Physic.
EDITED BY
WILLIAM RAMSAY K.CXIL.RE.S,
AN;D
F.G, DQNNAN, CB.E., MA, PH.D., FJLC, F.R.S.
UNIVERSITY OF CA T TFORNT\ LIBRARY
TEXTBOOKS OF PHYSICAL CHEMISTRY
EDITED BY SIR WILLIAM RAMSAY, K.C.B., D.Sc., F.R.S.
AND
F. G. DONNAN, C.B.E., M.A., PH.D., F.I.C., F.R.S.
A SYSTEM OF PHYSICAL CHEMISTRY
TEXTBOOKS OP PHYSICAL CHEMISTRY.
Edited by Sir WILLIAM RAMSAY, K.C.B., D.Sc., F.R.S.,
AND
F. G. DONNAN, C.B.E., M.A., Ph.D., F.I.C., F.R.S.
STOICHIOMETRY. By SYDNEY YOUNG, D.Sc., F.R.S., Professor of
Chemistry in the University of Dublin. With 93 Figures in the Text.
THE PHASE RULE AND ITS APPLICATIONS. By ALEX. FINDLAY,
M.A., Ph.D., D.Sc., Professor of Chemistry in the University of Aberdeen.
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SPECTROSCOPY. By E. C. C. BALY, C.B.E., F.R.S., Grant Professor of
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THERMOCHEMISTRY By JULIUS THOMSEN, late Professor of Chemistry
in the University of Copenhagen. Translated by KATHARINE A. BURKE,
B.Sc., Assistant in the Department of Chemistry, University College,
London.
ELECTRO-CHEMISTRY. Part I. General Theory. By R. A. LEHFELDT,
D.Sc. Including a Chapter on the Relation of Chemical Constitution to
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Birmingham.
STEREOCHEMISTRY. By ALFRED W. STEWART, D.Sc., Professor of
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THE THEORY OF VALENCY. By J. NEWTON FRIEND, D.Sc., Ph.D.,
F.I.C., Carnegie Gold Medallist, Head of the Chemistry Department of
the Birmingham Technical School.
METALLOGRAPHY. By CECIL H. DESCH, D.Sc., Ph.D., F.I.C., F.C.S.,
Professor of Metallurgy in the University of Sheffield. With 14 Plates
and 108 Diagrams in the Text.
A SYSTEM OF PHYSICAL CHEMISTRY. By WILLIAM C. McC. LEWIS,
M.A. (R.U.I.), D.Sc. (Liv.X Brunner Professor of Physical Chemistry in
the University of Liverpool. With Diagrams. 3 vols. Vol. I. Kinetic
Theory. Vol. H. Thermodynamics. Vol. III. Quantum Theory.
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A SYSTEM OF
PHYSICAL CHEMISTRY
WILLIAM C. McC. LEWIS, M.A.(R.U.I.). D.Sc.(Liv.)
BRUNNER PROFESSOR OF PHYSICAL CHEMISTRY IN THE
UNIVERSITY OF LIVERPOOL
IN THREE VOLUMES
VOLUME III
QUANTUM THEORY
(WITH TWO APPENDICES BY JAMES RICE, M.A., LECTURER IN
PHYSICS IN THE UNIVERSITY OF LIVERPOOL)
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BIBLIOGRAPHICAL NOTE
First Edition. Two Volumes. February, 1916.
The Second Edition was issued in Three Volumes.
Volume III. February, 1919.
New impression, June, 1921.
PREFATORY NOTE.
IN this volume an attempt is made to deal with some physico-
chemical applications of the principles of statistical mechanics.
