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 CATTFORNT\ LIBRARY

•

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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.
With 134 Figures in the Text.

SPECTROSCOPY. By E. C. C. BALY, C.B.E., F.R.S., Grant Professor of
Inorganic Chemistry in the University of Liverpool. With 180 Illustrations.

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
Conductivity, by T. S. MOORE, B.A., B.Sc., Lecturer in the University of
Birmingham.

STEREOCHEMISTRY. By ALFRED W. STEWART, D.Sc., Professor of
Chemistry in the Queen's University of Belfast. With 58 Illustrations.

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 Plt and
that of the second is P2, 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 Pb where Px = a/(a -I- b), and the chance
of drawing, say, a black ball from the other bag is P2, where P2 =
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.

2C/., 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 na molecules of the substance A, and nb 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 = (na + 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/(na + nb). Since there are na 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 na/(na + 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
nbl(na + nb). 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 ( - - — ) .

\«« + nbj n

.
nb na + nb

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 T8. Hence, as far as the radius
affects the question, the chance of one molecule encountering another
is 7T/-2///3 or 7T/-2//2. If the average velocity is u, the time of a journey
between two successive collisions is Iju. Hence the number of en-
counters per second is (ufl) x chance of collision = u-n-r2//3. This

6 A SYSTEM OF PHYSICAL CHEMISTRY

quantity is obviously a constant for a particular gas at constant tempera-
ture, and hence is really included in the proportionality factor or velocity
constant k.

Another illustration of the application of probability considerations
is afforded by the phenomenon of coagulation of colloids by electrolytes
(cf. Whetham, Phil Mag., [v.], 48, 474 (1899). It is supposed that
the coagulation is due solely to the neutralisation of the electrical
charge of the colloid, and that one trivalent ion is exactly equivalent to
i *5 divalent ions, or to three univalent ions as far as coagulating efficiency
is concerned. This is, of course, too simple an assumption, as we
have already seen in the criticism of Schultze's rule (Vol. II., Chap. VIII.).
Adopting this simple view of the effect, however, Whetham has shown
how the numerical values obtained experimentally may be accounted
for approximately. The coagulating power of an ion is defined as the
reciprocal of the number of gram-equivalents of the ion which are just
capable of bringing about coagulation of a given quantity of colloid.
Linder and Picton found that negatively charged colloidal arsenic
sulphide could be coagulated by tri-, di-, or uni-valent cations, the
relative coagulating powers being in the ratios —

1023 : 35 : i.

Let us suppose that in order to produce coagulation a certain
minimum electric charge has to be brought into contact with the
colloidal particle, and that such conjunctions must occur with a certain
minimum frequency throughout the solution. We shall get equal
charges by the conjunction of 2n trivalent ions, or $n divalent ions,
or 6n univalent ions, where n is any whole number. In a solution
where ions are moving freely the probability that an ion is at any instant
at a certain position is represented by a fraction which is proportional
to the ratio between the volume occupied by the sphere of influence of
the ion and the total volume of the solution. The probability is there-
fore proportional to the concentration of the ion, and may be written
as (Ac), where A is any constant. The probability that two such ions
are present together is (A<:)2, and the chance that n such ions are
present simultaneously at the position is (A<r)M. Let us suppose that
three solutions containing respectively tri-, di-, and uni-valent ions at
concentrations C3, C2, and Cj possess exactly the same coagulating
power upon a given amount of a given colloid. Then, since the coagu-
lating power is the same in these three cases —

A2«Ca2« = A3»C23M = A^Cj6" = a constant = B.
Hence, the critical concentration of the trivalent ions is given by —

and similarly C2 = —r- and Q = —r-.

WHETHAM'S RULE OF COLLOID COAGULATION 7
The ratios in which these three concentrations stand to one another are —

Cf~* • f** "D^W • ~D3^ • 15 2#

! . U2 . C8 = ±> . -D . &

= i : B^ : B3*.
Putting B6w = i/jc, the ratios can be written —

These represent numbers which indicate relative concentrations of equal
coagulating power. Hence the relative coagulating powers PI} P2, P3 of
equal concentrations of these three ions are given by the reciprocal of
the above numbers. That is —

P. "D • "P ._ T • <V • T^
-, . Jr 2 • -^3 ~~ •"• ••*•*»«

The value of x, which depends upon a number of unknown factors
characteristic of the colloid considered, cannot be found on a priori
grounds. If we take Linder and Picton's experiments into account and
set x = 32, we get for the relative coagulating powers of univalent,
divalent, and trivalent ions respectively, the values i : 32 : 1024.
It will be seen that these numbers are of quite the same order of magni-
tude as those observed. Whetham predicted on this basis that the
coagulating power of a tetravalent ion on the above colloid should be
a large number, approximately 33000. Recent measurements have
corroborated this result in so far as an extraordinarily large coagulating
power is actually obtained.

The Law of Error. — It is a familiar fact in physico-chemical
measurements that repetitions of a certain measurement give rise to a
series of numbers which are not identical. The variations we speak of
as experimental errors. The measurements are as likely to be too high
as too low, that is, the errors are as often positive as negative, provided
we make a very large number of determinations. (It is assumed that
there is no systematic error in the apparatus or in the method of
measurement.) The treatment of such results so as to obtain the most
probable result, i.e. the most accurate determination, is a further illus-
tration of the application of the theory of probability, somewhat more
complex in nature than that hitherto considered.

Thus, it is possible to construct a probability curve, by means of
the probability equation given below, which has been found to agree
closely with the actual results obtained in a series of experimental de-
terminations of a given quantity. Examples will be found in a text-
book of mathematics, e.g. Mellor's Higher Mathematics. The large
majority of the readings will fall very closely together, i.e. they will
not be far removed from the true result, a smaller number of readings
will be farther away on either side of the true result, and only a very
small number will be much to one side or the other. This distribution
of values may be represented by an expression of the form —

A SYSTEM OF PHYSICAL CHEMISTRY

This is known as the normal law of errors. The curve is shown in
Fig. i. x denotes the error and y the probability of its occurrence.
As x increases numerically, positively, or negatively, y decreases rapidly,
and when x becomes large y becomes vanishingly small. It will be
observed that the curve is symmetrical.

Maxwell's Law. — Maxwell has applied the principle of probability
to the problem of the distribution of velocities among the molecules of
a gas, the gas being in a condition of statistical equilibrium at a uni-
form temperature throughout. A gas is to be regarded as a molecular
chaos, the speed of any molecule varying from zero to infinity, its in-
stantaneous value being the result of chance collisions with its neigh-
bours. Although all values are theoretically possible for the speed of
a molecule, it is found that in a system containing a large number of
molecules, there are very few possessing either very great or very small

Negative Errors. Positive Errors.
FIG. i.

speeds. The majority of the molecules possess speeds which lie within
relatively restricted limits. A numerical illustration taken from Meyer's
Kinetic Theory of Gases is given in Appendix I.

There are, as a matter of fact, two ways of expressing Maxwell's
law of distribution. One of these ways has already been stated in
Chap. I., Vol. I. This way of expressing the law may be put in the
form —

dn = constant x N x

dc

where N is the total number of molecules in the system, and dn is the
number whose speeds lie between the limits c and c + dc. It is to
be clearly understood that the speed here referred to is simply a velocity
magnitude, and no restriction has been introduced as to direction of
motion. The term c is known as the root-mean-square speed (or
r-m-s speed). At a given temperature the system is characterised by
a certain mean or average kinetic energy of its molecules, which is

MAXWELL'S LAW 9

maintained as long as the temperature is maintained constant. This
average kinetic energy may be written as -^Nm?2, where f is the sum
of the squares of the velocities of all the molecules at any given instant
divided by N. This r-m-s speed is not the same thing as the mean or
average speed, which is simply the sum of all the velocities divided by
N, though the two quantities are not very different numerically. The
above way of expressing Maxwell's law, i.e. the above expression, is of
the general form : y = x2e~*2, which, it is to be observed, is not identical
with the normal law of errors. The resulting curve is, in fact, not
symmetrical. The alternative mode of expressing Maxwell's law where-
in the curve is symmetrical, and the distinction between the two modes,
is considered in Appendix I.

Maxwell's, law expresses the distribution of velocities as a continuous
function of the number of molecules present. That is, the speed of one
molecule may differ by any amount (down to the infinitesimally small)
from the speed of any other molecule. When we come to consider
the quantum theory we shall find continuous functions replaced by dis-
continuous ones, i.e. abrupt changes in finite small steps in place of
gradual change in infinitely small steps.1

It will be observed that Maxwell's expression involves the squares of
velocities. Since the kinetic energy of a molecule depends upon the
square of its velocity, it should be possible, on similar lines, to express
the distribution of kinetic energy amongst the molecules constituting
a gas system. If we denote by dn the number of molecules which
possess kinetic energy lying between the limits E and E + <£E, it can be
shown that at a given temperature —

dn = constant x N x e~ElkT . E^ . dE

where N is the total number of molecules, and k the gas constant per
molecule (i.e. k = R/N0, where R is the gas constant per gram-molecule,
and N0 is the number of molecules in i gram-molecule).

Further, potential energy of molecules may be considered as dis-
tributed in an analogous manner. We are here considering the
potential energy of a molecule, free from intermolecular forces (as in
the case of a perfect gas), the molecules being placed, however, in a
field of force and the potential energy being due to this field of force,
which might be conceived of as originating in some external body. A
good example is the earth's gravitational field which acts on a column
of gas in such a way that the molecules are more dense in the lower
portion of the column than in the upper. Denoting potential energy

aThis statement must not be taken as meaning that the distribution of kinetic
energy of translation takes place in the discontinuous manner postulated by the
quantum theory. The quantum theory applies only to energy of the vibrational or
rotational type, i.e. motion with respect to some fixed centre of gravity. Free
translational motion — such as that considered above — must be treated in a con-
tinuous manner, the velocity being capable of changing by an infinitely small
amount.

io A SYSTEM OF PHYSICAL CHEMISTRY

by W, we can state that the number of molecules, the potential energy
of which lies between W and W + <AV, is given by —

dn = constant x N x <rw/*T . dW.

We can integrate this to obtain the number «x of molecules of a
perfect gas (in equilibrium in a field of force at a uniform temperature)
which possess potential energy W15 that is all values from zero up to Wr

If the total number of molecules in the system be N, the expression

is «! = N(i - *-wi/*T).

If we introduce the Avogadro constant N0, i.e. the number of mole-
cules in one gram-molecule, the above expression becomes —

«! = N(i - *-Vi/RT),
where R is the gas constant per gram-molecule.

It follows from this that the number of molecules which possess
potential energy between Wl and infinity is (i - «J which is equal to

N<rwi/*T or N<rNowi/RT.

Distribution of Molecular Velocities and Temperature. — On the kin-
etic theory it is to be expected that the temperature of a gas should
be expressible in purely mechanical terms. We are already familiar
with the concept that temperature is measured by the kinetic energy of
the molecules. In view of the distribution of velocities and therefore
of kinetic energy, among molecules, as expressed in Maxwell's law, it
is evident that the kinetic energy of a given individual molecule may be
very different from that possessed by another molecule of the same
system. Further, the kinetic energy of one and the same molecule
varies from moment to moment as a result of collisions. The tempera-
ture of the system — measured in the ordinary way, by means of a ther-
mometer— is a perfectly definite quantity for the gas system as a whole
in the steady state. The temperature, in fact, is determined by the
average kinetic energy. It is therefore meaningless to speak of the
temperature of a single molecule in a gas. Temperature is essentially
a statistical effect due to the presence of a large number of molecules
each contributing its own share to the total effect. Two independent
systems are at the same temperature when the average kinetic energy
of each is the same. This is true whether the systems be gaseous,
liquid, or solid, homogeneous or heterogeneous.

It will be appreciated at the same time that pressure is likewise a
statistical effect. A single gas molecule cannot be conceived of as
exerting observable pressure, though each molecule exerts a certain
force against the walls of the containing vessel, the total effects of which,
when numerous molecules take part, is manifested as a uniform gas
pressure.

ENTROPY AND THERMODYNAMIC PROBABILITY.

It is proposed to indicate how the second law of thermodynamics
can be deduced on the basis of statistical mechanics. This was first

ENTROPY AND THERMODYNAMIC PROBABILITY n

demonstrated by Boltzmann. Hitherto we have regarded the second
law as a law of experience, its validity depending upon the fact that no
contradiction to it has been met with in nature. It is important to see
that this law possesses at the same time a mechanical basis. The
demonstration consists in showing the connection between the entropy
of a system — the concept of entropy involving necessarily the concept
of the second law — and a statistical quantity known as the thermo-
dynamic probability of the system.

It is necessary to recall first of all what is meant by thermodynamical
equilibrium, as stated in terms of entropy, that is as stated in terms of
the second law. Planck's definition of such equilibrium is as follows
(cf. Planck, Theory of Heat Radiation, English ed., p. 22) : "A system
of bodies of arbitrary nature, shape, and position, which is at rest and
is surrounded by a rigid cover impermeable to heat, will, no matter what
its initial state may be, pass in the course of time into a permanent
state in which the temperature of all bodies in the system is the same.
This is the state of thermodynamic equilibrium, in which the entropy
of the system has the maximum value, compatible with the total energy
of the system as fixed by the initial conditions. This state being reached,
no further increase in entropy is possible."

We know that heat, from the kinetic molecular point of view, is
represented by the kinetic energy of the molecules of a system, the
molecules moving about in a completely chaotic manner as a result of
collisions. Owing to collisions any ordered arrangement which the mole-
cules might be conceived of as possessing initially would be quickly
annulled, and completely disordered distribution, both as to position
and to molecular velocities, would ensue. This represents the direction
of change in any spontaneous or naturally occurring process. That is,
from the molecular standpoint a system always changes from an ordered
to a chaotic state, and the change will go on until the molecular motion
has become as disordered as possible. When this stage is reached,
there is no longer any reason for further change. When equilibrium is
reached the system has at the same time reached a maximum disorder
or " mixed-up-ness ". This involves the idea that a system in equili-
brium possesses a maximum value of the probability of the state, the
probability here referred to dealing with possible modes of molecular
arrangement and velocity. We may call this the thermodynamic prob-
ability.

According to Boltzmann the thermodynamic probability of an ideal
monatomic gas is a number which denotes by how many times or by
how much the actual state of a gas system is more probable than a
state of the same gas system (i.e. possessing the same total energy and
volume) in which the molecules are equally spaced and all possess the
same velocity. This "standard" state represents perfect order or
arrangement of the molecules. It is of course never realised in practice
owing to the disorder brought about as a result of collisions. The
standard state represents the stage farthest away from the equilibrium
state finally attained by the gas, in which final state the system is

12

A SYSTEM OF PHYSICAL CHEMISTRY

completely chaotic. The probability of disorder is very much greater
than the probability of complete order.

The thermodynamic probability of this "standard" state (that is
the probability of complete order) is taken to be unity. It follows there-
fore that the thermodynamic probability of a real equilibrium state is
an integral number usually much greater than unity. Thermodynamic
probability differs therefore from mathematical probability in that the
latter is always a fractional quantity, i.e. it denotes the ratio of the
number of favourable cases to the total number of possible cases.
Thermodynamic probability is proportional to but is not identical with
mathematical probability.

It has just been stated that a system in equilibrium possesses a dis-
tribution or arrangement which is characterised by a maximum value for
the thermodynamical probability of the state. It is necessary before
going further to give a somewhat more concrete idea of what we mean
by states or arrangements and the probability of arrangements. Let us
leave the problem of molecules and turn to a very simple kind of
system which can undergo various arrangements.

Let us suppose that we have two squares or areas denoted by the
symbols I and II, and further let us suppose that we have five letters,
a, b, c> d, e, and we wish to distribute or arrange these letters between
the squares in every possible way. It is evident that all possible ways
are included in the following : —

I.

II.

Nature of Arrangement.

First arrangement : —

ist way of arranging the letters is

abcde

—

All letters in I none in II.

Second arrangement : —

2nd way of arranging the letters is
3rd ,, ,, ,, „

abed
abce

e
d

Four letters in square I ; one
letter in II. There are five

4th „ „. „

abde

c

ways of producing this ar-

5th „ „

acde

b

rangement or distribution.

6th

bcde

a

Third arrangement : —

7th way of arranging the letters is
8th
9th

abc
abd
acd

de
ce
be

Three letters in square I ; two
letters in square II. There
are ten ways of producing

loth „ ,

bed

ae

this arrangement or dis-

nth „ ,

abe

cd

tribution.

i 2th „ ,

ace

bd

I3th

bee

ad

I4th „ ,

ade

be

I5th „ ,

bde

ac

i6th

cde

ab

We now begin with an arrangement similar to the first, but place
all the five letters in square II and none in square I. Similarly we

ENTROPY AND THERMODYNAMIC PROBABILITY 13

can interchange squares I and II in the other two arrangements, so
that in all, there are six arrangements possible in this system, and in
all, thirty-two different ways in which these six arrangements or dis-
tributions may be carried out. Each of these ways is analogous to
a " complexion " on Boltzmann's nomenclature. Every way or com-
plexion is to be regarded as equally probable. It is very necessary to
distinguish between arrangement or distribution and complexion. Thus
in the first case, the arrangement is five letters in square I, none in
square II. There is only one way or complexion of doing this. In
the second arrangement, which consists of four letters in square I and
one letter in square II, there are five ways or complexions of doing
this. It is simply a question of combinations. Thus there are five
letters to be divided in such a way that four are in one square, one in
another square. The number of possible ways of doing this particular

distribution is -=— = 5. In the third arrangement or distribution

we have to distribute the letters so that there are always three in
square I and two in square II. There are ten ways or complexions
corresponding to this single distribution. This number is given by

-i=— =10. Note that the problem is not which letters are in the

2

squares, but how many different ways can they bed ivided to cor-
respond with any particular arrangement, such as three letters in one
square, two in the other.

If instead of five letters we had N letters and divided them between
two squares, in such a way that n letters are in square I and N-« letters
in square II, the possible ways or complexions possible to this particular
distribution are —

|N
|N - n n

If, instead of two squares, we had m squares, the total number of
complexions in a particular distribution would be given by —

IN

where n-^ + n^ + n% + . . . nm = N.

To return to the simple case of five letters arid two spaces. We
have seen that there are six possible arrangement or distributions, viz.
(5) (o); (4) (I); (3) (2); (o) (5); (i) (4); (a) (3). .Each of these
arrangements has its own number of ways or complexions. Thus for
arrangement (5) (o), the number of complexions is one. For arrange-
ment (4) (i), the number of complexions is five ; for arrangement
(3) (2), the number of complexions is ten. Similar numbers of
complexions are found in the remaining arrangements or distributions.

A SYSTEM OF PHYSICAL CHEMISTRY

It is evident that the arrangement (3) (2) has the same number of
complexions as the arrangement (2) (3), and that either of these two
arrangements or distributions possesses the maximum number of com-
plexions. But the number of complexions is identical with the thermo-
dynamic probability of the particular distribution under consideration.
Either of the systems (3) (2) or (2) (3) possesses the maximum prob-
ability. Either distribution represents therefore the equilibrium dis-
tribution. The distribution (3) (2) or (2) (3) is the most likely
distribution of five letters scattered at random between two squares.
The equilibrium distribution in the above system is ten times more
likely than the distribution (5) (o) or the distribution (o) (5), for the
latter only possesses one complexion whilst the equilibrium distribution
possesses ten.

The choice of five letters has somewhat obscured an important
point in that we are led to two distributions of equal maximum
probability. To find a closer analogue to the actual case of molecular
systems containing an enormous number of molecules it is better to
choose as the simplest model the distributions of six letters between
two squares. With this system we have the following distributions
and corresponding complexion-number for each distribution : —

Complexion-
Number.

ist distribution or arrangement

6 letters in I

o letters in II

I

2nd

5

i

6

3rd

4

2

15

4th

3

3

20

5th

2

4

15

6th

I

5

6

7th

«

O

6

i

Total number of complexions = 64

The single arrangement or distribution which contains the maximum
number of complexions is that represented by three letters in each
square. This distribution has therefore the maximum probability, i.e.
it is the one most likely to occur if six letters be scattered at random
over two squares. This distribution is consequently the " equilibrium "
distribution of the constituents of this system which consists of six
letters.

Now let us apply this idea to a system of molecules. In this case
also that distribution or arrangement which has the greatest number
of complexions or ways is the most probable arrangement, its degree
of probability, or simply its probability, being measured numerically
by the number of complexions possible to it. The most probable
distribution is the equilibrium distribution or state or condition of the
system. A system changes in the sense that it tends to approach and
finally reach an equilibrium state. In this state the number of possible

ENTROPY AND THERMODYNAMIC PROBABILITY 15

complexions is a maximum, i.e. greater than the number of complexions
in any other distribution of the constituents of the system. A maximum
number of complexions is identical with the idea of maximum disorder
or maximum molecular chaos.

In all spontaneous processes the thermodynamical probability tends
to reach a maximum. But on purely thermodynamical grounds we know
that in spontaneous processes the entropy of a system tends towards
a maximum value consistent with the total energy of the system. It
follows therefore that there must be some close relation between the
thermodynamical probability of a state and the entropy of the state.
We can express this by writing —

S= F(w),

where S is the entropy of the system in any state, not necessarily the
equilibrium state, w the thermodynamic probability of the same state,
and F is some function still to be determined. To determine the nature
of F, let us suppose that we have two independent systems, each one
in a definite state, the entropy of the first being denoted by Si, the
probability of the state or arrangement of the first system being w^, the
entropy of the second system being S2, and the probability of the state
of the second system being ze/2. We then have the relations —

51 = F(wl)

52 = F(wz).

The total entropy S of the two systems taken together is the sum of
the separate entropies. That is —

S = Sj + S2 = F(w1) + F(w2).

Since the particular state or arrangement of the first system can be
realised by selecting any one of the wl complexions (contained in or
characteristic of that arrangement or state) and similarly for the second
system, it follows that the state or arrangement of the combined
system can be realised by selecting any one of the w\ complexions of
the first and combining them with the w2 complexions of the second.
That is, the compound arrangement is obtained by selecting any one
of the wl and zv2 complexions. That is the probability w of the
compound state is w: x wz.

But for the compound system we have the relation : S = F(w).
Hence from the above we get : S = F(z£/j . w2). But we have already
seen that S = F(w1) + F(wz). Hence, F(wl . w2) = Ffa) + F(o;2).

The only function which will satisfy this relation is the logarithmic
one, i.e. log xy = log x + log y.

Hence, the connection between the thermodynamic probability and
the entropy of a system is given by the relation : —

Entropy oc loge Probability
or S = k log, W,

where k is a constant independent of the chemical nature of the system

1 6 A SYSTEM OF PHYSICAL CHEMISTRY

and independent of the conditions under which the system is examined.
A more general form of the above expression is —

S = k log W + constant.

We have now to find out the significance of the universal constant k.

To do this we make use of a statistical expression arrived at by
Boltzmann for the entropy of a perfect monatomic gas. A monatomic
gas is one in which the total or internal energy U is due entirely to the
kinetic energy of the molecules. Boltzmann's expression is —

S = 3/2£N log U + m log v + K

where U and k are defined above, N is the number of molecules in the
system, v the volume of the system, and K is a constant independent of
the energy and volume but involving the number and mass of the
molecules. It has already been shown in Volume II. that the following
purely thermodynamical relation holds good : —

*S J W

yr ~ T Vr

Differentiating Boltzmann's expression for the entropy of a perfect
monatomic gas we obtain : —

as , ,i au

Hence U = 3/2/&NT.

If N be taken as denoting the number of molecules in one gram-mole-
cule, then U denotes the total energy of one gram-molecule of monatomic
gas.

But we have already seen (Vol. I.) that in the case of a perfect gas,
the total kinetic energy of all the molecules forming one gram-molecule
is 3/2RT, where R is the gas constant per gram -molecule. Further
in the case of a monatomic gas the internal energy is entirely kinetic.
Hence for one gram-molecule of a monatomic gas : U = 3/2 RT.

It follows therefore that —

N£ = R,

or k is the gas constant per single molecule.

Further, in the case of a perfect gas, 3/2 R = Cv where Cw is the
gram-molecular heat at constant volume. Hence the equation of
Boltzmann for the monatomic gas becomes —

S = Cw log T + R log v + K!

where S now denotes the entropy of one gram-molecule. This expres-
sion is in complete agreement with that already deduced in Volume II.
on thermodynamical grounds, viz. : —

S = C Jog T + R log v + Si

if we identify Si with Kj. This constant represents the value of the
entropy under certain conditions. On purely thermodynamical grounds

ENTROPY AND THERMODYNAMIC PROBABILITY 17

it is legitimate to consider that the entropy may assume in general any
value whatsoever, positive or negative, and that therefore all that we can
measure is the change in entropy resulting from a physical or chemical
process. It will be pointed out later in dealing with Nernst's Heat
Theorem, that according to Planck, the Heat Theorem itself is equivalent
to regarding the entropy of all substances as zero at the absolute zero
of temperature, and possessing therefore a positive value at all other
temperatures.1 This gives us a starting point from which to calculate
not only change in entropy but its absolute value under given conditions.
This likewise agrees with the simplified expression : S = k log W in
which W has been defined as a quantity greater than unity, and con-
sequently S is essentially positive. Of course if we retain the constant
in the expression : S = k log W + constant, the value of S may be
positive or negative depending upon the magnitude and sign of the
integration constant whether VV itself is greater than unity or not.
Classical statistical mechanics, which did not attempt to assign any
particular limit to the value of the entropy, is represented by the above
expression. If we assume with Planck that the integration constant is
zero, and remembering that W as defined above is greater than unity, it
follows that S is a positive term becoming zero at absolute zero. This
is equivalent to assuming the quantum hypothesis.

The general position which we have now reached as a result of the
considerations dealt with in this section may be summarised as follows : —

The second law of thermodynamics, regarded as a law of experience,
states that, whilst work may always be completely converted into heat,
heat on the other hand cannot be completely converted into work. In
other words, all natural spontaneous processes are thermodynamically
irreversible. In mechanics we deal only with reversible processes, and
from the standpoint of mechanics alone we would expect heat to be as
readily convertible into work as work into heat. Since this is not the
case there must be something characteristic of molecular systems to
which the irreversibility is due. This " something " is discovered in the
fact that heat consists of a chaotic motion of the molecules, and that as
a result of collisions this motion tends to become as chaotic or disordered
as possible. In other words, the irreversibility which finds expression in
the second law of thermodynamics is due essentially to the fact that
ordered motion always tends, of its own accord, to become disordered,
and chaotic motion never tends, of its own accord, to become ordered.

This statement is a statement of the second law of thermodynamics
not expressed simply as a result of experience but in terms of statistical
mechanics. We have therefore found a mechanical basis for the second
law.

It is obvious at the same time why Gibbs gave the significance

1 According to Planck this assumption is "the very quintessence of the hypo-
thesis of quanta ". It must be pointed out that whilst this assumption makes the
theory of quanta and Nernst's Heat Theorem agree, it is not essential to the deduc-
tion of the heat theorem itself, which only requires that the entropy of all substances
at absolute zero shall be the same, but not necessarily zero.
VOL. IIL 2

1 8 A SYSTEM OF PHYSICAL CHEMISTRY

" mixed- up-ness" to the concept of entropy; for maximum mixed-up-
ness means on the mechanical view maximum stability or equilibrium,
and hence when equilibrium exists the entropy is a maximum. The
same idea is involved in the term "run-down-ness" employed by
Tolman as a descriptive mechanical term for entropy. The latter term
appeals to the mind as being particularly applicable to a chemical re-
action which proceeds in order to attain an equilibrium state at which
the system will, chemically speaking, have completely " run down ".

THE PRINCIPLE OF EQUIPARTITION OF KINETIC ENERGY.

This principle, which has been deduced on the basis of statistical
mechanics by Maxwell and by Boltzmann (cf. Jeans, Dynamical Theory
of Gases), has already been employed in Chap. I. of Vol. I. in dealing
with Perrin's method of determining the Avogadro Constant from
measurements made upon emulsions. The principle states that in a
system consisting of a large number of particles (e.g. molecules) the
kinetic energy is on the average equally distributed amongst the various
degrees of freedom possessed by these particles. The first point with
which we have to deal is the term degree of freedom.

A degree of freedom is represented by a co-ordinate. We may
take the term as meaning an independent mode in which a body may
be displaced or a possible mode or direction of motion. We shall con-
sider briefly the problem of the number of degrees of freedom possessed
by bodies in the gaseous and solid states of matter respectively.
First of all a word about energy in general. We are familiar with the
two kinds of energy which material systems may possess, namely, kinetic
and potential. These " kinds " of energy are not to be confused with
the "types" of energy of which we are about to speak. A "type"
may consist of kinetic and potential energy together, or simply kinetic
alone. A gas molecule can possess theoretically three types of energy,
each of which is a function of the temperature : (i) Energy of Transla-
tion ; (2) Energy of Vibration, and (3) Energy of Rotation.

1. Energy of Translation. — This type of energy is possessed in virtue
of the free translational motion of the molecules along free paths
throughout the whole of the system. The energy in this case is entirely
kinetic. It has been represented diagrammatically in Chap. I. of Vol. I.
in connection with Perrin's work on Brownian movement. Every gas
molecule possesses translational motion and consequently translational
energy. Since the direction of any movement of this kind can be repre-
sented by the three space co-ordinates, we conclude that translational
energy corresponds to three degrees of freedom. The energy of transla-
tion is the same for any moving molecule at a given temperature.
Further, such a molecule possesses just three degrees of freedom in re-
spect of translation whether the molecule be monatomic or polyatomic ;
the constitution of a molecule does not enter into the question of its
translational energy.

2. Energy of Vibration. — This type of energy exists in virtue of the

EQUIPARTITION OF KINETIC ENERG Y 19

oscillations of particles with respect to a fixed centre of gravity. Vibra-
tion is only possible when there is a restoring force acting upon the
particle which tends to make it take up a mean position. The existence
of the restoring force is the factor which distinguishes vibration from
translation and also from rotation to be mentioned later. We meet
with vibration in the case of the atoms inside a molecule of a gas, and
likewise in the case of the atoms composing a solid. Vibration is
always a constrained movement. We can conceive of vibration as
corresponding either to one or to two or to three degrees of freedom.
When we speak of vibration in connection with molecules it is always
to be understood that we are referring to the vibrations of the atoms
inside the molecule. One such atom can vibrate with respect to the
other in the case of a diatomic molecule. In this case the vibration
is linear, that is, it is along the line joining the two atom centres.
Since the vibration is linear there is one degree of freedom in this case.
Linear vibration is represented in diagram (a) Fig. 2.

Figure Axis.

Mean Position.

Mean Position.

FIG. 2 (a). — Linear vibration of atoms in a diatomic FIG. 2 (6). — Circular

molecule. One degree of freedom. Energy, libration or spin of

kinetic + potential. an atom round a

centre of gravity.
Two degrees of free-
dom. Energy, ki-
netic + potential.

It is also conceivable that the vibration of the particle, an atom
or an electron, may be circular. That is, the particle may describe a
circular path about a centre of gravity, the orbit being traced out on a
surface. Hence in such a case there are two degrees of freedom to be
attributed to the vibration. This is represented by the spinning motion
shown in diagrams (b) and (c), Fig. 2. Further, in the case of a mon-
atomic solid (such as a metal) the only type of motion which can be
ascribed to the atom is vibration. Free translation cannot exist, for
if it did the solid would not retain its crystalline form. In this case
the vibration of every atorn can take place in three directions in space
with respect to the centre of gravity or mean position of the atom.
There are therefore three degrees of freedom to be attributed to the
vibrations of the ultimate particles in a solid. This is represented in
diagram (d), Fig. 2. The characteristic feature of all vibrations is the
existence of potential as well as kinetic energy. As will be shown later

20 A SYSTEM OF PHYSICAL CHEMISTRY

each complete vibration on the average contains just as much potential
energy as it does kinetic.

To return to the vibrations of the atoms in a gas molecule. In the
case of a diatomic molecule the molecule is said to possess one degree
of freedom in respect to the to-and-fro vibration of one atom with
respect to the other. The molecule also possesses three degrees of
freedom in respect of translation of the molecule as a whole. In a
triatomic molecule the atoms ABC probably vibrate in pairs, AB,
BC, CA, each pair functioning like a diatomic molecule, so that the
molecule as a whole has three degrees of freedom in respect of atomic
vibrations. The molecule possesses in addition three degrees of
freedom in respect of translation. Hence, in the case of a triatomic
molecule there are three degrees of freedom in respect of translation
and three in respect of vibration.

It is obvious that no atomic vibration is possible in the case of a
monatomic molecule in a gas. If such an atom is displaced there is no
restoring force ; the atom is not connected to any other as in the case

Figure Axis^
also Axis of Spin"
^' (Circular Vibration.)

\ i

FIG. 2 (c). — •* Spinning molecule." Two degrees of freedom. Energy, FIG. 2 (d). — Trilinear
kinetic +' potential. (Motion not maintained by collisions.) vibration of an atom

in a solid. Three
degrees of freedom.
Energy, kinetic +
potential.

of diatomic molecules, and any displacement would simply be identical
with tree translation. Contrast this with the behaviour of monatomic
molecules in solids in which any displacement can only take place
against a restoring force, with the result, as already mentioned, that- the
vibration in the case of a solid possesses three degrees of freedom. In
the case of a monatomic gas vibration is impossible, and, so far as we
have gone, we can only ascribe to this kind of molecule energy of the
translational kind. A difficulty crops up when we come to consider
a monatomic gas molecule in the light of the third possible type of
motion, viz. molecular rotation.

The circular vibration represented by diagram (c), Fig. 2, requires
a little more consideration. The amount of energy represented by a
circular vibration or spin depends upon the square of the rate of spin
and upon the moment of inertia of the spinning particle.1 If a di-

1 If a particle of mass m moving in a circle of radius r round a fixed position
with an angular velocity o> (u> being the number of radians swept out by the
particle per second), then the speed of the particle is <ar and its kinetic energy
E = 1/2 m . «ara. This expression can also be written : E = 1 . 1/2 . w2, where i = mr*.
The quantity I is called the moment of inertia of the particle. The dimensions of

EQ UIPARTITION OF KINETIC ENERG Y 2 1

atomic molecule were really represented by the diagram (<:), Fig. 2,
i.e. if the atoms were massive spheres extending a sensible distance
from the axis, the energy of the motion represented in the diagram
would be sufficiently large to make its presence felt in the molecular
heat of the gas. As a matter of fact this type of circular vibration does
not affect the molecular heat sensibly. This arises from the now
accepted conclusion that practically the entire mass of any atom is
concentrated in a nucleus situated at the centre of the atom, the di-
mensions of the nucleus being small even compared with those of the
atom (cf. Chap. V., the Rutherford-Bohr atomic model). Hence, in
the motion represented in diagram (c\ Fig. 2, the mass of the two atoms,
i.e. the mass of the whole molecule is practically all concentrated upon
the axis of circular vibration (the figure axis in this case), and the
moment of inertia of the atoms, and consequently the energy in respect
of this motion is negligible, because the r term referred to in the foot-
note is practically zero. Further, as shown in Chap. V., each molecule
possesses a number of electrons likewise spinning, as in diagram (c),
Fig. 2. In the case of the electrons the distance r is not negligible,
but on the other hand the mass of the electron is so small a fraction of
the total mass of the molecule that again the moment of inertia is
small and the energy of electronic spin does not enter sensibly into the
" ordinary" energy content of the molecule, the variation of which
(energy) with temperature is given by the molecular heat. The mole-
cular heat term is due to energy of translation, of linear vibration and
of molecular rotation or its equivalent, precessional vibration (cf. infra).
The electron spin enters into the question of the ultra-violet spectrum
of the gas, cf. Chap. V. It must be assumed, of course, that such a
spin as that represented in the diagram (c)t Fig. 2, takes place in all
cases. It is scarcely affected by temperature, however, and conse-
quently does not enter into molecular heat values, except in the limit
when the temperature is very high.

3. Energy of Molecular Rotation. — If a molecule resemble a solid
sphere we would expect it to rotate in the manner indicated in Fig. 3,
diagram (a}. The rotation of a sphere can be referred to three axes
of rotation, i.e. there are three degrees of freedom. The energy is
entirely kinetic. Molecules, however, are not necessarily spherical
unless they contain a number of atoms. It is believed that at least
three atoms must be present in a molecule before we can possibly
ascribe to the molecule as a whole the limiting number (3) of degrees
of freedom in respect of rotation.

It is a remarkable fact that monatomic gas molecules do not appear
to possess rotational energy. This conclusion rests upon the experi-
mental fact that the molecular heat of argon and other monatomic
gases and metallic vapours can be accounted for by simply assuming
translational energy. This point will be dealt with later. A monatomic

the particle in the above case are supposed to be small compared with r. In the
case of a solid sphere the moment of inertia of the sphere can be shown to be
2/5 . M . r2, the axis of rotation passing through the centre of the sphere.

A SYSTEM OF PHYSICAL CHEMISTRY

gas molecule appears to function simply as a massive point, and not
as a massive sphere. This is a remarkable conclusion. It is self-
evident that vibration of the atom cannot occur in the case of a mon-

Circular Orbit

FIG. 3 (a). — Sphere with three com- FIG. 3 (6). — Molecular rotation of diatomic molecule
ponent rotations. Three degrees (due to collisions). Two degrees of freedom,
of freedom. Energy, kinetic. Energy, kinetic.

c'>cular Orbit.

FIG. 3 (c).—Precessional
vibration (due to col-
lisions). The molecule
possesses gyroscopic
properties. Two de-
grees of freedom. En-
ergy, kinetic.

C/rcular Orbit

FIG. 3 (d). — Extreme precessional vi~
bration (due to collisions at high
temperatures). Merging into or-
dinary molecular rotation.

atomic molecule. It is by no means clear why rotation of the monatomic
molecule as a whole apparently does not occur. It is possible that
rotation is actually occurring but that the energy term corresponding
thereto is negligibly small for the same reasons as those advanced in

EQ UIPARTITION OF KINETIC ENERG Y 23

considering the circular vibration of diagram (c\ Fig. 2. This point
has not yet been settled.

Molecular rotations are ascribed to collisions with other molecules.
This distinguishes molecular rotation from atomic vibration as far as
origin is concerned. Atomic vibration can indeed be affected by
collisions, but the origin of atomic vibration is more deeply seated, so
to speak, than that of molecular rotations. Presumably atomic vibra-
tion is set up as a result of absorption of radiant energy on the part of
the molecule. This might occur at a collision, but not necessarily so ;
unless the collision be very inelastic (cf. Chap. VI., the section dealing
with resonance and ionisation potentials).

When a molecule consists of more than one atom it seems reason-
able at first sight to ascribe to the molecule as a whole a certain amount
of rotational energy. The reason for the qualifying clause will not be
given now ; the point is taken up in Chap. IV. in connection with the
theory of the molecular heats of gases. As Kriiger has shown, in place
of true molecular rotations we may have to substitute another kind of
motion, namely, precessional vibrations.1 The possible kinds of rota-
tions and precessional vibrations in the case of a diatomic molecule
are illustrated in Fig. 3, diagrams (b) to (d). It must be understood
that rotation or precessional vibration always refers to the molecule as
a whole. This is in contrast with the view taken of true atomic vibra-
tions which have been discussed.

As regards the rotation of a diatomic molecule the diagram (£),
Fig. 3, shows us that there are two degrees of freedom, i.e. there are two
co-ordinates at right angles defining the surface over which the rota-
tion of such a molecule can take place. The axis of rotation is at right
angles to the plane indicated in the diagram.

From the point of view of the internal molecular energy the signifi-
cance of the rotation of the molecule depends upon its moment of in-
ertia, i.e. the moment of inertia of each of the atoms with respect to the
axis of rotation and the number of revolutions v which the molecule
makes per second (as a result of collisions). The rotational energy is
given by the expression 1/2 . I . (27rr)'2, where I is the moment of
inertia and v has already been defined.

1 The term " precessional vibration " requires perhaps a word of explanation.
The type of motion represented by the term is shown in diagrams (c) and (c/), Fig. 3.
It is similar to the true precession of a gyroscope, but with this difference that a di-
atomic gas molecule does not precess in a definite field of force. It is assumed, in
fact, that there is no field of force. Precessional vibration (which must not be con-
fused with nutation) is brought about in the case considered by the collisions of
molecules with one another, each molecule being assumed to possess gyroscopic
properties in virtue of the electrons which it contains, and which spin [circular
vibration] around the figure axis as already indicated in diagram (c), Fig. 2. If no
collision occurred a diatomic molecule would spin with a fixed axis. As a result of
collision the spin is disturbed and the figure axis itself describes the circular motion
represented in diagram (c), Fig. 3. It maintains this type until a further collision
occurs which sets up a new vibrational precession, the figure axis now precessing
round a circle which has a greater or less circumference than before. Such pre-
cessional vibrations -might be more accurately described as Poinsot movement, dealt
with in rigid dynamics. The introduction of the idea into molecular motion and
molecular heat is due to Kruger, cf. Chap. IV.

24

A SYSTEM OF PHYSICAL CHEMISTRY

In the case of a diatomic molecule we have therefore three degrees
of freedom in respect of translation, one degree of freedom in respect
of linear vibration of the atoms, and two degrees of freedom in respect
of molecular rotation ; six degrees in all. If we regard rotation as im-
possible we have to substitute precessional vibration of the molecule
which corresponds to two degrees of freedom ; again six degrees of
freedom in all.

In the case of a molecule containing more than two atoms, if rota-
tion be possible at all, it is reasonable to ascribe three degrees of
freedom in respect of such rotation, i.e. the limiting number of degrees
of freedom. At the same time the molecule possesses three degrees of
freedom in respect of translation and three degrees of freedom in respect
of atomic vibrations in pairs ; in all nine degrees of freedom. In the
case of translation and of rotation l the energy is entirely kinetic ; in
the case of vibration it is potential as well as kinetic. The possible
degrees of freedom in various kinds of gas molecule are summarised in
the following table which is due to Bjerrum (Zeitsch. Elektrochem., 17,
731, 1911) ; the general correctness of the assigned values is borne out
to a certain extent by the values of the molecular heats of the respective
gases : —

No. of Atoms in
the Molecule.

Number of Degrees of Freedom.

In Virtue of
Translation.

In Virtue of
Rotation.

In Virtue of
Vibration.

I
2

3

4

3
3
3
3

O
2

3
3

O

I
6

Our knowledge of the liquid state is so scanty that it has not been
possible hitherto to assign any definite value to possible number of
degrees of freedom. A liquid molecule must, of course, possess three
degrees of freedom in respect of translation ; it must likewise possess

1 In regarding the rotation of a diatomic molecule as representing kinetic
energy alone, we are assuming that the bond between the atoms is quite firm. lit
general, however, we may expect a certain amount of " give " due to the atoms
being " centrifuged " from one another in the rotation. If this occurs it will
involve a certain amount of potential energy. Bjerrum (loc. cit.) finds that the kinetic
energy EK of molecular rotation is related to the potential energy EP by the
equation —

Ep/Ek = RT/4lrVMr2

where v is the frequency of internal vibration, M is the mass, and r the radius of the
molecule. Taking the case of oxygen at ordinary temperatures where v — io14,
r = io~8, and M = 32, it is found that —

That is, the potential energy of rotation is negligible compared with the kinetic
energy.

EQ UIPART1TION OF K1NE TIC ENERG Y 2 5

an unknown number in respect of rotation and vibration. We pass on,
therefore, to the solid state. In the case of a solid, i.e. a crystalline
substance, and probably also in the case of a super-cooled liquid like
glass, it is necessary to regard translational energy (and also rotational
energy) as absent, the energy possessed by the molecules of a solid
being vibrational. As already stated, vibration of each atom can take
place along all three axes, so that even the simplest type of solid
possesses three degrees of freedom. We shall return to this later.

For the present we have to take up the application of the principle
of equipartition of (kinetic) energy amongst degrees of freedom. Ac-
cording to this principle, when a system is in statistical equilibrium, such
equilibrium being determined by a number of variables, i.e. degrees of
freedom, to each such variable one must attribute the same quantity of
(kinetic) energy. It is to be remembered that there is no restriction as
regards the physical state of the system — the principles applies equally
well to gaseous, liquid, or solid systems and to systems embracing two
or more such states simultaneously. It is supposed to hold equally
well also for degrees of freedom in respect of translation, vibration, or
rotation. To see how much this energy amounts to per degree of
freedom, let us consider the kinetic energy of translation of a perfect
gas at a given temperature. First of all we have the relation —

PV = RT.

Further, we have seen in the theoretical deduction of Boyle's Law
that —

P = JP*2

where p is the density of the gas and u the root-mean-square velocity. If

molecular weight (M)
we are considering i gram-molecule of the gas, p = ^ -— ,

where V is the molecular volume, and hence we can write —
PV = JMw2 = JNww2

where m is the mass of a single molecule and N the number of mole-
cules in i gram-mole. Taking into the consideration the first equation,
we obtain —

RT = jMw2 = JNww2.

Now the kinetic energy of a single molecule is \mu^, and hence the
kinetic energy of i gram- mole is -£N#m2. That is the kinetic energy
of i gram-mole is f RT, and this holds good whether the gas be mon-
atomic or polyatomic. Now a gas molecule has three degrees of
freedom in virtue of translation, and hence by applying the equiparti-
tion principle, EACH DEGREE OF FREEDOM POSSESSES KINETIC ENERGY

EQUAL TO £RT.

This is the quantitative form of the principle of equipartition of
energy between degrees of freedom. The principle does not mean that
in a gas system, for example, the kinetic energy of translation of every

26 A SYSTEM OF PHYSICAL CHEMISTRY

molecule shall be exactly 3/2 £T, where k is the gas constant per single
molecule. We know from Maxwell's distribution law that the kinetic
energy varies in general from molecule to molecule. What the equi-
partition principle does mean is that on the average the kinetic energy
of a molecule is 3/2/£T.

Let us now consider the special case of a monatomic gas. As we have
already seen, it can possess no vibrational energy : whether it possesses
rotational energy is at this stage a doubtful point. Its energy, due to
translational motion, for i gram-mole at T° abs. is fRT, when R = 1*985
cals. Now the specific heat Ct, at constant volume is simply the increase
in the total energy per degree. If we are dealing with i gram-mole or
gram atom as unit of mass, the heat term will be the so-called molecular
or atomic heat Cv. It is clear from definition that the increase in the

(kinetic) energy of translation1 per i° rise in temperature = ^p(fRT)

= f R = 2-98 cals. per mole. Recent experiments of Pier in Nernst's
Laboratory (Zettsch. Elektrochem.^ 15, 546, 1909; 16, 897, 1910) have
shown that the molecular heat of argon is 2 -98 cals. per mole, and fur-
ther that this is independent of temperature. This agreement suggests
that one should neglect the rotational energy of the monatomic mole-
cule' (vibration of atoms inside the molecule is naturally impossible
since the gas is monatomic). In fact the monatomic molecule seems
to function as a massive point. Agreement of this order between cal-
culated and experimental values is, however, not found in other cases.
Thus, taking the case of a diatomic gas, the number of degrees of free-
dom in virtue of translation is again 3. The number of degrees of
freedom in virtue of vibration we have considered as i ; that is 4 degrees
in virtue of translation and vibration. The corresponding kinetic energy
of such a molecule will be 4 x -£RT == 4-0 T cal./mole, if the law
of equi partition be assumed. Of course this does not represent all the
energy due to translation and vibration. In vibrations we have poten-
tial energy as well as kinetic which must be taken account of. It can
be shown by a simple calculation that the potential energy of a particle
undergoing what we might call " circular vibration " is just equal to the
kinetic energy of the vibration. The calculation is as follows : —

Consider a particle whose mass is m travelling in a circle round a
centre of gravity with velocity u. Suppose r is the radius of the circular
path. A motion of this kind involves the action of two opposing forces,
one tending to draw the particle towards the centre, its position of rest,
the other due to the motion of the particle tending to make it fly off at
a tangent. The two forces must just balance in order to make the
circular movement permanent. Let us suppose that the diameter of
the circle, i.e. the amplitude of the vibration, to be so small that the
force tending to draw the particle back to the centre is proportional to
the distance of the particle from the centre. This very simple law of

1 Energy of translation is necessarily entirely kinetic, and it is to the distribution
of kin> tic energy amongst degrees of freedom that the law of equipartition is properly
to be applied.

EQUIPARTITION OF KINETIC ENERGY

27

attraction can only hold when the particle is not far removed from the
centre,/.«?. when r is small. If A is this attractive force per unit dis-
tance from the centre, then on the assumption just made, the force
acting inwards at a distance r is Ar. We have now to calculate this
force in terms of the motion of the particle. Consider the particle
traversing the circular path shown in the figure (Fig. 4).

Suppose that the particle is at the point a, travelling with a velocity
u in the direction ai. After a short interval of time 8t it is at b travel-
ling with a velocity u in the direction T£. The arc ab = «S/, and so the
radian measure of the angle aob or nd is 86 = u8t/r. If TV = id= u, then
the velocity has changed from one represented in magnitude and direc-
tion by ic to one represented by id. By the triangle of velocities, the
change in velocity is represented in magnitude and direction by the line
cd. The direction of cd is the same as that of TO ; and its magnitude is
2U sin 80/2. In the limit when S/ is infinitesimally small sin 80/-2 =- 80/2,
or the change of velocity is u86, /.<?., «28//r.

Hence the acceleration inward (viz. the velocity inward divided by
8t is equal to u2/r. But force = mass x acceleration. Hence the
force acting inwards and preventing the particle from flying off is mu*lr.
This must be identical with Ar. That is— r

mu2/r = Ar.

Further, the potential energy of the particle at a (namely, the work
which must be done upon the particle to bring it from the position of
rest, the centre of the circle, to the point a on the circumference) is the
product of the force acting into the distance traversed. The force varies
at every stage of the radius, so that it is necessary to integrate the work
expression for each increment dr in order to obtain an expression for the
potential energy of the particle at the point a. That is the potential

energy of the particle when it is on the circumference is f hrdr which

Jo

28 A SYSTEM OF PHYSICAL CHEMISTRY

Ar2
is "—. But it has just been shown that Ar = mu^/r. Hence the po-

Ar2

tential energy, viz, - — is equal to (i/2)mu2. But the term (i/2)mu2 is

the average kinetic energy of the particle. Hence in a complete circular
vibration the kinetic energy is just equal in magnitude to the potential energy.

We return now to the question of the diatomic molecule.

The linear vibration executed by one atom with respect to the other
involves one degree of freedom, i.e. the line of junction of the two atoms.
On the equipartition principle the kinetic energy involved per mole is
|RT. Since there is likewise an equal amount of potential energy, the
total energy due to vibration is iRT. Adding the amount due to the
(kinetic) energy of translation of the diatomic molecule as a whole, viz.
f RT, we obtain f RT as the total energy due to translation and vibra-
tion. That is, the rise in this energy per i° is fR = 5*0 cal. per mole
per degree, taking R = 2 cals. If now we take rotation of the molecule
as a whole into account we again have two degrees of freedom, to which
one must assign RT units of kinetic energy. The total energy of a
diatomic molecule, provided the law of equipartition is true, and pro-
vided ALL the degrees of freedom are effective, should be ^RT, and the
molecular heat therefore JR = 7*0 approx.

Experiment shows, however, very different values. For hydrogen
at o° C. the molecular heat Cy = 4*9 to 5*2 cal. per degree, and at
2000° C., Cv = 6*5 cal. (cf. Nernst, Zeitsch. Elektrochem., 17, 272, 1911).
For nitrogen at o° Cv = 4 '84, and at 2000° Cv = 67. For chlorine
at o° Cv = 5-85, at 1200° Cv = 7*0. For oxygen at o° Cv = 4-9, at
2000° Cv = 6*7. These values are only approximately correct. It is
evident, however, that not only is there lack of agreement in the numeri-
cal values between those observed and those calculated at lower tem-
peratures, but the fact that the molecular heat of diatomic gases varies
.considerably with the temperature is quite unaccounted for by the theory of
equipartition unless, indeed, the number of degrees of freedom is a
function of the temperature ; which is difficult to believe.

In the case of triatomic gas molecules the degrees of freedom in
respect of translation are three, the [kinetic] energy corresponding being
|RT. As regards vibration, there are possibly three vibrating pairs
each with one degree of freedom, corresponding to the quantity
3 x £RT of kinetic energy. To this has to be added an equivalent
amount of potential energy, making 3RT as the total energy term in
respect of vibration. Hence translation and vibration apparently entail
f RT units of energy, and the increase in this for i° rise in temperature
is fR = 9*0 cal. per mole per degree. Logically we should likewise
add a term for rotation of the molecule as a whole, which we have seen
amounts to |RT. The observed molecular heat Cv for CO2 at 18° is
7-09, and this becomes 10-47 at 2210° (Pier, I.e.). For water vapour
Cwat 50° C. = 5-96 (Nernst and Levy), and at 2327° C., Cv = 9-68
(Pier). Again the discordance between theory and experiment is very
apparent since theory predicts a constant molecular heat of either 9 or
12 cals. per mole per degree.

ATOMIC HEAT OF SOLIDS 29.

The Ratio y for Gases, and the number of " effective " Degrees of Freedom.

It may easily be shown (cf. Meyer's Kinetic Theory of Gases, p. 140

/-»
sea.) that the ratio f=r or y can be expressed approximately in the form —

where n is the number of degrees of freedom of the gas molecule. In
the case of monatomic gases the value of y is 1-666, and this is the
quantity which is obtained on putting n = 3 in the above expression.
That is, a monatomic gas molecule possesses three degrees of freedom,
in respect of translation only. This agrees with the conclusion we
came to above (though, of course, it must be remembered that the actual
number of degrees is possibly greater than this, i.e. degrees in virtue of
rotation, but for some unknown reason only some are effective in regard
to heat capacity, for after all an atom is not a point in the mathematical
sense). In the case of a diatomic molecule the value of y, found by
experiment, is in many cases 1*4, and this will correspond to putting
n = 5 in the above equation. In triatomic gas molecules y = i -3 in
general, and this makes n = 7. These numbers are certainly less than
the actual number of degrees of freedom possessed by di- and triatomic
gases, as is indeed shown by the fact that there are some diatomic
gases with as small a value as 1*29 for y (iodine vapour) and some
triatomic gases, e.g. CS2, for which y = 1-2. Further, the values of-y
are not constant but vary with the temperature. The whole problem of
the number of degrees of freedom is, therefore, in a very unfinished;
state.

The doubt which exists, in the case of polyatomic molecules, re-
garding the, true number of degrees of freedom, takes away from the
force of the criticism levelled against the principle of the equipartition,
on the ground of the lack of agreement between observed and calculated
molecular heats. The soundest criticism of the principle rests on the
experimental observation that the molecular heat varies with the tem-
perature, whilst the principle leads us to expect it to be constant, no
matter how many degrees of freedom be present.

Atomic Heat of Solids.

