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INTERNATIONAL SERIES IN PHYSICS
LEE A. DtrBRIDGE, CONSULTING EDITOR

INTRODUCTION

CHEMICAL PHYSICS

INTERNATIONAL SERIES IN
PURE AND APPLIED PHYSICS

G. P. HARNWELL, Consulting Editor

BRILLOUIN— WAVE PROPAGATION IN PERIODIC STRUCTURES

CADY— PIEZOELECTRICITY

CLARK— APPLIED X-RAYS

CURTIS— ELECTRICAL MEASUREMENTS

DAVEY— CRYSTAL STRUCTURE AND ITS APPLICATIONS

EDWARDS— ANALYTIC AND VECTOR MECHANICS

GURNEY— INTRODUCTION TO STATISTICAL MECHANICS

HARDY AND PERRIN— THE PRINCIPLES OF OPTICS

HARNWELL— ELECTRICITY AND ELECTROMAGNETISM

HARNWELL AND LIVINGOOD— EXPERIMENTAL ATOMIC PHYSICS

HOUSTON— PRINCIPLES OF MATHEMATICAL PHYSICS

HUGHES AND DUBRIDGE— PHOTOELECTRIC PHENOMENA

HUND— HIGH-FREQUENCY MEASUREMENTS

INGERSOLL, ZOBEL, AND INGERSOLL — HEAT CONDUCTION

KEMBLE— PRINCIPLES OF QUANTUM MECHANICS

KENNARD— KINETIC THEORY OF GASES

KOLLER— THE PHYSICS OF ELECTRON TUBES

MORSE— VIBRATION AND SOUND

PAULING AND GOUDSMIT— THE STRUCTURE OF LINE SPECTRA

RICHTMYER AND KENNARD— INTRODUCTION TO MODERN PHYSICS

RUARK AND UREY— ATOMS, MOLECULES, AND QUANTA

SCHIFF— QUANTUM MECHANICS

SEITZ— THE MODERN THEORY OF SOLIDS

SLATER— INTRODUCTION TO CHEMICAL PHYSICS

MICROWAVE TRANSMISSION
SLATER AND FRANK — ELECTROMAGNETISM

INTRODUCTION TO THEORETICAL PHYSICS

MECHANICS

SMYTHE— STATIC AND DYNAMIC ELECTRICITY
STRATTON— ELECTROMAGNETIC THEORY
WHITE— INTRODUCTION TO ATOMIC SPECTRA
WILLIAMS— MAGNETIC PHENOMENA

Dr. Lee A. DuBridge was consulting editor of the series from 1939 to 1946.

INTRODUCTION

CHEMICAL PHYSICS

J. C. SLATER

Professor of Physics
Massachusetts Institute of Technology

FIRST EDITION"
SEVENTH IMPRESSION

McGRAW-HILL BOOK COMPANY, INC.

NEW YORK AND LONDON
1939

(\)PYRK8HT, 1939, BY TIIK

i BOOK COMPANY, INC.

PRINTED IN" THE UNITED STATES OP AMERICA

parts thereof, may not be reproduced

in any form without permission of

the publishers.

THE MAPLE PRESS COMPANY, YORK,

PREFACE

It is probably unfortunate that physics and chemistry over were
separated. Chemistry is the science of atoms and of the way they com-
bine. Physics deals with the interatomic forces and with the large-scale
properties of matter resulting from those forces. So long as chemistry
was largely empirical and nonmathematical, and physics had not learned
how to treat small-scale atomic forces, the two sciences seemed widely
separated. But with statistical mechanics and the kinetic theory on the
one hand and physical chemistry on the other, the two sciences began to
come together. Now that statistical mechanics has led to quantum theory
and wave mechanics, with its explanations of atomic interactions, there is
really nothing separating them any more. A few years ago, though their
ideas were close together, their experimental methods were still quite
different : chemists dealt with things in test tubes, making solutions, pre-
cipitating and filtering and evaporating, while physicists measured every-
thing with galvanometers and spectroscopes. But even this distinction
has disappeared, with more and more physical apparatus finding its way
into chemical laboratories.

A wide range of study is common to both subjects. The sooner we
realize this the better. For want of a better name, since Physical
Chemistry is already preempted, we may call this common field Chemical
Physics. It is an overlapping field in which both physicists and chemists
should be trained. There4 seems no valid reason why their training in it
should differ. This book is an attempt to incorporate some of the
material of this common field in a unified presentation.

What should be included in a discussion of chemical physics? Logi-
cally, we should start with fundamental principles. We should begin
with mechanics, then present electromagnetic theory, and should work
up to wave mechanics and quantum theory. By means of these wre
should study the structure of atoms and molecules. Then we should
introduce thermodynamics and statistical mechanics, so as to handle
large collections of molecules. With all this fundamental material we
could proceed to a discussion of different types of matter, in the solid,
liquid, and gaseous phases, and to an explanation of its physical and
chemical properties in terms of first principles. But if we tried to do all
this, we should, in the first place, be writing several volumes which would
include almost all of theoretical physics and chemistry; and in the
second place no one but an experienced mathematician could handle the

vi PREFACE

theory. For both of these reasons the author has compromised greatly
in the present volume, so as to bring the material into reasonable com-
pass and to make it intelligible to a reader with a knowledge of calculus
and differential equations, but unfamiliar with the more difficult branches
of mathematical physics.

In the matter of scope, most of the theoretical physics which forms a
background to our subject has been omitted. Much of this is considered
in the companion volume, " Introduction to Theoretical Physics," by
Slater and Frank. The effort has been made in the present work to pro-
duce a book which is intelligible without studying theoretical physics
first. This has been done principally for the benefit of chemists and
others who wish to obtain the maximum knowledge of chemical physics
with the minimum of theory. In the treatment of statistical mechanics
only the most elementary use of mechanics is involved. For that reason
it has not boon possible to give a complete discussion, although the parts
used in the calculations have been considered. Statistical mechanics has
been introduced from the standpoint more of the quantum theory than of
classical theory, but the quantum theory that is used is of a very elemen-
tary sort. It has seemed desirable to omit wave mechanics, which
demands more advanced mathematical methods. In discussing atomic
and molecular structure and the nature of interatomic forces, descriptive
use has boon made of the quantum theory, but again no detailed use of it.
Thus it is hoped that the reader with only a superficial acquaintance with
modern atomic theory will be able to read the book without great diffi-
culty, although, of course, the reader with a knowledge of quantum
theory and wave mechanics will have a great advantage.

Finally in the matter of arrangement the author has departed from
the logical order in the interest of easy presentation. Logically one
should probably begin with the structure of atoms and molecules, crystals
and liquids and gases; then introduce the statistical principles that
govern molecules in large numbers, and finally thermodynamics, which
follows logically from statistics. Actually almost exactly the opposite
order has been chosen. Thermodynamics and statistical mechanics come
first. Then gases, solids, and liquids are treated on the basis of thermo-
dynamics and statistics, with a minimum amount of use of a model.
Finally atomic and molecular structure are introduced, together with a
discussion of different types of substances, explaining their interatomic
forces from quantum theory and their thermal and elastic behavior from
our thermodynamic and statistical methods. In this way, the historical
order is followed roughly, and, at least for chemists, it proceeds from
what are probably the more familiar to the less familiar methods.

It is customary to write books either on thermodynamics or on
statistical mechanics; this one combines both. It seems hardly necessary

PREFACE Vll

to apologize for this. Both have their places, and both are necessary in a
complete presentation of chemical physics. An effort has been made to
keep them separate, so that at any time the reader will be clear as to
which method is being used. In connection with thermodynamics, the
method of Bridgman, which seems by far the most convenient for prac-
tical application, has been used.

There is one question connected with thermodynamics, that of
notation. The continental notation and the American chemical notation
of Lewis and Randall are quite different. Each has its drawbacks. The
author has chosen the compromise notation of the Joint Committee of
the Chemical Society, the Faraday Society, and the Physical Society (all
of England), which preserves the best points of both. It is hoped that
this notation, which has a certain amount of international sanction, may
become general among both physicists and chemists, whose poblems are
similar enough so that they surely can use the same language.

In a book like this, containing a number of different types of material,
it is likely that some readers and teachers will want to use some parts,
others to use other parts. An attempt has been made to facilitate such
use by making chapters and sections independent of each other as far as
possible. The book has been divided into three parts: Part I, Thermo-
dynamics, Statistical Mechanics, and Kinetic Theory; Part II, Gases,
Liquids, and Solids; Part III, Atoms, Molecules, and the Structure of
Matter. The first part alone forms an adequate treatment of thermo-
dynamics and statistical theory, and could be used by itself. Certain of
its chapters, as Chap. V on the Fermi-Dirac and Einstein-Bose Statistics,
Chap. VI on the Kinetic Method and the Approach to Thermal Equilib-
rium, and Chap. VII on Fluctuations, can be omitted without causing
much difficulty in reading the following parts of the book (except for the
chapters on metals, which depend on the Fermi-Dirac statistics). In
Part II, most of Chap. IX on the Molecular Structure and Specific Heat
of Polyatomic Gases, Chap. X on Chemical Equilibrium in Gases, parts
of Chap. XII on Van der Waals' Equation and Chap. XIII on the
Equation of State of Solids, Chap. XV on The Specific Heat of Com-
pounds, Chap. XVII on Phase Equilibrium in Binary Systems, and
Chap. XVIII on Phase Changes of the Second Order are not necessary
for what follows. In Part III, Chap. XIX on Radiation and Matter,
Chap. XX on lonization and Excitation of Atoms, and Chap. XXI on
Atoms and the Periodic Table will be familiar to many readers. Much
of the rest of this part is descriptive; one chapter does not depend on
another, so that many readers may choose to omit a considerable portion
or all, of this material. It will be seen from this brief enumeration that
selections from the book may be used in a variety of ways to serve the
needs of courses less extensive than the whole book.

viii PREFACE

The author hopes that this book may serve in a minor way to fill the
gap that has grown between physics and chemistry. This gap is a
result of tradition and training, not of subject matter. Physicists and
chemists are given quite different courses of instruction; the result is
that almost no one is really competent in all the branches of chemical
physics. If the coming generation of chemists or physicists could receivo
training, in the first place, in empirical chemistry, in physical chemistry,
in metallurgy, and in crystal structure, and, in the second place, in
theoretical physics, including mechanics and electromagnetic theory, and
in particular in quantum theory, wave mechanics, and the structure of
atoms and molecules, and finally in thermodynamics, statistical
mechanics, and what we have called chemical physics, they would be
far better scientists than those receiving the present training in either
chemistry or physics alone.

The author wishes to indicate his indebtedness to several of his
colleagues, particularly Professors B. E. Warren and W. B. Nottingham,
who have read parts of the manuscript and made valuable comments.
His indebtedness to books is naturally vory great, but most of thorn arc
mentioned in the list of suggested references at the end of this volume.

J. C. SLATER.
CAMBRIDGE, MASSACHUSETTS,
September, 1939.

CONTENTS

PA an

PREFACE . . . v

PART I

THERMODYNAMICS, STATISTICAL MECHANICS,
AND KINETIC THEORY

CHAPTER I

HEAT AS A MODE OF MOTION

INTRODUCTION .... .3

1. THE CONSERVATION OF ENERGY .... . .3

2. INTERNAL ENERGY, EXTERNAL WORK, AND HEAT FLOW ... 6

3. THE ENTROPY AND IRREVERSIBLE PROCESSES . . 9

4. THE SECOND LAW OF THERMODYNAMICS . . . ... 12

CHAPTER II

THERMODYNAMICS

INTRODUCTION . . 16

1. THE EQUATION OF STATE ... . 16

2. THE ELEMENTARY PARTIAL DERIVATIVES .... 18

3. THE ENTHALPY, AND HELMHOLTZ AND GIBBS FREE ENERGIES 20

4. METHODS OF DERIVING THERMODYNAMIC FORMULAS 23

5. GENERAL CLASSIFICATION OF THERMODYNAMIC FORMULAS 27

6. COMPARISON OP THERMODYNAMIC AND GAS SCALES OF TEMPERATURE 30

CHAPTER III

STATISTICAL MECHANICS

INTRODUCTION . . . .32

1. STATISTICAL ASSEMBLIES AND THE ENTROPY ... . 32

2. COMPLEXIONS AND THE PHASE SPACE 36

3. CELLS IN THE PHASE SPACE AND THE QUANTUM THEORY .... 38

4. IRREVERSIBLE PROCESSES . . . . 43

5. THE CANONICAL ASSEMBLY 46

CHAPTER IV

THE MAXWELL-BOLTZMANN DISTRIBUTION LAW

INTRODUCTION 52

1. THE CANONICAL ASSEMBLY AND THE MAXWELL-BOLTZMANN DISTRIBUTION . 52

2. MAXWELL'S DISTRIBUTION OF VELOCITIES . 55

3t THE EQUATION OF STATE AND SPECIFIC HEAT OF PERFECT MONATOMIC GASES 58

4. THE PERFECT GAS IN A FORCE FIELD .... 62

X CONTENTS

PAOB
CHAPTER V

THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS

INTRODUCTION . 65

1. THE MOLECULAR PHASE SPACE 65

2. ASSEMBLIES IN THE MOLECULAR PHASE SPACE . . 68

3. THE FERMI-DIRAC DISTRIBUTION FUNCTION 72

4. THERM ODYNAMIC FUNCTIONS IN THE FERMI STATISTICS 76

5. THE PERFECT GAS IN THE FERMI STATISTICS. ... 80

6. THE EINSTEIN-ROSE DISTRIBUTION LAW 83

CHAPTER VI
THE KINETIC METHOD AND THE APPROACH TO

THERMAL EQUILIBRIUM
INTRODUCTION 86

1. THE EFFECT OF MOLECULAR COLLISIONS ON THE DISTRIBUTION FUNCTION IN

THE BOLTZMANN STATISTICS 86

2. THE EFFECT OF COLLISIONS ON THE ENTROPY ... ... 89

3. THE CONSTANTS IN THE DISTRIBUTION FUNCTION 92

4. THE KINETIC METHOD FOR FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS . 96

CHAPTER VII
FLUCTUATIONS
INTRODUCTION .101

1. ENERGY FLUCTUATIONS IN THE CANONICAL ASSEMBLY . . 101

2. DISTRIBUTION FUNCTIONS FOR FLUCTUATIONS 104

3. FLUCTUATIONS OF ENERGY \ND DENSITY 107

PART II
GASES, LIQUIDS, AND SOLIDS

CHAPTER VIII
THERMODYNAMIC AND STATISTICAL TREATMENT OF THE PERFECT

GAS AND MIXTURES OF GASES
INTRODUCTION 115

1. THERMODYNAMICS OF A PERFECT GAS 115

2. THERMODYNAMICS OF A MIXTURE OF PERFECT GASES 120

3 STATISTICAL MECHANICS OF A PERFECT GAS IN BOLTZMANN STATISTICS . . .124

CHAPTER IX
THE MOLECULAR STRUCTURE AND SPECIFIC HEAT OF

POLYATOMIC GASES

INTRODUCTION 130

1. THE STRUCTURE OF DIATOMIC MOLECULES 130

2. THE ROTATIONS OF DIATOMIC MOLECULES 134

3. THE PARTITION FUNCTION FOR ROTATION 138

4. THE VIBRATION or DIATOMIC MOLECULES 140

5. TH» PARTITION FUNCTION FOR VIBRATION 142

6. THE SPECIFIC HEATS OF POLYATOMIC GASES 145

CONTENTS xi

PAGB
CHAPTER X

CHEMICAL EQUILIBRIUM IN GASES

INTRODUCTION 150

1. RATES OP REACTION AND THE MASS ACTION LAW 151

2. THE EQUILIBRIUM CONSTANT, AND VAN'T HOFF'S EQUATION 154

3. ENERGIES OF ACTIVATION AND THE KINETICS OF REACTIONS .158

CHAPTER XI

THE EQUILIBRIUM OF SOLIDS, LIQUIDS, AND GASES

INTRODUCTION . . 100

1. THE COEXISTENCE OF PHASES 166

2. THE EQUATION OF STATE 169

3. ENTROPY AND GIBBS FREE ENERGY 170

4. THE LATENT HEATS AND CLAPEYRON'S EQUATION 174

5. THE INTEGRATION OF CLAPEYRON'S EQUATION AND THE VAPOR PRESSURE

CURVE 176

6. STATISTICAL MECHANICS AND THE VAPOR PRESSURE CURVE . .178

7. POLYMORPHIC PHASES OF SOLIDS 180

CHAPTER XII

VAN DER WAALS' EQUATION

INTRODUCTION ... . 182

1. VAN DER WAALS' EQUATION 182

2. ISOTHERMALS OF VAN DER WAALS' EQUATION . 184

3. GIBBS FREE ENERGY AND THE EQUILIBRIUM OF PHASES FOR A VAN DER WAALS

GAS 187

4. STATISTICAL MECHANICS AND THE SECOND VIRIAL COEFFICIENT 190

5. THE ASSUMPTIONS OF VAN DER WAALS' EQUATION . . 194

6. THE JOULE-THOMSON EFFECT AND DEVIATIONS FROM THE PERFECT GAS LAW 196

CHAPTER XIII

THE EQUATION OF STATE OF SOLIDS

INTRODUCTION . 199

1. THE EQUATION OF STATE AND SPECIFIC HEAT OF SOLIDS 199

2. THERMODYNAMIC FUNCTIONS FOR SOLIDS. . . ... . 205

3. THE STATISTICAL MECHANICS OF SOLIDS. . .... 211

4. STATISTICAL MECHANICS OF A SYSTEM OF OSCILLATORS 215

5. POLYMORPHIC TRANSITIONS 220

CHAPTER XIV

DEBYE'S THEORY OF SPECIFIC HEATS

INTRODUCTION . 222

1. ELASTIC VIBRATIONS OF A CONTINUOUS SOLID . 222

2. VlBRATIONAL FREQUENCY SPECTRUM OF A CONTINUOUS SOLID 225

3. DEBYE'S THEORY OF SPECIFIC HEATS 234

4. DEBYE'S THEORY AND THE PARAMETER y 238

CHAPTER XV

THE SPECIFIC HEAT OF COMPOUNDS

INTRODUCTION 241

1. WAVE PROPAGATION IN A ONE-DIMENSIONAL CRYSTAL LATTICE 241

2. WAVES IN A ONE-DIMENSIONAL DIATOMIC CRYSTAL 247

xii CONTENTS

PAGE

3. VIBRATION SPECTRA AND SPECIFIC HEATS OF POLYATOMIC CRYSTALS . . . 252

4. INFRARED OPTICAL SPECTRA OP CRYSTALS .... . 254

CHAPTER XVI

THE LIQUID STATE AND FUSION

INTRODUCTION ... . 256

1. THE LIQUID PHASE . . . 256

2. THE LATENT HEAT OF MELTING 258

3. THE ENTROPY OF MELTING 260

4. STATISTICAL MECHANICS AND MELTING 265

CHAPTER XVII

PHA8K EQUILIBRIUM IN BINARY SYSTEMS

INTRODUCTION . . . . 270

1. TYPES OF PHASES IN BINARY SYSTEMS. . 271

2. ENERGY AND ENTROPY OF PHASES OF VARIABLE COMPOSITION . 275

3. THE CONDITION FOR EQUILIBRIUM BETWEEN PHASES 278

4. PHASE EQUILIBRIUM BETWEEN MUTUALLY INSOLUBLE SOLIDS 283

5. LOWERING OF MELTING POINTS OF SOLUTIONS . 288

CHAPTER XVIII

PHASE CHANGES OF THE SECOND ORDER

INTRODUCTION . 291

1. ORDER-DISORDEU TRANSITIONS IN ALLOYS 293

2. EQUILIBRIUM IN TRANSITIONS OF THE Cu-ZN TYPE . 296

3. TRANSITIONS OF THE Cu-ZN TYPE WITH ARBITRARY COMPOSITION . 301

PART III
ATOMS, MOLECULES, AND THE STRUCTURE OF MATTER

CHAPTER XIX

RADIATION AND MATTER

INTRODUCTION . . . . 307

1. BLACK BODY RADIATION AND THE STEFAN-BOLTZMANN LAW . . 307

2. THE PLANCK RADIATION LAW 313

3. EINSTEIN'S HYPOTHESIS AND THE INTERACTION OF RADIATION AND MATTER 316

CHAPTER XX

IONIZATION AND EXCITATION OF ATOMS

INTRODUCTION . . . 32 J

1. BOHR'S FREQUENCY CONDITION . ... . 321

2. THE KINETICS OF ABSORPTION AND EMISSION OF RADIATION . . 324

3. THE KINETICS OF COLLISION AND IONIZATION .... 326

4. THE EQUILIBRIUM OF ATOMS AND ELECTRONS . . . 333

CHAPTER XXI
ATOMS AND THE PERIODIC TABLE

INTRODUCTION . . 336

1. THE NUCLEAR ATOM .... 336

CONTENTS xiii

PAGE

2. ELECTRONIC ENERGY LEVELS OF AN ATOM ... 338

3. THE PERIODIC TABLE OF THE ELEMENTS . 344

CHAPTER XXII

INTERATOMIC AND INTERMOLECULAR FORCES
INTRODUCTION ... 352

1. THE ELECTROSTATIC* INTERACTIONS BETWEEN RIGID ATOMS OR MOLECULES AT

LARGE DISTANCES 353

2. THE ELECTROSTATIC: OR COULOMB INTERACTIONS BETWEEN OVERLAPPING

RIGID ATOMS 361

3. POLARIZATION AND INTERATOMIC FORCES . 363

4. EXCHANGE INTERACTIONS BETWEEN ATOMS AND MOLECULES 367

5. TYPES OF CHEMICAL SUBSTANCES. ... ... . 375

CHAPTER XXIII
IONIC CRYSTALS
INTRODUCTION . . . 377

1. STRUCTURE OF SIMPLE BIN \RY IONIC COMPOUNDS 377

2. IONIC RADII 382

3. ENERGY AND EQUATION OF STATE OF SIMPLE IONIC LATTICES AT THE ABSOLUTE

ZERO . 385

4. THE EQUATION OF STATE OF THE ALKALI HALIDES 390

5. OTHER TYPES OF IONIC CRYSTALS . 396

6. POL ARIZ ABILITY AND UNSYMMETRICAL STRUCTURES . . 398

CHAPTER XXIV

THE HOMOPOLAR BOND AND MOLECULAR COMPOUNDS

INTRODUCTION 400

1. THE HOMOPOLAU BOND . 400

2. THE STRUCTURE OF TYPICAL HOMOPOLAR MOLECULES. . 402

3. GASEOUS AND LIQUID PHASES OF HOMOPOLAR SUBSTANCES 407

4. MOLECULAR CRYSTALS . 414

CHAPTER XXV

ORGANIC MOLECULES AND THEIR CRYSTALS

INTRODUCTION .... 420

1. CARBON BONDING IN ALIPATIIIC MOLECULES . 420

2. ORGANIC RADICALS 425

3. THE DOUBLE BOND, AND THE AROMVTIC COMPOUNDS 428

4. ORGANIC CRYSTALS . 432

CHAPTER XXVI

HOMOPOLAR BONDS IN THE SILICATES

INTRODUCTION 435

1. THE SILICON-OXYGEN STRUCTURE . 435

2. SILICON-OXYGEN CHAINS. . . . . . 436

3. SILICON-OXYGEN SHEETS 439

4. THREE-DIMENSIONAL SILICON-OXYGEN STRUCTURES . . . 441

CHAPTER XXVII
METALS

INTRODUCTION . ... 444

1. CRYSTAL STRUCTURES OF METALS . . 444

xiv CONTENTS

PAOB

2. ENERGY RELATIONS IN METALS ... 450

3. GENERAL PROPERTIES OF METALS . 456

CHAPTER XXVIII

THERMIONIC EMISSION AND THE VOLTA EFFECT

INTRODUCTION 460

1. THE COLLISIONS OP ELECTRONS AND METALS. . 460

2. THE EQUILIBRIUM OP A METAL \ND AN ELECTRON GAS 463

3. KINETIC DETERMINATION OP THERMIONIC EMISSION . . . . 464

4. CONTACT DIFFERENCE OF POTENTIAL 467

CHAPTER XXIX

THE ELECTRONIC STRUCTURE OF METALS

INTRODUCTION . . . 472

1. THE ELECTRIC FIELD WITHIN A MKTA.L 472

2. THE FREE ELECTRON MODEL OF A METAL 475

3. THE FREE ELECTRON MODEL AND THERMIONIC EMISSION 480

4. THE FREE ELECTRON MODEL AND ELECTRICAL CONDUCTIVITY . 484

.5. ELECTRONS IN A PERIODIC FORCE FIELD . ... 489

6. ENERGY BANDS, CONDUCTORS, AND INSULATORS . . 495

PROBABLE VALUES OF THE GENERAL PHYSICAL CONSTANTS 503

SUGGESTED REFERENCES 505

INDEX 507

PART I

THERMODYNAMICS, STATISTICAL MECHANICS,
AND KINETIC THEORY

CHAPTER I
HEAT AS A MODE OF MOTION

Most of modern physics and chemistry is based on three fundamental
ideas: first, matter is made of atoms and molecules, very small and very
numerous; second, it is impossible in principle to observe details of atomic
and molecular motions below a certain scale of smallncss; and third,
heat is mechanical motion of the atoms and molecules, on such a small
scale that it cannot be completely observed. The first and third of these*
ideas are products of the last century, but the second, the uncertainty
principle, the most characteristic result of the quantum theory, has arisen
since 1900. By combining these three principles, we have the theoretical
foundation for studying the branches of physics dealing with matter and
chemical problems.

1. The Conservation of Energy.- -From Newton's second law of
motion-one can prove immediately that the work done by an external
force cm a system during any nuitioiMHj^i^s Uin increase of kinetic emvrgy
of the system. This can be stated in the form

(1.1)

where KE stands for the kinetic energy, dW the infinitesimal clement of
work done on the system. Certain forces are called conservative; they
have the property that the work done by them when the system goes from
an initial to a final state depends only on the initial and final state, not on
the details of the motion from one state to the other. Stated technically,
we say that the work done between two end points depends only on the
end points, not on the path. A typical example of a conservative force
is gravitation; a typical nonconservative force is friction, in which the
longer the path, the greater the work done. For a conservative force,
we define the potential energy as

PE, = ~*W. (1.2)

This gives the potential energy at point 1, as the negative of the work done
in bringing the system from a certain state 0 where the potential energy
is zero to the state 1, an amount of work which depends only on the points
1 and 0, not on the path. Then we have

(1.3)

4 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. I

and, combining with Eq. (1.1),

+ PEl = KE* + PE* = E, (1.4)

where, since 1 and 2 are arbitrary points along the path and KE + PE
is the same at both these points, we must assume that KE + PE remains
constant, and may set it equal to a constant E, the total energy. This is
the law of conservation of energy.

To avoid confusion, it is worth while to consider two points connected
with the potential energy: the negative sign which appears in the defi-
nition (1.2), and the choice of the point where the potential energy is
zero. Both points can bo illustrated simply by the case of gravity acting
on bodies near the earth. Gravity acts down. We may balance its
action on a given body by an equal and opposite upward force, as by
supporting the body by the hand. We may then define the potential
energy of the body at height h as the work done by this balancing force
in raising the body through this height. Thus if the mass of the body is
m, and the acceleration of gravity g, tho force of gravity is —mg (positive
directions being upward), the balancing force is + nig, and the work dono
by the hand in raising the mass through height h is mgh, which we define
as the potential energy. Tho negative sign, then, comes because the
potential energy is defined, not as the work done by the force we are
interested in, but the work done by an equal and opposite balancing
force. As for the arbitrary position whero wo choose the potential energy
to bo zero, that appears in this example because \ve nan measure our
height h from any level \\o choose. It is important to notice that tho
same arbitrary constant appears essentially in the energy E. Thus, in
Eq. (1.4), if we chose to redefine our zero of potential energy, we should
have to add a constant to the total energy at each point of the path.
Another way of stating this is that it is only the difference E — PE whose
magnitude is determined, neither the total energy nor the potential energy
separately. For E — PE is the kinetic energy, which alono can be deter-
mined by direct experiment, from a measurement of velocities.

Most ftf*t.iifl.l f(frf»pK fiTn Tint, fionfiftf vn.ii vfi* f^p in fl.ln^ost fill n|*Rot,ir*fl.l
cases there is friction of one sort or another- _ And yo.t thor la^f rontnrv
has seen tho conservation of 'onofgy -built. np_fio that, H iff HOW regarded as
the most important principle of physics. The first step in this develop-
merit was the mechanical theory of heat, the sciences of thermodynamics
and statistical mechanics. Heat had for many years been considered as a
fluid, sometimes called by the name caloric, which was abundant in hot
bodies and lacking in cold ones. This theory is adequate to explain
calorimetry, the science predicting the final temperature if substances of
different initial temperatures are mixed. Mixing a cold body, lacking in
caloric, with a hot one, rich in it, leaves the mixture with a medium

SEC. 1] HEAT AS A MODE OF MOTION 5

amount of heat, sufficient to raise it to an intermediate temperature.
But early in the nineteenth century, difficulties with the theory began to
appear. As we look back, we can sec that these troubles came from tho
implied assumption that the caloric, or heat, was conserved. In a
calorimetric problem, some of the caloric from the hot body flows to the
cold one, leaving both at an intermediate temperature, but no caloric is
lost. It was naturally supposed that this conservation was universal.
The difficulty with this assumption may be soon as clearly as anywhere in
Rumford's famous observation on the boring of cannon. Rumford
noticed that a great deal of heat was given off in the process of boring.
The current explanation of this was that the chips of metal had their heal,
capacity reduced by the process of boring, so that tho heat which was
originally present in them was able to raise them to a higher temperature.
Rumford doubted this, and to demonstrate it ho used a very blunl tool,
which hardly removed any chips at all and yet produced oven moio heat
than a sharp tool. He showed by his experiments beyond any doubt
that heat could be produced continuously and in apparently unlimited
quantity, by the friction. Surely this was impossible if heat, or caloric,
were a fluid which was conserved. And his conclusion stated essentially
our modern view, that heat is really a form of energy, convertible into
energy. In his words:1

What is Heat? Is there any such thing as an igneous fluid? Is there any
thing that can with propriety be called caloric? ... In reasoning on this subject,
we must not forget to consider that most remarkable circumstance, that the source
of Heat generated by friction, in these Experiments, appeared evidently to be
inexhaustible.

It is hardly necessary to add, that any thing which any insulated body, or
system of bodies, can continue to furnish without limitation, cannot possibly be
a material substance; and it appears to me to be extremely difficult, if not quite
impossible, to form any distinct idea of any thing, capable of being excited and
communicated, in the manner the Heat was excited and communicated in these
experiments, except it be MOTION.

From this example, it is clear that both conservation laws broke down
at once. In a process involving friction, energy is not conserved, but
rather disappears continually. At the same time, however, heat is not
conserved, but appears continually. Rumford essentially suggested that
the heat which appeared was really simply the energy which had dis-
appeared, observable in a different form. This hypothesis was not really
proved for a good many years, however, until Joule made his experiments
on the mechanical equivalent of heat, showing that when a certain
amount of work or mechanical energy disappears, the amount of heat

1 Quoted from W. F. Magie, "Source Book in Physics," pp. 160-161, McGraw-
Hill Book Company, Inc., 1935.

6 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. I

appearing is always the same, no matter what the process of trans-
formation may be. The calorie, formerly considered as a unit for
measuring the amount of caloric present, was seen to be really a unit of
energy, convertible into ergs, the ordinary units of energy. And it
became plain that in a process involving friction, there really was no loss
of energy. The mechanical energy, it is true, decreased, but there was
an equal increase in what we might call thermal energy, or heat energy,
so that the total energy, if properly defined, remained constant. This
generalization was what really established the conservation of energy as a
great and important principle. Having identified heat as a form of
energy, it was only natural for the dynamical theory of heat to be
developed, in which heat was regarded as a mode of motion of the mole-
cules, on such a small scale that it could not be observed in an ordinary
mechanical way. The extra kinetic and potential energy of the molcules
on account of this thermal motion was identified with the energy which
had disappeared from view, but had reappeared to be measured as heat.
With the development of thermodynamics and kinetic theory, conser-
vation of energy took its place as the leading principle of physics, which
it has held ever since.

2. Internal Energy, External Work, and Heat Flow. — We have seen
that the theory of heat is based on the idea of conservation of energy,
on the assumption that the; total energy of the universe is conserved, if
we include not only mechanical energy but also the mechanical equivalent
of the heat energy. JLt is not very convenient to talk about the whole
universe every time we wish to work a problem, however. Ordinarily,
thermodynamics deals with a finite system, isolated from its neighbors
by an imaginary closed surface. Everything within the surface belongs
to the system, everything outside is excluded. Usually, though not
always, a fixed amount of matter belongs to the system during the
thennodynamic processes we consider, no matter crossing the boundary.
Very often, however, we assume that energy, in the form of mechanical
or thermal energy, or in some other form, crosses the boundary, so that
the energy of the system changes. The principle of conservation, which
then becomes equivalent to the first law of thermodynamics, simply
states that the net increase of energy in the system, in any process,
equals the energy which has flowed in over the boundary, so that no
energy is created within the system. To make this a precise law, we
must consider the energy of the body and its change on account of flow
over the boundary of the system.

The total energy of all sorts contained within the boundary of the
system is called the internal energy of the system, and is denoted by U.
From an atomic point of view, the internal energy consists of kinetic
and potential energies of all the atoms of the system, or carrying it

SEC. 2] HEAT AS A MODE OF MOTION 7

further, of all electrons and nuclei constituting the system. Since
potential energies always contain arbitrary additive constants, the
internal energy U is not determined in absolute value, only differences of
internal energy having a significance, unless some convention is made
about the state of zero internal energy. Macroscopically (that is, viewing
the atomic processes on a large scale, so that we cannot see what indi-
vidual atoms are doing), we do not know the kinetic and potential energies
of the atoms, and we can only find tlm p.lm.ny> nf internal nnnrgy by
the amounts of ClierV added to tr> {^stpm n.nrn«K t.hn

and by making use of the law of conservation of energy. Thermo-
dynamics, which is a macroscopic science^ makes no attempt to analyze?
internal energy into its parts, as for Dimple mechanical energy and heat,
onotgy. It simply deals with t.hn totnl iiit.p.riml onprgy n.nH wifji i|[ti
change^.

Energy can (Miter the system in many ways, but most methods can bo
classified easily and in an obvious way into mechanical work and heat.
Familiar examples of external mechanical work are work done by pistons,
shafts, belts and pulleys, etc., and work done by external forces acting at a
distance, as gravitational work dono on bodies within the system on
account of gravitational attraction by external bodies. A familiar
example of heat flow is heat conduction across the surface. Convection
of hrat into the system is a possible form of energy interchange if atoms
and molecules are allowed to cross the surface, but not otherwise. Elec-
tric and magnetic work done by forces between bodies within the system
and bodies outside is classified as external work; but if the electro-
magnetic energy enters in the form of radiation from a hot body, it is
classified as heat. There are casos whore the distinction between the
two forms of transfer of energy is not clear and obvious, and electro-
magnetic radiation is one of them. In ambiguous cases, a definite
classification can be obtained from the atomic point of view, by means of
statistical mechanics.

In an infinitesimal change of the system^ the energy which has_
entered the system as heat flow is called dQ^ and the energy which has
left the system as mechanical work is called dW (so that the energy which
has entered as mechanical work is called — rfTF). The reason for choosing
this sign for dw is simply convention; thermodynamics is very often
used in the theory of heat engines, which produce work, so that the
important case is that in which energy leaves the system as mechanical
work, or when dW in our definition is positive. We see then that the1
total energy which enters the system in an infinitesimal change is
dQ — dW. By the law of conservation of energy, the increase in internal
energy in a process equals the energy which has entered the system:

dU = dQ - dW. _ (2.1)

8 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. I

Equation (2.1) is the mathematical statement of the first law of thermo-
dynamics. It is to be noted that both sides of the equation should be
expressed in the same units. Thus if internal energy and mechanical
work are expressed in ergs, the heat absorbed must be converted to ergs
by use of the mechanical equivalent of heat,

1 calorie = 4.185 X 107 ergs = 4.185 joules.

Or if the hoat absorbed is to bo measured in calories, the work and
internal energy should be converted into that unit.

It is of tho utmost importance to realize that the distinction between
heat flow n.nrf mop.tin.ninn.1 work, which WQ have madn in talking about
pnnrgry in fmn^f infr) n system, does not apply to the onergy once it is in
the system. It is completely fallacious to try to break down tho stato-
mont of Kq. (2.1) into two statements: "Tho iiicroaso of hoat onorgy of a
body oquals tho hoat which has fiowod in," and "Tho doorcase of mechan-
ical onorgy of a body oquals tho work clono by tho body on its surround-
ings. " For theso statements would correspond just to separate
conservation laws for hoat and moohanioal onorgy, and we havo soon in
the last section that such separate laws do not exist. To return to tho
last section, Rumford put a groat deal of mechanical work into his
cannon, produced no moohanioal results on it, but succeeded in raising
its temperature greatly. As we havo statod before, the energy of a
system cannot be differentiated or separated into a mechanical and a
thermal part, by any method of thermodynamics. Tho distinction
between heat and work is made in discussing energy in transit, and
only there.

The internal energy of a system depends only on tho state of the
system; that is, on pressure, volume, temperature, or whatever variables
arc used to describe tho system uniquely. Thus, the change1 in internal
energy between two states 1 and 2 depends only on these states. This
change of internal energy is an integral,

f/» - I/i = f*dU = f'dQ - f*dW. (2.2)

Since this integral depends only on the end points, it is independent of the
path used in going from state 1 to state 2. But the separate integrals

J> and £dW,

representing the total heat absorbed and the total work done in going
from state 1 to 2, are not independent of the path, but may be entirely
different for different processes, only their difference being independent
of path. Since these integrals are not independent of the path, they
cannot be written as differences of functions Q and W at the end points,
as / dU can be written as the difference of the C77s at the end points.

SEC. 3] HEAT Ati A MODE OF MOT JON 9

Such functions Q and W do not exist in any unique way, and we are not
allowed to use them. W would correspond essentially to the negative of
the potential energy, but ordinarily a potential energy function does no1
exist. Similarly Q would correspond to tho amount of hoat in the body,
but we have seen that this function also dors not exist. The fact that
functions Q and W do not exist^ or that f dQ and I dW are not independent
of patht really is only another way of saying that mechanical and heat
energy are interchangeable, and that the internal energy cannot T?
dividftfl into n. mnnhanical and a thermal part by thermodynamics.

At first sight, it seems too bad that / dQ is not independent of path,
for some such quantity would be useful. It would be pleasant to be able
to say, in a given state of the system, that the system had so and so much
heat energy. Starting from the absolute zero of temperature, where we
could say that the heat energy was zero, we could heat tho body up to the
stato we wore interested in, find J dQ from absolute zero up to this state,
and call that the heat energy. But the stubborn fact remains that
we should get different answers if we heated it up in different ways. For
instance, we might heat it at an arbitrary constant pressure until we
reached the desired temperature*, then adjust the pressure at constant
temperature to the desired value; or we might raise it first to the desired
pressure, then heat it at that pressure to the final temperature; or many
other equally simple processes. Each would give a different answer,
as we can easily verify. There is nothing to do about it.

It is to avoid this difficulty, and obtain something resembling the
" amount of heat in a body/' which yet has a unique meaning, that we
introduce the entropy. If T is the absolute temperature, and if the heat
dQ is absorbed at temperature T in a reversible way, then / dQ/T proves
tcTbe an integral independent of path, which evidently increases as the
body is heated: that is. as heat flows into it. This integral, from a fixed
zero point (usually taken to be the absolute zero of temperature), is
called the entropy. Like the internal energy, it is determined by the
state of the system, but unlike the internal energy it measures in a
certain way only heat energy, not mechanical energy. We next take up
the study of entropy, and of the related second law of thermodynamics.

3. The Entropy and Irreversible Processes. — Unlike the internal
energy and the first law of thermodynamics, the entropy and the second
law are relatively unfamiliar. Like them, however, their best inter-
pretation comes from the atomic point of view, as carried out in statistical
mechanics. For this reason, we shall start with a qualitative description
of the nature of the entropy, rather than with quantitative definitions
and methods of measurement.

The entropy is a quantity characteristic of thft state of ft system.
Tnpfl.siiring the randomness or disorder in the atomic arrangement of that

10 INTRODUCTION TO CHEMICAL 1'UYMCti [CHAP. 1

state. It increases when a body is heated, for then the random atomic
motion increases. But it ftlso innrnn^fia wfrpn a regular, orderly motion is
f npvnrfnH info n fftpdom motion. Thus, consider an enclosure containing
a small piece of crystalline solid at the absolute zero of temperature, in
a vacuum. The atoms of the crystal are regularly arranged and at rest;
its entropy is zero. Heat the crystal until it vaporizes. The molecules
are now located in random positions throughout the enclosure and have
velocities distributed at random. Both types of disorder, in the coordi-
nates and in the velocities, contribute to the entropy, which is now
large. But we could have reached the same final .state in a different way,
not involving the absorption of heat by the system. We could have
accelerated the crystal at the absolute zero, treating it as a projectile and
doing mechanical work, but without heat How. We could arrange a
target, so that the projectile would automatically strike the target,
without external action. If the mechanical work which we did on the
system were equivalent to the heat absorbed in the other experiment,
the final internal energy would be the same in each case. In our second
experiment, then, when the projectile struck the target it would be heated
so hot as to vaporize, filling the enclosure with vapor, and the final state
would be just the same as if the vaporization were produced directly.
The increase of entropy must then be the same, for by hypothesis the
entropy depends only on the state of the system, not on the path by
which it has reached that state. In the second case, though the entropy
has increased, no heat has been absorbed. Rather, ordered mechanical
energy (the kinetic energy of the projectile as a whole, in which each
molecule was traveling at the same velocity as every other) has been
converted by the collision into random, disordered energy. Just this
change results in an increase of entropy. It is plain that entropy cannot
be conserved, in the same sense that matter, energy, and momentum are.
For here entropy has been produced or created, just by a process of
changing ordered motion into disorder.

Many other examples of the two ways of changing entropy could
be given, but the one we have mentioned illustrates them sufficiently.
We have considered the increase of entropy of the system; let us now ask
if the processes can be reversed, and if the entropy can be decreased again.
Consider the first process, where the solid was heated gradually. Let us
be more precise, and assume that it was heated by conduction from a hot
body outside; and further that the hot body was of an adjustable temper-
ature, and was always kept very nearly at the same temperature as the
system we were interested in. If it were just at the same temperature,
heat would not flow, but if it were always kept a small fraction of a degree
warmer, heat would flow from it into the system. But that process can
be effectively reversed. Instead of having the outside body a fraction

SBC. 3] HEAT AS A MODE OF MOTION 11

of a degree warmer than the system, we )et it be a fraction of a degree
cooler, so that heat will flow out instead of in. Then things will cool
down, until finally the system will return to the absolute zero, and every-
thing will be as before. In the direct process heat flows into the system;
in the inverse process it flows out, an equal amount is returned, and when
everything is finished all parts of the system and the exterior are in
essentially the same state they were at the beginning. But now try to
reverse the second process, in which the solid at absolute zero was
accelerated, by means of external work, then collided with a target, and
vaporized. The last steps were taken without external action. To
reverse it, we should have the molecules of the vapor condense to form a
projectile, all their energy going into ordered kinetic energy. It would
have to be as shown in a motion picture of the collision run backward,
all the fragments coalescing into an unbroken bullet. Then we could
apply a mechanical brake to the projectile as it receded from the target,
and get our mechanical energy out again, with reversal of the process.
But such things do not happen in nature. The collision of a projectile
with a target is essentially an irreversible process, which never happens
backward, and a reversed motion picture of such an event is inherently
ridiculous and impossible. The statement that such events cannot be
reversed is one of the essential parts of the second law of thermodynamics.
If we look at the process from an atomic point of viow, it is clear why it
cannot reverse. The change from ordered to disordered motion is an
inherently likely change, which can be brought about in countless ways;
whereas the change from disorder to order is inherently very unlikely,
almost sure not to happen by chance. Consider a jigsaw puzzle, which
can be put together correctly in only one way. If we start with it put
together, then remove each picco and put it in a different place on tho
table, we shall certainly disarrange it, and we can do it in almost countless
ways; while if we start with it taken apart, and remove each pioce and
put it in a different place on the table, it is true that we may happen to
put it together in the process, but the chances arc enormously against it.
The real essence of irreversibility, however, is not merely the strong
probability against the occurrence of a process. It is something deeper,
coming from the principle of uncertainty. This principle, as we shall
see later, puts a limit on the accuracy with which we can regulate or
prescribe the coordinates and velocities of a system. It states that any
attempt to regulate them with more than a certain amount of precision
defeats its own purpose : it automatically introduces unpredictable pertur-
bations which disturb the system, and prevent the coordinates and
velocities from taking on the values we desire, forcing them to deviate
from these values in an unpredictable way. But this just prevents us
from being able experimentally to reverse a system, once the randomness

12 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. 1

has reached the small scale at which the principle of uncertainty operates,
To make a complicated process like a collision reverse, the molecules
would have to be given very definitely determined positions and velocities
so that they would just cooperate in such a way as to coalesce and become
unbroken again; any errors in determining these conditions would spoi
the whole thing. But we cannot avoid these errors. It is true that b>
chance they may happen to fall into lino, though the chance is minute
But the important point is that we cannot do anything about it.

From the preceding examples, it is clear that we must consider twr
types of processes: reversible and irreversible. Tjie essential feature oi
reversible processes is that things are almost balanced, almost in equilib-
rium, at every stage, so that an infinitesimal change will swing the motioi
from one direction to the other. Tjrflveraihle processes, on the othei
hand, involve complete departure from equilibrium, as in a collision. II
will be worth while to enumerate a few other common examples oi
irreversible processes. Heat flow from a hot body to a cold body at mon
than an infinitesimal difference of temperature is irreversible, for the heal
never flows from the cold to the hot body. Another example is viscosity
in which regular motion of a fluid is converted into random moleculai
motion, or heat. Still another is diffusion, in which originally unmixec
substances mix with other, so that they cannot bo unmixed again withoul
external action. In all these cases, it is possible of course to bring tin
system itself back to its original state. Even the projectile which ha*
been vaporized can be reconstructed, by cooling and condensing the
vapor and by recasting the material into a new projectile. But th<
surroundings of the system would have undergone a permanent change
the energy that was originally given the system as mechanical energy, tc
accelerate the bullet, is taken out again as heat, in cooling the vapor, sc
that the net result is a conversion of mechanical energy into heat in the
surroundings of the system. Such a conversion of mechanical energy
into heat is often called degradation of energy, and it is characteristic oi
irreversible processes. A reversible process is one which can be reversed
iu such a way that the system itself and its surroundings both return tc
their original condition; while an irreversible process is one such that th(
system cannot be brought back to its original condition without requir-
ing a conversion or degradation of some external mechanical energy intt
heat.

4. The Second Law of Thermodynamics. — We are now ready to give
a statement of the second law of ^hermnflyria^nica, in n^p of its many
forms: The entropy. a function only of the state of fl. system, increases in a
reversible

absorbed, T the absolute temperature at which it is absorbed) and
increases by a larger amount than dQ/T in an irreversible process.

SEC. 4] HEAT AS A MODE OF MOTION 13

This statement involves a number of features. First, it gives a way
of calculating entropy. By sufficient ingenuity, it is always possible to
find reversible ways of getting from any initial to any final state, pro-
vided both arc equilibrium states. Then we can calculate / dQ/T for
such a reversible path, and the result will bo the change of entropy
between the two states, an integral independent of path. We can then
measure entropy in a unique \vay. If we now go from the same initial to
the same final stale by an irreversible path, the change of entropy must,
still be the same, though now / dQ/T must necessarily be smaller than
before, and hence smaller than the change in entropy. We see that,
the heat absorbed in an irreversible path must be less than in a reversible
path between the same end points. Since ^hfc change in internal energy
must be the same in either ease, the first law then tells us that the external
work done by the system is less for the irreversible path than for the
reversible one. If our system is a heat engine, whose object is to absorb
heat and do mechanical work, \\e see that the mechanical work accom-
plished will be less for an irreversible engine than for a reversible one,
operating between the same end points.

It is interesting to consider the limiting case of adiabatic processes,
processes in which the system interchanges no heat witkihc surroundings,
the only changes in internal energy coming from mechanical work. We
see that in a. reversible adiabatic process the entropy does not change
(a convenient way of describing such processes). In an irreversibly
fljin.hn.fip prr>n^^ foe entropy increases. In particular, for a system
entirely isolated from its surroundings, the; entropy increases whenever
irreversible processes occur within it. An isolated system in which
irreversible processes can occur is surely not in a steady, equilibrium state;
the various examples which we have considered are the rapidly moving
projectile, a body with different temperatures at different parts (to allow
heat conduction), a fluid with mass motion (to allow viscous friction),
a body containing two different materials not separated by an impervious
wall (to allow diffusion). All these systems have less entropy than the
state of thermal equilibrium corresponding to the same internal energy,
which can be reached from the original state by irreversible processes
without interaction with the outside. This state of thermal equilibrium
is one in which the temperature is everywhere constant, there is no mass
motion, and where substances are mixed in such a way that there is no
tendency to diffusion or flow of any sort. A condition for thermal
equilibrium, which is often applied in statistical mechanics, is that the
equilibrium state is that of highest entropy consistent with the given
internal energy and volume.

These statements concerning adiabatic changes, in which the entropy
can only increase, should not cause one to forget that in ordinary changes,

14 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. I

in which heat can be absorbed or rejected by the system, the entropy
can either increase or decrease. In most thermodynamic problems, wre
confine ourselves to reversible changes, in which the only way for the
entropy to change is by heat transfer.

We shall now state the second law in a mathematical form which is
very commonly used. We let S denote the entropy. Our previous
statement is then dS ^ dQ/T, or T dS ^ dQ, the equality sign holding
for the reversible, the inequality for irreversible, processes. But now we
use the first law, Eq. (2.1), to express dQ in terms of dU and dW. The
inequality becomes at once

J^dS ^ dU + dW, ' (4.1)

the mathematical formulation of the second law. For reversible . pror-
cssesf which we ordinarily consider, the ecmalitv sign is to be used.

The second law may be considered as a postulate. We shall see in
Jhap. II that definite consequences can be drawn from it, and they
prove to be always in agreement with experiment. We notice that
in stating it, we have introduced the temperature without apology, for
the first time. This again can be justified by its consequences: the
temperature so defined proves to agree with the temperature of ordinary
experience, as defined for example by the gas thermometer. Thermo-
dynamics is the science that simply starts by assuming the first and
second laws, and deriving mathematical results from them. Both laws
are simple and general, applying as far as \vc know to all sorts of processes.
As a result, we can derive simple, general, and fundamental results from
thermodynamics, which should be independent of any particular assump-
tions about atomic and molecular structure, or such things. Thermo-
dynamics has its drawbacks, however, in spite of its simplicity and
generality. In the first place, there are many problems which it simply
cannot answer. These are detailed problems relating, for insTanC^to
the equation of state and specjfi^ hfigf nf r^t.i/nilar types of substances.
Thfirnqftdynprnifta must assume JJiat_thp«p qnanfjfog «TP determined by
experiment* once they are known, it can predict certain relationships
between observed quantities, but it is unable to say what values the
quantities must have. IrT addition to this, thermodynamics is limited
to the discussion of problems in equilibrium. This is on account of the
form of the second law, which can give only qualitative, and not quanti-
tative, information about processes out of equilibrium.

Statistical mechanics is a much more detailed science than thermo-
dynamics, and for that reason is in some ways more complicated. It
undertakes to answer the questions, how is each atom or molecule of the
substance moving, on the average, and how do these motions lead to
observable large scale phenomena? For instance, how do the motions

SEC. 4] HEAT AS A MODE OF MOTION 15

of the molecules of a gas lead to collisions with a wall which we interpret
as pressure? Fortunately it is possible to derive some very beautiful
general theorems from statistical mechanics. In fact, one can give
proofs of the first and second laws of thermodynamics, as direct conse-
quences of the principles of statistical mechanics, so that all the results
of thermodynamics can be considered to follow from its methods. But
it can go much further. It can start with detailed models of matter
and work through from them to predict the results of large scale experi-
ments on the matter. Statistical mechanics thus is much more powerful
than thermodynamics, and it is essentially just as general. It is some-
what more complicated, however, and somewhat more dependent on the
exact model of the structure of the material which we use. Like thermo-
dynamics, it is limited to treating problems in equilibrium.

Kinetic theory 'is a study of the rates of atomic and molecular proc-
esses, treated by fairly direct methods, without much benefit of general
principles. If handled properly, it is an enormously complicated subject,
though simple1 approximations can be made in particular cases. It is
superior to statistical mechanics and thermodynamics in just two respects.
In the first place, it makes use only of well-known and elementary
methods, arid for that reason is somewhat more comprehensible at first
sight than statistical mechanics, with its more advanced laws. In the
second place, it can handle problems out of equilibrium, such as the rates
of chemical reactions and other processes, which cannot be treated by
thermodynamics or statistical mechanics.

We see that eadi of our three sciences of heat has its own advantages.
A properly trained physicist or chemist should know all three, to be able
to use whichever is most suitable in a given situation. We start witli
thermodynamics, since it is the most general and fundamental method,
taking up thermodynarnic calculations in the next chapter. Following
that we treat statistical mechanics, and still later kinetic theory. Only
then shall we be prepared to make a real study of the nature of matter.

CHAPTER II
THERMODYNAMICS

In the last chapter, we became acquainted with the two laws of
thermodynamics, but we have not seen how to use them. In this
chapter, we shall learn the rules of operation of thermodynamics, though
we shall postpone actual applications until later. It has already been
mentioned that thermodynamics can give only qualitative information
for irreversible processes. Thus, for instance, the second law may be
stated

dW Z TdS - dU, (1)

giving an upper limit to the work done in an irreversible process, but not
predicting its exact amount. Only for reversible processes, where the
equality sign may bo used, can thermodynamics make definite predictions
of a quantitative sort. Consequently almost all our work in this chapter
will deal with reversible systems. We shall find a number of differential
expressions similar to Eq. (1), and by proper treatment we can convert
these into equations relating one or more partial derivatives of one
thcrmodynamic variable with respect to another. Such equations,
called thermodynamic formulas, often relate different quantities all of
which can be experimentally measured, and hence furnish a check on the
accuracy of the experiment. In cases where1 one of the quantities is
difficult to measure, they can be used to compute one of the quantities
from the others, avoiding the necessity of making the experiment at all.
There are a very great many thermodynamic formulas, and it would be
hopeless to find all of them. But we shall go into general methods of
computing them, and shall set up a convenient scheme for obtaining any
one which we may wish, with a minimum of computation.

Before starting the calculating of the formulas, we shall introduce
several new variables, combinations of other quantities which prove to be
useful for one reason or another. As a matter of fact, we shall work with
quite a number of variables, some of which can be taken to be inde-
pendent, others dependent, and it is necessary to recognize at the outset
the nature of the relations between them. In the next section we consider
the equation of state, the empirical relation connecting certain thermo-
dynamic variables.

1. The Equation of State. — In considering the properties of matter,
our system is ordinarily a piece of material enclosed in a container and

SEC. 1] THERMODYNAMICS 17

subject to a certain hydrostatic pressure. This of course is a limited type
of system, for it is not unusual to have other types of stresses acting, such
as shearing stresses, unilateral tensions, and so on. Thermodynamics
applies to as general a system as we please, but for simplicity we shall limit
our treatment to the conventional case where the only external work is
done by a change of volume, acting against a hydrostatic pressure. That
is, if P is the pressure and V the volume of the system, we shall have

dW = PdV. (1.1)

In any case, even with much more complicated systems, the work done
will have an analogous form; for Eq. (1.1) is simply a force (P) times a
displacement (dV), and we know that work can always be put in such
a form. If there is occasion to set up the thermodynamic formulas for a
more general type of force than a pressure, we simply set up dW in a form
corresponding to P]q. (1.1), and proceed by analogy with the derivations
which we shall give here.

We now have a number of variables : P. V} rl\ U . and AS. How many
of these T we mav ask, are independent.? The_ answer is, any two. For
example, with a given system, we may fix the pressure and temperature
Then in general the volume is determined, as we can find experimentally
The experimental relation giving volume as a function of pressure and
temperature is called the equation of state. Ordinarily, of course, it is
not a simple analytical equation, though in special cases like a perfect gas
it may be. Instead of expressing volume as a function of pressure and
temperature, we may simply say that the equation of state expresses a
relation between these three variables, which may equally well give, pres-
sure as a function of temperature and volume, or temperature as a
function of volume and pressure. Of these three variables, two are inde-
pendent, one dependent, and it is immaterial which is chosen as the
dependent variable.

The equation of ata.t.e. doos not, include all the experimental informa-
tion which wo n^ist have about a system or substance. We need 'to know
also its hftftt pfl.pu>it.y or specific heat, as a. function of temperature. Sup-
pose, for instance, that we know the specific heat at constant pressure
CP as a function of temperature at a particular pressure. Then we can
find the difference of internal energy, or of entropy, between any two
states. From the first state, we can go adiabatically to the pressure at
which we know Cp. In this process, since no heat is absorbed, the change
of internal energy equals the work done, which we can compute from the
equation of state. Then we absorb heat at constant pressure, until we
reach the point from which another adiabatic process will carry us to the
desired end point. The change of internal energy can be found for the
process at constant pressure, since there we know CP, from which we can

18 INTRODUCTION TO CHEMICAL PHYSICS |CiiAi>. II

find the heat absorbed, and since the equation of state will tell us the
work done; for the final adiabatic process we can likewise find the work
done and hence the change of internal energy. Similarly we can find the
change in entropy between initial and final stale. In our particular case,
assuming the process to be carried out reversibly, the entropy will not
change along the adiabatics, but the change of entropy will be

dQ CrdT
T T

in the process at constant pressure. We see, in other words, that the
difference of internal energy or of entropy bnt,ween fl,Tiy twn «t"^« can
be found if we know equation of state and specific heat, and since both
these quantities have arbitrary additive constants, this is all the informa-
tion which we can expect to obtain about them anyway.

Given the equation of state and specific hoatr we snn that we can
obtain all but two of the quantities P, Fy T, U, S, provided those two are
known. We have shown this if two of the three quantities P, F, T an*
known; but if U and S are determined by these quantities, that means
simply that two out of the five quantities are independent, the rest
dependent. It is then possible to use any two as independent variables.
For instance, in thermodynamics it is not unusual to use T and S, or V
and >S, as independent variables, expressing everything else as functions
of them.

2. The Elementary Partial Derivatives. — We can set up a number of
familiar partial derivatives and thermodynamic formulas, from the
information which we already have. We have five variables, of which
any two are independent, the rest dependent. We can then set up the
partial derivative of any dependent variable with respect to any inde-
pendent variable, keeping the other independent variable constant. A
notation is necessary showing in each case what are the two independent
variables. This is a need not ordinarily appreciated in mathematical
treatments of partial differentiation, for there the independent variables
are usually determined in advance and described in words, so that there
is no ambiguity about them. Thus, a notation, peculiar to thermody-
namics, has been adopted. In any partial derivative, it is obvious that
the quantity being differentiated is one of the dependent variables, and
the quantity with respect to which it is differentiated is one of the inde-
pendent variables. It is only necessary to specify the other independent,
variable, the one which is held constant in the differentiation, and the
convention is to indicate this by a subscript. Thus (dS/dT)Py which is
ordinarily read as the partial of S with respect to T at constant P, is the
derivative of S in which pressure and temperature are independent vari-
ables. This derivative would mean an entirely different thing from the
derivative of S with respect to T at constant V, for instance.

SEC. 2| THERMODYNAMICS 19

There are a number of partial derivatives which have elementary
meanings. Thus, consider the thermal expansion. This is the fractional
increase of volume per unit rise of temperature, at constant pressure:

Thermal expansion — TJ -r-= 1 • (2.1)
.. V\dl/

Similarly, the isothermal compressibility is the fractional decrease of
volume per unit increase of pressure, at constant temperature:

Isothermal compressibility = ^ilv

This is the compressibility usually employed; sometimes, as in considering
sound waves, we require the adiabatic compressibility, the fractional
decrease of volume per unit increase of pressure, when no heat flows in or
out. If there is no heat flow, the entropy is unchanged, in a reversible
process, so that an adiabatic process is one at constant entropy. Then
we have

Adiabatic compressibility = — TT( ~7; 1 • (2.3)

v V \ol Js

The specific heats have simple formulas. At constant volume, the heat
absorbed equals the increase of internal energy, since no work is done.
Since the heat absorbed also equals the temperature times the change of
entropy, for a reversible process, and since the heat capacity at constant
A^olume CV is the heat absorbed per unit change of temperature at constant
volume, we have the alternative formulas

To find the heat capacity at constant pressure Cp, we first write the form-
ula for the first and second laws, in the case we are working with, where
the external work comes from hydrostatic pressure and where all processes
are reversible:

dU = TdS -PdV,
or

TdS = dU + PdV. (2.5)

From the second form of Kq. (2.5), we can find the heat absorbed, or
T dS. Now Cr is the heat absorbed, divided by the change of tempera-
ture, at constant pressure. To find this, we divide Eq. (2.5) by dTy
indicate that the process is at constant P, and we have

20 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. 11

Here, and throughout the book, we shall ordinarily mean by CV and CV
not the specific heats (heat capacities per gram), but the heat capacities
of the mass of material with which we are working; though often, where
no confusion will arise, wo shall refer to them as the specific heats.

From the first and second laws, Eq. (2.5), we can obtain a number of
other formulas immediately. Thus, consider the first form of the equa-
tion, dU = T dS — PdV. From this we can at once keep the volume
constant (set dV = 0), and divide by dS, obtaining

'dt/\-r. (2.7)

Similarly, keeping entropy constant, so that we have an adiabatic proc-
ess, we have

/*TT\

= -P. (2.8)

But we could equally well have used the second form of Eq. (2.5), obtain-
ing

(dS\ _ 1

\w)v ~ T'

dS\ _P

~ r (2-9)

From these examples, it will be clear how formulas involving partial
derivatives can be found from differential expressions like Eq. (2.5).

3. The Enthalpy, and Helmholtz and Gibbs Free Energies. — Wo
notice that Eq. (2.6) for the specific heat at constant pressure is rather
complicated. We may, however, rewrite it

\a(u+PV)\ / , n

Cp = [ — dT—\p'( (3>1)

for I ^TJJT- I = P[ -^TJ ) ) since P is held constant in the differentiation

L d' > wA
The quantity U + PV comes in sufficiently often so that it is worth giving
it a symbol and a name. We shall call it the enthalpy, and denote it by
H. Thus we have

H = U + PV^
dH = dU +PdV + VdP

= TdS+ VdP, (3.2)

using Eq. (2.5). From Eq. (3.2), we see that if dP = 0, or if the process
is taking place at constant pressure, the change of the enthalpy equals the
heat absorbed. This is the feature that makes the enthalpy a useful
quantity. Most actual processes are carried on experimentally at con-

SBC. 3] THERMODYNAMICS 21

stant pressure, and if we have the enthalpy tabulated or otherwise known,
we can very easily find the heat absorbed. We see at once that

Cp = \JT)P' (3-3)

a simpler formula than Eq. (2.6). As a matter of factT the enthalpy fills,
essentially the role for processes at constant pressure which the internal
e tie rgy docs for processes at constant volume. Thus the first form of
Eq. (2.5), dU = T dS - P dV, shows that the heat absorbed at constant
volume equals the increase of internal energy, just as Eq. (3.2) shows that
the heat absorbed at constant pressure equals the increase of the enthalpy.
In introducing the entropy, in the last chapter, we stressed the idea
that it measured in some way the part of the energy of the body bound up
in heat, though that statement could not be made without qualification.
The entropy itself t of course, has not the dimcnsigns of energy, hut, the
product TS has. This quantity TS is sometimes called the bound energy,
and in a somewhat closer wav it represents the o.no.rgy bound RS VIPHJ. ln
any process, the change in TS is given by T dS + SdT. If now the
process is reversible and isothermal (as for instance the absorption of
heat by a mixture of liquid and solid at the melting point, whore heat can
be absorbed without change of temperature, merely melting more of the
solid), dT = 0, so that d(TS) = T dS = dQ. Thus the increase of bound
energy for a reversible isothermal process really equals the heat absorbed.
This is as far as the bound energy can be taken to represent the energy
hound as heat; for a nonisothermal process the change of hound energy
np Inngpr equals the hoat absorbcdj and as we have seenr no quantity
which is ji function of the state alone can represent the total heat absorbed

If the bound energy TS represents in a sense the energy bound fls
heat, the remaining part of the internal energy, U — TS^ should be in
the same sense the mechanical part of the energy, which is available to do
mechanical jwork. We sjiall call this part of the energy the Helmholtz
free energy, and denote it by A. Let us consider the change of the Helm-
holtz free energy in any process. We have

A = U - TS,
dA =dU- TdS- SdT. (3.4)

By Eq. (1) this is

dA ^ -dW - SdTy
or

-dA 2 dW + SdT. (3.5)

22 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. II

For a system at constant temperature, this tells us that the work done
is less than or equal to the decrease in the Helmholtz free energy. The
Helmholtz free energy then measures the maximum ^rk which can be
done by foe system in nfn isothcrma] flhfl^g0 For a process at constant
temperature, in which at the same time no mechanical work is done, the
right side of Eq. (3.5) is zero, and wo see that in such a process the Holm-
holtz free energy is constant for a reversible process, but decreases for an
irreversible process. The Holmholtz free energy will decrease until tho
system roaches an equilibrium state, whon it will have reached tho mini-
mum value consistent with the temperature and with the fact that no
external work can be done.

For a system in equilibrium under hydrostatic pressure, wo may
rewrite Kq. (3.5) as

<IA = -PdV - Nrf7\ (:«))

.suggesting that the convenient variables in which to express the4 Holm-
holtz free energy are the volume} and the temperature. In the1 case1 of
equilibrium, wo find from Eq. (3.6) tho important relations

(dA\

Ul = ^

Tho first of those1 shows that, at constant temperature, the Holmholtz
free energy has some of the properties of a potential energy, in that its
negative derivative} with respect to it coordinate (tho volume) givos tho
force (the) pressure). If A is known as a function of V and T, the first
Eq. (3.7) givos a relation between P, V, and ?T, or the equation of state.
From the second, we> know entropy in terms of temperature and volume,
and differentiating with respect to temperature at constant volume, using
Kq. (2.4), we can find the specific heat. Thus a knowledge of tho Holm-
holtz free energy as a function of volume and temperature gives both the
equation of state and specific heat, or complete information about the
system.

Instead of using volume and tofnipfirfl.tiire as independent variables,
however^ wo more often wish to use pressure and temperature. In this
case, instead of using. [fa* Hnlmhnlt? froo energy, it is more convenient to
use tho Gibbs free .energy G} defined bv the equations

f / = // - TS = U + PV - TS = A + PV- (3.8)

It will be seen that this function stands in the same relation to the*
enthalpy that the Helmholtz free energy does to the internal energy. We
can now find the change of the Gibbs free energy G in any process. By
definition, we have dG = dH - T dS - S dT. Using Eq. (3.2), this is
dO = dU +PdV + VdP - TdS-SdT, and by Eq. (1) this is

dG :£ V dP - S dT. (3.9)

SEC. 4] THERMODYNAMICS 23

For a system at constant pressure anc} temperature, we see that the Gibbs
free energy is constant for a reversible process but decreases for an irrever-
sible process, reaching a minimum value consistent with the pressure and
temperature for the equilibrium state; just as for a system at constant
volume the Holmholtz free energy is constant for a reversible process but
decreases for an irreversible process. As with A, we can get the equation
of state and specific heat from the derivatives of (7, in equilibrium. We
have

the first of these giving the volume as a function of pressure and tempera-
ture, the second the entropy as a function of pressure and temperature,
from which we can find CP B^mFSFrts of Eq. (2.6).

Thp (^rilifys jjffi^ nnnrgy O is pfl.yt(jf?ularlv iniDOrfaftUti nn gM^-fmntj nf f]rtiMM 1

physical processes that occur at constant pressure and temperature The
most important of these processes^ a change of phase, as the melting of a
solid or the vaporization of a liquid If unit mass of a substance chan&es
phase reycrsiblv.tit constant pro^n™ *"<\ ^^"p'^+urffj thr tnf.n.1 (li^lm
frecMMiergy must be unchanged. That is. in equilibrium, the Gibbs free
energy per unit mass must hr> thp. sn.mr> For both plm.sps. On tlie other
hand, at a temperature and pressure which do not correspond to equilib-
rium between two phases, the Gibbs free energies per unit mass will be
different for the two phases. Then the st.n.hlf» plm«o nmW t.h<^» rttpfo-
t in|]«N| fpijaf \\r> ^.Vinf wliif'h fr^.s thp. low<*r Gibbs free energy. If the svstem
is actually found in the phase of higher Gibbs free energy, it will be
unstable and will irreversibly change to the other phase. Thus for
instance, the Gibbs free energies of liquid and solid as functions of the
temperature at atmospheric pressure are represented by curves whici i
cross at the melting point! Below the melting point the solid has the
lower Gibbs free energy. It is possible to have the liquid boln^y jLlif>
melting point: it is in the condition known as supercooling. But any
Alight disturbance is enough to produce a sudden and irreversible solidi-
fication, with reduction of Gibbs free energy, the final stable state being
the solid. It is evident from these examples that the Gibbs free energy is
uf gyeat importance in discussing physical and chemical processes Thp
Helmholtz free energy ^f>ers nnt. WVP *my annh ^p0rf,jinnf> Wo tt|ml| see
later, however, that the methods of fltatfctjf»«i.l mpnlmnW WH pa.rf.if»i|1«.rly
simply to a calculation of the Helmholtz free energy, and its principal
valueTc'omeg about IhTKiTway. " " k

4. Methods of Deriving Thermodynamic Formulas. — We have now
introduced all the thermodynamic variables that we shall meet: P, V,
T, S, U, H, A, G. The number of partial derivatives which can be formed

24 INTRODUCTION TO CHEMICAL PHY SIC $ [CHAP. II

from these is 8 X 7 X 6 = 336, since each partial derivative involves one
dependent and two independent variables, which must all be different. A
few of these are familiar quantities, as we have seen in Sec. 2, but the great
majority are unfamiliar. It can be shown,1 however, that a relation can
be found between any four of those derivatives, and certain of the thermo-
dynamie variables. Those rotations are the thormodynarnic formulas.
Since there are 336 first derivatives, there aro 336 X 335 X 334 X 333
ways of picking out four of these, so that tho numbor of independent rela-
tions is this number divided by 4!, or 521,631,180 separate formulas. No
other branch ot physics is so rich in mathematical formulas, and some
systematic method must be used to bring order into the situation. No
one can be expected to derive any considerable number of the formulas or
to keep them in mind. There are four principal methods of mathematical
procedure used to derive those formulas, and in the present section wo
shall discuss them. Then in the next section we shall describe a system-
atic procedure for finding any particular formula that we friay wish. Tho
four mathematical methods of finding formulas are

1. We have already seen that there are a number of differential rela-
tions of the form

dx = Kdy +Ldz, (4.1)

where K and L are functions of the variables. The most important rela-
tions of this sort which we have met aro found in Kqs. (2.5), (3.2), (3.6),
and (3.9), and aro

dU = -PdV + TdS,
dll = VdP + T dflf,

dA = -PdV - S dT,

dG = VdP - SdT. (4.2)

We have already seen in Eq. (2.6) how wo can obtain formulas from such
an expression. Wo can divide by tho differential of one variable, say
du, and indicate that the process is at constant value of another, say w.
Thus wo have

In rlninpr fjijg WP mnwf Ko giirp fh^ fa is tho diffnrnntin.1 of a function

of tho-Statfi_Qf the &v^tojii_-£Qii-£iiil v in that case is it proper to write R DRT-
tial derivative like (Ax/Au^^ Thus in particular we cannot proceed in

snporfu'.mllv t.hf>y look

1 For the method of classifying thermodynamic formulas presented in Sees. 4 and
5, see P. W. Bridgman, "A Condensed Collection of Thermodynamic Formulas,"
Harvard University Press.

SEC. 4]

THERMODYNAMICS

like Eq. (4.1). Using the method of Eq. (4.3), a very large number of
formulas can be formed. A special case has been seen, for instance, in the
Eqs. (2.7) and (2.8). This is the case in which u is one of the variables
y or z, and w is the other. Thus, suppose u = y, w = z. Then we have

(!),=«-

and similarly

= L.

(4.4)

It is to be noted that, using Eq. (4.4), we may rewrite Kq. (4.1) in the
form

(4.5)

a form in which it becomes simply the familiar mathematical equation
expressing a total differential in terms of partial derivatives.

2. Suppose we have two derivatives such as (dx/du)t, (dy/du)s, taken
with respect to t^e same variable and with the same variable held con-
stant. Since z is held fixed in both cases, they act like ordinary deriva-
tives with respect to the variable u. But for ordinary derivatives we

should have T , = --,— Thus, in this case we have

dy/du ay

(**\

(4.6)

We shall find that the relation in Eq. (4.6) is of great service in our sys-
tematic tabulation of formulas in the next section. For to find all partial
derivatives holding a particular z constant, we need merely tabulate the
six derivatives of the variables with respect to a particular u, holding this
z constant. Then we can find any derivative of this type by Eq. (4.6).
3. Let us consider Eq. (4.5), and set x constant, or dx = 0. Then
we may solve for dy/dz, and since x is constant, this will be (dy/dz)x.
Doing this, we have

x (-}

dy\ _ \dz/

dZ/x /dx\

(4.7)

26 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. II

Using Eq. (4.6), we can rewrite Eq. (4.7) in either of the two forms

(4.8)
(4.9)

I (?*\ (**}(*} __!
\\dyj \te/\dxL

The reader should note carefully the difference between Eq. (4.8) and
Eq. (4.6). At first glance they resemble each other, except for the differ-
ence of sign; but it will be noted that in Eq. (4.6) each of the throe deriva-
tives has the same variable held constant, while in Eq. (4.8) each one has a
different variable held constant.

4. We Start with Eq. (4.4). Then we use the fundamental theorem
regarding second partial derivatives:

Substituting from Eq. (4.4), this gives us

/a*\ /a/A

\dz)y \dy)z

(4.10)

(4.11)

In Eq. (4.10), it is essential that .r be a function of the state of the system,
or of y and z. Four important relations result from applying Eq. (4.11)

to the differential? expressions (4.2). These are

1 t

~ jtSJv

dP/.«

dTv \dV

(4.12)

The Eqs. (4.12) are pjcnerallv called Maxwell's relations.

We have now considered the four processes used in deriving thermo-
dynamic formulas. By combinations of them, any desired relation
connecting first derivatives can be obtained. In the next section we
consider the classification of these formulas.

SEC. 5] THERMODYNAMICS 27

5. General Classification of Thermodynamic Formulas. — Bridgman1
has suggested a very convenient method of classifying all the thermody-
namic formulas involving first derivatives. As we have pointed out, a
relation can be found between any four of the derivatives. Bridpman's
method is then to write each derivative in terms of three standard deriva-
tives, for which he chooses (dV/dT)p_, (dV/dP)T, and CP = (dH/dT)P
These are chosen because they can DC found immediately from expori-
ment, the first two being closely related to thermal expansion and
compressibility [see Eqs. (2.1) and (2.2)]. If now we wish a relation
between two derivatives, we can write each in terms of these standard
derivatives, and the relations will immediately become plain. Our task,
then, is to find all but these three of the 336 first partial derivatives, in
terms of these three. As a matter of fact, we do not have to tabulate
nearly all of these, on account of the usefulness of Eq. (4.6). We shall
tabulate all derivatives of the form (dx/dT)^ (dx/BP)r. (dx/dT)v. and
(dx/dT)s^ Then by application of Eq. (4.6), we can at once find any
derivative whatever at constant P, constant 77, constant V, or constant
S. We could continue the same thing for finding derivatives holding the
other quantities fixed; but we shall not need such derivatives very often,
and they are very easily found by application of methods (2) and (3) of
the preceding section, and by the use of our table. Wo shall now tabulate
these derivatives, later indicating the derivations of the only ones that
are at all involved, and giving examples of their application. We shall
be slightly more general than Bridgman, in that alternative forms are
given for some of the equations in terms either of CP or (7, .

TABLE 1-1. — TABLE OF TIIBUMODYNAMH; RELATIONS

(dS\ =CV
\dT/P T

= -P - 8

r \dTjp

1 See reference under Sec. 4.

28 INTRODUCTION TO CHEMICAL PHYSICS [€HAP. II

TABLE 1-1. — TABLE OF THERMODYNAMIC RELATIONS (Continued]

dI>/

OPT dP

(ov\

= _ \dT/P
/QV\

\dP/T

(dS\ = Cv = CP
\d'r)v T T +

dTP

dP/T

("}
aH

r
= '

SEC. 5] THERMODYNAMICS 29

TABLE 1-1. — TABLE OF THERMODYNAMIC RELATIONS (Continued)

fav\~
\af)P

<± + -<

rii ~T~ >

("'-V

•c, , \#rj,

\dPjr

-T +

/dH\ __ VCP

W- = JV

2-1

-s

The formulas of Table 1-1 all follow in very obvious ways from the
methods of See. 4, except perhaps the relation between Cv and CV, used
in the derivatives at constant V and constant *S. To prove the relation
between these two specific heats, we proceed as follows. We have

(5.1)

Tds = r(^

+ T\M)TdV = CvdT + T( ^

= T(^ dT + I

,(dS
\dp

dP = CPdT- T(~

We subtract the first of Eqs. (5.1) from the second, and sot dV = 0,
obtaining

(CP-Cv)dT= T(^-r)dP.

30 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. II

Dividing by dT7, this is

the result used in the formulas of Table 1-1. The result of Eq. (5.2) is
an important formula, and is the generalization holding for all substances
of the familiar formula CP — Cv = nR holding for perfect gases. It
serves in any case to find Cp — Cv from the equation of state. Since1
(dV/dP)r is always negative, we see that CV is always greater than C v .
"^ o. Comparison of Thermodynamic and Gas Scales of Temperature.-
In Chap. I, Sec. 4, we have discussed the therrnodynamic temperature,
introduced in the statement of the second law of thermodynamics, and tt e
have mentioned that it can be proved that this is the same temperature
that is measured by a gas thermometer. We arc now in position to prove
this fact. First, we must define a perfect gas in a way that could be
applied experimentally without knowing any temperature scale except
that furnished by the gas itself. Wo can define it as a gas which in the
first place? obeys Boyle's law: PV — constant at constant temperature, or
PV = f(T), where T is the thermodynamic temperature, and / is a func-
tion as yet unknown. Secondly, it obeys what is called Joule's la\v: the
internal energy 1T is independent of the volume at constant temperature.
These assumptions can both be proved by direct experiment. We can
certainly observe1 constancy of temperature without a temperature scale,
so that we can verify Boyle's law. To chock Joule's law, we may consider
the free expansion of the* gas. We let the gas expand irreversibly into
a vacuum. It is assumed that the process is carried out aeliabatically,
and since there is ne> external work done, the internal energy is unchanged
in the process. We allow the gas to come te) equilibrium at its new
volume, and observe whether the temperature is the? same that it was
originally, or different. If it is the same, then the gas is said to obey
Joule's law. To che»ck the mathematical formulation of this law, we note
that the experiment of free expansion tells us directly that the tempera-
ture is independent erf volume, at constant internal energy: (3T/dV)u — 0.
But by Eq. (4.9), we have

/d[A (dlf\(dT\ (dT\ , n

\av)r = ~\dT)\dv)v - ~CvWA = °* ((U)

Equation (6.1) states that the internal energy is independent of volume
at constant temperature, the usual statement of Joule's law.

Without further assumption about the gas, we can now prove that
f(T) = constant X T, so that the pressure of a perfect gas at constant

SEC. G] THERMODYNAMICS 31

volume is proportional to the thermodynamic temperature, and if we uae
proper units, the gas scale of temperature is identical with the thermo-
dynamic scale. Using Table 1-1 we have the important general relation

. _ _ __ /dP

av- -

/dV\
(dp)T

giving the change of internal energy with volume of any substance at
constant temperature. We set this equal to zero, on account of Joule's
law. From the equation of state,

f(T) J'(T)

Substituting Eq. (6.3) in Kq. (6.2), and canceling out a factor P, we have

Tf(T) =

d In/ = d In r, ln/(T) = In T + const.,

f(T) - const. X T, (6.4)

which was to be proved.

Instead of defining a perfect gas as we have done, by Boyle's law and
Joule's law, we may prefer to assume that a thermodynamic temperature
scale is known, and that the perfect gas satisfies the general gas law
PV = const. X T. Then we can at once use the relation (6.2) to calcu-
late the change of internal energy with volume at constant temperature,
and find it to be zero. That is, we show directly by thermodynamics that
Joule's law follows from the gas law, if that is stated in terms of the
thermodynamic temperature.

CHAPTER III
STATISTICAL MECHANICS

Thermodynamics is a simple, general, logical science, based on two
postulates, the first and second laws of thermodynamics. We have seen
in the last chapter how to derive results from these laws, though we have
not used them yet in our applications. But we have seen that they are
limited. Typical results are like Kq. (5.2) in Chap. II, giving the differ-
ence of specific heats of any substance, Cp — CV, in terms of derivatives
which can be found from the equation of state. Thermodynamics can
give relations, but it cannot derive the specific heat or equation of state
directly. To do that, we must go to the statistical or kinetic methods.
Even the second law is simply a postulate, verified because it leads to
correct results, but not derived from simpler mechanical principles as far
as thermodynamics is concerned. We shall now take up the statistical
method, showing how it can lead not only to the equation of state and
specific heat, but to an understanding of the second law as well.

1. Statistical Assemblies and the Entropy. — To apply statistics to
any problem, we must have a great many individuals whose average
properties we are interested in. We may ask, what are the individuals to
which we apply statistics, in statistical mechanics? The answer is, they
are a great many repetitions of the same experiment, or replicas of the
same system, identical as far as all large-scale, or macroscopic, properties
are concerned, but differing in the small-scale, or microscopic, properties
which we cannot directly observe. A collection of such replicas of the
same system is called a statistical assembly (or, following Gibbs, an
ensemble). Our guiding principle in setting up an assembly is to arrange
it so that the fluctuation of microscopic properties from one system to
another of the assembly agrees with the amount of such fluctuation which
would actually occur from one repetition to another of the same
experiment.

Let us ask what the randomness that we associated with entropy in
Chap. I means in terms of the assembly. A random system, or one of
large entropy, is one in which the microscopic properties may be arranged
in a great many different ways, all consistent with the same large-scale
behavior. Many different assignments of velocity to individual mole-
cules, for instance, can be consistent with the picture of a gas at high
temperatures, while in contrast the assignment of velocity to molecules
at the absolute zero is definitely fixed: all the molecules are at rest. Then

SEC. 1] STATISTICAL MECHANICS 33

to represent a random state we must have an assembly which is dis-
tributed over many microscopic states, the randomness being measured
by the wideness of the distribution. We can make this idea more precise.
Following Planck, we may refer to a particular miprosnopir. stn.t.o. nf thu
system as a complexion. We may describe an assembly by stating what
fraction of the systems of the assembly is found in each possible com-
plexion. We shall call this fraction, for the it\\ complexion. /V. and shall
refer to the sot of /t's as the distribution function describing tho assembly.
Plainly, since all systems must be in one complexion or another,

Then in a random assembly, describing a system of large entropy, there
will be systems of the assembly distributed over a great many complex-
ions, so that many fl's will be different from zero, each ono of these frac-
tions being necessarily small. On the other hand, in an assembly of low
entropy, systems will be distributed over only a small number of com-
plexions, so that only a few /t's will be different from zero, and these will
bo comparatively large.

We shall now postulate a mathematical definition of entropy, in terms
of the/?s, whicTTlij lafiafm the case of a random distribution, small other-
wise! This definition is

(1.2)

Hero k is a constant, called Boltzmann's constant, which will appear fre-
quently in our statistical work. It has the same dimensions as entropy.
or specific heat, that is. energy divided by tnmpnratiim. ^Its value in
absolute units is 1.379 X 1Q~16 er er degree. This value is derived

indirectly; using Eq. (1.2)? for thn pnt.rnpyj <™<* ^«" flfTJv^ tl

gas law and the gas constant, in torms of fc, thereby determining k from

It is easy to see that Eq. (1.2) has the required property of being
large for a randomly arranged system, small for one with no randomness.
If there is no randomness at all, all values of fl will be zero, except one,
which will be unity. But the function / In / is zero when / is either zero or
unity, so that the entropy in this case will be zero, its lowest possible
value. On the other hand, if the system is a random one, many com-
plexions will have /t different from zero, equal to small fractions, so that
their logarithms will be large negative quantities, and the entropy will be
large and positive. We can see this more clearly if we take a simple
case: suppose the assembly is distributed through W complexions, with

34 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

equal fractions in each. The value rf *y>.nh f- in titlfififi finTnplpr;/™° is
1/W, while for other complexions /»• is zero. Then we have

Q — _J&.T/F_ in —

o — /t rr jrr "I ™-
L = fc In W (1.3)

Tho^p j^py, in «nr»h a r»flfflj j<g nrflpprtional to the logarithm of the number
nf nnrnplpviflna in whipli ay^ff TUB of thfi assembly can be found. AsTtnis
number of complexions increases, the distribution becomes more random
or diffuse, and the entropy increases.

Boltzmann1 based his theory of t.hn relation of probability to entropy
on Eq. (1.3), rather than using the more general relation M .2V HP pnllod
)V the thermodynamic probability "f * «^tf r P-rtPI^ much as wo have
that a random state, which is inherently likely to be realized, will have a
large value of W. Planck2 has shown by the following simple argument
that the logarithmic form of Eq. (1.3) is reasonable. Suppose the system
consists of two parts, as for instance two different masses of gas, riot con-
nectod with each other. In a given state, represented by a given assem-
bly, let there be Wi complexions of the first part of the system consistent
with the macroscopic description of the state, and Wo complexions of the
second part. TV™ , «j"™> *h* *™" p?rf s nf f1"* *v*tf ™ f^re independonTof
each otherTthere must be W^W^ comnlexioiis of the combined system,
since each complexion of the first part can be joined to any one of the
complexions of the second part to give a complexion of the combined
system. We shall then find for the entropy of the combined system

S *= k In WiWz

= k\nWi + k\n W2. (1.4)

But if we considered the first part of the system by itself, it would have
an entropy Si-» k In Wi, and the second part by itself would have an
entropy /S>2 = Arin TF2. Thus, on account of the relation (1,3), we have

S = S! + S2. (1.5)

But surely this relation must bo true; in thermodynamics, the entropy
of two separated systems is tho sum of the entropies of the parts, as wo
can see directly from the second law, since the changes of entropy, dQ/T,
in a reversible process, are additive. Then we can reverse the argument
fl-bovp. Fjfliiation (\.&\ must be trno. &iicl if_tliii-£iiiiti*Qny is a, function of
W} it can be shown that the only possible function consistent with the
additivity of the entropy is tho logarithmic function of ECL fl.SV

1See for example, L. Boltzmarm, "Vorlesungen uber Gastheorie," Sec. 6, J. A.
Barth.

2 See for example, M. Planck, "Heat Radiation," Sec. 119, P. Blakiston's Sons &
Company.

SEC. 1] STATISTICAL MECHANICS 35

Going back to our more general formula (1.2), we can show that if
the assembly is distributed through W complexions, the entropy will have
its maximum value when the /»'s are of equal magnitude, and is reduced
by any fluctuation in /t from cell to cell, verifying that any concentration
of systems in particular complexions reduces the entropy. Taking the
formula (1.2) for entropy, wo find how it changes when the/t's aro varied.
Differentiating, wo havo at onco

dS = -*](l + In /.MA- (1-6)

But we know from Eq. (1.1) that X/t = 1, from which at. onco

, = 0. (1.7)

Thus the first term of Kq. (1.6) vanishes; and if we assume that the
density is uniform, so that ln/» is really independent of /„ we can take it-
out of the summation in Eq. (1.6) as a common factor, and the remaining
term will vanish too, giving dS = 0. That is. fof uniform ffan^yj fl™
variation of the entropy for small variations of the asscn^^fv vfl-fllfihfifti **-
necessary condition for a maximum of the entropy. A little further
investigation would convince us that this really gives a maximum, not a
minimum, of entropy, and that in fact Eq. (1.3) gives the absolute maxi-
mum, the highest value of which S is capable, so long as only W complex-
ions are represented in the assembly. The only way to get a still greater
value of S would be to havo more terms in the summation, so that each
individual /» could bo even less.

We have postulated a formula for the entrypy. How can we expeot,
to prove that it is correct? We can do this onlv bv goiny bank to thn_
second law of thermodynamics^ showing that our entropy has the prop-
erties ciemanciea Dy ttiat law, and that in forma of it thn Uw i« «a.t.i«finfj.
We have already shown that, our formula for the entropy has one of the
properties demanded ot the entropy: it is determined bv the Ht.fl.te of thn
system. In statistical mechanics, the only thing we can mean by the
state of the system is the statistical assembly, since this determines
average or observable properties of all sorts, and our formula (1.2) for
entropy is determined by the statistical assembly. Next we must show
that our formula, represents a quantity that increases in an irreversible*
process. This will be done by qualitative but valid reasoning in a later
section. It will then remain to consider thermal equilibrium and reversi-
ble processes, and to show that in such processes the change of entropy is
dQ/T.

36 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

2. Complexions and the Phase Space. — We wish to find how our
formula for the entropy changes in an irreversible process. To do this,
we must find how the/t's change with time, or how systems of the assem-
bly, as time goes on, change from one complexion to another. This is a
problem in kinetics, and we shall not take it up quantitatively until the
chapter on kinetic methods. For the present we shall be content with
qualitative discussions. The first thing that we must do is to get a more
precise definition of a complexion. We have a certain amount of informa-
tion to guide us in making this definition. We are trying to make our
definition of entropy agree with experience, and in particular we want the
state of maximum entropy to be the; stable, equilibrium state. But we
have just seen that for an assembly distributed through W complexions,
the state of maximum entropy is that in which equal numbers of systems
are found in each complexion. This is commonly expressed by saying
that complexions have equal a priori probability; that is, if we have no
specific information to the contrary, we are as likely to find a system of an
assembly in one complexion as in another, in equilibrium. Our definition
of a complexion, then, must be consistent with this situation.

The method of defining complexions depends on whether we are treat-
ing our systems by classical, Newtonian mechanics or by quantum theory.
First we shall take up classical mechanics, for that is more familiar. But
later, when we describe the methods of quantum theory, we shall observe
that that theory is more correct and more fundamental for statistical
purposes. In el^ssien.1 ^elianics. a system is f]esr*rihpfl by giving th^
coordinates and velocities of all its particles. -Instead of the velocities,
it proves to be more desirable to use the momenta. With rectangular
coordinates, the momentum associated with each coordinate is simply the
mass of the particle times the corresponding component of velocity; with
angular coordinates a momentum is an angular momentum ; and so on. If
there are N coordinates and N momenta (as for instance the rectangular
coordinates of JV/3 particles, with their momenta), we can then visualize
the situation by setting up a 2N dimensional space, called a phase space,
in which the coordinates and momenta are plotted as variables, and a
single point, called a representative point, gives complete information
about the system. An assembly of systems corresponds to y| p.nllop.finn nf
representative points, and we shall generally assume that ty™ QTO san
many systems in the assembly that the distribution of representative
points is practically continuous in tliejjhase space. Now a complexion,

Or microSCOPic State, of thp system mn«f. nopiypnnH t.n n

or small region, of the phase space; to bemore precise,
spond to a small volume of the phase space. We subdivide the whole
phase space into small volume elements and call each volume element a
complexion, saying that ftt the fraction of systems of the assembly in a

SBC. 2] STATISTICAL MECHANICS 37

particular complexion, simply equals the fraction of all representative
points in the corresponding volume clement. The only question that
'q fllp ghqpg and SJEP of Y^^mc elements representing

complexions.
"""To answer this question, we must consider how points move in the

phase space. We must know the time rates of chuufiQ of all coordinates
and momenta, in terms of the coordinates and momenta themselves.
Newton's i second law gives us the time rate of change of each momentum,
stating that it equals the corresponding component of force, which is a
function of the coordinates in a conservative system. The time rate of
change of each coordinate is simply the corresponding velocity com-
ponent, which can be found at onee from the momentum. Thus we can
find what- is essentially the 2N dimensional velocity vector of each repre-
sentative point. This velocity vector is determined at each point of
phase space and defines a rate of flow, the representative points streaming
through the phase space as a fluid would stream through ordinary space.
We are thus in a position to find how many points enter or leave each
element of volume, or each complexion, per unit time, and therefore to
find the rate at which the fraction of systems in that complexion changes
with time. It is now easy to prove, from the equations of motion, a
general theorem callec} LiouvihVs theorem. } This theorem states, in
mathematical language, the following fact : the swarm of points moves in
such a wfl.y f.Vm|, the H?*ic"fy "f p™***^ «*•* «™ fallow along with the swarm,
never changes^ The flow is like a streamline flow of an incompressible
fluid, each particle of fl*^] nlwnya proMf»rvingr it.M r>Wn density. This does
not mean that the rlnnsitv ut, a. given point of spn.o.n dons not rhginge with
time; in general it does, for in the course uf the flow, first a. f|on«o pnrf W
the swarm, then a less dense onn, m,«i.y woll be swept, by f-.hn point, jjj_
question, as if we had an incompressible fluid, but one whose density
changed from point to point. It does mean, however, that we can find a
very simple condition which is necessary and sufficient for the density
at a given point of space to be independent of time: the density of points
must be constant all along each streamline, or tube of flow, of the points.
For then, no matter how long the flow continues, the portions of the
swarm successively brought up to the point in question all have the same
density, so that the density there can never change.

To find the condition for equilibrium, then, we must investigate
the nature of the streamlines. For a periodic motion, a streamline will be
closed, the system returning to its original state after a single period.
This is a very special case, however; most motions of many particles are
not periodic and their streamlines never close. Rather, they wind around

1 For proof, see for example, Slater and Frank, " Introduction to Theoretical
Physics," pp. 365-366, McGraw-Hill Book Company, Inc., 1933.

38 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

in a very complicated way, coming in the course of time arbitrarily close
bo every point of phase space corresponding to the same total energy (of
course the energy cannot change with time, so that the representative
point must stay in a region of constant energy in the phase space). Such
a motion is called quasi-ergodic? and it can be shown to be the general
hypfi of rnofirm pnrinHif*. motions being a rare pv^^n|(jpT^ Then, from the
statement in the last paragraph, we see that to have a distribution inde-
pendent of time, we must have a density of DOII^S in phase apftf.e whinh
is constant for all regions of the same energy, _ But on the 0ther hand
thermal equilibrium must correspond to a distribution inHp.pondnnt. nf
time, and we have seen that the state of maximum entropy is one in which
all complexions have the same number of avstgiflfi, Tfrpsp lwn «*«-fg-
menta are only compatible if each complexion corresponds to the same
volume of phase space. For then a constant volume density of points,
which by Liouville's theorem corresponds to a distribution independent
of time, will at the same time correspond to a maximum entropy. We
thus draw the important conclusion that regions of equal volume in phase
space have equal a priori probability, pr that. ^ nomple^on norr^ponds
to a quite definite volume of phase space. Classical mechanics, however,
does not lead to any way of sayi_ri^hQw 1Qrp?° thiil Arolume is. Thi^g it
cannot lead to any unique definition of the entropy; fr>r tJhfi Ajfl dfiPftnr* on
how large a volume each complexion corrggpQiifo t.n;. «.nH they jp turn
determine the entropy.

3. Cells in the Phase Space and the Quantum Theory. — Quantum
mechanics starts out quite differently from classical mechanics. It does
not undertake to say how the coordinates and momenta of the particles
change as time goes on. Rather, it is a statistical theory from the begin-
ning: it sets up a statistical assembly, and tells us directly how that
assembly changes with time, without the intermediate step of solving for
the motion of individual systems by Newton's laws of motion. And it
describes the assembly, from the outset, in terms of definite complexions,
so that the problem of defining the complexions is answered as one of the
postulates of the theory. It sets up quantum states, of equal a priori
probability, and describes an assembly by giving the fraction of all sys-
tems in each quantum state. Instead of giving laws of motion, like
Newton's second law, its fundamental equation is one telling how many
systems enter or leave each quantum state per second. In particular, if
?qual fractions of the systems are found in all quantum states associated
with the same energy, we learn that these fractions will not change with
time; that is, in a steady or equilibrium state all the quantum states are
equally occupied, or have equal a priori probabilities, ye are then

jiiat.ififtfl in iHpnt.ifvinp; t.hpgP qimniiiinj af[ft.f.ofl

innc ^yV.^V. Wg ViQir*> moyif^™/^ When we deal witL quantum statistics,

SBC. 3) STATISTICAL MECHANICS 39

A will refer to the fraction of all svstmns in tfif tt.h niianfnm g+Qfo This
gives a definite meaning |.n t.hn pnmpWinnfi BT1fJ i™^c ^ o ^»finjt.P nni^n^
ical value for the entropy.

Quantum tfteory provides no unique way of setting up the quantum
states, or the complexions. We can understand this much better by
considering the phase space. iyrn.nv feature^ of the Quantum theory can
be described by dividing the phase space into cells of equal volume, and
associating each cell with a quantum state. The volume of these cells is
uniquely fixed by the quantum theory, but not their shape. We can, for
example, take simply rectangular cells, of dimensions A#i along the axis
representing the first coordinate, A<?2 for the second coordinate, and so on
up to A#.v for the Nth coordinate, and Api to ApN for the corresponding
momenta. Then there is a very simple rule giving the volume of such a
cell: we have

A?tAp, = A, (3.1)

where h is Planck's constant, equal numerically to 6.61 X 10~~7 absolute
units. Thus, with N coordinates, the 2Ar-dimensional volume of a cell
is hN.

We can equally well take other shapes of cells. A method which is
often useful can be illustrated with a problem having but one coordinate q
and one momentum p. Then in our two-dimensional phase space we can
draw a curve of constant energy. Thus for instance consider a particle*
of mass m held to a position of equilibrium by a restoring force propor-
tional to tho displacement, so that its energy is

E - SL + 27r2w*>y, (3.2)

where v is the frequency with which it would oscillate in classical
mechanics. The curves of constant energy are then ellipses in the p-q
space, as we see by writing the equation in the form

the standard form for the equation of an ellipse of semiaxes -\/2mE and
\/E/2ir2mv'2. Such an ellipse is showrn in Fig. III-l . Then we can choose
cells bounded by such curves of constant energy, such as those indicated
in Fig. III-l. Since the area between curves must be A, it is plain that the
nth ellipse must have an area nA, where n is an integer. The area of an
ellipse of semiaxes a and b is irdb\ thus in this case we have an area of
ir\/2mE\/E/2'ir2mvz = E/v, so that the energy of the ellipse connected

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. Ill

with a given integer n is given by

En = nhv.

(3.4)

Another illustration of this method is provided by a freely rotating wheel
of moment of inertia 7. The natural coordinate to use to describe it is
the angle 0, and the corresponding momentum pQ is the angular momen-

'TrrTT

Kiu. 1II-1 Cells in phase spare, for the linear oscillator The shaded area, between
two ellipses of constant energy, has an area h in the qinntum theory

turn, /co, \\licre co = dO/dt is the angular velocity. If no torques act, the
energy is wholly kinetic, equal to

E = ^

(3.5)

Then, as shown in Fig. III-2, lines of constant energy are straight lines at
constant value of po. Since 0 goes from zero to 2?r, and then the motion

3H/27T

repeats, we* use only values of the coordinate
in this range. Then, if the cells an* set up so
that the area of each is h, we must have them
bounded by the lines

(3.6)

so that the energy associated with the nth
line is

E. = Sr (3-7)

Km. III-2 —Cells m phase j[u forms of these cells, we can now under-

space, for the rotator. The xr j xix± x

shaded area has an area of h stand one of the most fundamental statements
in the quantum theory. of ^ jie quantum theory, the principle of uncer-

tainty : it is impossible to regulate the coordinates and momenta of a system
more accurately than to require that they lie somewhere within a given cell.
Any attempt to be more precise, on account of the necessary clumsiness
of nature, will result in a disturbance of the system just great enough to

SEC. 3] STATISTICAL MECHANICS 41

shift the representative points in an unpredictable way from one part of
the cell to another. The best we can do in setting up an assembly, in
other words, is to specify what fraction of the systems will be found in
each quantum state or complexion, or to give the /,'s. This does not.
imply by any means, however, that it does not make sense to talk about
the coordinates and momenta of particles with more accuracy than to
locate the representative point in a given cell. Xh^re is nothing inher-
ently impossible in knowing the coordinates and momenta of a system as
accurately as wo please; the restriction is only that we cannot prepare a
system, or an assembly of systems, with as precisely determined coordi-
nates and momenta as we might please.

Since we may be interested in precise values of the momenta and
coordinates of a system, there must bo something in the mathematical
framework of the theory to describe them. We must be able to answer
questions of this sort: given, that an assembly has a given fraction of its
systems in each coll of phase space, what is the probability that a certain
quantity, such as ono of the coordinates, lies within a certain infinitesimal
range of valuos? Put in another way, if we know that a system is in a
given coll, what is the probability that its coordinates and momenta lie
in definite ranges? The quantum theory, and specifically the wave
ID' chanics, can answer such questions; and because it can, wo aro justified
ia regarding it as an essentially statistical theory. By experimental
methods, we can insure that a system lies in a given cell of phase space.
That is, we can prepare an assembly all of whose representative points lie
in this single cell, but this is the nearest we can come to setting up a sys-
tem of quite definite coordinates and momenta. Having prepared such
an assembly, however, quantum theory says that the coordinates and
momenta will be distributed in phase space in a definite way, quite inde-
pendent of the way we prepared the assembly, and therefore quite unpre-
dictable from the previous history of the system. In other words, all
that the theory can do is t.o p;ivo. ijs statistical information about a system.
not jrlp^ily/* krmwlodge of exactly what it will do. This is in striking
contrast to. the classical mechanics, which allows precise prediction of the
future of a system if we know its past history.

The cells of the type described in Figs. III-l and III-2 have a special
property: all the systems in such a quantum state have the same energy.
The momenta and coordinates vary from system to system, roughly as if
systems were distributed uniformly through the cell, as for example
through the shaded area of either figure, though as a matter of fact the
real distribution is much more complicated than this. But the energy is
fixed, the same for all systems, and is referred to as an energy level. It is
equal to some intermediate energy value within the cell in phase space, as
computed classically. Thus for the oscillator, as a matter of fact, the

42 INTRODUCTION TO CHEMICAL PHYMCti [CHAP. Ill

energy levels arc

E» = (* + JV' (3.8)

\ */

which, as we see from Eq. (3.4), is the energy value in the middle of the
cell, and for a rotator the energy value is

Kn = n(n + IJg^j* (3.9)

approximately the mean value through the cell. The integer n is called
the quantum number. The distribution of points in a quantum state of
fixed energy is independent of time, and for that reason the state is called
a stationary state. This is in contrast to other ways of sotting up cells.
For instance, with rectangular cells, we find in general that tho systems in
one state have a distribution of energies, and as time goes on systems jump
at a certain rate from one state to another, having what are called quan-
tum transitions, so that the* number of systems in each state changes with
time. One can draw a certain parallel, or correspondence, between the
jumping of systems from one quantum state to another, and the uniform
flow of representative points in the phase space in classical mechanics.
Suppose we have a classical assembly whose density in the phase space
changes very slowly from point to point, changing by only a small amount
in going from what would be one quantum cell to another. Then we can
set up a quantum assembly, the fraction of systems in each quantum state
being given by tho fraction of the classical systems in the corresponding
cell of phase space. And the time rate of change of the fraction of sys-
tems in each quantum state will be given, to a good approximation, by
the corresponding classical value. This correspondence breaks down,
however, as soon as the density of the classical assembly changes greatly
from cell to cell. In that case, if we set up a quantum assembly as before,
we shall find that its time variation does not agree at all accurately with
what we should get by use of our classical analogy.

Antp.1 ftfomif systems pbev thf fljjQ-nfiim *Vnry n/rt r»]flssif»fll
mechanicsTso that we shall be concerned with auaptum statistics. The
only cases in whinfi we can use classical theory as an app^oxinmf.inii art*
tHbse.in which the density in phase varies only a, lif^lr fr-nm gfn.fn ^ state,
—the case we have mentioned in the last paragraph As a matter of fact,
as we shall see later, this corresponds roughly to the limit of high tempera-
ture. Thus, we shall often find that classical results are correct at high
temperatures but break down at low temperature. \ f-v™nnil jfflftTnnlf- of
this is the theory of specific heat; we shall find others as we go on. We
now understand the qualitative features of quantum statistics well enough
so that in the next section we can go on to our task of understanding the

SBC. 4] STATISTICAL MECHANICS 43

nature of irreversible processes and the way in which the entropy increases
with time in such processes.

4. Irreversible Processes. — We shall start our discussion of irreversi-
ble processes using classical mechanics and Liouville's theorem. Let us
try to form a picture of what happens when we start with a system out of
equilibrium, with constant energy and volume, follow its irreversible
change into equilibrium, and examine its final steady state. To have a
specific example, consider tho approach to equilibrium of a perfect gas
having a distribution of velocities which originally does not correspond
to thermal equilibrium. Assume that at the start of an experiment, a
mass of gas is rushing in one direction with a large velocity, us if it had just
been shot into a container from a jet. This is far from an equilibrium
distribution. The random kinetic energy of the molecules, which we
should interpret as heat motion, may be very small and the temperature
low, and yet they have a lot of kinetic energy on account of their motion
in the jet. In the phase space, the density function will be large only in
the very restricted region where all molecules have almost the same
velocity, the velocity of the jot (that is, the equations

Pi! _ Prf . _ y H(,

- — ~ - — r j , I I C . .

m i w2

where Vx is the x component of velocity of the jot, are almost satisfied by
all points in the assembly), and all have coordinates near tho coordinate
of the center of gravity of the rushing mass of gas (that is, tho equations
Xi = x2 = - - - = A', where X is the x coordinate of the center of gravity
of the gas, are also approximately satisfied). We see, then, that the

entropy, as defined by — kf, In/,, will be small under these conditions.

But as time goes on, the distribution \\ill change. The jet of molecules
will strike the opposite wall of tho container, and after bouncing back
and forth a few times, will become moro and more dissipated, with irregu-
lar currents and turbulence sotting in. At first we shall describe those
things by hydrodynamics or aerodynamics, but wo shall find that tho
description of the flow gets more and more complicated with irregularities
on a smaller and smaller scale. Finally, with the molecules colliding
with the walls and with each other, things will become extremely involved,
some molecules being slowed down, some speeded up, the directions
changed, so that instead of having most of the molecules moving with
almost the same velocity and located at almost the same point of space,
there will be a whole distribution of momentum, both in direction and
magnitude, and the mass will cease its concentration in space and will be
uniformly distributed over the container. There will now be a great

44 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

many points of phase space representing states of the system which could
equally well be this final state, so that the entropy will be large. And the
increase of entropy has come about at the stage of the process where we
cease to regard the complication in the motion as large-scale turbulence,
and begin to classify it as randomness on a microscopic or atomic scale.
Finally the gas will come to an equilibrium state, in which it no longer
changes appreciably with time, and in this state it will have reached its
maximum entropy consistent with its total energy.

This qualitative argument shows what we understand by an irreversi-
ble process and an increase of entropy: an assembly, originally concen-
trated in phase space, changes on account of the motion of the system in
such a way that the points of the assembly gradually move apart, filling
up larger and larger regions of phase space. This is likely, for there are
many ways in which it can happen ; while the reverse process, a concentra-
tion of points, is very unlikely, and we can for practical purposes say that
it does not happen.

The statement we have just made seems at first to be directly con-
trary to Liouville's theorem, for we have just said that points originally
concentrated become dispersed, while Liouville's theorem states that as
we follow along with a point, tho density never changes at all. We can
give an example used by Gibbs1 in discussing this point. Suppose we
have a bottle of fluid consisting of two different liquids, one black and
one white, which do not mix with each other. We start with one black
drop in the midst of the white liquid, corresponding to our concentrated
assembly. Now we; shake or stir the liquid. Tho black drop will become
shaken into smaller drops, or be drawn out into thin filaments, which will
become dispersed through the white liquid, finally forming something like
an emulsion. Each microscopic black drop or filament is as black as
ever, corresponding to the fact that the density of points cannot change in
the assembly. But eventually the drops will become small enough and
uniformly enough dispersed so that each volume element within the bottle
will seem uniformly gray. This is something like what happens in the
irreversible mixing of the points of an assembly. Just as a droplet of
black fluid can break up into two smaller droplets, its parts traveling in
different directions, so it can happen that two systems represented by
adjacent representative points can separate and have quite different
histories; one may be in position for certain molecules to collide, while the
other may be just different enough so that these molecules do not collide
at all, for example. Such chance events will result in very different,
detailed histories for the various systems of an assembly, even if the
original systems of the assembly were quite similar. That is, they will

!J. W. Gibbs, "Elementary Principles in Statistical Mechanics," Chap. XII,
Longmans, Green & Company.

SEC. 4] STATISTICAL MECHANICS 4/'»

result in representative points which were originally close together in
phase space moving far apart from each other.

From the example and the analogy we have used, we see that in an
irreversible process the points of the original compact and orderly assem-
bly gradually get dissipated and mixed up, with consequent increase of
entropy. Now let us see how the situation is affected when we consider
the quantum theory and the finite size of cells in phase space. Our
description of the process will depend a good deal on the scale of the
mixing involved in the irreversible process. So long as the mixing is on a
large scale, by Liouville's theorem, the points that originally wore in one
roll will simply be moved bodily to another cell, so that the contribution
of these points to — fc2/t ln/t will be the same as in the original distribu-
tion, and the entropy will be unchanged. The situation is very different,
however, when the distribution as we should describe it by classical
mechanics involves a set of filaments, of different densities, on a scale
small compared to a cell. Then the quantum /t, rather than equaling the
classical value, will be more nearly the average of the classical values
through the cell, leading to an increase of entropy, at the samo time that
the average or quantum density begins to disobey Liouville's theorem.

It is at this same stage of the process that it becomes really impossible
to reverse the motion. It is a well-known result of Newton's laws that
if, at a given instant, all the positions of all particles are left unchanged,
but all velocities are reversed in direction, the whole motion will reverse1,
and go back over its past history. Thus every motion is, in theory, rever-
sible. What is it that in practice makes some motions reversible, others
irreversible? It is simply the practicability of setting up the system with
reversed velocities. If the distribution of velocities is on a scale large
enough to see and work with, there is nothing making a reversal of the
velocities particularly hard to set up. With our gas, we could suddenly
interpose perfectly reflecting surfaces normal to the various parts of the
jet of gas, reversing the velocities on collision, or could adopt some such
device. But if the distribution of velocities is on too small n scale to see
and work with, we have no hope of reversing the velocities experimentally.
Considering our emulsion of black and white fluid, which we have pro-
duced by shaking, there, is no mechanical reason why the fluid could not
be unshaken, by exactly reversing all the motions that occurred in shaking
it. But nobody would be advised to try the experiment.

It used to be considered possible to imagine a being of finer and more
detailed powers of observation than ours, who could regulate systems on a
smaller scale than we could. Such a being could reverse processes that
we could not; to him, the definition of a reversible process would be differ-
ent from what it is to us. Such a being was discussed by Maxwell and is
often called "Maxwell's Demon." Is it possible, we may well ask, to

46 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

imagine demons of any desired degree of refinement? If it is, we can
make any arbitrary process reversible, keep its entropy from increasing,
and the second law of thermodynamics will cease to have any signifi-
cance. The answer to this question given by the quantum theory is
No. An improvement in technique can carry us only a certain distance,
a distance practically reached in plenty of modern experiments with
single atoms and electrons, and no conceivable demon, operating accord-
ing to the laws of nature, could carry us further. The quantum theory
gives us a fundamental size of cell in the phase space, such that we cannot
regulate the initial conditions of an assembly on any smaller scale. And
this fundamental cell furnishes us with a unique way of defining entropy
and of judging whether a given process is reversible or irreversible.

5. The Canonical Assembly. — In the preceding section, we have*
shown that our entropy, as_de_fined in Kg. (1.2), has one f}f fdlfi p^p^ti^
of the physical entropy: it increases in an irreversible process T for it
incrcascTwhonovor the assembly becomes ftffused or sc^ttero^ and this
happens in irreversible processes. We must next take up thermal eauilib-
ium. finding tirst the correct assembly to describe the fiensity fungtion in
thermal equilibrium, and then proving from this fonaitv fnnct.ionr thnf,
our entropy satisfies the condition dS = dO IT for a reversible, proress.
From Liouville's foeorem, we have one piece of information aboqt the,
assembly: in or^ler that it ffiav be independent of time, the quantity /,
must^be a function only of the energy of the system. We^et A\ be the
energy of a system in the ii\\ ccll? choosing for this purpose the; type of
quantum cells representing stationary states oy ^"p-rp[y H™lg
wish to haye/t a function of J?,, but we do not, vot, xr how-to
this r function.

The essential method which we use is the following: We have seen that
in an irreversible process, the entropy tends to increase to a maximum, for
an assembly of isolated systems. If all systems of the assembly have the
same energy, then the only cells of phase space to which systems can
travel in the course of the irreversible process are cells of this same energy,
— a finite number. The distribution of largest entropy in such a case, as
we have seen in Sec. 1, is that in which systems are distributed with
uniform density through all the available cells. This assembly is called
the microcanonical assembly, and it satisfies our condition that the
density be a function of the energy only: all the /t's of the particular
energy represented in the assembly are equal, and all other /t's are zero.
But it is too specialized for our purposes. For thermal equilibrium, we
do not demand that the energy be pr^p|y "dp.tenninpd WP dpnmnd
rather that the tempcrnturfl nf ^1] yretems of th* a^mblv be flie same.
This can be interpreted most properly in the .following way. We allow
eacfr system of the assembly to be in contact with a temperature bath of

SBC. 51 STATISTICAL MECHANICS 4?

the required temperature, a body of very large heat capacity held at the
desired temperature. The systems of the assembly are then not isolated.
Rather, they can change their energy by interaction with the temperature
bath. Thus, even if we started out with an assembly of systems all of the
same energy, some would have their energies increased, some decreased,
by interaction with the bath, and the final stable assembly would havo a
whole distribution of energies. There would certainly be a definite aver-
age energy of the assembly, however; with a bath of a given temperature,
it is obvious that systems of abnormally low energy will tend to gain
energy, those of abnormally high energy to lose energy, by the interaction.
To find the final equilibrium statcf then, wo may ask this question: what
is the assembly of systems wrhich has the maximum entropy, subject onT7
to the condition that its moan energy havo a given value?.. It seems most
reasonable that this will he the assembly which will.. bo tho final result, of
the irreversible contact of any group of systems with a largo temperature

The1 assembly that results from these conditions is called tho canonical
assembly! Lot us formulate tho conditions which it must satisfy. It

must bo the assembly for which S = ~k2\ft In /t is a maximum, subject

(o a constant mean energy. But we can find the mean energy immedi-
ately in terms of our distribution function/,. In tho ?th coll, u system has
energy E%. The fraction fv of all systems will be found in this coll. Hence
the weighted mean of the energies of all systems is

(5.1)

This quantity must be held constant in varying the /t's. Also, as we saw-
in Eq. (1.1), the quantity ]^/t equals unity. This must always be satis-

fled, no matter how the /t's vary. We can restate the conditions, by
finding dS and dU: we must have

, (In/. + 1), (5.2)

for all sets of d//s for which simultaneously

dU = 0 = 2///,A\, (5.3)

and

48 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

On account of Eq. (5.4), we can rewrite Eq. (5.2) in the form

dS = 0 - -kdfi ln/i. (5.5)

The set of simultaneous equations (5.3), (5.4), (5.5) can be handled by the
method called undetermined multipliers: the most general value which
In/* can have, in order that dS should be zero for any set of d/Vs for which
Eqs. (5.3) and (5.4) are satisfied, is a linear combination of the coeffi-
cients of dfi in Eqs. (5.3) and (5.4), with arbitrary coefficients:

In fi = a + bE%. (5.6)

For if Eq. (5.6) is satisfied, Eq. (5.5) becomes

= -ka^dfi - kb^dfrft, , (5.7)

% %

which is zero for any values of cZ/t for which Eqs. (5.3) and (5.4) are satis-
fied.