An account is first given of the application of what is now known
as the classical statistical mechanics, more particularly to the
problem of the energy content of substances and its variation
with temperature, on the one hand, and to the problem of
radiation on the other. It will be seen that in both these
branches classical statistical mechanics makes it possible for us
to advance very considerably beyond the limits set by the
elementary kinetic theory employed in Volume I., but at the
same time it will be found that the classical statistical mechanics
does not furnish us with a complete and adequate basis for all
the observed phenomena. It is therefore necessary to enlarge
or modify the fundamental concepts of statistical mechanics,
and it is precisely with this object in view that Planck has been
led to introduce the idea of ^quanta. Planck's quantum theory
is, therefore, properly speaking, a new or modified system of
statistical mechanics. It happens, however, that Planck was
led to his revolutionary changes by considerations based upon
the observed facts of radiation, and for this reason it is usual
to speak of the quantum theory of radiation. Radiation affords,
as a matter of fact, one illustration, and a very striking one,
of the applicability of the new mechanics. But even the success
which has attended Planck's treatment of radiation problems
would scarcely have sufficed to gain for his views that prominence
which they now have, had it not been for the satisfactory
explanation which his theory offers at the same time for the
heat content of substances and the variation of the heat content
with temperature. The idea of energy quanta has been applied
in recent years to other types of physico-chemical phenomena,
some of which will be considered. It will be assumed, in the
treatment of the subject-matter dealt with in this volume, that
373
vi PREFA TOR Y NOTE
the reader is familiar with the principles of elementary kinetic
theory and the principles of thermodynamics already discussed
in Volumes I. and II. Such of these principles and results as
may be required will be introduced therefore without further
explanation.
It will be observed that there is a change of attitude in the
mode of dealing with the experimental material in this volume
as compared with the attitude adopted in the preceding volumes.
The theoretical concepts made use of in Volumes I. and 1 1. have
become classical to a large extent, and the treatment resolves
itself into a brief account of underlying principles followed by
a systematic application of these principles to phenomena char-
acteristic of systems which had attained equilibrium or were
tending towards equilibrium. In the present volume, however,
the underlying ideas especially those involved in the quantum
theory have not as yet been fully accepted, at least in their
present form. The position of the quantum theory is to a certain
extent undefined. The physical significance of what is meant
by a quantum of energy or, in a stricter sense, the quantum of
action, is still vague. The present position has been summarised
by Professor Bragg in the words : " His [Planck's] hypothesis is
not so much an attempt to explain as a focussing of all the
difficulties into one ; so that, if this master difficulty is overcome,
a number of others melt away ". In view of what has just been
said, it will be readily appreciated that many of the experimental
investigations referred to in the present volume have been carried
out primarily with the object of testing the validity of the
quantum hypothesis itself, and as this can be done most effec-
tively by the intensive examination of certain relatively restricted
fields of research, the information available at the present time
is of a somewhat detached character as compared with the
variety and generality of the phenomena to the interpretation of
which the simple kinetic theory and the principles of thermo-
dynamics have been applied. For this reason, therefore, rela-
tively little stress is laid upon the system of classification already
adopted in Volumes I. and II. Naturally with the progress of
investigation it will become feasible ultimately provided the
quantum hypothesis becomes generally accepted to classify
phenomena as has been done in the preceding volumes ; but for
the present the interest centres elsewhere, viz. on the validity of
the underlying hypothesis itself. It is well that the reader
should appreciate this state of affairs at the outset.
TABLE OF CONTENTS OF VOL. Ill
CONSIDERATIONS BASED UPON THE QUANTUM THEORY
PAGE
PREFATORY NOTE v
CHAPTER I
Introductory Definitions Probability Statistical mechanics Entropy and
thermodynamic probability Principle of equipartition of energy among
degrees of freedom Application of the equipartition principle to specific
heats and to radiation phenomena Necessity of modifying the principle
of equipartition i
CHAPTER II
Introductory Planck's concept of quanta Planck's radiation formula . . 35
CHAPTER III
(Physical equilibrium in solids) Theory of atomic heat of solids Equations
of Einstein, Nernst-Lindemann, and Debye . . . . , . 50
CHAPTER IV
(Physical equilibrium in gaseous systems) Molecular heats of gases Bjerrum's
theory Kriiger's theory 80
CHAPTER V
(Physical equilibrium, continued) Distribution of electrons in atoms Struc-
ture of the atom from the standpoint of the quantum theory Rutherford-
Bohr atom-model Parson's magneton High-frequency spectra
Moseley's relation . . . .96
CHAPTER VI
(Systems not in equilibrium) Quantum theory and the photo-electric effect
Photochemical reactions Einstein's law of the photochemical equivalent
Thermal reactions Reaction velocity from the standpoint of the
quantum theory .... 123
viii TABLE Of CONTENTS OF VOL. Ill
CHAPTER VII
PAGE
(Systems in chemical equilibrium) Relation between Nernst's heat theorem
and the quantum theory Mass action equilibrium and heat of reaction in
terms of the quantum theory 144
APPENDICES
I. Maxwell's distribution law and the principle of equipartition of energy . 158
II. Foundations of the quantum theory . . 172
III. Kruger's theory of gyroscopic molecules .193
SUBJECT INDEX 205
AUTHOR INDEX 208
CHAPTER I.