Since the equipartition principle is considered to hold equally well
for all states of matter, the term -JRT must represent the kinetic, energy
per degree of freedom of i gram-mole of any system at the tempera-
ture T. In solid elements — the metals — it is generally agreed that the
gram-molecule and gram-atom are identical, so that in such a case
•£RT is the kinetic energy per degree of freedom for each gram-atom,
R being put equal to i '985 cals. The orientation of the atom of a
metal is such, that it is free to vibrate in any direction which can be

A SYSTEM OF PHYSICAL CHEMISTRY

resolved in terms of the three space co-ordinates X, Y, Z. Such a
particle possesses three degrees of freedom l (cf. Fig. 2, diagram (</)).

The kinetic energy in virtue of this " space " vibration is evidently
|RT, and since any vibration 2 involves an amount of potential energy
equivalent to the kinetic, the total energy of i gram-atom of a mon-
atomic solid is 3RT, according to the principle of equipartition of energy.
Now the atomic heat C» at constant volume is simply the change of
total atomic energy per degree, that is —

C, =

= 3R = 5-955 cals.

The application of the equipartition principle has therefore led to the
conclusion that the atomic heat of monatomic solids should be a constant^
viz. 5*955, the same for all monatomic solids and independent of tem-
perature (the term 3R does not contain T ; though, of course, the
total energy present in the solid at any given temperature depends on
this temperature, viz. 3RT in the above case). This result is practi-
cally identical with Dulong and Petit's experimentally discovered law,
and this agreement is one of the most striking pieces of evidence in
favour of the equipartition principle. The following table gives the
values of atomic heats for a number of elements at 20° C. (cf. G. N.
Lewis, Journ. Amer. Chem. Soc., 29, 1168, 1907): —

3R = 5*955*

Element.

Atomic Heat at
Constant Volume.

Element.

Atomic Heat at
Constant Volume.

Na

6-4

Cd

5*9

Mg

5-8

Sn

6-1

Al

5'6

Sb

5 '9

K

6'5

I

6*0 •

Fe

5'9

Pt

5*9

Ni

5*9

Au

5 '9

Cu

5-6

Tl

6-i

Zn

57

Pb

5*9

Pd

5'9

Bi

6-2

Ag

5'8

The approximation of these values to that of 3 R is obviously very
close. Nevertheless, too great stress cannot be laid upon this, for // has
been shown by experiment that the atomic heat of solids (as in the case of
gases] is a function of the temperature, decreasing to very small values at
very low temperatures. We shall take this question up later on in dis-
cussing the work of Einstein and of Nernst. Sufficient has been said,

1 In the case of metals each atom vibrates independently of others, at least as
long as its amplitude is less than the mean distance of the atoms apart, and hence
three degrees of freedom must be ascribed to it.

3 That is, any vibration of the simple harmonic kind, in which the restoring
force is proportional to the distance from the centre.

APPLICATION OF MECHANICS TO RADIATION 31

however, in this connection to show that while the principle of equi-
partition is partia ly true, in the form given by Boltzmann it is not
sufficiently comprehensive. It might be thought that a sufficient ex-
planation of the observed increase in atomic heat with rise in tempera-
ture lies in the supposition of new degrees of freedom coming into
existence. We cannot, however, imagine a " fractional " degree of
freedom. It must either exist definitely, or not at all. One would
expect, therefore, that the atomic heat should rise by steps as the
temperature rises. All observations, however, have shown that the in-
crease in atomic heat is a perfectly continuous function of the tempera-
ture. Leaving the problem of specific or atomic heats, let us turn to
another important problem, namely, that of thermal radiation ; for it
was through investigation carried out in this field that the modifications
of the principle of equipartition were eventually introduced, which in
the hands of Planck and Einstein have permitted a satisfactory explana-
tion to be given of the discrepancies hitherto existing between theory
and experiment, not only in the domain of radiation itself, but likewise
in that of the heat content of solids. Whether these modified views
form the ultimate solution of the problem, it is at present impossible to
say. They represent, at any rate, a fundamental stage in the develop-
ment of the subject.

APPLICATION OF [CLASSICAL] STATISTICAL MECHANICS TO RADIATION.

We are here concerned with temperature radiation only. A defini-
tion of this term has already been given together with a short account
of the radiation laws in Chap. XIV. of Vol. II.

In studving the question of radiation, that is of the exchange of
radiant eneigy between matter and ether, it is necessary, of course, to
limit our consideration to the equilibrium state. If an enclosed material
system is maintained at a temperature T, the interior of the system
contains energy constantly radiated to and from the boundary. When
these energy exchanges arrive at equilibrium each cubic centimetre of
the system contains energy in what we may call the undulatory form.
The problem is how to calculate the most probable distribution of the
energy between the various wave-lengths not only for the single tempera-
ture T but for any temperature; for the equilibrium state may be
defined as that for which the distribution of energy (at the given tem-
perature) between the various wave-lengths is the most probable. To
work out this statistical problem we must know something about the
number of degrees of freedom possessed by the matter and by the ether
(present throughout the matter) respectively. It will now be shown
that absolutely different results are arrived at, according as to whether
we Regard the ether as continuous (i.e. structureless) or regard it as
having a structure. Let us first consider the ether as continuous. In
this case the ether is a medium capable of vibrating in an infinite number
of ways the wave-lengths propagated throughout it having all possible
values between o and oo . This is the same thing as saying that the

32 A SYSTEM OF PHYSICAL CHEMISTRY

ether possesses an infinite number of degrees of freedom. Matter
imbedded in the ether has on the other hand a finite number of degrees
of freedom in virtue of its discontinuity, i.e. in virtue of its discrete or
heterogeneous structure. If now we apply the principle of equipartition
of energy to such a system composed of both matter and ether, it is clear
that the ether will take all the energy (since its number of freedoms is
infinite) leaving none at all for the matter.

The following mechanical analogy is suggested by Jeans. Let us
suppose that we have a number of corks held together by elastic and
floated upon the surface of still water. Cause the corks to move
violently to and fro. Waves will be formed, i.e. energy will be given
out from the cork system to the water by this kind of "radiation," as
well as by friction, until finally the corks come to rest. All the energy
originally given to the corks has now passed to the water. Permanent
movement of the corks is unthinkable. It will be observed that the
cause of this lies in the fact that the corks are large units compared
with water molecules ; in fact, the water is a sensibly continuous medium
compared with the coarse-grained structure consisting of corks. It is
impossible therefore to conceive of an equilibrium being set up as
regards energy interchange between matter and ether except at the
absolute zero of temperature. In other words, we cannot deduce any
radiation law. It is clear, therefore, we must follow out some other line
of reasoning. Let us take the second case, namely, the assumption that
the ether does possess a structure. On this basis the number of degrees
of freedom of the ether is no longer infinite, and it is possible to con-
ceive of equilibrium states being reached at different temperatures as a
result of energy transfer between matter and ether. If we think of the
ether as possessing a fine-grained structure, it follows that the waves
which can be propagated by such a medium must not become shorter
than a certain limiting size \Q. It is easy to see this by analogy.
Waves which can be transmitted by a material system (sound waves, for
example) must be great compared with the distance of the molecules
which compose the system, as otherwise the waves would not be trans-
mitted at all. Similarly, we must suppose that even the shortest light
waves which we know of must be large compared with the grain structure
of the ether itself. If this structure exists it ceases to be legitimate to
speak of infinitely short waves in the mathematical sense. The shortest
conceivable waves must be of the order of magnitude of the distance
apart of the " molecules of the ether ".

Starting out with the idea that the number of degrees of freedom
possessed by the ether is finite, Jeans has shown that by applying the
principle of equipartition of energy, the energy distributes itself in the
normal spectrum in such a way that the intensity — corresponding to a
region lying between A. and X + dX — is proportional to the temperature
and inversely proportional to the fourth power of the wave-length.
That is, the energy in the spectrum will so distribute itself that it will
be almost entirely confined to the region of extremely short wave-lengths.
As a matter of fact, however, this is not the real distribution of energy

APPLICATION OF MECHANICS TO RADIATION :S3

in the spectrum. It has been shown experimentally that the intensity
of energy radiating from a black body and in equilibrium with the body
shows a maximum for waves in the infra-red region at ordinary tempera-
tures, and vanishes almost completely for very long or very short waves.
Representing it graphically we obtain a curve containing a maximum,
as already shown in Fig. 53, Vol. II. On the other hand, the theoretical
expression for the distribution of radiation energy, referred to above
(which expression is identical with Rayleigh's radiation law, already
stated in Chap. XIV., Vol. II., viz. 47T/&TA-4. d\\ yields a curve
which contains no maximum but rises rapidly as the wave-length
diminishes.

We are considering the case in which wave-lengths of all magnitudes
exist between the infinitely long denoted by the symbol A^ and the
limiting value A0, but none shorter than this. If we integrate the Ray-
leigh- Jeans expression between the limits A0 and A^, we obviously obtain
an expression for the total radiant energy in the enclosure. The in-
tegration leads to the expression —

It follows from this expression that only one-eighth of the total energy
will be of wave-length greater than 2A0, whilst seven-eighths reside in
wave-lengths between 2X0 and AO itself. In the case of the ether — if we
give it a structure at all — the size of its grains must be considerably less
than io~7 cm. (the order of magnitude of a gas molecule). Taking this
value, however, as applying to the ether grains, it follows that no wave-
length shorter than io~7 cm. can be transmitted. (As a matter of fact
we know that X-rays are only about one-tenth of this.) Even with this
limit to the w#ve-length it can be shown on the basis of the Rayleigh-
Jeans formula that only one-millionth of the total energy would reside
in wave-lengths of the order io~5 cm. and longer. This is quite con-
trary to what is observed, for the greater part of the energy is known to
lie in the region io~4 cm. We conclude, therefore, that the assumption
that the number of degrees of freedom of the ether is not infinite, is
in itself insufficient to yield an expression (involving the equipartition
principle) which will agree with experiment. We are forced to the con-
clusion, therefore, that the principle of equipartition itself is not appli-
cable to the problem of distribution of radiant energy between matter
and the ether. The result obtained on the equipartition basis always
gives a partition of energy of such a kind that the energy is almost
entirely confined to the short wave-lengths, and in the limit, i.e. if the
ether be structureless, the energy goes completely into the ether and no
distribution is possible at all. Experiment shows, on the other hand,
that distribution certainly does exist, and furthermore, the distribution
does not require that all the energy shall be located in the shortest
wave-lengths ; on the contrary, the distribution is such that very little
energy is distributed among the very short or the very long waves, the
greater part of it belonging to waves of intermediate magnitude. This
VOL. III. 3

34 A SYSTEM OF PHYSICAL CHEMISTRY

is indicated by the maximum in the intensity curve obtained in measure-
ments of the amount of energy radiated from a heated body.

In the foregoing we have considered a number of problems, atomic
heats of solids, molecular heats of gases, and radiation phenomena,
from the point of view of classical statistical mechanics as expressed in
the principle of equipartition of energy between degrees of freedom.
In each case we find that the conclusions arrived at on this basis hold
over a limited region, but fail to give a complete explanation of the
phenomena observed. . It is obvious that some additional hypothesis
must be introduced, or rather, we must be prepared to discard the
principle of i^u/partition and substitute for it some other kind of
partition law. .An attempt to do this has been made by Planck in his
quantum hypothesis, which will be discussed in the following chapter.

CHAPTER II.

Introductory — Planck's concept of quanta — Planck's radiation tormula.

THE fundamental modification introduced by Planck consists in dis-
carding the principle of e^wpartition of kinetic energy. The equi-
partition principle assumes interchange of energy in a continuous
manner, i.e. without any lower limit to the amount of energy trans-
ferable. Planck assumes, on the other hand, that the exchanges of
energy between matter and ether, instead of taking place in any pro-
portion whatsoever, can only take place by steps, that is, in multiples
of some small energy unit, the energy unit itself being a function of
the vibration frequency concerned. We might regard radiant energy
itself as possessing a structure. Planck himself regards the matter from
a different standpoint. It would be more correct to say that the de-
finite concept of radiant energy itself being discrete in nature is that
upheld by Einstein, a view which has been given a clearer physical
basis by J. J. Thomson, who considers a radiant-energy unit or
"quantum" as a region of periodic disturbance travelling along a
Faraday tube. In place of a continuous ether, we have therefore to
substitute a number of stretched strings of ether, each string being a
Faraday tube differentiated in some (practically unknown) way from
the " space " surrounding it. This statement, is, however, to be
taken rather as a rough material analogy than as an exact description
or formulation, for any exact description is at the present time im-
possible. Planck, on the other hand, lays stress not on the question of
the ultimate structure of radiant energy itself so much as on the mode
of its absorption and emission by matter. The energy radiated by one
element of a black body is partially absorbed by other elements.
Each one of the vibrators, or "resonators," as Planck calls them, which
constitute the material element in question, can only emit (and absorb)
energy in certain fractions. (This at any rate is Planck's first position ;
we shall see later that he has modified the above view.) A word here
about the Planck resonators. It was pointed out in the chapter on
photochemistry in Vol. II. that the vibrating particles which produced
visible and ultra-violet light were much smaller than the atoms of the
substance and were therefore in all probability the electrons. The
amount of energy contributed to the "total" spectrum by the visible
or ultra-violet region has been shown by experiment (cf. Lummer and

35 3'

3 6 A SYSTEM OF PHYSICAL CHEMISTRY

Pringsheim's curve, Fig. 53, Vol. II.) to be a very small fraction of the
whole (even at fairly high temperatures). The main part of the energy
is confined to the infra-red region. These long waves are considered
as emitted and absorbed by the atoms of the substance being set in vibra-
tion. These statements apply only to the continuous spectrum given
out by a black body in the first instance and in general by heated
metals. Planck's resonators for the infra-red may therefore be identi-
fied with the atoms (electrically charged) ; for the ultra-violet, with the
electrons. But Planck in the theoretical treatment of the subject has
regarded his resonators as linear, that is, he only considers the energy
caused by vibration in a single direction. Such a vibration entails one
degree of freedom. If we were to apply the equipartition principle to
a system of such linear resonators, to each resonator, we would ascribe
RT units of energy made up of ^RT kinetic energy and an equivalent
quantity of potential. It is only for extremely long waves that the
equipartition principle holds (as is shown by the degree of applicability
of Lord Rayleigh's formula for radiation), and that only for a limited
temperature range. The essence of Planck's view is that it discards the
equipartition principle. It will be noted that the linear resonators of
Planck are considered to possess only one-third of the total vibrational
energy of the actual atoms (of a solid), each of which possesses three
degrees of freedom — as already pointed out in connection with the
values for atomic heat.

According to Planck, the material resonators considered do not
react with, or are not influenced by, infinitely small quantities of radia-
tion energy, using the word infinitely in its strict sense. Planck's
hypothesis may be stated thus : // is necessary that the energy attains a
finite value € in order that the resonators composing the material system
may be able to absorb it or emit it. (As already mentioned, Planck
later modified this statement by supposing absorption to be continuous,
but emission discontinuous, i.e. in quanta.) It will be clear how this
hypothesis modifies the principle of equipartition of energy among
various degrees of freedom. To any degree of freedom which actually
possesses energy we cannot ascribe less than one quantum (e), and the
actual quantity possessed will be an integral multiple of one quantum.
With this distribution of energy some of the degrees of freedom may
possess no energy at all, i.e. the d^w/partition idea breaks down.

The term " the quantum " requires now to be considered. Although
we speak of this as the unit of (radiant) energy, it must be clearly
understood that it is NOT a fixed and constant quantity of energy.
According to Planck, the quantum e, i.e. its size or magnitude, is a
function of the vibration frequency (either of the radiation, supposing
this to be monochromatic, or what amounts to the same thing, the
frequency of the resonator). In fact, according to Planck, there is
direct proportionality between the magnitude of e and the frequency v,
this proportionality being expressed in Planck's fundamental relation —

c = hv

THE QUANTUM THEORY 37

where h is a universal constant (Planck's constant) having the numeri-
cal value 6 -5 x io~27 erg- seconds.1

Now the smaller the unit the greater the probability that a resonator
will possess at least one or some quanta. If we consider a material
system, made up of molecules, atoms, and electrons, such a system
possesses resonators of various dimensions, i.e. capable of vibrating with
different frequencies. Such a system can absorb or emit a range (or
spectrum) of vibration frequencies. Considering the very short waves,
i.e. large vibration frequency, the quantum e corresponding to this is
large, and hence the chance that a resonator possesses even one
quantum of this size is less than in the case of longer waves, where
each quantum is a smaller magnitude. Less energy of the short wave
type will therefore be emitted than that of the longer wave type.
That is, the energy of the radiation emission curve falls off in the short
wave region. In this way Planck explains the observed diminution in
energy emitted in the visible and ultra-violet region, as shown in
Lummer and Pringsheim's curves. Further, in the region of extremely
long waves v is relatively very small, and hence the size of the unit e is
small, so that for extremely long waves the actual energy contribution
made by this region will be small. We should therefore expect on
Planck's view the energy wave-length curve to pass through a maxi-
mum, as is actually the case.

Starting out with Planck's hypothesis of the discrete nature of ab-
sorption and emission of radiation, it is now necessary to see what
radiation formula may be deduced; in other words, what theoretical
expression can be deduced for the distribution of energy in the spectrum
of a body emitting temperature-, i.e. black-body-radiation. For an
exact and complete account, the reader is referred to Planck's Theory
of Heat Radiation. We can only here attempt an abbreviated and
approximate deduction, based upon a new method employed by Jeans
(Phil. Mag., 20, 953, 191°)-

If a vibration — that is, a very small spectral region lying between
A and X+ rfA, which corresponds experimentally to monochromatic
radiation — can possess the following amounts of energy, viz. o, €, 2c
. . . etc., then the ratio of the probabilities of these events, as in the
usual gas theory calculations, is —

where e is the base of natural logarithms, k a constant, namely, the
gas constant per molecule, and T the absolute temperature. This
means that if we represent by " i " the number of vibrations possessing
no energy at all, then the number of vibrations, each of which possesses
one unit of magnitude €, will be <?~e^T, and so on. Instead of thinking
of vibrations in "space," let us think of the resonators or vibrations of

1 Planck (Annalen der Physik, [4], 4, 553, 1901) has shown that the magnitude
of e is a function of v by applying Wien's displacement law to an expression obtained
by him for the entropy of a system. The reader should also consult Planck's
Theory of Heat Radiation, 2nd edition.

38 A SYSTEM OF PHYSICAL CHEMISTRY

matter which can emit or absorb such vibrations. Let us suppose that
the material system consists of a great number of similar vibrators
(emitting and absorbing the wave-length A), each acting independently,
then if we represent by unity the number of such resonators which
possess no energy, the term *~«/*T gives the number of resonators
possessing e units each, etc. If out of N such resonators under con-
sideration M have zero energy, the number of resonators, each of
which has energy e, is M>-e/*T, the number having energy 2e is M*~2e'*T,
and so on. Hence, for the total —

N = M + M<?-'/*T + M<?-2e/*T + M*-3e/*T + etc.

= M

~ (i - <?-«/**)'

And if 2 U is the total energy of the N resonators —

Substituting the value for N already given, we obtain —

If we now substitute hv for e, we obtain —

And for the energy of one such resonator we obtain the mean value —

V-*£±T

The same expression may be obtained in a different manner (cf.
Appendix II.).

So much for the energy of a single vibrating resonator in radiation
equilibrium with its surroundings, the expression being based on the
hypothesis that energy is made up of units, the magnitude of a unit
being directly proportional to the vibration frequency of the resonator.
We have now to see the connection between this term U and the radia-
tion density uv, that is, the quantity of monochromatic radiation (fre-
quency v) energy per c.c. of radiated space. Although uv is usually
referred to as the radiation density, i.e. energy per unit volume, the term
uv itself has not got the dimensions of energy/volume. The correct
expression for radiation density is Uydv, where dv represents a narrow
strip of the spectrum virtually monochromatic. In using the symbol
uv for radiation density the quantity dv is implied. Physically a single
wave-length has no meaning ; we always work with a very narrow strip
of spectrum containing the wave-length or frequency denoted by A. or v.

THE QUANTUM THEORY 39

[In actual practice a heated body does not give out monochromatic
radiation, but a complete spectrum. We shall define uv in different
terms later, though still equivalent to the above.]

Planck has shown on the basis of the classical electro- magnetic
theory (which therefore introduces no unitary hypothesis) that uv and
U are connected by the relation —

" ...... <'>

[Cf. Planck, Annalen der Physik, [4], 4, 560, 1901 ; Lindemann,

Brit. Ass. Rep., 1912. If we take A. = SOO/A/A as a mean value for a

wave-length in the red part of the spectrum, it is easily calculated from

this formula that U~ = Su (approx.). Similarly for the violet end of the

•spectrum, taking \ = 400^, one finds U = 2u (approx).]

By combining the two expressions obtained above, the mean value
of the density of energy radiated from a single resonator in a system
consisting of a large number of similar resonators, all emitting mono-
chromatic light of frequency v (i.e. between the limits v and v + dv],
is given by the expression — /

.___..

This is one of the forms of Planck's Radiation Formula.

We wish now to change the shape of this expression a little, in order
to be clear about the relation of the term «„ and the term EA, which
latter we have already met with in Wien's radiation formula (cf. Chap.
XIV., Vol. II.).

The total energy (say in ergs) radiated per second from unit area of
a black body emitting a continuous spectrum covering the wave-lengths
o to oo has been denoted by S. This is the term which appears in
Stefan's Law, viz. S = o-T4, the temperature of the source being- T and
the radiation purely " temperature " radiation. Now S is the quantity
of energy which would be present in an imaginary cylinder, i square
centimetre base and length 3 x io10 cms. (since 3 x io10 cms. is the
distance which the radiation will travel in one second). The cylinder
is supposed to be placed with its base on the radiating body and extend-
ing out into space. The volume of this cylinder is 3 x io10 c.c., and
since this contains S ergs of energy the space density of the radiation,

S
i.e. the amount of energy per c.c. 'is - - j-0 erg. Denoting the space

g
density of the energy by E, we have E = - — j0 orS = 3 x io10E,

the term E referring, of course, to the entire range of wave-lengths
between o and oo emitted by the body. This total density E may be
expressed thus —

40 A SYSTEM OF PHYSICAL CHEMISTRY

Let us write EA = — , then E = \ EA</X.
0* Jo

It will thus be seen that E^ is a rate (the rate of change of E with
X). We might also define it as the density of the energy radiated from
a spectral region the wave-length limits of which differ by unity. This,
though correct, is physically inconceivable. It would entail the existence
of a spectrum the wave-length limits of which differ by i cm., a difference
enormous compared to any actual limits reached. It is therefore much
less confusing to think of EA as being the small energy-density incre-
ment AE divided by the correspondingly small spectral width AX (or

, dE c)E ,

more accurately -^- as above). The expression EA«X or ^- d\ is there-
fore the energy-density of the radiation between X and X + d\. Planck
gives an analogous significance to the term uvdv. Thus the energy-
density of the total spectrum may^6 written —

E = I ^— -dv

or writing «„=-',£ = nvd\>.

^ Jo

The term uv is also a rate, or it may be defined as the density of the
energy radiated from a spectral region, the limits of the vibration fre-
quency differing by unity. Numerically uv is of quite a different order

of magnitude from EA. Thus, since X = -, where c is the velocity of

light, it is evident that -=• = -• -g, so that on increasing v by unity
(i.e. corresponding to the production of uv x i energy-density units)
the wave-length decreases by the amount ^. This gives a result of the

order io~18 in the case in which the infra-red wave-length region X = I/A
is considered. Since EA is the energy-density corresponding to a wave-
length difference of unity, the term EA is io18 uv for the region in which
the vibration frequency differs by unity.

The shortest wave-length physically possible, i.e. the shortest wave-
length capable of being emitted or absorbed by an atomic-electronic
mechanism (such as that associated with a molecule), is of the order of
magnitude of very hard X-rays, namely, io~9 cm. This sets a limit,
therefore, to the width of spectrum which can be regarded as possessing
a physical meaning. It is interesting to see what this limiting width is

THE Q UANTUM THEOR Y 4 1

when expressed in terms of the difference of the frequencies between
the "edges" of the strip. Let us suppose that the strip is chosen in
the middle of the visible spectrum. The mean value of A is then o*6/A
or 6 x io~5 cm., the limiting value to be ascribed to dX being io~9 cm.
The frequency difference dv which corresponds to dX is given by —

dv = — r-~ . d \.

3 x io10 x 10"

That is, for the case considered,

- dv = - ~~ .

36 x io-1

Hence, in the region of the spectrum referred to, the smallest width
to which we can attach a physical meaning corresponds to a frequency
difference of the order io10. The actual or absolute frequency in the
middle of this strip is of the order 5 x io14.

In bolometric or spectro- photometric determinations we measure
the energy AS radiated per second by a small region or strip of a continu-
ous spectrum which lies between X and A. + AX. Since S = 3 x io10E)

and therefore AS = 3 x. io10 AE, we can calculate — which in turn is

identical with EA. We now wish to express the radiation formula of
Planck in such a way as to allow of convenient comparison between EA
observed and its value as given by the formula. To do this let us re-
write the formula already given : —

Remembering that uvdv = E\d\, and further that d\ = - -$v, we
obtain directly —

F - **ch I (9\

5 *A*T _

which is the more usual form of Planck's equation for the distribution
of energy throughout the spectrum. As already pointed out, we can
compare the observed and calculated values of EX for a whole series of
small regions and thus test the equation over the complete spectrum.
It has been found that the above formula reproduces the actual energy
distribution from a black body in an extremely accurate manner, being
in fact the most satisfactory equation yet proposed. The principal ex-
perimental investigations are those of Lummer and Pringsheim ( Verh.
d. Deutsch. physik. GeselL, 1, 23 and 215, 1899; 2, 163, 1900), Rubens
and Kurlbaum (Sitzungsber. d. K. Akad. d. Wissenschaft zu Berlin, 41,
929, 1900 ; Annalen d. Physik, [4], 4, 649, 1901), Paschen (Annalen d.
Physik, [4], 4, 277, 1901), and the more recent measurements of W.
Coblentz (Bull. Bur. Standards, 1914, 10, p. i).

A SYSTEM OF PHYSICAL CHEMISTRY

By way of illustration a few of Paschen's results may be quoted.
The extent of the spectrum examined was from i to 8-8^, (i.e. the
infra-red region), measurements being carried out by means of the
bolometer. Using a fluorite prism (which is more transparent than
quartz) successive parts of the spectrum could be isolated. The
radiating body consisted of a hollow vessel or cylinder, electrically
heated to any desired temperature, the cylinder being either of plati-
num or porcelain with or without a coating of copper oxide. In the
following table / denotes the magnitude of the swing of the galvano-
meter needle, a certain swing corresponding to a certain amount of
energy communicated by the radiation to the bolometer. After cali-
brating the apparatus, Planck's formula can be used to calculate values
of / for different wave regions, and these may be compared with experi-
ment.

TABLE I. — \= 1-0959^. Width of spectrum isolated
Temperature of surroundings = 9'8° C.

3'-

T = Temperature of Radiator =

•333-4.

I553-I.

I328-3.

I038-7.

[Observed
i -I Calculated from Wien's equation
( Calculated from Planck's equation .

60-6
6l-44
6I-27

252-6

252^
249H

59^

59-14
58^7

3^5

3'68

TABLE II. — \ = 8-7958 /t. Width of spectrum isolated = 6'.
Temperature of surroundings = 15*1° C.

Temperature of Radiator =

1458-6.

1069-8

844-4.

625-6

483-1.

( Observed ....

131-7

74H3

44-5I

20-46

9-22

i -1 Wien's equation

90-40

59-79

39-52

19-86

9-08

( Planck's equation

129-1

7375

20-90

9'22

It will be observed that Planck's formula is much more applicable
than Wien's for high temperatures and long waves. Besides the infra-
red region, the experiments of Paschen and Wanner, as well as those
of Lummer and Pringsheim, on radiation from the visible spectral
region, have shown that Planck's formula holds here also within the
limits of experimental error. In addition to the direct method of test-
ing Planck's formula outlined above, its validity is further shown by
the fact that we can deduce from it directly Wien's displacement law
(viz. Amax> x T = constant, and Emax. = constant x T5), and Stefan's
total radiation law —

= const, x T4, where. S = 3 x io10E = 3 x lo1

THE QUANTUM THEORY 43.

both of which have been experimentally verified. Planck's expression-
is thus a very comprehensive one.1

Some Numerical Values.

At this point it is of interest to calculate the values of the two-
fundamental constants h and k which occur in Planck's equation, h
is the universal proportionality-factor connecting the energy e of a
quantum with the frequency v ; k, as will be shown later, is the gas
constant R, reckoned not for a gram-mole but for a single molecule. .
The calculation may be carried out as follows.

Kurlbaum (Wied. Ann., 65, 759, 1898) has found by experiment
that the total energy emitted from i square centimetre of a "black
body" in i second, the temperature of the body being 100° C. and
that of the air being taken as o° C., amounts to 0*0731 watt/cm.2, that
is —

S = S100oC.- S0oC.= 4'2 x 7-31 x io5.ergs/cm.2-sec.
By applying Stefan's Law we obtain —

S = o-(3734 ~ 2734).
_ 4-2 x 7-31 x io5 ergs

(373* - 2734) ' cm.2-sec.-degrees4.

The physical significance of a- is evidently the total radiation
emitted from a black body per second, when the temperature differ-
ence between the black body and the surroundings is i degree, the
temperature of the black body being i° absolute, the surroundings
being at o° absolute. Corresponding to this emission at i° absolute,

1 To emphasise further the fact that Planck's formula is at variance with the
principle of ^wjpartition of energy among various degrees of freedom, it is inter-
esting to calculate the energy of an electron vibrating with a frequency identical
with that of the ultra-violet region when light is emitted, and compare this energy
with the energy of an atom or of a gaseous molecule possessing the mean kinetic
energy characteristic of ordinary temperature (say 300° absolute), (cf. J. Stark,
Zeitsch. physik. Chem., 86, 53, 1913). The frequency v for the ultra-violet region
will be about 5 x io14 per second. Planck's expression for the mean energy of a
single resonator is, as we have already seen —

' U

For ordinary (low) temperatures — T = 300° — this expression reduces to

hv

D = hve &T, since v is large and T small. The mean energy of an electron
resonator is therefore U, where U = i x io~16 ergs. The mean kinetic energy of

a gas molecule at the ordinary temperature = ~£T = 6 x io~14 ergs, so that instead

of equipartition of energy we find that the energy of the electron vibrating in the
atom or molecule is only one five-hundredth part of the mean kinetic energy of
translation of the molecule itself.

44 A SYSTEM OF PHYSICAL CHEMISTR Y

we see that E, i.e. the total energy density, is ^, that is —

4-2 x 7-31 x io5 ergs

El0abs = - ' ° 4 —rr = 7-061 X I0~15 - — .

3 x io10(3734 - 2734) c.c.

pOO pOO

Now E = EA*/X, or u.4v and E\ or uv is given by Planck's

Jo Jo

•expression. Using Planck's expression in the form in which uv occurs,
we get —

-

the term T being omitted in the final expression, since it is simply
unity. The integration may be effected by series, and we obtain
.finally —

Ei'abs. = ^r x 1-0823.

Setting this equal to the "observed" value of EP abs., viz.
7-061 x io~16, we obtain —

~= 1-1682 x io" .... (3)

Further, Lummer and Pringsheim ( Verh. d. Deutsch. physik. GeselL,
2, 176, 1900) have determined the value of Xmax. T, where Xmax. is the
wave-length corresponding to the maximum value of Ex from a black
body radiating at a given temperature. The expression Xmax. x T is
a constant independent of temperature as Wien has shown, the numeri-
cal value found by Lummer and Pringsheim being 0-294 cm. degiees.
Now, by differentiating Planck's formula (equation (2)) with respect
to X, and putting the differential equal to zero, when X = Xmax. we
obtain —

whence Xmax. x T =

4-9651^

h 4-0651 x 0-204 f ^

•or -1F^2 = 4'866 x lo"11 . . (4)

R 3 x io

By combining equations (3) and (4) we obtain finally—

h = 6-55 x io~27 erg/sec.
k = 1-346 x io~16 erg/degree.

Millikan (Proc. Nat. Acad. Sa'., 3, 3T4 (1917)) gives as the
most accurate value of the charge e on an electron, the quantity
4-774 ± 0-005 x io~10 electrostatic units. Knowing this quantity and
the value of e/m, viz. 1-767 x io7, and also knowing Rydberg's con-

THE QUANTUM THEORY 45

stant for the Balmer series in the hydrogen spectrum,1 viz. 3-290 x io15
according to Millikan, the value of Planck's constant h may be calcu-
lated with a high degree of precision. The value given by Millikan is—

h = 6'547 ± o'oii x io~27

which is in close agreement with that "observed" by Millikan (6-56)
and by Webster (6-53).

The Significance of the Constant k, and a Determination of the Number
of Molecules in one gram-molecule.

Consider once more the Planck expression for the energy of vibra-
tion of a large number of similar resonating particles emitting mono-
chromatic radiation, viz. —

2 U =

The expression £e/*T - i may be expanded thus —
~ + ~( •&*) + higher powers.

K 1 2 \K 1 /

If now we are dealing with a system vibrating at very high tempera-
ture it will be seen that the above expression becomes 7^. That is,.

at high temperatures —

^ Ne

Exactly the same result is obtained at less high temperatures if the
system is vibrating very slowly, for in this case v is small (relatively),
and since € = hv the quantity c is likewise small. In both cases the
quantity e vanishes from the expression for the sum of the energies of
the vibrating particles, this energy being simply proportional to the
absolute temperature. Under these conditions we reobtain the rt suits
of the ordinary kinetic theory, i.e. the principle of equipartition of
energy, it being no longer necessary to consider the energy as other
than continuous. The principle of equipartition of energy is therefore
true as a limiting case for large values of T, and for small values of v.
Suppose that we are dealing with a solid radiating energy at a tempera-
ture sufficiently high that the energy of vibration of the resonators could
be represented by N£T. The resonators, as employed by Planck, are
linear, i.e. they possess i degree of freedom, to which one would ascribe
(if the equipartition principle applied, i.e. if T is sufficiently high) ^RT
kinetic and ^RT potential, in all RT units of energy per gram- mole or

1 Cf. Bohr's theory of the atom (Chap. V.), according to which Rydberg's con-
stant =

46 A SYSTEM OF PHYSICAL CHEMISTRY

gram-atom, if a monatomic solid be considered, and if the atoms be
identified with the resonators.1

Hence, the term 2U reckoned for N " linear " atoms where N is
now regarded as the number of atoms in a gram-atom (or molecules in
a gram-mole) should be identical with RT. That is, RT = £NT, or
R = £N. The constant k is therefore simply the gas constant reckoned
for a single molecule.

We may see in another way that k has this significance. As has
been pointed out already, Jeans 2 has shown that for very long waves
the law of the partition of energy, or of energy-density between waves
of different lengths, as expressed in a formula, must contain the wave-
length term to the inverse fourth power. A similar formula was de-
duced by Lord Rayleigh on the classical theory, and holds well for
the very long wave- length region. Jeans' equation is —

Ex =

where Ry is here the gas constant reckoned for a single molecule. Now
for this very long wave region we have seen that Planck's expression

may be simplified, the term hv _ — becoming -- or

— i b hv ' ' he

That is, Planck's equation becomes —

Ex = ^~ • - - = S^TX-4.
A° nc

On comparing this with Jeans' equation, it is at once evident that
the two become identical — and they must be identical if they are
equally to reproduce experimental results in the long-wave region — if
k is identical with Ry, i.e. k is the gas constant per single molecule.
This may be tested at once by calculating N from the known value R
(per mole), and from the value of k calculated from radiation data,
using Planck's formula. It this way it is found that N = 6-175 x io23,
a number which agrees very well indeed with the values obtained by
Perrin and Millikan (cf. Vol. I., Chap. I.). This "radiation" method
may therefore be regarded as a new and independent method of cal-
culating the Avogadro constant. The value of N just given leads to
a value for the charge on an electron, viz. 4-69 x io~10 electrostatic
units, which agrees well with Millikan's value, 4*774 x io~10.3

1 The actual atoms in solids vibrate with three degrees of freedom. For the
present we are considering "linear" atoms or atoms possessing i degree of
freedom.

2 Jeans, Phil. Mag., [6], 17, 229, 1909.

3 In the above calculation the value of k was obtained from radiation data.
The most accurate because the most direct method of obtaining k is simply to divide
Rby N.

According to Millikan the most accurate value for N is §-062 x io23.

THE QUANTUM THEORY

47

Numerical Values of the Size of the Quantum e in different Spectral

Regions.

In the following table are collected some values of (hv) extending
over a wide range. The longest wave measured by Rubens, in 1910,
was A, = QO/X, though still longer waves have been measured more
recently. The shortest wave-length measured is of the order O-I/A, or
IOO/A/X (Schumann).

Wave-length A.

'-*r.

,-kv.

A = 90/t .

3-3 X IO12

2*1 x io— 14 ergs

\ = 6fJL

5 x io13

3-3 x io-13 ,,

A = 2/u. (This is the region of maxi"|
mum intensity when the body is at j-

1*5 x io14

9-9 x io-13 ,,

1646° absolute) . . . . I

A = O'8w. (This is approximately the\
limit of the visible red) . . ./

375 x io14

2'47 x io— 12 „

A = o'4/*. (This is approximately the\
limit of the visible violet) . . /

7-5 x io14

4-95 x io-1* ,,

A = o'2)ii. (This is the limit obtainable \
with a quartz prism) . . . /

i '5 x iois

9-9 x io— 12 „

A = o'i/x. Schumann rays

3'o x io15

1-9 x io— u „

The value of € can thus vary from io"11 to io~14 ergs, according to
the vibration frequency of the resonator, i.e. the vibration frequency of
the radiation. As already pointed out, the resonators emitting short
waves are probably electrons, those emitting long waves are probably
the atoms. As regards the total energy emitted from a heated body,
we can neglect the portion due to wave-lengths shorter than o'8/x, i.e.
we can neglect the visible and ultra-violet regions. This is true, of
course, only for bodies the temperature of which is not higher than
1600° C. If the temperature of the radiator be raised much higher
than this, the contribution from the visible may no longer be negligible,
since in accordance with Wien's displacement law the position of maxi-
mum energy emission shifts towards the shorter wave region the higher
the temperature. A body with the temperature of the sun (say
6000° C.), radiating purely thermally (without chemical effects in ad-
dition, as is the case with the sun), would have its maximum emission
of energy at X = o'5/x approximately, i.e. in the blue-green region. It
is of course quite impossible to realise a temperature so high as this
under any experimental conditions.

In connection with Einstein's view of the quantum hypothesis,
namely, the view that radiation itself consists of small units travelling
through space, it is necessary to point out one important phenomenon
which it is very difficult to reconcile with this view. The phenomenon
is that of interference of light, when two beams starting in the same
phase have traversed paths which differ in length. The difficulty of

48 A SYSTEM OF PHYSICAL CHEMISTRY

expecting interference on the Einstein view is very considerable, espe-
cially when it is recalled that interference can be produced — as in
certain experiments of Michelson — with a difference of path of 40 cms.,
a distance which is enormously great compared with one wave-length.

PLANCK'S LATER QUANTUM HYPOTHESIS.

Planck's Modified Equation for the Energy of an Oscillator.

(Cf. Max Planck, Ber. d. Deut. physik. GeselL, 13, 138, 1911.)

The essential modification here introduced is that while emission
of radiant energy takes place in quanta, and therefore discontinuously,
absorption can take place continuously. Planck was led to consider
this as more probable than the older view, one important reason
having to do with the question of time. Supposing absorption took
place in quanta only, then an oscillator might be exposed to radiation
so weak as not to yield even one quantum, under which condition the
oscillator would absorb no energy at all. This would be especially true
for large values of v (large units). Further, even in those cases in
which absorption could occur, it is conceivable that in weak radiation
there might be a time interval.

On the former view the energy U of an oscillator could be expressed
by U = «e, where n is an integer. If absorption is continuous, the
total energy can no longer be regarded as an exact multiple of e, but
could in general be regarded as consisting of nt units plus a quantity p
over. That is —

U = «e + p.

Now the limits for the value of p are zero and one e. If we are
considering a large number of oscillators p will have an average value

€ hv

- or — , so that we can write —

2 2

TT kv

\J = ?l( + .

2

On the older view Planck's equation for U is —

u = ^_ i

which evidently correspondsjo «e.

The new expression for tJ can therefore be written —

hv hv hv

U = -T-71- h — = —

Planck points out several consequences of this new expression.

PLANCK'S LATER QUANTUM HYPOTHESIS 49

When T = o, U is not = o, but = - This " residual energy " re-
mains with the oscillator even at o° absolute. It cannot lose it since
it cannot emit anything less than one hv. " At high temperatures and
for long waves in the region where the Jeans- Rayleigh Law holds the
new formula passes into the old."

For short waves (visible and ultra-violet region) ehvlkr becomes
large compared with unity, and we have —

U = -".

2

As regards the question of the specific heats of solids considered in
the following chapter, it is pointed out by Planck that measurements of
specific heat cannot be used to compare the older and newer forms of
U since Cv is </U/*/T, and differentiation removes the additional term

— . Planck doubts whether the' new expression can be tested experi-
mentally by any direct means.1

Up to this point the most important experimental evidence which
we have considered in support of the quantum theory is the fact that
Planck's radiation formula quantitatively reproduces experimental values
over the entire range of the spectrum investigated. In the next chapter
we shall deal with further experimental evidence in connection with the
atomic heats of solids which likewise is strongly in favour of the" con-
cept of quanta.

1 Planck refers to a paper by Stark (Phys. Zeitsch., g, 767 (1908)), on the
application of the quantum theory to canal rays. As regards an attempt to dis-
tinguish between the old and new form of the quantum theory, see A. Einstein and
O. Stern (Annalen der Physik, [4], 40, 551, 1913).

VOL. III.

CHAPTER III.

(Physical equilibrium in solids)— Theory of atomic heats of solids — Equations
of Einstein, Nernst-Lindemann, and Debye.

Einstein s Extension of Planck's Quantum Theory to the Calculation of
Specific Heats of Solids (Crystalline Substances) and Supercooled
Liquids ( * ' Amorphous Solids "). ( Cf. Einstein, Annalen der Physik,
[4], 22, 180, 1907.)

THE specific heat of a substance at constant volume is defined as the
increase in total- energy when the substance rises i° in temperature.
Let us take as our unit of mass the gram- mole or gram-atom (in the
case of monatomic substances), and we then can write —

CD = .

NOTE. — -U here stands for total energy possessed by the substance at
•a given temperature. It is not to be confused with the significance
•attached to U in the " elementary thermodynamical treatment " (Chap. I.,
Vol. II.), in which " U " stood for decrease in total energy due to chemical
^reaction.

Let us now restrict our attention to solids (or supercooled liquids),
taking as particular instances the metallic elements. These, as we have
seen, are regarded as monatomic, so that the gram-atom and gram-
molecule are identical terms in these cases. Now we want to find out
to what the total or internal energy of a metal is due. It is usual to
regard the solid state as characterised by vibrations of the atoms about
their respective centres of gravity. Such vibrations can, of course, take
place in the three dimensions of space, i.e. each atom possesses three
•degrees of freedom. As we have seen, each vibration represents energy,
one-half of which is kinetic, one-half potential, as long as the amplitude
of the vibration is not too great. This vibrational energy is regarded
as representing all the internal energy possessed by the atom, at least
at low temperatures (at high temperatures the energy of vibration of
the electrons inside each atom would have to be considered, but at
ordinary temperature and at lower temperatures the total energy of
the solid may be ascribed to the vibration of the atoms). We have
already discussed this, and we have seen that on applying the principle
of equipartition of energy the atomic heat of metals should be
^R = 5 '9 5 5 cals. per degree, and that this should be independent of
temperature. As already pointed out, this numerical value is certainly

50

A TO MIC HE A TS Of SOLIDS 5 1

approximated to at ordinary temperatures, but instead of being in-
dependent of temperature, it varies, becoming continuously smaller as
the temperature is lowered. The question therefore which arises is
how this variation with temperature is to be accounted for. Einstein
in 1907 made the first successful attempt at the solution of this problem
by suggesting that Planck's quantum theory — which Planck himself had
applied with so much success to the problem of the emission of radiant
energy — could also be applied to the vibrational energy of the atoms,
i.e. to the total internal energy of the solid, the temperature coefficient
of which is identical with the specific or atomic heat of the substance
in question : Planck's expression for the average energy of vibration of
a linear resonator (i.e. an atom vibrating along one of the dimensions of
space) is, as we seen —

_ hv

U = ghvlkt _ r'

[It will be observed that we are employing Planck's earlier hypothesis.]
The energy of vibration of an atom capable of vibrating along the
three dimensions of space will be three times this quantity, and if we
denote this average vibrational energy per gram-atom by U, we get —

U, 3Nfr .

gtolkr _ !

The significance of U is identical with that which has been ascribed to
it in Chap. II., Vol. II., namely, the total energy per mole or gram-
atom. It% the case considered one-half of U is kinetic, one-half po-
tential energy. N denotes the number of atoms in one gram-atom.
On Planck's earlier view the vibrational energy possessed by each atom
must be an even multiple of one quantum. On the " classical "
view (" structureless energy," so to speak) we should say that all
possible differences in energy content would manifest themselves in
a system made up of a large number of vibrating particles. On apply-
ing the unitary theory of energy we must recognise that a number of
atoms have no vibrational energy at all, i.e. are at rest. Of those
vibrating the energy content cannot fall below the quantum e', where «'
is three times Planck's quantum e. We have, therefore, sets of atoms
containing energy of the following amounts : —

so on-

In order to bring the expression for U given above into the form
used by Einstein, Nernst, and others, we shall make a slight change in
the symbols. If we denote the ratio of Planck's two fundamental con-

stants k and h by /Jo,1 we can write -7 = /30 = 4-87 x io~n C.G.S. units.

K

R R

Also, since k = ~, where R = i '985 calories, we can write hv = xr/Sov.

1 This is frequently written as j8. The slight change is here introduced to pre-
vent any confusion with 0, one of the terms in Nernst's " heat theorem " equations
of A and U.

A SYSTEM OF PHYSICAL CHEMISTRY
The expression for U then becomes —

U-*R. . ftv

It will be observed that this equation differs from the expression
3RT (obtained by applying the (^/partition principle, energy being

regarded as continuous), by the substitution of the term -jjjj- - in

place of T.

Differentiating U with respect to T we obtain Einstein's equation
for the atomic (or molecular) heat of a solid at constant volume, viz —

On the " classical " view, the factor multiplied by 3R would have been

1;

i-o

0-5

FIG. 5.

unity. The " correction " term will be seen to contain T and is there-
fore a function of temperature. On Einstein's theory one would expect
the atomic heat itself to be a function of temperature, as is experimentally
the case. Qualitatively, therefore, this marks a considerable advance over
the older theory. It still remains to be seen whether the expression
given really reproduces the values of Cv at various temperatures quantita-
tively. The shape of the curve for -=—, as given by Einstein's formula

is shown in the accompanying figure (Fig. 5), in which the ordinates
represent atomic heats and the abscissae the temperature expressed in

T
terms of -5— . For any given substance the vibration frequency is taken

to be a constant.

It will be seen from the figure that when -=— >o-o the term

ATOMIC HEATS OF SOLIDS

53

approximates to unity, so that the atomic heat becomes

equal to 3R. Dulong and Petit's Law is, therefore, a consequence of
Einstein's theory when the temperature is not too low. This region of
temperature is evidently reached at room temperature in the case of
the majority of metallic elements. As in radiation phenomena we see
that the equipartition principle applied in the ordinary way yields results
(in connection with atomic heat) which are in agreement with experi-
ment when the temperature reaches a certain magnitude. Einstein
tested his equation not on the data available in the case of a metal but
on the diamond. Some of the results are given in the subjoined table.
They are also indicated by circles in the figure. Einstein chose the
experimentally determined value for Cw at T = 331*3° and hence
calculated v, using the value so obtained to calculate the values of Cv at
other temperatures. The experimental data (quoted) refer, as a matter
of fact, to C/,, i.e. the atomic heat at constant pressure. In the case of
the diamond — though not so in. other cases — the difference between
Cp and Ct, is small. In the following section we shall consider the
question of the independent determination of v and Cv respectively.

ATOMIC HEAT OF CARBON (DIAMOND).

T (absolute).

T//V-

C^ calculated from
Einstein's equation.

Cfr observed
(Weber).

222'4

0-1679

0-762

0-76

262*4

0-1980

1-146

I-I4

2837

0-2141

1-354

1-35

306-4

0*2312

I-582

I-58

331*3

0*2500

1-838

I-84

358-5

0-2705

2-118

2-12

413-0

0-3II7

2-661

2-66

479-0

0-3615

3-280

3-28

520-0

0-3924

3'63I

3-63

8797

0*6638

5-290

5-29

1079-7

0-8147

5-387

5 '39

1258-0

0-9493

5-507

5'5i

Measurements of C? of the diamond at low temperatures down to
abs. have been carried out by Nernst, who discovered the remark-
able fact that the thermal capacity of this substance tends practically to
zero, even at the temperature + 50° abs. Between this temperature

and absolute zero, Cv or — = o. We have already had occasion to

point this out in connection with Nernst's heat theorem. From the
shape of Einstein's curve (Fig. 5), it will be seen that Cv is tending
towards zero at a temperature higher than o° abs. It thus appears that,

54 A SYSTEM OF PHYSICAL CHEMISTRY

as far as the diamond is concerned, Einstein's theory reproduces experi-
mental values with very considerable fidelity. While giving full weight
to such general agreement, it is necessary to point out that this agree-
ment is far from being complete in many other cases. We shall return
to this after having described in outline the experimental methods of
determining Cr, and the characteristic vibration frequency v.

Experimental Measurements of the Specific Heats of Solids, especially
at Low Temperature.

(Cf. Nernst, Journ. de Physique, [4], 9, 1910; Nernst, Koref, and
Lindemann, Sitzungsber. Berl. Akad., 1910, vol. i, p, 247 ; ibid., Nernst,
p. 262 ; Nernst, Annalen der Physik, [4], 36, 395, 1911-)

One form of calorimeter consisted of a heavy vessel of copper (about
400 grams in weight), the good thermal conductivity of such a mass
of copper doing away with the necessity of stirring the substance, which
is, of course, impossible in the case of solids. This was enclosed in a
Dewar vacuum vessel, placed in a bath, the temperature of the calori-
meter being measured by means of thermo-couples. The substance to
be investigated was heated or cooled to a known temperature, and in-
troduced into the calorimeter, the change in temperature of which was
observed. Knowing the heat capacity of the calorimeter, the specific
heat of the substance could be obtained. This method worked admir-
ably, but of course it is limited to the determination of mean values of
Cv, holding over a considerable temperature range. For the purposes
in view, it was necessary, however, to be able to determine Cp for small
temperature ranges, that is for a consecutive series of "points " on the
temperature scale. To accomplish this, a different procedure had to
be adopted.

The principle of this second method consists briefly in making the
investigated substance itself act as its own calorimeter. The substance
could be heated electrically by means of a platinum spiral, through
which a known quantity of electrical energy was passed, as measured
by an ammeter and voltmeter in the circuit. The rise in temperature
of the substance was obtained by using the heating spiral itself as a
resistance thermometer, i.e. its resistance was observed by means of a
bridge, before and after the heating of the substance, the alteration in
resistance giving the temperature change. Knowing the mass of sub-
stance employed, the electrical energy supplied, and the rise in tempera-
ture, it is easy to calculate the specific heat of the substance. This
extremely ingenious method was worked out by Eucken, in Nernst's
laboratory (Physik. Zeitschr., 10, 586, 1909), and has been employed
by Nernst and his collaborators for the accurate determination of specific
heats of substances at various temperatures, extending over a wide range,
even down to the temperature of liquid hydrogen. A few details may
be given here. A pear-shaped vessel of glass (shown in the accom-
panying figure, Fig. 6) contains the calorimeter K suspended by the

ATOMIC HEATS OF SOLIDS

55

wires which are connected to the heating spiral. By means of a Gaede
pump, together with a vessel of charcoal cooled in liquid air, a very
good vacuum is maintained in the glass vessel, so as to prevent any loss
of heat by convection. The calorimeter K (i.e. the substance itself),
together with the connecting wires, is brought to a certain temperature,
which is rapidly brought about by substituting hydrogen gas in place
of air (since the hydrogen conducts thermally so much better). The
hydrogen is subsequently removed, the vessel being evacuated. The
spiral resistance is of purest platinum. Its change in resistance with
.temperature was calibrated by comparison with an oxygen gas thermo-
meter. The calorimeter K varied in construction, according as to
whether the substance investigated conducted heat well or badly. The
first type shown in the figure is suitable when the substance is a metal.

U

Jl

FIG. 6.

It consists of a small cylinder of the metal, having a cylindrical hole
drilled almost throughout its length. Into this a plug of the same
metal fits loosely, the heating spiral occupying the space between. To
insulate the spiral thin paraffined paper was employed, liquid paraffin
itself being finally poured in. The upper part of the plug is somewhat
thicker than the remainder, good thermal contact being thus obtained.
For substances which conduct heat badly the second form of calori-
meter was employed. A vessel of silver was used, the platinum spiral
being wound round a cylindrical sheet of silver, which served to spread
the heat rapidly. Electrical insulation of the spiral was maintained by
the help of shellac and thin silk. The platinum wire, on entering the
calorimeter, passes through a small insulating tube, and finally, after
passing through the body of the calorimeter, ends on the silver vessel,
to which a lead is attached. It was found that some air must be left
in the calorimeter, itself (not in the pear-shaped vessel), so as to ensure

56 A SYSTEM OF PHYSICAL CHEMISTRY

better thermal conduction, and attainment of equilibrium. Nernst also
describes a third form of calorimeter, suitable for liquids, details of
which will be found in the paper cited. The heat capacity of the fittings
of the calorimeter K — other than the actual substance examined — was
determined, and found to agree well with that calculated from the
known masses and specific heats of the substances employed. When
the pear-shaped vessel was well evacuated, the temperature of the sub-
stance, i.e. the calorimeter K (as measured by the resistance of the
spiral) was found to remain remarkably constant for a considerable time,
even when K had been previously cooled down to very low temperatures,,
by placing the non-evacuated pear-shaped vessel in contact with liquid
air. Similarly, after cutting off the current the new (final) temperature
was again found to remain constant, and be accurately measurable.
Nernst considers that the probable error in the results does not exceed
i per cent, and may be much less.

Some of the experimental data, obtained by the above means, have
already been given in dealing with Nernst's heat theorem, and other
examples will be given when we take up in greater detail the question
of the applicability of Einstein's expression to the atomic heat of solids
in general.

METHODS OF DETERMINING THE CHARACTERISTIC VIBRATION
FREQUENCY OF A SOLID.

First Method.