The values of /t for the canonical assembly are determined by Eq.
(5.6). It may be rewritten

/, = fe6*.. (5.8)

Clearly b must bo negative; for ordinary systems have possible states of
infinite energy, though not of negatively infinite energy, and if b wore
positive, fi would become infinite for the states of infinite energy, an
impossible situation. We may easily evaluate the constant a in torms of

6, from the condition 5/» = 1. This gives at once

so that

J>K.

(5.9)

If the assembly (5.9) represents thermal equilibrium, the change of
entropy when a certain amount of heat is absorbed in a reversible process

SEC. 5] STATISTICAL MECHANICS 49

should be dQ/T. The change of entropy in any process in thermal
equilibrium, by Eqs. (5.5) and (5.9), is

dS = -k^df> In /, = -k^dMbEt - In

i l J

(5.10)

using Eq. (5.4). Now consider the change of internal energy. This is

E^ (5.11)

The first term in Ea. (5.11^ arises when the external forces stay constant,
resulting in constant values of Et) but there is a change m tne assembly,
meaning a shift of molecules from one position and velocity to another.
This change of course is different from that considered in Eq. (5.3), for
that referred to an irreversible approach to equilibrium, while this refers,
to a change from one equilibrium state to another of different energy.!
Such a change of molecules on a molfif.nlfl.r sp.fl.lft is t.q \}& interpreted as ail
absorption of heat. The second term, however, comes about when the
/Vs and the entropy do not change, but the energies of the cells themselves
change, on account of changes in external forces and in the potential
energy. This is to be interpreted as external work donft on thn s
the negative of the work done by the system. Thus we^have

dQ = Etdf dW = -

(5.12)

Combining Eq. (5.12) with Ea. (5.11^ yivns us the first law,

dU = dO - dW.
Combining with Eq. (5.10)T we have

dS = -kbdQ.^ (5.13)

Equation (5.13), stating the proportionality of dS and dQ for a reversible
process, is a statement of the second law of thermodynamics for a reversi-
ble process, if we have

- (5.14)

Using Eq. (5.14), we can identify the constants in Eq. (5.9), obtain-
ing as the representation of the canonical assembly

~"— • (5.15)

50 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. Ill

It is now interesting to compute the Helmholtz free energy A = (7 — TS.
This, using Eqs. (5.1) and (5.15), is

A = 2/t( #t - E\ - kT In
t \

= -ArTMn JV**, - (5.16)

^ t

(5.17)

j»

Using Eq. (5.17), we may rewrite the formula (5.15) as

= 6 *y • (5.18)

The result of Eq. (5.17) is, for practical purposes, the most importan t
result of statistical mechanics. For it gives a perfectly direct and
strajghtforwftff] way nf (jpfiving the Helmholtz fyee energy, find hence
the equation of state and specific heat, of any system, if we know its
energy as a function of coordinates and Hininmtft — The sum of Eq.
(5.17), which we have denoted bv Z. is often called the partition function.

Oftnn it fc flfiflfill ffft fro flh^ **^a™* l^o pn|rnnv[ ^ptP.i^^pnP.rpv, and

specific heat directly from the partition function, without separately
computing the Helmholtz free energy. For the entropjt, using

we have

S = \ — (kT In Z) / =• k In Z H ( — I (5.19)

For the internal energy, U = A + TS, we have

U = -*r In Z + kT]nZ + ^-fe

(5.20)
where the last form is often useful. For the specific heat at constant

SBC. 5| STATISTICAL MECHANICS 51

volume, we may use either CV = (dV/dT)v or Cv = T(dS/dT)r. From
the latter, we have

Jd*(kTluZ)\
Cv = 1—- --

(5.21)

We have stated our definitions of entropy, partition function, and
other quantities entirely in terms of summations. Often, however, the
quantity /t changes only slowly from cell to cell; in this case it is con-
venient to replace the summations bv integrations over the ohaso space.
We recall that all cells are of the same volume, An, if there are n coordi-
nates and n momenta in the phase space. Thus the number of cells in a
volume element dcji . . . da*. dv\ . . . dvn of phase space is

Then the partition function becomes —

e kT dqi . . . dpn, \ (5.22)

a very convenient form for such problems as finding the partition function
of a perfect gas.

CHAPTER IV
THE MAXWELL-BOLTZMANN DISTRIBUTION LAW

In most physical applications of statistical mechanics, we deal with a
system composed of a great number of identical atoms or molecules, and
are interested in the distribution of energy between these molecules. The
simplest case, which we shall take up in this chapter, is that of the perfect
gas, in which the molecules exert no forces on each other. We shall be
led to the Maxwell-Boltzmann distribution law, and later to the two forms
of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose
statistics.

1. The Canonical Assembly and the Maxwell-Boltzmann Distribu-
tion.— Let us assume a gas of N identical molecules, and let each molecule
have n degrees of freedom. That is, n quantities are necessary to specify
the configuration completely. Ordinarily, three coordinates are needed
to locate each atom of the molecule, so that n is three times the; number
of atoms in a molecule. In all, then, Nn coordinates are necessary to
describe the system, so that the classical phase space has 2Nn dimen-
sions. It is convenient to think of this phase space as consisting of N
subspaces each of 2n dimensions, a subspace giving just the variables
required to describe a particular molecule completely. Using the quan-
tum theory, each subspace can be divided into cells of volume hn, and a
state of the whole system is described by specifying which quantum state
each molecule is in, in its own subspace. The energy of the whole system
is the sum of the energies of the N molecules, since for a perfect gas there
are no forces between molecules, or terms in the energy depending on
more than a single molecule. Thus we have

N
#=2<(f), (1.1)

where 6(t) is the energy of the zth molecule. If the ith molecule is in the
Atth cell of its subspace, let its energy be €j£. Then we can describe
the energy of the whole system by the set of kjs. Now, in the canonical
assembly, the fraction of all systems for which each particular molecule, as
the fth, is in a particular state, as the fctth, is

SBC. 1] THE MAXWELL-BOLTZMANN DISTRIBUTION LAW 53

kl kN

,—t^ki/kT

(1.2)

ki kN

It is now interesting to find the fraction of all systems in which a particu-
lar molecule, say the ith, is in the fctth state, independent of what other
molecules may be doing. To find this, we merely sum the quantity (1.2)
over all possible values of the fc's of other molecules. The numerator of
each separate fraction in Eq. (1.2), when summed, will then equal the
denominator and will cancel, leaving only

(1.3)

as the fraction of all systems of the assembly in which the zth molecule is
in the /btth state, or as the probability of finding the ith molecule in the
A;tth state. Since all molecules are alike, we may drop the subscript i
in Eq. (1.3), saying that the probability of finding any particular molecule
in the kih state is

2e~Ff

(1.4)

Equation ^1.4) expresses what is called the Maxwell-Boltzmann distribu-
tion law. If Eq. (1.4) gives the probability of finding any particular
molecule in the fcth state, it is clear that it also gives the fraction of ^11
molecules to be found in that state, averaged through the assembly.

The Maxwell-fioltzmann distribution law can be used for many
calculations regarding gases ; in a later chapter we shall take up its appli-
cation to the rotational and vibrational levels of molecules. For the
present, we shall describe only its use for monatomic gases, in which there
is only translational energy of the molecules, no rotation or vibration. In
this case, as we shall show in the next paragraph, the energy levels «* of a
single molecule are so closely spaced that we can regard them as continu-
ous and can replace our summations by integrals. We shall have three
coordinates of space describing the position of the molecule, and three
momenta, px, py, pt, equal to the mass ra times the components of velocity,
vx, vv, vg. The energy will be the sum of the kinetic energy, %mv2 = p */2m,

54 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

and the potential energy, which we shall denote by <£(z, y, z) and which
may come from external gravitational, electrostatic, or other types of
force field. Then the fraction of molecules in the range of coordinates
and momenta dx dy dz dpx dpv dp, will be

_?_ kT ^dx dy dz dpx dpy dpz

'" (pV2m+*) * ' U-O;

J7JJ J J e kT dx dy dz dpx dpv dpz

In the next section we shall derive some simple consequences from this
form of the Maxwcll-Boltzmann distribution for perfect moriatomic
gases.

In the last paragraph we have used the fact that the translational
energy levels of a perfect monatomic gas are very closely spaced, accord-
ing to the quantum theory. We can sec this as follows, limiting ourselves
to a gas in the absence of an external force field. Each molecule will have
a six-dimensional phase space. Consider one pair of variables, as x and
px. Since no forces act on a molecule, the momentum px stays constant
during its motion, which must take placo in a range X along the x axis,
if the gas is confined to a box of sides X, F, X along the three coordinates.
Thus the area enclosed by the path of the particle in the x — px section of
phase space is pxX, which must be equal to an integer nx times h. Then
we have

nxh nvh nji ,, „,

where the n's are integers, which in this case can be positive or negative
(since momenta can be positive or negative). The energy of a molecule
is then

2m ~ 2m\X* F2 Z*

To get an idea of the spacing of energy levels, let us see how many levels
are found below a given energy e. We may set up a momentum space, in
which px, pvi pz are plotted as variables. Then Eq. (1.6) states that a
lattice of points can be set up in this space, one to a volume

where V = XYZ is the volume of the container, each point corresponding
to an energy level. The equation

SEC. 2] THE M AX W ELL-BOLT Z MANN DISTRIBUTION LAW 55

is the equation of a sphere of radius p = \/2we in this space, and the
number of states with energy less than 6 equals the number of points
within the sphere, or its volume, ^7r(2mc)^, times the number of points
per unit volume, or V/h*. Thus the number of states with energy less
than e is

, (1.9)

and the number of states between e and e + rfe, differentiating, is

(/c. (1.10)

The average energy between successive states is the reciprocal of Eq.
(1.10), or

Let us see what this is numerically, in a reasonable case. We take a
helium atom, with mass 6.63 X 10~24 gm., in a volume of 1 cc., with an
energy of k = 1.379 X 10~~16 erg, which it would have at a fraction of a
degree absolute. Using h = 6.61 X 10~27, this gives for the energy
difference between successive states the quantity 8. 1 X 10~38 erg, a com-
pletely negligible energy difference. Thus we have justified our state-
ment that the energy levels for translational energy of a perfect gas are so
closely spaced as to be essentially continuous.

2. Maxwell's Distribution of Velocities. — Returning to our distribu-
tion law (1.5), let us first consider the case where there is no potential
energy, so that the distribution is independent of position in space. Then
the fraction of all molecules for which the momenta lie in dpx dpv dpx is

e 2mhTdpx dpy dp, (e> - v

_ , \^' * )

fffe~*"*Tdpxdpvdpz

where p2 stands for pi +' p2 + p2. The; integral in the denominator can
be factored, and written in the form

/•oo P'* /•« ?"*- /• QO P«*

I g 2mkT ^px i g %mkT dpyl e %mkT dpg. (2.2)

a/ •• • oo J — w •/ ™ oo

/OO 1

e~au* du, where a = *„,• We shall

meet many integrals of this type before we are through, and we may as

56 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

well give the formulas for them here. We have

I une~QH* du =
Jo

Wa f°r n = ° 5? f°r W = *

1 [?r ' = 2 L for ;/ s 3

= 4 4-} for ?i = 5, etc. (2.3)

3 It
sVo1

Starting with fe~au* du, for the even powers of n, or with fue~au* du for the
odd powers, each integral can be found from the one above it by differen-
tiation with respect to — a, by means of which the table can be extended.
To get the integral from — oo to <x> , the result with the even powers is
twice the integral from 0 to oo t and for the odd powers of course it is
zero.

Using the integrals (2.3), the quantity (2.2) becomes (2wmkT)^'.
Thus we may rewrite Eq. (2.1) as follows: the fraction of molecules with
momentum in the range dpx dpy dpg is

(ZirmkT) e 2mkT dpt dpu dp,.

(2.4)

Equation (2.4) is one form of the famous Maxwell distribution of
velocities.

Often it is useful to know, not the fraction of molecules whose vector
velocity is within certain limits, but the fraction for which the magnitude
of the velocity is within certain limits. Thus let v be the magnitude of
the velocity:

Then we may ask, what fraction of the molecules have a speed between
i) and v + dv, independent of direction? To answer this question we
consider the distribution of points in momentum space. The volume of
momentum space corresponding to velocities between v and v + dv is
the volume of a spherical shell, of radii mv and rn(v + dv). Thus it is
4w(mvy d(mv). We must substitute this volume for the volume dps dpv
dpz of Eq. (2.4). Then we find that the fraction of molecules for which
the magnitude of the velocity is between v and v + dv, is

This is the more familiar form of Maxwell's distribution law. We give a

SEC. 2] THE M AXW ELL-BOLT Z MANN DISTRIBUTION LAW

graph of the function (2.6) in Fig. IV-1. On account of the factor y'2, tho
function is zero for zero speed; and on account of the exponential it is
zero for infinite speed. In between, there is a maximum, which is easily
found by differentiation and comes at v — \/2kT/m. That is, the maxi-
mum, and in fact the whole curve, shifts outward to larger velocities as
the temperature increases.

From Maxwell's distribution of velocities, either in the form (2.4) or
(2.6), we can easily find the moan kinetic energy of a molecule at tem-
perature T. To find this, we multiply the kinetic energy p2/2m by tho
fraction of molecules in a given range dpx dpy dpg, and integrate over all
values of momenta, to get the weighted mean. Thus we have

-54 C C Cv* —

Mean kinetic energy = (2irmkT) III s—e 2mkT dp* dpu dpz

- v ( C "° 2 - PT*

\ £—e~2"'^'dpx(2TrmkT)

(J- » 2m

+ similar terms in p*, pi /

*($kT)(2TrmkT)**
= tfcZ7. (2.7)

The formula (2.7) for the kinetic energy of a molecule of a perfect gas
leads to a result called the equipartition of onorgy. Each molecule has
three coordinates, or three degrees of
freedom. On the average, each of
those will have one-third of the total
kinetic energy, as we can see if we find
the average, not of (pi + p% + pi) /2m,
but of the part pl/2m associated with
the x coordinate. Thus eqg|i of thcao
degrees of frccdom^haa nn the average*
tJ^gj^gjgyA-fcT7. Thy energy, in other
words, is equally distributed between
each one having the

1.0

2.0

and this is called

the eauipartition of energy. Mathe- ., ... , *,v ^7 *. » .

. i " Yft .. n^ -— FIG. IV-1. —Maxwell's distribution of

matlCally, as an examination Of Our velocities, giving the fraction of molecules

proof shows ftniiinfl.rtit.inn is fl. result whose velocity is between v and v + dv, in

«— — • •••iM^^rftaBrfbBi^ « fc^Mfc^M— ii—M^ a gas a^ temperature T.
of th(^fact^thaJ^th£1J^BJ|2^^

with each degree of freedom is nronprtional to the square

^rro<

rifii

or coordinate.

which is found in the energy only as a square, will be found to have a
mean energy of ^feTy provided the energy levels are continuously

58 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

that anmmg.f.inng fian be replaced bv integrations. This
applies not only to the momenta associated with translation of a single
atom, but to such quantities as the angular momentum of a rotating
molecule (whose kinetic energy is p2/2/), if the rotational levels are spaced
closely. It applies as well to the coordinate, in the case of a linear oscilla-
tor, whose restoring force is — kx, and potential energy fcx2/2; the mean
potential energy of such an oscillator is ffiT, if the levels are spaced closely
enough, though in many physical cases which wo shall meet the spacing is

not close enough to replace summations by
integrations.

3. The Equation of State and Specific Heat
of Perfect Monatomic Gases. — Having found
the distribution of molecular velocities, we can
calculate the equation of state and specific
heat of perfect monatomic gases by elementary
methods, postponing until a later chapter the
iv-2. -Diagram to direct statistical calculation by means of the
illustrate collisions of mole- partition function. We shall again limit our-

cules with 1 sq. cm. of surface 0_i * 4.1 • i L , i

Of waji. selves to the special case where there is no

potential energy and where the distribution

function is independent of position. This is the only ease where we
should expect the pressure to be constant throughout the container. We
shall find the pressure by calculating the momentum carried to the wall
per second by molecules colliding with it. The momentum transferred
to 1 sq. cm. of wall per second is the force acting on the wall, or the
pressure.

For convenience, let us choose the x axis perpendicular to the square
centimeter of wall considered, as in Fig. IV-2. A molecule of velocity
v = p/m close to the wall will strike the wall if px is positive, and will
transfer its momentum to the wall. When it is reflected, it will in general
have a different momentum from what it 6ngmally~had, and will come
away with momentum p', the component p'x being negativeT^-STEer
collision, in other words, it will again belong to the group ofmolecules
near the wall, but now corresponding to negative px, and it wilFhave
taken away from the wall the momentum p', or will" have given the wall
the negative of this momentum. We can, then, get all the momentum
transferred to the wall by considering all molecules, both with positive
and negative px's. Consider those molecules contained in "the element
dpxdpydpz in momentum space, and lying in tHe prism drawn in Fig.
IV-2. Each of these molecules, and no others, will strike the square
centimeter of wall in time dt. The volume of the prism is px/m dt. The
average number of molecules per unit volume in the momentum element
dpx dpv dpzj averaged through the assembly, will be denoted by fm dpxdpv

SBC. 3] THE M AXW ELL-BOLT Z MANN DISTRIBUTION LAW 59

dp,. For Maxwell's distribution law, we have

fm = y(2irrofcrr V5^ (3.1)

using Eq. (2.4), where N/V is the number of molecules of all velocities
per unit volume. We shall not explicitly use this form for fm at the
moment, however, for our derivation is more general than Maxwell's
distribution law, and holds as well for the Fermi-Dirac and Einstcin-Bose
distributions, which we shall take up in later sections. Using the func-
tion fm, the average number of molecules of the desired momentum, in
the prism, is p^/m/m dpx dpi, dp,. Each such molecule takes momentum
of components px, py, pz to the surface. Hence the total momentum given
the surface in time dt by all molecules is the integral over momenta of the
quantity with components (pl/rn, pxp1t/m, pxpz/m)dtfmdpxdpvdp*.
Dividing by dt, we have the force exerted on the square centimeter of
surface, which has components

f f (V

x component of force = I I I —fm dpx dpy dpz,

y component of force = I I I — - fm dpx dpu dpz,

z component of force = I I I fm dp* dpy dpt. (3.2)

Now we shall limit our distribution function. We shall assume that
a molecule with a given value of px is equally likely to have positive or
negative values of py and pz, so that fm(px> py} = f™(pxj — py), etc.
Plainly the Maxwell law (3.1) satisfies this condition. Then the second
and third integrals of Eq. (3.2) will be zero, since the integrands with a
given px, py will have opposite sign to the integrands at px, —py, and will
cancel each other. The force on the unit area, in other words, is along the
normal, or corresponds to a pure pressure, without tangential forces.
Thus we finally have

///*

» dpx dpv dpz. (3.3)

Now fm dpx dpy dpz is the number of molecules per unit volume in the
range dpx dpv dpz. Multiplying by pl/m and integrating, we have simply
the sum of pl/m for all molecules in unit volume. This is simply the sum
of pl/m for all molecules of the gas, divided by V. Let us assume that the
distribution function is one for which all directions in space are equivalent.
This is the case with the Maxwell distribution, Eq. (3.1), for this depends
only on the magnitude of p, not on its direction. Then the sum of

60 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

equals that of pj/m or pl/m. We have, moreover,

rf + Py + Sf.P^^X kinetic energy. (3.4)

m m m m

Hence, the sum of pl/m is two-thirds the total kinetic energy, and since
there is no potential energy, this is two-thirds the internal energy.
Finally, then, we have

p-2^,

^ ~3F'
or

PV = f U. (3.5)

Equation (3.5) gives the relation predicted by kinetic theory for the
pressure of a perfect monatomic gas, in terms of its internal energy and
volume.

We can now combine Eq. (3.5), and Eq. (2.7), giving the mean kinetic
energy of a monatomic gas, to find the equation of state. From Eq.
(2.7), we have at once for N molecules

U = $NkT. (3.6)

Combined with Eq. (3.5), this gives at once

PV = NkT, (3.7)

as the equation of state of a perfect gas, as derived by elementary meth-
ods. We should compare Eq. (3.7) with the gas law as ordinarily set up
from experimental measurements. Let us suppose that we have n moles
of our gas. That is, the mass of gas wo are dealing with is nM, where M
is the molecular weight. Then the usual law is

PV = nRT, (3.8)

where R, the gas constant per mole, is given alternatively by the numerical
values

R = 8.314 X 107 ergs per degree
= 0.08205 l.-atm. por degree

(3-9)

The law (3.8) expresses not only Boyle's and Charles's laws, but also
Avogadro's law, stating that equal numbers of moles of any two perfect
gases at the same pressure and temperature occupy the same volume.
Now let JV0 be the number of molecules in a gram molecular weight, a
universal constant. This is ordinarily called Avogadro's number and is

SBC. 3] THE M AXW ELL-BOLT Z MANN DISTRIBUTION LAW 61

given approximately by

#o - 6.03 X 1028, (3.10)

by methods based on measurement of the charge on the electron. Then
we have

N = rcAT0, (3.11)

MO that Eq. (3.8) is replaced by

PV = N-^-T. (3.12)

-tVo

Equation (3.12) agrees with Eq. (3.7) if

k = -^-i or R = Nok,
NO

so that

8.314 X 107
6.03 X 102*
= 1.379 X 10-lfi erg per degree, (3.13)

as was stated in Chap. Ill, Sec. 1.

From the internal energy (3.6) we can also calculate the specific heat.
We have

The expression (3.14), as we have mentioned before, gives the heat capac-
ity of n moles of gas. The specific heat is the heat capacity of 1 gin., or
1/M moles, if M is the molecular weight. Thus it is

3 R

Specific heat per gram = HT>' (3.15)

Very often one also considers the molecular heat, the heat capacity per
mole. This is

Molecular heat, per mole = f 72 = 2.987 cal. per mole. (3.16)

To find the specific heat at constant pressure, we may use Eq. (5.2),
Chap. II. This is

(-Y

(3.17)

^P/r

which holds for any amount of material. Substituting V = nRT/P, we
have (dV/dT)f = nR/P, (dV/dP)T - -nRT/P\ so that

CP = CV + nR = -Jnff = 4.968 cal. per mole, (3.18)

62 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

and

^ = 7=f = 1.667. (3.19)

v/ V O

The results (3.14), (3.18), and (3.19) hold theoretically for monatomic
perfect gases, and actually they are approximately true for real monatomic
gases.

4. The Perfect Gas in a Force Field. — For two sections we have been
considering the distribution of velocities included in the Maxwell-Boltz-
mann distribution law. Next, we shall take up the distribution of coordi-
nates in cases where there is an external field of force. First, we should
observe that on account of the form of Eqs. (1.4) and (1.5), the distribu-
tions of coordinates and velocities are independent of each other. These

equations show that the distribution function contains one factor e 2mkT

depending on velocities, another factor e kT depending on coordinates.
This has important implications. The Maxwell distribution of velocities.
wfafoh we l^ave discu^snd. is the *ffiflic at any point of a gas, even in an
external field; and the variation of density with position is the same for
the whole density, or for the particular class of molecules having an v
chosen velocity. We wish, then, to discuss the variation of density com-
ing from the factor e k^. The most familiar example of this formula is
found in the decrease of density of the atmosphere as we go to higher
altitudes. The potential energy of a molecule of mass m at height ft
above the earth is mgft, where g is the acceleration of gravity. Then the
density of gas at height ft, assuming constant temperature throughout
(which is not a good assumption for the actual atmosphere), is given by

mgh

Density proportional to e kr . (4.1)

Formula (4.1) is often called the barometer formula, since it gives the
variation of barometric pressure with altitude. It indicates a gradual
decrease of pressure with altitude, going exponentially to zero at infinite
height.

The barometer formula can be derived by elementary methods, thus
checking this part of the Maxwell-Boltzmann distribution law. Con-
sider a column of atmosphere 1 sq. cm. in cross section, and take a section
of this column bounded by horizontal planes at heights ft and ft + dh.
Let the pressure in this section be P; we are interested in the variation of
P with ft. Now it is just the fact that the pressure is greater on the lower
face of the section than on the upper one which holds the gas up against
gravity. That is, if P is the upward pressure on the lower face, P + dP
the downward pressure on the upper face, the net downward force is dPy

SBC. 4] THE MAXWELL-BOLTZMANN DISTRIBUTION LAW 6

the net upward force — dP, and this must equal the force of gravity on th
material in the section. The latter is the mass of the gas, times g
The mass of the gas in the section is the number of molecules per uni
volume, times the volume dh, times the mass m of a molecule. The num
ber of molecules per unit volume can be found from the gas law, whicl
can be written in the form

P = ^*r, (4.2

where (N/V) is the number of molecules per unit volume. Then we fine
that the mass of gas in the volume dh is (P/kT)m dh. The differentia
equation for pressure is then

In P- const. -, (4.3;

from which, remembering that at constant temperature the pressure if-
proportional to the density, we have the barometer formula (4.1).

It is possible not only to derive the barometer formula, but the whole
Maxwell-Boltznuum distribution law, by an extension of this method,
though we shall not do it.1 One additional assumption must be made,
which we have treated as a consequence ot the distribution law rather thin
as an independent hypothesis: that the mean kinetic o^gy flf ft ™nW"]'
is 4ferT indepn^Hnnt of whore it may hn found. Assuming it and con-
sidering the distribution of velocities in a gravitational field, we seem at
first to meet a paradox. Consider the molecules that are found low in
the atmosphere, with a certain velocity distribution. As any one oi
these molecules rises, it is slowed down by the earth's gravitational field,
the increase in its potential energy just equaling the decrease in kinetic-
energy. Why, then, do not the molecules at a great altitude have lowci
average kinetic energy than those at low altitude? The reason is not
difficult to find. The slower molecules at low altitude never reach the
high altitude at all. They follow parabolic paths, whose turning points
come at fairly low heights. Thus only the fast ones of the low molecules
penetrate to the high regions; and while they are slowed down, they slow
down just enough so that their original excessive velocities are reduced
to the proper average value, so that the average velocity at high altitudes
equals that at low, but the density is much lower. Now this explanation

^cc for instance, K. F. Herzfeid, "Kinetische Theorie der Warme," p. 20,
Vieweg, 1929.

64 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IV

puts a very definite restriction on the distribution of velocities. By the
barometer formula, we know the way in which the density decreases from
h to h + dh. But the molecules found at height A, and not at height
h + dhj are just those which come to the turning point of their paths
between these two heights. This in turn tells us the vertical components
of their velocities at any height. The gravitational field, in other words,
acts to spread out molecules according to the vertical component of their
velocities. And a simple calculation based on this idea proves to lead
to the Maxwell distribution, as far as the vertical component of velocities
is concerned. No other distribution, in othnr wnrdsr would have this
special property of giving the same distribution of velocities at anv
height.

The derivation which we have given for the barometer formula in
Eq. (4.3) can be easily extended to a general potential energy. Let the
potential energy of a molecule be <t>. Then the force acting on it is d<t>/ds,
where ds is a displacement opposite to the direction in which the force
acts. Take a unit cross section of height ds in this direction. Then, as
before, we have

In P = const. - pp, (4.4)

general formula for pressure, or density^ as it depends on potential
energy.

CHAPTKR V
THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS

The Maxwell-Boltzmann distribution law, which we have* derived
and discussed in the last chapter, seems like a perfectly straightforward
application of our statistical methods. Nevertheless, when we come to
examine it a little more closely, we find unexpected complications, arising
from the question of whether there really is any way of telling the mole-
cules of the gas apart or not. We shall analyze these questions in the
present chapter, and shall find that on account of the quantum theory r t.lm
Maxwell-Boltzinann distribution law is really only an approximation
valid for gases at comparatively low density, a limiting case? of two other
distributions, known by the names of the Fcrmi-Dirac and the KlinsteiiY
Bosc statistics. _ Real gases obey one or the other of these latter forms of
statistics, some being governed by one, some by the other. As a matter
of fact, for all real fiases the nnrrnp.t.innH t.n t.ho MnYwp1]-fin]tzm.inn dis-
tribution law which result from the quantum statistics an* negligibly
small except at the very lowest temperatures, and helium is the only fi^s
remaining in the vauor state at low enough temperature for the correc-
tions to be important. Thus the reader who is interested only in molecu-
lar gases may well feel that these forms of quantum statistics are
unnecessary. There is one respect in which this feeling is not justified:
we shall find that in the calculation of the entropy, it is definitely wrong
not to take account of the identity of molecules. But the real iiimortanr»n

of the nnfUltllTn fo^T^fi nf Jttfq.f.l*jt.lf».H Pompfi from f.hfl ffU'.f i.ltn.f, f.lw* nl^nfrnnw

in solids satisfy the Fcrmi-Uirac statistics, and for them thf* nn moving]
quantities are such that the behavior is completely different frorp what
would be predicted bv the M^wn11-**"1*"™""" '1^+™^"+:™ law. The
^""tHn-RrtfiP ^-p*™*™^ tho11^ *f MS applications to black-body radia-
tion, does not have the general importance of the Fermi-Dirac statistics.
1. The Molecular Phase Space. — In the last chapter, we pointed out
that for a gas of N identical molecules, each of n degrees of freedom, the
phase space of 2Nn dimensions could be subdivided into N subspaces,
each of 2n dimensions. We shall now consider a different way of describ-
ing our assembly. We take simply a 2/i-dimensional space, like one of our
previous subspaces, and call it a molecular phase space, since a point
in it gives information about a single molecule. This molecular phase
space will be divided, according to the quantum theory, into cells of
volume hn. A given quantum state, or complexion, of the whole gas of N

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. V

molecules, then corresponds to an arrangement of N points representing
the molecules, at definite positions, or in definite cells, of the molecular
phase space. But now we meet immediately the question of the identity
of the molecules. Are two complexions to be counted as the same, or as
different, if they differ only in the interchange of two identical molecules
between cells of the molecular phase space? Surely, since two such
complexions cannot be told apart by any physical means, they should
not be counted as different in our enumeration of complexions. Yet they
correspond to different cells of our general phase space of 2Nn dimen-
sions. In one, for example, molecule 1 may be in cell a of the molecular
phase space, molecule 2 in cell 6, while in the other molecule 1 is in cell
bj molecule 2 in cell a. Tlm&JiLJj_S¥!stem. of identical molecules, it is
incorrect to assume that every complexion that we can set up in the
general phase space, by assigning each molecule to a particular cell of
its ffiKgpapp yq Hi.gf.iti/»+, 1™™ /vir^y ^hor Qfi ^yp tanitly assumed in Chap.

By considering the molecular phase space, we can see how many
apparently different complexions really are to be grouped together as
one. Let us describe a complexion by numbers Nv, representing the
number of molecules in the Mi cell of the molecular phase space. This
is a really valid way of describing the complexion; interchange of identical
molecules will not change the Nfs. How many ways, we ask, are there
of setting up complexions in the general phase space which lead to a given
set of JWs? We can understand the question better by taking a simple
example. Let us suppose that there are three cells in the molecular phase
space and three molecules, and that we arc assuming Ni — 1, N% = 2,

TABLE V-l

Cell

,..

Complexion

2 3
3 2

1 3

3 1

1 2

2 1

NB = 0, meaning that one molecule is in the first cell, two in the second,
and none in the third. Then, as we see in Table V-l, there are three
apparently different complexions leading to this same set of Ni's. In
complexion a, molecule 1 is in cell 1, and 2 and 3 are in cell 2; etc. We

SBC. 1] THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS 67

see from this example how to find the number of complexions. First, we
find the total number of permutations of N objects (the N molecules).
This, as is well known, is TV!; for any one of the N objects can come first,
any one of the remaining (N — 1 ) second, and so on, so that the number
of permutations is N(N — i)(N — 2) . . . 2.1 = Nl In the case of
Table V-l, there are 3! = 6 permutations. But some of these do not
represent different complexions, even in the general phase space, as
we show by our brackets; as for example the arrangements 1,23 and 1,32
grouped under the complexion (a). For they both lead to exactly the
same assignment of molecules to cells. In fact, if any N% is greater than
unity, Nil (in our case 2! = 2) different permutations of the N objects
will correspond to the same complexion. And in general, the number of
complexions in the general phase space which lead to the same TVYs, and
hence are really identical, will be

Remembering that 0! = 1! = 1, we see that in our example we have
3 1/1 12 10! =|= 3.

If then we wished to find the partition function for a perfect gas, using
the general phase space, we should have to proceed as follows. We
couldset uu cells in phase space, each of volume hNn. but we could not
assume that each of these represented a different complexion, or thftt we
were to sum over all these flnlla in nnrnpiifing +^ poi.*^™* 1,,^;,^
Rather, each cell would be one of Nl/N-dNf] - . . similar cells, all taken
together fa ygpreseiyfr one single complexion. We could handle this .11
we chose^ bv summing, or in the case of continuous energy levels fry
i ntegrathifi. over all cells, but dividinp: the contribution of each cell by
the number (1.1). computed for that cell. Since this number can change
from cell to cell, this is a very inconvenient procedure and cannot be
carried out without rather complicated mathematical methods. There
is a special case, however, in which it is very simple. This is the case
where the gas is so rare jfchat we are very unlikely! in our assembly, to find
any appreciablo^iiiinib£r of systems with more than a single molecule in
any cell. In this case, each of the JV^s^iiLJJifi_denQminatQr of formula
(1.1) will be 0 or 1, each of the JVJ's will be 1. and the number (1.1)
becomes simply Nl. Thus, in this case, we can find the partition function
by carrying out the summation or integration in the general phase space
in the usual way, but dividing the result by JVI, and using the final value
to find the Helmholtz free energy and other thermodynamic quantities.
This method leads to the Mn.Y^p|1-Rn1t.7.mnnn HiHf.Hhiitjpn IRW. and it is
the method which we shall use later in Chap. VIII, dealing with thermo-
dynamic and statistical properties of ordinary perfect gases. When we

68 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

are likely to find JVYs greater than unity, however, this method is imprac-
ticable, and we must adopt an alternative method based directly on the
use of the molecular phase space.

2. Assemblies in the Molecular Phase Space. — When we describe a
system by giving the JWs, tho numbers of molecules in each coll of the
molecular phase space, we automatically avoid the difficulties described
in the last section relating to the identity of molecules. We now meet
immediately the distinction between the Fcrmi-Dirac, the Einstein-Bose,
and the classical or Boltzmann statistics. In the Einstcin-Bose statistics.
the simplest form in thenrv WP «^f up a comnlexion hv giving a. set Of
Ni's, and we say that any possible set of JWs, subject only to the obvious
restriction

represents a. possible e.orflplexion. all florflplexions having equal A priori
probability. The Fermi-Dirac statistics differs from the Einstein-Bose
in that there is ayi n^ti™™! p^^pl^, nnllpH fh» pv»1i"=jjnn prj^piQ
superposed on the principle of identity of molennles.. The exclusion
nrTneiple states that no two molecules mav he in the same nail nf the
molecular phase space at the same time: that is. no one of t.hp. /y.'« nrm.v
be greater than unity. This principle gains its importance from the fact
that electrons are found experimentally to obey it. It is a principle
which, as we shall see later, is at the foundation of the structure of the
atoms and the periodic table of the elements, as well as having the greatest
importance in all problems involving electrons. In the Fermi-Dirac
statistics, then, any possible set of JWs. subject to ECL (2.1) and to the

nrn OT unitvr forms a

possible complexion nf the ^ysteni. Finally the Boltzmann statistics is
the limiting case of cither of the other types, in the limit of low density,
where so few molecules ^ajiL-disijibutod amiing— ^o-uiisiiiszL-cells thfl/h the
chance of finding two in the saino_c^ll js negligible anvwftyl ^i\c\ thn difYnr-
ence between the Fcrmi-Dirac and the Einstein-Bo*^ fftitjtfif ^ ^«*»pp»«^
Let us consider a single complexion, represented by a set of A/Ys, in
the molecular phase space. We see that the JVYs are likely to fluctuate
greatly from cell to cell. For instance, in the limiting case of the Boltz-
mann statistics, where there are many fewer molecules than cells, we shall
find most of the JVYs equal to zero, a few equal to unity, and almost none
greater than unity. It is possible in principle, according to the principle
of uncertainty, to know all the A/Ys definitely, or to prepare an assembly
of systems all having the same JVYs. But for most practical purposes
this is far more detailed information than we require, or can ordinarily
give. We have found, for instance, that the translational energy levels

SEC. 2) THE FERMI-DIRAC AND El N STEIN-BOISE STATISTICS 69

of an ordinary gas are spaced extremely close together, and while there is
nothing impossible in principle about knowing which levels contain
molecules and which do not, still practically we cannot tell whether
a molecule is in one level or a neighboring one. In other words, for
this case, for all practical purposes the scale of our observation is much
coarser than the limit set by the principle of uncertainty. Let us, then,
try to set up an assembly of systems reflecting in some way the actual
errors that we are likely to make in observing molecular distributions.
Let us suppose that really we cannot detect anything smaller than a
group of G cells, where G is a rather large number, containing a rather
large number of molecules in all the systems of our assembly. And let us
assume that in our assembly the average number of molecules in the ith
cell, one of our group of G, isffit, a quantity that ordinarily will b<» a frac-
tion rather than an integer In the particular case 01 ttie uoit/mami
statistics. N . ^jll b^ n. f motion much less than unity \ in the Feriui-Dirac .
statistics it will be les^ than unity, but not necessarily much less; while in
the Einstein-Bose statistics it can have any value. We shall now try to
set up an assembly leading to these postulated values of the N^. To do
this, we shall find all the complexions that lead to the postulated $Ys, in
the sense of having NtG molecules in the group of G cells, and we shall
assume that these complexions, and these only, are represented in the
assembly and with equal weights. Our problem, then, is to calculate the
number of complexions consistent with a given set of JVYs, or the thermo-
dynamic probability W of the distribution, in Boltzinann's sense, as
described in Chap. Ill, Sec. 1. Having found the thermodynamic prob-
ability, we can compute the entropy of the assembly by the fundamental
relation (1.3) of Chap. Ill, or

8 = * In W. (2.2)

For actually calculating the thermodynamic probability, we must
distinguish between the Fermi-Dirac and the Einstein-Bose statistics.
First we consider the Fermi-Dirac case. We wish the number of ways of
arranging N£f molecules in G cells, in such a way that we never have
more than one molecule to a cell. To find this number, imagine G coun-
ters, of which N£f are labeled 1 (standing for 1 molecule), and the remain-
ing (1 — ftt)G are labeled 0 (standing for no molecules). If we put one
counter in each of the G cells, we can say that the cells which have a
counter labeled 1 in them contain a molecule, the others do not. Now
there are G\ ways of arranging G counters in G cells, one to a cell, as we
have seen in the last section. Not all of these G\ ways of arranging the
counters lead to different arrangements of the molecules in the cells,
however, for the N$ counters labeled 1 are identical with each other, and

70 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

those labeled 0 are identical with each other. For a given assignment of
molecules to cells, there will be (N£f) 1 ways of rearranging the counters
labeled 1, and [(1 — Ni)G]\ ways of rearranging those labeled zero, or
(NlG)l[(l — Nl)G]\ arrangements in all, all of which lead to only one
complexion. Thus to get the total number of complexions, we must
divide G\ by this quantity, finding

Number of complexions of NjB atoms in G cells = — _

-• -- (2.3)

We shall now rewrite Eq. (2.3), using Stirling's theorem. This states
that, for a large value of N,

Nl = V'lirN ~ • (2.4)

Stirling's formula is fairly accurate for values of N greater than 10; for
still larger JV's, where N\ and (N/e)N are very large numbers, the factor
is so near unity in proportion that it can be omitted for most

purposes, so that we can write Nl simply as (N/e)N. Adopting this
approximation, we can rewrite Eq. (2.3) as

Number of complexions of N1G atoms in G cells

[1
#/'(! -

Equation (2.5) is of an interesting form: being a quantity independent
of G, raised to the G power, we may interpret it as a product of terms, one
for each cell of the molecular phase space. Now to get the whole number
of complexions for the system, we should multiply quantities like (2.5), for
each group of G cells in the whole molecular phase space. Plainly this
will give us something independent of the exact way we divide up the
cells into groups, or independent of (?, and we find

where JJ indicates a product over all cells of the molecular phase space.

SBC. 2] THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS 71

Using Eq. (2.2), we then have
S = -*5[#< In

- # In 1 -

(2.7)

as the expression for entropy in the Formi-Dirac statistics in terms of the
average number Ni of molecules in each cell.

For the Einstein-Bose statistics, we wish the number of ways of
arranging Nfr molecules in G cells, allowing as many molecules as we
please in a cell. This number ran be shown to be

Number of complexions of N1G atoms in G cells

_ (#/?+(? -1)1

- 1)!

We can easily make Eq. (2.8) plausible,1 though without really proving
it, by an example. Let us take N1G = 2, G = 3, and make a table, as
Table V-2, showing the possible arrangements:

TABLE V-2

Cell 1

' 3

In the first three columns of Table V-2, we indicate the three colls, and
indicate each of the two molecules by a figure 1, showing the six possible

arrangements 6 = - ^ ~ — -v .-" ; following that, we give a scheme with
L *H«J — l;i J

four columns (4 = NtG + G — 1 =2 + 3 — 1) in which we give all the
possible arrangements of the two 1's, two O's, with one in each column
(2 = NtG, 2 = G — 1). In the general case, the number of such arrange-
ments is given just by Eq. (2.8), as wo can see by arguments similar to
those used in deriving formula (1.1). But the four-column arrangement
of Table V-2 corresponds exactly witli the three-column one, if we adopt
the convention that two successive IV in the four-column scheme belong
in the same cell. It is not hard to show that the same sort of correspond-
ence holds in the general case and thus to justify Eq. (2.8).

Applying Stirling's theorem to Eq. (2.8) and neglecting unity com-
pared to G, we now have

1 For proof, as well as other points connected with quantum statistics, see L.
Brillouin, "Die Quantenstatistik," pp. 129 iT , Julius Springer, 1931.

72 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

Number of complexions of N&f atoms in G cells

[

As with the Fermi-Dirac statistics, this is a product of terms, one for
each of the G cells; to find the whole number of complexions, or
the thermodynamic probability, we have

w =

N,»-
and

S = -fc[#, In ft, - (1 + #,) In (1 + #,)]• (2.11)

Iii Eqs. (2.7) and (2.11), we have found the general expressions for
the entropy in the Fermi-Dirac and Ernst ein-Bose statistics. From either
one, we can find the entropy in the Boltzmann statistics by passing to the
limit in which all JVVs are very small compared to unity. For small
Nt, In (1 ± Nt) approaches ±Ni, and (1 ± JV») can be replaced by unity.
Thus either Eq. (2.7) or (2.11) approaches

, ln ^* ~ ^*). (2.12)

Equation (2.12) expresses the form of the ontropy for the Boltzmann
statistics.

3. The Fermi-Dirac Distribution Function. — In the preceding section,
we have set up assemblies of systems satisfying either the Formi-Dirac
or the Einstein-Bose statistics and having an arbitrary average number of
molecules Nl in the ith coll of molecular phase space. We have found the
thermodynamic probability and the entropy of such an assembly. These
assemblies, of course, do not correspond to thermal equilibrium and, as
time goes on, the effect of collisions of molecules will be to change the
numbers fft gradually, with an approach to a steady state. In the next
chapter we shall consider this process specifically, really following in
detail the irreversible approach to a steady state. We shall verify then,
as we could assume from our general knowledge, that during the irreversi-
ble process the entropy will gradually increase, until finally in equilibrium
it reaches the maximum value consistent with a constant value of the

SBC. 3] THE FERMI-D1RAC AND EIN8TE1N-BO8E STATISTICS 73

total energy. But, for the moment, we can assume this final condition
and use it to find the equilibrium distribution. We ask, in other words,
what set of Ni's will give the maximum entropy, subject to a constant
internal energy? We can solve this problem, as we solved a similar one in
Chap. Ill, Sec. 5, by the method of undetermined multipliers.

For the Fermi-Dirac statistics, we wish to make the entropy (2.7) a
maximum, subject to a constant value of the energy. Rather than
impose just this condition, we employ the thermodynamically equivalent
one of making the function A = U — TS a minimum for changes at
constant temperature, as discussed in Chap. II, Sec. 3. This is essentially
a form of the method of undetermined multipliers, the constants multiply-
ing U and S being respectively unity and — T. As in Chap. IV, Sec. 1, we
let the energy of a molecule in the ith state be «». Then the average
energy over the assembly is clearly

U =

the summation being over the cells of the molecular phase space. Using
Eq. (2.7) for the entropy, we then have

A = ][#tct + kTNi In Nl + kT(\ - #,) In (1 - Nt)]. (3.1)

Now we find the change of A, when the $Vs are varied, keeping tem-
perature and the e»'s fixed. We find at once

dA = 0 = + kT ln — T^ (3-2)

as the condition for equilibrium. This must be satisfied, subject only
to the condition

i = 0, (3.3)

expressing the fact that the changes of the Ni's are such that the total
number of molecules remains fixed. The only way to satisfy Eq. (3.2),
subject to Eq. (3.3), is to have

e< + kT In - *— = €<> = const., (3.4)

1 - Ni

independent of i. For then the bracket in Eq. (3.2) can be taken outside
the summation sign, and Eq. (3.3) immediately makes the whole expres-
sion vanish.

74 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

Using Eq. (3.4), we can immodiatcly solve for N*. We have

/
N\

l + e kT = e ~kT
<?-*>

o'kT

(3'5)

Equation (3.5) expresses the Fermi distribution law, which we shall now
proceed to discus:,.

First, let us see that the Fermi-Dirac distribution law reduces to the
Maxwell-Boltzmann law in the limit when the NSs are small. In that
case, it must be that the denominator in Eq. (3.5) is large compared to

the numerator. But if e k? +1 is large compared to 1, the numera-
tor, it must be that 1 can be neglected compared to the exponential.
Thus, in this limit, we can write

Ni = e^, (3.6)

which is the Maxwell-Boltzmann law, Eq. (1.4) of Chap. IV, if €0 is prop-
erly chosen. We notice that even if the temperature is low, so that some
JVVs are not small, still the states of high energy will have large values of

e»

ckT. Thus the argument we have just used will apply to these states,
and for them the Maxwell-Boltzmann distribution will be correct, even
though it is not for states of low energy.

The quantity «o is to be determined by the condition that the total
number of particles is N. Thus we have

N-^?,Ni = ^-Trrl (3-7)

Since the e»'s are determined, Eq. (3.7) can be satisfied by a proper choice
of €0. Unfortunately, Eq. (3.7) cannot be solved directly for c0, and it is a
matter of considerable difficulty to evaluate this important quantity. It
is not hard, however, to see how N* behaves as a function of €t, particu-
larly for low temperatures. When ct — eo is negative, or for energies
below €o, the exponential in Eq. (3.5) is less than unity, becoming rapidly
very small as the temperature decreases. Thus for these energies the
denominator is only slightly greater than unity, and ft* only slightly less
than unity. On the other hand, when a — eo is positive, for energies
above €0, the exponential is greater than unity, becoming rapidly large as

SEC. 3] THE FERMI-DIRAC AND EINSTEIN-ROSE STATISTICS

the temperature decreases. In this case we can almost neglect unity
compared to the exponential, and we have the case of the last paragraph,
where the Boltzmann distribution is approximately correct, in the form

(3.6). In Fig. V-l we show the function -j—( oy as a function of e,

e w' + 1

for several temperatures. At T = 0, the function drops from unity to
zero sharply when e = e0, while at higher temperatures it falls off
smoothly. For large values of €, it approximates the exponential falling
off of the Boltzmann distribution.

-7 -6 -5 -4 -3 -2

FIG. V-l. — Fermi distribution function, as function of energy, for several temperatures.
Curve a, kT = 0; 6, kT = 1; c, kT = 2.5

One important feature of the function (3.5) is the following:

1 -

1-1

(3.8)

+ 1

That is, in Fig. V-l, the distribution function at any point to the right
of €0 is equal to the difference between the function and unity, the same
distance to the left of e0, and vice versa. The curve, in other words, is
symmetrical with change of sign about the point € = €o and ordinate £.
From this it follows that €0 is approximately constant, for small tempera-
tures at least. For the summation in Eq. (3.7), which must give N
independent of temperature, is found as follows. Along the axis of
abscissae in Fig. V-l, we mark the various energy levels of the problem.
At each energy level we erect a line, extending up to the distribution
curve. The sum of the lengths of all these lines is the summation desired.
We must now adjust €0, moving the curve to the left or right, so that the
sum equals N. At the absolute zero this is perfectly simple: we simply
count up to the Nth energy level from the bottom, and put €0 somewhere
between the Nth and the (N -f l)st levels. At a higher temperature,

76 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

suppose we try the same value of «0 and see if it is correct. Then the sum-
mation will change by subtraction of the part of the lines above the distri-
bution curve to the left of €o and by addition of those below the distribution
curve to the right. These areas are equal by the result we have just found.
Thus, if levels come with the same density to the left and to the right of
co, the summation will again be N, and the same value of €o will be correct.
In the next section we find how much co will change with temperature if
the density of levels changes with the energy, as of course it will to some
extent.

4. Thermodynamic Functions in the Fermi Statistics. — Having
derived the Fermi-Dirac distribution law, we shall go on to find some of
its properties in the case of low temperatures, the important case in the
practical applications to electrons in metals, where for the present pur-
poses temperatures even of several thousand degrees can be regarded as
low. The distribution function, as we see in Fig. V-l, changes with
temperature mostly in the immediate neighborhood of the energy c0. If,
then, we know the distribution of energy levels in the neighborhood of
c0, we can find the variation of the thermodynamic functions with tem-
perature. We carry out that analysis in the present section, assuming
that the energy levels are distributed continuously in energy, an approxi-
mately correct assumption in the cases with which we shall deal.

Let the value of c0 at the absolute zero of temperature be c0o. We
know how to find it from Sec. 3, simply by counting up N levels from the
lowest level. First we shall try to find how eo depends on temperature.
We shall assume that the number of energy levels between c and e + d* is

« - [(£). + $),<• - •••> + • • • >• <">

a Taylor's expansion of the function dN/de about the point e = e0o, at
which the derivatives (dN/dt)^ and (dW/dc2)0 are evaluated. With this
assumption, all our summations can be converted into integrations. We
write the summation of Eq. (3.7) in the form of an integration; instead of
using just this form, we find the difference between the summation and
that at the absolute zero, which should give a difference of zero. Thus
we have

0 =

The term —1 in the first integral takes care of the summation at the
absolute zero, where the Fermi function is unity for energies less than «oo,

SEC. 4] THE FUHM1-D1HAV AND E1NSTJK1N-BOSE STATISTICS 77

zero for higher energies. In the first integral, we use Eq. (3.8). Then in
the first we make tho change of variables u = — (e — €0), and in the
second u = (e — €0). The two integrals then combine. We retain only
the terms necessary for a first approximation; this means, it is found,
that in the term (dW/d€2)0 we can neglect the distinction between c0 and
€0o, though this distinction is essential in tho term in (dN/di)^ In this
way we find

~< u 7 t4»\

-u --- du. (4.3)

In the first integral, through the very small range from c0o — to to c0 — e0o,
we can replace the integrand by its value when u = 0, or £. Thus the
first term becomes (rfJV/de)o(e0 — e0o). To reduce the second integral

__ u

to a familiar form, we let x = e *r, u = — kT In x, du = —kTdx/x.
The integral then becomes

(4.4)

f *-^—du = -(*T)> f 'iqr^r - g

*/o *r */ ^

the integral in Eq. (4.4) being tabulated for instance in B. O. Peirce's
"Short Table of Integrals." Then Eq. (4.3) becomes

(4.5)

Equation (4.5) represents €0 by the first two terms of a power series in the
temperature, and the approximations we have made give the term in
T2 correctly, though we should have to be more careful to get higher
terms. We see, as we should expect from the last section, that if
(dW/de2)o = 0, so that the distribution of energy levels is uniform at
€0, c0 will be independent of temperature to the approximation we are
using.

Next, let us find the internal energy in the same sort of way. Written
as a summation, it is

u = — (4-6)

+ 1

78 INTRODUCTION*™ CHEMICAL PHYSICS [CHAP. V

Here again, in converting to an integration, we shall find, not [/, but
U — f/o, where £70 is the value at the absolute zero. Then we find at
once that the integral expression for it is exactly like the integral in Eq.
(4.2), only with an additional factor e in the integrand of each integral.
The leading term hero, however, comes from the term (dAT/de)o, and we
can neglect the terms in (rfW/ck'2)0. Furthermore, in the integrals we
retain, we can neglect the difference between €o and eoo. Then we have

and

u-u' + (f )l(fcr)2' (4-7)

again correct to terms in T2. From the internal energy we can find the
heat capacity CV, by the equation

c" - (wl

-tk'T. (4.8)

We notice that at low temperatures the specific heat of a system with
continuous energy levels, obeying the Fermi statistics, is proportional
to the temperature. We shall later see that this formula has applications
in the theory of metals.

Let us next find the entropy. We can get a general formula from
Eq. (2.7). This can be rewritten

5 - -kN> In -r - k In (1 - ft.)

(4'9)

where we have used the Fermi distribution law. Replacing the summa-
tion in Eq. (4.9) by an integration and using Eqs. (4.5) and (4.7) for c0
and £7, we can compute S. The calculation is a little involved, however,
and it is easier to use the relation

SEC. 4] THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS 79
from which

8 - (f XT** «•"»

In the integration leading to Eq. (4.10), we have used the fact that S - 0
at the absolute zero. This follows directly from Eq. (2.7) for the entropy.
For each term in the entropy is of the form N In N or (1 — N) In (1 — N),
and at the absolute zero each value of N is cither 1 or 0, so that each of
these terms is 1 In 1 or 0 In 0, either of which is zero.

From the internal energy and the entropy we can find the Helmholtz
free energy

A = U - T8

_-A _>-«)

In (e. *T~ + 1) (4.11)

- (f

where Eq. (4.11) is derived from Eq. (4.9), and Eq. (4.12) from Eqs. (4.7)
and (4.10). By differentiating the function A with respect to tempera-
ture at constant volume, we get the negative of the entropy, as we should.
By differentiating with respect to the volume at constant temperature, we
get the negative of the pressure, and hence can find the equation of state.
So far, we have not mentioned the dependence of any of our functions on
the volume. Surely, however, the stationary states and energy levels of
the particles will depend on the volume, though not explicitly on the
temperature. Hence Uo and (rf#/rfe)0 are to be regarded as functions of
the volume. The functional dependence, of course, cannot be given in a
general discussion, applicable to all systems, such as the present one.
Using this fact, then, we have

The first term, the leading one ordinarily, is independent of temperature,
and the second, a small additional one which can be of either sign, is
proportional to the square of the temperature. Thus, the equation of
state is very different from that of a perfect gas on Boltzmann statistics.
It is to be borne in mind, however, that this formula, like all those of the
present section, applies only at low temperatures and is only the beginning
of a power series. At high temperatures the statistics reduce to the
Boltzmann statistics, as we have seen, and the equation of state of a Fermi

80 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

system and the corresponding system obeying Boltzmann statistics must
approach each other at high temperature.

5. The Perfect Gas in the Fermi Statistics. — As an example of the
application of the Fermi statistics, we can consider the perfect gas. In
Chap. IV, Sec. 1, we have found the number of energy levels for a mole-
cule of a perfect gas, in the energy range de. Rewriting Eq. (1.10) of that
chapter, we have at once

**L = ZLL(%m\*& (51)

ae /r

and

If we substitute €0o for e in Eqs. (5.1) and (5.2), we have the quantities
(dN/de)o and (rfW/dc2)0 of the previous section. To find eoo, we note
that from Eq. (1.9), Chap. IV, the number of states with energy less than
€ is (4irV/3Ji8)(2w€)w. Remembering that there are just N states with
energy less than eoo, this gives

We notice, as is natural, that eoo, the highest occupied energy level at the
absolute zero, increases as the number of particles N increases. It is
important to notice, however, that it is the density of particles, N/V, that
is significant, not the absolute number of particles in the system. In a
gas obeying the Fermi statistics, the particles cannot all have zero energy
at the absolute zero, as they would in the Boltzmann statistics; but since
there can be only one particle in each stationary state, there is an energy
distribution up to the maximum energy eoo. Let us see how large this
is, in actual magnitude. We can hardly be interested in cases where
N/V represents a density much greater than the number of atoms of a
solid per unit volume. Thus for example let N/V be one in a cube of
3 X lO"8 cm. on a side, or let it equal & X 1024. Let us make the
calculation in kilogram calories per gram mole (1 kg.-cal. equals 1000 cal.
or 4.185 X 1010 ergs, one mole contains 6.03 X 1023 molecules), and let
us do it first for an atom of unit molecular weight, for which one molecule
weighs 1.66 X 10~~24 gm. Then we have

6.03 X 1023

coo —

[3 X (6.61 X 1Q-27)3T
10-"L 47r X 27 X 10-24 J

4.185 X 10102 X 1.66 X 10-
= 0.081 kg.-cal. per gram mole. (5.4)

There do not seem to be any ordinary gases in which the energy calculated
from Eq. (5.4) is appreciable. Hydrogen H2 and helium He both satisfy

SBC. 5J THE FERMI-DIRAC AND E1NSTEIN-BOSE STATISTICS 81

the Einstein-Bose statistics instead of the Fermi, and so are not suitable
examples. With any heavier gas, the mass that comes in the denominator
of Eq. (5.3) would reduce the value to a few gram calories per mole, a
value small compared to the internal energy which the gas would acquire
in even a few degrees with normal specific heat as given by the Boltzmann
statistics, in Chap. IV, Sec. 3. The one case where the Fermi statistics
is of great importance is with the electron gas, on account of the very
small mass of the electron. The atomic weight of the electron can be
taken to be TSTV- Then to get coo for an electron gas of the density
mentioned above, we multiply the figure of Eq. (5.4) by 1813, obtaining

€0o = 148 kg.-cal. per gram mole = 6.4 electron volts. (5.5)

The value (5.5), instead of being small in comparison with thermal magni-
tudes like (5.4), is of the order of magnitude of large heats of dissociation
or ionization potentials, and enormously large compared with thermal
energies at ordinary temperatures. Thus in an electron gas at high den-
sity, the fastest electrons, even at the absolute zero, will be moving with
very high velocities and very large energies.

We can next use Eq. (4.5) to find c0 at other temperatures. We have
at once

From Eq. (5.6) we note that in ordinary gases, where too is of the order
of magnitude of kT for a low temperature, the term in 712 will be large,
showing that the series converges slowly. On the other hand, in an elec-
tron gas, where eoo is very large compared to fcT, the series converges
rapidly at ordinary temperatures, and €0 is approximately independent of
temperature, decreasing slightly with increasing temperature.

The quantity C/o, the internal energy at the absolute zero, is easily
found, from the equation

(5.7)

using Eq. (5.1). Thus the mean energy of a particle at the absolute zero
is three-fifths of the maximum energy. From Eq. (4.7) we can then find
the internal energy at any temperature, finding

82 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. V

The heat capacity is given by

Cv « M£— • (5.9)

* coo

For an electron gas at ordinary temperature, this is a small quantity. An
ordinary gas would have a heat capacity fJVfc. This value is the ordinary
value, multiplied by (7r2/3)(fc!T/€oo). Now at 1000° abs., for instance, a
fairly high temperature, kT is about 2 kg.-cal. per gram mole, whereas we
have seen in Eq. (5.5) that e0o is of the order of 148 kg.-cal. per gram mole.
Since ir2/3 is about 3, this means that the electronic specific heat at 1000°
abs. is about four per cent of that of a perfect gas on the Boltzmann
statistics, while at room temperature it would be only a little over 1 per
cent.

The other thermal quantities arc easily found. As we see from
Eqs. (4.8) and (4.10), the entropy equals the specific heat, up to terms
linear in the temperature, so that the entropy of the perfect gas is given
by Eq. (5.9) at low temperatures. And the function A, using Eq. (4.12),
is

Q ir2 IcT

A = ^€00 - NkT^-r—' (5.10)

O 4 €00

To find the pressure, we wish to know A explicitly as a function of volume.
Substituting for €0o from Eq. (5.3), we have

A -

We can now differentiate to get the pressure:

-GO

- (5-12)

as we can see by substituting for €00 in Eq. (5.8). Equation (5.12), stating
that PV = ft/, is the equation found in Eq. (3.5), Chap. IV, by a kinetic
method, without making any assumptions about the distribution of
velocities. It must therefore hold for the Fermi distribution as well as
for the Boltzmann distribution, and it was really not necessary to make a
special calculation of the equation of state at all. It is obvious, however,
that the final equation of state is very different from that of the perfect
gas on the Boltzmann statistics, on account of the very different relation
which we have here between internal energy and temperature. Since

SEC. 6] THE FERMI-DIRAC AND EINSTEIN-BOSE STATISTICS 83

the internal energy is very large at the absolute zero, but increases only
slowly with rising temperature, with a term proportional to T72, the same
is true here of PV. Thus, in the example used above, the pressure is
149,000 atm. at the absolute zero. We note that, in contrast to the
Boltzmann statistics, the internal energy here depends strongly on the
volume, as Eqs. (5.8) and (5.3) show; thus the gas does not obey Boyle's
law. This dependence of internal energy on volume is an interesting
thing, for it does not indicate in any way the existence of forces between
the particles, which we are neglecting here just as in the Boltzmann theory
of a perfect gas. The kinetic energy is what depends on the volume, on
account of the dependence of eoo on volume.

6. The Einstein-Bose Distribution Law. — We can find the Einstein-
Bose distribution law, proceeding by exact analogy with the methods of
Sec. 3 but using the expression (2.11) for the entropy. Thus for the
function A we have

A = 2)L/V\«< + kTNv In Ni - kT(\ + NJ In (1 + ft.)]. ,g ^
i

Varying the NS* and requiring that ^4 be a minimum for equilibrium, we
have

dA = 0 = ^5( €. + kT In TTVV^»- (6-2)

^MM V 1 — T- TV /

As in Sec. 3, Eq. (6.2) must be satisfied, subject to the condition

_ V, = 0,

leading to the relation

et + kT In - -j-^r = €o = const, (6.3)

1 ~r ^Vi

Solving for Ni, as in the derivation of Eq. (3.5), we have

N* = ~K-3 (6.4)

e kT - 1

Equation (6.4) expresses the Einstein-Bose distribution law. As with
the Fermi-Dirac law, the constant co is to be determined by the condition

^' = 2 "^^

' '

kT ' _

INTRODUCTION TO CHEMICAL I'/IYtiWS

[CHAP. V

We can show, as we did with the Fermi-Dirac statistics, that the
distribution (6.5) approaches the Maxwell-Boltzmann distribution law at.
high temperatures. It is no easier to make detailed calculations with the
Einstein-Bose law than with tho Fermi-Dirac distribution, and on account
of its smaller practical importance we shall not carry through a detailed

-2

FIG. V-2. — Distribution functions for Fcrmi-Dirac statistics (a) ; Maxwell-Boltzmann
statistics (&); and lOinstom-Bose statistics (f).

discussion. It is interesting, however, to compare the three distribution
laws. This is done in Fig. V-2, where we plot the function /^x— -—

representing the Einstcin-Bose law, l/e kT representing the Maxwell-
Boltzmann, and -(,_«,) representing the Fermi-Dirac, all as functions

• z.y i -t

e kl +1

of €. We observe that the curve for the Einstein-Boso distribution
becomes asymptotically infinite as c approaches eo. From this and from
Eq. (6.5), it follows that e0 must lie lower than any of the energy levels of a
system, in contrast to the case of the Fermi-Dirac distribution. We see
that the Maxwell-Boltzmann distribution forms in a certain sense an
intermediate case between the two other distributions. The Fermi-Dirac
statistics tends to concentrate the molecules more in the higher energies,
having fewer molecules in proportion in the lower energies than in the
Maxwell-Boltzmann statistics. On the contrary, the Einstein-Boso
statistics tends to have more molecules in the lower energies. As a mat-
ter of fact, more elaborate study of the Einstein-Bose distribution law
shows that the concentration of molecules in the low states is so extreme
that at low enough temperatures a phenomenon of condensation sets
in, somewhat analogous to ordinary changes of phase of a real gas. From

SBC. 6J THE FERMI-DIRAC AND E1NSTEIN-BOSE STATISTICS 85

these properties of the distribution laws, we can see that in some super-
ficial ways the effect of the Fermi-Dirac statistics is similar to that of
repulsive forces between the molecules, leading to a large pressure even
at the absolute zero, while the effect of the Einstein-Bose statistics is
similar to that of attractive forces, leading to condensation into a phase
resembling a liquid.

The real gases hydrogen and helium obey the Einstein-Bose statistics,
and there are indications that at temperatures of a few degrees absolute
the departures from the Maxwell-Boltzmann statistics are appreciable.
Of course, the molecules have real attractive forces, but the effect of the
statistics is to help those forces along, producing condensation at some-
what higher temperature than would otherwise be expected. The sugges-
tion has even been made that the anomalous condensed phase He II, a
liquid persisting to the absolute zero at ordinary pressures and showing
extraordinarily low viscosity, may bo tho condensed phase of tho Einstoin-
Bose statistics. For other gases than hydrogen and helium, the inter-
molecular attractions are so much greater than the effect of tho
Einstein-Bose statistics that they liquefy at temperatures too high to
detect departures from the Maxwell-Boltzmann law. Aside from these
gases, the only important application of the Einstein-Bose statistics
eomes in the theory of black-body radiation, in which it is found that
photons, or corpuscles of radiant energy, obey the Einstein-Bose statistics,
leading to a simple connection between the Einstein-Bose distribution
law and the Planck law of black-body radiation, which we shall discuss
in a later chapter.

CHAPTER VI

THE KINETIC METHOD AND THE APPROACH TO THERMAL

EQUILIBRIUM

In the preceding chapters, we have taken up in a very general v\ay
thermodynamics and statistical mechanics, including some applications
to perfect gases. Both, as we have seen, are very general and powerful
methods, hut both are limited, as far as quantitative predictions are
concerned, to systems in thermal equilibrium. The kinetic theory, some
of whose methods we shall use in this chapter, is not so limited. It can
handle the rates of molecular processes and incidentally treats thermal
equilibrium by looking for a steady state, in which the rate of change of
any quantity is zero. But it has disadvantages compensating this groat
advantage: it is much more complicated and much less general than
thermodynamics and statistical mechanics. For this reason we shall not
pretend to give any methods of handling an arbitrary problem by kinetic
theory. We limit ourselves to a very special case, the perfect monatomic
gas, and wo shall not oven make any quantitative calculations for it.
Later on, in various parts of the book, we shall handle other special
problems by the kinetic method. Always, we shall find that an actual
calculation of the rate of a process gives us a better physical insight into
what is going on than the more general methods of statistical mechanics.
But generally we shall find that the kinetic methods do not go so far, and
always they are more complicated. Our problem in this chapter is to
' investigate thermal equilibrium in a perfect monatomic gas by the kinetic
method. We sot up an arbitrary state of a gas and investigate how it
changes as time goes on. We compute its entropy at each stage of the
process, showing that in fact the entropy increases in the irreversible
process by which the arbitrary distribution changes over to thermal
equilibrium, and we can actually find how fast it increases, which we
could not do by our previous methods. Finally by looking for the final
state, in which the entropy can no longer increase, we get the condition
for thermal equilibrium and show that it agrees with the condition derived
from statistical mechanics and the canonical assembly.

1. The Effect of Molecular Collisions on the Distribution Function
in the Boltzmann Statistics. — Let us set up a distribution in the molecular
phase space, as described in Chap. V, Sec. 1. We consider, not a single
state of the gas, but an assembly of states, as set up in Sec. 2 of that same

SBC. 1] THE KINETIC METHOD 87

chapter, defined by the average number JVt of molecules in the ith cell
of the molecular phase space, and having the entropy given in Eq. (2.7),
(2.li), or (2.12) of Chap. V. We start with an arbitrary set of JVVs, and
ask how they change as time goes on. The changes of $Vs arise in two
ways. First, there are thoso changes that would be present even if there
were no collisions between molecules. A molecule with a certain velocity
moves from point to point, and hence from cell to cell, on account of
that velocity. And if the molecule is actod on by an external force
field, which changes its momentum, it goes from cell to cell for that
reason too. These changes are in the nature of streamline flows of the
representative points of the molecules in the molecular phase space. Wo
shall discuss them later and shall show that they do not result in any
change of entropy. Secondly, there are changes of $Ys on account of
collisions between molecules. These are the changes resulting in irrever-
sible approach to a random distribution and in an increase of entropy.
Since they are for our present purposes the most interesting changes, we
consider them first.

Consider two molecules, one in the itli cell, one in thejth, of molecular
phase space. If these cells happen to correspond to the same value of the
coordinates, though to different values of the momenta, there is a chance
that the molecules may collide. In the process of collision, the represen-
tative points of the molecules will suddenly shift to two other cells, say
the kth and Zth, having practically the same coordinates but entirely
different momenta. The momenta will be related to the initial values; for
the collision will satisfy the conditions of conservation of energy and
conservation of momentum. These relations give four equations relating
the final momenta to the initial momenta, but since there are six com-
ponents of the final momenta for the two particles, the four equations
(conservation of energy and conservation of three components of momen-
tum) will still leave two quantities undetermined. For instance, we may
consider that the direction of one of the particles after collision is undeter-
mined, the other quantities being fixed by the conditions of conservation.

We now ask, how many collisions per second are there in which mole-
cules in the ith and jth cells disappear and reappear in the kth and Ith
cells? We can be sure that this number of collisions will be proportional
both to the number of molecules in the ith and to the number of molecules
in the jth cell. This is plain, since doubling the number of either type of
molecule will give twice as many of the desired sort that can collide, and
so will double the number of collisions per unit time. In the case of the
Boltzmann statistics, which we first consider, the number of collisions will
be independent of the number of molecules in the kth and Ith cells, though
we shall find later that this is not the case with the Fermi-Dirac and
Einstein-Bose statistics. We can then write the number of collisions of

88 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

the desired type in unit time, averaged over the assembly, as

A^fWi. (1.1)

The coefficient A\{ will of course depend on the momenta associated with
all four cells, and in particular will be zero if these momenta do not satisfy
tho conditions of conservation. It will also depend on the properties
of the atom. For instance, it is obvious that the larger the molecules
are, the more likely they are to collide, and the larger the A's will be. We
do not have to go more into details of the A's for our present purposes,
however.

In addition to these collisions, we shall have to consider what we shall
call an inverse collision. This is one in which the molecules before colli-
sion are in the cells k and Z, and after collision are in cells i and j. The
number of such collisions per unit time, by the same argument as before,
will be

A"ftjti. (1.2)

Now we ask, what relation, if any, is there between the two coefficients
Atf and A1** of the direct and inverse collisions? The answer to this
question is simple but not very easy to justify. It is this: if the cells are
all of the same size, as we are assuming, we have simply

AH = A?f (1.3)

In case the collision takes place according to Newtonian mechanics, the
relation (1.3) can be proved by means of Liouville's theorem. In quan-
tum mechanics, Eq. (1.3) is practically one of the postulates of the theory,
following directly from quantum mechanical calculations of transition
probabilities from one state to another. For our present purpose, con-
sidering that this is an elementary discussion, we shall simply assume the
correctness of relation (1.3). This relation is sometimes called the prin-
ciple of microscopic reversibility.

We are now in position to find how our distribution function changes
on account of collisions. Let us consider a certain cell i} and ask how the
average number Ni of molecules in this cell changes with time, on account
of collisions. In the first place, whenever a molecule in the cell collides
with another molecule in any other cell, the first molecule will be removed
from the ith cell, and tho number of molecules in this cell will be dimin-
ished by one. But the whole number of collisions of a molecule in cell
ij with all other molecules, per second, is

where we are summing over all other types of molecule j with which the

SBC. 2] THE KINETIC METHOD 89

original molecule can collide, and over all possible states k and I into which
the molecules can be sent by collision. On the other hand, it is possible
to have a collision of two molecules having quite different momenta, such
that one of the molecules after collision would be in cell i. This would
result in an increase of unity in the number of molecules in the coll i. The
number of such collisions per second is

(1.5)
3ki

where we have used the result of Eq. (1.3). Thus the total change in ffl
per second is given by

(1.6)

2. The Effect of Collisions on the Entropy. — Equation (1.6) represents
the first part of our derivation of the effect of collisions in producing an
irreversible change in the distribution and hence in increasing the entropy.
Now we must go back to the definition of the entropy in Eq. (2.12) in
Chap. V, and find how much S changes per unit time on account of the
collisions. Differentiating that equation with respect to the time, we
have at once

In AT.f '. (2.1)

t
Substituting from Eq. (1.6), this becomes

I*'/ In AT (W N . A/. 3v.^ fO 9^

,. — /v s- , tljci ill IV i\l\ ilV j — IMjelMi). \£i.L)

at <***

We notice that the fourfold summation over i, j, ky I is perfectly sym-
metrical in i and j; they are simply the indices of the two colliding particles
before collision. We could interchange their names, and could equally
well write

ln R>(8& - #*#')• (2-3)

90 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

By Eq. (1.3), this could also be written

But in Eq. (2.4), we can interchange the names of i, j with k, Z, obtaining

dS ^^

dt <-*— J kl l 3

lyi In Nk(fflNj - NkNi). (2.5)

Finally, interchanging the role of the fcth and Ith atoms, we have
dS "SJ , . . , i

Tt = -*.^*«ln'

We now have, in Eqs. (2.2), (2.3), (2.5), and (2.6), four equivalent
ways of writing dS/dt. We add these equations and divide by 4, obtain-
ing the final form

J.Q ^'%n

\JUkJ Iv ^W ' A • • /» -jTV , i -Tr i iTV i

__^_ — — ^^ v4 ' ( ITI W • I In 7V • - \\\ /Vt - ifi

dt 4-*— J kl * 7

- ln N*Ni)(ff<fli - ^*^»)- (2-7)

The result of Eq. (2.7) is a very remarkable one. Each term of the
summation is a product of a coefficient A (which is necessarily positive),
and a factor of the form

(In z - In T/) (*-</), (2.8)

where x = NiNj, y = NkNi. But the factor (2.8) is necessarily positive.
If x > y, so that x — y is positive, then In x > In y, so that the other
factor In x — In y is positive as well. On the other hand, if x < y, so
that x — y is negative, In x — In y is also negative, so that the product
of the two factors is again positive. Thus, every term of the summation
(2.7) is positive, and as a result the summation is positive. The only
way to avoid this is to have each separate term equal to zero; then the
whole summation is zero. But if the summation is positive, this means
that the entropy S is increasing with time. Thus we have proved Boltz-
mann's famous theorem (often called the H theorem, because he called
the summation of Eq. (2.12), Chap. V, by the symbol H, setting
S = —kH) : the entropy S continually increases, on account of collisions,

SBC. 2] THE KINETIC METHOD 91

unless it has already reached a steady state, for which the condition is

N.Nj - RkNt - 0 (2.9)

for every set of cells i, ,7, &, I between which a collision is possible (that
is, for which Aft ^ 0).

By comparison with Eq. (1.6), we see that Eq. (2.9) leads to the
condition that dN%/dt should be zero, by demanding that each separate
term of Eq. (1.6) should be zero. That is, in equilibrium, the collision
in which atoms i and j collide to give atoms k and Z, together with the
inverse to this type of collision, by themselves give no net change in the
numbers of atoms in the various states, the number of direct collisions
just balancing the number of inverse collisions. This condition is called
the condition of detailed balancing. It is a general characteristic of
thermal equilibrium that this detailed balancing should hold and, as we
have seen, it follows directly from the second law in its statistical form.

We may now rewrite Eq. (2.9) in the form

fftffi = NkNi,
or

In #t + In Nj = In Nk + In NI. (2.10)

This holds for every transition for which A& 7* 0; that is, for every colli-
sion satisfying the laws of conservation of energy and momentum. Using
the notation of Sec. 3 in Chap. IV, we can let the average number of
molecules in an element dx dy dz dpx dpy dpz of the molecular phase space
be/m dx dy dz dpx dpy dpz. According to Chap. Ill, Sec. 3, the volume of
molecular phase space associated with one cell is A8. Then we have the
relation

ffi = h*fm, (2.11)

where /m is to be computed in the ith cell. We now substitute Eq. (2.11)
in Eq. (2.10), writing that equation in terms of the/m's. Since all four of
our cells must refer to the same point of coordinate space, since molecules
cannot collide unless they are in contact, we write fm merely as
fm(pxpypz), or/(p) for short. Then we have

ln/(p,) + ln/(p/) = ln/(p*) + ln/(p«). (2.12)

Equation (2.12) states that there is a certain function In /of the momen-
tum of a molecule, such that the sum of the functions of the two molecules
before collision equals the sum of the functions of the two after collision.
That is, the total amount of this function is conserved on collision. But
there are just four quantities that have this property: the energy and the
three components of momentum. Any linear function of these four
quantities will also be conserved, and it can be proved that this is the most

92 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

general function which has this property. Thus we may conclude that

In /(p) » A^ + Bpx + CPy + Dpz + E, (2.13)

where A, B, C, Z), E are arbitrary constants. Substitution of Eq. (2.13)
in Eq. (2.12) shows at once that Eq. (2.12) is satisfied. We have been
able, in other words, to get a general solution of the problem of the dis-
tribution function in equilibrium.

The linear combination of Eq. (2.13) can be rewritten in the form

[(PX ~~ Pxo)* + (PV "" Pvo)2 + (PZ " p§o)f]

. _
2m / " w*y roM' " mkT

where T (later to be identified with the temperature), px0, py0, p«o, /o are
arbitrary constants, whose relation to the constants of Eq. (2.13) is evi-
dent from Eq. (2.14). Thus we have

-((p»-pxo)2 + (7>v-pyo)2 + (pe-p.o)2) , ex

/(p) =/oe ~~2^r~ " (2.15)

3. The Constants in the Distribution Function. — In Eq. (2.15), we
have found a distribution function satisfying the condition of thermal
equilibrium and containing five arbitrary constants, T, px0, pyQj pzQ, /0.
Since our calculation has been entirely for a single point of ordinary space,
these five quantities, for all we know, may vary from point to point or be
functions of position. Shortly we shall find how the quantities must vary
with position in order to have thermal equilibrium. We may anticipate
by stating the results which we shall find and giving their physical
interpretation.

In the first place, we shall find that the four quantities T, px^ pv0, pzQ
must be constant at all points of space, for equilibrium. By comparison
with Eq. (2.4) of Chap. IV, the formula for the Maxwell distribution of
velocities, we see that T must be identified with the temperature, which
must not vary from point to point in thermal equilibrium. The quan-
tities PXO, pVQ, pZQ are the components of a vector representing the mean
momentum of all the molecules. If they are zero, the distribution (2.15)
agrees exactly with Eq. (2.4) of Chap. IV. If they are not zero, however,
Eq. (2.15) represents the distribution of velocities in a gas with a certain
velocity of mass motion, of components pXQ/m, p^o/w, pzo/in. The quan-
tities px — p*o, etc., represent components of momentum relative to this
momentum of mass motion, and the relative distribution of velocities is as

SBC. 3] THE KINETIC METHOD 93

in Maxwell's distribution. Since ordinarily we deal with gas without
mass motion, we ordinarily set pxQ, pyQ, and p*o equal to zero. In general,
we shall not have thermal equilibrium unless the velocity of mass motion
is independent of position; otherwise, as we can seo physically, there
would be the possibility of viscous effects between different parts of the
gas. We have now considered the variation of 77, pxo, pyQ, pz0 with posi-
tion and have shown that they are constants. Finally wo consider /0.
Our analysis will show that if the potential energy of a molecule is <£, a
function of position, we must have

— tf>
/o = const. X ek~f. ' (3-1)

Equation (3.1) shows that the density varies with position just as
described in Chap. IV, Sec. 4. Thus, with these interpretations of the
constants, we see that P]q. (2.15), representing the distribution function
which we find by the kinetic method for the bteady state distribution, is
exactly the same that we found previously, in Chap. IV, by statistical
methods.

We shall now prove the results that we have mentioned above, regard-
ing the variation of T, px0, py^ pzQ, /0 with position. We stated in Sec. 1
that a molecule could shift from point to point in phase spare not only on
account of collisions, which we have considered, but also on account of the
velocity and of external force fields, and that these shifts were in the
nature of streamline flows in the phase space and did not correspond to
changes of entropy. We must now analyze these motions of the mole-
cules. We shall assume classical mechanics for this purpose; the energy
levels of a perfect gas, as we have seen in Chap. IV, Sec. 1, are spaced so
closely together that the cells in phase space can be treated as continuous.
Let us, then, take a volume element dx dy dz dpx dpv dpz in molecular
phase space, and find the time rate of change of fm dx dy dz dpx dpy dpg,
the number of molecules in the volume clement, for all reasons except
collisions. First, we consider the number of molecules entering the
element over the surface perpendicular to the x axis, whose (five-dimen-
sional) area is dy dz dpx dpy dpz. The component of velocity of each
molecule along the x axis is px/m. Then, using an argument similar
to that of Chap. IV, Sec. 3, the number of molecules entering the element
over this surface per second is the number contained in a prism of base
dy dz dpx dpy dpz and altitude px/m. This is the volume of the prism
[(Px/rn)dy dz dpx dpy dpz] times the number of molecules per unit volume
in phase space, or /m. Hence the required number is

(P.\

•w

\dy dz dpx dpy dpg.

94 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

But a similar number of molecules will leave over the face at x + dx.
The only difference will be that in computing this number we must use
fm computed at x + dx, or must use

fm(x + dx, yzpxpypg) = fm(xyzpxpypz) + dx-J£(xyzpxpvpt) + - - • ,

the first two terms of a Taylor's expansion. Thus the net number enter-
ing over the two parallel faces perpendicular to x is

y dz dpx dpv dpz,

fit uX

and for the three sets of faces perpendicular to x, y, z, we have the net
number

-fe -/; + 5 £ + £ If)"* * * *• <"» *- <M>

In a similar way we can consider the face perpendicular to the px axis in
the phase space. The component of velocity of a representative point
along the px axis in this space is by definition simply the time rate of
change of px; that is, by Newton 's second law, it is the x component of
force acting on a molecule. If the potential energy of a molecule is <£,
this component of force is —d<t>/djc. Thus, the number of molecules
entering over the face perpendicular to the px axis is

/ml -fa)dx d1l dz dPy dP*>
and the net number entering over the faces at px and at px + dpx is

f^JL 3f

~dx dp X dy <iZ dpx dj)y flp"

We have three such terms for the three components of momentum, and
combining with Eq. (3.2), we have for the total change of the number of
molecules in the volume element per second the relation

*- <3-8)

Having found the change in the distribution function, in Eq. (3.3), we
shall first show that it involves no change of entropy. The physical
reason is that it corresponds to a streamline motion in phase space, result-
ing in no increase of randomness. We use Eq. (2.1) for the change of

SEC. 3]

THE KINETIC METHOD

entropy with time, Eq. (2.11) for the relation between
Eq. (3.3) for dfm/dt. Then we have

and /„,, and

\dx • dp9. (3.4)

Each of the integrals over the coordinates is to be carried to a point out-
side the container holding the gas, each integral over momenta to infinity.
In Eq. (3.4), each term can be integrated with respect to one of the vari-
ables. Thus the first term, as far as the integration with respect to x is
concerned, can be transformed by the relation

• '

= Infndfm = (/wln/w-,

(3.5)

At both limits of integration, fm = 0, since the limits lie outside the con-
tainer, so that the integral vanishes. A similar transformation can be
made on each term, leading to the result that the changes of fm we are now
considering result in no change of entropy. This justifies our analysis of
Sec. 2, in which we treated the change of entropy as arising entirely from
collisions.

Now we can use our condition (3.3) to find the variation of our quanti-
ties /o, T, pxo, py0, pz() of Sec. 2 with position. In thermal equilibrium, we
must have dfm/dt = 0. Thus Eq. (3.3) gives us a relation involving the
various derivatives of fm. We substitute for/m from Eq. (2.15), treating
the quantities just mentioned as functions of position. Then Eq. (3.3)
becomes, canceling the exponential,

0 = — £l?^L0
m dx

P*:
m

Pzdfo
m dz

dpXQ

-F + <"• - *•> «,

da-

+ P.

^ +IP.- r^

(Px ~

tg + (p, _ ^

~ P«o)2 + (py -

+ (Pz -

- P-O + (PV " p*o) + (p* " p*o) (3-6)

Equation (3.6) must be satisfied for any arbitrary values of the momenta.
Since it is a polynomial in px, pv) p«, involving terms of all degrees up to

96 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

the third, and since a polynomial cannot be zero for all values of its
argument unless the coefficient of each term is zero, we can conclude that
the coefficient of each power of each of the p's in Eq. (3.6) must be zero.
From the third powers we see at once that

dT - i\ — - c\ dT - ft tt 7^

te ~ °' ay " 0| °ei ~ °' (3'7)

or the temperature is independent of position. From the second powers
we then see that the derivatives of the form dpxQ/dx must all be zero, or
the average momentum is independent of position. We are left with only
the first and zero powers. From the zero powers, we sec that either

PxO = PV0 = PzQ = 0,

d<t> d<t> d<t> _

,- ~v

(3-8)

That is, if there is an external force field, there can be no mass motion of
the gas, for in this case the external field would do work on the gas and its
energy could not be constant. Then we are left with the first power
terms. The coefficient of px, for instance, gives

fadx = ~~kfdx' '3'9'

with similar relations for the y and z components. Equation (3.9) can
be rewritten

"-*

dx dx '

In /o = ^ + const.,

-»
/o = const. ekT ,

or Eq. (3.1). Thus we have proved all the results regarding our distribu-
tion function that we have mentioned earlier in the section, and have
completed the proof that the Maxwell-Boltzmann distribution law is the
only one that will not be affected by collisions or the natural motions of
the molecules and, therefore, must correspond to thermal equilibrium.

4. The Kinetic Method for Fermi -Dirac and Einstein -Bose Statistics.
The arguments of the preceding sections must be modified in only two
ways to change from the Boltzmann statistics to the Fermi-Dirac or
Einstein-Bose statistics. In the first place, the law giving the number
of collisions per unit time, Eq. (1.1), must be changed. Secondly, as

SBC. 4] THE KINETIC METHOD 97

we should naturally expect, we must use the appropriate formula for
entropy with each type of statistics. First, we consider the substitute
for the law of collisions. Clearly the law (1.1), giving AHftJtj collisions
per second in which molecules in states i and j collide to give molecules in
states k and I cannot be correct for Fermi-Dirac statistics. For the
fundamental feature of Fermi-Dirac statistics is that if the fcth or Ith
stationary states happen to be already occupied by a particle, there is no
chance of another particle going into them. Thus our probability must
depend in some way on the number of particles in the kill and Ith states,
as well as the ith and jth. Of course, the kth state can have either no
particles in it, or one; never more. Thus in one example of our system,
chosen from the statistical assembly, Nk may be zero or unity. If it is
zero, there is no objection to another particle entering it. If it is unity,
there is no possibility that another particle can enter it. Averaging over
the assembly, the probability of having a collision in which a particle
is knocked into the fcth state must clearly have an additional factor equal
to the fraction of all examples of the assembly in which the kth state is
unoccupied. Now Nk is the mean number of particles in the kth state.
Since the number of particles is always zero or one, this means that Nk
is just the fraction of examples in which the fcth state is occupied. Then
1 — Nk is the fraction of examples in which it is unoccupied, and this is
just the factor we were looking for. Similarly we want a factor 1 — Nt
to represent the probability that the Zth state will bo unoccupied and
available for a particle to enter it. Then, finally, we have for the number
of collisions per second in which particles in the ith and jth cells are
knocked into the fcth and Ith cells, the formula

AllftJt,(l - #*)(! - tf,). (4.1)

In the Einstein-Bose statistics, there is no such clear physical way
to find the revised law of collisions as in the Fermi-Dirac statistics. The
law can be derived from the quantum theory but not in a simple enough
way to describe here. In contrast to the Fermi-Dirac statistics, in which
the presence of one molecule in a cell prevents another from entering the
same cell, the situation with the Einstein-Bose statistics is that the
presence of a molecule in a cell increases the probability that another
one should enter the same cell. In fact, the number of molecules going
into the fcth cell per second turns out to have a factor (1 + $*), increasing
linearly with the mean number ffk of molecules in that cell. Thus, the
law of collisions for the Einstein-Bose statistics is just like Eq. (4.1), only
with + signs replacing the — signs. In fact, we may write the law of
collisions for both forms of statistics in the form

ffi), (4.2)

98 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

where the upper sign refers to the Einstein-Bose statistics, the lower to
the Fermi-Dirac.

Next, we must consider the change of the mean number of molecules
in the fth state, with time. Using the law of collisions (4.2) and proceed-
ing as in the derivation of Eq. (1.6), we have at once

± Ni}(l ± Nj} - M(1 ± ^)(1 ± Ni)]- (4-3)

Having found the number of collisions, we can find the change in entropy
per unit time. Using the formulas (2.7) and (2.11) of Chap. V for the
entropy in the case of Fermi-Dirac and Einstein-Bose statistics, we find at,
once that

where again the upper sign refers to Einstein-Bose statistics, the lower
one to Fermi-Dirac statistics. Substituting from Eq. (4.3), we have

AHlhi B* - ln
ijkl

± #*)(! ± Ni)]. (4.5)

As in Sec. 2, we can write four expressions equivalent to Eq. (4.5), by
interchanging the various indices t, jf, fc, I. Adding these four and divid-
ing by four, we obtain

2? = £§^H{ In [#*#i(i ± #0(1 ± #0] - ln[ftjt,(l ± #*)

Hid
(1 ± Ni)]}[N*Nl(\ ± #.)(! ± #0 - ftjt,(l ± N.)(l ± Ni)]. (4.6)

But, as in Sec. 2, this expression cannot be zero as it must be for a steady
state, unless

NkNi(l ± #0(1 ± Ni) = NiNj(l ± Nk)(l ± #0,

and if it is not zero, it must necessarily be positive. Thus we have demon-
strated that the entropy increases in an irreversible process, and have
found the condition for thermal equilibrium.

From Eq. (4.7) we can find the distribution functions for the Einstein-
Bose and Fermi-Dirac statistics. We rewrite the equation in the form

± #0

Sue. 41 THE KINETIC METHOD 99

N
As in Sec. 2, we may now conclude that the quantity In — must be

(1 ± N)

a function which is conserved on collision, since the sum for the two par-
ticles before and after collision is constant. And as in that section, this
quantity must be a linear combination of the kinetic energy and the
momentum, the coefficients in general depending on position. Also, as
in that section, the momentum really contributes nothing to the result,
implying merely the possibility of choosing an arbitrary average velocity
for the particles. Neglecting this, we then have

In -^~ = a + 6ekin, (4.9)

where a and 6 are constants, €km is the kinetic energy of a molecule in the
ith cell. That is, we have

ft*

As we see by comparison with the formulas (3.5) and (6.4) of Chap. V,
the quantity b is to be identified with —1/kT. Thus we have

*' = — i- •— (4'U)

e aekT + 1

In formula (4.11), as in (2.15), there are certain quantities a and T
which are constant as far as the momenta are concerned, but which might
vary from point to point of space. We can investigate their variation
just as we did for the Boltzmann statistics in Sec. 3. The formula (3.3)
for the change of the distribution function with time on account of the
action of external forces holds for the Einstein-Bose and Fermi-Dirac
statistics just as for the Boltzmann statistics, and leads to a formula very
similar to Eq. (3.6) which must be satisfied for equilibrium. The only
difference comes on account of the different form in which we have
expressed the constants in Eq. (4.11). Demanding as before that the
relation like (3.6) must hold independent of momenta, we find that the
temperature must be independent of position, and that the constant a
of Eq. (4.11) must be given by

100 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VI

a - ^t (4.12)

where eo is a constant, </> is the potential energy of molecule. Thus,
finally, we have

N* = "3 ' (4.13)

where c» is the total energy, kinetic and potential, of a molecule in the
ith cell, in agreement with Eqs. (3.5) and (6.4) of Chap. V.

CHAPTER VII

FLUCTUATIONS

A statistical assembly contains many replicas of the same system,
agreeing in large-scale properties but varying in small-scale properties.
Sometimes these variations, or fluctuations, are important. Thus, two
repetitions of the same experiment may disclose different densities of a
gas at a given point, though the average density through a large volume
may be the same in each case. Such fluctuations of density can be of
experimental importance in such problems as the scattering of light, which
is produced by irregularities of density. Again, in the emission of elec-
trons from a hot filament, there are fluctuations of current, which are
observable as the shot effect and which are of great practical importance
in the design of amplifying circuits. We shall take up some of the simpler
sorts of fluctuations in this chapter. We begin by considering the fluc-
tuations of energy in a canonical assembly. We recall, from the argu-
ments of Chap. Ill, Sec. 5, that an assembly of systems in equilibrium
with a temperature bath must be assumed to have a variety of energies,
since they can interchange energy with the bath. We can now show,
however, that actually the great majority of the systems have an energy
extremely close to a certain mean value and that deviations from this
mean are extremely small in comparison with the total energy. This can
be shown by a perfectly straightforward application of the distribution
function for the canonical assembly, in the case w^here our system is a
sample of perfect gas obeying the Boltzmarm statistics, and we start with
that example.

1. Energy Fluctuations in the Canonical Assembly. — Let E be the
energy of a particular system in the canonical assembly, U being the
average energy over the assembly, or the internal energy. We are now
interested in finding how much the energies E of the individual systems
fluctuate from their average value. The easiest way to find this is to
compute the mean square deviation of the energy from its mean, or
(E — £7)2. This can be found by elementary methods from the Maxwell-
Boltzmann distribution law. Referring to Eq. (1.1) of Chap. IV, we
can write the energy as

E = ««, (i.D

1-1
101

102 INTRODUCTION TO CHEMICAL PHYSICS (CHAP. VII

where «(i) is the energy of the ith molecule, and

U = 2«« (1.2)

where c(t) is the average energy of the ith molecule over the assembly.
Thus we have

(E-U) = ]£(€<•> -€<«), (1.3)

t-i
and

N N

(?» - e(i))(7» - €<»). (1.4)

We must now perform the averaging in Eq. (1.4). We note that
there are two sorts of terms: first, those for which i = j; secondly, those
for which i 7^ j. We shall now show that the terms of the second sort
average to zero. The reason is the statistical independence of two mole-
cules i and j in the Boltzmann distribution. To find the average of such
a term, we multiply by the fraction of all systems of the assembly in which
the fth and jth molecules have tho particular energies c(£ and ej^, where
kl and &, are indices referring to particular cells in phase space, and sum
over all states of the assembly. From Eq. (1.2) of Chap. IV, giving the
fraction of all systems of the assembly in which each particular molecule,
as the ith, is in a particular state, as the &»th, we see that this average is

(€« - €<«)(€<*> - €<'>)

J* hi

= (^>""-niW)(?jrirgO>) = (gW _ e<»>)(eC») - g(0) = 0. (1.5)

Having eliminated the terms of Eq. (1.4) for which i ^ j, we have left only

(E - C/)2" = (e<'> -!<•">)*. (1.6)

That is, the mean square deviation of the energy from its mean equals
the sum of the mean square deviation of the energies of the separate
molecules from their means. Each molecule on the average is like every
other, so that the terms in the summation (1.6) are all equal, and we may

SEC. 1] FLUCTUATIONS 103

write

(E - £7)2 = tflc - O2, (1.7)

where « represents the energy of a single molecule, i its mean value.

We can understand Eq. (1 .7) better by putting it in a slightly different
form. We divide the equation by f72, so that it represents the fractional
deviation of the energy from the mean, squared, and averaged. In
computing this, we use Eq. (1.2) but note that the mean energy of each
molecule is equal, so that Eq. (1.2) becomes

V = Nl. (1.8)

Using Eq. (1.8), we then have

Equation (1.9) is a very significant result. It states that the fractional
mean square deviation of energy for N molecules is 1/JVth of that for a
single molecule, in Boltzmann statistics. The greater the number of
molecules, in othor words, the less in proportion are the fluctuations of
energy from the mean value. The fractional deviation of energy of a
single molecule from its mean is of the order of magnitude of its total
energy, as we can see from the wide divergence of energies of different
molecules to be observed in the Maxwell distribution of velocities and a.s
we shall prove in the next paragraph. Thus the right side of Eq. (1.9)
is of the order of magnitude of 1/N. If N is of the order of magnitude of
1024, as with a large scale sample of gas, this means that practically all
systems of the assembly have energies departing from the mean by some-
thing whose square is of the order of 10~24 of the total energy, so that the
average deviation is of the order of 10~12 of the total energy. In other
words, the fluctuations of energy in a canonical assembly are so small
as to be completely negligible, so long as we are dealing with a sample of
macroscopic size.

To evaluate the fluctuations of Eq. (1.9) exactly, we must find the
fluctuations of energy of a single molecule. We have

(6 - g)2 = 5 _ 2c€ + (e)2

= T* - (i)2, (1.10)

a relation which is often useful. We must find the mean square energy of
a single molecule. Using the distribution function (2.4) or (2.6) of Chap.
IV, we find easily that

(1.11)

104 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VII

Remembering that

i = ikT, (1.12)

as shown in Eq. (2.7) of Chap. IV, this gives

(e - €-)2 = (v - |)(fcr)« = |(*r)', (1-13)

from which Eq. (1.9) becomes

(E-u\_2i_ a 14)

\~U~J ~3AT' (L14)

fixing the numerical value of the relative mean square deviation of the
energy.

2. Distribution Functions for Fluctuations. — In the preceding section
we have given an elementary derivation of the energy fluctuations in
Boltzmann statistics. The derivation we have used is not applicable to
many interesting fluctuation problems, and in the present section we shall
develop a much more general method. Suppose we have a quantity
x, in whose fluctuations from the mean we are interested. This may be
the energy, as in the last section, or many other possible quantities. Our
method will be to set up a distribution function /(x), such that f(x)djc
gives the fraction of all systems of the assembly for which x lies in the
range dx. Then it is a simple matter of integration to find the mean of
any function of #, and in particular to find the mean square deviation, for
which the formula is

(x - z)2 = f(x - xYf(x)dx. (2.1)

We shall assume that the energy levels of the problem are so closely
spaced that they can be treated as a continuous distribution. For each
of the energy levels, or states of the system, there will be a certain value
of our quantity x. We shall now arrange the energy levels according to
the values of x and shall set up a density function, which we shall write

in the form e k , such that

dx (2.2)

is the number of energy levels for which x is in the range dx. We shall
see later why it is convenient to write our density function in the form
(2.2). Now we know, from the canonical assembly, as given in formula
(5.15) of Chap. Ill, that the fraction of all systems of the assembly in a

_ E
given energy level is proportional to e **. Thus, multiplying by the

number (2.2) of levels in dx, we find that the fraction of systems in the

SBC. 2] FLUCTUATIONS 105

range dx is given by

f(x)dx = const, e kT dx, (2.3)

where E(x) is the energy corresponding to the levels in dx. We may
immediately evaluate the constant, from the condition that the integral
of f(x) over all values of x must be unity, and have

(E(r)-Ts(x)]

(2.4)

In the problems we shall be considering, f(x) has a very sharp and
narrow maximum at a certain value .TO, rapidly falling practically to zero
on both sides of the maximum. This corresponds to the fact that jc
fluctuates only slightly from XQ in the systems of the assembly. The
reason for this is simple. The function E(x) — Ts(x) must have a mini-
mum at £0, in order that/(x) may have a maximum therc\ The function
f(x) will then be reduced to l/e of its maximum value when E(x) — Ts(x)
is greater than its minimum by only kT. But E is the energy of the whole
system, of the order of magnitude of NkT, if there arc N atoms or mole-
cules in the system, and wo shall find likewise that TS(JC) is of this same
order of magnitude. Thus an exceedingly small percentage change in x
will be enough to increase the function E(x) — Ts(x) by kT or much
more.

We can get a very useful expression for/(r) by assuming that

E(x) - 7'sCr)

can be approximated by a parabola through the very narrow range in
which f(x) is appreciable. Let us expand E(x) — Ts(x) in Taylor's
series about x0. Remembering that the function has a minimum at
z0, so that its first derivative is zero there, we have
E(x) - Ts(x) = E(x») - TS(XO)

* x. . /0 -.

x-^ 2+ "• (2'5)

The second derivatives in Eq. (2.5) are to be computed at x = XQ. Then
the numerator of Eq. (2.4) becomes

where

a ~ 2icT\dx*

106 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VII

The function e~€(*~*g)l is called a Gauss error curve, having been used
by Gauss to describe distribution functions similar to our f(x) in the
theory of errors. It equals unity when x = z0, and falls off on both sides
of this point symmetrically, being reduced to l/e when

x — Xo = ±l/\/a.

Using formulas (2.6) and (2.7), and the integrals (2.3) of Chap. IV, we
can at once compute the denominator of Eq. (2.4) and find

f(x) = /JV«(*-*0)>. (2.8)

Formula (2.8) is an obviously convenient expression for the distribution
function. From it one can find the mean square deviation of XQ. Obvi-
ously the mean value of x is x0l from the symmetry of Eq. (2.8). Then,
using Eq. (2.1), we have

(x - *o)2 = ~ (2.9)

In Eq. (2.9), we have a general expression for mean square fluctua-
tions, if only we can express E — Ts as a function of x. This ordinarily
can be done conveniently for the internal energy E. We shall now show
that, to a very good approximation, s equals the entropy S, so that it
also can be expressed in terms of the parameter x, by ordinary thermo-
dynamic means. To do this, we shall compute the partition function Z
of our assembly and from it the entropy. To find the partition function,

_ V

as in Eq. (5.17) of Chap. Ill, we must sum ekT over all stationary states.
Converting this into an integral over x and remembering that Eq. (2.2)
gives the number of stationary states in dx, we have

-(/?— -}
Z = je kT dx

kT fe-*(f-f^dx

Ts(ro)} r

= e ' kT '~JZ- (2.11)

Then, using Eq. (5.16) of Chap. Ill, we have
A = U - TS = -kTlnZ

= E(x<>) - Ts(xQ) - kT In J^- (2.12)

Now if the peak of f(x) is narrow, E(XO) will be practically equal to C7, the
mean value of E, which is used in thermodynamic expressions like Eq.

SBC. 3] FLUCTUATIONS 107

(2.12). It and TS are proportional to the number of molecules in the
system, as we have mentioned before. But a, as we see from Eq. (2.7),
is of the order of magnitude of the number of molecules in the system, so
that the last term in Eq. (2.12) is of the order of the logarithm of the
number of molecules, a quantity of enormously smaller magnitude than
the number of molecules itself (In 1023 = 23 In 10 = 53). Hence we can
perfectly legitimately neglect the last term in PJq. (2.12) entirely. We
then have at once

*(x0) = S. (2.13)

This expression, relating the entropy to the density of energy levels by
use of Eq. (2.2), is a slight generalization of what is ordinarily called
Gibbs's third analogy to entropy [his first analogy was the expression
- k S/t In /t, his second was closely related to Eq. (2.13)]. UsingEq. (2.13),
we can then write the highly useful formula

(/*• /*. V2 —
•*• •*(>; — y-,rt~,

3. Fluctuations of Energy and Density.- Using the general formula
(2.14), we can find fluctuations in many quantities. Let us first find the
fluctuation in the total energy of the system, getting a general result whose
special caso for the perfect gas in Boltzmann statistics was discussed in
Sec. 1. In this case x equals E, so that d2E/dx2 = 0. The derivative
of S with respect to E is to be taken at constant volume, for all the states
represented in the canonical assembly are computed for the same volume
of the system. Then we have, using the thermodynamic formulas of
Chap. II, Sec. 5,

(as\ i (svs\ ^i

\dU/v T' \dU2Jv T

___

TdU/v T2CV'

Substituting in Eq. (2.14), we find for the fluctuation of energy

(E - U)2 = kT*Cv. (3.2)

We can immediately see that this leads to the value we have already
found for the perfect gas in the Boltzmann statistics. For the perfect
gas, we have Cv = %Nk, so that

(E - C7)2 = $AT(A;T)2, (3.3)

agreeing with the value found from Eqs. (1.6) and (1.13). The formula
(3.2), however, is quite general, holding for any type of system. Since
Cv is of the same order of magnitude for any system containing N atoms

108 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VII

that it is for a perfect gas of the same number of atoms, we see that the
energy fluctuations of any type of system, in the canonical assembly, are
negligibly small. The heat capacity CV is proportional to the number
of atoms in the system, so that the mean square deviation of energy from
the mean is proportional to the number of atoms, and the fractional mean
square deviation of energy for N atoms is proportional to 1/JV, as in
Eq. (1.9).

As a second illustration of the use of the general formula (2.14), we
take a perfect gas and consider the fluctuations of the number of molecules
in a group of G cells in the molecular phase space. Two important
physical problems are special cases of this. In the first place, the G
cells may include all those, irrespective of momentum, which lie in a
certain region of coordinate space. Then the fluctuation is that of tho
number of molecules in a certain volume, leading immediately to tho
fluctuation in density. Or in the second place, we may be considering
tho number of molecules striking a certain surface per second and the
fluctuation of this number. In this case, the G colls include all those
whoso molecules will strike the surface in a second, as for example the colls
contained in prisms similar to those shown in Fig. IV-2. Such a fluctua-
tion is important in the theory of the shot effect, or the fluctuation of tho
number of electrons omitted thcrmionically from an olement of surface
of a heated conductor, per second; we assumo that the number omitted
can be computed from the number striking tho surface from inside tho
metal.

To take up this problem mathematically, wo oxpross tho energy and
the entropy in terms of the 2VYs, the average numbers of molecules in
the various cells. The energy is

U = #,ft, (3.4)

where ct is the energy of a molecule in the tth cell of tho molecular phaso
space. For the entropy, combining Eqs. (2.7) and (2.11) of Chap. V, we
have

8 = -kfti In Ni + (1 ± #.) In (1 ± #,)], (3.5)

where the upper sign refers to Einstein-Bose statistics, the lower to Fermi-
Dirac, and where we shall handle tho Boltzmanu statistics as the limiting
case of low density. In this case, the quantity x is the number of mole-
cules in the particular 0 cells we have chosen, which we shall call No, so
that

#t = x/G a* No/G, (3.6)

SBC. 3) FLUCTUATIONS 109

where #» is the average number of molecules in one of the cells of our
group of G, which we assume are so close together that the numbers #t
and energies et- are practically the same for all G cells. In terms of this
notation, we have

U = Noc* + terms independent of No, (3.7)

and

8 = -kN0 In §? ± k(G ± NG) In (l ±

+ terms independent of No. (3.8)

Then we have

and

Substituting in Eq. (2.14), we have

AT + " > A7 \ ~~ ^ /^ \7 \ ' (3.10)

Mo .J+ , MG\ \r I i , fVo

± ~-

and

in which we remember that the upper sign refers to the Einstein-Bose
statistics, the lower to the Fermi-Dirac, and where we find the Boltzmaim
statistics in the limit where Noo/G approaches zero, so that the right side
becomes merely l/NaQ. Thus, we see that in the Boltzmann statistics
the absolute mean square fluctuation nf the ni]TnKfir nf mnWnlog in Q
volume of phase space equals the mean number in the volume, and the
relative fluctuation is the reciprocal of the mean number, becoming very
small if the number of molecules is large. These are important results,
often used in many applications. We also see that the fluctuations in
Einstein-Bose statistics are greater than in Boltzmann statistics, while in
Fermi-Dirac statistics they are less, becoming zero in the limit

N*/G - ffi - 1,

110 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VII

since in that limit all cells in the group of G are filled and no fluctuation is
possible.

As a third and final illustration of Eq. (2.14), we consider the fluctua-
tion of the density of an arbitrary substance, really a generalization of the
result we have just obtained. Instead of the fluctuation of the density
itself, we find that of the volume occupied by a certain group of molecules;
the relative fluctuations will be the same in either case, since a given
proportional increase in volume will give an equal proportional decrease
in density. In this case, then, the quantity x is the volume V of a small
mass of material selected from the whole mass. The derivatives in Eq.
(2.14) are those of the whole internal energy and entropy of the system, as
the volume of the small mass is changed. Since the part of the system
exterior to the small mass is hardly changed by a change of its volume, we
can assume that only the internal energy and entropy of the small mass
itself are concerned in Eq. (2.14). Furthermore, we are interested merely
in the fluctuation in density, neglecting any corresponding fluctuation in
temperature, so that the derivatives of Eq. (2.14) are to be computed at
constant temperature. Then we can rewrite Eq. (2.14) as

where A = U — TS. Using the thcrmodynainic formulas of Chap. II,
we have

so that

_ _P

9Vr ~ '

and

, /*T/\ 1 ^8)

The quantity in brackets is the isothermal compressibility, which is inde-
pendent of V0. We see, then, that the relative mean square fluctuation
of the volume is inversely as the volume itself, becoming small for large
volumes. This is in accordance with the behavior of the other fluctua-
tions we have found.

Let us check Eq. (3.13) by application to the perfect gas in the Boltz-
mann statistics. Using PV - NkT, we have (~l/Vd(dV/dP)T

SBC. 3] FL UCT UA TIONS 1 1 1

Thus

TrTZTvA2 /.T i

(3.14)

where No is the mean number of molecules in F0. This value checks
Eq. (3.11), for the case of the Boltzmann statistics, giving the same
value for the relative fluctuation of volume which wo have already found
for the fluctuation of the number of molecules in a given volume, as we
have seen should be the case. For substances other than perfect gases,
the compressibility is ordinarily less than for a perfect gas, so that Eq.
(3.13) predicts smaller relative fluctuations of density; a perfectly incom-
pressible solid would have no density fluctuations. ( )M the other hand, in
some cases the compressibility can be greater than for a perfect gas. An
example is an imperfect gas near the critical point, where the compressi-
bility approaches infinity, a finite change in volume being associated with
no change of pressure; here the density fluctuations are abnormally great,
being visible in the phenomenon of opalescense, the irregular scattering
of light, giving the material a milky appearance. Below the critical
point, in the region where liquid and gas can coexist, it is well known that
the material maintains the same pressure, the vapor pressure, through the
whole range of volume from the volume of the liquid to that of the gas.
Thus here again the compressibility is infinite. Formula (3.13) cannot be
strictly applied in this case, but the fluctuations of density which it would
indicate are easily understood physically. A given volume in this case
can happen to contain vapor, or liquid drops, or both, and the fluctuation
of density is such that the density can be anywhere between that of the
liquid and the vapor. Such problems arc hardly suitable for a fluctuation
theory, however; we shall be able to handle them better when we take up
equilibrium between phases of the same substance.

PART II
GASES, LIQUIDS, AND SOLIDS

CHAPTER VIII

THERMODYNAMIC AND STATISTICAL TREATMENT OF THE
PERFECT GAS AND MIXTURES OF GASES

In Chap. IV, we learned some of the simpler properties of perfect gases
obeying the Boltzinann statistics, using simple kinetic methods. We
can go a good deal further, however, and in the present chapter we apply
thermodynamics and statistical mechanics to the problem, seeing how far
each can carry us. The results may seem rather formal and uninteresting
to the reader. But we are laying the groundwork for a great many
applications later on, and it will be found very much worth while to under-
stand the fundamentals thoroughly before we begin to apply them to such
problems as the specific heats of gases, the* nature of imperfect gases,
vapor pressure, chemical equilibrium, thermionic emission, electronic
phenomena, and many other subjects depending direct 1 y on the properties
of gases. For generality, we shall include a treatment of mixtures of
perfect gases, a subject needed particularly in discussing chemical equilib-
rium. We begin by seeing how much information thermodynamics alone,
plus the definition of a perfect gas, will give us, and later introduce a
model of the gas and statistical methods, obtaining by statistical
mechanics some of the results found by kinetic theory in Chap. IV.

1. Thermodynamics of a Perfect Gas. — By definition, a perfect gas
in the Boltzmann statistics is one whose equation of state is

PV = nRT, (l.l)

which has already been discussed in Sec. 3 of Chap. IV. Furthermore,
from the perfect gas law, using Eq. (6.2) of Chap. II, we can prove that a
perfect gas obeys Joule's law that the internal energy is independent of
the volume at constant temperature. For we have

(6U\ _ (dP\

WA Wr ( '

This is a reversal of the argument of Chap. II, Sec. 6, where we used
Joule's law as an experimental fact to prove that the gas scale of tempera-
ture was identical with the thermodynamic temperature. Here instead
we assume the temperature T in Eq. (1.1) to be the thermodynamic
temperature, and then Joule's law follows as a thermodynamic conse-
quence of the equation of state.

115

116

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

No assumption is made thermodynamically about the specific heat of
a perfect gas. Some results concerning it follow from thermodynamics
and the equation of state, however. Wo have already seen that

CV - Cv

(1.3)

where Cp and Cv are the heat capacities of n moles of gas, at constant
pressure and volume respectively. Furthermore, we can find thermo-
dynamically how CP changes with pressure*, or C\ with volume, at con-
stant temperature. We have

jLjf*a\

dP\dT}P

^OF),

dPOT dTdP

(1.4)

But from the Table of Thermodynamic Relations in Chap. II we have

(dH^

Thus, we find

(1.5)

(1.6)
By an exactly analogous proof, substituting (7 for H, V for P, wo can prove

f*a\ = T(^I . (i.7)

Substituting the perfect gas law in Eqs. (1.6) and (1.7), we find at once

for a perfect gas. That is, both specific heats are independent of pressure,
or volume, at constant temperature, meaning that they are functions of
the temperature only.

Thermodynamics can state nothing regarding the variation of specific
heat with temperature. It actually happens, however, that the heat
capacities of all gases approach the values that we have found theoret-
ically for monatomic gases in Eq. (3.18), Chap. IV, namely,

f nR, CP = f nRt

(1.9)

SBC. 1] THE PERFECT GAS 117

at low enough temperatures. This part of the heat capacity, as we know
from Chap. IV, arises from the translational motion of the molecules.
The remaining heat capacity arises from rotations and vibrations of the
molecules, electronic excitation, and in general from internal motions, and
it falls to zero as the tomperaturo approaches the absolute zero, on account
of applications of the quantum theory which wo shall make in later
chapters. Sometimes it is useful to indicate this internal heat capacity
per mole as Ct, so that we write

Cv = %nR + nCi, CV - $nR, + nC,, (1. 10)

where experimentally C\ goes to zero at the absolute zero, liquation
(1.10) may be taken as a definition of C,.

