---\redacted\--------
in material bodies are to be considered as disappearing. Electrodynamics
and mechanics are not so remarkably far apart, as is
considered to be the case by many people, who already speak of a
conflict between the mechanical and the electrodynamic views
of the world. Mechanics requires for its foundation essentially
nothing more than the ideas of space, of time, and of that which
is moving, whether one considers this as a substance or a state.
The same ideas are also involved in electrodynamics. A sufficiently
generalized conception of mechanics can therefore also
well include electrodynamics, and, in fact, there are many indications
pointing toward the ultimate amalgamation of these two
subjects, the domains of which already overlap in some measure.

If, therefore, the gulf between ether and matter be once bridged,
what is the point of view which in the last analysis will best
serve in the subdivision of the system of physics? The answer
to this question will characterize the whole nature of the further
development of our science. It is, therefore, the most important
among all those which I propose to treat today. But for the
purposes of a closer investigation it is necessary that we go somewhat
more deeply into the peculiarities of physical principles.

We shall best begin at that point from which the first step was
made toward the actual realization of the unified system of
physics previously postulated by the philosophers only; at the
principle of conservation of energy. For the idea of energy is
the only one besides those of space and time which is common to
all the various domains of physics. In accordance with what I
have stated above, it will be apparent and quite self evident to
you that the principle of energy, before its general formularization
by Mayer, Joule, and Helmholz, also bore an anthropomorphic
character. The roots of this principle lay already in the recognition
of the fact that no one is able to obtain useful work from
nothing; and this recognition had originated essentially in the
experiences which were gathered in attempts at the solution of a
technical problem: the discovery of perpetual motion. To this
extent, perpetual motion has come to have for physics a far
%-----File: 020.png---\redacted\--------
reaching significance, similar to that of alchemy for the chemist,
although it was not the positive, but rather the negative results
of these experiments, through which science was advanced.
Today we speak of the principle of energy quite without reference
to the technical viewpoint or to that of man. We say that the
total amount of energy of an isolated system of bodies is a
quantity whose amount can be neither increased nor diminished
through any kind of process within the system, and we no longer
consider the accuracy with which this law holds as dependent
upon the refinement of the methods, which we at present possess,
of testing experimentally the question of the realization of
perpetual motion. In this, strictly speaking, unprovable generalization,
impressed upon us with elemental force, lies the emancipation
from the anthropomorphic elements mentioned above.

While the principle of energy stands before us as a complete
independent structure, freed from and independent of the accidents
appertaining to its historical development, this is by no
means true in equal measure in the case of that principle which
R.~Clausius introduced into physics; namely, the second law
of thermodynamics. This law plays a very peculiar rôle in the
development of physical science, to the extent that one is not
able to assert today that for it a generally recognized, and therefore
objective formularization, has been found. In our present
consideration it is therefore a matter of particular interest to
examine more closely its significance.

In contrast to the first law of thermodynamics, or the energy
principle, the second law may be characterized as follows. While
the first law permits in all processes of nature neither the creation
nor destruction of energy, but permits of transformations only,
the second law goes still further into the limitation of the possible
processes of nature, in that it permits, not all kinds of transformations,
but only certain types, subject to certain conditions.
The second law occupies itself, therefore, with the
question of the kind and, in particular, with the direction of any
natural process.
%-----File: 021.png---\redacted\--------

At this point a mistake has frequently been made, which has
hindered in a very pronounced manner the advance of science
up to the present day. In the endeavor to give to the second
law of thermodynamics the most general character possible, it
has been proclaimed by followers of W.~Ostwald as the second
law of energetics, and the attempt made so to formulate it that
it shall determine quite generally the direction of every process
occurring in nature. Some weeks ago I read in a public academic
address of an esteemed colleague the statement that the import
of the second law consists in this, that a stone falls downwards,
that water flows not up hill, but down, that electricity flows from
a higher to a lower potential, and so on. This is a mistake which
at present is altogether too prevalent not to warrant mention
here.

The truth is, these statements are false. A stone can just as
well rise in the air as fall downwards; water can likewise flow upwards,
as, for example, in a spring; electricity can flow very well
from a lower to a higher potential, as in the case of oscillating discharge
of a condenser. The statements are obviously quite correct,
if one applies them to a stone originally at rest, to water at
rest, to electricity at rest; but then they follow immediately from
the energy principle, and one does not need to add a special second
law. For, in accordance with the energy principle, the kinetic
energy of the stone or of the water can only originate at the
cost of gravitational energy, \ie, the center of mass must descend.
If, therefore, motion is to take place at all, it is necessary
that the gravitational energy shall decrease. That is, the
center of mass must descend. In like manner, an electric current
between two condenser plates can originate only at the
cost of electrical energy already present; the electricity must
therefore pass to a lower potential. If, however, motion and
current be already present, then one is not able to say, a priori,
anything in regard to the direction of the change; it can take
place just as well in one direction as the other. Therefore, there
is no new insight into nature to be obtained from this point of
view.
%-----File: 022.png---\redacted\--------

Upon an equally inadequate basis rests another conception of
the second law, which I shall now mention. In considering the circumstance
that mechanical work may very easily be transformed
into heat, as by friction, while on the other hand heat can only
with difficulty be transformed into work, the attempt has been
made so to characterize the second law, that in nature the transformation
of work into heat can take place completely, while
that of heat into work, on the other hand, only incompletely and
in such manner that every time a quantity of heat is transformed
into work another corresponding quantity of energy must necessarily
undergo at the same time a compensating transformation,
as, \eg, the passage of heat from a higher to a lower
temperature. This assertion is in certain special cases correct,
but does not strike in general at the true import of the matter,
as I shall show by a simple example.

One of the most important laws of thermodynamics is, that
the total energy of an ideal gas depends only upon its temperature,
and not upon its volume. If an ideal gas be allowed to
expand while doing work, and if the cooling of the gas be prevented
through the simultaneous addition of heat from a heat reservoir
at higher temperature, the gas remains unchanged in temperature
and energy content, and one may say that the heat furnished
by the heat reservoir is completely transformed into work without
exchange of energy. Not the least objection can be urged
against this assertion. The law of incomplete transformation
of heat into work is retained only through the adoption of a
different point of view, but which has nothing to do with the
status of the physical facts and only modifies the way of looking
at the matter, and therefore can neither be supported nor contradicted
through facts; namely, through the introduction ad~hoc
of new particular kinds of energy, in that one divides the energy
of the gas into numerous parts which individually can depend
upon the volume. But it is a~priori evident that one can never
derive from so artificial a definition a new physical law, and it is
with such that we have to do when we pass from the first law,
the principle of conservation of energy, to the second law.
%-----File: 023.png---\redacted\--------

I desire now to introduce such a new physical law: ``It is not
possible to construct a periodically functioning motor which in
principle does not involve more than the raising of a load and the
cooling of a heat reservoir.'' It is to be understood, that in one
cycle of the motor quite arbitrary complicated processes may
take place, but that after the completion of one cycle there shall
remain no other changes in the surroundings than that the heat
reservoir is cooled and that the load is raised a corresponding
distance, which may be calculated from the first law. Such a
motor could of course be used at the same time as a refrigerating
machine also, without any further expenditure of energy and
materials. Such a motor would moreover be the most efficient
in the world, since it would involve no cost to run it; for the
earth, the atmosphere, or the ocean could be utilized as the heat
reservoir. We shall call this, in accordance with the proposal of
W.~Ostwald, perpetual motion of the second kind. Whether in
nature such a motion is actually possible cannot be inferred from
the energy principle, and may only be determined by special
experiments.

Just as the impossibility of perpetual motion of the first kind
leads to the principle of the conservation of energy, the quite
independent principle of the impossibility of perpetual motion of
the second kind leads to the second law of thermodynamics,
and, if we assume this impossibility as proven experimentally,
the general law follows immediately: \emph{there are processes in
nature which in no possible way can be made completely reversible}.
For consider, \eg, a frictional process through which mechanical
work is transformed into heat with the aid of suitable
apparatus, if it were actually possible to make in some way such
complicated apparatus completely reversible, so that everywhere
in nature exactly the same conditions be reestablished as existed
at the beginning of the frictional process, then the apparatus
considered would be nothing more than the motor described
above, furnishing a perpetual motion of the second kind. This
appears evident immediately, if one clearly perceives what the
%-----File: 024.png---\redacted\--------
apparatus would accomplish: transformation of heat into work
without any further outstanding change.

We call such a process, which in no wise can be made completely
reversible, an irreversible process, and all other processes reversible
processes; and thus we strike the kernel of the second
law of thermodynamics when we say that irreversible processes
occur in nature. In accordance with this, the changes in nature
have a unidirectional tendency. With each irreversible process
the world takes a step forward, the traces of which under no
circumstances can be completely obliterated. Besides friction,
examples of irreversible processes are: heat conduction, diffusion,
conduction of electricity in conductors of finite resistance,
emission of light and heat radiation, disintegration of the atom
in radioactive substances, and so on. On the other hand, examples
of reversible processes are: motion of the planets, free
fall in empty space, the undamped motion of a pendulum,
the frictionless flow of liquids, the propagation of light and
sound waves without absorption and refraction, undamped
electrical vibrations, and so on. For all these processes are
already periodic or may be made completely reversible through
suitable contrivances, so that there remains no outstanding
change in nature; for example, the free fall of a body whereby
the acquired velocity is utilized to raise the body again to its
original height; a light or sound wave which is allowed in a suitable
manner to be totally reflected from a perfect mirror.

What now are the general properties and criteria of irreversible
processes, and what is the general quantitative measure of
irreversibility? This question has been examined and answered
in the most widely different ways, and it is evident here again
how difficult it is to reach a correct formularization of a problem.
Just as originally we came upon the trail of the energy
principle through the technical problem of perpetual motion, so
again a technical problem, namely, that of the steam engine,
led to the differentiation between reversible and irreversible
processes. Long ago Sadi Carnot recognized, although he utilized
%-----File: 025.png---\redacted\--------
an incorrect conception of the nature of heat, that irreversible
processes are less economical than reversible, or that in
an irreversible process a certain opportunity to derive mechanical
work from heat is lost. What then could have been
simpler than the thought of making, quite in general, the measure
of the irreversibility of a process the quantity of mechanical
work which is unavoidably lost in the process. For a reversible
process then, the unavoidably lost work is naturally to be set
equal to zero. This view, in accordance with which the import
of the second law consists in a dissipation of useful energy, has
in fact, in certain special cases, \eg, in isothermal processes,
proved itself useful. It has persisted, therefore, in certain of
its aspects up to the present day; but for the general case, however,
it has shown itself as fruitless and, in fact, misleading. The
reason for this lies in the fact that the question concerning the
lost work in a given irreversible process is by no means to be
answered in a determinate manner, so long as nothing further is
specified with regard to the source of energy from which the work
considered shall be obtained.

An example will make this clear. Heat conduction is an
irreversible process, or as Clausius expresses it: Heat cannot
without compensation pass from a colder to a warmer body.
What now is the work which in accordance with definition is
lost when the quantity of heat~$Q$ passes through direct conduction
from a warmer body at the temperature~$T_{1}$ to a colder body, at
the temperature~$T_{2}$? In order to answer this question, we make
use of the heat transfer involved in carrying out a reversible
Carnot cyclical process between the two bodies employed as
heat reservoirs. \label{png25lab1}In this process a certain amount of work
would be obtained, and it is just the amount sought, since it is
that which would be lost in the direct passage by conduction;
but this has no definite value so long as we do not know whence
the work originates, whether, \eg, in the warmer body or in the
colder body, or from somewhere else. Let one reflect that the
heat given up by the warmer body in the reversible process is certainly
%-----File: 026.png---\redacted\--------
not equal to the heat absorbed by the colder body, because
a certain amount of heat is transformed into work, and that we
can identify, with exactly the same right, the quantity of heat~$Q$
transferred by the direct process of conduction with that which in
the cyclical process is given up by the warmer body, or with that
absorbed by the colder body. As one does the former or the latter,
he accordingly obtains for the quantity of lost work in the process
of conduction:
\[
Q · \frac{T_{1} - T_{2}}{T_{1}} \quad  \text{or} \quad
Q · \frac{T_{1} - T_{2}}{T_{2}}.
\]
We see, therefore, that the proposed method of expressing mathematically
the irreversibility of a process does not in general effect
its object, and at the same time we recognize the peculiar reason
which prevents its doing so. The statement of the question is
too anthropomorphic. It is primarily too much concerned with
the needs of mankind, in that it refers directly to the acquirement
of useful work. If one require from nature a determinate
answer, he must take a more general point of view, more disinterested,
less economic. We shall now seek to do this.

Let us consider any typical process occurring in nature. This
will carry all bodies concerned in it from a determinate initial
state, which I designate as state~$A$, into a determinate final
state~$B$. The process is either reversible or irreversible. A
third possibility is excluded. But whether it is reversible or
irreversible depends solely upon the nature of the two states $A$
and~$B$, and not at all upon the way in which the process has been
carried out; for we are only concerned with the answer to the
question as to whether or not, when the state~$B$ is once reached, a
complete return to~$A$ in any conceivable manner may be accomplished.
If now, the complete return from $B$ to~$A$ is not
possible, and the process therefore irreversible, it is obvious that
the state~$B$ may be distinguished in nature through a certain
property from state~$A$. Several years ago I ventured to express
this as follows: that nature possesses a greater ``preference'' for
state~$B$ than for state~$A$. In accordance with this mode of
%-----File: 027.png---\redacted\--------
expression, all those processes of nature are impossible for whose
final state nature possesses a smaller preference than for the
original state. Reversible processes constitute a limiting case;
for such, nature possesses an equal preference for the initial and
for the final state, and the passage between them takes place as
well in one direction as the other.

We have now to seek a physical quantity whose magnitude
shall serve as a general measure of the preference of nature for
a given state. This quantity must be one which is directly
determined by the state of the system considered, without
reference to the previous history of the system, as is the case with
the energy, with the volume, and with other properties of the
system. It should possess the peculiarity of increasing in all
irreversible processes and of remaining unchanged in all reversible
processes, and the amount of change which it experiences
in a process would furnish a general measure for the irreversibility
of the process.

R.~Clausius actually found this quantity and called it
``entropy.'' Every system of bodies possesses in each of its
states a definite entropy, \label{png27lab1}and this entropy expresses the preference
of nature for the state in question. It can, in all the
processes which take place within the system, only increase and
never decrease. If it be desired to consider a process in which
external actions upon the system are present, it is necessary
to consider those bodies in which these actions originate as
constituting part of the system; then the law as stated in the
above form is valid. In accordance with it, the entropy of a
system of bodies is simply equal to the sum of the entropies of
the individual bodies, and the entropy of a single body is, in
accordance with Clausius, found by the aid of a certain reversible
process. Conduction of heat to a body increases its
entropy, and, in fact, by an amount equal to the ratio of the
quantity of heat given the body to its temperature. Simple
compression, on the other hand, does not change the entropy.

Returning to the example mentioned above, in which the
%-----File: 028.png---\redacted\--------
quantity of heat~$Q$ is conducted from a warmer body at the
temperature~$T_{1}$ to a colder body at the temperature~$T_{2}$, in
accordance with what precedes, the entropy of the warmer body
decreases in this process, while, on the other hand, that of the
colder increases, and the sum of both changes, that is, the change
of the total entropy of both bodies, is:
\[
-\frac{Q}{T_{1}} + \frac{Q}{T_{2}} > 0.
\]

This positive quantity furnishes, in a manner free from all
arbitrary assumptions, the measure of the irreversibility of the
process of heat conduction. Such examples may be cited
indefinitely. Every chemical process furnishes an increase of
entropy.

We shall here consider only the most general case treated by
Clausius: an arbitrary reversible or irreversible cyclical process,
carried out with any physico-chemical arrangement, utilizing
an arbitrary number of heat reservoirs. Since the arrangement
at the conclusion of the cyclical process is the same as that at
the beginning, the final state of the process is to be distinguished
from the initial state solely through the different heat content
of the heat reservoirs, and in that a certain amount of mechanical
work has been furnished or consumed. Let $Q$~be the heat given
up in the course of the process by a heat reservoir at the temperature~$T$,
and let $A$~be the total work yielded (consisting,
\eg, in the raising of weights); then, in accordance with the first
law of thermodynamics:
\[
\tsum Q = A.
\]
In accordance with the second law, the sum of the changes in
entropy of all the heat reservoirs is positive, or zero. It follows,
therefore, since the entropy of a reservoir is decreased by the
amount~$Q/T$ through the loss of heat~$Q$ that:
\[
\tsum \frac{Q}{T} \leq 0.
\]
This is the well-known inequality of Clausius.
%-----File: 029.png---\redacted\--------

In an isothermal cyclical process, $T$~is the same for all reservoirs.
Therefore:
\[
\tsum Q \leq 0, \quad \text{hence:}\quad A \leq 0.
\]
That is: in an isothermal cyclical process, heat is produced and
work is consumed. \label{png29lab1}In the limiting case, a reversible isothermal
cyclical process, the sign of equality holds, and therefore the
work consumed is zero, and also the heat produced. This law
plays a leading rôle in the application of thermodynamics to
physical chemistry.

The second law of thermodynamics including all of its consequences
has thus led to the principle of increase of entropy.
You will now readily understand, having regard to the questions
mentioned above, why I express it as my opinion that in the
theoretical physics of the future the first and most important
differentiation of all physical processes will be into reversible
and irreversible processes.

In fact, all reversible processes, whether they take place in
material bodies, in the ether, or in both together, show a much
greater similarity among themselves than to any irreversible
process. In the differential equations of reversible processes
the time differential enters only as an even power, corresponding
to the circumstance that the sign of time can be
reversed. This holds equally well for vibrations of the pendulum,
electrical vibrations, acoustic and optical waves, and
for motions of mass points or of electrons, if we only exclude
every kind of damping. But to such processes also
belong those infinitely slow processes of thermodynamics which
consist of states of equilibrium in which the time in general
plays no rôle, or, as one may also say, occurs with the zero power,
which is to be reckoned as an even power. As Helmholtz has
pointed out, all these reversible processes have the common
property that they may be completely represented by the principle
of least action, which gives a definite answer to all questions concerning
any such measurable process, and, to this extent, \label{png29lab2}the theory
of reversible processes may be regarded as completely established.
Reversible processes have, however, the disadvantage that
%-----File: 030.png---\redacted\--------
singly and collectively they are only ideal: in actual nature there
is no such thing as a reversible process. Every natural process
involves in greater or less degree friction or conduction of heat.
But in the domain of irreversible processes the principle of least
action is no longer sufficient; for the principle of increase of
entropy brings into the system of physics a wholly new element,
foreign to the action principle, and which demands special
mathematical treatment. The unidirectional course of a process
in the attainment of a fixed final state is related to it.

I hope the foregoing considerations have sufficed to make clear
to you that the distinction between reversible and irreversible
processes is much broader than that between mechanical and
electrical processes and that, therefore, this difference, with better
right than any other, may be taken advantage of in classifying
all physical processes, and that it may eventually play in the
theoretical physics of the future the principal rôle.

However, the classification mentioned is in need of quite an
essential improvement, for it cannot be denied that in the form
set forth, the system of physics is still suffering from a strong
dose of anthropomorphism. In the definition of irreversibility,
as well as in that of entropy, reference is made to the possibility
of carrying out in nature certain changes, and this means, fundamentally,
nothing more than that the division of physical processes
is made dependent upon the manipulative skill of man in
the art of experimentation, which certainly does not always
remain at a fixed stage, but is continually being more and more
perfected. If, therefore, the distinction between reversible and
irreversible processes is actually to have a lasting significance
for all times, it must be essentially broadened and made independent
of any reference to the capacities of mankind. How this
may happen, I desire to state one week from tomorrow. The
lecture of tomorrow will be devoted to the problem of bringing
before you some of the most important of the great number of
practical consequences following from the entropy principle.
%-----File: 031.png---\redacted\--------


\Chapter{SECOND LECTURE.}{%
Thermodynamic States of Equilibrium in Dilute
Solutions.}\label{Lect2}

In the lecture of yesterday I sought to make clear the fact
that the essential, and therefore the final division of all processes
occurring in nature, is into reversible and irreversible processes,
and the characteristic difference between these two kinds of
processes, as I have further separated them, is that in irreversible
processes the entropy increases, while in all reversible processes
it remains constant. Today I am constrained to speak of some
of the consequences of this law which will illustrate its rich fruitfulness.
They have to do with the question of the laws of thermodynamic
equilibrium. Since in nature the entropy can only
increase, it follows that the state of a physical configuration
which is completely isolated, and in which the entropy of
the system possesses an absolute maximum, is necessarily a
state of stable equilibrium, since for it no further change is
possible. How deeply this law underlies all physical and chemical
relations has been shown by no one better and more completely
than by John Willard Gibbs, whose name, not only in
America, but in the whole world will be counted among those of
the most famous theoretical physicists of all times; to whom, to
my sorrow, it is no longer possible for me to tender personally
my respects. It would be gratuitous for me, here in the land
of his activity, to expatiate fully on the progress of his ideas,
but you will perhaps permit me to speak in the lecture of today
of some of the important applications in which thermodynamic
research, based on Gibbs works, can be advanced beyond
his results.

These applications refer to the theory of dilute solutions, and
%-----File: 032.png---\redacted\--------
we shall occupy ourselves today with these, while I show you
by a definite example what fruitfulness is inherent in thermodynamic
theory. I shall first characterize the problem quite
generally. It has to do with the state of equilibrium of a material
system of any number of arbitrary constituents in an arbitrary
number of phases, at a given temperature~$T$ and given
pressure~$p$. If the system is completely isolated, and therefore
guarded against all external thermal and mechanical
actions, then in any ensuing change the entropy of the system will
increase:
\[
dS > 0.
\]
But if, as we assume, the system stands in such relation to
its surroundings that in any change which the system undergoes
the temperature~$T$ and the pressure~$p$ are maintained
constant, as, for instance, through its introduction into a calorimeter
of great heat capacity and through loading with a piston
of fixed weight, the inequality would suffer a change thereby.
We must then take account of the fact that the surrounding
bodies also, \eg, the calorimetric liquid, will be involved in the
change. If we denote the entropy of the surrounding bodies by~$S_{0}$,
then the following more general equation holds:
\[
dS + dS_{0} > 0.
\]
In this equation
\[
dS_{0} = -\frac{Q}{T},
\]
if $Q$~denote the heat which is given up in the change by the
surroundings to the system. On the other hand, if $U$~denote
the energy, $V$~the volume of the system, then, in accordance
with the first law of thermodynamics,
\[
Q = dU + p dV.
\]
Consequently, through substitution:
\[
dS - \frac{dU + p dV}{T} > 0
\]
%-----File: 033.png---\redacted\--------
or, since $p$~and~$T$ are constant:
\[
d \left(S - \frac{U + pV}{T} \right) > 0.
\]
If, therefore, we put:
\[
S - \frac{U + pV}{T} = \Phi,
\Tag{(1)}
\]
then
\[
d \Phi > 0,
\]
and we have the general law, that in every isothermal-isobaric
($T = \const.$, $p = \const.$) change of state of a physical system
the quantity~$\Phi$ increases. The absolutely stable state of
equilibrium of the system is therefore characterized through
the maximum of~$\Phi$:
\[
\delta \Phi = 0.
\Tag{(2)}
\]
If the system consist of numerous phases, then, because $\Phi$, in
accordance with~\Eq{(1)}, is linear and homogeneous in $S$,~$U$ and~$V$,
the quantity~$\Phi$ referring to the whole system is the sum of the
quantities~$\Phi$ referring to the individual phases. If the expression
for~$\Phi$ is known as a function of the independent variables for
each phase of the system, then, from equation~\Eq{(2)}, all questions
concerning the conditions of stable equilibrium may be
answered. Now, within limits, this is the case for dilute solutions.
By ``solution'' in thermodynamics is meant each homogeneous
phase, in whatever state of aggregation, which is composed of a
series of different molecular complexes, each of which is represented
by a definite molecular number. If the molecular
number of a given complex is great with reference to all the
remaining complexes, then the solution is called dilute, and the
molecular complex in question is called the solvent; the remaining
complexes are called the dissolved substances.

Let us now consider a dilute solution whose state is determined
by the pressure~$p$, the temperature~$T$, and the molecular numbers
$n_{0}$,~$n_{1}$, $n_{2}$, $n_{3}$,~$\cdots$, wherein the subscript zero refers to the solvent.
Then the numbers $n_{1}$,~$n_{2}$, $n_{3}$,~$\cdots$ are all small with respect to~$n_{0}$,
%-----File: 034.png---\redacted\--------
and on this account the volume~$V$ and the energy~$U$ are linear
functions of the molecular numbers:
\begin{align*}
  V &= n_{0}v_{0} + n_{1}v_{1} + n_{2}v_{2} + \cdots,\\
  U &= n_{0}u_{0} + n_{1}u_{1} + n_{2}u_{2} + \cdots,
\end{align*}
wherein the $v$'s and $u$'s depend upon $p$~and $T$ only.

From the general equation of entropy:
\[
dS = \frac{dU + p dV}{T},
\]
in which the differentials depend only upon changes in $p$~and~$T$,
and not in the molecular numbers, there results therefore:
\[
dS = n_{0} \frac{du_{0} + p dv_{0}}{T} + n_{1} \frac{du_{1} + p dv_{1}}{T} + \cdots,
\]
and from this it follows that the expressions multiplied by $n_{0}$,~$n_{1}$~$\cdots$,
dependent upon $p$~and $T$ only, are complete differentials.
We may therefore write:
\[
\frac{du_{0} + p dv_{0}}{T} = ds_{0}, \quad
\frac{du_{1} + p dv_{1}}{T} = ds_{1},\ \cdots
\Tag{(3)}
\]
and by integration obtain:
\[
S = n_{0}s_{0} + n_{1}s_{1} + n_{2}s_{2} + \cdots + C.
\]
The constant~$C$ of integration does not depend upon $p$~and~$T$,
but may depend upon the molecular numbers $n_{0}$,~$n_{1}$, $n_{2}$,~$\cdots$.
In order to express this dependence generally, it suffices to know
it for a special case, for fixed values of $p$~and~$T$. Now every
solution passes, through appropriate increase of temperature and
decrease of pressure, into the state of a mixture of ideal gases,
and for this case the entropy is fully known, the integration
constant being, in accordance with Gibbs:
\[
C = - R (n_{0} \log c_{0} + n_{1} \log c_{1} + \cdots),
\]
wherein $R$~denotes the absolute gas constant and $c_{0}$,~$c_{1}$, $c_{2}$,~$\cdots$
%-----File: 035.png---\redacted\--------
denote the ``molecular concentrations'':
\[
c_{0} = \frac{n_{0}}{n_{0} + n_{1} + n_{2} + \cdots}, \quad
c_{1} = \frac{n_{1}}{n_{0} + n_{1} + n_{2} + \cdots} ,\ \cdots.
\]
Consequently, quite in general, the entropy of a dilute solution is:
\[
S = n_{0}(s_{0} - R \log c_{0}) + n_{1}(s_{1} - R \log c_{1}) + \cdots,
\]
and, finally, from this it follows by substitution in equation~\Eq{(1)}
that:
\[
\Phi = n_{0}(\varphi_{0} - R \log c_{0}) + n_{1}(\varphi_{1} - R \log c_{1}) + \cdots,
\Tag{(4)}
\]
if we put for brevity:
\[
\varphi_{0} = s_{0} - \frac{u_{0} + pv_{0}}{T}, \quad
\varphi_{1} = s_{1} - \frac{u_{1} + pv_{1}}{T},\ \cdots
\Tag{(5)}
\]
all of which quantities depend only upon $p$~and~$T$.