This depends on the direct measurement of the "Reststrahlen
(Residual Rays) ". Many substances possess the property of selective
reflection, that is they powerfully reflect rays of certain wave-length.
The rays which are most strongly reflected in the case of incident light
are those which are most strongly absorbed when the light is transmitted.
The wave-lengths strongly absorbed are identical with the characteristic
wave-lengths corresponding to the natural vibration frequencies of the
substance, for it is when the light has the same frequency as that of the
vibrating atom that it is most strongly absorbed. If a beam of incident
light of all wave-lengths, that is to say, a beam emitted by a heated
black body, is reflected successively from a number of pieces of the
solid in question, at each reflection the beam will become purer, that is
the reflected light tends more and more to consist of the vibration
frequency characteristic of the solid. The rays which survive after
having suffered a number of such reflections are called the " Residual
Rays ". The measurement of the wave-length and the energy of such
Reststrahlen has been principally carried out by Prof. Rubens of Berlin
and his collaborators, although the idea was that of Beckmann in the
first instance (cf. Rubens and Nichols, Annalen der Physik, 60, 418,
1897 ; Rubens and Aschkinass, ibid.^ 65, 241, 1898 ; Rubens, ibid.^ 69,
576, 1899; Rubens and Kurlbaum, ibid.y [4], 4, 649, 1901; Rubens
and Hollnagel, Phil. Mag., [6], 19, 761, 1910). These rays correspond-

ATOMIC HEATS OF SOLIDS.

57

ing to the vibration frequency of atoms are naturally far in the infra-red
region. The apparatus employed is shown diagram matically in the
figure (Fig. 7). L is the electrically heated black body emitting waves
of all frequencies. Dx and D2 are diaphragms of small aperture. S is
a screen. The diaphragms and screen are kept cool with water at room
temperature. The substance under examination is represented by
Pl5 P2, PS, P^ The surfaces of these pieces are polished as highly as
possible. M is an adjustable mirror and T the thermopile. A diffrac-
tion grating of special make could be inserted in the path of the beam,
thereby analysing it into a spectrum, the position of the energy maxima
corresponding to the characteristic vibration frequencies of the substance
P being determined by means of the thermopile. In later experiments
the grating had to be abandoned owing to the large amount of absorp-

FIG.

tion suffered by the rays in passing through it. Instead, an interference
method has been devised — see the paper by Rubens and Hollnagel,
Phil. Mag,, I.e. The following table contains some of the results
summarised by Rubens and von Wartenberg (Sitzungsber. konigL preuss.
Akad., 1914, p. 169). In actual experiment two bands are usually ob-
served fairly close together. One of these, however, is due to the
presence of water vapour (Rubens, ibid., 1913, p. 513)-

RESIDUAL RAYS CHARACTERISTIC OF CERTAIN SOLID SALTS.

Substance.

Wave-Length in fx.

Substance.

Wave-Length in /x.

NH4C1

5i'5

AgCN

93 ap; rox.

NaCl

52-0

KI .

94-1

NH4Br

59-3

HgCl2

95 approx.

KC1

63-4

CaCO,

987

AgCl

81-5

HgoCl2

98-8

KBr

82-6

A-Br

112-7

PbCl2

91*0

TIBr

117*0

T1C1

91-6

Til

151-8

Second Method.

This is an indirect method due to Einstein (Annalen d. Physik, [4],
34, 170, 1911). From consideration of the distribution of atoms in
metals and the forces acting upon them when they are displaced from their
centres of gravity during vibration, Einstein obtained an approximate

A SYSTEM OF PHYSICAL CHEMISTRY

expression for the characteristic vibration frequency v in terms of
the compressibility K, the atomic (or molecular) volume V, and the
atomic (or molecular) weight M, which may be written as follows : —

v = 2-54 x io7j^i^-r

Using Griineisen's recent data for the compressibility (Ann. der^
Physik, 25, 848, 1908), Einstein calculated the following characteristic
vibration frequencies : —

SOME OF EINSTEIN'S VALUES FOR v.

Substance.

v.

Substance.

*•

Al

6-6 x ioia

Fe

6'5

Cu

57

Pt

4-6

Ag

4'i

Pb

2'2

Au

3'8

Cd

2-6

Ni

6-6

Bi

1-8

Einstein himself does not claim that this method is more than a
rough approximation. Nevertheless the values . agree moderately well
with those obtained by other methods.

Third Method.

This method, also an indirect one, is due to F. A. Lindemann-
(Physikal. Zeitsch., 11, 609. 1910). It is based upon the assumption
that at the melting point Ts of a solid the amplitude of vibration of the
atoms is approximately equal to the mean distance of the atoms apart. Let
us denote by rs the radius of the circular vibration of an atom (in a>
metal, say), then the mean velocity u with which it vibrates is given by
u = 27rrsv, where v is, of course, the number of such vibrations per
second.

The mean kinetic energy of vibration is \mu^t where m is the mass
of the atom, and since there is likewise present an equal quantity of
potential energy, the total mean energy of a vibrating atom is mu1 or
4;#7rVs2i/2. Now at the melting point the metals may be considered as
very closely obeying Dulong and Petit's Law, i.e. their atomic heat is
practically ,6. That is, the expression for the vibrating energy which is
given by —

U

3R

- i

reduces to

N

for large values of T as already pointed out.

Hence

or

N

v oc

ATOMIC HEATS OF SOLIDS

59

If we now regard the atomic volume V as proportional to the cube
of the mean distance apart of the atoms (which distance is identical with
rs on Lindemann's assumption), we can write —

i/oc

The proportionality factor obtained by comparison with experiment
is 2 -80 x io12, and hence Lindemann's formula is —

r

= 2'80 X IO12

It is thus possible to calculate v from measurements of melting
temperature, molecular weight of the atoms, and the atomic volume.
This is an extremely convenient method, but has the disadvantage of
not giving v very accurately. It appears, however, to be somewhat
more accurate than Einstein's method (Method 2). The following
table is given by Nernst and Lindemann (Z. Elektrochem., 17, 822,
1911). The observed values of v are those given by a formula of Nernst
and Lindemann which is a modification of Einstein's expression for
atomic heat, which at the same time represents experimental values
more closely. We shall discuss this expression later, but in the mean-
time it may be taken as giving the values of v from atomic heat measure-
ments.

LINDEMANN'S FORMULA FOR "v".

Substance.

Molecular
Weight.

Ts
Melting
Point.

V

Molecular
Volume.

Calculated by Lin-
demann's Formula.

V

" Observed ".

Al

23-1

930

IO'O

7-6 x io12

8-3 x IO12

Cu .

63-6

1357

7-1

6-8

6-6

Zn .

65-4

692

9-2

4*4

4-8

Ag -

107-9

1234

10-3

4*4

4*5

Pb

206*9

600

18-3

1-8

1-9

Diamond

I2'O

3600?

3'4

32-5

40*0

Iodine

127*0

386

257

1-7

2*o

NaCl .

29-2

1083

I3-5

7-2

5'9

KC1 .

37'2

1051

18-9

5'6

4'5

Lindemann's formula, like Einstein's, gives only a single mean
characteristic vibration frequency. For this nurpose, therefore, the
molecule of NaCl, KC1, etc., is treated as a single atom having an
atomic weight one-half of that usually assigned to the molecule of these
substances.

Fourth Method.

This method has also been suggested by Lindemann (Ber. Deutsch.
phys. Ges., 13, 1107, (1911)). Lindemann has shown (ibid., p. 482)
that the characteristic ultra-violet frequency v which enters into the
selective photo-electric effect1 is given by the expression —

1 C/. Chap. VI.

6o

A SYSTEM Of PHYSICAL CHEMISTRY

= ^\^

where n is the " valency," e the charge on an electron- (in electostatic
units), m the mass of an electron, and r one half of the distance between
the centres of two neighbouring atoms.

Lindemann introduces an expression due to Haber and discussed
later (p. 133), which connects the ultra-violet frequency with the char-
acteristic infra-red frequency, vred, the relation being —

Vviolet = /M
Vred \ m

where m is the mass of an electron, and M the actual mass of the
atom. Combining this expression with that given above we obtain —

e I n

Vred — A / V/n

Further, if A is the atomic weight, N the number of atoms in one gram-
atom, and d the density of the substance, then Lindemann shows
that —

r —

Hence,

4/2 XT Jdn

V o N

— . t . 1> . .

JT\

But <?N = F = the Faraday = 2-9 x i o14 electrostatic units.

Hence,

1-095 x io1

By means of a suitable choice of n we can calculate the infra-red
frequency of an element from this formula. The calculated values
agree with those observed, in some cases better than do those values
obtained by the melting point method. There is a certain arbitrariness,
however, in the choice of the valency «, as will be seen from the follow-
ing table which is quoted from Lindemann's paper : —

Element.

Lindemann Melting-

I'l X I0l4\/»'rf

v " Observed "

point Formula.

A '

Al

•
r

7-6 x io12

8'2 X IO12

8-3 x io12

Cu

2

6-8

7-2

6-6

Ag

2

4*4

.4-6

4'4

Hg

I

i'3

2'0

2-2

Pb

2

1-8

2'5

i*9

Diamond

4

>32'5

34'I

40 'o

S

2

3'9

6-6

7-3

I

I

i'7

fg

2'0

The weakness of the method lies in the selection of the value of the
valency n of the element under consideration. Of the four methods
mentioned, Lindemann's melting-point method is probably the most

ATOMIC HEATS OF SOLIDS

6 1

convenient ; the most accurate method is undoubtedly the direct;
measurement of the frequency of the residual rays.

The System of Elements and the Periodicity of the Vibration
Frequency with the Atomic Weights.

(Cf. W. Biltz, Zeitsch. Elektrochem., 17, 676, 1911.)

Employing Lindemann's formula, Biltz has calculated the character-
istic vibration frequency v for the elements (in the solid state) as far as
the existing data upon melting point and density permitted, and has
traced out a relation between the frequency and the atomic weight. In
the following table are given the values of v for each element
calculated by the help of Lindemann's formula1 : —

as

Element.

Density.

Melting-point
Temperature
T Absolute.

Atomic Weight.

r

V X IO

H .

0-0763

14

I'OI

4-36

He (liquid)

0'I54

Ca2

3'99

0-66

Li .

0'59

459

6-94

IO'O

Be .

173

Ca 1200

9-1

18-5

C (graphite)

2-3

„ 3600

I2'O

277

C (diamond)

3 '52

„ 3600

I2'0

317

N .

i -03

62-5

I4-0

2 '5

O .

i '43

28

16-0

1-7

F (liquid) .
[Ne (liquid)

1-14
1-24

50
Ca2

19-0

2O'2

1-8
0'34

Na .

0-98

371

23-0

3-96

Mg . .

!'74

924

24*3

7-2

Al .

2-66

930

27-1

7'5

Si .

2-49

1700

28-3

9-6

P (red) .

2'34

903

31-0

6-3

S (rhombic)

2-07

388

32-1

3-95

Cl (liquid)

i'66

171

35 '5

2-24

Ar (liquid)

1-42

83

39'9

1-32

K .

0-87

335

39'i

2-3

Ca .

i'59

1050

40-1

4'9

Ti .

4*50

2120

48-1

8-4

Vd .

5'8

1950

5i'i

8'3

Cr .

6-74

1790

52-0

8'3

Mn .

7*39

I52O

54'9

7'5

Fe .

7'85

1820

55-9

8'3

Co .

8-72

1780

59'0

6*2

Ni .

8-90

1760

587

8'2

Cu .

8-93

1357

63-6

67

Zn .

7-12

692

65H

4'36

Ga .

5 '95

303

69-9

2*5

Ge .

5 '47

Ca noo

72-5

4-6

As .

573

„ 1000

75'o

4-36

Se (grey) .
Br (liquid)

4-8
3'i9

490
266

79-2
79 '9

27
17

Kr (liquid)

2'l6

104

82-9

0-9

Rb .

i'52

3"

85-5

i*45

Sr .

2-63

[1080]

87-6

3-0

I

1 The values of v are slightly different from those given by Biltz, as Biltz em-
ployed the older and less correct proportionality factor 2'I2 x io12 in Lindemann's
formula instead of the later one, 2-80 x io12.

•62

A SYSTEM OF PHYSICAL CHEMISTRY

Element.

Density.

Melting-point
Temperature
T Absolute.

Atomic Weight.

V X IQ-12.

[Zr

6-4

Ca 2700

90-6

[Nb

7-8

222O

93*5

Mo

9-01

2400

96*0

6-3

[Ru

12-3

Ca 2250

I02'0

Pd .

11-9

1860

107-0

5 '74

Ag

10-5

1234

108-0

4'36

Cd .

8-65

595

II2'0

3-0

In .

7-12

428

115*0

2*1

Sn .

7'3<>

505

119-0

2-24

Sb .

6-62

904

120-0

3*o

Te.

6-23

723

127-5

2*45

I .
X (liquid

4-66
3-52

386
133

I27'0

130-0

1-6

0-85

Cs .

1-89

299

133-0

0*95

Ba.

378

II2O

137-0

2*4

La.

6-16

1085

139-0

2-8

Ce.

7-04

896

140-2

2*64

Ta .

16-6

2575

181

4*75

Os .

22-5

2700

191

5'i

Ir .

22-4

2500

193

4*9

Pt .

22-5

20l8

195

4-36

Au.

I9-3

1337

197

3*4

Hg

14-2

234

200

1*25

Tl .

n-8

575

204

1-84

Pb.

11-4

600

207

1-84

Bi .

978

54L

208

1-6

[Th

I2'2

Ca 2100

232

3*2]

Biltz points out that in the characteristic frequency we have a very
fundamental atomic constant. Further, since its measurement involves
two properties, namely, specific volume and melting point, it is a step
further than the stage reached, for example, in Lothar Meyer's curve,
in which one property only is brought into relation to the atomic
weight. Some doubt, of course, exists regarding the values in the case
of a few elements, especially those possessing allotropic forms. Biltz
has plotted the frequencies against the atomic weights, and has
found the periodic variation shown in the figure (Fig. 8). It will be
seen to be quite analogous to Mendeleef's original scheme. The
majority of the elements occupy places on the curve, exceptions being
fluorine, manganese, tin, and tellurium. Argon and potassium occupy
as usual their anomalous position. The position of hydrogen, which
was for a long time in doubt, is clearly that of the first member of the
halogen series, as Ramsay had insisted on many years ago. The
unique position of carbon, with extremely high value of v, brings out
very clearly its unique chemical character as regards its immense
capability of forming compounds. An atom possessing high frequency
no doubt is the most likely type of atom to allow of the transfer of
electrons to and from itself, i.e. valency electrons.

ATOMIC HEATS OF SOLIDS 63

Further Consideration of Einstein's Expression for Atomic Heat.
Einstein's expression, viz. —

has been applied to several metals and solid salts. The following two

w Os

Atomic
FIG. 8.

examples, at low temperatures, given by Nernst and Lindemann (Zeitsch.
ElektrocJiem., 17, 818, 1911), are instructive: —

COPPER. /?0v = 240.

Temperature,
Absolute.

Atomic Heat
(Observed).

Atomic Heat
(Calculated).

86-0
33*4

3-38
0-538

3'3i
0-234

22-5

0-223 •

0-023

A SYSTEM OF PHYSICAL CHEMISTRY
POTASSIUM CHLORIDE. /?0v =218.

Temperature,

Atomic Heat

Atomic Heat

Absolute.

(Observed).

(Calculated).

86-0

436

3-54

52-8

2'8o

170

30-1

0-98

0-235

22-8

0-58

0-039

It is clear that while Einstein's formula gives the qualitative course
of the variation of d, with temperature, it is far from reproducing the
variation quantitatively. It might be thought that this is due to an
error in using a single frequency value where in the actual case there
might be a broad band. From Rubens' results already quoted it will
be seen, however, that the characteristic frequency of each salt is
fairly isolated, and at lower temperatures this isolation would become
still more perfect. In metallic elements themselves, where it is usually
regarded that we are dealing with uncombined (unpolymerised) atoms
themselves, the characteristic frequency should be quite sharp and well
denned.

As a way out of the difficulty — prior to the publication of Debye's
formula of 1912, to which reference will be made later — Nernst and
Lindemann proposed an empirical expression for Ct,, which differs
somewhat from that of Einstein.

The Nernst -Lindemann Formula for the Atomic Heat of Solids.
This expression takes the following form : —

that is, the atomic heat expression of Einstein has been split up into
two portions. One portion represents the energy corresponding to
the characteristic frequency v ; the second portion represents the energy
due to the harmonic vibration, which is one " octave " lower than the

characteristic, namely, -.

Why this " octave " should be introduced

is as yet practically unexplained. The expression, however, reproduces
the experimental value of C» with extraordinary accuracy. Its im-
portance is therefore not to be questioned. Some illustration of its
applicability may be given here.

First of all, we have to notice that the direct experimentally de-
termined values of the atomic heat, such as those of Nernst and his
collaborators, hold for constant pressure, i.e. they are (Cp) values. In
the formula, however, it is the atomic heat at constant volume, viz. Ci>,
which occurs. They are nearly identical, but not quite. We have,

ATOMIC HEATS OF SOLIDS

therefore, to convert the one into the other. The procedure is as
follows. On the basis of thermodynamics l we have the relation —

where a is the linear coefficient of expansion, V the molecular or atomic
volume, and K the compressibility. Now experiment has shown that

V

it is allowable to consider = as a constant, independent of T (especially
K.

as it occurs in a small correction term), and further, making use of
Gruneisen's relation (Verhand. d. d. physik. GeselL, 1911, 491) re-
garding the proportionality between Cp and the expansion coefficient a,
we can rewrite the above expression in the form —

where A is a constant characteristic of each substance. The values
of A are given in the following table for some metals and salts : —

VALUES OF THE CONSTANT "A".

Substance.

Atomic Volume
V.

3a x 106.

K x i0l2.

c^.

A KIO&.

A.1

IO'O

72

I'48

•042

2'2

Cu .

7-1

48

0785

•025

1*3

Ag .

10-3

55

0775

•047

2*5

Pb

18-3

82

•055

3-0

Pt

9-1

27

0*40

•019

I'O

NaCl .

1 3 '5

121

4-28

•051

27

KC1 .

18-9

"4

7-6

•038

2'O

In the case of substances for which the compressibility K and the
coefficient of expansion a are known, it is easy to convert Cv values
into O, or vice versa. For other substances, however, the procedure
is somewhat different. The individual constant A can be shown to
be inversely proportional to the melting point Ts (cf. Nernst and

A

Lindemann, loc. cit.\ We can thus write A = =£ where A0 is a
universal constant having the value 0-0214 (C.G.S. units). Hence —
CP = CV+ C'-l A0.

1G. N. Lewis, y. Amer. Chem. Soc., 29, 1165 (1907).

* This expression has been recently transformed into a still simpler one, viz.
Cp = Cv + aT '2> where a is an empiric constant characteristic of the substance in
question (Lindemann and Magnus, Z. Elektroc^em., 16,269, 1910; cf. also Nernst,
Ann. Physik, 36, 395, 1911). Since direct measurement always gives us Cp, this is
a very convenient way of getting Cv.

VOL. III. 5

66

A SYSTEM OF PHYSICAL CHEMISTRY

The general method adopted by Nernst and Lindemann to test
their formula, equation i, was, first of all, to calculate Cv by its means
for substances for which v is known (either by means of Reststrahlen
measurements, or by means of Lindemann's melting-point formula).
The "calculated" value of Cv was then converted into terms of Q> by
means of the expression —

CP = CV+ C'TA

and the result finally compared with the observed values of Cp.

The following tables give the results obtained in the case of alu-
minium, copper, silver, lead, and the salts KC1, NaCl, KBr, and the
diamond. These are all taken from the paper of Nernst and Lindemann
(loc at.) : —

ALUMINIUM.

= 4°5-

T Absolute.

Cp Calculated.

Cp Calculated.

Cp Observed.

Observer.

32-4

0*23

0*23

0-25

Nernst.

35'1

0-31

0-31

0-33

,,

83*0

2-42

2-43

II

86-0

2-52

2-53

2-52

,,

88-3

2-61

2-62

2-62

M

3-99

4*05

3'97

Koref.

235-0

5-15

530

532

Koref, Schimpff.

331'0

576

5-82

Magnus, Schimpff.

433*0

5-70

6-06

6-10

Magnus.

555-o

5-80

6'3°

6-48

" .

COPPER.

321.

T Absolute.

Cv Calculated.

Cp Calculated.

Cp Observed.

Observer.

23-5

0-15

0-15

O'22

Nernst.

277

0-3I

0-3I

0-32

,,

33*4

0'59

°"59

0-54

,,

87-0

3'35

3*37

3-33

»t

88-0

3*37

339

3-38

M

137-0

4 '60
5'42

4-65

4*57
5-59

Koref.
Koref, Schimpff.

290-0

5-60

575

579

Gaede.
Bartoli, Stracciati,

323-0

5*66

5-81

5-90 |

Schimpff.

450-0

5-8i

6-03

6-09

Magnus.

ATOMIC HEATS OF SOLIDS
SILVER. /?0v = 221.

T Absolute.

Cv Calculated.

C^ Calculated.

Cp Observed.

Observer.

35'0

I'59

i '59

I-58

Nernst.

39'i

1-92

1-92

I-go

,,

42-9

2'22

2'22

2-26

,,

45 '5

2'43

2'44

2-47

it

5i'4

2-81

2-82

2-81

11

53-8

2-97

2-98

2*90

,,

77-0

4-07

4-11

4-07

»

IOO*0

4-72

477

4-86

Koref.

200-O

5-60

577

578

»

273-0

577

6 -02

6-00

Koref, Schimpff.

33 1 'o

5-82

6-12

6-01 I

Bartoli, Stracciati,
Sc impff.

535'Q

5 '9°

6-45

6-46

Magnus.

589-0

5*92

6-57

6-64

M

LEAD. /J0v = 95.

T Absolute.

Cv Calculated.

Cp Calculated.

Cp Observed.

Observer.

23-0

2'95

2-96

2-96

Nernst.

28-3

3-63

3-64

3-92

,,

36-8

4'35

4 '37

4-40

,,

38-1

4'43

4'45

4 '45

,,

85-5

5-60

5'68

5-65

,,

90-2

5-62

570

571

5»

200-0

5 '90

6-12

6-l3

Koref.

273-0

5-92

6-24

6-3I

Koref, Gaede.

290*0

5 '92

6-26

6-33

Gaede.

332-0

5'93

6-31

6-41

Magnus, Schimpff.

409-0

5'94

6-40

6-61

Magnus.

ATOMIC HEAT OF DIAMOND.

= 1940 (obtained by using Lindemann's formula, the melting point being taken

as 3600°).

T Absolute.

Cv Calculated.

C p Calculated.

Cp Observed.

Observer.

30

o-ooo

o-ooo

o-oo

Nernst.

42

o-ooo

o ooo

o-oo

88

0-006

0-006

0*03

92

0*009

0-009

0-03

205

0-62

0-62

O'02

209

0-65

0*65

0*66

220

0-74

0-74

0-72

222

0*78

078

0*76

Weber.

232

0-87

0-87

0*86

Koref.

' 243

0-97

0-97

0'95

Dewar.

262

rx6

1-16

1-14

Weber.

284

i'37

i*37

i'35

J}

306

i'59

i'59

i'58

it

331

1*82

r83

1*84

358

2*07

2-08

2'12

»>

413

2'53

2'55

2-66

n

Il6g

5 19

5'4I

5*45

•>

68

A SYSTEM OF PHYSICAL CHEMISTRY

The agreement between theoretical and observed values is good.
The important point about the diamond, as already remarked, is
that it ceases to have any measurable heat capacity at all for about 40°
above absolute zero. This is in agreement both with the thermo-
dynamic theorem of Nernst and the Einstein theory of specific heat.

We now pass to the consideration of the salts KC1, NaCl, and KBr.
Each of these possesses a single well-defined band in the far infra-red
measured experimentally by Rubens. This value has been employed
in the equation. The values of Cv and C/> are mean atomic heats, i.e.
one half the molecular heats.

KC1. /V=2i8.

T Absolute.

Cv Calculated.

C . Calculated.

C. Observed.

Observer.

22'8

O'6l

0-61

o-58

Nernst

26-9

0' 0

0-70

0-76

tt

30-1

1-23

1-23

0-98?

,j

337

i'53

i'53

1-25?

39-0

1-98

1-98

i-83

Jf

48-3

2-66

2-66

2-85

i

52'8

2- 6

2-97

2'8o

t .

57-6

3-25

3-26

3-06

,,

63-2

3'57

3'59

3-36

»

70-0

3-85

3-87

379

76-6
86-0

4-10
4-40

4'J3
4'43

4-II
4-36

"

137-0

5-26

5'33

5 '25

Koref

235-0

570

5-86

5-89

ii

33 ^o

5-83

6-06

6-16

Magnus

416-0

5-87

6-21

6-36

,,

55o-o

5'9i

6-36

6-54

»»

NaCl. fay = 287.

T Absolute.

Cv Calculated.

C . Calculated.

C . Observed.

Observer.

25-0

0-32

0-32

0'29

Nernst

25-5

0-34

o-34

0-31

28-0

048

0-48

0-40

67-5

2-87

2-88

3-06

69-0

2-94

2*95

3*13

81-4

3'47

3'49

3'54

83-4
138-0

3-61
4-82

3*64
4-90

37^
4-87

Koref

235'o

5'52

573

576

"

ATOMIC HEATS OF SOLIDS

69

KBr. /?0v =177 (O was calculated from Ct, by the usual formula, the
value of A being taken as identical with that for NaCl).

T Absolute.

Cv Calculated.

C p Calculated.

C p Observed.

Observer.

787

4-67

470

474

Nernst

82-5

477

4-80

476

»i

85M

4-82

4'85

4-82

»

89-2

4-91

4*94

5*03

137-0

5'47

5'56

5H2

Koref

234-0

579

6'O2

6'io

5f

The agreement between the observed mean atomic heat of these
three salts and that calculated on the basis of the Nernst- Lindemann
equation would appear to be thoroughly satisfactory. This, however,
is really to a large extent accidental. Nernst found later that the
Nemst-Lindemann equation would not apply to other salts. He was
led ultimately to a different treatment of the problem of the mean
atomic heat (or half molecular heat) of polyatomic substances which
will be discussed briefly when we have become acquainted with the
equation of Debye.

It should be pointed out that no very clear theoretical significance
can be attached to the Nernst-Lindemann equation even when applied
to monatomic substances. Its merit is that it reproduces experimental
data in such cases with a high degree of accuracy, and in this respect
is decidedly to be preferred to the equation of Einstein. The Nernst-
Lindemann equation resembles Einstein's in that a single frequency v,
obtained by one of the methods mentioned previously, is employed,
along with one half of this quantity.

The Specific Heat of Solids at High Temperatures and the Significance
of the Energy possessed by the Electrons.

It will be remembered that the expression for the atomic heat (at
constant volume) of a solid, whether stated in the original form of
Einstein or in the Nemst- Lindemann modification, reaches as a limit
the value 3R (5-95 cals.). As a matter of fact, however, the atomic
heat (at constant volume] has been found to exceed this limit consider-
ably when the temperature to which the value of the atomic heat refers
is very high. At these temperatures there must be some other source
of internal energy content not taken account of in simply ascribing the
total energy to the vibrations of the atoms. At high temperatures the
body becomes " white hot," that is it is capable of emitting short waves
in the visible, which are due in all probability (as we have already seen)
to the vibrations of the electrons in the atoms. Although the energy
of the electrons can be neglected at low temperatures compared with
that possessed by the atoms, this is no longer the case at high tempera-
tures, and hence such must be taken into account in any expression

A SYSTEM OF PHYSICAL CHEMISTRY

for specific (or atomic) heat. An attempt in this direction has been
made by J. Koenigsberger (Zeitsch. Elekirochem.^ 17, 289, 1911), who
considers that the atomic heat of metals (at constant volume) should
reach 9 instead of 6, by taking the vibrating energy of the electrons into
account. No great stress, however, can be laid upon the significance
of the actual numerical limit quoted (cf. Nernst and Lindemann, loc. cit.).
It is interesting, however, to see the numerical values for the atomic
heat Cv actually obtained at high temperatures (cf. Koenigsberger, loc.
cit.) :_

ATOMIC HEAT (AT CONSTANT VOLUME) AT HIGH TEMPERATURES.

Metal.

Region of Temperature.

cv.

Observer.

Silver

907-1100° C.

67

Pionchon (1887). ,

Tin .

1100°

9-2

»»

Iron

1200°

9-6

Nickel

II500

7*9

Copper

900°

7-1

Richards (1893).

Aluminium

650°

77

Boutchew.

DEBYE'S EQUATION FOR THE ATOMIC HEAT OF SOLIDS.

(P. Debye, Ann. Physik, [iv.], 39, 789, 1912.)

We have seen that Einstein's specific (atomic) heat formula only
holds qualitatively, and that whilst that of Nernst and Lindemann
accords much better with experiment, their expression suffers from the
very serious drawback of lacking a sound theoretical basis. Debye has
recently developed an expression having a theoretical basis — employing
the quantum hypothesis — and has found that the expression reproduces
experimental values with even greater fidelity than that of .Nernst and
Lindemann. Debye's formula may therefore be regarded as the most
significant of all specific heat formulae yet proposed. The general line
of thought pursued by Debye is as follows : — *

In Einstein's investigation it was assumed for the sake of simplicity
that the vibrating particle (the atom) only gave rise to monochromatic
radiation (frequency v) and only absorbed such. Instead of this limita-
tion to a single characteristic frequency v, Debye works out the expres-
sion for atomic heat on the basis of absorption (and emission) of a
number of vibration frequencies, in fact a whole spectrum. It will be
seen at once that this is far more likely to coincide with what actually
occurs than the simpler assumption of Einstein. Debye starts out with
the very plausible assumption that a vibrating atom in a solid cannot
be vibrating simply harmonically with a single frequency, but owing to
the proximity of other atoms and probably to collisions with them the

1 For details see Debye's paper. Also Appendix II.

DEB YES EQ UA T2ON 7 1

mode of vibration will be complex. Such complex modes of vibration
can be treated, as Fourier showed, as theoretically made up of a series
of true simple harmonic motions, and we have thus to integrate over a
spectrum of frequencies if we wish to calculate with greater exactness
the total energy content of a vibrating atom. The first point to be
noted is, however, that we do not deal with a spectrum extending from
v = o to v = oo . If a body consists of N atoms — treated as massive
points— the system possesses 3N degrees of freedom. The system will
therefore in general exercise 3N different periodic vibrations, i.e. 3N
different vibration frequencies. If the older view — consonant with the
principle of ^iifpartition of energy — were true, namely, that the energy
was the same for each degree of freedom over the entire spectrum, then
we could state directly that each vibration frequency corresponded to the

R

energy /£T (where k = ^, N being now the number of atoms in i gram-
atom), so that the total energy of the body would be simply 3NRT per
gram-atom as Dulong and Petit's Law requires.

We have seen, however, that the whole point of the quantum theory
is the negation of the principle of equipartition throughout a spectrum,
and that instead the energy per vibration varies with the type of vibra-
tion. This lack of equipartition is expressed as we have already seen in

hv
the Planck expression fe(//feT , which gives the true mean energy of

any single vibration v. To obtain the total energy of the vibrating
system it is necessary to sum this expression over the spectrum (of
absorption or emission), the spectrum not containing an infinite number
of frequencies but limited to 3N, as already pointed out. The spectrum
is characterised by two factors, (a) its boundaries, (b) the density of the
lines, i.e. the number of lines in any given vibration region dv. There
is, according to Debye, a certain definite limiting frequency vm beyond
which the spectrum does not extend. Debye has reached a number of
important conclusions in the course of his investigation. The first is
this : If the temperature T be regarded as a multiple (or submultiple] of a
temperature 9 (a characteristic constant for any given substance], then the
atomic heat for all monatomic substances can be represented by the same

T

curve, i.e. the atomic heat is a universal function of ' ^.

u

A second relation which is not confined to monatomic substances
states that the number of lines spread over a region dv is proportional to
v^dv (a relation for black-body radiation already obtained by Jeans).
From this Debye concludes that at sufficiently low temperatures the
atomic heat of all solids is proportional to T3, that is to the third power
of the absolute temperature. This conclusion differs (theoretically) widely
from the conclusion of the Einstein and the Nernst-Lindemann equa-
tions. It is borne out by the experimental data, but experiment at low

72 A SYSTEM OF PHYSICAL CHEMISTRY

temperature bears out the Nernst-Lindemann view equally. As a
corollary it follows that the atomic energy content at low temperatures is
proportional to the fourth power of the absolute temperature (cf, Stefan's
Law for total radiation). Debye's expression for the energy U of a
solid of volume V which contains N atoms is —

---**

the energy not being ascribed to a single harmonic vibration v, but to
a spectrum of such extending from o to vm. If we now define the
temperature 6 (the characteristic constant of the substance1) by the
expression —

e-^--ftym . . (*)

and introduce into equation (i) as a new dimensionless variable — the
magnitude —

we can write —

rhvm/kT

CbT\s I #*//£
3 4^7 • • • w

or, substituting 6 for ——-, we obtain —

/T

Dulong and Petit's Law, if true, would require the relation —

U = 3N^T.

Now if we write _ = x, and differentiate equation (5) with respect to
temperature, we obtain finally —

The magnitude 3^ has the value 5-955 cals. and may be denoted
by C^ since this numerical limit is reached at high temperatures. Equa-
tion (6) may therefore be rewritten in the form —

Cl) = if [VJ^1 _3* (7)

Equation (6) or (7) is Debye's atomic heat formula.

DEBYE'S EQUATION 73

For the case in which x is small, i e. T is large (compared to 0),

- becomes £2, and — * becomes — or simply 3. Hence,

— — = — — 3 = i, in other words, Dulong and Petit 's Law holds at high
temperatures. On the other hand, when x is large, i.e. T is small, we

obtain by expanding I ^ _ ' the series &(e~* + e~^ + e~^ + .. .)</£,
or —

Hence equation (7) yields — since the series expression in the bracket is
1-0823—

C,; 12 x 6 x 1-0*23 0T3

-

That is, at sufficiently low temperatures, the atomic heat is proportional
to the cube of the absolute temperature T. (The above relation can be

£

used to calculate 0, employing the experimental value of f£- at some

Uoo

low temperature.)

Debye has tested his formula (equation (6) or (7)) and found it to
agree very well with experiment. It has further been tested inde-
pendently by Nernst and Lindemann (Sitzungsber. Berl. Akad., p. 1160,
1912) with the following results, in which the values of C^, (atomic heat
at constant pressure) are obtained from the values of Cv given by
Debyc's formula by employing one of the relations between Cp and Cv
mentioned at an earlier stage. Nernst and Lindemann slightly alter
the form of Debye's equation (in respect of the symbols) so as to bring
it more into line with previous expressions. The expression as used by
them takes the form —

- -

ALUMINIUM (6 = /?v = 398 ; earlier value employed = 405).

T.

C^ Observed.

C p Calculated.

Difference
Obs.-Calc.
(Formula of
Debye).

Difference Obs.-Calc.
(Formula of Nernst
and Lindemann).

32-4

0'25

0-25

O'OO

+ 0-02

35'I

0'33

0-32

+ O-OI

+ 0-02

830

2-41

2-36

+ 0-05

- 0-O2

86-0

2-52

2*50

+ O'O2

- O-OI

88-3

2-62

259

+ 0-03

O'OO

137-0

3 '97

4-10

- 0-13

- 0*08

235-0

5-32

5'34

- O-O2

+ O-O2

33I*o

5-82

5'78

+ 0-04

+ 0'06

433'0
555'o

6*10
6-48

6-07
6-30

+ 0-03

+ 0-18

+ 0-04

+ 0-18

DIAMOND (ftv = 1860; earlier value 1940).

T.

C Observed.

C . Calculated.

Difference
Obs.-Calc.
(Formula of
Debye).

Difference Obs.-Calc.
(Formula of Nernst
and Lindemann).

88

0-028

0-049

- O-O2I

+ 0-022

92

0-033

0*058

- o 025

+ 0-024

205

0-618

0-61

4- 0-008

o-oo

209

0-662

0-66

+ O-OO2

+ o-oi

220

0-722

0-74

- 0-018

- 0-64

222

0-76

o-75

+ O'OI

- O-O2

243

°*95

0-925

+ 0-025

- O-02

262

1-14

i-io

+ 0-04

- 0-02

284

1-32

+ 0-03

- O'O2

306

1*58

1-54

+ 0-04

- o-oi

331

1-84

1-82

4- 0-02

+ o-oi

358

2*12

2-07

+ 0-05

•f 0-04

413

2-66

2*61

+ 0-05

+ o-ii

1169

5 '45

5-49

- 0-04

+ 0-04

SILVER (/3v = 215 ; earlier value employed = 221).

T.

Cp Observed.

Cp Calculated.

Difference
Obs.-Calc.
(Formula of
Debye).

Difference Obs.-Calc.
(Formula of Nernst
and Lindemann).

35*0

I-58

1-50

+ 0-o8

- o-oi

39-i

1-90

1-88

+ O'02

- O-02

42-9

2-26

2-14

+ O'I2

+ 0*04

45-5

2-47

2-42

+ 0-05

+ 0-03

5i-4

2-81

2-85

- 0*04

- o-oi

53-8

2-90

3-02

- O'I2

- 0-08

77-0

4-07

4-19

- O"I2

- 0-04

IOO

4-86

4-83

+ 0-03

+ 0-09

200

5-78

5-8o

- 0'O2

+ O'OI

273

6*00

6*04

- 0-04

— O'O2

33i

6-01

6-13

- O'I2

- O'll

535

6-46

6-46

0*0

-}- O'OI

589

6-64

6-57

+ 0-07

+ 0-07

DEB YES EQUATION 75

Copper gave an equally satisfactory result. On the whole Debye's
equation applies more exactly even than the equation of Nernst and
Lindemann. Debye's equation has of course the further advantage that
it possesses a theoretical basis, whilst that of Nernst and Lindemann is
semi-empirical.

NOTE on the method of evaluating the summation term of the Debye formula in.
the form given by Nernst and Lindemann.
To evaluate —

Set - equal to x. Also set e~l = a. Then e-** = a"* = (a*)« - yn if y be set

equal to a*.

The sum can therefore be written :

n = :
or the same expression may be written :

First term. Second term. Third term. Fourth term.

First term = ipL + ^ + Z_ + . . . adinf7\.
xLi 23 J J

Second term = —^^ + ~z+^+^~z + ' ' ' ad inf'\-
Third term = — -^ - + ? + 2L. , + £_ + . . ad in '. \.

A:3Ll 2s 3 4

Fourth term = — f^ +^--|.tL+^L + . . . ad inf. \.
x*[_i* 24 34 44 J

Each of these series is convergent, so that only the first three or four expressions
in each need be taken into account.

Characteristic Infra-red Frequencies Obtained from Debye's Formula.
—By means of equation (2) of the preceding section it is possible to
calculate the characteristic infra-red frequency vmax. (which represents
the upper limit of the spectrum of vibrations) from a determination of
the characteristic temperature 6. This temperature can be obtained in
turn from the elastic constants of the substance as has been shown by
Debye. Making use of Debye's values of (9, H. S. Allen (Proc. Roy.
Soc., A, 94, ioo (1917)) has calculated vmax. for the elements given in
the following table. It will be observed that the frequencies so obtained
agree very well with those calculated by methods previously mentioned : —

7 6 A SYSTEM OF PHYSICAL CHEMISTRY

Atomic Frequencies by Debye's Formula.

Element. ymax- x io-«-

Al 8-26

Fe 9-67

Ni g-oi

Cu 6-81

Pd 4-22

Ag 4'39

Cd 3-48

Sn 3-83

Pt 4-68

Au 3-44

Pb 1-49

Bi 2-30

MOLECULAR HEATS OF POLYATOMIC SOLIDS, E.G. SALTS. .

It has already been mentioned that the Nernst-Linderaann equation
does not take account of the molecular heats of salts in general, although
it appears to do so for KC1, NaCl, and KBr. The Debye equation is
equally inapplicable. Nernst proceeds thereforein the folio wing manner: —

In the case of a polyatomic molecule in a solid there are two types
of vibration to be considered ; first, the vibration of the molecule as a
whole, and secondly, the vibration of the atoms inside each molecule.
This view, it will be observed, is difficult to reconcile with the con-
clusion arrived at, as a result of Br gg's work on crystals by the X-ray
method, according to which molecules lose their identity in the solid
state. Nevertheless the method employed by Nernst in connection
with molecular heats of salts, etc., is highly successful, and must be
considered, although an open mind has to be kept regarding certain of
the assumptions. According to Nernst the mode of vibration of the
molecules is taken account of by Debye's formula, whilst that of the
atoms is taken account of by Einstein's formula. It is reasonable to
treat the motion of atoms inside a molecule as closely analogous to that
of the atoms in a monatomic solid, to which case Einstein's formula
was intended to apply in the first instance. Nernst's method amounts
therefore to a combination of Einstein's and Debye's formulae. The
frequency vb i.e. the upper frequency employed in the Debye portion
of the expression, is obtained from the melting-point formula of
Lindemann, whilst the frequency v2 employed in the Einstein portion
of the expression is that given directly by the Reststrahlen method.1
On adding the two portions together the resulting expression gives the
whole molecular heat of the solid. If Cv denotes, as f. efore, the mean
atomic (or half molecular) heat of a diatomic solid such as KC1, then
2Ci, denotes the whole molecular heat of the solid at constant volume,
and according to Nernst —

1 1/2 is greater than i/x ; hence at low temperatures there are practically no
resonators possessing the quantum hv2. That is, at low temperatures the atoms
possess scarcely any energy of vibration whilst the molecules as a whole do possess
energy of vibration.

MOLECULAR HEATS OF POLYATOMIC SOLIDS 77

where Fj and F2 are the Debye and Einstein functions or atomic heat
expressions respectively.

In the case of KC1, fivl = 166 (obtained from the melting-point
formula), and /3vz = 213*5 (obtained directly from the Reststrahlen
observations). The following table, taken from Nernst's Theory of the
Solid State, gives the values of the Einstein and Debye portions of the
molecular heat in the case of KC1. The column headed "correction n
contains the quantities which must be added to 2C» in order to give
2Cp, the molecular heat at constant pressure : —

KCl.£v2 = 213-5; ftvi = 1 66.

T.

Einstein.

Debye.

Correction.

Sum
(2Cp Calc.).

aCp Obs.

22-8

0-046

1-04

_

1-086

1-16

26'9

0'I3

1-48

—

l-6l

1-52

30'I

0'25

I'87

—

2T2

1-96

337

0'43

2-25

—

2'68

2*50

48-3

I '43

3*52

—

4'95

570

57*6

2-13

4'06

O'O2

6'2I

6-12

70-0

2-89

4'57

0*04

7-50

• 7-58

86-0

3-66

4*97

0-06

879

8-72

-35

5'55

3'8l

0-32

n-68

11-78

416

5-83

5*9i

0-68

12*42

12-72

550

5'87

5*93

0-90

1270

13-18

NaCl .

282 ; fa = 229 (215).

T.

Einstein.

Debye.

Correction.

Sum.

zCp Obs.

25

O'OIO

O'6io

_

0*62

0-58

25-5

O'OI2

0-637

—

0-65

0-62

28-0

O-O27

0-794

—

0-82

0-80

67-5

I-65

3-56

0-02

5-23

6*12

6g'o

173

3-63

O"O2

5-38

6-26

81-4

2'39

4*13

0-04

6-56

7-08

83-4

2'49

4*20

0*06

675

7'5°

138

4-26

5*22

0-16

9-64

974

235

5 '29

5-68

0-42

11-39

11-52

7 8 A SYSTEM OF PHYSICAL CHEMISTRY

AgCl./?v2 = 179; ^=102(114).

T.

Einstein.

Debye.

Correction.

Sum.

zCp Obs.

23-6

0-17

278

2*95

2-g8

26-4

0-31

3'l8

—

3 '49

3'44

32-8

076

3 '89

0*04

4-69

479

45-6

I-QO

4-69

0-07

678

7*25

87-0

4-24

- 5-58

0-18

lO'OO

974

116

4'92

575

0-28

10-95

10-34

207-5

5 "59

5'9o

0-52

I2'OI

11-81

330

5'8i

5'93

1 *3

I3-04

13-01

405

5*85

5'94

1-8

I3'59

13-60

430

5'86

5*94

1*9

1370

1372

The numbers in brackets denote the values of /fri actually obtained
by the melting-point method. They are very close to the values of
ftvl which are arbitrarily chosen.

An examination of the data given shows that at very low tempera-
tures the contribution made by the Debye portion is much greater than
that made by the Einstein portion, as we would expect.

It will be recalled that the Nernst-Lindemann formula applies fairly
well to the case of KC1 and NaCl, the frequency employed being that
obtained by the Reststrahlen method. "This agreement must be
regarded as half accidental, and may be explained on algebraic grounds
from the approximation —

In the case of KC1 and NaCl the fundamental [molecular] frequency
[vi in the above table] is in fact about 0-75 times the internal [atomic]
frequency [v2 in the above case]."

An interesting case is that of mercurous chloride. Taking the
molecular formula to be HgCl and calculating the molecular heat by
the above method, discrepancies are observed which lie outside experi-
mental error. Nernst concludes therefore, that in solid calomel the
molecule Hg2Cl2 (and possibly a still more complex molecule) exists,
which is in agreement with conclusions arrived at concerning the mole-
cular formula of calomel in the dissolved state. "The exact complexity
cannot yet be decided since the melting point of the salt is not open to
observation on account of decomposition."

A further interesting case is that of ice. In order to account for
the molecular heat of ice by the above method it is necessary to regard
the molecular weight as 36 approximately (cf. Nernst, The Theory of
the Solid State). Graphite and sulphur likewise show evidence of
considerable polymerisation.

Although the method of treatment which has just been outlined is
very ingenious, it is obviously of a somewhat hypothetical character in

MOLECULAR HEATS OF POLYATOMIC SOLIDS 79

its present formulation. It is not improbable that considerable modi-
fications may eventually be introduced. It represents, however, a
partially successful attempt to solve the problem of the heat capacity of
polyatomic substances.

We shall conclude this chapter by quoting some numerical data
obtained by Eueken (Ber. Deutsch. phys. Ges., 18, 4 (1916)) in con-
nection with the heats of fusion and vaporisation of certain condensed
gases at low temperatures : —

Molecular Heat

Substance. of Fusion in

Calories.

Argon 267-9

Nitrogen 168*7

Oxygen 105-5

Carbon monoxide . 224*1

Nitrogen was found to exist in two solid forms, the temperature of
transition being 35 '5° abs. and the molecular heat of transformation
53-8 cals. Oxygen exhibits three solid forms ; transition temperatures,
23 '5° and 42*5° abs., the corresponding molecular heats of transforma-
tion being 17*5 and 167*4 cals. respectively. Carbon monoxide exists
in two solid forms ; transition temperature, 60*4° abs., molecular heat
of transformation, 144*1 cals.

Molecular Heat

Substance. of Vaporisation

in Calories.

Hydrogen 29

Nitrogen . 1363

Oxygen 1599

Argon 1501

. Carbon monoxide 1414

Eucken (loc. tit.} has likewise measured the specific heats of solid
and liquid argon, nitrogen, oxygen, and carbon monoxide ; the specific
heats of solid carbon dioxide and of liquid hydrogen, and likewise the
specific heat of highly compressed gaseous helium.

CHAPTER IV.

(Physical equilibrium in gaseous systems) — Molecular heats of gases — Bjerrum's-
theory — Kriiger's theory.

BJER RUM'S THEORY.

IN Chapter I. we have already had occasion to discuss briefly the
problem of the molecular heats of gases. It has been pointed out
that the observed values cannot be accounted for on the basis of the
equipartition principle, especially the fact that the molecular heat varies
with the temperature. In view of the considerable advance which has
been made possible in the analogous case of solids by the application
of the quantum theory, it is of interest to see how far the same con-
siderations can be applied in the present case. This problem was
first taken up by Bjerrum (Zeitsch. Elektrochem., 17, 731 (1911) ; ibid., 18,
101 (1912)).

In Chapter I. we have given a table showing the number of possible
degrees of freedom, as estimated by Bjerrum, which are possessed by
mono- , di-, tri-, and tetra-atomic gas molecules in respect of translation,
rotation, and vibration. In the case of monatomic gases, which appear
to possess translational energy only, the equipartition principle of
classical statistical mechanics gives a satisfactory explanation of the
observed values, e.g. the case of argon already discussed. So long
as we restrict ourselves to translational movement the equipartition
principle necessarily holds good, whether the molecule be monatomic
or polyatomic; the distribution of energy in terms of the quantum
theory only enters when we deal with vibrations or rotations, i.e. move-
ment with respect to a centre of gravity.

As already pointed out, the result obtained in the case of argon
leads us to regard a mona.tomic gas as possessing no energy other than
that of translation. This is somewhat unexpected, and we shall return
to it later in connection with Kriiger's theory. For the present, how-
ever, we are discussing Bjerrum's treatment.

As regards rotation of the molecule as a whole Bjerrum shows that
the potential energy of rotation is negligible compared with the kinetic
energy. (The case is quite different of course for the vibrations of the
atoms inside the molecules.) In regard to vibration of one atom with
respect to the other in a diatomic gas molecule, according to the
quantum hypothesis the sum of the kinetic and potential energies, in-
stead of being RT, is a fraction <£ of this quantity. That is the total
vibrational energy is <£RT. <£ is a function of the temperature T and

80

BJERRUM'S THEORY Si

of the vibration frequency v. According to Nernst and Lindemann we
have —

fr/T

Bjerrum has investigated the Einstein function as well. It is found to
hold fairly well, but is not so exact as that of Nernst and Lindemann.
At absolute zero <f> is zero, and at high temperatures </> approximates
to unity, i.e. at high temperatures the conclusions based on classical
statistical mechanics ought to hold good.

In the case of a triatomic molecule, containing the atoms A, B, and C,
these may vibrate with respect to one another in pairs, viz. A with respect
to B, B with respect to C, and C with respect to A. There are there-
fore three different vibrations to be taken account of, and the total
vibrational energy is —

+ + .

The following tables contain the data obtained in the case of a number
of gases. The wave-length A, is that chosen by Bjerrum for substitution
in the Nernst- Lindemann equation. The temperature T is in degrees
absolute. Cv(x°, T0,) denotes the mean molecular heat (at constant.
volume) over the temperature range between x° abs. and T° abs.

Molecular Heat of Hydrogen.

The formula employed by Bjerrum is : —

C«,(o°, T°) = (5/2). R+ R<£[2-0ya]
where 5/2R covers the translational and rotational energy * of the mole-

1 Bjerrum, as already pointed out, ascribes two degrees of freedom to a diatomic
molecule, such as H2, in respect of rotation of the molecule as a whole. By writing
the corresponding energy term as RT (which then added to the 3/2RT, due to
translation, makes in all 5/2RT), Bjerrum is here assuming that rotational energy may
be treated from the standpoint of the equipartition principle. To get the contribution
which translation and rotation make to the total molecular heat it is of course neces-
sary to differentiate with respect to T, i.e. we obtain the term 5/2R. The same as-
sumption of equipartition is employed in the other gases, the molecular heats of which
are considered. In a later paper, however (Nernst Festschrift, 1912, p. 90), Bjerrum
regards rotational energy of the molecule as a quantity which has to be treated
from the quantum standpoint, the characteristic wave-length of molecular rotation
lying far in the infra-red region (about 30/4 in the case of hydrogen), whilst the
characteristic wave-length of atomic vibration (inside the molecule) lies in the short
infra-red region, between i and lo/m approx. In other words, molecular rotations are
much less violent than atomic vibrations and require a much smaller quantum, i.e.
a much smaller v or greater A. It has been observed that the atomic vibrations,
instead of manifesting themselves as sharp lines in the short infra-red portion of the
spectrum, actually exhibit bands more or less broad. An explanation of this has
been suggested by Bjerrum on the following lines. The broadening is regarded as
due to the presence of a number of " secondary " lines situated on either side of the
principal atomic-vibration lire, the latter being identified with the " head " of the
band. These secondary lines are regarded as multiples or overtones of the funda-
mental rotational frequency or wave-length which occurs in the longer wave region.
As the frequency of rotation is small its consecutive multiples will lie close together
VOL. III. 6

82

A SYSTEM OF PHYSICAL CHEMISTRY

cule and 2*o//, = X is the characteristic wave-length of atomic vibration,
which is of course connected with the frequency of atomic vibration
by means of the relation c = Xi/, where c is the velocity of light. The
following table indicates the degree of agreement between the observed
and calculated molecular heats of hydrogen : —

T Abs.

Cv (291°, TO,)

Cv (291°, TO,)

Calculated.

Observed.

1734

5'39

5-23

2083

5'5I

5'44

2431

5-62

5-68

2647

5-69

576

Molecular Heat of Nitrogen. (Oxygen and carbon monoxide have
the same molecular heat as nitrogen.)

Formula employed: Ci»(o°, T°,) = (5/2) . R +

T Abs.

Cy (273°, TO,)
Calculated.

Ctf (273°, TO,)
Observed.

473

4'97

473

903

5-12

4-91

1273

5'32

5*25

1620

5'47

5-3I

Cy (agio, T0))

Cv (291°, T°.)

Calculated

Observed

1792

5'55

5'43

2057

5-65

5'58

2225

571

579

2446

578

5-87

2640

5-84

5 '93

In the case of hydrogen the characteristic wave-length assumed for
the atomic vibration is X = 2 '0/4 ; in the case of nitrogen, X = 2 '4/4.
One would expect that if the vibrating particles are electrically charged
that an absorption band should occur at these positions in the case of
hydrogen and nitrogen. No bands, however, have been observed in
this region of the spectrum. On the other hand, oxygen and carbon
monoxide have the same molecular heat as nitrogen, and direct
measurement has shown that oxygen possesses bands at 3*2/4 and 47/4,
whilst carbon monoxide exhibits bands at 2*4/4 and 4*6/4. The bands

when they occur at all, and, in fact, by measuring the distance apart of the
"secondary" lines referred to, it is quite possible to calculate the actual funda-
mental frequency of the rotation itself. On the basis of a calculation made by
Rayleigh it can be shown that under certain conditions, if v^ is the vibration fre-
quency of the atomic vibrations, and v2 is the much smaller frequency of tte mole-
cular rotations, then in the region ot the spectrum in which the atomic vibration
manifests itself we may expect in general three absorption lines occurring at the
positions d efined byvl + vz, vlt and j/x - vz.

BJERRUM'S THEORY

of oxygen are weak, and it is possible that hydrogen would exhibit a
band if its absorption were measured when the gas itself were excited,
i.e. at a fairly high temperature.

Molecular Heat of CO2. (SO2 has the same molecular heat.)
As this is a triatomic molecule we have three atomic vibrations to
take account of. Two of these, however, may be expected to be the
same, as the oxygen atoms are presumably linked symmetrically to the
carbon atom. The translational energy plus the rotational energy are
in this case, according to Bjerrum, represented by the term 3RT, i.e.
3/2 RT translational and 3/2RT rotational. The wave-length of vibra-
tion of each of the oxygen atoms with respect to the carbon is taken
by Bjerrum to be 5'ctyi; the wave-length of vibration of the two oxygen
atoms against each other is taken to be 8 -I/A. The formula employed
by Bjerrum is —

C^o0, T°,) = 3R + 2R4>[5-o//.] + R^S-i^].