Next we take up the internal energy, entropy, Holmholtz free energy,
and Gibbs free energy of a perfect gas From Joule',1- law, the internal
energy is a function of the temperature alone, independent of volume.
We let (/n be the internal energy per mole* at the absolute zero, a quantity
which cannot be determined uniquely since there is always an arbitrary
additive constant in the energy, as we have pointed out in ('hap. I, Sec. 1.
Then the change of internal energy from the absolute zero to temperature
T is determined from the specific heat, and we have

r = wr» + (TcvdT

a/0

(1.11)

We find the entropy first as a function of temperature and pressure,
using the relations, following at once from the Table of Thermodynamic
Relations of Chap. II, and the equation of state,

(es\ = Cj. (*&\ = Jsv\ _ _UR

\dTjf T' \dPjT \dTjP P (IAZ)

Substituting for CP from Eq. (1.10) and integrating, we have

r fT 7m

S = ±nR In T - ntt In P + n\ C^ + const. (1.13)

^ Jo '

The constant of integration in Kq. (1.13) cannot bo determined by thermo-
dynamics. It is of no practical importance when we are considering tho
gas by itself, for in all cases we have to differentiate the entropy, or take
differences, in our applications. But when we come to the equilibrium of
different phases, as in the problem of vapor pressure, and to chemical
equilibrium, we shall find that the constant in the entropy is of great
importance. Thus it is worth while devoting a little attention to it here.
There is one piece of information which we can find about it from thormo-

118 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

dynamics: we may assume that it is proportional to the number of moles
of gas. To see this, consider first two separate masses of gas, one of n\
moles, the other of H* moles, both at the same pressure and temperature,
with a partition between their containers. The total entropy of the two
masses is certainly the sum of the separate entropies of the two. Now
remove the partition between them. This is a reversible process, involv-
ing no heat flow, and hence no change of entropy, so long as the gases on
the two sides of the partition are made of identical molecules; if there
had been different gases on the two sides, of course diffusion would have
occurred when the partition was removed, resulting in irreversibility.
Thus the entropy of the combined mass of (n\ + n%) moles is the sum of
the separate entropies of the masses of HI and n2 moles. But this will be
true only if the entropy is proportional to the number of moles of gas.
We know that this is true with every term of Eq. (1.13) except the con-
stant, and we see therefore that it must be true of the constant as well.
Thus the constant is n times a quantity independent of n, P, 77, and
hence depending only on the type of gas. This constant must have the
dimensions of /£, so that it must be nil times a numerical factor. For
reasons which we shall understand shortly, it is convenient to write it in
the form n(i + %)R, where i is called the chemical constant of the gas.
Thus we have

S = ^nR In T - nR In P + n I C^- + nRli + ~| (1 .14)

It is often useful as well to have the entropy as a function of tempera-
ture and volume. We can find this by integrating the equations

f^\ = ^ f^\ = f^\ = ^

V .nT* I W * V Z± \7 I V UHl I 17" ' N LtHjJ

\O ./ / V •*• \O w / T \U J. / V V

or by substituting for P in terms of T and V from the perfect gas law in
Eq. (1.14). The latter has the advantage of showing the connection
between the arbitrary constants in the two equations for entropy, in
terms of T and P, and in terms of T and V. Using this method, we have
at once

3 CT dT f 5

S = ~nR In T + nR In V + n I d-^- + nR\ i + ^ • — In (nl

n L C^Y

From Eq. (1.16) we note that the additive constant in the entropy, in
the form involving the temperature and volume, has a term — nR In n,
which is not proportional to the number of moles. This is as we should
expect, however, as we can see from rewriting Eq. (1.16) in the form

S =

ln T - nR\n~ + n ( C.~ + nfiti + | - In R\ (1.17)

SBC. 1] THE PERFECT GAS 119

In the form (1.17), each term is proportional to n, except the one

This involves the ratio n/F, the number of moles per unit volume, which
is proportional to the density. We thus see from Eq. (1.17) that if two
masses of gas of the same temperature and the same density are put in
contact, the total entropy is independent of whether they have a partition
between them or not. This statement is entirely analogous to the previ-
ous one about two masses at the same temperature and pressure.

Next we find the Helm holt z free energy A = U — • TS as a function
of temperature and volume. We can find this directly from Eqs. (1.11)
and (1.16) or (1.17). Wo havo at once

= n\ I/o - |«r In T - RT In -
L & n

. dT - TJ*CJjr\ - KT(i + 1 - In «)]• (1.18)

The two terms depending on C, can bo written in two other forms by
integration by parts. Those aro

rT __ , c1 dT _ _ rv CT dT\ ,,

Jo Jo T J0 \Jo l T /

CT IT CT

--rj.#j,(?"ir o-20'

To prove Eq. (1.19), we apply the formula Jw dv — uv — Jv du to the
right side, setting
»r

iajL j» i^iui ,- , Arr

,%-Y> du = Y~> v = *•> dv = dl,

and Eq. (1.19) follows at once. To prove Eq. (1.20), we integrate the
expression fC%/T dT on the left side of Eq. (1.19) by parts, setting
u = 1/r, du = -l/T2dT, v = fCidT, dv = Ct dT, from which Eq.
(1.20) follows. Thus we have the alternative formulas

To y

A = n t/o - ^RT In T - /2'f In ~

+ 1 - In /Z) (1.21)

- fi77 In T - B5T In -

C* dT - RW + 1 - In «) • d-22)

- In «) ]• d-

120 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

Finally, we wish the Gibbs free energy G = U + PV — TS as a
function of pressure and temperature. Using Eqs. (1.11), (1.14), (1.19),
and (1.20), this has the alternative forms

G = n(u« - ^RT In T + RT In P

7t dT - T f C\ ~ - flW) (1.23)
Jo V /

= n\ f/0 - ^RT In

- f (f c^)"1 - ««] <i-

T7 + /M' In P

-24>

- #7' Jn T + K77 In P

We see that the term proportional to T in Eqs. (1.23), (1.24), and (1.25),
— RTi, has a particularly simple form. It is for this reason that the
additive constant in the entropy, nR(i + f), in Eq. (1.20), is chosen in
the particular form it is. For practical purposes, the appearance of this
quantity in the Gibbs free energy is mon* important than it is in the
entropy.

2. Thermodynamics of a Mixture of Perfect Gases. — Suppose we
have a mixture of HI moles of a gas 1, n2 moles of gas 2, and so on, all in a
container of volume V at temperature 7\ First we define the fractional
concentration c*7 of the ith substance as the ratio of the number of moles
of this substance to the total number of moles of all substances present:

£t r= -- - — - . V^- U

n\ + nz + - • •

We also define the partial pressure Pt of UK; ?th substance as the pressure
which it would exert if it alone occupied the volume V. That is, since all
gases are assumed perfect,

r>//T

^ = n^- (2.2)

Then the equation of state of the mixture of gases proves experimentally
to be just what we should calculate by the perfect gas law, using the
total number of moles, (n\ + W& +•••)> that is, it is

P - fa + n2 + - • • )~ (2.3)

SEC. 2] THE PERFECT GAS 121

Equation (2.3) may be considered as an experimental fact; it follows, how-
ever, at once from our kinetic derivation of the equation of state in Chap.
IV, for that goes through without essential change if we have a mixture
instead of a single gas. Then from Eqs. (2.1), (2.2), and (2.3), we have

(2.4)

From Eq. (2.4), in other words, the fractional concentration of one gas
equals the ratio of its partial pressure to tho total pressure. Plainly as a
corollary of Eq. (2.4) we have

Pi + /J* + • • • - ir^r-i— -> - p> (2-5)

ni -f- HZ •+• • • •
and

ci + c, + • • • = I. (2.6)

Equation (2.5) expresses the fact that the sum of the partial pressures
equals the total pressure.

We next consider the entropy, Helmholtz free energy, and Gibbs free
energy of the mixture of gases. We start with the expression (1.14) for
the entropy of a single gas. In a mixture of gases, it is now reasonable to
suppose that the total entropy is the sum of the partial entropies of each
gas, each one being given by Eq. (1.14) in terms of the partial pressure of
the gas. If we have a mixture of n\ moles of the first gas, n^ of the second,
and so on, the total entropy is then

s = n'fi ln T + c'~r~ ~ R ln P] + 5R + i}R' (2'7)

where Cj is the internal heat capacity per mole of the jth gas, ij its chem-
ical constant. We can express Eq. (2.7) in terms of the total pressure.
Then we have

s = n/j? ln T + C; ~ R ln p + /e + *

jlnc,. (2.8)

In Eq. (2.8), the first summation is the sum of the entropies of the various
gases, if each one were at the same pressure P. The second summation
is an additional term, sometimes called the entropy of mixing. Since the
c/s are necessarily fractional, the logarithms are negative, and the entropy

122

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

of mixing is positive. It is such an important quantity that we shall
examine it in more detail.

Suppose the volume V were divided into compartments, one of size
CiF, another c2F, otc., and all the gas of the first sort were in the first
compartment, that of the second sort in the second, and so on. Then
each gus would have the pressure P, and the entropy of the whole system,
being surely the* sum of tho entropies of the separate samples of gas, would
be given by the first summation of Eq. (2.8). Now imagine tho partitions
between the compartments to be removed, so that the, gases can^irrcversi-
bly diffuse into o.'irlj nthi*r. This diffusion, being an irreversible process,
muafr result in rm increase uf entropy, and the second -term of Eg. (2.8). the
entropy of mixing, represents just this increase. To verify its correctness,
we must find an alternative reversible path for getting from the initial
state to the final one, and find / dQ/T for this reversible path. That will

i Gas
1+2

Gas 2

_J Gasl+2 p

(b) (c)

Flu VIII-1. — Reversible mixing oi two gases

give the change of entropy, whether the actual path is reversible or not.
We shall set up this reversible process by moans of semi-permeable mem-
branes, membranes allowing molecules of one gas to pass through them,
but impervious to the other gas. Such membranes actually exist in a few
cases, as for instance heated palladium, which allows hydrogen to pass
through it freely, but holds back all other gases. There is no error
involved in imagining such membranes in other cases as well.

We simplify by considering only two gases. Originally let the parti-
tion separating the compartments c\V and c2V be two semipermeable
membranes in contact, one permeable to molecules of type 1 but not of
type 2 [membrane (1)], the other permeable to type 2 but not type 1
[membrane (2)]. The two together will not allow any molecules to pass.
Each of the membranes will be subjected to a one-sided pressure from the
molecules that cannot pass through it. Thus, in Fig. VIII-1 (a), mem-
brane (1) is pushed to the left by gas 2, membrane (2) to the right by gas
1. Each of the membranes then really forms a piston, and if rods are
attached to them as in Fig. VIII-1, they are capable of transmitting force
and doing work outside the cylinder. Now let membrane (1) move slowly
and reversibly to the left, as in (b), doing work on some outside device.
If the expansion is isothermal, we know that the internal energy of the
perfect gases is independent of volume, so that heat must flow in just equal

SEC. 21

THE PERFECT GAS

123

to the work done. We can then find tho boat flowing in in the process by
integrating P <1V for membrane (1). The pressure exerted on it, when
the volume to the right of it is T, is n-£RT/V9 sinee only the molecules 2
exert a pressure on it. Thus the work done when the volume increases
from c%V to V is

Cv RT I V \

ri2-±dV = n2RT In (- ) = -RTni In c2.

JctV ' \^2r /

(2.9)

This equals the heat flowing in. Since the corresponding increase of
entropy is the heat divided by the temperature, it is

-/en2lnc2.

(2.10)

Now in a similar way we draw the membrane (2) to the right, extracting
external work reversibly and letting heat flow in to keep the temperature
constant. By similar arguments, the
increase of entropy in this process
is —Rni In Ci. And the total change of
entropy in this reversible mixing is

ICal -

AS =

In ci + H2 In r2), (2.11)

just the value given foi the entropy of
mixing in Eq. (2.8).

It is interest ing to see hon the entropy
of mixing of two gases depends on the
concentrations. Let n = /M + "2 ~ the

t_ i i 1 p 1 /• rm

total number ol moles oi gas. J hen,
remembering that c\ + c2 = 1, [Eq. (2.G)], we have

AS = ~nfl[ci In ci + (1 - ci) In (1 - ci)].

o o 10

vni-2. Entropy of mixing of

tWO KU*U*S.

(2.12)

In Fig. VIII-2 we plot AS from Eq. (2.12), as a function of ci, which
can range from zero to unity. We see that the entropy of mixing has its
maximum value when c± = ^, or with equal numbers of the two types of
molecules. At this concentration its value is given by

-n#(i In 4 + i In i) = nR In 2

= 0.6931n/2 = 1.375 cal. per mole per degree.

Having found the entropy of a mixture of gases in Eq. (2.8), it is a sim-
ple thing to find the Gibbs free energy, from the relation

G = U + PV - TS.
We have

+ f RT

124 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

PV = ty-RT, (2.14)

so that

G = jJnXZ, + flrjn, In c,-, (2.15)

y y

wliere

K rr rr ,rr

= U, - 5/22' In 71 + C/dr - 27 C^ + RT In P - t,*!?
2 Jo Jo J

P CTf?T CT

= Uj-^RTlnT - T\ %±\ C, dT + RT In P - {,-RT. (2.
* Jo 1 Jo

16)

In Eqs. (2.15) and (2.16), [// represents the arbitrary constant in the
energy for the jth type of molecule, corresponding to f/o of Eq. (1.11),
and the last step in Eq. (2.16) is made by the same integration by parts
used in Eqs. (1.19) and (1.20). The quantity (?/ represents the Gibbs
free energy per mole of the jth gas at temperature T and pressure P.
Thus Eq. (2.15) indicates that the Gibbs free energy of the mixture is the
sum of the free energies of the constituents, at the final pressure arid
temperature, plus a mixing term which is always negative.

3. Statistical Mechanics of a Perfect Gas in Boltzmann Statistics. ~
Since the internal energy of a perfect gas is independent of volume, by
Joule's law, it is obvious that there can be no forces acting between the
molecules, for if there were, they would result in an internal energy
depending on the volume. Thus the molecular model of a perfect gas,
which we make the basis of our statistical treatment, is a collection of N
molecules, each of mass m, exerting no forces on each other. If the gas is
monatomic, each molecule requires only three coordinates, the rectangular
coordinates of its center of gravity, and the three conjugate momenta, to
describe it completely, so that the phase space contains QN dimensions.
When the gas is polyatomic, additional coordinates are necessary to
describe the orientation and relative distances of separation of the atoms
in the molecules. We assume there are s such coordinates, 5 momenta, so
that in all there are (3 + s) coordinates, (3 + s) momenta, for each
molecule, or (6 + 2s)N dimensions in the general phase space. We shall
call the coordinates of the jth molecule

and the momenta

Here Xjyfif are the coordinates of the center of gravity, pxj, pui, pzj the
components of total momentum, of the molecule.

SBC. 3] THE PERFECT GAS 125

The first step in applying statistical mechanics to our gas is to compute
the partition function Z, given by Eq. (5.17) or (5.22) of Chap. III. To
do this, we must first know the energy of the gas, E, as a function of the
coordinates and momenta. Since there are no forces between the mole-
cules, this is a sum of separate terms, one for each molecule. Now it is a
general theorem of mechanics that the energy of a structure like a mole-
cule, composed of particles exerting forces on each other but not actod on
by an external force field, is the sum of tho kinetic energy of the structure
as a whole, determined by the velocity of tho center of gravity, and an
additional term representing the energy of the internal motions. Thus.
for the energy of the gas, we have

(3..)

In Eq. (3.1), e' represents the energy of internal motions of a molecule.
In evaluating the partition function, we must take exp ( — E/kT),
whore E is given in Kq. (3.1), and integrate over all coordinates and
momenta. We observe in the first place that, since? E is a sum of terms
for each molecule, exp ( — E/kT) will be a product of such terms, and the
whole partition function will be a product of N factors, one from each
molecule, each giving an identical integral, which we can refer to as the
partition function of a single molecule. We next observe that the parti-
tion function of a single molecule factors into terms depending on the
center of gravity of the molecule and terms depending on the internal
motion. Thus we have

/*
j

(3.2)

The integration over x, y, z is to be carried over the volume of the con-
tainer and gives simply a factor V. The integrations over px, py, p~ are
carried from — <*> to <*> , and can be found by Eqs. (2.3) of Chap. IV. The
integral depending on the internal coordinates and momenta will not be
further discussed at present; we shall abbreviate it

^« = i^ I ' I e kT dqi- dp,

_LL

T. (3.3)

126 INTRODUCTION TO CHEMICAL PHYSICS [CiiAP. VIII

In Eq. (3.3), Z» is the internal partition function of a single molecule.
The second way of writing it, in terms of a summation, by analogy with
Eq. (5.17) of Chap. Ill, refers to a summation over all cells in a 2s-dimen-
sional phase space in which q\ • p8 are the dimensions. We note, for
future reference, that the quantity Z> depends on the temperature, but
not on the volume of the gas.

Using the methods just described, Eq. (3.2) becomes

Z, I

Z = -^wmkT^Z, (3.4)

There is one thing, however, which wo have neglected in our derivation,
and that is the fact that the gas really is governed by Fermi- Dirac or
Kinstcin-Bose statistics, in the limit in which they load to tho Boltzrnann
statistics. As wo havo soon in Chap. V, Sec. 1, on account of the identity
of the molecules, thoro aro roally AM different colls of tho general phase
space corresponding to one state or complexion of the system. The
reason is that there are A"! different permutations of the molecules, each
of which would lead to the same number of molecules in each cell of the
molecular phase space, and each of which therefore would correspond to
the same complexion. In other words, by integrating or summing over
all values of the coordinates and momenta of each molecule, we have
counted each complexion AH times, so that the expression (3.4) is Nl
times as great as it should be, and we must divide by AH to get the correct
formula. Using Stirling's formula, I/AM = (e/N)N approximately, and
multiplying by this factor, our amended partition function is

(3.5)

We shall use Eq. (3.5) as the basis for our future work.

From the partition function (3.5), we can now find the Helmholtz free
energy, entropy, and Gibbs free* energy of our gas. Using the equation

A = -kT In Z, we have

A = -\NkT In T - NkT In V - NkT In Z<

_ NkThl + ! _ lu

L A8

From A we can find the pressure by the equation P — — (dA/dV^T* We
have at once

NIcT
p = iHi, or PV = NkT. (3.7)

SBC. 3] THE PERFECT GAS 127

Thus we derive the perfect gas law directly from statistical mechanics.
We can also find the entropy, by the equation S = —(dA/dT)r. Using
the relation Nk = n/Z, we have

S = |nfl In T + nR In V + nR^(T In Z%)

, _J. (ZwmW* ,5 , x Dxl ,0 ^
+ nfll In YS ------ h 2 - in (n/Z) I- (3.8)

From S we can find the specific heat GY by the equation CV = T(dS/dT)v.
We have

CF = j*nfi + nRTJ~(T In Zt). (3.9)

The specific heat given in Eq. (3.9) is of the form given in Eq. (1.10); by
comparison we see that the internal heat capacity per mole, Ct, is given by

,72

Ct = RT~(TlnZJ, (3.10)

similar to Eq. (5.21) of Chap. III. We shall use Eq. (3.10) in the next
chapter to compute the specific heat of polyatomic gases. For mon-
atomic gases, for which there are no internal coordinates, of course Ct
is zero.

Using Eq. (3.10), we can rewrite T In Zl and its temperature deriva-
tives in terms of Ct. First, we consider the behavior of T In Zz and its
derivatives at the absolute zero. Let there be go cells of the lowest energy
in the molecular phase space, gi of the next higher, and so on. It is cus-
tomary to call these g's the a priori probabilities of the various energy
levels, meaning merely the number of elementary eells which happen to
have the same energy. Then we have, from Eq. (3.3),

_/? -Jl1 __!l°

Zi=g0e kT + ffle **•+••• = ^ w as y^0> (3<uj

Hence T In Z% approaches T In go — ~ as T approaches zero. In the

derivative of T In Zt with respect to temperature, the only term which
does not approach zero with an exponential variation is the term In 0o.
Using these values, then, we have

p(T In Zt) =\ng, + ct? (3.12)

T In Z< = + T In ,. 4- cT. (3.13)

128 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. VIII

Using Eq. (3.13), we can rewrite Eq. (3.6) as

A = n\ f/o - l/Zr In T - RT In f H

whore

and

i = /

AT'

No being Avogadro's number, the number of molecules in a mole, given
in Eq. (3.10) of Chap. IV. We observe that Eq. (3.14) is exactly the
same as (1.21), determined by thermodynamics, except that now we have
found the quantities UQ, tho arbitrary constant in the energy, and z, the
chemical constant, in terms' of atomic constants. Similarly, wo can show
that all the other formulas of Sec. 1 follow from our statistical mechanical
methods, using Eqs. (3.15) and (3.16) for the constants which could not be
evaluated from thermodynamics.

If we have a mixture of NI molecules of one gas, N2 of another, and
so on, the general phase space will first contain a group of coordinates
and momenta for the molecules of the first gas, then a group for the
second, and so on. The partition function will then be a product of terms
like Eq. (3.5), one for each type of gas. The entropy will be a sum of
terms like Eq. (1.14), with n» in place of n, and Pt, the partial pressure, in
place of P. But this is just the same expression for entropy in a mixture
of gases which we have assumed thcrmodynamically in Eq. (2.7). Thus
the results of Sec. 2 regarding the thermodynamic functions of a mixture
of gases follow also from statistical mechanics.

It is worth noting that if we had not made the correction to our parti-
tion function on account of the identity of particles and had used the
incorrect function (3.4) instead of the correct one (3.5), we should not
have found the entropy to be proportional to tho number of molecules.
We should then have found an entropy of mixing for two samples of the
same kind of gas: the entropy of (n\ + n2) moles would be greater than
the sum of the entropies of HI moles and n2 moles. It is not hard to show
that the resulting entropy of mixing would be just the value found in
Eq. (2.8) for the mixing of unlike gases. This is natural; if we forgot
that the molecules were really alike, we should think that the diffusion
of one sample of gas into another was really irreversible, since surely we
cannot separate the gas again into two samples containing the identical

SEC- 31 THE PERFECT GAS 129

molecules with which we started. But the molecules really are identical,
and it is meaningless to ask whether the molecules we find in the final
samples are the same ones we started with or not. Thus mixing two
samples of unlike gases increases the entropy, while mixing two samples of
like gases does not. It might seem paradoxical that these two results
could he simultaneously true. For consider the mixing of two unlike
gases, with increase of entropy, and then let the molecules of the two kinds
of gas gradually approach each other in properties. When do they
become sufficiently similar so that the process is no longer irreversible, and
there no longer is an increase of entropy on mixing? This paradox is
known as Gibbs's paradox, and it is removed by modern ideo.s of the struc-
ture of atoms and molecules, based on the quantum theory. Jn the
quantum thenrvr ^re is a. perfectly clear-cut distinction: cither two
particles are identical or they are not. There is no such thing as a gradual
change from one to the other, for identical particles are things like elec-
trons, of fixed properties, which we cannot change gradually at will.
With this clear-cut distinction, it is no longer paradoxical that identical
particles are to be handled differently in statistics from unlike particles.

CHAPTER IX

THE MOLECULAR STRUCTURE AND SPECIFIC HEAT
OF POLYATOMIC GASES

We have seen in the preceding chapter that the equation of stale, of a
perfect gas is independent of the nature of the molecule. This is not
true, however, of the specific heat; the quantity G\, which we called the
internal specific heat, results from molecular rotations and vibrations and
is different for different gases. For monatomic gases, where Ct is zero,
we have found CV = fn/2, CP = §nR. Using the value R = 1.987 cal.
per degree per mole, from Eq. (3.9) of Chap. IV, we have found the
numerical values to be Cv — 2.980 cal. per degree per mole, and
Cp = 4.968 cal., values which are correct within the limits of experimental
error for the specific heats of He, Ne, A, Kr, Xe, and of monatomic vapors
of metals, when extrapolated to zero pressure, so that they obey the
perfect gas law. But for gases which are not monatomic, the additional
term C» in the specific heat can only bo found from a rather careful study
of the structure of the molecule. This study, which we shall make in
the present chapter, is useful in two ways, as many topics in this book will
be. In the first place, it throws light on the specific problem of the heat,
capacity of gases. But in the second place, it leads to general and valu-
able information about molecular structure and to theories which can be
checked from the experimentally determined heat capacities.

1. The Structure of Diatomic Molecules. —Many of the, most impor-
tant molecules are diatomic and furnish a natural beginning for our study.
The atoms of a molecule are acted on by two types of forces, fundamen-
tally electrical in origin, though too complicated for us to understand in
detail without a wide knowledge of quantum theory. First, there are
forces of attraction, the forces which are concerned in chemical binding,
often called valence forces. We shall look into their nature much more
closely in later chapters. These forces fall off rapidly as the distance r
between the atoms increases, increase; rapidly with decreasing r. Being
attractions, they are negative forces, as shown in Fig. IX-1 (a), curve I.
Secondly, there are repulsive forces, quite negligible at large distances, but
increasing even more rapidly than the attraction at small distances.
These repulsions are just the mathematical formulation of the impene-
trability of matter. If two atoms are pushed too closely into contact,
they resist the push. The repulsion, a force of positive sign, is shown in

130

SEC. 1]

POLYATOMIC GASES

131

Curve II, Fig. IX-1 (a). If the atoms were rigid spheres, this repulsion
would be zero if r were greater than the sum of the radii of the spheres,
and would become infinite as r became less than this sum of radii. The
fact that it rises smoothly, not discontinuously, shows that the atoms do
not really have sharp, hard boundaries; they begin to bump into each
other gradually, though quite rapidly. Now when we add the attractive
and repulsive forces, wo got a curve liko III of our figure. This represents
a negative, attractive force at largo distances, changing .sign and becoming
positive at small distances, ««• the repulsion begins to outweigh tho attrac-
tion. At the distance rf, where the force changes sign, there is a position
of equilibrium. The attraction and repulsion just balance, and the atoms
can remain at that distance apart indefinitely. This, then, is the normal
distance of separation of the atoms in the molecule. For small deviations

(a) (b)

Fia. IX-1 — Force (a) and energy (6) of interaction at two atoms in a molecule
I, attractive term, II, repulsive term, III, resultant curve.

of r from rt, the curve of force against distance can be approximated by a
straight line: the force is given by — constant (r — /v), a force propor-
tional to the displacement, the sort found in elastic distortion, and leading
to simple harmonic motion. Under some circumstances, the atoms
vibrate back and forth through this position of equilibrium, the amplitude
increasing with temperature. At the same time, the molecule as a whole
rotates about its center of gravity, with an angular velocity increasing
with temperature, and of course finally it moves as a whole, the motion
of the center of gravity being just as with a single particle.

Rather than using the force, as shown in Fig. IX-1 (a), we more
often need the potential energy of interaction, as shown in (6) of the same
figure. Here we have shown the potential energy of the attractive force
by 1, that of the repulsive force by II, and the total potential energy by
III. At the distance rf, where the force is zero, the potential energy has a
minimum; for we remember that the slope of the potential energy curve
equals the negative of the force. The potential energy rises like a parah-

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. IX

ola on both sides of the minimum; if the force is
*(r-

potential energy is

where k is a constant.

-k(r — re), then the
It continues to rise

indefinitely as /• decreases toward zero, since it requires infinite work to
force the atoms into contact. At large values of r, however, it approaches
an asymptotic value; it requires only a finite amount of work to pull the
atoms entirely apart from each other. This amount of work, indicated
by D on the figure, is the work required to dissociate the molecule, and is
important in thermodynamic applications.

In Table IX-1 we list values of re and D for a number of important
diatomic molecules. A few of these, as the hydrides of carbon, nitrogen,

TABLE IX-1. — CONSTANTS OF DIATOMIC MOLECULES

Substance

7> r, A
(electron volts)) °

a, A-1

H2 . . 103

4 454

0.75

1.94

3 5

1 12

1.99

NH. : 97

1 2

1.08

1.96

102 ''

4 4

0.96

2.34

HC1

102

4.40

1.27

1.91

123

5.3

15 i 3.06

02 . .

117

5 09

20 2.68

170

7 35

09 3.11

CO 223

9 fi

13 2.48

C2 128

5 (i

31 2.32

2 47

98 2 05

Br2

1.96

2 28 1 1.97

I2.

1 53

2 66

1.86

Li2

1 14

2 67

0.83

Na2 18

0 7(i

3 07

0.84

K, . ; 12

0 51

3.91

0 78

The data are taken from Spom-r, " Molekulspekticri und

ihie AnwendutiKcn auf chemische Piob-

lerne," Spnnger, Berlin, 1933, which tabulates Din electron volts, /*,, and vihiational frequencies. The

values of a in the table above are computed u-mg Eq. (4.5) of the pit-sent chapter, solved for a in terms

of the vibrational frequency and D, as tabulated by Sponer

Thus a calculation of the vibrational

frequency fiorn data of the present table, using Kq. (4 o),

\vill iiiitornutically give the right value

Sponer'a data are taken from band spectra.

and oxygen, do not ordinarily occur in chemistry, but they are formed in
discharge tubes and are stable molecules. The values of re are given in
angstrom units (abbreviated A), equal to 10 8 cm. The values of D are
given in kilogram-calories per gram mole, where we remember that
1 kg.-cal. is 1000 cal., or 4.185 X 1010 ergs. We also give D in electron
volts. One electron volt by definition is the energy acquired by an
electron in falling through a difference of potential of one volt. This is
the charge on the electron, 4.80 X 10~l° e.s.u., times one volt, or ?fa e.s.u.

SBC. 1] POLYATOMIC GASES 133

Thus one electron volt is 4.80 X 10~10/300 = 1.60 X 10~12 erg. To
compare with the other unit, we note that the value of D in kilogram-
calories is computed for a gram mole, that in electron volts for a single
molecule. Thus we have

1 electron volt per molecule = 1.60 X 10~12 erg per molecule
= 1.60 X 10- ia X 6.03 X 1023 ergs per mole
1.60 X 10~12 X 6.03 X 1023 ,

= 23.05 kg.-cal. per mole. (1.1)

A very useful empirical approximation to the curves of Fig. IX-1 has
been given by Morse, and it is often called a Morse curve. As a matter of
fact, Fig. IX-1 was drawn from Morse's equation. This approximation is

Force =
Energy = C + D(e~^r~-^ - 2r n(> ->•>). (1.2)

Here C is a constant fixing the zero on the scale of ordinates and therefore
arbitrary, since there is always an arbitrary additive constant in the
energy. D is the energy of dissociation tabulated in Table IX-1, and
finally a is a constant determining the curvature about the minimum of
the curve, given in the last column of Table IX-1. Thus from the data
given in Table IX-1 and the function (1.2), calculations can be made for
the interatomic energy or force. In Eq. (1.2), the first term, the positive
one, represents the repulsive part of the potential energy between the
two particles, important at small distances, while the second, negative
term represents the attraction at larger distances. While the Morse
curve has no direct theoretical justification, still it proves to represent
fairly accurately the curves which have been calculated in a few cases
from quantum mechanics. Such calculations have shown that it is
possible to explain in detail the interatomic energy curves, the magni-
tudes of D, r(, etc. Nevertheless, the explanations are so complicated
that it is better simply to treat the constants of Table IX-1 as empirical
constants, without trying to understand why some molecules have greater
/)'s, some less, etc., in terms of any model. For future reference, how-
ever, it is worth while pointing out that the smaller D is, the less energy
is required to dissociate the molecule, and therefore the lower the tem-
perature needed for dissociation. We shall later talk about thermal
dissociation of molecules; from the table it is clear that the best molecules
to use as examples, the ones which will dissociate at lowest temperatures,
will be iodine and the alkali metals lithium, sodium, and potassium.
Conversely, N2 and CO require such a high energy for their dissociation
that they do not dissociate under ordinary circumstances.

134 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

2. The Rotations of Diatomic Molecules. — If molecules were governed
by classical mechanics, the motions of their atoms would have the follow-
ing nature. First, the molecules as a whole would have a uniform motion
of translation, the mean kinetic energy being ffcT, on account of the
equipartition of energy, discussed in Chap. IV, Sec. 2. Secondly, the
molecules would rotate with uniform angular momentum about an arbi-
trary axis passing through the center of gravity. Two coordinates are
necessary to specify the rotation of the molecule; for instance, the latitude
and longitude angles of the line joining the centers of the atoms. Thus,
from equipartition, the mean kinetic energy of rotation would be
(%)kT = kT. Finally, the atoms would vibrate back and forth along
the line joining them. One coordinate, the interatomic distance r, deter-
mines this vibration. Thus, from equipartition the mean kinetic energy
of vibration would be %kT. At the same time, in simple harmonic motion,
there is a mean potential energy equal to the mean kinetic energy and
hence equal also to \kT, so that the oscillation as a whole would con-
tribute kT to the energy. We should then find a mean energy of rotation
and vibration of 2kT, with a contribution to the heat capacity per mole
of 2Nok = 2R cal. per degree. This would bo the value of C,, the heat
capacity of internal motions mentioned in Chap. VIII, Sec. 1, if the gas
obeyed classical mechanics. Actually, the observed values are less than
this, increasing from small values at low temperatures to something
approaching 2R at very high temperatures, and the discrepancies come
from the fact that the quantum theory, rather than the classical theory,
must be used.

We have seen in Chap. IV, Sec. 2, that equipartition of energy is found
only when a distribution of energy levels is so closely spaced as to be
practically continuous. The translational levels of a gas are spaced as
closely as this, as we have seen in Chap. IV, Sec. 1, so that we are per-
fectly justified in assuming equipartition for the translational motion,
resulting in the heat capacity Cv = InR. But the rotational and vibra-
tional levels are not so closely spaced, and we must use the quantum
theory to get evon an approximately correct value for this part of the
specific heat. Our first problem, then, is to find what the energy levels
of a diatomic molecule really are. This can be done fairly accurately by
quite elementary methods. To a good approximation we can treat the
rotation and vibration separately, assuming that the total energy is
the sum of a rotational and a vibrational term. We can treat the vibra-
tion as if the molecule wore not rotating, and the rotation as if it were not
vibrating, but as if the atoms were fixed at the interatomic distance re.

Let us consider the rotation first. In Chap. Ill, Sec. 3, we have
found that the energy of a rotating body of moment of inertia /, angular
momentum p$, is p02/27, which is equal to the familiar expression i/co2,

SBC. 2] POLYATOMIC OASES 135

where co is the angular velocity. Furthermore, we have found that in
the quantum theory the angular momentum is quantized : that is, it can
take on only certain discrete values, pe — nh/2ir, where n is an integer.
Thus, the energy according to elementary methods can have only the
values En = n2/i2/87r2/, as in Eq. (3.7), Chap. III. As a matter of fact,
as we saw in Eq. (3.9), Chap. Ill, we must modify this formula slightly.
To agree with the usual notation used for molecular energy levels, we
shall denote the quantum number by K, rather than n. Then it turns
out that the energy, instead of being given by K2h~/8w*I, is given by the
slightly different formula

E^ = K(K + 1)^, (2.1)

where K can take on the values 0, 1, 2, ... To evaluate these rotational
energy levels, we need the moment of inertia /, in terms of quantities that
we know. This is the moment of inertia for rotation about the center of
gravity of the molecule. Let the masses of the two atoms be Wi, nii, and
let nil be at a distance TI from the center of gravity, ra2 at a distance r2.
Then we have

Using Eq. (2.2), we find at once

I = - r—*, * = - -p — e
mi + mo mi + m2

But we have 7 = Srar2 = m\r\ + //^l- Thus

= rf,

= ra2r2. (2.2)

„> 0\

re. (2.3)

t = rf, (2.4)

l * e e1 v y

where

miniz 111 /0 Kv

M = - - , or - = -- --- (2.5)

nil + nit » mi m2

That is, the moment of inertia is that of a mass /x (sometimes called the
reduced mass) at a distance re from the axis.

Having found the rotational energy levels, we wish first to find how
closely spaced they are, to see whether we can use the classical theory to
compute the specific heat. The thing we are really interested in is the
spacing of adjacent levels as compared with kT] if the spacing is small
compared with fc!F, the summation in the partition function can be

136 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

replaced by an integration and equipartition will hold. Let us consider
the two lowest states, for K = 1 and 0, and find the energy difference
between them. We have

- #<, = (1 X 2 - 0 X

= -A2-

47T2/'

(2.6)

We wish to compare this quantity with kT; it is more convenient to define
a quantity which we can call a characteristic temperature 6rot by the
equation

and then our condition for the applicability of the integration is T » 9rot.
We now give, in Table IX-2, values for the characteristic temperatures

TABLE IX-2. — CHARACTERISTIC TEMPERATURE FOR ROTATION, DIATOMIC MOLECULES

Orot,

Substance °abs.

H2 . 171

CH 41 4

NH. 44 1

OH . 55 0

HC1 30 5

NO 4.93

O2 4 17

N2 5 78

CO 5 53

C2 4 70

C12 . . .0 693
Br2 . 0 233

h 0 108

Li2 I 96

Xa2 0 447

K5 0 162

By EIJ (2 7), we have defined Orot by the relation that fcO,,,t equals the eneigy difference between
the two lowest rotational energy levels of the molecule The method of calculation from the value of
re in Table IX-1 is illustrated in the text.

for the same diatomic molecules listed in Table IX-1. These values are
calculated, using Eq. (2.7), from the masses of the atoms, known from
the atomic weights and Avogadro's number, and the values of re in Table
IX-1. Thus, for instance, for H2 we find

ft2 (6.61 X 10-27)2(6.03 X 1023)

X 10-8)2(1.379 X

171° abs.

SBC. 21 POLYATOMIC GASES 137

1.008

2
Here /* = — = 6 Q3 x 1023' and r° = °-75 X 10~* cm-

From Table IX-2, we see that the gases are divided distinctly into
three types. In the first place, hydrogen stands entirely by itself, on
account of its small mass. The characteristic temperature 0rot, having
the value 171° abs., is the only one at all comparable with room tempera-
ture. Next are the hydrides, with characteristic temperatures between
20 and 60° abs. Finally, the characteristic temperatures of all gases
not containing hydrogen lie below 6° abs. Now it is not easy to calculate
the specific heat of a rotating molecule in the quantum theory on account
of mathematical difficulties, but the result is qualitatively simple. The
specific heat rises from zero at low temperatures, comes to the classical
value at high temperatures, and the range of temperature in which it is
rising is in the neighborhood of the characteristic temperature 6rot which
we have tabulated in Table IX-2. We may then infer from Table IX-2
that for molecules not containing hydrogen, the rotational specific heat will
have attained its classical value at a very low temperature, so that we
are entirely justified in using the classical value in our calculations. As
an illustration, we give values computed for the specific heat of NO at low
temperatures. We remember that the translational part of (7/» is
f/2 = 4.97 caL, whereas if the rotational specific heat is added we have
%R = 6.96 cal. The specific heat is actually computed1 to be 4.97 at
0.5° abs., 5.12 at 1.0°, 6.91 at 5.0°, and 6.95 at 10°. Of course, NO at
atmospheric pressure liquefies at a higher temperature than this, but at
sufficiently reduced pressure the boiling point can be reduced as far as
desired, so that there is nothing impossible about having the vapor at a
temperature as low as desired.

The hydrides have a decidedly higher temperature range in which
the rotational specific heat is less than the classical value. And for hydro-
gen, the quantum theory value is appreciably less than the classical value
even at room temperature. Thus, at 92° abs., we have the value 5.28 for
CP; at 197°, 6.30; at 288°, 6.78. The specific heat of hydrogen presents
complications not occurring with any other substance. It turns out, for
reasons which are too complicated to go into here, that in the energy levels
of hydrogen and of other diatomic molecules made of two like atoms, we
can make a rather sharp separation between the energy levels

1 For these values, and much other data relating to thermal properties of gases, see
Landolt-Bornstein, "Physikalisch-chernische Tabellen," Dritter Erganzungsband,
Dritter Teil, pp. 2315-2364, Springer, 1936.

138 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

in which K is even, and those in which K is odd. As a matter of fact, if a
molecule is in a state with K even, for instance, almost no physical agency,
such as collisions with other molecules, seems to have any tendency to
transfer it to a state with K odd. It is almost as if the gas wore a mixture
of two gases, one with K even, the other with K odd. Actually names
have been given to these gases, the case of K even being called parahydro-
gen, that of K odd being called orthohydrogen. At high temperatures,
such as we ordinarily have, we have molecules of both types in thermal
equilibrium. At first sight we should expect that there would be about
equal numbers of molecules of both sorts, but an additionally complicating
feature concerning the a priori probabilities of the states results in there
being three times as many molecules of orthohydrogen as of parahydrogcn
at high temperatures. When specific heat measurements are made at
low temperatures, it has always boon the practice to start with hydrogen
at room temperature, and cool it down. On account of the slow rate of
conversion of one type of hydrogen into the other, the two types of hydro-
gen appear in the same ratio of three to one \\hen the low temperature
measurements are made. This is not the equilibrium distribution cor-
responding to low temperature. At the very lowest temperature, we
should expect all the molecules to be in the lowest possible state, that of
K — 0, a state of parahydrogen. Thus to compute the observed specific
heat, we must assume a mixture of ortho- and parahydrogen in the ratio
of three to one, find the specific heat of each separately, and add. When
this is clone, the result agrees with experiment. To get the true equilib-
rium mixture at low temperature, we must either wait a period of a
number of days, or employ certain catalysts, which speed up the trans-
formation from one form of hydrogen to the other.

3. The Partition Function for Rotation. — Though we shall not be
able to find the rotational specific heat on account of mathematical
difficulties, still it is worth while setting up the partition function for
rotation and showing the limiting value which it approaches at high
temperatures. To do this, we must sum exp ( — Elot/kT\ where Elot is
given in Eq. (2.1), for all values of K. There is one point, however, which
we have not yet considered. That is the fact that the energy levels are
what is called degenerate: each level really consists of several stationary
states and several cells in the phase space. The reason for this is what is
called space quantization. We merely describe it, without giving the
justification in terms of the quantum theory. It is natural that the
angular momentum, Kh/2ir, of the rotating molecule can be oriented in
different directions in space. As a matter of fact, it turns out that in
quantum theory there are just (2K + 1) allowed orientations, each
corresponding to a different stationary state and a different cell. One
simple way of describing these orientations is in terms of a vector model,

SEC. 3]

POLYATOMIC GASES

139

as shown in Fig. IX-2. Here we have a vector of length Kh/2ir. Then
it can be shown that the projection of this vector along a fixed direction
is allowed to have just the values Mh/2ir, where M is an integer, cor-
responding to the various orientations shown in the figure. Obviously,
the maximum value of M is K, coming when the angular momentum is
oriented along the fixed direction, and the minimum is — M when it is
opposite. But there are just (2K + I) integers in the, group K, K — 1,
K — 2, • • • —(K — 1), — K, justifying us in our statement that there
are (2K + 1) allowed orientations. One says thnt the
state is (2K + l)-fold degenerate.

Considering this degeneracy, we see that the term in
the partition function corresponding to a given K must
really be counted (2K + 1) times, since all these stationary
states, corresponding merely to different orientations in
space, obviously have the same energy. Thus, we have

(3.1)

*'u;. IX-2.—

where Zrot is the factor in the partition function of a

single molecule, Z, of Eq. (3.3) of Chap. VIII, coming from

rotation. It is this summation which unfortunately can- quantization

not be evaluated analytically. But we can handle it in

the limit of high temperature, for then the terms corre-

spending to successive K's will differ so little that the

summation can be replaced by an integration. We then have

2
1, o, -i, -2,

""*•"•

/•oo -(+l\h'

lim Zrot = I (2K + })c~**'ikT- <1K.
•/o

(3.2)

T »0rot

The bulk of the integral in Eq. (3.2) will come from high quantum num-
bers or high values of K. For these, we can neglect unity compared to
K, obtaining

lim

T »0rot

(3.3)
(3.4)

using the integrals (2.3) of Chap. IV.

From the expression (3.4), we can in the first place find the rotational
heat capacity, using Eq. (5.21) of Chap. III. This may be written

140 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

where we must multiply by NQ because our quantity Zrot refers to a single
molecule. Thus we have, substituting Eq. (3.4) in Eq. (3.5),

Crot = Nok, (3.6)

in accordance with equipartition. It is also interesting to compute tho
contribution of tho rotation to the entropy, as given in Eq. (3.8) of Chap.
VIII. From that equation, the contribution is

*R*T(T In /„,) = wft(ln ~- + 1 + In T\ (3.7)

Thus, using Eq. (3.8) of Chap. VIII, the entropy in the temperature range
where the rotation can be treated classically, but where the vibration is
not excited enough to contribute appreciably to the entropy, is

S = %nR In T + nR In V + nR[i' + 1 - In (n/Z)], (3.8)

where

., (2rm)*'k''* . tar2/ A' ,Q ,

i' = In v~ — j/8 h In - jp - (3.9)

The quantity i1 of Eq. (3.9) can be considered as tho chemical constant
of a diaTf"Tni'p f'"*r *" fionnontion with the formula (3.8) tor the entropy.
We must remember, however, that Eq. (3.8) holds only in a restricted
temperature range, as stated above; with some gases, the vibration begins
to contribute to the entropy even at room temperature, as we shall see
in the next section. It is sometimes useful to have the formula for Gibbs
free energy of a diatomic gas in the range where Kq. (3.8) is correct. This
is easily found to be

(f = n(U» - iRT In T + RT in P - RTi'), (3.10)

where i' is given in Eq. (3.9).

4. The Vibration of Diatomic Molecules. — In addition to their
rotation, we have seen that diatomic molecules can vibrate with simple
harmonic motion if the amplitude is small enough. We shall use only
this approximation of small amplitude, and our first stop will be to calcu-
late the frequency of vibration. To do this, we must first find the linear
restoring force when the interatomic distance is displaced slightly from its
equilibrium value rr. We can get this from Eq. (1.2) by expanding the
force in Taylor's series in (r — rt). We have

Force = 2aD[l - 2a(r - rf) • • • - I -f a(r - r.) • • • ]

= -2a*D(r - r.), (4.1)

neglecting higher terms. Now we can find the equations of motion for
the two particles of mass m\ and m^ at distances r\ and r^ from the center

SBC. 4] POLYATOMIC GASES 141

of gravity, where TI + r2 = r, under the action of the force (4.1). These
are

r2 -

~ i r2 - rc). (4.2)

We divide the first of these equations by MI, the second by W2, and add,
obtaining

M~ = -2a'D(r - r.), (4,3)

where /x is given by Eq. (2.5). The vibration, then, is like that of a
particle of mass /x, with a force constant — 2a2D. By elementary mechan-
ics, we know that a particle of mass M, acted on by a linear restoring force
— kx, vibrates with a frequency

1 /* (AA\

v = 2^\t (4<4)

Thus the frequency of oscillation of the diatomic molecule is

(4.5)

We have found in Eq. (3.8), Chap. Ill, that the energy levels of an
oscillator of frequency i>, in the quantum theory, are given by

fe\,b = (v +%)hv, (4.6)

where v is an integer (called n in Chap. III). The spacing of successive*
levels is hv. We may then expect, as with the case of rotation, that for
temperatures T for which hv/kT is small, or temperatures large compared
with a characteristic temperature

the classical theory of specific heats, based on the use of the integration
to find the partition function, is applicable, while for temperatures small
compared with 6vlb we must use the quantum theory. To investigate
this, we give in Table IX-3 the characteristic vibrational temperatures
of the molecules we have been considering. The values of Table IX-3

142 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

can be found from D and a as tabulated in Table IX-1. Thus for H2 we
have

6.61 X_10^ /2(1.95 X 1Q8)2(103)(4.185 X 1010)
*" ar(1.379 X 10-") / 1.008

V ~2~

= 6140° abs. (4.8)

We see from Tal?le IX-3 tlmt fnr prn.p.f.inp.l]y all thft molecules the charac-
i« Inrpro pnmpfljpd to room tempcratyrn,. so thftl; at all

ordinary tttnippfatdip^ wo ™n^ 11gr> *-hf> quantum theory of specific heat.
We also note that in every case the characteristic temperature for vibra-

TABLE IX-3. — CHARACTERISTIC TEMPERATURE FOR VIBRATION, DIATOMIC MOLECULES

Substnnoo ^V1b, ° a^>s.

H2 6140

CH . . 4100

NH. . . . 4400

OH .. 5360

HC1 4300

NO . . 2740

O2 . . 2260

N2 . . 3380

CO . . 3120

C2 .. 2370

C12 ... 810

Br2 470

I2 . . 310

Li2 . . 500

Na, 230

K2 .... 140

These values are calculated as in Eq. (4.8).

is:i£firv large compare^] lift i^n-*' nf rnt itinn^ That is. the rotational

"^ ^"fh m^rfi filoafilv spn.p.P<l tlin.n f.lio

This is a characteristic feature of molecular energy levels, which is of
great importance in the study of band spectra, the* spectra of molecules.
5. The Partition Function for Vibration. —First, wo shall calculate
the partition function and specific heat of our vibrating molecule by classi-
cal theory, though we know that this is not correct for ordinary tempera-
tures. Using the expression (4.1) for the force, we have the potential
energy given by

- r.)», (5.1)

and the kinetic energy is

SBC. 5] POLYATOMIC GASES 143

where pr is the momentum associated with r, equal to /* dr/dt. Then, by
analogy with Eq. (5.22) of Chap. Ill, the vibrational partition function
Zvib can be computed classically as an integral,

1 r °° <'*D(r - r,)« /» *

z- = /J '~"~*r "*J_«

In the integral over r, we can approximately replace by an integral from
— oc to oo , for the exponential in the integrand is practically zero for
negative values of r. If we do this, we have

kT f .

= TV' (5'4)

The use of an equation analogous to Eq. (3.5) gives the value R for the
vibrational contribution to the specific heat, as mentioned at the begin-
ning of Sec. 2.

Next we calculate the specific heat in the quantum theory. The
partition function is

= Ye \ ^2jkT

_JLL/ _*i _?*r \

= e a"\i + ^ *r + ^ ^ + • • • ;

a ~ V&T

(5.5)

using the formula for the sum of a geometric series,

1 + x + z2 +.--=- - i- - (5.6)

We note that at high temperatures, the numerator of Eq. (5.5) can be set

equal to unity, the denominator becomes I 1 — f 1 — TTTJ • • • ) = r™'

L \ Kl /J Ki

so that the partition function reduces to kT/hv, in agreement with the
classical value (5.4). Using Eqs. (5.5) and (3.5), we have for the vibra-
tional specific heat per mole the value

144

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. IX

This result was first obtained by Einstein and is often called an Einstein
function. Introducing the characteristic temperature from Eq. (4.7), we
have

Ovlh

Cr
vib ~~~

v (;¥-_-,

(5.8)

It is also interesting to find the internal energy associated with the vibra-
tion; proceeding as in Eq. (5.20) of
Chap. Ill, we see this at once to be

Nhv

~T~

(5.9)

JOL
ht;

The average energy and heat capacity
per oscillator, from Eqs. (5.9) and (5.8),
are plotted as functions of temperature
in Fig. IX-3. It will be seen that the
energy is ^hv at the absolute zero and
increases from this value quite slowly.
The slow rise, with horizontal tangent
of the energy curve at- the absolute zero,
is what leads to the vanishing specific
hczit at the absolute zero. At higher
temperatures, however, the energy
approaches the classical equipartition
value kT, and the heat capacity ap-
proaches the classical value k.

As an example of the application of
FIG. ix-;*. Average energy and Eq. (5.8), we compute the vibrational

heat capacity of an oscillator, according spocific heat for CO, ill Table IX-4,
to the quantum theory. * '

then find the total specific heat by

adding the vibrational heat to %R = 6.96 cal., which is the sum of
f R for translation, R for rotation, and compare with the correct value.
The agreement between our calculations and the "correct" values of
CP in Table IX-4 is good but not perfect. More accurate calculation
agrees practically perfectly with experiment; in fact, calculation is in
general a more accurate method than the best experiments for finding the
specific heat of a gas, and the "correct" values of Table IX-4 are really
simply the results of more exact and careful calculation than we have
made. It is worth while discussing the errors in our calculation. In the
first place, the frequency v which we have found from the constants of
the Morse curve is correct, for as a matter of fact the constants a in
Table IX-1 were computed from the frequencies and values of D observed

SEC. 6] POLYATOMIC GASES 145

from band spectra, using Eq. (4.5) solved for a. But the Einstein specific
heat formula (5.7) is not exactly correct in this case, for the actual inter-
atomic potential is not simply a linear restoring force, as we have assumed
when we use the theory of the linear oscillator. Not only that, but as wo
have mentioned there is interaction between the vibration and the rota-
tion of the molecule. These effects make a small correction, which can
be calculated and which accounts for part of the discrepancy between the
last two columns in Table IX-4. They do not account, however, for the

TABLE IX-4.-- COMPUTED SPECIFIC HEVT OF CO

r« o , i Vib rational ; Total specific, Cp

J i specific heat , heat Cp correct

500

i
0.18

7 14

7 12

1000

0.94

7 90

7 94

2000

1 63

8 59

8 67

3000

1.81

8.77

8 90

4000

1 81)

8 8T>

9 02

5000

1 92

8 88

9 10

Cp is given in calorics per mole. Tho calculation is made by Eq. (5 8). Vtiluos tabulated in last
column, "Cp correct," are from Landolt-Boinstem, " Physikahach-chemische Tabellen," Drifter
Erganzungsband, Dnttor Teil, p. 2324, Springer, 1936.

fact that the correct specific heat rises above the value %R — 8.94 cal.,
for the quantum vibrational specific heat never rises above the classical
value. This effect comes in on account of a ne\v feature, electronic
excitation, which enters only at very high temperature. We can explain
itrbnefiy by stating that electrons, as well as linear oscillators, can exist
in various stationary states, as_a result of which they contribute to the
specific heat. Their specific heat curves are somewhat similar to an Ein-
stem curve^ but with extremely high p.hflTRo.t^rist.if. tempfimfrirraTso that
even at 5000° they are at the very low part of the curve and contribute
onlv slightly to the specific heat. When these small contributions are
computed and added to the values found from rotation and vibration, the
final results agree very accurately with observation.

6. The Specific Heats of Polyatomic Gases. — We shall now discuss
the specific heats of polyatomic gases, without going into nearly the detail
we have used for diatomic molecules. In the first place, the rotational
kinetic enerpry is different, from thai, in Hi«.toinin molecules. It requires
three, rather than two, coordinates to describe the orientation of a. poly-
atomic molecule in space. Thus, imagine an axis rigidly fastened to the
molecule. Two coordinates, say latitude and longitude angles, are
enough to fix the orientation of this axis in space. But the molecule can
still rotate about this axis, and an additional angle must be specified to
determine the amount which it has rotated about the axis. These three

146 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. IX

coordinates all have their momenta and their terms in the kinetic energy.
And when we find the mean kinetic energy of rotation, the new variable
contributes its $kT, according to equipartition. ThiiH ||»A ^r^y, t,rQr>«-
lational and rotational, amounts to 3kT per mnWuluffir 3nRT for n moles,
and the transnational and rotational heat capacity is CV = 3nB = 5.96
cal. per degree per mole, C/> = 4nR = 7.95 cal. per degree per mole. In
addition to translational and rotational energy, the polyatomic molecules
like the diatomic ones can have vibrational energy. As a matter of fact,
they can havo considerably more vibrational energy than a diatomic
molecule, for they havo more vibrational degrees of freedom. A diatomic
molecule has only one mode of vibration, but a triatomic molecule; has
throe. Thus the water molecule can vibrate in the three ways indicated

(a) 5170° Abs (b) 5400° Abs (c) 2290° Abs

FIG. IX-4.- Modes ol vibration of the 1LO molnculc Tho arro\vs indicate the direc-
tion of vibration ot each atom, for the normal mode whose characteristic temperature IP
indicated. For similar information on a variety of molecules, see 11. Sponer, " Molekulspek-
tren und ihre Anwfndungen aiif chomisrhe Problems," Springer, Berlin, 1935.

by the arrows of Fig. IX-4. The arrowy show the directions of displace-
ment of the throe atoms in the vibration. In general, to find the number
of vibrational degrees of freedom, it can be shown that one takes all the
degrees of freedom of the atoms of the molecule, regarded as free. This is
3N, if there are N atoms, each having three rectangular coordinates.
Then one subtracts from this total the number of other degrees of free-
dom: the translational degrees of freedom of the molecule as a whole,
three, and the rotational degrees of freedom (none when N = 1, two when
N = 2, three when N ^ 3). Thus the number of vibratiomil degrees of
freedom is

3(1) —3 — 0=0 for a monatomie gas,

3(2) — 3 — 2 = 1 for a diatomic gas,

3(3) — 3 — 3 = 3 for a triatomic gas, and in general

3N - Gfortf ^ 3. (6.1)

Each of the vibrational degrees of freedom given by Eq. (6.1) would
have a mean kinetic and potential energy of kT, according to equiparti-
tion, and would contribute an amount R to the specific heat. As with
diatomic molecules, however, the quantum theory tells us, and we find

SEC. 6]

POLYATOMIC GASES

147

experimentally, that the vibrational specific heat is practically zero at
low temperatures. We give a single example, the specific heat of water
vapor, which will show what actually happens. Since this is a triatomic
molecule, the specific heat should be Cp = ±nR = 7.95 cal. per degree per
mole for translation and rotation, plus three Einstein terms, as given by
Eq. (5.8), for characteristic temperatures which are 5170° abs., 5400° abs.,
and 2290° abs., for the modes of vibration (a), (b), mid (e) respectively
in Fig. IX-4.1 Calculations are given in Table IX-5, where the columns

TABLE IX-5. — COMPUTED SPECIFIC HEAT OF WATEK VAI-OR

T, ° abs.

(«)

(c)

: CP,

calculated

0-,
correct

300

0 00

0 10

8 05

8 (X)

400

0 00

0 25

8 20

8 16

500

0 00

0.40

8 35

8 38

600

0 02

0 66

8 63

8 64

800

0 16

0 15

1 05

9 31

9 20

1000

0 31

0 30

1 30

9.86

9 80

1500

0 80

0 78

1 65

11 18

11 15

2000

1 18

1 15

1 79

12 07

12 09

3000

1 58

1 52

[ 90

12 90

13 10

See fomments under Table IX-4

headed (a), (b), (c) are the vibrational heat capacities for the three modes
of vibration, the next column gives the calculated CP, and the last one the
correct (V, found from more accurate calculation and agreeing well with
experiment. As with CO, the slight discrepancies remaining between our
calculation and the correct values can be removed by more elaborate
methods, including the interaction between vibration and rotation, and
electronic excitation.

The calculation we have just made is based on the assumption that
the vibrations of the molecule are simple harmonic, the force being prft-
portional to the displacement and the potential energy to the square of
the displacement. Ordinarily this is a fairly good approximation for the
amplitudes of vibration met at ordinary temperatures, but there are some
important cases where this is not true. An example is found in the
so-called phenomenon of hindered rotation. There are some molecules, of
which ethane CHa-CHa, shown in Fig. IX-5, is an example, in which one
part of the molecule is almost free to rotate with respect to another part.
Thus, in this case, one CHs group can rotate with respect to the other
about the line joining the carbons as an axis. The rotation would be

1 See Sponer, " Molekiilspektren. und ihre Anwendungen auf chcmische Probleme,"
Vol. I, Springpr, 1935, for vibrational frequencies of this and other molecules.

148 INTRODUCTION TO CHEMICAL I'HYSICti [CHAP. IX

perfectly free if the potential energy were independent of the angle of
rotation 0, so that there were no torques. Then as far as this degree of
freedom was concerned, there would be no potential energy, so that the
mean energy on the classical theory would be ?kT, the kinetic energy,
rather than kT, the sum of the kinetic and potential energies, as in an
oscillator. Actually in such cases, however, there are slight torques, with
a periodicity in the case of ethane of 120° in 6. These arise presumably
from repulsions between the hydrogen atoms of the two methyl groups,
suggesting that the potential energy might have a maximum value when
the hydrogens in the two groups were opposite each other, and a minimum
when the hydrogens of one group were opposite the spaces between hydro-
gens in the other. In such a case, for small energies, the motion would

Fio. IX-5 Tho ethane molecule, OII3-CH.,

be an oscillation about one of the minima of the potential energy curve,
while for larger energies, greater than the maximum of the potential
energy curve, the motion would be a rotation, but not with uniform angu-
lar velocity. In such a case, in the classical theory, the mean kinetic
energy would equal \kT in any case. The mean potential energy, how-
ever, would increase as \kT for low temperatures, where the motion was
oscillatory, but would approach a limiting value, equal to the mean
potential energy over all angles, which it would practically reach at the
temperatures at which most of the molecules were rotating rather than
oscillating. Thus the heat capacity per molecule connected with this
degree of freedom would be k at low temperatures, but would fall to
?k at higher temperatures where the rotation became more nearly free.
With the quantum theory, of course the heat capacity would resemble
that of an oscillator at low temperatures, starting from zero at the abso-
lute zero, then rising to the neighborhood of &, but falling to $k at high
temperatures as in the classical case. Measurements and calculations of
specific heat and entropy of molecules which might be expected to show
free rotation of one part with respect to another generally seem to indicate

SEC. 6] POLYATOMIC GASES 149

that at ordinary temperatures the rotations are really hindered by periodic
torques in this way, the heat capacity being more like that of an oscillator
than that of a rotator. It is clear that from measurements of the specific
heat one can work backward and find useful information about the magni-
tude of the torques hindering the rotations, and hence about the inter-
atomic forces.

CHAPTER X

CHEMICAL EQUILIBRIUM IN GASES

In Chap. VIII we treated mixtures of gases in which the concentra-
tions were determined. Now we take up chemical equilibrium, or the
problem of mixtures of gases which can react with each other, so that the
main problem is to determine the concentrations of the various gases in
equilibrium. In this problem, as in all cases of chemical reactions, there
are two types of question that we may ask. In the first place, there is the
rate of reaction. Given two gases capable of reacting and mixed together,
how fast will the reaction occur and how will this rate, depend on pressure
and temperature? In the second place, there is the question of equilib-
rium. To every reaction there is a reverse reaction, so that the final
state of equilibrium will represent a balance between the direct and the
reverse reactions, with definite proportions of all the substances in the
equilibrium mixture. We may wish to know what these proportions are.
The first type of problem, the rate of reaction, can be answered only by
kinetic methods. Gas reactions take, place only when the reacting mole-
cules are in collision with each other, and only when the colliding mole-
cules happen to have a good deal more than the average energy. Thus
to find the rate of reaction we must investigate collisions in detail and
must know a great deal about the exact properties of the molecules. In
almost no case do we know enough to calculate a rate of reaction directly
from theory. We can, however, find how the rate of reaction depends
on the concentrations of the various substances present in the gas, and
even this small amount of information is useful. It allows us to use the
kinetic method to find the concentration of substances in equilibrium, for
we can simply apply the condition that the concentrations are such that
they do not change with time, and this gives us equations leading to the
so-called mass action law. The results we find in this way, however, are
incomplete. They do not tell us how the equilibrium changes with tem-
perature, a very important part of the problem. Fortunately, these*
questions of equilibrium can be answered completely by the method of
thermodynamics and statistical mechanics. For in equilibrium, the
Gibbs free energy of the mixed gas must have the minimum value possible,
and this condition leads not merely to the mass action law but to complete
information about the variation of the equilibrium with temperature. As
usual, thermodynamics gives us more complete and satisfactory informa-

150

SBC. 1] CHEMICAL EQUILIBRIUM IN GASES 151

tion, but about a more restricted problem, that of thermal equilibrium.
In our discussion to follow, we shall start with the kinetic method, speak-
ing about the mechanism of gas reactions and carrying the method as
far as we can. Then we shall take up the thermodynamic treatment,
deriving the conditions of equilibrium, and finding the interesting fact
that the chemical constants of gases, introduced previously in connection
with the entropy, are fundamental in the study of chemical reactions.
1. Rates of Reaction and the Mass Action Law. — Let us write a simple
chemical equation; for instance,

2H2 + 02^2H20, (1.1)

describing the combination of hydrogen and oxygen to form water, and
the reverse, the dissociation of water into hydrogen and oxygen. The
equation expresses the fact that when two molecules of H2 and one of ()2
disappear, two of H2O appeal-; or vice versa. Now lot us form the sim-
plest kinetic picture of the reaction that we can. For the combination of
two hydrogens and an oxygen to form two water molecules, we suppose in
the first place that a triple collision of the two hydrogens and the one
oxygen molecule is necessary; we suppose further that in a certain fraction
of such collisions, a fraction which may depend on the temperature, the
three molecules react. Thus the number of sets of molecules reacting
per unit time will be proportional to the number of triple collisions per
unit time. This number of collisions in turn will be proportional to the
number of oxygen molecules per unit volume and to the square of
the number of hydrogens per unit volume. For plainly if we double the
number of oxygens, we double the chance that one will be found at tho
point where the collision will take place; while if we double the number
of hydrogens, we double the chance that one hydrogen will be found at
the location of the collision, and furthermore we double the chance that,
if one is there, another will also be found on hand. Since, at a given
temperature, the number of molecules per unit volume is proportional to
the pressure, we find for the number of sets of molecules that react per
second

CCDPnVV (1.2)

Here (K^m^a^CQefficient depending on the size of the molecules,, their
velocities, the probability that if they collide they will react, etc. The
quantities PU2 and Pot are the partial pressures of H2 and 02; that is,
they are the pressures which these gases would exert by themselves, if
their molecules only were occupying the volume. It is the evaluation of
C(T) as a function of temperature which, as we have previously suggested,
is almost prohibitively difficult by purely theoretical methods.

152 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

At the same time that direct reactions are taking place, there will be
reverse reactions, dissociations of water molecules to produce hydrogens
and oxygons. From the chemical equation (1.1) we see that two water
molecules must be present in ordor to furnish the necessary atoms to break
up into hydrogen and oxygen molecules. Thus, by the type of argument
we have1 just used, the rate of the4 reverse reaction must be proportional to
the square of the number of water molecules per unit volume or to the
square of the partial pressure of water; we may write it as

C'(T)P?M. (1.3)

Suppose we start with only hydrogen and oxygen in the container, with
no water vapor. Reactions will occur at a rate given by Kq. (1.2),
producing water. As this happens, the oxygen and hydrogen will be
gradually used up, so that their partial pressures will decrease and the
number of molecules reacting per unit time will diminish. At the same
time, molecules of water will appear, so that the partial pressure of water
will build up and with it the number of dissociation reactions given by
Kq. (1.3), in which water dissociates into hydrogen and oxygen. This
will tend to diminish the amount of water and increase the amount of
hydrogen and oxygen, until finally an equilibrium will occur, with sta-
tionary amounts of the various gases, though individual molecules an;
reacting, changing from water vapor to oxygen and hydrogen and back
again with great rapidity, l,p equilibrium, frhp ^ymhnr ^f reactions of
type (1.2) must just equal the number of type (1.3) per unit time. Thus
we must have

where Kp(T) is a function of temperature, the subscript P indicating the
fact that Eq. (1.4) is stated in terms of partial pressures (we shall pres-
ently state it in a slightly different way). Eq. (1.4) expresses the law of
mass fie ti on for tKB-Dnrtip.uliir reaction in Question.

From Eq. (1.4) we can derive information about the effect of adding
hydrogen or oxygen on the equilibrium. Thus suppose at a given tem-
perature there is a certain amount of water vapor in equilibrium with a
certain amount of hydrogen and oxygen. Now we add more hydrogen
and ask what happens. In spite of adding hydrogen, the left side of
Eq. (1.4) must stay constant. If the hydrogen did not combine with
oxygen to form water, JPH2 would increase, the other P's would stay con-
stant and the expression (1.4) would increase. The only way to prevent
this is for some of the added hydrogen to combine with some of the

/ / P&Po* C'(T)

' - ^

SEC. 1] CHEMICAL EQUILIBRIUM IN GASES 153

oxygen already present to form some additional water. This will decrease
both terms in the numerator of Eq. (1.4), increase the denominator, and
so bring back the expression to its original value. Information of this
type, then, can be found directly from our kinetic derivation of the mass
action law. But we should know a groat doal more if we could calculate
KP(T), for thru we could find the actual amount of dissociation and its
variation with pressure and temperature.

It is easy to formulate the mass action law in the general case, by
analogy with what we have done for our illustrative reaction. In the
first place, let us write our chemical equations in i\ standard form.
Instead of Eq. (1.1), we write

2H2 + 02 - 2H2O = 0. (1.5)

We understand Eq. (1.5) to mean that two molecules (or moles) of hydro-
gen, one of oxygen, appear in the reaction, while two molecules (or moles)
of water disappear. The reverse reaction, according to this convention,
would be written with opposite sign. We write our general chemical
equation by analogy with Eq. (1.5), each symbol having an integral
coefficient v, giving the number of molecules (or moles) of the correspond-
ing substance appearing in the reaction, negative r's corresponding to
the disappearance of a substance. Let there be a number of substances,
denoted by 1, 2, . . . (as 112, O2, H2O in the example), with correspond-
ing v\, j>2, . . . , and partial pressures PI, P2, . . . Then it is clear by
analogy with our example that the general mass action law can be stated

7V'/Y« ' ' ' = Kp(T). (1.6)

Here the terms with negative v'$ automatically appear in the denomina-
tor, as they should from Eq. (1.4).

It is often convenient to restate Eq. (1.6), not in terms of partial
pressures, but in terms of the number of moles of each substance present,
or in terms of fractional concentrations. Thus let there be HI moles
of the first substance, ?i2 of the second, etc. Then we have

by the perfect gas law, where V is the volume occupied by the mixture of
gases. Substituting in Eq. (1.6), we have

(1-8)

EQ nation (1.8.) is con^ciiiQiilJJiiLfinding the effect iif_a_£iiaiige of volume
on the equilibrium. For example, in our case of water, from Eq. (1.5),
v\ + *>2 + • • • = )>H2 + ^02 + ''Hao = 2+1—2=1. Thus we have

154 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

aftQa _ py

2 ~~~ '

a quantity proportional to the volume. Now let the volume be changed
at constant temperature. If the volume increases, the numerator must
increase, showing that there must be dissociation of water into hydrogen
and oxygen. On the other hand, decrease of the volume produces recom-
bination. This is a special case of the general rule which is seen to follow
from Eq. (1.8) : decrease of volume makes the reaction run in the direction
to reduce the total number of moles of gas of all sorts. In our special
case, if two moles of hydrogen and one of oxygen combine to give two
moles of water vapor, there is one mole of gas less after the process than
before. It seems reasonable that decrease of volume should force the
equilibrium in this direction.

It is also useful to write the mass action law in terms of the relative
concentrations of the gases. From Eq. (1.6), using Eq. (2.4) of Chap.
VIII, or Pl = tftP, where c, is the relative concentration of the M\ gas, we
have

'i'1*'1 ' ' ' = /£^T} = K(l\ T). (1.10)

Equation (1.10) is convenient for finding the effect of pressure on the
equilibrium, as Eq. (1.8) was for finding the effect of volume. Thus in
the case of the dissociation of water vapor, we have

showing that increasing pressure increases the concentration of water
vapor. Of course, this is only a different form of stating the result (1.9),
but is generally more useful.

2. The Equilibrium Constant, and Van't Hoff's Equation. — In the
preceding section we derived the mass action law, but have not evaluated
the equilibrium constant KP(T) or K(P, T). Now WTC shall carry out
our thermodynamic discussion, leading to a derivation of this constant.
The method is clear: we remember from Chap. II, EW 3

free energy is a minimum for a system at constant pressure and tempera-
tureT Then we find the Gibbs free energy G of the mixture of gases, and
vfljv fhfTpnnnpnf rations to rrmke it a minimum. From Eq. (2.15) of
Chap. VIII, we have

(J - w/r'/ + KTn, In c,. (2.1)

For equilibrium, we must find the change of G when the numbers of moles
of the various substances change, and set this~shange equat'To "zero.

SEC. 21 CHEMICAL EQUILIBRIUM IN GASES 155

Using Eq. (2.1), we have

dG = 2J(G, + RT In a)dn, + Jn, d(G] + RT In c,) = 0. (2.2)
y ;

The second sum is zero. In the first place, the Or/s do not depend on the
n/s, so that they do not change when the n/s are varied. For the con-
concentrations, we have d In Cj = dcjcj. Hence the last term becomes

5jfir(n//c,)<fc,. But by Eq. (2.1), Chap. VIII, n,/c, = ni + n, + • • • ,

independent of j, so that the summation is really

Furthermore, by Eq. (2.6), Chap. VIII, Sc/ = 1, so that

dSc/ = 2 dc; = 0,
being the change in a constant. Hence, we have finally

dG = (G, + fiZ1 In c,)dn/ = 0 (2.3)

as the condition of equilibrium. But from the chemical equation wo
know that the piimi^ej:jiLmQle^iii[es yf thq ^tJrJzyiKi_aiiDcai'iii^ in an actual
reaction must be proportional to vt* the coefficient appearing in tho
chemical equation. Hejp.fi the dn.'* ipnst ho nroportional to tho v/s, and
we may rewrite Eq. (2.3) as

(0, + RT In c,) = 0. (2.4)

Taking the exponential and putting all terms involving the c's on the left,
the others on the right, we have

= K(P, T), whew

(2.5)

\ J

In Eq. (2.5) weJiave found the saine^na^a_actiQrL-IawLJia in Ea. (1.1UL
but^^ilJi^^complete evaluatioiLJiLJJie equilibrium constant K. Using
Eq. (2.16), Chap. VIII, for G,, we verify at once that K(P, T) varies with
P as in Eq. (1.10), and we find

In Kf(T) - - > - In T - T ^ dr - ^. (2.6)

156 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

In Eq. (2.6), U} is the arbitrary additive constant jciying the energy of
the /th gas per mole at the absolute zero, C, is the heat capacity per mole
of the jth gas coming from rotations and vibrations, and ?'; is tho chemical
constant of ihc j't

There is an important relation connecting the change of either K
or KP with temperature and a quantity called the heat of reaction. By
definition, the heat of reaction is the heat absorbed. ™» tihp ""*™*QS^ of
enthalpy A/7, when the reaction proceeds reversibly so that v\ moles of
the_first type of molecule are produced, v* of tho second, etc.T at constant
pressure? and temperature. From Eqs. (2.13), (2.14) of Chap. VIII, this
is at once seen to be

c, dT).

(2.7)

Now let us find the change of In K(P, T) or In Kr(T) with temperature.
Differentiating Eq. (2.6), we have

/a
\

_
W~ ~ " i VRT*

Equation (2.8) is called Van't Hoff's equation and is a very imnortant
one in physical chemistry. It can be shown at once that the same equa-
tion holds for K(P, T).

Van't Hoff's equation can be used in cither of two ways. First, we
may know the heat of reaction, from thermal measurements, and we may
then use that to find the slope of the curve giving K(P, T) against tem-
perature. Let us see which way this predicts that the equilibrium should
be displaced by increasing temperature. Suppose that heat is absorbed
in the chemical reaction, so that A// is positive. Then the constant
K (P, T) will increase with temperature. That means that at high
temperatures more of the material is in the form that requires heat to
produce it. For instance, to dissociate water vapor into hydrogen and
oxygen requires heat. Therefore increase of temperature increases the
amount of dissociation. In the second place, we may use Van't HofFs
equation to find the heat of reaction, if the change of equilibrium constant
with temperature is known. This, as a matter of fact, is one of the com-
monest ways of measuring heats of reaction in physical chemistry.

The heat of reaction at the absolute zero, from Eq. (2.7), is

E/,. (2.9)

SBC. 2] CHEMICAL EQUILIBRIUM IN GASES 157

It is interesting to see that this can be calculated from the quantities D
of Sec. 1, Chap. IX. From Table IX- 1 we know the value of Z), the
energy required to dissociate various diatomic molecules, and similar
values can be given for polyatomic molecules. Thus lot us consider our
case of the dissociation of water vapor. To remove one hydrogen atom
from an H^O molecule requires 118 kg.-cal. per mole (not given in the
table), and to remove the second hydrogen from the remaining OH
molecule requires 102, a total of 220 kg.-cal. In our reaction, there are
two 1120 molecules, requiring 440 kg.-cal. to dissociate them into atoms.
That is, 440 kg.-cal. are absorbed in this process. But now imagine the
four resulting hydrogen atoms to combine to form t\\o Ha molecules and
the two oxygens to combine to form ( )2. Each pair of hydrogens liberates
103 kg.-cal. in recombining, a total of 200 kg.-cal.. and the two OXV^MIS
liberate 117 kg.-cal., so that 200 + 1 17 = 323 kg.-cal. are liberated in this
part of the process. The net result is an absorption, of 440 — 323 == 117
kg.-cal., so that A//0 is 117 kg.-cal. This is in fairly good agreement
with the experimental value of about 113 kg.-cal. It is interesting to
notice that the final result is the difference of two fairly large1 quantities,
so that relatively small errors in the 7)'s can result in a rather large error
in the heat of reaction.

The calculation which we have just made for A//0 does not follow
exactly the pattern of Eq. (2.9). To see just how that equation is to be
interpreted, we must give values to the various ^/s. In general, since
there is an undetermined constant in any potential energy, we can assign
the f//s at will. But there is a single relation between them, on account
of the possibility of formation of water from hydrogen and oxygen. Let
Uut be the energy per mole of hydrogen at. tin* absolute zero, f/o_. of
oxygen, UK& of water, all of course in the vapor state1. Then from the,
last paragraph we know that the energy of two moles of hydrogen, plus
that of one mole of oxygen, is 117 kg.-cal. greater than the energy of two
moles of water vapor. That is,

2Um + l;o> = 2tV,o + 117 kg.-cal. (2.10)

Statements like Eq. (2.10) are sometimes written in combination with
the chemical equation, in a form like

2H2 + 02 = 2H2O + 117 kg.-cal. (2.11)

Aside from Eq. (2.10), the U,'s can be chosen freely; that is, any two of
them can be chosen at will and then the third is determined. Now let
us compute A//0, using Eqs. (2.9) and (2.10). It is

A//o = 217,1, + C7o2 - 2C7Hlo

= 2C7H20 + 117 kg.-cal. - 2Ulli0

= 117 kg.-cal., (2.12)

158 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

in agreement with our previous value. From this example it is clear
that all the undetermined constants among the t/o's cancel from the sum
in Eq. (2.9), leaving a uniquely determined value of A//0.

We have just seen that a knowledge of the heats of dissociation, or
constants D, of the various molecules concerned in a reaction allows us to
find A//o, the heat of reaction at the absolute zero. A further knowledge
of the specific heats of the molecules gives us all the information we need
to find the equilibrium constant KP(T), according to Kq. (2.6), except for
the final constant Siv';. In other words, this knowledge is enough to find
the rate of change of In K /•('/') with temperature, according to Eq. (2.8),
but not enough to determine the constant of integration of the integrated
equation (2.6). But we have seen in Chap. VIII how to find the con-
stants i theoretically, find later we shall see how to find them experimen-
tally from vapor pressure measurements. We now see why these
constants are so important and why they are railed chemical constants:
they determine the constants of integration for problems of chemical
equilibrium. For this reason, a great deal of attention has gone to finding
accurate values for them.

3. Energies of Activation and the Kinetics of Reactions. A curious
fact may have struck the reader in connection with the example which
we have used, the equilibrium between water vapor and hydrogen and
oxygen. Calculating the equilibrium, we find that at room temperature
the amount of hydrogen and oxygen in equilibrium with water vapor is
entirely negligible; even at several thousand degrees only a few per cent
of the water vapor is dissociated. This certainly accords with our usual
experience with steam, which does not dissociate into hydrogen and
oxygen in steam engines. And yet if hydrogen and oxygen gases are
mixed together in a container at room temperature, they will remain
indefinitely without anything happening. A spark or other such dis-
turbance is required to ignite them; as a result of ignition, of course, a
violent explosion results, the hydrogen and oxygen being practically
instantaneously converted into water vapor. The heat of combustion,
which is the sume thing as the heat of reaction of 117 kg.-cal. which we
have just computed, is a very large one (one of the largest for any common
reaction); since an explosion is an adiabatic process, this heat cannot
escape, but will go into raising the temperature, and consequently the
pressure*, of the resulting water vapor enormously. It is this sudden
rise of pressure and temperature that constitute the explosion. But now
we ask, wrhy was the spark necessary? Why do not the hydrogen and
oxygen combine immediately when they are placed in contact?

Our first supposition might be that the triple collisions of two hydro-
gen molecules and one oxygen, which we have postulated as being neces-
sary for the reaction, were rare events. But this is not the case.

SEC. 3] CHEMICAL EQUILIBRIUM IN GASES 159

Calculation, taking into account the cross section of the molecules, shows
that at ordinary temperatures and pressures there will be a tremendous
number of such collisions per unit time. The only remaining hypothesis
is that even when two hydrogen molecules and an oxygen are in the
intimate contact of a collision, still it is such a rare thing for their atoms
to rearrange themselves to form two water molecules that for all practical
purposes it never happens. This is indeed the case. The proportion of
all such triple collisions in which a reaction takes place is excessively
small, at ordinary temperatures, though it is finite; if we waited long
enough, equilibrium would be attained, but it might take thousands of
years. But the probability of a reacting collision increases enormously
with the temperature, which is the reason why a spark, a loeali/ed region
of exceedingly high temperature, can start the reaction. Once it is
started, the heat liberated by the reaction near the spark raises the gas in
the neighborhood to such a high temperature that it in turn can react,
liberating more heat and allowing gas still further away to react, and so
on. In this way a sort of wave or front of reaction is propagated through
the gas, with a very high velocity, and this is characteristic of explosion
reactions.

It is true in general that, given a collision of the suitable molecules
for a reaction, the probability of reaction increases enormously rapidly
with the temperature. When measurements of rates of reaction are
made, it is found that the probability of reaction can be expressed approxi-
mately by the formula

Probability of reaction = const. X c k'*\ (3.1)

where Qi is a constant of the dimensions of an energy. Equation (3.1)
suggests the following interpretation: suppose that out of all the collisions,
only those in which the colliding molecules taken together have an
energy (translational, rotational, and vibrational) of Qi or more can
produce a reaction. By the Maxwcll-Boltzmann distribution, the frac-

-5i
tion of molecules having this energy will contain the factor e kT. (The

fraction having an energy greater than Qi can be shown to contain this
factor, as well as the fraction having an energy between Q\ and Qi + dQi,
which is what we usually consider.) Thus we can understand the varia-
tion of rate of reaction with temperature. We must next ask, why do the*
molecules need the extra energy Qi, in order to react? This energy is
ordinarily called the energy of activation, and we say that only the
activated molecules, those which have an energy at least of this amount,
can react.

To understand why an energy of activation is required for a reaction,
we may think about a hypothetical mechanism for the reaction. In our

160 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

particular case of hydrogen and oxygen combining to form water, we
imagine a collision in which two hydrogen molecules hit an oxygen
molecule (this will be the way the collision will appear, for on account of
their light mass the hydrogens will be traveling much faster than the
oxygen in thermal equilibrium). During the collision, the atoms rear-
range themselves to form two water molecules, which then fly apart with
very groat energy (on account of the heat of reaction). Now, obviously,
those particular collisions will be favored for reaction in which the atoms
need the minimum rearrangement in the reaction, and a little reflection
shows that the most favorable configuration is that shown in Fig. X-l
(a), in which the velocities of the various atoms are shown by arrows. As
we follow the successive sketches of Fig. X-l showing the progress of the
collision, we see that the hydrogens approach the oxygens, attaining in (c)
a shape very much like two water molecules. In the first part of the
collision, (a) arid (6), the hydrogens have most of the kinetic energy. For
a favorable reaction, however, the relations between the velocities of
hydrogens and oxygons on collision must be such that the hydrogen gives
up most of its kinotic energy to the oxygon. The condition for this
can bo found from elementary considerations of conservation of momen-
tum and kinetic energy on collision, and demands that the oxygen atoms
be moving in the same direction as the hydrogens on collision but with
considerably smaller velocity. That is, the oxygen molecule must have
had considerable vibnition.il kinetic energy and the correct phase of
vibration, while the hydrogens must have had large translational kinetic
energy. Now in the second part of the collision, (W), (c), (/), and (g), the
oxygens have most of the kinetic energy. They fly apart, carrying the
hydrogens with them, and form the atoms into two water molecules. The
hydrogens end up bound to the oxygens, but with some vibrational
kinetic energy in the mode of vibration indicated by (g).

We can now follow the energy relations in the reaction by drawing a
suitable potential energy curve. The potential energy of the whole
system, of course, depends on the positions of all the atoms and would
have to be plotted as a function of many coordinates. We can simplify,
however, by considering it as a function only of the distance r between
the oxygen atoms. For each value of r, there will be a particular position
for the hydrogen atoms that will correspond to a minimum of energy.
Thus in (a), where the oxygen atoms are forming an oxygen molecule, the
hydrogen molecules and the oxygen molecule will attract each other
slightly, provided the oxygen-hydrogen distance is considerable, but will
repel provided they come too close together, as we shall learn later when
we consider intcrmolecular forces in imperfect gases. There will be a
position of equilibrium, with the hydrogens a considerable distance — three
or four angstroms — away from the oxygen, and with an energy of perhaps

SEC. 3]

CHEMICAL EQUILIBRIUM IN GASES

161

162

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. X

a fraction of a kilogram calorie lower than the energy at infinite separation
of hydrogen from oxygen. Similarly in (g) the atoms are formed into two
water molecules, and the minimum of energy of the hydrogens comes
when they are at the distances and angles with respect to the oxygen
which we find in a water molecule. We now show, in Fig. X-2, a sketch
of the potential energy of the whole system, when the oxygens are at
distance r, and the hydrogens are in their positions of minimum energy
for each value of r.

First, we ask how Fig. X-2 was constructed. When the oxygens are
close together forming an oxygon moloculo, the energy of the hydrogens,
being only interinolecular attraction, is small and the curve is practically

10 20

r (Angstroms)

Fr<; X-2. — Potential onorgy of 4H + 2O, as function of < M) distanro r

the interatomic energy for the oxygen molecule. This is the curve (a)
of Fig. X-2, going to an energy at infinite separation which is greater by
Z)( = 117 kg.-cal.) than at the minimum, which comes at 1.20 A. On
the other hand, when the oxygens are far apart they form two water
molecules. At infinite separation of these two molecules, the energy of
the whole system is less by 117 kg.-cal. (= A//0, which only happens to
be equal to the D of the oxygen molecule by a coincidence), than when
two hydrogen and one oxygen molecule are formed. The curve (6)
shows the interaction between these water molecules. Starting with the
asymptotic energy just mentioned, the curve rises with decreasing dis-
tance, because the two water molecules, set with their negative oxygen
ions facing each other, repel each other on account of electrostatic repul-
sion of like charges. As the distance decreases, to something of the order
of three angstroms, the molecules begin to hit each other, causing the
curve (6) to rise steeply. Curves (a) and (6) both form limiting cases.
For small r's, curve (a) must be correct, and for larger r's curve (6). The

SEC. 3] CHEMICAL EQUILIBRIUM IN GASES 163

full line in Fig. X-2 represents a sketch of the way the actual curve may
look, reducing to these two limiting curves. It will be noted that at
intermediate distances the actual curve lies below either curve (a) or
(b). Essentially, the reason is as follows: When we have quite separated
molecules, as the two water molecules at large distances, each atom of one
molecule repels each atom of the other. But as they approach, as in
configuration (c) of Fig. X-l, there is a little uncertainty as to whether
they form two water molecules, or two hydrogens and an oxygen. As a
consequence, the oxygen atoms make a compromise between repelling
each other, as they would in two water molecules, and attracting, as they
would in an oxygen molecule. That is, the repulsion which causes the
rise in curve (6), Fig. X-2, is diminished and the actual curve does not
continue to rise as curve (6) does.

Now that we have the curve of Fig. X-2, we can apply it to the reac-
tion as shown in Fig. X-l. The first part of the reaction, diagrams (a),
(6), and (c) of Fig. X-l, cannot be represented directly on Fig. X-2, for
in it the hydrogen molecules have a great deal of kinetic energy and aro
by no means in the position of minimum potential energy. But by (r)
of Fig. X-l the hydrogen atoms have given up most of their kinetic
energy to the oxygens, and during the rest of the process the curve of
Fig. X-2 applies fairly accurately. As far as the, first part of the process
is concerned, we can interpret it as a process in which the oxygens had
their vibrational energy increased from such a value as E} in Fig. X-2 to
Et, symbolized by the arrow in the figure. When they had the energy K\
they simply vibrated back and forth for a short range about the distance
re of minimum energy. But with the energy E2 the motion changes
entirely: the oxygens fly apart, carrying the hydrogens with them to form
water molecules and ending up with infinite separation of the molecules
and a very high kinetic energy. And now we see the need of the energy
of activation. From Fig. X-2 wo see that there is a maximum of potential
energy between the minimum at i\ and the still lower value at infinite
separation. For the water molecules to separate, the energy E% must
lie higher than this maximum. But this energy is supplied, as we have
seen, by the combined energy of all the colliding molecules before collision.
We thus sec that the energy of activation Qi is to be interpreted as the
height of this maximum above the minimum at r€.

A minimum of potential energy such as that at re in Fig. X-2, separated
by a maximum from a still lower region, is often met in atomic and
molecular problems and is called a position of metastablc equilibrium, or
a metastable state. It is stable as far as small displacements are con-
cerned, but a large displacement can push the system over the maximum,
after which it does not return to the original position but to the entirely
different configuration of really lowest potential energy. In all such

164 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. X

cases, the rate of transition from the metastable to the really stable con-
figuration, at temperature T, depends on a factor exp (— Qi/kT), where
Qi is the energy of activation, or height of the maximum above the mini-
mum, for in all such cases it is only the molecules with energy greater than
Qi that can react. Let us see how rapidly such a factor can depend on
temperature. From Fig. X-2 it seems reasonable that in that case Q\
could bo of the order of magnitude of 40 kg.-cal. Then the factor
exp (-QiAT) will become exp (-40,000/tf 5T) = exp (- 40,000/1. 98 T) =
exp (-20,000/T) approximately. For T = 300° abs., this factor is
exp (-66.7) = 1Q-*29 approximately, while for T = 3000° abs. it is
exp ( — 6.67) = 10~3 approximately. Thus an increase of temperature
from room temperature at 3000° abs., which could easily be attained in a
spark, might make a difference of 1C26 in the* rate of reaction. A process
that would take 10~16 sec. at the high temperature might take 1010 sec., or
3 X 108 yrs., at the low temperature, and would for all practical purposes
never happen at all. This is an extreme but by no means an unreasonable
example.

We are now in position to see why, though the energy of activation
enters into the rate of reaction in such an important way, it does not affect
the final equilibrium. The factor C(T\ in Eq. (1.2), determining the
rate of combination of hydrogen and oxygen molecules to form water, will
contain a factor exp ( — Qi/kT), as we have seen. But a glance at Fig.
X-2 shows that in the reverse reaction, in which two water molecules
combine to form hydrogen and oxygen, the water molecules must have
an energy at least equal to Q\ + A//0, so that they can climb over the
maximum of potential energy and approach closely enough to form an oxy-
gen molecule. Thus the probability of the reverse collision, given by Eq.
(1.3), contains the f actor C'(7T) with the exponential exp [-«?i+A//0)/*7T].
Finally in the coefficient KP(T)j given by Eq. (1.4), we must have

C' (T}

with the factor exp {[-(Qi + A//0) + Q\]/kT\, which equals

exp [ — (£Ho/kT)], in agreement with Eqs. (2.6) and (2.9), the energy of
activation canceling out. Of course, in this simple argument we have
neglected such things as specific heats, so that we have not reproduced
the whole form of Eq. (2.6) from a kinetic point of view, but this could
be done if sufficient care were taken.

There is one point about a reaction like the combination of two water
molecules to form oxygen and hydrogen, which we have just mentioned,
that is worth discussion. From Fig. X-2, if the molecules approach with
energy E% sufficient to pass over the maximum of potential, they will not
be trapped to form oxygen and hydrogen molecules unless the energy of
the oxygens drops from J?2 to some value like EI during the collisions.
This of course can happen by giving the excess energy to the hydrogen

SEC. 3]

CHEMICAL EQUILIBRIUM IN (JASES

165

atoms, sending them shooting off as hydrogen molecules. But there are
sometimes other reactions in which this cannot happen. For instance,
consider the simple recombination reaction of two oxygen atoms to form
an oxygen molecule, shown in Fig. X-3. Here if the atoms approach with

FIG. X-.3. — Kerombmation of atoms to form a molecule.

energy E%, there is nothing within the system itself able to absorb the
necessary energy to make them fall down to the energy E\, and be bound
to form a molecule. Such a recombination of two atoms can only occur
if they happen to be in collision with a third body, atom or molecule, at
the same time, which can absorb the excess energy and leave the scene of
collision with high velocity.

CHAPTER XI
THE EQUILIBRIUM OF SOLIDS, LIQUIDS, AND GASES

We have ho far studied only perfect gases and have not taken up
imperfect gases, liquids, and solids. Before we treat them, it is really
necessary to understand what happens when two or more phases are in
equilibrium with each other, and the familiar phenomena of melting, boil-
ing, and the critical point and the continuity of the liquid and gaseous
states. We shall now proceed to find the thermodynamic condition for
the coexistence of two phases and shall apply it to a general discussion'
of the forms of the various thermodynamic functions for matter in all
three states.

1. The Coexistence of Phases. — It is a matter of common knowledge
that at the melting point, a solid and a liquid can exist in equilibrium with
each other in any proportions, as can a liquid and vapor at the boiling
point. There is no tendency for the relative proportions of the two
phases, as they are called, to change with time. On the other hand, if
we are not at the melting or boiling point, there is no such equilibrium.
At 100°C., for instance, water vapor above atmospheric pressure will
immediately start to condense, enough liquid forming so that the remain-
ing vapor and the liquid will come to atmospheric pressure; while if water
at this temperature is below atmospheric pressure, enough liquid will
evaporate or boil away to raise the pressure to one atmosphere, so that
only at atmospheric pressure can the two coexist at 100°C. in arbitrary
proportions. For each temperature the equilibrium takes place at a
definite pressure; that is, we can give a curve, called the vapor pressure
curve or in general the equilibrium curve, in the P-T plane, along which
equilibrium occurs. This curve separates those parts of the P-T plane
where just one, or just the other, of the phases can exist. Thus in gen-
eral, where a number of phases occur in different regions of the P-T plane,
equilibrium lines separate the regions where each phase occurs separately.
Along a line two phases exist; where three lines join at a point, three
phases can exist, and such a point is called a triple point.

The resulting diagram is called a phase diagram. In the figures
below, such a diagram is drawn for water, a familiar and in some ways a
remarkable example. Figure XI-1 shows the diagram for a scale of
pressures on which the critical point is represented by a reasonable value.
The ordinary melting and boiling points, at 1 atm. and at 0°C., and
100°C., respectively, are easily found. We see that the boiling point

166

SBC. 1]

THE EQUILIBRIUM OF SOLIDS

167

rises rapidly to higher temperatures as the pressure is raised, until finally
the critical point is reached, above which there is no longer discontinuity
between the phases. The melting point, on the other hand, is almost
independent of pressure, decreasing as a matter of fact very slightly with
increasing pressure.

Figure XI-2 gives a different pressure scale, on which small fractions
of an atmosphere can be noted. The triple point is immediately observed,
corresponding to a low pressure and a temperature almost at 0°C., at
which ice, water, and water vapor can exist at the same time, so that if a
dish of water is cooled to this temperature in a suitable vacuum, a coating
of ice will form and steam will bubble up from below the ice. Below this
temperature, liquid water does not occur, but as we can sec an equilibrium
is possible between solid and gas. If the solid is reduced below the

£200

8
Jioo

Ice

Water

rpor

/ce

Wafer!

Vapor

100 200 300 400
Deg.C

FIG. XI-1. — Phase diagram of
water. Critical point * PC = 218
atm., Tc = 374°O

-20 0 20

Deg.C

FIG. XI-2. — Low-pressure
phase diagram of water. Triple
point: P = 4..5S mm., T =
().0075°0.

pressure corresponding to equilibrium, it will evaporate directly into
water vapor. This is the way snow and ice disappear in weather below
freezing; and it is a familiar fact that solid carbon dioxide, whose triple
point lies at a pressure greater than atmospheric;, evaporates by this
method without passing through the liquid phase.

In Fig. XI-3, the pressure scale is changed in the other direction, so
that we show up to 12,000 atm. Here the gaseous phase, which exists
for pressures only up to a few hundred atmospheres, cannot be shown on
account of the scale. On the other hand, a great deal of detail has
appeared in the region of the solid. It appears that, in addition to the
familiar form of ice, there are at least five other forms (the fifth exists at
higher pressures than those shown in the figure). These forms, called
polymorphic forms, presumably differ in crystal structure and in all
their physical properties, as density, specific heat, etc. The regions
where these phases exist separately are divided by equilibrium lines, on

168

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XI

10,000

-40 -20

0
Deg.C.

FIG. XI-3. — HiKh-pressuro phase diagram of water.

200-

a
</>
o

100-

10 20

cc/gm

FIG. XI-4. — P-V-T surface for water.

SEC. 2]

THE EQUILIBRIUM OF SOLIDS

169

which two of them can coexist in equilibrium, and a number of triple
points are shown. Transitions from one phase to another along an
equilibrium line are called polymorphic transitions. There has never
been found any suggestion of a critical point, or termination of an equilib-
rium line with gradual coalescence of the two phases in properties, for
any equilibrium between liquid and solid or between two solid phases.
Critical points appear to exist only in the liquid-vapor transition.

2. The Equation of State. — The three figures that we have drawn
give only part of the information about the phase equilibrium; for greater

10,000-

<b
a

5,000-

'/ce anef vapor
Icc/gm

Fi«. XI-5. — P-V-T Hiirftu'o for water, high pressure.

completeness, we should show the whole equation of state1, the relation
between pressure, temperature, and volume. This is done in Figs. XI-4
and XI-5, where P-V-T surfaces arc shown in perspective, for the case of
the liquid-vapor equilibrium and for the polymorphic forms in equilibrium
with the liquid. A number of isothermals, lines of constant temperature,
are drawn on each surface to make them easier to interpret. Some simple
facts arc immediately obvious from these surfaces. For instance, water
is exceptional in that the solid, ice, has a larger volume than the liquid.
As we see, this is true of ice I, the phase existing at low pressure, but it is
not true of the other phases II, III, V, VI, all of which have smaller
volumes than water at the corresponding pressure. Furthermore, we see
that though water seems quite incompressible as far as the low pressure

170 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XI

surface in the first figure is concerned, the second figure shows that a
pressure of 12,000 atm. produces a diminution of volume of about 20 per
cent. Again, the melting point of ice I is hardly affected by the pressures
indicated in the low pressure surface, but the other surface shows that a
pressure of about 2000 atm. lowers the melting point by more than 20°C.
One interesting fact to notice is that a vertical lino cutting either of tho
surfaces will cut it in just one point; that is, for a given volume and tem-
perature, the pressure is uniquely determined. We shall see shortly
that this can be shown to be true quite generally.

As we see, the P-V-T surfaces are divided into a number of different
regions with sharp edges separating them. In some of these regions one
phase alone can exist, while in others two phases can coexist. The regions
of the second type, when projected onto the P-T surface, become the
equilibrium lines that we have previously mentioned; thus they are ruled
surfaces, the rulings being parallel to the volume axis. At a given pres-
sure and temperature on an equilibrium line, in other words, tho volume
ran have any value between two limiting values, tho volumes of tho two
phases in question. The meaning of this is that there is a mixture of the
two phases, so that the volume of the mixture depends on tin; relative
concentrations of the two arid is not really a property of either phase, but
is a measure of the relative concentrations.

3. Entropy and Gibbs Free Energy. Tho equation of state does not
alone determine the thermodynamic behavior of a substance; wo must
also know its specific heat, or its entropy or Gibbs free energy. Wo shall
first give the entropy as a function of pressure and temperature. This
can of course be determined entirely by experiment. We start with the
solid at the absolute zero. There, according to tho quantum theory, as
we have seen in Chap. Ill, the entropy is zero. The entropy of tho solid
at a higher temperature can bo found from tho specific heat, for at con-
stant pressure we have

Since, according to the quantum theory, the specific heat goes to zero at
the absolute zero, tho integral in Eq. (3.1) behaves properly at the abso-
lute zero. By means of Eq. (3.1), we find the entropy of the solid at any
temperature1, at a given pressure; since tho specific heat depends only
slightly on pressure, this means practically that the entropy of the solid
is a function only of temperature, not of pressure, on the scales used in
Figs. XI-1 and XI-2, though not in Fig. XI-3. Next, we wish the
entropy of the liquid and vapor. If the pressure is below that at the
triple point, a horizontal line, or line of constant pressure, in the phase
diagram will carry us from the region of solid into that of vapor. There

SEC. 3] THE EQUILIBRIUM OF SOLIDS 171

is a discontinuous change of entropy as we cross the line, equal to the heat
absorbed (the latent heat of vaporization) divided by the temperature
(the temperature of sublimation for the pressure in question). This
change of entropy, which we may call the entropy of vaporization and
denote by A/SV, is

AS. - Y: (3-2)

where Lv, Tv are the latent heat and temperature of vaporization at the
given pressure. Adding this change of entropy to the entropy of the solid
(3.1) just below the sublimation point, we have the entropy of the vapor
just above this point. Then, applying Eq. (3.1) to the vapor rather than
the solid, we can follow to higher temperatures at constant pressure and
find the entropy of the gas, as

CTvn i CT

= --'-dT + J'° +

JO ^ 1 v J T,

(3.3)

In Eq. (3.3), the first term represents the entropy of the solid at the
sublimation point, the second the increase of entropy on vaporizing, and
the third the further increase of entropy from the sublimation point up to
the desired temperatures.

If the pressure is above the triple point, the solid will first melt, then
vaporize. In this case, we can proceed in a similar way. On melting,
the entropy increases by the entropy of fusion, determined from the latent
heat of fusion and temperature of fusion by the relation

AS/ = '-£, (3.4)

analogous to Eq. (3.2). Then the entropy of the liquid at any tempera-
ture is

(*Tic T l*Tr

Si- <£ur+^ + L*'dT.

JO * If JT, 1

(3.5)

As the temperature rises further, the liquid will vaporize and the entropy
will increase by the entropy of vaporization. The gas above this tem-
perature will then have the entropy

8. = r^dT + % + F'CfdT + (7 + f^f'dT. (3.6)
Jo :/ // jTf I I* JT, /

It is interesting to note that a relation between the latent heat of vapori-
zation of the solid, the heat of fusion, and the heat of vaporization of the
liquid, at the triple point, arises from the fact that Eqs. (3.3) and (3.6)
must give identical values for the entropy of the gas at the triple point.

172

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XI

Since in this case Tf = Tv = T, the integrals involving the specific heats
of the liquid and gas drop out, and wo have at once the relation

Lv of solid = Lf + Lv of liquid, at the triple point. (3.7)

In Fig. XI-6 we show the entropy of water in its three phases, as a
function of pressure and temperature, computed as we have described
above. We are struck by the resemblance of this figure to that giving
the volume, Fig. XI-4; the entropy, like the volume, increases with
increase of temperature or decrease* of pressure. Lines of constant pres-

10 20 30 40 50
Ceil per Mole Degree

FIG. XI-0. -Kntropy of Wiiter iia function ot pressure und temperature.

sure are drawn in Fig. XI-6. The regions of coexistence of phases are
shown in Fig. XI-6 as in Fig. XI-4, and the latent heat is given by the
length of the horizontal line lying in the region of coexistence, multiplied
by the temperature. Graphs of the form of Fig. XI-6 (generally pro-
jected onto the T-S plane) are of considerable1 practical importance in
problems involving thcrmodynamic cycles, as heat engines and refrigera-
tors, on account of the fact that the isothermals are represented by lines
of constant T and adiabatics by lines of constant S, so that the diagram
of a Carnot cycle in such a plot is simply a rectangle. Furthermore, the
area of a closed curve representing a cycle in the T-S diagram gives
directly the work done in the cycle, just as it does in the P-V diagram.

SEC. 3] THE EQUILIBRIUM OF SOLIDS 173

This is seen at once from the first and second laws in the form

Tdft = dU + PdV.

Integrating around a closed cycle, we must have u) dV — 0, since (r is a
function of the state of the system. Hence

<j>T dS = fP dV, (3.8)

where CD indicates an integral taken about a complete cycle, and since
(foP dV equals the work done, u)T dS must equal it also.

I 23456789 10 II

-G (Kg. coil/mole)
FIG. XI-7. — Gihbs froo energy of water, as function of pressure and temperature

The Gibhs free energy G as a fund ion of pressure and temperature is
sketched in Fig. XI-7. It can also be found directly from experiment.
At constant pressure, we have ft = — (d(?/d77)p, or G = — J/SrfT7, and
since V = (dG/dP)r, we have G = J V dP at constant temperature, from
a combination of which the Gibbs free energy can be found from equation
of state and specific heat. The surface of Fig. XI-7 looks quite different
from those for volume and entropy; for while the volume and entropy
change discontinuously with a change of phase, resulting in the ruled
surfaces indicating coexistence of phases wrhich are so characteristic of
Figs. XI-4, XI-5, and XI-6, the Gibbs free energy must be equal for the
two phases in equilibrium. This has already been discussed in Sec. 3,
Chap. II and follows from the fundamental property of the Gibbs free
energy, that its value must be a minimum for equilibrium at given pres-

174 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XI

sure and temperature. Thus, if there is equilibrium between two phases
at a given pressure and temperature, a transfer of some material from
one phase to the other cannot change the Gibbs free energy, so that the
value of G must be the same for both phases. As we observe from Fig.
XI-7, each phase has a different surface for G as a function of P and T,
and the intersection of two of these surfaces gives the condition for
equilibrium. It is interesting to notice the behavior of the surface near
the critical point: the lines of constant pressure, which are drawn on the
surface, have discontinuities of slope below the critical point but merely
continuous changes of slope above this point. We shall see in a later
chapter how such lines can come about mathematically.

4. The Latent Heats and Clapeyron's Equation. — There is a very
important thermodynamie relation concerning the equilibrium between
phases, called Clapeyron's equation, or sometimes the (1ln,peyron-(Mausius
equation. By way of illustration, let us consider the vaporization of
water at constant temperature and pressure. On our P-V-T surface, the
process we consider is that in which the system is carried along an iso-
thermal on the ruled part of the surface, from the .state \\here it is all
liquid, with volume Vi, to the state \\here it is all gas, with volume Va.
As we go along this path, we wish to find ihe amount of heat absorbed.
We can find this from one of Maxwell's relations, Kq. (4.12), Chap. II:

dVjr ~ \dT/ (4'1}

The path is one of constant temperature, so that if we multiply by T
this relation gives the amount of heat absorbed per unit increase of
.volume. >But on account of the nature of the surface, (dP/dT)v is the
same for any point corresponding to the same temperature, no matter
what the volume is; it is simply the slope of the equilibrium curve on the
P-T diagram, which is often denoted simply by rtP/dT (since in the P-T
diagram there is only one independent variable, and we do not need
partial derivatives). Then we can integrate and have the latent heat L
given by

L = T^(Va - Vi), ' (4.2)

which is Clapeyron's equation. It is often written

dP L

_
dT ~ T(Vt - Vi)'

SBC. 4] THE EQUILIBRIUM OF SOLIDJS 175

W - T(V. - 7Q. . .

dP - I (4<3)

Clapeyron's equation holds, as we can see from its method of derivation .
for any equilibrium between phases. In the general case, the difference
of volumes on the right side of the equation is the volume after absorbing
the latent heat L, minus the volume before absorbing it.

There is another derivation of Clapeyron's equation which is very
instructive. This is based on the use of the Gibbs free energy G. In the
last section we have seen that this quantity must be equal for two phases
in equilibrium at the same pressure and temperature, and that if one
phase has a lower value of G than another at given pressure and tempera-
ture, it is the stable phase and the other one is unstable. We can verify
these results in an elementary way. We know that in going from liquid
to vapor, the latent heat L is the difference in enthalpy between gas and
liquid, or L = //„ — HI. Bui if the change is carried out in equilibrium,
the heat absorbed will also equal T (1$, so that the latent heat will be
T(SU — Si). Equating these values of HK latent heat, we have

//. - Hi - L = T(Sa - ,SU
or

Hg - TKtt = Hi - TStj Gt, = <;,, (4.4)

or our previous condition that the Gibbs free energy should be the same
for the two phases in equilibrium. Since this must be true at each point
of the VcMpor pressure line in the P-T plane, we can find the slope of the
vapor pressure curve from the condition that, as P and T change* in the
same way for both phases, Ga and GI must undergo equal changes. That
is to say, we set

d(G» ~

n \ — n i v ° l) \ AT _L

0-Gi) - o = ^ — -^ — ^^y +

= ~-(Sa - Si)dT + (V0 - Vt)dP
= -L^~ + (V0- VfidP,

dP _ L

W = (V~- Vd~T' (4'5)

which is Clapc^yron's equation. In deriving Eq. (4.5), we used the rela-
tions (dG/dT), = -S, (dG/dP)T = F, from Chap. II.

Hap^yron's equation, as an exact result of thermodynamics, is useful
in several ways. In the first place, we may have measurements of the
equation of state but not of the latent heat. Then we can compute the
latent heat. This is particularly useful for instance at high pressures,

176 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XI

where measurements of volume and temperature are fairly easy, but
where ealorimetric measurements such as would be required to find the
latent heat are very difficult. Or, in the second place, we may know the
latent hoat and then we can find the slope of the equilibrium curve, and by
integration we may find the whole curve. We shall discuss the applica-
tion of this method to tho vapor pressure curve in the next sections.
Finally, we may have measurements of both the latent heat and tho
equilibrium curve, but may not be sure of their accuracy. We can test
them, and perhaps improve them, by soring whether the experimental
values satisfy Chipeyron's equation exactly. If they do not, it is certain
evidence that they are in error.

5. The Integration of Clapeyron's Equation and the Vapor Pressure
Curve. — The integration of Clapoyron's equation to got the vapor pres-
sure curve over a liquid or solid from a measurement of the latent hoat is
one of its principal uses. We may write the integral of Eq. (4.3) in tho
form

- f

~ J

LtiT

This can be evaluated exactly if \ve know the latent hoat L, tho volume
of tho gas, and tho volume of tho solid, as functions of temperature. In
many actual oases we know these1 only approximately, but we can use
thorn to got an approximate; vapor pressure curve. For instance, the
simplost approximation is to assume that tho latent heat is a constant,
independent of temperature. Furthermore, in the ease of low tempera-
ture, where tho volume of tho gas will bo very large compared with the
volume of the solid or liquid, we may neglect tho latter and furthermore
assume that the gas obeys tho perfect gas law. Then Vg — Vs = nRT/P,
approximately, and Eq. (4.3) becomes

JP T

(5.2)

PdT

Equation (5.2) holds whether L is constant or not. Assuming it to be
constant, we can integrate and find

In P = --T>TT + const.,

L__

P = const, e nRT. (5.3)

Equation (5.3) giving P in terms of T gives a first approximation to a

SBC. 5] THE EQUILIBRIUM OF SOLIDS 177

vapor pressure curve. Plainly it approaches e~°° = 0 at low tempera-
ture, while as the temperature increases the pressure rapidly increases,
agreeing with the observed form of the curve. By making more elaborate
assumptions, taking account of the variation of L with pressure and
temperature and the deviation of the volume of the gas from that for a
perfect gas, the equation of the vapor pressure curve can be obtained as
accurately as we please. Tho formula (5.3) in particular becomes
entirely unreliable near the critical point. Since the latent heat
approaches zero as we approach the critical point, and the volumes of
liquid and gas approach each other at the same place, f,he ratio dP/dT
becomes indeterminate and more accurate work is necessary to find just
what the slope of the vapor pressure curve is. In spite of this difficulty,
the formula (5.3) is a very useful one at temperatures well below the
critical point. The constant factor, of course, must be obtained as far as
thermodynamics is concerned from a measurement of vapor pressure at
one particular temperature.

To find the correct vapor pressure equation, we shall determine the
variation of latent heat with temperature4. In introducing the* enthalpy
H = U + PV in Chap. 11, we saw that the change in enthalpy in any
process equalled the heat absorbed at constant pressure. Now the latent
heat is absorbed at constant pressure; therefore it equals the change of
the enthalpy between solid and gas. That is,

L = II, - //, (5.4)

Now we can find the change* in L, for an arbitrary change of pressure
and temperature. We have

I'.- V. - Tf + 7< <//' + (Cfl - CMT, (5.5)

when1 we have used thermodynamic relations from the table in Chap. II.
Now we assume? as in the last paragraph that the ve>lume of the solid can
be neglected and that the gas obeys the perfect gas law. The* gas law
gives at once Va — T(dVti/dT}p = 0. Thus the first bracket is zero, and
we have

^ = Cf. - CPf (5.0)

In Chap. VIII \ve have expressed the specific heat of a gas as

Cf, = %nR + nC,. (5.7)

178 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XI

Introducing this value into Eq. (5.6) and working with one mole, we can
integrate Eq. (5.6) to get the latent heat L, in terms of L0, the latent heat
of vaporization at the absolute zero:

L = Lo 4- %RT + (C* - Cf.)dT. (5.8)

Dividing by RT~ and integrating with respect to 7T, wo have from Eq.
(5.2)

j *; CT jr CT

In P = -~ ° + ? In T - ~ (Cf, - CJdT + const. (5.9)

~ T
This expression, or the corresponding one

"'* _*£. fT*LfT(t. -

P = const. T c ~RTc JoJmJo(Cl" ' (5 10)

is the complete formula for a vapor pressure curve, as obtained from
thermodynamics, in the region where the vapor behaves like a perfect
gas. We shall see in the next section that statistical mechanics can
supply the one missing feature of Eqs. (5.9) and (5.10): it can give the
explicit value of the undetermined constant.

It is interesting to note the behavior of the latent heat of vaporization,
as given by Eq. (5.8), through wide temperature ranges. At low tempera-
tures, since CV. and Ct both are very small, the latent heat increases with
temperature. This tendency is reversed, however, as the specific heat
of the solid becomes greater than that of the gas, which it always does.
The latent heat then begins to fall again. At the triple point, as we have
stated, the latent heat of vaporization of the solid just belo\\ the triple
point equals the sum of the latent heat of fusion and the latent heat of
vaporization of the liquid, directly above the triple point. Above this
temperature, Eq. (5.8) is to be replaced by one in which C/> of the liquid
appears rather than that of the solid. This allows us to use the same
sort of method for finding the vapor pressure over a liquid. Finally, as
the temperature approaches the critical point, it is no longer correct to
approximate the vapor by a perfect gas, so that neither Eq. (5.2) nor
(5.8) is applicable, though we already know that the latent heat
approaches zero at the critical point, and of course Clapeyron's equation
can be applied here as well as elsewhere.