With the aid of the expression obtained for~$\Phi$ we are enabled
through equation~\Eq{(2)} to answer the question with regard to
thermodynamic equilibrium. We shall first find the general
law of equilibrium and then apply it to a series of particularly
interesting special cases.

Every material system consisting of an arbitrary number of
homogeneous phases may be represented symbolically in the
following way:
\[
n_{0}  m_{0},\   n_{1}  m_{1},\ \cdots \mid
{n_{0}}' {m_{0}}',\  {n_{1}}' {m_{1}}',\ \cdots \mid
{n_{0}}''{m_{0}}'',\ {n_{1}}''{m_{1}}'',\ \cdots \mid \cdots.
\]
Here the molecular numbers are denoted by~$n$, the molecular
weights by~$m$, and the individual phases are separated from one
another by vertical lines. We shall now suppose that each
phase represents a dilute solution. This will be the case when
each phase contains only a single molecular complex and therefore
represents an absolutely pure substance; for then the concentrations
of all the dissolved substances will be zero.

If now an isobaric-isothermal change in the system of such
kind is possible that the molecular numbers
\[
n_{0},\   n_{1},\   n_{2},\ \cdots,\quad
{n_{0}}',\  {n_{1}}',\  {n_{2}}',\ \cdots,\quad
{n_{0}}'',\ {n_{1}}'',\ {n_{2}}'',\ \cdots
\]
%-----File: 036.png---\redacted\--------
change simultaneously by the amounts
\[
\delta n_{0},\   \delta n_{1},\   \delta n_{2}, \cdots,\quad
\delta {n_{0}}',\  \delta {n_{1}}',\  \delta {n_{2}}', \cdots,\quad
\delta {n_{0}}'',\ \delta {n_{1}}'',\ \delta {n_{2}}'', \cdots
\]
then, in accordance with equation~\Eq{(2)}, equilibrium obtains with
respect to the occurrence of this change if, when $T$~and~$p$ are held
constant, the function
\[
\Phi + \Phi' + \Phi'' + \cdots
\]
is a maximum, or, in accordance with equation~\Eq{(4)}:
\[
\tsum (\varphi_{0} - R \log c_{0})\delta n_{0}
   + (\varphi_{1} - R \log c_{1})\delta n_{1} + \cdots = 0
\]
(the summation~$\tsum$ being extended over all phases of the system).
Since we are only concerned in this equation with the ratios of
the~$\delta n$'s, we put
\begin{multline*}
\delta n_{0}   : \delta n_{1}   : \cdots :
\delta {n_{0}}'  : \delta {n_{1}}'  : \cdots :
\delta {n_{0}}'' : \delta {n_{1}}'' : \cdots \\
  = \nu_{0}   : \nu_{1}   : \cdots
  : {\nu_{0}}'  : {\nu_{1}}'  : \cdots
  : {\nu_{0}}'' : {\nu_{1}}'' : \cdots,
\end{multline*}
wherein we are to understand by the simultaneously changing~$\nu$'s,
in the variation considered, simple integer positive or negative
numbers, according as the molecular complex under consideration
is formed or disappears in the change. Then the condition
for equilibrium is:
\label{png36lab1}
\[
\tsum \nu_{0} \log c_{0}
   + \nu_{1} \log c_{1} + \cdots
  = \frac{1}{R} \tsum \nu_{0} \varphi_{0} + \nu_{1} \varphi_{1} + \cdots
  = \log K.
\Tag{(6)}
\]
$K$ and the quantities $\varphi_{0}$,~$\varphi_{1}$, $\varphi_{2}$,~$\cdots$\ depend only upon $p$~and~$T$,
and this dependence is to be found from the equations:
\begin{align*}
\frac{\dd \log K}{\dd p} &= \frac{1}{R} \tsum \nu_{0} \frac{\dd \varphi_{0}}{\dd p} + \nu_{1} \frac{\dd \varphi_{1}}{\dd p} + \cdots,\\
\frac{\dd \log K}{\dd T} &= \frac{1}{R} \tsum \nu_{0} \frac{\dd \varphi_{0}}{\dd T} + \nu_{1} \frac{\dd \varphi_{1}}{\dd T} + \cdots.
\end{align*}
Now, in accordance with~\Eq{(5)}, for any infinitely small change of $p$~and~$T$:
\[
d \varphi_{0} = ds_{0} - \frac{du_{0} + p dv_{0} + v_{0} dp}{T} + \frac{u_{0} + pv_{0}}{T^{2}} · dT,
\]
%-----File: 037.png---\redacted\--------
and consequently, from~\Eq{(3)}:
\[
d \varphi_{0} = \frac{u_{0} + pv_{0}}{T^{2}} dT - \frac{v_{0} dp}{T},
\]
and hence:
\[
\frac{\dd \varphi_{0}}{\dd p} = -\frac{v_{0}}{T},\quad
\frac{\dd \varphi_{0}}{\dd T} =  \frac{u_{0} + pv_{0}}{T^{2}}.
\]
Similar equations hold for the other~$\varphi$'s, and therefore we get:
\begin{gather*}
\frac{\dd \log K}{\dd p}
  = -\frac{1}{RT} \tsum \nu_{0}v_{0} + \nu_{1}v_{1} + \cdots, \\
\frac{\dd \log K}{\dd T}
  = -\frac{1}{RT^{2}} \tsum \nu_{0}u_{0} + \nu_{2}u_{2} + \cdots + p(\nu_{0}v_{0} + \nu_{1}v_{1} + \cdots)
\end{gather*}
or, more briefly:
\[
\frac{\dd \log K}{\dd p} = -\frac{1}{RT} · \Delta V, \quad
\frac{\dd \log K}{\dd T} =  \frac{\Delta Q}{RT^{2}},
\Tag{(7)}
\]
if $\Delta V$~denote the change in the total volume of the system and
$\Delta Q$~the heat which is communicated to it from outside, during
the isobaric isothermal change considered. We shall now investigate
the import of these relations in a series of important
applications.


\Section{I.}{Electrolytic Dissociation of Water.}

The system consists of a single phase:
\[
n_{0}H_{2}O,\quad n_{1}\Hplus,\quad n_{2}\HOminus.
\]
The transformation under consideration
\[
\nu_{0} : \nu_{1} : \nu_{2} = \delta n_{0} : \delta n_{1} : \delta n_{2}
\]
consists in the dissociation of a molecule~$H_{2}O$ into a molecule~$\Hplus$
and a molecule~$\HOminus$, therefore:
\[
\nu_{0} = -1,\quad \nu_{1} = 1,\quad \nu_{2} = 1.
\]
Hence, in accordance with~\Eq{(6)}, for equilibrium:
\[
-\log c_{0} + \log c_{1} + \log c_{2} = \log K,
\]
%-----File: 038.png---\redacted\--------
or, since $c_{1} = c_{2}$ and $c_{0} = 1$, approximately:
\[
2 \log c_{1} = \log K.
\]
The dependence of the concentration~$c_{1}$ upon the temperature
now follows from~\Eq{(7)}:
\[
2 \frac{\dd \log c_{1}}{\dd T} = \frac{\Delta Q}{R T^{2}} .
\]
$\Delta Q$,~the quantity of heat which it is necessary to supply for the
dissociation of a molecule of~$H_{2}O$ into the ions $\Hplus$~and~$\HOminus$, is,
in accordance with Arrhenius, equal to the heat of ionization in
the neutralization of a strong univalent base and acid in a
dilute aqueous solution, and, therefore, in accordance with the
recent measurements of Wörmann,\footnote
  {Ad Heydweiller, Ann.\ d.~Phys.,~28, 506, 1909.}
\[
\Delta Q = 27,857 - 48.5 T \ \gr.\ \cal.
\]
Using the number~$1.985$ for the ratio of the absolute gas constant~$R$
to the mechanical equivalent of heat, it follows that:
\[
\frac{\dd \log c_{1}}{\dd T}
  = \frac{1}{2·1.985} \left(\frac{27,857}{T^{2}} - \frac{48.5}{T}\right),
\]
and by integration:
\[
\logten c_{1} = - \frac{3047.3}{T} - 12.125 \logten T + \const.
\]
This dependence of the degree of dissociation upon the temperature
agrees very well with the measurements of the electric
conductivity of water at different temperatures by Kohlrausch
and Heydweiller, Noyes, and Lundén.


\Section{II.}{Dissociation of a Dissolved Electrolyte.}

\label{png38lab1}Let the system consists of an aqueous solution of acetic acid:
\[
n_{0}H_{2}O,\quad n_{1}H_{4}C_{2}O_{2},\quad n_{2}\Hplus,\quad n_{3}\overset{-}{H_{3}C_{2}O_{2}}.
\]
The change under consideration consists in the dissociation of a
%-----File: 039.png---\redacted\--------
molecule $H_{4}C_{2}O_{2}$ into its two ions, therefore
\[
\nu_{0} = 0, \quad \nu_{1} = -1, \quad \nu_{2} = 1, \quad \nu_{3} = 1.
\]
Hence, for the state of equilibrium, in accordance with~\Eq{(6)}:
\[
-\log c_{1} + \log c_{2} + \log c_{3} = \log K,
\]
or, since $c_{2} = c_{3}$:
\[
\frac{{c_{2}}^{2}}{c_{1}} = K.
\]
Now the sum $c_{1} + c_{2} = c$ is to be regarded as known, since the
total number of the undissociated and dissociated acid molecules
is independent of the degree of dissociation. Therefore $c_{1}$~and~$c_{2}$
may be calculated from $K$~and~$c$. An experimental test of the
equation of equilibrium is possible on account of the connection
between the degree of dissociation and electrical conductivity of
the solution. In accordance with the electrolytic dissociation
theory of Arrhenius, the ratio of the molecular conductivity~$\lambda$ of
the solution in any dilution to the molecular conductivity~$\lambda_{\infty}$
of the solution in infinite dilution is:
\[
\frac{\lambda}{\lambda_{\infty}} = \frac{c_{2}}{c_{1} + c_{2}} = \frac{c_{2}}{c},
\]
since electric conduction is accounted for by the dissociated molecules
only. It follows then, with the aid of the last equation, that:
\[
\frac{\lambda^{2} c}{\lambda_{\infty} - \lambda} = K · \lambda_{\infty} = \const.
\]
With unlimited decreasing~$c$, $\lambda$~increases to~$\lambda_{\infty}$. This ``law of
dilution'' for binary electrolytes, first enunciated by Ostwald, has
been confirmed in numerous cases by experiment, as in the case
of acetic acid.

Also, the dependence of the degree of dissociation upon the
temperature is indicated here in quite an analogous manner to
that in the example considered above, of the dissociation of water.
%-----File: 040.png---\redacted\--------


\Section{III.}{Vaporization or Solidification of a Pure Liquid.}

In equilibrium the system consists of two phases, one liquid,
and one gaseous or solid:
\[
n_{0}m_{0} \mid {n_{0}}'{m_{0}}'.
\]

Each phase contains only a single molecular complex (the
solvent), but the molecules in both phases do not need to be the
same. Now, if a liquid molecule evaporates or solidifies, then
in our notation
\[
\nu_{0} = - 1,\quad {\nu_{0}}' = \frac{m_{0}}{{m_{0}}'},\quad c_{0} = 1,\quad {c_{0}}' = 1,
\]
and consequently the condition for equilibrium, in accordance
with~\Eq{(6)}, is:
\[
0 = \log K.
\Tag{(8)}
\]
Since $K$ depends only upon $p$~and~$T$, this equation therefore
expresses a definite relation between $p$~and~$T$: the law of dependence
of the pressure of vaporization (or melting pressure)
upon the temperature, or vice versa. The import of this law is
obtained through the consideration of the dependence of the
quantity~$K$ upon $p$~and~$T$. If we form the complete differential
of the last equation, there results:
\[
0 = \frac{\dd \log K}{\dd p} dp + \frac{\dd \log K}{\dd T} dT,
\]
or, in accordance with~\Eq{(7)}:
\[
0 = -\frac{\Delta V}{T} dp + \frac{\Delta Q}{T^2} dT.
\]
If $v_{0}$~and~${v_{0}}'$ denote the molecular volumes of the two phases, then:
\[
\Delta V = \frac{m_{0}{v_{0}}'}{{m_{0}}'} - v_{0},
\]
consequently:
\[
\Delta Q = T\left(\frac{m_{0}{v_{0}}'}{{m_{0}}'} - v_{0}\right) \frac{dp}{dT},
\]
%-----File: 041.png---\redacted\--------
or, referred to unit mass:
\[
\frac{\Delta Q}{m_{0}}
  = T \left(\frac{{v_{0}}'}{{m_{0}}'} - \frac{v_{0}}{m_{0}}\right) · \frac{dp}{dT},
\]
the well-known formula of Carnot and Clapeyron.


\Section{IV.}{The Vaporization or Solidification of a Solution of Non-Volatile
Substances.}

Most aqueous salt solutions afford examples. The symbol of
the system in this case is, since the second phase (gaseous or solid)
contains only a single molecular complex:
\[
n_{0}m_{0},\ n_{1}m_{1},\ n_{2}m_{2},\ \cdots \mid {n_{0}}'{m_{0}}'.
\]
The change is represented by:
\[
\nu_{0} = -1,\quad
\nu_{1} =  0,\quad
\nu_{2} =  0,\quad \cdots\quad
{\nu_{0}}' = \frac{m_{0}}{{m_{0}}'},
\]
and hence the condition of equilibrium, in accordance with~\Eq{(6)}, is:
\[
-\log c_{0} = \log K,
\]
or, since to small quantities of higher order:
\begin{align*}
c_{0} = \frac{n_{0}}{n_{0} + n_{1} + n_{2} + \cdots}
  &= 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}},\\[1ex]
\frac{n_{1} + n_{2} + \cdots}{n_{0}} &= \log K.
\Tag{(9)}
\end{align*}

A comparison with formula~\Eq{(8)}, found in example~III, shows
that through the solution of a foreign substance there is involved
in the total concentration a small proportionate departure from
the law of vaporization or solidification which holds for the pure
solvent. One can express this, either by saying: at a fixed pressure~$p$,
the boiling point or the freezing point~$T$ of the solution
is different than that~($T_{0}$) for the pure solvent, or: \label{png41lab1}at a fixed
temperature~$T$ the vapor pressure or solidification pressure~$p$ of the
solution is different from that~($p_{0}$) of the pure solvent. Let us
calculate the departure in both cases.
%-----File: 042.png---\redacted\--------

1. If $T_{0}$~be the boiling (or freezing temperature) of the pure
solvent at the pressure~$p$, then, in accordance with~\Eq{(8)}:
\[
(\log K)_{T = T_{0}} = 0,
\]
and by subtraction of~\Eq{(9)} there results:
\[
\log K - (\log K)_{T = T_{0}} = \frac{n_{1} + n_{2} + \cdots}{n_{0}}.
\]
Now, since $T$~is little different from~$T_{0}$, we may write in place of
this equation, with the aid of~\Eq{(7)}:
\[
\frac{\dd \log K}{\dd T} (T - T_{0})
  = \frac{\Delta Q}{RT_{0}^{2}} (T - T_{0})
  = \frac{n_{1} + n_{2} + \cdots}{n_{0}},
\]
and from this it follows that:
\[
T - T_{0} = \frac{n_{1} + n_{2} + \cdots}{n_{0}} · \frac{RT_{0}^{2}}{\Delta Q}.
\Tag{(10)}
\]

This is the law for the raising of the boiling point or for the
lowering of the freezing point, first derived by van't~Hoff: in the
case of freezing $\Delta Q$~(the heat taken from the surroundings during
the freezing of a liquid molecule) is negative. Since $n_{0}$~and~$\Delta Q$
occur only as a product, it is not possible to infer anything from
this formula with regard to the molecular number of the liquid
solvent.

2. If $p_{0}$~be the vapor pressure of the pure solvent at the
temperature~$T$, then, in accordance with~\Eq{(8)}:
\[
(\log K)_{p = p_{0}} = 0,
\]
and by subtraction of~\Eq{(9)} there results:
\[
\log K - (\log K)_{p = p_{0}} = \frac{n_{1} + n_{2} + \cdots}{n_{0}}.
\]
Now, since $p$~and~$p_{0}$ are nearly equal, with the aid of~\Eq{(7)} we may
write:
\[
\frac{\dd \log K}{\dd p} (p - p_{0})
  = - \frac{\Delta V}{RT} (p - p _{0})
  = \frac{n_{1} + n_{2} + \cdots}{n_{0}},
\]
%-----File: 043.png---\redacted\--------
and from this it follows, if $\Delta V$~be placed equal to the volume of
the gaseous molecule produced in the vaporization of a liquid
molecule:
\begin{gather*}
\Delta V = \frac{m_{0}}{{m_{0}}'} \frac{RT}{p}, \\
\frac{p_{0} - p}{p} = \frac{{m_{0}}'}{m_{0}} · \frac{n_{1} + n_{2} + \cdots}{n_{0}}.
\end{gather*}
This is the law of relative depression of the vapor pressure,
first derived by van't~Hoff. Since $n_{0}$~and~$m_{0}$ occur only as a
product, it is not possible to infer from this formula anything
with regard to the molecular weight of the liquid solvent. Frequently
the factor~${m_{0}}'/m_{0}$ is left out in this formula; but this is
not allowable when $m_{0}$~and~${m_{0}}'$ are unequal (as, \eg, in the
case of water).


\Section{V.}{Vaporization of a Solution of Volatile Substances.}

\begin{center}(\textit{\Eg., a Sufficiently Dilute Solution of Propyl Alcohol in Water.})\end{center}

The system, consisting of two phases, is represented by the
following symbol:
\[
n_{0} m_{0},\  n_{1} m_{1},\  n_{2} m_{2},\ \cdots \mid
{n_{0}}'{m_{0}}',\ {n_{1}}'{m_{1}}',\ {n_{2}}'{m_{2}}',\ \cdots,
\]
wherein, as above, the figure~$0$ refers to the solvent and the
figures $1$,~$2$, $3$~$\cdots$ refer to the various molecular complexes of
the dissolved substances. By the addition of primes in the case
of the molecular weights (${m_{0}}'$,~${m_{1}}'$, ${m_{2}}'$~$\cdots$) the possibility is
left open that the various molecular complexes in the vapor
may possess a different molecular weight than in the liquid.

Since the system here considered may experience various sorts
of changes, there are also various conditions of equilibrium to
fulfill, each of which relates to a definite sort of transformation.
Let us consider first that change which consists in the vaporization
of the solvent. In accordance with our scheme of notation,
the following conditions hold:\label{png43lab1}
\[
\nu_{0} = - 1,\ \nu_{1} = 0,\ \nu_{2} = 0,\ \cdots\
\nu_{0}' = \frac{m_{0} }{ {m_{0}}'},\ {\nu_{1}}' = 0,\ {\nu_{2}}' = 0,\ \cdots,
\]
%-----File: 044.png---\redacted\--------
and, therefore, the condition of equilibrium~\Eq{(6)} becomes:
\[
-\log c_{0} + \frac{m_{0}}{{m_{0}}'} \log {c_{0}}' = \log K,
\]
or, if one substitutes:
\begin{gather*}
c_{0} = 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}} \quad \text{and} \quad
{c_{0}}' = 1 - \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'},\\
\frac{n_{1} + n_{2} + \cdots}{n_{0}} - \frac{m_{0}}{{m_{0}}'} · \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'} = \log K.
\end{gather*}
If we treat this equation upon equation~\Eq{(9)} as a model, there
results an equation similar to~\Eq{(10)}:
\[
T - T_{0}
  = \left(\frac{n_{1} + n_{2} + \cdots}{n_{0}m_{0}}
        - \frac{{n_{1}}' + {n_{2}}' + \cdots}{{n_{0}}'{m_{0}}'}\right) \frac{RT_{0}^{2}m_{0}}{\Delta Q}.
\]

Here $\Delta Q$~is the heat effect in the vaporization of one molecule
of the solvent and, therefore, $\Delta Q/m_{0}$~is the heat effect in the
vaporization of a unit mass of the solvent.

We remark, once more, that the solvent always occurs in the
formula through the mass only, and not through the molecular
number or the molecular weight, while, on the other hand, in the
case of the dissolved substances, the molecular state is characteristic
on account of their influence upon vaporization. Finally, the
formula contains a generalization of the law of van't~Hoff, stated
above, for the raising of the boiling point, in that here in place
of the number of dissolved molecules in the liquid, the difference
between the number of dissolved molecules in unit mass of the
liquid and in unit mass of the vapor appears. According as the
unit mass of liquid or the unit mass of vapor contains more
dissolved molecules, there results for the solution a raising or
lowering of the boiling point; in the limiting case, when both
quantities are equal, and the mixture therefore boils without
changing, the change in boiling point becomes equal to zero.
Of course, there are corresponding laws holding for the change
in the vapor pressure.
%-----File: 045.png---\redacted\--------

Let us consider now a change which consists in the vaporization
of a dissolved molecule. For this case we have in our notation
\[
\nu_{0} = 0,\  \nu_{1} = -1,\  \nu_{2} = 0\ \cdots, \
{\nu_{0}}' = 0,\  {\nu_{1}}' = \frac{m_{1}}{{m_{1}}'},\  {\nu_{2}}' = 0,\ \cdots
\]
and, in accordance with~\Eq{(6)}, for the condition of equilibrium:
\[
-\log c_{1} + \frac{m_{1}}{{m_{1}}'} \log {c_{1}}' = \log K
\]
or:
\[
\frac{{{c_{1}}'}^{\frac{m_{1}}{{m_{1}}'}}}{c_{1}} = K.
\]
This equation expresses the Nernst law of distribution. If
the dissolved substance possesses in both phases the same
molecular weight ($m_{1} = {m_{1}}'$), then, in a state of equilibrium a
fixed ratio of the concentrations $c_{1}$~and~${c_{1}}'$ in the liquid and in the
vapor exists, which depends only upon the pressure and temperature.
But, if the dissolved substance polymerises somewhat in
the liquid, then the relation demanded in the last equation appears
in place of the simple ratio.


\Section{VI.}{The Dissolved Substance only Passes over into the Second
Phase.}

This case is in a certain sense a special case of the one preceding.
To it belongs that of the solubility of a slightly soluble salt,
first investigated by van't~Hoff, \eg, succinic acid in water. The
symbol of this system is:
\[
n_{0}H_{2}O,\ n_{1}H_{6}C_{4}O_{4} \mid {n_{0}}'H_{6}C_{4}O_{4},
\]
in which we disregard the small dissociation of the acid solution.
The concentrations of the individual molecular complexes are:
\[
c_{0} = \frac{n_{0}}{n_{0} + n_{1}}, \quad
c_{1} = \frac{n_{1}}{n_{0} + n_{1}}, \quad
{c_{0}}' = \frac{{n_{0}}'}{{n_{0}}'} = 1.
\]
For the precipitation of solid succinic acid we have:
\[
\nu_{0} = 0, \quad \nu_{1} = -1, \quad {\nu_{0}}' = 1,
\]
%-----File: 046.png---\redacted\--------
and, therefore, from the condition of equilibrium~\Eq{(6)}:
\[
-\log c_{1} = \log K,
\]
hence, from~\Eq{(7)}:
\[
\Delta Q = - RT^{2} \frac{\dd \log c_{1}}{\dd T}.
\]
By means of this equation van't~Hoff calculated the heat of
solution~$\Delta Q$ from the solubility of succinic acid at~$0°$ and at $8.5°$~C.
The corresponding numbers were $2.88$ and $4.22$ in an arbitrary
unit. Approximately, then:
\[
\frac{\dd \log c_{1}}{\dd T} = \frac{\ln 4.22 - \ln 2.88}{8.5} = 0.04494,
\]
from which for $T = 273$:
\[
\Delta Q = -1.98 · 273^{2} · 0.04494 = -6,600\ \cal.,
\]
that is, in the precipitation of a molecule of succinic acid, $6,600~\cal.$
are given out to the surroundings. Berthelot found, however,
through direct measurement, $6,700$~calories for the heat
of solution.

The absorption of a gas also comes under this head, \eg\
carbonic acid, in a liquid of relatively unnoticeable smaller
vapor pressure, \eg, water at not too high a temperature. The
symbol of the system is then
\[
n_{0}H_{2}O,\ n_{1}CO_{2} \mid {n_{0}}'CO_{2}.
\]
The vaporization of a molecule~$CO_{2}$ corresponds to the values
\[
\nu_{0} = 0,\quad \nu_{1} = -1,\quad {\nu_{0}}' = 1.
\]
The condition of equilibrium is therefore again:
\[
-\log c_{1} = \log K,
\]
\ie, at a fixed temperature and a fixed pressure the concentration~$c_{1}$
of the gas in the solution is constant. The change of the concentration
%-----File: 047.png---\redacted\--------
with $p$~and~$T$ is obtained through substitution in equation~\Eq{(7)}.
It follows from this that:
\[
\frac{\dd \log c_{1}}{\dd p} =  \frac{\Delta V}{RT} ,\quad
\frac{\dd \log c_{1}}{\dd T} = -\frac{\Delta Q}{RT^{2}}.
\]

$\Delta V$~is the change in volume of the system which occurs in the
isobaric-isothermal vaporization of a molecule of~$CO_{2}$, $\Delta Q$~the
quantity of heat absorbed in the process from outside. Now,
since $\Delta V$~represents approximately the volume of a molecule of
gaseous carbonic acid, we may put approximately:
\[
\Delta V = \frac{RT}{p},
\]
and the equation gives:
\[
\frac{\dd \log c_{1}}{\dd p} = \frac{1}{p},
\]
which integrated, gives:
\[
\log c_{1} = \log p + \const., \quad c_{1} = C · p,
\]
\ie, the concentration of the dissolved gas is proportional to the
pressure of the free gas above the solution (law of Henry and
Bunsen). The factor of proportionality~$C$, which furnishes a measure
of the solubility of the gas, depends upon the heat effect in
quite the same manner as in the example previously considered.

A number of no less important relations are easily derived as
by-products of those found above, \eg, the Nernst laws concerning
the influence of solubility, the Arrhenius theory of isohydric
solutions,~etc. All such may be obtained through the
application of the general condition of equilibrium~\Eq{(6)}. In
conclusion, there is one other case that I desire to treat here.
In the historical development of the theory this has played a
particularly important rôle.


\Section{VII.}{Osmotic Pressure.}

We consider now a dilute solution separated by a membrane
(permeable with regard to the solvent but impermeable as
regards the dissolved substance) from the pure solvent (in the
%-----File: 048.png---\redacted\--------
same state of aggregation), and inquire as to the condition of
equilibrium. The symbol of the system considered we may again
take as
\[
n_{0}m_{0},\ n_{1}m_{1},\ n_{2}m_{2},\ \cdots \mid {n_{0}}'m_{0}.
\]

The condition of equilibrium is also here again expressed by
equation~\Eq{(6)}, valid for a change of state in which the temperature
and the pressure in each phase is maintained constant. The
only difference with respect to the cases treated earlier is this,
that here, in the presence of a separating membrane between
two phases, the pressure~$p$ in the first phase may be different from
the pressure~$p'$ in the second phase, whereby by ``pressure,'' as
always, is to be understood the ordinary hydrostatic or manometric
pressure.