Introductory Definitions Probability Statistical mechanics Entropy and ther-
modynamic probability Principle of equipartition of kinetic energy among
degrees of freedom Application of the equipartition principle to specific heats
and radiation phenomena Necessity of modifying the principle of equipartition.
DEFINITION OF STATISTICAL MECHANICS.
IN what we may call classical mechanics, developed in the first instance
by Newton, we become acquainted with the concepts of mass, length,
and time as the fundamental physical quantities, and from these we
pass on to derived concepts, such as velocity, acceleration, force, and
energy, by means of which we arrive at certain principles and laws
which govern physical phenomena. We say that we have " explained "
a physical or chemical phenomenon, when we can restate it in. terms of
mechanics ; that is, when we can show that the phenomenon in question
is to be anticipated on the basis of a number of mechanical principles
logically applied. In Volume I. we have seen how the application of
mechanics to the small discrete particles, which we recognise as mole-
cules and atoms, leads to a reasonable explanation of many physico-
chemical phenomena. We have restricted ourselves, however, hitherto
by certain simplifying assumptions, i.e. we have dealt with systems of
molecules as though all the molecules possessed exactly the same value
for their velocity and therefore for their kinetic energy, throughout the
given mass of material, an elementary gas, for example. It is known,
however, that such an assumption is by no means true. We have
already indicated this in Chap. I., Vol. I., when referring to the distri-
bution of velocities among a large number of gas molecules in terms,
of Maxwell's distribution law. It is true that all our experimental;
measurements deal with average effects, and hence by regarding every/
molecule as in an average state and applying the principles of mechanics,
we are able to arrive at a number of very important and useful con-
clusions in terms of the elementary kinetic theory, for which we find
experimental evidence.
This mode of treatment, however, has its limitations. Certain prob-
lems present themselves which we are quite unable to solve on the
basis of the elementary kinetic theory. We have already met a number
of these in Volume II., and have shown how they may be dealt with
from the standpoint of thermodynamics. By way of illustration we
may cite : the relation between the lowering of vapour pressure, lower-
ing of freezing point, and rise of boiling point of a liquid as a result of
VOL. III. I
2 A SYSTEM OF PHYSICAL CHEMISTRY
dissolving some substance in the liquid ; the variation of the pressure
of saturated vapour with temperature in terms of the latent heat of
vaporisation ; the variation of the equilibrium constant of a chemical
reaction with temperature and pressure ; the electromotive force of
reversible cells; the relation between the heat of reaction and the
chemical affinity of the process.
The laws of thermodynamics are not, however, in the first place
mechanical laws though we shall see later that the second law pos-
sesses a statistical mechanical basis but are founded essentially upon
experience; they are taken to be true, because no phenomenon or
process in nature contradicts them. The characteristic feature of
thermodynamical treatment is, as we have already seen, that the results
obtained involve no assumption regarding the molecular structure
of the system under consideration. Thermodynamical deductions are
therefore perfectly general, in the sense that they hold good quite
apart from specific molecular theories. It is important to point out
that mechanics likewise furnishes a general mode of treatment, known
as generalised dynamics, by which certain physico-chemical results may
be obtained, the line of reasoning followed being to a certain extent
analogous to thermodynamic reasoning in that no particular assumption
is made regarding the molecular structure and molecular properties of
the system investigated. We treat the system as a whole and do not
attempt to deal with each molecule individually.
The best account of the applications of generalised dynamics to
physico-chemical problems is that of Sir J. J. Thomson, in his book
Applications of Dynamics to Physics and Chemistry. Among the prob-
lems solved by this method are : the process of evaporation (includ-
ing the effect of electric charge upon the vapour pressure), the effect of
an inert gas upon the value of the saturated vapour pressure ; certain
properties of dilute solutions, such as the lowering of the vapour pres-
sure of the solvent by addition of solute, and the lowering of the
freezing point by the solute ; the deduction of the law of mass action,
as expressed in the equilibrium constant, the equilibrium constant
being shown to be an explicit function of the temperature ; the
principle of mobile equilibrium ; the effect of pressure upon the freezing
point of a pure liquid, and the effect of pressure upon the solubility of
one substance in another ; the phenomenon of adsorption ; the relation
between the electromotive force and "the chemical change producing
it " ; and other problems.