6. Statistical Mechanics and the Vapor Pressure Curve.— From
statistical mechanics, we know how to write down the Gibbs free energy
of the solid and perfect gas explicitly. All we need do, then, to find the
complete equation of the vapor pressure curve is to equate these quanti-
ties. Thus, remembering that (dG/dT)P = -S, we can write the Gibbs

SEC. 6] THE EQUILIBRIUM OF SOLIDS 179

free energy of the solid

G. = f/o - f 8 dT

(*T fTr,

= Uo - \ dT\ ±*dT. (6.1)

Jo Jo l

Using Kqs. (1.19) and (1.20) of Chap. VIII, this can bo rewritten

CTdT CT

G. = £/„- H ~\ CP.dT. (6.2)

Jo I Jo

Here UQ is the internal energy, or free energy, at absolute zero, a function
of pressure only. Next we wish the Gibbs free energy of the gas. We
use Eq. (1.25) of Chap. VIII. We note, however, that l/0 is used in that
equation in a different sense from what it has been here; for there it
means the internal potential energy of the gas at absolute zero, while
here we have used it for the internal energy of the solid at absolute zero.
It is plain that the energy of the gas at absolute zero must be greater than
that of the solid by just the latent heat of vaporizal ion at the absolute zero,
orLo. Using this fact, we have

Gg = C/o + Lo - ^RT In T + RT In P

Equating Eqs. (6.2) and (6.3), we have

ln p - - + ln T ~ ' (Cft - c')rfr + '• (6'4)

Equation (6.4) is the general one for vapor pressure, and it shows that the
undetermined constant in In P, in Eq. (5.9), is just the chemical constant
that we have already determined in Eq. (3.16) of Chap. VIII. The
simplest experimental method of finding the chemical constants is based
on Eq. (6.4) : one measures the vapor pressure as a function of the tem-
perature, finds the specific heats of solid and gas, so that one can calculate
the term in the specific heats, and computes the quantity

as a function of temperature. Plotting as a function of 1/T, Eq. (6.4)
says that the result should be a straight line, whose slope is — L0 and
whose intercept on the axis 1/T = 0 should be the chemical constant /.
This gives a very nice experimental way of checking our whole theory of
vapor pressure and chemical equilibrium: the same chemical constants
obtained from vapor pressure measurements should correctly predict the

180 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XI

results of chemical equilibrium experiments. It is found that in fact they
do, within the orror of experiment.

7. Polymorphic Phases of Solids. Kxpcrimontally, polymorphism
at high pressures is ordinarily observed by discontinuous changes of
volume. As the pressure is changed at constant temperature, the volume
changes smoothly as long as we are dealing with one phase only. At jhe
equilibrium pressures, however, the volume suddenly changes discontinu-

OUslv to another V^luer whjf.fr nf nmirsn jfi always fijTmllor for fho ]} igJi

pressure modification. Then another smooth change continues from the
transition pressure. The measurement thus gives not only the pressure
and temperature of a point on the equilibrium line, but the change of
volume as well. Clapeyron's equation of course applies to equilibrium
lines between solids, and that moans that from the observed slope of the
transition line and the observed change of volume, we can find the latent
heat of the transition, even though a direct thermal measurement of this
latent heat might be very difficult. Thus we can find energy and entropy
differences between phases.

It is very hard to say anything of value theoretically about polymor-
phic transitions. The changes of internal energy and entropy between
phases are ordinarily quite small. Any calculation that we should tiy to
make of the thermodynamics properties of each phase separately would
have errors in both those quantities, at least of the order of magnitude
of the difference which is being sought. Thus it is almost impossible to
predict theoretically which of two phases should bo stable under given
conditions, or whore tho equilibrium line between thorn should lie.
Nature apparently is faced with essentially the same problem, for in many
cases polymorphism scorns to be a haphazard phenomenon. It has been
impossible to make any generalizations or predictions as to what sub-
stances should be polymorphic and what should not, and in many cases
substances that are similar chemically show quite different behavior as to
polymorphism. We can, however, say a little from thermodynamics as
to the stability of phases and the nature of equilibrium lines.

We can think of two limiting sorts of transitions: one in which the
transition always occurs at tho saino tcmporaturo independent of pros-
sure, the other whore; it is always at the same pressure independent of
temperature. These would correspond to vortical and horizontal lines
respectively in Fig. XI-3. In Clapoyron's equation dP/dT = L/TAV,
these correspond to the case dP/dT = oo or 0 respectively. Thus in the
first case we must have AF = 0, or the two phases have the same volume,
in which case pressure does not affect the transition. And in the second
case L — 0, or AS = 0, there is no latent heat, or the two phases have
the same entropy, in which case temperature does not affect the transition.
Put differently, increase of pressure tends to favor the phase of small
volume, increase of temperature favors the phase of large entropy.

SEC. 7] THE EQUILIBRIUM OF SOLIDS 181

Of course, in actual cases we do not ordinarily find two phases witli
just the same volume or just the same entropy. Qn_accaunt_oL_the
parallelism between the entropy and _the_vplumc. thore is a tendency for n
phase of larger volume also to haveji Jarger_on topy Thus t.ho fonHpuny
isjfor_the latent heat and theHiange of vojumMpJha/vo the same sign. so
that byjClapeyron's equation dP/dT tends to be positive, or the equilib-
rium lines tend to slope upward tcTflie " ngliTTTi the phase diagram. A
statistical study of the phase diagrams of many substances shows that in
fact this is the case, though of course there an1 many exceptions. In
fact, there is^evenji tendency joward a fairly definite slope dP/dT charac-
teristic of many substances, which according; To Bridgman1 is a change of
somcfhing less ilian 12,000 atm. for a temperature range'oT 200°.

In each region of the P-T diagram then1 Ts only one stable phase,
except on equilibrium lines or at triple points where there arc1 two or three
respectively. But a phenomenon analogous to supercooling is very
widespread in transitions between solids. Particularly at temperatures
well below the melting point, transitions occur very slowly. A stable
phase can often be carried into a region where it is unstable, by change of
pressure or temperature, and it may take a very long time to change over
to the phase stable at- that pressure and temperature. This makes it-
very hard in many cases to determine equilibrium lines with great accu-
racy, for near equilibrium the transitions tend to be slower than far from
equilibrium. It also makes it hard to continue investigations of poly-
morphism to low temperatures. In Fig. XI-3, for instance, the lines are
continued about as far toward low temperatures as it is practicable to go.
Sometimes these slow transitions can be of practical value, as in the case
of alloys. It often happens that a modification stable at high tempera-
ture, but unstable at room temperature, has properties that arc; desirable
for ordinary use. In such a case the material can often be quenched and
cooled very rapidly from the high temperature1 at which the desired modi-
fication is stable. The material is almost instantly cooled so far down
below its melting point that the transition to the phase; stable at room
temperature is so slow as to be; negligible for practical purposes. Thus
the desired phase is made practically permanent at room temperature,
though it may not be thermodynamically stable. The* ordinary process
of hardening or tempering steel by quenching is an example of this
process. In some cases such unstable phases change; ove;r to the stable
form in a period of years, but in the really valuable; cases the rate is so
slow that it can be disregarded e;ven for many years. A moderate heat--
ing, however, can accelerate the* process so much as to change the prop-
erties entirely, as a moderate heating can destroy the temper or hardness
of steel.

1 P. W. Bridgman, Pmc. Am. Acnd., 72, 45 (1937); soe p. 129

CHAPTER XII
VAN DER WAALS' EQUATION

Real gases do not satisfy the perfect gas law PV = nRT, though they
approach it more and more closely as they become less and less dense.
There is no simple substitute equation which describes them accurately.
There is, however, an approximate equation called Van der Waals'
equation, which holds fairly accurately for many gases and which is so
simple and reasonable that it is used a great deal. This equation is not
really one that can be exactly derived theoretically at all. Van der
Waals, when he worked it out, thought he was giving a very general and
correct deduction, but it has since been seen that his arguments were not
conclusive. Nevertheless it is a plausible equation physically, and it is
so simple and convenient that, it is very valuable just as an empirical
formula. We shall give, first, simply a qualitative argument for justify-
ing the equation, then show to what extent it really follows from statistical
mechanics. Being an equation of state, thermodynamics by itself can
give no information about it ; we remember that equations of state have
to be introduced into thermodynamics as separate postulates. Only
statistical mechanics can be of help in deriving it.

1. Van der Waals' Equation. — Van der Waals argued that the perfect
gas law needed revision for real gases on two accounts. In the first place,
he considered that the molecules of real gases must attract each other,
exerting forces on each other which are neglected in deriving the perfect
gas law. The fact that gases condense to form liquids and solids shows
this. Surely the only thing that could hold a liquid or solid together
would be intermolecular attractions. These attractions he considered
as pulling the gas together, just as an external pressure would push it
together. There is, in other words, an internal pressure which can assist
the external pressure. In a liquid or solid, the internal pressure is great
enough so that even with no external pressure at all it can hold the
material together in a compact form. In a gas the effect is not so great
as this, but still it can decrease the volume compared to the corresponding
volume of a perfect gas. To find the way in which this internal pressure
depends on the volume, Van der Waals argued in the following way.
Consider a square centimeter of surface of the gas. The molecules near
the surface will be pulled in toward the gas by the attractions of their
neighbors. For, as we see in Fig. XII-1, these surface molecules are
subjected to unbalanced attractions, while a molecule in the interior will

182

Ssr. 1) VAN DER WAALS' EQUATION 183

have balanced forces from the molecules on all sides. Now the range of
action of these intermolecular forces is found to be very small. Thus
only the immediate neighbors will be pulling a given molecule to any
extent. To indicate this, we have drawn a thin layer of gas near the
surface, including all the molecules exerting appreciable forces on the
surface molecules. Now the total force on one surface molecule will be
proportional to the number of molecules that pull it. That is, it will be
proportional to the number of molecules
per unit volume, times the volume close
enough to the molecule to contribute
appreciably to the attraction. The total
force on all the molecules in a square
centimeter of the surface layer will be
proportional to the number of molecules
in this square centimeter times the force Fi«. xil-i. intermolecular attra«-
011 each, so that it will be proportional tlons>

to the square of the number of molecules per unit volume, or to (JV/F)2, if
there are N molecules in the volume V, or to (n/V)'2, where n is the num-
ber of moles. But the force on the molecules in a square centimeter of
surface area is just the internal pressure, so that

Internal pressure = a(~t7y ' (1-1)

where a is a constant characteristic of the gas.

The second correction which Van dcr Waals made was on account of
the finite volume of the molecules. Suppose the actual molecules of a gas
were rather large and that the density was such that they filled up a good
part of the total volume. Then a single one of the molecules which we
might consider, batting around among the other molecules, would really
not have so large a space to move in as if the other molecules were not
there. Instead of having the whole volume V at its disposal, it would
move much more as if it were in a smaller volume. If there are n moles
of molecules present, and the reduction in effective volume is b per mole,
then it acts as if its effective volume were

Effective volume = V — nb. (1.2)

If the volume were reduced by this amount, the pressure would be cor-
respondingly increased, since the molecule would collide with any element
of surface more often.

Making both these corrections, then, Van dor Waals assumed that
the equation of state of an imperfect gas was

+ a(F - nb) = nRT. (1.3)

184

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XII

This is Van der Waals' equation. We shall later come to the question
of how far it ran be justified theoretically by statistical mechanics. First,
however, we shall study its properties as an equation of state and see how
useful it is in describing the equilibrium of phases.

2. Isothermals of Van der Waals' Equation. — In Fig. XII-2 we give
isothermals as computed by Van der Waals' equation. At first glance1,
they are entirely different from the actual isothermals of a gas, as shown in
perspective in Fig. XI-4, because for low temperatures the isothermals
show a maximum and minimum, the minimum corresponding in some
cases to a negative pressure. But a little reflection shows that this
situation is not alarming. We note that there is one isothermal at which
the maximum and minimum coincide, so that there is a point of inflection

FIG. XII-2. — Isothermals of Van dor Waata* equation.

of the curve here. This is the point marked C on Fig. XII-2. At every
lower temperature, there arc three separate volumes corresponding to
pressures lying between the minimum and maximum of the isothermal.
This is indicated by a horizontal line, corresponding to constant pressure,
which is drawn in the figure and which intersects one of the isothermals at
V\, V%, tind F». We may now ask, given the pressure and temperature
as determined by this horizontal line and this isothermal respectively,
which of the three volumes will the substance really have? This is a
question to which there is a perfectly definite answer. Thermodynamics
directs us to compute the Gibbs free energy of the material in each of
the three possible states, and tells us that the one with the lowest
Gibbs free energy will be the stable state. The material in one of the
other states, if it existed, would change irreversibly to this stable state.
We shall actually compute the free energy in the next section, and shall
find which state has the lowest value. The situation proves to be the

SEC. 2]

VAN DEK WAALS9 EQUATION

185

following. Suppose we go along an isothermal, increasing the volume at
constant temperature, and suppose the isothermal lies below the tempera-
ture corresponding to C. Then at first, the smallest volume V\ has the
smallest Gibhs free energy. A pressure is reached, however, at which
volumes V\ and F3 correspond to the same free energy. At still lower
pressures, F3 corresponds to the lowest free energy. The state F2 has a
higher free energy than either FI or Fa under all conditions, so that it is
never stable. We see, then, that above a certain pressure (below a certain
volume) the state of smallest volume is stable, at a definite pressure this
state and that of largest volume can exist together in equilibrium, and
below this pressure only the phase of largest volume can exist. But this
is exactly the behavior to be expected from experience with actual changes
of phase.

The isothermals of Van dor Waals' equation, then, correspond over
part of their length to states that are not thermodynamieally stable, in

Fm. XII-3. — Isothcrmals of Van der Waals' equation, showing equilibrium of liquid and

gas.

the sense that their free energy is greater than that of other states, also
described by the same; equation, at the same pressure and temperature.
In Fig. XII-3 we give revised isothormals, taking this change of phase into
account. In this figure, corresponding to each pressure and temperature,
only the stable phase is shown. In the region whore two phases are in
equilibrium, we draw horizontal lines, as usual in such diagrams, indicat-
ing that the pressure and temperature are constant over the whole range
of volumes between the two phases in which the stable state of the system
is a mixture of phases. The isothermals of Fig. XII-3 are plainly very
similar to those of actual gases and liquids.

From Fig. XII-3, it is plain that the critical point is the point C of
Fig. XII-2, at which the maximum and the minimum of the isothermal
coincide. We can easily find the pressure, volume, and temperature; of
the critical point in terms of the constants a and 6, from this condition.
The most convenient way to state the condition analytically is to demand
that the first and second derivatives of P with respect to V for an iso-
thermal vanish simultaneously at the critical point. Thus, denoting the

186 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XII

critical pressure, volume, and temperature by Pc, Ve, Tc, we have

- n = _
~

e~ nb Vs,'

nRT,

' '(Fe - n6)
_ _ 2nRTc _

3FVr (V, - »6)* "FJ ' (F; - nbY ~ V*' ( }

We can solve Kqs. (2.1), (2.2), (2.3) simultaneously for Pc, Tc, and Fc.
Dividing Eq. (2.3) by Kq. (2.2) we at once find Ff. Substituting this in
Eq. (2.2), we can solve for T.. Substituting both in Eq. (2.1), we find
Pc. In this way we obtain

P« = 27F v° = 3nb> RT''WH (2'4)

Equations (2.4) give the critical point in terms of a and b. Con-
versely, from any two of the Eqs. (2.4) we can solve for a and b in terms
of the critical quantities. Thus, from the first and third, we have

'-i£-

These equations allow us to make a calculation of the critical volume:

Fc (Van der Waals) = 3nb = c- (2.6)

o rc

If Van der Waals' equation were satisfied exactly by the gas, the critical
volume determined in this way from the critical pressure and temperature
should agree with the experimentally determined critical volume. That
this is not the case will be shown in a later chapter. The real critical

volume and ^ p are not far different, but the latter is larger. This is
o rc

one of the simplest ways of checking the equation and seeing that it really
does not hold accurately, though it is qualitatively reasonable.

Using the values of PCJ Vc, and Tc from Eq. (2.4), we can easily write
Van der Waals' equation with a little manipulation in the form

Ip_ 3_¥Z _ 11 = § L
p< (-IV LFc 3J g7'-
\ CJ \

(2.7)

This form of the equation is expressed in terms of the ratios P/PC, V/VC,
T/Tc, showing that if the scales of pressure, volume, and temperature are
adjusted to bring the critical points into coincidence, the Van der Waals'

SBC. 3] VAN DER WAAL& EQUATION 187

equations for any two gases will agree. This is called the law of cor-
responding states. Real gases do not actually satisfy this condition at all
accurately, so that this is another reason to doubt the accuracy of Van
der Waals' equation.

3. Gibbs Free Energy and the Equilibrium of Phases for a Van der
Waals Gas. — We have seen that the equilibrium between the liquid and
vapor phase is determined by setting the Gibbs free energy equal for the
two phases. Let us carry out this calculation for Van der Waals' equa-
tion. From Eq. (4.2), Chap. II, we have dO = V (IP - ASY dT. Thus
we can calculate the Gibbs free energy by integrating this expression. Wo
arc interested only in comparing free energies at various points along an
isothermal, however, and for constant temperatures we can set the last
term equal to zero, so that dG = V dP along an isothermal. This is not
a convenient form for calculation, unfortunately, for Van der Waals'
equation cannot be solved for the volume in terms of the pressure con-
veniently. It involves the solution of a cubic equation, and this can
usually be avoided by some means or other. To avoid this difficulty, we
shall instead compute the Ilelmholtz free energy A = (V — PV, and then
find the Gibbs free energy from it. We have

dA = -PdV - SdT = -PdV
for an isothermal process. Thus

A — — JP dV + function of temperature

nRT M2a\ ,„ , , .. , .
v ^T~ / ~~ \ri r* ~*~ funrtlon °f temperature

-/(

= —nRT In (V — nb) — ~^- + function of temperature, (3.1)

and for the Gibbs free energy we have
G = A +PV

= PV —nRT In (V — nb) y- + function of temperature

nRTV 2n2a nrri, /T, 1X , r A. f,

= ^ £ Tr nRTm (V — nb) + function of temperature.

V — no V

(3.2)

Equation (3.2) expresses G as a function of volume arid temperature. We
wish it as a function of pressure and temperature, and it cannot con-
veniently be put in this form in an analytic way, on account of the
difficulty of solving Van der Waals' equation for the volume. It is an
easy matter to compute a table of values, however. We plot curves, like
Fig. XII-3, for pressure as a function of volume in Van der Waals'
equation, compute values of G from Eq. (3.2) for a number of values of the

188

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XII

volume, and read off the corresponding pressures from the curves of
pressure against volume. In this way Hie curve of Fig. XII-4 was
obtained. In this, the Gihbs free energy is plot ted as a function of pres-
sure, for a particular temperature (T = 0.95TC in this particular case).
It is seen that for a range of pressures, which in this case runs from about
P = 0.74PC to P = 0.84PC, there are three values of G for each pressure, of
which the lowest one represents the stable state. The lowest curves cross
at about P = 0.82PC, which therefore represents the vapor pressure or
point of equilibrium between the phases, at this temperature. Compari-
son with Fig. XI-7 shows that Fig. XII-4 really represents tho correct
form of this function; it corresponds to a section of the solid shown in
Fig. XI-7 cut at constant temperature with suitable rotation of axes.

0.7

0.8

P/Pc

FIG. XII-4.-- Gibbs free energy
vs. pressure, at constant tempera-
ture, for Van der Waals' equation, at
T = 0.957V.

T/TC

Fits. XI 1-5. Vapor pressure by
Van der Waals' equation compared
with values for HjO, CO*.

We remember that dG = V dP at constant temperature. That is,
(3G/dP)T = V, or the slope of the curve in Fig. XII-4 measures the
volume. Clearly at smaller pressure the stable state is that with greater
slope or greater volume, while the state of smallest volume is stable at the
high pressures. The figure makes it clear why the phase of intermediate
volume (Vz on Fig. XII-2) is not stable at any pressure, since its free
energy is never lower than that of the other two phases. The discon-
tinuity in slope of the Gibbs free energy at the point of equilibrium
between phases measures the change of volume in vaporization. This
discontinuity becomes less and loss as the temperature approaches the
critical point, and the small pointed loop in the G curve diminishes, until
finally at the critical point it disappears entirely, and the curve becomes
smooth.

In the way we have just described, we can find the Gibbs free energy
for each temperature and determine the vapor pressure, and hence the

SEC. 31

VAN DER WAALS' EQUATION

189

correct horizontal line to draw on Fig. XII-2. This gives us the vapor
pressure curve, and we show this directly in Fig. XII-5. For comparison,
we have plotted on a reduced scale the vapor pressures of water and
carbon dioxide. We see that while the
general form of the curve predicted by
Van der Waals' equation agrees with the
observed curves, the actual gases show a
vapor pressure which diminishes a good
deal more rapidly with decreasing pres-
sure than the Van der Waals gas.

Our methods also allow us to calculate
the latent heat of vaporization from Van
der Waals' equation. We have

- Ul + P(V{, -

XIl-(i.~- Latent heat as func-

Furthermore, we can see by migrating t^fU|^;u£^ dpr Waala R1U,
the equation

and H2O and OO2

— I =r- T\ I — P

~av)r~ 1\^T)^

(hat U — -an-/V + function of temperature for a Van der Waals gas.
Thus

(3.3)

All the quantities of Eq. (3.3) can be found when \\e have carried out the
calculation above, for that gives us the volumes of gas and liquid. Thus
we can compute the latent heat as function of temperature. To express
it in terms of dimensionless quantities, we can write Eq. (3.3) in terms of
P/Pc, etc., and find

•J_ „_ '/ _^ \

(3.4)

showing that the latent heats of two gases at corresponding temperatures
should be in the proportion of a/6 to each other. In Fig. XI 1-6 we plot
the latent heat L/(o/6), as a function of temperature, as derived from Van
der Waals' equation. For comparison we give the latent heats of water
and carbon dioxide. Both Van der Waals' equation and experiment
agree in showing that the latent heat decreases to zero at the critical point
and the curves are of similar shape. However, the scale is quite different,
Van der Waals' equation predicting much too small a value for the latent
heat.

190 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XII

We have now examined Van der Waals' equation enough to see that
it is very useful as an empirical equation, even if it has no theoretical
justification at all. As a matter of fact, it can be justified from statistical
mechanics as a first approximation, though no further. In the next
sections we shall take up this justification, considering the problem of
the equation of state of a gas whose molecules attract each other at large
distances but have a finite size, so that they repel each other if they are
pushed too closely into contact. We begin by taking up by statistical
mechanics the general case of a gas with arbitrary intermolecular forces,
then specializing to agree with Van der Waals' assumptions about the
nature of the forces.

4. Statistical Mechanics and the Second Virial Coefficient. — The way
to derive the thermodynamic properties of an imperfect gas theoretically
is clear: we find the energy in terms of the coordinates and momenta,
compute the partition function, and derive the equation of state and
specific heat from it. The only trouble is that the calculation is almost
impossibly difficult, beyond a first approximation. In this section we
shall just derive that first approximation, which can be carried through
without a great deal of trouble. To understand the nature of the approxi-
mation, we write the equation of state in a series form which is often useful
experimentally. At infinite volume, we know that the gas will approach
a perfect gas, with an equation of state PV = nRT, or PV/nRT = 1.
At smaller volumes, the equation will begin to deviate from this. That
is, we can expand the quantity PV/nRT in series in 1/F; the term inde-
pendent of 1/F will be unity, but the other terms, which are different
from zero for imperfect gases, will bo functions of the temperature. Thus
we can write

PV I n\ //A2

•* r i i rwrrrxl'1'! i s'i/rn\l'l'\ i /A -j \

nRT A '

Here the quantity PV/nRT is often called the virial; and the quantities
1, B(T), C(7T)> etc., the coefficients of its expansion in inverse powers of
the volume per mole, V/n, are called the virial coefficients, so that B(T)
is called the second virial coefficient, C(T) the third, etc. The experi-
mental results for equations of state of imperfect gases are usually stated
by giving #(7"), C(T), etc., as tables of values or as power series in the
temperature. It now proves possible to derive the second virial coeffi-
cient B(T) fairly simply from statistical mechanics.

The first thing we must know is the energy of the gas as a function
of its coordinates and momenta. We use the same coordinates and
momenta as in Chap. VIII, Sec. 3: the coordinates of the center of gravity
of each molecule and other coordinates determining the orientation and
vibration of the molecule. The difference between our present problem

SEC. 4] VAN DEtt \YAALS' EQUATION 191

and the previous one of the perfect gas is that now we must add a term in
the potential energy depending on the relative positions of the molecules,
coming from intermolecular attractions and repulsions. Strictly speak-
ing, these forces depend on the orientations of the molecules as well as on
their distances apart, as is at once obvious if the molecules are very
unsymmetrical in shape, but we shall neglect that effect in our approxi-
mate treatment here. That allows us to write the energy, as before*, as a
sum of terms, tho first depending only on the coordinates of the centers of
gravity (the kinetic energy of the molecules as a whole, and the potential
energy of intermolecular forces), and the second depending only on orien-
tations and vibrations. Then the partition function will factor, the part
connected with internal motions separating off as before and, since it is
independent of volume, contributing only to the internal specific heat, and
not affecting the equation of state. For our present purposes, then, we
can neglect these internal motions, treating the gas as if it were mon-
atomic and simply adding on the internal specific heat at the end, using
the value computed for the perfect gas. We must remember, however,
that this is only an approximation, neglecting the effect of orientation on
intermodular forces.

Neglecting orientation effects, then, we deal only with the centers of
gravity of the molecules. We must now ask, how does the potential
energy depend on those* centers of gravity? We have seen the general
nature of Van dcr Waals's answer to this question. For the moment, let
us simply write the total potential energy of interaction between two
molecules i and j, at a distance rty apart, as <t>(rlt). Then we may reason-
ably assume that the whole potential energy of the gas is

(4.2)

pairs t,y

We now adopt Eq. (5.22), Chap. Ill, for the partition function, but
remember that, as in Sec. 3, Chap. VIII, we must multiply by 1/Nl, or
approximately by (c/N)N, in order to take account of the identity of
molecules. We then have

*'•*•''• (4-3)

The energy E is like that of Eq. (3.1) of Chap. VIII, except that for
simplicity we are leaving the internal part of the energy out of account,
and we have our potential energy Eq. (4.2). The integral (4.3) still
factors into a part depending on the momenta and another on the coor-
dinates, however, and the part depending on the momenta is exactly as
with a perfect gas and leads to the same result found in Chap. VIII.

192 INTRODUCTION TO CHEMICAL PHYSICS [CIIAF. XII

Thus we have

'*» -5- ,, .,

• (4-4)

*»r f f -5-^ .

J J Je </?i

The integral over coordinates is the one that simply reduced to VN in
the case of the perfect gas. The variables dq\ . . . can be written more
explicitly as dxL dyi dz\ . . . dx» dy\ dzN.

Tin1 integration over the coordinates can be carried out in steps.
First, we integrate over the coordinates of the Nth molecule. The

quantity e kT can be factored; it is equal to

~"kf ~/7,r~Ar"
c e

(4.5)

where 2' represents all those pairs that do not include the Nth molecule.
The first factor then does not depend on the coordinates of the Nth
molecule and may be taken outside the integration over its coordinates,
leaving

y, 0(f,Af)

-*TT
fffe ' dxNdyNdzN. (4.6)

We rewrite this as

rfrrv dyN dz, - J//(l - c~~^)dxN dyN dz» = V - W, (4.7)

///

where the first term is simply the volume, the second an integral to be
evaluated, which vanishes for a perfect gas. To investigate W, imagine
all the molecules except the Nth to be in definite positions. If the gas is
rare, the chances are that they will be well separated from each other.
Now if the point xNy^N is far from any of these molecules, the interatomic
potentials <£(rtV) will all be small, and the integrand will be practically
1 — e° = 0. Thus we have contributions to this integral only from the
immediate neighborhood of each molecule. Each of these will be equal to

/ *fr'*)\

= J/J\1 - e kT )dx dy dz.

w = J/J\1 - e kT dx dy dz. (4.8)

For simplicity we put the ith molecule at the origin of coordinates and
integrate to infinity instead of just through the container; the integrand
becomes small so rapidly that this makes no difference in the answer.
Then we have »

/. / »(r)\

w - f 4rrr2Vl - e kT )dr. (4.9)

In terms of this, we then have

W - (N - l)u>. (4.10)

SEC. 4] VAN DER WAALS' EQUATION 193

Now when we integrate over the coordinates of the (N — l)st mole-
cule, we have the same situation over again, except that there are only
(N — 2) remaining molecules, and so on. Thus finally we have for the
integral over coordinates in Eq. (4.4)

[V - (N - l)w][V - (N - 2)w] • • • V. (4.11)

To evaluate the quantity (4.11), we can most easily take its logarithm;
that is,

N-l

In (l - ™y (4.12)

8 = 0

Replacing the sum over s by an integral, this becomes

f In (l -

NlnV

*T 1 T7 ""^ l

= AT In V - - In 1 -

(4-13)

Our assumptions are only accurate if Nw/V is small; for it is only in this
case that we can assume that all molecules are well separated from each
other. In this limit, wo ran expand tho logarithm as

, /
111

Nw\ Nw
--

Substituting in Eq. (4.13) and retaining only the leading term, we

Nln V - l^N2~ - - • (4.15)

The quantity (4.15) for the logarithm of the integral over coordinates
in Eq. (4.4) can now be substituted in the expression for Helmholtz free
energy, giving at once

A = -WlnZ

= -^NkT In T - NkT In V + ^^~
z zv

- NkT\ lu (2^*1* + l - In (Nk) I- (4.16)

L A3 J

Equation (4.16) agrees exactly with Eq. (3.6), Chap. VIII, except for the
internal partition function Z», which we are here neglecting for simplicity,
and for the extra term N2kTw/2V. This represents the effect of inter-

194 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XII

atomic forces and is characteristic of the imperfect gas. Differentiating
A with respect to volume, we at once have for the equation of state

p — —I

^ ~ Wr

=. J™T_+.™I}?.

V + 2V* '
or, substituting JVfc = nft, AT = nJVo,

Equation (4.18) is in the form of Eq. (4.1 ) and shows that the second virial
coefficient is given by

B(T) = ; (4.19)

where w is given by Eq. (4.9). This deduction of the second virial coeffi-
cient is exact, in spite of the approximations wo have made; if further
terms are retained, they prove to affect only the third and higher virial
coefficients. But the calculation of these higher coefficients is much
harder than the treatment we have given here.

5. The Assumptions of Van der Waals' Equation. — The formula
(4.19) for the second virial coefficient, together with Eq. (4.9), furnishes
a method for deriving this quantity directly from any assumed inter-
molecular potential function, though generally the integration is so diffi-
cult that it must be carried out numerically. With the assumptions of
Van der Waals' equation, however, the problem is simplified enough so
that we can treat Eq. (4.9) analytically at high temperatures. We
assume that the molecules attract each other with a force increasing
rapidly as the distance decreases, so long, as they are not too close together.
We assume, however, that the molecules act like rigid spheres of diameter
r0, so that if the intermodular distance is greater than r0 the attraction
is felt, but if the distance r is equal to r0 a repulsion sets in, which becomes

__ ±

infinitely great if the distance becomes less than r0. Then e kT is zero, if
r is less than ro, so that Eq. (4.9) becomes

fr°4irr2dr + f °°47rr2(l - e~^)dr.

»/0 «/ro

w = 4irr2dr + 47rr2l - e~^dr. (5.1)

»/0 «/ro

The first term is simply -jTrrj!, the volume of a sphere of radius ro, or eight
times the volume of the sphere of diameter ro which represents a molecule.
In the second integral, we may expand in power series, since <£ is relatively

SBC. 5] VAN DEE WAALS' EQUATION 195

small. The bracket is * ~ ~ ' ' ' = ' Thus the term is

is * ~ ^ ~ tt ' ' ' )\ = pr'
± \

dr + • • •
Then for the second virial coefficient we have

~ '• (5-2)

We may write this

B(T) ^b-~>

where

, #o4

(5.3)

Here b is four times the volume of No spheres of radius r*o/2, or four times
the volume of all the molecules in a gram mole. Since the force repre-
sented by the potential <t> is attractive, <£ is negative and the quantity a
is positive and measures the strength of the intermolecular attractions.
It is found experimentally that the formula (5,3) for the second virial
coefficient is fairly well obeyed for real gases, showing that the assump-
tions of Van der Waals are not grc'atly in error. This formula leads to the
equation of state

Equation (5.4) indicates that for high temperatures (where a/RT is less
than 6) the pressure should be greater than that calculated for a perfect
gas, while at low temperatures (a/RT greater than 6) the pressure should
be less than for a perfect gas. The temperature

r. = & (5-5)

at which the second virial coefficient is zero, so that Boyle's law is satisfied
exactly as far as terms in l/V arc concerned, is called the Boyle tempera-
ture.

We can now take Van der Waals' equation (1.3), expand it in the form
of Eq- (4.1), and see if the second virial coefficient agrees with the value

196 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XII

given in Eq. (5.4). We have

nRT /n\2 nRT,

n&V1 /nV

-T) -a(v)

1 /"V

" \~a(v)'

nRT

agreeing with Eq. (5.4) as far as the second virial coefficient. In our
theoretical deduction, we have not found the third virial coefficient, but
this can be done by a good deal more elaborate methods than we have
used. When this is done, it is found that it does not agree with the cor-
responding quantity in Eq. (5.6). In other words, Van der Waals'
equation is correct as far as the second virial coefficient is concerned but
no further, as a theoretical equation of state for a gas whose molecules act
on each other according to Van der Waals' assumptions.

6. The Joule -Thomson Effect and Deviations from the Perfect Gas
Law. — The deviations from the perfect gas law are rather hard to measure
experimentally, since they represent small fractions of the total pressure
at a given temperature and volume. For this reason, another method of
detecting the departure from the perfect gas law, called the Joule-Thom-
son effect, is of a good deal of experimental importance. This effect is a
slight variation on the Joule experiment. That experiment, it will be
recalled, is one in which a gas, originally confined in a given volume, is
allowed to expand irreversibly into a larger evacuated volume. If the
gas is perfect, the final temperature of the expanded gas will equal the
initial temperature, while if it is imperfect there will be slight heating or
cooling. This experiment is almost impossible to carry out accurately, for
during the expansion there are irreversible cooling effects, which com-
plicate the process. The Joule-Thomson effect is a variation of the
experiment which gives a continuous effect, and a steady state.

Gas at a relatively high pressure is allowed to stream through some
sort of throttling valve into a region of lower pressure in a continuous
stream. The expansion through the throttling valve is irreversible, as in
the Joule experiment, and the gas after emerging from the valvr is in a
state of turbulent flow. It soon comes to an equilibrium state at the
lower pressure, however, and then it is found to have changed its tempera-
ture slightly. To make the approach to equilibrium as rapid as possible,
the valve is usually replaced by some sort of porous plug, as a plug of
glass wool, which removes all irregular currents from the gas before it.
emerges. Then all one has to do is to get a steady flow and measure the
difference of pressure and the difference of temperature, on the two sides
of the plug. If AP is the change of pressure, AT7 the change of tempera-

SEC. 0] VAN DER WAALS' EQUATION 197

ture, on passing through the plug, the Joule-Thomson coefficient is defined
to be AjT/AP. It is zero for a perfect gas and can be either positive or
negative for a real gas. Wo shall now evaluate the Joule-Thomson
coefficient in terms of the equation of state.

It is easy to show that the enthalpy of unit mass of gas is unchanged
as it flows through the plug. Let a volume V\ of gas be pushed into the
pipe at pressure PI] then, since P\ is constant through this pipe, work
/Pi (IV i — P\V\ is done on this sample of gas. After passing through
the plug, the same mass has a volume F2, and does work P2F2 in passing
out of the pipe. Thus the external work done by the gas in the process is
PzVz — PiTi- It is assumed that no heat is absorbed, so that if f/i is the
internal energy when the gas enters, C72 when it leaves, the first law gives

Ui + P,Fi = lJ\ + P2K2, 7/t = /72. (6.1)

Thus the change is at constant #, and the Joule-Thomson coefficient is
(3T/dP)n. But this can be evaluated easily from our Table of Thermo-
dynamic Relations in Chap. II. It is

(dT\ =
WA

(dg\

\dTjr

71 |D - V

(6.2)

From Eq. (6.2), we sec that for a perfect gas, for which V is proportional
to T at constant P, the Joule-Thomson coefficient is zero. For an imper-
fect gas, we assume the equation of state (5.4). We have

(dP\
WA

T7 , 2na
V ~ nb + ~

198 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XII

where we have regarded (b— nm)( 17) &s a small quantity compared
with unity, neglecting its square. Substituting, we then have

dT\ n 2a

From Eq. (6.4), we see that the Joule-Thomson coefficient gives
immediate information about a and b. If we measure the coefficient and
know CP, so that we can calculate the quantity (Cp/n)(dT/dP)n, we can
plot the resulting function as a function of l/T and should get a straight
line, with intercept — 6, and slope 2a/R, so that both b and a can be
found from measurements of the Joule-Thomson effect as a function of
temperature. We notice that at high temperatures the coefficient is
negative, at low temperatures positive. That is, since AP is negative in
the experiment, corresponding to a decrease of pressure, the change
of temperature is positive at high temperatures, loading to a heating of
the gas, while it is negative at low temperatures, cooling the gas. The
temperature 2a/Rb, whore the effect is zero, is called the temperature of
inversion; we see by comparison with Eq. (5.5) that, if our simple assump-
tions are correct, this should be twice the Boyle temperature. The Joule-
Thomson effect is used practically in the Lindc process for the liquefaction
of gases. In this process, the gas is first cooled by some method below
the temperature of inversion and then is allowed to expand through a
throttling valve. The Joule-Thomson effect cools it further, and by a
repetition of tho process it can be cooled enough to liquefy it.

CHAPTER XIII
THE EQUATION OF STATE OF SOLIDS

Next to perfect gases, regular crystalline solids are the simplest form
of matter to understand, being less complicated than imperfect gases near
the critical point, or liquids. Unlike perfect gases, there is no simple
analytic equation of state which always holds; we are forced either to use
tables of values or graphs to represent the equation of state, or to expand
in power scries. But the theory is far enough advanced so that we can
understand the simpler solids fairly completely. As with gases, we shall
start our discussion from a thermodynamic standpoint, asking how one
can find information from experiment, and then later shall go on to the
theory, seeing how far one can go by statistical mechanics in setting up a
model of a solid and predicting its properties. Of course, it is obvious
that in one respect the subject of solids is a much wider one than that of
gases: there is tremendous variety among solids, whereas all gases act
very much alike. This comes from the different types of forces holding
the atoms together and the different crystal structures. We shall put off
most of the discussion of the different types of solids until later in the
book, when we take up chemical substances and their properties. When
we come to that, we shall see to what a large extent the fundamental
atomic arid molecular properties of a solid are brought out in the behavior
of its solid state.

1. Equation of State and Specific Heat of Solids. — To know the
equation of state of a solid, we should have its pressure as a function of
volume and temperature. Really we should know more than this: a
solid can support a more complicated stress than a pressure, and can have
a more complicated strain than a mere change of volume. Thus for
instance it can be sheared. And in g£n_eraljthp "equation of state " kjL
set nfjvjfttfions giving the stress at every point of the solid^as a jujicliaiL
of the^ strains and the temperature. TBut we shall" not concern ourselves
with these general stresses "a'mPsTraihs, though they are of great impor-
tance both practically and theoretically; we limit ourselves instead to the
case of hydrostatic pressure, in which the volume and temperature are
adequate independent variables. Let us consider what we find from
experiment on the compression of solids to high pressures. At zero pres-
sure, the volume of a solid is finite, unlike a gas, and it changes with
temperature, generally increasing as the temperature increases, as given
by the thermal expansion. As the pressure is increased at a given tern-

199

200

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

perature, the volume decreases, as given by the compressibility. Com-
bining these pieces of information, we have a set of curves of constant
temperature, or isothormals, as given in Fig. XIII-1. These are plainly
very different from the isothormals of a perfect gas, which are hyperbolas,
the pressure being inversely proportional to tho volume. If we know
nothing experimentally but tho thermal expansion and the compressi-
bility, we should have to draw the lines as straight lines, with equal spac-
ing for equal temperature changes. Fortunately the measurements are
more extensive. The pressure is known as a function of volume over a
wide pressure rango, enough in most solids to change tho volumo by a
per cent, and with very compressible solids by many per cont, and the
volume is known as a function of tomporaturo for wido ranges of tompera-
turo. Tho curves must stop experimentally at zoro pressure1, but wo can

4QOOO

30,000

20,000

10,000

i i i I

275°obs

05 x<:

V/V0

FIG. XIII-1. — Isothermals for a solid (sodium) giving pressure as a function of volumo at

constant temperature.

imagine that they could bo extrapolated to negative pressures, as indi-
cated by the dotted lines in the figure.

To carry out any calculations with the equation of state, we wish to
approximate it in some analytic way. First, lot us consider the most
convenient variables to use. The results of experiment are usually
expressed by giving the volume as a function oTpressurcltnd temperature.
Thus thelhermal expansion Is" investigated as a function of temperature
at atmospheric pressure, and in measurements of compressibility the
volume is found as a function of pressure at certain fixed temperatures.
On the other hand, for deriving results from statistical mechanics, it is
convenient to find the Helmholtz free energy, and hence the pressure, as a
function of volume and temperature. We shall express the equation of
state in both forms, and shall find the relation between the two. We let
F0 be the volume of our solid at no pressure and at the absolute zero of
temperature. Then we shall assume

V = Fn(l

- a,(7')P

(1.1)

SEC. 1] THE EQUATION OF STATE OF SOLIDS 201

where a0, ai, a2, etc., are functions of temperature, the signs being chosen
so that they are positive for normal materials. Tho meaning of the q's
is easily found. Thus, first at zero pressure (which for practical purposes
is identical with atmospheric pressure, since the volume of a solid changes
so slowly with pressure) the volume is F0[l + ao(T)]. __ The coefficient
of thermal expansion at zero pressure is then

_ l(dV\
a - V\dTj

I 1 dtto rftto • ,1 /* rk\

'A " TT^df - dT approximately. (1.2)

If the material has a constant thermal expansion, so that the change in
volume is proportional to temperature, we should have approximately
dao/dT = a, where a is constant, leading to a0(71) = aT. This is a
special case, however; it is found that for real materials the coefficient
of thermal expansion becomes smaller at low temperatures, approaching
zero at the absolute zero; for this reason we prefer to leave a0(T) as an
undetermined function of the temperature, remembering only that it,
reduces to zero at the absolute zero (by the definition of F0), and that it
is very small compared to unity, since the temperature expansion of a
solid is only a small fraction of its whole volume.

The meaning of ai is simple: rt is almost exactly equal jto^jhc com-
pimsibilit/^^ The compressibility x is ordinarily defined

as — (\/V)(dV/dP)r, to be computed at zero pressure. From Kq. (1.1),
remembering that the volume at zero pressure is given by rn[l + a0(77)],
we have

where in the last form we have again neglected ao compared to unity.
The compressibility ordinarily increases with increasing temperature, so
that a\(T) must increase with temperature, enough to produce a net
mcrease'ih spite of the increase of the factor 1 + ao in the denominator of
Eq. (1.3). The increase is not very great, however; most compressi-
bilities do not change by more than 10 per cent or so between absolute
zera and hifih temperatures. I'h6 flUfthtity a* measures essentially the
change of comgrejsgibility with pressure .^JLtittkJs^ known experimentally
about its temperature variation, though it presumably increases with
temperature in some such way as ai does. The terms of the series in
P written down in Eq. (1.1) represent all that are required for most
materials and the available pressure range. Most measurements of
solids at high pressures have been carried out by Bridgman,1 who has
measured changes of volume up to pressures of 12,000 atm. with many

lSee P. W. Bridgman, "Physics of High Pressures," Chap. VI, The Macmillan
Company, 1931, and later papers.

202 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

solids and to 45,000 atm. with a few solids. At these highest pressures,
the most compressible solid, Cs, caesium, has its volume reduced to less
than half the volume at atmospheric pressure, and the other alkali metals,
Li, lithium, Na, sodium, K, potassium, and Rb, rubidium, have reductions
in volume of from 20 to 50 per cent. To represent these large changes of
volume accurately requires a considerable number of terms of such a
series as (1). Those are extreme cases, however; most solids are much
less compressible, and changes of volume of only a few per cent can be
produced with the available pressure, so that we can approximate quite
accurately by a quadratic function of pressure, as in Eq. (1.1). The
experimental results are usually stated by giving the relative change of
volume as a power scries in the pressure. That is, in our notation, we
have

70(1 + qQ) — V = aiP ___ az p2 n 4N

^ F0(l + a0) 1 + a0 1 + a0 ' V ;

The constants ai/(l + a0) and «2/(l + «o) are given as the result of
experiments on compressibility. If a0 is known from measurements of
thermal expansion, we can then find a\ and a2 directly from experiment.
The equation of state (1.1) is expressed in terms of pressure and
temperature as independent variables. We shall next express it in terms
of volume and temperature. We shall do this in the form

P = Pn(T) + P}(T)- + P2(T)- • • • (1.5)

Here Po(T), P\(T), and Pz(T) are functions of temperature, again chosen
to be positive. The meaning of PO is simple: it is the pressure that must
be applied to the solid to reduce its volume to F0, the volume which it-
would have at the absolute zero under no pressure. Obviously P0 goes
to zero at the absolute zero. At ordinary temperatures, while it repre-
sents a very considerable pressure, still it is small compared to the quan-
tities Pi and Pz, so that it can be treated as a small quantity in our
calculations and its square can be neglected. We shall see in a moment
that PI is approximately the reciprocal of the compressibility, or equals
the pressure required to reduce the volume to zero, if the volume decreased
linearly with increasing pressure (which of course it does not). Obvi-
ously this is much greater than the pressure required to reduce the volume
to Fo.

We shall now find the relations between the a's of Eq. (1.1) and the
quantities P0, PI, Pz of Eq. (1.5), assuming that we can neglect the squares
and higher powers of a0 and P0. To do this, we write Eq. (1.1) in the form

~ • • • , (1.6)

SBC. 1] THE EQUATION Of STATE OF SOLIDS 203

substitute in Eq. (1.5), and equate the coefficients of different powers of
P. We have

P - Po + Pi(-a0 +

ofP2 • • • ), (1.7)

where we have neglected a%. Equating coefficients, we have the equations
0 = Po -

- 2P2a0a,
0 = -P,a2 + 2P2aoa2 + P^al (1.8)

Solving for the a's, AVO have

„ Po
a«=p-

- 2P2a» ~ P

Similarly solving for the P's we have

Pi - -

1 «? /
at

(1.10)

Since we know how to find the a's from experiment, Eqs. (1.10) tell us
how to find the P's. We observe from Eqs. (1.10) that, as mentioned
before, PI is equal, apart from small terms proportional to a0, to the
reciprocal of the compressibility given in Eq. (1.3).

In addition to the equation of state, we must find the specific heat
from experiment. Ordinarily ono finds the specific heat at constant
pressure, CP) at atmospheric pressure, or practically at zero pressure. We
shall call this C£, to distinguish it from the general value of CPj which
can depencHm pressure. Le^usjSndJjic . jgp^dence^pn pressure. From
Eq. (1.6), Chap. VIII, we have ~ ~~ ~

>*P/T

Substituting for V from Eq. (1.1) and integrating with respect to pressure

204 INTRODUCTION TO CHEMICAL PHYSICS [Cinr. XIII

from P — 0 to P, we have

- v r*p - ip* + *p* • • - n in

VoTP ^P +^P (LH)

In case an, a\t and «2 can be approximated by linear functions of tempera-
ture, as we considered earlier for a0, the second derivatives in Eq. (L.ll)
will be zero and CP will be independent of pressure. Since da^/dT is
essentially the coefficient of thermal expansion, we see that the term in
Eq. (l.ll) linear in the pressure depends on the change of thermal expan-
sion with the temperature. We have mentioned that the thermal
expansion is zero at the absolute zero, increasing with temperature to an
asymptotic value. Thus we may expect d2aQ/dT2 to be positive, falling
off to zero at high temperatures, so that from Eq. (1.11) the specific heat
will decrease with increasing pressure, particularly at low temperature.
For theoretical purposes, it is better to use the specific heat at constant
volume, CV, computed for the volume V0 which the solid has at zero pres-
sure and temperature. We shall call this C°. CV will depend on the
volume as indicated by Eq. (1.7) of Chap. VIII:

Using Eq. (1.5) for the pressure, we obtain

1 • • (1.12)

1 3 dT

From Eq. (1.9), P0 is proportional to a0, so that its second derivative will
likewise; be positive, and we find that CV will decrease with decreasing
volume or increasing pressure, just as we found for Cp.

Since it is impracticable to find CV, or Cy, from direct experiment, it is
important to be able to find these quantities from CP. From Eq. (5.2),
Chap. II, we know how to find CP — CV: it is given by the formula
T(dV/dT)P(dP/dT)v. This gives the difference of specific heats at a
given pressure and temperature. We are more interested, however, in
the difference Cp — Cy, in which CP is computed at zero pressure, Cy at
the volume Fo. To find this difference, let us carry out a calculation of
CV at zero pressure, from Eq. (1.12). Here we have Fo — F = — F0a0,

from Eq. (1.1). Then Eq. (1.12) gives us CV = C°v + VQTad~~ Using
this value and the; equation for CP — CV, which we calculate for zero

SBC. 2] THE EQUATION Ob' STATK OF SOLIDS 205

pressure, we have

fK\ fin v Tn ^2^° - v Tda0dPQ
Op — or — Kn/a<r2 — ol ~'

, .,

(L13)

In tho derivation of Eq. (1.13), we have neglected the variation of a^
with temperature. In case the thermal expansion is constant, so that
ao = ctT, and the specific heat is independent of volume or pressure, Eq.
(1.13) takes the simple form

where we remember that a is Hie coefficient of thermal expansion, ai the
compressibility, to a good approximation. When numerical values are
substituted in Eq. (1.14), it is found that the difference of specific heats
for a solid is much less than for a gas, so that no great error is committed
if we use one in place of the other. This can be seen from the fact that
the difference of specific heats depends on aji, as we see in Eq. (1.13),
whereas elsewhere we have considered «n as being so small that its square-
could be neglected.

We have now discussed all features of the specific heats, except for
the dependence of C£ or CfJ themselves on temperature. Experimentally
it is found that the specific heat is zero at the absolute zero and rises to an
asymptotic valiuTat high temperatures, much like the specific heat of an
oscillator, as shown in Fig. IX-a. We shall see later that the t.frqrmal
energy oi a solid comes trom iheTscillations of the molecules, so that there
is a fundamental reason tor this behavior of the specific heat. We shall
also find theoretical formulas later which express the specific heat with
fairly good accuracy as a function of the temperature, formulas that differ
in some essential respects from Eq. (5.8), Chap. IX, from which Fig. IX-3
was drawn. For the present, however, where we are discussing thermo-
dynamics, we must simply assume that the specific heat is given by
experiment and shall treat C? and Cv as unknown functions of the tem-
perature, which however always reduce to zero at the absolute zero of
temperature.

2. Thermodynamic Functions for Solids. — In the preceding section
we have seen how to express the equation of state and specific heat of a
solid as functions of pressure, or volume, and temperature. Now we shall
investigate the other thermodynamic functions, the internal energy,
entropy, Helmholtz free energy, and Gibbs free energy. For the internal

206

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

energy as a function of volume and temperature, we have the relations
(dU/dT)v = Cv, (dU/dV)T = T(dP/dT)v - P. Let the energy of the
solid at volume Fo and zero temperature be C/oo. Then we find the
energy at any temperature and volume by starting at VQ at the absolute
zero, raising the temperature at volume VQ to the desired temperature,
and then changing the volume at this temperature. Using Eq. (1.5), we
find at once

U - t/oo +

(2.1)

The internal energy of metallic sodium is shown as a function of volume in
Fig. XIII-2, as an illustration. On account of the large compression
that can be attained with sodium, more terms of the power series must
be retained than are given in Eq. (2.1), but it is easier to show the prop-
erties of this metal than of a less compressible one. Lot us consider the
behavior of the internal energy as a function of volume at fixod tempera-
ture. If the thermal expansion is independent of temperature, so that P0

is proportional to the temperature and
dPQ/dT is a constant, the coefficient of
the term in (F0 - V) in Eq. (2.1) is
zero and the term in (Vo — V)'2 is the
principal term in U. In this term, PI,
which is the reciprocal of the compressi-
bility, is large compared to T dPi/dT, so
that the coefficient of (F0 — F)2 is posi-
tive and the internal energy has a mini-
mum at Fo, just as it does at the absolute
zero. If the thermal expansion depends

,.5

1.0

0.5

0.9

10
V/V0

for various temperatures. The dotted ordinarily to smaller volumes.

line connects points at zero pressure. . .

is an interesting consequence

12 on temperature, the term in (F0 — V)
will have a small coefficient different

FIG. XIII-2. — Internal energy of a . . 1.1,1

solid (sodium) as function of volume from zero, shifting the minimum slightly,

There
of the

fact that the minimum of U is approximately at F0. At ordinary
temperatures, the volume of the solid at zero pressure, which as we have
seen is Fo(l + «o), will be greater than Vo. Then on compressing the
solid, its internal energy will decrease until we have reduced its volume
approximately to Fo, when it will begin to increase again. Of course,
work is constantly being done on the solid during the compression, but
so much heat flows out to maintain the temperature constant that the
total energy decreases, with moderate compressions. The internal

SBC. 2] THE EQUATION OF STATE OF SOLIDS 207

energy of course increases as the temperature is raised at constant volume,
as we see from the obvious relation (dU/dT)v = CV, so that the curves
corresponding to high temperatures lie above those for low temperature.
Furthermore, since the specific heat is higher at large volume, as we saw
from Eq. (1.12), the spacing of the curves is greater at large volume,
resulting in the slight shift of the minimum to smaller volume with
increasing temperature.

The entropy is most easily determined as a function of volume and
temperature from the equation (dS/dT)v = CV/T. At the absolute zero
of temperature, the entropy of a solid is zero independent of its volume or
pressure. The reason goes back to our fundamental definition of entropy

in Chap. Ill, 8 — ~^2/t In /t> where/,- represents the fraction of all

systems of the assembly in the ith state. At the absolute zero, according
to the canonical assembly, all the systems will be in the state of lowest
energy, which will then have/ = 1, all other states having/ = 0. Thus
automatically S = 0. We can then find the entropy at any temperature
and volume as follows. First, at absolute zero, wo change the volume*
to the required value, with no change of entropy. Then, at this constant
volume, we raise the temperature, computing the chango of entropy from
JCv/TdT. We can use Eq. (1.12) for the specific heat at arbitrary
volume. Carrying out the integration from that equation, we have at
once

_ fVr- v\^(Y^L

Jo *T y°[dT\ V,

YVo -2 , «jL-

+ 2'dT\ V, ^' " ( }

The entropy of sodium, as computed from Eq. (2.2), is plotted in Fig.
XIII-3 as a function of temperature, for several volumes. Starting from
zero at the absolute zero, the entropy first rises slowly, since its slope,
Cv/Tj goes strongly to zero at the absolute zero. As tho temperature
rises, the curve goes over into something more like the logarithmic form
which it must have at high temperature, where CV becomes constant, and
S = JCV dT/T = CV In T + const. From the curves, it is plain that
the entropy increases with increasing volume, at constant temperature.
This can be seen from Eq. (2.2), in which the leading term in the variation

7jp

with volume can be written -j7ft(V — F0), where from Eq. (1.10) we see

»p

that -j~ is approximately the thermal expansion divided by the com-

pressibility. It can also be seen from the thermodynamic relation

208

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

which is seen, if we multiply and divide by 1/F, to be exactly the thermal
expansion divided by the compressibility. The reason for the increase of
entropy with increasing volume is simple: if the volume increased, or the
pressure decreased, adiabatically, the material would cool; to keep the
temperature constant heat must flow in, increasing the entropy.

The Helmholtz free energy A = U — • TS can be found from Kqs.
(2.1) and (2.2) for U and S, or can be found by integration of the equations

0 100 200

Deg. Abs.

Flu. XIII-3. — Entropy of a solid (sodium) as function of the temperature tit constant

volume.

(dA/dT)v = —S, (dA/dV)T = -P. The latter method is perhaps a
little more convenient. At the absolute zero and volume To, the Helm-
holtz free energy equals the internal energy and is given by f/oo, as in Eq.
(2.1). From that point we increase the temperature at volume VQ to
the desired temperature, and then change the volume at this temperature.
We find at once

A =

In Fig. XIII-4 we show A as a function of volume for a number of tem-
peratures. At the absolute zero, as we have mentioned above, the
Helmholtz free energy equals the internal energy, as given in Fig. XIII-2.

SEC. 2]

THE EQUATION OF STATE OP SOLIDS

209

From the equation (dA/dV)T = — P, we see that the negative slope of the
Helmholtz free energy curve is the pressure, and the change of Helmholtz
free energy between two volumes at constant temperature gives the
external work done in changing the volume. It is for this reason, of
course, that it is called the free energy. Thus the minimum of each curve
corresponds to the volume where the pressure is zero. It is obvious from
the graph that this minimum moves outward to larger volumes with
increase of temperature; this represents
the thermal expansion. In particular,
it is plain that this shift of the mini-
mum is very small for low temperatures,
corresponding to the small thermal
expansion at low temperatures. Since
the slope of the free energy curve gives
t he negative pressure, it is only the part
of the curve to the left of the minimum
that corresponds to positive pressure _£•
and has physical significance.

Finally, we consider the Gibbs free
energy,

G = U +PV - TS = A + Plr,

as a function of pressure and temper-
ature. This is most conveniently found
from the relations (dU/dP)? •= T,
(dG/dT)?=-S. Starting at the abao- „ VTTT ^ U1 . 14 ,

' ' ** FIG. XIII-4. — Helmholtz free energy

lute zero and zero pressure, where the of a solid (sodium) as function of tho
value of G is Z70o, we first increase the volume at c»n«*ant temperature,
temperature at zero pressure, then increase the pressure at constant
temperature, finding

•r j

- f(

PV.(l

a,) -

(2.4)

111 Fig. XIII-5, we plot G as a function of pressure, for a number of tem-
peratures. The term PVo(l + a0) is by far the largest one in (7, resulting
approximately in straight lines proportional to P. The spacing of the
curves is determined by the entropy: (dG/dT)P = — S, showing that G
decreases with increasing temperature at constant pressure and that the
decrease is greater at low pressure (large volume) than at high pressure.
These details of the change of the Gibbs free energy with temperature
are not well shown in Fig. XIII-5, however, on account of scale, and

210

I \TRODUCTWN TO CHEMICAL I'HYMC'ti [CHAP. Xii(

this sort of plot does not give a great deal-of useful information. Before
leaving it, it is worth while pointing out the resemblance to Fig. XII-4,
where we plotted G as a function of pressure for a liquid and gas in equilib-
rium, as given by Van der Waals' equation, and found again almost
straight lines.

The more useful way to give G graphically is to plot it as a function of
temperature for constant pressure, as we do in Fig. XIII-6. The slope of
these curves, being —5, is zero at the absolute zero, negative at all higher
temperatures, so that the curves slope down. The Gibbs free energy
decreases more slowly with temperature at high pressure, where the
entropy is lower, than at zero pressure. At zero pressure, the term PV

10,000 2QOOO

P Atmospheres

FIG. XIII-5. — Gibbs free energy of a solid (sodium) as function of the pressure at constant
J temperature.

is zero, so that the Gibbs free energy G equals the Helmholtz free energy
A. The difference between the two functions is small at low pressures, so
that at pressures of a few atmospheres the two functions can be used
interchangeably for solids. This of course does not hold for gases, for
which the volume V is much greater, and the term PV is very large even
at small pressures. As we can see from Chap. XI, Sees. 3 and 4, this
diagram, of G as a function of 77, is the important one in discussing the
equilibrium of phases, since the condition of equilibrium is that the two
phases should have the same Gibbs free energy at the same pressure and
temperature. Thus if we draw G for each phase, as a function of tem-
perature, for the pressure at which the experiment is carried out, the point
of intersection will give the equilibrium temperature of the two phases at
the pressure in question.

SEC. 3)

THE EQUATION OF STATE OP' SOL1DX

211

We have already shown, in Figs. XI-4, XI-5, XI-6, and XI-7, the
equation of state, entropy, and Gibbs free energy of a substance in all of
its three phases. Examination of the parts of those figures dealing with
solids will show the similarity of those curves to tho ones found in tho
present section in a more explicit and detailed way.

3. The Statistical Mechanics of Solids. — The first step in discussing
a solid according to statistical mechanics is to set up a model, describing
its coordinates and momenta, finding its energy levels according to the1
quantum theory, and computing the partition function. This represents
an extensive program, of which only the outline can bo given in the
present chapter. The typical solid is a crystal, a regular repeating

s.
3

P- 20. 000 atm

P- 10,000 atm

0 100 200 300

Deg Abs.

FIG. XIII-6. — Gibbs free energy of a solid (sodium) as function of temperature at constant

pressure.

structure composed of molecules, atoms, or ions. The repeating unit is
called the unit cell. The crystal is hold together by forces between the
molecules, atoms, or ions — forces that resemble those between atoms in
diatomic molecules, as discussed in Chap. IX, in that they lead to attrac-
tion at large distances, repulsion at small distances, with equilibrium
between. The interplay of the attractive and repulsive forces of all
atoms of the crystal leads to a state of equilibrium in which each atom
has a definite position, in which no forces act on it. At the absolute zero
of temperature, the atoms will be found just in these positions of equilib-
rium. At higher temperatures, however, they will vibrate about the
positions of equilibrium, to which they are held by forces proportional to
the displacement, if the displacements are small. We shall divide our
discussion of the model into two parts: first, the crystal at the absolute

212 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

zero with its atoms at rest in their equilibrium positions; secondly, the
thermal vibrations of the atoms about these positions.

Let us first consider the crystal at the absolute zero. The, energy
will depend on the state of strain of the crystal; as was mentioned in
Sec. 1, we omit discussion of shearing strains and types of deformation
other than change of volume. Thus we are interested simply in the
dependence of energy on volume. As the volume is changed, of course
each unit cell changes in the same proportion and the atoms change their
positions in the crystal. The interatomic energies also change, and the
change in energy of the whole crystal is simply the sum of the changes
of the energies of interaction of the various atoms. We cannot say
anything further about the energy as a function of volume, without
investigating specific examples, as we shall do in later chapters. But at
least we may assume that the energy of interaction of two atoms is most
conveniently expressed as a function of the distance of separation, and if
the whole energy is a sum of these energies of interaction, it also may be
expected to be particularly simple when regarded as a function of a linear
dimension of the crystal, rather than as a function of the volume. We
shall, therefore, express the energy of the crystal at the absolute zero as a
function of a quantity r, which may be a distance between atoms, a side of
a unit cell, or some other linear dimension of the crystal, and shall find
results that will later be useful to us, when we knowr more about the nature
of the interatomic forces.

Consider a crystal of volume V, containing N atoms or molecules.
(We purposely leave the description slightly vague, so as to allow more
generality in the result.) Then V/N is the volume per atom or molecule,
a quantity which of course can be changed by application of external
pressure. We shall limit the present discussion to cubic crystals, in
which only the volume, and not the shape, changes under pressure; many
crystals do not have this property, but the ones that we shall discuss
quantitatively happen to be cubic. Then V/N will be a numerical
constant times r3, the volume of a cube of side r, since in a uniform com-
pression the whole volume and the volume r3 will change in proportion.
Thus let

jj = <**, (3-1)

where c will be a definite number for each structure, which we can easily
evaluate. We define a quantity r0 in terms of F0, the values respectively
of r and V when the crystal is under no pressure at the absolute zero.
Thus we have

Fo -V r* -r3 _
V~o ~ rjj

SBC. 3] THE EQUATION OF STATE OF SOLIDS 213

We now take the expression (2.1) for the internal energy as a function
of volume, set the temperature equal to zero, and use Eq. (3.2), finding
the internal energy at absolute zero as a function of the linear dimensions.
Calling this quantity t/o, we have

where PJ, P£ are the values of the quantities PI, P2 of Eq. (2.1) at the
absolute zero of temperature. (We note that P0 == 0 at the absolute
zero.) Substituting from Eq. (3.2) and retaining terms only up to the
third, we have

[/00 + NcrP°~-~ - 9(P? - pojTir. (3.4)

Equation (3.4) will later prove to bo convenient, in cases where we have
a theoretical way of calculating t/o from assumed interatomic forces.
In these cases, P? and PI can be found directly from the theory, using
Eq. (3.4).

Our next task is to consider the solid at a higher temperature than
the absolute zero. The molecules and atorps will have kinetic energy and
will vibrate. We can get a simple, but incorrect, picture of the vibrations
by thinking of all the atoms but one as being fixed and asking how that
one would move. It is in a position of stable equilibrium at the absolute
zero, being held by its interactions with its neighbors in such a way that
it is pushed back to its position of equilibrium with a force proportional
to the displacement. Thus it will execute simple harmonic motion, with
a certain frequency v. To discuss the heat capacity of this oscillation,
we may proceed exactly as in Chap. IX, Sec. 5, where we were talking
about the heat capacity of molecular vibrations. Each atom can vibrate*
in any direction, so that its #, ?/, and z coordinates separately can execute
simple harmonic motion. It is then found easily that the classical
partition function for vibration for a single atom is (kT/hv)*, similar to
Eq. (5.4), Chap. IX, but cubed on account of the three dimensions. This
corresponds to a heat capacity of 3k per atom, or 3R per mole, if the
material happens to be monatomic, with corresponding values for poly-
atomic substances. This law, that the heat capacity of a monatomic
substance should be 3R or 5.96 cal. per mole, at constant volume, is called
the law of Dulong and Petit. It is a law that holds fairly accurately at
room temperature for a great many solids and has been known for over a
hundred years. It was first found as an empirical law by Dulong and
Petit. At lower temperatures, however, the specific heats of actual solids
are less than the classical value, and decrease gradually to zero at the
absolute zero.

214 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

It was to explain these deviations from the law of Dulong and
Petit that Einstein developed his theory of specific heats. He treated the
vibrations of the separate atoms by quantum theory, just as we did in
Sec. 5, Chap. IX, and derived the formula

/ hv \2

(«w-i)

- 1;
e

(3.5)

where

__
y
- l)

0 = ~ (3.6)

Equation (3.5) is analogous to Eq. (5.7), Chap. IX, but is multiplied by
3 on account of the three degrees of freedom. As we have seen in Fig.
IX-3, this gives a specific heat rising from zero at the absolute zero to the
classical value 3/2 at high temperatures. It is found that values of the
frequency i>, or of the corresponding characteristic temperature 9, of
Eq. (3.6), can be found, such that the Einstein specific heat formula (3.5)
gives a fairly good approximation to the observed specific heats, except at
very low temperatures. Close to the absolute zero, the Einstein formula
predicts a specific heat falling very sharply to zero. The actual specific
heats do not fall off so rapidly, but instead are approximately propor-
tional to T3 at low temperatures. Thus, while Einstein's formula is
certainly a step in the right direction, we cannot consider it to be correct.
The feature that we must correct is the one in which we have already
rioted that this treatment is inadequate: the atoms are not really held to
positions of equilibrium but merely to each other. In other words, we
must treat the solid as a system of many atoms coupled to each other, arid
we must find the vibrations of these atoms. This is a complicated prob-
lem in vibration theory, something like the problems met in the vibrations
of polyatomic molecules. We shall not take it up until the next chapter.
In the meantime, however, there are certain general results that we can
find regarding such vibrations, which are enough to allow us to make
considerable progress toward understanding the equation of state of
solids.

A system of N particles, held together by elastic forces, has in general
3AMJ vibrational degrees of freedom, as we saw in Eq. (6.1), Chap. IX,
where we were talking about polyatomic molecules. Really a whole
crystal, or solid, can be regarded as an enormous molecule, and for large
values of N we can neglect the 6, saying merely that there are SN vibra-

SEC. 4] THE EQUATION OF STATE OF SOLIDS 215

tional degrees of freedom. In general, there will then be 3N different
normal modes of vibration, as they are called. Each normal mode
consists of a vibration of all the atoms of the crystal, each with its own
amplitude, direction, and phase, but all with the same frequency. Each
atom then finds itself surrounded, not by a stationary group of neighbors,
but by neighbors which are oscillating with the same frequency as its own
motion. At each point of its path, it will always find the neighbors in
definite locations, so that the forces exerted on it by its neighbors will
depend only on its position; but the forces will not be the same as if the
neighbors remained at rest, for the positions will be different. Thus the
frequency will not be the same as assumed in Einstein's theory. Our
problem in the next chapter will be to consider these 3N modes of vibra-
tion and to find their frequencies, which in general will all be different.
For the present, however, we may simply assume the frequencies to be
known, and equal to vi . . . PS.V. The most general motion of the
atoms, of course, is not one of these normal modes but a superposition of
all of them, with appropriate amplitudes. This has a simple and in fact
a very fundamental analogy in the theory of sound. The normal modes
of a vibrating string or other musical instrument are simply the different
harmonic overtones in which it can vibrate. Each one consists of a purely
sinusoidal vibration, in which the string is divided up by nodes into cer-
tain vibrating segments. The simplest type of vibration of the string is
an excitation of only one of these overtones, so that it vibrates with a
pure musical tone. But the more general and common type of vibration
is a superposition of many overtones, each with an appropriate amplitude
and phase; it is such a superposition which gives a sound of interesting
musical quality. As a matter of fact, if we ask about the 3N vibrations
of a piece of matter, for studying its specific heat, we find that the vibra-
tions of low frequency are exactly those acoustical vibrations which are
considered in the theory of sound. As we go to higher and higher fre-
quencies and shorter and shorter wave lengths, however, the vibrations
begin to depart from the simple ones predicted by the ordinary theory of
sound, and finally when the wave length begins to be comparable with the
interatomic distance, the departure is very great. We shall investigate
the nature of these vibrations, as well as their frequencies, in the next
chapter.

4. Statistical Mechanics of a System of Oscillators. — I^jiamically,
we have seen that a.jprystal can be approximated by a set of 3N vibra-
tions, if there are jV^atonaa in the, crystal. These vibrations have fre-
quencies which we may label vi . . . VZN, varying through a wide range of
frequencies. To the approximation to which the restoring forces can be
treated as linear, these oscillations are independent of each other, each one
corresponding to a simple harmonic oscillation whose frequency is inde-

216 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

pendent of its own amplitude or of the amplitudes of other harmonics.
This is only an approximation, but it is sufficient for most purposes.
Then the energy is the sum of the energies of the various oscillators, and
each of these is quantized. That is, the energy of the jth oscillator
ran take on the values (n/ + ?)hvj, where n/, an integer, is the quantum
number associated with this oscillator. We see that 3N quantum num-
bers aro necessary to describe the total energy and to define a stationary
state. All these quantum numbers should then appear as subscripts of
the energy, and we have the relation

where U'Q is the energy which the lattice would have if the amplitudes
of all oscillations were zero. Actually, even at the absolute zero of
temperature, however each oscillation has a half quantum of energy.
Thus we may write

where

3tf

f/o = U'0 + 5iAiv. (4.3)

The quantity t/o is the same as that givon in Eq. (3.3), representing the
energy of the lattice, as a function of volume, at the absolute zero of tem-
perature. The subscripts n\ . . . HS.V take the place of the single index i
which we ordinarily use in defining the partition function. Thus we find
for the partition function

_ __ -

z = ekT = 2 • • • 2c jkT ekT (4*4)

HI U3N

We can write the exponential as a product of terms each coming from a
single value of j, and can carry out the summations separately, obtaining

(4.5)

Each of the summations in Eq. (4.5) is of the form already evaluated in
Sec. 5, Chap. IX. Thus we have

SBC. 4] THE EQUATION OF STATE OF SOLIDS 217

a product of 3N terms. Taking the logarithm, we have at once

A « C7o + 2*!T In (l - c~$). (4.7)

Differentiating with respect to 7\ we have

(hv, v
/ _^\ IT \

-1» U -'' "7 +-J- — )• (4-8)

e*r - I/
Finally, from Eq. (5.20), Chap. Ill, we hav«

By differentiating Eq. (4.9), we find the specific heat in agreement with
the value previously found, Eq. (3.5), for the special case whoro all 3N
frequencies are equal.

Having found the Helmholtz free energy (4.7), we can find the pressure
by differentiating with respect to volume. We have

p = — I —

\dV

where

The first two terms in Eq. (4.10) are the* pressure at the absolute zero of
temperature, which wo have already discussed. The summation repre-
sents the thermal pressure. It is different from zero only because the
7/s are different from zero; that is, because the vibrational frequencies
depend on volume. Wenaturally expect this dependence; as the crystal
is compressed it becomes harder, the restoring forces become greater, and
vibrational frequencies increase, so that the P/S increase with decreasing
volume and the 7/5 are positive. If we consider the 7/3 to be inde-
pen9ehEDf~ temperature, each term of the summation in Eq. (4.10) is
proportional to the energy of the corresponding oscillator as a function of
temperature, given by Eq. (4.9), divided by the volume V.

At high temperatures, we know that the quantum expression for the

i hv-

energy of an oscillator, ^hvj + -A- — - -> approaches the classical value

~ — — -~— - ••IIIIIMH_ imi n miiii a i« ••«II"W«B<II« nil < >ui«iM»M«»»mMk « „«,,, , _

\ ekT — 1

218 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

kT. Thus, at high temperature the thermal pressure approaches^

The first term is an expression similar to the pressure of a perfect gas,
NkT/V, except that the constant of proportionality is now Vy/ instead of

N, the number of molecules. We shall find that the 7/s are generally
between 1 and 2, so that, since there are 3N terms in the summation
ovorj, whore N is the number of atoms, the thermal pressure asjndicated
by Eq. (4.12) has a term which is from three to six times as greatjts the
corresponding pressure of a perfect gas of the same number of atoms, at
the same temperature and volume as the solid. The second term of
Eq. (4.12), coming from the zero point energy, gives a decrease of thermal
pressure compared to this gas pressure which is independent of tempera-
ture, until wo go down to low temperatures. There the thermal pressure
does not decmise so rapidly with decreasing temperature as we should
estimate from the gas law, but instead falls to the value zoro at the
absolute zero. The change of thermal pressure with temperature, involv-
ing the derivative of Eq. (4.10), approaches zoro very strongly at the
absolute zoro, just as the specific boat does, and sinoo this derivative
enters the formula for thermal expansion, this quantity goes to zero at
the absolute zoro. In fact, as we shall soo in the noxt paragraph, there is a
close connection between the thermal expansion and the specific heat.

To got a complete equation of state from statistics, we need only
take the expression (4.10) for P and expand tho summation, the thermal
pressure, as a power series in (Fo — F)/Fo. This can be donqjf we can
nniiJj£ de£)endence of .the i//s on volume, from the theory. Then we can
identify tho resulting equation with Eq. (1.5), equating coefficients, and
findPo, -Pi, and P^ which determine the pressure, in terms of the structure
of the crystal. Knowing the moaning of P0, PI, and PZ from our earlier
discussion, this allows us to find the thermal expansion, compressibility,
and change of compressibility with pressure, as functions of temperature.
Since very fow experiments are available dealing with the changes of these
quantities with temperature, we shall confine our attention to the thermal
expansion at zero pressure;. From Kq. (1.2), this is approximately
ddo/dT, where from Eq. (1.9) we have a0 = Po/Pi. Comparing with
Eq. (4.10) above, we soe that P0 is the value of the summation when
V = Fo. The quantity PI, which we have seen to be the reciprocal of
the compressibility, equals PJ plus a small term coming from the summa-
tion, which we can neglect for a very rough discussion, though of course it
would have to be considered for accurate work. Since P? is independent

SEC. 4] THE EQUATION OF STATE OF XOLIDS 219

of temperature, this gives us

k "%ri \kT/
Thermal expansion = X- >j y, \u. — ry (4.13)

u. — r
^- 1/

where x is the compressibility. Comparison of Eq. (4.13) with the
formula for heat capacity of linear oscillators, for example Eq. (3.5),
shows at once the close relation between the heat- capacity and the thermal
expansion. Each term in Eq. (4.13) is proportional to the term in the
heatjjapacity arising from the same oscillator, so that the thermal expan-
sion shows qualitatively the same sort of behavior, becoming constant at
high temperatures but reducing to zero as the temperature approaches
the absolute zero. While experimental data for thermal expansions are
not ne^lj^ojBxUinsive jis those for specific heats, still they are sufficient
tojjjhow that this is actually the observed behavior.

To allow the construction of a simple theory of thermal expansions,
Grlineisen assumed that the quantities 7, were all equal to each other and
to a constant 7, which he regarded as an empirical constant. To see the
meaning of 7, we assume that the frequencies v} are given in terms of the
volume by the relation

", = |~> (4.14)

where c3 is a constant, so that the frequencies are inversely proportional
to the 7 power of the volume. Since surely the p/s increase with decreas-
ing volume, this is a reasonable form of dependence to assume. Then we
find at once that

-l1^ - y (Al&)

d In V " 7' C4-15'

so that the 7 defined in Eq. (4.14) is the same as the 7, of Eq. (4.J1).
We see that 7 = 1 or 7 = 2 respectively corresponds to the frequencies
being inversely proportional to the volume or to the square of the volume.
If we assume with Griineiseri that 7,- = 7, we then have from Eq. (4.13)

Cv ]
Thermal expansion = -^-- ' (4.16)

y o

Equation (4.16) is a relation between the thermal expansion, compressi-
bility, specific heat, volume, and the parameter 7. If we have an inde-
pendent theoretical way of finding 7, we can use it to compute the thermal
expansion. Otherwise, we can use measured values of thermal expansion,
compressibility, specific heat, and volume, to find empirical values of y.
Both types of discussion will be given in later chapters, where we discuss

220 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIII

specific types of solids. We shall find that the agreement between the
various methods of finding 7 is rather good, and that values for most
ordinary materials are between 1 and 3, generally in the neighborhood
of 2.

6. Polymorphic Transitions. — It has been mentioned in Chap. XI,
Sec. 7, that the transition lines in the P-T diagram between polymorphic
phases of the same substance tend to have positive slope, a change of
pressure of something less than 12,000 atm. corresponding to a change of
temperature of about 200°. With our present knowledge of the equation
of state of solids, we can attempt a theoretical explanation of this relation.
To work out the slope, we note that Clapeyron's equation can be written
dP/dT = AS/A7, where AS is the difference of entropy between one
phase and the other, AF the difference of volume. We have seen in the
present chapter that the entropy of a single phase increases when its vol-
ume increases, and we have found quantitative methods of calculating the
amount of change. Let us, then, tentatively assume that the relation
of change of entropy to change of volume in going from one phase to
another is about the same as when we change the volume of a single
phase. This is certainly a very crude assumption, but we shall find that
it gives results of the right order of magnitude. The assumption we have
just made amounts to replacing AS/AF by dS/dV, computed for a single
phase. Now from Eq. (4.8) we can write the entropy of a substance per
gram atom as

f / _±A hv/kT I

S = 3Nok\ - In VI - e kT) + -jr - , (5.1)

L ** - ij

where No is Avogadro's number, and we assume for simplicity that all
frequencies vt are the same. We then have

rfS _ dS dv^ _ __ v_ dS f

dV ~ dvdV ~ 7F dv (b'Z)

where 7 has the same significance as in the last section. Differentiating
Eq. (5.1), this leads to

dS _ 3AT0fc7AA2 F* _ _ yCv ,. ^

dV V \kf) ^ Ji)*~~V '

Now Cv is about 3R per gram atom, and from the preceding section we see
that 7 is about 2 for most substances. Furthermore, examination of
experimental values shows that V is of the order of magnitude of 10 cc.
per gram atom, for most substances. Putting in these values and putting
proper units in Eq. (5.3), we find that dS/dV is approximately 50 atm. per
degree, or 10,000 atm. for 200 degrees, just about the value that Bridgman

SBC. 5] THE EQUATION OF STATE OF SOLIDS 221

finds to be most common experimentally. Thejaot 1ftftt' this fifl.)milfl.t.inn
agrees so welLwithjbhc average behavior of many materials is some justi-
fication_for thinking that the major part of the entropy change from Qwe
polymorphic phase .to another is simply that associated with the change of
volume. The individual variations are so great, however, that no groat
claim for accuracy can be made for such a calculation as we have just
made.

In the present chapter, we have laid the foundations for a statistical
study of the equation of state of solids, though we have not made any use
of a model, and hence have not been able to compute the thcrmodynamic
quantities we have been talking about. We proceed in the next chapter
to a discussion of atomic vibrations in solids, with a view to finding more
accurate information about specific heats and thermal expansion. Later,
when we study different types of solids more in detail, we shall make
comparisons with experiment for many special cases.

CHAPTER XIV
DEBYE'S THEORY OF SPECIFIC HEATS

We have seen in the last chapter that the. essential step in investigating
the specific heat and thermal expansion of solids is to find the frequencies
of the normal modes of elastic vibration. We shall take this problem
up, in its simplest form, in the present chapter. The vibrations of a solid
are generally of two sorts: vibrations of the molecules as a whole and
internal vibrations within a molecule. This distinction of course can be
found only in molecular crystals and is lacking in a crystal, like that of
a metal, whore all the atoms are of the same sort. For this reason solids
of the elements have simpler specific heats than compounds, and we take
them up first, postponing discussion of compounds to the next chapter.
At first sight, on account of the large number of atoms in a crystal, it
might seem to be impossibly hard to solve the problem of their elastic
vibrations, but as a matter of fact it is just the largo number of atoms
that makes it possible to handle the problem. For the vibrations of a
finite continuous piece of matter can bo handled by the theory of elas-
ticity, and for waves long compared to atomic dimensions this theory is
correct. We start therefore by considering the elastic vibrations of a
continuous solid, and later ask how the vibrations arc affected by the
fact that the solid is really made1 of atoms. We have already mentioned
briefly in Chap. XIII, Sec. 3, the close relation between the normal modes
of vibration of a solid composed of atoms and the harmonic or overtone
vibrations of acoustics.

1. Elastic Vibrations of a Continuous Solid. - It is well known that
elastic waves can be propagated through a solid. The waves are of two
sorts, longitudinal and transverse, having different velocities of propaga-
tion. The longitudinal waves are analogous to tin; sound waves in a fluid,
while the transverse waves, which cannot exist in a fluid, also have many
properties similar to sound waves and are ordinarily treated as a branch
of acoustics. The velocities of both sorts of waves are determined by the
elastic constants of the material and are independent of the frequency, or
wave length, of the waves, within wide limits. The waves with which we
are familiar have frequencies in the audible range, less than 10,000 or
15,000 cycles per second. The velocities of clastic waves in solids are of
the order of magnitude of several thousand meters per second (something
like ten times the velocity in air). Since we have

222

SBC. 1] DE BYE'S THEORY OF SPECIFIC HEATS 223

\v = y, (1.1)

where X is the wave length, v the frequency or number of vibrations per
second, and v the velocity, the {shortest sound waves with whieh we are
familiar have a wave length of the order of magnitude of

(5 X 105) _

— Q- =50 cm.,

taking the velocity to be 5000 m. per second, the frequency 10,000 cycles.
By methods of supersonics, frequencies up to 100,000 cycles or more can
be investigated, corresponding to waves of something like 5 cm. length.
There is every reason to suppose, however, that this is not the limit for
elastic, waves. In fact, we have every reason to believo that waves of
shorter and shorter wave length, and higher and higher frequency, arc
possible, up to the limit in which the wave length is comparable with the
distance between atoms. It is obvious that the wave length cannot be
appreciably shorter than interatomic distances. In fact, if the wave
length were just the interatomic distance*, successive atoms would be in
the same phase of the vibration, and there would not really be. a vibration
of one atom with respect to another at all. The shortest wave which
we can really have comes when successive atoms vibrate opposite to each
other, so that the \\ave length is twice the distance between atoms. It is
interesting to find the corresponding order of magnitude of the frequency
of vibration. If we set X = 5 X 10~~s, of the order of magnitude of twice1
an interatomic distance in a metal, we have1

(5 X 105) inll . i

= (5~X 10~:~s) = cycles per second.

This is a frequency of an order of magnitude of those found in the infrared
vibrations of light waves. There is good experimental evidence that
such frequencies really represent the maximum possible frequencies of
acoustical vibrations.

The situation, then, is that there is a natural upper limit set to fre-
quencies, and lower limit to wave lengths, of elastic waves, by the atomic
nature* of matter. It can be shown theoretically that as this limit is
approached, the velocity of the waves no longer is independent of wave*
length. However, the change is not great; it changes by something not
more than a factor of two. This change is the only important difference
between a vibration theory based on the theory of elasticity and a theory
based directly on interatomic forces, provided only that we recognize the
lower limit to wave lengths. Our first approach to a theory, the one made
by Debye, takes account of the lower limit of wave lengths but neglects
the change of velocity with frequency.

224 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

In a finite piece of solid, such for instance as a rod, standing waves
are set up. These arise of course from constructive interference between
direct and reflected waves, and they exist only when the wave length and
the dimensions of the solid have certain relations to each other. For the
transverse vibrations of a string, these relations are very familiar: tho
length of the string must be a whole number of half wave lengths. For
other shapes of solid, the relations are similar though not so simple.
Arranging the standing waves in order of decreasing wave length or
increasing frequency, we have a series of frequencies of vibration, often
called characteristic frequencies, or harmonics. For the string, these
are simply a fundamental frequency and any integral multiple of this
fundamental. The resulting harmonics or overtones form the basis of
musical scales and chords. For other shapes of solids, the relations are
not simple, and the overtones do not form pleasing musical relations
with the fundamental. Now for all one knows in ordinary acoustical
theory, the number of possible overtones is infinite, though of course few
of them can be heard on account of the limitations of the ear. Thus if we
have a string, with frequencies which are integral multiples of a funda-
mental, there seems no reason why the integer cannot be as large as wo
please. This no longer holds, as we can immediately see, when we con-
sider the atomic nature of the solid. For we have just mentioned that
there is an upper limit to possible frequencies, or a lower limit to possible
wave lengths, set by interatomic distances. The highest possible over-
tone will have a frequency of the order of this limiting frequency. That
means that the solid has a finite number of possible overtone vibrations.
And now we see the relation between our acoustical treatment and the
vibration problem we started with: these overtone vibrations are just the
normal modes of vibration of the atoms in the crystal, which we wanted to
investigate. If there are N atoms, with 3N degrees of freedom, we have
mentioned in Chap. XIII that we should expect 3N modes of oscillation;
when we work out the number of overtones, we find in fact that there are
just 3AT allowed vibrations.

The most general vibrational motion of our solid is one in which
each overtone vibrates simultaneously, with an arbitrary amplitude and
phase. But in thermal equilibrium at temperature T, the various vibra-
tions will be excited to quite definite extents. It proves to be mathe-
matically the case that each of the overtones behaves just like an
independent oscillator, whose frequency is the acoustical frequency of the
overtone. Thus we can make immediate connections with the theory of
the specific heats of oscillators, as we have done in Chap. XIII, Sec. 4. If
the atoms vibrated according to the classical theory, then we should have
equipartition, and at temperature T each oscillation would have the mean
energy kT. This means that each of the N overtones would have equal

SEC. 2] DE BYE'S THEORY OF SPECIFIC HEATS 225

energy, on the average, so that the energy of all of them put together
would be 3NkT, just as we found in Chap. XIII, Sec. 3, by considering
uncoupled oscillators. The fundamental and first few harmonics, which
are in the audible range, would have the average energy kT, just like the
harmonics of higher frequency. This does not mean that we should be
able to hear the rod in thermal equilibrium, because kT is such a small
energy that the amplitude of each overtone would be quite inappreciable.
Of the 3N harmonics, by far the largest number come at extremely high
frequencies, and it is here that the thermal energy is concentrated. The
superposition of these high frequency overtone vibrations, each with
energy proportional to the temperature, is just what we mean by tem-
perature vibration, and the energy is the ordinary internal energy of the
crystal. Actually the oscillations take place according to the quantum
theory rather than the classical theory, and we have seen in Chap. XIII,
Sec. 4, how to handle them. Each frequency v} can have a characteristic
temperature 6, associated with it, according to the equation

(1.2)
Then the heat capacity is

so that the heat capacity associated with each oscillator will be zero at
temperatures much below 9,, rising to the classical value at temperatures
considerably above O/. For the lower harmonics, the characteristic
temperatures are extremely low, so that these vibrations are excited in a
classical manner at any reasonable temperature. The highest harmonics,
however, have values of 0, in the neighborhood of room temperature, and
since many of the harmonics come in this range, the specific heat does not
attain its classical value until temperatures somewhat above room
temperatures are reached.

2. Vibrational Frequency Spectrum of a Continuous Solid. — To find
the specific heat, on the quantum theory, we must superpose Einstein
specific heat curves for each natural frequency vn as in Eq. (1.3). Before
we can do this, we must find just what frequencies of vibration are
allowed. Let us assume that our solid is of rectangular shape, bounded by
the surfaces x = 0, x = X, y = 0, y = Y, z = 0, z = Z. The fre-
quencies will depend on the shape and size of the solid, but this does not
really affect the specific heat, for it is only the low frequencies that art*
very sensitive to the geometry of the solid. As a first step in investigating
the vibrations, let us consider those particular waves that are propagated
along the x axis.

226 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

It is a familiar fact that there arc two sorts of waves, traveling and
standing waves, and that a standing wave can be built up by superposing
traveling waves in different directions. We start with a traveling wave
propagated along the x axis. Let us suppose that the point which was
located at x, yy z in the unstrained medium is displaced by the wave to
the point .r + £, // + r/, z + f , so that £, 17, f are the components of dis-
placement. To describe the wave, we must know £, r/, f as functions of
x, y, z, t. For a wave propagated along the x axis, £ represents a longi-
tudinal displacement, r? and f transverse displacements. For the sake of
definiteness lot us consider a longitudinal wave. Then the general expres-
sion for a longitudinal wave traveling along the x axis, with velocity v,
frequency vy amplitude1 A, and phase a, is

•HH-

£ = A sin 27r pit - ^ ) - a • (2.1)

Rather than using the phase constant a, it is often convenient to use both
sine and cosine terms, with independent amplitudes .4 and B, obtaining

£ = A cos 2irp( t - - ) + B sin 2irv(t - -Y (2.2)

\ v/ \ v)

an expression equivalent to Eq. (2. 1) if the constants A and B of Eq. (2.2)
have the proper relation to the A and a of Eq. (2. 1). Still another
way to write such a wave, this time using complex notation, is

£ - Ae\~\ (2.3)

where the A of Eq. (2.3) is still another constant, which may be complex
and so take care of the phase. In Eq. (2.3) it is to be understood that
the real part of the complex expression is the value to be used.

/ A

By writing expressions in l^+~) analogous to Eqs. (2.1), (2.2),

and (2.3), we get waves propagated along the negative x axis. Adding
such a wave to the one along the positive x axis, we have a standing
wave. As a simple example, we take the case of Eq. (2.2) and let B = 0.
Then we have

= A cos 2wp(t - ~ J + A cos 2Trv\t + *

A ( rt ,. 2irvx . . n J . 2irvx
= A I cos 2wpt cos h sin 2wvt sin

. rt . . n . . 2irvx\

+ cos 2wvt cos sm 2vvt sin I

v /

= 2A cos 27Tpt cos - (2.4)

SEC. 2] DEBYE'S THEORY OF SPECIFIC HEATS 227

By using different combinations of functions, we can get standing waves

P , ,> „ • 2irvx . rt A 2irvx i • rt , -

of the form cos 2irvt sin - > sin 2irvt cos - ; and sin 2^vt sm - as

' v v v

well. The particular characteristic of a standing wave is that the dis-
placement is the product of a function of the time and a function of the
position x. As a result of this, the shape, given by the function of x, is
the same at any instant of time, only the magnitude of the displacement
varying from instant to instant.

Certain boundary conditions must be satisfied at the surfaces of the
solid. For instance, the surface may be hold rigidly so that it cannot
vibrate, or it may be in contact with the air so that it cannot develop a
pressure at the surface. The allowed overtones will depend on the
particular conditions we assume, but again this is important only for tho
low overtones and is immaterial for the high frequencies. To bo specific,
then, let us assume that the surface is hold rigidly, so that the displace-
ment £ is zoro on the surface*, or when x = 0, x = X. The first con-
dition can be satisfied by using a standing wave containing the factor

sin - j rather than cos - > since sin 0 = 0. Then for the second condi-
v v

tion we must have

. 2irvX _ /rt _

sin - = 0. (2.5)

Condition (2.5) can be satisfied in many ways, for we know that the
sine of any integer times IT is zero. Thus we satisfy our boundary condi-
tion if we make

"; = 0, 1, 2, •••=*, (2.6)

where s is an integer. Using the relation v/v = I/A, this can be written

1 C tt\

I « *A v- /rfc _x

A = 2X' °r -2 - X< (2'7)

showing that a whole number of half wave lengths must be contained
in the length of the solid. Equation (2.6) or (2.7) solves entirely the
problem of the allowed vibrations of a continuous solid, so long as we
limit ourselves to longitudinal waves propagated along the x direction.
If we introduce the additional condition, demanded by the atomic nature
of the medium, that the minimum wave length is twice the distance
between atoms, we can immediately find the number of such possible
overtones. Let there be No atoms in a row in the length X. Then the
distance between atoms, along the x axis, is X/No. Our condition for
the maximum possible overtone is then X/2 = X/N*, or AT0X/2 = X,
showing that there are just NQ overtones corresponding to propagation

228 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

along the x axis. If we investigate transverse vibrations, but propagation
along the x axis, we obtain exactly analogous results, but with t\ or f
substituted for £. The allowed wave lengths for transverse vibrations
are the same as for longitudinal ones, but on account of the fact that
the velocity of transverse waves is different from that of longitudinal
waves, the frequencies are different. There are No possible vibrations
for each of the two directions of transverse vibration, giving 3No vibra-
tions in all corresponding to propagation along the x axis.

We can now use the results that we have obtained as a guide to the
general problem of waves propagated in an arbitrary direction. To
describe the direction of propagation, imagine a unit vector along the
wave normal. Let the x, y, and z components of this unit vector be
I, m, n. These quantities are often called direction cosines, for it is
obvious that they are equal respectively to the cosines of the angles
between the direction of the wave normal and the #, y, z axes. Then in

place of the quantity sin 2wvlt ),' or similar expressions, appearing in

Eqs. (2.1), (2.2), and (2.3), we must use the expression

sin 2irAt - --J-^-l- -I- (2.8)

Let us verify the fact that Eq. (2.8) represents the desired plane wave.
At time t, the expression (2.8) is zero when

rt /. Ijc + My + nz\ vv . ,

2irv(t — — - —I = TT X an integer,

Ix + my + nz = — (- X — 2") ~*~ **" ^'^

Now

/j + ra?/ + nz = a (2.10)

is the equation of a plane whose normal is a vector with components
proportional to Z, rrc, n, and whose perpendicular distance from the origin,
measured along the normal drawn through the origin, is a. Thus the
surfaces given, by putting different integers in Eq. (2.9), are a series of
equidistant parallel planes with normal I, m, n, the distance apart being
$v/v, and the distance from the origin increasing linearly with the time,
with velocity v. This is what we should expect for the zeros of a traveling
wave of wave length X = v/v, so that the zeros come half a wave length
apart.

By superposing traveling waves of the nature of Eq. (2.8), we can set
up the standing waves that we wish. We must superpose eight waves,

SBC. 2] DE BYE'S THEORY OF SPECIFIC HEATS 229

having all eight possible combinations of ± signs for the three terms
Ix, my, nz. One of the many typos of standing waves which wo can sol,
up in this way has the form

A . 0 . 2-irvlx . Zirvmy . Zirvnz f .

A sm 2wt sin - sin - — - sin - > (2. 11)

V V V

and this proves to be the one that we nood. Wo impose the boundary
condition that the displacement bo zero whon x = 0, x = X, y = 0,
y = }r, z — 0, z = Z. The conditions at a* = 0, y = 0, z — 0, arc* auto-
matically satisfied by the function we have chosen in Eq. (2.11). To
satisfy those at x = X, y — Y, z = Z, wo must make

2vlX _ 2vmY _ 2mZ _ (f> .

' 7T ~~ 'SiM 7, ~~ '9'" ?> ~ **' v^'l^y

where s^, sy, sa are integers. From Eq. (2.12), wo have

I = «-TO-V' m = 5^ov; w = 's*o^> whrro X = • (2.13)

L\ £tl &/J V

Since I, m, n arc* tho components of unit voctor, we must have

P + m- + n- = 1,

K\2 / • \2 / • V21/\V2
j) + (T) +\t) M = '• (2-u)

Equation (2.14) can be used to find the allowed wave lengths, in terms
of the integers sx, sy, sz:

1 (2.15)

i //TV / o \2 7~TV

(2.16)

We can now introduce the condition demanded by the atomic nature;
of the medium. We shall do this only for the simplest case of a simple
cubic lattice, but similar results hold in general. Let the atoms bo spaced
with lattice spacing d, such that X = JV*d, Y = Nyd, Z = Nzd, ami

NJfJf. = N (2.17)

is the total number of atoms in the crystal. We assume as the condition
for the maximum overtone that the minimum distance between nodes,

230 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

along any one of the three axes, is d. That is, considering for instance the
x direction and referring to Eq. (2.11), we assume that increasing x by the
amount d increases the argument of the sine, or 2irvlx/v^ by TT. Expressed
otherwise, this states that for the maximum overtone we must have
sx = X/d, and nimilarly sy = F/d, sg = Z/d. This means that all values
of sx, sv, sz are possible up to the values sx = Nx, sv = Nv, sz = Ng. Wo
can visualize the situation most easily by considering a three-dimensional
space in which sx, suy sz are plotted as three rectangular coordinates.
Then the integral values of sx, sy, sz, which represent overtones, form a
lattice of points, one per unit volume of the space. The overtones
allowed for an atomic crystal are represented by the points lying between
sx = 0, sx — Nx, sv = 0, sy = Ny, sz = 0, sx = Nz. The volume of space
filled with allowed points is thus NXNVNZ = N, and since there is ono
point per unit volume there are N allowed overtones. In place of using
this three-dimensional space, it is often more convenient to use what is
called a reciprocal space. This is one in which sx/X, sy/Y, sz/Z are
plotted. The allowed points in the reciprocal space then form a lattice
with spacings I/ A", 1/Y, l/Z, so that there is one point in volume
l/XYZ = 1/F, if F = XYZ is the volume of the crystal. For the maxi-
mum overtone we have sx/X = sy/Y = sz/Z = 1/d, so that the allowed
overtones fill a cube of volume 1/d3, or the reciprocal of the volume of unit
cell in the crystal. The number of allowed overtones, given by the
volume (l/dtj) divided by the volume (1/F) per allowed overtone, is
F/d3 = N, as before. It is plain why this space is called a reciprocal
space, since distances, volumes, etc., in it are reciprocals of the cor-
responding quantities in ordinary space.

We have so far omitted discussion of the fact that we have both longi-
tudinal and transverse vibrations. For a single traveling wave like
Eq. (2.8), there are of course three possible modes of vibration, one
longitudinal along the direction Z, m, n, and two transverse in two direc-
tions at right angles to this direction. The longitudinal wave will travel
with velocity vi, the transverse ones with velocity vt. We can superpose
eight longitudinal progressive waves to form a longitudinal standing
wave, and by superposing transverse progressive waves we can form two
transverse standing waves. Three standing waves can be set up in this
way for each set of integers sx, sy, sz. These three waves will all corre-
spond to the same wave length, according to Eq. (2.16), but to different
frequencies, according to Eq. (1.1). Considering the three modes of
vibration, there will be in all 3N allowed overtones, just the same as in
the theories of Dulong-Petit and Einstein, discussed in Chap. XIII,
Sec. 3.

From Eqs. (1.1) and (2.16), we can now set up the frequency distribu-
tion, or spectrum, as it is often called from the optical analogy, of our

SBC. 2] DE BYE'S THEORY OF SPECIFIC HEATS 231

oscillations. We have at once

In our reciprocal space, where sx/X, sy/Y, sz/Z are the three coordinates,

the quantity V(W^)M:T^7F)"r+ (sz/Z)2 is simply the radius vector,
which we may call r. Thus we have

the frequency being proportional to the distance from the center. Now
we can easily find the number of overtones whose frequencies lie in the
range dv of frequencies. For the points in the reciprocal space represent-
ing thorn must lie in the shell between r and r + dr, where r is given by
Eq. (2.19). This shell has the volume 4wr2 dr, or 32-jrv* dv/v\ Only
|X)sitivo values of the integers sx, sy, $z are to be used, however, so that we
have overtones only in the octant corresponding to all coordinates being
positive. This means that the part of the shell containing allowed
overtones is one eighth of the value above or 4irv2 dv/tf. We have seen
that the number of overtones per unit volume in the reciprocal space is
V. Thus the number of allowed overtones in dv, for one direction of
vibration, is

dN = . v. (2.20)

Considering the three directions of vibration, the number of allowed
overtones in dv is

dN = ±Trv*dvV( ~ + -I- (2.21)

\vi vt/

Formulas (2.20) and (2.21) hold only when sx/X, sv/Y, sz/Z are less than
1 /d; that is, when the spherical shell lies entirely inside the cube extending
out to sx/X — ±l/d, etc. For larger values of the frequency, the shell
lies partly outside the cube, so that only part of it corresponds to allowed
vibrations. It is a problem in solid geometry, which we shall not go into,
to determine the fraction of the shell lying inside the cube. When this
fraction is determined, we must multiply the formula (2.20) by the
fraction to get the actual number of allowed states per unit frequency

range. In Fig. XIV-1 we plot the quantity y-j-> tbe number of

vibrations per unit volume per unit frequency range, computed in this
way, for one direction of vibration. The curve starts up as a parabola,

232

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

(1 \dN /43T\
,r J-J- «= ( -y ]v2j but starts down when v = v/2d, the point where the

sphere is inscribed in the cube, and reaches zero at v = \/3v/2d, where
the sphere is circumscribed about the cube. The area under the curve of
Fig. XIV-1 of course is N/V, where N is the total number of atoms.
When we consider both types of vibration, the longitudinal and the
transverse, we must superpose two curves like Fig. XIV-1, with different
scales on account of the difference between vi and vt. This results in a
curve similar to Fig. XIV-2, having two peaks, the one at lower fre-
quencies corresponding to the transverse vibration, which has a lower
velocity than the longitudinal vibration.

In Fig. XIV-2 we have a representation of the frequencies of vibration
of an clastic solid, under tho assumption that the waves are propagated
as in an isotropic solid,' the velocity of propagation boing independent

§13

FIG. XIV-1. — Number of vibrations of one direction of polarization, per unit frequency
range, in a simple cubic lattice with lattice spacing d, and constant velocity of propagation v.

of direction and wave length, but the number of overtones being limited
by the conditions that the maximum values of sx/X, sy/Y, sz/Z are 1/d,
where d is the interatomic spacing. This is tho condition appropriate to
a simple cubic arrangement of atoms, an arrangement which does not
actually exist in the real crystals of elements. It is not hard to make
the changes in the conditions that are necessary for other types of struc-
ture, such as body-centered cubic, face-centered cubic, and hexagonal
close-packed structures, which will be discussed in a later chapter. The
general situation is not greatly changed. We can still describe a wave by
the three integers sx, syj sz and the frequency is still given by Eq. (2.19),
in terms of the radius vector in tho reciprocal space. Thus the number of
overtones in dv is still given by Kq. (2.20) or (2.21), provided the fro-
quency is small enough so that the spherical shell lies entirely within the
allowed region in the reciprocal space. The only difference comes in the
shape of this allowed region. Instead of being a cube, it can be shown
that the region takes the form of various regular polyhedra, depending

SBC. 2]

DE BYE'S THEORY OF SPECIFIC HEATS

233

on the crystal structure. These polyhedra, which are often called zones
or Brillouin zones, are important in the theory of electronic conduction
in metals as well as in elastic vibrations. The volumes of those zones in
each case are such that they allow just N vibrations of each polarization.
The zones for the three crystal structures mentioned resemble each other
in that they are more nearly like a sphere than the cubical zone of the
simple cubic structure. That is, the radii of the inscribed and circum-
scribed spheres are more nearly equal than for a cube. That means that

the region in which the curve of ( -T? )~7~ ^s falling from its maximum to

FIG. XIV-2. — Number of vibrations per unit fit-quonry range, in u simple cubic lattice
with constant velocity of propagation. It is assumed that the \elocity of the longitudinal
wave is twice that of the transverse waves. Dotted curve indicates Debye's assumption.

zero is more concentrated than in Fig. XI V-l, and corresponds to a higher
maximum and more precipitate fall. If the zone were a sphere instead of
a polyhedron, the fall would be perfectly sharp, as shown by the dotted
line iii Fig. XIV-2, the distribution being given by a parabola below a
certain limiting frequency j>niax, and falling to zero above this frequency.
The calculation which we have carried out in this section has been
limited in accuracy by our assumption that the velocity of propagation of
the elastic waves was independent of direction and of wave length.
Actually neither of these assumptions is correct for a crystal. Even for
a cubic crystal, the elastic properties are more complicated than for an
isotropic solid and the velocity of propagation depends on direction.

234 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

And we have stated that on account of the atomic nature of the. material
the velocity depends on wave length, when the wave length becomes of
atomic dimensions. These limitations mean that the frequency spectrum
which we have so far found is not vory close to the truth. Nevertheless,
our model is good enough so that valuable results can be obtained from
it, and we go on now to describe the approximations made by Debye,
loading to the specific heat curve known by his name.

3. Debye's Theory of Specific Heats. — Debye's approximation con-
sists in replacing the actual spectral distribution of vibrations by the
dotted line in Fig. XIV-2. That is, he assumed that dN is given by Eq.
(2.21) as long as v is less than a certain v^^ and is zero for greater v's. It
is obvious that this is not a very good approximation. Nevertheless, it
reproduces the form of the correct distribution curve at low frequencies
and has the correct behavior of predicting no vibrations above a certain
limit. To find the proper j>max to use, Debye simply applied the condi-
tion that the area under his dotted curve, giving the total number of
allowed overtones, must be 3N to agree with the correct curve. That is,
he assumed

* 4. (l J. 2\ f""" 2

,- — 4x1 — , + -; I I V

\»i Wo

from which

_/9A* 1 . . n

"-"' ~ 4xF T~2 ' (AA>

\ vl + v*J

In terms of the assumed frequency distribution and the formula (1.3),
we can now at once write down a formula for the specific heat. This is

The integration in Eq. (3.2) cannot be performed analytically. It is
worth while, however, to rewrite the expression in terms of a variable

x " EF' with Xn ** TT' (3'3)

We then have

1 CXQ ~i»x

.dx. (3.4)

SEC. 3] DEBYE'S THEORY OF SPECIFIC HEATS 235

It is customary to define a so-called Dobye temperature 6/> by the
equation

9* = *!=*• (3.5)

Then we have

1 T

-- = — (3.6)

XQ 00

so that Eq. (3.4) gives the specific heat in terms of T/&D, the ratio of tho
actual temperature to the Dobye temperature. That means that tho
specific heat curve should bo the same for all substances, at temperatures
which are tho same fraction of the corresponding Dobye temperatures.

Whon integrated numerically, the function (3.4) proves to bo not
unlike an Einstein specific heat curve, except at low temperatures. To
facilitate calculations with tho Dobye function, wo give in Table XIV-1

TABLE XIV-1. — SPECIFIC' HEAT AS A FUNCTION OF .r,, = O/>/7T, ACCORDING TO

DEBYE'S THEORY

XQ C\

0 5.955

1 5 670

2 4 918

3 3 948

4 2 996

5 2 197

6 1 582

7 1 137

8 0 823

9 0.604

10 0 452

11 0.345

12 0 267

13 0 211

14 0 169

15 0 137

16 0 113

17 0.0945

18 0 0796

19 0 0677

20 0.0581

A more extended table will be found in NeniHt, "Die GiundltiKcn des neucn Waiinenataes "

the specific boat per mole, calculated by Debye's theory, as a function of
XQ. We also give in Fig. XIV-3 a graph of the Debye specific heat curve
as a function of temperature. We can easily investigate the limit of low
temperatures analytically. If T « 6z>, we have XQ > > 1. Then,
approximately, we can carry the integration in Eq. (3.4) from 0 to oo,
since the integrand becomes very small for large values of x anyway. It

236 INTRODUCTION TO CHEMICAL PHYSICS (CHAP. XIV

can be shown1 that

f * ~4*x A

(3.7)

f * 4/# J.

Jo ^lAr ^4>

Then we have, for low temperatures

r - 12,

Cr = -=-n

(3.8)

From Eq. (3.8), we see that the specific heat should be proportional to

6 1

02 04 06 08 10 12 1-1 K>

T/0

FIQ. XIV-3. — Specific heat of a solid as a function of the temperature, according to Debyc's

theory.

the third power of the absolute temperature, for low temperatures. This
feature of Debye's theory proves to be true for a variety of substances.

TABLE XIV-2. — OBSERVED SPECIFIC HEAT OF ALUMINUM, COMPARED WITH DEBYE'S

THEORY

T, °abs.

CP observed

Cv observed

Cv, Debye

54.8

1.129

1.127

1.11

70.0

1.856

1.851

1 88

84.0

2.457

2.446

2 51

112.4

3.533

3.502

3 54

141.0

4.239

4.183

4.23

186.2

4.932

4.833

4 87

257.5

5.558 ; 5.382

5.35

278 9

5.698 ! 5.499

5.42

296 3

5.741

5 526

5 48

Data for this table are taken from the article by Eucken, in "Handbuch der Experimentalphysik,'
Vol. 8, a useful reference for the theory and experimental discussion of specific heats.

1 See Debye, Ann. Physik, 39, 789 (1912).

SEC. 3]

DEBYE'S THEORY OF SPECIFIC HEATS

237

Considering the crudeness of the assumptions, Debye's theory works
surprisingly well for a considerable number of elements. Thus we give-
in Table XIV-2 the observed specific heat of aluminum, and the values
calculated from Debye's theory, using GD = 385° abs. In Table XIV-2,
specific heats are given in calories per mole. The value of CV observed is
computed from Cp observed by the use of Eq. (1.14) of Chap. XIII. It
is interesting to note how much less the difference CP — CV is for such a
solid than the value R = 1.98 cal. per mole for a gas. Calculations for
many other substances show agreement with experiment of about the
accuracy of Table XIV-2. We have already pointed out the shortcom-
ings of Debye's treatment, and the remarkable thing is that it agrees as
well with experiment as it does.

TABLE XIV-3. — DEB YE TEMPERATURES FOR ELEMENTS

Substance

Oz>, high tem-
perature, °ahs.

6/> (T»)

9y> calculated

C (diamond)

1840

2230

159

398

385

399

420

428

467

315

321

329

235

205

379

215

212

Cd.

160

129

168

160

127

225

226

180

162

•

Data for this table, as for Tablp XIV-2, are from Eucken's attiele in Vol. 8 of the "Handbuch der
Experimentalphysik. ' '

In Table XIV-3, we give Dcbye temperatures for a number of elements
for which the specific heat has been accurately determined. We give
three columns, and the agreement of these three is a fair indication of
the accuracy of Debye's theory. The first column, 0/, (high tempera-
ture), gives temperatures determined empirically from the specific heat, so
as to make the agreement between theory and experiment as good as
possible through the temperature range in the neighborhood of 6/>/2,
where the specific heat is fairly large. The second column, 6i>(r3), is a
Debye temperature determined to make the T3 part of the curve, at very
low temperatures, fit as accurately as possible. If the Debye curve
agreed perfectly with experiment, these two temperatures would of course
be equal. Finally, in the third column we give 9z> (calc.), calculated

238 INTRODUCTION* W ffHEM/(&L PHYSICS [CHAP. XIV

from the elastic constants. To find these, Eq. (3.1) is used to find vmax
in terms of the velocity of propagation of longitudinal and transverse
waves, and Eq. (3.5) to find QD in terms of i>inftx. We shall not go into
the theory of elasticity to find the velocity of propagation in terms of the
elastic constants, but shall merely state the results, in terms of x the vol-
ume compressibility, <r Poisson's ratio, and p the density. In terms of
these quantities, it can be shown1 that

Using Eq. (3.9), the velocity can be found in terms of tabulated quantities.
The agreement between the columns in Table XIV-3 is good enough so
that it is plain that Debye's theory is a good approximation, but far from
perfect. It is interesting to note from the table the inyer,s£_rclatioii
between compressibility and Debye temperature, which can be seen
analytically "from Kqs. (3.1), (3.5), and (3.9)'. Thus cITamohd7'a suHstnnce
with extremely low compressibility, has a very high Debye temperature,
while lead, with very high compressibility, has a very low Dobye tempera-
ture. This moans that at room temperature the specific heat of diamond
is far below the Dulong-Petit value, while that of lead has almost exactly
the classical value.

4. Debye's Theory and the Parameter y — Debye's theory furnishes
us with an approximation to the frequency spectrum of a solid, and we
can use this approximation to find how the frequencies change with

volume, arid hence to find the parameter 7 = — j-. — ^ which is important

in the theory of thermal expansion, as we saw in Chap. XIII, Sec. 4.
According to Debye's theory, the frequency spectrum is entirely deter-
mined by the limiting frequency j>max, and if this frequency changes, all
other oscillations change their frequencies in the same ratio. Thus
Grimeisen's assumption that y is the same for all frequencies is justified,
and we can set

d In J>max / A . v

From Eq. (3.1) we see that the Debye frequency vmax varies proportion-
ally to the velocity of elastic waves, divided by the cube root of the
volume, and from Eq. (3.9) we see that the velocity of either longitudinal
or transverse waves varies inversely as the square root of the compressi-
bility times the density, if we assume that Poisson's ratio is independent
of the volume. As we shall see later, this assumption can hardly be

1 For a derivation, see for instance, Slater and Frank, "Introduction to Theoretical
Physics," McGraw-Hill Book Company, Inc., 1933. Combine results of paragraphs
109, 110 with result of Prob. 3, p. 183.

SEC. 4] DE 'BYE'S THEORY OF SPECIFIC HEATS 239

justified; Poisson's ratio presumably increases as the volume increases.
But for the moment we shall assume it to be constant. Then we have

(4.2)

On the other hand, the density is inversely proportional to the volume, so
that we have

In *>max = & In V — £ In x + const.,

7 " ~6 + 2 d In V (4'3)

The compressibility concerned in Eq. (3.9) is that computed for the*

actual volume of solid considered; that is, it is — v( Yp ) , where V is

\ \&r/7
the actual volume, rather than the volume at zero pressure. Thus

In x = - In V -
and

/ ''

din V

(4.5)

To evaluate the derivative in Kq. (4.5), we express P as a function of V
according to Eq. (1.5), Chap. XIII:

= ?P2;

from which, computing for V =

This simple formula will be compared with experiment in later chapters,
computing PI and P2 both by theory from atomic models, and by experi-
ment from measurements of compressibility. We may anticipate by
saying that in general the agreement is fairly good, certainly as good as we

240 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIV

could expect from the crudity of the approximations made in deriving
the equation.