The proof of the applicability of equation~\Eq{(6)} is found in the
same way as this equation was derived above, proceeding from the
principle of increase of entropy. One has but to remember that,
in the somewhat more general case here considered, the external
work in a given change is represented by the sum~$p dV + p' dV'$,
where $V$~and~$V'$ denote the volumes of the two individual phases,
while before $V$~denoted the total volume of all phases. Accordingly,
we use, instead of~\Eq{(7)}, to express the dependence of the
constant~$K$ in~\Eq{(6)} upon the pressure:
\[
\frac{\dd \log K}{\dd p}  = -\frac{\Delta V}{RT}, \quad
\frac{\dd \log K}{\dd p'} = -\frac{\Delta V'}{RT}.
\Tag{(11)}
\]
We have here to do with the following change:
\[
\nu_{0} = -1,\quad \nu_{1} = 0,\quad \nu_{2} = 0,\quad \cdots,\quad {\nu_{0}}' = 1,
\]
whereby is expressed, that a molecule of the solvent passes out
of the solution through the membrane into the pure solvent.
Hence, in accordance with~\Eq{(6)}:
\[
-\log c_{0} = \log K,
\]
or, since
\[
c_{0} = 1 - \frac{n_{1} + n_{2} + \cdots}{n_{0}}, \quad
\frac{n_{1} + n_{2} + \cdots}{n_{0}} = \log K.
\]
%-----File: 049.png---\redacted\--------
Here $K$~depends only upon $T$,~$p$ and~$p'$. If a pure solvent were
present upon both sides of the membrane, we should have
$c_{0} = 1$, and $p = p'$; consequently:
\[
(\log K)_{p = p'} = 0,
\]
and by subtraction of the last two equations:
\[
\frac{n_{1} + n_{2} + \cdots}{n_{0}}
  = \log K - (\log K)_{p = p'}
  = \frac{\dd \log K}{\dd p} (p - p')
\]
and in accordance with~\Eq{(11)}:
\[
\frac{n_{1} + n_{2} + \cdots}{n_{0}} = -(p - p') · \frac{\Delta V}{RT}.
\]
Here $\Delta V$~denotes the change in volume of the solution due to the
loss of a molecule of the solvent ($\nu_{0} = -1$). Approximately
then:
\[
-\Delta V · n_{0} = V,
\]
the volume of the whole solution, and
\[
\frac{n_{1} + n_{2} + \cdots}{n_{0}} = (p - p') · \frac{V}{RT}.
\]
If we call the difference $p - p'$, the osmotic pressure of the
solution, this equation contains the well known law of osmotic
pressure, due to van't~Hoff.

The equations here derived, which easily permit of multiplication
and generalization, have, of course, for the most part not been
derived in the ways described above, but have been derived,
either directly from experiment, or theoretically from the consideration
of special reversible isothermal cycles to which the
thermodynamic law was applied, that in such a cyclic process
not only the algebraic sum of the work produced and the heat
produced, but that also each of these two quantities separately, is
equal to zero (first lecture, p.~\pageref{png29lab1}). The employment of a cyclic
process has the advantage over the procedure here proposed,
%-----File: 050.png---\redacted\--------
that in it the connection between the directly measurable quantities
and the requirements of the laws of thermodynamics
succinctly appears in each case; but for each individual case a
satisfactory cyclic process must be imagined, and one has not
always the certain assurance that the thermodynamic realization
of the cyclic process also actually supplies all the conditions
of equilibrium. Furthermore, in the process of calculation
certain terms of considerable weight frequently appear as
empty ballast, since they disappear at the end in the summation
over the individual phases of the process.

On the other hand, the significance of the process here employed
consists therein, that the necessary and sufficient conditions
of equilibrium for each individually considered case appear
collectively in the single equation~\Eq{(6)}, and that they are derived
collectively from it in a direct manner through an unambiguous
procedure. The more complicated the systems considered are,
the more apparent becomes the advantage of this method, and
there is no doubt in my mind that in chemical circles it will be
more and more employed, especially, since in general it is now
the custom to deal directly with the energies, and not with cyclic
processes, in the calculation of heat effects in chemical changes.
%-----File: 051.png---\redacted\--------

\Chapter{THIRD LECTURE.}{The Atomic Theory of Matter.}\label{Lect3}

The problem with which we shall be occupied in the present
lecture is that of a closer investigation of the atomic theory of
matter. It is, however, not my intention to introduce this
theory with nothing further, and to set it up as something apart
and disconnected with other physical theories, but I intend above
all to bring out the peculiar significance of the atomic theory as
related to the present general system of theoretical physics; for
in this way only will it be possible to regard the whole system
as one containing within itself the essential compact unity, and
thereby to realize the principal object of these lectures.

Consequently it is self evident that we must rely on that sort
of treatment which we have recognized in last week's lecture as
fundamental. That is, the division of all physical processes into
reversible and irreversible processes. Furthermore, we shall be
convinced that the accomplishment of this division is only possible
through the atomic theory of matter, or, in other words,
that irreversibility leads of necessity to atomistics.

I have already referred at the close of the first lecture to the
fact that in pure thermodynamics, which knows nothing of an
atomic structure and which regards all substances as absolutely
continuous, the difference between reversible and irreversible
processes can only be defined in one way, which a priori carries
a provisional character and does not withstand penetrating analysis.
This appears immediately evident when one reflects that
the purely thermodynamic definition of irreversibility which
proceeds from the impossibility of the realization of certain
changes in nature, as, \eg, the transformation of heat into
work without compensation, has at the outset assumed a definite
limit to man's mental capacity, while, however, such a
%-----File: 052.png---\redacted\--------
limit is not indicated in reality. On the contrary: mankind is
making every endeavor to press beyond the present boundaries
of its capacity, and we hope that later on many things will be
attained which, perhaps, many regard at present as impossible
of accomplishment. Can it not happen then that a process,
which up to the present has been regarded as irreversible, may
be proved, through a new discovery or invention, to be reversible?
In this case the whole structure of the second law would undeniably
collapse, for the irreversibility of a single process conditions
that of all the others.

It is evident then that the only means to assure to the second
law real meaning consists in this, that the idea of irreversibility
be made independent of any relationship to man and especially of
all technical relations.

Now the idea of irreversibility harks back to the idea of entropy;
for a process is irreversible when it is connected with an increase
of entropy. The problem is hereby referred back to a proper
improvement of the definition of entropy. In accordance with
the original definition of Clausius, the entropy is measured by
means of a certain reversible process, and the weakness of this
definition rests upon the fact that many such reversible processes,
strictly speaking all, are not capable of being carried out in
practice. With some reason it may be objected that we have
here to do, not with an actual process and an actual physicist,
but only with ideal processes, so-called thought experiments, and
with an ideal physicist who operates with all the experimental
methods with absolute accuracy. But at this point the difficulty
is encountered: How far do the physicist's ideal measurements
of this sort suffice? It may be understood, by passing to the
limit, that a gas is compressed by a pressure which is equal to
the pressure of the gas, and is heated by a heat reservoir which
possesses the same temperature as the gas, but, for example,
that a saturated vapor shall be transformed through isothermal
compression in a reversible manner to a liquid without at any
time a part of the vapor being condensed, as in certain thermodynamic
%-----File: 053.png---\redacted\--------
considerations is supposed, must certainly appear
doubtful. Still more striking, however, is the liberty as regards
thought experiments, which in physical chemistry is granted the
theorist. With his semi-permeable membranes, which in reality
are only realizable under certain special conditions and then
only with a certain approximation, he separates in a reversible
manner, not only all possible varieties of molecules, whether or
not they are in stable or unstable conditions, but he also separates
the oppositely charged ions from one another and from the
undissociated molecules, and he is disturbed, neither by the
enormous electrostatic forces which resist such a separation, nor
by the circumstance that in reality, from the beginning of the
separation, the molecules become in part dissociated while the
ions in part again combine. But such ideal processes are necessary
throughout in order to make possible the comparison of
the entropy of the undissociated molecules with the entropy of
the dissociated molecules; for the law of thermodynamic equilibrium
does not permit in general of derivation in any other way,
in case one wishes to retain pure thermodynamics as a basis. It
must be considered remarkable that all these ingenious thought
processes have so well found confirmation of their results in
experience, as is shown by the examples considered by us in the
last lecture.

If now, on the other hand, one reflects that in all these results
every reference to the possibility of actually carrying out each
ideal process has disappeared---there are certainly left relations
between directly measurable quantities only, such as temperature,
heat effect, concentration,~etc.---the presumption forces
itself upon one that perhaps the introduction as above of such
ideal processes is at bottom a round-about method, and that
the peculiar import of the principle of increase of entropy with
all its consequences can be evolved from the original idea of
irreversibility or, just as well, from the impossibility of perpetual
motion of the second kind, just as the principle of conservation
of energy has been evolved from the law of impossibility of
perpetual motion of the first kind.
%-----File: 054.png---\redacted\--------

This step: to have completed the emancipation of the entropy
idea from the experimental art of man and the elevation of the
second law thereby to a real principle, was the scientific life's
work of Ludwig Boltzmann. Briefly stated, it consisted in
general of referring back the idea of entropy to the idea of
probability. Thereby is also explained, at the same time, the
significance of the above (p.~\pageref{png27lab1}) auxiliary term used by me;
``preference'' of nature for a definite state. Nature prefers the
more probable states to the less probable, because in nature
processes take place in the direction of greater probability. Heat
goes from a body at higher temperature to a body at lower
temperature because the state of equal temperature distribution
is more probable than a state of unequal temperature distribution.

Through this conception the second law of thermodynamics
is removed at one stroke from its isolated position, the mystery
concerning the preference of nature vanishes, and the entropy
principle reduces to a well understood law of the calculus of
probability.

The enormous fruitfulness of so ``objective'' a definition of
entropy for all domains of physics I shall seek to demonstrate
in the following lectures. But today we have principally to do
with the proof of its admissibility; for on closer consideration we
shall immediately perceive that the new conception of entropy
at once introduces a great number of questions, new requirements
and difficult problems. The first requirement is the introduction
of the atomic hypothesis into the system of physics. For, if one
wishes to speak of the probability of a physical state, \ie, if he
wishes to introduce the probability for a given state as a definite
quantity into the calculation, this can only be brought about, as
in cases of all probability calculations, by referring the state back
to a variety of possibilities; \ie,~by considering a finite number
of a~priori equally likely configurations (complexions) through
each of which the state considered may be realized. The greater
the number of complexions, the greater is the probability of the
state. Thus, \eg, the probability of throwing a total of four
%-----File: 055.png---\redacted\--------
with two ordinary six-sided dice is found through counting the
complexions by which the throw with a total of four may be
realized. Of these there are three complexions:
\begin{center}
with the first die, $1$, with the second die, $3$,\\
with the first die, $2$, with the second die, $2$,\\
with the first die, $3$, with the second die, $1$.
\end{center}
On the other hand, the throw of two is only realized through
a single complexion. Therefore, the probability of throwing a
total of four is three times as great as the probability of throwing
a total of two.

Now, in connection with the physical state under consideration,
in order to be able to differentiate completely from one another
the complexions realizing it, and to associate it with a definite
reckonable number, there is obviously no other means than to
regard it as made up of numerous discrete homogeneous elements---for
in perfectly continuous systems there exist no reckonable
elements---and hereby the atomistic view is made a fundamental
requirement. We have, therefore, to regard all bodies in nature,
in so far as they possess an entropy, as constituted of atoms, and
we therefore arrive in physics at the same conception of matter as
that which obtained in chemistry for so long previously.

But we can immediately go a step further yet. The conclusions
reached hold, not only for thermodynamics of material
bodies, but also possess complete validity for the processes of
heat radiation, which are thus referred back to the second law
of thermodynamics. That radiant heat also possesses an entropy
follows from the fact that a body which emits radiation into a surrounding
diathermanous medium experiences a loss of heat and,
therefore, a decrease of entropy. Since the total entropy of
a physical system can only increase, it follows that one part
of the entropy of the whole system, consisting of the body and the
diathermanous medium, must be contained in the radiated heat.
If the entropy of the radiant heat is to be referred back to the
notion of probability, we are forced, in a similar way as above, to
%-----File: 056.png---\redacted\--------
the conclusion that for radiant heat the atomic conception
possesses a definite meaning. But, since radiant heat is not
directly connected with matter, it follows that this atomistic conception
relates, not to matter, but only to energy, and hence,
that in heat radiation certain energy elements play an essential
rôle. Even though this conclusion appears so singular and even
though in many circles today vigorous objection is strongly urged
against it, in the long run physical research will not be able
to withhold its sanction from it, and the less, since it is confirmed
by experience in quite a satisfactory manner. We shall return
to this point in the lectures on heat radiation. I desire here
only to mention that the novelty involved by the introduction
of atomistic conceptions into the theory of heat radiation is by no
means so revolutionary as, perhaps, might appear at the first
glance. For there is, in my opinion at least, nothing which makes
necessary the consideration of the heat processes in a complete
vacuum as atomic, and it suffices to seek the atomistic features at
the source of radiation, \ie, in those processes which have
their play in the centres of emission and absorption of radiation.
Then the Maxwellian electrodynamic differential equations can
retain completely their validity for the vacuum, and, besides,
the discrete elements of heat radiation are relegated exclusively
to a domain which is still very mysterious and where there is
still present plenty of room for all sorts of hypotheses.

Returning to more general considerations, the most important
question comes up as to whether, with the introduction of atomistic
conceptions and with the reference of entropy to probability,
the content of the principle of increase of entropy is exhaustively
comprehended, or whether still further physical hypotheses are required
in order to secure the full import of that principle. If this
important question had been settled at the time of the introduction
of the atomic theory into thermodynamics, then the
atomistic views would surely have been spared a large number of
conceivable misunderstandings and justifiable attacks. For it
turns out, in fact---and our further considerations will confirm
%-----File: 057.png---\redacted\--------
this conclusion---that there has as yet nothing been done with
atomistics which in itself requires much more than an essential
generalization, in order to guarantee the validity of the
second law.

We must first reflect that, in accordance with the central
idea laid down in the first lecture (p.~\pageref{png17lab1}), the second law must
possess validity as an objective physical law, independently of
the individuality of the physicist. There is nothing to hinder
us from imagining a physicist---we shall designate him a ``microscopic''
observer---whose senses are so sharpened that he
is able to recognize each individual atom and to follow it in
its motion. For this observer each atom moves exactly in
accordance with the elementary laws which general dynamics
lays down for it, and these laws allow, so far as we know, of an
inverse performance of every process. Accordingly, here again
the question is neither one of probability nor of entropy and its
increase. Let us imagine, on the other hand, another observer,
designated a ``macroscopic'' observer, who regards an
ensemble of atoms as a homogeneous gas, say, and consequently
applies the laws of thermodynamics to the mechanical and thermal
processes within it. Then, for such an observer, in accordance
with the second law, the process in general is an irreversible
process. Would not now the first observer be justified in saying:
``The reference of the entropy to probability has its origin in
the fact that irreversible processes ought to be explained through
reversible processes. At any rate, this procedure appears to me
in the highest degree dubious. In any case, I declare each change
of state which takes place in the ensemble of atoms designated
a gas, as reversible, in opposition to the macroscopic observer.''
There is not the slightest thing, so far as I know, that one can
urge against the validity of these statements. But do we not
thereby place ourselves in the painful position of the judge who
declared in a trial the correctness of the position of each separately
of two contending parties and then, when a third contends that
only one of the parties could emerge from the process victorious,
%-----File: 058.png---\redacted\--------
was obliged to declare him also correct? Fortunately we find ourselves
in a more favorable position. We can certainly mediate
between the two parties without its being necessary for one or
the other to give up his principal point of view. For closer
consideration shows that the whole controversy rests upon a misunderstanding---a
new proof of how necessary it is before one
begins a controversy to come to an understanding with his
opponent concerning the subject of the quarrel. Certainly, a
given change of state cannot be both reversible and irreversible.
But the one observer connects a wholly different idea with the
phrase ``change of state'' than the other. What is then, in
general, a change of state? The state of a physical system cannot
well be otherwise defined than as the aggregate of all those physical
quantities, through whose instantaneous values the time
changes of the quantities, with given boundary conditions, are
uniquely determined. If we inquire now, in accordance with
the import of this definition, of the two observers as to what
they understand by the state of the collection of atoms or the
gas considered, they will give quite different answers. The
microscopic observer will mention those quantities which determine
the position and the velocities of all the individual atoms.
There are present in the simplest case, namely, that in which
the atoms may be considered as material points, six times as many
quantities as atoms, namely, for each atom the three coordinates
and the three velocity components, and in the case of combined
molecules, still more quantities. For him the state and the
progress of a process is then first determined when all these
various quantities are individually given. We shall designate
the state defined in this way the ``micro-state.'' The macroscopic
observer, on the other hand, requires fewer data. He will
say that the state of the homogeneous gas considered by him is
determined by the density, the visible velocity and the temperature
at each point of the gas, and he will expect that, when these
quantities are given, their time variations and, therefore, the progress
of the process, to be completely determined in accordance
%-----File: 059.png---\redacted\--------
with the two laws of thermo-dynamics, and therefore accompanied
by an increase in entropy. In this connection he can call upon
all the experience at his disposal, which will fully confirm his expectation.
If we call this state the ``macro-state,'' it is clear that
the two laws: ``the micro-changes of state are reversible'' and
``the macro-changes of state are irreversible,'' lie in wholly
different domains and, at any rate, are not contradictory.

But now how can we succeed in bringing the two observers to
an understanding? This is a question whose answer is obviously
of fundamental significance for the atomic theory. First of all,
it is easy to see that the macro-observer reckons only with mean
values; for what he calls density, visible velocity and temperature
of the gas are, for the micro-observer, certain mean values, statistical
data, which are derived from the space distribution and from
the velocities of the atoms in an appropriate manner. But the
micro-observer cannot operate with these mean values alone, for,
if these are given at one instant of time, the progress of the process
is not determined throughout; on the contrary: he can easily
find with given mean values an enormously large number of
individual values for the positions and the velocities of the atoms,
all of which correspond with the same mean values and which, in
spite of this, lead to quite different processes with regard to the
mean values. It follows from this of necessity that the micro-observer
must either \label{png59lab1}give up the attempt to understand the unique
progress, in accordance with experience, of the macroscopic
changes of state---and this would be the end of the atomic theory---or
that he, through the introduction of a special physical
hypothesis, restrict in a suitable manner the manifold of micro-states
considered by him. There is certainly nothing to prevent
him from assuming that not all conceivable micro-states are
realizable in nature, and that certain of them are in fact thinkable,
but never actually realized. In the formularization of such a
hypothesis, there is of course no point of departure to be found
from the principles of dynamics alone; for pure dynamics leaves
this case undetermined. But on just this account any dynamical
%-----File: 060.png---\redacted\--------
hypothesis, which involves nothing further than a closer specification
of the micro-states realized in nature, is certainly permissible.
Which hypothesis is to be given the preference can only
be decided through comparison of the results to which the
different possible hypotheses lead in the course of experience.

In order to limit the investigation in this way, we must obviously
fix our attention only upon all imaginable configurations and
velocities of the individual atoms which are compatible with
determinate values of the density, the velocity and the temperature
of the gas, or in other words: we must consider all the
micro-states which belong to a determinate macro-state, and
must investigate the various kinds of processes which follow in
accordance with the fixed laws of dynamics from the different
micro-states. Now, precise calculation has in every case always
led to the important result that an enormously large number of
these different micro-processes relate to one and the same macro-process,
and that only proportionately few of the same, which are
distinguished by quite special exceptional conditions concerning
the positions and the velocities of neighboring atoms, furnish
exceptions. Furthermore, it has also shown that one of the
resulting macro-processes is that which the macroscopic observer
recognizes, so that it is compatible with the second law
of thermodynamics.

Here, manifestly, the bridge of understanding is supplied. The
micro-observer needs only to assimilate in his theory the physical
hypothesis that all those special cases in which special exceptional
conditions exist among the neighboring configurations of interacting
atoms do not occur in nature, or, in other words, that the
micro-states are in elementary disorder. Then the uniqueness
of the macroscopic process is assured and with it, also, the fulfillment
of the principle of increase of entropy in all directions.

Therefore, it is not the atomic distribution, but rather the
hypothesis of elementary disorder, which forms the real kernel of
the principle of increase of entropy and, therefore, the preliminary
condition for the existence of entropy. Without elementary
%-----File: 061.png---\redacted\--------
disorder there is neither entropy nor irreversible process.\footnote
  {To those physicists who, in spite of all this, regard the hypothesis of
  elementary disorder as gratuitous or as incorrect, I wish to refer the simple
  fact that in every calculation of a coefficient of friction, of diffusion, or of heat
  conduction, from molecular considerations, the notion of elementary disorder
  is employed, whether tacitly or otherwise, and that it is therefore essentially
  more correct to stipulate this condition instead of ignoring or concealing it. But
  he who regards the hypothesis of elementary disorder as self-evident, should
  be reminded that, in accordance with a law of H.~Poincaré, the precise investigation
  concerning the foundation of which would here lead us too far,
  the assumption of this hypothesis for all times is unwarranted for a closed
  space with absolutely smooth walls,---an important conclusion, against which
 can only be urged the fact that absolutely smooth walls do not exist in nature.}
Therefore, a single atom can never possess an entropy; for we
cannot speak of disorder in connection with it. But with a
fairly large number of atoms, say $100$ or~$1,000$, the matter is
quite different. Here, one can certainly speak of a disorder, in
case that the values of the coordinates and the velocity components
are distributed among the atoms in accordance with the
laws of accident. Then it is possible to calculate the probability
for a given state. But how is it with regard to the increase of
entropy? May we assert that the motion of $100$~atoms is irreversible?
Certainly not; but this is only because the state of
$100$~atoms cannot be defined in a thermodynamic sense, since the
process does not proceed in a unique manner from the standpoint
of a macro-observer, and this requirement forms, as we have seen
above, the foundation and preliminary condition for the definition
of a thermodynamic state.

If one therefore asks: How many atoms are at least necessary
in order that a process may be considered irreversible?, the answer
is: so many atoms that one may form from them definite mean
values which define the state in a macroscopic sense. One must
reflect that to secure the validity of the principle of increase of
entropy there must be added to the condition of elementary disorder
still another, namely, that the number of the elements
under consideration be sufficiently large to render possible the
formation of definite mean values. The second law has a
meaning for these mean values only; but for them, it is quite
%-----File: 062.png---\redacted\--------
exact, just as exact as the law of the calculus of probability, that
the mean value, so far as it may be defined, of a sufficiently large
number of throws with a six-sided die, is~$3\frac{1}{2}$.

These considerations are, at the same time, capable of throwing
light upon questions such as the following: Does the principle of
increase of entropy possess a meaning for the so-called Brownian
molecular movement of a suspended particle? Does the kinetic
energy of this motion represent useful work or not? The entropy
principle is just as little valid for a single suspended particle as
for an atom, and therefore is not valid for a few of them, but
only when there is so large a number that definite mean values
can be formed. That one is able to see the particles and not
the atoms makes no material difference; because the progress of a
process does not depend upon the power of an observing instrument.
The question with regard to useful work plays no rôle
in this connection; strictly speaking, this possesses, in general, no
objective physical meaning. For it does not admit of an answer
without reference to the scheme of the physicist or technician
who proposes to make use of the work in question. The second
law, therefore, has fundamentally nothing to do with the idea of
useful work (cf.\ first lecture, p.~\pageref{png25lab1}).

But, if the entropy principle is to hold, a further assumption is
necessary, concerning the various disordered elements,---an
assumption which tacitly is commonly made and which we
have not previously definitely expressed. It is, however, not
less important than those referred to above. The elements must
actually be of the same kind, or they must at least form a number
of groups of like kind, \eg, constitute a mixture in which each
kind of element occurs in large numbers. For only through the
similarity of the elements does it come about that order and law
can result in the larger from the smaller. If the molecules of a
gas be all different from one another, the properties of a gas can
never show so simple a law-abiding behavior as that which is
indicated by thermodynamics. In fact, the calculation of the
probability of a state presupposes that all complexions which
%-----File: 063.png---\redacted\--------
correspond to the state are a priori equally likely. Without
this condition one is just as little able to calculate the probability
of a given state as, for instance, the probability of a given throw
with dice whose sides are unequal in size. In summing up we
may therefore say: the second law of thermodynamics in its
objective physical conception, freed from anthropomorphism,
relates to certain mean values which are formed from a large
number of disordered elements of the same kind.

The validity of the principle of increase of entropy and of the
irreversible progress of thermodynamic processes in nature is
completely assured in this formularization. After the introduction
of the hypothesis of elementary disorder, the microscopic
observer can no longer confidently assert that each process considered
by him in a collection of atoms is reversible; for the
motion occurring in the reverse order will not always obey the
requirements of that hypothesis. In fact, the motions of single
atoms are always reversible, and thus far one may say, as before,
that the irreversible processes appear reduced to a reversible
process, but the phenomenon as a whole is nevertheless irreversible,
because upon reversal the disorder of the numerous
individual elementary processes would be eliminated. Irreversibility
is inherent, not in the individual elementary processes
themselves, but solely in their irregular constitution. It is
this only which guarantees the unique change of the macroscopic
mean values.

Thus, for example, the reverse progress of a frictional process
is impossible, in that it would presuppose elementary arrangement
of interacting neighboring molecules. For the collisions between
any two molecules must thereby possess a certain distinguishing
character, in that the velocities of two colliding molecules
depend in a definite way upon the place at which they meet.
In this way only can it happen that in collisions like directed
velocities ensue and, therefore, visible motion.

Previously we have only referred to the principle of elementary
disorder in its application to the atomic theory of matter. But
%-----File: 064.png---\redacted\--------
it may also be assumed as valid, as I wish to indicate at this
point, on quite the same grounds as those holding in the case of
matter, for the theory of radiant heat. Let us consider, \eg,
two bodies at different temperatures between which exchange of
heat occurs through radiation. We can in this case also imagine
a microscopic observer, as opposed to the ordinary macroscopic
observer, who possesses insight into all the particulars
of electromagnetic processes which are connected with emission
and absorption, and the propagation of heat rays. The microscopic
observer would declare the whole process reversible
because all electrodynamic processes can also take place in the
reverse direction, and the contradiction may here be referred
back to a difference in definition of the state of a heat ray. Thus,
while the macroscopic observer completely defines a monochromatic
ray through direction, state of polarization, color, and
intensity, the microscopic observer, in order to possess a complete
knowledge of an electromagnetic state, necessarily requires the
specification of all the numerous irregular variations of amplitude
and phase to which the most homogeneous heat ray is actually
subject. That such irregular variations actually exist follows
immediately from the well known fact that two rays of the same
color never interfere, except when they originate in the same source
of light. But until these fluctuations are given in all particulars,
the micro-observer can say nothing with regard to the progress
of the process. He is also unable to specify whether the exchange
of heat radiation between the two bodies leads to a decrease or
to an increase of their difference in temperature. The principle
of elementary disorder first furnishes the adequate criterion of
the tendency of the radiation process, \ie, the warming of the
colder body at the expense of the warmer, just as the same principle
conditions the irreversibility of exchange of heat through conduction.
However, in the two cases compared, there is indicated
an essential difference in the kind of the disorder. While in
heat conduction the disordered elements may be represented
as associated with the various molecules, in heat radiation there
%-----File: 065.png---\redacted\--------
are the numerous vibration periods, connected with a heat ray,
among which the energy of radiation is irregularly distributed.
In other words: the disorder among the molecules is a material
one, while in heat radiation it is one of energy distribution. This
is the most important difference between the two kinds of disorder;
a common feature exists as regards the great number of
uncoordinated elements required. Just as the entropy of a body
is defined as a function of the macroscopic state, only when the
body contains so many atoms that from them definite mean
values may be formed, so the entropy principle only possesses
a meaning with regard to a heat ray when the ray comprehends
so many periodic vibrations, \ie, persists for so long a time, that
a definite mean value for the intensity of the ray may be obtained
from the successive irregular fluctuating amplitudes.