[Note that the second term on the right-hand side of this expression
contains the number 2, to allow for the fact that there are two
similar kinds of atomic vibrations present. It may also be pointed
out that if CO2 were a linear molecule, O = C = O, the rotational
energy term would be RT (as in a diatomic gas). The fact that the
value 3/2RT for the rotational energy agrees with experiment (cf. the
following table) means that CO2 is not a linear molecule, its spatial

constitution being represented approximately by

The con-

stitution of the CO2 molecule is considered in detail by Bjerrum (Verh.
d.D.phys. Ges.,16, 737, 1914).]

T Abs.

Cv (273°, I0,)
Calculated.

Cv (273°, T°,)
Observed.

Cv (273°, T°,) Calculated

ing the Wave-Lengths
and I47f*.

Employ-
27, 4 '3.

473

7*44

7-48

7-67

903

8-67

8-60

8-47

I273

9*35

9'33

9-00

1637

9-80

9-84

9-40

1884

10-03

9-98

9-62

2112

IO'2I

10-28

9-81

2383

10-43

10-47

10-04

The agreement between observed and calculated values is satisfactory.
We have now to see what evidence is available from the absorption
spectrum of CO2 as regards the choice of the wave-lengths employed.
Direct measurement has shown that CO2 possesses bands at 14*7, 4*3,
and 2 "jfji. These are of the same order of magnitude as those used
in the above formula. As a matter of fact, Bjerrum has also used the
three observed values of X and has obtained values for Cv which agree
moderately with the observed. The formula in this case is —
C (o°, T°,) = 3R

84

A SYSTEM OF PHYSICAL CHEMISTRY

The resulting values are given in the last column of the preceding
table.

Molecular Heat of Water Valour.
The formula employed by Bjerrum is —

C,(o°, T0,) = 3R + 2R«i-3rf + R^[3-6rf +
The following values are thereby obtained : —

T Abs.

Cv Calculated.

Cv Observed.

Remarks.

These values are true mole-

cular heats, i.e. instan-

323

6'02

taneous values for the

543

6-34

temperatures referred to,

723

6-67

not mean values over a

wide range of tempera-

ture.

Cv (383°, T0,)

Cv (383°, T°.)

Calculated

Observed

893
1273 -
1600

6-55
6'95
7.36

6-51 (
6-95 \
7-40 (

Mean values for the mole-
cular heat over the tem-
perature range indicated.

Cv (291°, T0,)

Cv (291°, TO,)

Calculated

Observed

2084

7-99

7-92 r

2383

8-54

8-54

2650

9-14

9'37

Ditto.

2936

9'9

io'o I

3181

107

10-5 1

3337

11-3

10-9

The absorption spectrum of water vapour contains many bands
over the range of X's employed above, i.e. bands at i 'i, i -5, 2 -o, 3-2, 47/4,
etc. Where there are so many bands almost any value might be re-
garded as borne out by the absorption spectrum. It appears from this
and other cases that the calculated molecular heat is not very sensitive

to error in the actual value chosen for X The term

\3ioo

a purely

empirical term introduced by Bjerrum to account for the fact that the
molecular heat of water vapour increases with the temperature more
rapidly than would be anticipated on the basis of the quantum theory.
The cause of this is by no means clear : possibly it is connected with a
depolymerisation of any bimolecules which may be present, though at
the higher temperatures this would be a negligible quantity.

The molecular heat of ammonia, a tetra-atomic body, may likewise
be calculated more or less satisfactorily by writing down terms for

BJERRUM' S THEORY 85

translation and rotation, and by applying the quantum theory to the
vibrations of the atoms, using the wave-lengths 2* I/A and 8'9/x, which
actually occur in the absorption spectrum of ammonia.

It is evident from the foregoing that to account for the observed
values of the molecular heats of gases it is essential to introduce the
quantum thebry in some form. Bjerrum's mode of treatment, though
marking a considerable advance, is open to criticism, more particularly
as regards the choice of the number of degrees of freedom.

We shall consider later the views put forward by Kriiger in connection
with the same problem.

One point remains to be emphasised in connection with the mole-
cular heat, or rather the energy content of gases. Owing to the fact that
the true atomic vibrations inside the molecule correspond to relatively
high frequencies (i.e. short infra-red region) such vibrations contribute
a relatively small amount to the total energy content ; for as the fre-
quency is high very few molecules will possess even one quantum oft
this type of energy. The so-called rotational frequencies (obtained
on applying the quantum theory to rotation) are much more important
as they occur in the longer infra-red region.

The Absorption Spectrum of Water Vapour and of Hydrogen Chloride
Gas in the infra-red region^ from the point of view of Molecular
Rotations^ treated on the basis of the Quantum Theory.

As already mentioned, Bjerrum has treated the rotational energy of
the di- or tri-atomic gas molecule from the point of view of the quantum
theory. That is, the rotational spectrum should exhibit a number of
lines related to one another by a constant frequency difference ; this, at
any rate, is the simplest possible statement. The spectrum due to such
rotations would be expected to lie in the farther infra-red (beyond IO/A),
the principle lines in the shorter infra-red region being due to the vibra-
tions of the atoms inside the molecule. Bjerrum assumes that the total
rotational energy varies in terms of quanta, and as a further simplification
he assumes that the moment of inertia of the molecule for all axes
through the centre of gravity is the same ; i.e. we have only to deal with
one such moment. The energy of rotation of a particle round an axis
is given by the expression 1/2 . 1 . (2?rj/)2, where v is the frequency of
rotation and I is the moment of inertia. On the older quantum theory
this energy must be represented by hv, 2h\>, etc., or in general by nhvt
where n is a whole number. Hence we have —

1/2 I . (27Ti/)2 = n . hv
or, v = n . /$/27r2 . I.

According to this expression we would expect a difference series in the
frequencies of the band heads in the spectrum of a gas. That is, har-
monics of the fundamental rotational frequency are to be expected in
the spectrum according to the value ascribed to n. From an examina-
tion of the spectrum of water vapour Bjerrum concludes that the fre-
quency difference in the series of rotational bands is 1*73 x io12. He

86

A SYSTEM OF PHYSICAL CHEMISTRY

then proceeds to consider the spectrum between the limits io/x and 20//,,
ascribing all the band heads therein to rotations, n being given con-
secutive values from 10 to 16. The corresponding wave-length series
(in /A) is given by the expression —

3_x ioa

V

3 *

n x 173 x 10

12-

The following table indicates the degree of concordance obtained by
this means : —

n.

=

16.

15-

14.

13-

12.

II.

IO.

\

Calculated.

10-8

n-6

12-4

13*3

14-4

15-8

i? *3/*

\

Observed.

IO'Q

n-6

12-4

i3'4

I4'3

157

iJ'SH

*With increasing wave-length the bands become deeper and tend to
overlap. For wave-lengths greater than iy'5/x. the absorption becomes
general and without structure unless very good dispersion is obtained in
the apparatus. Whilst these results are to be regarded as a considerable
step in our understanding of the problem of the source of bands, the
numerical results just quoted are, according to E. von Bahr, open to a
certain amount of doubt (Verh. d. D. phys. Ges., 15, 731 (1913)).

At sufficiently high temperatures the energy of rotation will be
directly proportional to the absolute temperature. Hence, the fre-
quency of rotation will be directly proportional to the square root of
the temperature, since the rotational energy term involves v2. Hence,
the product of the rotational wave-length into */T should be a con-
stant. The rotational wave-length is obtained in the following way. In
the short infra-red region (i.e. the region of the spectrum in which the
bands are due to ordinary vibrations of atoms inside the molecule) it is
to be expected, on the basis of Rayleigh's considerations, already re-
ferred to, that, instead of a single well-defined band head due to vibra-
tion, we should find a triple-headed band, viz. a centre one due to
atomic vibration and two others due to compounded rotational and
vibrational motion. As a matter of fact, what is frequently found with
a certain degree of dispersional power in the spectrometer, is a double-
headed band. E. von Bahr suggests that the middle band is not really
missing, but only apparently so, on account of insufficient dispersion.
The two wave-lengths A^ and X2, which in general lie fairly close together,
correspond to two frequencies v± and i/2, and their difference (v2 — vi) is
just twice the most probable frequency vr of the rotation. vr is- of course
considerably smaller than either vl or i/2 and its normal position is far in
the infra-red. On expressing vr in terms of its wave-length \r we find

that \r = 2 *' * . The above is a convenient way of measuring the

A! - AS

rotational wave-length without having to investigate the far infra-red

ABSORPTION SPECTRUM OF WATER VAPOUR 87

region. The validity of the expression can of course be tested by com-
parison with actual measurements in the far infra-red where the bands
are due to rotation only. From what has been said above, it follows

that the product

* ' * . JT should be a constant, provided the

- A2

theoretical considerations are sound. Making use of some data ob-
tained by Paschen in the region 5 to 6/x, in the case of water vapour,
E. von Bahr has found that the two heads of the double band (due to
molecular rotations compounded with atomic vibration) alter their
relative position in accordance with the above expression.

tc

^1

Ag.

. A! . A2

VT • A! - A2 ]

17

6-512

5-948

•17

IOO

6-527

5-900

•19

600

6-563

5-607

•14

IOOO

6-597

5'4i6

•08

1470

6-620

5'377

•20

In view of the fact that the temperature measurements were not very
exact, the constancy of the expression in the last column is very satis-
factory.

E. von Bahr has investigated more closely the constitution of the
bands of water vapour between 5 and 6/x and has found that each ex-
hibits a number of sub-maxima. Similar behaviour is shown by other
gases. That is, the discontinuous nature of the absorption required by
the quantum theory is fully borne out. From measurements of the
positions of the successive sub- maxima it is found in the case of water
vapour that the largest value for \Y is of the order 4oo/x, i.e. water
vapour should exhibit absorption at various positions down to this
region, a result in agreement with the observations of Rubens. On the
other hand, the experimental results obtained by E. von Bahr indicate
a lack of symmetry with respect to the principal maximum of a given
band, which indicates in turn that a single frequency difference is in-
sufficient to account for the whole series of band heads.

The question of the rotational spectrum of water vapour, i.e. the
infra-red spectrum beyond IO/A, hasmlso been investigated by Eucken
(Verh. d. D. phys. Ges., 15, 1159 (1913)), who adopts Bjerrum's expres-
sion for the series of frequencies to be expected, viz.

v = n . £/2ir2 . I.

This expression is based upon the assumption that the molecule is
symmetrical, and possesses therefore but one moment of inertia. Eucken,
however, emphasises the lack of symmetry of the water molecule and
ascribes to it at least two principal moments of inertia, which would lead
us to expect two different series in the spectrum. Sackur (Ann. physik,
[iv.], 40, 95 (1913)) has arrived at the conclusion — on quite other

88

A SYSTEM OF PHYSICAL CHEMISTRY

grounds — that the water molecule possesses two main moments of inertia,
which are in the ratio of i : 2. Eucken finds, on taking two values for
I, one of which is approximately twice the other, and calculating the
corresponding series as given by the Bjerrum expression, that satisfactory
agreement is obtained between the calculated and observed band
maxima. The following table contains Eucken's results : —

INFRA-RED ABSORPTION BANDS OF WATER VAPOUR.

A Observed (in jx).

A Calculated (in /*).

Direct.l

i

Indirect.^

First Series.
I = 0-96 x io-40. n.

Second Series.
1= 2'2i x to"40, n.

j

IO'9/A

—

iQ'SfjL 16

ir6

—

n-6 15

— —

12-4

—

12-4 14

— —

i3'4

—

I3'3 13

— —

i4'3

—

14-4 12

— —

157

—

I5'8 ii

— —

J7'5

—

17*3 10

— —

—

—

i9'3 9

— —

—

—

21-6 8

__

—

25M

24-8 7

25-0,11 16

—

11

28-9 6

267 15
28-5 14

—

30

— —

307 13

—

33

347 5

33*3 12

—

36

36-5 ii

—

39

— —

40-0 io

—

42

43*3 4

— —

—

45

— —

44'5 9

50

47-5°

— —

50-0 8

ca. 58

54-56

57*8 3

57'2 7

66-67

64-69

— —

66-6 6

78-80

79-81

—

80-0 5

—

91-92

87 2

— —

ca. 103

109

—

100 4

—

124-128

— —

133 3

—

170-172

173 I

— —

—

240-250

—

2OO 2

—

385-398

— —

40O I

E. von Bahr (ibid., 15, 1154) has examined very carefully a simpler
case than that of water vapour, namely, the infra-red spectrum of HC1
gas. The subdivision of each part of the double band (found by Bur-
meister and shown by the dotted line in the figure) into twelve separ-
ate maxima is very clearly shown in Fig. 9.

The data corresponding to Fig. 9 are given in the following table.
The wave-lengths Aj and A2 of the right and left parts of the double
absorption band, as well as the corresponding frequencies, are given.

1 Rubens and Aschkinass, Wied. Ann., 64, 584, 1898; Rubens, Ber. Berliner^
Akad.t 1913, p. 513.

2 E. von. Bahr, Verh. D. phys. Ges., 15, 731, 1913.

ABSORPTION SPECTRUM OF HYDROGEN CHLORIDE 89

The first column contains the value of nt i.e. the order of that maximum
reckoned from the middle of the absorption band. The middle of the

3-4 /A 3-3 f* 3-2/t,

80

FIG. g. — Hydrogen chloride gas (E. von Bahr). S denotes slit width.
Dotted curve due to Burmeister. Upper full curve corresponds to
760 mm. ; lower full curve corresponds to 380 mm.

absorption band is characterised by the wave-length
the frequency v0 = 8-636 x io13.

HYDROGEN CHLORIDE GAS.

3'474/x, and

ft.

A! in /x.

\z in IJL.

1 x io-13-

f2 X jo"13'

I

3*444

3*504

8711

8-562

2

3-416

3-528

8-782

8-503

3

3*394

3-556

8-839

8-436

4

3'372

3'583

8-897

8-373

5

3'352

3'6lo

8-950

8-310

6

3-332

—

9-004

—

7

33I7

~

9-044

—

In the following table are given the values of (i/j - v0) and (i/0 - v2),
which, according to Bjerrum, should be equal to the rotational fre-
quencies of the molecule of HC1 : —

„.

n^»-».

-11

VQ — v>2 x io

-11

v xio mean.

\ in /*.

v'10"11'

I

7-5

7'4

7*45

403

7*45

2

14-6

13-3

I3-95

215

6-98

3

20-3

2O-O

20-15

149

6-72

4

26-1

26-3

26-20

"5

6-55

5

3 1 '4

32-6

32-0

94

6-40

6

36*8

36-8

82

6-13

7

40-8

*"*"_

40-8

74

5*83

9o A SYSTEM OF PHYSICAL CHEMISTRY

On Bjerrum's theory we would expect vrjn to be constant. As
a matter of fact it is moderately constant over a considerable range but
shows a definite tendency to diminish as n increases. E. von Bahr re-
gards the variation as outside the limit of experimental error, although
Eucken (loc. tit.) considers the question as not yet definitely settled.
By employing the Bjerrum theory it is seen that HC1 should absorb at
several positions down to A. = 400^. Rubens and von Wartenberg
(Verh. d. D. phys. Ges., 13, 796 (191 1)) have found considerable absorp-
tion exhibited by this substance in the residual-ray range, the absorption
being especially strong at 150^.

Bjerrum (Verh. d. D. phys. Ges., 16, 640 (1914)) recalculates the
rotation frequencies of HC1 in a manner somewhat different from that
employed by E. von Bahr, and concludes that the agreement is as good
as can be expected in view of the unavoidable experimental error in de-
termining the position of the successive sub-maxima.

Whilst we are dealing with inter-relations between different portions
of the infra-red spectrum, it is convenient to refer briefly to certain ap-
plications of Bjerrum's theory of molecular rotations which have been
made by Baly (Phil. Mag., 27, 632 (1914) ; ibid., 29, 223 (1915) ; ibid., 30,
510 (1915)) in connection with the structure of bands exhibited by sub-
stances in the ultra-violet region. According to Baly the frequency of
the central position of such a band must be a multiple of the principal
frequency exhibited in the infra-red region by the same substance.
Further, the breadth which such ultra-violet bands possess is due to
a number of lines lying close together, and the distances apart of
these lines, when expressed in terms of frequencies, are found to have
constant values, and are to be ascribed, according to Baly, to the exist-
ence of certain basis constants characteristic of the molecule, these
basis constants being also capable of accounting for the infra-red ab-
sorption of the molecule. This method of linking up the infra-red and
ultra-violet absorption spectra has found considerable support in the
spectra of benzene and other organic compounds, and particularly so in
connection with the absorption of gaseous sulphur dioxide which has
been examined by Garrett (Phil. Mag., 31, 505 (1916)) and by Miss
Lowater (As trophy s. Journ. , 31, 311 (1910)). Thus Garrett has found
that the optical centre of the marked absorption band of SO2 in the
near ultra-violet occurs at A, = 296'i/A/x, the corresponding wave-number
(i.e. reciprocal of wave-length) being 3378. On applying the quantum
hypothesis as Baly has done in this case we would expect that this wave-
number should be an even multiple of the principal infra-red band of
SO2. The infra-red spectrum of SO2 has been examined by Coblentz
(Pub. Carnegie Inst., Washington, 1905) and has found a pronounced
band at 7 '4/n or wave-number 135*1. If this wave-number be multiplied
by 25 we obtain the number 33 7 7 '5, which is almost identical with the
centre of the ultra-violet band as determined by Garrett. For details of
the method of determining the basis constants the papers referred to
must be consulted. At the present time the mechanical significance of
these constants is not clear. The accuracy, however, with which they

KRUGERS THEORY 9t

permit of even very complex spectra being calculated in detail indicates
quite clearly that there is a very close connection between the infra-red
and the ultra-violet absorption of any substance. This question is taken
up briefly at a later stage (in Chap. VI.) in connection with the selective
photo-electric effect.

KRUGER'S THEORY.

(Cf. Kriiger, Ann. Physik, [iv.], 50, 346; ibid., 51, 45° (iQ1^).)

Kriiger assumes on the basis of the now generally accepted Ruther-
ford-Bohr structure of atoms and molecules (if. Chap. V.) that gaseous
molecules must be gyroscopic in nature; that is, such molecules are
quite incapable of rotations, but are capable of carrying out precessio nal
vibrations (cf. Chap. I.). These vibrations are totally distinct from the
ordinary vi oration of two atoms along the line joining their centres, to
which we have already referred in dealing with Bjerrum's treatment.
As a result of collisions with other molecules Kriiger considers that the
ring of rotating electrons in a molecule, which serves to unite the two
atoms together (in the case of a diatomic molecule), suffers displace-
ments perpendicular to its direction of motion, with the result that the
atoms themselves describe small circular orbits and the molecule as a
whole exhibits motion of the precessional type. Such precession has
already been shown diagram matically in Chapter I. Kriiger points out
that such precessional motion is entirely kinetic and necessarily involves
two degrees of freedom. In the temperature range in which the
principle of equipartition holds good the energy term corresponding to-
these two degrees of freedom will be RT, which is identical in magni-
tude with that postulated by Bjerrum for a diatomic molecule on the
basis of rotation of the molecule as a whole over the same temperature
region. Over this range therefore the numerical values for the energy
term is the same on either view in the case of a diatomic molecule.

The chief advantage of Kriiger's theory lies however in the explana-
tion which it offers of the behaviour of monatomic gases. Monatomic
molecules, such as that of argon, possess energy which is completely
taken account of by the free translation alone. An atom is therefore
incapable of rotation. If molecular rotations were possible in general
we would expect monatomic molecular rotations to be possible also.
The theory of molecular rotations is not very satisfactory in this con-
nection. On the other hand, Kriiger shows that precessional vibrations
in the case of a spherical monatomic molecule become practically
infinitely rapid, and may therefore be left out of account in regard ta
the energy content. This conclusion becomes clearer when we intro-
duce the quantum hypothesis.

At low temperatures it is necessary to treat all kinds of constrained
motion, such as ordinary atomic vibrations, precessional vibrations or
rotations from the standpoint of the quantum theory. The molecular
heat in virtue of any of these types of motion decreases with falling

92 A SYSTEM OF PHYSICAL CHEMISTRY

temperature and converges to zero at absolute zero of temperature. In
the special case of a monatomic molecule in which the precessional
vibrations are exceedingly rapid the corresponding quantum must be
exceedingly great, and consequently the likelihood of any molecule
possessing even one such quantum is negligibly small. In other words,
the observed energy content can only be due to the free translational
motion which is always directly proportional to the absolute temperature,
i.e. the molecular heat in such a case is 3/2R, in complete agreement
with experiment. In the case of diatomic gases Kriiger has shown
that the vibration frequencies of precession correspond to the farther
infra-red region, about 30^ in the case of hydrogen, and that the so-
called rotation spectrum is really due to precessional vibrations. It is
only at high temperatures that the precessional vibration becomes so
violent as to merge into a rotation (cf. Chap. I.). It must be clearly
understood that the conflicting views of Bjerrum and Kriiger have
nothing to do with the ordinary atomic to-and-fro vibration inside the
molecule, which Bjerrum has shown gives rise to bands in the short
infra-red region, i.e. at wave-lengths shorter than io//»

On the whole, Kriiger's view serves to bring the behaviour of mon-
atomic and diatomic gases into much closer accord than had hitherto
been the case. Considerable experimental work is still necessary, how-
ever, before we can decide between the theory of molecular rotations
and the theory of precessional vibrations.

A translation of the major part of Kriiger's papers will be found in
Appendix III. for those desirous of following Kriiger's treatment in
greater detail.

THE MOLECULAR HEAT OF HYDROGEN GAS AT Low TEMPERATURES.

(Cf. A. Eucken, Sitzungsber. kon. preuss. Akad. Wissenschaft ,
p. 141, 1912.)

The molecular heat of hydrogen is of particular interest because
hydrogen is the simplest type of diatomic molecule. The most striking
result obtained by Eucken is that at low temperatures hydrogen behaves
as a monatomic gas, i.e. it exhibits no energy of a rotational or of a
gyroscopic kind. The experimental details will be found in the paper
Teferred to. It is sufficient in the present instance to quote certain of
the results. Eucken employed different quantities of hydrogen in the
calorimeter (internal volume, 39 c.cs.), and his experiments are suf-
ficiently exact to show that the molecular heat varied with the concen-
tration of the gas. The following table gives the value of the molecular
heat observed corresponding to a particular temperature,1 and the mass
of hydrogen employed : —

1 The molecular heat values are " instantaneous," or true molecular heats at a
tgiven temperature; not mean values over a temperature range.

MOLECULAR HEAT OF HYDROGEN GAS

TAbs.

Gram-moles of HJ.

Molecular Heat.

273-1

0-0966

4*94

194

0-1710

4'39

91

0-3550

3 '44

91

82

0-1315
0-1268

332
3'30

The data obtained at still lower temperatures are given in the follow-
ing table : —

TAbs.

Gram-moles
Ha.

Molecular
Heat.

Gram-moles
Ha.

Molecular
Heat.

Gram-moles
H2.

Molecular
Heat.

35

0-1909

3'42

0-1794

0-1040

3*20

40

3-28

—

3-18

45

3*30

3*28

3-13

50

—

—

3*13

60

—

—

3-09

65

3-21

—

3*14

70

3-26

—

80

0-2585

3-29

0-1295

3-23

3*20

85

3*34

3-27

3*32

90

3*42

, 3*31

—

IOO

3*64

3'37

—

no

3-82

3-62

It will be seen that the lower the concentration of the gas the lower
the molecular heat.1 This effect is taken account of by the thermo-
dynamic relation —

?

To make use of this expression Eucken employs a relation of D. Ber-
thelot which connects the molecular heat at any given concentration
with that of the substance in the ideal gas state.2 In this way Eucken
arrives at the following values for the molecular heat of hydrogen in the
ideal gas state : —

1 This is not true universally for hydrogen, i.e. at very high pressures there is a
tendency for Cv to diminish.

2 Berthelot's expression involves the critical temperature and critical pressure of
the gas.

•94 A SYSTEM OF PHYSICAL CHEMISTRY

MOLECULAR HEAT OF HYDROGEN IN THE IDEAL GAS STATE.

T Abs. Molecular Heat.
35 2-g8

40 2-98

45 3'oo

50 3-01

60 2-99

65 3'°4

70 3-10

80 3-14

85 3'2i

90 3-26

ioo 3 '42

no 3*62

I96-5 439

273-1 4-84

These values show that the molecular heat of hydrogen falls rapidly
•with a fall in temperature, finally (at about 60° abs.) attaining a value
2-98, identical with that of a monatomic gas. Eucken finds that the
molecular heat curve shows even a more rapid fall than that given by
the Einstein formula for monatomic solids. The reason of the rapid
fall is not clear ; Eucken discusses the question in the paper referred to.
The most striking fact, as already stated, is that at low temperatures the
molecule of hydrogen possesses only energy of translation. Possibly
other diatomic gases behave similarly ; nothing is known on this point.

Eucken (Ber. Deutsch. phys. Ges., 18, 4 (1916)) has measured the
molecular heats of hydrogen and helium at high pressures and at low
temperatures, and has found that when the gases are strongly compressed,
the molecular heat falls below the value 3/2R(2'98). This is discussed
by Nernst (Zeitsch. Elektrochem., 22, 185 (1916)), who regards this
" degradation of the ideal gaseous state " as due to the substitution of
circular rotational motion of the molecules at very low temperatures in
place of free translation. Thus, instead of the energy of the gas being
expressed by the relation : E = 3/2 RT (due to translation), we have to
substitute the expression —

F - 3R ft*

- 7* • ^O"/T - j

where v is the frequency of rotation. This expression can give rise to a
molecular heat which is less than 3/2R. Experiment shows, however,
that even at fairly low temperatures the relation E = 3/2 RT holds good,
so that the second expression is only of importance at very low tem-
peratures.

As a result of this modified expression for the energy of the gas
which is functioning as a monatomic one, it follows that, under the same
extreme conditions, in place of the perfect gas law P = RT/V we must
write —

p = R &v
V ' i - e

MOLECULAR HEAT OF HYDROGEN 95

The most striking conclusion arrived at by Nernst (loc. at. ; also
Verh. Deutsch. phys. Ges., p. 99 (1916)) is that this "degradation phe-
nomenon " stands in close causal connection with the chemical con-
stant of the gas, i.e. the quantity — ; — where / is the integration

constant of the Clapeyron equation for the vapour pressure of the
liquified substance. We have already discussed the significance of the
chemical constant in connection with the application of Nernst Js Heat
Theorem to gaseous reactions.

Nernst finds theoretically that the chemical constant of a monatomic
substance, or a substance which functions as such, can be expressed by
a relation of the form —

C = C0 + i'5 log10M,

where M is the gram- molecular weight of the substance, C is the
chemical constant (identical with the C0 of Chap. III., Vol. II.), and C0
in the present instance is a constant independent of the nature of the
substance considered. Nernst shows from an examination of the most
reliable data existing at present that an expression of this form is in
agreement with the experimental values of C. Nernst's theoretical expres-
sion is likewise in agreement with those 01 Sackur (Nernst Festschrift,
p. 405 ; Ann. Physik, 40, 67 (1913)), and of Tetrode (Ann. Physik, 38,
434 j 39, 255 (1912); Proc. Acad. Amsterdam (1915)).

By introducing the quantum theory into a vapour pressure relation
based upon his Heat Theorem, Nernst calculates Goto be — 1*590, the
unit of pressure being the atmosphere, which is in good agreement with
the mean observed value, namely, — 1*62 obtained from a consideration
of the data available in connection with hydrogen, argon, monatomic
iodine gas, and mercury vapour. Tetrode's (calculated) value for C0 is
- i -608.

CHAPTER V.

(Physical equilibrium, continued) — Distribution of electrons in atoms — Structure
of the atom from the standpoint of the quantum theory — The Rutherford-
Bohr atom-model — Parson's magneton and the structure of the atom — High
frequency spectra of the elements — Moseley's relation.

The Rutherford Atom-model.
(Rutherford, Phil. Mag., [vi.], 21, 669 (1911).)

THE underlying idea in this atom-model is that the atom consists of a
central charge, concentrated on a nucleus, which is surrounded by an
" atmosphere " of electrons, rotating in certain orbits. Practically the
whole mass of the atom is to be ascribed to the nucleus. The number
of "atmosphere " electrons is not large. The dimensions of the nucleus
are considerably smaller than the dimensions of the atom as a whole.
Rutherford estimates the diameter of the nucleus to be of the order
io~~12 cm., whilst the diameter of the atom as a whole is of the order
io~8 cm. The nucleus itself possesses a structure about which, how-
ever, nothing is known. It, the nucleus, contains probably quite a
large number of electrons, but these are bound firmly, except in the
case of radioactive materials which are capable of expelling one of these
bound electrons in the form of y3-rays. The outermost ring of the
"atmosphere'' electrons contains those which give the property of
valency to the atom. The innermost electron of the atmosphere is the
source of X-rays, the shortest wave-length which the atom is capable
of emitting. The necessary disturbance of the innermost electron is
brought about by collision with a /2-ray (cathode ray) which jerks the
innermost electron out ; on its return journey to the innermost ring it
causes the emission of the extremely short waves which we call X-rays.
Rutherford was led to this planetary view of the atom in order to ex-
plain the scattering of a and (3 particles. When such particles pass
very close to the centre of an atom they are violently deflected, an
effect which could be produced by a nuclefls of the kind referred to.
The angle of scattering has been measured by Geiger and Marsden in
the case of a-rays, and by Crowther in the case of /3-rays. The outer
" atmosphere " of electrons rotating in rings do not appreciably affect
the direction of the motion of an a particle travelling through the atom.
It is only when the a particle comes into close contact with the nucleus
that its path is abruptly altered, the path becoming hyperbolic.

RUTHERFORD-BOHR ATOM-MODEL 97

Bohr's Application of the Quantum Theory to the Rutherford Atom-model.

(Bohr, Phil. Mag., [vi.], 26, i., 4?6, 857 (1913) ; ibid., 27, 5°6 (1914) ;

30, 394 (iQ^)-)

The following is a brief account of the line of thought pursued by
Bohr, given to a large extent in his own words.

Bohr first of all points out that the Rutherford atom-model, which
has just been referred to, suffers from the serious drawback that the
system of "atmosphere" electrons is unstable; that is, unstable from
the standpoint of classical electro-dynamics. If, however, we introduce
Planck's concept, the instability may no longer exist from the theoretical
point of view. The problem of atomic structure affords therefore a
further instance of the necessity of introducing some new concept, such
as that of Planck, into electro-dynamics in order to account for the
observed facts. Bohr first attempts to apply the quantum theory to
the process whereby a free electron — such for example as exists in
a vacuum tube when a discharge is passed — may be conceived of as
attaching itself to a positively charged nucleus. It will be shown
that it is possible from the point of view taken to account, in a simple
way, for the Balmer law of line* spectra of hydrogen and helium, and
possibly the theory will eventually be capable of dealing with the spectra
of more complicated atomic structures.

According to Rutherford, the hydrogen atom consists of a nucleus
with a single electron describing a closed orbit around it The first as-
sumption is that the mass of the electron is negligible compared with the
mass of the nucleus, and that the velocity of the electron is small compared
with that of light. Bohr considers that whilst the electron is rotating in
this orbit or stationary state it is neither radiating or absorbing energy.
Such rotation represents an equilibrium condition of the system as a
whole. The production of spectra must be due therefore to some kind
of departure from an equilibrium state. In general Bohr considers the
possibility of a series of such stationary states corresponding to electron
orbits of different radius, and emission or absorption of radiation is due
to the electron passing from one stationary or equilibrium state to* another*
The quantum idea is introduced in the further hypothesis that during
the passage of the electron from one equilibrium state to another,
homogeneous radiation of a certain frequency v is emitted or absorbed,
the amount of radiant energy so emitted or absorbed being hv, where
h is Planck's constant. It will be realised at once how very different
these considerations are from those based on classical electrodynamics.
On the classical view a charged particle rotating steadily in a closed
orbit is giving off radiation continuously, and in so doing its orbit
gradually becomes smaller so that the electron would finally drop into
the nucleus. Furthermore, with the alteration in orbit the wave-length
of the light emitted would gradually change, a conclusion which is op-
posed to direct observation, for spectra are clearly characteristic and
unchangeable properties of an atom. Bohr assumes that no radiation
at all is given out when the electron is rotating in an orbit.
VOL. III. 7

98 A SYSTEM OF PHYSICAL CHEMISTRY

Let us now consider the hydrogen atom — one in which there is a
single nucleus carrying unit positive charge with one valency electron
rotating round it — in a state of equilibrium, the electron traversing
an elliptical orbit with a frequency of revolution o>, the major axis of
the orbit being 20. The amount of energy W which must be given
to the system (i.e. which the system must absorb) in order to remove
the electron to an infinitely great distance (i.e. to dissociate the atom
into a positively charged nucleus and a free electron), is connected
with w and 2a by the two following relations : —

W3/2 eE

a = -

where e is the charge on an electron, E the charge on the nucleus, and
m the mass of an electron. Further it can be shown that the kinetic
energy of the electron taken for a complete revolution when it is rotating
in one of it orbits is equal to W, the work required to eject the electron
entirely from the system. Note that removal of the electron necessitates
absorption of radiant energy.

Now let us consider the reverse process, namely, the act of binding
a free electron to the nucleus. At the beginning the electron may be
regarded as possessing no sensible velocity with respect to the nucleus,
i.e. its frequency of revolution is zero relatively to the nucleus. The
electron, after interaction has taken place, settles down to a station-
ary orbit — the word stationary refers to the orbit, the electron itself
is in rapid rotation round the orbit. Bohr now takes the orbit as circular
for reasons given later in connection with the physical significance of h.
The initial free state of the electron represents the extreme limit of a
series of stationary states through which the electron is capable of
passing. The other limit is given by the smallest value of za or the
greatest value of o> which necessitates the maximum value of W.

Let us assume that during the act of binding of the free electron to
the nucleus, a homogeneous radiation of frequency v is emitted. This
frequency is, according to Bohr, just half the frequency of rotation o>
of the electron in its final orbit. On the basis of Planck's theory we
would expect that the total amount of radiant energy thus emitted
would be rhv where T is a whole number. The assumption that the
frequency v is just w/2 suggests itself, since the frequency of revolution
at the beginning of the binding process is zero and at the end of the
process is w, the mean or average of the two being <*>/2. Bohr gives
a more rigorous treatment of this point in the first paper referred to.
Since rhv is the amount of energy emitted whilst the electron is
approaching the nucleus from an infinite distance, and W is the amount
of energy which has to be absorbed to make it reverse the operation,
it follows that —

W = rhv = T/&- . . . . (2)

RUTHERFORD-BOHR ATOM-MODEL 99

Combining thus with the equation (i) we obtain —

If in these expressions we give T different values, we get a series of
values for W, o>, and a, corresponding to a series of configurations of
the system, such configurations being stationary states in which there
is no radiation of energy, and in which the electron will remain as long
as the system is not disturbed from outside. We see that the value of
W is greatest if T has its smallest value ', viz. unity. This will therefore
correspond to the most stable state of the system : i.e. it will correspond
to the binding of the electron for the removal of which the greatest
amount of energy is required. Substituting in the above expressions
T = i, and E = e = the charge on a single electron, and introducing
the experimental values, viz. e — 47 x io-10, elm = 5*31 x io17,
h = 6*5 x io 27 we obtain :— -

2a = i'i x io-8 cm. ; o> = 6*2 x io15 per sec. ; W/e = 13 volt.
We see that these values are of the accepted order of magnitude for the
linear dimensions of the atoms, the optical frequencies, and the ionisa-
tion potentials respectively.

It is to be clearly borne in mind that the results so far obtained
rest on two main hypotheses : —

1. That the dynamical equilibrium of the systems in the stationary
states can be discussed by help of the ordinary electro-dynamics, whilst
the passage of the systems between different stationary states cannot be
treated on that basis.

2. That the latter process is followed by the emission of homogeneous
radiation, for which the relation between the frequency and the amount
of energy emitted is the one given by Planck's theory.

The next problem which Bohr takes up is to show that his theory
is capable of accounting for the line spectra of hydrogen.

If we consider the act of binding a free electron to a hydrogen
nucleus carrying a positive charge equivalent to the charge on a single
electron so that the electron finally rotates in one of the stationary
states, the energy radiated out in the formation of this stationary state
is given by equation (3), viz. —

WT =

The amount of energy emitted by the passing of the electron from a
state where T — T: to a state where T = r2 is consequently —

The state corresponding to r2 is one in which the electron is closer to
the nucleus than in the state corresponding to rx ; in short, TS is less
than T1} and the linear dimension a of the atom is smaller when T = r2
than when T = n.

7*

ico A SYSTEM OF PHYSICAL CHEMISTRY

If we now suppose that the radiation in question is homogeneous,
and that the amount of energy emitted is equal to hv, we get —

WT2 - WTX = hv
and hence

27T me

-X? - *) •

This expression gives the frequency of the homogeneous radiation
emitted by the gas when the atomic system changes from the stationary
state denned by rx to the stationary state defined by r2. If these two
states were the only possible ones the gas could only give rise to a single
frequency v. The quantities r2 and TI are whole numbers, and a single
atom is capable of existing in a corresponding number of stationary
states, so that in general a number of lines will be emitted as is borne
out by experiment. The above expression is capable of accounting for
series of lines emitted by incandescent hydrogen. If we put TZ = 2
and allow TJ to vary, i.e. TJ can take on the values 3, 4, 5, etc., we get
the well-known Balmer series of lines in the hydrogen spectrum. It is
noteworthy that to account for the Balmer hydrogen line spectra we
have to assume that the electron from an outer orbit passes to the
second orbit (TS = 2) and not to the first or innermost orbit, which re-
presents maximum stability. If we put r2 = 3 and allow TJ to vary
(TI = 4, 5, 6, etc.), we get the infra-red series of lines observed by
Paschen (Ann. Physik, 27, 565 (1908)). If we put r2 = i and allow
TJ to vary we get a series in the extreme ultra-violet, the furthest Lyman
region. In the case of elements heavier than hydrogen, e.g. platinum,
the kind of radiation to be expected on putting r2 = i corresponds to
X-rays. If we put r2 = 4, 5, etc., we get series of lines in the extreme
infra-red not yet observed. It will be observed that Bohr's theory
accounts excellently for those series of lines which have been observed
in the case of hydrogen and is even capable of predicting other series
in regions not yet examined. Further, the agreement is quantitative
as well as qualitative. Putting e — 4*78 x io~10 electrostatic units;
e\m — 5*31 x io17; and h = 6^55 x io~27, we get —

= 3-26 x io15,

whilst the value obtained experimentally for the factor (the so-called
Rydberg constant) outside the bracket in formula (4) is 3*290 x io15.
The agreement between the observed and calculated value is within the
error due to experimental determination of the various quantities in-
volved.

It may be pointed out that it has not been possible to observe more
than twelve lines of the Balmer series in experiments with vacuum tubes
whilst thirty-three lines are observed in the spectra of certain celestial
bodies. This may likewise be anticipated on the basis of Bohr's
theory. According to equation (3) the diameter of the orbit of the

RUTHERFORD-BOHR ATOM-MODEL 101

electron in the different stationary states is proportional to T2. For
T = 12, the diameter is equal to 1-6 x io~6 cm., or equal to the mean
distance apart of the molecules of a gas at a pressure of about 7 mm. of
mercury; for T = 33 the diameter is equal to 1*2 x io~5 cm., corre-
sponding to the mean distance apart of the molecules at a pressure of
about 0-02 mm. of mercury. According to the theory, the necessary
condition for the appearance of a great number of lines is therefore a
very small density of the gas, and if this were realised in a vacuum tube
the whole mass of gas present would be insufficient to give rise to a
measurable intensity of emission. In solar conditions we have large
masses of incandescent gas at extreme rarefaction, and hence lines are
observed under these conditions which cannot be observed in a vacuum
tube,

Bohr next considers the spectrum of helium.

The neutral atom of helium consists, according to Rutherford, of a
single positive nucleus of charge 2e, with two electrons rotating in the
same orbit around it. Now, considering the binding of a single electron
by a helium nucleus, we get, on putting E = 2<?, in expression (3) —

If we put T2 = i or T2 = 2 in this formula we get series of lines in the
extreme ultra-violet. If we put r2 = 3 and let TI vary we get a series
of lines which' includes two of the series observed by Fowler and
ascribed by him to hydrogen. If we put r2 = 4 we get the series observed
by Pickering in the spectrum of £ Puppis. Every second one of the
lines of this series is identical with a line in the Balmer series of the
hydrogen spectrum. It is not surprising therefore that these lines from
£ Puppis were ascribed to hydrogen. In Fowler's experiments hydrogen
was mixed with helium in the vacuum tube. With helium alone in a
vacuum tube the lines given by putting r2 = 3 are not shown, as the
ionisation of helium is so small. The presence of hydrogen appears to
increase the ionisation of helium, due very probably to the fact that a
positively charged hydrogen nucleus has a considerable affinity for
electrons and consequently causes the helium atom to lose both of its
electrons more easily than it would do in the absence of ionised hydrogen.
The lines predicted by Bohr's theory refer to the binding of an electron
by the helium nucleus which carries a positive double charge. Further,
hydrogen atoms are known to be capable of acquiring a negative
charge, so that their " catalytic " effect on the helium may be partly due
to the affinity of neutral hydrogen atoms for electrons. Bohr explains
in this manner a certain set of the lines obtained from a vacuum tube
containing hydrogen and helium as due to the presence of the doubly
charged helium nuclei combining with electrons. Finally, if we put
T2 = 5> 6, etc., in the above formula we get series, the strong lines of
which are to be expected in the infra-red region.

102 A SYSTEM Of PHYSICAL CHEMISTRY

We are now in a position to give something like a physical interpre-
tation of Planck's constant h. Consider equation (2). If we denote
the angular momentum of an electron rotating round a nucleus by M, it

T
follows that for a circular orbit TrM = — where w is the frequency of

O)

revolution and T the kinetic energy of the electron. For a circular
orbit we further have T = W, as we have already seen. Hence —

M = rM0
where M0 = h^w = 1*04 x io~27.

If we assume therefore that the orbit of the electron in the stationary
state is circular we can say : the angular momentum of the electron
round the nucleus in any one of the stationary states of the system is
equal to an entire multiple of a universal value M0 or ^/2ir, independent
of the charge on the nucleus. The possible^importance of the angular
momentum in relation to Planck's theory ha's also been emphasised by
Nicholson.

We can only observe a great number of different stationary states by
investigating emission and absorption of radiation. In most of the
other physical phenomena we deal with atoms in a single state, the state
which corresponds to low temperatures. From these considerations we
are led to the assumption that the permanent state is the one among
the stationary states during the formation of which (from free electron
and nucleus) the greatest amount of energy is emitted. According to
equation (3) this state corresponds to T = i.

As regards the process of absorption of radiation, the existence of
Kirchhoff's law (viz. that a body absorbs radiation of the same wave-
length as it emits), naturally suggests that the same mechanism is to be
attributed to absorption as to emission, absorption taking place when
an electron passes from an inner orbit to an outer.

Bohr's theory is likewise in agreement with Einstein's expression
for the photo-electric effect, viz. T = hv — W, where T is the kinetic
energy of the electron ejected under the influence of light of frequency
v, and W is the total amount of energy which would be emitted during
the act of binding a free electron to the nucleus of the atom (cf.
following chapter).

We have now to consider in more detail what has been called the
permanent state of an atomic system. We have seen that in the case of
a system consisting of a nucleus and a single electron, the permanent
state is defined by the condition that the angular momentum of the
electron is h\2ir. The atom of hydrogen corresponds to this system,
but the atom of hydrogen is difficult to deal with because of the very
slight dissociation of gaseous molecular hydrogen. In order to get a
closer comparison with experiment it is necessary to consider more
complex systems. Let us consider a ring of n electrons rotating round
a nucleus of charge E, the electrons being arranged at equal angular
intervals round the circumference of a circle of radius a.

RUTHERFORD-BOHR ATOM-MODEL

103

The total potential energy of the system consisting of the electrons
and the nucleus is —

where

sn = -

1 The above expressions require probably a little explanation. Let us consider a
positively charged nucleus, charge E surrounded by n electrons, each of charge e,
the electrons being arranged at equal distances on the circumference of a circle of
radius a. We shall number the electrons i, 2, 3, /«»

etc., up to n. The problem is to find the expression -^\**'

for the potential energy of this entire system, the
potential energy being measured by the work
which it is necessary to do upon the system in
order to remove all the electrons to an infinite dis-
tance from the nucleus and from each other. First
of all consider the nucleus with respect to electron
i. There is a force of attraction between the elec-
tron and the nucleus amounting to ^E/a2. The
potential of the electron with respect to the nucleus
being - e-^E/a, the minus sign being introduced
because, in pulling the electron away, we oppose
the natural direction in which the electron tends to
move (viz. towards the nucleus). For electron 2
we have a similar term, so that for the n electrons
we have the potential energy of the electrons with
respect to the nucleus given us by the expression
- neE/a. We have now to consider the effect of the
electrons upon each other. The force is now one
o£ repulsion. It therefore aids the process of pulling
apart. The expression for the potential energy of
each electron with respect to every other appears
therefore with a positive sign. Consider electron

electron 2. The force of repulsion is

where

first of all with respect to
2 is the chord connecting

the two electrons. The potential of electron i with respect to 2 is therefore ^2/
The potential of electron i with respect to electron 3 is given by e^\r^ 3 and so on
up to the nth electron. The total potential energy due to the (n - i) electrons
acting upon electron i is thus given by a sum of (n - i) terms : —

*2/'l, 2 + *2/'l> 3 + ''/'I. 4 + • • • + '2/'l> «•

Now from the figure it is seen that —

rlt 2 = 2a sin ir\n where 2ir/w is the angle between successive electrons.
rlt 8 = 2a sin 2w/» „ „ „ „ „

r j, 4 ~ aa sin 3*> „ „ „ „ ,,

fj, n = 2a sin (n- i) ir\n „ ,, ,, ,,

Hence the sum of the potential energy terms of electron i with respect to all the
others is—

— [cosec IT In + cosec 2ir/n + cosec

. . . + cosec (n -

For electron 2 we can write down an analogous set of terms and so on up to the
nth electron. The sum of all such terms is therefore n times the series given. That
is, for all electrons mutually acting upon one another we have —

- [cosec IT/M + cosec zirjn + cosec

. . . + cosec (n - i)v/n].

104 A SYSTEM OF PHYSICAL CHEMISTRY

For the radial force exerted on an electron by the nucleus and the other
electrons we get —

Denoting the kinetic energy of an electron by T and neglecting the
electro-magnetic forces due to the motion of the electrons, we get on
equating the centrifugal force on an electron to the radial force —

or T = — (E - esn).

From this we get for the frequency of rotation—

e(E - ^

The total amount of energy W which must be given to the system
in order to remove the electrons to infinite distances apart from the
nucleus and from each other is —

W = - P - «T = — (E - esn) - — (E - esH) = — (E - esn) = «T . (6)

But it is evident at once that in adding up all the terms which go to make up the
above expression we have counted each potential term twice over. Thus, the
potential between i and 3 appears when we are considering electron i by the term
*2/rif 3- Similarly when we consider electron 3 with respect to the others we have
amongst other terms the term *2/y3» i> which is simply the potential 0a/ri» 3 over again*
Hence to get the true potential of all the electrons with respect to one another we
have to divide the above expression by two. That is we get—

— [cosec irfn + cosec 2ir/» + cosec 3?r/w + . . . + cosec (n - i )*•/»].

Now the total potential P of the system is made up of terms due to the nucleus
acting upon the electrons and the electrons acting upon on 2 another.
That is—

P = : — H [cosec ir\n + cosec 2ir/» -f . . . + cosec (n - i)»/»],

s = n - i
If we now write the bracket in the form J cosec sir In

and further write the symbol sn for £ 2, cosec sir/n

s = i
it follows that we can write —

nEe ne*
P = + — sn

a a

\

or P «= (E - ««)

a

which is the expression given by Bohr.

RUTHERFORD-BOHR ATOM-MODEL 105

That is, the energy W is equal to the kinetic energy taken for a whole
revolution of all the electrons when rotating in the stationary orbit.
This is the same relation as holds for the case of a single electron
rotating in an orbit. Notice also that the total potential energy P of
the system is just double the kinetic energy of the electrons.

We see that the only difference in the above formula (6) and that
holding for the motion of a single electron in a circular orbit round a
nucleus is the exchange of E for (E - esn). We are therefore led to
suppose that the kinetic energy per electron rotating in a stationary

state is again rA-. The permanent configuration, that is, the configura-
tion in the formation of which the maximum amount of energy is
emitted, is the one for which r = i. The angular momentum of each
single electron in this state is again hJ2ir. Also, we can express angular
momentum in all cases as mva where m is the mass of the rotating
particle travelling with a velocity v round a circle of radius a. That is —

i h . .

v = (7)

a 2irm

There may be many stationary states which a single ring of electrons may
assume. This seems necessary to account for the line spectra of sub-
stances containing more than one electron per atom. Further there
may be stationary configurations of a system of n electrons and a
nucleus of charge E in which all the electrons are not arranged in a
single ring.

As regards the stability of a ring of electrons, two problems arise.
First as regards the stability for displacement of the electrons in the
plane of the ring, and secondly, as regards displacements perpendicular to
this plane. Nicholson has shown that the question of stability is very
different in the two cases. While the ring for the latter displacements
(displacements perpendicular to the ring) is in general stable, the ring
is in no case stable for displacements in the plane of the ring. This
conclusion is based upon classical electro-dynamics, and Bohr avoids
the difficulty by making use of the quantum idea. In fact he shows
that the stability of a ring of electrons rotating round a nucleus is
secured through the condition of the universal constancy of the angular
momentum hJ2ir of each electron together with the further condition
that the configuration of the electrons is that in the formation of which
— from free electrons and nucleus — the amount of energy, W, emitted
is a maximum ; in other words, the stable configuration possesses the
least energy. That this assumption leads to the condition for stability
may be shown as follows.

Consider a ring of electrons rotating round a nucleus and let us
assume that the system is in dynamical equilibrium, the radius of the
ring being a0> the velocity of the electrons #0, their total kinetic energy
T0, and the total potential energy of the entire system P0. We have
already seen from equation (6) that P0 =* - 2T0. Next' consider a

106 A SYSTEM OF PHYSICAL CHEMISTRY

configuration of the system in which the electrons, under the influence
of extraneous forces, rotate with the same angular momentum round
the nucleus in a ring of radius a where a = aaQ, a being greater than

unity. In this case we have P = - P0 (as follows from the definition of

a

potential energy given by equation (5)) ; also on account of the con-
stancy of the angular momentum it follows that v — - VQ (compare equa-

a

tion (7)), and T = ^T0, for T involves the square of v and therefore
the square of a. It follows then that —

P + T - £P0 + I2T0 - (P0 + TO) + T0

We see that the total energy (P + T) of the new configuration is
greater than the (P0 + T0) of the original configuration. The system
is therefore stable for the displacement considered, that is it will tend
to revert to the first configuration, for the first configuration corresponds
to a maximum value of W as defined previously.

The condition of stability which we have just been considering refers
to possible displacements of the electrons in the plane of the ring. It
is necessary to consider the problem of stability in more detail howevef
before we take up the problem of actual atoms containing several
electrons.

Configuration and Stability in Systems Possessing one Nucleus.

Let us consider an electron of charge e and mass m moving in a
circular orbit of radius a with a velocity v, which is small compared
with the speed of light. Let us denote the radial force acting on the

e2

electrons by -^ F0 ; F0 will be dependent in general upon a. The con-
dition of dynamical equilibrium gives —

Introducing the condition of universal constancy of the angular
momentum of the electron we have —

mva — ti/27r.
From these relations we get —

v =
and for the frequency of revolution —

RUTHERFORD-BOHR ATOM-MODEL 107

[f F° is known, the dimensions and frequency of the corresponding
orbit are determined by these equations.

We have seen already (equation (50)) that the radical force F is
given by —

F = - 1(E - «„)

where F is the force acting upon a single electron in a system contain-
ing n electrons, E being the charge on the nucleus. Now E = N<?,
where N denotes the number of unit l charges carried by the nucleus.
It follows therefore that —

i

But we have just seen that —

*-*•*-

Hence F0 = (N - jn).

Further, we have seen that the energy term, W, is given by —

W = - P/2 = £ (E - «„) = ^(N* - «.)- ^(N - Sn)

ne*

= — Fo.

2d

W refers of course to all the electrons in the system. Further —

a =
Hence W =

If we now consider the simplest type of atom, viz. the hydrogen atom,
the energy emitted, W0, in bringing the single electron from infinity up
to the most stable position is given by —

W0 = 2Tr2e*m/tf

(compare equation (3), remembering that the most stable or permanent
condition means that T = i).

Hence for a system containing n electrons with nuclear charge N^
we can write —

W = W0«F02.

Also, for the simplest atom in its permanent state we have seen already
(equation (3)) that —

a0 = /&2/47T2^2.

Hence for a system containing n electrons, the charge on the nucleus
being N<?, we can write —

a = a0/F0 ; and w = o>0F02.

1 The charge on a single electron being here regarded as the " unit " of (nega-
tive) electricity.

io8

A SYSTEM OF PHYSICAL CHEMISTRY

Using the numerical values already employed for the hydrogen atom we
see that —

W0 = 2'Q x io

0

ao = o*55
o)0 = 6-2 x

~n

erg.

io~8 cm.
io18 revolutions per second,

and therefore, for the system containing n electrons and charge N* on
the nucleus —

iV = 2-0 x io-11 x «F02 . . (i)

= 0-55 x io-8/F0 .... (ii)

= 6-2 x io15F02

(in)

In order to use these expressions in actual calculation it is necessary to
know F0, i.e. (N - sn). Bohr gives the following table of values for sn
for various values of n up to n — 16. For n = i, sn is of course zero : —

n.

Minimum Value

n.

s

Minimum Value

of N.l

of N.

I

0

O

9

3-328

13-14

2

0-25

0-25

10

3-863

18-13

3

0*577

0-58

ii

4-416

23-60

4

o-957

1-41

12

4-984

30-8

5

1*377

2*43

13

5-565

38-57

6

1-828

4-25

14

6-159

48-38

I

-2-305
2-805

6-35

15

16

6-764
7*379

71-65

The numbers of electrons represented by n refer to the number of
electrons in a single ring. This is in general a smaller number than
that represented by N, for in a neutral atom N is necessarily equal to the
total number of electrons present, some of which are present in the
innermost ring, others in outer rings. We see from the table that the
number of electrons which can rotate in a single ring round a nucleus
of charge Ne increases only very slowly with increase in N ; for N = 20,
the maximum value of n is io; for N = 40, n = 13; for N = 60,
n = 15. We see further that a single ring of n electrons cannot rotate
in a single ring round a nucleus of charge ne unless n is less than 8.
That is, if we consider an atom which has but one ring of electrons
(and therefore n = N), the maximum number of electrons in this ring is
7. If more electrons be present at least two rings must be formed.

Hydrogen.