In the calculations we have just made, we have assumed that Poisson's
ratio was independent of volume. We have mentioned, however, that
actually Poisson's ratio increases with the volume. We recall the mean-
ing of Poisson's ratio: it is the ratio of the relative contraction in diameter
of a wire, to the relative increase in length when it is stretched. For most
solids, it is of the order of magnitude of £. For a liquid, however, it-
equals \. One cannot see this directly, since a wire cannot be made out of
a liquid, but the value $ indicates that a wire has no change of volume
when it is stretched, and this is the situation approached by a solid as it
becomes more and more nearly like a liquid. Now as the volume of a
solid increases, either on account of heating or any other agency, it
becomes more and more like a liquid, the atoms moving farther apart so
that they can flow past each other more readily. Thus we may infer that
Poisson's ratio increases, and experiments on the variation of Poisson's
ratio with temperature indicate that this is in fact the case. If we look
at Eq. (3.9), we see that an increase of Poisson's ratio decreases the
velocities of both longitudinal and transverse waves. In fact, the change
of vt is so great that for a liquid, for which a = £, the velocity of transverse
waves becomes zero, in agreement with our usual assumption that trans-
verse waves cannot be propagated through a liquid. Thus, when we
consider Poisson's ratio, we find that it provides an additional reason why
the velocity of elastic waves and the Debyc frequency should decrease
with increasing volume. In other words, it tends to increase 7 over the
value found in Eq. (4.3). The exact amount of increase is impossible to
calculate, since the available theories do not predict how Poisson's ratio
should vary with volume, and there are not enough experimental data
available to compute the variation from experiment.

In considering thermal expansion, we must remember that Debye's
theory is but a rough approximation, and that really the elastic spectrum
has a complicated form, as indicated in Fig. XIV-2. If it were not for
the variation of Poisson's ratio with volume, our discussion would still
indicate that the whole spectrum should shift together to higher fre-
quencies with decrease of volume, since the velocities of both transverse
and longitudinal waves would then vary in the same way with volume,
according to Eq. (3.9). When we consider the Poisson ratio, however, we
see that the velocity of the transverse waves should increase more rapidly
than that of the longitudinal waves with decreasing volume, so that the
shape of the spectrum would change. Thus Griineisen's assumption, that
the Vs should be the same for all frequencies, is not really justified, and
we cannot expect a very satisfactory check between his theory and
experiment.

CHAPTER XV
THE SPECIFIC HEAT OF COMPOUNDS

In the preceding chapter, where we have been considering the specific
heat of elements, there was no need to speak of internal vibrations within
a molecule. In considering compounds, however, this is essential. A
real treatment of the mathematical problem of the vibrations is far beyond
the scope of this book. Nevertheless, we can take up a simple one-
dimensional model of a molecular crystal, which can furnish a guide to
the real situation. Suppose we have a one-dimensional chain of diatomic
molecules. That is, we have an alternation of two sorts of atoms, with
alternating spacings and restoring forces. The vibrations, transverse
or longitudinal, of such a chain have analogies to the vibrations in a
molecular crystal, and yot they form a simple enough problem so that we
can carry it through completely. As a preliminary, we take up the
simpler case of a chain of like atoms, equally spaced, analogous to the case
of an elementary crystal. This preliminary problem in addition will give
justification for the discussion of the preceding chapter, in which we have
arbitrarily broken off the vibrations of a continuum at a given wave
length and have said that that resulted from the atomic nature of the
medium. Also it will allow us to investigate the change of velocity of
propagation with wave length, which we have mentioned before but have
not been able to discuss mathematically.

1. Wave Propagation in a One -dimensional Crystal Lattice. — Let
us consider N atoms, each of mass m, equally spaced along a lino, with
distance d between neighbors. Let the x axis be along the line of atoms.
We may conveniently take the positions of the atoms to be at x = rf,
2d, 3dj . . . Nd, with y = 0, z = 0 for all atoms. These are the equilib-
rium positions of the atoms. To study vibrations, we must assume that
each atom is displaced from its position of equilibrium. Consider tho
jth atom, which normally has coordinates x = jd, y = z = 0, and assume
that it is displaced to the position x = jd + £/, y = ?;/, z = f „ so that
£;> 'nn f ; are the three components of the displacement of the atom. If
the neighboring atoms, the (j — l)st and the (j + l)st, are undisplaced,
we assume that the force acting on the jth atom has the components

Fx - -a*/, Fv - -&,,, F, - -ify, (1.1)

each component being proportional to the displacement in that direct ion

241

242 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

and opposite to the displacement. We assume that the force constant a
for displacement longitudinally, or along the line of atoms, is different
from tho constant 6 for displacement transversely, or at right angles to
tho lino of atoms. This can well happen, for in a longitudinal displace-
ment tho jth atom changes its distance from its neighbors considerably,
while in a transverse displacement it moves at right angles to the linos
joining it to its neighbors and stays at an almost constant distance from
the neighbors.

Instead of assuming that the force (1.1) depends only on the position
of the jih particle, wo now assume that it is really tho sum of two forces,
exerted on the jih atom by its neighbors the (j — l)st and (j + l)st
atoms. The force exerted by the (j — l)st on thojth, whon the (j — l)st
is at its position of equilibrium, has components ( — a/2)(f/, ( — 6/2)77;,
( — 6/2)f/. But we must suppose that this force doponds only on the
relative positions of the two atoms in question, not on their absolute
positions in space. Thus it must dcporid on tho differences of coordinates
of the j'th and (j — l)st atoms, so that the general expression for the force
has components (— a/2)(£, — £/_i), ( — b/2)(rjj — r7;_i), ( — 6/2)(f/ — f/-i).
Similarly, the force exerted on the jth atom by the (j + l)st must have
components (— a/2)(£/ — £/+i), ( — 6/2)(tj/ — T?,+I), ( — 6/2)(f7- — £/+i).
Combining, the total force acting on the jth atom is

p = _& + &( +

F, - -6f, +g(f,-i + f,+i). (1-2)

Using the expressions (1.2) for the force, we can set up the equations
of motion for the particles, using Newton's law that the force equals the
mass times the acceleration. Thus we have

a
m%j = — Q>%J + ^

, , &

(1-3)

where £/ indicates the second time derivative of {/. We now inquire
whether we can solve the equations (1.3) by assuming that the displace-
ments form a standing wave of the sort discussed in the preceding chapter.
Let us consider a longitudinal vibration, for which £ is different from

SEC. 1] THE SPECIFIC HEAT OF COMPOUNDS 243

zero, vf and f equal to zero, so that only the first equation of (1.3) is
significant, and let us assume

£ = A sin 2wvt sin-^-> (1.4)

by analogy to the standing waves in a continuous medium. In particular,
for thojth particle, whose undisplaced position x is equal tojW, wo assume

£/ = A sin 2irvt sin — ~~- (1.5)

Then we have

£j*i = A sin 2irvt sin 2ir(j ± 1)-

A • o • i •

= A sin 27ml sin cos -^- ± cos —^-- sin ~- ;

A A A

i > >« • o ,/« • 27r;d 2?rd\

_i + fy+i = A sm 2?r^( 2 sm -y- cos — 1

„ _N
(1.6)

Substituting Eqs. (1.5) and (1.6) in Eq. (1.3), we find that the factor

A sin 2wvt sin -
factor, we have

A sin 27r^ sin -~- is common to each term. Canceling this common

A 9 9 i n

47r2i>2rw = —a + o 2 cos

& A

aA 27rd\ 2a .

— I 1 — COS -r- ) = - SI

m\ X / w

sin* — '

1 /2a . ird ,, -N

p = 7T--V/— sm T" (J-7)

2?r \ m X

If the frequency ? and ihe wave length X are related by Ea. (1.7). the
values of f, in Eq. (1.5) will satisfy the equations (1.3). If the velocity
wcro constant, we should have v = v/\. the frequency being inversely
proportional to the wave length. Fronf Eq. (1.7) we can see that this is
the case for long waves, or low frequencies, where we can approximate the
sine by the angle. In that limit we have

*2awd
X'

. - X, - (1.8)

a value that can easily be shown to agree with what we should find b}'
Elasticity theory. For higher frequencies, however, since sin wd/\ is

244 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

less than ird/\, we see that the velocity of propagation must be less than
the value (1.8). This is the dependence of velocity on wave length which
was mentioned in the preceding chapter.

NcxtL we must impose. boundary conditions on our chain pf atoms as
WG. did wjf-h t.hp ftoflfarmona solid. We are assuming N atoms, with
undisplaced positions at x = d, 2d, . . . Nd. We shall assume that the
chain is held at the ends, and to be precise we assume hypothetical atoms
at x = 0, x = (N + l)d, which are held fast. As with the continuous
solid, the precise nature of the boundary conditions is without effect on
the higher harmonics. In Eq. (1.3), then, we assume that the equations
can be extended to include terms £0 and &\r+i, but with the subsidiary
conditions

fo = fr+i = 0. (1.9)

The first of these is automatically satisfied by the assumption (1.5),
setting,; = 0. For the second, we have the additional condition

sin 2ir(N + 1)^ = 0. (1.10)

Since sin TTS = 0, where s is any integer, Eq. (1.10) can be satisfied by
2(N + 1)- = s, where s is an integer. 0-H)

The significance of s is the same as in Eq. (2.7), Chap. XIV: s = 1 corre-
sponds to the fundamental vibration, s = 2 to the first harmonic, etc.
Introducing the condition (1.11), we may rewrite Eqs. (1.5) and (1.7) for
displacement and frequency. For the vibration described by the integer
s, we introduce a subscript s, obtaining

A9 sin 2irv,t sin

1 I2a . ITS /« «rt\

* = 8in (L12)

On examining Eq. (1.12), we see that both £/, and vt are periodic in
s. Aside from a question of sign, which is trivial, £/, repeats itself when ,s
increases by (N + 1) or any integral multiple of that quantity, and vs
similarly repeats itself. That is, all the essentially different solutions are
found in the range between s = 0 and s = N + 1. These two limiting
values, by Eq. (1.12), correspond to all (•' s equal to zero, so that they are
not really modes of vibration at all. The essentially different values then
correspond to s = 1, 2, ... N, just N possible overtones. This verifies

SEC. 1]

THE SPECIFIC HEAT OF COMPOUNDS

245

the statement of the previous chapter that the number of allowed over-
tones, for one direction of vibration, equals the number of atoms in the
crystal. It is interesting to consider the periodicity in connection with
the reciprocal space of Sec. 2, Chap. XIV. In that section, we imagined
s/X to be plotted as an independent variable. Hero, since X, the length

of the chain of atoms, equals (N + l)d, we should plot ,AT--r +\~, — T

\J\ ~T~ 1 )Cl A

as variable. Rather than plotting v9 as a function of this quantity, ue
prefer to plot its square, *>*. This is done in Fig. XV-1, where the perio-
dicity is clearly shown. We see, furthermore, that the region from
2/X = 0 to 2/X = \/d includes all possible values of 1%. This funda-
mental region corresponds to those described in See. 2, Chap. XIV. In
addition to the sinusoidal curve giving v2, we plot a parabola v2 = 02/X2,
where v is the velocity of propagation for long waves. Tins parabola is

FIG. XV-l. — v- vs. 2/X, for one-dimensional crystal.

l/d 2/X=S/X

Dotted parabola, case of no dispersion.

the curve which we should find if there were no dependence of velocity
on wave length or no dispersion of the waves. We see from Fig. XV-1
that the effect of dispersion is to reduce tho frequencies of the highest
overtones, compared to the values which we should find from the theory
of vibrations of a continuum. While we cannot at once apply this result
to the three-dimensional case, it is natural to suppose that, for instance in
Fig. XIV-1 of the previous chapter, the effect will be to shift the peak of

1 dtf

— -r— ); the number of overtones per unit frequency range, to lower

frequencies, and at the same time to make the peak higher, so as to keep
the number of overtones the same. This is a type of change that makes
the curve resemble Debye's assumption (the dotted curve of Fig. XIV-2)
more closely than before. Very few actual calculations of specific heat
have yet been made using the more exact frequency spectrum that we
have found.

246

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XV

In Fig. XV-1 we have seen graphically the way in which the square
of the frequency, v2, is periodic in the reciprocal space. It is informing
to see as well how the actual displacements ?/, repeat themselves. From
Eq. (1.12) we see that £/, contains the factor sin irjs/(N + 1). This of
course equals zero for s = N + 1, and two values of s which are greater
than N + 1 and less than N + 1 by the same amount respectively
will havo equal and opposite values of the sine. In other words, the
values of the sine which wo have when s goes from 0 to N + 1 repeat each

(a) s« N + l

(b)s =

N-H

Cc) s almost N-M

(d) s slightly greater thanN+1. Displacements as in
(c) with opposite sign

Ce) s = "2 (N+1). Displacements as in (b),with opposite sign

(f) s almost 2 (N+1). Displacements as i'n(a),with opposi4esign
FIG. XV-2. — Displacements of atoms, for different overtones.

other iii opposite order as s goes from N + 1 to 2(N + 1). As far as we
can tell from the displacements of the particles, a value of s/X greater
than !/(/, in other words, doos not correspond to a shorter value of wavo
length than 2d but to a longer wave length, and when s/X equals 2/d the
actual wave length is not (/ but infinity, corresponding to 110 wave at all.
These paradoxical results are illustrated in Fig. XV-2, where we show
TTJS

curves of sin

indicated as functions of a continuous variable

SBC. 2] THE SPECIFIC HEAT OF COMPOUNDS 247

j, with the integral values of j shown by dots, for several values of s. It is
clear from the figure that increase of 5 does not always mean decrease of
the actual wave length, but that there is a definite minimum wave length
for the actual disturbance, oqual as we have previously stated to twice the
interatomic spacing. From the fact which we have pointed out that the
range of s/X from \/d to 2/d repeats the range from 0 to 1/d in opposite
order, we understand why the curve of i>2 vs. s/X, in Fig. XV-1, has a
maximum for s/X = 1/d, falling to zero again for 2/d.

2. Waves in a One -dimensional Diatomic Crystal. — In the preceding
section we havo seen that the atomic nature of a one-dimensional mon-
atomic crystal lattice leads to a dependence of the velocity of elastic
waves on wave length and to a limitation of the number of allowed over-
tones to the number of atoms in the crystal. Now wo attack our real
problem, the vibrations of a diatomic one-dimensional crystal, using
analogous methods. Wo assume each molecule to have two atoms, one
of mass m, the other of mass m'. Lot there bo N molecules, and in
equilibrium let the atoms of mass m bo at the positions X = d, 2rf, • • •
Nd, and those of mass m' at x = d + rf', 2ii + d', - • • Nd + dr, whoro
d' is loss than d. Wo formulate only tho problem of longitudinal vibra-
tions, understanding that tho transverse vibrations can bo handled in a
similar way. By analogy with Sec. 1, wo assume that tho foroos on oach
atom come from its two neighboring atoms. Those neighboring atoms
are both of tho opposite typo to the one we are considering, but are at

- e - 9 - e - 0 - e - 0 -

XJ-1 XJ-! XJ XJ Xj+l X'j*'

Flo. XV-3. — Arrangement of atoms in ono-dimeiiHioritil diatomic molc-culnr lattice.

different distances, one being at distance d' (in tho same molecule) the
other at distance d — dr (in an adjacent molecule). The arrangement of
atoms will be clearer from Fig. XV-3. We assume two force constants:
a for tho interaction between atoms in different molecules, a1 for inter-
action between atoms in tho same molecule. Thus the equations of
motion are

m'% = -a(tf - UO ~ a'U,' - fc), (2.1)

where £/, {J represent the displacement of the atoms of mass m and m'
respectively in the jth molecule.

To solve Eq. (2.1), we assume sinusoidal standing waves for both types
of atoms, but with different phases:

248 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

£; = A sin 2Trvt sin »

£ = /T sin 27r^ sin + B' sin 2r*< cos - (2.2)

A A

Substituting Eq. (2.2) in Eq. (2.1), we find that the factor sin 2irvt cancels
from all forms. The remaining equations are

[-4irVffi + (a + a')] A sin ^~ = .4' a sin 2*0' - 1)^ + a' sin ^

A L A A

A J

cos '

[-4irVm' + (n + a')l/T sin + B' cos

X ' X /

= A] a si

sin 2x(j + 1) + «' sin

A A

(2.3)

In Eqs. (2.3) wo expand the sines of 2ir(j ± 1) by the formulas for the

sines of sums and differences of angles and collect terms in sin — ^~ and

cos -^- Then Eqs. (2.3) become

+ (a + n')] - X Ya cos -~ + a'J

sn, -~

sin - ^a cos + «'= 0,

/A'[-4r*v*m' + (a -f- a')) - ^l(a cos ^ + a'H

'JB'f-igrVm' + (a + a')] - --ifa sin ^H = 0. (2.

+ cos B'f-igrVm' + (a + a')] - --ia sin = 0. (2.4)

If our assumptions (2.2) arc to furnish solutions of Eq. (2.1), we must
have Eqs. (2.4) satisfied independent of j; that is, for each atom in tin-
chain. The only way to do this is for each of the four coefficients of sin

-^- or cos -~ in Eq. (2.4) equal to zero. We thus have four simul-

A A

taneous equations for the four unknowns A, A', B', and v. Really there
are only three unknowns, however, for we can determine only the ratios
A' /A, B' I A, and not the absolute values of the three amplitudes A, A',

SBC. 2]

THE SPECIFIC HEAT OF COMPOUNDS

249

Bf. Thus we should not expect to find solutions for our equations, since
there are more equations than unknowns, but it turns out that the equa-
tions are just so set up that they have solutions. We have the four
equations

J[~4irV7tt + (a + a')] - ;t'(a cos -^ + a'j - B'(a sin —-) = 0

,,/ . 27r<A ,J 2ird . \

A'la Hiu -r- 1 — B '( a cos — — h a ) = 0,

A'[-4irVm' + (a + a')] - A\a cos ~ + a') = 0,
B'[-4**v*m' + (a + a')] - A\a sin

= 0.

(2.5)

To solve them, we first determine Br in terms of A' from the second. Sub-
stituting in the other thrco, wo have equations relating A and A'. We
find, however, that tho third and fourth equations load to tho same rosult,
so that there arc only two independent equations for A in terms of A'.
It is this which makes solution possible. These two equations arc at
once found to be

A[ - 47rVw + (a + a')] - A'

( 2ird . \

(a cos -— — h a 1

, ,
a cos ~ -- h a

= 0,

(2.6)

From oach of Eqs. (2.6) wo can solve for tho ratio A /A'. Equating those
ratios wo get an equation for tho frequency:

A\a cos ~ + a'J - A'(-Wv*m' + (a + a')] = 0.

r f • 2

9/7 \ a SU1 ~
af/^c — — 1 ft -L. ^

r)"

-4rrVm + (a + a')

X 27rd
a cos -r-

+ «'

2m' + (a + a')

a cos

(2.7)

250 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

From Eq. (2.7) we have at once

[4*-Vm - (a + a')][4a-Vm' - (a + a')]

/ 2nd ^ ,V . ( . 27r<A2

= I a cos -r— + a' ] + (a am — )

= a2 + a'5 + 2aa' cos ^

= (a + a')2 - 2oa'(l - cos ^\

= (a + a')2 - 4aa' sin2^- (2.8)

Expanding the left side of Eq. (2.8),

(27n/)4mm' - (2Trv)*(m + m')(a + a') + (a + a')2

= (a + a')2 - 4aa; sin2 ^- (2.9)

Equation (2.9) is a quadratic in (27rv)2. Solving it, we have

mm
where

/ /\ if" / / /\ ~~1 o

m /a + a \ / m -f- m /a + a \ r 4aa . „ ird

I _ 1 _|_ ^ M [ i ..... i i _ sin

' \ 2 / ~ \ I mm' \ 2 / J mm' X

_ . , «[\
X/'

' ~ (2'1())

m + r/i'/fl

(2.11)
/

4mm'

(a + a')2 (w + m')2

Equation (2.11) is the desired equation giving frequency in terms of wave

length, for longitudinal vibrations of a chain of diatomic molecules.

As in Eq. (1.7), we see that the frequency depends on the quantity

sin — , so that we go through all possible values when - goes from zero

A A

to l/2d. Thus there is the same sort of periodicity seen in Fig. XV-1.
Furthermore, when we introduce boundary conditions for a crystal of N
molecules, we find as before that there are N allowed overtones in this
fundamental region of reciprocal space. In the present case, however,
Eq. (2.11) has two solutions for each value of 1/X, coming from the ±
sign. That is, there are two branches to the curve, two allowed types of
vibration for each wave length, and consequently 2N vibrations in all.
This is natural, for while there are just N molecules, each of these has
two atoms, so that there are 2N atoms and 2N degrees of freedom for
longitudinal vibration. In Fig. XV-4 we give curves of v2 vs. 2/X, for
several different values of the constant K. From Eq. (2.11) we see that

SEC. 2]

THE SPECIFIC HEAT OF COMPOUNDS

251

except for scale the curve depends on only one constant K. This constant,
equals unity when a = a', m = m'. Under all other circumstances it is
easy to show from its definition that it is less than unity; when a is very
different from a', or m very different from m', or both, it is very small.
Thus the limit K = 0 corresponds to very unlike atoms in the molecule,
with very unlike forces between the atoms in the molecule and atoms in
adjacent molecules. This is the case of strong molecular binding, with
weak interatomic forces. The limit K = 1 corresponds to the ease where
the atoms are similar and the binding within
the molecule is almost the same as that between
molecules. This limit, for instance, would be
found approximately in the case of an ionic
crystal like the alkali halides, where there is
no molecular structure in the proper sense awl
where the two typos of atom have approxi-
mately the same mass. In (a), Fig. XV-4,
we have the case of strong molecular binding.
Here one branch of the curve* corresponds to
low frequency vibrations, going to zero fre-
quency in the limit of infinite wave length.
These vibrations are those in which the mole-
cules as a whole vibrate, and they are acous-
tical vibrations of the same sort we have
discussed in the preceding chapter. The other
branch, however, corresponds to a much higher
frequency, even at infinite \\avci length.
These vibrations are vibrations of the two

0
(a) K « \

2/\

/wv

0 l/d 2/d 2/X

atoms in the molecule with respect to each (c) K Shgh% Less Thorn Unity
other, almost exactly as the vibrations would FIG. xv-4.— »>2 vs. 2/x, for Hia-
occur in the isolated molecule. These are tonne molecular lattice.
often called optical vibrations, for as we shall see later they can be
observed in certain optical absorptions in the infrared.

In (6), Fig. XV-4, there is an intermediate case between strong and
weak binding. The forces between molecules are, here not much greater
than those within a molecule, and the result is that the optical branch
of the spectrum is not at much greater frequency than the acoustical
branch, Finally in (c) we show almost the limiting case of equal atoms
and equal binding. The exactly equal case would correspond to a crystal
with 2N equal atoms with a spacing of rf/2. This is the case of Sec. 1, and
we should expect the curve of v* against 2/X to correspond to Fig. XV-1,
except that the first maximum of the curve should come at 2/d instead of
l/d, on account of the spacing of d/2. In Fig. XV-4 it is shown how
this limit is approached. We have already pointed out that on account

252 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

of periodicity we repeat all the overtones found between 2/X = 0 and
2/X = l/d in the next, period, from l/d to 2/d. Let us then take the
lower branch of the curve in the range 0 to 1 A/, and the upper branch from
l/d to 2/rf, as shown by the heavy line in (c), Fig. XV-4. If K were exactly
unity, these two branches would join smoothly, and would give exactly
the curve we expect. When K is slightly less than unity, so that the two
types of atom arc slightly different from each other or there is a slight
tendency to form diatomic molecules, there is a slight discontinuity in
the curve at 1/rf, as shown in the figure, but still it is better to regard
them as both parts of a single curve. It is clear from this discussion how
the case of the present section reduces to that of Sec. 1 as the molecular
lattice reduces to an atomic lattice.

3. Vibration Spectra and Specific Heats of Polyatomic Crystals. —
The simplified one-dimensional model which wre have taken up in the last
two sections is enough so that we can readily understand what happens in
the case of real crystals. First, we consider definitely molecular crystals,
with strong binding within the molecules, and relatively weak inter-
molecular forces. The vibration spectrum in this case breaks up
definitely into two parts. First there is the acoustic spectrum, connected
with vibrations of the molecules as a whole. This reduces to the ordinary
elastic vibrations at very low frequencies, and extends to a high enough
frequency in the infrared to include 3N modes of oscillation, where there
are N molecules in the crystal. If the intermolecular forces are weak, so
that the compressibility of the crystal is large, this limiting frequency will
be low, in the far infrared. Then there is the optical spectrum, connected
with vibrations within the molecules. As we see from Fig. XV-4 (a),
these vibrations in different molecules are coupled together to some
extent, so that their frequencies depend on the particular way in which
the vibrations combine to form a standing wave in the crystal. Never-
theless, this dependence of frequency on wave length is relatively small, as
the upper branch of Fig. XV-4 (a) shows. The important point is that
the optical vibrations in molecular crystals come at a good deal higher
frequency than the acoustical vibrations, lying in the part of the infrared
near the visible. And these optical frequencies are only slightly different
from what they would be in isolated molecules. This can be seen from
our simple model. Thus in Eq. (2.12) let a', the force constant for
molecular vibration, be large compared to «, the force constant between
molecules. Then K will be very small, and if we neglect a compared to a'
we have (2irv)2 = [(m + m')/mm']a', just the value for a diatomic mole-
cule. We may see this, for instance, from Chap. IX, Eqs. (2.5) and
(4.4), where we obtained the same result. Of course, with complicated
molecules, there will be many optical vibration frequencies of the isolated
molecule, and all of these will appear in the spectrum of the crystal with
slight distortions on account of molecular interaction.

SEC. 3] THE SPECIFIC HEAT OF COMPOUNDS 253

With a spectrum of this sort, it is clear how to handle the specific heat
of such a molecular crystal. The acoustical vibrations can be handled
by a Debye curve, using the elastic constants, or an empirical charac-
teristic temperature, and using the number of molecules as N. Then we
add a number of Einstein terms to take care of the molecular vibrations.
These again can be found empirically, or they can be deduced from
vibration frequencies observed in the spectrum. Those frequencies
would be exported to bo approximately the samo as in tho molecules, so
that this part of tho specific heat should agree with the vibrational part
of the .specific heat of the corresponding gas. In some eases, as wo shall
mention later, the; vibration frequencies can be found directly by optical
investigation of the solid. In addition to the molecular vibrations,
corresponding to tho Einstein terms, and tho molecular translation, cor-
responding to tho Debye terms, in tho specific heat of the crystal, there
must be something corresponding to the molecular rotation. In most
solids, the molecules cannot rotate but are only free to change their
orientation through a small angle, being held to a particular orientation by
linear restoring forces. In their vibration spectrum, this will lead to
vibwtional terms like the upper branch of Fig. XV-4 (a), and there will
be additional Einstein terms in the specific heat coming from thi>s hindered
rotation. These terms of course cannot bo predicted from tho properties
of the separate molecules, but ordinarily must be found empirically to fit
the observed specific heat curves. There are certain cases, on the other
hand, in which the molecules at high temperatures really can rotate in
crystals, though at low temperatures they cannot. In such a case, at
high temperature, there would be a term in the specific heat of the solid
much like tho rotational term in a gas. At low temperatures whore the
rotation changes to vibration, the transition is more complicated than any
that we have taken up so far and is really more like a change of phase.

A crystal such as those of tho alkali halides. formed from a succession
of equally spaced ions of alternating sign, is quite different, from JLho
molecular crystals. The spectrum, as indicated in Fig. XV-4 (<:). is much
more like that of an element, the distinction between the two types of
ions being unimportant. Thus we can treat it bv methods of tho proced-
ing chapter, using only a Dobye curve, but taking N to bo tho total
number of atoms, not the total number of molecules. This is commonly
done for the alkali halides, and it is found that one gets as good agreement
between theory and experiment as for the metals. For more complicated
ionic crystals, such as carbonates or nitrates, which aro formed of positive
metallic ions and negative carbonate or nitrate radicals, the situation is
midway between the ionic and molecular cases. In CaCOar for instance,
we should expect a Debye curve coming from vibrations of tho Cn.4"*- a.nrl
CO»— ions as a whole, and also Einstein terms from the internal vibrations
of the carbonate ions.

254 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV

4. Infrared Optical Spectra of Crystals. — Though we have said
nothing about the interaction of molecules and light, we may mention
the infrared optical spectra of crystals, related to their optical vibrations.
Light can be emitted or absorbed by an oscillating dipole; that is, by two
particles of opposite charge oscillating with respect to each other. We
should then expect that such vibrations as involved the relative motion
of different charges could bo observed in the spectrum. This is never
tho case with the acoustic branch of the vibration spectrum, for there
we have molecules vibrating as a whole, and they are necessarily
uncharged. But in the optical branches, for instance with the alkali
halides, wo have just tho nooossary circumstances. The rase 1/X = 0, in
the optical branch of an alkali halide, corresponds to a rigid vibration
of all the positive ions with respect to all tho negative ions, so that each
pair of positive and negative ions in the crystal can radiate light, and all
these sources of radiation are in phase with each other. If such an
oscillation were excited, then, the crystal would emit infrared radiation of
tho frequency of the vibration. Only the case 1/X - 0 corresponds
to radiation, for if 1/X were different from zero, different parts of the
crystal would be vibrating in different phases and tho emitted radiation
from the various atoms would cancel by interference. Ordinarily the
radiation is not observed in emission but in absorption, for there is no
available way to excite this typo of vibration strongly. There is a general
law of optics, however, stating that any frequency that can bo omitted by
a system can also be absorbed by the same system. Thus light of this
particular infrared wave length can be absorbed by an alkali halide
crystal. A still further optical fact is that light of a frequency which is
very strongly absorbed is also strongly reflected. Hence alkali halide
crystals have abnormally high reflecting power for this particular wave
length. This is observed in the experiment to measure residual rays, or
" Reststrahlon." In this experiment, infrared light with a continuous
distribution of frequencies, as from a hot body, i.s reflected many times
from the surfaces of alkali halide crystals. For most frequencies, the
reflection coefficient is so low that practically all the light is lost after
the multiple reflection. Tho characteristic frequencies have such high
reflecting power, however, that a good deal of the light of these wave
lengths is reflected, and the emergent beam is almost monochromatic,
corresponding to the absorption frequency. These beams which are left
over, called residual rays, form a convenient way .of getting approxi-
mately monochromatic light in the far infrared.

By measurement of the wave length of tho residual rays, for instance
with a diffraction grating, it is possible to get a direct measurement of the
maximum frequency in the optical band of the vibration spectrum of an
alkali halide. If we treat the spectrum by the Debye method, regarding

SEC. 4] THE SPECIFIC HEAT OF COMPOUNDS 255

the crystal as an atomic rather than a molecular crystal, this frequency
should agree approximately with tho Dehye characteristic frequency,
which can be found from the characteristic temperature 9#. Such an
agreement is in fact found fairly accurately, as will be shown in a later
chapter on ionic crystals, in which we shall make comparison with experi-
ment. For molecular lattices, it is also possible to got residual ray
frequencies, in case the molecules contain ions which can vibrate with
respect to each others. These frequencies have no connection with the
Debye frequencies, however, and they have been much loss studied than
in the case of ionic crystals.

CHAPTER XVI
THE LIQUID STATE AND FUSION

For several chapters wo have been taking up the properties of solids.
Earlier, in Chap. XI, we discussed the equilibrium of solids, liquids, and
gases in a general way but without- using much information about the
liquid or solid states. In Chap. XII, discussing imperfect gases and Van
der Waals1 equation, we again touched on the properties of liquids but
again without much detailed study of them. Now that we understand
solids better, we can again take up the problem of liquids and of melting.
The liquid phase1 forms a sort of bridge between the solid and the gaseous
phases. It is hard to treat, because it has no clear-cut limiting cases, such
as the crystalline phase of the solid at the absolute zero and the perfect
gas as a limiting case of the real gas. The best we can do is to handle it
as an approximation to a gas or an approximation to a crystalline solid.
The first is essentially the approach made in Van der Waals' equation,
which we have already discussed. The second, the approach through the
solid phase and through fusion, is the subject of the present chapter.

1. The Liquid Phase.-- We ordinarily deal with liquids at tem-
peratures and pressures well below the critical point, and it is here that
they resemble solids. When studied by x-ray diffraction methods, it is
found that the immediate neighbors of a given atom or molecule in a
liquid are arranged very much as they are in the crystal, but more* distant
neighbors do not have the same regular arrangement. We shall meet a
particularly clear case of this later, in discussing glass. A glass is simply
a supercooled liquid of a silicate, or other similar material, held together
by bonds extending throughout the structure, just as in the crystalline
form of the same substance. These materials supercool particularly
easily, presumably because the atoms or ions of the liquid are so tightly
bound together that they do not rearrange themselves easily to the
positions suitable for the crystal. Thus we can observe their liquid
phases at temperatures low enough so that they take on most of the
elastic properties of solids. They acquire rigidity, resistance to torque.
Nevertheless they never lose entirely their properties of fluidity. A rod
of glass at room temperature, subjected to a continuous stress which is not
great enough to break it, will gradually deform or flow over long periods of
time. The study of materials which behave in this way, showing both
fluidity and elasticity, is called rheology, and it shows that such a com-

256

SEC. 1] THE LIQUID STATE AND FUSION 257

bination of properties is very widespread. The fluidity of the glasses
is a result of the fact that there is no unique arrangement of the atoms, as
there is in a perfect crystal. Certain atoms may be in a situation where
there are two possible positions of equilibrium, near to each other. That
is, the atom is froe to move from tho position whore it is io an adjacent
hole in the structure, with only a small expenditure of energy. Tho
maximum of potential energy between tho minima is closoly analogous
to the energy of activation in chemical reactions, mentioned in C -hap. X,
Sec. 3. In Fig. XVI-1 (a) we show schematically how the potential
energy acting on this atom might appear, as we pass from one position of
equilibrium to the other. There will ordinarily not be much tendency
for the atom to go from one position to the other. But if the material is
under stress and one of the positions tends to relieve tho stress, the other

(oO

Irir,. XVI-1. Schematic potential energy acting on an atom in n KhiHH. (a) Unstressed
material, (//) material under stress

not, the energy relations will be shifted so that the position which relieves
tho stress, as shown in (16), Fig. XVI-1, will have lower energy than (26),
which does not. Then if the atom is in position (2), it will have; a good
chance of acquiring the energy c2, enough to pass over the; potential hill
and fall to the position (1), simply by thermal agitation. By tho Max-
vvell-Boltzmaim relation, wo should expect the probability of finding
an atom with this energy to bo proportional to the quant ity exp ( — €2/&77),
increasing rapidly with increasing temperature. On tho other hand, the
probability that an atom in (1) should have the energy t\ necessary to
pass over the hill to position (2) would contain tho much smaller factor
exp ( — €\/kT). The net result would bo that atoms in positions like (2)
would move to positions like (1), relieving the stress, but tho opposite;
type of transition would not occur. This would amount to a flow of
the material, if many such transitions took place. Furthermore, the rate;
of the process woujd be proportional to exp ( — e2/M7), anel this woulel
be expected to be^roportional to the rate of flow under fixed stress, or to
the coefficient of" yisoosity. The* actual viscosities of glasses show a
dependence on temperature of this general nature, the flow becoming
very slow at low temperatures, so slow that it is ordinarily not observed
ut all. But there is no sudden change between a fluid and a solid state.

258 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

The glasses are particularly informing fluids, because we can observe
them over such wide temperature ranges. Other types of liquids do not
supercool to any such extent, so that ordinarily they cannot be observed
at temperatures low onough so that they have begun to lose their charac-
teristically fluid properties. We may infer, however, that the process of
flow in all liquids is similar to what we have described in the glasses.
That is, the atoms surrounding a given atom arc approximately in the
arrangement correct for a crystal, but there arc so many flaws in the
structure that there are many atoms which can move almost freely from
the position where they happen to bo, to a nearby vacant place where
they would fit into the structure almost as well. One rather informing
approximation to the liquid state treats it straightforwardly as a mixture
of atoms and holes, the holes simply representing atoms missing from the
structure, and it treats flow as the motion of atoms next the holes into the
holes, leaving in turn empty holes from which they move. The essential
point is that the structure is something like the solid, but like an extremely
imperfect, disordered solid. And it is this disorder that gives the possi-
bility of flow. A perfect crystal cannot flow, without becoming deformed
and imperfect in the very process of flow.

2. The Latent Heat of Melting.— With this general picture of the
liquid state, let us ask what we expect to find for the thermodynamic
properties of liquids. We shall consider two thermodynamic functions,
the internal energy and the entropy, and shall ask how we expect these
quantities to differ from the corresponding quantities for the solid. This
information can be found experimentally from the latent heat of fusion
Lw, which gives directly the change in internal energy between the two
phases (for solids and liquids the small quantity P(Vi — V8) by which
this should be corrected is negligible), and from the melting point Tm, for
the* entropy of fusion is given by Lm/Tm. As a matter of general orienta-
tion, we first give in Table XVI-1 the necessary information about a
number of materials. In the first column we give the latent heat of
fusion. We shall find it interesting to compare it with the latent heat of
vaporization; therefore we give that quantity in the next column, and the
ratio of heat, of fusion to heat of vaporization in the third. Next wo
tabulate the molting point and finally the entropy of fusion.

Referring to Table XVI-1, let us first consider the latent heat of fusion.
We observe that in practically every case it is but a small fraction of the
boat of vaporization. That is, the atoms or molecules are pulled apart
only slightly in the liquid state compared with the solid, while in the
vapor they are completely separated. Of course, this holds only for
pressures low compared to the critical pressure; near the critical point, the
heat of vaporization reduces to zero. To be more specific, we notice
that in the metals the heat of fusion is generally three or four per cent of

2] THE LIQUID STATE AND FUSION

TABLE XVI-1. — DATA REGARDING MELTING POINT

259

Lm, kg.-cal.
per mole

Tm, ° abs.

ASm, cal. per
degree

Metals:
Na

0 63

26 2

0 024

371

1 70

1 16

34 4

0 034

923

1 26

2 55

67 6

0 038

932

2 73

0 58

21 9

0 020

330

1 72

3 93

89 4

0 044

1823

2 15

3 45

G9 7

0 050

14'J3

2 31

3 50

96 5

0 037

1802

1 97

3 GO

1763

2 08

4 20

98 1

0 043

1725

2 44

3 11

81 7

0 038

1357

2 29

Zn ....
Ga
Se

1 00
1 34
1 22

31 4

0 051

(192
303
490

2 32
4 42
2 49

0 53

20 6

0 020

312

1 70

Ag ...

2 70

G() 4

0 039

1234

2 19

1 40

°7 0

0 054

594

2 46

In
Sn
Sb

0 78
1 72
4 77

08 0
54 4

0 025
0 088

429
505
903

1 82
3 40
5 29

0 50

18 7

0 027

302

1 GO

5 33

125

0 043

2028

2 03

Au
Hg

3 03
0 58

90 7
15 5

0 033
0 037

1330
234

2 27
2 48

Ti : :: :.

Pb
Bi

Ionic substances
NaF
NaCl
KF
KC1
KBr
AgCl

asr .

TlBr
LiNOs .
NaNOs
KN03
A«N03
NaClOa
NaOH . .
KOH
K2Cr2O;
BaCli
CaCh
PbCh
PbBrz
Pblj
HgBrz
HgI2
Molecular substances.
H,
NO
H:0
O2
A
NHj . .

0 70
1 22
2 51

7 81
7 22
(\ 28
G 41
2 84
3 15
2 18
4 2(1
3 '.)!)
G 00
3 70
2 57
2 70
5 29
1 00
1 01
8 77
5 75
G 03
5 05
4 29
5 18
4 02
4 50

0 028
0 551
1 43
0 096
0.280
1 84

43 0
40 7
47 8

213
183
190
105
159

0 22
3 82
11 3
2 08
1.88
7 )4

0 018
0 02G
0 053

0 037
0 039
0 033
0 039
0 018

0 13
0 14
0 13
0 05
0 15
0 20

570
001
544

1 205
1073
1133
1043
Oil

728
703
700
733
523
583
581
481
528
591
033
071
1232
1047
771
761
648
508
523

14
110
273
54
83
198

I 32
2 03
4 01

0 19
0 72
5 53
0 15
4 05
4 33
3 10
0 09
8 18
11 G
6 45
4 42
5 72
9 90
2 71
2 54
13 07
4 65
5 77
7 32
5 63
8 00
9 09
8 60

2 0
5 02
5 25
1 78
3 38
9 30

0 218

1 69

0 13

3 46

0 200

1 90

0 11

2 04

HC1

0 506

4 85

0 10

159

3 20

CO*

1 99

6 44

0 31

217

9 16

CH4

0 224

2 33

0 10

2 49

HBr

0 620

5.79

0 11

187

3 31

1 63

7 43

0 22

170

9 59

ecu

0 577

8 0

0 07

250

2 30

CHsOH

0 525

0 2

0 06

176

2 08

CzH6OH

1 10

10 4

0 11

156

7 10

CHsCOOII

2 64

20 3

0 13

290

9 21

C«H«

2 35

8 3

0 28

278

8 45

Data are from Landolt's Tables. The heats of vaporization tabulated for alkali hulide* are the
energies required to break the crystal up into ions, rather than into atoms.

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

the heat of vaporization. This receives a ready explanation in terms of
our model of a liquid as a crude approximation to a solid but with many
holes or vacant places. Suppose there are three or four chances out of a
hundred that there will be a vacant space instead of an atom at a given
point in the imperfect lattice. Then the energy of the substance will bo
a, corresponding amount less than if all points wore occupied, for each
atom will have a correspondingly smaller number of neighbors on the
average, and the energy of the crystal comes from the attraction of neigh-
boring atoms for each other. This is about the right order of magnitude
to account for the latent heat of fusion, then, and at the same time it
indicates a density three or four per cent less for the liquid than for the
solid, which is about the right order of magnitude. From Table XVi-1
we see that the situation is about the same for the alkali halides (and
presumably for other ionic crystals) as for the metals, if we understand by
the latent heat of vaporization the energy required to break up the sub-
stance into a gas of ions.

In the case of molecular substances, the latent heat of fusion is a
larger fraction of the latent heat of vaporization, from 10 to 20 per cent
or even higher, so that it seems clear that the liquid differs from the solid
more in these cases than with metals and ionic substances. In many of
the molecular crystals, the molecules are fitted together in a regular
arrangement, whereas in the liquid there is presumably more tendency to
rotation of the molecules, and they do not fit so perfectly. This tendency
would make considerable change in the energy and in the volume, and
would represent a feature which is absent with metals, where the atoms
are effectively spherical. For example, in ice, as we shall see later, the
triangular water molecules are arranged in a definite structure, each
oxygen being surrounded by four others, with the hydrogens in such an
arrangement that the dipoles of adjacent molecules attract each other.
In liquid water, on the other hand, the arrangement is far less perfect
and precise, the molecules are farther apart on centers, and one can
understand the latent heat of fusion simply as the work necessary to
increase the average distance between the dipoles, against their attrac-
tions. Presumably in all the molecular substances, it is more accurate to
think simply of the increased interatomic distance as leading to the
heat of fusion, rather than postulating holes in the structure as definitely
as one does with a metal. However one looks at it, the liquid is a more
open, less well-ordered structure than the solid, and the latent heat
represents the work necessary to pull the atoms or molecules apart to this
open structure.

3. The Entropy of Melting. — When we look at the entropies of
melting, in Table XVI-1, we see that there is a certain amount of regu-
larity in the table. For most of the metals, the entropy of fusion is

ftEC.

TUB

STATE AMU FUS1UJN

between two and three calories. For the diatomic ionic crystals it is
about twice as much, so that if we figure the entropy per atom instead of
per molecule it is about the same as for the metals. As a first step toward
understanding the entropy of molting, we might use a rough argument
similar to that of Chap. XIII, Soc. 5, where we discussed the entropy
changes in polymorphic transitions. We were them interested in tho
slope of the transition curves between phases, but the calculation we made
was one of A$/AF, the change of entropy between two phases divided by
the change of volume, and we assumed that the change of entropy with
volume in going from one phase to another was the same as in changing
the volume of a single phase. Tn this case, using the thermodynamic
relation

(es\ _

\dVjr 7

(ev\
Wi-

we have

AS
AV

thermal expansion
compressibility

(3.1)

(3.2)

From the relation (3.2), and tho observed change of volume on molting,
we can compute the change of entropy. In Table XVI-2 we give values
of volume of the solid per mole (extrapolated from room temperature to
tho melting point by use of the thermal expansion), volume4 of the liquid

TABLE XVI-2. CALCULATION OF ENTKOPY OK MELTINCJ

Molecular
volume of
solid, cc.

Molecular
volume oi
liquid

,. Thermal A/S',,,
expansion computed

A&.
observed

24 2

24 6

0 4

21 f> X 10 * ' 0 13

I 70

14 6 15 2

0 6

7 3 0 36

1 26

10 6

11 0

0.4

6 8

0.49

2.73

46.3 ! 47.2

0 9

23.0

0 13

1.72

7 50

8 12

0 6

3 36

0 86

1.97

A*~

10 8

11 3

0 f>

3 7

0 69

2.19

13 4

14 0

0 6

9 3

0.93

2.46

NaCl

29.6

37 7

8 1

12 1

5.6

6.72

KC1

40 5

48.8

8.3

11 4

4 0

6.15

KBr

45.0

56.0

11 0

12 6

4.9

4 65

AgCl

27.0

29.6

2 6

9.9

2.6

4 33

AgBr

30.4

33 6

3 2

10.4

3.2

3 10

Molecular volumes of the solid are calculated from observed densities at room temperature (as
tabulated in Landolt's Tables), extrapolated to the melting point by using the thermal expansion. For
the ionic ciystals, data on densities of liquids and solids are taken from Lorenz and Herz, X anory.
allffem Chetn., 145, 88 (1025).

262 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

at the melting point per mole, change of volume, thermal expansion, and
AS as computed from them by Eq. (3.2), for a number of solids for which
the required quantities are known, and we compare with the values of A/S
as tabulated in Table XVI-1, which we repeat for convenience. We
see that the calculated entropies of melting are of the right order of
magnitude, but that in most cases they are decidedly smaller than tho
observed ones. In other words, wo must assume that the entropy actu-
ally increases in melting more than we should assume simply from the
change of volume, though as a first approximation that gives a useful and
partially correct picture of what happens. The fact that the ratio of
change of entropy to change of volume is approximately given by this
simple picturo shows that the calculation of Chap. XIII, Sec. 5, on the
slope of transition lines, will apply roughly to the slope of the melting
curves, and this is found to be true experimentally. Of course, as with
transitions between solids, there is great variation from one material to
another, arid occasional materials, of which water is the most conspicuous
example, actually have a decrease of volume on melting, though their
entropy increases, so that in such cases relation (3.2) is obviously entirely
incorrect.

The simple argument we have given so far does not give a very ade-
quate interpretation of the entropy of melting, and as a matter of fact no
very complete theory is available. We can analyze the problem a little
better, however, by considering it more in detail. We can imagine that
the entropy of the liquid should be greater than that of the solid for two
reasons. First, at the absolute zero, the solid has zero entropy. If we
could carry the liquid to the absolute zero by supercooling, however, we
should imagine, at least by elementary arguments, that its entropy would
be greater than zero. The reason is that the atoms or molecules of the
liquid are arranged in a much more random way than in the crystal, and
since entropy measures randomness, this must lead to a positive entropy
for the liquid. We shall be able to estimate this contribution to the
entropy in the next paragraph and shall see that while it is appreciable, it
is not great enough to account for nearly all of the entropy of fusion.
Secondly, there aro good reasons for thinking that the specific hoat of the
liquid, at temperatures between the absolute zero and the melting point,

would be greater than that of the solid. Thus the integral I -=• dT

Jo *

measuring the difference of entropy between absolute zero and the melting
point will be greater for the liquid than for the solid, giving an additional
reason why the entropy of the liquid should be greater than that of the
solid. It is reasonable to think that this effect is fairly large, and the
whole entropy of fusion can be regarded as a combination of the two
effects we have mentioned.

SEC. 3] THE LIQUID STATE AND FUSION 263

Let us try first to estimate the contribution to the entropy of fusion on
account of the randomness of arrangement of the atoms in a liquid. This
calculation can be carried out at the absolute zero, and we can get some-
thing of the right order of magnitude by taking our simple picture of the
liquid as a mixture of atoms and holes. Let us assume N atoms and Na
holes, where a is a small fraction of unity. We may consider that these
form a lattice of N(l + a) points and may say very crudely that any
arrangement of the N atoms and the Na holes on these N(\ + a) points
will constitute a possible arrangement of the substance having the same,
lowest energy Vo. By elementary probability theory, the number of
ways of arranging N things of one sort, and Na of another, in N(l + «)
spaces, one to a space, is

[JVQ +«)]! ,,,.

W ~ Nl(Na)\" (At6)

Using Stirling's formula, AM = (N/e)N approximately, the expression
(3.3) becomes

w _

the powers of N and e canceling in numerator and denominator. Now
we can calculate the entropy by Boltzmann's relation S = k In W, of
Eq. (1.3), Chapter III. We have immediately

S = Nk[(l + a) In (1 + a) - a In a]. (3.5)

In the expression (3.5) let us put a. — 0.04, the value which we roughly
estimated from the latent heat. Then calculation gives us at once

S = O.l&SNk = 0.33 cal. per degree per mole. (3.6)

The value (3.6), while appreciable compared with the values of Table
XVI-1, which are of the order of magnitude of two or more calories
per degree, is definitely less, so much less that it cannot possibly account
for the whole entropy of fusion. Let us see what value of a we should
have to take to get the whole entropy of fusion from this term. If we set
a = 1, for instance, we have

S = l.38Nk = 2.75 cals. per degree per mole, (3.7)

about the right value. But this would correspond to an equal mixture
of atoms and holes, a substance with a density only half that of the solid,
which is clearly impossible. It is unlikely that the crudity of our calcula-
tion could make a very large difference in the result, so that we may
conclude that the effect of randomness on the entropy of fusion is impor-

264 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

tant, but not the only significant effect. In this connection it is interest-
ing to note that in one or two cases supercooled liquids have been carried
down practically to the absolute zero and their specific heats measured, so
that the entropy could be determined at the absolute, zero. To within
an error of a fow tenths of a unit, tho entropy was found to be zero. This
would not exclude an entropy at tho absolute zero of tho order of magni-
tude of Kq. (3.5), which seems possible, but it definitely would show that
tho out ropy difference between solid and liquid comes mostly at tempera-
tures above the absolute1 zero.

It appears from the previous paragraph that the larger part of tho
entropy of fusion must arise because the liquid has a larger specific heat
than tho solid in the hypothetical state of supercooling, so that its entropy
difference1 between absolute zero and the melting point is greater than for
tho solid. An indication as to why this should bo tnio is soon in the
preceding paragraphs of the present section, whore we have discussed
tho change of entropy with volume. This is represented graphically in
Fig. XIII-3, whore tho entropy is shown as a function of temperature, for
several volumes. At larger volumes (as V = VQ in Fig. XIII-3), the
natural frequencies of molecular vibration are lower, so that tho specific
heat rises to its classical value at lower temperatures, and the specific,
heat, and consequently the entropy, are greater than at the smaller
volume (V — 0.7Fo in the figure, for instance). In tho particular case
shown in tho figure, the entropy difference between tho two volumes
shown, which differ by 30 per cent, amounts to about throe entropy units
at temperatures above 300° abs. Something of the same effect, though on
a smaller scale, would be expected in comparing solids and liquids, as wo
have mentioned at the beginning of this section. The liquid is n more
open structure, having therefore lower frequencies of molecular vibration
and a more nearly classical specific heat at low temperatures. Thus its
entropy difference between the absolute zero zmd the melting point, if
the liquid really could bo carried to absolute zero, would bo greater than
for the solid. By itself, however, us we ran judge from Table XVI-2, it
seems unlikely that this effect would be large1 enough. If 30 per cent
difference in volume amounts to throe entropy units, wo should need
something like 15 per cent difference in volume to account for the approxi-
mately 1.5 entropy units needed, when we take account of possible
entropy of the liquid coming from randomness. And this is more than
the actual difference in volume, in most cases. Nevertheless, there is an
additional feature of difference between the liquid and solid that might
lead to still higher specific heat and entropy for the liquid. In Fig. XVI-1
we have seen the type of potential to be expected for an appreciable
number of atoms, those that are capable of shifting to a near-by position
of equilibrium with small expenditure of energy. This is so far from the

SBC. 4] THE LIQriD STATE AND FUSION 265

potential of a linear restoring force that our whole discussion of specific
heats, which rests on simple harmonic motion, does not apply to it. As a
matter of fact, tho energy levels in a potential of the type shown in Fig.
XVI-1 lie closer together than we should suppose from our study of linear
restoring forces. But in general, the closer together the energy levels of
any problem are, the lower the temperature at which its specific heat
becomes approximately classical. This reason to expect a high specific
heat for a supercooled liquid, in addition to those already discussed, is
probably enough to account for the entropy difference between the liquid
and tho solid. It is very hard to get a satisfactory way of calculating tho
magnitude of this entropy difference, however, and we must remain con-
tent with a qualitative explanation of the values of Table XVI-1. One
thing is clear from our discussion: it will hardly bo possible to understand
fusion without studying the liquid state as woll as tho solid state from
the standpoint of the quantum theory, and this is a field that has hardly
been explored at all. No treatment based puroly on classical theory can
be expected to be very good.

4. Statistical Mechanics and Melting. — Objection might bo made to
our argument of tho preceding section, in which we considered a hypo-
thetical supercooled state of tho liquid down to tho absolute zero, on tho
ground lhat that state is not one of thermal equilibrium and that wo can-
not proporly consider it by itself at all. A correct statistical treatment
should yield the equilibrium state at any temperature; that is, below
the melting point it should give the solid, above the molting point tho
liquid, with a discontinuous change in properties at that point. Wo
shall now show by a simple example that the statistical treatment really
will give such a discontinuous change, but that nevertheless our method
of treatment was entirely justified. We shall calculate the partition
function, and from it the free energy, of a simple model of solid and liquid,
and shall show that the free energy as a function of temperature is a
function with practically a discontinuous slope at a given temperature, the
molting point, below which one phase, the solid, is stable, and abovo
which another, the liquid, is tho stable phase.

To describe our model, we shall give its energy levels, so that we can
calculate the partition function directly. The simplest model that shows
the properties we wish is the following. We assume a single level, at
energy NE8, whore N is the number of atoms, corresponding to tho solid.
At a higher energy, NEi, we assume a multiple level corresponding to tho
liquid. The energy is higher on account for example of the greater
interatomic distance in liquids. The multiplicity of the level arises, for
example, on account of the randomness of molecular arrangement. Wo
assume that the multiple energy level at NEi really consists of WN coin-
cident levels, where w is a constant. Then the partition function is

266

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

NE.
~ kT ,

NEi

~ kT'

(4.1)

and the Helmholtz free energy, which is equal to the Gibbs free energy
in this case where there is no pressure, is

/ NE, NEj\

G = -kTln (e kT+w"e~' kT).

(4.2)

If we had only the solid, or only the single level at NEa, the partition
function would have contained only the first term in Eq. (4.2), and the
free energy would have been

G, = NE.. (4.3)

If we had only the liquid, or the multiple level at NEi, we should have had
only the second term in Eq. (4.2), and the free energy would have been

- (Nklnw)T,

(4.4)

in which the first term, NEi, is the internal energy, and Nk In w is the
entropy, exactly analogous to Kq. (1.3), Chap. Ill (the number of states

NE,

Tm T

FIG. XVI-2. — Gibbs free energy as function of temperature, for simplified model of solid
and liquid, illustrating change of phase on melting.

is here WN, so that the entropy should be k In (WN) = Nk In w). For free
energy as a function of temperature we should then have the two straight
lines of Fig. XVI-2, the horizontal one representing the solid, the sloping
one the liquid. The slope of the curve measures the negative of the
entropy, as we see at once from Eqs. (4.3) and (4.4), where the solid has
zero entropy, the liquid the positive entropy Nk In w. This accords at
once with the thermodynamic relation S = —(dA/dT)v = — (dG/dT)P.
From Fig. XVI-2, we see that the solid has the lower free energy at tem-
peratures below the intersection, on account of its lower internal energy,
while the liquid has lower free energy above the intersection, its greater
entropy resulting in a downward slope which counteracts its greater
internal energy. The melting point comes at the intersection, at the

SBC. 4] THE LIQUID STATE AND FUSION 267

temperature where G6 = Gi, or at

m ~T-\ - — A 0 > 'W

kin w ASm x '

where the latent heat of melting, Lm, equals N(Ei — Et) and the entropy
of melting, A/SW, equals JV/fc In w.

The calculation we have just made, considering the solid and liquid
separately, drawing a free energy curve for oach, for all temperatures,
whether they are stable or not, and finding which free energy curve is
lower at any given temperature, is analogous to the method used in
this chapter to discuss fusion and also to the method used in Chap. XII,
for example in Fig. XII-4, in discussing vaporization by Van der Waals'
equation. Properly, however, we should have used directly the single
free energy formula (4.2), and plotted it as a function of temperature.
This almost precisely equals the function G9 when T < Tm, and GI when
T > Tm. For if T « Tm, the first term in the bracket of Eq. (4.2)
is much larger than the second, and Eq. (4.3) is a good approximation to
Eq. (4.2), while if T > > Tm the second term is much larger than the
first, and Eq. (4.4) is the correct approximation. The formula (4.2),
however, represents a curve which joins these two straight lines con-
tinuously, bending sharply but not discontinuously through a small range
of temperatures, in which the two terms of Eq. (4.2) are of the same
order of magnitude. For practical purposes, this rango of temperatures is
so small that it can be neglected. Let us compute it, by finding the
temperature T at which the second term in the bracket of Kq. (4.2) has
a certain ratio, say c, to the first term. That is, we have

-JVEi

c kT

, ., . N(Ei - E.)
In c = N In w pr

= JV hi w( } ^J

= N In wf — T^"/' (4'6)

using Eq. (4.5). Thus we have

T - Tm In c

Nlnw

Here In u; is of the order of magnitude of unity. If we ask for the tem-
perature when the second term of Eq. (4.2) is, say, ten times the first, or

268 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVI

one tenth the first, we have c equal respectively to 10 or -fa, so that In c
is of the order of magnitude of unity, and positive or negative as the case
may be. Thus (T - Tm)/T is of the order of I/TV, showing that the
range of temperature in which the correct free energy departs from the
lower of the two straight lines in Fig. XVI-2 is of the order of 1/N of
the melting temperature, a wholly negligible range if N is of the order of
the number of molecules in an ordinary sample. Thus, our procedure
of finding the intersection of the two curves is entirely justified.

The real situation, of course, is much more complicated than this.
There are many stationary states for the solid, corresponding to different
amounts of excitation of vibrational energy; when we compute the parti-
tion function considering all these states, and from it the free energy, we
get a curve like those of Fig. XIII-6, curving down as the temperature
increases to indicate an increasing entropy and specific heat. Similarly
the liquid will have not merely one multiple level but a distribution of
stationary states. There is even good reason, on the quantum theory, to
doubt that the lowest level of the liquid will be multiple at all; it is much
more likely that it will be spread out into a group of closely spaced, but
not exactly coincident, levels, so that the entropy will really be zero at
the absolute zero, but- will rapidly increase to the value characteristic
of the random arrangement as the temperature rises above absolute zero.
However this may be, there will surely be a great many more levels for
the liquid than for the solid, in a given range of energies, and the liquid
levels will lie at definitely higher energies than those of the solid. This is
all we need to have a partition function, like Eq. (4.1), consisting of two
definite parts : a set of low-lying levels, which arc alone of importance at
low temperatures, and a very much larger set of higher levels, which are
negligible at low temperatures on account of the small exponential Boltz-
mann factor but which become much more important than the others at
high temperature, on account of their great number. In turn this will
lead to a free energy curve of two separate segments, joined almost with
discontinuous slope at the melting point.

It is not impossible, as a matter of fact, to imagine that the two
groups of levels, those for the solid and for the liquid, should merge
partly continuously into each other. An intermediate state between solid
and liquid would be a liquid with a great many extremely small particles
of crystal in it, or a solid with many amorphous flaws in it that simulated
the properties of the liquid. Such states are dynamically possible and
would give a continuous series of energy levels between solid and liquid.
If there were enough of these, they could affect our conclusions, in the
direction of rounding off the curve more than we should otherwise expect,
so that the melting would not be perfectly sharp. We can estimate
very crudely the temperature range in which such a hypothetical gradual

SEC. 4] THE LIQUID STATE AND FUSION 269

change might take place, from our formula (4.7). Suppose that instead
of considering the melting of a large crystal, we consider an extremely
small crystal containing only a few hundred atoms. Then, by Eq. (4.7),
the temperature range in which the gradual change was taking place
might be of the order of a fraction of a per cent of the melting temperature,
or a degree or so. Crystals of this size, in other words, would not have a
perfectly sharp melting point, and if the material breaks up into as fine-
grained a structure as this around the melting point, even a large crystal
might melt smoothly instead of discontinuously. The fact, however, that
observed melting points of pure materials are as sharp as they are, shows
that this effect cannot be very important in a large way. In any case, it
cannot affect the fundamental validity of the sort of calculation which
we have made, finding the melting point by intersection of free energy
curves for the two phases; for mathematically it simply amounts to an
extremely small rounding off of the intersection. We shall come back to
such questions later, in Chap. XVIII, where we shall show that in certain
cases there can be a large rounding off of such intersections, with corre-
sponding continuous change in entropy. This is not a situation to be
expected to any extent, however, in the simple problem of melting.

CHAPTER XVII
PHASE EQUILIBRIUM IN BINARY SYSTEMS

In the preceding chapter wo have been considering the equilibrium
of two phases of the same substance. Some of the most important cases
of equilibrium come, however, in binary systems, systems of two com-
ponents, and we shall take thorn up in this chapter. We can best under-
stand what is meant by this by some examples. The two components
mean simply two substances, which may bo atomic or molecular and
which may mix with each other. For instance, they may be substances
like sugar and water, one of which is soluble in the other. Then the study
of phase equilibrium becomes the study of solubility, the limits of solu-
bility, the effect of the solute on the vapor pressure, boiling point, molting
point, etc., of the solvent. Or the components may be metals, like copper
and zinc, for instance. Then we meet the study of alloys and the whole
field of metallurgy. Of course, in metallurgy one often has to deal with
alloys with more than two components — ternary alloys, for instance, with
three components — but they are considerably more complicated, and we
shall not deal with them.

Binary systems can ordinarily exist in a number of phases. For
instance, the sugar-water system can exist in the vapor phase (practically
pure water vapor), the liquid phase (the solution), and two solid phases
(pure solid sugar and ice). The copper-zinc system (the alloys that form
brasses of various compositions), can exist in vapor, liquid, and five or
more solid phases, each of which can exist over a range of compositions.
Our problem will be to investigate the equilibrium between these phases.
We notice in the first place that we now have three independent variables
instead of the two, which we have ordinarily had before. In addition to
pressure and temperature, we have a third variable measuring the com-
position. We ordinarily take this to be the relative concentration of ono
or the other of the components, N\/(N\ + AT2) or N2/(Ni + N2), as
employed in Chap. VIII, Sec. 2; since these two quantities add to unity,
only one of them is independent. Then we can express any thermo-
dynamic function, as in particular the Gibbs free energy, as a function
of the three independent variables pressure, temperature, and composi-
tion. We shall now ask, for any values of pressure, temperature, and
composition, which phase is the stable one; that is, which phase has the
smallest Gibbs free energy. In some cases we shall find that a single

270

SEC. 1] PHASE EQUILIBRIUM IN BINARY SYSTEMS 271

phase is stable, while in other cases a mixture of two phases is more stable
than any single phase. Most phases are stable over only a limited range
of compositions, as well as of pressure and temperature. For instance, in
the sugar and water solution, the liquid is stable at a given temperature,
only up to a certain maximum concentration of sugar. Above this
concentration, the stable form is a mixture of saturated solution and
solid sugar. The solid phases in this case, solid sugar and solid ice, are
stable only for quite definite compositions; for any other composition,
that is for any mixture; of sugar and water, the stable form of the solid is a
mixture of sugar and ice. On the1 other hand, we have stated that the
solid phases of brass are stable ovor a considerable range of compositions,
though for intermediate compositions mixtures of two solid phases are
stable. In studying these subjects, the first thing is to get a qualitative
idea of the various sorts of phases that exist, and we proceed to that in
the following section.

1. Types of Phases in Binary Systems. — A two-component system,
like a system with a single component, can exist in solid, liquid, and
gaseous phases. The gas phase, of course, is perfectly simple: it is simply
a mixture of the gas phases of the two components. Our treatment of
chemical equilibrium in gases, in Chap. X, includes this as a special
case. Any two gases can mix in any proportions in a stable way, so long
as they cannot react chemically, and we shall assume only the simple case
where the two components do not react in the gaseous phase;.

The liquid phase of a two-component system is an ordinary solution.
The familiar solutions, like that of sugar in water, exist only when a
relatively small amount of one of the components, called the solute (sugar
in this case) is mixed with a relatively large amount of the other, the*
solvent. But this is the case mostly with components of very different
physical properties, like sugar and water. Two similar components often
form a liquid phase stable over large ranges of composition, or even for all
compositions. Thus water and ethyl alcohol will form a solution in any
proportion, from pure water to pure alcohol. And at suitable tempera-
tures, almost any two metals will form a liquid mixture stable at any
composition . A liquid solution is similar in physical properties to any
other liquid. We have; seen that an atom or molecule in an ordinary
liquid is surrounded by neighboring atoms or molecules much as it would
be in a solid, but the ordered arrangement does not extend beyond nearest
neighbors. When we have a mixture of components, it is obvious that
each atom or molecule will be surrounded by some others of the same;
component but some of the other component. If an atom or molecule of
one sort attracts an unlike neighbor about as well as a like neighbor, and
if atoms or molecules of both kinds fit together well, the solution may well
have an energy as low as the liquids of the individual components. In

272 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

addition, A golntjon. like p, mixture of gases t has ail entropy of mixing, so
that the entropy of the solution will be greater than that of the com-
ponents. In such a case, then, the Gibbs free energy of the solution will
he lower than for the pure liquids and it will be the stable phase. On
1 he other hap^r if thn fttoms or molecules of the two sorts fit together badlv
or do not attmr.fr ^q-eh nf.hor, the rmerfiy of the solution may well be greater
1 1) an that of the j>ure liquids, enough greater to make the free energy
greater in spite of the entropy of mixing, so that the stable situation will
be a mixture of the pure liquids, or a liquid and solid, depending on the
temperature. Thus oil and water do not dissolve in each other to any
extent. Their molecules are of very different sorts, and the energy is
lower if water molecules are all close together and if the oil molecules
are congregated together in another region. That is, the stable situation
will be a mixture of the two phases, forming separate drops of oil and
water, or an emulsion or suspension. A very little oil will presumably be
really dissolved in the water and a very little water in the oil, but the drops
will be almost pure.

As we have just seen, the condition for the existence of a liquid phase
stable for a wide range of concentrations (that is, for a large solubility
of one substance in another), is that the forces acting between atoms or
molecules of one component and those of the other should be of the same
order of magnitude as the forces between pairs of atoms or molecules of
either component, so that the internal energy of the solution will be at
least as low as that of the mixture, and the entropy of mixing will make
the free energy lower for the solution. Let us consider a few specific
cases of high solubility. In the first place, we are all familiar with the
large1 solubility of many ionic salts in water. The crystals break up into
ions in solution, and these ions, being charged, orient the electrical
dipoles of the water around them, a positive ion pulling the negatively
charged oxygen end of the water molecule toward it, a negative ion pulling
the positively charged hydrogen. This leads to a large electrical attrac-
tion, and a low energy and free energy. The resulting free energy will be
lower than for the mixture of the solid salt and water, unless the salt is
very strongly bound. Water is not the only solvent that can form ionic
solutions in this way. Liquid ammonia, for instance, has a large dipole
moment and a good many of the same properties, and the alcohols, also
with considerable dipole moments, are fairly good solvents for some ionic
crystals.

Different dipole liquids, similarly, attract each others' molecules by
suitable orientation of the dipoles and form stable solutions. We have
already mentioned the case of alcohol and water. In ammonia and
water, the interaction between neighboring ammonia and water molecules
is so strong that they form the ammonium complex, leading to NH4OH

SEC. 1] PHASE EQUILIBRIUM IN BINARY SYSTEMS 273

if the composition is correct, and a solution of NH4OH in either water or
ammonia if there is excess of either component. The substance NH^)!!
is generally considered a chemical compound; but it is probably more
correct simply to recognize that a neighboring water and ammonia mole-
cule will organize themselves, whatever may be the composition of the
solution, in such a way that one of the hydrogens from the water and
the three hydrogens from the ammonia form a fairly regular tetrahedral
arrangement about the nitrogen, as in the ammonium ion. There are no
properties of the ammonia- water system which behave strikingly differ-
ently at the composition NH<OH from what they do at neighboring
concentrations.

Solutions of organifi substances are almost invariably made in organi(
solvents^ simply because here again the attractive forces between twi
different types of molecule are likely to be large if the molecules an
similar. Different hydrocarbons, for instance, mix in all proportions, ai-
one is familiar with in the mixtures forming kerosene, gasoline, etc. The
forces between an organic solvent and its solute, as between the molecule?
of an organic liquid, are largely Van der Waals forces, though in some
cases, as the alcohols, there are dipole forces as well.

In the metals, the same type of interatomic, force1 acts bet ween atom,-
of different metals that acts between atoms of a single element. We have
already stated that for this reason liquid solutions of many metals with
each other exist in wide ranges of composition. There, are many other
cases in which two substances ordinarily solid at room temperature are
soluble in each other when liquefied. Thus, a great variety of molten
ionic crystals are soluble in each other. And among the silicates and
other substances held by valence bonds, the liquid phase permits a wide
range of compositions. This is familiar from the glasses, which can have
a continuous variability of composition and which can then supercool to
essentially solid form, still with quite arbitrary compositions, and yet
perfectly homogeneous structure.

Solid phases of binary systems, like the liquid phases, are very com-
monly of variable composition. Here, as with the liquid, the stable range
of composition is larger, the more similar the two components are. This
of course is quite contrary to the chemists' notion of definite chemical
composition, definite structural formulas, etc., but those notions are
really of extremely limited application. It happens that the solid phases
in the system water — ionic compound are often of rather definite com-
position, and it is largely from this rather special case that the idea of
definite compositions in solids has become so firmly rooted. In such a
system, there are normally two solid phases: ice and the crystalline ionic
compound. Ice can take up practically none of any ionic compound, so
that it has practically no range of compositions. And many ionic crystals

274 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

take up practically no water in their crystalline form. But there are
many ionic crystals which are said to have water of crystallization.
Water molecules form definitely a part of the structure. And in some of
these the proportion of water is not definitely fixed, so that they form
mixed phases of variable composition.

Water and ionic compounds are very different types of substances, and
it is not unnatural that they do not form solids of variable composition.
The reason why water solutions of ionic substances exist is that the water
molecules can rotate so as to be attracted to the ions; this is not allowed
in the solid, where the ice structure demands a fairly definite orientation
of the molecule. But as soon as we think about solid phases of a mixture
of similar components, we find that almost all the solid phases exist over
quite a range. Such phases are often called by the chemists solid solu-
tions, to distinguish them from chemical compounds. This distinction
is valid if we mean by a chemical compound a phase which really exists
at only a quite definite composition. But the chemists, and particularly
the metallurgists, are not always careful about making this distinction; for
this reason the notation is misleading, and we shall not often use it.

Solid phases of variable composition exist in the same cases that we
have already discussed in connection with liquids. Thus mixtures of
ionic substances can often form crystals with a range of composition.
The conditions under which this range of composition is large are what we
should naturally suppose: the ions of the two components should be of
about the same size and valence, and the two components should be
capable of existing in the same crystal structure. We shall meet many
examples of such solids of variable composition later, when we come to
study different types of materials. The best-explored range of solid
compounds of variable composition comes in metallurgy. Here an atom
can replace another of the same size quite freely but not another of rather
different size. Thus the copper and nickel atoms have about the same
size; they form a phase stable in all proportions. On the other hand,
calcium and magnesium have atoms of quite different sizes, normally
existing in different crystal structures, and they cannot bo expected to
substitute for each other in a lattice. They form, as a matter of fact, as
close an approach to phases of definite chemical composition as wre find
among the metals. They form three solid phases: pure magnesium, pure
calcium, and a compound CasMg4, and no one of these is soluble to any
extent in any of the others; that is, each exists with almost precisely fixed
composition. Most pairs of elements are intermediate between these.
They form several phases, each stable for a certain range of compositions,
and often each will be centered definitely enough about some simple
chemical composition so that it has been customary to consider them as
being chemical compounds, though this is not really justified except in

SEC. 2] PHASE EQUILIBRIUM IN BINARY SYSTEMS 275

such a definite case as CaaMg4. Each of the phases in general has a
different crystal structure. Of course, the crystal cannot be perfect, for
ordinarily it contains atoms of the two components arranged at random
positions on the lattice. It is the lattice that determines the phase, not
the positions of the metallic atoms in it. But if the two types of atom in
the lattice are very similar, they will not distort it much, so that it will
be practically perfect. For compositions intermediate between those in
which one of the phases can exist, the stable situation will be a mixture of
the two phases. This occurs, in the solid, ordinarily as a mixture of tiny
crystals of the two phases, commonly of microscopic size, with arbitrary
arrangements and sizes. It is obvious that the properties of such a
mixture will depend a great deal on the size and orientation of the crystal
grains; these are things not considered in the thermodynamical theory at
all.

2. Energy and Entropy of Phases of Variable Composition. — The
remarks we have just made about mixtiires of crystalline phases raise the
question, what is a single phase anyway? We have not so far answered
this question, preferring to wait until we had some examples to consider.
A single phase is a mixture that is homogeneous right down to atomic
dimensions. If it has an arbitrary composition, it is obvious that really
on the atomic scale it cannot be homogeneous, but if it is a single phase
we assume that there is no tendency for the two types of atom to segregate
into different patches, even patches of only a few atoms across. On the
other hand, a mixture of phases is supposed to be one in which the two
types of atoms segregate into patches of microscopic or larger size. These
two types of substance have quite different thermodynamic behavior,
both as to internal energy and as to entropy. We shall consider this
distinction, particularly for a metallic solid, but in a way which applies
equally well to a liquid or other type of solid.

Suppose our substance is made of constituents a and b. Let the
relative concentration of a be ca = £; of 6, c& = 1 — x. Assume the
atoms arc arranged on a lattice in such a way that each atom has s neigh-
bors (s = 8 for the body-centered cubic structure, 12 for face-centered
cubic and hexagonal close packed, etc.). In a real solid solution, or
homogeneous phase, there will be a chance x of finding an atom a at any
lattice point, a chance 1 — .r of finding an atom 6. We assume a really
random arrangement, so that the chance of finding an atom at a given
lattice point is independent of what happens to be at the neighboring
points. This assumption will be examined more closely in the next chap-
ter, where we take up questions of order and disorder in lattices. Then
out of the 5 neighbors of any atom, on the average sx will be of type a,
s(l — x) of type b. If we consider all the pairs of neighbors in the crystal,
we shall have

276 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIT

Nsx* . , L

- ^ - pairs of type na

£t

Ns(\ - .r)2 . , 4 M
-- £ pairs of type 60

Nsx(l — a:) pairs of type ab. (2.1)

Hero N is the total number of atoms of both sorts. The factors £ in
Eq. (2.1) arise4 because each pair must be counted only once, not twice
as we should if \\e said that the number of pairs of type aa equaled the
number of atoms of type a (Nx) times the number of neighbors of type a
which each one had (s.r). Now we make a simple assumption regarding
the energy of the crystal at the absolute zero. We assume that the total
energy can be written as a sum of terms, one for each pair of nearest neigh-
bors. We shall be interested in this energy only at the normal distance
of separation of atoms. At this distance, we shall assume that the energy
of a pair aa is Eaa, of a pair bb is 7?w>, and of a pair ab, Eab. All these
quantities will be negative, if we assume the zero state of energy is the
state of infinite separation of the atoms (the most convenient assumption
for the present purpose) and if all pairs of atoms attract each other.
Then the total energy of the crystal, at the absolute zero, will be

~E(Ul + N*(}~ ^Ebh + Nsx(l - x)Eab

+ 2r(l - x)(Eab - ^-f^)} (2.2)

According to our assumptions, the energy of a crystal wholly of a is
(Ns/2)Eaa, and wholly of 6 is (Ns/2)Ehb. Thus the first two terms on the
right side of Eq. (2.2) give the sum of the energies of a fraction x of
the substance «, and a fraction (1 — x) of 6. These two would give the
whole energy in case wre simply had a mixture of crystals of a and b. But
the third term, involving x(l — x), is an additional term arising from the
mutual interactions of the two types of atoms. The function x(} — x) is
always positive for values of x between 0 and 1, being a parabolic function
with maximum of £ when x = ?, and going to zero at x = 0 or 1 . Thus
this last term has a sign which is the same as that of Eab — (Eatt + Ebb)/2.
If Eab is more positive than the mean of Eaa and EM (that is, if atoms a
and b attract each other less strongly than they attract atoms of their own
kind), then the term is positive. In this case, the solution will have
higher energy than the mixture of crystals, and if the entropy term in
the free energy does not interfere, the mixture of crystals will be the more
stable. On the other hand, if Eab is more negative than the mean of Eau
and EM, so that atoms of the opposite sort attract more strongly than
either one attracts its own kind, the term will be negative and the solution

SBC. 2] PHASE EQUILIBRIUM IN BINARY SYSTEMS 277

will have the lower energy. In order to get the actual internal energy at
any temperature, of course we must add a specific heat term. We shall
adopt the crude hypothesis that the specific heat is independent of
composition. This will be approximately true with systems made of two
similar components. Then in general we should have

rr Ns\ T1 . ,. >. „ io/i \i v H<in + ^MM
"2" a + ^ ~~ ^ + ^ "~ x\Aab ~~ 2 )

f*T

+ CpdT. (2.3)
Jo

Next, lot us consider tho entropy of the homogeneous phase and
compare it with the entropy of a mixture of two pure components. Tho
entropy of tho puro oompononts will bo just the part determined from the

specific heat, or I -ni dT. But in tho solution there will bo an additional
Jo J

term, the entropy of mixing. This, as a, matter of fact, is just the same*
term found for gases in Kq. (2.12), Chap. VIII: it is

Art - -Nk[x In x + (1 - x) hi (1 - jr)],

in the notation of the present chapter. We can, however, justify it
directly without appealing to the theory of gases, which certainly cannot
be expected to apply directly to tho present case. Wo use a method like
that used in deriving Eq. (3.5), Chap. XVI. We have a lattice with
N points, accomodating NX atoms of one sort, N(l — x) of another sort.
There are then

AM
W = ______ - ' (9 4

(Nx)\[N(l - (

ways of arranging the atoms on the lattice points, almost all of whirl)
have approximately the same energy. Using Stirling's formula, this
becomes

w -

By Boltzmann's relation S = k In W, this gives for the entropy of mixing
Atf = -Nk[x In x + (1 - x) In (1 - x)]. (2.6)

The other thermodynamic functions are also easily found. If we confine
ourselves to low pressures, of the order of atmospheric pressure, we can
neglect the term P V in the Gibbs free energy of liquid or solid. Then the
Helmholtz and Gibbs free energies are approximately equal and are given
by

278

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

(a) Eab - (Kaa +

> 0;

Ox,

0.5
(q)Eqb-E*°*Ebb>0

x, I

1 x

A = (? = U + NkT[x In x + (1 - x) In (1 - x)] - T \ ^ dT, (2.7)

Jo i

where U is given in Eq. (2.3).

We have now found approximate values for the thermodynamic
functions of our homogeneous phase. The entropy is greater than that
of the mixture by the term (2.6), and the internal energy is either
greater or smaller, as we have seen. To illustrate our results, we give
in Fig. XVII-1 a sketch of G as a function of a*, for three cases:

(b) Eab - (Eaa + E^/2 = 0; (c) Eab -
(Eon + Ebb)/2 < 0. We observe that in
each case the free energy begins to
decrease sharply as x increases from
zero or decreases from unity. This is
on account of the logarithmic function
in the entropy of mixing (2.6). But in
case (a), whero the atoms prefer to
segregate rather than forming a homo-
geneous phase, the free energy then rises
for intermediate concentrations, while in
case (6), where tho atoms are indifferent
to their neighbors, or in (c) where they
definitely prefer unlike* neighbors, the
free energy falls for intermediate con-
centrations, moro strongly in case (c).
In each case we have drawn a dotted line
connecting the points x = 0 and x = 1.
This represents the free energy of tho
mixture of crystals of tho pure compo-
nents. We see that in cases (6) and (c)
the solution always has a lower free
energy than the mixture of crystals, but
< 0 in (a) there is a range where it does not.

FIG. xvn-i.-GibbB free energy of However, we shall see in the next section

a binary system, as function of that \VC must look a little more Carefully
concentration. into the situation before bc}ng surc what

forms the stable state of the system.

3. The Condition for Equilibrium between Phases. — Suppose we
have two homogeneous phases, one with composition XL and free energy
Gij the other with composition z2 and free energy Gr2. By mixing these
two phases in suitable proportions, the resulting mixture can have a
composition anywhere between x\ and rr2. And the free energy, being
simply the suitably weighted sum of the free energies of the two phases, is
given on a G vs. x plot simply as a straight line joining the points Gi, x\

(b)Eab-

• = 0

SKC. 31 PHASE EQUILIBRIUM IN BINARY SYSTEMS 279

and (72, #2- That is, for an intermediate composition corresponding to a
mixture, the free energy has a proportional intermediate value between
the free energies of the two phases being mixed. We saw a special case
of this in Fig. XVII-1, where the dotted line represents the free energy
of a mixture of the two phases with x = 0, x = i respectively.

Now suppose we take the curve of Fig. XVII-1 (a) and ask whether
by mixing two phases represented by different points on this curve, we
can perhaps get a lower Gibbs free energy than for the homogeneous
phase. It is obvious that we can and that the lowest possible lino
connecting two points on the curve is the mutual tangent to the two
minima of the curve, shown by (?i(?2 in Fig. XVII-1 (a). The point G\
represents the free energy of a homogeneous phase of composition Xi and
the point (?2 of composition #2. Between these compositions, a mixture
of thffije f,wo homogeneous phases represented by the dotte^ line will
have lower Gibbs free energy than the homogeneous phase, and will
represent the stable situation. For x loss than x\, or greater than xz, the
straight line is meaningless; for it would represent a mixture with more
than 100 per cent of one phase, less than zero of the other. Thus for x
less than Xi, or greater than xz, the homogeneous phase is the stable
one. In other words, we have a case of a system that has only two
restricted ranges of composition in which a single phase is stable, while
between these ranges we can only have a mixture of phases.

We can now apply the conditions just illustrated to some actual
examples. Ordinarily we have two separate curves of G against xt to
represent the two phases; Fig. XVII-1 (a) was a special case in that a
single curve had two minima. And_in a region where a common tangent
to the curve lies lower than either curve between the points of tangcncy,
the mixture of the two phases represented by the points of tanpencv will
be the stable phase. First we consider the equilibrium between liquid
and solid, in a case where the components are soluble in all proportions,
both in solid and liquid phases. For instance, we can take the case of
melting of a copper-nickel alloy. To fix our ideas, let copper be consti-
tuent a, nickel constituent 6, so that x — 0 corresponds to pure nickel,
.r = 1 to pure copper. We shall assume that in both liquid arid solid the
free energy has the form of Fig. XVII-1 (6), in which the bond between a
copper and a nickel atom is just the mean of that between two coppers
and two nickels. This represents properly the case with two such similar
atoms. Such a curve departs from a straight line only by the entropy of
mixing, which is definitely known, the same in liquid and solid. Thus it
is determined if we know the free energy of liquid and solid copper and
nickel, as functions of temperature. From our earlier work we know how
to determine these, both experimentally and theoretically. In particular,
we know that the free energy for liquid nickel is above that for solid

280

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

nickel at temperatures below the melting point, 1702° abs. but below at
iemperatures above the melting point, and that similar relations hold for
copper with melting point 1356° abs. Wo further know that the rate of
change of the difference between the free energy of liquid and solid nickel,
with temperature, is the difference of thoir entropies, or T times the latent
heat of fusion. Thus we have enough information to draw at least
approximately a set of curves like those shown in Fig. XVII-2. Hero

Ni Cu Ni Cu Ni x, x2 Cu

T, < 1,356° Abs T2 = 1,356° Abs 1,356<T3 < 1,702°

Ni Cu Ni Cu

T4 = 1,702° Abs T5 => 1,702° Abs

KUJ. XVII-2.- (ribbs free energy for solid and liquid as function of concent ration, for
different temperatures, Ni-Cu system.

\ve give G for UK; solid, G1 for tho liquid phase, as functions of composition,
for five temperatures: Ti below 1356°, 7\ at 1356°, 773 between 1356 and
1702°, T± at 1702°, and 7T5 above 1702°. Below 1356°, the free energy
for the solid is everywhere below that for the liquid, so that the former
is the stable phase at any composition. At 1356°, the liquid curve
touches the solid one, at 100 per cent copper, and above this temperature
the curves cross, the solid curve lying below in systems rich in nickel,
the liquid curve below in systems rich in copper. In this case, T* in the
figure, we can draw a common tangent to the curves, from G\ to (?2 at

SEC. 3]

PHASE EQUILIBRIUM IN BINARY SYSTEMS

281

1700

concentrations x\ and x%. In this range, then, for concentrations of
copper below x\, a solid solution is stable; above x$, a liquid solution is
stable; while between x\ and x2 there is a mixture of solid of composition
Xi and liquid of composition .r2. These two phases, in other words, are
in equilibrium with each other in any proportions. At 1702°, the range
in which tho liquid is stable has extended to the whole range of com-
positions, the curve of G for the liquid lying lower for all higher
temperatures.

The stability of the phases can be shown in a diagram like Fig. X VII-3,
railed a phase diagram. In this, temperature is plotted as ordinate,
composition as abscissa, and lines separate the regions in which various
phases are stable. Thus, at high temperature, the liquid is stable for all
compositions. Running from 1702° to 1356° are iwo curves, one called
the liquidus (the upper one) and
the other called the solidus. For
any T-x point lying between the
liquidus and solidus, the stable
state is a mixture of liquid and solid.
Moreover, we can read off from the
diagram the compositions of the
liquid and solid in equilibrium at
any temperature. The horizontal
line drawn in Fig. XVII-3, at tem-
perature Z'8 (see Fig. XVII-2),
cuts the solidus at composition x±
and the liquidus at :r2, agreeing with
the compositions for 7T3 in Fig. XVII-2. Then .ri represents the
composition of the solid, o*2 of the liquid, in equilibrium with eaeh other at
this temperature. Finally, below the solidus the stable phase is always
the solid.

From the phase diagram we can draw information not only about
equilibrium but about the process of solidification or melting. Suppose
we have a melt of composition or2, at a temperature above the liquidus, and
suppose we gradually cool the material. The composition of course,
will not change until we reach the liquidus, and solid begins to freeze
out. But now the solid in equilibrium with liquid of composition .r2 has
the composition Xi, much richer in nickel than the liquid. This will be
frozen out, and as a result the remaining liquid will bo deprived of nickel
and will become richer in copper. Its concentration will then lie farther
to the right in the diagram, so that it will intersect tho liquidus at a lower
temperature. As the temperature is decreased, then, some of this liquid,
perhaps of composition z2, will have solid of composition x( freeze from
it, further enriching the liquid in copper. This process continues, more

Fl°*

XVII-'J — Phiiso diagram
system.

for

282 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

and more liquid freezing out until the temperature reaches 1356°, when
the last portion of the liquid will freeze out as pure copper. There are
two interesting results of this process. In the first place, the freezing
point is not definite; material freezes out through a range of temperatures,
all the way from the temperature corresponding to the point xz on the
graph to the freezing point of pure copper. In the second place, the
material which has frozen out is by no moans homogeneous. Ordinarily
it will freeze out in tiny crystal grains. And we shall observe that the
first grains freezing out are rich in nickel, while successive grains are
more and more rich in copper, unt il the last material frozen is pure copper.
The over-all composition of the solid finally left is of course the same
as that of the original liquid, but it is not of homogeneous composition
and henco is not a stable material. We can see this from Fig. XVII-2,
where the curves of G vs. composition for the solid are convex downward.
With such a curve, the G at a definite composition is necessarily lower
than the average G's of two compositions, one richer and the other poorer
in copper, which would contain the same total amount of each element.
By an extension of this argument, the inhomogeneous material freezing
out of the melt must have a higher free energy than homogeneous material
of the same net composition, and on account of its thermodynamic insta-
bility it will gradually change over to the homogeneous form. This
change can be greatly accelerated by raising the temperature, since as
mentioned in Sec. 1, Chap. XVI, the rate of such a process, involving the
changing place of atoms, depends on a factor exp ( — e/kT), increasing
rapidly with temperature. Accordingly, material of this kind is often
annealed, held at a temperature slightly below the melting point for a
considerable time, to allow thermodynamic equilibrium to take place
and at the same time to allow mechanical strains to be removed.

The reverse process of fusion can be discussed much as we have con-
sidered solidification. Of course, if we take the solid just as it has solidi-
fied, without annealing, there will be crystal grains in it of many different
compositions, which will melt at different temperatures, the liquids
mixing. But if we start with the equilibrium solid, of a definite composi-
tion, it will begin to melt at a definite temperature. The liquid melting
out will have a higher concentration of copper than the solid, however,
leaving a nickel-rich material of higher melting point. The last solid to
melt will be rich in nickel, of such a composition as to be in equilibrium
with the liquid. It is interesting to notice that the process of melting
which we have just described is not the exact reverse for solidification.
This is natural when we recall that the solid produced directly in solidi-
fication is not in thermodynamic equilibrium, so that when the proc-
ess is carried on with ordinary, finite velocity it is not a reversible
process.

SEC. 4]

PHASE EQUILIBRIUM IN BINARY SYSTEMS

283

4. Phase Equilibrium between Mutually Insoluble Solids. — In the
preceding section we have considered phase equilibrium between solid and
liquid, in the case where the components were soluble in each other in any
proportions, in both liquid and solid. Now we shall consider the case
where there is practically no solubility of one solid in the other; that is,
there are two solid phases, each one stable in only a very narrow range of

FIG. XVII-4. — Gibbs free energy for solids and liquid a& function of concentration, for
different temperatures, in a system with almost mutually insoluble solids.

concentration about x = 0 or x = 1. The free energy for the solid will
have much the form given in Fig. XVII-1 (a). But we shall assume that
the minima of the curve are extremely sharp and we shall not assume that
both minima belong to the same curve. In a case like this, it is most
likely that the pure phase of one component will have different crystal
structure and other properties from the pure phase of the other, and there
will be no sort of continuity between the phases, as in Fig. XVII-1 (a).
For the liquid we shall again assume the form of Fig. XVII-1 (b). Then
we give in Fig. XVII-4 a series of curves for G against x at increasing
temperatures, and in Fig. XVII-5 the corresponding phase diagram. The

284

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

method of construction will be plain by comparison with the methods
used in Figs. XVII-2 and XVII-3. At low temperatures like 7\, there is
a very small range of compositions from 0 to Xi, in which phase a is stable,
and another small range, from .r2 to 1, in which phase ft is stable. Here
we have given the name a to the phase composed of pure a with a little
b dissolved in it, and ft to pure b with a little a dissolved in it. If the
substanc.es a and 6 are really mutually insoluble in the solid, the two
curves representing G vs. x for solids a and ft will have infinitely sharp
minima in Fig. XVII-4, rising to groat heights for x infinitesimally greater
than zero, or infinitesimally less than 1. For the whole range of composi-
tions between x\ and Xz, at these low temperatures, the stable form will

be a mechanical mixture of crystals of a
and 0.

At a somewhat higher temperature,
between rl\ arid 7T2, the G curve for the
liquid will fall low enough to be tangent
to the straight line representing the mix-
ture of Xi and a>2. This is for the com-
position denoted by x^^ in Fig. XVII-5
and will be discussed later. At higher
temperatures, as T2, there is a range of
compositions from 0*2 to x3, in which the
liquid is the stable phase, while for com-
a * b x positions from :TI to xz the stable form is

Fi<} xvn-5. Ph!seICoquihbrium a mixture of liquid and phase a, and from
diagram for a system with almost #3 to #4 it is a mixture of liquid and phase

mutually insoluble solids, as given in „ A ,, , . • , rn .1

Fig. XVII-4. P* As the temperature rises to 7 3, the

melting point of pure material b, the

phase ft disappears, and at 7T&, the melting point of pure a, the phase
a disappears, leaving only the liquid as the stable phase above this
temperature.

The process of freezing is similar to the previous case of Fig. XVII-3.
Suppose the liquid has a composition between x = 0 and #eutoetic. Then
as it is cooled, it will follow along a line like the dotted line in Fig. XVII-5,
which intersects the line marked #2 in the figure at temperature T3. At
this temperature it will begin to freeze, but the material freezing out will
be phase a with the composition x\ appropriate to that temperature, very
rich in component a. The liquid becomes enriched in 6, so that it has a
lower melting point, and we may say that the point representing the
concentration and temperature of the liquid on Fig. XVII-5 follows down
along the curve x*. When the composition reaches the eutectic composi-
tion and the temperature is still further reduced, a liquid phase is no
longer possible, and the remaining liquid freezes at a definite temperature

SEC. 4] PHASE EQUILIBRIUM IN BINARY SYSTEMS 285

and composition. It is to bo noticed, however, that the resulting solid is
still a mixture of phases a and ft. With the usual methods of freezing, the
two phases freeze out as alternating layers of platelike crystals. Such a
solid is called a eutectic and is of importance in metallurgy. It is inter-
esting to observe that if the composition of the original liquid is just the
eutectic composition, it will all freeze at a single temperature, which
will be the lowest possible freezing point for any mixture of a and 6. If
the original composition is between xeutoctit. and x = 1, the situation will
be similar to what we have described, only now the point representing
the liquid will move down curve .Ta to tho eutectic composition, and the
solid freezing out will be phase ft, and tho liquid will become enriched in
component a until it reaches the eulectic composition, when it, will all
freeze as the cutcctie mixture of a and ft. The temperature where this
freezing of the eutectir occurs, we; notice, represents a triple point : phases
a, ft, and the liquid are all stable at this temperature in any proportions,
corresponding to the fact that a single tangent can be drawn in the (f-x
diagram to the curves representing all three phases. For every pressure,
there is a temperature at which there is such a triple point, in contrast
to the situation with a one-component system, where triple points exist
only for certain definite combinations of pressure and temperature. The
difference arises because there are more independent variables, the
composition as well as pressure and temperature.

Familiar examples of the situation wo have just described are found in
the solubility of substances in water and other solvents. Thus in Fig.
XVII-6 we give the phase diagram for the familiar system NaCl-water.
This diagram is not carried to a very high concentration of salt, for then
the curve corresponding to x3 would rise to such high temperatures that
we should be involved with the vaporization of the water, which we have
not wished to discuss. In this system, as we have already mentioned, the
solid phases are practically completely insoluble in each other, the phase
corresponding to a being pure ice, ft being pure solid NaCl, combined with
the water of crystallization at low temperature. Thus the curves Xi and
24 of Fig. XVII-5 do not appear in Fig. XVII-6 at all, coinciding prac-
tically with the lines x = 0 and x = 1. We can now find a number of
interesting interpretations of Fig. XVII-6. In the first place, if water
with a smaller percentage of salt than the eutectic mixture is cooled
down, the freezing point will be below 0°C., the freezing point of pure
water, showing that the dissolved material has lowered the freezing point.
We shall calculate the amount of this lowering in the next section. At
these compositions, the solid freezing out is pure ice. This is familiar
from the fact that sea water, which has less salt than the eutectic mixture,
forms ice of pure water without salt. On the other hand, if the liquid
has a larger percentage of salt than the eutectic mixture, the solid freezing

286

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

out as the temperature is towered is pure salt. Under these circum-
stances we should ordinarily describe the situation differently. We
should say that as the temperature was lowered, the solubility of the salt
in water decreased enough so that salt precipitated out from solution. In
other words, the curve separating the liquid region in Fig. XVII-6 from
the region where liquid and NaCl are in equilibrium may be interpreted
as the curve giving tho percentage of salt in a saturated solution or the
solubility as a function of temperature. The rise to the right shows that
the solubility increases rapidly with increasing temperature.

100
90
80
70
60

</) 50

$ 40

g* 30

^ 20

-10
-20

Liquid

Ice + Liquid"

NaCl

NaCl-
2H20
-+Liq.

5 10 15 20 25 30

Grams NaCl per 100 Grams Solution

FIG. XVII-6. ---Equilibrium between NaCl Jiud water.

From Fig. XVII-6 wo can also understand the behavior of freezing
mixtures of ice and salt. Suppose ice and salt, both at approximately
0°C., are mixed mechanically in approximately the right proportions to
give the eutcctic mixture. We see from Fig. XVII-6 that a solid of this
composition is not in thennodynamic equilibrium at this temperature; the
stable phase is the liquid, which has a lower free energy than the mixture
of solids. Thus the material will spontaneously liquefy, the solid ice and
salt dissolving each other at their surfaces of contact and forming brine.
If the process were conducted isothermally, we should end up with a
liquid. But in the process a good deal of h^at would have to be absorbed,
the latent heat of fusion of the material. Actually, in using a freezing
mixture, the process is more nearly adiabatic than isothermal: heat can
flow into the mixture from the system which is to be cooled, but that
system has a small enough heat capacity so that its temperature is rapidly
reduced in the process. In order to get the necessary latent heat, in other
words, the freezing mixture and external system will all cool down below
0°C., falling to lower and lower temperatures as more and more of the
freezing mixture melts. The process can continue, if the proportion of ice

SBC. 4]

PHASE EQUILIBRIUM IN BINARY SYSTEMS

287

to salt is just the eutectic proportion, down to the temperature — 18°, the
lowest temperature at which the liquid can exist.

The most important examples of the phase diagrams we have discussed
are found in metallurgy. There, in alloys of two metals with each other,

1,000°

800° -

600° -

400°

Cu MGJ

FIQ. XVII-7. — Phase equilibrium diagram for the system Cu-Mg, in which tho two metals
are insoluble in each other, forming mtermetallie compounds of definite composition.

1000 ~

400 -

Cu Zn

FIG. XVII-8. — Phase equilibrium diagram for the system Cu-Zn, in which a number
of phases of variable composition arc formed, mixtures of the phases being stable between
tho regions of stability of the pure phases. The phase a is face centered cubic, as Cu is,
/3 is body centered, 7 is a complicated structure, € and 17 are hexagonal. The transition
between ft and 0' is an order-disorder transition, (3 being disordered, and /8' ordered, as
discussed in the following chapter.

we generally find much more complicated cases than those take up so
far, but still cases which can be handled by the same principles. Thus in
Fig. XVII-7 we show the phase diagram for the system Cu-Mg, two

288

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

metals that are almost entirely insoluble in each other. In this case
there are four solid phases, each having its own crystal structure, and
each stable in only an extremely narrow range, about the compositions
Cu, MgCu2, Mg2Cu, and Mg. The free energy of each composition will
then have an extremely sharp minimum, so that the construction neces-
sary to derive the phase diagram will be similar to Fig. XVII-4, but with
four sharp minima instead of two, so that there are three regions, rather
than one, in which a mixture of two phases is the stable solid, and three
eutectics. For contrast, we give in Fig. XVII-8 the phase diagram for
the system Cu-Zn, or brass. In this case there are a number of phases,
again each with its own crystal structure but each with a wide range of
possible compositions. The free energy curves of the various phases in
this case are then not sharp like the case of Cu-Mg but have rather flat

minima, more as in Fig. XVII-2. We
11 shall not try to follow the construction
of the phase diagram through in detail
but shall merely state that it can be
derived from hypothetical free energy
curves according to the type of reason-
ing already used in this section and the
preceding one.

5. Lowering of Melting Points of
Solutions. — We have just seen that the
lowering of the melting point of a
solvent by action of the solute can

0 1 x easily be explained in terms of the

FKJ. xvil-9.™ Gibbs free energy aa phase diagram, and it is an easy matter

function of ronoontrntion, for lowering to find a numerical Value for this
of freezing point. i . T T^- VTTTT r» i_

lowering. In rig. XVII-9 we have a

diagram of G against x, appropriate to this case. The solid solute corre-
sponds to the point G8, with rr = 0, and the liquid is given by the curve.
We wish to find the value of x at which a straight line through x = 0,
G = Ga is tangent to the liquid curve. To do this, we must first find the
equation of the liquid curve. We assume the liquid to correspond to case
(6) of Fig. XVII-1, the internal energy being a linear function of concen-
tration. Then, if GIO is the free energy of the liquid for x = 0, G^ for
x = 1, we have

Gi = Gh + x(Gh - Gto) + NkT[x In x + (1 - x) In (1 - x)], (5.1)

where Gi is the free energy for the liquid. The desired tangent is now
determined by the condition

(5.2)

SEC. 51 PHASE EQUILIBRIUM IN BINARY SYSTEMS 289

the geometrical condition that the tangent to the curve GI at the point
Gi should pass through the point G8 when x = 0. Differentiating Eq.
(5.1), this gives

Gi. + NkT In (1 - ar) = G9. (5.3)

Wo can now find the difference (7jfc — G8 in terms of the latent heat of
fusion of the solvent. From fundamental principles we have

(d(Gio — Gs)\ ,„ ~, — Lm

- ST --- )P = ~(Sl "~ &) = ~~f~' (5'4)

where Si is the entropy of the liquid, Sa of the solid, and Lm the latent
heat of fusion. Computed just at the melting point, the quantity in
Eq. (5.4) becomes —Lm/Tm. Now we shall not use the result except for
temperatures very close to the melting point, so that we may assume that
(Gio — Gs) can be expanded as a linear function of temperature. Just
at the melting point, by the fundamental principle of equilibrium, it is
zero. Thus we have

Gln -G8 = ^(Tm - T). (5.5)

J- m

Inserting in Eq. (5.3), setting T = Tm approximately, and writing
Nk = /£, this gives us

_ In (1 - x) = (Tr* - T). (5.6)

For dilute solutions, to which alone we shall apply our results, x is very
small, and we may write In (1 — x) — — x. Then we have

— (T _ T\

\* m •*• />

(Tm ~ T) - x. (5.7)

JUm

Equation (5.7) gives the lowering of the freezing point, Tm — T7, by
solution of another substance with relative concentration x. We note
the important fact that the result is independent of the nature of the
solute: all its properties have canceled out of the final answer. Thus the
lowering of the freezing point can be used as a direct method of measuring
x, the relative number of molecules of solute in solution. This is some-
times a very important thing to know. Suppose one knows the mass of a
certain solute in solution but does not know its molecular weight. By
measuring the depression of the freezing point, using Eq. (5.7), we can
find the number of moles of it in solution. By division, we can find at
once the mass per mole, or the molecular weight. This method is of

290 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVII

practical value in finding the molecular weights of complicated substances.
It is also of importance in cases where there is association or dissociation
of a solute in solution. Some materials form clusters of two, three, or
more molecules in solution, each cluster traveling around as a single
molecule. Each cluster will count as a single molecule in the entropy of
mixing, and consequently in the depression of the freezing point. Thus
really there are fewer molecules than one would suppose from the known
amount of material in solution and the usual molecular weight, so that tho
depression of the freezing point is smaller than we should suppose. On
the contrary, in some cases substances have their molecules dissociated in
solution. The well-known case of this is ionic substances in water solu-
tion, in which the ions, rather than molecules, form the separate objects
in the solution. In these cases there are more particles in solution than
we should suppose, and the freezing point is depressed by an abnormally
large amount.

From Eq. (5.7) we can find at once the amount of depression of the
freezing point of different solvents. Thus for water, Tm = 273° abs.,
Lm = 80 X 18 cal. per mole, giving Tn - T = 103° for x = 1. To get
useful figures, we calculate for what the chemists denote as a normal
solution, containing 1 mole of solute in 1000 gm. of water, or nnhr of a
mole of solute in 1 mole of water. Thus in a normal solution we expect a
lowering of the* freezing point of 103 X .018 = 1.86°C., provided the
solute is neither associated nor dissociated.

CHAPTER XVIII
PHASE CHANGES OF THE SECOND ORDER

In an ordinary change of phase, there is a sharp transition tempera-
ture, for a given pressure, at which the properties change discontinuously
from one phase to a second one. In particular, there is a discontinuous
change of volume and a discontinuous change of entropy, resulting in a
latent heat and allowing the application of Clapeyron's equation to tin*
transition. In recent years, a number of cases have been recognized
in which transitions occur which in most ways resemble real changes of
phase, but in which the changes of volume and entropy, instead of being
discontinuous, are merely very rapid. Volume and entropy change
greatly within a few degrees' temperature range, with the result thai- there
is an abnormally large specific; heat in this neighborhood, but no latent
heat. Often the specific heat rises to a peak, then discontinuously falls
to a smaller value. To distinguish these transitions from ordinary
changes of phase, it has become customary to denote ordinary phase
changes as phase changes of the first order, and these sudden but not dis-
continuous transitions as phase changes of the second order. Sometimes
the discontinuity of the specific heat is regarded as the distinguishing
feature of a phase change of the second order, but we shall not limit our-
selves to cases having such discontinuities.

There is one well-known phenomenon which might well be considered
to be a phase change of the second order, though ordinarily it is not.
This is the change from liquid to gas, at temperatures and pressures above
the critical point. In this case, as the temperature is changed at con-
stant pressure, we have a very rapid change of volume from a small
volume characteristic of a liquidlike state to the larger volume charac-
teristic of a gaslike state, yet there is no discontinuous change as then*
is below the critical point. And there is a very rapid change of entropy,
from the small value characteristic of the liquid to the large value charac-
teristic of the gas, as we can see from Fig. XI-6, resulting in a very
abnormally high value of CP at temperatures and pressures slightly above
the critical point. At the critical point, where the curve of S vs. T
becomes vertical, so that (6S/dT)P is infinite, CP becomes infinite. At
this temperature and below, we cannot use the specific heat to find the
change of entropy, but must use a latent heat instead, representing, so to
speak, the finite integral under the infinitely high, but infinitely narrow,
peak in the curve of T(dS/dT)P vs. T.

291

292 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

Although the liquid-gas transition above the critical point, as we
have seen, has the proper characteristics for a phase change of the second
order, that name is ordinarily used only for phaso changes in solids. Now
it seems hardly possible that there could be a continuous transition from
ono solid phase to another one with different crystal structure. There
have boon some suggestions that such things are possible; that, for
instance, ordinary equilibrium lines in polymorphic transitions, as shown
in Fig. XI-3, might terminate in critical points, above which one could
pass continuously from one phase to another. But no such critical
points have been found experimentally, and there is no experimental
indication, as from a decreasing discontinuity in volume and entropy
between the two phases as we go to higher pressure and temperature, that
such critical points would be reached if the available ranges of pressure
and temperature could be increased. Thus it seems that our na'ivc sup-
position that two different crystal structures are definitely different, and
that no continuous series of states can be imagined between thorn, is
really correct, and that phase changes of the second ordor arc impossible
between phases of different structure and must be looked for only in
changes within a single crystal structure.

Then* fire at least three types of change known which do not involve
changes of crystal structure and which show the properties of phase
changes of the second order. The best known ono is the ferromagnetic
change, between the magnetized state, for instance of iron or nickel, at
low temperatures, and the unmagnotizod state at high temperatures.
There is no change of crystal structure associated with this transition, at.
least in pure metals, no discontinuous change of volume, and no latent
hoat. The magnetization decreases gradually to zero, instead of changing
discontinuously, though then* is a maximum temperature, called the
Curio point, from P. Curio, who investigated it, at which it drops rather
suddenly to zero. And there is no latent hoat, the entropy increasing
rather rapidly as wo approach the Curio point, but nowhere changing
discontinuously, so that there is an anomalously largo specific heat. This
anomaly in the specific hoat is sometimes concentrated in a small enough
temperature range so that it almost seems liko a latent hoat to crude
observation; the metallurgists, who aro accustomed to determining phaso,
changes by cooling curves, which essentially measure discontinuities or
rapid changes in entropy, have sometimes classified these ferromagnetic
changes as real phaso changes. As a matter of fact, mathematical analy-
sis shows that under some, circumstances in alloys, it is possible for the
ferromagnetic change to bo associated with a change of crystal structure
and a phase change of the first ordor, one phase being magnetic up to its
transition point, above which a new nonferromagnetic phase is stable, but
this is a complication not found in pure metals. Though this ferro-

efcsc. 1] PHASE CHANGES OF THE SECOND ORDER 293

magnetic change is the most familiar example of phase changes of the
second order, we shall not discuss it hero.

A second type of phase change of the second order is found with
certain crystals like NH4C1 containing ions (NH4* in this rase) which
might be supposed capable of rotation at high temperature but not at
low. The ammonium ion, being tot rahcd rally symmetrical, is not far
from spherical, and we can imagine it to rotate freely in the crystal if
it is not packed too tightly. At low temperatures, however, it will fit
into the lattice best in one particular orientation and will tend merely
to oscillate about this orientation. The rotating state, it is found, has
the higher entropy and is preferred at high temperatures. The change
from one state to the other comes experimentally in a nither narrow
temperature range, giving a specific heat anomaly but no latent heat, and
forming again a phase change of the second order. Unfortunately the
theory is rather involved and we shall not try to give it here.

The third type of phase change of the second order is fortunately
easy to treat theoretically, at least to an approximation, and it, is the one
which will be discussed in the present chapter. This is what is known
as an order-disorder transition in an alloy, and can be better understood
in terms of specific examples, which we shall mention in the next section.

1. Order -Disorder Transitions in Alloys. — The best-known example
of order-disorder transitions comes in the 0 phase of brass, Cu-Zn, a
phase which is stable at compositions in the neighborhood of 50 per cunt
of each component. The crystal structure of this phase is body-centered
cubic, an essential feature of the situation. In this type of lattice, the
lattice points are definitely divided into two groups: half the points an;
at the corners of the cubes of a simple cubic lattice, the other half at
the centers of the cubes. It is to be noticed that, though they are dis-
tinct, the centers and corners of the cubes are interchangeable. Now we
can see the possibility of an ordered state of Cu-Zn in the neighborhood
of 50 per cent composition: the copper atoms can be at the corners of the
cubes, the zinc at the centers, or vice versa, giving an ordered structure in
which each copper is surrounded by eight zincs, each zinc by eight cop-
pers; whereas in the disordered state which we have previously considered,
each lattice point would be equally likely to be occupied by cither a copper
or a zinc atom, so that each copper on the average would be surrounded by
four coppers and four zincs.

Just as the body-centered cubic structure can be considered as made
of two interpenetrating simple cubic lattices, the face-centered cubic
structure can be made of four simple cubic lattices. There are some
interesting cases of ordered alloys with this crystal structure and ratios of
approximately one to three of the two components. An example is found
in the copper-gold system, where such a phase is found in the neighbor-

294 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

hood of the composition CuaAu. Evidently the ordered phase is that
in which the gold atoms are all on one of the four simple cubic lattices,
tho copper atoms occupying the other three.

We shall now investigate phase equilibrium of the Cu-Zn type, start-
ing with the simple case of equal numbers of copper and zinc atoms, later
taking the general case of arbitrary composition. We shall make the
same assumptions about internal energy that we have made in Sec. 2,
Chap. XVII, so that the problem in computing the internal energy is to
find the number of pairs of nearest neighbors having the type aa, a6, and
66; a and b being the two types of atoms. We assume that the only
neighbors of a given atom to be considered are the eight atoms at the
corners of a cube surrounding it, so that all the neighbors of an atom
on one of tho simple cubic lattices lie on the other simple cubic
lattice.

We shall now introduce a parameter w, which we shall call the degree
of order. We shall define it so that w = 1 corresponds to having all the
atoms a on one of the simple cubic lattices (which we may call the lattice
a), all the atoms b on the other (which we call ft), w = 0 will correspond
to having equal numbers of atoms a and 6 on each lattice; w = — 1 will
correspond to having all the atoms 6 on lattice a, all the atoms a on lattice
ft. Thus 10 = ±1 will correspond to perfect order, w = 0 to complete
disorder. Let us now define w more completely, in terms of the number
of atoms a and 6 on lattices a and ft. Let there be N atoms, N/2 of each
sort, and N lattice points, N/2 on each of the simple cubic lattices. Then
we assume that

XT 1 e 9 1 J.J.' (1 +

Number ot as on lattice a = -

Number of a's on lattice ft =

(1 - w)N

Number of 6's on lattice a = T-

Number of fr's on lattice ft = (1 . (1.1)

Clearly the assumptions (1.1) reduce to the proper values in the cases
w = ±1, 0, and furthermore they give w as a linear function of the
various numbers of atoms.

To find the energy, we must find the number of pairs of neighbors of
types aa, a&, 66. The number of pairs of type aa equals the number of
a's on lattice a, times 8 /(N/2) times the number of a's on lattice ft. This
is on the assumption that the distribution of atoms on lattice ft surround-
ing an atom a on lattice a is the same proportionally that it is in the whole
lattice ft, an assumption which is not really justified but which we make

SEC. 1] PHASE CHANGES OF THE SECOND ORDER 295

for simplification. Thus the number of pairs aa is - — , times
4(1 — w), or N(l — w>2). Similarly we have

Number of pairs aa = Number of pairs 66 = N(l — w2)
Number of pairs ab = N(l + w)2 + N(l — w)2

w2). (1.2)

To find the internal energy at the absolute zero, we now proceed as in
Sec. 2 of Chap. XVII, multiplying the number of pairs aa by Ean, etc.
Then we obtain

Energy = f/0 = N(l ~ to2)(Ea(t + E») + 2N(l + w*)E*. (1.3)
rfhis can be rewritten in the form

xEa<l + (1 ~ X)Eu + 2X(1 -

+ $Nx2w2(Eab - ~aa^~\ (1.4)
\ /

where x = ^ is the relative composition of the components. Wo use this
form (1 .4) because it turns out to be the correct one in the general case
where x 7* % and because it is analogous to Eq. (2.2), Chap. XVII. We
note that for w = 0, the disordered state, Eq. (1.4) reduces exactly to
Eq. (2.2), Chap. XVII, as it should. To find the energy at any tempera-

ture, we assume as in Chap. XVII that we add an amount I Cp dT, where

•/o

Cp is the specific heat of a completely disordered phase. The actual
specific heat will bo different from this CP, because w in Eq. (1 .4) will
prove to depend on temperature, giving an additional term in the deriva-
tive of energy with respect to temperature. With these assumptions, we
then have

U = Uo + fJCpdT, (1.5)

vvhero Uo is given in Eq. (1.4).

Next we consider the entropy. We have ^— - atoms a and

(1 — w)N . , , ... , (1 — w)N . i (1 + w)Ar

.^ — L — atoms 6 on lattice a, and ~ — atoms a and - '

4 4 4

atoms 6 on lattice ft. The number of ways of arranging these is

(1.6)

296

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

each of the lattices a and 0 furnishing an equal factor, resulting in the
square in Eq. (1.6). Using Boltzmann's relation, this leads to an entropy

S - -Nk(-

+ w . 1 + w 1 — w , 1 — w

___ 111 ~ -f" rt m =-

TCP

Nkr

= Nk In 2 - -^[(1 + w) In (1 + w) + (1 - w) In (1 - w)}

T<£dT. (1.7)

T-Tc/2

When w = 0, the second term of Eq. (1.7) reduces to zero, leaving
S = Nk In 2, agreeing with the value of Kq. (2.6), Chap. XVII, when we
sot x = -J, checking the correctness of Eq. (1.7) in this special case.