Now, after the principle of elementary disorder has been
introduced and accepted by us as valid throughout nature, the
fundamental question arises as to the calculation of the probability
of a given state, and the actual derivation of the entropy
therefrom. From the entropy all the laws of thermodynamic
states of equilibrium, for material substances, and also for
energy radiation, may be uniquely derived. With regard to
the connection between entropy and probability, this is inferred
very simply from the law that the probability of two independent
configurations is represented by the product of the individual
probabilities:
\[
W = W_{1} · W_{2},
\]
while the entropy~$S$ is represented by the sum of the individual
entropies:
\[
S = S_{1} + S_{2}.
\]
Accordingly, the entropy is proportional to the logarithm of the
probability:
\[
S = k \log W.
\Tag{(12)}
\]
$k$~is a universal constant. In particular, it is the same for atomic
as for radiation configurations, for there is nothing to prevent
%-----File: 066.png---\redacted\--------
us assuming that the configuration designated by~$1$ is atomic,
while that designated by~$2$ is a radiation configuration. If $k$~has
been calculated, say with the aid of radiation measurements,
then $k$~must have the same value for atomic processes. Later
we shall follow this procedure, in order to utilize the laws of heat
radiation in the kinetic theory of gases. Now, there remains, as
the last and most difficult part of the problem, the calculation of
the probability~$W$ of a given physical configuration in a given
macroscopic state. We shall treat today, by way of preparation
for the quite general problem to follow, the simple problem: to
specify the probability of a given state for a single moving
material point, subject to given conservative forces. Since the
state depends upon $6$~variables: the $3$~generalized coordinates
$\varphi_{1}$,~$\varphi_{2}$,~$\varphi_{3}$, and the three corresponding velocity components
$\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\dot{\varphi}_{3}$, and since all possible values of these $6$~variables constitute
a continuous manifold, the probability sought is, that
these $6$~quantities shall lie respectively within certain infinitely
small intervals, or, if one thinks of these $6$~quantities as the
rectilinear orthogonal coordinates of a point in an ideal six-dimensional
space, that this ideal ``state point'' shall fall within
a given, infinitely small ``state domain.'' Since the domain is
infinitely small, the probability will be proportional to the magnitude
of the domain and therefore proportional to
\[
\int d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\dot{\varphi}_{1} · d\dot{\varphi}_{2} · d\dot{\varphi}_{3}.
\]

But this expression cannot serve as an absolute measure of
the probability, because in general it changes in magnitude with
the time, if each state point moves in accordance with the laws
of motion of material points, while the probability of a state
which follows of necessity from another must be the same for
the one as the other. Now, as is well known, another integral
quite similarly formed, may be specified in place of the one
above, which possesses the special property of not changing in
value with the time. It is only necessary to employ, in addition
to the general coordinates $\varphi_{1}$,~$\varphi_{2}$,~$\varphi_{3}$, the three so-called momenta
%-----File: 067.png---\redacted\--------
$\psi_{1}$,~$\psi_{2}$,~$\psi_{3}$, in place of the three velocities $\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\dot{\varphi}_{3}$ as the determining
coordinates of the state. These are defined in the
following way:
\label{png67lab1}
\[
\psi_{1} = \left(\frac{\dd H}{\dd \dot{\varphi}_{1}}\right)_{\varphi},\quad
\psi_{2} = \left(\frac{\dd H}{\dd \dot{\varphi}_{2}}\right)_{\varphi},\quad
\psi_{3} = \left(\frac{\dd H}{\dd \dot{\varphi}_{3}}\right)_{\varphi},
\]
wherein $H$~denotes the kinetic potential (Helmholz). Then, in
Hamiltonian form, the equations of motion are:
\label{png67lab2}
\[
\dot{\psi}_{1} = \frac{d\psi_{1}}{dt} = -\left(\frac{\dd E}{\dd \varphi_{1}}\right)_{\psi},\ \cdots,\quad
\dot{\varphi}_{1} = \frac{d\varphi_{1}}{dt} =  \left(\frac{\dd E}{\dd \psi_{1}}\right)_{\varphi},\ \cdots,
\]
($E$~is the energy), and from these equations follows the ``condition
of incompressibility'':
\[
\frac{\dd \dot{\varphi}_{1}}{\dd \varphi_{1}} + \frac{\dd \dot{\psi}_{1}}{\dd \psi_{1}} + \cdots = 0.
\]
Referring to the six-dimensional space represented by the coordinates
$\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\psi_{1}$, $\psi_{2}$, $\psi_{3}$, this equation states that the magnitude
of an arbitrarily chosen state domain,~viz.:
\label{png67lab3}
\[
\int d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\psi_{1} · d\psi_{2} · d\psi_{3}
\]
does not change with the time, when each point of the domain
changes its position in accordance with the laws of motion of
material points. Accordingly, it is made possible to take the
magnitude of this domain as a direct measure for the probability
that the state point falls within the domain.

From the last expression, which can be easily generalized for
the case of an arbitrary number of variables, we shall \label{png67lab4}calculate
later the probability of a thermodynamic state, for the
case of radiant energy as well as that for material substances.
%-----File: 068.png---\redacted\--------


\Chapter{Fourth Lecture.}{The Equation of State for a Monatomic Gas.}\label{Lect4}

My problem today is to utilize the general fundamental laws
concerning the concept of irreversibility, which we established
in the lecture of yesterday, in the solution of a definite problem:
the calculation of the entropy of an ideal monatomic gas in a
given state, and the derivation of all its thermodynamic properties.
The way in which we have to proceed is prescribed for us
by the general definition of entropy:
\[
S = k \log W.
\Tag{(13)}
\]
The chief part of our problem is the calculation of~$W$ for a given
state of the gas, and in this connection there is first required a
more precise investigation of that which is to be understood as
the state of the gas. Obviously, the state is to be taken here
solely in the sense of the conception which we have called macroscopic
in the last lecture. Otherwise, a state would possess
neither probability nor entropy. Furthermore, we are not
allowed to assume a condition of equilibrium for the gas. For
this is characterized through the further special condition
that the entropy for it is a maximum. Thus, an unequal distribution
of density may exist in the gas; also, there may be
present an arbitrary number of different currents, and in general
no kind of equality between the various velocities of the molecules
is to be assumed. The velocities, as the coordinates of the
molecules, are rather to be taken a~priori as quite arbitrarily
given, but in order that the state, considered in a macroscopic
sense, may be assumed as known, certain mean values of the
densities and the velocities must exist. Through these mean
%-----File: 069.png---\redacted\--------
values the state from a macroscopic standpoint is completely
characterized.

The conditions mentioned will all be fulfilled if we consider
the state as given in such manner that the number of molecules
in a sufficiently small macroscopic space, but which, however,
contains a very large number of molecules, is given, and furthermore,
that the (likewise great) number of these molecules is
given, which are found in a certain macroscopically small velocity
domain, \ie, whose velocities lie within certain small intervals.
If we call the coordinates $x$,~$y$,~$z$, and the velocity components
$\dot{x}$,~$\dot{y}$,~$\dot{z}$, then this number will be proportional to\footnote
  {We can call $\sigma$ a ``macro-differential'' in contradistinction to the micro-differentials
  which are infinitely small with reference to the dimensions of a
  molecule. I prefer this terminology for the discrimination between ``physical''
  and ``mathematical'' differentials in spite of the inelegance of phrasing, because
  the macro-differential is also just as much mathematical as physical and the
  micro-differential just as much physical as mathematical.}
\[
dx · dy · dz · d\dot{x} · d\dot{y} · d\dot{z} = \sigma.
\]
It will depend, besides, upon a finite factor of proportionality
which may be an arbitrarily given function $f(x, y, z, \dot{x}, \dot{y}, \dot{z})$ of
the coordinates and the velocities, and which has only the one
condition to fulfill that
\[
\tsum f · \sigma = N,
\Tag{(14)}
\]
where $N$~denotes the total number of molecules in the gas.
We are now concerned with the calculation of the probability~$W$
of that state of the gas which corresponds to the arbitrarily
given distribution function~$f$.

The probability that a given molecule possesses such coordinates
and such velocities that it lies within the domain~$\sigma$ is
expressed, in accordance with the final result of the previous lecture,
by the magnitude of the corresponding elementary domain:
\[
d\varphi_{1} · d\varphi_{2} · d\varphi_{3} · d\psi_{1} · d\psi_{2} · d\psi_{3},
\]
therefore, since here
\[
\varphi_{1} = x,\quad \varphi_{2} = y,\quad \varphi_{3} = z,\quad
\psi_{1} = m\dot{x},\quad \psi_{2} = m\dot{y},\quad \psi_{3} = m\dot{z},
\]
%-----File: 070.png---\redacted\--------
($m$~the mass of a molecule) by
\[
m^{3} \sigma.
\]
Now we divide the whole of the six dimensional ``state domain''
containing all the molecules into suitable equal elementary
domains of the magnitude~$m^{3} \sigma$. Then the probability that a
given molecule fall in a given elementary domain is equally
great for all such domains. Let $P$~denote the number of these
equal elementary domains. Next, let us imagine as many dice
as there are molecules present, \ie,~$N$, and each die to be
provided with $P$~equal sides. Upon these $P$~sides we imagine
numbers $1$,~$2$, $3$,~$\cdots$ to~$P$, so that each of the $P$~sides indicates
a given elementary domain. Then each throw with the $N$~dice
corresponds to a given state of the gas, while the number of
dice which show a given number corresponds to the molecules
which lie in the elementary domain considered. In accordance
with this, each single die can indicate with the same probability
each of the numbers from $1$ to~$P$, corresponding to the circumstance
that each molecule may fall with equal probability in any
one of the $P$~elementary domains. The probability~$W$ sought,
of the given state of the molecules, corresponds, therefore, to
the number of different kinds of throws (complexions) through
which is realized the given distribution~$f$. Let us take, \eg,
$N$~equal to $10$~molecules (dice) and $P = 6$ elementary domains
(sides) and let us imagine the state so given that there are
\begin{center}
\begin{tabular}{l@{\ }l@{\ }l}
3 molecules & in 1st & elementary domain \\
4 molecules & in 2d  & elementary domain \\
0 molecules & in 3d  & elementary domain \\
1 molecule  & in 4th & elementary domain \\
0 molecules & in 5th & elementary domain \\
2 molecules & in 6th & elementary domain,
\end{tabular}
\end{center}
then this state, \eg, may be realized through a throw for which
the 10 dice indicate the following numbers:
\[
\begin{array}[b]{@{\qquad}*{10}{c}}
\rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} &
\rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} & \rule{1.6em}{0pt} \\[-2ex]
1\text{st} & 2\text{d} & 3\text{d} & 4\text{th} & 5\text{th} & 6\text{th} & 7\text{th} & 8\text{th} & 9\text{th} & 10\text{th} \\
2 & 6 & 2 & 1 & 1 & 2 & 6 & 2 & 1 & 4.
\end{array} \Tag{(15)}
\]
%-----File: 071.png---\redacted\--------
Under each of the characters representing the ten dice stands
the number which the die indicates in the throw. In fact,
\begin{center}
3 dice show the figure 1\phantom{.} \\
4 dice show the figure 2\phantom{.} \\
0 dice show the figure 3\phantom{.} \\
1 die shows the figure 4\phantom{.} \\
0 dice show the figure 5\phantom{.} \\
2 dice show the figure 6.
\end{center}
The state in question may likewise be realized through many other
complexions of this kind. The number sought of all possible
complexions is now found through consideration of the number
series indicated in~\Eq{(15)}. For, since the number of molecules
(dice) is given, the number series contains a fixed number of
elements ($10 = N$). Furthermore, since the number of molecules
falling in an elementary domain is given, each number, in all
permissible complexions, appears equally often in the series.
Finally, each change of the number configuration conditions a
new complexion. The number of possible complexions or the
probability~$W$ of the given state is therefore equal to the number
of possible permutations with repetition under the conditions
mentioned. In the simple example chosen, in accordance with
a well known formula, the probability is
\[
\frac{10!}{3!\; 4!\; 0!\; 1!\; 0!\; 2!\;} = 12,600.
\]
Therefore, in the general case:
\[
W = \frac{N!}{\prod(f · \sigma)!}.
\]
The sign~$\prod$ denotes the product extended over all of the $P$~elementary
domains.

From this there results, in accordance with equation~\Eq{(13)}, for
the entropy of the gas in the given state:
\[
S = k \log N! - k \tsum \log (f · \sigma)!.
\]
%-----File: 072.png---\redacted\--------
The summation is to be extended over all domains~$\sigma$. Since
$f · \sigma$ is a large quantity, Stirling's formula may be employed for
its factorial, which for a large number~$n$ is expressed by:
\[
n! = \left(\frac{n}{e}\right)^{n} \sqrt{2 \pi n},
\Tag{(16)}
\]
therefore, neglecting unimportant terms:
\[
\log n! = n (\log n - 1);
\]
and hence:
\[
S = k \log N! - k \tsum f \sigma (\log [f · \sigma] - 1),
\]
or, if we note that $\sigma$~and $N = \tsum f \sigma$ remain constant in all changes
of state:
\[
S = \const - k \tsum f · \log f · \sigma.
\Tag{(17)}
\]
This quantity is, to the universal factor~$(-k)$, the same as that
which L.~Boltzmann denoted by~$H$, and which he showed to
vary in one direction only for all changes of state.

In particular, we will now determine the entropy of a gas in a
state of equilibrium, and inquire first as to that form of the law of
distribution which corresponds to thermodynamic equilibrium.
In accordance with the second law of thermodynamics, a state
of equilibrium is characterized by the condition that with given
values of the total volume~$V$ and the total energy~$E$, the entropy~$S$
assumes its maximum value. If we assume the total volume
of the gas
\[
V = \int dx · dy · dz,
\]
and the total energy
\[
E = \frac{m}{2} \tsum (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2} )f \sigma
\Tag{(18)}
\]
as given, then the condition:
\[
\delta S = 0
\]
must hold for the state of equilibrium, or, in accordance with~\Eq{(17)}:
\[
\tsum (\log f + 1) · \delta f · \sigma = 0,
\Tag{(19)}
\]
%-----File: 073.png---\redacted\--------
wherein the variation~$\delta f$ refers to an arbitrary change in the
law of distribution, compatible with the given values of $N$,~$V$
and~$E$.

Now we have, on account of the constancy of the total number
of molecules $N$, in accordance with~\Eq{(14)}:
\[
\tsum \delta f · \sigma = 0
\]
and, on account of the constancy of the total energy, in accordance
with~\Eq{(18)}:
\[
\tsum (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}) · \delta f · \sigma = 0.
\]
Consequently, for the fulfillment of condition~\Eq{(19)} for all permissible
values of~$\delta f$, it is sufficient and necessary that
\[
\log f + \beta (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}) = \const,
\]
or:
\[
f = \alpha e^{-\beta (\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2})},
\]
wherein $\alpha$~and~$\beta$ are constants. In the state of equilibrium,
therefore, the space distribution of molecules is uniform, \ie,
independent of $x$,~$y$,~$z$, and the distribution of velocities is the
well known Maxwellian distribution.

The values of the constants $\alpha$~and~$\beta$ are to be found from those
of $N$,~$V$ and~$E$. For the substitution of the value found for~$f$
in~\Eq{(14)} leads to:
\[
N = V \alpha \left(\frac{\pi}{\beta}\right)^{\tfrac{3}{2}},
\]
and the substitution of~$f$ in~\Eq{(18)} leads to:
\[
E = \tfrac{3}{4} Vm \frac{\alpha}{\beta}\left(\frac{\pi}{\beta}\right)^{\tfrac{3}{2}}.
\]
From these equations it follows that:
\[
\alpha = \frac{N}{V} · \left(\frac{3mN}{4\pi E}\right)^{\tfrac{3}{2}},\quad
\beta = \frac{3mN}{4E},
\]
and hence finally, in accordance with~\Eq{(17)}, the expression for the
%-----File: 074.png---\redacted\--------
entropy~$S$ of the gas in a state of equilibrium with given values
for $N$,~$V$ and~$E$ is:
\[
S = \const + kN (\tfrac{3}{2} \log E + \log V).
\Tag{(20)}
\]
The additive constant contains terms in $N$~and~$m$, but not in
$E$~and~$V$.

The determination of the entropy here carried out permits
now the specification directly of the complete thermodynamic
behavior of the gas, viz., of the equation of state, and of the
values of the specific heats. From the general thermodynamic
definition of entropy:
\[
dS = \frac{dE + p dV}{T}
\]
are obtained the partial differential quotients of~$S$ with regard
to $E$~and~$V$ respectively:
\label{png74lab1}
\[
\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{1}{T},\quad
\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{p}{T}.
\]
Consequently, with the aid of~\Eq{(20)}:
\label{png74lab2}
\[
\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{3}{2} \frac{kN}{E} = \frac{1}{T},
\Tag{(21)}
\]
and
\[
\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{kN}{V} = \frac{p}{T}.
\Tag{(22)}
\]
The second of these equations:
\[
p = \frac{kNT}{V}
\]
contains the laws of Boyle, Gay~Lussac and Avogadro, the latter
because the pressure depends only upon the number~$N$, and not
upon the constitution of the molecules. Writing it in the
ordinary form:
\[
p = \frac{RnT}{V},
\]
%-----File: 075.png---\redacted\--------
where $n$~denotes the number of gram molecules or mols of the
gas, referred to $O_{2} = 32g$, and $R$~the absolute gas constant:
\[
R = 8.315 · 10^{7} \frac{\erg}{\deg},
\]
we obtain by comparison:
\[
k = \frac{Rn}{N}.
\Tag{(23)}
\]
If we denote the ratio of the mol number to the molecular
number by~$\omega$, or, what is the same thing, the ratio of the
molecular mass to the mol mass:
\[
\omega = \frac{n}{N},
\]
and hence:
\[
k = \omega R.
\Tag{(24)}
\]
From this, if $\omega$~is given, we can calculate the universal constant~$k$,
and conversely.

The equation~\Eq{(21)} gives:
\[
E = \tfrac{3}{2} kNT.
\Tag{(25)}
\]
Now since the energy of an ideal gas is given by:
\[
E = Anc_{v} T,
\]
wherein $c_{v}$~denotes in calories the heat capacity at constant
volume of a mol, $A$~the mechanical equivalent of heat:
\[
A = 4.19 · 10^{7} \frac{\erg}{\cal},
\]
it follows that:
\[
c_{v} = \frac{3kN}{2An},
\]
and, having regard to~\Eq{(23)}, we obtain:
\[
c_{v} = \frac{3}{2} \frac{R}{A} = 3.0,
\Tag{(26)}
\]
%-----File: 076.png---\redacted\--------
the mol heat in calories of any monatomic gas at constant volume.

For the mol heat~$c_{p}$ at constant pressure we have from the
first law of thermodynamics
\[
c_{p} - c_{v} = \frac{R}{A},
\]
and, therefore, having regard to~\Eq{(26)}:
\[
c_{p} = 5,\quad \frac{c_{p}}{c_{v}} = \tfrac{5}{3},
\]
a known result for monatomic gases.

The mean kinetic energy~$L$ of a molecule is obtained from~\Eq{(25)}:
\[
L = \frac{E}{N} = \tfrac{3}{2} kT.
\Tag{(27)}
\]
You notice that we have derived all these relations through the
identification of the mechanical with the thermodynamic expression
for the entropy, and from this you recognize the fruitfulness
of the method here proposed.

But a method can first demonstrate fully its usefulness when
we utilize it, not only to derive laws which are already known,
but when we apply it in domains for whose investigation there
at present exist no other methods. In this connection its
application affords various possibilities. Take the case of a
monatomic gas which is not sufficiently attenuated to have the
properties of the ideal state; there are here, as pointed out by
J.~D. van~der Waals, two things to consider: (1)~the finite size of
the atoms, (2)~the forces which act among the atoms. Taking
account of these involves a change in the value of the probability
and in the energy of the gas as well, and, so far as can now be
shown, the corresponding change in the conditions for thermodynamic
equilibrium leads to an equation of state which agrees
with that of van~der Waals. Certainly there is here a rich field
for further investigations, of greater promise when experimental
tests of the equation of state exist in larger number.
%-----File: 077.png---\redacted\--------

Another important application of the theory has to do with
heat radiation, with which we shall be occupied the coming
week. We shall proceed then in a similar way as here, and shall
be able from the expression for the entropy of radiation to derive
the thermodynamic properties of radiant heat.

Today we will refer briefly to the treatment of polyatomic
gases. I have previously, upon good grounds, limited the treatment
to monatomic molecules; for up to the present real difficulties
appear to stand in the way of a generalization, from
the principles employed by us, to include polyatomic molecules; in
fact, if we wish to be quite frank, we must say that a satisfactory
mechanical theory of polyatomic gases has not yet been found.
Consequently, at present we do not know to what place in the
system of theoretical physics to assign the processes within a
molecule---the intra-molecular processes. We are obviously confronted
by puzzling problems. A noteworthy and much discussed
beginning was, it is true, made by Boltzmann, who introduced
the most plausible assumption that for intra-molecular
processes simple laws of the same kind hold as for the motion of
the molecules themselves, \textit{\ie}, the general equations of dynamics.
It is easy then, in fact, to proceed to the proof that for a monatomic
gas the molecular heat~$c_{v}$ must be greater than~$3$ and that
consequently, since the difference $c_{p} - c_{v}$ is always equal to~$2$,
the ratio is
\[
\frac{c_{p}}{c_{v}} = \frac{c_{v} + 2}{c_{v}} < \tfrac{5}{3} .
\]
This conclusion is completely confirmed by experience. But this
in itself does not confirm the assumption of Boltzmann; for,
indeed, the same conclusion is reached very simply from the
assumption that there exists intra-molecular energy which
increases with the temperature. For then the molecular heat
of a polyatomic gas must be greater by a corresponding amount
than that of a monatomic gas.

Nevertheless, up to this point the Boltzmann theory never leads
%-----File: 078.png---\redacted\--------
to contradiction with experience. But so soon as one seeks to
draw special conclusions concerning the magnitude of the specific
heats hazardous difficulties arise; I will refer to only one of them.
If one assumes the Hamiltonian equations of mechanics as
applicable to intra-molecular motions, he arrives of necessity at
\label{png78lab1}the law of ``uniform distribution of energy,'' which asserts that
under certain conditions, not essential to consider here, in a
thermodynamic state of equilibrium the total energy of the gas
is distributed uniformly among all the individual energy phases
corresponding to the independent variables of state, or, as
one may briefly say; the same amount of energy is associated
with every independent variable of state. Accordingly, the
mean energy of motion of the molecules~$\frac{1}{2} kT$, corresponding to a
given direction in space, is the same as for any other direction,
and, moreover, the same for all the different kinds of molecules,
and ions; also for all suspended particles (dust) in the gas, of
whatever size, and, furthermore, the same for all kinds of motions
of the constituents of a molecule relative to its centroid. If
one now reflects that a molecule commonly contains, so far as
we know, quite a large number of different freely moving
constituents, certainly, that a normal molecule of a monatomic
gas, \eg, mercury, possesses numerous freely moving
electrons, then, in accordance with the law of uniform energy
distribution, the intra-molecular energy must constitute a much
larger fraction of the whole specific heat of the gas, and therefore
$c_{p}/c_{v}$~must turn out much smaller, than is consistent with the
measured values. Thus, \eg, for an atom of mercury, in
accordance with the measured value of $c_{p}/c_{v} = 5/3$, no part
whatever of the heat added may be assigned to the intra-molecular
energy. Boltzmann and others, in order to eliminate this contradiction,
have fixed upon the possibility that, within the time
of observation of the specific heats, the vibrations of the constituents
(of a molecule) do not change appreciably with respect
to one another, and come later with their progressive motion so
slowly into heat equilibrium that this process is no longer capable
%-----File: 079.png---\redacted\--------
of detection through observation. Up to now no such delay in
the establishment of a state of equilibrium has been observed.
Perhaps it would be productive of results if in delicate measurements
special attention were paid the question as to whether
observations which take a longer time lead to a greater value of
the mol-heat, or, what comes to the same thing, a smaller value
of~$c_{p}/c_{v}$, than observations lasting a shorter time.

If one has been made mistrustful through these considerations
concerning the applicability of the law of uniform energy distribution
to intra-molecular processes, the mistrust is accentuated
upon the inclusion of the laws of heat radiation. I shall make
mention of this in a later lecture.

When we pass from stable atoms to the unstable atoms of
radioactive substances, the principles following from the kinetic
gas theory lose their validity completely. For the striking
failure of all attempts to find any influence of temperature
upon radioactive phenomena shows us that an application here of
the law of uniform energy distribution is certainly not warranted.
It will, therefore, be safest meanwhile to offer no definite conjectures
with regard to the nature and the laws of these noteworthy
phenomena, and to leave this field for further development
to experimental research alone, which, I may say, with every
day throws new light upon the subject.
%-----File: 080.png---\redacted\--------


\Chapter{Fifth Lecture.}{Heat Radiation. Electrodynamic Theory.}\label{Lect5}

Last week I endeavored to point out that we find in the
atomic theory a complete explanation for the whole content of
the two laws of thermodynamics, if we, with Boltzmann, define
the entropy by the probability, and I have further shown, in the
example of an ideal monatomic gas, how the calculation of the
probability, without any additional special hypothesis, enables
us not only to find the properties of gases known from thermodynamics,
but also to reach conclusions which lie essentially
beyond those of pure thermodynamics. Thus, \eg,
the law of Avogadro in pure thermodynamics is only a definition,
while in the kinetic theory it is a necessary consequence;
furthermore, the value of~$c_{v}$, the mol-heat of a gas, is
completely undetermined by pure thermodynamics, but from the
kinetic theory it is of equal magnitude for all monatomic gases
and, in fact, equal to~$3$, corresponding to our experimental
knowledge. Today and tomorrow we shall be occupied with
the application of the theory to radiant heat, and it will appear
that we reach in this apparently quite isolated domain conclusions
which a thorough test shows are compatible with experience.
Naturally, we take as a basis the electro-magnetic
theory of heat radiation, which regards the rays as electro-magnetic
waves of the same kind as light rays.