We have already considered the values of W, a, and <•> (i.e. W0, aot
and w0) which satisfy the condition of stability for the electrically neutral
atom of hydrogen. It is of interest, however, to consider the case of a

1 Of course N cannot be fractional. The nearest whole number greater than
the numbers quoted will give the smallest value for N.

RUTHERFORD-BOHR ATOM-MODEL 109

negatively charged hydrogen atom, that is, one which has picked up an
extra electron. In this case n = 2 and N = i, for N* = the nuclear
charge = e in the case of the hydrogen nucleus. On substituting n = 2
and the corresponding value of sn from the above table in the expres-
sions (i), (ii), and (iii), we find that this system — which can be repre-
sented by the symbol 1, (2), where (2) refers to the total number of
electrons in the atmosphere and 1 refers to N, the number of unit
charges on the positive nucleus — will give rise to a permanent state
conditioned by the following values of a, w, and W : —

a = i'33 ao\ <* = 0-563 <D;; W = 1-13 W0.

Since W is here greater than it is when the atom is neutral, it follows
that the hydrogen atom is actually capable of taking on a negative
charge, for in doing so, the term W is increased from W0 to 1-13 W0,
and one of the conditions for stability is, as we have seen, a maximum
value for W. The fact that the hydrogen atom can take on a negative
charge probably accounts for the fact that in the periodic table we
usually place hydrogen with the halogens partly because hydrogen can
replace chlorine in organic compounds.

Bohr has also calculated the value of W which would result if the
hydrogen atom acquired. two extra electrons, thus becoming (1, (3)).
It is found that W now becomes equal to 0-54 W0. This value of
W is now less than W0, its value for the neutral atom, and consequently
still less than for the atom carrying one extra electron. Hence such a
doubly charged hydrogen atom is unstable and will not exist. Experi-
ments on positive rays show that the hydrogen atom can acquire a single
negative charge, but it is doubtful if it can acquire a double negative
charge.

Helium.

In this case N = 2, and the neutral atom has therefore two elec-
trons. We have already considered Bohr's theory in relation to the
spectrum of helium. We have now to take up the process whereby first
one, then a second, and finally a third electron can be added to the
doubly charged helium nucleus. First, for the formation of the system,
(2, (i)), by bringing an electron from infinity up to the nucleus, since
n = i and therefore sn = o, and further since N = 2, the permanent
state is given by —

a = 0-5 a0 ; w = 4<o0 ; W = 4W0.

If we now bring the second electron up to the (2, (i)) system we find
for the permanent state — since sn = 0-25 for n equal to 2 —

a = 0-571 a0\ o> = 3'o6w0; W = 6'i3W0.

Further, since

(Wa>(2) - W2)(I)) = 2-i3W0

no A SYSTEM OF PHYSICAL CHEMISTRY

it follows that each electron in a helium atom is more firmly held than
the single electron in the hydrogen atom. It follows also that the
ionisation potential of helium should be correspondingly high, as we
would expect from the chemical inertness of helium. Bohr regards the
two electrons of helium as being in the same ring.

If we now consider the process of adding an extra electron to the
neutral helium atom so as to produce the configuration (2, (3)), it
follows that the permanent state of the resulting system (since
.$•„ = o§577 when n = 3) would be given by —

a = 0*703^0; o) = 2'O2w0; W = 6*o7\V0.

Since for this configuration W is less than it was for the neutral atom,
the theory indicates that a helium atom cannot acquire a negative
charge. This is in agreement with experiment, for Franck (Ber. Detitsch.
phys. Ges., 12, 613 (1910)) has shown that helium atoms have no
affinity for electrons.

Lithium.

In this case N = 3, the nuclear charge being 3*. If one electron
be brought up so as to form the system (3, (i)), i.e. an atom of lithium
carrying effectively two positive charges, we get for the permanent state
of the configuration (3, (i)) —

a = 0-33^0 ; u> = 9<o0 ; W = 9\V0.

If a second electron be brought up and enters the ring containing the
first, the. stable state for the resulting system (3, (2)) is —

a = 0-364^; G> = 7-560)0 ; W = i

Since W3i (X) = 9W0, and since (W3t (2> — W3i (X)) = 6*i3\V0, it follows
that the first two electrons of the lithium atom are bound much more
firmly than the electron of the hydrogen atom.

If we now consider the process of adding a third electron to the
lithium nucleus so as to form a neutral atom, chemical considerations,
i.e. considerations of valency, would suggest that the third electron
should lie on a ring outside the ring containing the first two electrons,
both rings being concentric. The single electron in the outer ring
would be the valency electron of the lithium atom. The arrangement
is represented by the symbol (3, (2) (i)), the large numeral indicating
the nuclear charge. Proceeding as before, Bohr calculates that the
stable state would correspond to the following : —

inner ring : a^ = o'$62a0; wj = 7'65<o0

outer ring : a% = i'i&2a0; <o2 — o'7i6o>0; W = i6'02W0.

Since (W3i(2)(l) - W3f (2)) = o-&gW0, we see that the outer electron of
the neutral atom is less firmly held than the electron of the hydrogen
atom.

For a negatively charged lithium atom, i.e. the configuration (3, (2)

RUTHERFORD-BOHR ATOM-MODEL in

(2)) we would have W = i6'i6W0. This large value of W would
suggest that negatively charged lithium might under certain circumstances
manifest itself.

Bohr also considers beryllium, but enough has been said in connec-
tion with light atoms, in which the number of electrons is small, to
indicate the line of reasoning.

We have now to consider the question of series of rings present in
neutral atoms which contain a considerable number of electrons.

These rings are regarded as co-axial, lying in the same plane.
Bohr calculates that only in the case of systems containing a very large
number will there be any tendency for the rings to separate into different
planes. It is unnecessary to discuss this case. Considering systems
of rings in the same plane Bohr is led to important conclusions regard-
ing the maximum number of electrons which can rotate in a single ring.
We have dealt with this point earlier. It must be pointed out, however,
that the value for n simply gives us the maximum number of electrons
which could exist in a stable single ring ; it does not follow that this
number is reached in actual atoms. The formation of a second or of a
third ring is possible, even when the nuclear charge does not exceed Se,
i.e. when the total number of electrons does not exceed 8. The possi-
bility of this becomes clear when we take into account the question of
valency, for valency will be due essentially to the external ring. Bohr
further shows that in the stable system the number of electrons in any
ring diminishes as we proceed outwards from the nucleus. In dealing
with actual atoms Bohr makes use of the now generally accepted relation
(which will be discussed later), that the atomic number gives us directly
the number of electrons in the atom. The atomic number of an
element represents the position of the element in the whole series of
elements, arranged in order of increasing atomic weight. It has been
found that the atomic number, and consequently the number of
electrons present in an atom is approximately one half of the atomic
weight, i.e. helium 2, lithium 3, beryllium 4, oxygen 8, and so on.

It has already been stated that considerations of valency are a guide
to the number of electrons in the outermost ring. For monovalent
atoms the number is unity and so on for other atoms. A different con-
sideration also enters into the problem, viz. the possibility of confluence
of rings, especially when a considerable number of electrons are present.

Two rings may run together to form a single ring, and Bohr shows
that this will occur most easily when the two rings contain the same
number of electrons. Considering the binding of successive electrons
by a positive nucleus we conclude from this that, unless the charge on
the nucleus be very great, rings of electrons will join together if they
contain the same number of electrons and rotate in the same direction,
and that accordingly the number of electrons in inner rings will be the
numbers, 2, 4, 8, 16, etc. This assumption in regard to the number of
electrons in the rings is strongly supported by the fact that the chemical
properties of the elements of low atomic weight vary with a period of
eight. Further, it follows that the number of electrons on the outermost

112

A SYSTEM OF PHYSICAL CHEMISTRY

ring will always be' odd or even according as the total number of
electrons in the atom is odd or even. This suggests a relation with the
fact that the valency of an element of low atomic weight is always odd
or even according as the number of the element in the periodic series
is odd or even.

The fact that several considerations have been introduced into the
problem shows that the criterion of stability — constancy of angular
momentum and a maximum value for W — is not sufficient to determine
completely the constitution of an atomic system. In very simple cases
the constitution of the atom is in relatively little doubt. It is possible,
however, by making use of all the foregoing considerations to arrive at
a moderately satisfactory statement of the constitution of atoms which
contain up to twenty-four electrons, that is, up to an atomic weight
of forty-eight approximately. Bohr gives the following table for these
atomic systems. The large numeral indicates the value of N, the total
number of electrons in the neutral atom, whilst the smaller figures in
brackets give the composition of the rings starting from the innermost.

Thus the symbol 9 (4, 4, i) denotes a neutral atom which contains in
all nine electrons, arranged in three concentric rings, the innermost ring
containing four electrons, the second likewise four, and the outermost one
electron, thereby indicating that the normal valency of this atom is one.
The atomic weight of the element in question is approximately 2x9
= 1 8, i.e. fluorine. It will be observed that Bohr places hydrogen
along with the halogens, and not with the alkali metals. Bohr does not
give the name of the corresponding element to these configurations but
the attempt is here made to do so.

TABLE SHOWING STRUCTURE OF ATOMS.

Probable

Element.

Structure of
Atom.

Probable
Element.

Structure of
Atom.

Probable
Element.

Structure of
Atom.

Hydrogen

MI)

Fluorine .

9 (4, 4» J)

Chlorine .

17 (8, 4, 4, i)

Helium .

2(2)

Neon

10 (8, 2)

Argon

18 (8, 8, 2)

Lithium .

3 (2, i)

Sodium

II (8, 2, I)

Potassium

19 (8, 8, 2, i)

Beryllium

4 (2, 2)

Magnesium

12 (8, 2, 2)

Calcium .

20 (8, 8, 2, 2)

Boron

S (2, 3)

Aluminium

13 (8. 2, 3)

—

21 (8, 8, 2, 3)

Carbon .

6 (2, 4)

Silicon

14 (8, 2, 4)

—

22 (8, 8, 2, 4)

Nitrogen

7 (4, 3)

Phosphorus

15 (8» 4, 3)

—

23 (8, 8, 4, 3)

Oxygen .

8 (4, 2, 2)

Sulphur

16 (8, 4, 2, 2)

""™™"

24 (8, 8, 4, 2, 2)

Elements in the same horizontal line belong to the same group,
the vertical columns indicate the first three periods of the elements.

As regards elements of higher atomic weight, Bohr points out that
at the end of the third period of eight elements we meet with the iron
group, which takes a particular position in the system of elements since
it is the first time that elements of neighbouring atomic weights show
similar chemical properties. This circumstance indicates that the con-
figurations of the electrons in the elements of this group differ only in

RUTHERFORD-BOHR ATOM-MODEL 113

the arrangement of the inner electrons. The fact that the period in the
chemical properties of the elements after the iron group is no longer
eight but eighteen suggests that elements of higher atomic weight con-
tain a recurrent configuration of eighteen electrons in the innermost
rings. The deviation from 2, 4, 8, 16 may be due to a gradual inter-
change of electrons between the rings as already indicated when speak-
ing of the confluence of rings. Since a single ring of eighteen electrons
will not be stable, the electrons may be arranged in two parallel rings.
Such a configuration of the inner electrons will act upon the outer
electrons in very nearly the same way as a nucleus of (N — 18) x e.
It is conceivable that with increase in N another configuration of the
same type will be found outside the first, such as is suggested by the
presence of a second period of eighteen elements.

Systems Containing Several Nuclei. Molecules.

According to Bohr a simple molecule such as the molecule of hydro-
gen consists of two positively charged nuclei, each carrying unit charge
with a ring of two electrons rotating between them, the rotation being
round the line joining the two nuclei. If the two nuclei are similar in
all respects the electron ring rotates at a position midway between the
nuclei, the plane of the ring being perpendicular to the line joining the
nuclei. If the nuclei be dissimilar the ring will rotate at some position
lying nearer the heavier and more complex nucleus.

The equilibrium of such a system is conditioned as before by the
constancy of the angular momentum of each electron, along with the
further condition that W — the energy emitted in binding an electron to-
the system — is a maximum.

Consider a system consisting of two positive nuclei of equal charges,
and a ring of electrons rotating round the line joining the nuclei. Let
the number of electrons be n, the charge on each be e, and the charge
on each nucleus be N*. It can be shown that the system will be in<
equilibrium if (i) the nuclei are equidistant from the ring and (2) if the
ratio between the diameter of the ring, 2a, and the distance apart of the
nuclei, 2^, be given by —

It is assumed that the frequency of revolution to is a magnitude such
that for each electron the centrifugal force balances the radial force (due
to the attraction of the nuclei and the repulsion of the electrons).

g3

Denoting this radial force by -^F0 we get from the condition of the

universal constancy of the angular momentum of the electrons as already
shown —

• • • (9)

0

VOL. III. 8

ii4 A SYSTEM OF PHYSICAL CHEMISTRY

and VV = total energy necessary to remove all the charged particles to
infinite distances from one another = 2n7r *2 ° . . . (10)
For the system in question we have —

•*-* - . (II)

where sn has its former significance. It will be observed that F0 is a
function of the radius of the ring.

As regards the stability of the ring of electrons with respect to dis-
placements perpendicular to the plane of the ring, Bohr shows that the
system will not be stable unless when N = i, n = 2 or 3. Throughout
the entire treatment of the problem it will be observed that Bohr regards
the question of stability with respect to such "perpendicular" displace-
ments as being capable of being dealt with by classical electrodynamics.
In the case of displacements of electrons in the plane of the ring classical
electrodynamics is incapable of indicating any position of stability, and
Bohr finds the condition by introducing the quantum theory in the form
of the constant angular momentum (/#/27r) possessed by each electron in
whatever orbit it may be rotating.

It is assumed that the motions of the nuclei with respect to one
another are so slow that the state of motion of the electrons at any
moment will not differ sensibly from that calculated on the assumption
that the nuclei are at rest. This assumption is permissible on account
of the great mass of the nuclei compared with that of the electrons,
which means that the vibrations resulting from a displacement of the
nuclei are very slow compared with those resulting from a displacement
of the electrons.

Let us now imagine that, by the help of extraneous forces acting on
the nuclei, we slowly alter the distance between them. During this dis-
placement the radius of the ring of electrons will alter in consequence
of the alteration in the radial force due to the attraction of the nuclei
for the electrons. During this alteration we suppose that the angular
momentum of the electrons remains constant. If the distance apart of
the nuclei increases, the radius of the electron ring will likewise increase.
It can be shown, however, that the radius of the ring will increase at a
slower rate than does the distance between the nuclei. On account of
this difference in the rate the attraction on one of the nuclei due to the
ring will be greater than the repulsion from the other nucleus. The
work done during the displacement by the extraneous forces acting upon
the nuclei will therefore be positive, and the system will be stable for
this displacement. The same result will also hold in the case in which
the distance between the nuclei diminishes.

For a system consisting of a ring of electrons and two nuclei of un-
equal charge the investigation becomes more complicated. During a
variation of the distance of the nuclei apart not only will the radius of
the electron ring vary, but also the ratio in which the plane of the ring

RUTHERFORD-BOHR ATOM-MODEL 115

divides the line connecting the nuclei. If we consider a neutral system,
containing two nuclei carrying large charges, in the stable configuration
the greater part of the electrons must be arranged round each nucleus,
approximately, as if the other nucleus were absent ; and only a few of
the outer electrons will be arranged differently, rotating in a ring round
the line connecting the nuclei. This ring which keeps the system
together represents the chemical bond.

Systems Containing few Electrons ; the Hydrogen Molecule.

As already stated, the neutral hydrogen molecule consists of two
similar nuclei each carrying unit charge e with two electrons rotating in
a ring between the nuclei. Denoting the radius of the ring by a, and
the distance apart of each nucleus from the plane of the ring by ^, we
get from equation (8), on putting N = i, and n = 2 —

From (n) we get further —

F° = 3— = 1-049.

From (9) and (io) we get — denoting the values of a, w, and W for a
system consisting of a single electron and single nucleus (i.e. the
hydrogen atom) by a0, o>0, and W0 — the following values for the hydro-
gen molecule : —

a = o'95<z0; &> = I'lowo ', W = 2'2oW0.

Since W is greater than 2\Y0, it follows that two hydrogen atoms
combine to form a molecule with emission of energy. That is a
hydrogen molecule dissociates with absorption of energy. This is in
agreement with experiment. Putting W0 = 2'o x io~n erg, and
NO = 6-2 x io23, where N0 is the number of molecules in one gram-
molecule, we get for the energy emitted during the formation of one
gram-molecule of hydrogen from two gram-atoms, the quantity
N0(W - 2 W^) = 2-5 x io12 erg, or 60,000 cals. This value is of the.
correct order of magnitude, the value observed by Langmuir being
approximately 80,000 cals.

In considering displacements of electrons perpendicular to the plane
of the ring, we have seen that the molecule will become unstable unless
when N = i, « «» 3 or 3, Hence if we remove one of the two elec-
trons from the hydrogen molecule the whole system will become un-
stable and will break up into a single nucleus positively charged and
a neutral hydrogen atom. This process corresponds to the change
from a stationary state represented by the complete neutral molecule
and a second stationary state represented by the dissociated molecule.
On Bohr's theory when such a change takes place from one stationary
to another there is involved homogeneous radiation of frequency v, the

n6 A SYSTEM OF PHYSICAL CHEMISTRY

energy per molecule " decomposed " being hv. In this case hv repre-
sents energy absorbed given by the relation —

hv= W - W0 = i-2oW0.
Hence v = 37 x io15.

The value for the frequency in the farthest ultra violet calculated from
dispersional measurements in the case of hydrogen is 3*5 x io15, so
that the agreement is good. It may be pointed out that the " chemi-
cal" process involved in the above is H2->H + H+ + 0 It is not
clear why this process should be identified with the dispersional
frequency.

The above calculation refers to the quantum involved in adding or
removing an electron. We have now to consider the frequency and
consequent magnitude of the quantum involved if one nucleus is removed
from the other. Bohr shows that this frequency is given by — l

m

where m is the mass of an electron and M the mass of the nucleus.
Putting M/m = 1835 and o>0 = 6*2 x io15, we get —

vr =1-91 x iou,

or \r = i -5 7/*, t.e. the short wave-length region of the infra-red spec-
trum. This frequency is of the same order of magnitude as that cal-
culated for the nuclear or atomic vibration of hydrogen on the basis of
molecular heat (cf. Chap. IV.). On the other hand, no absorption of
radiation in hydrogen gas corresponding to this frequency is observed.
According to Bohr this is to be expected on account of the symmetri-
cal structure of the hydrogen molecule and the great ratio between the
frequencies corresponding to the displacements of the electrons and the
nuclei respectively.

A system of two nuclei, each of charge <?, and a ring of 3 electrons
rotating between them — that is, a negatively charged hydrogen molecule
— will be stable for displacements perpendicular to the ring, for the
system is stable if N — i and n = 2 or 3. Bohr calculates that for
such a system —

a = 1-14^0 ; w = 0770)0 ; W = 2'32W0.

Since the W is greater in this case than it is in the case of the neutral
hydrogen molecule (W = 2'2oW0), the negatively charged molecule
should be capable of existence. Proof of the existence of negatively
charged hydrogen molecules has been obtained by Sir J. J. Thomson
in experiments on positive rays (Phil. Mag., 24, 253 (1912)).

We have now to consider the mechanism whereby a molecule of
hydrogen is formed from its constituents. We shall restrict ourselves
to the union of two neutral hydrogen atoms. Suppose that two such

* This expression obtained by Bohr is obviously the same as that found by Haber
in igri and referred to in the following chapter.

RUTHERFORD-BOHR ATOM-MODEL 117

atoms are approaching one another at a rate which is sufficiently slow
that the dynamical equilibrium of the electrons for every position of
the nuclei is the same as if the latter were at rest. Suppose that the
electron of atom i is rotating in the same plane as that of atom 2, the
difference in phase being one-half of a revolution. During the ap-
proach, the direction of the planes of the orbits of the electrons and
the difference in phase will be unaltered. The planes of the orbits
will, however, at the beginning of the process approach each other at
a higher rate than do the nuclei, for two electrons rotating in the same
direction attract one another. Finally, at a certain distance of the
nuclei apart the electron planes will coincide, the two electrons being
now arranged in a single ring. During the further approach of the
nuclei the ratio between the diameter of the electron ring and the
distance apart of the nuclei will increase,1 and the system will pass
through a configuration in which it will be in equilibrium without the
application of extraneous forces on the nuclei. That is, a permanent
neutral hydrogen molecu'e is formed.

Now let us suppose that we are dealing with two helium atoms, i.e.
systems consisting of a nucleus of charge 2<? surrounded by a ring of
two electrons, and let us imagine the above process repeated. Assume
that at the beginning the helium atoms are orientated to each other like
the hydrogen atoms, with the exception that the electrons in the helium
atoms differ by one-quarter of a revolution instead of one-half. The
planes of the electrons will again approach one another at a higher rate
than do the nuclei, and for a certain position of the latter the planes
will coincide. During the further approach of the nuclei, the 4 elec-
trons will be arranged at equal angular intervals in a single ring. As
in the former case it may be shown that at any moment during this
operation the system will be stable for a displacement of the electrons
perpendicular to the plane of the ring. Contrary, however, to what
took place in the case of hydrogen, the extraneous forces to be applied
to the nuclei in order to keep the system in equilibrium will always be
in a direction to diminish the distance apart of the nuclei, and the
system will never pass through a position of equilibrium. The helium
atoms will in fact repel one another and no molecule will be perman-
ently formed.

As regards molecular systems containing a considerable number of
electrons the treatment becomes much more complicated. Bohr
attempts to deal with such cases, and for details his paper must be
consulted. It may be mentioned that, following an approximate
method of calculation, Bohr is led to expect an absorption band in the
infra-red spectrum of hydrochloric acid gas at v = 13*7 x io13 whilst
the observed band lies at v = 8*5 x io13.

Bohr regards the water molecule as consisting of an oxygen nucleus
surrounded by a ring of four electrons, and the two hydrogen nuclei
situated on the axis of the ring at equal distances apart from the oxygen

1 Of course the actual diameter of the electron ring diminishes but not so fast as
does the distance between the nuclei.

n8 A SYSTEM OF PHYSICAL CHEMISTRY

nucleus and kept in equilibrium by two rings of greater radius than the
one just mentioned, each of these two rings containing 3 electrons ; the
latter rotate in parallel planes round the axis of the system and are
situated relatively to each other in such a way that the electrons in the
one ring are opposite the intervals between the electrons in the other.
If we imagine such a system broken up by slowly removing the hydro-
gen nuclei we should obtain two positively charged hydrogen atoms
and an oxygen atom with a double negative charge in which the outer-
most electrons will be arranged in two rings of three electrons each, the
two rings being parallel to one another.

Parson's Magneton and the Structure of the Atom.

Hitherto, in dealing with the problem of atomic structure, we have
regarded the electron as a minute discrete particle of electricity or
matter, assumed to be spherical, and capable of movement " inside "
the atom in certain orbits, the velocity of movement being relatively
small compared with the velocity of light. A new concept has been
introduced, however, in Connection with the electron which must be
very briefly discussed. According to this view the electron itself is a
tiny ring or annulus carrying a negative charge. The ring is regarded
as rotating continuously with a peripheral velocity which is of the same
order as that of light. This is equivalent to a circuit current, analogous
to the Ampere currents assumed to account for magnetism, and conse-
quently such a structure possesses magnetic properties. For this reason,
A. L. Parson, to whom this concept of the electron is due (Smithsonian
Inst. Pub., 65, No. n, 1915), substitutes the word magneton in place of
electron.

The term magneton was first introduced by Weiss in 1911 (cf. N.
Campbell's Modern Electrical Theory], On the basis of the electron
theory of paramagnetism an expression has been deduced for the sus-
ceptibility according to which this quantity should be proportional to
M'2/T, where M is the magnetic moment of a molecule and T is the
absolute temperature. Weiss observed that in the case of magnetite
over a certain temperature range the curve connecting the susceptibility
with the reciprocal of the temperature was not a straight line through
the origin, as it should have been, but consisted of a series of lines
separated by definite differences in the value of the susceptibility.
This was ascribed by Weiss to abrupt, discontinuous variations in the
value of the molecular magnetic moment M, and Weiss further showed
that the various observed values were all even multiples of a certain
quantity to which he gave the name the magneton. The magnitude
(i.e. the moment) of the magneton is the same as that which would be
produced by a rotating electron revolving in an orbit of radius io~8 cm.
with a frequency of io14 per second approximately. Weiss of course
kept to the classical concept of the electron.

In the case of Parson's ring electron or circuit current the radius of
the ring is estimated to be about 1-5 x io~9 cm., that is a quantity

PARSON'S A TOM-MODEL 1 1 9

somewhat -less than the accepted value for the diameter of an atom.
Parson's theory of atomic structure based upon the magneton hypothesis
may be summarised as follows : —

The positive charge in the neutral atom, which must be present to
balance the negative charges of the magnetons is regarded as a sphere
of uniform positive electrification possessing a volume which is pro-
portional to the number of magnetons which it contains. That is, the
distribution of the positive charge is once more treated as in the
original theory of atomic structure first advanced by Sir J. J. Thomson
(cf. Chap. I., Vol. I.). This positive sphere is regarded as having the
properties of an elastic solid and " is surrounded by an atmosphere or
envelope of very low charge density which is also elastic ". Given this
distribution, Parson shows that a group of eight magnetons may be
airanged symmetrically round the sphere so as to give a stable configura-
tion ; and even with atoms possessing a larger number of magnetons,
a similar distribution of eight is assumed. Such an arrangement ob-
vi3usly recalls the Abegg-Bodlander theory of valency, according to
waich the natural number of valencies is eight, though all these are not
effective as far as another atom is concerned. To explain the existence
of the long periods in the period classification as well as the properties
of certain elements contained therein, it is necessary to assume a certain
" hindrance " to the formation of the normal number of eight magnetons.

On the basis of these assumptions Parson discusses the single mag-
netic attraction between two such magnetons, the tendency to form the
group of eight, the residual valency effect and the electrical polarisation
set up by a magneton of the above type. On the basis of this atomic
structure Parson has made a rough calculation of the heat of dissocia-
tion of hydrogen, obtaining the value 135,000 cals. per gram- molecule;
Langmuir's value being of the order 80,000 cals.

One of the most obvious advantages of the Parson magneton is that
it affords a rational basis for the assumption made by Bohr, viz. that
an electron rotating in a stable orbit does not radiate. A rotating
electron — as hitherto envisaged — suffers an effective inward acceleration
in rotating, and hence, on the basis of the classical electromagnetic
theory, it should emit radiation continuously, its orbit becoming smaller
until it falls into the centre of the atom. On the basis of the classical
electromagnetic theory it is also known that the greater the number of
electrons rotating in a ring the less the radiation from the ring. The
Parson ring magneton is equivalent to a continuous series of charges,
and consequently such a ring, howsoever great its velocity, will not
radiate at all. This carries out Bohr's idea of the absence of radiation
from an electron in an orbit.

On the other hand one of the most obvious disadvantages of Parson's
theory of atomic structure lies in the assumption of a uniform positive
sphere of electricity, for on such a basis it is quite impossible to account
for the large angle scattering of a rays observed by Rutherford. The
point is therefore whether we can retain the advantage of the ring
electron or magneton and at the same time get rid of the hypothesis of

120 A SYSTEM OF PHYSICAL CHEMISTRY

the positive sphere, by substituting for it the nucleus idea of Rutherford.
The difficulty, as already pointed out, consists in the fact that an atom
consisting of a nucleus surrounded by an atmosphere of magneton?
would be an unstable system. The problem has been discussed in a
preliminary manner by D. L. Webster (/ Amer. Chem. Soc., 40, 375,
1918).

From what has been said it will be obvious that the magneton
theory of atomic structure as developed by Parson is still in a rudiment-
ary stage. Its advantages, however, are such, particularly in regard to
its applicability to chemical phenomena, that it cannot be discarded
without a serious attempt being made to overcome the main difficulty
of the scattering of the a particles.

High Frequency Spectra of the Elements. Moseley's Relation,

By the high frequency spectrum is meant the X-ray spectrum which
is obtained from a substance by bombarding the substance with /3-rays.
It has been pointed out in Vol. I., Chap. II., that the investigation of the
structure of crystals by means of their X-ray spectra affords a means of
determining with precision the wave-length of any homogeneous X-ray
employed. X-rays only differ from light in possessing very much shorter
wave-lengths (of the order io~8 cm.). The substance to be used as a
source of X-rays forms the anticathode in a vacuum tube, and the beam
of X-rays emitted as a result of bombardment by the cathode rays is
characteristic of the substance forming the anticathode. In general
the X-rays thus excited are not homogeneous. They consist of two
or more wave-lengths which can be analysed and measured by means
of the crystal acting as a grating. As a rule, however, one frequency
is more prominent than the others, that is, the intensity of the result-
ing spectrum is greatest for one type, and for the sake of simplicity we
shall consider these single frequencies as characteristic of the X-rays
emanating from the substance examined. When the anticathode is
of nickel, the characteristic wave-length of the X-ray produced is
i -io x io~8 cm. It follows, therefore, that a single quantum hv of
this type contains energy in amount 178 x io~8 ergs. It is of in-
terest to compare this quantity with the energy of the electron of the
cathode stream which gave rise to the X-ray. Whiddington (Proc.
Camb. Phil. Soc., 1910) has shown that an electron of the cathode
stream will not excite X-ray radiation unless its velocity exceeds a
certain critical value, which varies with the nature of the material
composing the anticathode. If the latter be of nickel the limiting
velocity of the electron striking it is 6 -17 x io9 cm. per sec., or about
one-fifth of the speed of light. Taking the mass of the electron to be
0-9 x io~27 gram, the kinetic energy of the electron possessing the
limiting speed is 1*7 x io~8 erg, a quantity which is very nearly the
same as the energy in a single quantum of the X-ray produced.
Further, the investigation of the reverse process, i.e. the emission of an

HIGH FREQ UENC Y SPECTRA 1 2 1

electron by means of X-rays, has shown that the kinetic energy of the
electron is approximately equal to that of the cathode ray which
excited the X-ray in the first instance. As Bragg remarks " we are
justified in assuming that these processes are actual examples of the
give-and-take of radiant energy with which Planck's hypothesis is con-
cerned ".

Whiddington has also found that the limiting velocity of an electron
just necessary to excite an X-ray from a particular element is proportional
to the atomic weight of the element. Now we have seen that the
frequency v of the X-ray is proportional to the kinetic energy of the
electron producing the ray. Hence the frequency of the X-ray is pro-
portional to the square of the velocity of the electron and therefore to
the square of the atomic weight. This serves as a useful guide in the
choice of a substance for the anticathode which will produce an X-ray
of a given kind.

In 1913 van den Broek (Physikal. Zeitsch., 14, 3 2) suggested, that in
comparing elements with one another, the atomic number, that is,
number which represents the position of the element in the periodic
table is a far more fundamental quantity than the atomic weight itself.
As a matter of fact the atomic number of an element is approximately
one-half of the atomic weight, so that any quantity proportional to
the atomic weight is likewise proportional to the atomic number.

The connection between the nature of an element and the frequency
of the X-ray produced from it has been systematically studied by
Moseley (Phil. Mag., [vi.], 26, 1024 ; ibid., 27, 7°3 (1914))- On plotting
the square root of the frequency of the X-ray produced from a given
element against the atomic number of the element, for a whole series
of elements, Moseley found that a simple linear relation connected the
two quantities. This was true both for the K and L types of X-rays,
types which differ from one another in intensity, the K type being pro-
duced from a "hard" bulb (high vacuum), the L type being pro-
duced from a "soft" bulb. It is evident that the atomic number is
not simply an ordinal but possesses some actual physical significance.
The Rutherford atom-model, already discussed, suggests what this signi-
ficance is.

The measurements of Rutherford and Geiger showed that the magni-
tude of the positive charge on the nucleus was N* where e is identical
with the magnitude of the charge on an electron, and N is a number
approximately one-half the atomic weight. N is in fact a number
identical with the atomic number. It is probable that the elements
differ from one another in the magnitude of the nuclear charge, and it is
reasonable to suppose that successive elements in a table of elements vary
by the amount e on the nuclear charge. Regarding the nuclear charge of
hydrogen atom as e itself, the succeeding elements will possess 2<?, 3^, and
so on, or in other words, the atomic number indicates the magnitude of
the nuclear charge. If this be so, the various relations, found between
atomic number and other quantities serves to relate these quantities to
the magnitude of the nuclear charge. The simplest form of Bohr's

122

A SYSTEM OF PHYSICAL CHEMISTRY

relation, already discussed, between the frequency of emitted radiation
and electronic charge involves the square of the latter. In other words,
the square root of the frequency emitted is proportional to e itself.
Bohr's relation, in fact, is capable of predicting the characteristic X-ray
radiation from an element of high atomic weight with considerable
exactness. In this connection it may be pointed out (cf. Millikan
Physical Review, Aug., 1917) that the orbit of the innermost electron
is inversely proportional to the atomic number or nuclear charge, i.e.
the greater the charge on the nucleus the smaller the radius of circular
path followed by the innermost electron.

Characteristic Infra-red Frequency and Atomic Number.

An interesting relationship has been pointed out by Allen (Proc.
Roy. Soc., A, 94, ioo (1917)) between the atomic number and the infra-
red frequency which accounts approximately for the atomic heat of the
element. Writing N for the atomic number and v for the frequency of
a given element, Allen's relationship is —

Nv = nv^

where n is an integer called the frequency number, and VA is a funda-
mental constant the same for all elements. The mean value for VA is
2 1 '3 x i o12. The following table illustrates the above relation. The
experimental values of v for the various elements (obtained from atomic
heat measurements of Nernst, Griffiths, and Keesom-Onnes) have
been multiplied by the atomic number N.

-12
Nv x IO

T-M

»*•

Nernst.

Griffiths.

Keesom-Onnes.

Al

13

5 x 21-6

5 x 21-4

5 x.21-5

Fe

26

—

10 X 20*9

—

10 x 20-9

Cu

29

9 x 213

9 x 21-9

9 X 21'2

9 x 21-3

Zn

30

7 x 20-6

7 x 20-6

—

7 x ao'6

Ag

47

IO X 21'2

TO X 21*1

—

IO X 2I'I

Cd

48

8 X 2I'I

—

8 X 2I'I

Hg.

80

8 X 20'2

—

—

8 X 2O'2

Pb .

82

7 X 22'2

7 x 22-5

7 x 21*6

7 X 22'O

The above relationship is empirical and its full significance has not
yet been determined. It may be pointed out, however, that Biltz has
shown the characteristic infra-red frequency v of the elements to be a
periodic function of the atomic weight, and therefore of the atomic
number. Since VA is a constant it follows that n must likewise vary in
a periodic manner with the atomic weight or number. This is not the
same thing, however, as explaining the significance of n.

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.

THE PHOTO-ELECTRIC EFFECT.

HALLWACHS in 1888 was the first to show that a body carrying a charge
of negative electricity loses its charge on being exposed to ultra-violet
light ; on the other hand a positively charged body is not discharged.
Later, Hallwachs and Righi showed, independently, that an insulated
metal exposed to ultra-violet light acquires a positive charge. That is
to say, negatively charged particles, electrons, are emitted from the
metal under the action of ultra-violet light. This phenomenon is
known as the photo-electric effect. Certain metals, such as the alkali
metals and zinc and aluminium, exhibit this effect even under the in-
fluence of ordinary " visible ' ' light, the effect being greatest in the case
of the alkali metals. It is essential in all these cases that the surface
should be fresh and clean ; if a layer of oxide forms, the effect is greatly
weakened. This is known as photo-electric fatigue.

This emission of electrons is bound up with an absorption of the
light. The periodic electric force in the radiation sets the electrons on
the surface into more violent vibration, so that some of them are ejected
from the metal into the neighbouring space. If v is the velocity
possessed by an ejected electron, its kinetic energy is ^mv2, where m
is the mass of the electron. If the surface be charged positively to a
potential of V volts, which potential is just able to prevent the escape
of the electron, it follows that W, where e is the charge on the electron,
is just equal to ^mv2, i.e. the kinetic energy which the electron would
have had if no potential had existed.

Experiment has shown that the photo-electric effect really comprises
two phenomena, apparently distinct from one another. These are
known as the normal photo-electric effect and the selective photo-electric
effect respectively. We shall consider the normal effect in the first
place.

The photo-electric current, produced by the moving electrons which
have been set free from the surface, depends upon two factors, (i) the
number of electrons emitted in unit time, and, (2) the speed of these
electrons. If the intensity of the light be increased, without altering
the colour or quality (i.e. without altering the vibration frequency), it

123

124

A SYSTEM OF PHYSICAL CHEMISTRY

has been found that the photo-electric effect increases proportionally
to this intensity, because the number of electrons emitted increases in
the same proportion. As long, however, as the frequency of the light
is maintained constant, the velocity of the electrons is also constant.
On the other hand, working with constant intensity of light but varying
the quality or frequency it has been found that, on increasing the
frequency of the light, the photo-electric effect increases, because the
speed of the electrons is now increased. According to Lenard and
Ladenburg, when the wave-length is thus shortened (at constant in-
tensity), the velocity of the electrons emitted increases proportionally
to the frequency. It is now known that the square of the speed is
proportional to the frequency of the incident light. Below a certain
wave-length or frequency, known as the threshold wave-length, the
normal photo-electric effect is not observable. That is, with wave-
lengths longer than a certain value A.0 — characteristic of the substance
under examination — no electrons are emitted. Hughes has determined
the value of X0 for a number of substances, using fresh surfaces, obtained
by distillation of the substance in vacua. These values are given in
the following table : —

NORMAL PHOTO-ELECTRIC EFFECT. THRESHOLD WAVE-LENGTHS (\o).

Substance.

AO.

Substance.

*».

Ca

370 MM

Sb

308 MM

Mg

330

As

236

Cd

3M

Se

220

Zn

302

O2

J35

Pb

312

C (soot)

255-260

Bi

323

The values obtained by Richardson and Compton with freshly scraped
surfaces correspond to somewhat longer wave-lengths than those given.

Starting from the above threshold wave-lengths and diminishing the
wave-length, the photo-electric effect increases in virtue of an increase
in the velocity of the electrons. According to Richardson and Compton
this goes on until a certain limiting value, corresponding to an ex-
tremely short wave-length is attained, beyond which the normal photo-
electric effect no longer increases.

One of the most striking features of the photo-electric effect is the
fact that the speed of the electrons is the same for a given quality
(frequency) of light quite independent of the intensity of the light.
Further, on keeping the intensity constant and varying the quality of
the light, the speed of the electrons increases as the frequency increases.
These two facts have found a satisfactory explanation on the basis of
the quantum hypothesis, in which the light is regarded as " hetero-
geneous". The following statements indicate the view held by Sir
J. J. Thomson in regard to the point under consideration (Proc.

THE PHOTO-ELECTRIC EFFECT 125

Camb. Phil. Soc., 14, 421, 1908): The [radiant] energy travelling
outwards [from the radiating source] with the wave is not spread
uniformly over the wave front, but is concentrated on those parts of
the front where the pulses are travelling along the lines of force ; 1
these parts correspond to the bright specks, the rest to the dark
ground. . . . The energy of the wave is thus collected into isolated
regions, these regions being the portions of the lines of force occupied,
by the pulses or wave motion. In fact, from this point of view, the
distribution of energy is very like that contemplated on the old emission,
theory, according to which the energy was located on moving particles,,
sparsely disseminated throughout space. The energy is, as it were,
done up into bundles, and the energy in any particular bundle does
not change as the bundle travels along the line of force. Thus, when
light falls upon, a metal plate, if we increase the distance of the source
of light, we shall diminish the number of these different bundles or units
falling on a given area of the metal, but we shall not diminish the
energy in the individual units ; thus any effect which can be produced
by a unit by itself, will, when the source of light is removed to a greater
distance, take place less frequently it is true, but when it "does take
place it will be of the same character as when the intensity of the light
was stronger. This is, I think, the explanation of the remarkable
result discovered by Lenard, that though the number of corpuscles
emitted by a piece of metal exposed to ultra-violet light increases [as
intensity increases], the velocity with which individual corpuscles come
from the metal does not depend upon the intensity of the light. If
this result stood alone, we might suppose that it indicated that the
forces which expel the corpuscles from the metal are not the electron
forces in the light wave incident on the metal, but that the corpuscles
are ejected by the explosion of some of the molecules of the metal
which have been put into an unstable state by the incidence of the
light; if this were the case the velocity of the corpuscle would be
determined by the properties of the atom of the metal, and not by
the intensity of the light, which merely acts as a trigger to start
the explosion. Some experiments made quite recently by Dr. E.
Ladenburg make, however, this last explanation exceedingly improbable.
Ladenburg has investigated the velocities of corpuscles emitted under
the action of ultra-violet light of different wave-lengths, and finds that
the velocity varies continuously with the frequency ; according to his
interpretations of his experiments, the velocity is directly proportional
to the frequency.2 Thus, though the velocity of the corpuscles is
independent of the intensity of the light, it varies in apparently quite a
continuous way with the quality of the light ; this would be very improb-
able if the corpuscles were expelled by an explosion of the molecule.

1 Thomson assumes that the ether has disseminated through it discrete lines of
electric force, these being in a state of tension. The light consists of transverse
vibrations travelling along these lines.

2 As already stated, the evidence now available goes to show that the square of
the velocity of the electron is proportional to the frequency of the light.

126 A SYSTEM OF PHYSICAL CHEMISTRY

It seems more reasonable to suppose that the velocity is imparted by
the light, and yet, as we have seen, the velocity is independent of
the intensity of the light. These results can, however, be reconciled
by the view stated above, that a wave of light is not a continuous
structure, but that its energy is concentrated in units (the places where
the lines of force are disturbed), and that the energy in each of these
units does not diminish as it travels along its line of force. Thus if a
unit by impinging on a molecule can at any place make it liberate a
corpuscle, it will do so and start the corpuscle with the same velocity
whatever may be the distance from the source when it strikes the
molecule; thus the velocity of the corpuscles would be independent
of the intensity of the light. Ladenburg found that the velocity of the
corpuscle increases with the frequency of the light ; this shows that, if
the view we are discussing is correct, the energy in the units will in-
crease with the frequency of the light. This latter statement is, of
course, in complete agreement with the quantum hypothesis, according
to which the energy in any unit is directly proportional to the frequency,
i.e. t = hv.

Einstein has given a quantitative application of the quantum
hypothesis to the photo-electric effect in the expression —

Ve = mv* = hv - hv.

The term $mv2 is the kinetic energy of the electron emitted by light of
frequency v. VQ is the threshold frequency already denned, character-
istic of the substance from which the electrons are emitted. The signi-
ficance of Ve has already been pointed out. It follows from the above
expression that when v = i/0 the electron will just not be emitted, v0
being therefore the lower limit of frequency capable of bringing about
the photo-electric effect at all. Further, the frequency is proportional
to the kinetic energy and therefore to the square of the velocity of the
electron.

The applicability of the quantum theory to the photo-electric effect
may be regarded as evidence of considerable value in favour of the
quantum hypothesis itself — especially in the form given to it by
Einstein.

Sir J. J. Thomson has likewise made an attempt to calculate the
amount of energy in each unit of light of given frequency from some of
Lenard's data. With the ultra-violet light used by Lenard the maxi-
mum velocity of the corpuscles was about io8 cms. per second; the
kinetic energy (^mv*) of a corpuscle moving with this speed is about
3 x io~12 ergs. If we suppose that the energy of the corpuscle liber-
ated is of the same magnitude as that of the light unit e which liberated
the corpuscle, we would expect e (for the given light) to be about io~12
ergs. It will be seen that this agrees very well with the values for e
already given as a result of calculation, with the help of Planck's] con-
stant h. A similar calculation, with equally satisfactory results, has been
recently carried out by J. Franck and G. Hertz ( Verh. d. d. physik.

THE PHOTO-ELECTRIC EFFECT 127

Gesell., 13, 967, 1911; ibid., 14, 167, 1012). The problem of the
probable structure, constitution, and origin of these light units, as well
as a criticism of Planck's views, has been considered by Sir J. J. Thom-
son (Phil. Mag., 1908-10), but it would be quite outside the scope of
this book to attempt anything further than this reference.

To show by an extreme case how few light units it is necessary to
postulate, say, in the visible region of the spectrum under ordinary con-
ditions, Sir J. J. Thomson has carried out the following calculation.
Light of such intensity that-io~4 ergs fall on unit area per second would
be very faint, but would still be visible. If we think of a cylinder of
ether, base i square centimetre and 3 x io10 cms. in length, then the
4bove-mentioned intensity is such that io~4 ergs are distributed through-
out this cylinder (at any moment), and the density of the light is there-

—4

fore jo ergs per c.c. Taking as a mean value for one quantun

the number io~12 ergs, we see that there will only be one such unit in
every 1000 c.c. of space. The structure of light is therefore of a coarse-
grained character. A further point still remains to be mentioned. The
idea of a point source not radiating uniformly (this being a result of the
heterogeneous nature of emission) seems to be in disagreement with the
ordinary laws of propagation of light equally in all directions. It must
be remembered, however, that any physical source of radiation consists
of a very large number of resonators, so that the total radiating effect
is on the average symmetrical about the source.

The phenomenon of ionisation of a gas or vapour by ultra-violet
light is a further instance of the photo-electric effect. As is well known,
ionisation of a gas may be brought about by means of cathode rays (i.e.
streams of electrons) provided these electrons have attained a certain
velocity, i.e. a kinetic energy ^mv*. In an electric discharge this velo-
city is reached as a rule before the electrons have travelled far, for they
move with a positive acceleration towards the anode, the act of ionisa-
tion being demonstrated by a glow at some portion of the tube. This
lower limit of kinetic energy which an electron must possess in order to
ionise a gas may be equated to W, where V is called the ionisation
•potential of the gas, and denotes the smallest voltage which is required
in order to give to an electron the kinetic energy ^mv2" which will just
enable the electron to ionise a neutral gas molecule by impact. Apply-
ing the quantum theory to this case we see that the frequency i/0 of the
light which will just ionise a gas is given by hv§ = V*.

The following table contains the most recent values for the ionisa-
tion potential of a number of gases as determined by Hughes and Dixon
(Phys. Rev., 10, 495

128

A SYSTEM OF PHYSICAL CHEMISI^RY

Gas.

V in Volts.

Gas.

V in Volts.

H2

I0'2

CO

7'2

02

9-2

C02

IO'O

N2

77

NO

9'3

S.

8-3?

CH4

9*5

Br2

lO'O

C2H6

lO'O

Hg
HC1

IO*2
9'5

C,H4
C2H2

9-9
9-9

Tate and Foote (Journ. Wash. Acad, Set., 7, 517 (I9i7)) have de-
termined the ionisation potential of sodium vapour to be 5*13 + OTO
volts ; when ionised, the sodium vapour emits a brilliant yellow light.
The theoretical value of the ionisation potential, using the wave-length
limit of the principle series in the sodium spectrum, is 5 -13 volts. These
authors remark that this agreement affords another instance of the funda-
mental correctness of the theory of Bohr (vide. Chap. V.).

On referring to the results obtained by Hughes and Dixon, it will
be observed that no general relation exists between the magnitude of
the potential and the chemical nature of the substance or its stability.
This is disappointing. It raises the question of what the actual signifi-
cance of ionisation from the chemical standpoint really is. This point is
further emphasised by certain results obtained by Richardson and Baz-
zoni (Phil. Mag., 34, 285 (1917)), which are briefly as follows : —

According to these authors, we may expect a limiting frequency in
the spectrum of a substance, i.e. the spectrum comes to an abrupt end
at a certain frequency or wave-length in the extreme ultra-violet region.
Richardson and Bazzoni have determined this limit in the case of a few
gases by allowing the radiation from the gas to strike a metallic target,
thereby causing the emission of electrons whose velocities depend upon
the frequency of the light. By employing a magnetic field it is pos-
sible to calculate the velocity of the fastest electrons, and hence by using
the expression \mv* = eV = hv it is possible to calculate the limiting
frequency v emitted by the gas. In this way it was found that the limit
of the helium spectrum occurs at 42 to 47/x/x. The hydrogen spectrum
extends down to a limit somewhere between 83 and 9 5 /A/* (probably near
QO/X/X,), whilst in the case of mercury vapour the limit lies between 100
and 120/Aju,. These limiting values correspond to certain ionisation
potentials which can be calculated from the above relation. These
potentials are decidedly higher than those found experimentally (e.g. the
measurements of Hughes just cited), but these higher values appear to
agree fairly well with the ionisation potentials calculated by Bohr (cf.
Chap. V.). Thus Bohr calculated the ionisation potential of hydrogen
to be 13 volts, whilst the potential corresponding to the above limiting
frequency is 13-5 volts, a result decidedly higher than that found by
direct experiment. According to Richardson the ionisation potentials

THE PHOTO-ELECTRIC EFFECT 129

thus calculated bear no obvious relation to those obtained by Hughes
and others.1

Let us return for a moment to the expression hv§ = W. To test
this expression we may take the case of oxygen. Hughes has found
that the ionisation of air sets in at a wave-length i35/x,/x. This is like-
wise the position of the absorption band of oxygen in the ultra-violet.
We may conclude, therefore, that the effect in air is due to the ionisa-
tion of oxygen, and hence we should be able to calculate the ionisation
potential of oxygen from the expression V = hvje. The value of V thus
obtained is 9*2 volts which is identical with the observed value.

In dealing with collisions between electrons and molecules or atoms,
it is necessary to distinguish between two kinds, namely, elastic collisions
and inelastic collisions. In the first, the time of contact or encounter
is very short compared with the second. No "chemical" effects are
produced as a result of such collisions, the particles separating in the
same chemical state and with the same momentum (though not, of
course, with the same direction) as they possessed prior to the encounter.
These are the kind of collisions postulated in kinetic molecular theory
to account for the gas laws, for example. In the second kind of colli-
sion the time of encounter is longer, the total momentum after collision
is less than it was before, and there is opportunity for transfer of energy
between the two particles, which may result in observable chemical or
physical change. This is consequently the more important kind from
the standpoint of atomic interaction, ionisation, and resonance, etc.
Tate and Foote (Proc. Nat. Acad. Set., 4, 9 (1918)) point out that "no
considerable transfer of energy from the light electron to the relatively
heavy gas can take place except when the time of encounter between
electron and atom bears some simple relation to the characteristic period
of one of the vibrational degrees of freedom in the atom ".

" Two types of /^elastic encounter between electrons and gas atoms
have been observed. One of these results in the emission of a radiation
of a single frequency, without ionisation of the gas, whilst the other ion-
ises the gas and causes it to emit a composite spectrum of radiations.
The potential giving the first type of encounter may be termed a reson-
ance potential, that giving the second type an ionisation potential. The
line which defines the shortest wave limit or " convergence frequency "
in the spectrum which is produced when the ionisation potential is ap-
plied, occurs much further in the ultra-violet than does the single line
produced by the resonance potential. Naturally the single line charac-
teristic of resonance also appears when the applied voltage causes
ionisation. The following table is that given by Tate and Foote, and
contains the values of resonance and ionisation potentials respectively.
The calculated values of V in each case are obtained by employing the
quantum expression eV = hv. Thus the conditions are chosen experi-
mentally under which the gas emits only the single line. Its frequency

1 These limiting wave-lengths appear, however, to lie fairly close to the d sper-
sional wave-lengths. Accurate values for the dispersional wave-lengths have been
obtained in a number of cases by Baly, Phil. Mag., 27, 632, 1914.

VOL. III. 9

130

A SYSTEM OF PHYSICAL CHEMISTRY

is determined and from the above expression the resonance potential is
calculated. Again the conditions are obtained under which the gas is
emitting a spectrum of lines, and from the shortest wave-length repre-
sented the ionisation potential is calculated by the aid of the formula.
It will be seen that the values thus calculated agree extremely well with
those observed. The observed values for the resonance and ionisation
potential of mercury vapour are an addition to the table, and are due
to Kunz (Phys.Rev., 11, 246, 1918).

RESONANCE AND IONISATION POTENTIALS.

Metal Vapour.

Resonance
Observed.

Potential.
Calculated.

A in /m/u..

Ionisation
Observed.

Potential.
Calculated.

A. in ^.,u.

Qadmium .

3-88 volts

3*79 volts

326

8 'Q2 volts

8-97 volts

I37-9

Zinc .

4-10 „

4'02 „

307-6

9'5o „

9'37 „

132

Potassium .

i"55 ».

1-65 „

768-5

4-10 „

4"33 M

285-6

Sodium

2'12 ,,

2-10 „

589'3

5-I3 M

5-13 M

241-3

Mercury .

4'9 „

10-4

It may be pointed out that the ionisation potential in the case of
the above monatomic metallic vapours is approximately 2-3 times the
resonance potential. No reason for this or any other relationship be-
tween the two has so far been suggested.

Whilst we are dealing with the question of /^elastic collisions be-
tween electrons and atoms or molecules, it may be remarked that
chemical effects produced, for example, in solutions are very probably
due to the same kind of collisions. Thus the catalytic effect of hydro-
gen and other ions is probably due to the inelastic nature of the collision
which permits of transfer of energy to the molecule catalysed. This
appears to be borne out by the fact that in such cases there is a tendency
for the formation of a compound between the two interacting individuals,
and presumably compound formation is simply an extreme case of in-
elastic collision.

To return to the case of metallic vapours. The observations made
in the case of mercury vapour are particularly interesting. From the
resonance potential 4-9 volts it is easily calculated with the help of the
quantum expression that the corresponding radiation has the wave-length
254/A/A. Experiment has shown that this is actually the wave-length of
the single line emitted by mercury vapour when the vapour is traversed
by electrons whose velocities correspond to a potential of 4*9 volts.
Further, there is a sharp band at the same position in the absorption
spectrum of non-luminous mercury vapour (cf. McLennan and Edwards,
Phil. Mag., 30, 695, 1915). Goucher (Phys. Rev., 8, 56l» *9l6) nas
found, as a matter of fact, three characteristic electron potentials in
mercury vapour, viz. 4-9, 10^25, and 19 volts respectively. The first
is evidently the resonance potential, the other two being apparently
ionisation potentials. McLennan (Proc. Roy. Soc., A, 91, 485> I9I5)

THE PHOTO-ELECTRIC EFFECT

had already suggested that more than one type of ionisation is to be ex-
pected. At the present time there is no general agreement as to the
actual significance of each of these types.

We now turn to the Selective Photo-Electric Effect.

Elster and Geitel in 1894 discovered that in certain cases the photo-
electric effect depended upon the plane of polarisation of the incident
light. The substance examined by
them was the liquid alloy of sodium
and potassium. It was found that
when light fell upon the surface at
an angle of 45°, a maximum photo-
electric current was produced when
the electric vector of the light was
vibrating in the plane of incidence.
(Compare, however, the conclusions
arrived at by Millikan, infra.)
Further, in this so-called selective

o

X = 300 400 500 fLfj.

FIG. ii. — The dotted line indicates the
normal effect ; the maximum, the
selective effect.

effect the number of electrons
emitted has a maximum value for
a particular frequency of the light.
This maximum value occurs at a
wave-length somewhat shorter than
the threshold wave-length X0 (for the normal effect) and is not to be
confused with X0 itself. For wave-lengths shorter than that correspond-
ing to the maximum emission the selective effect rapidly falls off. The
relative magnitude of the normal and selective effects are shown in the
above diagram (Fig. n).