When w = ± 1 , however, the expression
(1.7) reduces to zero, showing that tho
ordered state has zero entropy. This is
as we should expect; there is only one
arrangement of the atoms, all the a's being
on ono lattice, all the b's on the other, so
that then* is no randomness at all.

2. Equilibrium in Transitions of the
Cu-Zn Type.— Having found the internal
energy and entropy as a function of the
degree of order and the temperature, in
Eqs. (1 .4) and (1 .7), we can at once set up
the free energy, and find which value of
the degree of order gives the stablest
phase at any given temperature. In Fig.
/ XVIII-1 we plot the Gibbs free energy G
KKJ. xvin-i. -Gibbs free energy as a function of w, for various tempcra-

:IH function of the degree of order, tulvs< Of course, since equal positive and
for various temperatures. . „ ,

negative values of w really correspond to

the same state, the curves are symmetrical about tho line w = 0. The,
curves are drawn on the assumption that #„?> — (Ean + Ebb)/2 is negative.
This ease, in which unlike atoms attract each other more than like atoms,
rase (c) of Fig. XVII-1, is the only one in which we may expect the
ordered state to be more stable than the disordered ono. For if like
atoms attract more than unlike, as in case (a), Fig. XVII-1, the case in
which atoms tend to segregate into two separate phases, we shall surely
find that the disordered state, iu which each atom has on the average four
neighbors of the same kind as well as four of the opposite kind, will be
more stable than the ordered state, where all neighbors are of the oppo-
site kind, even at the absolute zero.

SEC. 2]

PHASE CHANGES OF THE SECOND ORDER

297

We see that at low temperatures the minimum of tho G curve, giving
the stable phase, comes at values of w different from zero, approaching
w = ±1 as the temperature approaches zero. As the temperature rises,
the minima move inward toward w = 0, and at a certain temperature
(Tc in the figure), there is a double minimum, with a very flat curve, at,
w = 0. Above this temperature there is a single minimum at w = 0. In
other words, the degree of order gradually decreases from perfect order at
T = 0, to complete disorder at and above a certain temporal uro 7\.
This limiting temperature corresponds to the Curio temperature in forro-
magnetism, and by analogy it is of ton referred to as tho ( 'urio temporal uro
in this case as well. To got tho minimum of tho curve, tho natural thing
is to differentiate G with respect to w, keeping T constant. Then wo have1

o =

in -

(2. i

Equation (2.1) is a transcendental equation for w and cannot bo solved
explicitly. We can easily solve it graphically, however, using the form

0 + «>) _
— "

8 Eaa + E,,,

" "* ----- 2

l-w

Wo plot In (1 + w)/(\ — w) as a func-
tion of w, and on the same graph
draw the straight line w(-8/kT)[Kalt - 2
(Eatl + ATM,)/2j. The intersections give
the required value of ir. As we see from 1
Fig. XVIII-2, at low temperatures the
straight line is steep and there will be 0
tliree intersections, one at w = 0 (evi-
dently corresponding to the maximum
of the curve, as wo see from Fig. "I
XVIII-1) and two others, which wo
desire, at equal positive and negative
values of w. As the temperature in-
creases and the slope of the4 straight
line decreases, those intersections move Flu- XVlil-a. Craphiral solution of
toward w = 0 and finally coalesce when

the slope of the straight line equals that of In (1 + w)/(\ — w) at the
origin. Now

(1 +w)

-2 -

(1 -w)

starts out from the origin like 2w, with a slope 2, so that for the Curie
point we must have

298 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

Egg + Ebb

= '

(2.3)

In terms of this, the straight line in Fig. XVIII-2 is 2Tcw/T.

By the graphical method of Eq. (2.2) and Fig. XVIII-2, the curve

of Fig. XVIII-3 is obtained for the stable value of w, as a function of

temperature. This shows the decrease
of w from 1 to 0, first very gradual, then
as the Curie point is approached very

w "" x rapid, so that the curve actually has a

vertical tangent at the Curie point.
The curve of Fig. XVIII-3 cannot be
T/Tc expressed analytically, though it can be

FIG. XVIII-3.— Degree of order as approximated in the two limits of T = 0

function of temperature. i 71 __ //r

Having found the variation of w with temperature, we can find the
specific heat anomaly, or the excess of specific heat over the value CP
characteristic of disorder. This excess is evidently

dU\ dw

dwrdT

E*« + E»\fa
-- g - )df

NkT . (1 +w)dw

2 (1 - w)dT

- -NkTcw^ (2.4)

using Eqs. (1.4), (1.7), (2.2), and (2.3). In Eq. (2.4), it is understood
that dw/dT is the slope of the curve of Fig. XVIII-3 and that it is to be
determined graphically. Since the slope is negative, the excess specific
heat is positive. We give the resulting curve for specific heat in Fig.
XVIII-4, where we see that it comes to a sharp peak at the Curie point
and above that point drops to zero.

From the discussion we have given, it is plain that the change from
the ordered to the disordered state occupies the whole temperature range
from zero degrees to Te, though it is largely localized at temperatures
slightly below Tc. Thus this change, a gradual one occurring over a
large temperature range, is just of the sort that we wish to call a phase

SEC. 2]

PHASE CHANGES OF THE SECOND ORDER

299

change of the second order. We can make the situation clearer by
plotting curves for G as a function of T. We do this for a number of
values of w, ranging from zero to unity. In a sense, we may consider that
we have a mixture of an infinite number of phases, corresponding to the
continuous range of w, and at each temperature that particular phase
(or particular w) will be stable whose curve of G against T lies lowest.
The resulting curves are shown in Fig. XVIII-5. To mako thorn clearer,

1.5

FIG. XVI 11-4. — Kxc-css specific* heat arising from the ordered state, in units of Nk, as func-
tion of temperature.

W=l

FIG. XVIII-5. — Gibhs free energy as function of temperature, for different degrees
of order, in the order-disorder transition. The envelope of the straight lines represents
the free energy of the stable state.

we leave out the terms coming from the specific heat Cp of the disordered
state, which are common to curves for all w's, and do not affect the
, relative positions of the curves. When this is done, the curves become
straight lines, since the internal energy and entropy are then independent
of temperature. At the absolute zero, the lowest curve is the one with
the lowest internal energy or the ordered state. The disordered states
have greater entropy, however, even at the absolute zero, so that their
curves slope down more, and at higher temperatures their free energies

300

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

lie lower than that of the ordered state. From Fig. XVIII-5 we see that
there is an envelope to the curves, and this envelope represents the actual
curve of G vs. T7, whose slope is the negative entropy and whose second
derivative gives the specific heat. The particular value of w whose curve
is tangent to the envelope at any temperature is the stable w at that
temperature, as given by Fig. XVIII-3.

Graphs like Fig. XVIII-5 show particularly plainly the difference
between phase changes of the first and second order. We can readily
imagine that, by slightly altering the mathematical details, the curves
could be changed to the form of Fig. XVIII-6, in which, though we have a

w=I

FIG. XV11I-0. - Gibbs free energy as function of temperature, for different degrees of
order, in a phase change of the first order, in which the ordered state is stable below a
temperature T0, the completely disordered state above this temperature.

continuous set of phases from w = 0 to w = 1, the envelope lies aboye
rather than below the axis of abscissas. In this case the stable state
is that with w = 1 up to a certain temperature, w = 0 from there on, all
other values of w corresponding to states that are never stable. This
would then be a phase change of the first order, as shown in Fig. XVI-2,
with a discontinuity in the slope of the G vs. T curve, or the entropy, and
hence with a latent heat. When we see the small geometrical difference
between these two cases, we see that in some cases the distinction between
phase changes of the first and the second order is not very fundamental.
In this connection, it is interesting to note that the rotation vibration
transition in NH4C1, which we mentioned in a preceding paragraph, is
clearly a phase change of the second order, the change occurring through
a considerable range of temperature or pressure. However, there is a
similar transition in NH4Br, undoubtedly due to the same physical cause,
which at least at high pressure takes place so suddenly that it certainly
seems to be a phase change of the first order. This is probably a case

SEC. 3] PHASE CHANGES OF THE SECOND ORDER 301

where the distinction is no more significant than in Figs. XVIII-5 and
XVIII-6. We must not forget, however, that there is one real and
definite distinction between most phase changes of the first order and all
those of the second order: in every phase change of the second order, we
must be able to imagine a continuous range of phases between the two
extreme ones under discussion, while in a phase change of the first order
this is not necessary (though, as we have seen in Fig. XVIII-6, it can
sometimes happen), and in the groat majority of cases it is not possible.
3. Transitions of the Cu-Zn Type with Arbitrary Composition. — It is
not much harder to discuss the general case of arbitrary composition
than it is the simple case of 50 per cent concentration taken up in the
two preceding sections. We assume that there are NX atoms a, JV(1 — x)
6's, and we shall limit ourselves to the case where x is loss than \\ the
same formulas do not hold for x greater than ^, but to get this caso we can
merely interchange the names of substances a and b. As before, we lot
the degree of order bo w. Then we assume

N
Number of atoms a on lattice a = ^ (1 + w)x

Number of atoms I) on lattice a = -~[l — (1+ w)x]

N
Number of atoms a on lattice ft = -^(1 — w)x

Number of atoms 6 on lattice ft = -0-[1 — (1 — w)x]. (3.1)

To justify these assumptions, we note that they lead to the correct num-
bers in the three cases w = 0, ±1, and that they give the numbers as
linear functions of w, conditions which determine Eqs. (3.1) uniquely.
Then for the numbers of pairs we find

Number of pairs aa: 4/Vz2(l — w*)

Number of pairs bb: 4JV[(1 — :r)2 — x'2w~]

Number of pairs ah: ZN[x(l - x) + x*w*l (3.2)

and for the internal energy we have

U = 4N\ xEaa + (1 - x)Ebb

+ 8Nx*w{Eab - E-^E^\ + (TCP dT. (3.3)
\ ^ / Jo

The steps in the derivation of Eqs. (3.2) and (3.3) have not been given
above, but the principles used in their derivation are just like those used
in Sec. 1. We note that Eq. (3.3) is the one already written in Eqs. (1.4)
and (1.5) but previously justified only for the case x = £.

302

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XV1U

The derivation of the entropy is also exactly analogous to that of Sec. 1
and the result is

S - --/{(I + w)x In (1 + w)x + (1 - w)x In (1 - w)x
Z

}- fl - (1 + w)x] In [1 - (1 + w)x] + [1 - (1 - w)x] In [1 - (1 - w)x\\

dT. (3.4)

It is easy to verify that in the case x = £ this leads to the value already
found in Eq. (1.7). From Eqs. (3.3) and (3.4) we can find tho froo
onorgy and carry out the same sort of discussion that we havo above, but,
for any concentration. To find the value of w for tho stable state, at any
value of x, we differentiate 0 with respect to w and sot it equal to zoro.

Then we havo

Q =

x In

O -w)[l - (1 +»)*]'

In ° +

ln ~(i~- w) [i

[1 - (1 -

- (1+

whore the Tc used in Eq. (3.6) is the ono
defined in Eq. (2.3), holding for the concon-
x=0J ^/\ Cation x — i.. Equation (3.6) can bo

solved as in the special case x = £, plotting
the left side of Eq. (3.6) against w and
finding the intersection with the straight
line given by the right side. Qualitatively
we find the same sort of result as in our
previous case, the dogree of order going
from unity at absolute zero to zero at a
compositions, T => Curie point. The Curie point, however,
depends on concentration. We find it, as
before, by lotting the slope of the straight line representing the right side
of Ea. (3.6) be the same as the slope of the left side at the origin, which is
2/(l — r). Equating these, we have

0
w

tor different
°'8 Ti'

Tcx =

(3.7)

where Tcx is the Curie temperature for concentration x, Tc for concentra-
tion x = 4. From Eq. (3.7) we see that Tc* is a parabolic function of

SEC. 3]

PHASE CHANGES OF THE SECOND ORDER

303

z, having its maximum for x = £, and falling to zero at the extreme
concentration x = 0. This is Of metallurgical interest, for on phase
diagrams in cases where there is a transition of the second order it is quite
common to draw a line of Curie temperature vs. composition to indicate
the transition, though thore is no real equilibrium of phases to be indicated
by it. In a case like the Cu-Zn transition, this curve should theoretically
have the form (3.7). The experimental data are hardly good enough to
soo whether this is verified or not.

w=08-

0.1

0.4

FIG. XVIII-8. Gibbs free energy as function of composition, for different decrees ot

order, T = 0.8 Te.

At a temperature below the Curie point TC1 it is plain from Eq. (3.7)
that for concentrations nearer £ than a certain critical concentration the
alloys will be below thoir Curie points and will be in partly ordered
phases, while for x loss than this critical concentration they will bo above*
their Curie points and will bo in the disordered state. This is indicated
in Fig. XVIII-7, where wo show G as a function of w for different values of
x, at a temperature of 0.8 Tc. The critical concentration for this tem-
perature is 0.277, as can bo found at once from Eq. (3.7); it is noted in
Fig. XVIII-7 that the curves for x = 0.1 and 0.2 definitely have thoir
minima at w = 0, indicating complete disorder, while that for x = 0.3
is very flat at the center, and those for 0.4 and 0.5 definitely have minima
for w j± 0, indicating a partly ordered state. Finally, in Fig. XVIII-8
we show G as a function of x, for different values of w, at this same tern-

304 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XVIII

perature T = 0.8 Tc. For compositions up to 0.277, as we have men-
tioned, the curve for w — 0 lies the lowest. At higher concentrations, the
other curves begin to cross it and the stable state corresponds to the
envelope of these curves, the lowest w rising from w = 0 to a maximum
of about w = 0.80 at x = \. This envelope is of interest, for it is the
curve of G vs. x which should really be used to represent the stable stato
in such a system and which should bo usod in investigating the equilibrium
between this phase and other phases, in the manner of Chap. XVII. We
notice that this envelope is a smooth curve, convex downward, just as
the curve for w — 0 is, and in fact it does not greatly differ from that for
w = 0. Thus our discussion of phase equilibrium of the preceding
chapter, where we entirely neglected the order-disorder transition, is not
seriously in error for a phase in which such a transition is possible. The
reason is that, though there is a considerable difference in energy and
entropy separately between the ordered and disordered states, these make
contributions of opposite sign in the free energy, so that it is only slightly
affected by the degree of order.

PART III
ATOMS, MOLECULES, AND THE STRUCTURE OF MATTER

CHAPTER XIX
RADIATION AND MATTER

In the development of quantum theory, light, or electromagnetic
radiation of visible wave lengths, has had a very special place. It was
the study of black-body radiation that first showed without question the
inadequacy of classical mechanics, and that led Planck to the quantum
theory. One of the first triumphs of quantum theory was Einstein's
prediction of the law of photoelectric emission, a prediction which was
beautifully verified by experiment. And in the development of the
theory of atomic and molecular structure, the most complicated and
involved test which has yet been given the quantum theory, the tool has
been almost entirely optical, the spectrum, the light emitted and absorbed
by matter. Some of the most difficult logical concepts of the quantum
theory have come in the field of light. The difficulty of reconciling prob-
lems like interference of light, which clearly indicate that it is an electro-
magnetic wave motion, with problems like the photoelectric effect, which
equally clearly* indicate that it is made of individual particles of energy,
or photons, is well known. And these difficulties, indicating that light
really has a sort of dual nature, gave the suggestion that matter might
have a dual nature too, and that the particles with which we were familiar
might also be associated with waves. This was the suggestion which
led to wave mechanics and which raised the quantum theory from a
rather arbitrary set of rules to a well-developed branch of mathematical
physics.

Throughout the development of modern ideas of light, black-body
radiation has played an essential role. This is simply light in thermal
equilibrium — the distribution of frequencies and intensities of light
which is in equilibrium with matter at a given temperature. Our study
in this chapter will be of black-body radiation, and we shall handle it by
direct thermodynamic methods, using the quantum theory much as we did
in the theory of specific heats. In the following chapter we shall take up
the kinetics of radiation, the probabilities of emission and absorption of
light by matter. This will lead us to a kinetic derivation of the laws of
black body radiation, and at the same time to a usable method of handling
the kinetics of radiation problems out of equilibrium, which we very
commonly meet in the laboratory.

1. Black-body Radiation and the Stefan-Boltzmann Law. — Light is
simply electromagnetic radiation, a wave motion in space, in which the

307

,308 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

electric and the magnetic fields oscillate rapidly with time. It can carry
energy, just as sound or any other wave can carry energy. We are all
familiar with this; most of the available energy on the earth was carried
here from the sun, by electromagnetic radiation. Like all waves, it can
be analyzed into sinusoidal or monochromatic waves, in which the
electromagnetic field oscillates sinusoidally with time, with a definite
frequency v\ the possibility of such an analysis is a mathematical one,
based on Fourier's series, and does not imply anything about the physics
of radiation. The velocity of light, at least in empty space, is inde-
pendent of the frequency of oscillation, and is ordinarily denoted by c,
equal to 2.998 X 1010 cm. per second. We can associate a wave length
with each frequency of oscillation, by the equation

\v = c, (1.1)

where A is the wave length. The mathematics of the light waves is
essentially like that of sound waves, given in Sec. 2 of Chap. XIV, and
we shall not repeat it here. In that section, however, we found that
elastic waves were of two sorts, longitudinal and transverse. Light on
the contrary is only transverse, with two possible planes of polarization, or
directions for the electric or magnetic field, at right angles to the direction
of propagation. We ordinarily deal, in discussions like the present, with
fairly short wave lengths of light. The long waves, as -found in radio
transmission, are of small significance thermodynamically or in atomic
structure. Among waves shorter, say, than a tenth of a millimeter, it is
customary to speak of those longer than 7000 A as infrared or heat waves,
those between 7000 and 4000 A as light (since the eye can see only these
wave lengths), those between 4000 A and perhaps 50 A as ultraviolet, and
those shorter than 50 A but longer than perhaps 0.01 A as x-rays. Waves
shorter than this are hardly met in ordinary thermodynamic or atomic
processes, though of course they are essential in nuclear processes and
cosmic rays. Although there is this classification of wave lengths, it is
purely a matter of convenience, and we shall not have to bother with it.
For our purposes, we may consider as light any radiation from perhaps
^ mm. to TV A; it is only in this range that the radiations we consider are
likely to have appreciable intensity.

Ordinary bodies at any temperature above the absolute zero auto-
matically emit radiation, and are capable of absorbing radiation falling on
them. Thus an enclosure containing bodies at a temperature above the
absolute zero cannot be in equilibrium unless it contains radiation as
well. In fact, in equilibrium, there must be just enough radiation so that
each square centimeter of surface of each body emits just as much radia-
tion as it absorbs. It seems clear that there must be a definite sort of
radiation in equilibrium with bodies at a definite temperature. For we

SBC. 1] RADIATION AND MATTER 309

know that all bodies at a given temperature are in thermal equilibrium
with each other, and if they are all in a container with the same thermal
radiation, this radiation must be in equilibrium with each body separately,
and must hence be independent of the particular type of body, and
characteristic only of the temperature, and perhaps the volume, of the
container. It is an experimental fact, one of the first laws of temperature
radiation, that the type of radiation — its wave lengths, intensities, and
so on — is independent of the volume, depending only on the temperature.
This type of radiation, in thermal equilibrium, is called black-body
radiation, for a reason which we shall understand in a moment.

The first and most elementary law of black-body radiation is Kirch-
hoff's law, a simple application of the kinetic method. To understand
it, we must define some terms. First we consider the emissive power
e\ of a surface. We consider the total number of ergs of energy emitted
in the form of radiation per second per square centimeter of a surface, in
radiation of wave length between X and X + d\, and by definition set it
equal to e\d\. Next we consider the absorptivity. Suppose a certain
amount of radiant energy in the wave length range d\ falls on 1 sq. cm.
per second, and suppose a fraction a\ is absorbed, the remainder, or
(1 — a\), being reflected. Then a\ is called the absorptivity, and
(1 — a\) is called the reflectivity. Now consider the simple requirement
for thermal equilibrium. We shall demand that, in each separate range
of wave lengths, as much radiation is absorbed by our square centimeter
in thermal equilibrium as is radiated by it. This assumption of balance
in each range of wave lengths is a particular example of the principle
of detailed balancing first introduced in Chap. VI, Sec. 2. Now let
I\d\ be the amount of black-body radiation falling on 1 sq. cm. per second
in the wave length range dX. This is a function of the wave length and
temperature only, as we have mentioned above. Then we have the
following relation, holding for 1 sq. cm. of surface:

Energy emitted per second = e\d\

= energy absorbed per second = I\a\d\,

— = I\ = universal function of X and T. ( 1.2)

G&X

Equation (1.2) expresses Kirchhoff s law: the ratio of the emissive power
to the absorptivity of all bodies at the same wave length and temperature
is the same. Put more simply, good radiators are good absorbers, poor
radiators are poor absorbers. There are many familiar examples of this
law. One, which is of particular importance in spectroscopy, is the
following : if an atom or other system emits a particularly large amount of

310 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

radiation at one wave length, as it will do if it has a line spectrum, it
must also have a particularly large absorptivity at the same wave length,
so that a continuous spectrum of radiation, passing through the body, will
have this wave length absorbed out and will show a dark line at that point.

A black body is by definition one which absorbs all the light falling
on it, so that none is reflected. That is, its absorptivity a\ is unity, for
all wave lengths. Then it follows from Eq. (1 .2) that for a black body the
emissive power ex is equal to /x, the amount of black-body radiation
falling on 1 sq. cm. per second per unit range of wave length. We can
understand the implications of this statement better if we consider what
is called a hollow cavity. This is an enclosure, with perfectly opaque
walls, so that no radiation can escape from it. It contains matter and
radiation in equilibrium at a given temperature. Thus the radiation is
black-body radiation characteristic of that temperature. Now suppose
we make a very small opening in the enclosure, not big enough to disturb
the equilibrium but big enough to let a little radiation out. We can
approximate the situation in practice quite well by having a well insulated
electric furnace for the cavity, with a small window for the opening. All
the radiation falling on the opening gets out, so that if we look at what
emerges, it represents exactly the black-body radiation falling on the area
of the opening per second. Such a furnace makes in practice the moist
convenient way of getting black-body radiation. But now by Kirchhoff 's
law and the definition of a black body, we see that if we have a small piece
of black body, of the shape of the opening in our cavity, and if we heat it
to the temperature of the cavity, it will emit exactly the same sort of
radiation as the opening in the cavity. This is the reason why our radia-
tion from the cavity, radiation in thermal equilibrium, is also called black-
body radiation. A black body is the only one which has this property
of emitting the same sort of radiation as a cavity. Any other body will
emit an amount a\I\ d\ per square centimeter per second in the range
d\, and since a\ must by definition be less than or equal to unity, the
other body will emit less light of each wave length than a black body. A
body which has very small absorptivity may emit hardly anything. Thus
quartz transmits practically all the light that falls on it, without absorp-
tion. When it is heated to a temperature at which a metal, for instance,
would be red or white hot and would emit a great deal of radiation, the
quartz emits hardly any radiation at all, in comparison.

Now that we understand the emissive power and absorptivity of
bodies, we should consider /x, the universal function of wave length and
temperature describing black-body radiation. It is a little more con-
venient not to use this quantity, but a closely related one, u*. This
represents, not the energy falling on 1 sq. cm. per second, but the energy
contained in a cubic centimeter of volume, or what is called the energy

SEC. 1] RADIATION AND MATTER 31 1

density. If there is energy in transit in a light beam, it is obvious that
the energy must be located somewhere while it is traveling, and that we
can talk about the amount of energy, or the number of ergs, per cubic
centimeter. It is a simple geometrical matter to find the relation between
the energy density and the intensity. If we consider light of a definite
direction of propagation, then the amount of it which will strike unit
cross section per second is the amount contained in a prism whose base
is 1 sq. cm. and whose slant height along the direction of propagation is
the velocity of light c. This amount is the volume of the prism (c cos 0,
if 0 is the angle between the direction of propagation and the normal to
the surface), multiplied by the energy of the light wave per unit volume.
Thus we can find very easily the amount of light of this definite direction
of propagation falling on 1 sq. cm. per second, if we know the energy
density, and by integration we can find the amount of light of all direc-
tions falling on the surface. We shall not do it, since we shall not need
the relation. In addition to this difference between uv and 7\, the former
refers to frequency rather than wave length, so that uvdv by definition is
the energy per unit volume in the frequency range from v to v + dv.

In addition to energy, light can carry momentum. That means that
if it falls on a surface and is absorbed, it transfers momentum to the
surface, or exerts a force on it. This force is called radiation pressure.
In ordinary laboratory experiments it is so small as to be very difficult
to detect, but there are some astrophysical cases where, on account of the
high density of radiation and the smallness of other forces, the radiation
pressure is a very important effect. The pressure on a reflecting surface,
at which the momentum of the radiation is reversed instead of just being
reduced to zero, is twice that on an absorbing surface. Now the radiation
pressure can be computed from electromagnetic theory, and it turns out
that in isotropic radiation (radiation in which there are beams of light
traveling in all directions, as in black-body radiation), the pressure against
a reflecting wall is given by the simple relation

P = £ X energy density

\,dv. (1.3)

From Eq. (1.3) we can easily piove a law called the Stefan-Boltzmann
law relating the density of radiation to the temperature.

Let us regard our radiation as a thermodynamic system, of pressure
P, volume 7, and internal energy [7, given by

U * V*u,d*. (1.4)

Then, from Eqs. (1.3) and (1.4), the equation of state of the radiation is

PV - W, (1.5)

312 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

which compares with PV « ft/ for a perfect gas. There is the further
fact, quite in contrast to a perfect gas, that the pressure depends only on
the temperature, being independent of the volume. Then we have

W)T * 3/>' (L6)

using the fact that P is independent of V. But by a simple thormo-
dynamic relation we know that in general

-'•

Combining Eqs. (1.6) and (1.7), we have

- (1.8)

The pressure in Eq. (1.8) is really a function of the temperature only, so
that the partial derivative can be written as an ordinary derivative, and
we can express the relation as

(IJ)

which can be integrated to give

In P = 4 In T + const.,

P = const. T\ (1.10)

or, using Kq. (1.3),

f*uvdv = const. T4. (1.11)

«/o

Equation (1.11), stating that the total energy per unit volume is propor-
tional to the fourth power of the absolute temperature, is the Stefan-
Boltzmann law. Since the intensity of radiation, the amount falling on a
square centimeter in a second, is proportional to the energy per unit
volume, we may also state the law in the form that the total intensity of
black-body radiation is proportional to the fourth power of the absolute
temperature. This law is important in the practical measurement of
high temperatures by the total radiation pyrometer. This is an instru-
ment which focuses light from a hot body onto a thermopile, which
absorbs the radiation energy and measures it by finding the rise of tem-
perature it produces. The pyrometer can be calibrated at low tempera-
tures that can be measured by other means. Then, by Stefan's law, at
higher temperatures the amount of radiation must go up as the fourth

SEC. 2] RADIATION AND MATTER 313

power of the temperature, from which we can deduce the temperature
of very hot bodies, to which no other method of temperature measurement
is applicable. Since Stefan's law is based on such simple and fundamental
assumptions, there is no reason to think that it is not perfectly exact, so
that this forms a valid method of measuring high temperatures.

2. The Planck Radiation Law. — The Stefan-Boltzmann law gives us a
little information about the function uvj but not a great deal. We shall
next see how the function can be evaluated exactly. There are many
ways of doing this, but the first way we shall use is a purely statistical
one. We can outline the method very easily. We consider a hollow
cavity with perfectly reflecting walls, making it rectangular for con-
venience. In such a cavity we can have standing waves of light; these
waves, in fact, constitute the thermal radiation. There will be a discrete
set of possible vibrations or overtones of a fundamental vibration, just
as we had a discrete set of sound vibrations in a rectangular solid in Chap.
XIV; only instead of having a finite number of overtones, as we did with
the sound vibrations on account of the atomic nature of the material, the
number of overtones here is infinite and stretches up to infinite fre-
quencies. As with the sound vibrations, each overtone acts, as far as the
quantum theory is concerned, like a linear oscillator. Its energy cannot
take on any arbitrary value, but only certain quantized values, multiples
of hv, where v is the frequency of that particular overtone. This leads
at once to a calculation of the mean energy of each overtone, just as we
found in our calculation of the specific heat of solids, and from that we
can find the mean energy per unit volume in the frequency range dv, and
so can find uv.

First, let us find the number of overtone vibrations in the range dv.
We follow Chap. XIV in detail and for that reason can omit a great deal
of calculation. In Sec. 2 of that chapter, we found the number of over-
tones in the range dp, in a problem of elastic vibration, in which the
velocity of longitudinal waves was i>/, that of transverse waves vt. From
Eqs. (2.20) and (2.21) of that chapter, the number of overtones of longi-
tudinal vibration in the range di>, in a container of volume F, was

dN = 4™» di. (2.1)

and of transverse vibrations

dN = 87TJ/2 drt- (2.2)

There was an upper, limiting frequency for the elastic vibrations, but
as we have just stated there is not for optical vibrations. In our optical
case, we can take over Eq. (2.2) without change. Light waves are only

314 INTRODUCTION TO CHEMICAL PHYSICS [€HAP. XIX

transverse, so that the overtones of Eq. (2.1) are not present but those of
Eq. (2.2) are. Since the velocity of light is c, we have

dN - 8™2 d (2.3)

as the number of standing waves in volume V and the frequency range dv.
Next we want to know the moan energy of each of these standing
waves at the temperature T. Before the invention of the quantum
theory, it was assumed that the oscillators followed classical statistics.
Then, being linear oscillators, the mean energy would have to be kT at
temperature T. From this it would follow at once that the energy density
u,9 which can be found by multiplying dN in Eq. (2.3) by the mean energy
of an oscillator, and dividing by V and dv, is

,0 A.
uv = — ^— - (2.4)

Kquation (2.4) is the so-called Kayleigh-Jcans law of radiation. It was
derived, essentially as we have done, from classical theory, and it is the
only possible radiation law that can be found from classical theory. Yet
it is obviously absuru, as was realized as soon as it was derived. For it
indicates that uv increases continually with v. At any temperature, the
farther out we go toward the ultraviolet, the more intense is the tempera-
ture radiation, until finally it becomes infinitely strong as we go out
through the x-rays to the gamma rays. This is ridiculous; at low tem-
peratures the radiation has a maximum in the infrared, and has fallen
practically to zero intensity by the time we go to the visible part of the
spectrum, while even a white-hot body has a good deal of visible radiation
(hence the "white heat"), but very little far ultraviolet, and practically
no x-radiation. There must be some additional feature, missing in the
classical theory, which will have as a result that the overtones of high
frequency, the visible and even more the ultraviolet and x-ray frequencies,
have much less energy at low temperatures than the equipartition value,
and in fact at really low temperatures have practically no energy at all.
But this is just what the quantum theory does, as we have seen by many
examples. The examples, and the quantum theory itself, however, were
not available when this problem first had to be discussed; for it was to
remove this difficulty in the theory of black-body radiation that Planck
first invented the quantum theory.

In Chap. IX, Sec. 5, we found the average energy of a linear oscillator
of frequency v in the quantum theory, and found that it was

A v . v /n ,.

Average qnergy = -TT + ~rz -- > (2.5)

»*

BBC. 2]

RADIATION AND MATTER

315

as in Eq. (5.9), Chap. IX. The term ?hv was the energy at the absolute
zero of temperature; the other term represented the part of the energy
that depended on temperature. The first term, sometimes called the
zero-point energy, arose because we assumed that the quantum condition
was En = (n + %)hi>, instead of En = nhv. We can now use the expres-
sion (2.5) for the average energy of an overtone, but we must leave out
the zero-point energy. For, since the number of overtones is infinite, this
would lead to an infinite energy density, even at a temperature of absolute
zero. The reason for doing this is not very clear, oven in the present state
of the quantum theory. We can do it quite arbitrarily, or we can say
that the quantum condition should be En = nhv (an assumption, how-
ever, for which there is no justification in wave mechanics), or we can

FIG. XIX-l.-

-Enorgy density from Planck's distribution law, for fom temperatures in thw
ratio 1.2.3.4.

say that the infinite zero-point energy is really there but, since it is inde-
pendent of temperature, we do not observe it. No one of these reasons is
very satisfactory. Unfortunately, though it was the branch of physics
in which quantum theory originated, radiation theory still has more
difficulties in it than any other parts of quantum theory. We shall then
simply apologize for leaving out the term ^hv in Eq. (2.5), and shall hope
that at some future time the theory will be well enough understood so
that we can justify it.

If we assume the expression (2.5), without the zero-point energy, for
the average energy of a standing wave, and take Eq. (2.3) for the number
of standing waves in volume V and frequency range dvy we can at once
derive uv, and we have

C3 *!

(2.6)

ekT -

316 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

Equation (2.6) is Planck's radiation law, and as far as available experi-
ments show, it is the exactly correct law of black-body radiation. Curves
of uv as a function of v, for different temperatures, are shown in Fig.
XIX-1. At low frequencies, even for room temperatures, the frequencies
are so low that the energy of an oscillator has practically the classical
value, and the Rayleigh-Jeans law is correct. At higher frequencies,
however, this is not the case, and the curves, instead of rising indefinitely
toward high frequencies, curve down again and go very sharply to negligi-
ble values. The maximum of the curve shifts to higher frequencies as
the temperature rises, checking the experimental fact that bodies look
red, then white, then blue, as their temperature rises. The area under
the curve rises rapidly with temperature. It is, in fact, this area that
must be proportional to the fourth power of the temperature, according
to the4 Stefari-Boltzmaiin law. We can easily verify that Planck's law is
in accordance with that law, and at the same time find the constant
in Eq. (1.11), by integrating uv from Eq. (2.6). We have

I uv dv = I —~ -j- dv

Jo Jo ekf _ i

= ^TiS-4 f YS) V1 ~d(lf)

/t C/ I (\ \lv J. I \ Iv JL I

«/° \ / ekT _ ] \ /

4T4 r « !

c3~Jo ** e* - 1 x

-, (2.7)

where we have used the relation

a = (l + ~ + ~ +'••) = 1.0823

(2.8)

3. Einstein's Hypothesis and the Interaction of Radiation and
Matter. — To explain the law of black-body radiation, Planck had to
assume that the energy of a given standing wave of light of frequency v
could be only an integral multiple of the unit hv. Thus this carries with
it a remarkable result: the energy of the light can change only by the quite
finite amount hv or a multiple of it. This is quite contrary to what the
wave theory of light indicates. The theory of emission and absorption of
energy has been thoroughly worked out, on the wave theory. An oscillat-
ing electric charge has oscillating electric and magnetic fields, and at
distant points these fields constitute the radiation field, or the light wave

SEC. 3] RADIATION AND MATTER 317

sent out from the charge. The field carries energy out at a uniform and
continuous rate, and the charge loses energy at the same rate, as one can
see from the principle of conservation of energy, and gradually comes to
rest. To describe absorption, we assume a light wave, with its alternating
electric field, to act on an electric charge which is capable of oscillation.
The field exerts forces on the charge, gradually setting it into motion with
greater and greater amplitude, so that it gradually and continuously
absorbs energy. Both processes, emission and absorption, then, are
continuous according to the wave theory, and yet the quantum theory
assumes that the energy must change by finite amounts hv.

Einstein, dearly understanding this conflict of theories, made an
assumption that seemed extreme in 1905 when he made it, but which has
later come to point the whole direction of development of quantum
theory. He assumed that the energy of a radiation field could not be
considered continuously distributed through space, as the wave theory
indicated, but instead that it was carried by particles, then called light
quanta, now more often called photons, each of energy hv. If this hypo-
thesis is assumed, it becomes obvious that absorption or emission of light
of frequency v must consist of the absorption or emission of a photon, so
that the energy of the atom or other system absorbing or emitting it must
change by the amount hv. Einstein's hypothesis, in other words, was the
direct and straightforward consequence of Planck's assumption, and it
received immediate arid remarkable verification in the theory of the
photoelectric effect.

Metals can emit electrons into empty space at high temperatures by
the thermionic effect used in obtaining electron emission from filaments in
vacuum tubes. But metals also can emit electrons at ordinary room
temperature, if they are illuminated by the proper light; this is called the
photoelectric effect. Not much was known about the laws of photo-
electric emission in 1905, but Einstein applied his ideas of photons to the
problem, with remarkable results that proved to be entirely correct.
Einstein assumed that light of frequency v, falling on a metal, could act
only as photons hv were absorbed by the metal. If a photon was
absorbed, it must transfer its whole energy to an electron. Then the
electron in question would have a sudden increase of hv in its energy.
Now it requires a certain amount of work to pull an electron out of a
metal; if it did not, the electrons would continually leak out into empty
space. The minimum amount of work, that required to pull out the most
easily detachable electron, is by definition the work function <£. Then if
hv was greater than the work function, the electron might be able to
escape from the metal, and the maximum possible kinetic energy which
it might have as it emerged would be

= hv - 0. (3.1)

318 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

If the electron happened not to be the most easily detachable one, it
would require more work than <£ to pull it out, and it would have less
kinetic energy than Eq. (3.1) when it emerged, so that that represents the
maximum possible kinetic energy.

Einstein's hypothesis, then, led to two definite predictions. In the
first place, there should bo a photoelectric threshold: frequencies less than
a certain limit, equal to <l>/h, should be incapable of ejecting photoelec-
trons from a metal. This prediction proved to be verified experimentally,
and with more and more accurate determinations of work function it
continues to hold true. It is interesting to see where this threshold comes
in the spectrum. For this purpose, it is more convenient to find the
wave length X = c/v corresponding to the frequency <t>/h. If we express
<£ in electron volls, as is commonly done, (see Eq. (1.1), Chap. IX), we
have the relation

__ he X 300_ _ _ _

4~80~X l(F10< ~ " ' ( '

All wave lengths shorter than the threshold of Eq. (3.2) can eject photo-
electrons. Thus a metal with a small work function of two volts (which
certain alkali metals have) has a threshold in the red and will react
photoelectrically to visible light, while a metal with a work function of
six volts would have a threshold about 2000 A, and would be sensitive
only in the rather far ultraviolet. Most real metals lie between these
limits.

The other prediction of Einstein's hypothesis was as to the maximum
velocity of the photoeleotrons, given by Eq. (3.1). This is also verified
accurately by experiment. There is a remarkable feature connected with
this: the energy of the electrons depends on the frequency, but not on the
intensity, of the light ejecting them. Double the intensity, and the
number of photoelectrons is doubled, but not the energy of each indi-
vidual. This can be carried to limits which at first sight seem almost
absurd, as the intensity of light is reduced. Thus let the intensity be so
low that it will require some time, say half a minute, for the total energy
falling on a piece of metal to equal the amount hv. The light is obviously
distributed all over the piece of metal, and we should suppose that its
energy would be continuously absorbed all over the surface. Yet that
is not at all what happens. About once every half minute, a single
electron will be thrown off from one particular spot of the metal, with an
energy which in order of magnitude is equal to all that has fallen on the
whole plate for the last half minute. It is quite impossible, on any
continuous theory like the wave theory, to understand how all this
energy could have become concentrated in a single electron. Yet it
is; photoelectric cells can actually be operated as we have just described.

SBC. 3] RADIATION AND MATTER 319

An example like this is the most direct sort of experimental evidence for
Einstein's hypothesis, that the energy in light waves, at least when it is
being emitted or absorbed, acts as if it were concentrated in photons.

For a long time it was felt that there was an antagonism between
wave theory and photons. Certainly the photoelectric effect and similar
things are most easily explained by the theory of photons. On the other
hand, interference, diffraction, and the whole of physical optics cannot be
explained on any basis but the wave theory. How could these theories
be simultaneously true? We can sec what happens experimentally, in a
case where we must think about both types of theories, by asking what
would happen if the very weak beam of light, falling on the metal plate
of the last paragraph, had previously gone through a narrow slit, so that
there was actually a diffraction pattern of light and dark fringes on the
plate, a pattern which can be explained only by the wave1 theory. We say
that there is a diffraction pattern; this does not seem to mean anything
with the very faint light, for there is no way to observe it. We mean
only that if nothing is changed but the intensity of the light, and if that is
raised far enough so that the beam can be observed by the eye, a diffrac-
tion pattern would be- seen on the plate. But now even with the weak
light, it really has meaning to speak of the diffraction pattern. Suppose
we marked off the light and dark fringes using an intense light, and then
returned to our weak light and made a statistical study of the points on
the plate from which electrons were ejected. We should find that the
electrons were all emitted from what ought to be the bright fringes, none
from the dark fringes. The wave theory tells us where photons will be
absorbed, on the average. This can be seen even more easily if we replace
the photoelectric plate by a photographic plate. This behaves in a very
similar way: in weak light, occasionally a process takes place at one single
spot of the plate, producing a blackened grain when the plate is developed,
and the effect of increasing the intensity is simply to increase the number
of developed grains, not to change the blackening of an individual grain.
Then a weak diffraction pattern, falling on a plate for a long time, will
result in many blackened grains in the bright fringes, none in the dark
ones, so that the final picture will be almost exactly the same as if there
had been a stronger light beam acting for a shorter length of time.

Nature, in other words, does not seem to be worried about which is
correct, the wave theory or the photon theory of light: it uses both, and
both at the same time, as we have just seen. This is now being accepted
as a fact, and the theories are used more or less in the following way. In
any problem where light is concerned, an electromagnetic field, or light
wave, is set up, according to classical types of theories. But this field is
not supposed to carry energy, as a classical field does. Instead, its
intensity at any point is supposed to determine the probability that a

320 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XIX

photon will be found at that point. It is assumed that there is no way
at all of predicting exactly where any particular photon will go; we cannot
say in any way whatever, in weak light, which electron of the metal will
be ejected next. But on the average, the wave theory allows us to pre-
dict. This type of statistical theory is quite different from any that has
been used in physics before. Whenever a statistical element has been
introduced, as in classical statistical mechanics, it has been simply to
avoid the trouble of going into complete detail about a very complicated
situation. But in quantum theory, as we have already mentioned in
Chap. Ill, Sec. 3, we consider that it is impossible in principle to go into
complete detail, and that the statistical theory is all that there is. When
one meets wave mechanics, one finds that the laws governing the motion
of ordinary particles, electrons, and atoms, are also wavelike laws, and
thai the intensity of the wave gives the probability of finding the particle
in a particular spot, but that no law whatever seems to predict exactly
where it is going. This is a curious state of affairs, according to our usual
notions, but nature seems to be made that way, and the theoiy of radia-
tion has been the first place to find it out.

CHAPTER XX
IONIZATION AND EXCITATION OF ATOMS

In the second part of this book, we have been concerned with the
behavior of gases, liquids, and solids, and we have seen that this behavior
is determined largely by the nature of the interatomic and intermolecular
forces. These forces arise from the electrical structure of the atoms and
molecules, and in this third part we shall consider that structure in a very
elementary way, giving particular attention to the atomic and molecular
binding in various types of substances. Most of the information which
we have about atoms comes from spectroscopy, the interaction of atoms
and light, and we must begin with a discussion of the excited and ioni-
zated states of atoms and molecules, and the relation between energy
levels and the electrical properties of the atoms.

1. Bohr's Frequency Condition. — We have seen in Chap. Ill, Sec. 3,
that according to the quantum theory an atom or molecule can exist only
in certain definite stationary states with definite energy levels. The
sparing of these energy levels depends on the type of motion we are con-
sidering. For molecular vibrations they lie so far apart that their
energy differences are largo compared to kT at ordinary temperatures, as
we saw in Chap. IX. For the rotation of molecules the levels are closer
together, so that only at very low temperatures was it incorrect to treat
the energy as being continuously variable. For molecular translation, as
in a gas, we saw in Chap. IV, Sec. 1, that the levels came so close together
that in all cases we could treat them as being continuous. Atoms and
molecules can also have energy levels in which their electrons are excited
to higher energies than those found at low temperatures. Ordinarily
the energy difference between the lowest electronic state (called the
normal state, or the ground state) and the states with electronic excitation
is much greater than the energy differences concerned in molecular
vibration. These differences, in fact, are so large that at ordinary tem-
peratures no atoms at all are found in excited electronic levels, so that we
do not have to consider them in thermal problems. The excited levels
have an important bearing, however, on the problem of interatomic
forces, and for that reason we must take them up here. Finally, an
electron can be given so much energy that it is entirely removed from
its parent atom, and the atom is ionized. Then the electron can wander
freely through the volume containing the atom, like an atom of a perfect

321

322

INTRODUCTION TO CHEMICAL PHYSICS

[CHAP. XX

gas, and its energy levels are so closely spaced as to be continuous. The
energy levels of an atom or molecule, in other words, include a set of dis-
crete levels, and above those a continuum of levels associated with
ionization. Such a set of energy levels for an atom is shown schematically
in Fig. XX-1. The energy difference between the normal state and the
beginning of the continuum gives the work required to ionize the atom or
molecule, or the ionization potential. The ionization potentials are
ordinarily of the order of magnitude of a number of volts. The lowest
excited energy level of an atom, called in some cases the resonance level
(sometimes the word resonance level is used for an excited level somewhat

higher than the lowest one, but one which is
reached particularly easily from the lowest
one by the absorption of radiation), is
ordinarily several volts above the normal
state, the energy difference being called the
resonance potential. This verifies our state-
ment that electrons will not be appreciably
excited at ordinary temperatures. We can
see this by finding the characteristic tempera-
luro associated with an energy difference of
the order of magnitude of a volt. If we let
fcO = energy of one electron volt, we have
0 = (4.80 X 10-10)/(300 X 1.379 X 10~1G) =
11,600° abs. Thus ordinary temperatures
tire very small compared with such a charac-
teristic temperature. If we consider the
possibility of an electronic specific heat com-
ing from the excitation of electrons to excited
levels, we see that such a specific heat will
be quite negligible, for ordinary substances,
at temperatures below several thousand
degrees. A few exceptional elements, how-
ever, such as some of the transition group metals, have excited energy
levels only a few hundredths of a volt above the normal state, and these
elements have appreciable electronic specific heat at ordinary
temperatures.

There are two principal mechanisms by which atoms or molecules can
jump from one energy level to another. These are the emission and
absorption of radiation, and collisions. For the moment we shall con-
sider the first process. An atom in a given energy level can have transi-
tions to any higher energy level with absorption of energy, or to any
lower level with emission, each transition meaning a quite definite energy
difference. By the conservation of energy, the same quite definite

Ionization
potent/a/

Resonance

potc

ihal

FIQ. XX-1. — Schematic set
energy levels for an atom.

SEC. 1] ION1ZATION AND EXCITATION OF ATOMS 323

energy must be converted into a photon if light is being emitted, or must
have come from a photon if it is being absorbed. But by Einstein's
hypothesis, the frequency of a photon is determined from its energy, by
the relation energy = hv. Thus a transition between two atomic energy
levels, with energies E\ and #2, must result in the emission or absorption
of a photon of frequency v, where

#2 - El = hv. (1.1)

With sharp and discrete energy levels, then, we must haw definite fre-
quencies emitted and absorbed, or must have a sharp lino spectrum. Tho
relation (1.1), as applied to the spectrum, is duo to Bohr, and is often
called Bohr's frequency condition; it is really tho foundation of the
theory of spec! roscopy. Regarded as an empirical fact, it states that the
frequencies observed in any spectrum can be written as the differences
of a sot of numbers, called terms, which are simply the energy levels of tho
system, divided by h. Since with a given table of terms wo can find a
groat many more differences than there are terms, this means that a given
complicated spectrum can bo greatly simplified if, instead of tabulating
all the spectral frequencies, wo tabulate only the much smaller number of
term values. And the importance of Bohr's frequency condition goes
much further than this. For by observing tho frequencies in the spec-
trum and finding the terms, we can get the energy levels of tho atom or
molecule emitting the spectrum. Wo can use these directly, with no
more theory, in such things as a calculation of tho specific heat. For
instance, the energy levels of molecular vibration and rotation, used in
finding tho specific heat of molecules in Chap. IX, arc the results of spoc-
troscopic observation. Furthermore, wo can use tho observed energy
levels to verify, in a very precise way, any theoretical calculation which
we have made on the basis of the quantum conditions. The relation
between the sharp linos observed in spectra, and the energy levels of the
atoms or molecules making the spoctra, has boon the most important fact
in the development of our knowledge of the structure of atoms and
molecules.

Bohr's frequency condition has one surprising feature. The fre-
quency of emitted light is related, according to it, to tho energy rather
than the frequency of the motion in the atom that produces it. This is
entirely contrary to classical theory. A vibrating charge, oscillating
with a given frequency, in classical electromagnetic theory, sends out light
of the frequency with which it vibrates. According to wave mechanics,
however, there is really not a contradiction here. For in wave mechanics,
the particles, ordinarily the electrons, which produce the light do not move
according to classical theory, but the frequencies actually present in their
average motions are those given by Bohr's frequency condition. Thus

324 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

the relation between the motion of the particles, and the light they send
out, is more nearly in accord with classical electromagnetic theory than
we should suppose at first sight.

2. The Kinetics of Absorption and Emission of Radiation. — With
Bohr's picture of the relation between energy levels and discrete spectral
lines in mind, Einstein gave a kinetic derivation of the law of black-body
radiation, which is very instructive and which has had a great deal of
influence. Einstein considered two particular stationary states of an
atom, say the fth and jth (whore for definiteness wo assume that the ith
lies above tho ji\i), and the radiation which could be omitted and absorbed
in going between these two states, radiation of frequency v»-/, where

hvt] = Ei - E3. (2.1)

Suppose the atom, or group of atoms of the same sort, capable of existing
in these stationary states, is in thermal equilibrium with radiation at
temperature T. Then for equilibrium, using the principle of detailed
balancing, the amount of energy of frequency vlt absorbed by the atoms
per second in making the transition from state j to state i must equal the
amount of the same frequency emitted per second in going from state i
to state j. Einstein made definite assumptions as to the probability of
making those two transitions. In the first place, consider absorption.
The number of atoms absorbing photons per second must surely be pro-
portional first to the number of atoms in the lower, ^th energy level, which
we shall call N,', and to the intensity of radiation of the frequency vlj,
which is uvij. Thus Einstein assumed that the number of atoms absorb-
ing photons per second was

NtB^u,,,, (2.2)

where /?»•,- is a constant characteristic of the transition. Next consider
emission. Quite clearly an atom in an excited state can emit radiation
and jump to a lower stato without any outside action at all. This is
called spontaneous emission, and the probability of it was assumed by
Einstein to be a constant independent, of the intensity of radiation. Thus
he assumed the number of atoms falling spontaneously from the ith to
the jth levels per second with emission of radiation was

NiAi,-, (2.3)

where AH is another constant. But at the same time there must be
another process of emission, as Einstein showed by considering very high
temperatures, where u,a is very large. In this limit, the term (2.2) is
bound to be very large compared to the term (2.3), so that with just these
two terms equilibrium is impossible. Guided by certain arguments
based on classical theory, Einstein assumed that this additional probabil-

SEC. 21 10NIZAT10N AND EXCITATION OA' ATOMS 325

ity of emission, generally called induced emission, was

,,* (2.4)

proportional as the absorption was to the intensity of external radiation.
For equilibrium, then, we must have equal numbers of photons emitted
and absorbed per second. Thus we must have

N*(A>, + J3tfUM-,) = N^u^ (2.5)

But at the same time, if there is equilibrium, we know that the number
of atoms in the ith and jth states must be determined by the Boltzmann
factor. Thus we must have

_Ei

Ni = const, e kT,

_Ej

NJ = const, e kr,

N _<!?,-.»,) _h^

jf = c kT = e kT- (2.6)

We can now solve Eqs. (2.5) and (2.6) to find u^. We have

A „ 1 ,~

( i

The energy density (2.7) would bo the Planck distribution law, if we had

Atl _ Sirhv?, fl> ft.

B^ ~ ~* ' (2'8)

as we see by comparison with Eq. (2.6), Chap. XIX. Einstein assumed
that Eq. (2.8) was true, and in that way had a partial derivation of the
Planck law.

Einstein's derivation of the black-body radiation law is particularly
important, for it gives us an insight into the kinetics of radiation processes.
Being a kinetic method, it can be used even when we do not have thermal
equilibrium. Thus if we know that radiation of a certain intensity is
falling on atoms, we can find how many will be raised to the excited state
per second, in terms of the coefficient By. But this means that we can
find the absorptivity of matter made of these atoms, at this particular
wave length. Conversely, from measurements of absorptivity, we
can deduce experimental values of B^. And from Eq. (2.8) we can find
the rate of emission, or the emissive power, if we know the absorptiv-

326 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

ity. Equation (2.8), saying that these two quantities are proportional
to each other, is really very closely related to Kirchhoff's law, discussed in
Chap. XIX, Sec. 1, and Einstein's whole method is closely related to the
arguments of Kirchhoff.

We can put Einstein's assumption of spontaneous and induced emis-
sion in an interesting light if we express Eq. (2.5), not in terms of the
energy density of radiation, but in terms of the average number of
photons Nv in the standing wave of frequency v. Let us see just what we
mean by this. We are assuming that the energy of this standing wave
is quantized, equal to nhv, as in Chap. XIX, Sec. 2; and by Einstein's
hypothesis we are assuming that this means that there are really n
(or Nv) photons associated with this wave. In terms of this, we see that
the cnorgy density uv is dotorminod by the relation

Total energy in dv = uvV dv

= number of waves in dv times number of photons in
each wave times energy in each photon

__ \7 A7 Ij .. /O Q\

- -jjj- vJMrnv, v.j)

using the result of Eq. (2.3), Chapter XIX. From this, we have

^ = TT (2-10)

I/ i V \f

Then we can rewrite Eq. (2.5), using Eq. (2.8), as

Number of photons emitted per second

- Al3Nt(Nv + 1)

= Number of photons absorbed per second

= AVN,N,. (2.11)

The interesting feature of Eq. (2.11) is that the induced and spontaneous
emission combine into a factor as simple as Nv + 1. This is strongly
suggestive of the factors Nj + 1, which we met in the probability of
transition in the. Einstein-Bose statistics, Eq. (4.2), of Chap. VI. As a
matter of fact, the Einstein-Bose statistics, in a slightly modified form,
applies to photons. Since it does not really contribute further to our
understanding of radiation, however, we shall not carry through a discus-
sion of this relation, but merely mention its existence.

3. The Kinetics of Collision and lonization. — In the last section we
have been considering the emission and absorption of radiation as a
mechanism for the transfer of atoms or molecules from one energy level
to another. The other important mechanism of transfer is that of
collisions with another atom, molecule, or more often with an electron.
In such a collision, the colliding particles can change their energy levels,

SEC. 3] IONIZATION AND EXCITATION OF ATOMS 327

and at the same time change their translational kinetic energy, which is
not considered in calculating their energy levels, by such an amount that
the total energy is conserved. A collision in which only the translational
kinetic energy changes, without change of the internal energy levels of
the colliding particles, is called an elastic collision; this is the typo of
collision considered in Chap. VI, where we were finding the effect of colli-
sions on the molecular distribution function. The type of collision that
results in a change of energy level, however, involving either excitation or
ionization, is called an inelastic collision, and it is in such collisions that
we are particularly interested here. Wo consider the kinetics of such
collisions in the present section, coming later to the treatment of thermal
equilibrium, in particular the equilibrium between ionization and recom-
bination, as treated by thermodynamics and kinetic theory. For
generality, we begin by considering the general nature of collisions
between atomic or electronic particles.

The processes which we consider are collisions, and most of them
arc collisions of two particles, which separate again after their encounter.
The probabilities of such collisions are described in terms of a quantity
called a collision cross section, which we now proceed to define. First
let us consider a simple mechanical collision. Suppose we have a small
target, of area A (small compared to 1 sq. cm.). Then suppose; we fire
many projectiles in its direction, but suppose they arc not well aimed, so
that they are equally likely to strike any point of the square centimeter
containing the target. Then we ask, what is the chance that any one
of the projectiles will strike the target? Plainly this chance will be the
ratio of the area of the target, A, to the area of the whole square centi-
meter, which is unity. In other words, A, which we call the collision
cross section in this particular case, is the fraction of all projectiles that
hit the target. If instead of one target we had N in the region traversed
by the beam of unit cross section, and if even the N targets filled only
a small fraction of the square centimeter, so that there was small chance
that one target lay behind another, then the chance that a particular
projectile would have a collision would be NA, and to get the collision
cross section of a single target we should have to take the fraction having
collisions, and divide by N.

In a similar way, in the atomic or molecular case, we allow a beam of
colliding particles to strike the atoms or molecules that we wish to investi-
gate. A certain number of the particles in the incident beam will pass
by without collision, while a certain number will collide and be deflected.
We count the fraction colliding, divide this fraction by the number of
particles with which they could have collided, and the result is the colli-
sion cross section. This can plainly be made the basis of an experi-
mental method of measuring collision cross sections. We start a beam

328 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

of known intensity through a distribution of particles with which they
may collide, and we measure the intensity of the beam after it has trav-
ersed different lengths of path. We observe the intensity to fall off
exponentially with the distance, and from that can deduce the cross
section in the following manner.

Lot the beam have unit cross section, and let a: be a coordinate
measured along the beam. The intensity of the beam, at point x, is
defined as the number of particles crossing the unit cross section at x
per second. We shall call it I(x), and shall find how it varies with x.
Consider the collisions in the thin sheet between x and x + dx. Let the
number of particles per unit volume with which the beam is colliding be
N/V. Then in the thin sheet between x and x + dx, with a volume dx,
there will be N dx/V particles. Let each of these have collision cross
section A. Then the fraction of particles colliding in the sheet will by
definition be NA dx/V. This is, however, equal to the fractional decrease
in intensity of the beam in this distance. That is,

Integrating, this gives

dl NA . ,01.

-y = -y dx. (3.1)

N A
In 7 = ~x + const., (3.2)

or, if the intensity is /0 when x = 0, we have

(3.3)

From Eq. (3.3) we see the exponential decrease of intensity of which we
have just spoken, and it is clear that by measuring the rate of exponential
decrease we can find the collision cross section experimentally.

The intensity of a beam falls to 1/e of its initial value, from Eq. (3.3),
in a distance

(3.4)

(ft

This distance is often called the mean free path. As we can see, it is
inversely proportional to the number of particles per unit volume, or
the density, and inversely proportional to the collision cross section.
The mean free path is most commonly discussed for the ordinary elastic
collisions of two molecules in a gas. For such collisions, the collision
cross sections come out of the order of magnitude of the actual cross
sectional areas of the molecules; that is, of the order of magnitude of

SEC. 3] IONIZATION AND EXCITATION OF ATOMS 329

10~~16 cm2. In a gas at normal pressure and temperature, there aro
2.70 X 1019 molecules per unit volume. Thus, the mean free path is
of the order of 1/(2.70 X 10') = 3.7 X 10™4 cm. As a matter of fact,
most values of A are several times this value, giving moan free paths
smaller than the figure above*. As the pressure is reduced, however,
the mean free paths become quite long. Thus at 0°C., but a pressure of
10~5 atm., the mean free paths become of the order of magnitude of 1 cm.;
with pressures several thousand times smaller than this, which are
easily realized in a high vacuum, the mean free path becomes many
meters. In other words, the probability of collision in the dimensions
of an ordinary vacuum tube becomes negligible, and molecules shoot from
one side to the other without hindrance.

The collision cross section is closely related to the quantities Afa, which
we introduced in discussing collisions in Sec. I, Chap. VI. We were
speaking there about a particular sort of collision, one in which one of the
colliding particles before collision was in cell i of the phase space, the
other in cell j, while after collision the first was in cell k, the second in
cell L The number of such collisions per unit time was assumed to be
AlkJiNlN]. In the present case, we are treating all collisions of two
molecules, one moving, the other at rest, irrespective of the velocities
after collision. That is, the present case corresponds to the case where
one of the two cells i or j corresponds to a molecule at rest, and where
we sum over all cells k and L Furthermore, there we were interested
in the number of collisions per unit time, here in the number per unit
distance of path. It is clear that if we knew the Aft's, we could compute
from them the collision cross section of the sort we are now using. Our
collision cross section gives less specific information, however. Wo
expect to find a different collision cross section for each velocity of
impinging particle, thoiigh our restriction that the particle with which
it is colliding is at rest is not really a restriction at all, for it is an easy
problem in mechanics to find what would happen if both particles
were initially in motion, if we know the more restricted case where one is
initially at rest. But the Aft's give additional information about the
velocities of the two particles after collision. They assume that the
total kinetic energy after collision equals the total kinetic energy before
collision; that is, they assume an elastic collision. Then, as mentioned
in Sec. 1, Chap. VI, there are two quantities which can be assigned at will
in describing the collision, which may be taken to be the direction of one
of the particles after collision. To give equivalent information to the Aft
in the language of collision cross sections, we should give not merely the
probability that a colliding particle of given velocity strike a fixed parti-
cle, but also the probability that after collision it travel off in a definite
direction. This leads to what is called a collision cross section for

330 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

scattering in a given direction: the probability that the colliding particle
have a collision, and after collision that it travel in a direction lying within
a unit solid angle around a particular direction in space. This cross
section gives as much information as the set of A$s, though in a different
form, so that it requires a rather complicated mathematical analysis,
which we shall not carry out, to pass from one to the other.

We are now ready to consider the collision cross sections1 for some of
the processes concerned in excitation and ionization. First we consider
the collision of an electron with a neutral atom. In the first place,
there are two possible types of collision, elastic and inelastic. If the
energy of the incident electron is less than the resonance potential of the
atom, then an inelastic collision is not possible, for the final kinetic energy
of the two particles cannot be less than zero. Thus below the resonance
potential all collisions arc elastic. The cross section for elastic collision
varies with the velocity of the impinging electron, sometimes in what
seems a very erratic manner. Generally it increases as the velocity
decreases to zero, but for some atoms, particularly the inert gas atoms,
it goes through a maximum at a velocity associated with an energy of the
order of 10 electron volts, then decreases again as the velocity is decreased,
until it appears to become zero as the velocity goes to zero. This effect,
meaning that extremely slow electrons have extremely long mean free
paths in these particular gases, is called the Ramsauer effect, from its
discoverer. The collision cross sections for elastic collision of electrons
and atoms have been investigated experimentally for all the convenient
atoms, and many molecules, disclosing a wido variety of behaviors.
They can also be investigated theoretically by the wave mechanics,
involving methods which cannot be explained here, and the theoretical
predictions agree very satisfactorily with the experiments, even to the
extent of explaining the Ramsauer effect.

Above the resonance potential, an electron has the possibility of
colliding inclastically with an atom, raising it to an excited level, as well
as of colliding elastieally. The probability of excitation, or the collision
cross section for inelastic collision, starts up as the voltage is raised above
the excitation potential, rising quite rapidly for some transitions, more
slowly for others, then reaches a maximum, and finally begins to decrease
if the electron is too fast. Of course, atoms can be raised not merely to
their resonance level, but to any other excited level, by an electron of
suitable energy, and each one of these transitions has a collision cross
section of the type we have mentioned, starting from zero just at the
suitable excitation energy. The probabilities of excitation to high
energy levels are small, however; by far the most important inelastic

1 For further information about collisions, see Massey and Mott, "The Theory of
Atomic Collisions," Oxford University Press, 1933.

SBC. 3] IONIZATION AND EXCITATION OF ATOMS 331

types of collision, at energies less than the ionization potential, are those
in which the atom is raised to one of its lowest excited levels.

As soon as the energy of the impinging electron becomes greater
than the ionization potential, inelastic collisions with ionization become
possible. Here again the collision cross section starts rather rapidly
from zero as the potential is raised above the ionization potential, reaches
a maximum at the order of magnitude of two or throe times the ionization
potential, and gradually falls off with increasing energy. The reason
for the falling off with rising energy is an elementary one: a fast electron
spends less time in an atom, and consequently has less time to ionize it
and less probability of producing the transition. A collision with ioniza-
tion is of course different from an excitation, in that the ejected electron
also leaves the scene of the collision, so that after the collision we have
three particles, the ion and two electrons, instead of two as in the previous
case. This fact is used in the experimental determination of resonance
and ionization potentials. A beam of electrons, of carefully regulated
voltage, is shot through a gas, and as the voltage is adjusted, it is observed
that the mean free path shows sharp breaks as a function of voltage at
certain points, decreasing sharply as certain critical voltages are passed.
This does not tell us whether the critical voltages are resonance or
ionization potentials, but if the presence of additional electrons is also
observed, an increase in these additional electrons is noticed at an
ionizatiou potential but not at a resonance potential.

Of course, each of these types of collision must have an inverse type,
and the principle of microscopic reversibility, discussed in Chap. VI,
shows that the probability, or collision cross section, for the inverse
collision can be determined from that of the direct collision. The oppo-
site or inverse to a collision with excitation is what is called a collision
of the second kind (the ordinary one being called a collision of the first
kind). In a collision of the second kind an electron of low energy collides
with an excited atom or molecule, the atom has a transition to its normal
state or some lower energy level than the one it is in, and the electron
comes off with more kinetic energy than it had originally. The inverse
to an ionization is a three-body collision: two electrons simultaneously
strike an atom, one is bound to the atom, which falls to its normal state
or some excited state, while the other electron, as in a collision of tho
second kind, is ejected with more kinetic energy than the two electrons
together had before the collision. Such a process is called a recombina-
tion; and it is to be noticed that we never have a recombination just of
an atom and an electron, for there would be no body to remove the extra
energy.

In addition to the types of collision we have just considered, where
an electron and an atom or molecule collide, one can have collisions of

332 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

two atoms or molecules with each other, with excitation or ionization.
It is perfectly possible to have two fast atoms collide with each other,
originally in their normal states, and result in the excitation or ionization
of one or both of the atoms. The collision of the second kind, inverse to
this, is that in which an excited atom and a normal one collide, the
excited one falls to its normal state, and the atoms gain kinetic energy.
Then one can have an exchange of excitation: an excited and a normal
atom collide, and after the collision the excited one has fallen to its normal
state, the. normal one is excited, and the discrepancy in energy is made
up in the kinetic energy of translation of the atoms. Or one can have an
interchange of ionization: a neutral atom and a positive ion collide, and
after collision the first one has become a positive ion, the second one is
neutral. We shall consider this same process from the point of view of
statistical mechanics in the next section. In these cases of collisions of
atoms, it is very difficult to calculate the probabilities of the various
processes, or the collision cross sections, and in most cases few measure-
ments have been made. In general, however, it can be said that the
probability of elastic; collision, with the collision of two atoms, is much
greater than the probability of any of the various types of inelastic
collision.

In Chap. X, we have taken up the kinetics of chemical processes, the
types of collisions between molecules which result in chemical reactions.
Thorn is no very fundamental distinction between those collisions and
the type we have just considered. In ordinary chemical reactions, the
colliding molecules are under all circumstances in their lowest electronic
state; they arc not excited or ionized. The reason is that excitation or
ionization potentials, of molecules as of atoms, arc ordinarily high enough
so that the chance of excitation or ionization is negligible at the tempera-
tures at which reactions ordinarily take place, or what amounts to the
same thing, the colliding molecules taking part in the reaction almost
never have enough energy to excite or ionize each other. This does not
mean, however, that excitation and ionization do not sometimes occur,
particularly in reactions at high temperature; undoubtedly in some cases
they do. It is to be noted that in the case of colliding molecules, unlike
colliding atoms, inelastic collisions are possible without electronic excita-
tion: the molecules ean lose some of their translational kinetic energy
in the form of rotational or vibrational energy. In this sense, an ordinary
chemical reaction, as explained in Sec. 3, Chap. X, is an extreme case of
an inelastic collision without excitation. But such inelastic collisions
with excitation of rotation and vibration are the mechanism by which
oquipartition is maintained between translational and rotational and
vibrational energy, in changes of temperature of a gas. Sometimes they
do not provide a mechanism efficient enough to result in equilibrium

SEC. 4] IONIZATION AND EXCITATION OF ATOMS 333

between these modes of motion. For example, in a sound wave, there
are rapid alternations of pressure, produced by the trauslational motion
of the gas, and these result in rapid alternations of the translational
kinetic energy. If equilibrium between translation and rotation can
take place fast enough, there will bo an alternating temperature, related
to the pressure by the adiabatic relation, and at each instant there will
be thermal equilibrium. Actually, this holds for low frequencies of
sound, but there is evidence that at very high frequencies the inelastic
collisions are too slow to produce equilibrium, and the rotation does not
partake of the fluctuations in energy.

Another interesting example is found in some cases in gas discharges
in molecular gases. In an arc, there are ordinarily electrons of several
volts' energy, since an electron must be* accelerated up to the lowest-
resonance potential of the gases present before it can have an inelastic*,
collision and reduce its energy again to a low value. These electrons
have a kinetic energy, then, which gas moleeules \\ould acquire only at
temperatures of a good many thousand degrees. The electrons collide*
elastically with atoms, and in these collisions the electrons tend to lose
energy, the atoms to gain it, for this is just the mechanism by which
thermal equilibrium and cquipartition tend to be brought about. If there
are enough elastic collisions before the electrons are slowed down by an
inelastic collision, the atoms or moleeules will tend to get into thermal
equilibrium, as far as their translation is concerned, corresponding to
an extremely high temperature. That such an equilibrium is actually
set up is observed by noticing that the fast; electrons in an arc have a
distribution of velocities approximating a Maxwellian distribution.
But apparently the inelastic collisions between molecules, or between
electrons and molecules, are not effective enough to give the molecules the
amount of rotational energy suitable to cquipartition, in the short length
of time in which a molecule is in the arc, before it diffuses to the wall or
otherwise can cool off. For the, rotational energy can bo observed by
band spectrum observation, and in some cases it is found that it corre-
sponds to rather cool gas, though the translational energy corresponds
to a very high temperature.

4. The Equilibrium of Atoms and Electrons. — From the cases we
have taken up, we see that the kinetics of collisions forms a complicated
and involved subject, just as the kinetics of chemical reactions does.
Since this is so, it is fortunate that in cases of thermal equilibrium, we
can get results by thermodynamics which are independent of the precise
mechanism, and depend only on ionization potentials and similarly easily
measured quantities. And as we have stated, thermodynamics, in the
form of the principle of microscopic reversibility, allows us to get some
information about the relation between the probability of a direct process

334 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX

and of its inverse, though we have not tried to make any such calculations.
To see how it is possible, we need only notice that every equilibrium con-
stant can be written as the ratio of the rates of two inverse reactions, as
we saw from our kinetic derivation of the mass action law in Chap. X,
so that if we know the equilibrium constant, from energy considerations,
and if we have experimental or theoretical information about the rate
of one of the reactions concerned, we can calculate the rate of the inverse
without further hypothesis.

A mixture of electrons, ions, and atoms forms a system similar to that
which we considered in Chap. X, dealing with chemical equilibrium in
gases. Equilibrium is determined, as it was there, by the mass action
law. This law can bo derived by balancing the rates of direct and inverse
collisions, but it can also be derived from thermodynamics, and the
equilibrium constant can be found from the heat of reaction and the
chemical constants of the various particles concerned. The heats of
reaction can be found from the various ionization potentials, quantities
susceptible of independent measurement, and the chemical constants
are determined essentially as in Chap. VIII. Thus there are no new
principles involved in studying the equilibrium of atoms, electrons, and
ions, and we shall merely givo a qualitative discussion in this section,
the statements being equivalent to mathematical results which can bo
established immediately from the methods of Chap. X.

The simplest type of problem is the dissociation of an atom into a
positive ion and an electron. By the methods of Chap. X, we find for
the partial pressures of positive ions, negative electrons, and neutral
atoms the relation

where P(+), -P(-), P(n) are the pressures of positive ions, electrons,
and neutral atoms respectively. From Eq. (2.6), Chap. X, we can find
the equilibrium constant KP explicitly. For the reaction in which one
mole of neutral atoms disappears, one mole of positive ions and electrons

appears, we have v+ = 1, i/_ = 1, vn = — 1. Then the quantity — ^VjUj

becomes — U+ — f/_ + Un = —I. P., where I. P. stands for the ionization
potential, expressed in kilogram-calories per mole, or other thermal unit
in which we also express KT. Then we have

« -—
KP(T) = e?++*--**r*e RT . (4.2)

From Eq. (4.2), we see that the equilibrium constant is zero at the
absolute zero, rising very slowly until the temperature becomes of the

SBC. 4] IONIZATION AND EXCITATION OF ATOMS 335

order of magnitude of the characteristic temperature LP./R, which as
we have seen is of the order of 10,000°. Thus at ordinary temperatures,
from Eq. (4.1), there will be very little ionization in thermal equilibrium.
This statement does not hold, however, at very low pressures. We can
see this if we write our equilibrium relation in terms of concentrations,
following Eq. (1.10), Chap. X. Then we have

**= - **&). (4.3)

CH *

From Eq. (4.3), we see that as the pressure is reduced at constant tem-
perature, the dissociation becomes greater, until finally at vanishing
pressure the dissociation can become complete, even at ordinary tem-
peratures. This is a result of importance in astrophysics, as has been
pointed out by Saha. In the solar atmosphere, there is spectroscopic
evidence of the existence of rather highly ionized elements, even though
the temperature of the outer layers of the atmosphere is not high enough
for us to expect such ionization, at ordinary pressures. However, the
pressure in these layers of the sun is extremely small, and for that reason
the ionization is abnormally high.

Another example that can be handled by ordinary methods of chemical
equilibrium is the equilibrium between an ion and a neutral atom of
another substance, in which the more electropositive atom is the one
forming the positive ion, in equilibrium. Thus, consider the reaction
Li + Ne+ <=± Li+ + Ne, in which Li has a much smaller ionization poten-
tial than Ne, or is more electropositive. The equilibrium will be given

v = Kp(T), (4.4)

CLI ^NO

the pressure canceling in this case. And the equilibrium constant
KP(T) is given by

-I.P.(Ne) + _
ef)-i(Li+)-»<Ne)0 RT ^ (^gj

Since the ionization potential of neon is greater than that of lithium, the
equilibrium constant reduces to zero at the absolute zero, showing that
at low temperatures the lithium is ionized, the neon unionized. In
other words, the element of low ionization potential, or the electropositive4
element, tends to lose electrons to the more electronegative element,
with high ionization potential. This tendency is complete at the absolute
zero. At higher temperatures, however, as the mass action law shows,
there will be an equilibrium with some of each element ionized.

CHAPTER XXI
ATOMS AND THE PERIODIC TABLE

Interatomic forces form the basis of molecular structure and chemis-
try, and we cannot understand them without knowing something about
atomic structure. We shall for this reason give in this chapter a very
brief discussion of the nuclear model of the atom, its treatment by the
quantum theory, and the resulting explanation of the periodic table.
There is of course not the slightest suggestion of completeness in our
discussion; volumes can be written about our present knowledge of atomic
structure and atomic spectra, and the student who wishes to understand
chemical physics properly should study atomic structure independently.
Since however there are many excellent treatises available on the subject,
we largely omit such a discussion here, mentioning only the few points
that wo shall specifically use.

1. The Nuclear Atom. — An atom is an electrical structure, whose
diameter is of the order of magnitude of 10~8 cm., or 1 angstrom unit,
and whoso mass is of the order of magnitude of 10~ 24 gm. More precisely,
an atom of unit atomic weight would have a mass 1.66 X 10~24 gm., and
the mass of any atom is this unit, times its atomic weight. Almost all
the* mass of the atom is concentrated in a small central body called the
nucleus, which determines the properties of the atom. The diameter of
the nucleus is of the order of magnitude of 10™13 cm., a quantity small
enough so that it can be neglected in practically all processes of a chemical
nature. The nucleus carries a positive charge of electricity. This
charge is an integral multiple of a unit charge, generally denoted by the
letter c, equal to 4.80 X 10~10 e.s.u. of charge. The integer by which
we must multiply this unit to get the charge on the nucleus is called the
atomic number, and is often denoted by Z. This atomic number proves
to be the ordinal number of the corresponding element in the periodic
table of the elements, as used by the chemists. Thus for the first few
elements wo have hydrogen H, Z = 1 ; helium He, Z = 2; lithium Li,
Z = 3; and so on, up to uranium U, Z = 92, the heaviest natural ele-
ment. The electric charge of the nucleus, or the atomic number, is the
determining feature of the atom chemically, rather than the atomic
weight. In a large number of cases, there are several types of nuclei,
all with the same atomic number but with different atomic weights.
Such nuclei arc called isotopes. They prove to have practically identical

336

SEC. 1] ATOMS AND THE PERIODIC TABLE 337

properties, since most properties depend on the nuclear charge, not its
mass. Almost the only property depending on the mass is the vibra-
tional frequency, as observed in molecular vibrations, specific heat, etc.
Thus, different isotopes have different characteristic temperature** and
specific heats, but since the masses of different isotopes of the same ele-
ment do not ordinarily differ greatly, these differences are not very
important. Almost the only exception is hydrogen, where the heavy
isotope has twice the maSvS of the light isotope, making the properties
of heavy hydrogen, or deuterium, decidedly different from those of
ordinary hydrogen. The atomic weights of isotopes are almost exactly
whole number multiples of the unit 1.66 X 10"24 gm., but the atomic
weight measured chemically is the weighted mean of those of its various
isotopes, and hence is not a very fundamental quantity theoretically.
For our purposes, which are largely chemical, we need not consider the
possibility of a change in the properties of a nucleus. But many reac-
tions are known, some spontaneous (natural radioactivity) and some
artificial (artificial or induced radioactivity), by which nuclei can be
changed, both as to their atomic; weight and atomic number, and hence
converted from the nuclei of one element to those of another. We shall
assume that such nuclear transformations are not occurring in the proc-
esses we consider.

In addition to the nucleus, the atom contains a number of light, nega-
tively charged particles, the electrons. An electron has a mass of
9.1 X 10~28 gm., iVrs- of the mass of a nucleus of unit atomic weight.
Its charge, of negative sign, has the magnitude of 4.80 X 10~10 e.s.u.,
the unit mentioned above. There seem to be no experiments which
give information about its radius, though there are some theoretical
reasons, not very sound, for thinking it to be of the order of 10~13 cm. , If
the atom is electrically neutral, it must contain just as many electrons
as the nucleus has unit charges; that is, the number of electrons equals
the atomic number. But it is perfectly possible for the atom to exist
with other numbers of electrons than this. If it loses electrons, becoming
positively charged, it is a positive ion. It can lose any number of
electrons from one up to its total number Z, and we say that it then forms
a singly charged, doubly charged, etc., positive ion. A positive ion is a
stable structure, like an atom, and can exist indefinitely, so long as it
does not come in contact with electrons or matter containing electrons,
by means of which it can neutralize itself electrically. On the other hand,
an atom can sometimes attach one or more electrons to itself, becoming
a singly or multiply charged negative ion. Such a structure tends to be
inherently unstable, for it is negatively charged on the whole, repelling
electrons and tending to expel its own extra electrons and become neutral
again. It is doubtful if any multiply charged negative ions are really

338 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

stable. On the other hand, a number of elements form stable, singly
charged, negative ions. These are the so-called electronegative elements,
the halogens F, Cl, Br, I, tho divalent elements O and S, arid perhaps a
few others. These elements have slightly lower energy in the form of a
negative ion, as F", than in the dissociated form of the neutral atom, as F,
and a removed electron. The energy difference between these two states
is called the electron affinity; as we see, it is analogous to a heat of reac-
tion, for a reaction like

F~<=>F + e. (1.1)

The energy required to remove an electron from a neutral atom is its
ionizatiori potential; that required to remove the second electron from a
singly charged positive ion is the second ionization potential; and so on.
In each case, the most easily removed electron is supposed to be the one
considered, some electrons being much more easily detachable than
others. Successive ionization potentials get rapidly larger and larger, for
as the ion becomes more highly charged positively, an electron is more
strongly hold to it by electrostatic forces and requires more work to
remove. Ionization potentials and electron affinities, as we have already
mentioned, are commonly measured in electron volts, since electrical
methods are commonly used to measure them. For the definition of the
electron volt and its relation to thcrmodynamic units of energy, the reader
is referred to Eq. (1.1), Chap. IX, where it is shown that one electron volt
per atom is equivalent to 23.05 kg.-cal. per gram mole, so that ionization
potentials of several electron volts represent heats of reaction, for the
reaction in which a neutral atom dissociates into an electron and a
positive ion, which are large, as measured by thermodynamic standards,
as mentioned in the preceding chapter.

2. Electronic Energy Levels of an Atom. — The electrons in atoms arc
governed by the quantum theory and consequently have various sta-
tionary states and energy levels, which arc intimately related to the
excitation and ionization potentials and to the structure of the periodic
table. We shall not attempt here to give a complete account of atomic1
structure, in terms of electronic levels, but shall mention only a few
important features of the problem. A neutral atom, with atomic num-
ber Z, and Z electrons, each acted on by the other (Z — 1) electrons a*-
well as by the nucleus, forms a dynamical problem which is too difficult
to solve except by approximation, either in classical mechanics or in
quantum theory. The most useful approximation is to replace the forcr
acting on an electron, depending as it does on the positions of all othei
electrons as well as on the one in question, by an averaged force, averaged
over all the positions which the other electrons take up during theii
motion. This on the average is a central force; that is, it is an attraction

SBC. 2] ATOMS AND THE PERIODIC TABLE 339

pulling the electron toward the nucleus, the magnitude depending only
on the distance from the nucleus. It is a smaller attraction than that
of an electron for a bare nucleus, for the other electrons, distributed
about the nucleus, exert a repulsion on the average . Nevertheless, so
long as it is a central force, quantum theory can quite easily solve the
problem of finding the energy levels and the average positions of the
electrons.

An electron in a central field has three quantum numbers, connected
with the three dimensions of space. One, called the azimuthal quantum
number, is denoted by I, and measures the angular momentum of the
electron, in units of h/2ir. Just as in the simple rotator, discussed in
Section 3, Chap. Ill, the angular momentum must be an integer times
h/2ir, and here the integer is I, taking on the values 0, 1, 2, ... For
each value of I, we have a series of terms or energy levels, given by
integral values of a second quantum number, called the principal or
total quantum number, denoted by n, and by convention taking on the
values I + 1, I + 2, • • • Following spectroscopic notation, all the
levels of a given I value are grouped together to form a series and are
denoted by a letter. Thus I = 0 is denoted by s (for the spectroscopic
Sharp series), I = 1 by p (for the Principal series), I — 2 by d (for the
Diffuse series), I = 3 by/ (for the Fundamental series), and I = 4, 5, 6,
. . . by 0, h, j, . . . , using further letters of the alphabet. A given
energy level of the electron is denoted by giving its value of n, and then
the letter giving its I value; as 3p, a level with n = 3, I = 1. The third
quantum number is connected with space quantization, as discussed in
Sec. 3, Chap. IX, and is denoted by mi. Not only the angular momentum
I is quantized, but also its component along a fixed direction in space,
and this is equal to mih/2ir. The integer mi, then, can go from the limits
of lh/2ir (when the angular momentum points along the direction in
question) to —lh/2ir (when it is oppositely directed), resulting in 21 + 1
different orientations. Being a problem with spherical symmetry, the
energy does not depend on the orientation of the angular momentum.
Thus the 21 + 1 levels corresponding to a given n and Z, but different
orientations, all have the same energy, so that the problem is degenerate,
and an « level has one sublevel, a p has three, a d five, an / seven, etc.

This number of levels is really doubled, however, by the electron spin.
An electron has an intrinsic permanent magnetism, and associated with
it a permanent angular momentum of magnitude %h/2ir. This can be
oriented in either of two opposite directions, giving a component of
±%h/2ir along a fixed direction. This, as will be seen, is in harmony
with the space quantization just described, for the special case I = £.
For each stationary state of an electron neglecting spin, we can have the
two possible orientations of the spin, so that actually an s level has two

340 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

sublevels, a, p has six, a d ten, an / fourteen. These numbers form the
basis of the structure of the periodic table.

The energies of these energy levels can be given exactly only in the
case of a single electron rotating about a nucleus of charge Z units, in the
absence of other electrons to shield it. In this case, the energy is given
by Bohr's formula

In Eq. (2.1), m is the mass of an electron (9.1 X 10~28 gm.), e is its charge
(4.80 X 10~10 c.s.u.), K is the so-called Rydberg number, 109,737 cm.-1,
so that Rhc, where c is the velocity of light (3.00 X 1010 cm. per second),
is an energy, equal to 2.17 X 10~"n erg, or 13.56 electron volts, or 313
kg.-cal. per gram mole. The zero of energy is the state in which the
electron reaches an infinite distance from the nucleus with zero kinetic
energy. In all the stationary states, the energy is less than this, or is
negative, so that the electron can never be entirely removed from the
atom. The1 smaller the integer n, the lower the energy, so that the
lowest states correspond to n = 1, 2, etc. At the same time, the lower
the energy is, the more closely bound to the nucleus the electron is, so
that the orbit, or the region occupied by the electron, is small for small
values of n. The tightness of binding increases with the nuclear charge
Z, as we should expect, and at the same time the size of the orbit decreases.
We notice that for an electron moving around a nucleus, the levels of
different series, or different I values, all have the same energy provided
they have the same principal quantum number n.

For a central field like that actually encountered in an atom, the
energy levels are quite different from those given by Eq. (2.1). They
are divided quite sharply into two sorts: low-lying levels corresponding
to orbits wholly within the atom, and high levels corresponding to orbits
partly or wholly outside the atom. For levels of the first type, the
energy is given approximately by a formula of the typo

E = -Rhc(Z--Z-Z«)\ (2.2)

In Eq. (2.2), Z0 is called a shielding constant. It measures the effect
of the other electrons in reducing the nuclear attraction for the electron
in question. It is a function of n and /, increasing from practically zero
for the lowest n values to a value only slightly less than Z for the outer-
most orbits within the atom. For levels outside the atom, on the other
hand, the energy is given approximately by

E = -Rhc- (2.3)

SEC. 2]

ATOMS AND THE PERIODIC TABLE

341

Here 6 is called a quantum defect. It depends strongly on Z, but is
approximately independent of n in a single series, or for a single I value.
The value of 5 decreases rapidly with increasing I] thus the s series may
have a large quantum defect, the p series a considerably smaller one, and
the d and higher series may have very small values of 6, for some particu-
lar atom. We may illustrate these formulas by Fig. XXI-1, in which
the energies of an electron in a contra! field representing ooppor, as a
function of n, are shown on a logarithmic scale. Tho sharp break between

FIG XXI-1. — ICnergies of electrons in tho copper atom, in Rydbcrg unit**, IM a function
of principal quantum number ?i. Eneigies are shown on. a logni ithmic scale. The energies
in the hydrogen atom are shown for comparison.

the two typos of energy levels is woll shown; Is, 2s, 2p, 3s, 3;;, 3d belong
very definitely to the orbits lying within the atom, while tho others are
outside and are governed approximately by Kq. (2.3).

It has been mentioned that the region occupied by tho electron's
orbit increases in volume, as tho binding energy becomes less or as the
quantum number n increases. For our later use in studying the sizes of
atoms, it is useful to know the size of the orbit quantitatively. These
sizes are not definitely determined, for the electron is sometimes found
at one point, sometimes at another, in a given stationary state, and all we
can give is the distance from the nucleus at which there is the greatest

342 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

probability of finding it. This is not given by a simple formula, though
it can be computed fairly accurately by wave mechanics, but to an
approximation the radius r of maximum charge density or probability is
given by

fW = a0~; (2.4)

for the case of an electron moving about a bare nucleus of charge Z, in
an orbit of quantum number n, where a0 = h*/4ir2me2 = 0.53 A. This
is the formula connected with the type of orbit whose energy is given by
Eq. (2.1). We observe the increase of size with increasing n, and the
decrease with increasing nuclear charge, which we have mentioned
before. Similarly, if the energy is given by Eq. (2.2), the radius is given
approximately by

n2
r-max = a^7 „ , (2.5)

and if the formula (2.3) holds for the energy, the radius is

rumx = «o(n - 6)2. (2.6)

We may expect that the radius of the atom, if that expression has a
meaning, will be of the order of magnitude of the radius of the largest
orbit ordinarily occupied by an electron in the neutral atom. In the
case of copper this is the 4s orbit, while in the copper ion it is the 3d. In
the next section we tabulate such quantities for the atoms, and in later
chapters we shall find these radii of interest in connection with the
dimensions of atoms as determined in other ways.

We can now use the energy levels of an electron in a central field in
discussing the structure of the atom. At the outset, we must use a
fundamental fact regarding electrons: they obey the Fermi-Dirac statis-
tics. That is, no two electrons can occupy the same stationary state.
The principle, stated in this form, is often called the Pauli exclusion
principle, and it was originally developed to provide an explanation
for the periodic table, which we shall discuss in the next section. As a
result of the Pauli exclusion principle, there can be only two Is electrons,
two 2s's, six 2p's, etc. We can now describe what is called the con-
figuration of an atom by giving the number of electrons in each quantum
state. In the usual notation, these numbers are written as exponents.
Thus the symbol (ls)2(2s)2(2/;)6(3s)2(3p)6(3rf)104s would indicate a state
of an atom with two Is electrons, two 2s, etc., the total number of elec-
trons being 2 + 2 + 6 + 2 + 6 + 10 + 1 = 29, the number appro-
priate to the neutral copper atom. If all the electrons are in the lowest
available energy level, as they are in the case above, the configuration

SEC. 2] ATOMS AND THE PERIODIC TABLE 343

corresponds to the normal or ground state of the atom. If, on the other
hand, some electrons are in higher levels than the lowest possible ones, the
configuration corresponds to an excited state. In the simplest case, only
one electron is excited; this would correspond to a configuration like
(ls)2(2s)2(2p)6(3s)2(3p)H3d)10(5p) for copper. To save writing, the two
configurations indicated above would often be abbreviated simply as 4s
and 5p, the inner electrons being omitted, since they are arranged as in
the normal state. It is possible for more than one electron to bo excited ;
for instance, we could have the configuration which would ordinarily
be written as (3d)9(4p)(5s) (the Is, 2s, 2p, 3s, 3p electrons being omitted),
in which one of the 3d electrons is excited, say to the 4p level, and the 4.x
is excited to the 5s level, or in which the 3d is excited to the 5s level, the
4s to the 4p. (On account of the identity of electrons, implied in the
Fermi-Dirac statistics, there is no physical distinction between these two
ways of describing the excitation.) While more than two electrons can
theoretically be excited at the same time, it is very unusual for this to
occur. If one or more electrons are entirely removed, so that we have an
ion, the remaining electrons will have a configuration that ran be indi-
cated by the same sort of symbol that would be used for a complete
atom. For example, the normal state of the Cu+ ion has the configura-
tion (ls)2(2«)2(2p)6(3«)2(3p)6(3d)10.

The energy values which we most often wish are excitation and
ionization potentials, the energies required to shift one or more electrons
from one level to another, or the differences of energy between atoms or
ions in different configurations. We can obtain good approximations
to these from our one-electron energy values of Eqs. (2.2) and (2.3). The
rule is simple: the energy required to shift an electron from one energy
level to another in the atom is approximately equal to the difference of
the corresponding one-electron energies. If two electrons are shifted,
we simply add the energy differences for the two. This rule is only
qualitatively correct, but is very useful. In particular, sinee the absolute
values of the quantities (2.2) and (2.3) represent the energies required to
remove the corresponding electrons from the central field, the same
quantities in turn are approximate values of the ionization potentials of
the atom. An atom can be ionized by the removal of any one of its
electrons. The ordinary ionization potential is the work required to
remove the most loosely bound electron; in copper, for instance, the
work required to remove the 4s electron from the neutral atom. But
any other electron can be removed instead, though it requires more
energy. If an atom is bombarded by a fast electron, the most likely
type of ionization process is that in which an inner electron is removed,
as for instance a Is, 2s, 2p, etc. For such an ionization, in which the
ionization potential may be many thousands of volts, the impinging

344 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

electron must of course have an energy greater than the appropriate
ionization potential. After an inner electron is knocked out, in this
way, a transition is likely to occur in which one of the outer electrons falls
into the empty inner shell, the emitted energy coming off as radiation
of very high frequency. It is in this way that x-rays are produced, and
on account of their part in x-ray emission, the inner energy levels are
known by a notation derived from x-rays. Thus the Is electrons are
known as the K shell (since the x-rays emitted when an electron falls
into the K shell are called the K series of x-rays), and 2s and 2p grouped
together are the L shell, the 3s, 3p, and 3d together are the M shell, and
so on.

In contrast to the x-ray ionization, which is not often important in
chemical problems, an impinging electron with only a few volts' energy is
likely to excite or ionize the outermost electron. This electron has an
energy given approximately by Kq. (2.3), which thus gives roughly the
energies of the various excited and ionized levels of the atom. As a
matter of fact, the real situation, with all but a few of the simplest atoms,
is very much more complicated than would be indicated by Eq. (2.3), on
account of certain interactions between the outer electrons of the atom,
resulting in what are called multiplets. A given configuration of the
electrons, instead of corresponding to a single stationary state of the
atom, very often corresponds to a large number of energy levels, grouped
more or less closely about the value given by our elementary approxima-
tion of one-electron energies. An understanding of this multiplet
structure is essential to a real study of molecular structure, but we shall
not follow the subject far enough to need it. One principle only will
be of value: a closed shell of electrons, by which we mean a shell contain-
ing all the electrons it can hold, consistent with the Pauli exclusion
principle [in other words, a group like (Is)2, (2p)6, etc.] contributes noth-
ing to the multiplet structure or the complication of the energy levels.
Thus an atom all of whose electrons are in closed shells (which, as we
shall see in the next section, is an inert gas) has no multiplets, and its
energy level is single. And an atom consisting mostly of closed shells,
but with one or two electrons outside them, has a multiplet structure
characteristic only of the electrons outside closed shells. Thus the alkali
metals, and copper, silver, arid gold, all have one electron outside closed
shells in their normal state (as we have found that copper has a 4s elec-
tron). As a result, all these elements have similar spectra.

3. The Periodic Table of the Elements. — In Table XXI-1 we list
the elements in order of their atomic numbers, which are given in addition
to their symbols. The atoms in the table are arranged in rows and
columns in such a way as to exhibit their periodic properties. The
diagonal lines are drawn in such a way as to connect atoms of similar

SBC. 3]

ATOMS AND THE PERIODIC TABLE

345

properties. Table XXI-2 gives the electron configuration of the normal
states of the elements. Table XXI-3 gives the ionization potentials
of the various electrons in the lighter atoms, in units of Rhc, the Rydberg
energy, mentioned in the preceding section. And Table XXI-4 gives
the radii of the various orbits, as computed by wave mechanics. We
can now use these tables, and other information, to give a brief discussion
of the properties of the elements, particularly in regard to their ability
to form ions, which is fundamental in studying thoir chemical behavior.
In this regard, we must remember that low ionization potentials cor-
respond to easily removed electrons, high ionization potentials to tightly
held electrons.

*
TABLE XXI-1. — THE PERIODIC TABLE OK THE ELEMENTS

Li 3 Na 11

Be 4 Mgl2

B 6 Al 13

C 6 Si 14

7 P 15

S 16
Cl 17

N
O 8
F 9

NelO-

-A 18

In a way, the most distinctive elements are the inert gases, He, Ne,
A, Kr, and Xe. As we see from Table XXI-2, all the electrons in these
elements are in closed shells. They form no chemical compounds and
have high ionization potentials, showing very small tendency to form
ions. The reason for their stability is fundamentally the fact that
electrons in closed shells are difficult to remove, as is shown by an exami-
nation of ionization potentials throughout Table XXI-3. That is, closed
shells form a very stable structure, difficult to deform in such a way as to
form ions or molecules. To see why the inert gases appear where they
do in the periodic table, we may imagine that we are building up the
periodic table, adding more and more electrons to a nucleus. The first

346

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

TABLE XXI-2. — ELECTRON CONFIGURATIONS OP THE ELEMENTS, NORMAL STATES

3-9

6<*

«,

Pel

SBC. 3]

ATOMS AND THE PERIODIC TABLE

347

TABLE XXI-2. — ELECTRON CONFIGURATIONS OF THE ELEMENTS, NORMAL STATES. —

(Continued)

AP.

5rf

La -

—

348 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

TABLE XXI-3. — IONIZATION POTENTIALS OP THE LIGHTER ELEMENTS, IN RYDBERGS

^
N

3«

4«

5«

1.00

1.81

4.80

0.40

(9.3)

0.69

(15.2)

1.29

0.61

(22.3)

1.51

0.83

(31.1)

1.91

1.07

(41.5)

2.10

1.00

(53.0)

2.87

1.37

(66.1)

3.56

1.59

(80.9)

(5.10)

2.79

0.38

96.0

(6.96)

3 7

0.56

114.8

(9.05)

5.3

0 78

0.44

135.4

(11.5)

7.2

1.10

0.60

157.8

(14.2)

9.4

(1 40)

(0 65)

181.9

(17.2)

11.9

1.48

0.76

207.9

(20.4)

14.8

1.81

0.96

235.7

(23.9)

(18.2)

2 14

1.15

265.6

(27 8)

21.5

(2.6)

1.2

0.32

297 4

(31 9)

25.5

(3.1)

1.9

0 45

331.2

(36 2)

30 0

(3 6)

2.7

0.54

0 .50

365.8

(41.0)

33.6

(4 2)

2.6

0.51

0.50

402 7

(46.0)

37 9

(4.8)

3.0

0.50

0.52

441 1

(51.2)

42.3

(5.4)

3.1

0 61

0 50

481 9

(56.7)

47 4

(6.7)

3.8

0.68

0.55

523.9

62 5

52 2

6.9

4.1

0.60

0.58

568.1

(68.5)

57.7

7.6

4.7

0.63

0 66

614.1

74 8

63 2

8.2

5.4

(0.68)

0 64

661 6

81 0

fi8.9

8 9

5.7

0.77

0.57

711 7

88.4

75.4

10.1

6.7

1.26

0.69

765.6

(96.0)

84.1

12.4

8.8

1.8

0.87

0.44

817.6

(104.0)

89.3

13.4

9.5

3.2

1.39

0.60

874.0

112.6

97.4

14.9

10.3

3.0

(1-0)

0.74

932.0

(121.9)

108.4

16.7

11.6

3.9

(1.7)

0 70

992.6

(131.5)

117.8

19.1

13.6

5.4

(1.9)

0.87

(1055)

(141.6)

(127.2)

(21.4)

(15.4)

(6.8)

(2.1)

1.03

1119.4

152.0

137.2

(23.7)

17.4

(8.3)

(2.3)

1.46

0.31

1186.0

162.9

147.6

26.2

19.6

9.7

2.5

(2.1)

0 42

1256.1

175.8

159 9

30.3

23.3

13 0

4.7

2.9

0 48

0 49

1325.7

186.6

170.0

31.8

24.4

13.3

3.8

2.1

0 53

0 51

1398.5

198.9

181.7

34.7

26.9

15.2

4.3

2 5

(0.5)

(0 5)

1473.4

211.3

193.7

37.5

29.2

17.1

5.1

2.9

(0.5)

0.54

The ionization potentials tabulated represent in each case the least energy required to remove the
electron in question from the atom, in units of the Rydberg energy Rhc (13.54 electron volts). Data for
optical ionization are taken from Bacher and Goudsmit, "Atomic Energy States,'1 McGraw-Hill Book
Company, Inc., 1932. Those for x-ray ionization are from Siegbahn, " Spektroskopie der Rontgen-
strahlen," Springer. Intermediate figures are interpolated. Interpolated or estimated values are
given in parentheses.

SEC. 3]

ATOMS AND THE PERIODIC TABLE

349

TABLE XXI-4. — RADII OF ELECTRONIC ORBITS IN THE LIGHTER ELEMENTS

(Angstrom units)

0.53

0 30

0.20

1.50

0 143

1.19

0.112

0.88

0.85

0.090

0.67

0.66

0.080

0.56

0 53

0.069

0.48

0.45

0 061

0 41

0.38

0.055

0.37

0.32

0.050

0.32

0 28

1.55

0 046

0.30

0.25

1.32

0 042

0.27

0.23

1.16

1 21

0 040

0 24

0.21

0.98

1.06

0.037

0.23

0.19

0.88

0.92

0 035

0 21

0.18

0.78

0.82

0 032

0 20

0.16

0.72

0.75

0 031

0.19

0,155

0.66

0.67

0 029

0 18

0 145

0.60

0.63

2.20

0.028

0 16

0 133

0.55

0.58

2 03

0.026

0.16

0.127

0 52

0 54

0.61

1.80

0.025

0.150

0 122

0.48

0.50

0.55

1.66

0 024

0 143

0.117

0.46

0.47

0.49

1 52

0.023

0.138

0.112

0.43

0.44

0.45

1.41

0.022

0.133

0 106

0.40

0.41

0.42

1 31

0.021

0 127

0 101

0 39

0.39

0 39

1.22

0.020

0.122

0.096

0.37

0.36

1.14

0.019

0.117

0.090

0.35

0.36

0.34

1.07

0.019

0 112

0.085

0.34

0.32

1.03

0.018

0.106

0.081

0.32

0.30

0.97

0.017

0.103

0 078

0.31

0.28

0.92

1.13

0.017

0 100

0.076

0.30

0 27

0.88

1.06

0.016

0.097

0.073

0.29

0.25

0.84

1.01

0.016

0.095

0.071

0.28

0,28

0 24

0.81

0.95

0.015

0.092

0.069

0 27

0.27

0.23

0.76

0 90

0.015

0.090

0.067

0.25

0 22

0.74

0 86

The radii tabulated represent the distance from the nucleus at which the radial charge density (the
charge contained in a shell of unit thickness) is a maximum. They are computed from calculations of
Hartree, in various papers in "Proceedings of the Royal Society," and elsewhere. Since only a few
atoms have been computed, most of the values tabulated are interpolated. The interpolation should
be fairly accurate for the inner electrons of an atom, but unfortunately is quite inaccurate for the outer
electrons, so that these values should not be taken as exact.

350 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXI

two electrons go into the K shell, resulting in He, a stable structure with
just two electrons. The next electrons go into the L shell, with its sub-
groups of 2s and 2p electrons. These electrons are grouped together, for
they are not very different from hydrogenlike electrons in their energy,
and as we see from Eq. (2.1), the energy of a hydrogen wave function
depends only on n, not on I, so that the 2s and 2p have the same energy
in this case. For the real wave functions, as we see from Fig. XXI-1, for
example, the energies of 2s and 2p are not very different from each other.
The L shell can hold two 2s and six 2p electrons, a total of eight, and is
completed at neon, again a stable structure, with two electrons in its K
shell, eight in its L shell. The next electrons must go into the still larger
M shell. Of its three subgroups, 3s, 3p, and 3d, the 3s and 3p, with
2 + 6=8 electrons, have about the same energy, while the 3d is defi-
nitely more loosely bound. Thus the 3s and 3p electrons are completed
with argon, with two K, eight L, and eight M electrons, again a stable
structure and an inert gas. It is in this way that the periodicity with
period of eight, which is such a feature of the lighter elements, is brought
about. After argon, the order of adding electrons is somewhat peculiar.
The next electrons added, in potassium and calcium, go into 4s states,
which for those elements have a lower energy than tho 3d. But with
scandium, the element beyond calcium, the order of levels changes, the
3d becoming somewhat more tightly bound. In all the elements from
scandium to copper the new electrons are being added to the 3d level,
the normal state having either one or two 4s electrons. For all these
elements, the 4s and 3d electrons have so nearly the same energy that the
configurations with no 4s electrons, with one, and with two, have approxi-
mately the same energy, so that there are many energy levels near the
normal state. At copper, the 3d shell is filled, so that the M shell con-
tains its full number of 2 + 6 + 10 = 18 electrons, and as we have seen
from our earlier discussion, there is one 4s electron. The elements
following copper add more and more 4s and 4p electrons, until the group
of eight 4s and 4p's is filled, at krypton. This is again a stable configura-
tion. After this, very much (he same sort of situation is repeated in the
atoms from rubidium and strontium through silver, which is similar to
copper, and then through xenon, which has a complete M shell, and
complete 4s, 4p, 4d, 5,s, and 5p shells. Following this, the two electrons
added in caesium and barium go into the 6s shell, but then, instead of
the next electrons going into the 5d shell as we might expect by analogy
with the two preceding groups of the periodic table, they go into the 4/
shell, which at that point becomes the more tightly bound one. The
fourteen elements in which the 4/ is being filled up are the rare earths, a
group of extraordinarily similar elements differing only in the number of
4/ electrons, which have such small orbits and are so deeply buried inside

Sfic. 3] ATOMS AND THE PERIODIC TABLE 351

the atom that they have almost no effect on chemical properties. After
finishing the rare earths, the 5d shell is filled, in the elements from hafnium
to platinum, and the next element, gold, is similar to copper and silver.
Then the 6s and 6p shells are completed, leading to the heaviest inert
gas, radium emanation, and finally the 7s electrons are added in radium,
with presumably the Qd or 5/ in the remaining elements of the periodic
table.

Now that we have surveyed the elements, we are in position to under-
stand why some atoms tend to form positive ions, some negative. The
general rule is simple: atoms tend to gain or lose electrons enough so
that the remaining electrons will have a stable structure, like one of the
inert gases, or some other atom containing completed groups or subgroups
of electrons. The reason is plain from Table XXI-3, at- least as far as the
formation of positive ions is concerned: the electrons outside closed shells
have much smaller ionization potentials than those in closed shells and
are removed by a much smaller amount of energy. Thus the alkali
metals, lithium, sodium, potassium, rubidium, and caesium, each have
one easily removed electron outside an inert gas shell, and this electron
is often lost in chemical processes, resulting in a positive ion. The
alkaline earths, beryllium, magnesium, calcium, strontium, and barium,
similarly have two easily removable electrons and become doubly charged
positive ions. Boron and aluminum lose three electrons. Occasionally
carbon and silicon lose four and nitrogen five, but these processes are
certainly very rare and perhaps never occur. The electrons become too
strongly bound as the shell fills up for them to be removed in any ordinary
chemical process. But oxygen sometimes gains two electrons to form the
stable neon structure, and fluorine often gains one, forming doubly and
singly charged negative ions respectively. Similarly chlorine, bromine,
and iodine often gain one electron, and possibly sulphur occasionally
gains two. In the elements beyond potassium, the situation is somewhat
different. Potassium and calcium tend to lose one and two electrons
apiece, to simulate the argon structure. But the next group of elements,
from scandium through nickel, ordinarily called the iron group, tend to
lose only two or three electrons apiece, rather than losing enough to form
a closed shell. Nickel contains a completed K, L, and M shell and is a
rather stable structure itself, though not so much so as an inert gas; and
the next few elements tend to lose electrons enough to have the nickel
structure. Thus copper tends to lose one, zinc two, gallium three, and
germanium four electrons, being analogous to a certain extent to sodium,
magnesium, aluminum, and silicon. Coming to the end of this row,
selenium tends to gain two electrons like oxygen and sulphur, and bromine
to gain one. Similar situations are met in the remaining groups of the
periodic table.

CHAPTER XXII
INTERATOMIC AND INTERMOLECULAR FORCES

One of the most fundamental problems of chemical physics is the study
of the forces between atoms arid molecules. We have seen in many
preceding chapters that these forces are essential to the explanation of
equations of state, specific heats, the equilibrium of phases, chemical
equilibrium, and in fact all the problems we have taken up. The exact
evaluation of these forces from atomic theory is one of the most difficult
branches of quantum theory and wave mechanics. The general prin-
ciples on which the evaluation is based, however, are relatively simple,
and in this chapter we shall learn what these general principles are, and
see at least qualitatively the sort of results they lead to.

There is one general point of view regarding interatomic forces which
is worth keeping constantly in mind. Our problem is really one of the
simultaneous motion of the nuclei and electrons of the atomic or molecular
system. But the electrons are very much lighter than the nuclei and
move very much faster. Thus it forms a very good approximation to
assume first that the nuclei are at rest, with the electrons moving around
them. We then find the energy of the whole system as a function of the
positions of the nuclei. If this energy changes when a particular nucleus
is moved, we conclude that there is a force on that nucleus, such that the
force times the displacement equals the work done, or change of energy.
This force can be used in discussing the motion of the nucleus, studying
its translational or vibrational motion, as we have had occasion to do in
previous chapters. Our fundamental problem, then, is to find how the
energy of a system of atoms changes as the positions of the nuclei are
changed. In other words, we must solve the problem of the motion of
the electrons around the nuclei, assuming they are fixed in definite posi-
tions. The forces between electrons arc essentially electrostatic; there
are also magnetic forces, but they are ordinarily small enough so that they
can practically be neglected. Then the problem of solving for the motion
of the electrons can be separated into several parts. It is a little difficult
to know where to start the discussion, for there is a sort of circular type
of argument involved. Suppose we start by knowing how the electrons
move. Then we can find their electrical charge distribution, and from
that we can find the electrostatic field at any point of space. But this
field is what determines the forces acting on the electrons. And those
forces must lead to motions of the electrons which are just the ones we

352

SEC. 1] INTERATOMIC AND INTERMOLECULAR FORCES 353

started with. An electric field of this type, leading to motions of the
electrons such that the electrons themselves, together with the nuclei,
can produce the original field, is sometimes called a self-consistent field.

As a first attempt to solve the problem, let us assume that each atom
is a rigid structure consisting of a nucleus and a swarm of electrons sur-
rounding it, not affected by tho presence of neighboring atoms. This
leads to a problem in pure electrostatics: the energy of the whole system,
as a function of the positions of the nuclei, is simply the electrostatic
energy of interaction between the charges of the various atoms. This
electrostatic energy is sometimes called the Coulomb energy, since it
follows directly from Coulomb's law stating that the force between two
charges equals the product of the charges divided by the square of the
distance between. This first approximation, however, is far from ade-
quate, for really tho electrons of each atom will be displaced by the
electric fields of neighboring atoms. We shall later, then, have to study
this deformation of the atoms and to find tho forces bo t ween tho distorted
atoms.

1. The Electrostatic Interactions between Rigid Atoms or Molecules
at Large Distances. — In this section, we are to find the forces between
two atoms or ions or molecules, assuming that each can be represented
by a rigid, undistortod distribution of charge. The discussion of these
electrostatic, or Coulomb, forces is conveniently divided into two parts.
First, we find the electric field of the first charge distribution at all points
of space; then, we find the force on the second charge distribution in this
field. By fundamental principles of electrostatics, tho force on tho
second distribution exerted by the first is equal and opposite to the force
on the first exerted by the second, if we make a corresponding calculation
of the field exerted by the second on the first. Let us first consider, then,
the field of a charge distribution consisting of a number of charges e»,
located at points with coordinates xlj y^ zt-. Rather than find the field, it
is more convenient to compute the potential, the sum of the terms et-/rt-
for the charges, where r» is the distance from the charge to the point x, y, z
where the potential is being found. That is, rt- is the length of a vector
whose components are x — xlt y — ?/t, z — zt, so that we have

-~*7)2 +72/~^7)2~ + '(z - *J2, (1.1)

and the potential is

(1.2)

There is a very important way of expanding the potential (1.2), in
case we wish its value at points far from the center of the charge distribu-

354

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

tion. This is the case which we wish in investigating the forces between
two atoms or molecules at a considerable distance from each other. Let
us assume, then, that all the charges ei arc located near a point which we
may choose to be the origin, so that all the z/s, y/s, and z/s are small, and
let us assume that the point #, y, z where we are finding the potential is
far off, so that r = \/x2 + y2 + z2 is large. Then we can expand the
potential in power series in zt-, y^ and z,, regarded as small quantities.
We have

• • (1.3)
= z, = 0.

(1.4)
(1.5)

The derivatives of (l/r») are to be computed when xt =
But from Eq. (1.1) we have

When xt = 0, this becomes

rt — x

L(i\

dx\rj

1 x

On the other hand, we have

d_ /A ^ _ 1 £
ax\r/ r2 r '

0.6)

Thus, comparing Eqs. (1.5) and (1.6), we can rewrite Eq. (1.3) as
d

(1.7)

From Eq. (1.7), the potential of the charge distribution depends on the
quantities Set-, I>elxiy Seti/t, SetZi, and higher terms such as Setz?, etc.,
which we have not written.

The quantity 2e* is simply the total charge of the distribution, and
the first term of Eq. (1.7) is the potential of the total charge at a distance
r. This term, then, is just what we should have if the total charge were
concentrated at the origin. The next three terms can be grouped

SBC. 1] INTERATOMIC AND INTERMOLECULAR FORCES 355

together. The quantities Ze0<9 2^-, Se^< form the three components
of a vector, which is known as the dipole moment of the distribution. A
dipole is a pair of equal charges, say of charge +q and —qy separated by
a distance d. For the sake of argument let the charges be located along
the x axis, at d/2 and — d/2. Then the three quantities above would be

qd> e*' ~ fi* = °- That is> the dipol°

moment is equal in this case to the product of the charge and the distance

--Line of force

FIG. XXII-1. —Lines of foice and equipotentials of a <lip<il«»

of separation, and it points along the axis of the dipole, from tin* negative
to the positive end. We now see that as far as the terms written in
Eq. (1.7) are concerned, any two distributions with the same net charge
and the same dipole moment will have the same potential. In the
particular case mentioned above, the potential, using Eqa. (1.6) and
(1.7), is (qd/r*)(x/r). Here x/r is simply the cosine of the angle between
the radius r and the x axis, a factor depending on the direction but not
the magnitude of r. As far as magnitude is concerned, then, the poten-
tial decreases as 1/r2, in contrast to the potential of a point charge,
which falls off as 1/r. Thus at large distances the potential of a dipole is
unimportant compared to that of a point charge. In Fig. XXII-1, we

356 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

show the equipotentials of this field of a dipole and the lines of force,
which are at right angles to the equipotentials, and indicate the direction
of the force on a point charge in the field. The lines of force are those
familiar from magnetostatics, from the problem of the magnetic field of a
bar magnet, which can be approximated by a magnetic dipole.

In addition to the terms of the expression (1.7), there are terms
involving higher powers of x^ y^ and Zi, and at tho same time higher
derivatives of 1 /r, so that these terms fall off more rapidly with increasing
distance. Tho next terms after the ones written, quadratic in the #/s,
and with a potential falling off as 1/r3, are called quadrupole terms, the
corresponding moment being called a quadrupolo moment. We shall
not have occasion to use quadrupole moments and hence shall not develop
their theory here, though sometimes they are important.