We shall utilize the time today in developing in bold outline
the important consequences which follow from the electro-magnetic
theory for the characteristic quantities of heat radiation,
and tomorrow seek to answer, through the calculation of the
entropy, the question concerning the dependence of these quantities
%-----File: 081.png---\redacted\--------
upon the temperature, as was done last week for ideal
gases. Above all, we are concerned here with the determination
of those quantities which at any place in a medium traversed
by heat rays determine the state of the radiant heat. The state
of radiation at a given place will not be represented by a vector
which is determined by three components; for the energy flowing
in a given direction is quite independent of that flowing in any
other direction. In order to know the state of radiation, we
must be able to specify, moreover, the energy which in the time~$dt$
flows through a surface element~$d\sigma$ for every direction in
space. This will be proportional to the magnitude of~$d\sigma$, to
the time~$dt$, and to the cosine of the angle~$\theta$ which the direction
considered makes with the normal to~$d\sigma$. But the quantity to
be multiplied by $d\sigma · dt · \cos \theta$ will not be a finite quantity;
for since the radiation through any point of~$d\sigma$ passes in all directions,
therefore the quantity will also depend upon the magnitude
of the solid angle~$d\Omega$, which we shall assume as the same for all
points of~$d\sigma$. In this manner we obtain for the energy which in
the time~$dt$ flows through the surface element~$d\sigma$ in the direction
of the elementary cone~$d\Omega$, the expression:
\[
K d\sigma dt · \cos \theta · d\Omega.
\Tag{(28)}
\]
$K$~is a positive function of place, of time and of direction, and is
for unpolarized light of the following form:
\[
K = 2 \int_{0}^{\infty} \frakK_{\nu} d\nu
\Tag{(29)}
\]
where $\nu$~denotes the frequency of a color of wave length~$\lambda$ and
whose velocity of propagation is~$q$:
\[
\nu = \frac{q}{\lambda},
\]
and $\frakK_{\nu}$~denotes the corresponding intensity of spectral radiation
of the plane polarized light.
%-----File: 082.png---\redacted\--------

From the value of~$K$ is to be found the space density of radiation~$\epsilon$,
\ie, the energy of radiation contained in unit volume. The
point~$0$ in question forms the centre of a sphere whose radius~$r$
we take so small that in the distance~$r$ no appreciable absorption
of radiation takes place. Then each element~$d\sigma$ of the surface
of the sphere furnishes, by virtue of the radiation traversing the
same, the following contribution to the radiation density at~$0$:
\[
\frac{d\sigma · dt · K · d\Omega}{r^{2} d\Omega · q dt} =  \frac{d\sigma · K}{r^{2} q}.
\]
For the radiation cone of solid angle~$d\Omega$ proceeding from a point
of~$d\sigma$ in the direction toward~$0$ has at the distance~$r$ from~$d\sigma$ the
cross-section~$r^{2} d\Omega$ and the energy passing in the time~$dt$ through
this cross-section distributes itself along the distance~$q dt$. By
integration over all of the surface elements~$d\sigma$ we obtain the
total space density of radiation at~$0$:
\[
\epsilon = \int \frac{d\sigma K}{r^{2} q} = \frac{1}{q} \int K d\Omega,
\]
wherein $d\Omega$~denotes the solid angle of an elementary cone whose
vertex is~$0$. For uniform radiation we obtain:
\[
\epsilon = \frac{4\pi K}{q} = \frac{8\pi}{q} · \int_{0}^{\infty} \frakK_{\nu} d\nu.
\Tag{(30)}
\]

The production of radiant heat is a consequence of the act of
emission, and its destruction is the result of absorption. Both
processes, emission and absorption, have their origin only in
material particles, atoms or electrons, not at the geometrical
bounding surface; although one frequently says, for the sake of
brevity, that a surface element emits or absorbs. In reality a
surface element of a body is a place of entrance for the radiation
falling upon the body from without and which is to be
absorbed; or a place of exit for the radiation emitted from
within the body and passing through the surface in the outward
%-----File: 083.png---\redacted\--------
direction. The capacity for emission and the capacity for
absorption of an element of a body depend only upon its own
condition and not upon that of the surrounding elements. If,
therefore, as we shall assume in what follows, the state of the
body varies only with the temperature, then the capacity for
emission and the capacity for absorption of the body will also
vary only with the temperature. The dependence upon the
temperature can of course be different for each wave length.

We shall now introduce that result following from the second
law of thermodynamics which will serve us as a basis
in all subsequent considerations: ``a system of bodies at rest
of arbitrary nature, form and position, which is surrounded by a
fixed shell impervious to heat, passes in the course of time from
an arbitrarily chosen initial state to a permanent state in which
the temperature of all bodies of the system is the same.''
This is the thermodynamic state of equilibrium in which the
entropy of the system, among all those values which it may assume
compatible with the total energy specified by the initial conditions,
has a maximum value. Let us now apply this law to a
single homogeneous isotropic medium which is of great extent
in all directions of space and which, as in all cases subsequently
considered, is surrounded by a fixed shell, perfectly reflecting as
regards heat rays. The medium possesses for each frequency~$\nu$
of the heat rays a finite capacity for emission and a finite capacity
for absorption. Let us consider, now, such regions of the medium
as are very far removed from the surface. Here the influence
of the surface will be in any case vanishingly small, because no
rays from the surface reach these regions, and on account of the
homogeneity and isotropy of the medium we must conclude that
the heat radiation is in thermodynamic equilibrium everywhere
and has the same properties in all directions, so that $\frakK_{\nu}$,~the
specific intensity of radiation of a plane polarized ray, is independent
of the frequency~$\nu$, of the azimuth of polarization, of the
direction of the ray, and of location. Thus, there will correspond
to each diverging bundle of rays in an elementary cone~$d\Omega$,
%-----File: 084.png---\redacted\--------
proceeding from a surface element~$d\sigma$, an exactly equal bundle
oppositely directed, within the same elemental cone converging
toward the surface element. This law retains its validity, as a
simple consideration shows, right up to the surface of the medium.
For in thermodynamic equilibrium each ray must possess
exactly the same intensity as that of the directly opposite ray,
otherwise, more energy would flow in one direction than in
the opposite direction. Let us fix our attention upon a ray
proceeding inwards from the surface, this must have the
same intensity as that of the directly opposite ray coming
from within, and from this it follows immediately that the
state of radiation of the medium at all points on the surface is
the same as that within. The nature of the bounding surface
and the spacial extent of the medium are immaterial, and in a
stationary state of radiation~$\frakK_{\nu}$ is completely determined by the
nature of the medium for each temperature.

This law suffers a modification, however, in the special case
that the medium is absolutely diathermanous for a definite
frequency~$\nu$. It is then clear that the capacity for absorption
and also that for emission must be zero, because otherwise no
stationary state of radiation could exist, \ie, a medium emits
no color which it does not absorb. But equilibrium can then obviously
exist for every intensity of radiation of the frequency considered,
\ie, $\frakK_{\nu}$~is now undetermined and cannot be found without
knowledge of the initial conditions. An important example of
this is furnished by an absolute vacuum, which is diathermanous
for all frequencies. In a complete vacuum thermodynamic
equilibrium can therefore exist for each arbitrary intensity of
radiation and for each frequency, \ie, for each arbitrary distribution
of the spectral energy. From a general thermodynamic
point of view this indeterminateness of the properties of thermodynamic
states of equilibrium is explained through the presence
of numerous different relative maxima of the entropy, as in the
case of a vapor which is in a state of supersaturation. But
among all the different maxima there is a special maximum, the
%-----File: 085.png---\redacted\--------
absolute, which indicates stable equilibrium. In fact, we shall
see that in a diathermanous medium for each temperature there
exists a quite definite intensity of radiation, which is designated
as the stable intensity of radiation of the frequency~$\nu$ considered.
But for the present we shall assume for all frequencies
a finite capacity for absorption and for emission.

We consider now two homogeneous isotropic media in thermodynamic
equilibrium separated from each other by a plane
surface. Since the equilibrium will not be disturbed if one
imagines for the moment the surface of separation between the
two substances to be replaced by a surface quite non-transparent
to heat radiation, all of the foregoing laws hold for each of the
% [Illustration: Fig. 1.]
two substances individually. Let the specific intensity of radiation
of frequency~$\nu$, polarized in any arbitrary plane within the
first substance (the upper in Fig.~1)\footnote
  {From my lectures upon the theory of heat radiation (Leipzig, J.~A. Barth),
  wherein are to be found the details of the above somewhat abbreviated
  calculations.},
be~$\frakK_{\nu}$ and that within the
second substance~${\frakK_{\nu}}'$ (we shall in general designate with a dash
%-----File: 086.png---\redacted\--------
those quantities which refer to the second substance). Both
quantities $\frakK_{\nu}$~and~${\frakK_{\nu}}'$, besides depending upon the temperature
and the frequency, depend only upon the nature of the two substances,
and, in fact, these values of the intensity of radiation
hold quite up to the boundary surface between the substances,
and are therefore independent of the properties of this surface.

\vspace{2\baselineskip}

\pngcent{illo085.png}{1263}

\vspace{2\baselineskip}

Each ray from the first medium is split into two rays at the
boundary surface: the reflected and the transmitted. The directions
of these two rays vary according to the angle of incidence
and the color of the incident ray, and, in addition, the
intensity varies according to its polarization. If we denote
by~$\rho$ (the reflection coefficient) the amount of the reflected
energy of radiation and consequently by~$1 - \rho$ the amount of
transmitted energy with respect to the incident energy, then $\rho$~depends
upon the angle of incidence, upon the frequency and
upon the polarization of the incident ray. Similar remarks hold
for~$\rho'$, the reflection coefficient for a ray from the second
medium, upon meeting the boundary surface.

Now the energy of a monochromatic plane polarized ray of
frequency~$\nu$ proceeding from an element~$d\sigma$ of the boundary
surface within the elementary cone~$d\Omega$ in a direction toward the
first medium (see the feathered arrow at the left in Fig.~1) is
for the time~$dt$, in accordance with \Eq{(28)}~and~\Eq{(29)}:
\[
dt · d\sigma · \cos \theta · d\Omega · \frakK_{\nu} d\nu,
\Tag{(31)}
\]
where
\[
d\Omega = \sin \theta d\theta d\varphi.
\Tag{(32)}
\]
This energy is furnished by the two rays which, approaching the
surface from the first and the second medium respectively, are
reflected and transmitted respectively at the surface element~$d\sigma$
in the same direction. (See the unfeathered arrows. The surface
element~$d\sigma$ is indicated only by the point~$0$.) The first ray proceeds
in accordance with the law of reflection within the symmetrically
drawn elementary cone~$d\Omega$: the second approaches
the surface within the elementary cone
%-----File: 087.png---\redacted\--------
\[
d\Omega' = \sin \theta' d\theta' d\varphi',
\Tag{(33)}
\]
where, in accordance with the law of refraction,
\[
\varphi' = \varphi\quad \text{and}\quad
\frac{\sin \theta}{\sin \theta'} = \frac{q}{q'}.
\Tag{(34)}
\]
We now assume that the ray is either polarized in the plane of
incidence or perpendicular to this plane, and likewise for the
two radiations out of whose energies it is composed. The radiation
coming from the first medium and reflected from~$d\sigma$ contributes
the energy:
\[
\rho · dt · d\sigma \cos \theta · d\Omega · \frakK_{\nu} d\nu,
\Tag{(35)}
\]
and the radiation coming from the second medium and transmitted
through $d\sigma$ contributes the energy:
\[
(1 - \rho') · dt · d\sigma \cos \theta' · d\Omega' · {\frakK_{\nu}}' d\nu.
\Tag{(36)}
\]
The quantities $dt$,~$d\sigma$,~$\nu$, and~$d\nu$ are here written without the
accent, since they have the same values in both media.

Adding the expressions \Eq{(35)}~and~\Eq{(36)} and placing the sum
equal to the expression~\Eq{(31)}, we obtain:
\[
\rho \cos \theta d\Omega \frakK_{\nu}
  + (1 - \rho') \cos \theta' d\Omega' {\frakK_{\nu}}'
  = \cos \theta d\Omega \frakK_{\nu}.
\]
Now, in accordance with~\Eq{(34)}:
\[
\frac{\cos \theta d\theta}{q} = \frac{\cos \theta' d\theta'}{q'},
\]
and further, taking note of \Eq{(32)}~and~\Eq{(33)}:
\[
d\Omega' \cos \theta' = d\Omega \cos \theta · \frac{q'^{2}}{q^{2}},
\]
and it follows that:
\[
\rho \frakK_{\nu} + (1 - \rho') \frac{q'^{2}}{q^{2}} {\frakK_{\nu}}' = \frakK_{\nu}
\]
or:
\[
\frac{\frakK_{\nu}}{{\frakK_{\nu}}'} · \frac{q^{2}}{q'^{2}} = \frac{1 - \rho'}{1 - \rho}.
\]
%-----File: 088.png---\redacted\--------

In the last equation the quantity on the left is independent
of the angle of incidence~$\theta$ and of the kind of polarization, consequently
the quantity upon the right side must also be independent
of these quantities. If one knows the value of these
quantities for a single angle of incidence and for a given kind of
polarization, then this value is valid for all angles of incidence
and for all polarizations. Now, in the particular case that the
rays are polarized at right angles to the plane of incidence and
meet the bounding surface at the angle of polarization,
\[
\rho = 0\quad \text{and}\quad \rho' = 0.
\]
Then the expression on the right will be equal to~$1$, and therefore
it is in general equal to~$1$, and we have always:
\[
\rho = \rho',\quad q^{2} \frakK_{\nu} = q'^{2} {\frakK_{\nu}}'.
\Tag{(37)}
\]
The first of these two relations, which asserts that the coefficient
of reflection is the same for both sides of the boundary surface,
constitutes the special expression of a general reciprocal law,
first announced by Helmholz, whereby the loss of intensity which
a ray of given color and polarization suffers on its path through
any medium in consequence of reflection, refraction, absorption,
and dispersion is exactly equal to the loss of intensity which a ray
of corresponding intensity, color and polarization suffers in
passing over the directly opposite path. It follows immediately
from this that the radiation meeting a boundary surface between
two media is transmitted or reflected equally well from both
sides, for every color, direction and polarization.

The second relation,~\Eq{(37)}, brings into connection the radiation
intensities originating in both substances. It asserts that in
thermodynamic equilibrium the specific intensities of radiation
of a definite frequency in both media vary inversely as the square
of the velocities of propagation, or directly as the squares of the
refractive indices. We may therefore write
\[
q^{2} \frakK_{\nu} = F(\nu, T),
\]
%-----File: 089.png---\redacted\--------
wherein $F$~denotes a universal function depending only upon $\nu$~and~$T$,
the discovery of which is one of the chief problems of the
theory.

Let us fix our attention again on the case of a diathermanous
medium. We saw above that in a medium surrounded by a
non-transparent shell which for a given color is diathermanous
equilibrium can exist for any given intensity of radiation of this
color. But it follows from the second law that, among all the
intensities of radiation, a definite one, namely, that corresponding
to the absolute maximum of the total entropy of the system,
must exist, which characterizes the absolutely stable equilibrium
of radiation. We now see that this indeterminateness is eliminated
by the last equation, which asserts that in thermodynamic
equilibrium the product~$q^{2}\frakK_{\nu}$ is a universal function. For it
results immediately therefrom that there is a definite value of~$\frakK_{\nu}$
for every diathermanous medium which is thus differentiated
from all other values. The physical meaning of this value is
derived directly from a consideration of the way in which this
equation was derived: it is that intensity of radiation which
exists in the diathermanous medium when it is in thermodynamic
equilibrium while in contact with a given absorbing and emitting
medium. The volume and the form of the second medium is
immaterial; in particular, the volume may be taken arbitrarily
small.

For a vacuum, the most diathermanous of all media, in which
the velocity of propagation $q = c$ is the same for all rays, we can
therefore express the following law: The quantity
\[
\frakK_{\nu} = \frac{1}{c^{2}} F(\nu, T)
\Tag{(38)}
\]
denotes that intensity of radiation which exists in any complete
vacuum when it is in a stationary state as regards exchange of
radiation with any absorbing and emitting substance, whose
amount may be arbitrarily small. This quantity~$\frakK_{\nu}$ regarded
as a function of~$\nu$ gives the so-called normal energy spectrum.
%-----File: 090.png---\redacted\--------

Let us consider, therefore, a vacuum surrounded by given
emitting and absorbing bodies of uniform temperature. Then,
in the course of time, there is established therein a normal energy
radiation~$\frakK_{\nu}$ corresponding to this temperature. If now $\rho_{\nu}$~be
the reflection coefficient of a wall for the frequency~$\nu$, then of
the radiation~$\frakK_{\nu}$ falling upon the wall, the part~$\rho_{\nu} \frakK_{\nu}$ will be reflected.
On the other hand, if we designate by~$E_{\nu}$ the emission
coefficient of the wall for the same frequency~$\nu$, the total radiation
proceeding from the wall will be:
\[
\rho_{\nu} \frakK_{\nu} + E_{\nu} = \frakK_{\nu},
\]
since each bundle of rays possesses in a stationary state the intensity~$\frakK_{\nu}$.
From this it follows that:
\[
\frakK_{\nu} = \frac{E_{\nu}}{1 - \rho_{\nu}},
\Tag{(39)}
\]
\ie, the ratio of the emission coefficient~$E_{\nu}$ to the capacity for
absorption $(1-\rho_{\nu})$ of a given substance is the same for all
substances and equal to the normal intensity of radiation for
each frequency (Kirchoff). For the special case that $\rho_{\nu}$~is equal
to~$0$, \ie, that the wall shall be perfectly black, we have:
\[
\frakK_{\nu} = E_{\nu},
\]
that is, the normal intensity of radiation is exactly equal to the
emission coefficient of a black body. Therefore the normal
radiation is also called ``black radiation.'' Again, for any given
body, in accordance with~\Eq{(39)}, we have:
\[
E_{\nu} < \frakK_{\nu},
\]
\ie, the emission coefficient of a body in general is smaller than
that of a black body. Black radiation, thanks to W.~Wien and
O.~Lummer, has been made possible of measurement, through
a small hole bored in the wall bounding the space considered.

We proceed now to the treatment of the problem of determining
the specific intensity~$\frakK_{\nu}$ of black radiation in a vacuum,
%-----File: 091.png---\redacted\--------
as regards its dependence upon the frequency~$\nu$ and the temperature~$T$.
In the treatment of this problem it will be necessary
to go further than we have previously done into those processes
which condition the production and destruction of heat rays;
that is, into the question regarding the act of emission and that
of absorption. On account of the complicated nature of these
processes and the difficulty of bringing some of the details into
connection with experience, it is certainly quite out of the question
to obtain in this manner any reliable results if the following
law cannot be utilized as a dependable guide in this domain: a
vacuum surrounded by reflecting walls in which arbitrary
emitting and absorbing bodies are distributed in any given
arrangement assumes in the course of time the stationary state
of black radiation, which is completely determined by a single
parameter, the temperature, and which, in particular, does not
depend upon the number, the properties and the arrangement of
the bodies. In the investigation of the properties of the state
of black radiation the nature of the bodies which are supposed
to be in the vacuum is therefore quite immaterial, and it is certainly
immaterial whether such bodies actually exist anywhere
in nature, so long as their existence and their properties are
compatible throughout with the laws of electrodynamics and of
thermodynamics. As soon as it is possible to associate with
any given special kind and arrangement of emitting and absorbing
bodies a state of radiation in the surrounding vacuum which
is characterized by absolute stability, then this state can be no
other than that of black radiation. Making use of the freedom
furnished by this law, we choose among all the emitting and
absorbing systems conceivable, the most simple, namely, a single
oscillator at rest, consisting of two poles charged with equal
quantities of electricity of opposite sign which are movable
relative to each other in a fixed straight line, the axis of the
oscillator. The state of the oscillator is completely determined
by its moment,~$f(t)$; \ie,~by the product of the electric charge of
the pole on the positive side of the axis into the distance between
%-----File: 092.png---\redacted\--------
the poles, and by its differential quotient with regard to the time:
\[
\frac{df(t)}{dt} = \dot{f}(t).
\]
The energy of the oscillator is of the following simple form:
\[
U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2},
\Tag{(40)}
\]
wherein $K$~and~$L$ denote positive constants which depend upon
the nature of the oscillator in some manner into which we need
not go further at this time.

If, in the vibrations of the oscillator, the energy~$U$ remain absolutely
constant, we should have: $dU = 0$ or:
\[
K f(t) + L \ddot{f}(t) = 0,
\]
and from this there results, as a general solution of the differential
equation, a pure periodic vibration:
\[
f = C \cos (2\pi \nu_{0} t - \theta),
\]
wherein $C$~and~$\theta$ denote the integration constants and $\nu_{0}$~the
number of vibrations per unit of time:
\[
\nu_{0} = \frac{1}{2\pi} \sqrt{\frac{K}{L}}.
\Tag{(41)}
\]
Such an oscillator vibrating periodically with constant energy
would neither be influenced by the electromagnetic field surrounding
it, nor would it exert any external actions due to radiation.
It could therefore have no sort of influence on the heat
radiation in the surrounding vacuum.

In accordance with the theory of Maxwell, the energy of
vibration~$U$ of the oscillator by no means remains constant in
general, but an oscillator by virtue of its vibrations sends out
spherical waves in all directions into the surrounding field and,
in accordance with the principle of conservation of energy, if no
actions from without are exerted upon the oscillator, there must
%-----File: 093.png---\redacted\--------
necessarily be a loss in the energy of vibration and, therefore, a
damping of the amplitude of vibration is involved. In order to
find the amount of this damping we calculate the quantity of
energy which flows out through a spherical surface with the
oscillator at the center, in accordance with the law of Poynting.
However, we may not place the energy flowing outwards in
accordance with this law through the spherical surface in an
infinitely small interval of time~$dt$ equal to the energy radiated
in the same time interval from the oscillator. For, in general,
the electromagnetic energy does not always flow in the outward
direction, but flows alternately outwards and inwards, and
we should obtain in this manner for the quantity of the radiation
outwards, values which are alternately positive and negative,
and which also depend essentially upon the radius of the
supposed sphere in such manner that they increase toward
infinity with decreasing radius---which is opposed to the fundamental
conception of radiated energy. This energy will, moreover,
be only found independent of the radius of the sphere
when we calculate the total amount of energy flowing outwards
through the surface of the sphere, not for the time element~$dt$,
but for a sufficiently large time. If the vibrations are purely
periodic, we may choose for the time a period; if this is not
the case, which for the sake of generality we must here assume,
it is not possible to specify a~priori any more general criterion
for the least possible necessary magnitude of the time than that
which makes the energy radiated essentially independent of the
radius of the supposed sphere.

In this way we succeed in finding for the energy emitted from
the oscillator in the time from $t$ to $t + \frakT$ the following expression:
\[
\frac{2}{3c^{3}} \int_{t}^{t + \frakT} \ddot{f}^{2}(t) dt.
\]
If now, the oscillator be in an electromagnetic field which has the
electric component~$\frakE_{z}$ at the oscillator in the direction of its axis,
%-----File: 094.png---\redacted\--------
then the energy absorbed by the oscillator in the same time is:
\[
\int_{t}^{t + \frakT} \frakE_{z} \dot{f} · dt.
\]
Hence, the principle of conservation of energy is expressed in
the following form:
\[
\int_{t}^{t + \frakT} \left(\frac{dU}{dt} + \frac{2}{3c^{3}} \ddot{f}^{2} - \frakE_{z} \dot{f}\right) dt = 0.
\]
This equation, together with the assumption that the constant
\[
\frac{4\pi^{2} \nu_{0}}{3c^{3} L} = \sigma
\Tag{(42)}
\]
is a small number, leads to the following linear differential equation
for the vibrations of the oscillator:
\[
Kf + L\ddot{f} - \frac{2}{3c^{3}} \dddot{f} = \frakE_{z}.
\Tag{(43)}
\]
In accordance with what precedes, in so far as the oscillator is
excited into vibrations by an external field~$\frakE_{z}$, one may designate
it as a resonator which possesses the natural period~$\nu_{0}$ and the
small logarithmic decrement~$\sigma$. The same equation may be
obtained from the electron theory, but I have considered it an
advantage to derive it in a manner independent of any hypothesis
concerning the nature of the resonator.

Now, let the resonator be in a vacuum filled with stationary
black radiation of specific intensity~$\frakK_{\nu}$. How, then, does the
mean energy~$U$ of the resonator in a state of stationary vibration
depend upon the specific intensity of radiation~$\frakK_{\nu_{0}}$ with the natural
period~$\nu_{0}$ of the corresponding color? It is this question which
we have still to consider today. Its answer will be found by expressing
on the one hand the energy of the resonator~$U$ and on
the other hand the intensity of radiation~$\frakK_{\nu_{0}}$ by means of the
component~$\frakE_{z}$ of the electric field exciting the resonator. Now
however complicated this quantity may be, it is capable of
%-----File: 095.png---\redacted\--------
development in any case for a very large time interval, from
$t = 0$ to $t = \frakT$, in the Fourier's series:
\[
\frakE_{z} = \sum\limits_{n = 1}^{n = \infty} C_{n} \cos \left(\frac{2\pi n t}{\frakT} - \theta_{n}\right),
\Tag{(44)}
\]
and for this same time interval~$\frakT$ the moment of the resonator
in the form of a Fourier's series may be calculated as a function
of~$t$ from the linear differential equation~\Eq{(43)}. The initial
condition of the resonator may be neglected if we only consider
such times~$t$ as are sufficiently far removed from the origin of
time $t = 0$.

If it be now recalled that in a stationary state of vibration
the mean energy~$U$ of the resonator is given, in accordance with
\Eq{(40)},~\Eq{(41)} and~\Eq{(42)}, by:
\[
U = K \bar{f}^{2} = \frac{16\pi^{4} \nu_{0}{}^{3}}{3 \sigma c^{3}} \bar{f}^{2},
\]
it appears after substitution of the value of~$f$ obtained from the
differential equation~\Eq{(43)} that:
\[
U = \frac{3 c^{3}}{64\pi^{2} \nu_{0}{}^{2}} \frakT \bar{C}_{n0}{}^{2},
\Tag{(45)}
\]
wherein $\bar{C}_{n0}{}^{2}$~denotes the mean value of~$C_{n}$ for all the series of
numbers~$n$ which lie in the neighborhood of the value~$\nu_{0} \frakT$, \ie,
for which $\nu_{0} \frakT$~is approximately~$= 1$.

Now let us consider on the other hand the intensity of black
radiation, and for this purpose proceed from the space density
of the total radiation. In accordance with~\Eq{(30)}, this is:
\[
\epsilon = \frac{8\pi}{c} \int_{0}^{\infty} \frakK_{\nu} d\nu
  = \frac{1}{8\pi} (\bar{\frakE}_{x}{}^{2} + \bar{\frakE}_{y}{}^{2} + \bar{\frakE}_{z}{}^{2}
                  + \bar{\frakH}_{x}{}^{2} + \bar{\frakH}_{y}{}^{2} + \bar{\frakH}_{z}{}^{2}),
\Tag{(46)}
\]
and therefore, since the radiation is isotropic, in accordance with~\Eq{(44)}:
\[
\frac{8\pi}{c} \int_{0}^{\infty} \frakK_{\nu} d\nu
  = \frac{3}{4\pi} \bar{\frakE}_{z}{}^{2}
  = \frac{3}{8\pi} \sum\limits_{n = 1}^{n = \infty} C_{n}{}^{2}.
\]
%-----File: 096.png---\redacted\--------
If we write $\Delta n/\frakT$ on the left instead of~$d\nu$, where $\Delta n$~is a large
number, we get:
\[
\frac{8\pi}{c} \sum\limits_{n = 1}^{n = \infty} \frakK_{v} \frac{\Delta n}{\frakT}
  = \frac{3}{8\pi} \sum\limits_{n = 1}^{n = \infty} C_{n}{}^{2},
\]
and obtain then by ``spectral'' division of this equation:
\[
\frac{8\pi}{c} \frakK_{\nu_{0}} \frac{\Delta n}{\frakT}
  = \frac{3}{8\pi} \sum\limits_{n_{0} - (\Delta n/2)}^{n_{0} + (\Delta n/2)} C_{n}{}^{2},
\]
and, if we introduce again the mean value
\[
\frac{1}{\Delta n} · \sum\limits_{n_{0} - (\Delta n/2)}^{n_{0} + (\Delta n/2)} C_{n}{}^{2} = \bar{C}_{n0}{}^{2},
\]
we then get:
\[
\frakK_{\nu_{0}} = \frac{3 c \frakT}{64\pi^{2}} · \bar{C}_{n 0}.
\]
By comparison with~\Eq{(45)} the relation sought is now found:
\[
\frakK_{\nu_{0}} = \frac{\nu_{0}{}^{2}}{c^{2}} U,
\Tag{(47)}
\]
which is striking on account of its simplicity and, in particular,
because it is quite independent of the damping constant~$\sigma$ of the
resonator.