It is evident from this enumeration that the methods of generalised
dynamics are of wide applicability. No attempt is made, however, to
pursue this method in the present volume. Suffice it to say that
generalised dynamics is based upon a general principle embodied in
Hamilton's and Lagrange's functions. These " hardly require a more
detailed knowledge of the structure of the system to which they are
applied than the conservation of energy the first law of thermody-
namics itself, and yet are capable of completely determining the
motion of the system". Thomson compares the thermodynamical
DEFINITION OF STATISTICAL MECHANICS 3
method of treatment with that of generalised dynamics, and points
out that the disadvantage of the latter compared with the former lies
in the fact that, "the results are expressed in terms of dynamical
quantities, such as energy, momentum, or velocity, and so require
further knowledge before we can translate them in terms of the physical
quantities we wish to measure, such as intensity of a current tempera-
ture, and so on ; a knowledge which in all cases we do not possess.
The second law of thermodynamics, on the other hand, being based on
experience, does not involve any quantity which cannot be measured in
the laboratory."
It is evident that generalised dynamics and thermodynamics have
the great merit in common that they are essentially generalisations, and
their application does not require any specialised information such as
that involved in the molecular kinetic theory. What has been said
therefore in Volume II. in regard to the advantage which this quality
confers in the case of thermodynamics applies to much the same ex-
tent to generalised dynamics. The same disadvantage manifests itself
of course, viz. that we do not get any clearer picture of the pheno-
menon in question in terms of the molecules taking part therein.
It is very necessary, however, to attempt to deal with processes
which are essentially molecular in terms of the molecules themselves.
The difficulty lies in the fact that when a system consists of a very large
number of individuals (e.g. the molecules in a gas), it is quite impossible
to follow out the extremely complicated path pursued by each single in-
dividual. In order to advance beyond the mode of treatment employed
in the elementary kinetic theory of Volume I.-* in which we got round
the difficulty here presented by making the certainly untrue assumption
that all molecules were identical in all respects it is necessary to proceed
in quite a different manner by introducing a new idea into mechanics
which will enable us to deal with physical and chemical problems in a
more exact and logical way. This new idea the introduction of which
into mechanics we owe principally to Maxwell and to Boltzmann is
embodied in the theory of Probability. When we bring probability con-
siderations into mechanics we arrive at a mode of treatment known as
statistical mechanics}- By treating molecular processes from the stand-
point of statistical mechanics we are able to take account of the fact
that all the molecules are not really identical but differ from one another
in general in respect of velocity, direction, and energy content. The
results obtained are indeed average results as they must be if they are
to be amenable to experimental test but such results represent the
combined effect of all the molecules present, due allowance being made
for the lack of equality in the actual contribution made by each in-
dividual molecule to the total observed effect. This must not be taken
as meaning that we have to calculate the particular position, velocity,
and energy of each individual molecule at various intervals of time.
1 For an account of this method of treatment, the reader is referred to the forth-
coming work on General Dynamics and Statistical Mechanics, by J. Rice, M.A.,
which is specially adapted to the needs of physicists and chemists.
4 A SYSTEM OF PHYSICAL CHEMISTRY
This, as already pointed out, would be quite impossible. Instead, we
take advantage of the fact that the number of the molecules involved
in any system with which we are concerned in physics or chemistry is
so enormous that we are justified in dealing with these aggregations of
molecules in a statistical manner, by introducing the principle of prob-
ability or chance into the mechanics of the process considered.
It is not proposed to attempt to give a systematic account of what
may be called the principles of statistical mechanics. We are con-
cerned mainly with one such principle, known as the principle of equi-
partition of kinetic energy among degrees of freedom. We shall state
and apply this principle later. For the present it is necessary to
familiarise ourselves with the idea of probability.
PROBABILITY.