Now that we have found the nature of the potential of a charge
distribution, we can ask what sorts of eases we are likely to find with real
atoms, molecules, and ions. First we consider a neutral atom. Since
there are as many electrons as arc necessary to cancel the positive nuclear
charge, 2et- is zero, and there is no term in the potential falling off as
1/r. The atom at any instant will have a dipole moment, however; the
electrons move rapidly from place to place, and it is unlikely that they
would be so arranged at a given instant that the dipole moment was
exactly zero, though it is likely to be small, since some electrons will be
on one side of the nucleus, others on the other side. On account of the
motion of the electrons, this dipole moment will be constantly fluctuating
in magnitude and direction. It is not hard to show by wave mechanics
that its average value must be zero. Now, for most purposes, we care
only about the average dipole moment, for ordinarily we are interested
only in the time average force between atoms or molecules, and the
fluctuations will average to zero. Thus, generally, we treat the dipole
moment of the atom as being zero. Only two important cases come up in
which the fluctuating dipole moment is of importance. One does not
concern interatomic forces at all: it is the problem of radiation. In
Chap. XIX, Sec. 3, we have mentioned that an oscillating electric charge
in classical theory radiates energy in the form of electromagnetic waves.
It turns out that the oscillating dipole moment which we have mentioned
here is closely connected with the radiation of light in the quantum
theory. The frequencies present in its oscillatory motion are those
emitted, according to Bohr's frequency condition, and there is a close
relation between the amplitude of any definite frequency in the oscillation
and the intensity of the corresponding frequency of radiation. The
other application of the fluctuating dipole moment comes in the calcula-
tion of Van der Waals forces, which we shall consider later. It appears
that the fluctuating external field resulting from the fluctuating dipole

SBC. 1] INTERATOMIC AND INTERMOLECULAR FORCES 357

moment can produce displacements of charge in neighboring atoms, in
phase with the fluctuations. The force exerted by the fluctuating field
on this displaced charge does not average to zero, on account of the1
phase relations, but instead results in a net attraction between the
molecules, which as we shall see is the Van der Waals attraction. It is
rather natural from what we have said that it is possible in wave mechan-
ics to give a formula for the Van der Waals force between two atoms
which depends on the probabilities of the various optical transitions
which the atoms can make, though we shall not be able to state this
formula since it involves too much application of quantum theory.

As far as the time average is concerned, we have seen that an atom
has no field coming from its net charge or from its dipolc moment. As a
matter of fact, in most important cases, an atom has no not field at all at
external points. The reason is that atoms, at least in the special ease
where all their electrons are in closed shells, as in inert gas atoms, arc
spherically symmetrical in their average charge distributions. This can
be proved from wave mechanics and is a property of closed shells. But
it is a familiar theorem of electrostatics that a spherically symmetrical
charge distribution has a field just equal to that which it would have if
all its charge were placed at the center. Thus a neutral atom has no
external field. The reason is seen easily from Eq. (1.7). Each term of
this expression after the first one depends on the angle between the radius
vector and the axes. This is plain for the terms written, where we have
seen that they vary as the cosines of the angles between the radius and
the x, y, and z axes respectively, but it proves to be true also for the
remaining terms. But a spherically symmetrical charge distribution
must obviously have a spherically symmetrical potential, so that all
these terms depending on angles must be zero. In other words, a spher-
ically symmetrical distribution, like an atom, not only has no average
dipole moment, but has no average quadrupole moment or moment of
any higher order.

Next after a neutral atom, we may consider a positive or negative
ion of a single atom, such as Na+, Ba++, or Cl~. As we have seen in the
preceding chapter, such an ion always has the configuration of an inert
gas, and hence is always spherically symmetrical on the average. Thufc
an ion has no dipole or higher moments, and its potential and field are
just as if its whole charge were concentrated at the nucleus. As a next
more complicated example, we take a molecule, charged or uncharged,
formed from two or more atoms or ions. If the molecule is charged,
forming an ion like NH4+, OH~, NO3~, S04 , etc., then in the first place
it has a term in the potential varying as 1/r, determined by the total
charge on the ion. In addition to this, the ion or molecule may have a
dipole moment. When we come to discussing specific ions and molecules,

358 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

in later chapters, we shall see which ones have dipole moments, which
do not; in general, for there to be a dipole moment different from zero, the
ion or molecule must be unsymmetrical in some way, with positive charge
localized on one side, negative on the other. The ions NHU*, NOa ", and
864 — , as we shall see, prove to be very symmetrical, and have no dipole
moment, while OH~ has a dipole moment, the negative charge being at
the oxygen end, the positive at the hydrogen end. Similarly there are
some unsymmetrical neutral molecules which have dipole moments.
An example is HC1, in which the H end tends to be positive, the Cl
negative. The dipole moments have been measured in many of these
cases and are generally found to be much less than one would suppose
from a crude ionic picture. One might at first think, for instance, that
HC1 was made of a H+ and a Cl" ion, joined together without distortion,
so that each was spherically symmetrical. Then the resulting charge
distribution would have a field at external points like a unit positive
charge at the position of the hydrogen nucleus, and a unit negative charge
at the chlorine nucleus, and the dipole moment would equal the product
of the electronic charge and the internuclear distance. The measured
dipole moment is only a small fraction of this, showing that there has
been a large distortion of the electronic distribution in the process of
forming the molecule. This is the sort of distortion that we must take
up in a later section.

We see, then, that at a considerable distance a single atom has no
electric field, an ion consisting of a single charged atom has a field like a
point charge concentrated at its center, and a molecule or ion consisting
of several atoms or ions may have in addition a dipole moment, with its
accompanying field, as well as having the field of its net charge, if it is an
ion. In addition, the molecule or molecular ion may have quadrupole
and higher moments. The effect of these is usually small compared to
the others, but in the case of an uncharged molecule with no dipole
moment, the quadrupole term would be the first important one in the
expansion of the field. Having found the nature of the field of an atom
or ion, our next problem is to find the forces exerted by this field on
another atom or ion, always assuming both to be rigid charge distribu-
tions. Fundamentally, the problem is very simple: the force exerted by
the field of one atom or ion on each element of charge of the second atom
or ion is simply the product of the field intensity and the charge, by
definition, and we need merely treat the problem as one in statics, adding
the forces vectorially to find the total force on the atom or ion, and
adding their moments about the center of gravity to get the resultant
moment or torque.

Thus, let the potential of the electrostatic field be <£, and let the
field strength have components Ex, Ev, E», where by well-known methods

SBC. 1] INTERATOMIC AND INTERMOLECULAR FORCES 359

of electrostatics the field is the negative of the derivative of <£ with respect
to displacements along the axes, so that the product of the force and the
displacement gives the work done, or the negative of the change of
potential. That is,

The components Ex, Ev, and Ez will be functions of position. Now
assume that the ion or molecule on which the force acts has charges e%
at positions #t, y^ zt-, whore the origin is chosen to be at the center of
gravity of the ion or molecule. Then, for example, the x component of
total force on the ion or molecule is the sum of the x components of force
on all its charges, and if we write Ex at an arbitrary position by the Taylor
expansion

Ex(xyz) = EM + ^ + d-^y + *jfz + • • - , (1.9)

whoro EX(Q) and the derivatives arc all to be computed at the origin, we
have the following expression for the total x component of force on the ion
or molecule :

.+ • • • (UO)

The first term in Eq. (1.10) represents the field at the center of gravity,
times the total charge. This term of course is zero if the molecule is
uncharged. The next three terms depend on the dipole moment arid the
rate of change of field strength with position. Their interpretation is
very simple. If the field strength is independent of position, the electro-
static forces on the two poles of a dipole will be equal and opposite and
will give no net force on the dipole as a whole. But if the field is stronger
at one end than at the other, one charge will be pulled more strongly in
one direction than the other one is in the other direction, and there will
be a net pull on the dipole as a whole. This pull depends on the orienta-
tion of the dipole with respect to the external field; if the dipole is reversed
in direction, so that each component of its dipole moment changes sign,
the dipole terms in the force expression (1.10) change sign, showing
that the force is reversed.

A dipole is acted on not only by a force, but also by a torque, in an
external field, and this torque is proportional to the field strength rather
than to its rate of change with position. The x component of this torque,
regarded as a vector, is seen to be

Mx = (Se#<)^ - (2«t*)«* (1-11)

360 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

showing that the torque is proportional to the dipole moment, the external
field, and the sine of the angle between them. To see this in an elemen-
tary way, we show in Fig. XXII-2 a simple dipole consisting of charges
±q at a distance of separation d, the line of centers making an angle
0 with the external field. Then we see that the field exerts a force of
magnitude qE on each charge, with a lever arm d/2 sin 0, so that the
torque exerted on each charge is q(d/2)E sin 0, and the total torque is
qdE sin 0, where qd is the dipole moment. The potential energy asso-
ciated with this torque is

Potential energy = — qdE cos 0, (1.12)

having a minimum when the dipole points along the direction of the
electric field. That is, the field tends to swing the dipole around so that
it is parallel to the field.

External field

Fio. XXII-2. — Illustrating the torque on a dipole in an external force field.

We are now in position to understand the forces between rigid ions or
molecules at a distance from each other. With two ions, of course the
largest term in the force is the Coulomb attraction or repulsion between
the net changes of the ions — an attraction if the ions have unlike charges,
repulsion if they have like charges. If the molecules are uncharged,
however, the largest term in the interaction comes from dipole-dipole
interaction. Each dipole is acted on by a torque in the field of the other,
and if we look into the situation, we see that these torques are in such
directions as to tend to place the dipoles parallel to each other, the posi-
tive end of one being closest to the negative end of the other. Also, the
dipoles exert a net force on each other, an attraction or repulsion depend-
ing on orientation. If the orientation is that of minimum potential
energy, with the positive end of one dipole opposite the negative end of
the other, the net force will be an attraction, for the attraction between
the close unlike charges will more than balance the repulsion of the more
distant like charges. We may anticipate by mentioning the sort of appli-
cation we shall make later to the force between two dipole molecules in a
gao. In this case, both dipoles will be rotating. If they rotated uni-
formly, they would be pointing in one direction just as often as in the

SEC. 2] INTERATOMIC AND 1NTERMOLECULAR FORCES 361

opposite direction, so that the net force between them would cancel, since
as we have seen this net force changes sign when the dipole reverses its
direction. But they will really not rotate uniformly, for there are
torques acting on them, tending to keep them in a parallel position.
These torques will result in a potential energy term, of the nature of
Eq. (1.12), between them, and if we insert this term into the Maxwell-
Boltzmann distribution law, we shall find that the dipoles will be oriented
in the position of minimum potential energy more often than in other
positions. Thus, on the average, the attractions between the dipoles will
outweigh the repulsions, and the net effect of dipole-dipole interaction is
an intermolecular attraction.

As two molecules or ions get closer and closer together, higher terms
in the expansion of the potential and the force become important, and we
must consider quadrupoles and higher multipoles. The whole expansion
in inverse powers of r, and direct powers of the xl's, becomes badly
convergent when the molocules approach to within a distance com-
parable to their own dimensions. When the charge distributions of two
atoms or molecules really begin to overlap each other, the situation
becomes entirely different and must be handled by different methods.
We shall take up in the next section the electrostatic or Coulomb inter-
action of two rigid charge distributions representing atoms, when they
approach so closely as to overlap.

2. The Electrostatic or Coulomb Interactions between Overlapping
Rigid Atoms. — We have seen in the preceding section that two neutral
spherically symmetrical atoms exert no forces on each other, so long as
they do not overlap and so long as we can treat their charge distributions
as being rigid, so that they do not distort each other. Once they overlap,
however, this conclusion no longer holds. A rigid neutral atom consists
of a positive nucleus surrounded by a spherical negative distribution of
charge, just great enough to balance the charge on the nucleus. Such a
distribution exerts no electrostatic force at outside points. At points
within the charge distribution, however, it does exert a force, determined
by a well-known rule of electrostatics : the electrostatic field at any point
in a spherical distribution of charge is found by constructing a sphere,
with center at the center of symmetry, passing through the point where
the field is to be found. The charge within the sphere is imagined to be
concentrated at the center, that outside the sphere is neglected. Then
the electric field is that computed by the inverse square law from the
charge concentrated at the center of the sphere, disregarding the outside
charge. At a point outside the atom, this reduces to the same result
already quoted: the net charge within the sphere is zero, so that there is no
field. But as we get closer to the nucleus, we penetrate into the negative
charge distribution, so that some of the negative charge lies outside

362 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

our sphere and is to be disregarded. The charge within the sphere, which
we are to imagine concentrated at the center, then has a net positive value,
becoming equal to the charge on the nucleus as the sphere grows smaller
and smaller. Thus the electric field approaches that of the positive
nuclear charge, in the limit of small distances. It is correct to consider
that the electrons shield the nucleus at external points, counteracting
its field, but this shielding effect decreases as we penetrate the electron
shells. At a given distance from the nucleus, the field is like that of a
charge of (Z — Z0) units concentrated at the center, where Z is the
nuclear charge, Z0 a shielding constant representing the amount of elec-
tronic charge within the sphere, a quantity which decreases from Z to
zero as we go from great distances in to the nucleus. This shielding
constant Z0 is essentially the same as that introduced in Eq. (2.2), Chap.

(a) (b) (c)

FIG. XXII-3. — Schematic representation of the overlapping of two atoms. The
points represent the nuclei, the circles the regions occupied most densely by negative
electronic charge distributions.

XXI, where we were considering the effect of electronic shielding on the
motion of one of the electrons of the atom.

It is now easy, at least in principle, to find the interatomic forces
between two rigid atoms whose charge distributions penetrate each other.
We simply find the force on each clement of the charge of one atom,
exerted on it by the field of the other. It is a difficult problem of integra-
tion actually to compute this force, but the results are qualitatively
simple. Suppose the distributions have only penetrated slightly, as
shown in (a), Fig. XXII-3. Then some negative charge of each atom is
within the distribution of the other, and hence is attracted by part of the
nuclear charge. Thus the first effect of overlapping is an attraction
between the atoms. This effect begins to be counteracted in the case
(6) in the figure, however, when the nucleus of one atom begins to pene-
trate the charge distribution of the other. For the nucleus will be
repelled, not attracted, by the other nucleus. Finally, in case (c), where
the atoms practically coincide, there will be great repulsion. For the
nuclei will repel very strongly, being very close together, and exposed to
all of each others' field, while the electronic distribution of each atom is
still at a considerable average distance from the nucleus of the other, and
hence is not very strongly attracted. Furthermore, part of the electronic

SEC. 3]

INTERATOMIC AND INTERMOLECULAR FORCES

363

distribution of each atom is on one side of the nucleus of the other, part
on the other side, so that the forces on it almost cancel, and exactly cancel
when the two atoms exactly coincide. The net effect of the Coulomb
forces, then, is a potential energy curve similar to Fig. XXII-4, with a
minimum, corresponding to a position of equilibrium, and an infinitely
high potential energy as the nuclei are brought into contact.

It might be thought at first sight that the curve of Fig. XXII-4, which
surely has close resemblance to the Morse curve of Fig. IX-1, would give
an adequate explanation of the interatomic forces that hold atoms
together into molecules. On closer examination, however, this proves
not to be the case. The attractions of Fig. XXII-4 are not nearly strong
enough to account for molecular binding, and the distances of separation

Internuclecir distance

Fio. XXII-4. — Schematic representation of the electrostatic 01 Coulomb energy of inter-
action of two overlapping rigid atoms, as shown in Fig. XXII-3.

are not what they should be. The reason is that our assumption of rigid
atoms breaks down completely when the electronic distributions begin to
overlap. The charge distribution becomes greatly distorted, and this
must be taken into account in calculating the energy and forces. We
shall now pass on to a discussion of this distortion, first taking up the
effect of polarization, the type of distortion met at large distances of
separation, then the effect that is usually called exchange interaction,
which is important when atoms overlap.

3. Polarization and Interatomic Forces. — An atom or molecule
in a uniform external electric field is polarized; that is, it acquires an
induced dipole moment, parallel and proportional to the field. This is
the phenomenon so well known from electrostatics, when a charge brought
near a conductor induces a charge of opposite sign on the near-by parts of
the conductor. It is not so marked with an insulator as with a conductor,
but it always occurs. It is illustrated in Fig. XXII-5, where we show
simply a sphere of matter in an external field, with induced positive
charge on the right hand part of it and negative charge on the left. We
see that the induced charge is similar to a dipole. The induced dipole
moment, as we have stated, is proportional to the external field, and the

364 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

constant of proportionality is called the polarizability and denoted by
a, so that the induced dipole moment is a times the external field. The
polarization can be brought about in either of two ways. In the first
place, the electrons can be displaced in the direction opposite to the field,
so that the electronic distribution is distorted or deformed. This is
the only mechanism for polarization with atoms or symmetrical molecules.
With dipole molecules, however, an additional form of polarization is
possible; on account of the Maxwell-Boltzmann distribution, the dipoles
can be oriented in such a way as to have a net dipole moment along the
direction of the field. We can easily compute the net dipole moment on
account of this effect.

Let the permanent dipole moment of a molecule be /*. Then, as wo
saw in Eq. (1.12), its potential energy in a field E is — pE cos 0. Its
component of dipole moment along the direction of the field is p cos 9. If

all orientations were equally likely,
the average component along the
field would be zero. But on ac-
count of the Maxwoll-Boltzmann
distribution, the probability of
finding the axis of the dipole in
unit solid angle about a given orien-

Fio. XXII-5. — Induced polarization of a ... . , . , , - — r»r-

sphere in an external field. tatlOn IS proportional to 6 kT .

Thus to find the mean moment, we

multiply /x cos 0 by the Boltzmann factor above and integrate over solid
angles. The solid angle contained between 6 and 8 + dd, or the fraction
of the surface of unit sphere between these angles, is 2?r sin 0 dB. Thus
we have

fV cos de kT 2?r sin 0 dd
Mean dipole moment = — ^ cos 0 (3.1)

C*i~Mr~2v sin 0 dS
Jo

The integrals in Eq. (3.1) can be evaluated at once by substituting
cos 0 = x, — sin 6 dd = dx, and introducing the abbreviation nE/kT = y,
from which at once we have

/i
_1XeXV dX ^ ey _J_ e-y ^ I

_ _ '..I ~ ^ - ^V »'

J-l

The function (3.2) is shown as a function of y, which is proportional to
the external field, in Fig. XXII-6. We see from the figure that at low
fields the mean dipole moment is proportional to the field, but at high

SEC. 3]

INTERATOMIC AND INTERMOLECULAR FORCES

365

fields there is a saturation, all the dipoles being parallel to the external
field. It is only at low fields, where there is proportionality, that we can
speak of a polarizability. To get the value of the polarizability, we
should find the initial slope of the function (3.2), We easily find, by
expanding in power series in yy that for small y the function (3.2) can be
approximated by the straight line y/3. Thus, remembering the definition
of y, we have the dipolc moment at low fields equal to »2E/3kT and the
polarizability equal to n*/3kT, decreasing with increasing temperature as
we should expect.

FIG. XX1I-6. — Function — - — _?/ — as function of y, giving mean dipole moment

arising from the rotation of dipole molecules, as a function of ?/ = nE/kT, where /z is the
dipole moment, K the field strength, according to Eq. (3.2).

If the part of the polarizability resulting from the electronic distortion
is ao, we then have

= ao 3kf

(3.3)

as the total polarizability of a molecule. This quantity can be found
experimentally, on account of its connection with the dielectric constant.
The molecular theory of dielectrics shows that a substance having N
molecules in a volume V, each having a polarizability a, will have a dielec-
tric constant equal to1

(3.4)

If we measure the dielectric constant as a function of temperature, then,
it should be a linear function of 1/T, and from the constants of the curve
we can find both the electronic polarizability ao and the dipole moment

1 See P. Debye, "Polare Molekeln," Hirzel, 1929, for further discussion of dielec-
tric constants.

366 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXH

M. It is in this way that the dipole moments of a great many molecules
have been determined.

We now understand the polarization of a molecule in an external field.
Next, we must ask about intermolccular forces resulting from this polari-
zation. If a molecule has an induced dipole moment equal to aE in the
direction of the field, then we see from Eq. (1.10) that it is acted on by a
force equal to aE(dE/dx)j if the x axis is chosen in the direction of the
external field and of the dipole. This can be written

showing that the force pulls the molecule in the direction in which the
magnitude of the field increases most rapidly. In the type of problem wo
are considering, this means a force of attraction toward the other mole-
cule. The attraction will depend on the nature of the field of the other
molecule. Thus suppose we are considering the force between a polar-
izable molecule and an ion. The field of the ion varies as 1/r2, so that E~
varies as 1/r4, its derivative with respect to x (which in this case is r)
is proportional to 1/r5, so that the force varies inversely as the fifth power
of the distance, and the potential energy inversely as the fourth power.
The commoner ease, however, is that in which a dipole molecule produces
a field that polarizes its neighbors, resulting in an attraction. A molecule
of dipole moment // produces a field proportional to ju'/r3 at another
molecule. This field results, according to Eq. (3.5), in a force on the
second molecule proportional to 3a/z'2/r7. The attractive energy will
then be proportional to 1/r6. This depends on the angle between the
dipole moment of the first molecule and the line of centers of the two, and
calculation shows that the average over all directions is given by1

Energy =-|^- (3.6)

Equation (3.6) is essentially the formula for Van der Waals attrac-
tions between molecules. There are two distinct cases: the attractions
between molecules with or without permanent dipole moments. First
let us consider molecules without permanent moments. Even in this
case, we have seen in Sec. 1 that the molecule will have a fluctuating
dipole moment, which will average to zero. Nevertheless it can polarize
another molecule instantaneously, producing an attraction, and the net
result, averaged over the fluctuations, will be an attraction given by Eq.
(3.6), where a is the electronic polarizability of a molecule, and /*'2 the

1See for instance Slater and Frank, " Introduction to Theoretical Physics," Sec.
301, McGraw-Hill Book Company, Inc., and Pauling and Wilson, "Introduction to
Quantum Mechanics/' Sec. 47, McGraw-Hill Book Company, Inc., 1935.

SEC. 4] INTERATOMIC AND INTERMOLECULAR FORCES 367

mean square dipole moment. It is significant that it is the mean square
moment that is concerned in the attraction, and this mean square is
different from zero even when the mean moment vanishes. This attrac-
tion is the typical Van der Waals attraction, a force whose potential is
inversely proportional to the sixth power of the interatomic distance and
is independent of temperature. On the other hand, if we are considering
the forces between two dipole molecules, there will be two changes. First,
the moan square dipole moment /*'2 of the first molecule will now include
two terms : the one coming from electronic fluctuations, which we have
already considered, and the one coming from the fixed average dipole
moment. Thus, in the first place, the external field will be greater than
before. Then, in the second place, the polarizability will be given by
Eq. (3.3), including both the electronic polarizability, and that coming
from orientation of the fixed dipoles. In many cases the second term
proves to bo several timos as large as the first, but it decreases with
increasing temporaturo. Thus we may expect the Van der Waals attrac-
tion between molecules with permanent dipoles to be several times as
large as that between similar molecules without the permanent dipoles,
and furthermore we may expect the Van der Waals force to decrease with
temperature in the dipole case. Both these predictions prove to be
borne out by experiment, as we shall see in a later chapter where we
take up Van der Waals forces numerically for a variety of molecules.
4. Exchange Interactions between Atoms and Molecules. — In the
preceding section, we have found how an atom or molecule is distorted
in a uniform electric field, and have used this to discuss its distortion in
the field of another atom or molecule. This is clearly an approximation,
for the field of a molecule is not uniform, though it approaches uniformity
at great distances. When two atoms or molecules approach closely, this
type of approximation, using merely an induced dipole, becomes very
inaccurate. We must consider in this section how the charge distribu-
tions of two atoms or molecules are really distorted when they come so
close together that they touch or overlap. We shall find that there
are two very different types of behavior possible: there may be forces of
attraction between the atoms or molecules, tending to bind them together,
or there may be forces of repulsion. The first case is that of valence
binding, the second the case of the type of repulsion considered in Van der
Waals constant 6. We shall first take up the simplest case, the interac-
tion of two atoms, then shall pass on to molecular interactions. The
problems we are now meeting are among the most complicated ones of the
quantum theory, and we shall make no attempt at all to treat the ana-
lytical background of the theory. When one studies that background,
one finds that there are two different approximate methods of calcu-
lation used in wave mechanics, sometimes called the Heitler-London

368

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

method and the method of molecular orbitals respectively. These differ,
not in their fundamentals, but in the precise nature of the analytical steps
used. For that reason, we shall not discuss them or their differences.
We shall rather try to focus our attention on the fundamental physical
processes behind the intermolecular actions and shall find that we can
understand them in terms of fundamental principles, without reference to
exact methods of calculation.

Let us, then, bogin with the simplest possible problem, the interaction
of two atoms. Unless they overlap, the only force between them will be
the Van der Waals attraction, coming from the polarization of each
atom by the fluctuating dipole moment of the other. This type of inter-
action persists even when the valence electrons of the atoms do overlap.

Po ten fior/ energy

(a) (b)

FIG. XXII-7. — Potential energy of an electron 111 (a) the central field representing an
atom; (b) the field representing two overlapping atoms.

It is simple to describe in words. When the valence electron of one atom
is at a given point, the valence electron of the other atom, which of course
is repelled by it, tends to stay away from it. Thus the electrons do not
approach each other so closely as if the repulsion were absent, and as a
result the interaction energy between them is lower than if we neglected
this type of interaction. This effect, tending to keep the electrons in the
pair of atoms, or the molecule, away from each other, is sometimes called
a correlation effect, since it depends on a correlation between the motions
of the two electrons. It results in a lowering of energy or an increase of
the strength of the binding between the atoms. As we see, it is the direct
extrapolation of the Van der Waals attraction to the case of close approach
of the atoms. But as we shall soon see, it is by no means the principal
part of the interatomic force but rather forms a fairly small correction
term.

In discussing the Coulomb interactions between overlapping atoms,
in Sec. 2, we saw that as the electronic charge of one atom begins to
penetrate into the electron shells of the other, it becomes attracted to
the nucleus of the other atom. That is, it is in a region of lower potential
energy than it otherwise would be. This is illustrated in Fig. XXII-7.
There, in part (a), we show the potential energy of an electron at different

SBC. 4] INTERATOMIC AND INTERMOLECULAR FORCES 369

points within an atom, taking account of the decrease of shielding as we
go closer to the nucleus. In (fc), the potential curves of two overlapping
atoms are superposed, the resulting curve showing the potential energy of
an electron in the combined molecule. We see that the potential energy
is lower in the region where they overlap than it would be in the cor-
responding part of either atom separately. This change of potential
energy would mean that the electrons from both atoms were attracted
to this region of overlapping, so that the tendency would be for extra
electronic charge to concentrate itself in this region. This in turn would
decrease the total energy, for it would mean the concentration of more
charge in a region of lower potential energy. Thus this would result in
an added attraction between the atoms. We could regard this as an
increase in the magnitude of the Coulomb attraction, if we chose, giving a
much deeper minimum to the potential energy curve than one finds in
Fig. XXII-4. This effect is different from the Van dor Waals attraction
or correlation effect, in that it depends on the average field rather than on
the fluctuating field, distorting or polarizing the charge distribution and
hence decreasing the energy.

Even this effect, however, is not the whole story. For we have
forgotten one essential fact : the electrons obey the Fermi-Dirac statistics
or the Pauli exclusion principle. Let us state the Fcrmi-Dirac statistics
in a very simple form, remembering the existence of the electron spin.
We set up a molecular, or rather an electronic, phase space, in which
the coordinates of each electron are given by a point. This phase space
has cells of volume A3 in it. Then the Fermi-Dirac statistics states that
no complexion of the system is possible in which more than one electron,
of a given spin orientation, is in the same cell. Since two orientations of
the spin are possible, as we saw in Chap. XXI, Sec. 2, this means that at
most we can have two electrons in a cell, one of each spin. There is, in
other words, a maximum possible density of electrons in phase space.
We can translate this statement into one regarding the maximum density
of electrons in ordinary coordinate space : for a given range of momenta,
there is a maximum possible density of electrons in coordinate space. If
we wish to pack more electrons into a region of coordinate space, they
must have a different momentum from those already there. If the elec-
trons already present have a low kinetic energy, this means that any
additional electrons must have higher kinetic energy, and hence higher
total energy, in order not to have the same momenta as those already
present. The exclusion principle in this form can be used immediately
to discuss the electronic interactions when two atoms begin to overlap.

We have already talked about the lowering of the potential energy
between two atoms when they begin to overlap, and have illustrated it in
Fig. XXII-7. And we have stated that there is a tendency for electrons

370 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

to be concentrated in this region, seeking a lower potential energy and
hence decreasing the energy of the whole system. But this would involve
an increasing density of electrons in the region between the atoms, and
from what we have just said, this might well make difficulties with the
Fermi-Dirac statistics. We now meet very different situations, depend-
ing on whether the electrons in the two atoms already are in closed shells,
or not. First lot us assume that they are in closed shells. This can be
interpreted in terms of the Fermi-Dirac statistics: we choose our cells in
the electronic phase space to coincide with the stationary states of the
one-electron problem of an electron in a central field. When we state that
the electrons are all in closed shells, we mean that there are two electrons
each, one of each spin, in the lowest cells or stationary states, while the
higher stationary states are empty. All the region of space occupied
by electrons of either atom, then, is filled to such a density with electrons
that no additional charge can enter the region, without having a higher
kinetic energy and total energy, than the charge already there. Now
let us see how this affects the situation. If two atoms begin to overlap, we
can certainly not have the charge shifting into the region between the
atoms, for this would involve an increase of charge density. We can not
even have the charge of the two atoms overlap without redistribution, for
the same reason. For this would involve such a large density of charge
between the atoms that the electrons would have to increase their kinetic
energy, and hence their total energy, considerably. The thing that hap-
pens is that some of the charge actually shifts away from the region
between the atoms, to the far sides of the nuclei. This involves some
increase of kinetic energy, for the electrons must increase the charge
density everywhere except between the atoms, to make up for the decrease
of density there; it involves increase of potential energy, since electrons
are moving away from the region of low potential energy between the
nuclei. Nevertheless, it does not mean so much increase of total energy
as if the electrons piled up between the atoms. The net effect of this
redistribution of charge is an increase of energy and hence a repulsion
between the atoms.

The effect of which we have spoken, giving a repulsion between atoms
all of whose electrons are in closed shells, is the origin of the impenetra-
bility of atoms and of the correction to the perfect gas law made by Van
der Waals constant b. It is illustrated in Fig. XXII-8, where in (a) we
show the charge distribution surrounding two repelling atoms, by means
of contour lines. It is clear that the charge has been forced out of the
region between atoms, by the effect of the exclusion principle. As a
matter of fact, we can get similar effects, even if the outer electrons are
not all in closed shells. Thus, consider two atoms of hydrogen, or of an
alkali metal, each with one valence electron. Suppose the electrons of

SEC. 4]

INTERATOMIC AND 1NTERMOLECULAR FOHCKti

371

both atoms have their spins oriented in the same way. Then as far as
electrons of that spin are concerned, the shells are filled, although they are
empty of electrons of the opposite spin. When the electrons begin to
overlap, then, there is the same difficulty about an increase of charge

(a) (b)

FIG. XXII-8. — Electronic charge density represented by contours*, foi (a) two repelling
atoms, (b) two attracting atoms.

density that there would be with really closed shells, the charge distribu-
tion becomes distorted as in Fig. XXII-8 (a) and the atoms repel. There
really is a mode of interaction of two hydrogen or two alkaline atoms
leading to a repulsion of this sort. We show it graphically in Fig. XXII-9
(a). In this figure, we plot the total energy of a pair of hydrogen atoms,

Interatomic
distance

FIG. XXII-9. — Interaction energy of two hydrogen atoms, as a function of the distance of
separation, (a) repulsive state, (b) attractive state, with molecular formation.

as a function of the distance of separation. At large distances, the
energy is negative, on account of the Van der Waals attraction. But at
small distances, when the atoms overlap, the curve (a), indicating the
case where the spins of the two electrons are parallel, gives a repulsion.
There is a minimum in this curve, leading to a position of equilibrium
between the atoms, but it corresponds to a large interatomic distance and

372 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX11

very small binding energy, and does not correspond in any way to the
binding of the two atoms to form a molecule.

The cases which we have taken up so far are those of repulsion between
atoms, resulting from the exclusion principle. But this principle can also
operate, in a somewhat less obvious way, to give an attraction between
atoms which is even greater than the other forms of attraction previously
considered. In the case of two hydrogen or alkaline atoms, which we
have just discussed, it may be that the two electrons will have opposite
spins. Then the exclusion principle does not operate directly; there is no
obstacle in the way of the electrons from the two atoms overlapping, as
much as they please, since their spins are different. The electrons are
then free to pile up in the region between the nuclei, as we have mentioned
before, thus decreasing the potential energy and leading to a binding
between the atoms. But the exclusion principle, or Fermi-Dirac statis-
tics, comes into wave mechanics in a more fundamental way than we have
indicated, a way that can hardly be explained at all without going much
further into wave mechanics than we can. The effect in this caso is
something of the following sort: if two electrons have the same spin, as we
have seen, the exclusion principle prevents them from being in the same
cell of phase space. But if they have opposite spins, it operates just in
the opposite direction, making electrons tend to occupy the same cell
rather than different cells. A hint as to why this should be so can be
found from Chap. V, Sec. 6, where we discussed the Einstein-Bose statis-
tics. We remember that that form of statistics resulted merely from
the identity of molecules, and that it could be qualitatively described as a
tendency for molecules to stick together or condense, as if there wore
attractive forces between them. If the exclusion principle is added to
the principle of identity, the Fermi-Dirac statistics results, leading to
something like a repulsion between electrons. Now two electrons of the
same spin must obey the exclusion principle, and we have already seen
the effect of the resulting repulsion. But two electrons of the opposite
spin no longer need to satisfy an exclusion principle, as far as their coor-
dinates and momenta are concerned, and yet they still satisfy a principle
of identity. Hence, in essence, they obey the Einstein-Bose statistics
and tend to crowd together as closely as possible.

The effect of which we have just spoken is often called exchange, on
account of a feature in the analytical calculation connected with it, in
which the essential term relates to an exchange of electrons between the
two atoms. The exchange interaction results in an additional piling up of
electrons in the region of lowest potential energy, between the nuclei, and
hence in an addition to the strength of binding. This is indicated in
Fig. XXII-8 (6), where the charge distribution for this case of attraction
is shown, and in Fig. XXII-9 (6), where we show the potential energy of

SBC. 4] INTERATOMIC AND INTERMOLECULAR FORCES 373

this type of interaction. The type of attraction which we have in this
case is what is generally called homopolar valence attraction, the word
"homopolar" meaning that the two atoms in question have the same
polarity, rather than one being electropositive and one electronegative, as
in attraction between a positive and a negative ion. We can now see the
various features involved in it: there is a tendency, from pure electro-
statics, for the outer or valence electrons of the atoms to concentrate in
the region between the atoms; if the electrons have the same spin this is
prevented by the exclusion principle, but if they have opposite spin the
exclusion principle indirectly operates to enhance the concentration of
charge; since this charge is concentrated in a region of low potential
energy, the net result is a binding of the two atoms together; and finally,
the correlation effect, analogous to the Van der Waals attraction, tends to
keep the electrons out of each others' way, still further decreasing the
energy. All those effects result in interatomic attraction at moderate
distances of separation. As the distance is further decreased, however,
two effects tend to produce repulsion. First, there is the simple effect of
the Coulomb forces, discussed in Sec. 2: as the nucleus of one atom
begins to penetrate inside the charge distribution of the other, the nuclei
begin to repel each other, the repulsion growing stronger as they approach.
But secondly, all atoms but the simplest ones have inner, closed shells.
It is only the outer electrons, which are not in closed shells, that can
take part in valence attraction. When the atoms come close enough
together so that the closed shells begin to overlap each other, the same
sort of repulsion produced by the exclusion principle sets in which we
have previously mentioned. In many cases this repulsion of closed shells
is the major feature in producing the rise of the potential energy curve
which is shown in Fig. XXII-9 (fe).

We have spoken of the effect of the exclusion principle and exchange
on the forces between two atoms. Now we shall see what happens with
more than two atoms. For the sake of illustration, let two hydrogen
atoms be bound into a molecule by the action of exchange forces, and
then ask what happens when a third hydrogen atom approaches the
molecule. The two electrons of the first two atoms have cooperated to
form the valence bond between them. One of these electrons has one
spin, the other the other, and they have shifted into the region between
the atoms, filling that region up to approximately the maximum density
allowed by the exclusion principle, with electrons of both spins. Such a
pair of electrons, shared between two atoms, is the picture furnished
by the quantum theory for the electron-pair bond. But now imagine a
third atom, with its electron, to approach. The spin of this third elec-
tron is bound to be the same as that of one of the two electrons of the
electron-pair bond. Thus this third atom cannot enter into the attrac-

374 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

tive, exchange type of interaction with either of the atoms bound into
the molecule. Instead, the exclusion principle will force its electron
away from the other atoms, and there will be repulsion between them.
This effect is what is called saturation of valence: two electrons, and no
more, can enter into an electron-pair bond, and once such a bond is
formed, the electrons concerned in it can form no more bonds. It might
have been, however, that one of the atoms concerned in the original
molecule had two valence electrons which it could share. In that case,
one of them would be used up in forming a valence bond with the first
hydrogen atom, leaving the other one to form a bond with another hydro-
gen atom. In this way, we can have atoms capable of forming two or
more valence bonds. If all the possible bonds are already formed, how-
over, the structure will act as if all its electrons were in closed shells, and
any additional atom or molecule approaching it will be repelled.

There is just one case in which the formation of a bond by one electron
does not prevent the same electron from taking part in another bond.
This is the case of the metallic bond. If two sodium atoms approach,
with opposite spins, their electrons form a valence bond botwoon
them. But if a third sodium atom approaches, it turns out that the first
valence bond becomes partly broken, so that the valence electrons of the
first two atoms spend only part of their time in the region between those
two atoms and have part of their time left over to form bonds with the
new atom. As more and more atoms are added, this effect can continue,
the electrons forming bonds which are essentially homopolar in nature
but spread out throughout the whole group of atoms, holding them all
together. The reason why metallic atoms behave in this way, while
nonmetallic ones do not, is probably largely the fact that the valence
electrons of metals are less tightly held than in non metals, as we can see in
Table XXI-3, giving the ionization potentials of the elements, and con-
sequently their orbits are larger, as we see in Table XXI-4. Then the
orbit of one atom overlaps other atoms more than in a nonmetal, and it is
easier for a number of neighbors to share in valence attraction. The
conspicuous features of the metallic bond are two: first, there is no satura-
tion of valence, so that any number of atoms can be held together, forming
a crystal rather than a finite molecule; and secondly, the electron density
is not so great as the maximum allowed by the exclusion principle. This
second fact makes it possible for electrons to move from point to point
without significant increase in their energy, whereas in a molecule held
by valence bonds this is impossible, since the electron would have to
acquire enough extra energy to rise to a higher quantum state, or more
excited cell in phase space, before it could enter regions which already had
their maximum density of electrons. This free motion of the electrons
in a metal is what leads to its electrical conductivity and its typical
metallic properties.

SBC. 5] INTERATOMIC AND 1NTERMOLECULAR FORCES 375

6. Types of Chemical Substances. — In the preceding sections we
have made a survey of the types of interatomic and intermolecular
forces. Now we can correlate these by making a brief catalog of the
important types of chemical substances and the sorts of forces found in
each case. Following this, the remaining chapters of this book will take
up each type of substance in more detail, making both qualitative and
quantitative use of the laws of interatomic and intermolecular forces
found in each particular case, and deriving the physical properties of the
substances as far as possible from the laws of force.

The simplest class of anhst,fl.m*Psr in ^ way, is the class of inorpjamn
salts, or ionic crystals. J?amiliar examples are NaCl, NaNO3, BaS04.
These substances are definitely constructed of ions, as Na+, Ba++, Cl~,
NOa~, and SO4 — . The ions act on each 'other by Coulomb attraction
or repulsion, and an ion of one sign is always surrounded more closely by
neighbors of the opposite sign than by others of the same sign, so that the
attractions outweigh the repulsions and the structure holds together.
As the distance between ions decreases and they begin to touch each other,
they repel, on account of the exclusion principle; for as we have seen, the
electrons in an ion form closed shells and the repulsion between them is
the typical repulsion of closed shells. Thus a stable structure can be
formed, the electrostatic attractions balancing the repulsions. There is
nothing of the nature of saturation of valence; even though two ions, as
Na+ and Cl~, may be bound into a molecule, the electrostatic effect of
their charges extends far away from them, since the molecule NaCl has a
strong dipole moment. Thus further ions can be attracted, and the
tendency is to form an extended structure. If the atoms in this structure
are arranged regularly, it is a crystalline solid, the most characteristic
form for ionic substances. On the other hand, at higher temperatures,
where there is more irregularity, the solid can liquefy, and at high enough
temperatures it vaporizes. It is only in the vapor state that we can say
that the substance is composed of molecules; in the liquid or solid, each
ion is surrounded at equal distances by a number of ions of the opposite
sign and there is no tendency of the ions to pair off to form molecules.
The electrostatic attractions met in ionic crystals are large, so that the
materials are held together strongly with rather high melting and boiling
points. We shall see in a later chapter that we can account for their
properties satisfactorily by quite simple mathematical methods.

The ionic substances are those in which an clectroposjtfivfi ftlftmftnf,, »«
Na. and an electronegative one? as Cl, are held together by electrostatic
forces. The other types of substances are compounds of two electro-
negative elements or of two electropositive ones. The first group, made
of electronegative elements, is the group of homopolar compounds.
These are held together by homopolar valence bonds, coming from shared
electron pairs, as we have described in the previous section. Since the

376 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXII

bonds have the property of saturation, we ordinarily have molecular
formation in such cases, the molecules being of quite definitely deter-
mined size. Two molecules attract each other only by Van der Waals
forces, and if they are brought too closely into contact, they repel each
other on account of their closed shells. The simpler compounds of this
type, like H2, 62, CO, etc., are the materials most familiar as gases. The
Van der Waals forces holding the molecules together are rather weak,
while the interatomic forces holding the atoms together in the molecule
are very strong. Thus a relatively low temperature suffices to pull the
molecules apart from each other, or to vaporize the liquid or solid, while
an extremely high temperature is necessary to dissociate the molecule or
pull its atoms apart. As we go to more complicated cases of compounds
held together by homopolar valence, we first meet the organic compounds.
They arise on account of the tendency of the carbon atom to hold four
other atoms by valence bonds, using its four available electrons in this
way. The carbon atoms, on account of their many valences, can form
chains and still have other valence bonds available for fastening other
atoms to them; in this way the great variety of organic compounds is
built up. Another very important class of materials held together by
valence bonds contains the minerals, silicates, and glasses, and various
refractory materials, like diamond, carborundum, or SiC, and so on. In
these cases, the valence bonds hold the atoms into endless chains, sheets,
or three-dimensional structures, so that the materials form crystals in
their most characteristic form and are held together very tightly. These
materials have very high melting and boiling points, since all the bonds
between atoms are the very strong valence bonds, rather than the weak
Van der Waal$ f^rnes as jp the iflolecniar substances. _

Finally we have the metals, made entirely of electropositive atoms.
We *"""» "»-»" that these atoms are hold together by the metallic bond,
similar to the valence hnnHa, W. without, the propqrftes of saturation.
Thus the metals, like the ionic crystals and the silicates, tend to form
indefinitely large structures, crystals or liquids, arid tend to have high
melting and boiling points and great mechanical strength. We have
already seen that the same peculiarity of the metallic bond which prevents
the saturation of valence, and hence which makes crystal formation
possible, also leads to metallic conduction or the existence of free electrons.

With this brief summary, we have covered most of the important
types of materials. In the next chapter we shall make a detailed study of
ionic substances, and in succeeding chapters of the various other sorts of
materials, interpreting their properties in terms of interatomic and inter-
molecular forces.

CHAPTP:R xxm

IONIC CRYSTALS

The ionic compounds practically all form crystalline solids at ordinary
temperatures, and it is in this form that they have been most extensively
studied. The reason for this crystal formation has been seen in the pre-
ceding chapter: ionic attractions are long range forces, falling off only as
the inverse square of the distance, so that more and more ions tend to be
attracted to a minute crystal which has started to form, and the crystal
grows to large size, held together by electrostatic attractions throughout
its whole volume. Above the melting point, the liquids of course are
held together by the same type of force, and in the gaseous phase undoubt-
edly the same sort of thing occurs, one molecule tending to attract others,
>so that presumably there is a strong tendency for the formation of double
and multiple molecules in the gas. Unfortunately, however, the liquids
and gases of ionic materials have been greatly neglected experimentally,
so that there is almost no empirical information with which to compare
any theoretical deductions. For this reason, we shall be concerned in
this chapter entirely with the solid, crystalline phase* of these substances,
but venture to express the suggestion that further experimental study of
the liquid and gaseous phases would be very desirable. In considering
the solids, the first step naturally is to examine the geometrical arrange-
ment of the atoms or the crystal structure. Then we shall go on to the
forces between ions, and the mechanical and thermal properties, first
taking up the case of the behavior at the absolute zero, then studying
temperature vibrations and thermal effects.

1. Structure of Simple Binary Ionic Compounds. — To be electrically
neutral, every ionic compound must contain some positive and some
negative ions. The simplest ones are those binary compounds that
contain one positive and one negative ion. Obviously the positive ion
must have lost the same number of electrons that the negative one has
gained. Thus monovalent metals form such compounds with monova-
lent bases, divalent with divalent, and so on. In other words, this group
includes compounds of Li+, Na+, K+ , Rb+, Cs+, Cu+, Ag+, and Au+ with
F-, C1-, Br-, I-; of Be++ Mg++ Ca++, Sr++ Ba++, Zn++, Cd++ Hg++
with 0— S— Se— , Te~; of B+++ A1+++ Ga+++, In+++ T1+++ with

N — , P , As , Sb , Bi . Even this list contains some negative

ions which ordinarily do not really exist, as As , Sb , Bi . Never-
theless some of the compounds in question are found. One can formally

377

378 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

go even further and set up such compounds as carborundum, SiC, as if

it were made of 81++++ and C , or C++++ and Si . But by the

time an atom has gained or lost so many electrons, it turns out that the
ionic description does not really apply very well, and we shall see later
that such compounds are really better described as homopolar compounds.
The positive ions which we have listed above by no means exhaust the
list of possibilities, on account of the fact that most of the elements of
the iron, palladium, and platinum groups are found as divalent, or tri-
valent positive ions, and consequently form binary oxides, sulphides,
selenides, and tellurides in their divalent form, and nitrides and phos-
phides in their trivalent form.

In addition to these single ions,
there .are a few complex positive
ions, which are so much like metallic
ions that they can conveniently be
grouped with them. Best known of
these is the ammonium ion, NH4+,
and somewhat loss familiar is the
analogous phosphonium ion PH4+.
Tic* The ammonium ion has ten elec-
trons: seven from nitrogen, one
from each of the four hydrogens,
FIG. xxiil-i.— The sodium chloride iess one because it is a positive ion.
structure. n^ , . ., , . , ,,

lhat is, it has just the same num-
ber as neon, or as Na+. Similarly the phosphonium ion has eighteen,
like argon, or K+. The hydrogen ions are presumably imbedded in the
distribution of negative charge, in a symmetrical tetrahedral arrangement,
and do not greatly affect the structure, so that these ions act surprisingly
like metallic ions. We shall group these compounds with the binary ones,
though really the positive ion is complex.

Most of the binary ionic compounds occur in one of four structures,
and by far the commonest is the sodium chloride structure. This is
shown in Fig. XXIII-1. It can be described as a simple cubic lattice, in
which alternate positions are occupied by the positive and negative ions.
Each ion thus has six nearest neighbors of the opposite sign, and the
electrostatic attraction between the ion and its oppositely charged neigh-
bors holds the crystal together. This illustrates a principle which we
mentioned in the preceding chapter and which obviously must hold for
stability in an ionic crystal, namely that for stability each ion must be
surrounded by as many ions of the opposite charge as possible, and the
nearest ions of the same sign must be as far away as possible.

The second common structure is the caesium chloride structure. In
this structure, ions of one type are located at the corners of a cubic lattice

SEC'. I)

IONIC CRYSTALS

379

and ions of the other sign at the centers of the cubes. In this structure
each ion has eight neighbors of the opposite sign. This structure is shown
in Fig. XXIII-2.

The third and fourth structures are sometimes called the zmchlcndr
and the wurtzite structures, on account of two forms of ZnS. The zinc-
blende structure is also often called the diamond structure, since it is
found in diamond and some other crystals. The fundamental features
of both structures are similar: each ion is tetrahedrally surrounded by four
ions of the opposite sign, as in Fig. XXIII-3 (a). There arc a number
of ways of joining such tetrahedra to form regular crystals, however. The
diamond, or zincblende, lattice is the simplest of these. In the first
place, tetrahedra like Fig. XXIII-3 (a), can be formed into sheets like

FIG. XXIII-2. — Tho caesium chloride structure.

Fig. XXIII-3 (6) and (c), the latter looking straight down along the
vertical leg of the tetrahedron, so that the ion at the center and that
directly above it coincide in the figure. Then in the diamond structure,
the next layer up is just like that shown, but shifted along so that atoms
like a, 6, c coincide with a', 6', c'. The wurtzite structure, on the other
hand, has the next layer looking just like the one shown in (6), as far as its
projection is concerned, but actually being a mirror image of it in a hori-
zontal plane. When examined in three dimensions, the wurtzite struc-
ture proves to be less symmetrical than the diamond structure, in that the
vertical direction stands out as a special axis in the crystal. For this
reason, the length of the vertical distance between ions in the wurtzite
structure does not have to equal the other three distances, while in the
diamond structure all distances must be the same. The diamond, or
zincblende, structure is shown in Fig. XXIII-3 (d), and the wurtzite
structure in (e). In addition to these two, a number of other structures
built of tetrahedra can exist, in which the two types of planes, the one

380

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

(e)

FIG. XXIII-3.— The zincblende and
wurtzite structures, (a) one ion tetra-
hedrally surrounded by four others. (6)
and (c) sheets formed from such tetrahedra,
in perspective and plan, (d) the zincblende
structure (which is identical with the dia-
mond structure if all atoms are alike).
(e) the wurtzite structure.

shown in (6) and its mirror image,
are arranged in various regular ways
through the crystal. Several of
these structures are found, for in-
stance, in different forms of carbo-
rundum, SiC.

We shall now list in Table XXIII-
1 some of the binary ionic compounds
crystallizing in the four structures
just discussed. Under each struc-
ture, we shall arrange the compounds
according to their valence, starting
with monovalent substances. We
note that some compounds exist in
several polymorphic forms, as ZnS
in both zincblende and wurtzite
structures. We tabulate not merely
the substances crystallizing in each
form, but also several quantities
characterizing each substance.
First we give the distance between
nearest neighbors, r0, in angstroms.
This is necessarily the distance be-
tween an ion of either sign and the
nearest ions of the opposite sign, of
which there are six in the sodium
chloride structure, eight in the
caesium chloride, and four in the
zincblende and wurtzite structures.
This is followed by a calculated
value of r0, discussed in Sec. 2.
Finally we give the melting point
where this is known. This is simply
an indication of the tightness of bind-
ing, since a strongly bound material
has a high melting point. We notice
that the divalent substances con-
sistently have a much higher melting
point than the monovalent ones.
This is a result of the tighter binding.
Since each ion in a divalent crystal
has twice as great a charge as in a
similar monovalent one, the inter-

SBC. 1J IONIC CRYSTALS 381

TABLE XX 1 1 1-1. — LATTICE SPACINGS OF IONIC CRYSTALS

Material

ro observed,
angstroms

ro calculated

Melting point, °C.

Sodium Chloride Structure

LiF

2.01
2.57
2.75
3.00

2.31
2.81
2.98
3.23

2.67
3.14
3.29
3 . 53

2.82
3.27
3.43
3.66

3.00

3.27
3.45
3 62

2.46
2.77
2.88

2.10
2.60
2.73

2.40
2 84
2.96
2.97

2.58
3.01
3.12
3.33

2.77
3.19
3.30
3.50

2.10
2.60
2.75
3.00

2.35
2.85
3.00
3.25

2.65
3.15
3.30
3 . 55

2.80
3.30
3.45
3.70

3 05

3.25
3.40
3 65

2.30
2.80
2 1)5

2.15
2 60
2.70

2.40
2.85
2.95
3 15

2.60
3.05
3.15
3.35

2.75
3.20
3.30
3.50

870
613
547
446

980
804
755
651

880
776
730
773

760
715
682
642

684

435
455
13 1

2800
2572

2430
882

1923

Lid

LiBr

Lil

NaF. .
NaCl
NaBr. .
Nal.

KC1.
KBr.
KI

RbF .
RbCl .
RbBr.
Rbl. .. .

CaF

NH4C1 . .
NH4Br .
NHJ

AgF. .. .
AgCl
AgBr .

MgO
MgS.. .
MgSe . .

CaO
CaS

CaSe
CaTe

SrO. .

SrS .. .
SrSe ...

SrTe

BaO
BaS
BaSe

BaTe

Caesium Chloride Structure

CsCl ....
CaBr

3.56
3.71
3.95

3.34

3.55
3.70
3.95

3.25

046
636
621

Csl . .
NH4C1

382 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

TABLE XXIII-1. — LATTICE SPACINGS OF IONIC CRYSTALS. — (Continued)

Material

ro observed,
angstroms

ro calculated

Melting point, °C.

Caesium Chloiide Stiucture — (Continued)

NH4Br
NHJ.

TICi .
TlBr

CuCl
CuBr
CuT

Bi-8

BeSe

BeTe

ZnS

ZnSe

ZnTe

CdS

CdSe

CdTe

HgS

HgSe

HgTe

3 51
3 78

3.33
3.44

3.40
3.65

Zincblende Structure

2.34
2.46
2 62

2 10
2 18
2 43

2 35
2 45
2 04

2 52
2. 02
2 80

2 53
2.62
2.79

2.30
2.45
2 70

2 10
2 20
2 40

2 35
2 45
2.65

2 50
2 60
2.80

2 50
2.60
2 80

430
460

422
504
005

1800

1750

Wurtzite Stiucture (Fir.sl distance is that to neighbor along axis, second to three neighbo

in same layer)

NH4F . «,

2 63, 2.76

2 75

BeO

1.64, 1.60

1 65

2570

ZnO

1.94,2.04

1.90

ZnS

2.36, 2.36

2.35

1850

CdS . .

2.52, 2.56

2.50

1750

CdSe

2.63, 2.64

2.60

Data regarding crystal stiucture, here and in other tables in this book, are taken from the "Struk-
turbencht," issued as a supplement to the Zettachrift fdr Krixtallographie in several volumes from 1931
onward This is the (standard reference for crystal structure data.

ionic forces are four times as great, with correspondingly large latent heats
of fusion. Since the entropy of fusion is not very different for a divalent
crystal from what it is for a monovalent one, this means that the melting
point of a divalent crystal must be several times as large as for a mono-
valent one, as Table XXIII-1 shows it to be.

2. Ionic Radii. — The first question that we naturally ask about the
crystals is, what determines the lattice spacings? Examination of the

SBC. 2] IONIC CRYSTALS 383

experimental values shows that these spacings are very nearly additive;
that is, we can assign values to radii of the various ions, such that the
sums of the radii give the distance between the corresponding ions of the
crystals. A possible set of ionic radii is given in Table XXIII-2. Using

TABLE XXIII-2.— IONIC RADII
(Angstroms)

Be++
0.20

0 80

Mg+H

Na+

F-

o—

0.70

1 05

1 30

1.45

Ca++

K+

ci-

Zn+4

Cu+

0.95

1 35

1.80

1.90

0.45

0.50

Sr++

Rb+

Br~

So—

Cd-* +

Ag+

1 15

I 50

1.95

2 00

0.60

1.00

Ba^+

Csf

I-

Te—

Hg++

1 30

1 75

2.20

2 20

0.60

1 45

these radii, which as will be observed are smoothed off to 0 or 5 in the last
place, the values r0 calculated of Table XXIII-1 are computed. The
agreement between calculated and observed lattice spacing is surely
rather remarkable. There have been a good many discussions seeking to
show the reasons for the small errors in the table, the departures from
additivity. In particular, there are good reasons for thinking that
different radii should be used for the sodium chloride structure, where
every ion is surrounded by six neighbors, from those used in the zinc-
blende and wurtzite structures, where there are four neighbors. But the
comparative success of the calculations of Table XXIII-1, where both
types of structure are discussed by means of the same radii, shows that
these corrections are comparatively unimportant and the significant fact-
is that the agreement is as good as it is.

There is one point in which our assumed valuqp for ionic radii are not
uniquely determined. The observed interionic spacings can determine
only the sum of the radii for a positive and negative ion of the same
valency. We can add any constant to all the radii of the positive ions
of one valency, subtract the same constant from the radii of the negative
ions, without changing the computed results. On the other hand, the
difference between the assumed radii of two ions of the same sign cannot
be changed without destroying agreement with experiment. We have
chosen this arbitrary additive constant in such a way as to make the
positive ion, as K+, an appropriate amount smaller than the negative
ion, such as Cl~, which contains the same> number of electrons, considering
the tendency for extra negative electrons to be repelled from an atom,
making negative ions large, positive ones small. Our estimate is probably

384

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

not reliable, however, and the absolute values should not be taken seri-
ously as representing in any way the real radii of the ions. In particular,
the ion of Be++ is pretty certainly not so small as its extremely small
radius, 0.20 A, would suggest.

It is interesting to compare the radii of Table XXIII-2 with those of
Table XXI-4. It will be seen that though they are of the same order of
magnitude, the ionic radii of Table XXIII-2 are several times the radii
of the corresponding orbits in Table XXI-4. Remembering that Table
XXI-4 gives the radius of maximum density in the shell, we see that
the region occupied by electrons, given by the ionic radii, is several times
the sphere whose radius is the radius of maximum density. This is surely
a natural situation, since the charge density falls off rather slowly from its
maximum.

In Table XXIII-2, we can interpolate between the monovalent posi-
tive and negative ions to get radii for the inert gas atoms. Thus we find
approximately the following: No 1.1 A, A 1.5 A, Kr 1.7 A, Xe 1.95 A. It
is interesting to compute the volumes of the inert gas atoms which we
should get in this way, and compare with the volumes which we find for
them from the constant b of Van der Waals' equation. For neon, for
instance, the volume from Table XXIII-2 would be

^r(l.l)3 = 5.53 X 10~24 cc. per atom

= 5.53 X 10~24 X 6.03 X 1023 cc. per mole = 3.33 cc. per mole.

In Table XXIII-3 we give the volumes computed this way, the values of
TABLE XXIII-3. — VOLUMES OF INERT GAS ATOMS

Volume from

Volume of

ionic radius

volume

liquid

3 33

17.1

5 1

16.7

8 6

32 2

3 8

28 1

12 5

39 7

3 2

38 9

18 8

50.8

2 7

47 5

The volumes computed from ionic radii are interpolated as described in the text from the ionic radii
of Table XXIII-2. Values of b are taken from Table XXI V-l. Volumes are in cubic centimeters per
mole.

Van der Waals 6, computed from the critical pressure and temperature,
the ratio of 6 to the volume computed from the ionic radius, and finally
the molecular volume of the liquid. From the table we see that the
values of b, and the volumes of the liquid, agree fairly closely with each
other, and are three to five times the volume of the molecule, as computed
from Table XXIII-2. Since the liquid is a rather closely packed struc-
ture, this seems at first sight a little peculiar. The explanation, however,
is not complicated. The molecules really are not hard, rigid things but

SEC. 3] IONIC CRYSTALS 385

are quite compressible. Thus when they are held together loosely, by
small forces, they have fairly large volumes, while when they are squeezed
tightly together they have much smaller volumes. Now the ions used in
computing Table XXIII-2 are held together by strong electrostatic forces,
equivalent to a great many atmospheres external pressure. Thus their
atoms are greatly compressed, as we saw in the preceding paragraph; by
interpolating, we find volumes for the inert gas atoms in a very com-
pressed state. On the other hand, the real inert gas liquids are held
together only by weak Van der Waals forces, which cannot squeeze the
atoms to nearly such a compressed state. Thus it is reasonable that
the volumes computed from the radii of Table XXIII-2 should be much
smaller than the volumes of the liquids. These remarks lead to a conclu-
sion regarding the divalent ions. Being doubly charged, the forces in the
divalent crystals are much greater than in the monovalcnt ones, as we
have mentioned earlier, and the ions are correspondingly more squeezed.
Thus the sizes of the divalent ions in Table XXIII-2 are really too small
in comparison with the monovalent ones, and we cannot get a correct idea
of the relative sizes of the ions by studying Table XXIII-2.

3. Energy and Equation of State of Simple Ionic Lattices at the Abso-
lute Zero. — The structure of the ionic lattices is so simple that we can
make a good deal of progress toward explaining their equations of state
theoretically. In this section we shall consider their behavior at the
absolute zero of temperature. Then the thermodynamic properties can
be derived entirely from the internal energy as a function of volume, as
discussed in Chap. XIII, Sec. 3, and particularly in Eq. (3.4) of that
chapter. The internal energy at the absolute zero is entirely potential
energy, arising from the forces between ions. As we saw in the preceding
chapter, these forces are of two sorts. In the first place, there are the
electrostatic forces between the charged ions, repulsions between like ions
and attractions between oppositely charged ones. The net effect of these
forces is an attraction, for each ion is closer to neighbors of the opposite
sign, which attract it, than to those of the same sign, which repel it. In
addition to this electrostatic force, there are the repulsive forces between
ions, resulting from the exclusion principle, vanishingly small at large
distances, but increasing so rapidly at small distances that they prevent
too close an approach of any two ions.

First we take up the electrostatic forces. We shall consider only
the sodium chloride structure, though other types are not essentially
more difficult to work out. Let the charge on an ion be ± zet where z = 1
for monovalent crystals, 2 for divalent, etc. We shall now find the
electrostatic potential energy of the crystal, by summing the terms
±z*e*/d for all pairs of ions in the crystal, where d is the distance between
the ions. We start by choosing a certain ion, say a positive one, and

386

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

summing for all pairs of which this is a member. Let the spacing between
nearest neighbors of opposite sign be r. Assume the ion we are picking
out is at the origin, and that the $, y, z axes point along the axes of the
crystal. Then other ions will.be found for x = n\r, y = n2r, z = n8r,
where n\, w2, w3 are any positive or negative integers. It is easy to see
that if n\ + n2 + n$ is even the other ion will be positive, and if it is odd
it will be negative; for this means that increasing one of the three integers
by unity, which corresponds to a translation of r along one of the three
axes, will change the sign of the ion, which is characteristic of the sodium
chloride structure. The distance from the origin to the ion at nir, n2r,
n$r is of course r\Ai? + n\ + n|, so that the potential energy arising from
the pair of ions in question is ±z2e2/(rv/^i + ™! + ™1)> where the sign
is + if n\ + r?2 + n3 is even, — if it is odd. Now there will be a number
of ions at the same distance from the origin, and since the potential is a
scalar quantity, these will all contribute equal amounts to the potential
energy and can be grouped together. We can easily find how many such
ions there are. They all arise from the same set of numerical values of
Wi, ^2, fta, but arranged in the different possible orders, with all possible
combinations of sign. If rii, n2, n3 are all different and different from
zero, there are six ways of arranging them, and each can have either sign,
so there are eight possible combinations of signs, making 48 equal terms.
If two out of the three n's are equal in magnitude, but not zero, there are
only three arrangements, but eight sign combinations, making 24 equal
terms. If all three arc equal in magnitude, there are still the eight com-
binations of signs, making 8 terms. If one of the rz's is zero, there is no
ambiguity of sign connected with it, so that there are only half as many
possible terms, and if two of the n's arc zero there are only a quarter as
many terms. By use of these rules, we can set up Table XXIII-4, giving

TABLE XXIII-4. — CALCULATION OP ELECTROSTATIC ENERGY, SODIUM CHLORIDE

STRUCTURE

n\n^n^

Number of
terms

Distance

Contribution to potential energy

1 0 0

rVT

(z*e*/r) X (-6/VT)

= -6.00C

1 1 0

rV2

(z*e*/r) X (12/V2)

8.48*

1 1 1

r\/3

(zV/r) X (-8/\/3)

= -4.62(

200

r\/4

<«V/r) X (6/V4)

3. OCX

2 1 0

r\/5

(zW/r) X (-24/V6)

= -10.73(

2 1 1

r\/6

(«V/r) X (24/\/6)

9.80C

220

r\/8

frV/r) X (12A/8)

4.244

2 2 1

r\/9

(rfe»/r) X (-24/V9)

- -8.00C

222

r\/12

(iV/r) X (8A/I5)

= 2.31C

SBC. 3] IONIC CRYSTALS 387

the values of n\n^n^ arranged in order so that n\ £: n* ^ ns; the number
of neighbors associated with various combinations of these n's; the
distance; and the total contribution of all these neighbors to the potential
energy. This table can be easily extended by analogy to any desired
distance.

On examining Table XXIII-4, we see that the terms show no tendency
to decrease as the distance gets greater, though they alternate in sign. It
is plainly out of the question to find the total potential energy simply by
adding terms, for the series would not converge. But we can adopt a
device that brings very rapid convergence.
As shown in Fig. XXIII-4, we set up a
cube, its faces cutting through planes of
atoms. Then if we count each ion on a
face of the cube as being half in the cube,
each one on an edge as being a quarter
inside, and each one at a corner as being
one-eighth inside, the total charge within
the cube will be zero. Enlarging the cube
by a distance d all around will then add

a volume that contains a net charge of zero, FIO. xxni-4.— Cube of ions
and so contributes fairly little to the total ,tak!n from sodmm cUoride type of

. , , . . T , , .„ lattice.

potential at the origin. In other words, if

we set up the total potential energy of interaction of the ion at the origin,
and all ions in such a cube, the result should converge fairly rapidly as the
size of the cube is increased. This is in fact the case. If the cube extends
from — r to +r along each axis, the points 100 will be counted as half in
the cube; those at 110 will be one quarter in; and those at 111 one-eighth
in. Thus the contribution of these terms to the potential energy will be

6.000 ,8.485 4.620

If it extends from — 2r to 2r, the points 100, 110, 111 will be entirely
inside the cube, those at 200, 210, 211 will be half inside, those at 220, 221
one quarter inside, and those at 222 one-eighth inside. Thus for the
potential energy we have

-6.000 + 8.485 - 4.620 + - +

The next approximation is found similarly to be —1.714, and successive
terms oscillate slightly but converge rapidly, to the value —1.742, the
correct value, which as we see is very close to the value (3.2).

388 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

As we have just seen, the sum of potential energy terms between one
positive ion and all its neighbors is

~2f>2

-1.742^-- (3.3)

The number 1.742 is often called Madelung's number, since it was first
computed by Madelung.1 We should have found just the same answer
if we had started with a negative ion instead of a positive one, since
the signs of all charges would have been changed, and each term involves
the product of two charges. Now to get the total energy, we must sum
over all pairs of neighbors. Let there be N molecules in the crystal
(that is, N positive ions and N negative ions). Then each of the positive
ions contributes the amount (3.3) to the summation, and each of the
negative ions contributes an equal amount, so that at first sight we should

/ z*c*\
say that the total energy was — 2AT( 1.742 — )• This is incorrect, how-

ever, for in adding up the terms in this way we have counted each term
twice. Each pair of ions, say ion a and ion 6, has been counted once when
ion a was the one at the origin, b at another point, and then again when b
was at the origin. To correct for this, we must divide our result by two,
obtaining

Electrostatic energy = -7VM.742— V (3.4)

Next we consider the repulsive forces between ions. The only thing
we can say about these is that they are negligible for large interionic dis-
tance, and get very large as the distance becomes small. A simple
function having this property is l/dm, where d is the distance between
ions, m is a large integer. We shall tentatively use this function to repre-
sent the repulsions. To give results agreeing with experiment, it is found
that m must be of the order of magnitude of 8 or 10 in most cases. Thus,
let the potential energy between two positive ions at distance d apart be
a++/dm, between two negatives a __ /dm, and between a positive and a
negative a+_/dm. The coefficients a are all assumed to be positive,
leading to repulsions. Then we can compute the total repulsive potential
energy, just as we have computed the electrostatic attraction, only now
the series converges so rapidly that we do not have to adopt any special
methods of calculation. Thus for the sum of all pairs of ions in which
one is a positive ion at the origin, we have

(3-5)

1 For other methods of computation, see for instance M. Born, "Problems of
Atomic Dynamics," Series II, Lecture 4, Massachusetts Institute of Technology, 1926.

SBC. 3] IONIC CRYSTALS 389

To have a specific case to consider, let us take w = 9, which works fairly
satisfactorily for most of the crystals. Then the series (3.5) becomes

0.530a++ + 0.0571a+_ + 0.0117a++ + 0.0171a+_
+ 0.00750++ + 0.0010a++ + 0.0012a+_ • • • )

0.550a++)- (3.6)

There is a similar formula for the sum of all pairs in which one ion is a
negative one at the origin, with a __ in place of a++. Then, as before, we
can get the total energy by multiplying each of these formulas by N/2.
That is, the total repulsive energy is

Repulsive energy = AT^)[6.075a+_ + O.OH»(«y + a.-jj

- £ (3.7)

where A is a constant. More generally,

^
Repulsive energy = — • (3.8)

We can now combine the electrostatic energy from Eq. (3.4) and the
repulsive energy from Eq. (3.8) to obtain the total internal energy at the
absolute zero,

f/0= -Ni.7*££ + A. (3.9)

To make connection with our discussion of the equation of state in Chap.
XIII, we should expand Eq. (3.9) in power series about r0, the value of r
at which UQ has its minimum value. First we have

df/o *71 -An**e* 4

-r- = Nl. 742— ~ -- w-^ri

ttT T2 f.m+1

= 0 when r = r0. (3.10)

From Eq. (3.10) we can write A in terms of other quantities, finding

We then have

A = N1.742z*e*- (3.11)

m ^ '

U. - -tfl.74a - . (3.12)

TOr

390 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

Expanding Eq. (3.12) in power series in r — r0, we have

U0 - -#1.742— 2 ( 1 - 1)
ro \ m/

Equation (3.13) is of the form of Eq. (3.4), Chap. XIII,

[700 + ATc7ips - 9(P? - Pto ' ' ' ' (3-14)

To see the significance of c in this equation, it will be remembered that the
volume per molecule is given by

jj = cr». (3-15)

In this case, consider a cube of edge r, with a positive or negative ion at
each corner. There are 8 ions at the corners, each counting as if it were
one eighth inside the cube, so that the cube contains just one ion, or half a
molecule. In other words, V/N = 2r3, or c = 2, for the sodium chloride
structure. It will also be remembered that the quantities PJ arid P£ in
Eq. (3.15) are coefficients in the expansion of the pressure as a function
of volume, at the absolute zero of temperature, as shown in Eq. (1.5) of
Chap. XIII. These can be found from experiment by Eqs. (1.10) of
Chap. XIII. Identifying Eqs. (3.13) and (3.14), we can then solve for
t/oo, the energy at zero pressure at the absolute zero, P? andPJj, finding

(3.16)
^ = ^~r(m-l), (3.17)

*-^%("-»(« + W- <3-18>

4. The Equation of State of the Alkali Halides.— The alkali halides,
the fluorides, chlorides, bromides, and iodides of lithium, sodium, potas-
sium, rubidium, and caesium have been more extensively studied experi-
mentally than any other group of ionic crystals. For most of these
materials, enough data are available to make a fairly satisfactory com-
parison between experiment and theory. The observations include the
compressibility and its change with pressure, at room temperature, from
which the quantities ai(T), a2(T) of Eq. (1.1), Chap. XIII, can be found

t/oo = -tfl.742^Yl - A

SEC. 4] IONIC CRYSTALS 391

for room temperature; very rough measurements of the change of com-
pressibility with temperature, giving the derivative of ai with respect to
temperature; the thermal expansion, giving the derivative of ao with
respect to temperature; and the specific heat. There are two sorts of
comparison between theory and experiment that can be given. In the
first place, we have found a number of relations between experimental
quantities, not involving the detailed theory of the last section, which we
can check. Secondly, we can test the relations of the last section and see
whether they are in agreement with experiment.

The relations between experimental quantities mostly concern the
temperature effects. First, let us consider the specific heat. In Chap.
XV, Sec. 3, we have seen that it should be fairly accurate to use a Debye
curve for the specific heat of an alkali halido, using the total number of
ions in determining the number of characteristic frequencies in that
theory. It is, in fact, found that the experimental values fit Debye
curves accurately enough so that we shall not reproduce them. We can
then determine the Debye temperatures from experiment, and in Table
XXIII-5 we give these values for NaCl and KC1, the two alkali halides

TABLE XXIII-5. — DEBYE TEMPERATURES FOR ALKALI HALIDES

NaCl, ° abs.

KC1, ° abs.

O/> from specific heat ...
Qr> from elastic constants
Bo from residual rays

281
305
277

230
227
227

Data are taken from the article by Schrodinger, "Spezifische Warme," in "Handbuch der Physik,"
Vol. X, Springer, 1926. This volume contains a number of articles bearing on topics taken up in this
book and is useful for reference.

which occur in a crystalline form in nature and for which most measure-
ments have been made. But from Eqs. (3.1), (3.5), and (3.9), Chap.

XIV, we have information from which the Debye temperature can be
calculated from the elastic constants and the density. These constants
are known for NaCl and KC1, and in the table we also give the calculated
Debye temperature found from the elastic constants. Finally, in Chap.

XV, Sec. 4, we have seen that the frequency of the residual rays should
agree with the Debye frequency. In the table we give the observed
frequency of the residual rays, in the form of a characteristic temperature.
We see that the agreement between the three values of Debye temperature
in Table XXIII-5 is remarkably good, indicating the general correctness
of our analysis of the vibrational problem. As a matter of fact, the agree-
ment is better than we could reasonably expect, on account of the approx-
imations made in Debye's theory, and there are many other crystals for
which it is not so good, so that we may lay the excellent agreement here
partly to coincidence. '

392 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

Next we may consider the equation of state. From the compressi-
bility and its change with pressure we have the quantities ai, a2 of Eq.
(1.1), Chap. XIII, as we have mentioned before, and from the thermal
expansion we have the derivative of ao with respect to temperature, but
not its value itself. Since the thermal expansion of most of the materials
has not been measured as a function of temperature we cannot integrate
the derivative to find values of a0. The quantity a0 comes in only as a
small correction term in applications, however, and if we are willing to
assume Griineisen's theory we can calculate it to a sufficiently good
approximation. From Eq. (1.9) or (1.10) of Chap. XIII, we can find a0
from the thermal pressure and the compressibility. The thermal pressure
PO, the pressure necessary to reduce the volume to the volume at the
absolute zero, is given by Eq. (4.12) of Chap. XIII. For the present
case, if there are N molecules, 2N atoms, and 6AT degrees of freedom of the
atoms, and if we assume according to Gruneisen that all the 7/8 are equal
to 7, this equation gives

where v is a suitable mean of the natural frequencies Vj. The Eq. (4.1)
is the limiting case suitable for high temperatures, where the thermal
expansion is constant, and can be regarded as the integral of Eq. (4.16),
Chap. XIII. If we assume as a rough approximation that v can be
replaced by the Debye frequency, Eq. (4.1) leads to

Po = ^Fl T - ^

v o

6Nk /„, 9D

p ,r D ,,0,

ao = aiP0 = -y^yx( T - -^l (4.2)

where 6Nk is the heat capacity at high temperatures, x the compressi-
bility, GD the Debye temperature. Equation (4.2) should hold for
temperatures considerably above half the Debye temperature and should
be fairly accurate at temperatures as high as the Debye temperature,
where the specific heat is fairly constant. From Table XXIII-5, we see
that the Debye temperatures for these materials are of the order of
magnitude of room temperature, so that we should expect Eq. (4.2) to
be fairly accurate at room temperature where the observations have been
made.

Using the approximation (4.2) and measured values of ai and a* at
room temperature, we can use Eqs. (1.10), Chap. XIII, to find PO, Pi, and
P2. We find, as a matter of fact, that the term in a0, in Eq. (1.10) for
Pi, is a small correction term, so that to a gojod approximation Pi and P2

SEC. 4]

IONIC CRYSTALS

393

can be found directly from the observed compressibility and its change
with pressure. In Table XXIII-6, we give values of PI and P^ computed

TABLE XXIIT-6. — QUANTITIES CONCERNED IN EQUATION OF STATE OF ALKALI

HALIDES

7
(Grttn-
eisen)

7
(Eq.

4.3)

P2
calculated

7
(Eq.
4.4)

LiF

0.652 X 1012

2.41 X 1012

1.34

3.02

5.80

1.72 X 1012

1.97

LiCl... .

0.293

0 815

1 52

2 11

6 75

0 819

2 12

LiBr

0.232

0.635

1.70

2.06

6.95

0.655

2.16

NaCl

0.238

0.600

1 63

1.85

7.66

0.700

2 27

NaBr

0.197

0.476

(1 56)

1.75

7.97

0.590

2.33

0.302

0 885

1.45

2.26

7.90

0.900

2.32

KC1

0.178

0 402

1.60

1.59

8.75

0.557

2.46

KBr

0 149

0 341

1 68

1 62

8 82

0 468

2 47

0.117

0.259

1.63

1.54

9.15

0.373

2.52

RbBr.. ..

0.126

0.268

(1.37)

1.46

8.82

0.395

2.47

Rbl .

0 104

0 226

(1.41)

1.50

9 37

0 335

2.56

Values of Pi and P* are computed from data of J. C. Slater, Phys. Rev., 23, 488 (1024). Values of y
by Gruneisen's method are taken from the article by Gruneisen, "Zustand des festen Korpers," in
"Handbuch der Physik," Vol. X, Springer, 1926. Values of m, Pz calculated and the two calculated
values of 7, are found as described in the text.

in this way, for those of the alkali halides for which suitable measurements
have been made. We can now make a calculation that will serve to check
Gruneisen's theory of thermal expansion. In the first place, from Eq.
(4.16), Chapter XIII or from Eq. (4.2) above, we see that y can be com-
puted from the thermal expansion, specific heat, compressibility, and
density. But if we assume Debye's theory and neglect the variation of
Poisson's ratio with volume, we have seen in Chap. XIV, Eq. (4.6), that
we can write y in terms of P\ and P*:

(4.3)

This gives us two independent ways of computing 7, and if they agree
with each other we can conclude that Gruneisen's theory is fairly accurate.
In Table XXIII-6, we use the thermal expansion, specific heat, and
volume per mole of the crystals at room temperature, in order to
compute y by Gruneisen's theory. In the next column we give y com-
puted by Eq. (4.3). It will be seen that the two sets of numbers agree in
order of magnitude, and for most of the crystals they are in rather close
quantitative agreement. Putting this in another way, if we knew merely

394 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

the compressibility and its change with pressure, w^ could make a good
calculation of thcTtherma^^^ the thermaTJ^pansion

and compressihilHy we could calculate jthei Changs of '£gngp^ssjil^^Y_yth
pressure. The only serious discrepancies come with thejluosdes, the
most^incompressible of the crystals, and it will be foundjin latcjr^clmprers
that thliT'situation holds also for metals: the more incompressible the
crystal* the poorer the agreement between the 7 computed from the
clastic congtants^'and that found from the thermal expansion. Experi-
mentally, the change of compressibility with pressure is greater than we
should conclude from the thermal expansion, or conversely the thermal
expansion is less than we should suppose from the change of compressi-
bility with pressure. The reason for this discrepancy is not understood,
but it presumably arises from inaccuracies in Griineisen's assumptions,
since there is no indication that the experimental error could be great
enough to explain the lack of agreement between theory and experiment.
The comparisons with experiment which we have made so far do not
involve the assumptions of the preceding section about interatomic forces.
We shall now see how far those assumptions are correct. From Eqs.
(3.17) and (3.18) we can find values of Pi and P2 at the absolute zero, in
terms of r0, which we can take from experiment, and the one parameter
m. For approximate purposes, we can replace the values of PI and P2
at the absolute zero by the values at room temperature. Then we can
ask whether it is possible to find a single value of m that will reproduce
both PI and P2. To test this, we have used PI to find a value of m and
then have substituted this in Eq. (3.18) to compute a value of P2, compar-
ing this computed value with experiment. These computed values are
given in Table XXIII-6, and it is seen that the values agree as to order of
magnitude but not in detail. In jpther jvyords^ pur assumptioji.jtha.t the
repulsive potential varies inversely as a power of r is not very accurate,
and to do better one would have .to. use a function with an extr^diappsable
constant. In the table we give values of m, and it is seen that they are
in the neighborhood of 9 for most of the crystals, as we have stated earlier.
Using the values (3.17) and (3.18) for Pi and P2, and Eq. (4.3), we at once
find

7=^ + 1. (4.4)

Values of 7 computed in this way are tabulated in Table XXIII-6, and
it is seen that the agreement with the value found from the thermal expan-
sion is only moderately good, much poorer than that found with values
computed by Eq. (4.3) from the experimental values of Pa/Pi. In other
words, if we had a theoretical formula for the repulsive potential which
gave a better value for the change of compressibility with pressure than

SBC. 4]

IONIC CRYSTALS

395

the inverse power function, it would at the same time give a better value
for the thermal expansion.

In the preceding paragraph, we have seen that the potential energy
curve (3.9) derived from theory, gives qualitative but not vory good quan-
titative agreement with experiment for the change of compressibility with
pressure, and the thermal expansion. From Eq. (3.16), we can also use
this curve to find the energy of the crystal at the absolute zero and zero
pressure, C/oo. The negative of this quantity is the heat of dissociation
of such a crystal into ions, which we may call D. Of course, a crystal
would not really dissociate in this way if it were heated. It would
dissociate into neutral molecules, for example of NaCl, or possibly into
atoms of Na vapor and molecules of CU, instead. Nevertheless, thermo-
chemical measurements are available from which we can get experimental
values for D. We may imagine that we go from the crystal to the ionized
gas in the following steps, each of which is understood experimentally:
(1) we vaporize the crystal, obtaining NaCl molecules, the necessary
energy being found from the heat of vaporization, which has been meas-
ured; (2) we dissociate the NaCl molecules into atoms, the energy being
the heat of dissociation of the diatomic molecule, as used in the Morse
curve; (3) we ionize the Na atom to form a positive Na+ ion, the energy
being the ionization potential; (4) we add the electron so obtained to the
chlorine atom, obtaining a Cl~ ion, releasing an amount of energy that
is called the electron affinity of the chlorine ion. By adding the amounts
of energy required for all these processes, we find experimental values that

TABLE XXI 11-7.— LATTICE KNEHGJES OF THE ALKALI
(Kilogrnm-enlorics per mole)

Observed

Calculated

LiF . .
LiCl

240
199

238
191

LiHr

NaCl .
NaBr . ...

KF . ..
KC1

188

183
175

190
165

180

179
169

189
163

KBr

159

156

RbBr

154

149

Rbl

145

141

The observed values are taken from Landolt-Bornstein's Tables, Dritter Erg&nzungsband, p. 2870,
Dritter Teil. Calculated values are found by Eq. (3.16), using numerical values from Tables XXIII- 1
and XXIII-6.

396

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

•should agree with the values calculated from Eq. (3.16). In Table
XXIII-7 we give a number of these values and the calculated ones. The
units are kilogram calories per mole. The agreement between theory
and experiment in these values is quite striking and is one of the most
satisfactory results of the theory of ionic crystals. It is not hard to see
why the agreement here is so much better than in the calculation of change
of compressibility with pressure. The heat of dissociation depends on
the value of C/o as a function of F, while the compressibility depends
essentially on the second derivative of this curve, and the change of
compressibility with pressure on the third derivative. It is a well-known
fact that differentiating exaggerates the errors of a curve which is almost,
but not quite, correct. It thus seems likely that the energy of Eq. (3.16)
is really quite accurate, but that its second and third derivatives are
slightly in error.

6. Other Types of Ionic Crystals. — In the preceding sections we have
been talking about simple binary crystals, formed from a positive and a
negative ion of the same valency. Of course, there are many other types
of ionic crystals, and we shall not take up the other sorts in nearly such
detail. We shall, however, list a number of crystal structures, with the

substances crystallizing in them.
First we may mention the fluorite
structure, named for fluorite, CaF2,
which crystallizes in it. This is one
of the simplest crystals, having
twice as many ions of one sort as
of the other. The structure is
shown in Fig. XXIII-5. It can be
considered as a cube of calcium ions,
with an ion at the center of each
face of the cube as well as at the
corners, and inside this a cube of
8 fluorine ions. It is really better,
however, to consider the neighbors of each ion. As we see from the
figure, each fluorine is surrounded tetrahcdrally by 4 calciums. On
the other hand, each calcium is at the center of a cube of 8 equally
distant fluorine ions. Thus each calcium has twice as many fluorine
neighbors as each fluorine has calciums, as the chemical formula demands.
It is plain that molecules have no more independent existence in such a
structure than they do in sodium chloride.

In Table XXIII-8 we give the crystals that exist in the fluorite struc-
ture and the distances between nearest neighbors. In addition, we
tabulate the sum of the ionic radii of Table XXIII-2. Though these
were computed from binary compounds of elements of equal valency, they

FIG. XXIII-5. — Fluorite structure.

IONIC CRYSTALS

397

give fairly good results for the ionterionic distances even in these rather
different compounds.

TABLE XXIII-8. — SUBSTANCES CRYSTALLIZING IN FLTJORITE STRUCTURE

Substance

Distance,
angstroms

Distance
computed

CaF2... .

2.36

2.25

SrF2

2.50

2.42

SrCl2

3.02

2.92

BaF2...

2.68

2.60

CdF2 . . .

2.34

PbF2. . .

2.57

Ce02

2.34

PrO2. ..

2.32

ZrO2. . . .

2.20

ThO2 . .

2.41

Li20

2.00

2.2,

Li2S..

2.47

2 70

Na2S. .

2.83

2 95

Cu2S.

2.42

Cu2Sc. ...

2.49

In addition to the fluorite structure, there are a number of other
structures assumed by similar compounds. We shall not attempt to
enumerate or describe them. Some of
them are considerably more complicated
than the fluorite structure, but they
resemble it in that there is no semblance
of separate molecules. Each positive ion
is surrounded by a number of negative
ions and each negative by a number of
positives, at equal distances, so that it is in

no sense correct to say that one ion is FIO. XXIII-G. - Tho rairite
bound to one or two neighbors more than ture-

to others.

It is rather interesting to consider the crystal structure of substances
containing more complicated negative ions. Simple examples are
nitrates, sulphates, and carbonates. These are all similar to each other
in that the negative ion exists as a structure by itself, like an ionized
molecule, while the positive and negative ions are arranged in a lattice
without suggestion of molecular structure, as in the other ionic crystals.
Thus in the calcite structure, CaC03, the CO8 — ion exists as a triangular
structure, with the carbon in the middle, the oxygens around the corners
of the triangle. This structure is built of hexagonal units, as shown in
Fig. XXIII-6, with a COa — ion at the center, surrounded by six Ca++

398

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIII

ions. Units like that of Fig. XXIII-6, and others which are the mirror
image of it in a horizontal plane through the carbonate ion, are built up
into a crystal. The substances which crystallize in this structure are
tabulated in Table XXIII-9. Several distances are necessary to describe

TABLE XXIII-9. — SUBSTANCES CRYSTALLIZING IN CALCITE STRUCTURE

Substance

C-0 distance,
angstroms

C-metal
distance

0-metal
distance

CaCO8
MgCO3

1.24

3.21
2 92

2.37

ZnCOa

2.93

MnCO3. .

1.27

3 00

2 14

FeCO,.. . ...
NaNO3

1 27
1.27

3.01
3.25

2.18
2 40

the structure, and these cannot be found so accurately from x-ray methods
as in the simpler crystals. In a few cases they are not known, and in any
case they are not very certain. We tabulate the carbon to oxygen (or
nitrogen to oxygen) distance, giving the size of the negative ion, and also
the distances from positive ion to carbon and oxygen. We see that
while the lattice spacing depends on the positive ion, the carbonate or
nitrate ion is of almost the same size in each case, forming a practically
independent unit.

The sulphate ion, in sulphates, is a tctrahedral structure, with the
sulphur in the center, the four oxygens at the corners of a regular tetra-
hedron surrounding it. The sulphur-oxygen distance is about 1.40 A
in all the compounds. Examples are CaS04 and BaSC>4. These form
different lattices, rather complicated, but as we should expect they are
structures formed of positive metallic ions and negative sulphate ions,
each ion being surrounded by a number of ions of the opposite sign.
There are a number of other compounds crystallizing in the BaSO4 struc-
ture: BaS04, SrS04, PbS04, (NH4)C104, KC104, RbClO4, CsClO4, T1C104,
KMnO4.

6. Polarizability and Unsymmetrical Structures. — In discussing the
energy of ionic crystals, we have assumed that the only forces acting were
electrostatic attractions and repulsions, and the repulsions on account
of the finite sizes of ions. But under some circumstances there can also
be forces and changes of energy arising from the polarizability of the
ions. Of course, just as in Chap. XXII, we can have Van der Waals
attractions between ions, but this is ordinarily a small effect compared
to the electrostatic attraction and can be neglected. There can be
other, larger effects of polarizability, however. We remember that
according to Sec. 3 of Chap. XXII, an atom or ion in an electric field E

SBC. 6] IONIC CRYSTALS 399

acquires a dipole moment aE, where a is the polarizability of the ion.
Furthermore, the force on the resulting dipole is equal to the dipole
moment, times the rate of change of electric field with distance. This
force and the resulting term in the energy can be large if a polarizable ion
is in an external field, such as can arise from other ions. Now in most
of the structures we have considered, this does not occur. In the sodium
chloride, caesium chloride, zincblende, wurtzite, and fluorite structures,
each ion is surrounded by ions of the opposite sign in such a symmetrical
way that the electric field at each ion is zero, so that it is not polarized.
But the calcite and barium sulphate structures are quite different. There
the oxygens in the carbonate or sulphate ions are by no means surrounded
symmetrically by other ions, and there is a strong field acting on them.
Furthermore, they are very polarizable, and the result is a large added
attraction between the parts of the complex ion. Adopting an ionic
picture of the structure of the C03 — and other ions, we should have it
made of C++++ and three O 's, giving the net charge of two negative
units. Similarly NOa" would be made of N+f~+H~ and three O — X and
SC>4 of S+4"f"l"f + and four 0 's. Each of the ions, in these cases, would
form a closed shell, the carbon and nitrogen being like helium, the sulphur
and oxygen like neon. The central ion of the complex ion in each case
would be very strongly positively charged and would polarize the oxygen
very strongly, adding greatly to its attraction to the carbon, nitrogen, or
sulphur. We shall not try to estimate the effect of this added attraction
at present, but we can easily get evidence of it. Thus we have mentioned
that the sulphur-oxygen distance in the sulphates is about 1.40 A. On the
other hand, the 0~ radius, from Table XXIII-2, is about 1 .45 A. Of
course, S~H~f+~H~ would have an extremely small radius, but still we should
expect that the sulphur-oxygen distance would be something like 1.50 A
in the absence of extra attraction. Even more striking is the carbon-
oxygen distance in the carbonates, about 1.27 A, well below the ionic
radius of oxygen alone. These facts suggest that some additional attrac-
tion is acting between the ions in question, decreasing the distance of
separation. One way of interpreting this added attraction is the polariza-
tion effect we have mentioned. In the next chapter, we shall see that
another interpretation is to suppose that the ions are not really as highly
charged as the ionic picture would suggest, but that in addition there are
homopolar bonds between the atoms making up these complex ions. The
homopolar binding would in this interpretation furnish the extra attrac-
tive force resulting in the small spacing between atoms. Thus we are
not perhaps forced to think about polarizability at all in such a case.
There are many cases, however, where it is definitely important, and
the effect of polarization can be calculated easily from known polariza-
bilities and charge distributions.

CHAPTER XXIV
THE HOMOPOLAR BOND AND MOLECULAR COMPOUNDS

Ionic compounds, as we have seen in the preceding chapter, exist most
characteristically in crystalline solids, for the electrostatic forces that hold
them together extend out in all directions, binding the ions together into a
structure that has no trace of molecular formation. Compounds held
together by homopolar bonds, on the contrary, form definitely limited
molecules, which are bound to each other only by the relatively weak
Van der Waals forces. Thus their most characteristic form is the gaseous
phase in which the molecules have broken apart from each other entirely.
The ordinary gases with which we are familiar, and the ordinary liquids,
belong to this group of compounds. They arc the only group to which the
idea of the molecule, so common in chemistry, really applies. We shall
begin our discussion by taking up some of the familiar molecular com-
pounds and discussing the homopolar bonds which hold their atoms
together, and the nature of homopolar valence. Then we shall go on to a
discussion of tho Van der Waals constants of these substances, as indicat-
ing their behavior in the gaseous and liquid phases, and finally we shall
take up the solid forms of the homopolar substances.

1. The Homopolar Bond. — The Qrincipa^glements sometimes form-

Te: H. F. CL Br.

In these groups of elements we must add four, three, two, or one electron
respectively to form a closed shell. But if two such elements combine
together, where are the extra electrons to come from? There is no
positive ion losing electrons and ready to donate them to help form nega-
tive ions. The expedient which these elements adopt in their effort to
form closed shells is the sharing of electrons, as we have discussed in
Chap. XXII. An electron can sometimes be held by two atoms in com-
mon, spending part of its time on one, part on the other, and part in the
region between; in so doing it helpslfill UP the shells of both atoms'. There
is just one conspicuous rule that holds for almost all such bonds, and
that is tbftt ordinarily two electrons are shared in a similar way, the two
together forming what is called a homopolar or electron-pair bond. The
reason why two cooperate, as we saw in Chap. XXII, is essentially the
electron spin in conjunction with the exclusion principle.

In the first place, we can symbolize the process of forming a homopolar
bond by a simple device used by G. N. Lewis, to whom many of the ideas
of homopolar binding are due. Most of the elements forming this type of
bond are trying to complete a shell, or subshell, of eight electrons, as we

400

SEC. 1] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 401

have explained in Chap. XXII. Lewis indicates the eight electrons by
eight dots surrounding the symbol of the element. Thus, for instance, the
neutral fluorine atom, which has only seven electrons in its outer shell,

would be symbolized as : F . . This does not have a completed shell. But

by combining two such atoms, we can form the structure :F:F:, contain-
ing fourteen electrons but sharing two of them in a homopolar bond, so
that each atom in a sense has a completed shell. It is clear from this
symbolization that the halogons, F, Cl, Br, I, can form one homopolar
bond; the divalent elements 0, S, Se, Te, can form two; and so on. In
this symbolization, hydrogen takes a special place, for by adding electrons
it forms a completed helium shell of two electrons. It thus forms one
homopolar bond, and in this type of bonding it is in many respects analo-
gous to a halogen. In such a way, for instance, we can indicate the

structure of hydrogen chloride as H : Cl : , the electron pair sharod between

the two atoms helping to fill up the hydrogen shell of two, and the chlorine
shell of eight. Similarly and illustrating also the valences of 0, N, and

C, we may write water, ammonia, and methane respectively as H:O:,

H
H

H : N : H, and H : C : H. In each of these compounds, we observe that the

H H

total number of electrons indicated is just the number furnished by the
outer shells of the atoms entering into the molecule. Thus, in NH3
the nitrogen furnishes five electrons, each of the hydrogens three, making
eight in all.

Hydrogen is in a very special position, in that it forms a closed shell
(heliumlike) by adding an electron, and also a closed shell (the nucleus
without any electrons) by losing an electron. It can act, in the language
of ions, like either a univalent positive or a univalent negative element.
This gives the possibility of an ionic interpretation of most of the hydrogen

compounds. Thus we may symbolize hydrogen chloride as H+(:C1:)~,

and water as H+(:6:)— H+, or as H+(:6:)~. We shall see later that

there is good reason to think, however, that in most of these cases the
homopolar way of writing the compound is nearer the truth than the ionic
method.

The elements carbon, nitrogen, and oxygen have a peculiarity rarely
shown by other elements: they form sometimes what is called a double

402 INTRODUCTION TO CHEMICAL PHYSICS [CHA*. XXIV

bond, and in the cases of carbon and nitrogen a triple bond. This means
that two or three pairs of electrons, rather than one, may be shared
between a pair of atoms. This is seen in its simplest form in the molecules
O2 and N2. If two oxygens shared only one pair of electrons, they would
not achieve closed shells; they must share two pairs, so that two electrons
of each atom count in the shell of the other as well. Similarly two nitro-
gen atoms must share three pairs. We can symbolize these compounds

by : O : : O : , where both pairs of electrons between the O's are counted
in each group of eight, and by :N: : :N:. Compounds having double or
1 riple bonds generally have a rather unsaturated nature; they tend to add
more atoms, breaking down the bonds into single ones and using the
valences left over in order to attach the other atoms. A familiar example
is the group of compounds acetylene C2H2, ethylene C2H4, and ethane
C2H6, in which the last named is the most stable. These are symbolized

HH
H:C:::C:H, ij-C:;C-H> H:C:C:H, formed with triple, double, and

HH
single bonds respectively.

One can derive a good deal of information about the three-dimensional
structure of a molecule in space from the nature of the homopolar bonds,
and it must not be supposed that the chemical formulas, written as we
have been writing them in a plane, express the real shape of the molecule.
We have written them in each case so as to approximate the shape as
closely as possible, but in many cases have not succeeded very well. In
general, the four pairs around an atom tend to be arranged in the only
symmetrical way they can be, namely at the four corners of a regular
tetrahedron. The vectors from the center to the corners of a tetrahedron
form angles of 109.5° with each other, often called the tetrahedral angle,
and in a great many cases it is found that when two or more atoms are
attached to another atom by homopolar bonds, the lines of centers
actually make approximately this angle with each other. We shall dis-
cuss this more in detail in the next section, in which we take up the struc-
tures of a number of homopolar molecules.

2. The Structure of Typical Homopolar Molecules. — Many of the
homopolar molecules are among the most familiar chemical substances.
In this section we shall describe a few of them, discussing the nature of
their valence binding and giving information about their shape and size.
For the diatomic, and some of the polyatomic, molecules, this information
is derived from band spectra. In other cases, it is found by x-ray diffrac-
tion studies of the solid, using the fact that homopolar molecules generally
are very similar in the solid and gaseous phases, or by electron diffraction
with the gas. We begin with some of the diatomic molecules listed in
Table IX-1, including H2, C12, Br2, I2, NO, 02, N2, CO, HC1, and HBr.

SBC. 2] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 403

The first molecule in the list, and the simplest diatomic molecule, is
hydrogen, H2. Its structure of course is H:H, the two electrons being
shared to simulate a helium structure about each atom. The internuclear
distance1 0.75 A is the smallest internuclear distance known for any
compound, as is natural from the small number of electrons in hydrogen.
As we have seen, hydrogen acts a little like a halogen when it forms homo
polar bonds, and we might consider next the molecules F2, C12, Br2, I2.
We have already mentioned their bonding, by a single electron pair
bond, in the previous section. The internuclear distances in these
molecules are large, being 1.98 A for C12, 2.28 A for Br2, and 2.66 A for
I2, as we saw in Table IX-1. It is interesting to compare these inter-
nuclear distances with the ionic radii, from Table XXIII-2. There we
found radii of 1.80 A, 1.95 A, and 2.20 A for C1-, Br~, and I~. If these
radii represented the sizes of the atoms in the diatomic molecules, the
internuclear distances would be twice the radii, or 3.60 A, 3.90 A, and
4.40 A, almost twice the observed distances. This is an illustration of the
fact, which proves to be quite general, that interatomic distances in
homopolar binding are decidedly less than in ionic binding. The reason is
simple. While the sharing of a pair of electrons is in a sense a way of
building up a closed shell of electrons, still the shell is really not filled to
capacity. There is, so to speak, a soft place in the shell just where the
bond is located, and the atoms tend to pull together closer than if the shell
were really filled.

The remaining molecules in our list are NO, 02, N2, CO, HC1, and
HBr. The first of these, NO, is the most peculiar compound in the list
and one of the most peculiar of the known compounds. We note that
nitrogen supplies five, and oxygen six, outer electrons to the compound,
making a total of eleven, an odd number. It is quite obvious that an odd
number of electrons cannot form closed shells, electron pairs, or anything
else associated with stable molecules. As a matter of fact, out of all the
enormous number of known chemical compounds, only a handful have an
odd number of electrons, and NO is almost the only well-known one of
these. We shall make no effort to explain it in terms of ordinary valence
theory, for it is in every way an exception, though it can be understood
in terms of atomic theory.

Oxygen, with a double bond, and nitrogen, with a triple one, have
already been discussed. The internuclear distances, from band spectra,
are 1.20 A for oxygen, and 1.09 A for nitrogen. In line with what we have
just said about the halogens, it is interesting to notice that the inter-
nuclear distance in oxygen is a great deal less than the double radius of
O — . That ionic radius was 1.45 A, so that if it represented the size of

1 See Sponer, " Moleklilspektren und ihre Anwendungen auf chemische Probleme,"
Vol. I, Springer, 1935, for interatomic distances of diatomic and polyatomic molecules
in this chapter.

404 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

the oxygen in O2, the internuclear distance would be 2.90 A, more than
twice the actual distance. In the case of 02, with its double bond, the
tendency of the atoms to pull together in homopolar binding is particu-
larly pronounced, for the shell is even less nearly filled than with a single
bond and can be even more compressed by the interaction forces. In
nitrogen with its triple bond, the interatomic distance is even smaller, in
line with this fact. The next molecule on the list, CO, does not fit in very
well with our rules. A clue to its structure is provided by the fact that
it has 4 + 6 == 10 outer electrons, just like N2, and that in many of its
properties it strongly resembles N2. We have stated that the internuclear
distance in nitrogen was 1.09 A; in CO it is 1.13 A. The suggestion is
very natural that in a sense a triple bond is formed in this case also, with
the structure : C : : : 0 : , though a triple bond is not usually formed by
oxygen.

We have already discussed the structure of the next two molecules,
HC1 and HBr, whose valence properties are indicated by the symbols

H:C1:, H:Br:. The internudear distances are 1.27 A and 1.41 A respec-
tively. We note, as before, how much smaller these are than the values
given by ionic radii. We have no ionic radius for H, but for Cl~ we have
1.80 A, and for Br~ the distance is 1.95 A. The internuclear distances in
these cases are actually less than the ionic radii. This is good evidence
for the homopolar, rather than the ionic, nature of these compounds.
Another reason comes from the magnitudes of the electric dipole moments
of these two molecules, which are found to be 1.03 X 10~18 and
0.78 X 10~18 e.s.u.-cm. If the molecules were really ionic, we should
expect that electrically they would consist of a unit positive charge on
the hydrogen nucleus, and a net negative charge of one unit, spherically
symmetrical, and therefore acting as if it were on the chlorine or bromine
nucleus. That is, there would be charges of one electronic unit located
1.27 A and 1.41 A apart respectively. This would give dipole moments,
equal to the product of charge and displacement, of

(4.8 X 10-10) X (1.27 X 10~8) = 6.1 X 10~18,

and of 6.8 X 10~~18 units, respectively. The observed dipole moment of
HC1, as we have seen, is only about one-sixth of this value, and of HBr
about one-ninth of the value given by the polar model. To explain this,
we must assume that the negative charge is not located symmetrically
about the Cl or Br but is displaced toward the hydrogen. This is what we
should expect if there is really a homopolar bond, for then the shared
electrons would be in the neighborhood of the hydrogen, displaced in
that direction from the halogen ion. The dipole moments then furnish
arguments for the homopolar nature of the bond and for thinking that it

SBC. 2] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 405

is less polar in HBr than in HC1. There is still another way of regarding
this situation : we may make use of the polarizability of the halide ion. A
hydrogen nucleus close to a halide ion would produce an extremely strong
electric field, which would polarize the ion, changing it into a dipole with
the negative charge displaced slightly toward the positive hydrogen ion.
This dipole moment would tend to cancel the moment produced by the
two undistorted ions, so that the net moment would be less than the figure
6.1 X 10~18 calculated above. As a matter of fact, calculations by
Debye1 show that the resulting dipole moment calculated in this way
would be of the right order of magnitude. We should not regard this
calculation as indicating that our homopolar shared electron theory is not
accurate, however. For in this case the polarizability is simply a rather
crude way of taking account of the shifting of electric charge which we
can describe more precisely as electron sharing between the atoms. The
situation, however we describe it, is essentially this: that the electrons,
instead of being arranged in a spherically symmetrical way about the
halogen nucleus, tend to be somewhat displaced toward the hydrogen, so
that it also is partly surrounded by electrons, rather than acting as an
entirely isolated ion.

We have now completed our list of diatomic molecules. Next wo
might well take up various hydrides: H2O, NH3, CH4, H2S, PH3, SiH4.
We have already given structural formulas for H2O, NH3, and CH4; the
others are analogous, H2S resembling H2O, PH3 being like NH3, and SiH4
like CH4. The hydrogens are bound, on the homopolar theory, by
single bonds, and the angles made by the radius vectors to different
hydrogens are very closely the tetrahedral angle 109.5°. Thus H2O is a
triangular molecule, the hydrogen-oxygen distance being 0.96 A, and tho
H-O-H angle being 104.6°. NH3 is pyramidal, the nitrogen-hydrogen
distance being 1.01 A and the H-N-H angle 109°. CH4 is tetrahedral, the
carbon-hydrogen distance being 1.1 A. H2S is triangular, like water, the
sulphur-hydrogen distance 1.35 A and the angle 92.1°, rather smaller
than we should have supposed. PH3 is presumably pyramidal like NH3
but the distances and angles do not seem to be known. SiH4 is tetra-
hedral but again the distances are not known.

There are only a few other common inorganic molecules to be
mentioned. CC>2 is a linear structure, with valences symbolized by

:O: :C: :O:. That is, there are double bonds between the carbon and
each oxygen. The C-0 distance is 1.16 A, slightly greater than the
value 1.13 A found in CO, where there is a triple bond. N2O is also a

linear molecule, presumably with the structure : N : : N : : 0 : , again formed
with double bonds like CO2, which has the same number of outer electrons.

1 See Debye, "Polare Molekel," Sec. 14, Hirzel, 1929.

406 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

The distance between end atoms is 2.38 A, slightly greater than the value
2 X 1.18 = 2.32 A between end atoms in C02. Here again we see the
resemblance between N2 and CO, in that they form similar molecules
when another oxygen atom is added. The molecule 802 is a triangular
molecule shaped something like water. Its structure presumably is

:0:S:. This is the first example we have met of a case where an atom

is not surrounded, even with the aid of shared electrons, by a closed
shell: the sulphur has only six electrons around it. We shall come to
other examples later, when we talk about inorganic radicals. The
sulphur-oxygen distance in 802 is 1.37 A. Carbon disulphide CS2
resembles CO2 in being a linear molecule, with carbon-sulphur distance of
1.6 A, decidedly larger than the value 1.16 A in C02, in accordance with
the fact that sulphur is a larger atom than oxygen.

The molecules taken up so far are all very simple, composed of very
few atoms. The more complicated examples of homopolar molecular
compounds are found almost entirely in the field of organic chemistry.
We shall postpone a discussion of organic compounds until the next
chapter, since they form a field by themselves. Before closing our discus-
sion, however, we shall take up a different sort of homopolar structure,
namely, a few inorganic negative ions, formed very much like molecules.
The most important ones are NOa~, COa , 804 , C104~, mentioned
in the preceding chapter. The first two, as stated in Chap. XXIII, Sec.
5, are triangular structures in a plane, the N or C being in the center, the
oxygens at the corners. Each has 24 electrons (when we take account
of the negative charge 011 the ion), so that it was possible in the last
chapter to treat them as ionic structures, the oxygen having a closed
shell of eight electrons, the nitrogen or carbon having no outer electrons.

.. :6: .. :6:

A structure much nearer the truth, however, is :O:N \\ and :0:C *.'. .

" :0: " :0:

These structures differ from the ionic one in that we have indicated two
electrons from each oxygen as being shared with the central nitrogen or
carbon. This case resembles that of SO2 in the preceding paragraph, in
that one of the atoms, in this case the central one, has only six rather
than eight electrons surrounding it. Another molecule showing similar
structure is SO3. We have stated in the preceding chapter, Table
XXIII-9, that the C-0 or N-O distance in the carbonates and nitrates is
about 1.27 A. This is decidedly greater than the C-0 distance in CO,
which is 1.13 A, and in C02, 1.16 A, but in the present case there is only
a single bond, rather than the triple or double bonds found in those two

SEC. 3] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 407

compounds. The agreement is close enough so that it is quite plain that
the bonds in these cases are homopolar and not ionic, as was stated in
Chap. XXIII, Sec. 6. In the sulphate and perchlorate ions, S04~ and
C1O4~, there are enough electrons, 32 outer ones per ion, to form com-
plete shells around the central atoms. The valence can be symbolized

: O : S : O : and : O : Cl : O : , and the compounds can be described as having
":6:" " :6:"

single valence bonds between the central atom arid each oxygen. They
have a tetrahedral form, the sulphur-oxygen distance in the sulphates
being about 1.40 A.

3. Gaseous and Liquid Phases of Homopolar Substances. — We have
already mentioned that the gaseous phase is the most characteristic one
for molecular substances with homopolar binding. We shall begin,
therefore, by examining the Van dor Waals constants a and 6 for a con-
siderable number of homopolar substances. The constant b will give us
information about the dimensions of the molecules, information that we
can correlate with the known interatomic distances, and a will lead to
information about the strength of the Van der Waals attraction.

In Table XXIV-1, we give Van der Waals constants for quite a series
of gases, arranged in order of their &'s. We include not merely the gases
mentioned in the preceding section, but the inert gases, for comparison,
and then a considerable number of organic substances, which as we have
mentioned furnish the largest and most characteristic group of homopolar
substances. In the table, the units" of a are (dynes per square centimeter)
times (cubic centimeters per mole)2. The units of 6 are cubic centimeters
per mole. These constants are obtained from the critical pressure and
temperature by Eq. (2.5), Chap. XII. They do not, therefore, have just
the same significance as the a and 6 of Eq. (5.3), Chap. XII, for those are
the constants that would lead to agreement between Van der Waals'
equation and experiment at low density, while the values we use are suit-
able to pressures and temperatures around the critical point, which will
disagree with the other ones unless Van der Waals' equation is really
applicable over the whole range of pressures and temperatures. We
note from Eq. (2.6) of Chap. XII, that if Van der Waals' equation were
correct, we should have Vc/3 = 6, where Ve is the observed critical
volume. To test this relation, Table XXIV-1 lists observed values of
Fc/3. We see that these values do not agree very closely with the values
of 6, as was stated in Chap. XII, though they are not widely different. In
addition to these quantities, we also tabulate the molecular volume of
the liquid, in cubic centimeters per mole, for the lowest temperature for
which figures are available, and also the dipole moment for dipole mole-

408 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

TABLE XXI V-l. — VAN DER WAALS CONSTANTS FOB IMPERFECT GASES

Gas

Formula

TV3

Molec-
ular vol-
ume of
liquid

Electric
moments

Neon

0 21 X 1012

17 1

14 7

16 7

0 X 10~»

Helium

0 035

23 f>

20 5

27 4

Hydrogen
Nitric oxide ....
Water ....
Oxygen .

H2
NO
H2O
O2

0 25
1 36
5.53
1 40

26 5
27 8
30 4
32 2

21 6
19.1
18.9
24 8

26 4
23 7
18.0
25 7

0
1.85

Argon
Ammonia
Nitrogen

A
NHs

1.36
4 22
1 36

32.2
36.9
38 3

26 1
24.2
30 0

28 1
24.5
32 8

0
1.44

Carbon monoxide

1 50

39 7

30 0

32 7

0 10

Krypton

2 35

39 7

36 0

38 9

Hydrogen chloride

HC1

3 72

40 7

20 8

30 g

1 03

Nitrous oxide ...
Carbon dioxide .....

N;O
COs

3 61
3.64

41.1
42 5

32.3
32 8

44 0
41 7

0.25
0

Methane

CH4

2 28

42 6

32 9

49 5

Hydrogen sulphide .
Hydrogen bromide

H2S
HBr

4 49
4 51

42.7
44 1

35.4
37 5

0.93
0 78

4.15

50 8

38 0

47 5

Acetylene . ....
Phosphtne ....

C2H2
P1I3

4 43
4 69

51.3
51 4

37.5
37 7

50.2
49 2

0
0 55

Chlorine

6.57

56 0

41 0

41 .2

Sulphur dioxide

SOa

6 80

56 1

41 0

43 8

1 61

Ethylene

C2H4

4 46

56 1

42 3

49 3

Silicon hydride

SiH4

4.38

57 6

Methylamine
Ethane .

CHaNH2
CHa — CHs

7 23
5.46

59.6
63 5

47 6

44.5
54 9

1.31
0

Methyl chloride
Methyl alcohol
Methyl ether
Carbon bisulphide

CHaCl
CHaOH
(CH3)20

CS2

7.56
9 65
8.17
11 75

64.5
66.8
72 2
76 6

45.4
39.0

67 5

49 2
40.1

59 0

1.97
1.73
1.29

Dimethylamine
Propylene

(CHa)aNH
CsHe

9 77
8.49

79.6
82 4

66 2
69 0

Ethyl alcohol

CaHsOII

12 17

83 8

41 0

57 2

1 63

Propane

CHs—CHz — CHd

8.77

84 1

75 3

Chloroform .

CHCh

15.38

102

77 1

80.2

1.05

Acetic acid . .
Trimethylamine . . .

CHaCOOH

(CHa)aN

17.81
13 20

106
108

57.0

56.1
89 3

iso-Butane . . ....

CH(CHa)3

13.10

114

96 3

Benzene

Cell 6

18 92

120

85 5

86 7

n-Butane
Ethyl ether

CH3(CH2)2CH3

(C2H5)2O

14.66
17.60

122
134

94 0

96 5
100

0
1.2

Triethylamirie ...

(C2H6)3N
CioH«

27.5
40.3

183
193

139
112

0.69

w-Octane •

CHa(CH2)aCHa

37 8

236

162

Decane

CIIa(CHa)8CHi

49.1

289

195

The unit of pressure in the constants above is the dyne per square centimeter, the unit of volume is
cubic centimeters per mole. The electric moments are expressed in absolute electrostatic units. Data
for Van der Waals constants and volumes are taken from Landolt's Tables; for the electric moments from
Debye, " Polare Molekeln," Leipzig, 1929.

SBC. 3] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 409

cules, in electrostatic units; as we have seen in Chap. XXII, this has a
bearing on the Van der Waals attraction.

The first thing which we shall consider in connection with the con-
stants of Table XXIV-1 is the set of b values. It will be remembered that
b represents in some way the reduction in the free volume available to a
molecule, on account of the other molecules of the gas. That is, it should
bear considerable resemblance to the actual volume of the molecules.
According to statistical mechanics, we have seen in Chap. XII, Sec. 5,
that the b appropriate to the limit of low densities should be four times
the volume of the molecules, but this prediction is not very accurately
fulfilled by experiment. The reason no doubt is that that prediction was
based on the assumption of rigid molecules, whereas we have scon in this
chapter and the preceding one that molecular diameters really depend a
great deal on the amount of compression produced by various forms of
interatomic attraction. This was made particularly plain in Table
XXIII-3, where wo computed volumes of the inert gas atoms by using
radii interpolated between the ionic radii of the neighboring positive
and negative ions and compared these volumes with the b values and the
volumes of the liquids. We found, as a matter of fact, that the b values
were from three to five times the computed volumes of the molecules, in
fair agreement with the prediction of statistical mechanics, but wo found
that the volumes of the liquids, in which the molecules are held together
only by Van der Waals forces, were of the same order of magnitude as the
b values, indicating that the Van der Waals forces, being very weak
compared to ionic forces, cannot compress the molecules very much. To
see if this situation is general, we give the molecular volume of the liquid
for each gas in Table XXIV-1. A glance at the table will show the
striking parallelism between the volume of the liquid and the constant b.
For the smaller, lighter molecules, b is of the same order of magnitude
as the volume of the liquid, as a rule somewhat larger, but not a great deal
larger. For the heavier molecules, there seems to be a definite tendency
for b to be larger than the volume of the liquid, but even here it is not so
great as twice as large.