This relation, found in a purely electrodynamic manner,
between the spectral intensity of black radiation and the energy
of a vibrating resonator will furnish us in the next lecture, with
the aid of thermodynamic considerations, the necessary means of
attack in deriving the temperature of black radiation together
with the distribution of energy in the normal spectrum.
%-----File: 097.png---\redacted\--------


\Chapter{Sixth Lecture.}{Heat Radiation. Statistical Theory.}\label{Lect6}

Following the preparatory considerations of the last lecture
we shall treat today the problem which we have come to recognize
as one of the most important in the theory of heat radiation:
the establishment of that universal function which governs the
energy distribution in the normal spectrum. The means for the
solution of this problem will be furnished us through the calculation
of the entropy~$S$ of a resonator placed in a vacuum filled
with black radiation and thereby excited into stationary vibrations.
Its energy~$U$ is then connected with the corresponding
specific intensity~$\frakK_{\nu}$ and its natural frequency~$\nu$ in the radiation
of the surrounding field through equation~\Eq{(47)}:
\[
\frakK_{\nu} = \frac{\nu^{2}}{c^{2}} U.
\Tag{(48)}
\]
When $S$~is found as a function of~$U$, the temperature~$T$ of the
resonator and that of the surrounding radiation will be given by:
\[
\frac{dS}{dU} = \frac{1}{T},
\Tag{(49)}
\]
and by elimination of~$U$ from the last two equations, we then
find the relationship among $\frakK_{\nu}$,~$T$ and~$\nu$.

In order to find the entropy~$S$ of the resonator we will utilize
the general connection between entropy and probability, which
we have extensively discussed in the previous lectures, and inquire
then as to the existing probability that the vibrating resonator
possesses the energy~$U$. In accordance with what we have seen
in connection with the elucidation of the second law through
%-----File: 098.png---\redacted\--------
atomistic ideas, the second law is only applicable to a physical
system when we consider the quantities which determine the
state of the system as mean values of numerous disordered
individual values, and the probability of a state is then equal
to the number of the numerous, a~priori equally probable, complexions
which make possible the realization of the state. Accordingly,
we have to consider the energy~$U$ of a resonator
placed in a stationary field of black radiation as a constant mean
value of many disordered independent individual values, and
this procedure agrees with the fact that every measurement of
the intensity of heat radiation is extended over an enormous
number of vibration periods. The entropy of a resonator is
then to be calculated from the existing probability that the energy
of the radiator possesses a definite mean value~$U$ within a certain
time interval.

In order to find this probability, we inquire next as to the
existing probability that the resonator at any fixed time possesses
a given energy, or in other words, that that point (the
state point) which through its coordinates indicates the state of
the resonator falls in a given ``state domain.'' At the conclusion
of the third lecture (p.~\pageref{png67lab3}) we saw in general that this probability
is simply measured through the magnitude of the corresponding
state domain:
\[
\int d\varphi · d\psi,
\]
in case one employs as coordinates of state the general coordinate~$\varphi$
and the corresponding momentum~$\psi$. Now in general, the
energy of the resonator, in accordance with~\Eq{(40)}, is:
\[
U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2}.
\]
If we choose $f$ as the general coordinate~$\varphi$ and put, therefore,
$\varphi = f$, then the corresponding impulse~$\psi$ is equal
\[
\frac{\dd U}{\dd \dot{f}} = L \dot{f},
\]
%-----File: 099.png---\redacted\--------
and the energy~$U$ expressed as a function of $\varphi$~and~$\psi$ is:
\[
U = \tfrac{1}{2} K\varphi^{2} + \frac{1}{2} \frac{\psi^{2}}{L}.
\]
If now we desire to find the existing probability that the energy
of a resonator shall lie between $U$ and $U + \Delta U$, we have to
calculate the magnitude of that state domain in the $(\varphi, \psi)$-plane
which is bounded by the curves $U = \const.$\ and $U + \Delta U = \const.$
These two curves are similar and similarly placed ellipses and
the portion of surface bounded by them is equal to the difference
of the areas of the two ellipses. The areas are respectively $U/\nu$
and $(U + \Delta U)/\nu$; consequently, the magnitude sought for the
state domain is:~$\Delta U/\nu.$ Let us now consider the whole state
plane so divided into elementary portions by a large number of
ellipses, such that the annular areas between consecutive ellipses
are equal to each other; \ie, so that:
\[
\frac{\Delta U}{\nu} = \const = h.
\]
We thus obtain those portions~$\Delta U$ of the energy which correspond
to equal probabilities and which are therefore to be designated
as the energy elements:
\[
\epsilon = \Delta U = h \nu.
\Tag{(50)}
\]

If the determination of the elementary domains is effected in
a manner quite similar to that employed in the kinetic gas theory,
there exist, with respect to the relationships there found, very
notable differences. In the first place, the state of the physical
system considered here, the resonator, does not depend as there
upon the coordinates and the velocities, but upon the energy
only, and this circumstance necessitates that the entropy of a
state depend, not upon the distribution of the state quantities
$\varphi$~and~$\psi$, but only upon the energy~$U$. A further difference
consists in this, that we have to do in the case of molecules with
spacial mean values, but in the case of radiation with mean values
%-----File: 100.png---\redacted\--------
as regards time. But this distinction may be disregarded when
we reflect that the mean time value of the energy~$U$ of a given
resonator is obviously identical with the mean space value at a
given instant of time of a great number~$N$ of similar resonators
distributed in the same stationary field of radiation. Of course
these resonators must be placed sufficiently far apart in order
not directly to influence one another. Then the total energy of
all the resonators:
\[
U_{N} = NU
\Tag{(51)}
\]
is quite irregularly distributed among all the individual resonators,
and we have referred back the disorder as regards time to a
disorder as regards space.

We are now concerned with the probability~$W$ of the state
determined by the energy~$U_{N}$ of the $N$~resonators placed in the
same stationary field of radiation; \ie,~with the number of
individual arrangements or complexions which correspond to the
distribution of energy~$U_{N}$ among the $N$~resonators. With this
in view, we subdivide the given total energy~$U_{N}$ into its elements~$\epsilon$
so that:
\[
U_{N} = P \epsilon.
\Tag{(52)}
\]
These $P$~energy elements are to be distributed in every possible
manner among the $N$~resonators. Let us consider, then, the
$N$~resonators to be numbered and the figures written beside
one another in a series, and in such manner that the number
of times each figure appears is equal to the number of energy
elements which fall upon the corresponding resonator. Then
we obtain through such a number series a representation of a
fixed complexion, in which with each individual resonator there
is associated a definite energy. For example, if there are $N = 4$
resonators and $P = 6$ energy elements present, then one of
the possible complexions is represented by the number series
\[
1\quad 1\quad 3\quad 3\quad 3\quad 4
\]
which asserts that the first resonator contains two, the second~$0$,
%-----File: 101.png---\redacted\--------
the third~$3$, and the fourth $1$~energy element. The totality of
numbers in the series is~$6$, equal to the number of the energy
elements present. The arrangement of figures in the series is
immaterial for any complexion, since the mere interchange of
figures does not change the energy of a given resonator. The
number of all the possible different complexions is therefore
equal to the number of possible ``combinations with repetition''
of $4$~elements with $6$~classes:
\[
W = \frac{(4 + 6 - 1)!}{(4 - 1)!\;6!} = \frac{9!}{3!\;6!} = 84,
\]
or, in our general case the probability sought is:
\[
W = \frac{(N + P - 1)!}{(N - 1)!\;P!}.
\]
We obtain, therefore, for the entropy~$S_{N}$ of the resonator system,
in accordance with equation~\Eq{(12)}, since $N$~and~$P$ are large
numbers,
\[
S_{N} = k \log \frac{(N + P)!}{N!\;P!}
\]
and with the aid of Sterling's formula~\Eq{(16)}:
\[
S_{N} = k \{(N + P) \log (N + P) - N \log N - P \log P\}.
\]
If, in accordance with~\Eq{(52)}, we now write $U_{N}/\epsilon$ for~$P$, $NU$~for $U_{N}$
in accordance with~\Eq{(51)}, and $h\nu$~for~$\epsilon$, in accordance with~\Eq{(50)},
we obtain, after an easy transformation, for the mean entropy
of a single resonator:
\[
\frac{S_{N}}{N} = S
  = k \left\{\left(1 + \frac{U}{h\nu}\right) \log \left(1 + \frac{U}{h\nu}\right)
           - \frac{U}{h\nu} \log \frac{U}{h\nu}\right\}
\]
as the solution of the problem in hand.

We will now introduce the temperature~$T$ of the resonator,
and will express through $T$ the energy~$U$ of the resonator and
also the intensity~$\frakK_{\nu}$ of the heat radiation related to it through a
%-----File: 102.png---\redacted\--------
stationary state of energy exchange. For this purpose we utilize
equation~\Eq{(49)} and obtain then for the energy of the resonator:
\[
U = \frac{h\nu}{e^{h\nu/kT} - 1}.
\]
It is to be observed that we have not here to do with a uniform
distribution of energy (cf.\ p.~\pageref{png78lab1}) among the various resonators.

For the specific intensity of the monochromatic plane polarized
ray of frequency~$\nu$, we have, in accordance with~\Eq{(48)}:
\[
\frakK_{\nu} = \frac{h\nu^{3}}{c^{2}} · \frac{1}{e^{h\nu/kT} - 1}.
\Tag{(53)}
\]
This expression furnishes for each temperature~$T$ the energy
distribution in the normal spectrum of a black body. A comparison
with equation~\Eq{(38)} of the last lecture furnishes us then
with the universal function:
\[
F(\nu, T) = \frac{h\nu^{3}}{e^{h\nu/kT} - 1}.
\]

If we refer the specific intensity of a monochromatic ray, not to
the frequency~$\nu$, but, as is commonly done in experimental physics,
to the wave length~$\lambda$, then, since between the absolute values of
$d\nu$~and~$d\lambda$ the relation exists:
\[
|d\nu| = \frac{c · |d\lambda|}{\lambda^{2}},
\]
we obtain from
\[
E_{\lambda} |d\lambda| = \frakK_{\nu} |d\nu|,
\]
the relation:
\[
E_{\lambda} = \frac{c^{2}h}{\lambda^{5}} · \frac{1}{e^{ch/k \lambda T} - 1}
\Tag{(54)}
\]
as the intensity of a monochromatic plane polarized ray of wave
length~$\lambda$ is emitted normally to the surface of a black
body in a vacuum at temperature~$T$. For small values of~$\lambda T$
%-----File: 103.png---\redacted\--------
\Eq{(54)}~reduces to:
\[
E_{\lambda} = \frac{c^{2} h}{\lambda^{5}} · e^{-(ch/k\lambda T)},
\Tag{(55)}
\]
which expresses Wien's Displacement Law. For large values of~$\lambda T$
on the other hand, there results from~\Eq{(54)}:
\[
E_{\lambda} = \frac{ckT}{\lambda^{4}},
\Tag{(56)}
\]
a relation first established by Lord Rayleigh and which we may
here designate as the Rayleigh Law of Radiation.

From equation~\Eq{(30)}, taking account of~\Eq{(53)}, we obtain for the
space density of black radiation in a \label{png103lab1}vacuum:
\[
\epsilon = \frac{48\pi h}{c^{3}} \left(\frac{kT}{h}\right)^{4} · \alpha = aT^{4},
\]
wherein
\[
\alpha = 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + \cdots = 1.0823.
\]
The Stefan-Boltzmann law is hereby expressed. In accordance
with the measurements of Kurlbaum, we have the constant
\[
a = \frac{48\pi k^{4}}{c^{3} h^{3}} · \alpha = 7.061 · 10^{-15} \frac{\erg}{\cm^{3} \deg^{4}}.
\]

For that wave length~$\lambda_{m}$ which corresponds in the spectrum
of black radiation to the maximum intensity of radiation~$E_{\lambda}$
we have from equation~\Eq{(54)}:
\[
\left(\frac{dE_{\lambda}}{d\lambda}\right)_{\lambda = \lambda_{m}} = 0.
\]
Carrying out the differentiation, we get, after putting for brevity:
\[
\frac{ch}{k\lambda_{m} T} = \beta,\quad
e^{-\beta} + \frac{\beta}{5} - 1 = 0.
\]
The root of this transcendental equation is:
\[
\beta = 4.9651;
\]
%-----File: 104.png---\redacted\--------
and $\lambda_{m} T = ch/k\beta = b$ is a constant (Wien's Displacement Law).
In accordance with the measurements of O.~Lummer and E.~Pringsheim,
\[
b = 0.294\ \cm · \deg.
\]
From this there follow the numerical values
\[
k = 1.346 · 10^{-16} \frac{\erg}{\deg},\quad \text{and}\quad
h = 6.548 · 10^{-27} \erg · \sec.
\]
The value found for~$k$ easily permits of the specification numerically,
in the C.G.S. system, of the general connection between
entropy and probability, as expressed through the universal
equation~\Eq{(12)}. Thus, quite in general, the entropy of a physical
system is:
\[
S = 1.346 · 10^{-16} \ln W.
\]

In the application to the kinetic gas theory we obtain from
equation~\Eq{(24)} for the ratio of the molecular mass to the mol mass:
\label{png104lab1}
\[
\omega = \frac{k}{R} = 1.62 · 10^{-24},
\]
\ie, to one mol there corresponds $1/\omega = 6.175 · 10^{23}$ molecules,
where it is supposed that the mol of oxygen
\[
O_{2} = 32\text{g}.
\]
Accordingly, the number of molecules contained in $1$~cu.~cm.\ of
an ideal gas at $0°$~Cels.\ and at atmospheric pressure is:
\[
N = 2.76 · 10^{19}.
\]
The mean kinetic energy of the progressive motion of a molecule
at the absolute temperature $T = 1$ in the absolute C.G.S. system,
in accordance with~\Eq{(27)}, is:
\[
L = \tfrac{3}{2} k = 2.02 · 10^{-16}.
\]
In general, the mean kinetic energy of progressive motion of a
%-----File: 105.png---\redacted\--------
molecule is expressed by the product of this number and the
absolute temperature~$T$.

The elementary quantum of electricity, or the free electric
charge of a monovalent ion or electron, in electrostatic measure is:
\[
e = \omega · 9658 · 3 · 10^{10} = 4.69 · 10^{-10}.
\]
This result stands in noteworthy agreement with the results of
the latest direct measurements of the electric elementary quantum
made by E.~Rutherford and H.~Geiger, and E.~Regener.

Even if the radiation formula~\Eq{(54)} here derived had shown itself
as valid with respect to all previous tests, the theory would still
require an extension as regards a certain point; for in it the
physical meaning of the universal constant~$h$ remains quite
unexplained. All previous attempts to derive a radiation formula
upon the basis of the known laws of electron theory, among which
the theory of J.~H. Jeans is to be considered as the most general
and exact, have led to the conclusion that $h$~is infinitely small,
so that, therefore, the radiation formula of Rayleigh possesses
general validity, but, in my opinion, there can be no doubt that
this formula loses its validity for short waves, and that the pains
which Jeans has taken to place\footnote
  {In that the walls used in the measurements of hollow space radiations
  must be diathermanous for the shortest waves.}
the blame for the contradiction
between theory and experiment upon the latter are unwarranted.

Consequently, there remains only the one conclusion, that
previous electron theories suffer from an essential incompleteness
which demands a modification, but how deeply this modification
should go into the structure of the theory is a question upon
which views are still widely divergent. J.~J. Thompson inclines
to the most radical view, as do J.~Larmor, A.~Einstein, and
with him I.~Stark, who even believe that the propagation of
electromagnetic waves in a pure vacuum does not occur precisely
in accordance with the Maxwellian field equations, but in
definite energy quanta~$h\nu$. I am of the opinion, on the other
hand, that at present it is not necessary to proceed in so revolutionary
%-----File: 106.png---\redacted\--------
a manner, and that one may come successfully through by
seeking the significance of the energy quantum~$h\nu$ solely in
the mutual actions with which the resonators influence one
another.\footnote
  {It is my intention to give a complete presentation of these relations in
  Volume~31 of the Annalen der~Physik.}
A definite decision with regard to these important
questions can only be brought about as a result of further
experience.
%-----File: 107.png---\redacted\--------


\Chapter{Seventh Lecture.}{General Dynamics. Principle of Least Action.}\label{Lect7}

Since I began three weeks ago today to depict for you the
present status of the system of theoretical physics and its
probable future development, I have continually sought to
bring out that in the theoretical physics of the future the most
important and the final division of all physical processes would
likely be into reversible and irreversible processes. In succeeding
lectures, with the aid of the calculus of probability and with the
introduction of the hypothesis of elementary disorder, we have
seen that all irreversible processes may be considered as reversible
elementary processes: in other words, that irreversibility does
not depend upon an elementary property of a physical process,
but rather depends upon the ensemble of numerous disordered
elementary processes of the same kind, each one of which individually
is completely reversible, and upon the introduction
of the macroscopic method of treatment. From this standpoint
one can say quite correctly that in the final analysis all processes
in nature are reversible. That there is herein contained no contradiction
to the principle regarding the irreversibility of processes
expressed in terms of the mean values of elementary processes
of macroscopic changes of state, I have demonstrated fully in
the third lecture. Perhaps it will be appropriate at this place
to interject a more general statement. We are accustomed in
physics to seek the explanation of a natural process by the method
of division of the process into elements. We regard each complicated
process as composed of simple elementary processes,
and seek to analyse it through thinking of the whole as the sum
of the parts. This method, however, presupposes that through
%-----File: 108.png---\redacted\--------
this division the character of the whole is not changed; in somewhat
similar manner each measurement of a physical process
presupposes that the progress of the phenomena is not influenced
by the introduction of the measuring instrument. We have
here a case in which that supposition is not warranted, and where
a direct conclusion with regard to the parts applied to the whole
leads to quite false results. If we divide an irreversible process
into its elementary constituents, the disorder and along with it
the irreversibility vanishes; an irreversible process must remain
beyond the understanding of anyone who relies upon the fundamental
law: that all properties of the whole must also be recognizable
in the parts. It appears to me as though a similar difficulty
presents itself in most of the problems of intellectual life.

Now after all the irreversibility in nature thus appears in a
certain sense eliminated, it is an illuminating fact that general
elementary dynamics has only to do with reversible processes.
Therefore we shall occupy ourselves in what follows with reversible
processes exclusively. That which makes this procedure
so valuable for the theory is the circumstance that all known
reversible processes, be they mechanical, electrodynamical or
thermal, may be brought together under a single principle which
answers unambiguously all questions regarding their behavior.
This principle is not that of conservation of energy; this holds, it
is true, for all these processes, but does not determine unambiguously
their behavior; it is the more comprehensive principle
of least action.

The principle of least action has grown upon the ground of
mechanics where it enjoys equal rank and regard with numerous
other principles; the principle of d'Alembert, the principle of
virtual displacement, Gauss's principle of least constraint, the
Lagrangian Equations of the first and second kind. All these
principles are equivalent to one another and therefore at bottom
are only different formularizations of the same laws; sometimes
one and sometimes another is the most convenient to use. But
the principle of least action has the decided advantage over all
%-----File: 109.png---\redacted\--------
the other principles mentioned in that it connects together in a
single equation the relations between quantities which possess,
not only for mechanics, but also for electrodynamics and for
thermodynamics, direct significance, namely, the quantities:
space, time and potential. This is the reason why one may
directly apply the principle of least action to processes other
than mechanical, and the result has shown that such applications,
as well in electrodynamics as in thermodynamics, lead to
the appropriate laws holding in these subjects. Since a representation
of a unified system of theoretical physics such as we
have here in mind must lay the chief emphasis upon as general
an interpretation as possible of physical laws, it is self evident
that in our treatment the principle of least action will be called
upon to play the principal rôle. I desire now to show how it is
applied in simple individual cases.

The general formularization of the principle of least action in
the interpretation given to it by Helmholz is as follows: among
all processes which may carry a certain arbitrarily given physical
system subject to given external actions from a given initial
position into a given final position in a given time, the process
which actually takes place in nature is that which is distinguished
by the condition that the integral
\[
\int_{t_{0}}^{t_{1}} (\delta H + A) dt = 0,
\Tag{(57)}
\]
wherein an arbitrary displacement of the independent coordinates
(and velocities) is denoted by the sign~$\delta$, and $A$~denotes the
infinitely small increase in energy (external work) which the
system experiences in the displacement~$\delta$. The function~$H$
is the kinetic potential. When we speak here of the positions,
the coordinates, and the velocities of the configuration, we understand
thereby, not only those special ones corresponding to mechanical
ideas, but also all the so-called generalized coordinates
with the quantities derived therefrom; and these may represent
equally well quantities of electricity, volumes, and the like.
%-----File: 110.png---\redacted\--------

In the applications which we shall now make of the principle
of least action, we must first decide as to whether the generalized
coordinates which determine the state of the system considered
are present in finite number or form a continuous infinite
manifold. We shall distinguish the examples here considered
in accordance with this viewpoint.


\Section{1.}{The Position (Configuration) is Determined by a Finite Number
of Coordinates.}

In ordinary mechanics this is actually the case in every system
of a finite number of material points or rigid bodies among whose
coordinates there exist arbitrary fixed equations of condition.
If we call the independent coordinates $\varphi_{1}$,~$\varphi_{2}$,~$\cdots$, then the
external work is:
\[
A = \Phi_{1} \delta \varphi_{1} + \Phi_{2} \delta \varphi_{2} + \cdots = \delta E,
\Tag{(58)}
\]
wherein $\Phi_{1}$,~$\Phi_{2}$,~$\cdots$ are the ``external force components'' which
correspond to the individual coordinates, and $E$~denotes the
energy of the system. Then the principle of least action is
expressed by:
\label{png110lab1}
\[
\int_{t_{0}}^{t_{1}} dt · \sum\limits_{1, 2, \cdots} \left(
   \frac{\dd H}{\dd \varphi_{1}} \delta \varphi_{1}
 + \frac{\dd H}{\dd \dot{\varphi}_{1}} \delta \dot{\varphi}_{1}
 + \Phi_{1} \delta \varphi_{1}\right) = 0.
\]
From this follow the equations of motion:
\[
\Phi_{1} - \frac{d}{dt} \left(\frac{\dd H}{\dd \dot{\varphi}_{1}}\right)
  + \frac{\dd H}{\dd \varphi_{1}} = 0,
\Tag{(59)}
\]
and so on for all the indices, $1$,~$2$,~$\cdots$. Through multiplication
of the individual equations by $\dot{\varphi}_{1}$,~$\dot{\varphi}_{2}$,~$\cdots$ addition and integration
with respect to time, there results the equation of conservation
of energy, whereby the energy~$E$ is given by the expression:
\[
E = \sum\limits_{1, 2, \cdots} \dot{\varphi}_{1} \frac{\dd H}{\dd \dot{\varphi}_{1}} - H.
\Tag{(60)}
\]
In ordinary mechanics $H = L - U$, if $L$~denote the kinetic and
%-----File: 111.png---\redacted\--------
$U$~the potential energy. Since $L$~is a homogeneous function of
the second degree with respect to the~$\dot{\varphi}$'s, it follows from~\Eq{(60)}
that:
\[
E = 2L - H = L + U.
\]
But this expression holds by no means in general.

We pass now to the consideration of the quasi-stationary
motion of a system of linear conductors carrying simple closed
galvanic currents. The state of the system is given by the
position and the velocities of the conductors and by the current
densities in each of the same. The coordinates referring
to the position of the first conductor may be represented by
$\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$, corresponding designations holding for the
remaining conductors. We inquire now as to the increase of
energy or the external work,~$A$, which corresponds to a virtual
displacement of all coordinates. Energy may be conveyed to
the system through mechanical actions and through electromagnetic
induction as well. The former corresponds to mechanical
work, the latter to electromotive work. The former will
be of the familiar form:
\label{png111lab1}
\[
\Phi_{1} \delta\varphi_{1}
  + {\Phi_{1}}' \delta\varphi_{1} + \cdots
  + \Phi_{2} \delta\varphi_{2} + \cdots.
\]
If we denote by $E_{1}$,~$E_{2}$,~$\cdots$ the electromotive forces which
are induced in the individual conductors through external
agencies (\eg,~moving magnets which do not belong to the
system), then the electromotive work done from outside upon
the currents in the conductors of the system is:
\[
E_{1} \delta\epsilon_{1} + E_{2} \delta\epsilon_{2} + \cdots,
\]
if $\delta\epsilon_{1}$,~$\delta\epsilon_{2}$,~$\cdots$ denote the quantities of electricity which pass
through cross sections of the conductors due to infinitely small
virtual currents. The finite current densities will then be denoted
by $\dot{\epsilon}_{1}$,~$\dot{\epsilon}_{2}$,~$\cdots$. The electrical state of the first conductor is
thus determined in general by the current density~$\dot{\epsilon}_{1}$, the
mechanical state (position and velocity) by the coordinates
%-----File: 112.png---\redacted\--------
$\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$ and the corresponding velocities $\dot{\varphi}_{1}$,~$\dot{\varphi}_{1}'$, $\dot{\varphi}_{1}''$,~$\cdots$.
The coordinates $\epsilon_{1}$,~$\epsilon_{2}$,~$\cdots$ are so-called ``cyclical'' coordinates,
since the state does not depend upon their momentary values,
but only upon their differential quotients with respect to time,
just as, for example, the state of a body rotatable about an axis
of symmetry depends only upon the angular velocity, and not
upon the angle of rotation. The scheme of notation adopted
permits of the direct application of the above formularization
of the principle of least action to the case here considered.
Thus $H = H_{\phi} + H_{\epsilon}$, where $H_{\phi}$, the mechanical potential, depends
only upon the $\varphi$'s~and~$\dot{\varphi}$'s, while the electrokinetic potential~$H_{\epsilon}$
takes the following form:
\[
H_{\epsilon}
  = \tfrac{1}{2} L_{11} \dot{\epsilon}_{1}{}^{2}
  + L_{12} \dot{\epsilon}_{1} \dot{\epsilon}_{2}
  + L_{13} \dot{\epsilon}_{1} \dot{\epsilon}_{3} + \cdots
  + \tfrac{1}{2} L_{22} \dot{\epsilon}_{2}{}^{2} + \cdots.
\]
The quantities $L_{11}$,~$L_{12}$,~$L_{13}$~$\cdots$ $L_{22}$,~$\cdots$ the coefficients of self
induction and mutual induction depend, however, in a definite
manner upon the coordinates of position $\varphi_{1}$,~${\varphi_{1}}'$, ${\varphi_{1}}''$,~$\cdots$, $\varphi_{2}$,~${\varphi_{2}}'$,
${\varphi_{2}}''$,~$\cdots$.