The position of the maximum in the selective effect occurs at the
wave-lengths given in the following table (quoted from Hughes' Photo-
electricity) : —

Substance.

Rb .
K

Na .
Li .
Ba

Maximum Selective

Effect Occurs at

. 470-510 fifj.

440 •

340

280

280

The same effect is observed when the metals are in the colloidal
state in photo-electric cells. Thus the maximum sensitivity of such
cells occurs at the following wave-lengths : —

Rb ........... 480 /i/i

Na
Cs

320
550

No general agreement has yet been arrived at regarding the relation
of the selective to the normal effect. As indicating the obscure nature
of the selective effect, it may be mentioned that Wiedmann (Ber. d.
Deutsch. phys. GeselL, 18, 333 (1916)) has found that the presence of

132 A SYSTEM OF PHYSICAL CHEMISTRY

hydrogen gas is essential to the production of the selective effect in the
case of potassium. If the metal be repeatedly distilled in vacua the
selective effect is removed. It would be erroneous, however, to argue
from this that the selective effect is accidental, due to some impurity.
It is possible that in Wiedmann's case a temporary and unstable
hydride is formed which, on decomposing, leaves the metal atom in a
chemically activated state— differing as regards its electronic arrange-
ment from the normal atom. Such chemical activation has been
assumed in other connections. In the present connection all this is
speculative however. According to Millikan and Souder (Proc. Nat.
Acad. Set., 2, 19 (1916)) there is an "essential identity" between the
selective and photo-electric effects, evidenced by the following facts •
the energy of emission is in all cases given by \mvz = hv - p, where/
is the work necessary to separate an electron from the surface of the
metal ; the amount of emission is not dependent on the angle of inci-
dence or on the azimuth of polarisation, but solely on the coincidence
of the impressed frequency with the natural frequency ; as an active gas
acts progressively on a fresh surface the normal effect merges gradually
into the selective.

As already pointed out in Chap. III., in connection with the fourth
method of determining the characteristic infra-red frequency of a metal,
Lindemann has suggested the following expression for the frequency
"violet which corresponds to the position of the maximum of the selec-
tive photo-electric effect : —

''violet == ~

where n is the valency of the atom to which the electron belongs, m the
mass of an electron and r one half of the distance between two neigh-
bouring atoms. It will be observed that r3 is proportional to the atomic
volume V of the metal. Hence the greater the atomic volume the
smaller, ceteris paribus, the frequency. This is in agreement with the
idea that a large atomic volume means that the electron is less firmly held
and can therefore be set into movement by a quantum of relatively
small magnitude, i.e. at a longer wave-length, than is the case with a
substance of small atomic volume. The above expression can be put
in a form which is convenient to use by first determining the value of
the constant outside the square root by comparison with a single experi-
mental determination. Thus Lindemann writes —

"violet = 4*594 x Iol6A/y.°r ^-violet in^ = 65-3 J -.

The following -table indicates the degree of applicability of the
formula : —

THE PHOTO-ELECTRIC EFFECT

133

Element

Atomic Volume.

Calculated A.

Observed X.

Cs

70*6

550 Wt

550/Ufl

Rb

56-3

490

480

K

45-0

438

440

Na

23-5

317

320-340

Li

II'9

225

280

Ba

367

280

280

Sr

34'5

271

—

Ca

25'4

233

—

Mg

14

172

Haber (Verh. d. Deutsch. physik. Gesell., 13, 1117, 1911) has given
a very simple and interesting relation between the 'Vvioiet" and the
Einstein infra-red characteristic frequency of the atoms of the same
substance, which we may denote by " ^infra-red "• This relation is —

"violet
"infra-red

M

where M is the mass of the atom and m the mass of an electron. Haber
has shown that this agrees extremely well with observed values. Thus,
using Lindemann's formula just referred to to calculate vvioiet, one can
make use of the Haber relation to calculate vinfra-red and compare it with
the values given by Lindemann's melting-point formula (for calculating
"infra-red)- The results are as follows : —

15

_„

"infra-red x I0

Substance.

"violet XI° '

"infra-red x IO
Calculated by Haber.

Calculated by Linde-
mann's Melting-point
Formula.

Li

1-27

11-26

10-78

Na

0-947

4-62

4*38

K .

0-685

2'55

2*57

Rb

0-612

i '55

1-56

Cs.

0-546

1-108

I .

0-906

1-87

1-85

A discussion of the above relation of Haber will be found in a paper
by the writer (Trans. Chem. Soc., HI, 1086, 1917). The relation may
be applied to compounds, but in some cases there is considerable doubt
as to the significance of the term M. The relation is, in fact, semi-
empirical.

Whilst we are dealing with the above relation, which demonstrates that
there is a close connection between infra-red and ultra-violet character-
istic vibration frequencies, it may be pointed out that Baly has recently
discovered a relation between the position of the absorption band or
bands in the ultra-violet spectra of certain substances and the frequency
of the infra-red band exhibited by the same substances (compare Chap.

i34 A SYSTEM OF PHYSICAL CHEMISTRY

IV.). Baly has likewise shown the connection between absorption bands
and fluorescence and phosphorescence bands. A summary of the latter
work is given by Baly in the Astrophys.Journ., July, 1915. Fluorescence
is touched on briefly at the close of Chapter VII. For further informa-
tion regarding the photo-electric effect, the reader is referred to Allen's
Photo-electricity or to Hughes' Photo-electricity.

PHOTOCHEMICAL REACTIONS. EINSTEIN'S LAW OF THE PHOTO-
CHEMICAL EQUIVALENT.

In Chap. XIV. of Vol. II. we have already discussed photochemical
reactions at some length. Our present purpose is to indicate briefly
the mode of application of the quantum theory to photochemical
change.

At the present time there appears to be a somewhat general tendency
to regard photochemical reactions as primarily due to a photo-electric
effect.1

As the photo-electric effect is itself independent of temperature, the
whole process is only slightly affected by temperature, i.e. photochemi-
cal reaction velocity constants have very small temperature coefficients.

The most important application of the quantum theory to photo-
chemical processes is that of Einstein as expressed in his law of the
photochemical equivalent.

(A. Einstein, Ann. d. Physik, [4] 37, 832, 1912.)

According to Einstein, a photochemical reaction takes place owing
to the absorption of radiation in terms of quanta, each single molecule of
a photo-sensitive substance requiring just one quantum hv§ (of the requisite
frequency VQ which the substance itself can absorb) in order that it may be
decomposed. So far as experiment has gone, the law has been approxi-
mately verified. Bodenstein (Zeitsch. Physik. Chem., 85, 329, 1913)
gives a table of several reactions from which it will be seen that if not
a single quantum, at least a very small number, 2 to 5, are required per
molecule of the substance photochemically decomposed. Such dis-
crepancy as does exist is probably to be attributed to the fact that the
experimental conditions did not correspond closely enough to those
postulated by Einstein, the principal one of which is that there shall be
the same intensity or density of radiation throughout the system. If the
system absorbs light very strongly, and is only illuminated in one direc-
tion, this condition will not be fulfilled, and instead of reaching a true
equilibrium governed, as Einstein supposes, by an expression analogous

1 In this connection, compare Landsberg (Journ. Russ. Phys. Chem. Soc.t 47, 908
(1915)) ; Nazarov, ibid., p. 943 ; (Chem. Soc. Abstracts, 108, ii., 718, 754 (1915)).
Volmer (Zeitschr. Elektrochem., 21, 113 (1915)) concludes that the close relation
between photo-electric conductivity and photochemical sensitiveness " confirms the
Luther theory that a loosening of electrons is the immediate cause of photochemical
processes ".

LAW OF THE PHOTOCHEMICAL EQUIVALENT 135

to the ordinary mass action law, we shall meet with photochemical
stationary states instead, such as those realised in the anthracene -di-
anthracene reaction.

Einstein's general argument is as follows. Consider a gas mixture
containing three different species, whose molecular weights are mlt m^ and
ms, and suppose there are «i, n2, and ns gram -molecules of each of these
present, the reaction considered is one in which i molecule of ml
decomposes photochemically into i molecule each of m% and m%, an
equilibrium point finally being reached. The first assumption made is
that the decomposition of m^ proceeds (owing to the absorption of
radiation) independently of the presence of the other species. The
second assumption is that the probability that an m^ molecule decom-
poses in a given time is proportional to the density p of a given mono-
chromatic radiation to which the system is exposed. From these two
assumptions it follows that the number Z of m1 molecules decom-
posing per unit time can be written —

Z

where A is a proportionality factor which depends only upon the tem-
perature T of the gas mixture.

As regards the recombination process, Einstein assumes it to be an
ordinary bimolecular reaction (which emits 1 the radiation first absorbed
by the ml molecules). If Z: denotes the number of ml molecules thus
re-formed per unit of time, we can write —

ZL = A'.V.-2.^

where V is the volume of the system ( ^ and ^ denoting the concen-
trations of each of these species). A' likewise only depends upon the
temperature T of the gas, not upon p (assumption 3), and if the tem-
perature be kept constant, A and A' are constant. When thermody-
namic equilibrium is reached, i.e. equilibrium between the matter
involved and the radiation itself, we can equate the two quantities,
supposing Z and Zl to refer to this case. That is, we obtain the
relation —

V ' V _ A_

^\_ A'P'
V '

Einstein now proceeds to consider a small virtual change of the

1 The object of making this assumption is to maintain the radiation equilibrium.
A more detailed examination of the mechanism of chemical change, discussed in
the concluding section of the present chapter and also in Chap. VII., leads to the
conclusion that the act of decomposition or of union involves first of all an absorp-
tion of energy followed by an evolution. The radiation equilibrium is, nevertheless,
maintained when mass action equilibrium has been reached owing to the equality of
the absorption and evolution effects in the system as a whole.

i36

A SYSTEM OF PHYSICAL CHEMISTRY

system which is accompanied by a small change in the entropy of the
matter and in the entropy of the radiation, finally obtaining as a
criterion of the above equilibrium state, the relation —

where a stands for a complex expression, involving the temperature T
of the gas (which is independent of the temperature Ts of the radiation),
e is the quantity of energy absorbed per molecule of substance m^ the
energy being in the form of monochromatic radiation v0, the density of
the radiation being p, and N the number of molecules in i gram-
molecule. " Since Ts and p are independent of the gas temperature T,

the magnitudes — — - and e must also be independent of T. Since these
A

quantities are also independent of Ts, we arrive at the same relation
between p and Ts as is expressed in Wien's radiation formula." Since
Wien's formula only holds for the short wave region, Einstein's relation
in the first instance is likewise restricted to this region. "If we write
Wien's radiation formula after introducing the Planck constants h and
k, we obtain —

and on comparing this with the previous equation, we see that —

and

A 'a

The most important consequence of the above is that e = hv^
which shows that one gas molecule which decomposes under radiation
of frequency v0 absorbs in its decomposition just hv§ of energy, i.e. i
quantum, as a mean value.

Bodenstein (foe. tit.) has collected the results of a number of in-
vestigations carried out by various authors, with the object of showing
how far the law of the photochemical equivalent is borne out by ex-
periment. That it is so at least approximately is shown by the follow-
ing table : —

Reaction Studied.

Number of hv Required to
" Decompose " one Molecule.

302 = 203 ....
2NH3 = N2 + 3H2 .
2 anthracene — > i dianthracene
S\ = Sju,
Oxidation of quinine by chromic \
acid J
2O3 = 3O2 (effected by means of\
chlorine) . . . ./
Nitro-benzaldehyde into nitro-)
benzoic acid . . . ./

i for 2O3 formed
4
0-7 to 3

4 to 5
i'5
0-8 to 1-7

9

LAW OF THE PHOTOCHEMICAL EQUIVALENT 137

Bodenstein gives other instances in which the law appears to break
down badly, one quantum decomposing many molecules. Even in the
cases quoted the results are subject to considerable uncertainty. One
of the principal reasons for this is, in the writer's opinion, the fact that
the wave-length of the light employed did not correspond sufficiently
closely to the head of the absorption band of the substance decomposing.
Einstein assumes that the decomposition is brought about by a single
definite frequency. As is well known, absorption bands exhibit width,
especially when the concentration of the substance is fairly large, and,
further, experiment goes to show that any wave-length inside the band
is capable of bringing about the photochemical effect. It is not to be
expected, however, that all wave-lengths inside a band will be equally
efficient. To test the Einstein expression in the most unequivocal
manner it would appear to be necessary to select the wave-length which
corresponds to the head of the band. In the case of other wave-lengths,
still inside the band, it is probable that a considerable fraction of the
radiation absorbed is not converted into chemical change, but is simply
degraded into quanta of smaller size, i.e. degradation into heat. This
indeed appears to be the case in a number of experiments of Warburg
(Sttzungsber. kon. preuss. Akad., 1912-15) on the ozonisation process,
in which the wave-length employed (209^) does not correspond to the
position of maximum absorption of oxygen. The fact that an absorp-
tion band exhibits width, and further, the fact that the width increases
with the concentration of the material is probably due to collisions
which set up forced vibrations inside the molecule thereby disturbing
the normal electronic period or frequency which an isolated molecule
would possess.

In the case of photochemical reactions in solution, effects due to
the solvent may be anticipated. This important problem has been
dealt with by Baly (Physikal Zeitsch., 14, 893 (1913)) in the following
manner : —

According to Baly, the chemical reactivity of an atom or molecule
is due to the existence of atomic electro-magnetic fields of force, which
differ from atom to atom in the density and location of the lines of
force. When an atomic or molecular species is activated in the chemical
sense, this is due to an " opening up of the atomic fields of force "
which are thus capable of interacting with the corresponding fields of
other atoms, thereby producing chemical change. This " opening up "
can be brought about essentially by two agencies, (i) by the absorption
of light, and (2) by the solvent (which presumably acts in an analogous
manner in virtue of its electro-magnetic properties). Different solvents
are capable of "opening " the field of a molecule to different extents,
these different extents or stages being manifested by the different posi-
tions of the absorption band or bands possessed by the dissolved sub-
stance in various solvents. On this basis we would expect that a
substance dissolved in a given solvent, being already partially activated,
would require less energy, say of the radiational type, to complete the
activation, than the same substance would require in the absence of the

138 A SYSTEM OF PHYSICAL CHEMISTRY

solvent. Baly's argument is, therefore, 'that in the case of a dissolved
substance reacting photochemically, less energy is required per mole-
cule than is required for the same substance in the gaseous state ; and
hence Einstein's law does not apply quantitatively in such a case. In
other words, one quantum can apparently activate numerous molecules.
Baly suggests that this affords an explanation of the results obtained by
Henri and,Wiirmser (Journ. de Physique, [5], 3, 305 (1913)) in connec-
tion with the decomposition of hydrogen peroxide and of acetone in
aqueous solution. In both of these cases the amount of energy actually
required is considerably less than that demanded by Einstein's law, in
fact, only i to o'i per cent, of the "theoretical" value. As Henri and
Wiirmser point out, the light in these cases acts as a catalyst, accelerat-
ing but not initiating the reaction. Baly's view, which gives the pos-
sible mechanism of such photo -catalysis, is further borne out by the
statement of Henri and Wiirmser that the above reactions will occur in
the dark, i.e. the solvent itself is evidently capable of bringing them
about. On the other hand, it is precisely in those cases in which sol-
vent is absent that we find the closest approximation to Einstein's law.
At the present time we do not possess as satisfactory experimental
evidence for Einstein's law as we could wish, but the law may be re-
garded as substantiated at least approximately.

Einstein's law depends upon Wien's radiation expression. It is
therefore restricted to regions in which the wave-length is not very long.
It will be shown in the next section that it is justifiable to regard
Einstein's law as applicable down to the short infra-red region, i.e. the
region which is responsible for so-called thermal reactions at ordinary
temperatures.

THERMAL REACTIONS. REACTION VELOCITY FROM THE STANDPOINT
OF THE QUANTUM THEORY.

By the term thermal reactions is meant reactions of the " ordinary "
kind which proceed without being exposed to any definite external
source of short-wave radiation such as we meet with in the so-called
photochemical reactions. The fundamental difference between the
two kinds of reaction is, as we shall see, the fact that in photochemical
reactions, as ordinarily carried out, the temperature of the active
radiation is very much higher than the temperature of the material
system affected, whilst in the case of thermal reactions, the temperature
of the effective radiation is identical with that of the material system
itself.

Photochemical phenomena have demonstrated the fact that
chemical change can be brought about as a result of absorption of
radiation in the short-wave region. We are led to generalise this and
to seek in radiation of longer wave-length the ultimate cause of ordin-
ary or thermal reactions as well. It is obvious that every material
system, in virtue of its temperature, contains radiation at the same

THERMAL REACTIONS 139

tempearture. Under ordinary conditions, i.e. at ordinary temperatures,
this radiation is mainly of the infra-red type. It is well known that
electro- magnetic properties of substances, e.g. the dielectric constant,
are intimately connected with the radiation density in the material, and
further, such properties have a marked effect upon the chemical re-
activity of molecules, e.g. the electrolytic dissociating power of solvents
which possess high dielectric constants. It will be readily accepted,
therefore, that infra-red radiation is the source of the energy which is
required to bring about ordinary or thermal reactions. The problem
before us is to discover how this concept may be applied in as quanti-
tative a manner as possible. An attempt in this direction, made by
the writer (inter alia, Trans. Chem. Soc., 109, 796 (1916)), will be con-
sidered briefly here.

For our present purpose, the most significant phenomenon associ-
ated with thermal reactions is the influence of temperature upon the
reaction velocity, the velocity constant increasing three or four fold for
a rise of 10° in the neighbourhood of room temperature. It has long
been known that this very great effect cannot be accounted for simply
on the basis of an increase in the kinetic energy of translation of the
molecule as a whole, for in the most favourable case, this increase could
not account for more than i or 2 per cent, of the observed effect, and
it is doubtful whether the change so produced would be even in the
right direction. It is evident, therefore, that the effect of temperature
upon the velocity constant has to do not with the kinetic energy of
translation but with the internal energy of the molecule.1

By treating the problem of reaction velocity from the point of view
of statistical mechanics it is possible to correlate the velocity constant
with the temperature, as has been done by Marcelin, and later in a
more exact manner by J. Rice (Brit. Ass. Rep.^ 1915, p. 397). The
relation between the observed velocity constant k and the absolute
temperature T for a simple type of gaseous reaction is shown by Rice
to be —

</log£A/T = E/RT2.

This expression is, as a matter of fact, a slightly simplified form, a
small term having been neglected. In deducing this expression it has
been assumed that a molecule reacts when its internal energy has been
raised to a certain critical value which is in general large compared with

1 From measurements of specific heat it is known that, on the average, the in-
crease in internal energy per degree rise in temperature is not large. In the simplest
cases the increase in internal energy varies directly as the absolute temperature,
and hence a rise of 10° would cause an increase of approximately 3 per cent, in the
average internal energy at ordinary temperatures. The influence of temperature on
velocity cannot be ascribed therefore to the average increase in internal energy.
When heat is added to a body, however, the energy does not distribute itself evenly
among the molecules ; instead, some receive no energy, others a moderate amount,
and a very few receive excessively great amounts of energy. It would seem there-
fore that the influence of temperature on reaction-speed is to be sought in the fact
that a very small number of molecules receive an abnormally large quantity of in-
ternal energy.

i4o A SYSTEM OF PHYSICAL CHEMISTRY

the average internal energy of the molecule. This critical limit is
known as the critical energy of the molecule. The term E denotes
the difference between the critical energy and the average or mean in-
ternal energy reckoned per gram-molecule. E may therefore be called
the critical increment ; E denotes the amount of energy which has to
be added to make an average [gram] -molecule react. It is at this point
that the radiation hypothesis of chemical reactivity enters, the assump-
tion being that the quantity E is contributed by the absorption of infra-
red radiation of a given frequency which the substance is capable of
absorbing. It is assumed that this energy is communicated in terms of
quanta, the same assumption as is made by Einstein in obtaining his
expression for the photochemical equivalent.

It will be observed that the Marcelin-Rice equation is identical in
form with the empirical equation of Arrhenius, already discussed in
Vol. I., which is known to be in good agreement with experiment. A
considerable advance has been made, however, in that the empirical
term A which occurs in Arrhenius' equation is now replaced by the
term E to which a definite physical significance can be attached. In
the case of polymolecular reactions the active molecular state actually
exists, as is shown by the experiments of Baly and F. O. Rice (Trans.
Chem. Soc., 101, 147 5, 1912); the rate of observed chemical reaction
depends then upon the frequency of collision bet ween activated molecules.

In the case of unimolecular change molecules in the active state do
not exist, for it is assumed that when a molecule becomes active it im-
mediately decomposes and becomes transformed into resultant. If the
reaction be reversible it follows, on the statistical basis of the expression,
that the activated resultant has exactly the same critical energy as the
activated reactant. This does not mean that the E term is the same
for both reactant and resultant, E being the difference between the critical
and mean internal energies.1

Having indicated in a general way the rdle which radiation plays in
regard to the activation of a molecule, it is now necessary to show how
an expression analogous to that of Marcelin and J. Rice may be obtained
on the basis of Planck's radiation theory.

In order that any of Planck's relations may be applied it is essential
that temperature radiation alone be considered, and further, that the
radiation and the matter composing the system be in temperature
equilibrium. On Planck's view, the interchange between matter and
radiation is effected through the medium of oscillators, situated on the
molecules of the substances composing the system. In the case of
thermal reactions occurring at constant temperature in a thermostat
(which defines the temperature of the system both with regard to the

1 Granted that the critical increment is due to radiation, and further assuming
that a single quantum is capable of accounting for the E of a single molecule, it
can be easily shown from a knowledge of the magnitude of the temperature co-
efficient of a reaction, that the effective radiation must belong to the short infra-red
region, i.e. of the order i to 3^4, in the case of reactions which occur with sensible
velocity at ordinary temperatures.

THERMAL REACTIONS 141

radiation and to the material), Planck's theory may be applied with
some degree of certainty. In this connection the following statement
made by Planck (Theory of Heat Radiation, Eng. ed., p. 7) is of im-
portance : "A special case of temperature radiation is the case of the
chemical nature of the emitting substance being invariable. Neverthe-
less, it is possible, according to what has been said, to have temperature
radiation while chemical changes are taking place, provided the chemical
condition is completely determined by the temperature."

Conditions for Reactivity on the Radiation Hypothesis.

According to the law of mass action the reactivity of a substance is
measured by its active mass. Since the time of Guldberg and Waage
it has been considered sufficient to regard the concentration of the. sub-
stance as an accurate measure of the active mass, and this has been
borne out by an accumulation of experimental results, which serve to
demonstrate the general truth of the hypothesis. At the same time
numerous facts are known which indicate that concentration per se is
not the only factor determining reactivity. To attempt to extend the
principle of mass action so as to allow for environment, it is necessary to •
take into account the energy exchanges which accompany, or rather
precede, molecular change, for as the fact of & finite velocity demon-
strates, all the molecules of a system are not at any one moment in the
same reactive state. From what has already been said regarding the
absorption of energy by an average molecule to bring its energy content
to the critical value, it will be evident that the amount and nature of
the radiation present in the system must play a definite part in the
reactivity of a given species. From this point of view, therefore, the
active mass of a species is not simply proportional to its concentration,
but is simultaneously dependent on the radiational environment, the
observed reactivity being due to the product of these two effects. The
active mass is, in the case of unimolecular decomposition, identical
with the number of molecules brought per second into the active state
by means of the radiation present in the system. The fundamental
assumption is that the radiational environment can be identified with
the radiation density of the absorbable type present throughout the
system. That is, the active mass of a species, spontaneously decom-
posing, is proportional to its concentration, multiplied by the radiation
density of the type which the substance can absorb, the radiation being
present in virtue of the temperature of the system. The radiation
density, uv,1 is the amount of radiation made up of quanta correspond-
ing with a certain frequency v, which is present in unit volume of the
system, when the radiation and the matter are in thermal equilibrium.
If the radiation density is too small the substance will not react, how-
soever great its concentration may be. On this view the function of the

1 As already pointed out uv has not the dimensions of energy density ; the
actual expression for energy density is uvdv. The term uv may be called the radia-
tion density factor.

1 42 A SYSTEM OF PHYSICAL CHEMISTRY

solvent acting as a positive catalyst is to increase the radiation density
of the type absorbable by the reactant, whereby a greater number
of the molecules of the reactant attain the critical value for their in-
ternal energy per second, with the result that the reaction-velocity is
increased.

Planck's expression for the radiation density factor may be written
in the following form : —

uv = constant x

where h is Planck's constant, k the gas constant per molecule, and T
the absolute temperature. For the short infra-red region the above
expression reduces to the following one,1 which is identical with the
expression of Wien : —

uv = constant e - h"lkr.

This expression may be regarded as holding for a gaseous system. To
simplify the treatment still further we shall assume that the reaction
considered is unimolecular, and that a single frequency is absorbed by
the substance in the process of activation. For this case the rate of
reaction can be written in the following way, in which a denotes as
usual the initial concentration of the substance and x is the amount
transformed after a time t: dx\dt is proportional to (a — x)uv, or

dx\dt = constant (a - x)e~hvlkr . . . (i)
On integrating this expression we obtain—

- log - - = constant . e - /ZF/*T . . . (2)

T Ct ~~ OC

At constant temperature this expression becomes identical with the
ordinary mass action expression. Further - log - - = kobs, where

&obs is the experimentally determined velocity constant. Hence —

kobs = constant x e~hvlkr . . . (3)

The " constant " which occurs here is practically independent of tem-
perature, especially for a gaseous system. We can find the effect of
temperature upon the observed velocity constant by simply differentiat-
ing the above expression with respect to T. The resulting expression
is —

</log,W<*T = ^/^T2 .... (4)

But kt the gas constant per molecule, can be written as R/N where R
is the gas constant per gram-molecule and N is the number of mole-
cules in one gram -molecule. Hence —

d log faJJT = N/&./RT2 . . . (5)

1 C/. W. C. M. Lewis, Brit. Assoc. Report, p. 394, 1915.

THERMAL REACTIONS 143

The term N/$i/ is a constant, characteristic of a given substance, and
independent of the temperature. It may therefore be denoted by the
symbol A, whence we obtain —

d log kobJcK = A/RT2 . . (6)

This expression is identical in form with the empirical relation of
Arrhenius which is known to be in agreement with experiment for a
great variety of reactions. The fact that this relation can be deduced
on the basis of the radiation hypothesis of chemical reactivity con-
stitutes strong evidence in favour of the hypothesis itself.

Let us now compare equation (5) with that obtained by Marcelin
and J. Rice, namely —

d log kldT = E/RT2.
On equating like terms it follows that —

E/N = hv.

But E/N represents the amount of energy which has to be added to
a single molecule to make it react, and by the above relation this is just
one quantum of the absorbable radiation. This is simply a statement
of the Einstein law of the photochemical equivalent, which is now shown
to be applicable to ordinary or thermal reactions caused by radiation of
the short infra-red type. That this result is to be expected follows from
the fact that for radiation of this type Planck's expression reduces to
that of Wien, upon which Einstein founded his law.

The radiation hypothesis has been extended to solute catalysis. It
is not proposed to enter into this question in the present instance ex-
cept to point out that the following general conclusion is arrived at : the
presence of a positive catalyst diminishes the temperature coefficient,
i.e. decreases E, whilst a negative catalyst increases the temperature
coefficient.

CHAPTER VII.

(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.

NERNST'S HEAT THEOREM AND THE QUANTUM THEORY.

WE have already seen (Chap. III.) that the quantum theory applied
to atomic heat ot solids leads us to expect that at very low temperatures
the atomic heat will become practically zero. If we think of two solids
A and B we can state that at low temperatures —

= o
or *T(UA - UB)A/T = o.

If the solid substances A and B are the reactant and resultant respec-
tively of a chemical reaction, the expression (UA - UB) represents the
change in internal energy, U accompanying the reaction. Hence the
quantum theory leads to the conclusion that in a chemical reaction
between solids at very low temperatures —

<fUATT = o.

This, as far as it goes, is in complete agreement with Nernst's heat
theorem which states (cf. Chap. XIII., Vol. II.) that —

limit, T. :r:,:ri°

To show that the relation dKjdT = o is likewise in agreement with
the Planck-Einstein hypothesis regarding the energy distribution in
solids, we may proceed as follows : As the molecules or atoms of a
solid at very low temperatures do not possess any sensible kinetic energy
of vibration, their mutual distance apart will change very little with
temperature over a limited range in this region. Consequently, the
mutual potential energy of the atoms or molecules will remain practically
unchanged, and hence their free or available energy, which depends
upon the potential energy, will also remain unchanged. That is, in the
limit, when T = o, dA/di? = o. The same conclusion may be arrived
at in a slightly different way. We know that the intensity of radiation,
and therefore the amplitude of the resonators (i.e. the atoms), increases
with the temperature according to a power law considerably greater than
the first. Hence, as temperature falls the amplitude will decrease much
more rapidly, finally becoming exceedingly small. Again, the potential

i44

NERNST S HEAT THEOREM 145

energy, and therefore the free energy, may be regarded as sensibly
independent of the temperature when the temperature is low. It cannot
be said, of course, that the validity of the Planck-Einstein theory is a
proof of the validity of Nernst's heat theorem. They are, however, in
agreement. This point will perhaps be rendered clearer by the follow-
ing considerations.

We know that in the case of any process carried out reversibly, the
Gibbs-Helmholtz equation states —

A - U = T .

Further, A - U = Q where Q is heat absorbed in the process.

Hence Q/T = S = (dAfdT)v, where S is the change in entropy of the
system as a result of the process. Applying Nernst's theorem we con-
clude that at very low temperatures the change in entropy due to the
process is zero. In other words, Nernst's theorem is identical with the
statement that at absolute zero the entropy of all substances is the same.
Planck has already pointed out (cf. Chap. I.) that the quantum hypothesis
is equivalent to the assumption that the entropy of all substances at
absolute zero is zero also. This is really a special case of the conclusion
involved in Nernst's heat theorem. The theorem and the quantum
theory are therefore in agreement, but the heat theorem is independent
of the quantum hypothesis.1

Calculation of the Affinity of a Chemical Process from Thermal Data by
the Simultaneous Application of Nernsfs Heat Theorem and the
Quantum Theory of Atomic Heats.

Equations (3) and (4) of Chap. XIII. of Vol. II., which express the
Nernst heat theorem, contain certain coefficients /?, y, etc., which are
calculated from measurements of the molecular or atomic heats of the
reactants and resultants of the reaction considered. These terms are,
in fact, the temperature coefficients of the molecular heats of the react-
ing substances. In principle, the introduction of the quantum theory
(say in the form of the Nernst-Lindemann equation) is simply a matter
of substitution of this equation in the original equations of Nernst for
A and U. To indicate how this is carried out in practice it is necessary
to put the Nernst equations into a more general form than that ex-
pressed by the equations (3) and (4) referred to.

The change, U, in the internal energy of a system as a result of a
chemical reaction is, by definition, the difference of the internal energies
of reactants and resultants, reckoned in terms of the stoicheiometrical
quantities required for the reaction. For any single substance Z we can
write : d\Jz/dT = cz> where cx denotes the molecular heat of the sub-

pT

stance Z at constant volume. Hence Uz = constant + c^dT. Hence

*9

1 The relation between the heat theorem and the quantum theory is further
considered by O. Sackur (Annalen der Physik, 34, 455, 1911).
VOL. III. 10

146 A SYSTEM OF PHYSICAL CHEMISTRY

for the quantity U, which involves the algebraic sum of a number of such
terms, we can write —

U = U0

where U0 is the change in internal energy accompanying the reaction
at absolute zero, and n denotes the number of moles of reactants and
resultants (with proper sign) which take part in the reaction.

We have now to express the molecular or atomic heat c of each re-
acting substance in terms of the Nernst-Lindemann equation. Having
done this we find that the above integral may be evaluated thus —

f

I

Hence-

f RS + ST - (,)

The final term is an empirical correction, which has to be added to
allow for the fact that the experimental measurements of molecular or
atomic heats refer to constant pressure, whilst the quantities to which
the Nernst-Lindemann expression refer are molecular or atomic heats
at constant volume. (Cf. Magnus and Lindemann (Zeitsch. Elektro-
chem., 16, 369 (1910)) ; a more exact correction is given by Nernst and
Lindemann (Zeitsch. Elektrochem.^ 17, 817 (1911)).)

We have now to consider the free energy term A. On the basis of
the first and second laws of thermodynamics we have already seen that —

A

Further, Nernst's theorem leads to the conclusion that a = 0, and
therefore we can write —

ru
£ - a.

Again writing —

U = U0

and substituting this in the above integral, and also substituting the
Nernst-Lindemann expression, we obtain —

(2)

NERNSTS HEAT THEOREM

»47

Equations (i) and (2) express the Nernst heat theorem in terms of the
quantum theory.

[NOTE. — The mode of obtaining equations (i) and (2) is given in detail in Pol-
litzer's article, Die Berechnung chemischer Affinitdten nach dem Nernstchen Theorem,
Ahrens Sammlung, 17, 1912. To facilitate calculation Pollitzer has included as an
appendix a number of tables giving the Einstein and Nernst-Lindemann functions
for various values of &0 and i/.]

We now proceed to consider an actual case, namely, the affinity of
silver for iodine investigated by U. Fischer (Zeitsch. Elektrochem., 18,
283, 1912). The treatment of a number of difficulties met with is
instructive and is therefore given in some detail.

The first question taken up by Fischer is the heat of the reaction—

2Ag +' I2 = 2AgI

(solid) (solid) (solid)

The direct measurements of Thomsen and of Berthelot being unsatis-
factory, recourse was had to the Gibbs-Helmholtz equation —

E-U-Tg

as applied to the measurement of the e.m.f. of the cell —
Ag | KI solution, saturated with Agl | I2

at several temperatures.

The results were as follows : —

N
Employing — KI solution, Evolts = 0-6948 + 0-000305?

N

„ - „ EvoUs = 0-6932 + 0-000305/

o

N

— „ Evolts = 0-6916 -f 0-000305/

the temperature / being given in degrees centigrade.

These values for the e.m.f. require a correction, however, owing to
the fact that some of the iodine from the right-hand electrode dissolves
in KI, giving tri-iodide (and other polyiodides, the quantity of which
may, however, be neglected). This causes the concentration of the
iodine ion I' on the iodine side of the cell to be diminished, and the
concentration difference of I' on the two sides of the cell give rise to
an e.m.f. <?, where —

e = RT log, ^

C%

- being the ratio of the I' ion concentrations. With the help of the

^2

data of von Ohlendorff, and some experiments of his own on the solu-

10*

148

A SYSTEM Of PHYSICAL CHEMISTRY

bility of iodine in potassium iodide, Fischer succeeded in calculating the

N N N

value of e for the three cases considered, viz. — , — , and — KI respec-

23 10

tively. The following table contains the corrected value of the e.m.f.

7-p

and the temperature coefficient — : —

• KI Solution.

Temperature,

tf

E Corrected.

dE
rfr

?:.-.{

13-1
23'8
35*3

0*01708
0*01869
0*02154

0*6817
0*6833
0*6839

L 2*419

*-•;'•{

14-9
24-6
38-6

0*01719
0-01938
0*02327

0*6805
0*6813
0*6817

1*267

a- ••'.: -1

13-1
23*9
35'4

0-01658
0*01848
0*02118

0*6791
0*6805
0*6807

I 1*66

Employing these data in the Gibbs-Helmholtz equation, Fischer
found for 15° C. (288° abs.) :—

-KI solution, U

2

10

0-6821 x 23046 - 288 x 2*419

= 15,023 cals.
„ U = 0*6806 x 23046 - 288 x 1*267

= 15,315 cals.
„ U = 0-6791 x 23046 - 288 x 1*66

= 15,216 cals.

The mean value of U is 15,169 cals.

Besides this indirect method of measuring U, Fischer also employed
a calorimetric method, precipitation of insoluble salt being avoided by
the following very ingenious procedure due to von Wartenberg. As
solvent a 3N.KCN solution was employed, and in this was placed a
known weight of silver powder. The silver is not attacked by the KCN.
To the liquid was added a known weight of iodine, the following reac-
tion taking place : —

Ag + I + 2KCN = AgK(CN)2 + KI ' . . (i)

The heat effect of this reaction was measured. Then using a similar
solution of KCN, a known weight of silver iodide was added (no silver
powder being present in this case), the resulting reaction being—

Agl + 2KCN = AgK(CN)2 + KI . . . (2)

NERNST' S HEAT THEOREM

149

The difference of the heat effects of the reactions (i) and (2) gives
the heat effect desired, viz. —

The mean value thus obtained was U = 14,280 cals., which agrees
pretty closely with the result of the first method. Fischer made use of
still another calorimeter method, in which some of Thomsen's data on
other salts were employed, the results being less trustworthy. [Note
that this is a heat effect at constant volume, since the system does not
alter appreciably in volume.]

Now, if we consider the cell —

Ag | Agl | I2

(solid) (solid) (solid)

this represents a condensed system, each phase being a single substance
(no solution or vapour being present), and we can apply Nernst's heat
theorem directly in the form given in equations (i) and (2) above.

[It should be noted that the A or E of this cell is not quite the
same as the E or A' of the cell in which KI solution is present. The
heat effect in both cases is, however, the same, for, as we have seen,
the heat effect as calculated (say by the first method) is independent
of the KI concentration, being simply the heat effect of the reaction
Ag + I = Agl.]

The values of /V and of a are known in the case of the three sub-
stances Ag, 1, and Agl from the atomic and molecular heat measure-
ments of Nernst.

By making use of the values of @Qv and a obtained in this manner,
Fischer calculated U for a temperature T = 288, finding the value
J5>°79 calories, a quantity which agrees extremely well with the former
values which were obtained for U quite independently of the Nernst
heat theorem, or the Nernst- Lindemann atomic heat expression. It is
to be remembered that the U and A terms here refer to the reaction
between pure solid Ag and solid I2, giving solid Agl. A is therefore
the affinity which we set out to calculate. The following table con-
tains the values of A and U thus obtained : —

AFFINITY OF SILVER FOR IODINE AT VARIOUS TEMPERATURES.

Temperature,
Degrees Absolute,

U in Calories per Gram-
Atom of Silver.

A in Calories per Gram-
Atom of Silver.

O

15,166

15,166

2O

15,153

15,173

40

I5»I36

15,201

60

15,124

15,235

80

I5,«9

15,274

100

I5.II4

15,314

140

15,108

15,394

180

15,101

15,477

220

15,093

15,560

260

15,085

15,650

300

15,074

15,733

150 A SYSTEM OF PHYSICAL CHEMISTRY

The, accompanying figure (Fig. 12) shows diagrammatically the
relative variation of U and A. It will be observed that, in this reac-
tion, the affinity increases with increase in temperature, whilst the heat
effect or total energy change decreases.

U

100

300°T

FIG. 12.

A. Magnus (Zeitsch. Elektrochem., 16, 273, 1910) has carried out a
similar investigation of the reaction —

Pb + 2AgCl-»PbCl2 + 2Ag

in which the affinity was found to decrease as the temperature was
raised.

MASS ACTION EQUILIBRIUM IN SOLUTIONS FROM THE STANDPOINT OF
INFRA-RED RADIATION.

(Cf. F. Kriiger, Zeitsch. Elektrochem., 17, 453, 1911.)

The problem to be considered is the relation of the ionisation con-
stant of the solute to the dielectric constant of the solvent.

The radiation density in a medium depends upon the cube of the
refractive index n as well as upon other factors. Further, on the basis
of the classical electro-magnetic theory of light, the refractive index is
connected with the dielectric constant D by the relation n* = D.
Hence the radiation density is proportional to the dielectric constant
raised to the power 3/2. It follows that in media of high dielectric con-
stant, such as water, the radiation density is correspondingly high.
Thus, D for water is approximately 81, and Si1'5 = 730. Hence for
long waves, the radiation density in water at ordinary temperatures is
730 times the radiation density in a vacuum, where D = i. Kriiger
considers the case of a dissolved electrolyte, and assumes that only a
fraction of all the ions vibrate, but that those which do vibrate at all
possess the theoretical amount of vibrational energy required by classical

MASS A CTION EQ UILIBRIUM IN SOL UT1ONS 1 5 1

statistical mechanics, i.e. 3RT per "active" gram-ion. The high
dielectric constant, according to Kriiger, also sets a number of the
charged atoms in the undissociated molecules of the solute into vibra-
tion again with the amount of energy per " active " gram-atom * required
by statistical mechanics. The higher the radiation density of a medium,
that is, the higher its dielectric constant, the greater will be the number
of molecules of solute activated in this sense, and hence the greater the
degree of ionisation of the solute. This is suggested as the basis, in
terms of radiation, of the relation known as the Nernst-Thomson rule.

We now pass to the consideration of the mass action equilibrium
resulting from such ionisation. Kriiger carries out a thermodynamic
work process in which allowance is made for the difference in radiation
density arising from the fact that two different solvents or media are
employed.

Let us consider two media, dielectric constants Dx and D2," the media
being capable of mixing in all proportions. We might imagine a
cylinder containing one solvent placed above the other with a mixture
layer between the two. In each solvent let us suppose that there is
one gram-molecule of a binary electrolyte dissolved, the ionisation con-
stant of the electrolyte being Kj in solvent I. and K2 in solvent II. ;
Kj > K2. The following cyclic process is then carried out. In medium
I. one mole of the undissociated electrolyte ionises. The ions are
brought by the aid of semi-permeable membranes into solvent II. where
they unite, the undissociated molecule being returned to solvent I. The
osmotic work of transferring the material is as follows : let C0, C1} C2
be the equilibrium concentrations of the undissociated molecules and
the ions respectively of the solute in solvent I. ; C'0, C'i, C'2 are the cor-
responding quantities in solvent II. The osmotic work of the cycle is
then given by —

In this process a certain amount of radiation is absorbed when
ionisation occurs in solvent I., and the same amount is given out when
the ions unite in solvent II.2 We have to deal therefore with a trans-
port of radiation at the same time as we deal with a transport of material.
The radiation is transported from a region of higher radiation density
(solvent I.) to a region of smaller radiation density (solvent II.). The
work of transporting the radiation may be calculated in the following
manner : —

Let us consider a certain amount of radiation, i.e. the amount re-
quired to " activate " one gram-molecule of the electrolyte, and let us
suppose that this radiation is enclosed in a membrane impermeable to
radiation. The radiation exerts a pressure — " the pressure of light " —

1 In the present case the " critical increment " of the undissociates molecule is
very small (cf. Chap. VI., section dealing with Thermal Reactions).

2 Kriiger assumes here that the heat of the reaction is the same in both solvents.
This, in general, will not be the case.

152 AS YSTEM OF PHYSICAL CHEMISTR Y

upon the membrane tending to expand it. This pressure is proportional
to the density of the radiation. It is known that the pressure TT of the
radiation is related to the volume <f> by an expression which is quite
analogous to Boyle's law for gasesv, viz. TT<£ = k.

[NOTE. — There is a serious difficulty here, as the expression assumes
the temperature of the radiation is constant. If it is isolated its tem-
perature cannot remain constant on expansion or compression.]

If we accept this expression it follows that the maximum work gained
in expanding the radiation from TTJ to 7r2, corresponding to the radiation
densities u^ and u2, is k log irj/irg. Further, on the electro-magnetic
theory of light, TT = «/3, so that the radiation work is k log uju^. The
density terms are proportional to the cube of the refractive indices or to
the i *5 power of the respective dielectric constants D! and D2. Hence
the radiation work can be written as —

We have now to determine the constant k. " Let us assume that each
vibrating ion [charged atom] in the undissociated molecule possesses
three degrees of freedom, that is, each ion possesses 3/2 RT of kinetic
energy and a like amount of potential energy, making in all 3RT per
ion, or 6RT for the two ions in the molecule reckoned per gram-mole-
cule — each gram-molecule when it dissociates requires on the average
an amount of energy 6RT which is- drawn from the radiation." Hence —

«<£ = 6RT,
and sinceiTr = 0/3, it follows that —

7T<f> = 2RT.

Substituting this in the expression for the work done in expanding
the radiation, we obtain —

A = 2RT log (D,3/'/D23/*) = RT log D^/D/.

The ionisation equilibrium is denned by the equality of the osmotic
and radiation work terms.1

We obtain therefore —

or Ki;'K2 = D*IDj>.

That is, on the basis of the assumptions made by Kriiger, we would
expect the ionisation constant to vary as the cube of the dielectric con-
stant of the system. In order to test this conclusion it is convenient to
put it into another form. It yl and y2 are the degrees of ionisation at
the dilutions VT and V2 respectively in the two solvents, then —

K 7i2 (i - ya)V» _ D!*

1 Naturally if the solvent had been the same in both cases, and if the solutions
are dilute, the radiation work term would be zero, and we are left with the osmotic
work terms. The cycle would then reduce to the familiar one employed to deduce
the law of mass action, i.e. Kl would be identical with K2.

MASS ACTION EQUILIBRIUM IN SOLUTIONS 153

If we choose the concentrations, i.e. the dilutions, of the solute in the
two solvents in such a manner that the degrees of dissociation are the
same, the above relation becomes —

or

V2/V, = Df/Df
D y V = constant for a given value of y.

This relation had already been discovered as a purely empirical one
by Walden (Zeitsch. .phys. Chem., 54, 129 (1906)) as a result of measure-
ments carried out with tetra-ethylammonium iodide in a number of
solvents. The following are a few of Walden's results. In each case
the dilution term V denotes the dilution at which the degree of ionisa-
tion has the value assigned to it at the head of the table : —

r-o

'S3

Solvent.

Dielectric
Constant.

Dilution V
in Litres.

Di/v:

Water .... .

80

32

254

N itromethane

40

2OO

234

Furfurol , .

39

200

230

Acetonitrile .

36

320

245

Methyl sulfocyanide

36

270

233

„ alcohol . . .

32-5

400

240

Proprionitrile

27'2

800

253

Acetyl acetone
Acetic anhydride . .

26
17-9

1000

2000

260
226

Benzyl cyanide

I6'7

3200

246

Amsaldehyde . .

15 '5

3100

226

Benzaldehyde

r4'5

4100

232

7 = c

•91.

Water .....

80

no

38q

Furfurol

39

800

j"j
365

Nitromethane .....

40

800

371

Acetonitrile

36

IOOO

358

Methyl alcohol

32-5

2OOO

365

As will be seen, the relation is borne out approximately. .Kriiger
deals with the possible causes of the deviations. Of these the most
important is connected with the relation ri* — D, which is known to
hold only for very long waves. It is possible that the wave-lengths
chiefly concerned in ionisation, although belonging to the infra-red
region, do not conform to this relation. It is known that n varies with
the wave-length when the latter is short, i.e. the ultra-violet, visible, and
the short infra-red regions. Hence the values of D are to a consider-
able extent arbitrary, the values not being strictly comparable. Thus
the value obtained for the dielectric constant will depend very largely
on whether there exists an absorption band at the wave-length employed

154 A SYSTEM OF PHYSICAL CHEMISTRY

in the dielectric constant apparatus. In the case of water it is con-
sidered that the high value obtained is partly due to the fact that water-
possesses a band for the electro- magnetic waves usually employed.
Further, Kriiger's assumptions are possibly not correct in all cases.
Thus, it is very surprising that the results quoted should be as satis-
factory as they are, for the electrolyte is strong. Nevertheless the
deduction, so far as it goes, indicates that radiation plays a r61e in
ordinary chemical processes. Kriiger shows further how an empirical
relationship of Walden, according to which " saturated solutions of one
and the same electrolyte in various solvents possess the same degree of
ionisation," can be deduced on the radiation basis. For details regard-
ing this and other matters the original paper of Kriiger must be con-
sulted.

THE HEAT OF REACTION. HABER'S RELATION.

In Chap. VI. we have dealt briefly with the rate of reaction and the
velocity constant from the point of view of the quantum theory. In
the case of a monomolecular reaction in a dilute gaseous system the
variation of the velocity constant with the temperature is given by the
expression —

d log kjdl = N^/RT2

where v is the characteristic vibration frequency, or head of the absorp-
tion band, of the decomposing substance. If the reaction be a re-
versible one, the resultant being characterised by the frequency v', the
corresponding velocity constant being /£', we have —

d log k'fdT = NA//RT2.

d log klk'
Hence -JL-

But klk' = K, the equilibrium constant. Hence we obtain the relation —
d log K/dT = N*(v - v')/RT2.

At the same time the statistical mechanical expression of Marcelin
and Rice gives the relation —

d log K/tfT = (E - E')/RT2

where E is the critical increment of the reactant, E' that of the resultant.
On comparing these expressions with the van 't Hoff isochore, viz. —

d log K/rfT = - QV/RT2

where - Qw is the heat absorbed per stoicheiometric quantity of the
reactant transformed or decomposed, we obtain the relation —

- Qv (heat absorbed) = N>&(v - v) = E - E'

or, heat evolved in the reaction = critical quantum of the resultant
minus the critical quantum of the reactant.

HABEAS HEAT RELATION

155

In the case of reactions in which catalytic effects are absent we may
generalise the above expression, and write —

Heat evolved = 2N/&i/resuitants - 2N/&i/reactants.

This expression, deduced however by a totally different method, was
first given by Haber (Ber. Deutsch. phys. Ges., 13, m7» iQ11)- It
may therefore be called the Haber expression for the heat effect. The
above mode of deduction is that given by the writer (Trans. Chem. Soc.,

Ill, 457> I9I7)-

In the above formulation the heat of reaction at constant volume
has been expressed as the difference of the critical quanta or the dif-
ference of the critical increments of the resultants and the reactants.
The heat effect at constant volume is also thermodynamically defined as
the difference of the mean internal energies of the reactants and re-
sultants. To see that these two definitions are concordant we may
proceed in the following manner.

Let us consider the simplest type of reaction, viz. A l£ B. The
various energy terms involved in the process are represented in the ac-
companying diagram (Fig. 13). The ordinates denote internal energy,
the length ab corresponding to the mean values Uj of the internal
energy possessed by one gram-molecule of the substance A, and similarly
the length de represents the mean energy U2 of the substance B. Be-
fore one gram-molecule of A, possessing the average internal energy
Ux, can become reactive its internal energy must rise to the point c by
the addition of the critical increment E^ At this stage the gram-
molecule possesses the energy (Ul 4- EJ. It may now change into
one gram-molecule of B, with an evolution of energy denoted by the
dotted line ce. The gram-molecule now possesses the mean energy de, or
LJ2 characteristic of the substance B. In passing from c to e the energy
emitted is (Ul + Ea) - U2. In passing from the mean state b to

the mean state e the total energy
evolved is (Uj + Ex - U2) - Elf
or Ux - U2. The total energy
evolved in passing from e to b is
(U2 + E2 - Uj) - E2,orU2 - Uj.
This expresses the fact that if the
reaction is exothermic in one direc-
tion it necessarily endothermic in
the reverse direction.1

When a molecule is in the
critical state it is impossible to say
whether it belongs to the system A
or to the system B. That is, the
critical state is common to both A
and B. If Ec denotes the total
critical energy — not the critical in-
crement— Ec will have the same
value for the A and B molecules

A

I
I

E,

eX

Exothermic Process.
FIG. 13.

1 Note that on the above mechanism the system never passes along the line be.

156 A SYSTEM OF PHYSICAL CHEMISTRY

alike. This is indicated in the diagram by the lines ac and df being of
equal length. From the above considerations it follows that —

Ec = Ux + E! = U2 + E2.

Hence Uj - U2 = E2 - Ex = Q, = Nh(/ - v).

It will be seen that the above mechanism is somewhat more complicated
than that assumed by Einstein in deducing his law of the photochemical
equivalent. It agrees with Einstein's view, however, in that the energy
evolutions just balance the energy absorptions when both reactions are
taken into account, and therefore the radiation equilibrium is not
altered when mass action equilibrium has once been attained. This
assumes that the evolution from c to e is in the form of a single quantum
if the absorption ef takes place in the form of a single quantum ; and
similarly for the quantities represented by fb and be respectively.

In writing Qw as the difference between the critical quantum of the
resultant and reactant we have really assumed that Qv is a constant
independent of temperature. This, of course, cannot be exact, as it
would mean that d\J/dT = o, or that the sum of the molecular heats
of reactants is equal to those of the resultants. The above radiation
expression is therefore an approximation. In this connection it may
be mentioned that a recent investigation of equilibria in the gaseous
state has led Scheffer (Proc. Akad. Wetenschap Amsterdam, 19, 636, 1 9 1 7)
to the conclusion that the variation in Qj, with temperature is a negligible
quantity in general, i.e. " the experimental error in the determination
of K is always such as to render meaningless any attempt to allow for
the variation of Q with T ". This practical conclusion does not affect
the question, however, of the true variation of Q with T.

The integrated expression for K on the above radiation basis takes
the approximate form —

loge K = - — ^ " v' + constant

Jx 1

log. K ~ + constant

Jx 1

where v refers to resultants, v to reactants, and K is so defined that the
equilibrium concentration terms of the resultants appear in the numerator.
An interesting application of the above considerations suggests
itself in connection with the phenomenon of fluorescence or phos-
phorescence. In this phenomenon we are dealing with the absorption
of one type of radiation and the emission of another. The effect is
regarded as due to an emission of energy which accompanies the
return of an electron to its original position in the atom, the initial
•disturbance from this position having been brought about by absorption
of radiation. According to Stokes' law the frequency of the emitted
radiation is always less than that of the radiation absorbed. The pro-
cess is therefore analogous to an endothermic change, represented in
Fig. 13 by the absorption ef and the evolution fb. Stoke's law means
that the process is irreversible. On the other hand, Baly (loc. cit^] has

THE HEAT OF REACTION 157

made the very important discovery that the fluorescence (emission)
wave-length exhibited by a substance in one solvent is identical with the
absorption band exhibited by the same substance in a second solvent.
That is, under one set of conditions the substance is capable of ab-
sorbing the wave-length which it emits as fluorescence under other
conditions.

In the " second " solvent it is conceivable that the conditions
exist for a reversal of the process (i.e. possibly Stokes' law is not
necessarily true in general). No information is available on this point
at the present time.

APPENDIX I.

MAXWELL'S DISTRIBUTION LAW AND THE PRINCIPLE OF EQUIPARTITION

OF ENERGY.

BY JAMES RICE, M.A.

THE aim of mechanics is the description of motion. We seek to
specify the position of every part of a system of bodies at every instant.
The most direct way of doing this would be to express all the necessary
geometrical co-ordinates of the system as known functions of the time.
To this end the laws of motion are applied to the special features and
environment of each system, and a series of differential equations are
obtained which, among other quantities, involve the first and second
differential coefficients of each co-ordinate with respect to the time,
i.e. the velocities and accelerations. If the mathematician can solve
for us the particular differential equations arrived at, we have attained
our object for that system. Many special cases of considerable interest
have proved amenable to mathematical treatment, but, at present, no
solution for the general case exists.