In a purely algebraic sense probability may be defined as follows :
If an event can occur in a ways and fail in b ways, each of these ways
being equally likely, then the chance or probability of its occurring is
a/(a + b), and the chance or probability of its failing to occur is
bl(a + b). The sum of these two terms is necessarily unity, for the sum
of the two probabilities covers all eventualities, i.e. the event must
either happen or fail, and the sum represents certainty. It follows that
mathematical probability is a fractional quantity which may be small or
large, but can never exceed unity, 1 i.e. certainty. We may illustrate
the idea by one or two examples. Suppose we have equal numbers of
black and white balls inside a bag, the bag being well shaken so as to
destroy any possible regularity or ordered arrangement of the balls,
what is the probability or chance that, say, a white ball will be drawn
from the bag ? It is evident that the chance of drawing a white is the
same as that of drawing a black. In other words, the probability is one
half, for here a b when a is the number of white and b the number of
black balls, and a/ (a + b) = 0-5 = b\(a + b). It is evident that in
the limit, if b becomes very small compared with a, the probability
of drawing a white increases almost to a certainty, i.e. the fraction*
aj(a + b) is nearly unity. We are here considering the probability of
a single event occurring. Let us now consider the probability that two
independent events may occur simultaneously. The probability in such
a case is easily shown 2 to be the product of the probabilities of the
separate events. That is, if the probability of the first event is P lt and
that of the second is P 2 , then the probability P of both events occurring
simultaneously is P = P^. Thus, if we have two bags, each con-
taining a white balls and b black ones, the chance of drawing a single
white from one bag is P b where P x = a/(a -I- b), and the chance
of drawing, say, a black ball from the other bag is P 2 , where P 2 =
bj(a + b). The chance of drawing a white ball from the first bag
1 Whilst this is {rue of mathematical probability we shall find later that there is
a quantity to which the term " thermodynamic probability" has been given, this
quantity being in general a large integral number.
2 C/., for example, Hall and Knight's Algebra.
PROBABILITY 5
and simultaneously a black ball from the second bag is P where P =
abl(a + ) 2 . This simple relation between the probabilities of two
separate events and the probability that both will occur simultaneously
is of great importance, and will be made use of later. Let us now
consider one or two cases which possess a more distinctly chemical
character.
The first which we shall take is the elementary deduction of the law
of mass action given in Vol. I., Chap. III. Suppose we have a gaseous
system containing n a molecules of the substance A, and n b molecules
of the substance B. The probability of a collision between a single
molecule of A and a single molecule of B is the product of the fractional
concentration of each, for a collision is analogous to two events
happening simultaneously, which we have just seen depends upon the
product of two single chances, each of which is represented by the
fractional concentration of A and B respectively.
We might regard the problem in the following way. Suppose that
v is the molecular volume, i.e. the actual volume " occupied " by or
allotted to any single molecule of A or B in the mixture. Let V be the
total volume. Then, V = (n a + n^v. Suppose for the moment that
there is only one molecule of A present. Then the chance that this
molecule would occupy a given volume v at a given instant of time
would be the ratio of this volume to the total volume, i.e. the ratio #/V
or i/(n a + n b ). Since there are n a molecules of the substance A actually
present, the chance that any one of them occupies a certain " position "
or space v is given by the ratio n a /(n a + nb). This term is likewise the
fractional concentration of the substance A. The chance that any
B molecule occupies the same position is given by the expression
n b l(n a + n b ). If a molecule of A and a molecule of B occupy the
same position together, this is equivalent to a collision, and hence the
chance of a collision is the product of the fractional concentrations.
If the reaction required say two molecules of A to meet one molecule
of B simultaneously the chance of this occurring is ( - - ) .
\ + n b j n
.
n b n a + n b
which finally takes the form : rate of collision = kC* a - C&, for the total
volume is proportional to the total number of molecules present.
These simple probability ideas may also be used to account for the
influence exerted upon the collision frequency by the fact that in actual
gaseous systems the molecules possess volume, Thus if r is the radius
of a molecule, and / the average distance between two molecules, then
when a molecule moves over a distance / it sweeps out a cylinder the
cross-section of which is irr^ and the length /. The volume of this cylinder
is therefore irr*l. Each molecule has, on the average, a free space
allotted to it which is a cube of volume T 8 . Hence, as far as the radius
affects the question, the chance of one molecule encountering another