In accordance with~\Eq{(59)}, we have for the motion of the first
conductor:\label{png112lab1}
\[
\Phi_{1} - \frac{d}{dt} \left(\frac{\dd H_{\phi}}{\dd \dot{\varphi}_{1}}\right)
  + \frac{\dd H_{\phi}}{\dd \varphi_{1}}
  + \frac{\dd H_{\epsilon}}{\dd \varphi_{1}} = 0,
\]
with corresponding equations for ${\varphi_{1}}'$,~${\varphi_{1}}''$,~$\cdots$, and for the electric
current in it:
\[
E_{1} - \frac{d}{dt} \left(\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}\right) = 0.
\]

The laws for the mechanical (ponderomotive) actions may be
condensed into the statement that, in addition to the ordinary
force upon the first conductor expressed by~$\Phi_{1}$, there is a mechanical
force
\[
\frac{\dd H_{\epsilon}}{\dd \varphi_{1}}
  = \frac{1}{2} \frac{\dd L_{11}}{\dd \varphi_{1}} \dot{\epsilon}_{1}{}^{2}
  + \frac{\dd L_{12}}{\dd \varphi_{1}} \dot{\epsilon}_{1} \dot{\epsilon}_{2}
  + \frac{\dd L_{13}}{\dd \varphi_{1}} \dot{\epsilon}_{1} \dot{\epsilon}_{3} + \cdots,
\]
which is composed of an action of the current upon itself (first
term) and of the actions of the remaining currents upon it
(following terms).
%-----File: 113.png---\redacted\--------

The laws of electrical action, on the other hand, are expressed
by the statement, that to the external electromotive force~$E_{1}$
in the first conductor there is added the electromotive force
\label{png113lab1}
\[
-\frac{d}{dt} \left(\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}\right)
  = -\frac{d}{dt} (L_{11} \dot{\epsilon}_{1} + L_{12} \dot{\epsilon}_{2} + L_{13} \dot{\epsilon}_{3} + \cdots)
\]
which likewise is composed of an action of the current upon itself
(self induction) and of the inducing actions of the remaining
currents, and that these two forces compensate each other.

The galvanic conductance or the galvanic resistance is not
contained in these equations because the corresponding energy,
Joule heat, is produced in an irreversible manner, and irreversible
processes are not represented by the principle of least action.
One can formally include this action, likewise any other irreversible
action, in accordance with the procedure of Helmholz,
by introducing it as an external force, in the present case as
the electromotive force due to the resistance~$w$, which operates
to cause a diminution in the energy of the system. For an
infinitely small element of time, the amount of this energy change
is:
\[
  -(w_{1} \dot{\epsilon}_{1}{}^{2}
  + w_{2} \dot{\epsilon}_{2}{}^{2}
  + w_{3} \dot{\epsilon}_{3}{}^{2} + \cdots) · dt
  = -(w_{1} \dot{\epsilon}_{1} d\epsilon_{1}
    + w_{2} \dot{\epsilon}_{2} d\epsilon_{2} + \cdots).
\]
Consequently, since the external work $E_{1} d\epsilon_{1} + E_{2} d\epsilon_{2} + \cdots$ now
includes the Joule heat, the external force components $E_{1}$,~$E_{2}$,~$\cdots$
in the electromotive equations must be increased by the additional
terms $-w_{1} \dot{\epsilon}_{1}$,~$-w_{2} \dot{\epsilon}_{2}$,~$\cdots$.

The application of the principle of least action to thermodynamic
processes is of special interest, because the importance
of the question relating to the fixing of the generalized coordinates,
which determine the state of the system, here becomes
prominent. From the standpoint of pure thermodynamics,
the variables which determine the state of a body can certainly
be quite arbitrarily chosen, \eg, in the case of a gas of invariable
constitution any two of the following quantities may be chosen
%-----File: 114.png---\redacted\--------
as independent variables and all others expressed through them:
volume~$V$, temperature~$T$, pressure~$P$, energy~$E$, entropy~$S$. In
the present case, the matter is quite different. If we inquire, in
order to apply the principle of least action, with regard to the
energy change or the total work~$A$ which will be done upon the
gas from without in an infinitely small virtual displacement, it
may be written in the form:
\[
A = -p · \delta V + T · \delta S.
\]
$T \delta S$ is the heat added from without, $-p \delta V$~the mechanical work
furnished from without. In order to bring this into agreement
with the general formula for external work~\Eq{(58)}:
\[
A = \Phi_{1} \delta \varphi_{1} + \Phi_{2} \delta \varphi_{2}
\]
it becomes necessary now to choose $V$~and~$S$ as the generalized
coordinates of state and, therefore, to identify with them the
previously employed quantities $\varphi_{1}$~and~$\varphi_{2}$. Then $-p$~and~$T$
are the generalized force components $\Phi_{1}$~and~$\Phi_{2}$. Now, since in
thermodynamics every reversible change of state proceeds with
infinite slowness, the velocity components $\dot{V}$~and~$\dot{S}$, and in general
all differential coefficients with respect to time, are to be placed
equal to zero, and the principle of least action~\Eq{(59)} reduces to:
\[
\Phi + \frac{\dd H}{\dd \varphi} = 0,
\]
and, therefore, in our case:
\[
-p + \left(\frac{\dd H}{\dd V}\right)_{S} = 0\quad \text{and}\quad
 T + \left(\frac{\dd H}{\dd S}\right)_{V} = 0.
\]
Further, in accordance with~\Eq{(60)}:
\[
E = -H.
\]
Now these equations are actually valid, since they only present
other forms of the relation
\[
dS = \frac{dE + p dV}{T}.
\]
%-----File: 115.png---\redacted\--------

The view here presented is fundamentally that which is given
in the energetics of Mach, Ostwald, Helm, and Wiedeburg. The
generalized coordinates $V$~and~$S$ are in this theory the ``capacity
factors,'' $-p$~and~$T$ the ``intensity factors.''\footnote
  {The breaking up of the energy differentials into two factors by the exponents
  of energetics is by no means associated with a special property of
  energy, but is simply an expression for the elementary law that the differential
  of a function~$F(x)$ is equal to the product of the differential~$dx$ by the derivative~$\dot{F}(x)$.}
So long as
one limits himself to an irreversible process, nothing stands in
the way of carrying out this method completely, nor of a generalization
to include chemical processes.

In opposition to it there is an essentially different method of regarding
thermodynamic processes, which in its complete generality
was first introduced into physics by Helmholtz. In accordance
with this method, one generalized coordinate is~$V$, and the other
is not~$S$, but a certain cyclical coordinate---we shall denote it,
as in the previous example, by~$\epsilon$---which does not appear itself
in the expression for the kinetic potential~$H$ and only appears
through its differential coefficient,~$\dot{\epsilon}$; and this differential coefficient
is the temperature~$T$. Accordingly, $H$~is dependent only
upon $V$~and~$T$. The equation for the total external work, in
accordance with~\Eq{(58)}, is:
\[
A = -p \delta V + E \delta\epsilon,
\]
and agreement with thermodynamics is obviously found if we
set:
\[
E \delta\epsilon = T \delta S,\quad \text{and also:}\quad
E d\epsilon = T dS,\quad
E dt = dS.
\]
The equations~\Eq{(59)} for the principle of least action become:\label{png115lab1}
\[
-p + \left(\frac{\dd H}{\dd V}\right)_{T} = 0\quad \text{and}\quad
 E - \frac{d}{dt} \left(\frac{\dd H}{\dd T}\right)_{V} = 0,
\]
or
\[
d\left(\frac{\dd H}{\dd T}\right)_{V} = E dt = dS,
\]
%-----File: 116.png---\redacted\--------
or by integration:
\[
\left(\frac{\dd H}{\dd T}\right)_{V} = S,
\]
to an additive constant, which we may set equal to~$0$. For the
energy there results, in accordance with~\Eq{(60)}:
\[
E = \dot{\epsilon} \frac{\dd H}{\dd \dot{\epsilon}} - H
  = T \left(\frac{\dd H}{\dd T}\right)_{V} - H,
\]
and consequently:
\[
H = -(E - TS).
\]
$H$~is therefore equal to the negative of the function which
Helmholz has called the ``free energy'' of the system, and the
above equations are known from thermodynamics.

Furthermore, the method of Helmholz permits of being carried
through consistently, and so long as one limits himself to the
consideration of reversible processes, it is in general quite impossible
to decide in favor of the one method or the other. However,
the method of Helmholz possesses a distinct advantage
over the other which I desire to emphasize here. It lends itself
better to the furtherance of our endeavor toward the unification
of the system of physics. In accordance with the purely energetic
method, the independent variables $V$~and~$S$ have absolutely
nothing to do with each other; heat is a form of energy which is
distinguished in nature from mechanical energy and which in
no way can be referred back to it. In accordance with Helmholz,
heat energy is reduced to motion, and this certainly indicates an
advance which is to be placed, perhaps, upon exactly the same
footing as the advance which is involved in the consideration of
light waves as electromagnetic waves.

To be sure, the view of Helmholz is not broad enough to include
irreversible processes; with regard to this, as we have earlier
stated in detail, the introduction of the calculus of probability
is necessary in order to throw light on the question. At the
same time, this is also the real reason that the exponents of
%-----File: 117.png---\redacted\--------
energetics will have nothing to do with the strict observance
of irreversible processes, and they either declare them as doubtful
or ignore them completely. In reality, the facts of the case are
quite the reverse; irreversible processes are the only processes
occurring in nature. Reversible processes form only an ideal
abstraction, which is very valuable for the theory, but which is
never completely realized in nature.


\Section{II.}{The Generalized Coordinates of State Form a Continuous
Manifold.}

The laws of infinitely small motions of perfectly elastic bodies
furnish us with the simplest example. The coordinates of state
are then the displacement components, $\frakv_{x}$,~$\frakv_{y}$,~$\frakv_{z}$, of a material
point from its position of equilibrium $(x, y, z)$, considered as a
function of the coordinates $x$,~$y$,~$z$. The external work is given
by a surface integral:
\[
A = \int d\sigma (X_{\nu} \delta \frakv_{x} + Y_{\nu} \delta \frakv_{y} + Z_{\nu} \delta \frakv_{z})
\]
($d\sigma$,~surface element; $\nu$,~inner normal). The kinetic potential
is again given by the difference of the kinetic energy~$L$ and the
potential energy~$U$:
\[
H = L - U.
\]
The kinetic energy is:
\[
L = \int \frac{d\tau k}{2} (\dot{\frakv}_{x}^{2} + \dot{\frakv}_{y}^{2} + \dot{\frakv}_{z}^{2}),
\]
wherein $d\tau$~denotes a volume element, $k$~the volume density.
The potential energy~$U$ is likewise a space integral of a homogeneous
quadratic function~$f$ which specifies the potential energy
of a volume element. This depends, as is seen from purely
geometrical considerations, only upon the $6$ ``strain coefficients:''
\begin{gather*}
\frac{\dd \frakv_{x}}{\dd x} = x_{x},\quad
\frac{\dd \frakv_{y}}{\dd y} = y_{y},\quad
\frac{\dd \frakv_{z}}{\dd z} = z_{z}, \\
\frac{\dd \frakv_{y}}{\dd z} + \frac{\dd \frakv_{z}}{\dd y} = y_{z} = z_{y},\quad
\frac{\dd \frakv_{z}}{\dd x} + \frac{\dd \frakv_{x}}{\dd z} = z_{x} = x_{z},\quad
\frac{\dd \frakv_{x}}{\dd y} + \frac{\dd \frakv_{y}}{\dd x} = x_{y} = y_{x}.
\end{gather*}
%-----File: 118.png---\redacted\--------
In general, therefore, the function~$f$ contains $21$~independent
constants, which characterize the whole elastic behavior of the
substance. For isotropic substances these reduce on grounds
of symmetry to~$2$. Substituting these values in the expression
for the principle of least action~\Eq{(57)} we obtain:
\begin{multline*}
\int dt  \biggl\{ \int d\tau k (\dot{\frakv}_{x} \delta\dot{\frakv}_{x} + \cdots)
  - \int d\tau \left(\frac{\dd f}{\dd x_{x}} \delta x_{x}
                   + \frac{\dd f}{\dd x_{y}} \delta x_{y} + \cdots\right)\\
  + \int d\sigma (X_{\nu} \delta\frakv_{x} + \cdots) \biggr\} = 0.
\end{multline*}
If we put for brevity:
\begin{align*}
-\frac{\dd f}{\dd x_{x}} &= X_{x},         &-\frac{\dd f}{\dd y_{y}} &= Y_{y},         &-\frac{\dd f}{\dd z_{z}} &= Z_{z},\\
-\frac{\dd f}{\dd y_{z}} &= Y_{z} = Z_{y}, &-\frac{\dd f}{\dd z_{x}} &= Z_{x} = X_{z}, &-\frac{\dd f}{\dd x_{y}} &= X_{y} = Y_{x},
\end{align*}
it turns out, as the result of purely mathematical operations in
which the variations $\delta\dot{\frakv}_{x}$,~$\delta\dot{\frakv}_{y}$,~$\cdots$ and likewise the variations
$\delta x_{x}$,~$\delta x_{y}$,~$\cdots$ are reduced through suitable partial integration with
respect to the variations $\delta\frakv_{x}$,~$\delta\frakv_{y}$,~$\cdots$, that the conditions within
the body are expressed by:
\[
k \ddot{\frakv}_{x}
  + \frac{\dd X_{x}}{\dd x}
  + \frac{\dd X_{y}}{\dd y}
  + \frac{\dd X_{z}}{\dd z} = 0,\ \cdots
\]
and at the surface, by:
\[
X_{\nu} = X_{x} \cos \nu x + X_{y} \cos \nu y + X_{z} \cos \nu z,\ \cdots
\]
as is known from the theory of elasticity. The mechanical significance
of the quantities $X_{x}$,~$Y_{y}$,~$\cdots$ as surface forces follows
from the surface conditions.

For the last application of the principle of least action we will
take a special case of electrodynamics, namely, electrodynamic
processes in a homogeneous isotropic non-conductor at rest, \eg,
a vacuum. The treatment is analogous to that carried out in the
foregoing example. The only difference lies in the fact that in
%-----File: 119.png---\redacted\--------
electrodynamics the dependence of the potential energy~$U$ upon
the generalized coordinate~$\frakv$ is somewhat different than in elastic
phenomena.

We therefore again put for the external work:
\[
A = \int d\sigma (X_{\nu} \delta\frakv_{x} + Y_{\nu} \delta\frakv_{y} + Z_{\nu} \delta\frakv_{z}),
\Tag{(61)}
\]
and for the kinetic potential:
\[
H = L - U,
\]
wherein again:
\[
L = \int d\tau \frac{k}{2} (\dot{\frakv}_{x}{}^{2} + \dot{\frakv}_{y}{}^{2} + \dot{\frakv}_{z}{}^{2})
  = \int d\tau \frac{k}{2} (\dot{\frakv})^{2}.
\]
On the other hand, we write here:
\[
U = \int d\tau \frac{h}{2} (\curl \frakv)^{2}.
\]
Through these assumptions the dynamical equations including
the boundary conditions are now completely determined. The
principle of least action~\Eq{(57)} furnishes:
\begin{multline*}\textstyle
\int dt \{ \int d\tau k (\dot{\frakv}_{x} \delta\dot{\frakv}_{x} + \cdots)
  - \int d\tau h (\curl_{x} \frakv \delta\curl_{x} \frakv + \cdots)\\
\textstyle  + \int d\sigma (X_{\nu} \delta\frakv_{x} + \cdots) \} = 0.
\end{multline*}
From this follow, in quite an analogous way to that employed
above in the theory of elasticity, first, for the interior of the
non-conductor:
\[
k \ddot{\frakv}_{x}
  = h\left(\frac{\dd \curl_{y} \frakv}{\dd z}
         - \frac{\dd \curl_{z} \frakv}{\dd y}\right),\ \cdots
\]
or more briefly
\[
k \ddot{\frakv} = -h \curl \curl \frakv,
\Tag{(62)}
\]
and secondly, for the surface:
\[
X_{\nu} = h(\curl_{z} \frakv · \cos \nu y - \curl_{y} \frakv · \cos \nu z),\ \cdots
\Tag{(63)}
\]
These equations are identical with the known electrodynamical
equations, if we identify $L$~with the electric, and $U$~with the
%-----File: 120.png---\redacted\--------
magnetic energy (or conversely). If we put
\[
L = \frac{1}{8\pi} \int d\tau · \epsilon \frakE^{2} \quad\text{and}\quad
U = \frac{1}{8\pi} \int d\tau · \mu \frakH^{2},
\]
($\frakE$~and~$\frakH$, the field strengths, $\epsilon$,~the dielectric constant, $\mu$,~the
permeability) and compare these values with the above expressions
for $L$~and~$U$ we may write:
\[
\dot{\frakv} = -\frakE · \sqrt{\frac{\epsilon}{4\pi k}},\quad
\curl \frakv =  \frakH \sqrt{\frac{\mu}{4\pi h}}.
\Tag{(64)}
\]
It follows then, by elimination of~$\frakv$, that:
\[
\dot{\frakH} = -\sqrt{\frac{\epsilon h}{\mu k}} · \curl \frakE,
\]
and further, by substitution of $\dot{\frakv}$~and~$\curl \frakv$ in equation~\Eq{(62)} found
above for the interior of the non-conductor, that:
\[
\dot{\frakE} = \sqrt{\frac{\mu h}{\epsilon k}} \curl \frakH.
\]
Comparison with the known electrodynamical equations expressed
in Gaussian units:
\[
\mu \dot{\frakH} = -c \curl \frakE,\quad
\epsilon \dot{\frakE} = c \curl \frakH
\]
($c$,~velocity of light in vacuum) results in a complete agreement,
if we put:
\[
\frac{c}{\mu} = \sqrt{\frac{\epsilon h}{\mu k}} \quad\text{and}\quad
\frac{c}{\epsilon} = \sqrt{\frac{\mu h}{\epsilon k}}.
\]
From either of these two equations it follows that:
\[
\frac{h}{k} = \frac{c^{2}}{\epsilon \mu},
\]
the square of the velocity of propagation.

We obtain from~\Eq{(61)} for the energy entering the system from
without:
\[
\textstyle dt · \int d\sigma (X_{\nu} \dot{\frakv}_{x} + Y_{\nu} \dot{\frakv}_{y} + Z_{\nu} \dot{\frakv}_{z}),
\]
%-----File: 121.png---\redacted\--------
or, taking account of the surface equation~\Eq{(63)}:
\[
\textstyle dt · \int d\sigma h \{(\curl_{z} \frakv \cos \nu y - \curl_{y} \frakv \cos \nu z) \dot{\frakv}_{x} + \cdots\},
\]
an expression which, upon substitution of the values of $\dot{\frakv}$ and~$\curl \frakv$
from~\Eq{(64)}, turns out to be identical with the Poynting energy
current.

We have thus by an application of the principle of least action
with a suitably chosen expression for the kinetic potential~$H$
arrived at the known Maxwellian field equations.

Are, then, the electromagnetic processes thus referred back to
mechanical processes? By no means; for the vector~$\frakv$ employed
here is certainly not a mechanical quantity. It is moreover not
possible in general to interpret~$\frakv$ as a mechanical quantity, for
instance, $\frakv$~as a displacement, $\dot{\frakv}$~as a velocity, $\curl \frakv$~as a rotation.
Thus, \eg, in an electrostatic field $\dot{\frakv}$~is constant. Therefore,
$\frakv$~increases with the time beyond all limits, and $\curl \frakv$~can
no longer signify a rotation.\label{png121lab1}\footnote
  {With regard to the impossibility of interpreting electrodynamic processes
  in terms of the motions of a continuous medium, cf.\ particularly, H.~Witte:
  ``Über den gegenwärtigen Stand der Frage nach einer mechanischen Erklärung
  der elektrischen Erscheinungen'' Berlin, 1906 (E.~Ebering).}
While from these considerations
the possibility of a mechanical explanation of electrical phenomena
is not proven, it does appear, on the other hand, to be undoubtedly
true that the significance of the principle of least
action may be essentially extended beyond ordinary mechanics
and that this principle can therefore also be utilized as the
foundation for general dynamics, since it governs all known reversible
processes.
%-----File: 122.png---\redacted\--------


\Chapter{Eighth Lecture.}{General Dynamics. Principle of Relativity.}\label{Lect8}

In the lecture of yesterday we saw, by means of examples,
that all continuous reversible processes of nature may be represented
as consequences of the principle of least action, and
that the whole course of such a process is uniquely determined
as soon as we know, besides the actions which are exerted upon
the system from without, the kinetic potential~$H$ as a function
of the generalized coordinates and their differential coefficients
with respect to time. The determination of this function
remains then as a special problem, and we recognize here a
rich field for further theories and hypotheses. It is my purpose
to discuss with you today an hypothesis which represents a magnificent
attempt to establish quite generally the dependency of
the kinetic potential~$H$ upon the velocities, and which is commonly
designated as the principle of relativity. The gist of this principle
is: it is in no wise possible to detect the motion of a
body relative to empty space; in fact, there is absolutely
no physical sense in speaking of such a motion. If, therefore,
two observers move with uniform but different velocities, then
each of the two with exactly the same right may assert that with
respect to empty space he is at rest, and there are no physical
methods of measurement enabling us to decide in favor of the one
or the other. The principle of relativity in its generalized form
is a very recent development. The preparatory steps were taken
by H.~A. Lorentz, it was first generally formulated by A.~Einstein,
and was developed into a finished mathematical system by
H.~Minkowski. However, traces of it extend quite far back
into the past, and therefore it seems desirable first to say something
concerning the history of its development.
%-----File: 123.png---\redacted\--------

The principle of relativity has been recognized in mechanics
since the time of Galilee and Newton. It is contained in the
form of the simple equations of motion of a material point, since
these contain only the acceleration and not the velocity of
the point. If, therefore, we refer the motion of the point,
first to the coordinates $x$,~$y$,~$z$, and again to the coordinates
$x'$,~$y'$,~$z'$ of a second system, whose axes are directed parallel
to the first and which moves with the velocity~$\nu$ in the direction
of the positive $x$-axis:
\[
x' = x - \nu t,\quad y' = y,\quad z' = z,
\Tag{(65)}
\]
and the form of the equations of motion is not changed in the
slightest. Nothing short of the assumption of the general validity
of the relativity principle in mechanics can justify the inclusion
by physics of the Copernican cosmical system, since through
it the independence of all processes upon the earth of the progressive
motion of the earth is secured. If one were obliged to take
account of this motion, I should have, \eg, to admit that the piece
of chalk in my hand possesses an enormous kinetic energy, corresponding
to a velocity of something like $30$~kilometers per~second.

It was without doubt his conviction of the absolute validity
of the principle of relativity which guided Heinrich Hertz
in the establishment of his fundamental equations for the electrodynamics
of moving bodies. The electrodynamics of Hertz
is, in fact, wholly built upon the principle of relativity. It recognizes
no absolute motion with regard to empty space. It speaks
only of motions of material bodies relative to one another. In
accordance with the theory of Hertz, all electrodynamic processes
occur in material bodies; if these move, then the electrodynamic
processes occurring therein move with them. To speak
of an independent state of motion of a medium outside of material
bodies, such as the ether, has just as little sense in the theory of
Hertz as in the modern theory of relativity.

\label{png123lab1}\pngcent{illo124.png}{1350}

But the theory of Hertz has led to various contradictions with
experience. I will refer here to the most important of these.
%-----File: 124.png---\redacted\--------
Fizeau brought (1851) into parallelism a bundle of rays originating
in a light source~$L$ by means of a lens and then brought it
to a focus by means of a second lens upon a screen~$S$ (Fig.~2).
In the path of the parallel light rays between the two lenses he
placed a tube system of such sort that a transparent liquid could
be passed through it, and in such manner that in one half (the
upper) the light rays would pass in the direction of flow of the
liquid while in the other half (the lower), the rays would pass in
the opposite direction.

If now a liquid or a gas flow through the tube system with the
velocity~$\nu$, then, in accordance with the theory of Hertz, since
light must be a process in the substance, the light waves must
be transported with the velocity of the liquid. The velocity
of light relative to $L$ and $S$ is, therefore, in the upper part
$q_{0} + \nu$, and the lower part $q_{0} - \nu$, if $q_{0}$~denote the velocity
of light relative to the liquid. The difference of these two
velocities,~$2\nu$, should be observable at~$S$ through corresponding
interference of the lower and the upper light rays, and quite independently
of the nature of the flowing substance. Experiment
did not confirm this conclusion. Moreover, it showed in gases
generally no trace of the expected action; \ie,~light is propagated
in a flowing gas in the same manner as in a gas at rest. On the
other hand, in the case of liquids an effect was certainly indicated,
%-----File: 125.png---\redacted\--------
but notably smaller in amount than that demanded by the theory
of Hertz. Instead of the expected velocity difference~$2\nu$, the
difference $2\nu(1 - 1/n^{2})$ only was observed, where $n$~is the refractive
index of the liquid. The factor $(1 - 1/n^{2})$ is called
the Fresnel coefficient. There is contained (for $n = 1$) in this
expression the result obtained in the case of gases.

It follows from the experiment of Fizeau that, as regards
electrodynamic processes in a gas, the motion of the gas is
practically immaterial. If, therefore, one holds that electrodynamic
processes require for their propagation a substantial
carrier, a special medium, then it must be concluded that this
medium, the ether, remains at rest when the gas moves in an arbitrary
manner. This interpretation forms the basis of the electrodynamics
of Lorentz, involving an absolutely quiescent ether.
In accordance with this theory, electrodynamic phenomena have
only indirectly to do with the motion of matter. Primarily all
electrodynamical actions are propagated in ether at rest. Matter
influences the propagation only in a secondary way, so far as it
is the cause of exciting in greater or less degree resonant vibrations
in its smallest parts by means of the electrodynamic waves
passing through it. Now, since the refractive properties of substances
are also influenced through the resonant vibrations of its
smallest particles, there results from this theory a definite connection
between the refractive index and the coefficient of Fresnel,
and this connection is, as calculation shows, exactly that demanded
by measurements. So far, therefore, the theory of
Lorentz is confirmed through experience, and the principle of
relativity is divested of its general significance.

The principle of relativity was immediately confronted by
a new difficulty. The theory of a quiescent ether admits the
idea of an absolute velocity of a body, namely the velocity
relative to the ether. Therefore, in accordance with this theory,
of two observers $A$~and~$B$ who are in empty space and who
move relatively to each other with the uniform velocity~$\nu$, it would
be at best possible for only one rightly to assert that he is at
%-----File: 126.png---\redacted\--------
rest relative to the ether. If we assume, \eg, that at the moment
at which the two observers meet an instantaneous optical signal,
a flash, is made by each, then an infinitely thin spherical wave
spreads out from the place of its origin in all directions through
empty space. If, therefore, the observer~$A$ remain at the center
of the sphere, the observer~$B$ will not remain at the center and,
as judged by the observer~$B$, the light in his own direction of
motion must travel (with the velocity $c - \nu$) more slowly than
in the opposite direction (with the velocity $c + \nu$), or than in a
perpendicular direction (with the velocity $\sqrt{c^{2} - \nu^{2}}$) (cf.\ Fig.~3).
Under suitable conditions the observer~$B$ should be able to
detect and measure this sort of effect.