In the phrase, "system of bodies," we must be definite as to the
meaning to be attached to the word " body ". In physics and chemistry,
a single body is a system, i.e. a collection of molecules, which are in
themselves discrete, if minute bodies. Indeed the molecule itself is a
system of atoms, and, on present views, the atom is a system of nuclei
and electrons. Even if we regard a body of fluid or solid material as
a system of molecules alone, without concerning ourselves about its
internal structure, the complexity of description involved in a complete
account of molecular motion is so great that it becomes necessary to
introduce the mathematical theory of probability. We are no longer
concerned with an exact solution of the dynamical problem, by which one
could predict with certainty the position and motion of each molecule
at a given instant ; instead we endeavour to find the law of distribution
of the co-ordinates and velocities of the molecules, so as to be able
to state, with but small possibility of error, that at a given instant such
and such a fraction of the molecules will occupy such and such a portion
of the space filled by the body, and have velocities lying between such
and such limits. It is this feature which characterises a problem as one
of statistical mechanics.

In dealing with a system of molecules, the co-ordinates referred to
above are naturally the Cartesian co-ordinates of the centres of each

158

APPENDIX I 159

molecule referred to a definite set of axes ; but it should be borne in mind
that in general dynamics the word " co-ordinate " is used to indicate
any geometrical quantity which serves to specify the position and con-
figuration of the parts of a system, the least number of such co-ordinates
which is necessary to specify the system completely being in fact its
number of "degrees of freedom". Thus, if we attempt to treat the
problem of a system of molecules which are themselves systems of
atoms, we should have to introduce further geometrical magnitudes to
those mentioned above, which would define the position of each atom
relative to the centre of its corresponding molecule.

As regards the velocities of the individual molecules, there are two
ways open to us to specify such a velocity, both as regards its magnitude
and its direction. We may give the three components of the velocity
parallel to the three axes of co-ordinates, or we may give the actual
speed combined with some convenient way of indicating its direction.
The following is a simple method of defining direction : —

Conceive of a sphere of unit radius. Every direction in space is
parallel to some radius of the sphere. Consequently a point P on the
surface of the sphere can be said to represent a direction in space, viz.
the direction parallel to OP where O is the centre of the sphere. Any
direction and its opposite are of course represented by diametrically
opposite points on the sphere. In statistical work we have to classify
directions into groups, and one convenient way of ' doing this is as
follows : Draw a series of plants perpendicular to a given diameter of
the unit sphere, and meeting the diameter in points which divide the
diameter into an exact number, say m> of equal parts. These planes
divide the surface into a series of zones, and it is well known that these
zones have equal areas, viz. 4Tr/m, 4-0- being in fact the entire area of
the sphere whose radius is unity. Still further subdivide the surface by
a number of planes, say n, each containing the diameter and each one
making, with its neighbouring planes, angles which are all equal to an
aliquot part of 2-n- radians, viz. 2ir/n radian. This new family of planes
cuts the surface of the sphere in great circles which subdivide each
zone in equal parts, each part being four-sided and equal in area to
4-ir/mn. (The parts round the ends of the diameter are three sided but
have the same area.) If we call this area w, then the number of such
elements of area is mn. or 47r/w. It will prove convenient presently to
classify directions into groups such that all the directions in one group
correspond to points in one superficial element of the representative
sphere. In the nature of things no finite limit can be assigned to the
number of directions in one group, no matter how small the elements of
area are made by closer and closer subdivision by the planes of " latitude
and longitude," but of course closer subdivision and an increase in the
number of groups involves closer and closer limits to the angular differ-
ences between any pair of directions in one group.

The problem in statistical mechanics which has received the greatest
attention concerns the statistics of a body of gas enclosed in a fixed
volume and in a steady state with regard to such properties as tempera-

160 A SYSTEM Of PHYSICAL CHEMISTRY

ture pressure and density. Here, among other things, answers have
been found to such questions as refer to distribution of the positions
and velocities of the molecules, the mean number of molecular colli-
sions per second, the mean free path, the connection between these
quantities and the pressure, temperature, viscosity, rate of diffusion,
etc., of the gas. We proceed to quote some known results concerning
the distribution of the molecules in position and in velocity.

As regards the positions of the molecules, we conceive the enclosure
to be subdivided into a great number, say n, "physically small" volumes
or cells. This means that each cell is supposed too small to be dealt
with separately by our experimental apparatus and yet large enough
to contain an enormous number of molecules. As it is known that the
number of molecules in one cubic mm. of a gas at N.T.P. is about
3 x io16, this condition is easily complied with. Suppose we have
altogether N molecules in the enclosure, then the average number of
molecules per cell is N/« which is assumed to be a large number.
Suppose we express the actual number of molecules in any cell as
N/« . (i +8) where 8 is the fractional variation of this molecular density
from the mean molecular density N/«, and may be positive or negative.

It can be proved that there is an enormous probability against the
possibility of 8 acquiring values of an order of magnitude greater than
the order of i/N. As N is enormous, this practically means that there
is an enormous probability in favour of uniform distribution of the
molecules in position. It should be noted as a feature of this statisti-
cal proposition that the proof of it does not prove the impossibility of
the number of molecules in any cell deviating seriously from the mean
number ; it proves that such a state is extremely improbable, and that
the dynamical conditions which would produce it occur so infrequently
and exist for so brief a time that actual demonstration of its existence
would elude our experimental arrangements.

When we come to deal with the distribution of the molecules in
terms of velocity, we do not find this uniformity of distribution. Tak-
ing the question of speed alone, apart from questions of direction, there
is certainly a theoretical upper limit to the possible speed attainable
by any one molecule ; it is in fact the speed which that molecule would
have if it possessed the entire energy of the gas, the other molecules
being absolutely at rest. Such a speed is, however, far beyond any
practical limit; although there is no dynamical impossibility in the
state of affairs pictured, there is an enormous probability against it.

The application of statistical methods to this problem leads to the
view that certain speeds are more privileged than others. Thus there
is one speed such that there are at a given instant more molecules
possessing velocities within, say, one foot per second of this speed, than
there are molecules possessing speeds within one foot per second of
any other speed ; and if we choose speeds smaller and smaller or larger
and larger than this " maximum probability " speed, these speeds are
less and less privileged, until, when we arrive at zero speed or at speeds
very great compared with the maximum probability speed, the proba-

APPENDIX 1 161

bility of their occurrence is very small indeed. There are, however,
no privileged directions, all directions of motion being equally likely.
This rough statement will serve to introduce the mathematical law of
the distribution of velocities, first formulated by Maxwell and often
stated in the words: "the law of distribution of the velocities among
the molecules of a gas in the steady state is the law of errors ".

As a matter of fact there are two ways of stating Maxwell's Law,
one of which assumes a velocity to be measured by its components
parallel to the Cartesian axes, the other way assumes that a velocity
is to be measured by its magnitude and direction on the plan explained
above (the unit sphere device). In the former case the law of the
distribution is for each component similar to the normal law of errors,
but in the latter case the law of distribution of the speeds is not the
normal law of errors. Let us take the first method of formulation.

Considering an enormous number of molecules, N, it can be shown
that m the steady state those of them which at one instant have veloci-
ties whose X components lie between narrow limits, say u and u + du,
are in number equal to —

NA e-v^du

where A and y are constants to be determined shortly. This number
constitutes a fraction Ke~tlftdu of the whole. It should be carefully
observed that the group of velocities here dealt with are not neces-
sarily close to each other in magnitude and direction : it is only their
X components which are close to each other in magnitude ; no restric-
tion is so far imposed on the Y and Z components.

Of this limited number there will be at the same instant a certain^
fraction which have their Y components between v and v + dv, this:
fraction being in fact A.e~yv2dv ; so that the number whose X and Y
components are between u and u + du, and v and v + dv respectively,
is —

or

A similar step gives the number of those whose X, Y, and Z components
are at one instant within the limits u to u + du, v to v + dv, and w to
w + dw respectively, as —

^&Ze-y(u* + v* + ^)dudvdw . . . (i)

We can calculate the constant A in terms of y ; for if we inte-
grate for all values of u, v, and w from - infinity to + infinity, we must
getN.

Hence N = NA*f+~*r7«V« . ^ e~^dv . I ^

J —oo J —oo J

Now it is known that the definite integral —

i:

VOL. Ill II

T 62

A SYSTEM OF PHYSICAL CHEMISTRY

Hence N = NA3( VV^)3.

Hence A = J^fc

We can also calculate y in terms of the total kinetic energy of the
molecular motion. For the fraction of the whole, i.e. fractional number
of the whole number of molecules, which is represented by—

N AV-r(«2 + v* + ^ dudvdw

have each kinetic energy %m(u2 + vz + w2) where m is the mass of a
molecule.

Hence the total kinetic energy is —

f-foo r+

du\

J —oo J —

dv

00 _y(M2+j;2+a,2)

dw . (ti2 + v2 + w2) . e dudiau,

Leaving aside the constant factor for the moment, we see that the. i.ple
integral is the sum of three triple integrals, one of which, for ins ; i ce,

r+oo r+oo r

u2e~yu<idu . e-i^dv .

J —oo J —oo J

Now it is known that —

J.

and the two* remaining integral factors in the above chosen integral havt
already been evaluated. Hence the selected member of the three triple
integrals has the value -J ^/yYy5, and the other two have each the same
value. Hence the total kinetic energy is —

/S/^/?; />• INw/y; since A

Now we could conceive the molecules all moving with one uniform
velocity as a rigid body, for instance, and that velocity such as to give
the same kinetic energy as that due to the gaseous motion. Denote
the magnitude of this hypothetical velocity by c, and we have —

or y = 32.

This particular speed c is not the true average speed, nor is it the
maximum probability speed referred to above (its relation with these
will be given presently) ; it is in fact a speed whose square is the aver-
age of the squares of the molecular speeds at one instant, and on that
account is called the root-mean-square or r-m-s-speed. Assuming that
this r-m-s-speed is known, we formulate Maxwell's Law in one way
thus : —

Of a great number N of gaseous molecules in a steady state with an
r-m-s-speed c, the number whose velocity components at one instant
lie between the limits u to u + du, v to v + dvt w to w + dw, is —

. e-sWW+^l^dudvdw .

. (2)

APPENDIX I 163

and we see that any one particular component, say u, enters into this
expression in a manner similar to the way the error x enters into the
expression for the normal law of errors, viz. fy-^^dx.

We can employ a geometrical method of representing this law.
Imagine that u, v, w, are chosen to be the Cartesian co-ordinates of a
point in a threp dimensional diagram, which we will refer to as the
velocity diagram. The origin O of the co-ordinates in this diagram
represents absolute rest, while any other point P represents a velocity
whose magnitude is given by the length OP and whose direction is the
direction of OP. Let us write c for the length OP, i.e. —

<= J u* + v* + ze/2)

gives the actual speed corresponding to the point P. Now suppose we
represent, as it were, every molecule by a point in this diagram, i.e.
each point represents the velocity — not the position — of some molecule,
so that we have therefore N points marked in the diagram ; these points
will, of course, move about with lapse of time, because of the changing
velocity of the corresponding molecule. The expression (2) states the
volume distribution of these points at a given instant. Thus, taking
dudvdw to be an element of volume of the velocity diagram the
" density " of the points around the point representing zero speed, i.e.
representing the condition of rest, is —

Around any point representing a speed c Cwith no restriction as to
direction), i.e. around a point lying anywhere on a sphere of radius c
with its centre representing rest, the density of the points is —

-3iW' .... (3)

So the density of the swarm of points diminishes in this exponential
manner as we recede from the point representing rest. The reader is
warned against drawing the erroneous conclusion that there are at any
moment more molecules in the gas at rest, or very nearly so, than there
are those possessing any other assigned speed or near to it. The fallacy
involved in such an inference will be pointed out presently.

An element of volume in the velocity diagram represents what is
called an " extension-in- velocity," just as an element of volume in actual
space occupied by a gas is an " extension-in-position ". Thus we may
refer to expression (3) as the " density-in-velocity " of the molecules
about any velocity of the magnitude c.

We can readily pass to the second manner of formulating Maxwell's
Law, and in so doing clear away the possibility of misconception referred
to a few lines back. Suppose we wish to find the number of molecules
whose speeds (i.e. merely velocity magnitudes) are at any moment
between the limits c and c + dc, but with no limits placed on their
directions of motion. We must obviously find the number of points in
our velocity diagram which lie between spherical surfaces of radii c and
£ + dc. The volume of the elementary region of this diagram is

1 64 A SYSTEM OF PHYSICAL CHEMISTRY

and the point density in the region is given by (3). Hence the number
sought is —

47rN V27/87T3;6 . &-&&* . dc . . . (4)

The expression (4) has, for a given value of the differential element of
speed, dc, its maximum value for a value of c which makes —

a maximum. By the usual methods of the calculus it appears that this
value of c is : >J\c or o'8i6?; so that there are more molecules at an
instant possessing speeds within, say, one foot per second of this speed
than of any other assigned speed. That is, 0-8 1 6c is the speed with
" maximum probability ".

We can now deal with the fallacious inference mentioned above.
Let us consider a velocity with speed c and a definite direction whose
direction angles are a, (3, and y. The velocity has then the com-
ponents —

c cos a, c cos ft, c cos y.

Now, undoubtedly, there are more molecules whose velocities have
components between o, o, o and du, dv, dw than there are molecules with
velocity components between c cos a, c cos ft, c cos y and c cos a 4- du,
c cos ft + dv, c cos y + dw. This is a just inference from equation (3) •
but this statement is not true if we remove the condition ol ccefiniteness
from the angles a, ft, and y and allow them to take any possible values.
The removal of this restriction does not alter the speed limits, but it
multiplies the number in the second group enormously. From the
point of view of the velocity diagram, the representative points do un-
doubtedly cluster more densely around the origin of the diagram, and
.less and less densely around a point gradually receding from the origin,
but the actual law of distribution provides for the number of points
which lie in a thin spherical shell of definite thickness, increasing as we
go out to a shell of average radius o'8i6£and then decreasing as we
go still further. From the origin to 0*816?, the increasing volume of
the shell more than compensates the decreasing density of the points ;
further on it fails to do so.

We can modify expression (4) so as to deal with velocities having
denned limits of direction. Expression (3) shows us that the density-
in-velocity is a function of speed only ; hence there are no privileged
directions. So of those molecules which at an instant have speeds
between c and c + dc, the number whose directions of motion are re-
stricted to the directions represented by points on a superficial element
w of the unit sphere constitute a fraction w/47r. Multiplying expres-
sion (4) by this fraction, we find that the number of molecules sc*
restricted as to speed and direction is —

cuN ^27/87^. c1 . e-v*!** . dc . (5)

If we wish to find the number of molecules which have speeds within,
limits finitely separated we must integrate (4) or (5) between the limits,

APPENDIX 1

165

Thus the number, unrestricted as to direction, with speeds between ^
and ^ is —

. . . (6)

Expression (6) can be much simplified by introducing instead of the
speed c the ratio which this speed bears to the maximum probability
speed ^/f . c.

Call this ratio x so that —

and hence

c=

dc =

. c . dx,

On introducing these expressions into (6) we find after a little re-
arrangement that the number of molecules whose speed ratios lie
between xl and x% is —

_ r**

4N/V7T. ;

J X:

(7)

[In (7) we might refer to x as the speed, if we took the maximum
probability speed as the unit of speed.]

The properties of the integral in (7) can be most readily exhibited
by means of Fig. 1 4, which is the graph of the curve —

y = *V**
The graph begins at the origin O; y rises to a maximum value i/* (or

M, Mn

FIG. 14.

°'37 approximately) when x = i ; the curve then descends asymptoti-
cally to the axis of x, y reaching as low a value as 0*02 when x = 2-5,
showing that all but a very small fraction of the molecules have speeds
at any moment less than two and a half times the maximum probability
speed (which corresponds to x equal to unity) or practically twice the
r-m-s-speed. In fact, the fraction of the molecules whose speed ratios

1 66

A SYSTEM OF PHYSICAL CHEMISTRY

lie between xl and x2 is given by : 4/ ^/TT area of PjMiMaPa, where
OMi = xl and OM2 = x2. By methods of analysis or quadrature of a
carefully drawn graph numerical values can be obtained. The following
table from Meyer's Kinetic Theory of Gases (translated by Baynes)
illustrates the point : —

Of 1000 molecules of oxygen at o° C., whose r-m-s-speed is known
(by methods to be explained presently) to be 461-2 metres per second —

13 to 14 molecules have speeds below 100 metres per second.

81 82 from 100 to 200

1 66
214

76

167

215

203

152

92

77

200
300
400
500
600

above 700

300
400
500
600
700

The table shows that all but about 10 per cent, of the molecules have
speeds between one-half and double the r-m-s-speed. It thus appears
that despite the possibility of any definite molecule acquiring a speed
enormous in comparison with the average, the probability of it doing so
is exceedingly remote. It should also be observed in illustration of the
point dealt with earlier, that the number possessing quite low speeds is
also very small.

The kinetic theory of gases explains the pressure of a gas as the
lesult of molecular bombardment, and connects the pressure and tem-
perature of the gas with its mean molecular kinetic energy. The actual
relations are as follows : —

If p grams per c.c. is the density of the gas, and/ dynes per square
cm. its pressure, it can be shown that —

where c, the r-m-s-speed, is measured in cms. per second. If p and
p are known, this enables us to calculate c, as, for example, in the table
just given. If E is the energy of motion of a molecule —

E

where m is the mass of a molecule ; and if E is the mean molecular
kinetic energy, then —

E = mc2.

Hence /

where n is the number of molecules per c.c.

The experimental gas laws are summarised in the statement that —

pv = RT

where v c.c. is the volume of one gram-molecule of the gas under the
pressure/ at the temperature T and R is the gram -molecular gas con-

stant.

APPENDIX I 167

Hence we derive —

But nv is the number of molecules in the gram-molecule ; so R/#z> is
the gas constant per molecule, usually denoted by k. Thus we obtain,
as the connection between the temperature of the gas and its mean
molecular motion —

E = f*T . . . . (8)

We have so far dealt with a gas composed of similar molecules, but
if we consider a gas mixture containing molecules of different masses,
little alteration is required in the formulae. It can be shown that among
each group of molecules of one type, the relative distribution in speed
and direction is similar to that already outlined for a simple gas, and is
unaffected by the presence of other types of molecules. This is a
statistical generalisation of the well-known law of Dalton concerning
gas mixtures. Thus, each group of molecules has a certain r-m-s-speed,
but the value of this varies from group to group ; however, a very simple
relation connects them. If we denote types of molecules by suffixes,
i, 2, 3, etc., then —

etc.

where mit m2, m& etc., are the masses of each type of molecule. In
terms of average kinetic energies per group of molecules this is —

Ej = E2 = E3 = etc.

Referring to (8) we see that this dynamical conclusion is the statistical
statement of the equality of temperature which obtains throughout the
gas mixture when a steady state has been attained. On account of this
relation it becomes more convenient to state the distribution law in
terms of limits of kinetic energy. This is easily done. Thus, revert-
ing to the case of a simple gas for a moment, we introduce the following
changes into (4) : —
Write E = \mc*

E = fync*
ffE = me . dc

and we obtain after a few steps —

47rN

as the number of molecules with kinetic energies between E and
E + dE at an instant.

We may avail ourselves of (8) and obtain instead of this expression
the following one : —

E*..-*.* .. . . (9)

as the number of molecules limited in the manner mentioned.

1 68 A SYSTEM OF PHYSICAL CHEMISTRY

Now it can be shown that for a gas mixture consisting of N! mole-
cules of type i, N2 molecules of type 2, etc., in a steady state, and
therefore with the same mean energy of motion (i.e. the same tempera-
ture) throughout, the number of molecules of type i, of type 2, etc.,
within the defined limits of kinetic energy are given by expression (9),
with N replaced by Nlf N2, etc., in succession. Briefly, the distribution
of kinetic energy ignores differences in molecular mass.

Hitherto we have considered the gas to be free from all external
forces such as gravity, and subject only to the forces arising from collision
with each other or with the comparatively fixed molecules in the solid
walls of the containing vessel. If we introduce external forces of a
conservative type the previous formulae must be modified as follows :—

Denote the co-ordinates of a point in the gas as X, Y, Z ; the poten-
tial energy of a molecule m of the gas at this point is a function of
X, Y, Z ; call it XXYZ or simply x.

We no longer have a uniform density of distribution imposition for
the molecules. The density, in fact, diminishes as we move to places
of greater potential energy. A good illustration of this is the progres-
sive decrease in the density of the atmosphere as the altitude increases.
It can be shown that the molecular density is proportional to —

e ~ X/*T.

Hence, the number of molecules in an element of volume dx, dy, dz at
the point XYZ is—

ne ~ xl^dxdydz

where n is a constant. As a matter of fact n is the molecular density
in a small volume surrounding a point where the potential energy, x, is
zero.

The distribution in kinetic energy of the molecules in this element
of volume is just as before. Hence we can write in full —

. dxdydz (i o)

as the number of molecules in an element of volume dxdydz, where the
potential energy of a molecule is x, and whose kinetic energies lie be-
tween E and E + <£E.

The extension of this result to the case of a gaseous mixture is easily
given. Let the potential energies of each type of molecule in the ex-
ternal field of force at the point XYZ be xi» X2> xs> etc- » tnen tne
numbers of each type of molecule in the volume element dxdydz, at
this point limited as in (10), are —

. (n)

and so on.

APPENDIX I 169

Here «lt nz are the molecular densities of each type of molecule
around a point where the potential energy is zero.

The formulae developed hitherto have postulated implicitly an
absence of intermolecular forces, except such as arise in a collision
between molecules, an event which, though absolutely very frequent
for a molecule, is relatively rare, since the time which a molecule
spends in its free paths is very much larger than the time spent in
collisions. When we introduce intermolecular forces, i.e. when we
approach the liquid or solid state, the formulae become more complex,
and as they are not required in the text of the chapter to which this
appendix refers, may be laid aside.

A more important case for our purpose is the case of a gas whose
molecules can no longer be considered as simple rigid bodies, but for
which an atomic and even a sub-atomic structure must be postulated.
It would take us too far afield to treat the formulae for this case in all
their generality, but we can deal with a few statements of service later.

As before, we take the number of molecules as N, and consider that
the general position of each molecule is given by the three Cartesian
co-ordinates of its centre of mass. Besides these three co-ordinates there
are other geometrical quantities (or co-ordinates in a more general sense)
necessary to define the structure of a molecule. These internal co-or-
dinates can be most readily thought of as the least number of lengths
and angles necessary to specify the position of every distinct and con-
ceivably separate part of the molecule with regard to its centre of mass.
The number of these co-ordinates constitute the internal " degrees of
freedom " of the molecule. Suppose there are n of them, the neces-
sary n co-ordinates being denoted by the letters —

The rates of change of these co-ordinates as the molecular parts move
relatively to one another are referred to as "components of velocity"
or simply "velocities," each corresponding to a definite degree of
freedom. Writing q for dq\dt^ these velocities are —

As the parts of the molecule act on each other by means of inter-
atomic and intra-atomic forces (of electric origin), the molecule pos-
sesses a certain amount of internal potential energy, as well as a certain
amount of internal kinetic energy due to the motions of the parts rela-
tive to the centre of the mass. This kinetic energy can be expressed
as a quadratic function of the velocities q^ . . . qn. The case which
has been most extensively treated, and which is very amenable to
mathematical analysis, permits of the potential energy also being ex-
pressed as a quadratic function of the co-ordinates q 1 . . . qn. This
assumption is possible when we have to deal with the small oscillations
of the various parts about positions of relative equilibrium, or about
states of steady relative motion, and these conditions have thrown a
great deal of light, e.g. on the radiating mechanisms of molecules. It

1 70 A SYSTEM OF PHYSICAL CHEMISTRY

is possible with this assumption so to chose the co-ordinates that the
total internal energy, kinetic and potential, involves only squares of the
co-ordinates and velocities, so that we can write the total internal energy
« as —

where al . . . an, bl ... bn are constants.

In this manner and under the conditions laid down we can, as it
were, partition the energy among the various degrees of freedom, a^q^
being, for example, the kinetic energy " belonging to " the first degree of
freedom, and b^q^ the potential energy belonging to the same degree of
freedom, and so on. There is in addition, of course, the kinetic energy
of translation of the molecule as a whole, viz. %mx2 + %my2 + ^mz2,
which we will write as K (kappa) involving three more degrees of free-
dom, with the energy as before partitioned between the three degrees.

We can now state some results of the application of statistical
mechanics to a system of such molecules. It must be understood,
however, that these results rest on the validity of the classical dyna-
mics. Of late these results have been impugned by Planck and others,
and the foundations of dynamics are undergoing a revision. It will,
however, enable the reader to grasp the modifications proposed if he
masters the following few statements and accepts their truth at all
events provisionally.

As the molecule moves about and its parts oscillate with regard to
each other, its energy changes, not only in toto but also in its several
terms. It can be shown, however, that the mean energy, kinetic or
potential, belonging to any co-ordinate, averaged over a considerable
period of the molecule's history, is the same for all the degrees of free-
dom and is equal to -££T. That is —

This is a statement of the famous theorem due to Maxwell and usually
referred to as the " equipartition of the energy among the various de-
grees of freedom ".

Further, it can be shown, as before, that the number of molecules
which have velocities between uvw (i.e. xyz) and u + du, v + dv,
w + dw is —

A . N . <rK/*T . dudvdw . . . (12)

where A is a constant.

Of this limited number a still more limited number exists at one
instant whose several total internal energies, kinetic and potential, tie
between c and c + de.

This latter number forms a fraction of the former whose value is—

B€»-i<re/fcT^e ...... (13)

where B is a constant.

Combining the two expressions, we have the number whose internal

APPENDIX I 17*

energies lie between c and c + dt, and whose general velocities of trans-
lation lie between uvw and u + du, v + dv, w + dw, as —

C . N .€«-'. e-(e + VlhTdidudvdw . . . (14)
where C is a constant.

As an important particular case, expression (13) reduces for one
internal degree of freedom to —

Be-*lkT<te ... , (15)

Equation (15) forms the basis of the expression of Marcelin and
Rice for the rate of decomposition of a molecule referred to in Chapter
VI. under the heading " Reaction velocity from the standpoint of the
quantum theory ". The expression for the velocity constant on the
basis of statistical mechanics is given by Rice (Brit. Ass. Report, 1915,
P- 397).

APPENDIX II.
FOUNDATIONS OF THE QUANTUM THEORY.

BY J. RICE, M.A.
(From the Transactions of the Faraday Society xi., p. i, 1915.)

DURING the nineteenth century certain facts had been well established
concerning the radiation contained in an enclosure maintained at con-
stant temperature. Balfour Stewart and Kirchhoff had shown that if
the material of the walls was. not perfectly reflecting for any quality of
radiation, any constituent of the radiation having a definite frequency
was present in an amount depending on that frequency, the temperature
and size of the enclosure, but independent of the nature of the wall-
material. Stefan and Boltzmann had proved that the energy of the
total radiation in one c.c. was proportional to the fourth power of the
absolute temperature. Wien had reached the conclusion that the
-energy-density of those constituents of the radiation, whose wave-lengths

lay between narrow limits X and X + 8X, was 5 8X, where the

function /in the numerator, although undetermined in form, was de-
pendent on the single variable XT, the product of the wave-length and
the absolute temperature. These statements had been arrived at by
reasoning of a purely thermodynamic nature, based on the existence
of a radiation pressure. It was only natural that the statistical methods,
already employed as a successful weapon of attack on problems arising
out of the kinetic theory of gases, should be presently pressed into ser-
vice in this new line of research. There is some resemblance between
a vessel containing molecules, individually uncontrollable but maintain-
ing a certain average condition of energy by encounters, and an en-
closure maintaining a statistically permanent condition of radiation by
the emission and absorption of its walls, while the history of any par-
ticular wave-train of definite frequency cannot be followed in detail.
There is one marked difference between the two cases : molecules can,
by direct encounter with each other, as well as by collision with the
walls, exchange energy; but in a temperature enclosure passage of
energy from one wave-train to another must take place through the
agency of the walls alone or of matter contained within them, since two
such trains can pass through the same element of volume (i.e. cross one

172

AP-PENU2X 11 173,

another without any change in the energy or quality of the trains). For
this reason, if the walls are perfectly reflecting, any arbitrary condition
of radiation supposed existing at one moment would remain permanently
unaffected in the energy and quality of its constituents, and only by the
introduction of a piece of non-reflecting matter into the enclosure could
the radiation be gradually brought to the condition of " full " or " com-
plete " radiation for the temperature, the condition considered by Balfour
Stewart and Kirchhoff.

Wien, himself, had made an attempt to determine the form of the
function in the numerator of his so-called displacement formula.1 He
considered the enclosure to contain a perfectly absorbing or "black"
body, which he assumed to be a gas with its molecules distributed ac-
cording to the Maxwell Law, and its temperature therefore proportional
to the mean-squared velocity. A further hypothesis (a very doubtful
one) was the assumption that those molecules whose velocities lie within
certain narrow limits at a definite instant are at that instant emitting
light within narrow frequency limits, with an intensity proportional to
the number of those molecules. By these means Wien arrived at the

c

form e~J^ for his function /(AT), c being a constant. Measurements
carried out shortly after by Lummer and Pringsheim, Beckmann, and
Rubens verified the formula as a good representation of the facts for
short wave-lengths, but found it completely at variance with the facts
for long wave-lengths.

It was by the application of statistical principles in another direction
that the next advance towards a correct radiation formula was made.
Wien had considered the molecules of the black body. Lord Rayleigh,
and afterwards Jeans, considered the radiation itself, assigned co-ordin-
ates to it and " degrees of freedom," and applied the results of Max-
well's distribution law directly to these concepts. Such applications
are certainly rather easier to apprehend " physically " in the case of gas
molecules than in that of constituent wave-trains of radiation ; there is
more " substantiality " about a molecule than a wave-train. The repre-
sentation of a molecule as a small, hard sphere with perfect resilience,
which is quite adequate for many purposes, and analogies with billiard-
balls, discs, etc., put one on fairly familiar terms with molecular motion
and exchange of energy. The degrees of freedom of such simple systems-
are easily calculable, being in fact six for a " rigid " molecule of any
shape. Even if we introduce atomic structure into the molecules, the
degrees can still be computed if one knows the parts and their connec-
tions. The energy of the system at a given temperature is then obtained1
by the principle of equipartition, which is derived from the law of dis-
tribution, and asserts that the kinetic energy can be calculated by as-
signing to each degree of freedom an amount -J^T ergs, where k is the
molecular gas constant (1-35 x io~16 ergs/degrees), an equal amount
of potential energy being also assigned to any degree of freedom, if there,
are " elastic " forces of the usual simple harmonic type.
1 Wied. Ann., 58, p. 662, 1896.

i74 A SYSTEM OF PHYSICAL CHEMISTRY

If such ideas are to be generalised and extended to the ether of an
enclosure, a definite notion as to the structure of the ether is imperative
before any headway can be made in calculating its degrees of freedom.
At the outset it is fairly evident that the usual conception of the ether
as a perfectly continuous medium indivisible into an enumerable number
of discrete parts, should lead to an infinite number of degrees of freedom,
with the resulting conclusion that in a state of equilibrium the ether in
the enclosure should contain all the energy and the walls none. The.
well-known illustration of the gradual loss of vibratory energy from the
particles of a sounding body to the surrounding air may serve to make
this point clearer. This conclusion is in fact reached by the Rayleigh-
Jeans analysis, and is very much at variance with the facts as we know
them. The formula arrived at, however, is a close approximation to
the truth for long wave-lengths, and the calculation of the number of
degrees of freedom has proved of signal service in itself. It is impos-
sible to reproduce the analysis here, but an analogy from sound waves
may serve to show the principles on which it is based. It is well known
that an organ-pipe will resound only to notes of definite frequencies, the
fundamental and its overtones. This is due to the fact that any state
of " stationary " wave-motion which will persist in the air of the pipe has
to satisfy certain end conditions — e.g. at a closed end there can be no
vibratory motion of the air particles, at an open end no change of pres-
sure. Any text-book on sound shows that from these conditions there
can exist in a very narrow pipe, closed at both ends or open at both

ends, only waves whose wave-lengths are 2/, —, — ...—, etc., where

v v
I is the length of the pipe. The frequencies are, of course, — -„ 2 — ,

3 — . . . n - etc., where v is the velocity of sound. Now if we write
v for n—p we see that the number of possible modes of vibration whose
frequencies are not greater than v is n = — . v. Remembering that

this is the result for a narrow pipe in which the wave-motion is parallel
to one direction, let us extend the method to a flat shallow box, enclosed
between two square ends so that the wave- motion takes place in any
direction lying in one of the ends. The number of modes of vibration
having v as the upper limit of their frequencies would now be propor-

C2/ \2
. v\ , a result not difficult to apprehend, since we can in a

rough sort of way suppose that with any one type of wave-motion
parallel to one edge of the flat ends, we can compound any of the types
which can exist parallel to the rectangular edges so as to obtain a pos-
sible wave- train travelling obliquely round and round the box, reflected

APPENDIX II 175

•continuously from the narrow long faces which also bound the box. If
we still further extend these considerations to a cubical box, the number

/2/ \3 T

of possible modes come out to be proportional to ( — . v \ , i.e. to 8-3 . v2,

T is the volume of the box. A differentiation shows that the number of
possible modes which lie between narrow limits of frequency v and

v + &v, is 24^ . v28v. As a matter of fact a complete analysis shows

that the numerical factor should be 4?r and not 24. Now the point of
this analogy lies in the fact that whatever be the nature of a beam of
radiant energy, a certain definite condition has to be obeyed at a per-
fectly reflecting surface — on the electro- magnetic theory the tangential
component of the electric intensity in the wave must be zero there — and
this condition limits the number of types of radiation wave-trains which
can. persist unchanged as stationary waves in an enclosure with reflecting
walls. Consequently, if we introduce a small portion of perfectly ab-
sorbing matter into a cube with perfectly reflecting walls, we can assert
that, despite the fact that the matter can radiate and absorb any type of
radiation, only those types will exist in the final state of equilibrium
which satisfy the boundary conditions referred to above, just as our box
organ-pipe would not resound to every small whistle introduced into it,
but only to a whistle having one of a definite series of pitches. These
types of radiation will constitute complete radiation at the temperature,
since experiment has certainly justified Kirchhoff s conclusion that the
frequencies and energies of the constituents of radiation in a temperature
enclosure are independent of the size or nature or shape of the walls, so
long as there is present a portion of non-perfectly reflecting matter.
The number of types of radiation, therefore, which exist in full radiation,
having frequencies between the narrow limits v and v + 8v, would ap-
pear to be 471- -g . v28»/, where c is the velocity of light ; but, as a matter

of fact, this number has to be doubled, since radiation waves are trans-
verse to the direction of propagation (not longitudinal to it, as in the
case of sound waves), and therefore any particular wave has to be re-
garded as due to the composition of two waves of the same period and
phase, each polarised in one of two definite rectangular planes. So the

final result for the number is -^-v28v. This means that in order to

specify the electro-magnetic condition (electric and magnetic intensities)
at any assigned point in the ether of the box, at any assigned instant,

expressions involving "a number — ^-V28v of terms each varying harmoni-
cally with the time would be required to designate that part of the effect due

1 76 A SYSTEM OF PHYSICAL CHEMISTRY

to waves of frequencies in the range v, v + Sv. This number is in fact the
number of independent co-ordinates or degrees of freedom for such quali-
ties of radiation. Ascribing the usual /fcT units of kinetic and potential
energy to each degree, we obtain for the energy of radiation in the range

o

v, v -f Si/, the amount -y-/£Ti/2 . Si> ergs, or for the energy-density,

o

O7T 7 ,

-pel i/- . Sv ergs per c.c. In terms of wave-lengths we obtain by put-
ting v = - and Si/ = - — ^SA, that the energy-density of radiation

A. A.

whose wave-lengths lie in the range A, A + SA is -^— SA ergs per
cc.1

It is an obvious drawback to the Rayleigh-Jeans expression that it
does not approach a finite limit, as A decreases to zero. In fact, the
ether would appear to contain an infinite amount of energy per c.c.
since —

is certainly infinite.

We might evade this objection by observing that exchange of energy
from one type to another must be effected by the material of the en-
closure (as was pointed out above), and that we might reasonably sup-
pose the radiating mechanisms in the atoms to be of such a nature that
they could not emit radiation of a quality higher than a certain limiting
frequency, and so the upper limit of the* integral would be a finite
quantity and not zero. But this would not meet the difficulty that for
the same amount of range in wave-lengths SA, there is a greater contri-
bution to the energy density from high-grade qualities than from low-
grade, according to the Rayleigh-Jeans formula; whereas experiment
shows that the factor of SA exhibits a maximum value for a certain wave-
length Xm (dependent on temperature), and approaches zero as a limit,
as A approaches zero or infinity.

It has been suggested that the tendency, expressed in the Rayleigh-
Jeans Law, of the radiation energy to pass more and more into the higher
qualities, is in reality a true phenomenon of nature, and that we fail to
appreciate it in our experimental tests because all temperature enclosures
fail to confine the energy of the highest frequencies; such energy leaks
out, as it were, through the walls and through the small opening facing
the radiometer, almost as fast as it is supplied from low-frequency radia-
tion by the agency of the walls ; and so the condition we actually ob-
serve is a compromise — a stage on the way to the final consummation
expressed in the ideal formula, but unattainable except after an enormous

1 Phil. Mag., 49, 539, 1900; io»gi, 1905; 17, 229, 1909. Nature, 72, p. 94 and *
p. 243, 1905.

APPENDIX II 177

lapse of time. The difficulty of accepting this suggestion is the difficulty
of believing that such a compromise should be found to be so com-
pletely independent of the nature of the walls as experiment demon-
strates. The formula, however, does agree with fact very well for long
wave-lengths, just as Wien's does for short.

Faced with this discrepancy, there are two alternatives which offer
themselves as obvious methods of escape from it. We may deny the
validity of the calculation of the degrees of freedom, or we may urge
objections to the law of equipartition of the energy. As a matter of
fact, Planck's quantum hypothesis arises from his adoption of the second
alternative, and a denial of the validity of equipartition in the case of
vibratory motion. The principle of equipartition was first deduced in
connection with the kinetic theory of gases. Now even in this original
and limited sphere the principle does not stand on absolutely undebatable
foundations. A great deal depends on the interpretation of the meaning
to be attributed to the word " average ". It is usually assumed that the
average condition, say, of a molecule, is a "time" average; the sum of
successive values of a quantity connected with the molecule, over a long
period of change in the system, divided by the number of such values.

But it is doubtful if the principle of equipartition based on such an
interpretation of the word " average " is really proved at all by dynamical
principles. The point is fully treated by Jeans in his Dynamical Theory
of Gases. It would appear that the " average " really referred to is an
average extended over all conceivable conditions or " complexions " of
a system (excepting a negligible number) and not merely over such condi-
tions or "complexions" through which the system passes on any
particular "path". The extension to "time" average cannot be made
unless by the introduction of Maxwell's assumption of " continuity of
path," viz. that the system will in process of time pass through all con-
ceivable " complexions ". This assumption is of doubtful validity, and
is known to be unsound in certain of the problems treated in general
dynamics, e.g. the periodic orbits of astronomy (a type of vibratory
motion).

To proceed, Planck, denying the applicability of equipartition to
radiation theory, works out from special considerations (to be dealt
with presently) that in the vibratory motion obtaining in wave-trains,
each degree of freedom corresponding to a frequency v should possess
on the average not £T ergs of kinetic and potential energy, but an
amount —

— I

(where h is a new universal constant, determined by experiment to be
approximately 6*55 x io~27). If we make use of the Rayleigh- Jeans
calculation of the degrees of freedom, we find that the energy-density
of the radiation in the range v, v + <5v is — >

VOL. III. 12

i78 A SYSTEM OF PHYSICAL CHEMISTRY

STT hv*

73" —„_ ^ ergs per c.c.,

ekT - i
or in terms of wave-lengths —

ergs per c.c

These expressions agree remarkably well with the most recent
measurements, for all values of v or A, enabling the calculation of h to
be made with some precision.1

It is only natural that, in adopting a more or less revolutionary
attitude to a principle hitherto enjoying the confidence of the scientific
world, Planck and his followers should have modified somewhat the
original presentation of their ideas to meet the various criticisms urged
by those who seek to retain the traditional standpoint. It is well
known that Planck himself has receded somewhat from the position
adopted in his earlier papers on the subject. This position is fully
expounded in the first (1906) edition of his book, Theorie der
Wdrmestrahlung (Leipzig, Barth) ; the later form of his theory is
contained in Deutsch. Phys, Gesell. Ver., 13, 3, 138, 1911, or in the
second (1913) edition of the above book. Einstein's speculations and
applications are to be found in various papers in the Annalen der
Physik from 1905 onwards.

It is impossible to reproduce the whole of Planck's line of argument
here, but the following brief account will bring out the essentially novel
principle introduced by him, and serve also to show to some extent
why and how he receded from the earlier to the later position.

He remarks that any radiating mechanism of frequency v, placed in
a temperature enclosure, will acquire an average energy depending on
the energy-density of the radiation of that frequency in the enclosure.
The radiating mechanism of which he avails himself is an electric
doubtlet, consisting of an electrified particle vibrating harmonically
under the attraction of an oppositely electrified nucleus to which it is
held by a force of the usual elastic, proportional-to-the-displacement
type. In the enclosure there exist a great number N of these doublets
or " oscillators," having their axes fixed in a certain direction, i.e.
endowed with one degree of freedom. By an application of the usual
equations of the electro-magnetic field, he shows that —

. (i)

where L(v) is the mean energy (kinetic and potential) of an oscillator
and U(v) . Si> is the energy-density of the radiation in range v to v •+ &v.
Of course if we put L(v) equal to /£T, as usual, we would arrive as
before at the Rayleigh-Jeans Law. Planck, however, avoids this by
assuming that his oscillators can only emit and absorb discontinuously,

1 La Theorie du Rayonnement (Gauthier-Villars, 1912;, pp. 78-92.

APPENDIX II 179

and that at any instant the energy of an oscillator can only be an
integral multiple of a finite unit or " quantum " of energy, the amount
of the unit depending on the frequency. It is to be carefully noted
that this does not imply, of necessity, an atomic structure for energy —
a view, indeed, repudiated by Planck ; it does imply that the oscillator
must emit or absorb whole quanta of radiation at one time (at present,
be it noted, we are discussing Planck's earlier views). Now such an
assumption necessarily involves a modification of Maxwell's distribution
law, for that law depends on the possibility of the infinite divisability
of the energy among the oscillators, atoms, molecules, and so forth.
The point is more easily followed with aid of a "condition-diagram".
Suppose we denote the charges of the doublets as ±e, the mass of the
vibrating particle as m, and x as its displacement along its axis at time /,
then the equation of its " free " motion is —

X + 47T2!'2 . X = O

or mx + 47rV . mx — o

where v is the frequency. If X is the resolved component of the
electric intensity of the field along x at time /, the equation of its
forced motion is —

mx + 47r2*2mx =tfX . . . (2)

If we denote the energy of the particle by e —

mx2 + AtirLvLmxfi = 2e,
or denoting its momentum by y( = mx) —

£ + 4«*i'V. **-'•• .... (3)

Now the condition of any oscillator as regards the displacement and
momentum of its vibrating particle can be clearly represented on a plane
diagram by a point with co-ordinates x and y. All those oscillators
whose energy is e have their representative points lying on the ellipse —

where

.and ft = 2 em.

The area of this ellipse is of course irab, i.e'. -.

The history of any oscillator can be represented by the motion of
a point on this diagram, travelling along a curve, passing outward from
the origin when the energy of the oscillator is increasing, and inwards
when the energy is decreasing. Maxwell's Law states that the number
of oscillators whose displacements and momenta lie between the limits
jc, y, and x + $xt y + by is —

€

A . e'iwdxdv CA being a constant).

12 *

i8o A SYSTEM OF PHYSICAL CHEMISTRY

If we integrate this over the narrow ring in the diagram between
two ellipses determined by values of the energy e and c +• Be, we have
for the number of oscillators whose energies lie between e and e + Se, the
expression —

Of course, on the assumption of infinite divisability of the energy, the
total number of oscillators, N, should be —

tf

v J0

-todt.

But the essence of Planck's idea is that this integral should be replaced
by a convergent series. For, according to his ideas, the representative
points can only be found on the ellipses corresponding to energies
«j, f2, c3 ... «n, etc., where €1 is the quantum of energy and en = n . el ;
they will lie nowhere between these ellipses. The points will, as it were,
jump from ellipse to ellipse. The areas of these ellipses are of course —

€1 €o €o 61 €1 €1

•i -, 3, etc., or A 2^, 3-', etc.,

V V V V V V

and the various rings between them have all a common area. It is a
feature of Planck's theory that he gives this area a constant value for
all oscillators of any period, and denotes it by the letter " h" so that —

€l = hv,

f2 = 2hv,

€3 = ^hv, etc.

The integral expression N above must now be replaced by the
series —

= Ar s g, n

v[_ *2 *2 J

_ Ar _>£ J£L n

~ ~v\_v j

p /tt« 2/t»/ -|

— A-^ I + <?~/fer + ^~ AT" + . . .

- AA^-^ • ..... (4)

I — £ *T

If we now wish to calculate the total energy E, we see that the oscil-
lators corresponding to the first term of the above series have no energy,
those corresponding to the second have each hv, those to the third have
each 2hv, and so on. Hence we have —

hv zhv $hv

E - Ah

r hv_ 2hv 3/tt/

e^'krhv + e~kt2hv + e~ kT$hv + . . .

r hv 2hv ~|

T\ i + 2e~kr + y~ k-r + . . .1

APPENDIX II 181

• • • • (5)

. . . . (6)
**- i
By equation (i) it follows that —

UW-?.-^- .... (7)

**T - I

an equation which agrees well with observed results, if, as has already
been noted, h is put equal to 6*55 x io~27. It should be evident now
that Planck escapes from the Rayleigh- Jeans conclusion entirely by
his hypothesis of discontinuity in the exchanges of energy between the
oscillators and the medium. Had he assumed that cv e2 - e1} e3 — e2)
etc., are infinitesimally small, he would have found for L(v) in (6) the
limit of the right-hand expression as hv approaches zero, which is just
/&T, and so have arrived back at equipartition and the previous dis-
crepancy. His essential point is, in fact, the assumption that h is
finite, i.e. that the " elementary region of the condition diagram " has
a finite size. So that in a sense Planck's theory is more an innovation
in the method of probabilities, and need not be confined to radiation
problems. This point will be touched on later.

Two obvious criticisms of this first line of argument have been
advanced. First, equation (i) is obtained by an application of electro-
magnetic equations which rest on the assumption of a continuous ex-
change of energy between oscillators and medium. But as Campbell
in his Modern Electrical Theory (2nd edition) points out, this may not
be a fatal flaw in the reasoning. Maxwell's equations may be quite
true for average values of the electric and magnetic quantities involved
without implying their absolute validity at all instants in the immediate
neighbourhood of one oscillator, and so equation (i) might still stand as
an equality of average values, even though the discontinuity assumed by
Planck existed for individual doublets and their near environment. It
should be noted, however, in passing, that at a high temperature, say
2000° G, the quantum of energy for high-grade light of the order
v — To15 is 30 times as large as the average energy of an oscillator in
these conditions.

The second objection is that to produce the enormously great
number of qualities which are known to exist in complete radiation, we

1 82 A SYSTEM OF PHYSICAL CHEMISTRY

should have to postulate the existence of a very great number of oscil-
lators of different frequencies. Now, all the evidence which we possess
on the nature of atomic structure negatives such a view, nor is there
any real evidence that any vibrating electron in an atom emits simple
harmonic waves at all. Indeed, one of the outstanding difficulties of
the Planck theory is just this difficulty of formulating a plausible
physical basis for it

There is a further difficulty awaiting us when we seek to apply the
notion of discontinuous absorption to the phenomena of ejection of
electrons from the surfaces of bodies on which is incident a beam of
ultra-violet light (the photo-electric effect), or a pencil of X-rays. All
the evidence favours the view that a definite amount of energy has to be
at least resident in the electron before it can be ejected. In the normal
photo-electric effect the energy of the electron after escape is found to
have a maximum value h'v — w, where v is the frequency of the incident
light, K a constant whose value agrees fairly closely with Planck's h,
and w is a constant characteristic of the metal from which the electrons
are escaping. A quite plausible hypothesis is that h'v represents the
amount of energy in the electron before its escape ; of this it loses an
amount w in leaving the atom against restraining forces which depend
on the particular structure of the atom ; and so, if it is on the surface
and has not to part with still more energy in escaping from the influence
of neighbouring atoms, it would retain an amount h'v - w after emer-
gence. Now this appears to fit very well indeed with Planck's original
assumption of the discontinuous and sudden absorption of a quantum
hv, which, if v is large enough, would supply the necessary energy for
the escape of the electron from those atoms which the laws of chance
brought at any definite instant into the suitable condition for absorption.
But there seems to be this fatal objection to such a simple hypothesis,
that the photo-electric effect has been observed without any delay in
cases where the intensity of the light, as calculated by the inverse
square law and on the assumption of continuous propagation through
the ether, could not have possibly supplied the necessary quantum to ah
electron in a molecule or atom in less than 10 or 15 minutes. Similar
difficulties arise in connection with the secondary /?-rays caused by
X-rays, if one considers the energy of the X-ray pulse as uniformly
distributed throughout the pulse. In fact, if one adheres to the
hypothesis of discontinuous absorption, one must admit that the energy
of any beam of radiation is not uniformly diffused throughout the space
occupied by the beam. This is indeed roughly the manner in which
Einstein escapes from the difficulties. He has advanced the view that
the energy of light is itself atomic in structure, and absorption and
emission are of necessity discontinuous, not because of the special
structure of the oscillators, etc., but because of the nature of light itself.
His view is, in reality, a quantum of energy view ; it has already been
observed that Planck's is not necessarily so ; indeed Planck dissociates
himself at present from so revolutionary a hypothesis. It is noteworthy
that J. J. Thomson had already been led to speculations somewhat

APPENDIX II 183

similar to Einstein's, by considerations based on his hypothesis of
Faraday tubes in the ether emanating from charged particles.1 It
should be noted that Thomson's view of a " filamental " structure in the
ether might well be expected to lead to a different result for energy-
density than that reached by Rayleigh and Jeans ; for their calculation
of the degrees of freedom in the ether, it will be recalled, depends on
taking the ether as a perfectly continuous medium. Such a view would
indeed be more like the adoption of the first of the alternatives referred
to previously as methods of escape from the Rayleigh- Jeans discrepancy.
In reference to such views, it must be admitted that, however useful
they may be for the immediate purpose of overcoming such difficulties
as those connected with the photo-electric effect, and X-rays, and how-
ever well they suit the needs of radiation formulae, they involve a return
to the corpuscular theory of light, a raising of all the old difficulties of
accounting for diffraction and interference on such a theory, and at
present they lack a system of fundamental postulates, such as those on
which stands the present edifice of dynamics and electro-dynamics. Of
course, there is no gainsaying that the construction of such a system
may be attempted and may be successfully carried out, and Newtonian
dynamics placed in the position of a good approximation to truth, but
without absolute validity.

As has been already suggested, Planck meets these difficulties in a
less revolutionary fashion ; he abandons the hypothesis of discontinuous
absorption, retaining discontinuous emission. By so doing he meets fairly
successfully (though not entirely, as I hope to show presently) the ob-
jection lodged against his method of obtaining equation (i). His new
hyppthesis concerning his oscillators is that they can emit only at definite
instants, viz. when they contain an integral number of quanta, hv. If
they contain any fraction of a quantum over an integral number they
cannot radiate. When radiation does take place it is " catastrophic,"
i.e. it cannot cease until the whole energy of the oscillator has been
lost. The particular number of quanta possessed by an oscillator when it
radiates are determined by the laws of chance, the chance of its reaching
a given number being less the greater the number and the smaller the
radiation-density. In fact, Planck's new line is based on three
postulates : —

(a) In any finite time the total emission equals the total absorption.

(b) For small frequencies (i.e. long wave-lengths), the quanta of
energy are so small that emission is practically continuous, and the
ordinary dynamical and electro-dynamical laws are a good approximation
to the facts even for individual oscillators and their immediate neigh-
bourhood.

(c) The chance of a particular oscillator emitting when arriving at a
" critical " state (i.e. containing an integral number of quanta) is less
the stronger the field. In particular, the chance of no emission taking
place to the chance of an emission is a ratio proportional to the radia-
tion-density.

1 Electricity and Matter (Constable, 1904). Phil. Mag., 19, 301, 1910.

184 A SYSTEM OF PHYSICAL CHEMISTRY

Concerning these, (a) is clearly a statement of fact in a state of
equilibrium ; (b) is a justifiable hypothesis, inasmuch as the Rayleigh-
Jeans formula is suitable for experiments on long wave-lengths. As
regards (<:), the first part of it is plausible enough, but the particular
statement of it in the second sentence appears to be an a posteriori
assumption, based on the necessity of arriving at a formula (already
experimentally proven) by the method employed.

To proceed to a brief outline of his new analysis, Planck replaces
equation (i) by another giving the rate of absorption of an individual
oscillator in a field of given radiation-density. With the same notation
as before, he finds on the basis of equation (2) (taken to be valid for
continuous absorption) that —

dt

(8)

As the right-hand expression is clearly positive, the equation refers to
an increase of the oscillator energy, i.e. absorption.

Adopting the same geometrical representation of the condition of
an oscillator as before, the history of an oscillator is not represented
now by the leaping of a point from one to another of the " critical "
ellipses corresponding to the energies, hv, 2fo, $hv, etc. Its behaviour
follows the continuous movement of a point from the origin outwards,
and a jump back to the origin when the point reaches some one of the
" critical " ellipses (the particular one being a matter of chance), the
outward journey being once more resumed. The number of oscillators
is as before given by —

N -

_

but the equation (5) for E must be altered. The energy of any oscillator
is no longer, at any instant, an integral number of quanta ; all amounts
can exist, and the average energy of an oscillator whose representative
point is known to lie in a particular ring must be taken as the mean of
the energies corresponding to the elliptical boundaries of the ring, i.e. it
must be an integral number of quanta plus half a quantum. On this
basis it is easy to show that (5) must be replaced by —

.... (9)

and therefore —

hv hv

L(y) = ,M*T _ r + 7 '

on the average.

Of course we cannot apply equation (i) to this ; that has been dis-
carded, and replaced by (8) ; but before proceeding further it may be
as well to remark that Planck's new hypothesis enables him to account
for the photo-electric effect taking place without delay, even for weak
light ; for obviously there are at any instant a number of oscillators just

APPENDIX II 185

within an infinicesimally small amount of energy from arrival in a
"critical" state, and of these a portion will radiate almost instantane-
ously, or eject an electron. Thus Planck avoids Einstein's revolutionary
postulate on the atomicity of energy.

We saw above that the number of oscillators whose energy lies be-
tween c and e + 8e is —

and so the number whose representative points lie between the nth and
the (n + i)th ellipse is A^*-<*+ *>*"/**; the number whose points are
in the next ring are Afo-(n + ^hvlkr. That is, of all those which acquire
an amount of energy between nhv and (n + i)fo, the fraction e—hvlkT
pass the (n + i)th "critical" state and acquire an amount of energy
between (n + i)hv, and (n + 2]hv. So e-hvlkT is the chance of no
emission taking place, and therefore i — e— M*T is the chance that an
emission will take place. By postulate (c) above, we have —

where p is a constant to be determined, f.e. —

therefore by equations (8) and (10) —

dt \mehv\k^ - i

hv
Now for small values of v, p= is small and L(i/) approaches the value

K> -L

^T, hv being therefore negligible in comparison with L(v) ; so for small
values of v —

Now Hertz has shown that in this case (for here by postulate (£)
the classical methods are applicable) the rate of emission of a vibrating

electric doublet is equal to the fraction — 3 — of its own energy.1 But

in equilibrium the rate of emission is equal to rate of absorption, there-
fore Hertz's result gives us for our case —

dL(v) frrW ...

~^r "~^r-

1 Wied. Ann., 36, p. 12, 1889.

1 86 A SYSTEM OF PHYSICAL CHEMISTRY

and so by (iza) —

or

and therefore by (n) we obtain once more —

TU \ 87r ^3
UW = -T _

Planck has thus succeeded in obtaining his radiation formula once
more, but with the help of a rather artificial postulate (<r), and also by
ignoring the fact that his adoption of the electro- magnetic equations
necessarily implies a continuous radiation from his vibrating electric
particle. Such a particle in its excursions backwards and forwards
along an ever-increasing amplitude must of necessity have a motion
which is not uniform, but now accelerated, now retarded, and must
during such variations of its motion be radiating ; at least, that is the
logical conclusion from traditional views. As a matter of fact, in a
series of interesting papers in the Phil. Mag., dealing with the line-
spectra of the elements, Bohr frankly admits this point, but makes a
most ingenious use of it, taking the first step, in fact, towards a system of
dynamics with which the behaviour of such systems as Planck's oscillators
would harmonise.1

It is doubtful if Planck's work would ever have received the con-
sideration bestowed on it, had it not happened that an ingenious appli-
cation of it by Einstein to the problem of specific heats initiated a
remarkable series of researches, theoretical and experimental, which
showed the hypothesis of discontinuity was fruitful of good results in
other branches of physics and chemistry than radiation. It is well
known that while high-grade radiation is exchanged by some mechanism
which is sub-atomic in nature (vibrating electrons for instance), infra-
red " heat " radiation is emitted and absorbed by means of the atoms
and molecules as a whole. Einstein's idea was that in a monatomic
solid each atom has a definite frequency of vibration, which, he as-
sumed, can be determined by optical methods (the "residual ray"
method of Rubens, for instance, or dispersion-formulae). If L(v) is the
expression used previously, each atom has on the average an amount of
energy L(v) for each degree of freedom, that is, $L(v) altogether, on
the basis of Planck's work, extending to "infra-red atom" oscillators.
Hence if there are N atoms per gram-atom in the body, the atomic

heat of the body would be 3NL(i>) or j*^ A differentiation of

this expression with respect to T would give the specific heat at the
temperature T (for constant volume). The result is-^-

lPhil. Mag., July, September, November, 1913 ; March, 1914.

APPENDIX II 187

where x = 7^. If x is small, i.e. if v is small or T large, this expression

approaches the value 3N£, which is known to be 5*95 calories, and is
indeed the Dulong-Petit value of the atomic specific heat. For small
values of T, however, x is large and the above expression grows smaller
and smaller, ultimately approaching the limit zero as T approaches zero
or x approaches infinity. That is, there should be a gradual decrease
to zero of the specific heat of a solid as the temperature approaches
absolute zero. This was known to be true in a general way. Further,

as C is a function of ^, the curves connecting specific heat and tem-
perature for various monatomic solids should be similar. In fact, for two
solids whose atomic frequencies are v\ and v2, the curve for the second
would be obtained from the curve for the first by maintaining the same
ordinates (representing C) and altering all the abscissae (representing T)
in the ratio vi : v& A series of now famous researches were undertaken
by Nernst and his pupils ; details of the work will be found in papers
in the Ann. der Phys.^ p. 395, 1911, in the Zeitsch. jiir Electrochem.,
1911 and 1912, in the Theorie du Rayonnement, and in Nernst's four
lectures delivered at University College, London, on the solid state.
The similarity of the curves was proved, while Einstein's actual ex-
pression, although following the general course of the change in specific
heat with the temperature, exhibited considerable numerical discrepan-
cies, especially at low temperatures. Nernst and Lindemann pro-
posed an alternative formula—

C = 2

** - lV»

This, although in good agreement with results, was a piece of lucky
guesswork, founded on no solid theoretical basis. In a paper in the
Ann. der Phys., 39, p. 789, 1912, Debye attacked the problem from
a broad standpoint, and his result, from its agreement with experiment
and the soundness of its premises, seems to have approached finality
in this domain.

Debye points out that there can be no single "characteristic"
vibration of the atoms of a monatomic solid. There are an enormous
number of such vibrations; they constitute an "acoustic" spectrum;
they are the fundamental and overtones of the body considered as
emitting a note. A knowledge of the elastic constants of the material
is sufficient to determine them, just as Rayleigh and Jeans determined
the frequencies possible in a cubical "block" of ether. For these

i88 A SYSTEM OF PHYSICAL CHEMISTRY

vibrations constitute waves of " sound " travelling through the body
with definite speeds. There are, for instance, " distortional " trans-
verse waves travelling with a velocity vlt and " compressional " longi-
tudinal waves travelling with a velocity v%. vl can be calculated if one
knows the density p and modulus of rigidity p of the material ; it is in

fact Vlm I^\ #2 can be calculated if one knows in addition the bulk-

\P
modulus K —

Now, the number of degrees of freedom for vibrations having frequen-
cies between v and v + Sv is, as previously, — xi>28v per unit volume for

vf

the transverse waves, and ^v28i/ for the longitudinal. (N.B. — The

factor STT occurs because the transverse waves are polarisable in two
rectangular planes, the longitudinal are not.) Consequently the
whole modes of vibration per unit volume between v and v + 8v are

( ~T + —i V2Sv. We saw above that the similar formula developed for
\#i3 0s /

the ether by Rayleigh and Jeans suffered from the drawback that, in in-
tegrating it, no finite upper limit could be assigned on account of the
assumed continuity of the ethereal medium. In this case, however,
there must be a finite upper limit to v, because the number of degrees
of freedom cannot exceed 3 N for a gram-atom of the substance.
Hence if vm is the highest frequency possible —

(where r is the volume of a gram-atom)

where F is a constant calculable from the elastic constants and density
of the body.

Since vm = ( y-J , vm is also calculable. For Al, Cu, Zn, Ag, Pb,

Diamond it is of the order io12 to io13. Now by ascribing, according
to the extension of Planck's work, the energy —

hv

- i

APPENDIX II 189.

to every degree of freedom, we obtain for the energy-content of body
due to wave- motion of frequencies v to v + Sv —

Hence the total energy —

hvm kv

where x = — , and y = ^

A differentiation of this expression with regard to T produces this
expression for the specific heat at constant volume —

c* =

^~' . i— - - - i vi _,

This is clearly a function of x or -7=?, and therefore suitably accounts

/£ J.

for the similarity of the specific heat curves of different solids. Further,
the expression agrees remarkably well with determinations of Nernst
and his co-workers, more so in fact if vwis calculated not from the
values of elastic constants, but from a formula discovered by Lindemann,1
connecting vm and the melting point (taking vm as the frequency of
vibration of an atom under the influence of its neighbours). The
computation is not easily carried out, for the integral in the above ex-
pression cannot be evaluated in finite terms, and so one has to resort
to the summation of series which can be obtained to suit various ranges
of temperature as regards their convergency. One striking result
emerges from the calculation, viz. that at very low temperatures the
specific heat varies as the cube of the absolute temperature ; this would
appear to be true to within i per cent, for the range of temperature o°

I hvm

r

Nernst has made a very happy combination of Einstein's and
Debye's formulae in the treatment of diatomic solids such as KC1>

1 Physikalische Zeitschrift, 11, 609 (1910).

1 90 A SYSTEM OF PHYSICAL CHEMISTRY

NaCl, AgCl, KBr, and the triatomic substance PbCl2. He supposes
the heat content of the body to be constituted of two parts, one due to
the vibrations of the molecules as a whole, and the other to the vibra-
tions of the atoms within the molecules. The first part can be obtained
by Debye's method, taking the upper limit of the frequency as deter-
mined by Lindemann's melting-point formula ; the second part can be
determined by Einstein's method, the characteristic frequency being
-determined by Rubens' " residual rays " method. The agreement with
observation is quite good ; details can be found in Vortrdge uber die
Kinetische Theorie der Materie und der Elektrizitiit (Teubner). Still
further extensions of Debye's ideas have been made by Born and
Karman and Thirring in papers contained in the Phys. Zeit., 14 (1913).
It was pointed out above that Planck's formula is derived from an
innovation in the treatment of statistical problems, the assumption, in
fact, that the elementary region of the two-dimensional condition dia-
gram for a system of oscillators is finite in area, and the replacement of
Maxwell's integral by a series. Planck has applied the same idea to the
six-dimensional condition diagram for an ideal monatomic gas, and
arrived at a result which had been already obtained by Nernst, who
bases his derivation of it on his well-known " heat theorem "^ There
are several ways of wording this theorem, according to the particular
thermodynamic function which one elects to make use of in the state-
ment. As expressed by Planck in his Thermodynamics, it is as follows :
"The entropy of a condensed system, which is chemically homogeneous,
is zero at the absolute zero of temperature ". This law removes from the
expression for the entropy of a body in a definite state that indefinite-
ness in value which arises in the usual thermodynamic treatment and
which is due to the entrance into the formula of an undetermined con-
stant on integrating the differential of the entropy. For clearly if the
entropy of a body can be given definitely for one state, it is determinate
for all other states which can be conceivably reached from the former
by a reversible path. So if we can conceive that the body can be led
to the condition of a chemically homogeneous solid or liquid at absolute
zero, by a reversible process, Nernst's heat theorem makes it possible to
calculate its entropy without ambiguity, if sufficient experimental data
are available. By such means it can be shown that the entropy of one
gram-molecule of an ideal gas is given by the equation —

S = Q,(i + log T) - R log/ + a (13)

(absolute units of temperature, pressure, and energy being used), where
the "chemical constant," a, indeterminate on pure thermodynamical
grounds, has, according to Nernst's theorem, the value—

•J • • • • - (i4)

in which ps is the vapour pressure of liquid condensed from the gas at
a very low temperature 0, and rQ is the heat of vaporisation of the liquid
at absolute zero.

1 Nachr. d. Ges. d. Wissensch zur Gotting&n Math. Phys. Kl. (1906).

APPENDIX 11 191

Planck arrives at this result on dynamical grounds in the following
manner. Conceiving a monatomic gas as an aggregate of particles with
three effective degrees of freedom, and representing the co-ordinates of
a particle by x, y, z, and its velocity components by u, v, w, the con-
dition diagram is a six- dimensional one in which the position and
momentum of a point representing a particle are given as x, y, z, mu,
mv, mw (m being the mass of a molecule). According to the Maxwell
Law the number of molecules whose representative points lie within
the limits x, y, z, mu, mv, mw, and x + dx . . . m(w + dw) is —

e

N Ke ~ *r dxdydzdudvdw . (15)

where N is number of molecules in a gram-molecule, c the energy of
a molecule in condition xyz, u, v, w, due to its position and velocity,
-and A is a number given by the equation —

f - —
i = A \e kr dxdydzdudvdw.

If one puts e = %m(tfi + v2 + w1), it follows that —

i / m \f

V l^TrJT/
V \ ^ TTK A /

V being the volume of the gas.

Now, according to Planck's view, instead of the differential element
of the six- dimensional diagram which occurs in (15), we ought to intro-
duce a finite element having a value G, definite in size for a system of
definite molecules. The form and position of these " elementary
regions" would, as in the two-dimensional case, be bounded by the
" surfaces " or loci of equal energy. If we number these regions i, 2,
3, ... n, we can say that a fraction^ of the representative points lie
in region i, etc.,/, in region n, where —

<n being the molecular energy corresponding to the nth region. With
this notation, it is possible to express the entropy of the gas, as Boltz-
mann has shown (see his Gastheorie, or Jeans' Dynamical Theory of
Gases), in the form —

Of course in Boltzmann's treatment, as in Maxwell's, f^, f%, etc., are in-
finitesimally small, since the differential element dxdydzdudvdw oc-

p
•curs instead of — 3, and so the expression for S contains a constant

of indeterminate (and infinite) value. By assuming G to be finite,

Planck avoids this conclusion, just as he avoids the Rayleigh- Jeans

. law in radiation, by assuming h to be finite and not infinitesimally

small. He finds, by using the value given above for A and./,,, that —

192 A SYSTEM OF PHYSICAL CUEMISTJtY

V(2jr«>&T)* E 1
-g- - + mj

where E is the kinetic energy of the molecules.

Writing R for k. N and putting E equal to -RT, we obtain —

S = RJJ (i + log T) + log V + $ log (2irmk) - log G] . (16)

Since /V = RT and cp = -R, equation (13) can be readily trans-
formed to —

logT) + logV - logR + + i , (17)

The formal similarity between (16) and (17) is obvious, and the
two results lead to the connection between Planck's G and Nernst's
" chemical constant," a, given by the equation —

- log (z-n-mk) - log G = - log R 4- ~ + i,
or —

^ R(:

\jr — —

The thermodynamic treatment can be found in the third edition of
Planck's Thermodynamics, Chap, VI. The development of the ideas
based on the quantum hypothesis is given in his Theorie der W'drmes-
trahlung (2nd edition, pp. 110-32).

APPENDIX III.
KRUGER'S THEORY OF GYROSCOPIC MOLECULES.

[This appendix represents the substance of two papers by F. Kriiger,
Annalen Physik, [iv.], 50, 346; ibid., 51, 450 (1916), on "The vibra-
tions of free gyroscopic molecules," and on " Molecular precessional
vibrations and specific heat."]

MAXWELL pointed out that a magnet must behave as a gyroscope if
the ampere molecular stream (the origin of magnetisation) be of a
material nature, and he carried out an experiment to decide this point
with an electromagnet, without, however, finding the expected result.
The electron theory has referred the ampere (magnetic) current to
rotating electrons. But the electrons possess mass, and therefore their
circulatory motion gives to the molecule containing them a certain,
angular momentum. O. W. Richardson (Phys. Rev., 26, 248, 1908)
showed in the case of a freely hanging magnetised iron wire that, owing
to the angular momentum N of the molecules, a torsion must manifest
itself, although he was unable at first to obtain a definite result owing
to experimental difficulties. Recently Einstein and J. W. de Haas
(Verh. d, D. phys. Ges., 17, 152, 203 (1915)) have succeeded in giving,
an exact calculation of the torsion which the wire should experience, and
have also succeeded in overcoming the great experimental difficulties.
They have in fact employed their method to calculate the ratio of the
charge to the mass of an electron. Further, S. J. Barnett (Phys. Rev.
[ii.], 6, 239 (1915)) has shown that magnetic effects can be produced in
an iron wire by rapid rotation of the wire. Also Maxwell's original
experiment has been taken up again by W. J. de Haas' and by G. L. de
Haas and Lorentz (Proc. K. Akad. Wetenschap., 24, 398 (1915)), who
have shown that the effect can scarcely be expected to be experimentally
detectable. The gyroscopic characteristics and properties of the atoms
of metallic iron due to the existence of molecular currents, i.e. the
rotations of electrons in the atems, can now be considered to be
definitely demonstrated.

In addition to magnetic investigations there have sprung into exist-
ence in the last few years certain optical theories in connection with the
laws of line spectra which have rendered very probable the existence of
rotating electrons even in the case of those molecules whose magnetic
character is by no means pronounced. Such electron planetary systems
have formed the basis of the atomic models of Rutherford and of
Nicholson. Based on such, Bohr has constructed a model atom by
VOL. III. I93 13

i94 A SYSTEM OF PHYSICAL CHEMISTRY

bringing in the quantum theory which not only accounts for the series
laws but also quantitatively for the Rydberg constant in terms of the
Planck constant h, the charge e, and the mass /x of the electron. Ac-
cording to Bohr, one or more electrons rotate round the nucleus of an
atom in a fixed path with the angular momentum ft/2ir or a whole
multiple thereof, the different angular momenta corresponding to
rotations of the electrons in paths of different diameter. This theory
has been applied with success by Warburg (Verh. d. D. phys. Ges., 15,
1259 (1913)) to explain the Stark electrical effect. Recently Debye
has employed a similar atomic model to calculate the dispersional
properties of hydrogen in particular. The general dispersion formula
has been given by Sommerfeld (Elster and Geitel Festschrift, p. 549

(1915)).

If these models correspond to reality, it follows that such atoms, and,
as a special case, the hydrogen molecule, must possess gyroscopic pro-
perties. In the following treatm'ent it will be shown that the gyroscopic
properties of gaseous molecules, in particular the hydrogen molecule,
-are of considerable importance for the theory of gases, especially in con-
nection with the problem of molecular heat of gases, and are capable of
solving certain difficulties which exist at the present time.

On Bohr's theory the hydrogen molecule consists of two positively
charged hydrogen nuclei with two electrons rotating round the line
joining the centres of the nuclei, each electron possessing an angular
momentum of hJ2Tr. The hydrogen molecules behave therefore as
small gyroscopes, which are at the same time free from forces such as
gravity, for the gas is almost perfect. What is particularly significant
from the standpoint of kinetic theory is the behaviour of such molecular
gyroscopes under collisions.

The translational motion of such freely moving molecules, each with
three degrees of freedom, is naturally the same whether the molecules
possesses gyroscopic properties or not. But we have now to consider
the nature of the remaining degrees of freedom which must be attributed
to the molecule in order to account for its molecular heat. There are
at least two degrees of freedom to be accounted for. These, as has
been pointed out in Chapter IV., have been ascribed by Bjerrum to rota-
tion of the molecule as a whole. But, as Kriiger goes on to say,
according to the fundamental equations of gyroscopic theory, a gyroscope
cannot carry out any rotations, but it can be put into a state of vibration
which, in the case of a symmetrical gyroscope free from external forces
— the case here considered — corresponds to regular precession, i.e. pre-
cessional vibrations. Kriiger's theory consists in substituting the idea
of precessional x vibrations of the atoms in place of rotation of the
molecule as a whole, in order to account for the two remaining degrees
of freedom. The advantage of this comes in. as we shall see later, in
connection with monatomic molecules. In the case considered the
precessional vibration ot the atoms is regarded as due to a vibration of
each electron perpendicular to its own orbit whilst it still keeps on fol-
1 See the footnote on p. 23 of Chap. I.

APPENDIX III 195

lowing the orbit as its " average " path. Kriiger points out that ac-
cording to gyroscopic theory the angular velocity of regular precessional
vibrations is —

C.i// i
~ A - C ' cos $

where C is the moment of inertia of the electron round the figure-axis,
/>.• the axis joining the centres of the two atomic nuclei together, A is
the moment of inertia of the atom nucleus perpendicular to the figure
axis, and <f> denotes the angle swept out by the figure axis as a result of
the precession. Further —

f"*

where «// is the angular velocity of a rotating electron or rate of spin
of the electron. The angle <f> depends naturally upon the vibrational
energy and the temperature. For low temperatures and molecules with
very small moments of inertia, as is probably the case with the
hydrogen molecule <j> will be small, and we can make the usual approxi-
mation that since <f> is small cos <f> =» i. Hence we obtain —

The product in the numerator, viz. Ci//, represents the angular
momentum N of the electron spin so that we can write —

N

The angular velocity of the precession is thus directly proportional to
the angular momentum of the spin of the electrons. Further v is
greater the smaller the denominator, that is, the more closely the
moments of inertia A and C approach one another, that is, the more
spherical the gyroscope. For a spherical gyroscope A = C and v =
infinity.

It is to be remembered that in the molecular gyroscope the preces-
sional vibrations are brought about as a result of collisions with other
molecules. In the case of the hydrogen molecule the two hydrogen
atoms or nuclei are enclosed by the path followed by the two electrons,
this path being fixed by the electrostatic and centrifugal forces. The
moment of inertia C is here that of the two electrons which rotate or
spin at a distance r round the figure-axis of the molecule.

The important conclusion now arrived at is that those molecules which
possess rotating electrons do not themselves rotate but carry out vibra-
tions of the precessional kind which are independent of temperature, or
nearly so. The two atoms at either end of the figure-axis follow a small
circular path as a result of the precessional " wobble " of the electrons.

The vibration of regular precession of symmetrical bodies possesses
the characteristic of having two degrees of freedom, each of which
represents kinetic energy but no potential energy. The hydrogen

13*

196 A SYSTEM Of PHYSICAL CHEMISTRY

molecule possesses therefore in all five degrees of freedom as on the older
theory. It will be observed that Kriiger does not take into account any
vibration of the nuclei with respect to one another along the path or
figure-axis joining their centres. Calling the molecular heat Cr, then, in
the region of temperature in which the equipartition principle holds, it
follows that Cv is given by the relation —

Q, = sR/2 = 4-96.

It will be recalled that the molecular heat of hydrogen in the neighbour-
hood of o° C. is 4*9 to 5 '2 cals., whilst at 2000° abs. it is of the order
6 '5 cals.

Kriiger further points out that the two degrees of freedom which cor-
respond to the precessional vibrations can be treated from the quantum
standpoint just as in the analogous case of the vibrations of a solid : that
is, the energy content decreases with falling temperature and converges
to zero at zero temperature. A marked decrease in energy content
with lowering of temperature has been shown experimentally to be the
case with hydrogen by Eucken (Sitzungs. Akad. Wiss., Berlin, 1912,
p. iAe(cf. Chap. IV.)). The curve obtained by Eucken for the molecular
heat points to a single frequency independent of the temperature.

For the special case of the hydrogen molecule in which the optical
properties yield quantitative values for the gyroscopic properties, the
angular velocity of precession may be calculated and compared with
that obtained from the specific heat data. For this purpose it is necessary
to use the numerical values which Bohr has calculated for the constants
of the hydrogen molecule (Phil. Mag., [vi.], 26, 487, 1913 (cf. Chap. V.)).
For the angular momentum of a single rotating electron, ftr2^' (where
r = the radius of the electron's path, i// = the angular velocity of its
rotation, and //, the electronic mass), we have the value —

h\2TT = 1*06 x io~27.
For the radius r of the electron's path Bohr finds the value — - x io~8cm.,

and for i// the value 3-86 x io16 per second. [The latter quantity is
just 27T(u, where o> has been defined in Chapter V. as the frequency of
revolution of the electron, and possesses the value 6-2 x io15 for the
electron of the hydrogen atom. Bohr has shown (cf. Chap. V.) that,
for the electrons of the hydrogen molecule, o> is i'i times the o> for the
single electron of the atom. We should therefore employ a slightly
greater value for if/' than that adopted by Kriiger.] Further, Bohr has
calculated that zd, the distance apart of the two atoms, is 0*635 x io~8 cm.,
the mass of the atom having the value 1*64 x io~24 gram, the mass
of the electron being 8-97 x io~28 gram. It follows therefore that A,
the moment of inertia of the atoms perpendicular to the figure axis,
is 2 x 1-64 x io~24 x (0-317 x io~8)2 or 3-30 x io~41. Similarly
moment of inertia, C, of the two electrons is given by —

2 x 8-97 x io-2G x (5-52 x io~9)2 or 5-43 x io~44.

APPENDIX III 197

C can therefore be neglected compared with A. Hence for the angular
velocity of precession we have —

1-06 x io -27

v = /&/2rrA = — n = 3'21 x io14.

3-30 x io - 41

The angular velocity v of precessional vibration is related to the fre-
quency v of precessional vibration in a very simple manner, viz. —

V = V/2TT.
V V2I X IO14

Hence v = — = - — -? — 5 = 5*1 1 x io12.

27T 6'28

This expression is obtained by equating the angular momentum to
/fyW. This, however, only holds for a single electron. Hence for the
two electrons of the hydrogen molecule we get just twice the value for v —

v = 1*006 x io13

(revised value given in Kriiger's second paper).

Thus, with the help of Bohr's results, i.e. results in agreement with the
optical properties of the hydrogen molecule, Kriiger has obtained a
value for the vibrational frequency exhibited by the molecule in virtue
of precessional vibrations. We have now to see how far this value
is borne out by a consideration of the molecular heat of hydrogen.
Eucken has applied the quantum theory to the problem of the mole-
cular heat of hydrogen in the simple form originally suggested by
Einstein for the case of solids, i.e. we have to do with a single vibration
frequency which ought to agree with the value of v obtained above,
if Kriiger's argument is correct.

The formula employed by Eucken is —

To make this agree with observed values it is necessary to write
ftv = 430, and, since ft = 4-86 x io-11, it follows that v = 8-85 x io12.
The agreement between the two values for v is satisfactory.

The equation obtained by Kriiger on the basis of the gyroscopic
theory of the molecule has also been obtained by Nernst ( Verh. d. D.
phys. Ges., 18, 83, (1916)) based on quite different considerations, in-
volving rotation of the molecule. Nernst points out that in the case
of hydrogen a more correct value for /3v' is 450, whence v is 9*3 x io12.
This agrees even more closely with the value obtained by Kriiger on
the basis of precessional vibrations.

The above considerations hold good, naturally, not only for hydrogen
but for all gas molecules in which rotating electrons exist, and, accord-
ing to Bohr, the series spectra, which are a general property of gases,
are due to such rotating electrons. The diminution of approximately
5 per cent, in the molecular heats of nitrogen and carbon monoxide
between 273° and 90° abs. observed by K. Scheel and W. Heuse (Ann.

198 A SYSTEM OF PHYSICAL CHEMISTRY

Physik, 40, 473, (I9I3)) ig also> according to Kriiger, to be referred to
precessional vibrations, which are obeying the quantum principle of
energy distribution as a function of temperature. Kriiger's conclusions
may be summed up as follows : —

Since, in general, we may expect all molecular systems to possess
rotating electrons, we may say that in the case of molecules which
contain two or more atoms such molecules do not exhibit rotation as
a whole, but on the other hand exhibit precessional vibrations on ac-
count of their gyroscopic properties. The number of degrees of freedom
on this view remains the same as on the older view of rotations.

it has already been pointed out that the molecular heats lead us
to the conclusion that in the ordinary range of temperature a diatomic
gas has five and not six degrees of freedom. That is, one of the expected
degrees is missing. Kriiger points out that the ordinary rotation theory
does not give any satisfactory explanation of why there is one degree of
freedom missing. By making use of the precessional vibration idea
the fact that five degrees are possessed is self-evident, for precessional
vibration gives rise to two degrees of freedom (purely kinetic), and
the free translation of the molecule gives rise to three degrees of
freedom also purely kinetic. It will be observed that Kriiger is
not considering the vibration of the ordinary kind of the two
atoms with respect to one another. Further, in the case of a mon-
atomic gas we have seen that the inertia terms A and C approximate
to one another with the result that the frequency of precession
becomes exceedingly great, and in the limit when the molecule is
spherical it becomes infinite. Hence monatomic molecules are unable,
according to the quantum theory, to take up any sensible amount of
energy over the temperature ranges hitherto examined, for the only
kind of radiant energy which such a system would be capable of picking
up would be that belonging to the extreme ultra-violet region. That is,
the monatomic gases, such as mercury, argon, etc., do not possess any
internal energy as far as their molecular heats are concerned ; the* mole-
cular heat is completely taken account of by the three degrees of freedom
of translation of the molecule as a whole. This explanation of the
behaviour of monatomic gases is one of the main advantages of Kriiger's
theory.

In the second of the papers cited Kriiger goes on to consider in
more detail the question of the infra-red spectra of gases in relation to
the concept of precessional vibrations.

It has been shown experimentally by Rubens and Wartenberg
(Verh. D. phys. Ges., 13, 796 (1911)) that the elementary gases such
as H2, O2, C12, Br2, which are electrically neutral structures, show no
absorption in the infra-red region between 23 and 314/4, whilst on the
other hand the markedly electrically-polar gases, such as HC1, HBr,
and H2O, possess strong absorption over this range. This fact has
hitherto been ascribed to the rotations of the dipoles, for rotating dipoles
must emit and absorb radiant energy. According to Bohr's atomic
model as extended by W. Kossel (Ann. Physik, 49, 359 (I9l6))» these

APPENDIX III 199

dipoles are surrounded by a swarm of rotating electrons which no
longer occupy a position in the middle of the figure-axis, but are
displaced towards one end. It follows that these molecules must
possess gyroscopic properties, and cannot rotate, but experience pre-
cessional vibrations. The infra-red bands referred to would correspond
therefore to precessional vibrations and not to rotations. [Note that
the infra-red bands produced by the " ordinary " vibration of atoms
with respect to one another (in a manner analogous to the vibration of
atoms in solids) occur in the short infra-red region, viz. 2 to 5^,. We
are at present dealing with bands in the much further infra-red.] In
general both ideas, rotation and precessional vibration, lead to the
same results; a distinction does indeed exist, viz. the position of the
absorption band on the basis of precessional vibration should be
practically independent of the temperature, whilst in the case of
rotations the position should be displaced towards the shorter wave-
length region as the temperature rises. Direct measurements upon
this point are wanting.

An indirect test of the question is possible, however. Lord Ray-
leigh (Phil. Mag., [v.], 21> 410 (1892)) has calculated that a linear
oscillator, vibrating with frequency v0 and rotating with frequency vl
must radiate or absorb at the three frequencies, v0, v0 + vj, and v0 — v\.
As a rule vl is small compared with v0. If, instead of rotation, we
substitute precessional vibration with frequency vv exactly the same
frequencies should be brought into play ; their intensities will, however,
be altered.

Rayleigh's considerations have been applied by Bjerrum (Nernst
Festschrift, 1912, p. 90; Verh. d. D. phys. Ges., 16, 640, 737 (1914))
to explain the infra-red absorption spectra of gases. He points out
that an absorption band in the short wave part of the infra-red spectrum,
which is due to linear vibrations, must be split into three owing to rota-
tion of the molecule. In this way he has explained the broadening of
the 3'5/A line of HC1, the broadening of the infra-red lines of CO2, and
the splitting of the 5*9/u. band of water vapour, as due to rotational
frequencies existing in the far infra-red region. [The rotational
frequency repeats itself, as it were, close to a true vibrational frequency,
the rotational frequency being separated from the central vibrational
frequency by the small quantity ± i/j.]

The absorption bands in the short infra-red region have been
measured very accurately by Burmeister (Verh. d. D. phys. Ges., 15,
589 (1913)) and by E. von Bahr (ibid., 731, 1150 (1913)), who found
that, in general, double bands are characteristic of this part of the
spectrum, which would correspond to the theory of Bjerrum provided
one assumes, as does E. von Bahr that, on account of insufficient dis-
persion, the middle sharp line does not appear.

The distance of the two decomposition bands must depend on
temperature on account of the variation of the rotational frequency with
the temperature — that is, assuming that rotation is the cause of the
phenomenon. On the other hand, this distance should be practically

200 A SYSTEM OF PHYSICAL CHEMISTRY

independent of the temperature if it be due to precessional vibrations.
According to von Bahr the product \r x ^T is a constant, where \r is
the rotational wave-length and is given by —

2X^2

A'-XT^

when A! and X2 are tne two wave-lengths of the double-headed band.

von Bahr's relation is to be expected if a rotation accompanied by
the amount of energy assumed on the classical mechanics were present
(cf. Chap. IV.). It is to be remarked at the same time that with very
energetic precessional vibrations the angle <£ swept out by the axis as a
result of precessions becomes greater, so that cos <£ cannot any longer
be set equal to unity. This will naturally occur at high temperatures.
In fact, the smaller the angular momentum compared with the difference
of .the moments of inertia A and C of a symmetrical gyroscope, the
more closely do precessional vibrations approximate to rotations.

It is fairly clear from an examination of the infra-red spectrum that
in the case of water vapour we are not dealing with a rotation obeying
the laws of classical statistical mechanics. Beyond 9/4 approximately
it shows a series of absorption bands with constant wave-length difference ;
on the shorter wave side of 9/4 it shows a number of bands symmetrically
placed with respect to the band at 6'24/x. Bjerrum considers the first
group, i.e. those beyond 9/x as single rotation frequencies in the sense
of the quantum theory ; those on the shorter side of Q/A as the " decom-
position" bands brought about by the interaction of the 6 -24^ band
with each of the rotational bands. As a matter of fact, the measurements
of Rubens and Hettner (Verh. d. D.phys. Ges., 18, 154 (J9l6)) show
that such a relation exists between the lines under and beyond Q/A.
The Rayleigh formula corresponds very closely to the frequencies
observed by Rubens and- Hettner between 9 and 2oo/x. To regard
these long wave-lengths as due to rotations in the sense of the original
quantum theory is, however, not very satisfactory. The original expres-
sion of Bjerrum, with a constant frequency difference of 173 x io12
gives far too few lines, so that Eucken introduced a second series with
a frequency difference of 075 x io12, which accounts for the majority
of the missing bands, though not for all (cf. Chap. IV.).

We have now to consider the spectrum of water vapour from the
standpoint of the precessional vibrations. Kriiger has employed the
Bohr model for the atom or molecule, and it will be remembered that
the angular momentum may not only be h\2Tr but also twice, thrice,
etc., this value. These different angular momenta correspond to dif-
ferent frequencies, which, since C is negligible compared with A, cor-
respond to twice, thrice, etc., the fundamental vibration frequency. We
would expect therefore, on Kriiger's theory, series of bands inter-related.
Kriiger, however, considers it doubtful whether these simple considera-
tions could give rise to a sufficiently great number of bands, for the
Bjerrum series is 20 and that of Eucken 40.

It must be pointed out, however, that the water molecule is not a

APPENDIX III 201

symmetrical gyroscope. Owing to this lack of symmetry more complex
relations are to be expected. In the case of an asymmetric gyroscope
with three different moments of inertia, we find in place of the regular
precession of the figure-axis describing a circular path on a surface, the
more complicated form of the general Poinsot vibration. Such move-
ment possesses three degrees of freedom as the molecular heat of water
vapour requires. In the case of the water molecule the electrons do
not all revolve in the same plane. These new modes of motion and
interaction may be expected to give rise to a number of bands so that
the spectrum of water vapour would be expected on these grounds to
be complex. There is a further point. The measurements of Rubens
and Hettner have shown that the bands are fairly broad. This is not
easy to reconcile with the Bjerrum rotation theory. The breadth of the
bands is attributed by Kriiger to certain of the molecules which possess
a value for the angle <£ which is far from the mean value.

If we consider the difficulties which surround the application of the
quantum theory to rotation frequencies, it would appear that precessional
movement with complex vibrational form is the more plausible hypo-
thesis. Possibly an intensive examination of the dependence upon
temperature of the wave-lengths of rotation bands, together with a com-
plete discussion of the form of vibration of the water molecule, might
serve to decide between the two views.

The Quantity <£.

The general equation for the frequency of the precessional vibrations
is, as we have seen —

, _ N_ _!_ ,,,

" ' cos

For small values of <f> the cosine term becomes unity. Kriiger now
considers more closely the question of the calculation of <f>. The angle
<f> is defined by the movement of the atom (in the hydrogen molecule)
which describes a circle of radius p with a velocity v. If m be the mass
of the atom the precessional energy of both atoms (or of the whole
molecule) is mv^ = E Now, v = 2-xpv and p = a sin <£, where a is
the distance of an atom from the middle point of the figure-axis.
Hence v = 2-rrav sin <f>.

Therefore mv* = ^awV2 sin2 <j> = 2rrAV2 sin2 <£

where A stands for 2 ma2.

Further 27r2Av'2 sin'2 <£ = E . . . . (2)

or sin24> = E/27r2Av'2 .... (3)

From equation (i) we have —

N2 i

COS2 <£ = - n-To . -To - . • (4)

2 l

From equations (3) and (4) we get —

v

202 A SYSTEM Of PHYSICAL CHEMISTRY

sin2 <f> 2EA

cos2 < N!

tan2 <f> =

or tan <£ = - . (5)

It follows that we can write cos <£ in the form —

cos d> == — — . (6)

If we denote by v'Q the value of v when cos <f> = i, we get —

cos <f> = v'ojv . . (7)

a formula which may be easily introduced into (6).

In order to make cos <£ = i it is necessary, from equation (6), to
have 2EA small compared with N2. This condition will be favoured
by a small moment of inertia A, and by a small value for the energy of
precessional movement ; this corresponds to moderately low tempera-
tures.

If we eliminate <£ from equations (3) and (4) we obtain the preces-
sional vibration frequency as a function of the energy. Thus we find —

N2
X2 = E + J£ . (8)

where / is written for 27r2A.

N
Since "0 = ^,

rp
we have v'2 = — + i/02

or E = /(v/2 - v 02) . (9)

If we calculate, as do Einstein and Stern (Ann. Physik, 40, 551 (1913))
with a mean value for the vibration frequency v at a given temperature
T, we find for the energy of the precessional movement of a molecule
the expression —

hv

E

I hv' N2

and hence v = -. ^^—^ + — . .

This equation allows us to calculate <£ from equations (6) or (7) for each
value of v corresponding to a given temperature. Kriiger gives a table
for the temperature range T = o° to T = 300°. The following is a
selection from Kriiger's values : —

APPENDIX III

203

T.

v'.

Cos 4..

*•

Srec.

O

1-0055 x io13

i-ooo

O

O

67-5

I -0060

0-9995

2° 45'

0-036

79'5

1-0070

0-9986

3°

0-159

103*1

I'OIO

0-9956

5° 21'

0-385

I57'8

1-028

0-9781

12°

0-871

299-6

I'lIO

0-9059

25° 3'

1-38

The last column denotes that part of the molecular heat which is due
to the precessional vibrations, i.e. —

It will be observed that the change in the mean vibration frequency v
due to the increase in the precessional energy amounts at 100° to about
0-5 per cent, whilst at 300° it is not quite io per cent.

Kriiger considers alternative modes of calculation of the molecular
heat of hydrogen at considerable length, for which the reader is referred
to the original paper.

Behaviour at High Temperatures.

Let us return to the expression —

cos < =

N

2EA

We have already considered the case in which 2EA is small com-
pared with N2. As the temperature is raised, however, E increases, so
that cos <£ diminishes and the frequency v increases. At very high
temperatures 2EA may be very large compared with N2. Under these
conditions we have —

cos <f> = N/ ,/2EA.
and for the vibration frequency —

N i

_ __

27TA COS

1

Since, on the basis of the equipartition principle, which holds at high
temperatures, we can write E = £T, we have for the vibration frequency
of precession —

k —

(12)

The frequency is thus proportional to the square root of T ; undet
these conditions the precessional movement has become a true rotation whose
vibration frequency is given by the same formula. The conclusions may
therefore be summed up as follows : Molecules, to which their revolving

204 A SYSTEM OF PHYSICAL CHEMISTRY

electrons give gyroscopic properties, possess, at low temperatures and
with small values for the moment of inertia (A), a vibration frequency
which is nearly independent of temperature ; with rising temperature the
frequency increases, and, at high temperatures, if the product 2EA
exceeds N2, the precessional movement passes into a true rotation. In
the case of hydrogen, if we put E = £T, then 2EA = N2 at T = 489° ;
that is, the gyroscopic angular momentum N is still sensible ; it is only
at much higher temperatures that hydrogen is capable of giving rise to
true rotations, as a result of molecular collisions.

In the case of molecules which possess moments of inertia A larger
than that of hydrogen, the transformation of precessional vibration into
true rotations occurs at correspondingly lower temperatures. Thus it
is very probable that the increase in distance apart of the double bands
of water vapour at A = 6*512/01 and X = 5'948/u, which Paschen has
observed (Ann. Physik^ 52, 215 (1894)), is due to the rotational char-
acter of the motion, since, according to E. von Bahr ( Verh. d. D. phys.
Ges., 15, 731, 1150 (1913)), the product of the rotation wave-length A,

(defined by X, = -r-^ — j— J into JT is approximately a constant (cf.
A! — A£ /

Chap. IV.).

On the other hand, the double bands which E. von Bahr has observed
in the short infra-red spectrum of carbon monoxide are characterised by
the fact that their distance apart, i.e. the quantity A,, is only very slightly
affected by temperature change over the range 15 to 145° C. The
quantity A, in this case is regarded by Kruger as being the wave-length
characteristic of true precessional vibration, and it is conceivable, as
Kruger points out, that on raising the temperature still further, the dis-
tance apart of the double bands, i.e. the quantity Ay, would then show
a definite temperature effect. Under such conditions we would be
observing the change from precessional vibrations into true rotations.

As regards the question of molecular heats, the chief point of in-
terest lies in the departure from the principle of equipartition at low
temperatures, and here the precessional vibrations of the molecules with
a certain gyroscopic moment of inertia possess the character of vibra-
tions with constant period.

SUBJECT INDEX.

AFFINITY of a chemical process, 145 seq.
Aluminium, atomic heat of, 66, 70, 74.
Ammonia (gas), infra-red bands of, 85.
Atomic heat. See Heat.
Atomic-number, 121.

— and infra-red frequency, 122.
Atom-model of Rutherford, 96 seq.
Avogadro constant, 46.

BOHR'S application of the quantum theory
to atomic structure, 97 seq., 193 seq.

Broadening of absorption bands, 81, 82,
199.

CARBON dioxide, infra-red bands of, 83.

— molecular heat of, 83.

— monoxide, molecular heat of, 82, 197.
Chemical constant of a gas, 95, 192.
Coagulation of colloids, application of

probability to the, 6 seq.
"Complex on," 13.
Copper, atomic heat of, 63, 66, 70.
Critical energy of a molecule, 140.

— increment of a molecule, 140 seq.,

154 seq.

DEGREE of freedom. See Freedom.
Diamond, atomic heat of, 53, 67, 74.
Distribution law, Maxwell's, i, 8 seq.,
also Appendix I.

— of energy between wave-lengths, 31

seq. See also Radiation formula.
Dulong and Petit's law, 30, 53, 187.

ELECTRON, angular momentum of an,
102.

— charge on, 46.

— emission of radiation from an, 97.
Electronic orbits in an atom, 97 seq.
Eqergy-density of radiation, 38.

— of an oscillator, residual, 49.
Entropy and thermodynamic probability,

10 seq.
Equations of Einstein, Nernst-Linde-

mann, and Debye. See under Heat,

atomic.
Equipartition of kinetic energy, principle

of the, 4, 18 seq., 25, 30, 31 seq., also

Appendix I.
Error, law of, 7 seq.

FREEDOM, degree of, 18, 159 seq.

— number of degrees of, in a gas mole-

cule, 24, 29.

solid, 25.

Fusion, molecular heats of, 79.

GAS constant per molecule, 45 seq.
Gyroscopic molecules, Appendix III.

HEAT of a gas, molecular, 26 seq., Chap.
IV. (p. 80).

reaction ; Haber's relation, 154

seq.

solid, atomic, 29 seq.

; application of the quan

turn theory to its cal-
culation, Chap. III.,
p. 50.

; equation of Debye, 70

seq., 187 seq.
; — of Einstein, 52, 187.

— — ; — ofNernstandLinde-

mann, 64 seq., 187.
solids, at high temperature, atomic,

69 seq.
experimental measurement of

specific, 54 seq.
molecular, 76 seq., 189 seq.

— theorem of Nernst, 95, Chap. VII.,

190 seq.

Helium atom, Bohr's theory of the struc-
ture of the, 109 seq.
High-frequency spectra. See Spectra.
Hydrogen atom, Bohr's theory of the

structure of the, 98 seq., 107, 108

seq.

— chloride, infra-red absorption spectrum

of, 89, 117.

— molecular heat of, 81 seq., 92 seq., 94.

— molecule, Bohr's theory of the struc-

ture of the, 115 seq.
dissociation of the,

Kruger's theory of " precessional

vibration " of the, 195 seq.

ICE, molecular heat of, 78.
" Ideal gaseous state, degradation of the,
94 seq.

205

206

A SYSTEM OF PHYSICAL CHEMISTRY

Infra-red absorption bands to ultra-violet
bands, relation of, go, 116, 133.

— radiation and reaction velocity. See

Thermal reactions.
the law of mass action, 150

seg.

lonisation potential, 99, 127 seq., 130.
Iron, atomic heat of, 70.

LEAD, atomic heat of, 67.
Limiting frequency of the spectrum, 128.
Line spectra of hydrogen, 99 seq.
Lithium atom, Bohr's theory of the, no.

MAGNETON and structure of the atom,

118 seq.

Mass action and probability, law of, 5.
Maxwell's law. See Distribution law.
Mechanics, statistical. See Statistical.
" Melting-point " formula of Lindemann,

58 seq.
Mercurous chloride, molecular heat of,

78.
Molecular heat. See Heat.

— velocities and temperature, 10.
Moseley's relation, 120.

NICKEL, atomic heat of, 70.
Nitrogen, molecular heat of, 82, 197.
" Nucleus " of an atom, 96 seq.
Number of molecules in one gram-mole-
cule. See Avogadro constant.

OSCILLATOR. See Resonator.
Oxygen, infra-red bands of, 82.

— molecular heat of, 82.

PHOTOCHEMICAL equivalent, Einstein's
law of the, 134 seq.

— reactions, 134 seq.
Photo-electric effect; Einstein's expres-
sion, 102, 126, 182.

" selective," 131 seq.

the " normal," 123 seq., 182.

Planck's constant, " h," 36 seq., 102.
numerical value of, 44, 45.

— later quantum hypothesis, 48 seq., 184.
Potassium bromide, (half)-molecular heat

of, 69.

— chloride, (half)-molecular heat of, 64,

68, 77.
" Precessional vibration," 23, 91 seq.,

also Appendix III.
Probability, 4 seq.

— thermodynamic, 10 seq.

QUANTUM in different spectral regions
size of, 47.

— theory, Chap. II., p. 35. also Ap-

pendix II.

Quantum theory and atomic heat of
solids, Chap. III. (p. 50).

structure, 97 seq.

— chemical affinity, 145 seq.

heat of reaction (Haber's rela-
tion), 154 seq.

theorem of Nernst, Chap.

VII.

molecular heat of gases, Chap.

IV., also Appendix III.

— — — optical absorption, 90, 116,

133-
photochemical reactions, 134

seq.
the photo-electric effect, 126

seq.
thermal reactions, 138 seq.

RADIATION and classical statistical me-
chanics, 31 seq.

— Bohr's theory of the mechanism of

absorption and emission of, 97 seq.

— formula of Planck, 39 seq., 41 seq.,

178, 181.

Rayleigh and of Jeans, 33,- 46,

176.

Residual rays (Reststrahlen), 56 seq.
Resonance potential, 129 seq.
Resonator, average energy of a, 38 seq.,

45, 48, 177 seq., 181.
Resonators, 35.
Rotation compounded with vibration in a

molecule, 81, 82, 199.
— , energy of, 18, 21 seq., So seq., 194.
Rydberg constant, Bohr's calculation of
the, 100.

SCATTERING of o and ft particles, 96. '
Selective photo-electric effect. See Photo-
electric effect.
Silver, atomic heat of, 67, 70, 74.

— chloride, molecular heat of, 78.
Sodium chloride, (half)-molecular heat of,

68, 77.
Spectra of the elements, high-frequency,

120 seq.

Spectrum, limiting frequency in the, 128.
Standard state, n seq.
Statistical mechanics, i seq.
Stefan's law, 39, 43.

Sulphur dioxide (gas), infra-red and ultra-
violet absorption bands of, 90.
molecular heat of, 83.

THERMAL reactions, application of the
quantum theory to. 138 seq., 171.

Thermodynamic probability. See Prob-
ability, thermodynamic.

Threshold-frequency (photo-electric effect),
124.

Tin, atomic heat of, 70.

SUBJECT INDEX

207

Transformation, heat of, 79.
Translation, kinetic energy of, 9, 18.

VAPORISATION, heat of, 79.

Vibration, energy of, 18 seq., 25, 80 seq.

— frequency (infra-red), periodicity of,

61 seq.
maximum (equation of Debye),

75 seq.

in the ultra-violet, characteristic,

60, 132.

Vibration- frequency of a solid, methods
of determining the infra-red
characteristic, 56 seq.

elements, 58, 59, 60 seq.

salts, 57.

WATER-VAPOUR, infra-red, absorption
spectrum of, 84, 85 seq., 88, 200 seq.
— molecular heat of, 84.
Work of binding an electron, 98 seq.

X-RAY spectra. See Spectra, high-fre-
quency.

AUTHOR INDEX.

ABEQG, 119.
Allen, 75, 122.
Arrhenius, 140, 143.
Aschkinass, 56, 88.
Avogadro, 18, 46.

VON BAHR, Eva, 86-90, 199, 200, 204.

Balmer, 45, 97, 100, 101.

Baly, 90, 129, 133, 134, 137, 138, 140,

157.

Barnett, 193.
Bazzoni, 128.
Beckmann, 56, 173.
Berthelot, D., 93.
Berthelot, M., 147.
Biltz, 6.1, 62, 122.
Bjerrum, 24, 80, 81, 83, 84, 85, 87-92,

194, 199, 200, 201.
Bodenstein, 134, 136, 137.
Bodlander, 119.
Bohr, 21, 45, 91, 96-102, 104, 105, 108-

117, 119, 121, 122, 128, 186, 193,

196-198.
Boltzmann, 3, n, 13, 16, 18, 31, 172,

191.

Born, 190.
Bragg, 76, 121.
van den Broek, 121.
Burmeister, 89, 199.

CAMPBELL, N., 181.
Clapeyron, 95.
Coblentz, 41, 90.
Compton, 124.
Crowther, 96.

DALTON, 167.

Debye, 50, 64, 69-73, 75*78, 187, 189,

190, 194.
Dixon, 127, 128.
Dulong, 30, 53, 58, 71, 72, 73, 187.

EDWARDS, 130.

Einstein, 30, 31, 35, 47-54, 56-59, 63, 64,
69-71, 76-78, 81, 94, 102, 123, 126,
133-138, 143-145, 147, 156, 178, 183,
185, 186, 189, 190, 193, 200, 202.

Elster, 131.

Eucken, 54, 79, 87, 88, 90, 92-94, 196,

197.

FISCHER, U., 147, 148, 149.
Foote, 128, 129.
Fourier, 71.
Fowler, 101.
Franck, no, 126.

GARRETT, C. S., go.
Geiger, 96, 121.
Geitel, 131.
Gibbs, 17, 145, 147.
Goucher, 130.
Griffiths, 122.
Gruneisen, 58, 65.
Guldberg, 141.

DE HAAS, 193.

Haber, 60, 116, 133, 154, 155.

Hallwachs, 123.

Hamilton, 2.

Helmholtz, 145, 147.

Henri, 138.

Hertz, G., 126.

Hertz, H., 185.

Hettner, 200, 201.

Heuse, 197.

van't Hoff, 154.

Hollnagel, 56, 57.

Hughes, 124, 127-129, 131.

JEANS, 18, 32, 33, 37, 46, 49, 71, 173, 174,
176-178, 181, 183, 184, 187, 188, 191.

KARMAN, 190.

Keesom, 122.

Kirchhoff, 102, 172, 173, 175.

Konigsberger, 70.

Koref, 54.

Kossel, 198.

Kruger, 23, 80, 85, 91, 92, 150-154, 193,

194-198, 200-204.
Kunz, 130.
Kurlbaum, 41, 43, 56.

LADENBURG, 124, 126.
Lagrange, 2.
Landsberg, 134.
Langmuir, 119.
Lenard, 124, 126.
Levy, 28.

Lewis, G. N., 30, 65.
208

AUTHOR INDEX

209

Lewis, W. C. McC., 142.

Lindemann, 39, 50, 54, 58-61, 63-66, 6g-

73, 75, 76, 78, 81, 132, 133, 146, 147,

187, 189, 190.
Linder, 6, 7.
Lorentz, 193.
Lo water, Miss, 90.
Lummer, 35, 37, 41, 42, 44, 173.
Luther, 134.
Lyman, 100.

MCLENNAN, 130.

Magnus, 65, 146, 150.

Marcelin, 139, 140, 143, 171.

Marsden, 96.

Maxwell, i, 3, 8-10, 18, 26, 161-163, 173,

177, 179, 190, 191, 193.
Mellor, 7.
Mendileef, 62.
Meyer, Lothar, 62.
Meyer, O. E., 8, 29, 166.
Michelson, 48.

Millikan, 44-46, 122, 131, 132.
Moseley, 120, 121.

NAZAROV, 134.

Nernst, 17, 28, 30, 50, 51, 53, 54, 56, 59,
63-66, 68-73, 75-78» 81, 94, 95, 122,
144-147, 149, 151, 187, 189, 190, 192,
197.

Newton, i.

Nichols, 56.

Nicholson, 102, 105, 193.

VON OHLENDORFF, 147.
Onnes, 122.

PARSON, A. L., 118-120.

Paschen, 41, 42, 87, 100, 204.

Perrin, 18, 46.

Petit, 30, 53, 58, 71, 72, 73, 187.

Pickering, 101.

Picton, 6, 7.

Pier, 26, 28.

Planck, Max, u, 17, 31, 34-37, 39-46, 48,
49, 51, 7i» 97, 98, 99, 102, 126, 127,
136, 140-145, 170, 177-186, 190, 191,
192, 194.

Poinsot, 23, 201.

Pollitzer, 147.

Pringsheim, 36, 37, 41, 42, 44, 173.

RAMSAY, Sir William, 62.

Rayleigh, Lord, 33, 36, 46, 49, 82, 86,

173, 174, 176-178, 181, 183, 184, 187,

188, 191, 199, 200.
Rice, F. O., 140.

Rice, J., 3, 139, 140, 143, 158, 171, 172.
Richardson, O. W., 124, 128, 193.
Righi, 123.
Rubens, 41, 47, 56, 57, 64, 68, 87, 88, 90,

173, 190, 198, 2OO, 2OI.

Rutherford, 21, 91, 96, 97, 99, 100, 101,

119, 120, 121, 193.
Rydberg, 44, 100, 194.

SACKUR, 87, 95, 145.
Scheel, 197.
Scheffer, 156.
Schultze, 6.
Schumann, 47.
Sommerfeld, 194.
Souder, 132.
Stark, 43, 49, 194.
Stefan, 39, 42, 43, 72, 172.
Stern, 49, 202.
Stewart, Balfour, 172, 173.
Stokes, 157.

TATE, 128, 129.

Tetrode, 95.

Thirring, 190.

Thomsen, 147, 149.

Thomson, Sir J. J., 2, 35, 116, 119, 124-

127, 151, 182, 183.
Tolman, 18.

VOLMER, 134.

WAAGE, 141.

Walden, 153.

Wanner, 42.

Warburg, 137, 194.

von Wartenberg, 57, 90, 148, 198.

Webster, 45, 120.

Weiss, 118.

Whetham, 6, 7.

Whiddington, 120, 121.

Wiedmann, 131, 132.

Wien, 39, 42, 47, 136, 138, 142, 143, 173,

177.
Wurmser, 138.

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