\label{png126lab1}\pngcent{illo126.png}{914}

This elementary consideration led to the celebrated attempt
of Michelson to measure the motion of the earth relative to the
ether. A parallel beam of rays proceeding from~$L$ (Fig.~4)
falls upon a transparent plane parallel plate~$P$ inclined at~$45°$,
by which it is in part transmitted and in part reflected. The
transmitted and reflected beams are brought into interference
by reflection from suitable metallic mirrors $S_{1}$~and~$S_{2}$, which are
removed by the same distance~$l$ from~$P$. If, now, the earth with
the whole apparatus moves in the direction~$PS_{1}$ with the velocity~$\nu$,
then the time which the light needs in order to go from $P$ to
$S_{1}$ and back is:
%-----File: 127.png---\redacted\--------
\[
\frac{l}{c - \nu} + \frac{l}{c + \nu}
  = \frac{2l}{c} \left(1 + \frac{\nu^{2}}{c^{2}} + \cdots\right).
\]
On the other hand, the time which the light needs in order to pass
from $P$ to $S_{2}$ and back to~$P$ is:
\[
\frac{l}{\sqrt{c^{2} - \nu^{2}}} + \frac{l}{\sqrt{c^{2} - \nu^{2}}}
  = \frac{2l}{c} \left(1 + \frac{1}{2} \frac{\nu^{2}}{c^{2}} + \cdots\right).
\]
If, now, the whole apparatus be turned through a right angle, a
noticeable displacement of the interference bands should result,
since the time for the passage over the path~$PS_{2}$ is now longer.
No trace was observed of the marked effect to be expected.

\pngcent{illo127.png}{1002}

Now, how will it be possible to bring into line this result,
established by repeated tests with all the facilities of modern
experimental art? E.~Cohn has attempted to find the necessary
compensation in a certain influence of the air in which
the rays are propagated. But for anyone who bears in mind the
great results of the atomic theory of dispersion and who does
not renounce the simple explanation which this theory gives for
the dependence of the refractive index upon the color, without
introducing something else in its place, the idea that a moving
%-----File: 128.png---\redacted\--------
absolutely transparent medium, whose refractive index is absolutely~$= 1$,
shall yet have a notable influence upon the velocity
of propagation of light, as the theory of Cohn demands, is not
possible of assumption. For this theory distinguishes essentially
a transparent medium, whose refractive index is~$= 1$, from a
perfect vacuum. For the former the velocity of propagation of
light in the direction of the velocity~$\nu$ of the medium with relation
to an observer at rest is
\[
q = c + \frac{\nu^{2}}{c},
\]
for a vacuum, on the other hand, $q = c$. In the former medium,
Cohn's theory of the Michelson experiment predicts no effect,
but, on the other hand, the Michelson experiment should give
a positive effect in a vacuum.

In opposition to E.~Cohn, H.~A. Lorentz and FitzGerald
ascribe the necessary compensation to a contraction of the whole
optical apparatus in the direction of the earth's motion of the
order of magnitude~$\nu^{2}/c^{2}$. This assumption allows better of the
introduction again of the principle of relativity, but it can first
completely satisfy this principle when it appears, not as a necessary
hypothesis made to fit the present special case, but as a
consequence of a much more general postulate. We have to
thank A.~Einstein for the framing of this postulate and H.~Minkowski
for its further mathematical development.

Above all, the general principle of relativity demands the
renunciation of the assumption which led H.~A. Lorentz to the
framing of his theory of a quiescent ether; the assumption
of a substantial carrier of electromagnetic waves. For, when
such a carrier is present, one must assume a definite velocity of a
ponderable body as definable with respect to it, and this is exactly
that which is excluded by the relativity principle. Thus the
ether drops out of the theory and with it the possibility of
mechanical explanation of electrodynamic processes, \ie, of referring
them to motions. The latter difficulty, however, does
%-----File: 129.png---\redacted\--------
not signify here so much, since it was already known before,
that no mechanical theory founded upon the continuous motions
of the ether permits of being completely carried through (cf.\ p.~\pageref{png121lab1}).
In place of the so-called free ether there is now substituted
the absolute vacuum, in which electromagnetic energy is independently
propagated, like ponderable atoms. I believe it follows
as a consequence that no physical properties can be consistently
ascribed to the absolute vacuum. The dielectric constant and the
magnetic permeability of a vacuum have no absolute meaning,
only relative. If an electrodynamic process were to occur in a
ponderable medium as in a vacuum, then it would have absolutely
no sense to distinguish between field strength and induction.
In fact, one can ascribe to the vacuum any arbitrary value of the
dielectric constant, as is indicated by the various systems of
units. But how is it now with regard to the velocity of propagation
of light? This also is not to be regarded as a property of
the vacuum, but as a property of electromagnetic energy which
is present in the vacuum. Where there is no energy there can
exist no velocity of propagation.

With the complete elimination of the ether, the opportunity is
now pre\-sent for the framing of the principle of relativity. Obviously,
we must, as a simple consideration shows, introduce
something radically new. In order that the moving observer~$B$
mentioned above (Fig.~3, p.~\pageref{png126lab1}) shall not see the light
signal given by him travelling more slowly in his own direction
of motion (with the velocity $c - \nu$) than in the opposite direction
(with the velocity $c + \nu$), it is necessary that he shall not identify
the instant of time at which the light has covered the distance
$c - \nu$ in the direction of his own motion with the instant of time at
which the light has covered the distance $c + \nu$ in the opposite
direction, but that he regard the latter instant of time as later.
In other words: the observer~$B$ measures time differently from
the observer~$A$. This is a~priori quite permissible; for the
relativity principle only demands that neither of the two observers
shall come into contradiction with himself. However, the
%-----File: 130.png---\redacted\--------
possibility is left open that the specifications of time of both
observers may be mutually contradictory.

It need scarcely be emphasized that this new conception of the
idea of time makes the most serious demands upon the capacity
of abstraction and the projective power of the physicist. It
surpasses in boldness everything previously suggested in speculative
natural phenomena and even in the philosophical theories
of knowledge: non-euclidean geometry is child's play in comparison.
And, moreover, the principle of relativity, unlike non-euclidean
geometry, which only comes seriously into consideration
in pure mathematics, undoubtedly possesses a real physical
significance. The revolution introduced by this principle into
the physical conceptions of the world is only to be compared in
extent and depth with that brought about by the introduction
of the Copernican system of the universe.

Since it is difficult, on account of our habitual notions concerning
the idea of absolute time, to protect ourselves, without
special carefully considered rules, against logical mistakes in the
necessary processes of thought, we shall adopt the mathematical
method of treatment. Let us consider then an electrodynamic
process in a pure vacuum; first, from the standpoint of an observer~$A$;
secondly, from the standpoint of an observer~$B$, who
moves relatively to observer~$A$ with a velocity~$\nu$ in the direction
of the $x$-axis. Then, if $A$~employ the system of reference $x$,~$y$,~$z$,~$t$,
and $B$~the system of reference $x'$,~$y'$,~$z'$,~$t'$, our first problem is to
find the relations among the primed and the unprimed quantities.
Above all, it is to be noticed that since both systems of reference,
the primed and the unprimed, are to be like directed, the equations
of transformation between corresponding quantities in the
two systems must be so established that it is possible through
a transformation of exactly the same kind to pass from the first
system to the second, and conversely, from the second back to
the first system. It follows immediately from this that the velocity
of light~$c'$ in a vacuum for the observer~$B$ is exactly the same
as for the observer~$A$. Thus, if $c'$~and~$c$ are different, $c' > c$,
%-----File: 131.png---\redacted\--------
say, it would follow that: if one passes from one observer~$A$ to
another observer~$B$ who moves with respect to~$A$ with uniform
velocity, then he would find the velocity of propagation of light
for~$B$ greater than for~$A$. This conclusion must likewise hold
quite in general independently of the direction in which $B$ moves
with respect to~$A$, because all directions in space are equivalent
for the observer~$A$. On the same grounds, in passing from~$B$ to~$A$,
$c$~must be greater than~$c'$, for all directions in space for the
observer~$B$ are now equivalent. Since the two inequalities contradict,
therefore $c'$~must be equal to~$c$. Of course this important
result may be generalized immediately, so that the totality
of the quantities independent of the motion, such as the
velocity of light in a vacuum, the constant of gravitation
between two bodies at rest, every isolated electric charge, \label{png131lab1}and
the entropy of any physical system possess the same values for
both observers. On the other hand, this law does not hold for
quantities such as energy, volume, temperature,~etc. For these
quantities depend also upon the velocity, and a body which is
at rest for~$A$ is for~$B$ a moving body.

We inquire now with regard to the form of the equations
of transformation between the unprimed and the primed coordinates.
For this purpose let us consider, returning to the
previous example, the propagation, as it appears to the two
observers $A$~and~$B$, of an instantaneous signal creating an infinitely
thin light wave which, at the instant at which the observers
meet, begins to spread out from the common origin of
coordinates. For the observer~$A$ the wave travels out as a
spherical wave:
\[
x^{2} + y^{2} + z^{2} - c^{2}t^{2} = 0.
\Tag{(66)}
\]
For the second observer~$B$ the same wave also travels as a
spherical wave with the same velocity:
\[
x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2} = 0;
\Tag{(67)}
\]
for the first observer has no advantage over the second observer.
%-----File: 132.png---\redacted\--------
$B$~can exactly with the same right as~$A$ assert that he is at rest
at the center of the spherical wave, and for~$B$, after unit time, the
wave appears as in Fig.~5, while its appearance for the observer~$A$
after unit time, is represented by Fig.~3 (p.~\pageref{png126lab1}).\footnote
  {The circumstance that the signal is a finite one, however small the time
  may be, has significance only as regards the thickness of the spherical layer
  and not for the conclusions here under consideration.}

\pngcent{illo132.png}{944}

The equations of transformation must therefore fulfill the
condition that the two last equations, which represent the same
physical process, are compatible with each other; and furthermore:
the passage from the unprimed to the primed quantities
must in no wise be distinguished from the reverse passage from
the primed to the unprimed quantities. In order to satisfy
these conditions, we generalize the equations of transformation~\Eq{(65)},
set up at the beginning of this lecture for the old mechanical
principle of relativity, in the following manner:
\[
x' = \kappa(x - \nu t),\quad y' = \lambda y,\quad z' = \mu z,\quad t' = \nu t + \rho x.
\]
Here $\nu$~denotes, as formerly, the velocity of the observer~$B$ relative
to~$A$ and the constants $\kappa$,~$\lambda$, $\mu$, $\nu$,~$\rho$ are yet to be determined. We
must have:
\[
x = \kappa' (x' - \nu' t'),\quad y = \lambda' y',\quad z = \mu' z',\quad t = \nu' t' + \rho' x'.
\]
It is now easy to see that $\lambda$~and~$\lambda'$ must both~$= 1$. For, if, \eg,
%-----File: 133.png---\redacted\--------
$\lambda$~be greater than~$1$, then $\lambda'$~must also be greater than~$1$; for the
two transformations are equivalent with regard to the $y$~axis.
In particular, it is impossible that $\lambda$~and~$\lambda'$ depend upon the
direction of motion of the other observer. But now, since, in
accordance with what precedes, $\lambda = 1/\lambda'$, each of the two
inequalities contradict and therefore $\lambda = \lambda' = 1$; likewise,
$\mu = \mu' = 1$. The condition for identity of the two spherical
waves then demands that the expression~\Eq{(66)}:
\[
x^{2} + y^{2} + z^{2} - c^{2}t^{2}
\]
become, through the transformation of coordinates, identical with
the expression~\Eq{(67)}:
\[
x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2},
\]
and from this the equations of transformation follow without
ambiguity:
\[
x' = \kappa (x - \nu t),\quad y' = y,\quad z' = z,\quad t' = \kappa \left(t - \frac{\nu}{c^{2}} x\right),
\Tag{(68)}
\]
wherein
\[
\kappa = \frac{c}{\sqrt{c^{2} - \nu^{2}}}.
\]
Conversely:\label{png133lab1}
\[
x = \kappa (x' + \nu t'),\quad y = y',\quad z = z',\quad t = \kappa \left(t' + \frac{\nu}{c^{2}} x'\right).
\Tag{(69)}
\]

These equations permit quite in general of the passage from the
system of reference of one observer to that of the other (H.~A.
Lorentz), and the principle of relativity asserts that all processes
in nature occur in accordance with the same laws and with the
same constants for both observers (A.~Einstein). Mathematically
considered, the equations of transformation correspond to
a rotation in the four dimensional system of reference $(x, y, z, ict)$
through the imaginary angle $\arctg (i(\nu/c))$ (H.~Minkowski).
Accordingly, the principle of relativity simply teaches that there
is in the four dimensional system of space and time no special
characteristic direction, and any doubts concerning the general
%-----File: 134.png---\redacted\--------
validity of the principle are of exactly the same kind as those
concerning the existence of the antipodians upon the other side
of the earth.

We will first make some applications of the principle of
relativity to processes which we have already treated above.
That the result of the Michelson experiment is in agreement
with the principle of relativity, is immediately evident; for, in
accordance with the relativity principle, the influence of a
uniform motion of the earth upon processes on the earth can
under no conditions be detected.

We consider now the Fizeau experiment with the flowing
liquid (see p.~\pageref{png123lab1}). If the velocity of propagation of light in
the liquid at rest be again~$q_{0}$, then, in accordance with the
relativity principle, $q_{0}$~is also the velocity of the propagation
of light in the flowing liquid for an observer who moves with
the liquid, in case we disregard the dispersion of the liquid;
for the color of the light is different for the moving observer. If
we call this observer~$B$ and the velocity of the liquid as above,~$\nu$,
we may employ immediately the above formulae in the calculation
of the velocity of propagation of light in the flowing
liquid, judged by an observer~$A$ at the screen~$S$. We have only
to put
\[
\frac{dx'}{dt'} = x' = q_{0},
\]
to seek the corresponding value of
\[
\frac{dx}{dt} = \dot{x}.
\]
For this obviously gives the velocity sought.

Now it follows directly from the equations of transformation~\Eq{(69)}
that:
\[
\frac{dx}{dt} = \dot{x} = \frac{\dot{x}' + \nu}{1 + \dfrac{\nu \dot{x}'}{c^{2}}},
\]
%-----File: 135.png---\redacted\--------
and, therefore, through appropriate substitution, the velocity
sought in the upper tube, after neglecting higher powers in $\nu/c$
and~$\nu/q_{0}$, is:
\[
\dot{x} = \frac{q_{0} + \nu}{1 + \dfrac{\nu q_{0}}{c^{2}}}
  = q_{0} + \nu \left(1 - \frac{q_{0}^2}{c^{2}}\right),
\]
and the corresponding velocity in the lower tube is:
\[
q_{0} - \nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right).
\]
The difference of the two velocities is
\[
2\nu \left(1 - \frac{q_{0}^{2}}{c^{2}}\right) = 2\nu \left(1 - \frac{1}{n^{2}}\right),
\]
which is the Fresnel coefficient, in agreement with the measurements
of Fizeau.

The significance of the principle of relativity extends, not only
to optical and other electrodynamic phenomena, but also to
all processes of ordinary mechanics; but the familiar expression~($\frac{1}{2} mq^{2}$)
for the kinetic energy of a mass point moving with
the velocity~$q$ is incompatible with this principle.

But, on the other hand, since all mechanics as well as the
rest of physics is governed by the principle of least action, the
significance of the relativity principle extends at bottom only to
the particular form which it prescribes for the kinetic potential~$H$,
and this form, though I will not stop to prove it, is characterized
by the simple law that the expression
\[
H · dt
\]
for every space element of a physical system is an invariant
\[
= H' · dt'
\]
with respect to the passage from one observer~$A$ to the other
%-----File: 136.png---\redacted\--------
observer~$B$ or, what is the same thing, the expression $H/\sqrt{c^{2} - q^{2}}$
is in this passage an invariant $= H'/\sqrt{c^{2} - q'^{2}}$.

Let us now make some applications of this very general law,
first to the dynamics of a single mass point in a vacuum, whose
state is determined by its velocity~$q$. Let us call the kinetic
potential of the mass point for $q = 0$, $H_{0}$, and consider now the
point at an instant when its velocity is~$q$. For an observer~$B$
who moves with the velocity~$q$ with respect to the observer~$A$,
$q' = 0$ at this instant, and therefore $H' = H_{0}$. But now
since in general:
\[
\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}},
\]
we have after substitution:
\[
H = \sqrt{1 - \frac{q^{2}}{c^{2}}} · H_{0}
  = \sqrt{1 - \frac{\dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2}}{c^{2}}} · H_{0}.
\]
With this value of~$H$, the Lagrangian equations of motion~\Eq{(59)}
of the previous lecture are applicable.

In accordance with~\Eq{(60)}, the kinetic energy of the mass point
amounts to:
\[
E = \dot{x} \frac{\dd H}{\dd \dot{x}}
  + \dot{y} \frac{\dd H}{\dd \dot{y}}
  + \dot{z} \frac{\dd H}{\dd \dot{z}} - H
  = q \frac{\dd H}{\dd q} - H
  = - \frac{H_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}},
\]
and the momentum to:
\[
G = \frac{\dd H}{\dd q} = -\frac{q H_{0}}{c \sqrt{c^{2} - q^{2}}}.
\]
$G/q$~is called the transverse mass~$m_{t}$, and $dG/dq$~the longitudinal
mass~$m_{l}$ of the point; accordingly:
\[
m_{t} = -\frac{H_{0}}{c \sqrt{c^{2} - q^{2}}}, \quad
m_{l} = -\frac{c H_{0}}{(c^{2} - q^{2})^{3/2}}.
\]
%-----File: 137.png---\redacted\--------
For $q = 0$, we have
\[
m_{t} = m_{l} = m_{0} = -\frac{H_{0}}{c^{2}}.
\]
It is apparent, if one replaces in the above expressions the constant~$H_{0}$
by the constant~$m_{0}$, that the momentum is:
\[
G = \frac{m_{0}q}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}
\]
and the transverse mass:
\[
m_{t} = \frac{m_{0}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}},
\]
and the longitudinal mass:
\[
m_{l} = \frac{m_{0}}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{3/2}},
\]
and, finally, that the kinetic energy is:
\[
E = \frac{m_{0} c^{2}}{\sqrt{1 - \dfrac{q^{2}}{c^{2}}}}
  = m_{0}c^{2} + \tfrac{1}{2} m_{0}q^{2} + \cdots.
\]
The familiar value of ordinary mechanics~$\frac{1}{2} m_{0}q^{2}$ appears here
therefore only as an approximate value. These equations have
been experimentally tested and confirmed through the measurements
of A.~H.~Bucherer and of E.~Hupka upon the magnetic
deflection of electrons.

A further example of the invariance of $H · dt$ will be taken from
electrodynamics. Let us consider in any given medium any
electromagnetic field. For any volume element~$V$ of the medium,
the law holds that $V · dt$~is invariant in the passage from the one
to the other observer. It follows from this that $H/V$~is invariant;
%-----File: 138.png---\redacted\--------
\ie, the kinetic potential of a unit volume or the ``\textit{space density
of kinetic potential}'' is invariant.

Hence the following relation exists;
\[
\frakE \frakD - \frakH \frakB = \frakE' \frakD' - \frakH' \frakB',
\]
wherein $\frakE$~and~$\frakH$ denote the field strengths and $\frakD$~and~$\frakB$ the
corresponding inductions. Obviously a corresponding law for
the space energy density $\frakE \frakD + \frakH \frakB$ will not hold.

A third example is selected from thermodynamics. If we
take the velocity~$q$ of a moving body, the volume~$V$ and the
temperature~$T$ as independent variables, then, as I have shown
in the previous lecture (p.~\pageref{png115lab1}), we shall have for the pressure~$p$
and the entropy~$S$ the following relations:
\[
\frac{\dd H}{\dd V} = p \quad\text{and}\quad \frac{\dd H}{\dd T} = S.
\]
Now since $V/\sqrt{c^{2} - q^{2}}$ is invariant, and $S$~likewise invariant
(see p.~\pageref{png131lab1}), it follows from the invariance of $H/\sqrt{c^{2} - q^{2}}$
that $p$~is invariant and also that $T/\sqrt{c^{2} - q^{2}}$ is invariant, and
hence that:
\[
p = p' \quad\text{and}\quad
\frac{T}{\sqrt{c^{2} - q^{2}}} = \frac{T'}{\sqrt{c^{2} - q'^{2}}}.
\]

The two observers $A$~and~$B$ would estimate the pressure of a
body as the same, but the temperature of the body as different.

A special case of this example is supplied when the body
considered furnishes a black body radiation. The black body
radiation is the only physical system whose dynamics (for quasi-stationary
processes) is known with absolute accuracy. That the
black body radiation possesses inertia was first pointed out by
F.~Hasenöhrl. For black body radiation at rest the energy
$E_{0} = a T^{4}V$ is given by the Stefan-Boltzmann law, and the entropy
$S_{0} = \int (dE_{0}/T) = \frac{4}{3} aT^{3}V$, and the pressure $p_{0} = (a/3)T^{4}$, and,
therefore, in accordance with the above relations, the kinetic
%-----File: 139.png---\redacted\--------
potential is:
\[
H_{0} = \frac{a}{3} T^{4} V.
\]
Let us imagine now a black body radiation moving with the
velocity~$q$ with respect to the observer~$A$ and introduce an
observer~$B$ who is at rest ($q = 0$) with reference to the black body
radiation; then:
\[
\frac{H}{\sqrt{c^{2} - q^{2}}} = \frac{H'}{\sqrt{c^{2} - q'^{2}}} = \frac{H'_{0}}{c},
\]
wherein
\[
H'_{0} = \frac{a}{3} T'^{4} V'.
\]
Taking account of the above general relations between $T'$~and~$T$,
$V'$~and~$V$, this gives for the moving black body radiation the
kinetic potential:
\[
H = \frac{a}{3} \frac{T^{4} V}{\left(1 - \dfrac{q^{2}}{c^{2}}\right)^{2}},
\]
from which all the remaining thermodynamic quantities: the
pressure~$p$, the energy~$E$, the momentum~$G$, the longitudinal and
transverse masses $m_{l}$~and~$m_{t}$ of the moving black body radiation
are uniquely determined.

Colleagues, ladies and gentlemen, I have arrived at the conclusion
of my lectures. I have endeavored to bring before
you in bold outline those characteristic advances in the present
system of physics which in my opinion are the most important.
Another in my place would perhaps have made another and better
choice and, at another time, it is quite likely that I myself
should have done so. The principle of relativity holds, not only
for processes in physics, but also for the physicist himself, in
that a fixed system of physics exists in reality only for a given
physicist and for a given time. But, as in the theory of relativity,
there exist invariants in the system of physics: ideas and
%-----File: 140.png---\redacted\--------
laws which retain their meaning for all investigators and for
all times, and to discover these invariants is always the real
endeavor of physical research. We shall work further in this
direction in order to leave behind for our successors where possible---lasting
results. For if, while engaged in body and mind
in patient and often modest individual endeavor, one thought
strengthens and supports us, it is this, that we in physics work,
not for the day only and for immediate results, but, so to speak,
for eternity.

I thank you heartily for the encouragement which you have
given me. I thank you no less for the patience with which you
have followed my lectures to the end, and I trust that it may be
possible for many among you to furnish in the direction indicated
much valuable service to our beloved science.

\newpage
\pagestyle{empty}
\begin{center}\Large % make the heading a bit more noticeable
\textsc{Typographical Errors corrected\\in Project Gutenberg
edition}\end{center}

\noindent p.~\pageref{png29lab2}.~In `the theory of reversible processes', `the' omitted (before line-break `the-ory').

\vspace{\baselineskip}

\noindent p.~\pageref{png36lab1}.~Eqn.~(6), first term $\tsum \nu_{0} \log c_{0}$
was printed as $\tsum \nu_{0} \log c_{1}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png38lab1}.~`Let the system consist'--`consists' in text.

\vspace{\baselineskip}

\noindent p.~\pageref{png41lab1}.~`at a fixed temperature~$T$' was printed `at a fixed pressure~$T$'.

\vspace{\baselineskip}

\noindent p.~\pageref{png43lab1}.~$\displaystyle\nu_{0}' = \frac{m_{0} }{ {m_{0}}'}$ was printed $\displaystyle v_{0}' = \frac{m_{0} }{ {m_{0}}'}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png59lab1}.~In `give up the attempt to understand'--`undertand' in text.

\vspace{\baselineskip}

\noindent p.~\pageref{png67lab1}.~Definitions of $\psi$, third expression $\displaystyle\psi_{3} = \cdots$
printed as $\displaystyle\dot{\varphi}_{3} = \cdots$.

\vspace{\baselineskip}

\noindent p.~\pageref{png67lab2}.~Equation beginning $\displaystyle\dot{\psi}_{1} = \frac{d\psi_{1}}{dt}$, this term
was printed as $\displaystyle\psi_{1} = \frac{d\psi_{1}}{dt}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png67lab4}.~In `we shall calculate later'--`calulate' in text.

\vspace{\baselineskip}

\noindent p.~\pageref{png74lab1}.~Eqn. before (21), the term $\displaystyle\left(\frac{\dd S}{\dd E}\right)_{V}$
was printed as $\displaystyle\left(\frac{dS}{\dd E}\right)_{V}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png74lab2}.~Eqn. (21), the term $\displaystyle\left(\frac{\dd S}{\dd E}\right)_{V}$
was printed as $\displaystyle\left(\frac{\dd S}{dE}\right)_{V}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png103lab1}.~`black radiation in a vacuum'--`vaccuum' in text.

\vspace{\baselineskip}

\noindent p.~\pageref{png104lab1}.~The mass ratio symbol $\omega$ was consistently printed as $\infty$ on this page.

\vspace{\baselineskip}

\noindent p.~\pageref{png110lab1}.~Eqn. after (58), the term $\displaystyle\frac{\dd H}{\dd \varphi_{1}}$
was printed as $\displaystyle\frac{\dd H}{\delta \varphi_{1}}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png111lab1}.~Term ${\Phi_{1}}' \delta{\varphi_{1}}'$ was printed as ${\Phi_{1}}' \delta\varphi_{1}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png112lab1}.~Term $\displaystyle\left(\frac{\dd H_{\phi}}{\dd \dot{\varphi}_{1}}\right)$
was printed as $\displaystyle\left(\frac{dH_{\phi}}{\dd \dot{\varphi}_{1}}\right)$.

\vspace{\baselineskip}

\noindent p.~\pageref{png113lab1}.~Eqn. after `The laws of electrical action',
the term $\displaystyle\frac{\dd H_{\epsilon}}{\dd \dot{\epsilon}_{1}}$
was printed as $\displaystyle\frac{dH_{\epsilon}}{\dd \dot{\epsilon}_{1}}$.

\vspace{\baselineskip}

\noindent p.~\pageref{png133lab1}.~Eqn. (69), the term $\kappa (x' + \nu t')$
was printed as $\kappa (x' + vt')$.

\vspace{\baselineskip}

\newpage

\begin{verbatim}
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### lprep configuration
@ControlwordReplace = (
  ['\eg',"e. g."],
  ['\ie',"i. e."],
  ['\Eg',"E. g."],
  );
@ControlwordArguments = (
  ['\\pngcent',1,0,"\n[Illustration","]\n",1,0,'',''],
  ['\\Chapter',1,1,''," ",1,1,"","\n\n"],
  ['\\Section',1,1,''," ",1,1,"","\n\n"],
  ['\\begin{spacing}',1,0,'',"",1,1,"",""],
  );

###
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