The actual magnitude of the constants 6 is of considerable interest.
In the first place, we may ask how much of the volume of a gas, under
ordinary conditions of pressure and temperature, is occupied by the
molecules. One mole of a gas at atmospheric pressure and 0°C. occupies
22.4 1., or 22,400 cc. The molecules, under the same circumstances,
occupy the volume tabulated in Table XXIV-1, in cubic centimeters.
For the common gases, these volumes are of the order of 30 or 40 cc. This
is the order of magnitude of two-tenths of 1 per cent of the actual volume,
so that it is correct to say that most of the volume occupied by a gas is
really empty. On the other hand, even under these circumstances, the

410 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

molecules are not very far apart in proportion to their size. We may
take an extreme case of helium, where at normal pressure and temperature
the molecules would occupy some 23 cc., or about one one-thousandth of
the volume. To get an idea of the spacing, we may imagine the atoms
spaced out uniformly, each one in the center of a cube (though of course
actually they will be distributed at random). Then a cube of the volume
of the atom would have one one-thousandth the volume of one of these
cubes containing an atom, but the side of the small cube would be one-
tenth [= (TWIT)^] *he s^e °f the large cube, meaning that the average
distance between atoms is only about 'ten times the diameter of a single
atom. This is for a small molecule; with the larger molecules in the table,
the molecules are twice as large or more in diameter and are correspond-
ingly closer together in proportion. We can imagine that under these
circumstances real gases depart quite appreciably from perfect gas condi-
tions. Furthermore, it is natural that collisions between atoms are
frequent and that they are of great importance in many phenomena.
This is all, however, for one atmosphere pressure. A pressure of 10"6 mm.
of mercury can easily be obtained in the laboratory. This is about
10~9 atm. and corresponds to atoms spaced a thousand times farther
apart, or something like 10,000 atomic diameters apart. It is clear that
a gas in this condition must be very much like a perfect gas and that
deviations from the gas law and interactions between molecules can be
neglected.

It is interesting to ask how much space, on the average, is occupied
by each atom or molecule. In a gram molecular weight, as we have said
before, there are about 6.03 X 1023 molecules. Thus if we divide b by
this figure, we shall get the volume per molecule. It is obvious from the
table that the molecules with many atoms have much larger volumes than
those with few atoms, and it appears very roughly that the volume is
something like 12 cc. per mole per atom, a figure which we could get by
dividing the b for a particular molecule by the number of atoms in the
molecule. Variations of more than 100% from this figure are seen in the
table, but still it is correct as to order of magnitude. This gives
12/(6.03 X 1023) = 2 X 10~23 cc. as the volume assigned to an atom.
This is the volume of a cube 2.7 X 10~8 cm. on a side; this figure seems
reasonable, being of the order of magnitude of the dimensions of most of
the atoms, within a factor of two at most.

From what we have said, the values of the Van der Waals constants
b for the gases of our table look very reasonable. Next we can consider
their a's. In Eq. (3.6) of Chap. XXII, we have seen that the Van der
Waals interaction energy between molecules of polarizability a, mean
square dipole moment /z*, at a distance r, is

Energy- -JJ2L*. (3.1)

SBC. 3] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS

411

In Eq. (5.3), Chap. XII, we have seen that the Van der Waals a for
molecules of radius r0/2, with an intermolecular attractive potential of

(3.2)

)

where NQ is Avogadro's number. Using Eqs. (3.1) and (3.2), we can
now write an explicit formula for the Van der Waals a. It is

(3.3)

The terms of Eq. (3.3) are all things that can be estimated. From Table
XXI V-l, we see that the permanent dipole moments of dipole molecules
are of the order of magnitude of 2 X 10~18 absolute units, and we may
expect the root mean square fluctuating moments of other molecules to
be of the same order of magnitude. The polarizability can be found from
the measured dielectric constant, as we have explained in Chap. XXII,
Sec. 3. In addition to these quantities, we need the volume of the sphere,
^TrrJ, which appears in the denominator of Eq. (3.3). This will certainly
be of the general order of magnitude of the molecular volume, and for the
present very crude calculation we may take it to be the same as 6, which
we have tabulated in Table XXI V-l. (For air we use a value inter-
mediate between oxygen and nitrogen.) We have now computed values
of ^ which, substituted into the formula (3.3), will give the correct value
of a, and have tabulated these in Table XXIV-2. We see that the
values of M necessary to give the observed a values are of the order of

TABLE XXIV-2. — CALCULATIONS CONCERNING VAN DER WAALS ATTRACTIONS

Gas

Dielectric
constant e

AT0a, cubic
centimeters

H2
Air

1.000264
.000590

0.470
1.05

1 . 7 X 10-"18
3 1

.000690

1.23

3.1

CO2

.000985

1 75

4 2

CH4

.000944

1 .68

3.4

C2H4

.00131

2.33

4 7

The dielectric constant c is given for gas at 0°C., one atmosphere pressure.
gas occupies 2.24 X 10* cc. at this pressure and temperature, we have

Thus, since a mole of

- 2.24 X 10*

using Eq. (3.4), Chap. XXII. The value of *i is calculated from Eq. (3.3), as described in the text, and
represents the dipole moment necessary to explain the observed Van der Waals a.

magnitude which we expected to find. As we should naturally expect,
they increase as we go to larger and more complicated molecules. Direct
calculations of /*, or rather of the whole Van der Waals force, have been

412 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

made for a few of the simple gases like hydrogen and helium, with fairly
close agreement with the experimental values. It seems likely, therefore,
that our explanation of these attractions is quite close to the truth.

The explanation just given for the magnitude of the Van der Waals
attractions must be modified for strongly polar molecules, those having
large dipole moments. From Chap. XXII, Sec. 3, we remember that in
such cases there is an extra term in the polarizability, on account of the
orientation of the molecule in an external field. We mentioned that this
would increase the Van der Waals attraction, because pairs of molecules
would tend to orient each other into the position of maximum attraction,
and suggested that it might result in a Van der Waals attraction several
times as great as for nonpolar molecules. Examination of Table XXIV-1
shows that, in fact, the strongly polar molecules have constants a which
are much greater than those of nonpolar molecules near them in the list.
Thus water has an a value about four times those of its neighbors, and
ammonia about three times. We can understand in detail what happens
by considering the most conspicuous case, water. The crystal structure
of ice is well known and will be described in the next section. It is a
molecular crystal, in which each oxygen is surrounded tetrahedrally by
four other oxygens. Between each pair of oxygens is a hydrogen. Each
oxygen thus has four hydrogens near to it. But two of these four hydro-
gens are close to the oxygen, forming with it a water molecule, with its
two hydrogens at an angle, just like the water molecule in the gas. The
other two hydrogens are attached to two of the four neighboring oxygens,
forming part of their water molecules. This structure puts each of the
hydrogens of one molecule near the oxygen of another, so that their oppo-
site electrical charges can attract each other, helping to hold the crystal
together. This arrangement undoubtedly persists to a large extent in
the liquid and even to some extent in the gas, though it undoubtedly
decreases as the temperature is raised, for at high temperatures the mole-
cules tend to rotate, spoiling any effect of orientation. And it is this
extra attraction, on account of the particular orientations of the molecules,
which results in the very largo value of a for water. Similar explanations
hold for the other molecules with large dipole moments, but examination
of their structure shows that the others cannot form such tightly bound
structures as water.

There is another feature of Table XXIV-1 that bears out the unusually
large forces between dipole molecules, and that is the molecular volumes
of the liquids. If the polar molecules have unusually large attractive
forces, we should expect that these forces, which after all hold the liquid
together, would bind it particularly tightly, so that the liquids would be
unusually dense. Consistent with this, we note that water and ammonia
conspicuously, and some of the other polar liquids to a lesser extent, have

SBC. 3] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 413

molecular volumes for their liquids decidedly smaller than their b values,
while the nonpolar molecules have molecular volumes rather closely equal
to their b's. To put these facts in somewhat more striking form, if water
had no dipole moment we should expect it to have a density only about
two-thirds what it does, and we should expect the intermolecular forces
to be so small that it would boil many degrees below zero, as its neighbors
NO and 02 in the table do, and to be known to us as a permanent gas
difficult to liquefy.

The extra intermolecular force in water resulting from dipole attrac-
tion, which we have just discussed, is closely tied up with one of the most
remarkable properties of water, its ability to dissolve and ionize a great
many ionic compounds. We have seen in the last chapter that it requires
a large amount of energy to pull a crystal of an ionic substance apart into
separated ions. Such a process surely will not occur naturally; if we
examined the equilibrium between the solid and the ionic gas, the heat of
evaporation would be so enormous that at ordinary temperatures there
would be practically no vapor pressure. If an ion is introduced into
water, however, there is a strong binding between the water molecules
and the ion, corresponding to a large negative term in the energy, with the
result that the heat of solution, or the work required to break up the
crystal into ions and dissolve the ions in water, is a small quantity. In
other words, referring back to the type of argument met in Chap. XVII,
an ion is about as strongly attracted to a water molecule as to the ions of
opposite sign in the crystal to which it normally belongs; this is the
necessary condition for solubility. We now ask, why is the ion so
strongly bound to the water molecules? The reason is simply that as we
have seen the hydrogens of the water molecule are positively charged, the
oxygen negatively, so that a positive ion can locate itself near the oxygens
of a number of neighboring water molecules, a negative ion near a number
of hydrogens, which attract it electrostatically approximately as much as
two ions would attract each other, or as the negative oxygen of one water
molecule would attract the positive hydrogens of its neighboring water
molecules. This effect is particularly strong in water, for the same reason
that the Van der Waals binding is strong in water; similar effects are
found in a lesser extent in liquid ammonia, which is also a powerful ioniz-
ing solvent, with many of the same properties as water.

We have mentioned several times that the characteristic of the
molecular compounds is that the Van der Waals forces between molecules
are small compared to the valence forces holding the atoms together to
form a molecule. Thus the substances vaporize at a low temperature,
whereas their molecules do not dissociate chemically to any extent except
at very high temperatures. For instance, the dissociation H2 ^ 2H is
a typical example of chemical equilibrium, to be handled by the methods

414

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

of Chap. X, and the forces holding the atoms together are the sort taken
up in Chap. IX. We saw in Table IX-1 that the heat of dissociation of a
hydrogen molecule was 103 kg.-cal. per gram mole, such a large value that
the dissociation is almost negligible at any ordinary temperature. On the
other hand, the latent heat of vaporization, the heat required to pull the
molecules of the liquid or solid apart to form the gas, is only 0.256 kg.-cal.
per gram mole in this case, so that a temperature far below 0°C. will
vaporize hydrogen. To illustrate how general this situation is, Table
XXIV-3 gives the latent heat of vaporization and the heat of dissociation

TABLE XXIV-3. — LATENT HEAT OF VAPOKIZATION AND HEAT OF DISSOCIATION

Substance

Latent heat,
kg.-cals. per
gram mole

Heat of dis-
sociation,
kg.-cals. per
gram mole

0.220

103

2.08

117

1.69

170

I 90

223

CO2

6 44

NH8

7.14

HC1

4.85

102

H2O

11.26

118

Latent heats are from Landolt's Tables, and in each case are for as low a temperature as possible,
since the latent heat of vaporization decreases with increasing temperature, going to zero at the critical
point. Heats of dissociation are from Sponer, " Molekulspektren und ihre Anwendungen auf chemische
Probleme," Berlin, 1935, some of them having been quoted m Table IX-1.

of a few familiar gases. The heat of dissociation in each case is the energy
required to remove the most loosely bound atom from the molecule. We
see that in each case the latent heat is only a few per cent of the heat of
dissociation; water is a distinct exception on account of its high latent
heat, ten per cent of the heat of dissociation, which of course is tied up
with the large Van der Waals attraction arising from the dipole moments
of the molecules.

4. Molecular Crystals. — The molecules which we have been discussing
in this chapter are tightly bound structures, held together by strong
homopolar forces. Mathematically, these forces can be described
approximately by Morse curves, as discussed in Sec. 1, Chap. IX. On
the other hand, the forces holding one molecule to another are simply the
Van der Waals forces, which we have spoken about in Chap. XXII, and
which are very much weaker than homopolar forces, as we saw from
Table XXIV-3. It thus comes about that the crystals of these materials
consist of compact molecules, spaced rather widely apart. Since the
forces between molecules are so weak, the crystals melt at low tempera-

SBC. 4] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 415

tures, are very compressible, and are easily deformed or broken, in con-
trast to the ionic crystals with their considerable mechanical strength, low
compressibility, and high melting points. In this section we shall discuss
the structure of a few of the molecular crystals.

We start with the inert gases. The atoms of these substances are
spherical, and we should naturally expect that their crystals would be
formed simply by piling the spheres on top of each other in the closest
manner possible. This is, in fact, the case. There are two alternative
lattices, corresponding to the closest packing of spheres. Of these, the
inert gases choose the type called the face-centered cubic structure. This
structure is shown in Fig. XXI V-l. In the first place, we can regard the
structure as arising from a simple cubic lattice, as shown in (a). There
is an atom at each corner of the cube and in the center of each face.
Comparison with Fig. XXIII-1, showing the sodium chloride structure,

(«) (b) (c)

FIG. XXI V-l. — The face-centered cubic structure, (a) atoms at the corners of a cube
and the centers of the faces. (6) the same atoms connected up in planes perpendicular to
the cube diagonal, (c) view of successive planes looking along the cube diagonal, illustrat-
ing the close-packed nature of the structure.

will show that in the latter type of structure the positive ions by them-
selves, or the negative ions by themselves, form a face-centered cubic
structure. Another way of looking at the face-centered cubic structure
can be understood from Fig. XXI V-l (6). In this, we have drawn the
atoms just as in (a) but have connected them up differently, so as to form
two parallel triangles of six atoms each, oppositely oriented, with two
extra atoms. If we had considered, not simply the cube of (a) but the
whole crystal, each of these triangles would have been part of a whole
plane of atoms. These same six atoms are shown by the heavy circles in
(c), where we now look down along the normal to the planes of the tri-
angles; that is, along the body diagonal of the cube, shown dotted in
(a). In (c), we have drawn the circles representing the atoms large
enough so that they touch, as if they were closely packed spheres. The
next layer of spheres is shown in (c) by dotted lines, and we see that it is a
layer similar to the first but shifted along, atoms of the second layer fitting
into every other one of the depressions between atoms in the first layer.
The third layer, of which only one atom is shown in (6), fits on top of (c)

416

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

in a similar manner, the one atom shown in (6) going in the common center
of the triangles of (c). After these three layers, the structure repeats, the
fourth layer being just like the one drawn in heavy lines in (c). From
this description in terms of Fig. XXI V-l, (c), it is clear that the face-
centered structure is a possible arrangement for close packed spheres.
Each atom has twelve equally spaced neighbors, six in the same plane
in (c), three each in the planes directly above and directly below.

As we have stated, the inert gases crystallize in the face-centered
cubic structure. The distances between nearest neighbors are given in
Table XXIV-4. In this table we give also the volume of the crystal per

TABLE XXIV-4. — CRYSTALS OF INERT GASES

Interatomic
distance, A

Volume, cc.
per mole

Volume of
liquid, cc.
per mole

3 20

14 0

16.7 '

3.84

24 3

28 1

Kr ... .

3.94

26.4

38 9

4 37

35.8

47 5

mole, computed in a simple way from the crystal structure, and finally we
give tho volume of the liquid per mole, from Table XXI V-l, for compari-
son. We see that the volumes of the crystals are somewhat but not a
great deal less than the volumes of the liquids, as we should probably
expect, since in the liquids the same atoms are packed in a less orderly
arrangement. The interatomic distances in these crystals are much
greater than interatomic distances in any cases where the atoms are held
by either homopolar or ionic bonds. This has already been commented
on in Sec. 3 and has been explained by saying that the large attractive
forces of the homopolar or ionic bonds pull atoms together, essentially
compressing them, so that they get much closer together than when held
only by the weak Van der Waals forces.

The inert gases are the only strictly spherical molecules, but a number
of the other gases have molecules nearly enough spherical to pack together
in similar ways. The hydrogen molecule, except at the very lowest
temperatures, is in continual rotation, so that while it is not spherical at
any instant, still it fills up a spherical volume on the average, the volume
swept out by its two atoms when they are pivoted at the midpoint of the
line joining them and are free to rotate in any plane about this point.
Hydrogen molecules, then, pack as rigid spheres, but they adopt the other
method of close packing, the so-called hexagonal close-packed structure.
This is shown in Fig. XXIV-2. It starts with the same layer of atoms
shown by the heavy lines in Fig. XXI V-l (c), then has the dotted layer of

SBC. 4] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS

417

Fia. XXIV-2.- -Hexagonal
riose-paekcd structure.

(c), but after these two layers it has another layer like the first one, and
so on, having just two alternating types of layer instead of three as in the
face-centered cubic structure. Part of the structure is shown in perspec-
tive in Fig. XXIV-2. This indicates plainly the hexagonal unit from
which the structure takes its name. As we have stated, hydrogen crystal-
lizes in this form, a hydrogen molecule being at each lattice point. The
distance between molecules, on centers, is 3.75 A, the volume per molo
is 21.7 cc., to be compared with 26.3 cc. in the liquid.

The molecules N2 and CO, though they are rather far from spherical,
still crystallize approximately though not exactly
like close-packed spheres. Their crystal is a
slightly distorted face-centered cubic structure,
one dumbbell-shaped molecule being located at
each lattice point. The molecules of those sub-
stances do not rotate enough to simulate a
spherical shape at ordinary temperature, but in-
stead they oscillate about definite directions in
space. These directions are determined for the
various molecules in the crystal in a rather com-
plicated though regular way, there being several
different orientations for different molecules.
The molecules are spaced about 3.96 A apart on centers, resulting in a
volume of 27.0 cc. per mole, both for N2 and CO, compared to 32.8 cc. per
mole in the liquid for N«, and 33 cc. for CO.

Many molecules that are not spherical still rotate enough at high
temperatures so that they simulate spheres, like the hydrogen molecule.
In some cases, such molecules oscillate about definite directions at low
temperatures, as with N2 and CO, but simulate spheres at higher tempera-
tures when they are rotating with more energy. In such cases, the
substance has two crystal forms, and there is a transition from one to
the other at a definite temperature. The low temperature phase is likely
to be complicated in structure, with the molecules pointing in definite
directions, while the high-temperature phase is one of the close-packed
structures. Hydrogen chloride HC1 is a case in point. Below 98° abs.,
the molecules are hindered from rotating and the structure is a compli-
cated one which has not been completely worked out. At this tempera-
ture there is a transition, and above 98° the molecules rotate freely and
the substances crystallize in a face-centered cubic structure. In many
cases, where we might expect such a transition, it does not occur in the
available temperature range. Thus we should expect that hydrogen,
which shows free rotation under ordinary conditions, might conceivably
show a transition to another structure with hindered rotation at low
enough temperatures, while CO and N2, with hindered rotation at ordi-

418 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIV

nary temperatures, might have a transition to a state of free rotation at
high enough temperatures. But the necessary temperatures might be
above the melting point, in which case the transition could not really be
observed.

The diatomic molecules which show hindered rotation in the solid
generally have quite complicated molecular crystals. This is true, for
instance, of the halogens. CU forms a crystal composed of molecules,
each of interatomic distance 1.82 A (compared to 1.98 A in the gas),
arranged in a complicated way which we shall not describe. Iodine I2
forms a layer lattice. In Fig. XXIV-3 we show one of the layers, showing

FIG. XXIV-3. — Layer of molecules in la structure.

the diatomic molecules arranged in two sorts of rows. The spacing
between atoms in a molecule is 2.70 A compared to 2.66 A in the gas.
The spacing between different molecules in the same row is 4.79 A;
between rows, 4.89 A on centers; between layers, 3.62 A. This structure
is typical of the sort that one finds in other cases.

Among the hydrides, water has received more attention than the
others and is fairly well understood. The hydrides are hard to analyze
by x-ray diffraction methods, because the hydrogens are not shown by the
x-ray photographs; we must use other evidence to find where they are
located. As we have mentioned earlier, we find that each oxygen is
tetrahedrally surrounded by four other oxygens. Between each pair of
oxygens is a hydrogen, two of the four hydrogens near an oxygen being
joined to it to form a water molecule, the other two being attached to two
of the four neighboring oxygens to form part of their molecules. This
structure, as we have mentioned in Sec. 3, puts each hydrogen of one
molecule near the oxygen of another, so that their opposite electrical

SBC. 4] HOMOPOLAR BOND AND MOLECULAR COMPOUNDS 419

charges can attract each other, helping to hold the crystal together almost
as if it were an ionic crystal. In Fig. XXIV-4 we show a layer of the
crystal, indicating the oxygens by spheres, with vectors drawn out to the
hydrogens. Three neighbors of the upper molecules in the layer, which
we have drawn, are shown; the fourth lies directly above and is not
shown. The molecules are spaced 2.76 A on centers. It is interesting to
notice that this is slightly less than twice the ionic radius 1.45 A of oxygen
given in Table XXIII-2, showing that ice is not entirely different from an
ionic crystal. The molecules have such a spacing that the volume of the
solid is 20.0 cc. per mole, compared to 18 cc. per mole for the liquid, agree-

FIG. XXIV-4 — Layer of ice structure.

ing with the well-known fact that ice is less dense than water. This is
because the molecules in ice are unusually loosely packed. In water, the
individual molecules are actually farther apart, about 2.90 A on centers
compared to 2.76 A in ice, but they are packed so much more efficiently
that there are more molecules in less space and a greater density. In
addition to showing how each oxygen is surrounded by four others, Fig.
XXIV-4 shows the hexagonal structure which is so characteristic of ice
crystals and which is well known from the form of snow flakes.

CHAPTER XXV
ORGANIC MOLECULES AND THEIR CRYSTALS

In Chap. XXIV, we have been talking about substances held together
by homopolar bonds. We have had a rather small list of compounds to
work on; but we have hardly touched the most fertile field for discussing
the homopolar bond. Organic chemistry of course presents the best
organized and most extensive field for the theory of homopolar valence.
The carbon atom can form four single bonds, which tend to be oriented
toward the four corners of a tetrahedron, and this furnishes the funda-
mental fact on which the chemistry of the aliphatic or chain compounds is
based. The great difference between organic and inorganic chemistry is
the way in which more and more carbons can be bonded together to form
great chains, resulting in molecules of great complexity. These carbon
chains form the framework of the organic compounds, the other atoms
merely being attached to the carbons in most cases. In the first section,
we discuss the ways in which carbons can be joined together.

1. Carbon Bonding in Aliphatic Molecules. — In the first place, two
carbon atoms can join together, as for instance in ethane, by a single

H H

bond. Thus ethane has the structure H:C:C:H, as indicated in Sec. 1,

Chap. XXIV. The carbon-carbon distance in this case is about 1 .54 A, a
value approximately correct for the carbon-carbon distance in all aliphatic
molecules with single bonds. The hydrogens are arranged around the
carbons so that the three hydrogens and one carbon surrounding either
carbon have approximately tetrahedral symmetry. The carbon-hydro-
gen distance is presumably about 1.1 A, as in methane. Unfortunately
this carbon-hydrogen distance is almost impossible to determine accu-
rately, since the hydrogen atom represents too small a concentration of
electrons to be shown in x-ray or electron diffraction pictures. Each of
the CH3 groups is able to rotate almost freely about the axis joining the
two carbons, as shown in Fig. XXV-1. Thus there is no fixed relation
between the positions of the hydrogens on one carbon and those on
another.

If more than two carbon atoms join together to form a chain, they
necessarily form a zigzag structure, on account of the tetrahedral angle
between bonds. Thus in Fig. XXV-2 we show propane

420

SBC. 1]

ORGANIC MOLECULES AND THEIR CRYSTALS

421

and butane CHaCI^CEUCHs. The carbon-carbon distances as before
are about 1.54 A and the carbon-hydrogen distances about 1.1 A. On
account of this zigzag nature, the chains with an even number of carbons
act differently from those with an odd number, and there is an alternation

FIG. XXV-l.— The ethane molecule, CH3-OH3.

in physical properties as we go up the scries of chain compounds, the
even-numbered compounds falling on one curve, the odd-numbered on
another. Chains of practically indefinite length can be built up, and it is
interesting to see how the physical properties of the substances change

Propane

Fia. XXV-2.-

Bufane
-The molecules of propane, CH»CH2CH3, and butane, CH8CH2CH2CH8.

as the chains get longer and longer. For example, in Fig. XXV-3 we show
the melting point and boiling point of the chain compounds as a function
of the number of carbon atoms in the chain. The alternation of which we
have spoken is obvious in the melting points, though not in the boiling

422

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

points. This can be explained as follows. In the solids, as we shall
mention later, the zigzag molecules are arranged with their carbons all
in a plane, so that the line joining carbons to their neighbors makes a
sort of saw-tooth shape. Then the molecules with an odd number of
carbons, in which the lines joining the two end carbons point in different
directions at the two ends of the chain, are definitely different from those
with an even number, in which the end lines point in the same direction
at the two ends of the chain. Thus we can expect the solids to show an
alternation in properties. But in the liquid, or the gas, the possibility of
*ree rotation about a carbon-carbon bond results in a great flexibility of

300 -

-200

345

6 7 8 9 10 11 12 13 14 15 16 17
Number of Carbons

FIG. XXV-3. — Melting and boiling points of chain compounds, as a function of the number

of carbon atoms in the chain.

the chain. It can turn and twist, forming anything but an approximately
straight chain, and the result is that the two ends will be oriented quite
independently of each other. Thus there will be no average difference, in
the liquid or gas, between the chains with even and odd numbers of car-
bons. Now the melting point measures the equilibrium between liquid
and solid, so that the alternation in the properties of the solid will show in
the curve, but the boiling point depends only on liquid and gas and will
not show the alternation.

In addition to this feature, Fig. XXV-3 is interesting in that it shows
that the melting and boiling points of the chain hydrocarbons increase
rapidly as the chain gets longer. Not only that, but the viscosity of the
liquids goes up as carbons are added to the chain. These effects are
qualitatively reasonable. Anything tending to hold molecules together
tends to increase the melting and boiling points. Now a chain hydro-
carbon, as far as the carbons are concerned, is much like a string of

SBC. 1] ORGANIC MOLECULES AND THEIR CRYSTALS 423

methane molecules fastened together like a string of beads. In a very
rough way, the valence forces hold the carbons and their hydrogens
together tightly, whereas the Van der Waals forces, the only ones opera-
tive in methane, hold that substance together only very loosely. It is
only reasonable, then, that the boiling points of these long chain hydro-
carbons should be higher than for the short ones. The increased viscosity
of the liquids is also reasonable. After all, a liquid made of long flexible
chains will certainly get tangled up, just as a mass of threads will get
tangled and knotted. Quite literally, if the chains are long enough, the
molecules will tie each other up in knots and prevent flow of one molecule
past another.

H H H H H M

H— C — C— C — C — C — C — H
I I I I I I
H H H H H H

H H

H-C-H H-C-H

H | H H H H H I H H

I I I I I I I I I I

H-C — C- C — C— C-H H-C — C— C — C— C— H
I I I I I I I I I I

HHHHH HHHHH

H H

H H HH H

I I I I I I I

H— C— C— C — C— H H— C — C — C— C— H

K J

Flo. XXV-4. — Isomers of hexane, C«HU.

We have spoken of the simple chain hydrocarbons, in which there is a
single chain of carbons, with their attached hydrogens. These arise when
each carbon of the chain, except the end ones, is joinod to only two other
carbons and two hydrogens. But there is nothing to prevent a carbon
being bonded to three or to four other carbons. In other words, branch
chains can be formed. In this way, new branching compounds are
formed, which in general will have the same chemical composition as some
one of the simple chains, but of course will be rather different in physical
properties. Two such compounds, having the same number of atoms of
each element but with different arrangements, are called isomers. As
an illustration, Fig. XXV-4 shows the formulas for the isomers of hexane.
Of course, on account of the possibility of rotation about C-C bonds, these

424 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

molecules can assume many complicated shapes. When we begin to get
complicated side chains, however, the possibility of rotation is somewhat
diminished, since different parts of the molecule can get into each other's
way with some orientations. This effect is called steric hindrance, and it
operates to stiffen the molecule to gome extent. It can hardly stiffen a
long chain to any great extent, however, and the various branches of a
hydrocarbon with long branches are presumably very flexible.

If the branching process extends very far, there is no reason why the
extremity of one branch cannot join onto another, forming a closed loop.
The simplest compound in which this occurs is cyclohexane, shown in
Fig. XXV-5. A geometrical investigation, made most easily with a
model, will show that with less than the six carbon atoms of cyclohexane

FIG. XXV-5. — The cyclohexane mole- FIG. XXV-6. — The diamond

cule (CH2)e» structure.

it involves considerable distortion of the tetrahedral bond angles to form
a closed ring, but that six carbons can join up with no distortion of the
tetrahedral angles. The atoms are rigidly held in position by their bond-
ing, in this case, so that cyclohexane is a much more rigid molecule than
the flexible chain hydrocarbons.

The branching process and the formation of closed loops can continue
much further than it is carried in cyclohexane. The ultimate in this
line is the diamond. This is the structure obtained when every carbon
is joined tetrahedrally to four other carbons. It makes a continuous lat-
tice filling space, as shown in Fig. XXV-6, which is essentially the same
as the zincblende structure of Fig. XXIII-3 except that it is formed of
only one type of atom. The type of rigidity present in cyclohexane is
found here in its most extreme form. The structure is braced in every
direction, and the result is that diamond is the hardest and most rigid
material known. One can trace out hexagons like cyclohexane in the

SEC. 2] ORGANIC MOLECULES AND THEIR CRYSTALS 425

diamond crystal; one only has to replace the hydrogens in cyclohexane by
carbons and continue the lattice indefinitely to get diamond. Not only
the arrangement but the lattice spacing of diamond is the same as in the
aliphatic chain compounds: the carbon-carbon distance in diamond is
1.54 A, just as in the chains.

The possibility of carbon chains is what leads to the richness of
organic chemistry. A diamond is really a molecule of visible dimensions,
held together by just the same forcoa actirig in small molecules. There is
no reason why there cannot be all intermediate stages between the small
molecules made of a few atoms which we usually think about and mole-
cules of enormous size. Obviously carbon atoms can link themselves
together in innumerable ways, if we only have enough of them. The
organic chemists have discovered a very great number of kinds of mole-
cules, but there seems no reason why they cannot go on forever without
exhausting the possibilities. For by the time a structure is built up of
carbon atoms, many atoms in length, one end can no longer be expected
to know what the other end is doing. There is no reason why one mole-
cule cannot add chains in one way, another in another, and form a con-
tinually increasing variety of new molecules. One gets to the point very
easily where it hardly pays to speak about molecules of a single type at
all, but where one may have chains of indefinite length and things of that
sort. Such situations are presumably met in problems of living matter,
where many molecules are of almost microscopic size. We shall meet one
other field in which we have similar chain formation, and therefore a great
variety of compounds: the silicates, which form a chain of alternating
silicons and oxygens. As the carbon chain leads to the great variety of
materials in organic chemistry and living matter, so we shall see that the
silicon-oxygen chain leads to the great variety of materials in the field of
mineralogy.

2. Organic Radicals. — The carbon chains form the skeleton, so to
speak, of aliphatic organic compounds. But in place of the hydrogens
which are attached to them in the simple hydrocarbons, there are many
organic radicals which can be bonded to the carbons in various positions
of the molecule, thus greatly increasing the complexity of the possible
compounds. We shall mention only a few of the simple radicals in this
section.

In the first place, a single electronegative atom can act like an organic
radical, being substituted for a hydrogen. The monovalent halogens,
F, Cl, Br, I, replace hydrogen freely. Like hydrogen, they can form a
single homopolar bond with carbon. For instance, methyl chloride has

the structure H : C : Cl : . Like methane, it is tetrahedral. The carbon-
H "

426

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

hydrogen distance is presumably about 1.1 A, as in methane, and the
carbon-chlorine distance is 1.77 A. Similarly two, three, or all four, of
the hydrogens can be replaced by a halogen atom, not necessarily all
the same halogens. In these compounds, the carbon-halogen distances
are always approximately the following:

Carbon-fluorine 1.36 A

Carbon-chlorine. . . . 1.77 A

Carbon-bromine 1.93 A

Carbon-iodine 2.28 A. (2.1)

These distances are not far from the ionic radii of Table XXIII-2, which
were 1.30, 1.80, 1.95, 2.20 A respectively for F", C1-, Br~, I". Since
the carbon certainly has nonvanishing dimensions, this means that in
these bonds there is considerable shrinkage from the atomic distances in
ionic crystals, but not so much shrinkage as in some other cases, so that
we should not be surprised to find that the halogen atoms have quite a
little of the properties of negative ions. As a matter of fact, these com-
pounds have rather strong dipole moments: in CH3C1, for instance, we see
from Table XXI V-l that the moment is 1.97 X 10~18 absolute units,
corresponding to about 0.23 of an electron at the distance 1.77 A. We
may conclude, then, that the halogen atoms pull the electrons that they
share with the methyl or other organic group rather strongly toward
them, so that they have quite a little the structure of negative ions. In
the matter of physical properties, we can see from Table XXIV-1 that
replacing hydrogen by halogens increases both Van der Waals a and 6, as
we should expect from the fact that the halogen atoms are much bigger
than hydrogen. Thus for the series CH4, CH3C1, CH2C12, CHC18, CC14,
we have the properties shown in Table XXV-1. The Vs increase fairly

TABLE XXV-1. — PROPERTIES OF SUBSTITUTED METHANES

Boiling
point, °C

CH4 ...
CH8C1 .

2.28 X 1012
7.56

42.6
64.5

-161.4
- 23 7

CH2C12

CHCla

15.38

102

61 2

ecu

20.65

138

76 0

regularly as more chlorines are added, and the amount of increase per
chlorine is not far from 28 cc. per mole, which is half the b value for
C12 (56.2, from Table XXIV-1). The increase in the a's and &'s leads to
an increase in boiling points, as is shown, and as is natural with larger
and heavier molecules.

SBC. 2] ORGANIC MOLECULES AND THEIR CRYSTALS 427

Divalent and trivalent as well as monovalent electronegative atoms
can also attach themselves to organic carbon-hydrogen chains. Thus, in
particular, oxygen plays a very important part in organic compounds. If
an oxygen atom attaches itself by a single bond to a carbon, it has another
bond free, with which to attach itself to something else. This second
bond may go to a hydrogen, in which case we have the organic OH group,

H
forming an alcohol. Thus methyl alcohol is H:C:0:. The OH group,

like the halogens, though it does not exist as a separate ion in the organic
molecules, still has a considerable tendency to draw negative charge to it,
pulling the shared electrons away from the methyl group. Thus the
dipole moment of methyl alcohol is 1.73 X 10~18, almost as large as that
of methyl chloride CH8C1. Instead of being bound to one organic group
and one hydrogen, as in the alcohols, the oxygen may join to two simple
organic groups, forming an ether, like dimethyl ether (CH3)O(CH3),
diethyl ether (C2H5)O(C2H5), etc. Here the organic groups come off
from the oxygen more or less at tetrahedral anglos. The oxygen in the
ethers also has some tendency to draw negative charge to itself, so that
the ethers have a dipole moment, though not quite so large as in the
alcohols. It is plain -from this discussion that the alcohols and ethers
can in a way be derived from water by replacing one or both of the
hydrogens by methyl, ethyl, or more complicated groups. The carbon-
oxygen distances in these compounds are about 1.44 A and the angles
between the bonds are roughly the tetrahedral angle, though there are
considerable variations both in distance and in angle from one compound
to another.

As the alcohols and ethers can be derived from water by replacing
one or both of the hydrogens, so the amines come from ammonia by
replacing one, two, or three of the hydrogens by organic groups. In
Table XXI V-l, for instance, we give properties of methylamine CHsNH2,
dimethylamine (CHs^NH, and trimethylamine (CHa^N. Evidently
a complicated set of compounds can be built up in this way, using more
and more complicated groups to tie to the nitrogen atom.

One of the most important organic radicals is the carboxyl group,
COOH. This group attaches itself by a single bond to any carbon atom,
forming an organic acid. For instance, acetic acid has the structure

H .0:
H:C:C" . That is, one oxygen is held to the carbon by a double

H:O:H

bond, the other by a single bond, so that this latter can also attach itself
to a hydrogen. Even simpler is formic acid, HCOOH. The conspicuous

428 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

tendency of the carboxyl group is for it to lose the H as a positive ion,
leaving the remainder of the molecule as a real negative ion, as for exam-
ple (CH8CQO)~. Then a metallic ion, for instance sodium, can attach
itself, forming for instance sodium acetate, (CH3COO)~Na+. While we
have written this as if it were a real ionic compound, to emphasize this
quality more than in most organic compounds, still such a substance is
not as definitely ionic as in the inorganic salts. For the sodium in this
case furnishes a single electron, just like hydrogen, so that we can consider
it as forming a homopolar bond with the oxygen. In sodium acetate,
for instance, we could write the structure just as in acetic acid above,
with the replacement of the hydrogen by sodium. The distinction
between this way of writing it and the ionic way is simply a matter of
degree, depending on how much the shared electrons between the sodium
and the oxygen are really shared (the homopolar interpretation) or how
much they are definitely held to the oxygen (the ionic interpretation).
The compounds formed in this way, by replacing the H in the carboxyl
group by a metal, are called the esters and are a very important group of
compounds.

It is obvious that in a section such as this, it is impossible to do more
than give a cursory notice to a very few of tho many important organic
radicals. All we have hoped to do is to give some idea of the principles
of valence leading to the bonding. Those same principles continue to
be a guide in the more complicated radicals as well. At least, one can get
some idea of the way in which, by attaching any one of a great number of
radicals to any of the possible points of attachment in a carbon chain
structure, one can get an enormous number of compounds, as one actually
finds in organic chemistry.

3. The Double Bond, and the Aromatic Compounds. — Ethylene, with

H. .H
structure C::C , and acetylene, with structure H:C:::C:H, are

H- 'H

elementary examples of the double and triple bonds between carbons.
The carbon-carbon distances are 1.34 A and 1.22 A, in comparison with
1.54 A for single bonds. This illustrates the general rule that inter-
nuclear distances are less with double bonds than with single, and still
less with triple bonds. There is one feature of interest in ethylene.
Unlike the situation with the single bond in ethane, there is not the
possibility of free rotation about the double bond. That is, the hydrogens
all tend to lie in a plane, as our structural formula would indicate. This
tendency is true in general with double bonds.

The best-known, and at the same time the most puzzling, double
bonds occur in the benzene ring, which is the foundation of the aromatic
compounds. Benzene, CeHc, is a plane hexagonal structure, as shown in

SEC. 3]

ORGANIC MOLECULES AND THEIR CRYSTALS

429

Fig. XXV-7. The carbon-carbon distance is 1.39 A, definitely less than
the 1.54 A spacing in the chain compounds, but slightly greater than the
1.34 found with double bonds in ethylene. Each carbon is bonded to
only three other atoms, rather than four,
and these three lie in a plane at angles
of 120° to each other, rather than being
arranged tetrahedrally in space. Car-
bon in such a compound seems to behave
definitely differently from its behavior
in the aliphatic compounds. Something
of a guide to the interpretation is fur-
nished, however, by the structure of
graphite, a form of carbon having the
same relation to the aromatic compounds
that diamond has to the aliphatic com-
pounds. The graphite structure is
shown in Fig. XXV-8. It will be seen
that the atoms are arranged in sheets,
forming regular hexagons similar to the
benzene ring in the sheets. Not only

are the hexagons of the same shape as the ring but they are of
almost the same size; the carbon-carbon spacing between neighbors in a
sheet is 1.42 A, very slightly larger than the value 1.39 A in benzene.

FIG. XXV-7.— The benzene molecule,
C6H6.

FIG. XXV-8. — The graphite structure.

The sheets, however, are 3.4 A apart, a much greater distance. This is
as if the valence bonds acted wholly within the separate sheets, and only
Van der Waals forces, or similar weak forces, held the sheets together and
were not able to pull them very close together. This impression is made
stronger by the physical properties of graphite. It is a very soft material

430 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

and a good lubricant, and the lubricating properties arise because one of
the sheets slides over another, indicating that the forces between sheets
are small and easily overcome. It is strongly suggested that the graphite
structure is similar to the benzene ring, the carbon being in the same form
in both substances. In harmony with this point of view, it is found that
benzene rings can join together the same way that the hexagons do in
graphite. Thus if we join two hexagons we have naphthalene, CioH8,
shown in Fig. XXV-9. More and more rings can continue to join up in
this way, so that we have the same possibility of continuous extension and
complication with the aromatic compounds that we have with the ali-
phatic ones with their carbon chains.

FIG, XXV-9. — The structure of naphthalene,

Not only can benzene rings join each other, but also the hydrogens
can be replaced by various organic radicals. Thus, for instance, one
hydrogen in benzene can be replaced by an OH group, as in the alcohols
among the aliphatic compounds. The resulting compound, CeH6OH, is
phenol, or carbolic acid. Or one hydrogen can be replaced by a carboxyl
group, COOH, resulting in benzoic acid C6H6COOH. If the hydrogen
in the carboxyl is replaced by a metallic atom, we have a benzoate of
the corresponding metal, A great number of other radicals can replace
the hydrogens. And we observe that if more than one of the hydrogens
of a ring is replaced by some other radical, there are a number of possible
positions for the radicals on the ring. Thus if two of the hydrogens of a
benzene ring are replaced by chlorines, we have dichlorbenzene, with
three possible structures, as indicated and named in Fig. XXV-10. The
three types are called orthodichlorbenzene, metadichlorbenzene, and
paradichlorbenzene, according to the position of the second chlorine with

SBC. 3]

ORGANIC MOLECULES AND THEIR CRYSTALS

431

respect to the first. Three such compounds can often have quite different
properties. In particular, it is plain that in the case shown the para-
compound is symmetrical and can have no dipole moment, while the
ortho- and meta- compounds are quite unsymmetrical and have a con-
siderable dipole moment, since the chlorine as usual pulls a good deal of
negative charge to itself.

(a) ortho-

Fio. XXV-10. — The three forms of dichlorbenzene. (a) ortho-, (6) mota-,
(c) paradichlorbenzene.

H. ./

•C*

• •

• c '

»H

.C.

H'.C/C..%C-'H

H
(a)

H
(b)

V
(c)

Fio. XXV-11. — Three alternative valence structures for benzene.

The valence properties of the carbon in the benzene ring do not at
first sight fit in with our rules for determining valence. To explain the
valence by homopolar bonds, we must assume that each carbon is bound
to one of its neighboring carbons by a single bond, to the other one by a
double bond, and to the hydrogen or other radical by a single bond. Two
possible ways of doing this are shown in Fig. XXV-11 (a) and (6). This
structure would suggest that the hexagon should not be a regular one, but

432

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

that instead the pairs of carbons held by double bonds should be closer
together than those held by single bonds. Experimentally this is not the
case, however. X-ray and electron diffraction methods show that the
hexagon is a regular one. Furthermore, if there were real difference
between the single and double bonds, a molecule with two radicals in the
ortho-position and a double bond between would be different from a
molecule with the same two radicals in the same position but with a
single bond between. No such difference is observed. For all these
reasons one concludes that the structures of Fig. XXV-11 (a) and (6) do
not correspond exactly with the facts. It is generally considered that
the true state of affairs is a sort of combination of the two structures of
(a) and (6), in which the pair of electrons forming a double bond is some-

FIG. XXV-12. — Structure of tho naphthalene crystal as determined by x-ray analysis.
(From J. Monteath Robertson, Science Progress 32, 246 (1937) by permission of the author.)

times found as in one of the structures, sometimes as in the other. This
may perhaps be symbolized as in (c) of the figure, where the extra electrons
are shown in a position where they could contribute somewhat to both
bonds. This valence structure of benzene, though it does not fit in with
ordinary ideas of valence very satisfactorily, proves to be very well
explained in terms of the quantum theory.

4. Organic Crystals. — Most organic compounds are definitely formed
of molecules, arranged in some fairly closely packed form in the crystals.
As with the simpler homopolar compounds, the molecules are held
together largely by Van der Waals forces, though in some cases where
the molecules have dipole moments, there are electrostatic forces as well.
The intermolecular forces, as with the simpler compounds, are small
compared to the energy of dissociation in most cases, and the distances
between molecules are large compared to the interatomic distances

SEC. 4]

ORGANIC MOLECULES AND THEIR CRYSTALS

433

within a single molecule. We shall not attempt any cataloguing of the
different types of crystal structure found in the crystals; almost every
compound has its own form of structure. Often they are complicated on
account of the complicated shape of the molecules. As a single example,
we show in Fig. XXV-12 a diagram of the naphthalene crystal. This is
the direct result of the x-ray diffraction experiments and shows a projec-
tion of the crystal along a certain axis, by giving contour lines indicating
the density of electric charge. The naphthalene molecules stand out
clearly, and we see by comparison with Fig. XXV-9, in which the double-
ring structure of this molecule is shown, that the plane of the molecule is
inclined to the plane of the paper. The other organic crystals which havo
been investigated' all have the same general properties of representing
fairly closely packed structures of distinct molecules.

The general remarks regarding energy which we have made in Chaps.
IX and XXII apply here also. The interatomic forces within the mole-
cules can be well represented by Morse curves, as in Chap. IX, Sec. 1.
In Table XXV-2 wo givo values of D and re for the most important bonds

TABLE XXV-2. — CONSTANTS OF BONDS IN ORGANIC MOLECULES

Substance

Bond

D, kg.-cal.

D, electron
volts

ret A

Chain hydrocarbon. . . .
Chain hydrocarbon
Alcohols and ethers
Amines . . . .

C— H
C— C
C— 0
C— N

101
83
83
67

4.34
3.60
3.6

2.88

1.1
1.54
1.44

Fluorine-subst. hydrocarbons
Chlorine-subst. hydrocarbons
Brominc-subst. hydrocarbons .
lodine-subst hydrocarbons

C— F
C— Cl
C— Br
C— I

125
79
66
51

5.40
3 41
2.83
2 2

1 36
1.77
1.93
2 28

Carbon double bond

C::C

150

6.46

1.34

Carbon triple bond

C:::C

200

8.7

1.22

Bond energies taken from Pauling, J. Am. Chem. floe., 54, 3570 (1932).

occurring in organic compounds, similar to the values for diatomic mole-
cules given in Table IX-1. In contrast to these high values of heats of
dissociation, the heats of vaporization of the simpler compounds are
small. Thus in Table XXV-3 we give heats of vaporization, again in
kilogram-calories per mole, of some of the organic compounds we have
mentioned. The latent heats, as we see from the table, are of the same
order of magnitude as those for inorganic compounds shown in Table
XXIV-3. The alcohols stand out as having rather high latent heats, for
the same reason that water does: their OH groups tend to hold the mole-
cules together by electrostatic attraction. In addition, there is a general
trend toward higher latent heats as we go down the list. This is natural,

434 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXV

TABLE XXV-3. — LATENT HEATS OF VAPORIZATION OF ORGANIC COMPOUNDS

Substance

Formula

Latent heat

Methane

CH4

2.3

Ethane

C2Hfl

3.9

Methyl chloride . . ....

CH8C1

4.7

Methyl alcohol

CHaOH

9 2

Ethyl alcohol

C2H6OH

10.4

Propane

C3H8

4.5

Chloroform

CHC1,

8.0

Acetic acid

CHsCOOH

5.0

iso-Butane

CH(CHs)3

5.5

T&-Butane

\u/4jtl.io

5.6

Diethyl ether

(C2H6)20'

7.0

Naphthalene

Ciolls

9.7

w^Decane

^IO'H'22

8.5

Latent heats are in kilogram-calories per mole, taken from Landolt's Tables, and are for the lowest
available temperature in each case.

for the substances of higher melting and boiling points lie toward the
end of the list, and if the entropy of melting is approximately constant for
the various substances, as it is found to be, we see that the greater the
melting or boiling point, the higher the corresponding latent heat. Aside
from these points, the important thing to observe in comparing Tables
XXV-2 and XXV-3 is the fact that heats of vaporization are much less
than heats of dissociation, so that molecules in organic compounds tend
to have separate and independent existence.

CHAPTER XXVI
HOMOPOLAR BONDS IN THE SILICATES

In the preceding chapters we have discussed the general nature of
the homopolar bond and have seen how it operates in certain inorganic
molecules and in the organic compounds. In the present chapter we
shall continue the discussion to include the silicates, the substances form-
ing the foundation of a great many of the rocks and minerals. Silicon and
oxygen, like carbon, can form continuous chains, and in this way can build
up a skeleton structure for a great variety of compounds. First we shall
describe this chain structure and then we shall go on to show how it is
found in substances of many different sorts.

1. The Silicon-Oxygen Structure. — Silicon has four outer electrons,
the same as carbon. Like carbon, it can form four homopolar bonds,
arranged in a tetrahedral structure. Thus we have SiH4, mentioned in
the preceding chapter, and substitution products like SiF4, SiCl4, etc.
But the characteristic compounds are those in which oxygen atoms are
held by the four bonds. Since oxygen is divalent, each oxygen must also
be bonded to something else, and the simplest case is that in which each
oxygen is bonded also to a hydrogen, forming Si(OH)4, with the structure

:0:

H:0:Si:O:H.

:9:
H

In this structure, orthosilicic acid, the oxygens surround the silicons
tetrahedrally, and a hydrogen is somewhat loosely attached to each
oxygen. If the hydrogens are detached as positive ions, we leave behind
the orthosilicate radical (SiO4)~~4, a tetrahedral structure similar geometri-
cally to the sulphate (804) — and perchlorate (C1O4)~ ions mentioned in
Chap. XXIII. The orthosilicate radical occurs as a negative ion in the
orthosilicates, typical ionic crystals in which the orthosilicate ion, and
certain positive ions, form an ionic lattice. An example is magnesium
orthosilicate, or olivine, Mg2(Si04), a regular structure of Mg++ ions and
orthosilicate radicals. In this, as in practically all silicates, the silicon-
oxygen distance is approximately 1.60 A. This can be compared with the
carbon-oxygen distance of 1.44 A found with single bonds in the alcohols

435

436

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XX VT

and ethers, and seems to indicate a similar bonding in this case, the silicon
atom of course being larger than the carbon atom.

There is no reason why, instead of losing its hydrogen, one of the oxy-
gons attached to a silicon cannot attach itself to another silicon. This

:0: :0:
leads to the structure (Si2O7)~6, with valence structure :O:Si:O:Si:O:.

(sio4y

In Fig. XXVI-1 we show the (SiO4)~4 and (8i2O7)-6 ions, which are found
in ionic crystals, occurring in nature as minerals. The positive ions
found associated with them include a variety of metallic ions, such as

Ca++, Zn++ Na+, Mg++, A1+++ etc. We
shall not go into the crystal structure of
these substances, for they are rather com-
plicated for the most part, and yet do not
show any new principles of structure that
we have not already met in our discussion
of ionic crystals, with one exception. This
exception is the fact that the natural
minerals show a rather strong tendency
to substitute one positive ion for another.
That is, the minerals are not always of the
same chemical composition. Often a posi-
tive ion is missing from the lattice, and its
place taken by another positive ion, with-
out seriously distorting the lattice. Such
substitutions take place only when the two
ions in question have about the same ionic
radii. For instance, from the table of ionic
radii, Table XXIII-2, we find that Ca++
has a radius of 0.95 A, and Na+ a radius of 1.05 A. These are near enough
alike so that calcium and sodium can substitute for each other in the
lattice. Of course, these ions have different charges, and such a substitu-
tion would upset the electrostatic equilibrium, resulting in a crystal
having a net electric charge. This is not allowed, and to balance it other
substitutions are always made. There is, for instance, a silicate contain-
ing Ca"1"* or Na+ interchangeably, and Mg4""*" and A1"H4' interchangeably.
By having suitable relations between the amounts of these four ions, the
crystal can always be kept electrically neutral and yet may have a variable
composition. Such a situation is very often the case in minerals.

2. Silicon-oxygen Chains. — In Fig. XXVI-1, we have seen how
two or more silicons can be joined together by oxygens. This process in
the ideal case can continue indefinitely, forming endless chains and in
practice forming chains of varying length. The simplest such chain is

FIG. XXVI-1. —Structure of silicate
ions.

SBC. 2]

HOMOPOLAR BONDS IN THE SILICATES

437

that in which each silicon is joined, through oxygens, to two other silicons.
This is shown in Fig. XXVI-2. The valence structure can be symbolized

:Si:6:Si::Si::Si etc. This structure, like the orthosilicate ion, is

:0:":0:":0:":0:

electrically charged and is sometimes called the metasilicate ion. A

single unit of the structure evidently has the structure (SiO3) .

FIG. XXVI-2. — Section of metasilicate chain, (SiO8) — . (a) View looking directly down.

(6) Perspective.

Each unit of the metasilicate ion has two oxygens carrying negative
charges, which are able to form homopolar bonds with other atoms. The
simplest way they can do this is to take up hydrogens, forming bonds
that are partly polar, but mostly homopolar, with the hydrogens, and
resulting in metasilicic acid, H2SiO3. It is evident that the single unit
H2SiO3 has no separate existence; the whole structure arises when the
silicates are joined together to form chains. Similarly the oxygens might
take up alkali ions, for example sodium, forming sodium metasilicate,
Na2SiOa, again made out of endless chains. If the chains are finite, as of
course they really would be, the proportion of Na increases, until finally
if the chains are only one silicon long, we have sodium orthosilicate,
Na4SiO4, which we have mentioned before. There is the possibility of a
continual variation from one of these limiting cases to the other, as the
average length of the chain changes. These substances, intermediate

438 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVI

between Na2SiO8 and Na4Si04, have remarkable properties. They are
commonly called water glass. If dissolved in water and left to set, they
form a viscous, jellylike mass. The explanation is the same that we gave
in the preceding chapter for the viscosity of the long chain hydrocarbons:
the long stringlike molecules, on account of the possibility of rotation
about the bonds joining neighboring atoms, are flexible and get tangled
up in each other. Similarly metasilicic acid, H2SiO8, in water, forms the
remarkable substance known as silica gel. This again is a jellylike
material, getting stiffer and stiffer as it is allowed to set. It is remark-
able, like all jellies, in the very small amount of material required to
stiffen up a large mass of water. Two per cent or so of EUSiOs in water
will transform the whole mass into a gel. Presumably the action is
somewhat more complicated than simply the entanglement of the stringy
molecules with each other. A hydrogen atom of one chain, carrying a
net positive charge, can happen to come close to an oxygen of another with
its negative charge. The resulting electrostatic attraction, holding the
chains rather tightly together, is analogous to the attraction of hydrogens
for oxygens found in water, discussed in Sees. 3 and 4, Chapter XXIV.
By such bonding of one chain to another, the structure could be stiffened
greatly, so that it would take on some of the properties of a solid. Of
course, with only occasional joinings such as we could have with a small
percentage of metasilicic acid in a great deal of water, the rigidity would
be much less than in a real solid, but that is just the characteristic
behavior of jellies, which act solid and yet can be greatly distorted by a
small force. These inorganic jellies are interesting in that they probably
show us what is the essential structure of the organic jellies like gelatin, for
instance. They are of great practical importance, in that they furnish
the fundamental structure in cement and concrete. They can be pre-
pared in water as fairly fluid materials but become gradually stiffer as the
chains join together, and as water evaporates out of them.

Metasilicate chains of varying length are found not only in jellylike
materials, but also in crystals and minerals. In such cases, the chains
are straightened out and really become indefinitely long, running parallel
to each other through the whole length of the crystal. Positive ions are
arranged in regular positions between the chains, so as to make the whole
structure electrically neutral. An example is diopside, CaMg(Si()3)2.
These materials are really ionic crystals, in which the negative ions,
instead of being of atomic dimensions, are lengthened out in one direction
to be of large-scale size.

More important than the simple metasilicate chain, in the structure
of crystals, is the double chain found in the amphiboles. This structure
is shown in Fig. XXVI-3, looking down on the top. A unit of the chain
contains four silicons, eleven oxygens, as we see by counting, and it has

SBC. 3] HOMOPOLAR BONDS IN THE SILICATES 439

six oxygens capable of forming bonds, so that its structure is
This chain, unlike that of the metasilicates, has a type of cross-bracing
which makes it quite rigid. Thus substances of this structure do not form
jellies or viscous fluids, since these depend on the flexibility of the chains.
The characteristic form of materials containing the double chain is the
crystalline form, the chains extending parallel to each other through the
crystal, and the interstices filled in with positive ions in such a way as to
make the structure neutral and hold it together. The ionic forces holding
the crystal together, however, are not so strong as the homopolar forces
acting between atoms within the chain. Thus mechanical forces insuffi-
cient to break the chains can pull them apart and separate them into
fibers. Such a fibrous structure is characteristic of the amphiboles and
the double chain compounds. The most familiar example is asbestos,

FIG. XXVI-3. — Double silicate chain, view looking directly down.

a very strongly bound material with high melting point, and yet one in
which the fibers can be separated from each other with ease. These
simple fibrous materials are interesting in that they suggest the way in
which the more complicated fibers of organic chemistry are built up.
They also contain chains, generally double or more complicated chains
to get the necessary mechanical rigidity, the atoms and bonds concerned
being typical of organic compounds. Since even the simplest of naturally
fibrous materials are quite complicated, we shall give no examples of
organic fibres.

3. Silicon-Oxygen Sheets. — The chain structure of Fig. XXVI-3 can
be extended to form a two-dimensional sheet, as shown in Fig. XXVI-4.
A unit of this structure has the composition (Si2OB) , but as with the
chains this individual unit has no existence by itself. Sheets of this kind
are found in the structure of mica. This mineral consists of silicon-
oxygen sheets piled parallel to each other, alternating with sheets contain-
ing metallic ions, as Al, Mg, etc., along with oxygen and hydrogen. The
sheets are stronger than the ionic bonds holding them together, so that
mechanical force easily cleaves the crystal into thin sheets. This very
characteristic property of mica thus follows directly from its valence

440 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVI

structure. Of course, it should not be thought that one can really cleave
the crystal down to single silicon-oxygen sheets; this would demand much
more delicate technique than can be used. But the tendency to split one
sheet from another is great enough so that cleavage easily occurs along
these planes.

Another very important group of materials, the clays, are formed
from silicon-oxygen sheets. In this case, however, the sheet is not the
simple one shown in Fig. XXVI-4, but each sheet is bound by homopolar
valences to another sheet of different composition. For instance, in
kaolinite, one form of clay, one starts with a silicon-oxygen sheet. Above
that, there is an aluminum layer, with as many aluminums as silicons,
bonded to the oxygens. The other bonds of the aluminums extend to a

FIG. XXVI-4. — Silicon-oxygen sheet.

still higher sheet of oxygens, and finally these oxygens hold hydrogens by
their unused valences. This makes an electrically neutral sheet with no
unshared homopolar bonds, so that successive sheets are held to each
other only by polarization and Van der Waals forces. In this they
resemble the sheets of graphite, described in Sec. 3 of the preceding chap-
ter. Like graphite, the clays have a soft and plastic structure, resulting
from the ease with which one sheet can slide over another. They are
quite different from mica, in which the bonds between sheets, being
ionic, hold the sheets comparatively firmly together. The clays are also
different from mica in that large crystalline sheets cannot be obtained.
It has been suggested that this may be because the two sides of the sheet,
the silicon side and the aluminum side, may have slightly different natural
sizes, so that one sheet may be under compression, the other under ten-
sion, in the material. This would mean that thin sheets would have a
tendency to pucker and break, preventing the formation of large sheets.

SBC. 4) HOMOPOLAR BONDS IN THE SILICATES 441

4. Three-dimensional Silicon-Oxygen Structures. — There are a
number of ways in which silicons and oxygens can be joined together into
three-dimensional structures, filling all space. These structures are all
alike in that each silicon is surrounded by four oxygens and each oxygen
by two silicons, so that the formula is Si02. The valence bonds are
completely used up in holding the silicons and oxygons together, so that
there is no possibility of adding other atoms to the structure. These
materials, then, being bound in all directions by valence bonds, are very
strong and rigid, something like diamond, though not quite so hard
Perhaps the simplest of these struc-
tures to visualize is cristobalite,
shown in Fig. XXVI-5. This
structure is very similar to the dia-
mond structure, as shown in Fig.
XXV-6. One can start with the
diamond structure, place a silicon
in the position occupied by each
carbon, and an oxygen midway
between each silicon and each of its
four neighbors, and one has the
cristobalite structure. This is a
slightly undesirable structure for
the atoms to form, however, from Fla' XXVI'5-Tho -istobaiite structure.

the standpoint of the oxygens. We note that it requires the two
valences of the oxygen to be diametrically opposite each other. On
the other hand, as we have seen earlier, oxygen prefers to have its
two valence bonds make roughly the tetrahedral angle 109° with each
other. It is likely, as a matter of fact, that the oxygens in cristobalite
are really somewhat out of the line joining neighboring silicons,
just in order to have their two valences at an angle with each other.
They oscillate around the midpoints of the lines joining silicon atoms,
however, rather than being found always on one side or the other,
as one tells from the crystal symmetry, which is the same as for the
diamond lattice. In the other crystalline forms of SiO2, in contrast to
cristobalite, the two bonds formed by each oxygen make much more
nearly the angle that is preferred. The best-known form, of course, is
quartz. This is a decidedly complicated crystal, atoms being arranged
with a screwlike symmetry, as is shown by the fact that quartz rotates
the plane of polarization of polarized light. We shall not give its
structure, but merely point out that as in other cases each silicon is
surrounded tetrahedrally by four oxygens, each oxygen by two silicons,
and that the angle between the two bonds from each oxygen is about
145°.

442

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVI

From the fact that SiOa exists in several forms, we see that the whole
structure of this material is not determined by the nearest neighbors of a
given atom. That is, one can have each silicon surrounded by four
oxygens, each oxygen by two silicons, in many different ways. This fact
is responsible for the amorphous nature of fused quartz, the simplest form
of glass. This is an irregular structure, with no indication of a repeating
regularity. And yet the immediate neighbors of any given atom are
arranged as they are in cristobalite. It is rather hard without a model

to see how this can be done, but we show in
Fig. XXVI-6 a two-dimensional schematic
representation of the same situation. In
this case, the black dots represent the silicons,
and each is surrounded by three circles repre-
senting oxygens (instead of four as in the
three-dimensional case), while each oxygen
is between two silicons. The diagram (6),
representing the glass, shows that by only
slight distortions of the bond angles of the
oxygens, the molecules can be joined in a
quite irregular structure. Some of the bonds
in the glass are under more strain than
others, and are fairly easy to break. Thus
as the temperature is raised, some bonds will
break and the material will lose a little
strength. If bonds break, it may be that
they can join again with a different arrange-
FIO. xxvi-6.— Schematic repre- ment, resulting in permanent distortion of
sentation of crystal and glass. t}ie material. It is in this way that the glass
flows at high temperatures. There is no sharp melting point as there is
with a crystal, for this demands that in all parts of the material the bonds
weaken at the same temperature. Rather we might say that on account
of strain some parts of the glass have higher melting points, some lower,
so that the softening is gradual.

The actual glasses generally contain Na20, or other constituents, in
addition to SiO2. Now if we look back, we notice that the proportion of
oxygen to silicon is higher in the compounds where the atoms form chains
or sheets than in the three-dimensional lattice of SiO2. The reason is
that the chains or sheets contain some oxygens bonded to only one silicon
rather than two, the other bond being attached to some other atom. By
analogy, we see that in a soda glass, the extra oxygens coming from the
Na20 will join onto silicons and will introduce some oxygens bonded to
only one silicon. Obviously the other bond will hold the oxygen to
sodium atoms. Such a glass, then, will have fewer cross links or bonds

SBC. 4] HOMOPOLAR BONDS IN THE SILICATES 443

between atoms than pure 8102 and consequently will be softer, and will
begin to soften at a lower temperature. These properties of course are
characteristic of the soft glasses. In any case, however, the glass struc-
tures, like the SiC>2 structures, have a very strong and rigid bracing, so
that it is natural that they, like diamond, which also has valenco binding
in three dimensions, should form very hard substances.

CHAPTER XXVII
METALS

Most of the elements are metals. There are six inert gases and
twelve or fourteen definitely electronegative elements, leaving approxi-
mately seventy metallic elements. We have already seen how they
combine with negative radicals to form ionic substances, and how on
occasion they become bound in organic compounds, by a bond partly
ionic and partly homopolar. But the elements themselves take the
familiar metallic form, and mixtures or compounds of one metal with
another form alloys, similar in most respects to pure metals. As we know,
the metals are fairly strongly bound substances. Their melting points
are rather high, mercury and gallium being the only ones that are liquid
near room temperature. Thus we must assume that there are fairly
strong bonds between the atoms of the metal. As we saw in Chap.
XXII, these bonds resemble the homopolar type, but it is generally
considered best to treat them as in a class by themselves, the metallic
bonds.

1. Crystal Structures of Metals. — Metals are divided fairly definitely,
though not perfectly sharply, into two groups: the ordinary metals and
a group of peculiar metals verging on the nonmetals. These peculiar
metals come at the ends of the rows of the periodic table and include
approximately the following: C, Si (both of which have some metallic
properties, though we have previously treated them as nonmetals), Ga,
Ge, As, Se (which is partly metallic), In, Sn, Sb, Te, Tl, Bi. The ones
we have just named are peculiar in that their bonds are very similar to
homopolar bonds. Thus Si, Ge, Sn, all crystallize in the diamond struc-
ture, like carbon, so that we can consider that they have four homopolar
valences binding them to their four nearest neighbors. Arsenic, anti-
mony, and bismuth have peculiar structures in which each atom is bound
to three nearest neighbors, corresponding to the three homopolar valences
which we should expect these elements, like nitrogen and phosphorus, to
form. The binding joins atoms in layers, resulting in brittleness and a
tendency to cleave easily along the layers. And selenium and tellurium,
with two valences like oxygen and sulphur, tend to form chains, each atom
being bound to two nearest neighbors, a formation found alsx> in sulphur,
though not in oxygen.

The ordinary metalsT in contrast to these peculiar elements, do na
form homopolar valence bonds in any ordinary sense. Each atom has to<

444

SBC. 1] METALS 445

many nearest neighbors to share electron pairs with each; they try^to
share electrons^with their neighbors but there are not enough to go around.
This has 9flY*!Tftl *p«"1+-g- In the first place, there is no such directional
property to the bonds as we have with real homopolar~compounds.
Instead of tending to come off in tetrahedral directions or something of
that sort, the metallic bonds can attract other atoms no matter in what
directions they may lie. Thus the atoms act as if they were spheres
without preferred directions, as far as the crystal structure is concerned,
and the ordinary metals crystallize in forms that are characteristic of
close packed spheres. Secondly, since the valence bonds arc shared,j*p
to speak, between several pairs of atomsT they are not so strong as real
homopolar valences. Thiy? a metallic crystal, though much stronger
than one held togcther_hy Van der ^Vqalfi fniwn, is wnn-lror t.lmn nno, Ifrn
diamond or quartz, held fry p^re frQnfiopolar y***™*™ Furthermore, the
fewer electrons an atom has to share with its neighbors, the weaker its
bonds, so that the alkali metals, with only one electron each outside
RlQaed-flhclls. are the weakest metals mechanically and the lowest molting
pnes, the alkaline earths are next, and sp HP, white thn iron, puling jnm, anH
platinum groups have thojrtrongest bonds. Finally, the electrons hold^
ing the metal together aTenot localized between definite pairs of atoms, as
they are in homopolar bondsT but can ^ove fl.rnimHJ

Found between one pain of atoms, sometimes between another.
result of this freedom of motion of the electrons is the property of
conductivity, the most charactgngtic distinguishing feature of f.hp.
With this preliminary view of metals, we shall now consider their crystal
structure and later go on to some of their other properties.

The ordinary metals crystallize in one of three structures: the facc-
centered cubic, hexagonal, or body-centered cubic. Of these, the face-
centered cubic and hexagonal close-packed structures have been described
in Chap. XXIV, Sec. 4. The face-centered cubic structure is shown in
Fig. XXIV-1 and the hexagonal close-packed structure in Fig. XXIV-2.
Sometimes in the metals the latter structure is slightly distorted, being
elongated along the vertical axis of Fig. XXIV-2, so that it is no longer
a close-packed structure. The body-centered cubic structure consists of a
simple cubic lattice, with atoms at the corners of the cubes and also at the
centers. Thus it resembles the caesium chloride structure, shown in
Fig. XXIII-2, except that its atoms are all alike, instead of their being
two sorts of atoms as in the caesium chloride structure. In the face-
centered cubic structure each atom has twelve equidistant neighbors. In
the hexagonal close-packed structure each atom also has twelve equi-
distant neighbors, but if the structure is distorted so that it is no longer
close packed, the six atoms in the basal plane will be at a different dis-
tance from the six neighbors in adjoining planes. In the body-centered

446 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

cubic structure each atom has eight equidistant neighbors. In addition
to these structures, several elements crystallize in the diamond structure,
with four equidistant neighbors. The other structures, in which the
unusual metals crystallize, will be described later.

We now give, in Table XXVII-1, the crystal structures and lattice
spacings of the metals. For each metal we give the structure, abbreviat-
ing face-centered cubic, hexagonal, body-centered cubic, and diamond
structures by f.c., hex., b.c., di., respectively. Those not crystallizing
in these structures are indicated by an asterisk and will be taken up later.
In addition, we give the distance to the nearest neighbor (or to the two
sets of nearest neighbors, in the case of hexagonal structures), in ang-
stroms. It will be observed that some of the metals crystallize in more
than one form; this is the phenomenon of polymorphism, mentioned in
Chap. XI, Sec. 7. The rare earths are omitted from our list, since
practically none of them can be obtained in the metallic form.

In examining Table XXVII-1, the first thing to notice is the regu-
larity of the interatomic distances from metal to metal. Going through a
row of the table, as from potassium to selenium, we observe that the
largest distance is for the alkali, the next largest for the alkaline earth,
with continuing decrease for one or two more elements. Then there is
surprising constancy for the rest of the elements of the series. We have
already mentioned that the alkalies, having the fewest electrons for
sharing, are the most loosely bound metals, and the alkaline earths the
next most loosely bound; the interatomic distances bear this out, since
large interatomic distance corresponds to loose binding. Of course, in
addition to this trend, there is the natural increase in size as we go to
heavier atoms of the same type, as from lithium to caesium. This
tendency is much less marked for the more tightly bound metals, however.
For instance, cobalt, rhenium, and iridium, similar elements in the three
long periods, have almost exactly the same interatomic distances.

It is interesting to observe that this constancy of interatomic distances
seems to hold irrespective of the particular crystal structure which the
element may have. Thus cobalt has exactly the same interatomic dis-
tance in its hexagonal and face-centered modifications, lanthanum and
thallium have almost exactly the same, and iron changes only slightly
from one structure to another. As we look over the table, while the
structures appear to be arranged almost at random, there seems to be no
correlation between spacing and crystal structure. For instance, in the
series chromium, manganese, iron, cobalt, we have a body-centered cubic
structure, a complex one (manganese), a face-centered and a body-
centered cubic form of iron, and a hexagonal and a face-centered form of
cobalt, and yet the spacings, 2.49, 2.50, 2.48, and 2.57, 2.71, show only a
general trend, rather than erratic variation. This fits in with the idea

SBC. 1] METALS 447

TABLE XXVII-1. — CRYSTAL STRUCTURE AND INTERATOMIC DISTANCES IN METALS

(Angstroms)
Abbreviations: b.c. body-centered cubic; f.c. face-centered cubic; hex. hexagonal;

di. diamond;* other structures

Li b.c. Na b.c. K b.c. Rb b.c. Cs b.c.

3.03 3.72 4.50 4.86 5.25

Be hex. Mg hex. Ca f.c.
2.28 3 20 3 93
2 24 3 19

Sr f.c.
4.29

Ba b.c.
4.35

B Al f.c. Sc
2.85

3.58

La hex., f.c.
3.72,3 73

Ti hex.
2.95
2.90

Zr hex.
3.23

3.18

Hf hex.
3 32
3 33

Vb.c.
2 63

Ta b.c.
2.88

Cr b.c.
2 49

Mo b.c.
2.72

Wb.c.
2 73

Mn *
2 50

Fe f.c., b.c.
2 57, 2 48

Ru hex.
2 69
2 65

Os hex.
2.71
2 67

Co hex., f <•.
2 71

Rh f.c.
2 69

Ir f.c.
2 70

Ni f.c.
2.49

Pd f.c.
2 74

Pt f.c.
2.76

Cu f.c.
2.55

Ag f.c.
2 88

Au f.c.
2.87

Zn hex.
2.65
2.94

Cd hex.
2.97
3 30

Hg*
2 99

Ga *
2 56

In *
3.24,
3.33

Tl hex., f.c.
3 45, 3 43

Si di. Ge di.
2.35 2.43

Sndi.
2.80

Pb f.c.
3.49

As *
2.50

Sb *
2.88

Bi *
3.10

Se *
2.32

Te *
2.88

448 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

that the metallic atoms act very much like spheres, as far as their packing
is concerned, so that the interatomic distances of Table XXVII-1 measure
the diameter of these spheres, irrespective of the particular way they
happen to be packed together. There is another conclusion to be drawn
from the fact that the lattice spacing seems to be independent of crystal
structure. Surely the energy of the crystal will depend principally on the
lattice spacing. Then we must expect that the same metal can exist in
different structures with almost exactly the same energy. This moans
that it should be possible to change from one structure to another very
easily, and that very slight and apparently trivial circumstances can
determine which structure really has the lowest energy and is stable.
This makes the polymorphism of the metals seem reasonable, and it
makes it comprehensible that there should be such apparent lack of order
in the crystal structure of the elements. Looking at Table XXVII-1, it-
is very difficult to draw any general conclusions as to why the various
elements should have the structures they do. There seems to be an
almost random distribution of the various structures among the elements.
But this is not unnatural if the structure depends, not on very fundamen-
tal properties like valence but on relatively minor details of the atomic
structure.

We must still say something about the structures indicated with a
star in Table XXVII-1. Manganese is a very peculiar and anomalous
exception to the general order of the elements. It is the only definite
metal, far from the nonmetals in the table, which has a complicated
structure. The structure is really complicated : there are 58 atoms in the
unit cell. No one has suggested a very convincing reason why this
should be so. It becomes a little less remarkable, however, when one
studies the crystal structure of alloys. It is found that some alloys of
quite ordinary metals have very complicated structures, as for instance
one of the forms of the copper-zinc alloy found in brass. These com-
plicated structures come from the mixture of two different kinds of
atoms. It has been suggested that possibly manganese atoms exist in two
different forms in its crystal, perhaps corresponding in some indirect
way to the two valences of manganese, and that its structure is really
more like that of an alloy than of a pure metal. However that may
be, it remains a peculiar and unexplained exception.

The other complicated structures come at the ends of the groups in
the periodic table, and as we have said they correspond to something
more like homopolar bonds than metallic bonds. We have already
commented on germanium and tin (the so-called gray modification of
tin), which crystallize in the diamond structure, corresponding to the
four homopolar bonds which they could form. They are of course very
different from diamond in their properties, though silicon is between a

SBC. 1J METALS 449

metal and a nonmetal in its behavior, forming a sort of bridge between
diamond and the others. Thus the melting points are 3500°C. (dia-
mond), 1420°C. (silicon), 959°C. (germanium), 232°C. (tin), indicating a
rapid but not discontinuous decrease of tightness of binding as we go
from diamond, with pure homopolar bonds, to germanium and tin, which
are much more metallic. In electrical properties, diamond is a very
good insulator, silicon is a substance of fairly high resistance but no
higher than some alloys, and germanium and tin are fairly good conduc-
tors. In appearance, diamond of course is transparent, while silicon,
germanium, and tin are all grayish materials, of fairly metallic appear-
ance. In mechanical properties, silicon possesses some of the hardness
of diamond but germanium and tin are soft.

FIG. XXVII-1. — Layer of atoms from tho bismuth crystal.

The next substances that we shall consider are arsenic, antimony, and
bismuth. Their crystals consist essentially of layers similar to that shown
in Fig. XXVII-1. In such a layer, each atom is bound to three neighbors,
by bonds making almost a right angle with each other, but slightly opened
out, so as to approximate the tetrahedral angles. If we now stack such
layers on top of each other, wo can arrange things so that each atom will
also have three neighbors in the layer above, which are only slightly
further away than the three neighbors in its own layer. These six neigh-
boring atoms, three in its own layer and three in the next, will surround
the atom very much like the six nearest neighbors in a simple cubic
lattice. In other words, the bismuth structure can be regarded as a
slightly distorted simple cubic lattice, distorted in such a way that each
atom has three nearest neighbors. As we have already mentioned, the
bonds to nearest neighbors have something of the character of homopolar
bonds, and each atom has just enough electrons to form three bonds.
But the homopolar character is not strong, the substance resembling
a metal more than a homopolar compound. Still, the layer structure has
an important effect on the properties of the substances. Single crystals
of arsenic, antimony, and bismuth cleave very easily, splitting between
layers, so that the metals are very brittle. And while the electrical
conductivity is fairly good in directions parallel to the layers, it is very

450 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

poor in the direction normal to the layer, as if the electrons had trouble
jumping from one layer to another.

Selenium and tellurium have structures in which, as we have said
before, the atoms form chains, each atom being joined to its two neighbors
in the chain. The chains are not straight, but helical, would up like
springs. The electrical conductivity is much poorer at right angles to the
directions of the chains than along them, as we should expect. There is
one remarkable mechanical property of tellurium: when it is put under
hydrostatic pressure, although of course the volume decreases, the length
along the direction of the chains increases. It is practically the only
substance known that expands in any direction under hydrostatic pres-
sure. The interpretation is that the sidewise pressure tends to straighten
out the springs, thus lengthening them, though they contract laterally
enough to decrease the volume as a whole.

2. Energy Relations in Metals. — In Chap. XXIII we have seen that
we can understand the equation of state of the ionic crystals in a good
deal of detail. We shall now show that similar progress can be made in
considering the energy relations in metals. As in that chapter, we shall
begin by considering the empirical values of compressibility and its
change with pressure, and shall see whether the value of the constant 7
computed from them agrees with the value found from the thermal expan-
sion, as Grtineisen's theory would indicate that it should. We give the
necessary information in Table XXVII-2. First we give the quantities
Pi and P2, of Chap. XIII, determined from the experimental values of
compressibility and its change with pressure. Then we give values of 7

2 P

found from Eq. (4.6), Chapter XIV: 7 = — ^ + -™ For comparison,

o /i

we give values of 7 found by Griineisen's relation (4.16) of Chap. XIII,
thermal expansion = 7xCV/T0. The agreement is definitely poorer than
with ionic crystals, though still correct as to order of magnitude.1

The relations shown in Table XXVII-2 tell nothing about the theory
giving the interatomic energy as a function of volume. We cannot get
nearly so far with this as we could with ionic crystals. There the attrac-
tive forces between ions were simple Coulomb forces, . which we could
calculate exactly, though we had to approximate the repulsive forces by
an empirical formula, which we took to be an inverse power term. For
metals, the theory of the metallic bond is so complicated that the forces

1 Even the order of magnitude is incorrect for the less compressible metals, most
conspicuously for W and Pt, according to Bridgman's original measurements of P\
and Pa. A revision of Bridgman's value of Fa for iron, however [see Bridgman, Phys.
Rev., 67, 235 (1940), Slater, Phys. Rev., (1940)], makes possible a revision of the P2's
for other incompressible metals, resulting in the figures tabulated in Table XXVII-2,
and bringing about agreement with Gruneisen's value of -y in almost all cases.

SBC. 2]

METALS

451

have been calculated for only a few of the simplest metals, the alkalies,
and even there the results are available only in the form of numerical
calculations, though there are analytic approximations that work fairly
well for sodium, which turns out to be the simplest of all metals theoret-
ically. We have pointed out in Chap. XXII, however, that the bonds
between atoms in a metal are not entirely different in their properties
from homopolar bonds. Thus it is not unreasonable to approximate the
TABLE XXVII-2. — QUANTITIES CONCERNED IN EQUATION OF STATE OF METALS

_2 P*

3 /*,

(Gruneisen)

0.115 X 10ia

0.149 X 101*

0 63

1 17

Be . .

1.17

6.2

4 63

0.064

0.160

1.83

1.25

0 338

0 77

1 62

0 757

1 46

1 27

2 17

K...

0 028

0.090

2.55

1.34

Ca. .

0 168

0.23

0.71

1 04

5.6

2.7

Cr. . . .

1 93

6.4

2.6

Mn.. .

1 24

8.1

5.8

2.42

Fe.. .

1 73

4.0

1.7

1.60

Co .

1.85

5.1

2.1

1.87

NL...

1.90

5 4

2 2

1.88

Cu...

1.39

3 6

1 9

1 96

Rb .

1 48

Sr.

0 122

0.13

0 40

Zr.

0 91

5 6

5 5

2.88

1.57

Pd .

1.93

5 9

2 4

2.23

1.01

3 2

2 5

2.40

Cd..

0.70

2 6

3 04

Cs..

1.29

0.098

0 123

0.60

0.284

0 34

0.53

1.11

3 2

2.22

2.09

1.75

3.40

1 7

1.62

2.78

11.

3 3

2.54

Au. . .

1.73

3 03

Pb * .

0.42

1 30

2 42

2.73

Measurements of compressibility, from which Pi and Pi are computed, are from Bridgman, Proc.
Am. Acad., 58, 165 (1923), 62, 207 (1927), 63, 347 (1928), 64, 51 (1929), 68, 27 (1933), 70, 71 (1935),
72, 207 (1938). Values of y by Grtineisen'e method are taken from the article by Grtlneisen, " Zustand
dee fasten Kcrpera," in "Handbuch der Physik." Vol. X, Springer, 1926.

452 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

internal energy of a metal by a Morse curve, as in Eq. (1.2) of Chap. IX.
That is, we write

t/o = I/(e-2o(r-ro) - 2e-a(r~ro)). (2.1)

In Eq. (2.1), C/o represents the energy of the crystal at the absolute zero
as a function of the distance r between nearest neighboring atoms, where
ro represents the value of this distance in equilibrium, when f/o has its
minimum value. The quantity L is the energy required to break up the
metal into atoms at the absolute zero, or the latent heat of vaporization
at the absolute zero, a quantity that can be found from experimental
measurements of vapor pressure, as we saw in Chap. XI. The quantity
a is an empirical constant. We shall determine L from the experimental
value of the vapor pressure, ro from the experimental density at the
absolute zero, and a from the compressibility. To compare with our
formulation of Chap. XIII, we expand Eq. (2.1) in Taylor's series, finding

[fr _ A2 /r _ A3 1

(ar0)2(~ ) + (ar0)3f-- ) • • • . (2.2)

Then, comparing with Eq. (3.4) of Chap. XIII, which is

f/o = t/oo + ATcrgjJ PI^—-^ ~ 9(F? - PJ)
we have

+ 2]. (2.4)

In terms of these expressions, we have

2 , JT2 CWO ,1 te* ~\

y == r J5~ == i o* \2-O)

To use these formulas, we must know the geometrical constant c,
given by the relation that the volume per atom is erg. In the body-
centered cubic structure, we have a cube with an atom at each of the
eight corners and one in the center. Each of the eight corners is shared
equally by eight cubes, so that each of these atoms counts one-eighth
in the cube in question. Thus the cube contains f + 1 = 2 atoms. The
semi-body diagonal is ro, so that the body diagonal is 2rp, and the side of
the cube is 2r0/\/3. Thus a cube of volume (2r0/-\/3)3 contains two
atoms, or a cube of volume 4r8/(3)** contains one atom, so that

c = — for the body-centered cubic structure. (2.6)

SEC. 2] MKTALS 453

In the face-centered cubic structure, a cube contains eight atoms at the
eight corners (counting as f = 1 atom within the cube), and six at the
centers of the six faces (counting as f = 3 within the cube). Thus each
cube contains four atoms in all. The semi-face diagonal is r0, the face
diagonal 2r0, the side of the cube \/2ro, the volume 2*VJ. Thus, since
this volume contains four atoms, the volume per atom is (l/\/2)rj!, and

c = — 7= for the face-centered cubic structure. (2.7)

By similar methods we can show

c = •$& for the diamond structure, (2.8)

- for the hexagonal close-packed structure. (2.9)

It is natural that the face-centered and hexagonal close-packed structures,
both corresponding to the close packing of spheres, should have the
same value of c. In case the hexagonal lattices are not close packed, there
is an additional correction factor in c, which we shall not evaluate.

We can now use the observed heat of evaporation, the observed r*o,
from Table XXVII-1, and the observed PI, or reciprocal of the compressi-
bility, from Table XXVII-2, to determine the values of L and of a. In
doing so, we neglect the difference between PI and PJ, or between the
compressibility at room temperature and its value at the absolute zero.
Then, using Eqs. (2.4) and (2.5), we can compute 7, Griineisen's constant,
in terms of these quantities. In Table XXVII-3 we give the necessary
information. It is seen from the table that the computed values of 7
agree fairly well with the observed values. This is an indication, there-
fore, that the Morse curve is not an entirely unsuitable potential energy
function for a metal. The agreement between computed and observed 7
is, in fact, rather surprisingly good and indicates the degree of accuracy
obtainable from a theoretical calculation of thermal expansion from the
compressibility and specific heat.

We have already mentioned, in connection with the lattice spacings,
that the alkali and alkaline earth metals are the most loosely bound, the
transition group metals like the iron group the most tightly bound.
This can be easily seen from Table XXVII-3. A tightly bound crystal
has a large latent heat and a low compressibility, or large value of PI. It
is striking to see how these quantities change, for instance from caesium
to tungsten. The latent heat of caesium is 18.7 kg.-cal. per mole, of
tungsten 203, almost the lowest and highest respectively for any two
metals. And the value of Pi is .014 X 1012 dynes per square centimeter
(or about 14,000 atm.) for caesium, and 3.40 X 1012 for tungsten. The

454 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

TABLE XXVII-3. — QUANTITIES CONCERNED IN ENERGY RELATIONS OP METALS

y (Eq. 2.5)

Li 36.0

Na 26.2

Mg 36.6

Al 67.6

K 21.9

Ca 42.8

Cr 89.4

Fe 96.5

Co 74.0

Ni 98.1

Cu 81.7

Zn 31.4

Rb 20.6

Mo 156.

Ag 69.4

Cd 27.0

Cs 18.7

W 203.

Pt 125.

Au 90.7

0.80

0.67
1.14
1.21

0.53
0.84
1.64
1.45
1.76
1.49
1.41
1.70

0.47
1.58
1.39
1.93

0.44
1.51
1.68
1.58

1.55

1.58
2.15
2.05

.98

2.37
2.20
2.72
2.19
2.13
2.68

1.48
2.48
2.33
3.35

,1.48
2.39
2.65
2.60

Values of latent heat of vaporization L are taken from Landolt's Tables. They are in kilogram-
calories per gram mole. Values of a, in reciprocal angstroms, are computed by Eq. (2.4), using data
from Tables XXVII-1 and XXVII-2. Values of y, computed from Eq. (2.5), are to be compared with
values computed by GrUneisen's method, tabulated in Table XXVII-2.

compressibility of caesium is the highest not only for any metal but for
any known solid at ordinary temperatures, while that of tungsten is
among the lowest known. The values of PI show a continuous increase
from caesium to tungsten, having the values .014 X 1012, .098, 288, 1.11,
2.09, 3.40; it is to be presumed that if the latent heats of the intermediate
elements were known, they would show a continuous increase in a similar
way. It is interesting to note that toward the ends of the periods in the
table the binding again becomes somewhat less tight. Thus the com-
pressibilities increase and the latent heats decrease quite strikingly, in
the series Ni-Cu-Zn, Pd-Ag-Cd, and even more in Pt-Au-Hg, the latent
heat of mercury, like its melting point, being the lowest for any metal, and
that of platinum being among the highest. Thus it is quite clear that the
transition groups of metals are much more strongly bound than their
neighbors either before or after them in the table.

While we are not prepared to say much about the quantitative details
of the forces holding a metal together, still it is not hard to understand in a

SEC. 2]

METALS

455

qualitative way the specially tight binding of the transition elements.
The alkali metals have only one outer electron per atom. This forms a
bond which must be shared with all eight neighbors of an atom. The
alkaline earths on the other hand have two outer electrons to be shared,
and the succeeding elements have three, four, etc. outer electrons. Thus
the number of electrons available to form metallic bonds increases rapidly
as we go through a period of the table. It is only natural that this
increases the tightness of binding. It is a phenomenon not entirely differ-
ent from that met in homopolar binding, where double and triple bonds
give considerably tighter binding than single bonds; only in the metallic
case, each bond is even less than a single bond in strength. As we go
beyond the transition elements, there is a reversal of this tendency. The
electrons added in tho transition elements tend to form completed shells
and to be no longer available for bonding, so that only a few electrons per
atom operate to hold the crystal together. This tendency has not pro-
gressed very far with the elements copper, silver, and gold. Though
these behave chemically to some extent similarly to the alkalies, they arc
obviously much more tightly bound, copper for instance having a latent
heat of 81.7 against 21.9 for potassium, and a value of Pl of 1.39 X 1012
against .028 for potassium. Plainly more than one electron per atom is
operative in the binding of copper, though the next elements, Zn, Cd, Hg,
are more nearly comparable to Ca, Sr, Ba.

The values of the constants a in Table XXVII-3 are of the same order
of magnitude as the values for diatomic molecules, given in Table IX-1,
indicating therefore that the binding is not entirely different from homo-
polar binding. It is particularly interesting to compare these constants,
and in fact all the properties, of the metals with the corresponding
properties of the diatomic molecules Li2, Na2, K2. These values are
given in Table XXVII-4. We give also the values for C2 for comparison

TABLE XXVII-4. — COMPARISON or CONSTANTS FOR MOLECULES AND SOLIDS

TO, A

L, kg.-cal.

a,A~i

Li2

2.67

26.4

0.83

Li metal

3.03

36.0

0.80

Na2

3.07

17.6

0 84

Na metal

3.72

26.2

0 67

3.91

11.8

0 78

K metal

4.50

21.9

0.53

1.31

128.

2.32

Diamond

1.54

199.

2.17

456 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

with the properties of diamond, though of course it is not a metal. While
there is no exact parallelism between the molecules and the solids, still
there are strong resemblances. The interatomic spacing in the molecule
is in each case between 80 and 90 per cent of the spacing in the crystal.
This is to be explained by the fact that the bond is concentrated between
the two atoms of the molecule, pulling them together, while in the lattice
it is shared among the neighbors. The latent heats of evaporation are
in every case greater than the heats of dissociation of the molecule but
less than twice the heat of dissociation. Finally, the values of the con-
stants a for the metal, while they do not agree exactly with those for the
molecules, are of the same order of magnitude, indicating rather close
similarity between the binding in the two cases.

3. General Properties of Metals. — In many ways the most con-
spicuous property of metals is the electrical conductivity. We have
mentioned that this results from free electrons, electrons not definitely
tied up in any definite atom or homopolar bond but free to wander from
one bond to another. We shall treat conductivity in detail later on. We
may mention now only one general fact, bearing on our picture of conduc-
tion. Surely, we should expect at first sight that the more electrons there
were to carry the current, the bigger the conductivity would be. This
being so, we might suppose that metals with two, three, or more valence
electrons per atom would conduct better than those with only one. The
opposite is the case, however. When reduced to proper units, the con-
ductivities of the alkali metals, and of copper, silver, and gold, the
elements with one electron per atom, are greater than for any other
substances. The interpretation of this is that, though the other elements
have more electrons, still these electrons form bonds which are more like
valence bonds, are localized more definitely between pairs of atoms, and
consequently are less free to travel around. The limiting case is diamond,
where there are just enough electrons to form bonds and there is no
conductivity. The alkalies, with the fewest electrons, have at the same
time the freest electrons, for they must continually circulate about to
produce the binding between different pairs of atoms.

Another characteristic property of metals is ductility. A metal can
be bent and deformed without breaking, much more than most other sub-
stances, as for instance ionic crystals. This is particularly striking with
the close-packed metals. In a metal, the bonds act quite indiscriminately
between any closely neighboring atoms. They do not depend greatly
on the exact orientation of the atoms, as the real homopolar valences do.
Thus a distortion of the lattice, so long as it docs not involve much net
change of the interatomic distances, will not greatly change the energy
and will not be opposed by a large force. And even a large distortion
does not weaken the lattice and may in some cases even strengthen

SEC. 3] METALS 457

it. This is shown in the phenomenon called cold-working. A single
crystal of a metal can often be deformed by a process called gliding. As
indicated in Fig. XXVII-2, showing a projection of a schematic lattice,
there are planes through a crystal which contain unusually many atoms,
and so are unusually smooth. It is easy for part of the crystal to slide
over the rest, slipping or gliding along one r> O O O O O
of these planes. The possibility of gliding, v Q O O O O
however, obviously depends on the per- o\ O O O O O
fection of the crystal. If there are local O^v O O O O

irregularities in the glide planes, these will Q O \ O O O O
act like roughnesses on two surfaces of a Q O\O O O

bearing and will increase the friction, pro- Q O O\ O O O

venting gliding. Now if a crystal glides, o O O \O O

while the sliding over the planes will be O O O O\O O

fairly smooth and will not result in much O O O O 'V O

distortion of the lattice, still some atoms O O O O O \ O

are bound to be pulled out of place. These FIG. xxvii-2.— Schematic repre-

.,i ,, , ri i ... . , sentation of a glide plane.

will then act like irregularities in prevent-
ing further gliding, so that the crystal will have become hardened by its
distortion. If this process is long continued, the metal can become very
much harder than in its crystalline form. This is very well known in
metallurgy, where metals are hardened by being hammered, drawn, and
otherwise distorted.

The most conspicuous examples of gliding, and hardening by cold
work, are found in those noncubic crystals that have only one set of

planes over which gliding is possible.
The best-known case is zinc. This is a
hexagonal crystal, but so far from close
packed that the separations between an
atom and its neighbors out of the basal

/\ Y A plane are about ten per cent greater than

between the atom and its neighbors in its
own plane. That is, it is almost a layer
structure, the binding perpendicular to
Unstrefched ^"* the layers being considerably weaker

S+ retched than the binding in the layers. Thus

Fio. XXVII-3.— Gliding of a zinc ,, . . f. .. , ..

crystai. the zinc crystal slips or glides very easily

parallel to the planes or at right angles

to the hexagonal axis. A single unstrained zinc crystal can be greatly
distorted by the application of very small forces, by such gliding. For
instance, a crystal in the form of a rod, with the layers inclined to the
axis of the rod, as in Fig. XXVII-3, can be stretched out, as shown in the
figure, by the application of a force that seems unbelievably small when

458 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVII

judged by our ordinary ideas of the strength of metals. The stretching
is plainly produced by gliding, for in the actual stretched material one
can see the surfaces along which gliding has occurred, in a way shown in an
exaggerated form in the figure. But once the gliding has occurred, the
rod is strengthened so much that it cannot be pushed back into its original
form by any small force. The gliding and hardening are easy to see and
understand in this case, because there is only one set of slip planes. In a
cubic crystal, with many sets of planes on account of symmetry, some
parts of the crystal will glide along one plane, other parts along others,
and the phenomenon is much harder to visualize and interpret. But the
essential feature is the same, that the undistorted crystal glides easily
but is distorted enough by gliding so that it is very much hardened. The
ordinary materials that we meet in practice are made up of many small
crystal grains and are ordinarily much distorted in the process of manu-
facture. Thus they are very hard compared to an undistorted single
crystal, which has properties that at first sight make it appear very
peculiar and unfamiliar.

Still another characteristic property of metals is their ability to form
alloys or compounds of different metals with variable composition. We
have already mentioned alloys to some extent in Chap. XVII, when we
were talking about equilibrium between phases. There we saw that some
alloys consist of a single phase, while others are mixtures of two phases,
each phase having characteristic crystal structure and other properties.
Generally in these cases each phase exists in small crystal grains, the
whole alloy consisting of a mixture of these two kinds of crystals, in close
juxtaposition to each other. We now consider the nature of one of the
pure phases, which itself generally contains two or more elements in
variable proportions. These phases are of two sorts, substitutional and
interstitial compounds. In a substitutional alloy, atoms of one type in
the lattice are simply absent and are replaced by atoms of the other
type, so that the final lattice is not much affected by the substitution.
Such alloys occur when the two (or more) types of atom in question are
about the same size and form similar crystals. Thus, from Table
XXVII-1, we see that the interatomic distance in nickel is 2.49 A and in
copper 2.55 A, quite similar distances, indicating that the atoms are
about the same size. Furthermore, nickel and copper both form face-
centered crystals. It is then not surprising that any fraction, from zero
to 100 per cent, of the atoms in a nickel lattice can be replaced by copper
atoms. There is, in other words, a complete series of substitutional alloys
of copper and nickel for any composition. All these alloys have the same
face-centered structure, with a lattice spacing that varies smoothly from
one limit to the other. The other extreme is the interstitial alloy. This
is found where an atom much smaller than the other atoms of the crystal

SBC. 3) METALS 459

enters into combination. In this case, the small atom does not substitute
for another atom but goes into a hole between other atoms, filling one of
the interstices between atoms, whence the name. A characteristic exam-
ple is the alloying of carbon with iron to form steel. The interatomic
distance in diamond is 1.54 A and in iron 2.57 A, showing that the carbon
atom is much smaller than that of iron. The carbon atoms fit between
iron atoms, distorting the lattice a good deal. This has one obvious
effect: by distorting the lattice, the interstitial atoms prevent gliding
and harden the metal. It is for this reason that steel is so much harder
than iron. The alloying atoms in an interstitial compound interfere
with the lattice much more than in a substitutional compound, with the
result that far fewer can be introduced into the lattice. Thus only a few
per cent of carbon can be introduced into iron, in contrast to the case of
nickel and copper where any amount of copper can be introduced into
nickel.

CHAPTER XXVIII
THERMIONIC EMISSION AND THE VOLTA EFFECT

In the preceding chapter we took up the properties of metals, but we
have said very little about their most characteristic feature, their electrical
behavior. An understanding of the electrical conductivity of metals, and
its bearing on the free electrons in the metal, is essential to a proper treat-
ment of the metallic bond and of the forces holding the metal together.
We shall take these questions up in the next chapter. Before doing
so, however, we shall take up a related problem, the thermionic emission
of electrons from hot metals. By studying the interaction between
electrons and metals we can get some information about electrons inside
metals, more or less in the same way that by studying the interaction
between electrons and atoms, in such problems as resonance and ioniza-
tion potentials, we can get information about atomic structure. The
information is not so detailed as with atoms, but nevertheless both the
practical importance of the problem itself and its bearing on the structure
of metals furnish justification for studying it at this point.

In Chap. XX, Sec. 3, we spoke about the detachment of electrons
from atoms, and in Sec. 4 of that chapter we took up the resulting chem-
ical equilibrium, similar to chemical equilibrium in gases. But electrons
can be detached not only from atoms but from matter in bulk, and par-
ticularly from metals. If the detachment is produced by heat, we have
thermionic emission, a process very similar to the vaporization of a solid
to form a gas. The equilibrium concerned is very similar to the equilib-
rium in problems of vapor pressure, and the equilibrium relations can be
used, along with a direct calculation of the rate of condensation, to find
the rate of thermionic emission. In connection with the equilibrium of a
metal and its electron gas, we can find relations between the electrical
potentials near two metals in an electron gas and derive information about
the so-called Volta difference of potential, or contact potential difference,
between the metals. We begin by a kinetic discussion of the collisions of
electrons with metallic surfaces.

1. The Collisions of Electrons and Metals. — When a slow electron,
with a few volts' energy, strikes a clean metallic surface, there is a very
large probability that it will be captured by the metal, and a very small
probability, depending on its energy and direction, that it will be reflected.
In other words, the most likely collisions are inelastic ones, in which the
electron loses all its energy and never gets out again. The mechanism is

460

SBC. 1] THERMIONIC EMISSION AND THE VOLTA EFFECT 461

simple. The electron can penetrate a few atoms deep, but it is likely to
have a collision without electronic excitation with each atom it strikes,
losing energy to produce thermal vibration of the atoms. A few such
collisions reduce its energy far enough so that it does not escape from
the metal. For it requires considerable kinetic energy for an electron to
leave a metal. It is attracted to the metal by the so-called image force,
which we shall discuss later, and it requires several volts' energy to escape.
This can be indicated graphically as in Fig. XXVIII-1. Here we show
schematically the potential energy of an electron, inside and outside a
metal. If an electron with one volt kinetic energy outside the metal
enters it, it will have a kinetic energy of a number of volts inside the
metal, since its total energy, kinetic plus potential, must remain constant,
and the potential energy is lower inside the metal. If the electron's
energy is lowered by inelastic collision from its original value E to a value

— Outside metah- — •* >

Fio. XXVIII-1. — Potential energy of an electron at a metallic surface.

E' below the potential at infinity, it will be unable to escape. This is
what usually happens. The exceptional case, reflection, comes when the
direction of the electron happens to be reversed at its first, or practically
its first, collision with an atom within the metal, so that it comes out again
without chances of further collisions. In such a case it will have lost only
a small amount of energy, and the collision will be almost elastic.

When faster electrons strike a metal, the situation is more complicated.
In the first place, secondary electrons can be emitted, electrons liberated
from the metal by the impact of the primary electron. In some cases it
is not possible to know whether the electron coming out of the metal is a
secondary or the primary, except in cases where both come out, so that
more electrons leave the metal than enter it, an important case prac-
tically. But generally from the velocities it is possible to draw conclu-
sions as to whether the emitted electron is primary or secondary. The
secondary electrons mostly have fairly low energies, in the neighborhood
of ten or twenty volts; of course, electrons of such energy cannot be
emitted unless the primary electrons had a suitably larger amount of
energy. Among the few primary electrons that are reflected, some are
elastically reflected, as with slow electrons, and have lost almost no
energy. But some have lost just about as much energy as the secondaries

462 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVIII

acquire, and this is easily interpreted. It is assumed in such cases that
the primaries have ionized atoms within the metal, losing energy in the
process, and have then happened to be reversed in direction and to escape
from the metal. The secondaries are supposed to be the electrons ionized
from the atoms. Since the probability of ionization, or the collision cross
section for ionization, has its maximum when the energy of the impinging
electron is something like twice the ionization potential, it is reasonable
that these secondaries should have energies of a number of volts.

In addition to these reflection phenomena, which are observed with
all types of surfaces, there are some very special effects observed when
the surface is a crystal face of a single crystal of metal. Electrons of
certain definite energies and angles of incidence have an abnormally
large reflection coefficient for elastic reflection, sometimes several times as
great as that found with neighboring angles and energies. This phe-
nomenon is called electron diffraction and has been of great importance in
developing wave mechanics, for in it the beam of electrons acts like a
wave, and the directions of abnormal reflection are determined by inter-
ference conditions, exactly analogous to the interference conditions in
x-ray diffraction. While electron diffraction is of great importance in
theory and is useful in determining the crystal structure of surface layers,
it is not very important for our present purpose.

Now let us ask what are the inverse processes to the collisions we have
considered. Of course, the inverse to an elastic collision is also an elastic
collision, so we do not need to consider this case further. The inverse
to the inelastic capture of a slow electron by a metal is obviously a process
by which an electron in the metal happens to receive a number of collisions
in succession by atoms of the metal, in which its energy increases instead
of decreasing in the normal way, thus giving it enough energy, as E in
Fig. XXVIII-1, so that it can escape from the metal. Speaking in a less
detailed way, it is a process in which the electron inside the metal, by
thermal agitation, happens to get an abnormal energy and escapes. The
inverse to the emission o'f a secondary is a complex process in which two
electrons, a fast and a slow one, strike the metal, the slow one recombines
with an atom of the metal, which gives up the extra energy to the fast
electron, throwing it out of the metal. Of these processes, the only
important one is the ejection of a fast electron by a metal, and this is
what is known as thermionic emission. The higher the temperature the
more likely one is to find electrons fast enough to escape, and the greater
the thermionic emission. Our task in the next sections will be to calculate
the rate of thermionic emission, the number of electrons per second
3mitted by a square centimeter of surface, for this is a very important
quantity in practice. But it is a very difficult thing to find directly, as so
many rates of reaction are difficult to compute. To find it directly, we

SBC. 2] THERMIONIC EMISSION AND THE VOLT A EFFECT 463

have to know a great deal about the structure of the metal. In the next
chapter we shall investigate this and shall be able to make a direct calcula-
tion. But for the present we can proceed otherwise. First we investigate
the equilibrium of a metal and a gas of electrons outside it. Then we
find the rate of the process by which an electron strikes the surface and
sticks, the inverse to thermionic emission. By combining those two
pieces of information, we can derive the rate of thermionic emission, with-
out knowing any details about the metal at all.

2. The Equilibrium of a Metal and an Electron Gas. — The problem of
equilibrium between a metal and the electron gas surrounding it is very
much like that of the vapor pressure of a solid, which we have considered
in Chap. XI, Sec. 5, except that it is not the solid itself which is evaporat-
ing, but only the electrons from it. Nevertheless, with proper interpreta-
tion, we can use the same formulas that were developed in that section.
For equilibrium, we must equate the Gibbs free energy first of a piece of
metal and the electron gas outside it, then of the same piece of metal
lacking a certain number of electrons, plus the gas formed outside the
metal by the original electrons plus those just separated from the metal.
The resulting formula for the pressure is just like Eq. (6.4), Chap. XI:

T T

^-Lo CTdT (T _

e RTe * ° RT*J ° (2.2)

Here the quantity i is the chemical constant of the electron gas, defined
in Eq. (3.16), Chap. VIII. The interpretation of the quantities L0 and
Cp must be examined in detail. The quantity Lo is clearly the work
required to remove a mole of electrons reversibly from the metal at the
absolute zero, leaving the metal in its lowest possible electronic energy
level. Thus Lo is the latent heat of vaporization of electrons from the
metal at the absolute zero, or the thermionic work function. For most
metals it is of the order of magnitude of four or five electron volts (or
around 100 kg.-cal. per gram mole). We shall find its interpretation
in terms of a model in the next chapter. The quantity Cp is the difference
between the heat capacity of the metal and its heat capacity when a mole
of electrons is removed. Thus it is really more exact to write the expres-
sion No(dCp/dN), where dCp/dN is the change in heat capacity per unit
change in the number of electrons. Of course, one cannot remove a mole
of electrons from a mole of metal, for that would give it an enormous
electric charge, but the quantity NQ(dCp/dN) is really merely the change
in specific heat per mole of electrons removed, and can be found theo-
retically by removing only a few electrons, resulting in a negligible charge.
Since changing the number of electrons in the metal would result in a

484 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVIII

corresponding surface charge on the surface of the metal, we may regard
our quantity as the heat capacity of the surface charge, per mole. Now
this is a very difficult quantity to find, either theoretically or experi-
mentally. Perhaps it is better to interpret it in terms of Eqs. (5.6), (5.7),
and (5.8), Chap. XI, which become

dL 5p Ar aCp
df = 2R " N'dN'

L = Lo + \RT - JQ AT0^ dT. (2.3)

The quantity L represents, here, the heat absorbed when a mole of elec-
trons is evaporated at temperature T. That is, it is the change in
enthalpy, or T times the change of entropy, on evaporation, or is tho
change of internal energy plus PV, since we can neglect the term PV for
the electrons in the metal. Thus it is the latent heat of evaporation
of electrons at temperature T. Then we can interpret the quantity
No(dCp/dN) merely in terms of the change of latent heat with tempera-
ture, a change that can actually depend on many factors. We may

expect in any case that the term I Nv(dCP/dN)dT will not be of an order

of magnitude greater than RT. For a temperature of 3000° abs., a high
temperature for thermionic experiments, it will then be of the order of
magnitude of 6 kg.-cal. per gram mol, small compared to I/0, which is of
the order of 100 kg.-cal. per gram mol, so that presumably the latent
heat does not change by a very large fraction of itself in the usable range
of temperatures.

Finally the quantity i in Eq. (2.2) is the chemical constant of an
electron gas. This is given by Eq. (3.16), Chap. VIII:

(2 4)

All the quantities in Eq. (2.4) are familiar except gQ, which we now dis-
cuss. As we saw in Chap. XXI, Sec. 2, the orientation of the electron
spin is quantized in space, so that it has two possible stationary states
of orientation, in which the spin is directed in either of two opposing
directions. This results in having 00, the a priori probability of the lowest
stationary state, equal to 2, so that we have

3. Kinetic Determination of Thermionic Emission. — We have now
found the pressure of an electron gas in equilibrium with a metal, at an

SEC. 3] THEUMION1C EMISSION AND THE VOLTA EFFECT 465

arbitrary temperature, by thermodynamic methods. Next we shall
investigate the same problem by the kinetic method. We shall find the
number of electrons per second that enter the metal from the electron
gas, at the equilibrium pressure. But for equilibrium this must equal
the number of electrons per second emitted from the metal, which is the
quantity we seek. To find the number of electrons entering the metal per
second, we first find the number striking the metal per second, which is a
simple calculation from the kinetic theory of gases. Then we assume that
a fraction r of these electrons will be reflected, (1 — r) captured. From
what we have said, for slow electrons such as we are dealing with, r is
small compared to unity, so that (1 — r) is almost unity. Actually the
reflection coefficient will presumably depend on the velocity of the imping-
ing electron, but for simplicity we shall ignore this dependence, proceeding
as if r were a constant.

Let us first find the number of molecules of a perfect gas, at pressure*
P, temperature T, striking a square centimeter of surface per second.
Since the electrons act like a perfect gas, this calculation will apply to
them as well as to an ordinary gas. The calculation is similar to that of
Sec. 3, Chap. IV, whore we found the pressure by tho kinetic method.
Consider the molecules whose momentum lies in the range dpx dpv dpz.
As in Fig. IV-2, the number of such molecules crossing one square centi-
meter perpendicular to the x axis per second will be the number in a prism
of base one centimeter, slant height along the direction of tho velocity
equal to p/m, or altitude equal to px/m. The volume of this prism is
Paj/ra, and the number of suoh molecules per unit volume, by Eq. (2.4),
Chap. IV, is

dpx dpy dp,. (3.1)

Thus the number of molecules crossing the square centimeter per second,
found by integrating over all values of pv and pg, but only over positive
values of px, is

™ pt* f " p*v f °° Pa*

^e 2mkT dpA e *™kT dpy\ e *mkT dp,

o Wl ,7 — 00 J — oo

)-H = P(2nntT)-Hf (3.2)

using the formulas Eq. (2.3) of Chap. IV. The number (3.2) gives the
number of electrons striking a square centimeter of metallic surface per
second, if the pressure is given by Eq. (2.2). Since the fraction (1 — r)
of these enter the metal, we have as the number of electrons entering the
metal per second

466 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVIII

For equilibrium, the number of electrons leaving the metal per second
must equal this value. Thus the electron current is this multiplied by tho

-V - CTdT tTNJ*!* dT

electronic charge e, or is A'T*e Te J ° RT** ° °~^ , where

A! = (1 — r) — p — = (1 — r) 120 amp. per sq. cm. per degree,2

V = ^ (3.4)

_£

If No(dCP/dN') were zero, F" (3.4) would have the form A'T*e T.
This is the familiar formula for thermiunic emission, and is one that shows
good agreement with experiment, except that the experimental value of
A' does not agree with the theoretical value given in Eq. (3.4). Actually
Ns(dCP/dN) is undoubtedly not zero, and the last term of Eq. (3.4) must
be retained. It varies slowly with temperature, however, while the

-5!
factor e T varies extremely rapidly. Thus we can expand it in serins.

What we shall do is to assume

a linear function of temperature, which presumably is sufficiently accurate
for the temperature range used. Then we have

TdT

f
e ~ J

For the thermionic emission, we then have

where

A - AV, b = V - a. (3.7)

Formula (3.7) is of the familiar form, often called a Richardson equation.
We notice in it that 6 does not represent the latent heat at the absolute
zero but a slightly different quantity without theoretical significance,
since the expression (3.5) is merely a convenient approximation for a
small temperature range. And A is not the value of Eq. (3.4) but has
the additional factor e&. This factor is of the order of magnitude of

BBC. 4] THERMIONIC EMISSION AND THE VOLT A EFFECT 467

unity, but from the observed values of the A's we gather that it varies
from something like 10~3 to 103, depending on circumstances.

4. Contact Difference of Potential. — Suppose we have two metals,
a and 6, in the same container at the same temperature. For each one,
we can calculate the vapor pressure of the electron gas in equilibrium with
it by Eq. (2.2). Since this equation depends on the properties of the
metal, we shall get a different answer in the two cases. That is, an elec-
tron gas in equilibrium with one metal, say one with a low work function,
will have too great a pressure to be in equilibrium with the second metal
with a larger work function. Let us use a kinetic argument to see what
will happen and what sort of final equilibrium we may expect. Suppose
the metal a, of low work function, has established its equilibrium pressure
in the electron gas, and then the metal b is introduced into the container
and brought to the same temperature. The gas pressure is too great for
equilibrium with 6, so that more electrons will strike its surface and

c* b

Fia. XXVIII-2. — Potential energy of an electron between two metals, in equilibrium.

condense than will be emitted. Thus there will be a net flow of electrons
into metal 6, and it will become negatively charged. It will then tend to
repel electrons near it, by electrostatic repulsion, and this will diminish
the electron current toward it, until finally the flow of electrons will
stop, the electrical difference of potential being just great enough to reduce
the number of electrons coming toward 6 to equality with the number
leaving it. The net result is that the metals have become charged in such
a way that there is a definite difference of potential between them. This
is known as the Volta effeet, and the difference of potential is the contact
difference of potential.

Let us now investigate the Volta effect more quantitatively. When
equilibrium is established, there will be a difference of potential between
the empty space outside the two metals. The potential energy of an
electron in this space will then be as in Fig. XXVIII-2. The jump in
potential at the surface of each metal is as in Fig. XXVIII-1, but now the
potential varies from one metal to another. Since metal 6 is negatively
charged, the potential in its neighborhood is less than near metal a, and
the potential energy of an electron, which is — e times the electrostatic
potential, will be greater. The difference of potential between empty

468 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVIII

space outside the two metals is the contact difference of potential and is
shown in Fig. XXVIII-2.

The electrons in the empty space now form a perfect gas in an external
force field. This problem has been discussed in Section 4, Chap. IV.
There we found that in such a case the temperature of the gas is constant
throughout, but the pressure and the density vary from point to point, the

_a-

number of molecules per unit volume being proportional to e kT . In
such a case, where the pressure is not constant throughout, we cannot use
the thermodynamic method of the Gibbs free energy, for that is based
on a single pressure, constant throughout the system, which can be used
as a thermodynamic variable. Fortunately, however, we can get as far

Epot

as we need to here by means of the Boltzmann factor e kT for the perfect
gas. It is just as well to remember how this factor comes about and to
see that it is closely related to our explanation of the Volta effect. On
account of the difference in potential between the neighborhood of a and 6,
an electron with energy shown in E\ in Fig. XXVIII-2 will be slowed
down and stopped as it emerges from a and tries to reach 6. Instead, it
will be turned around and will fall back to a again. Only electrons with
energies like Eo will be able to enter the metal 6. It is this stopping of
the slower particles by the regions of high potential energy which keeps
the density smaller there, and which in this case keeps too many electrons
from striking metal 6.

It is now clear what conditions we must have for equilibrium between
metal a, metal 6, and the gas. The vapor pressure outside metal a must
be the correct one for thermal equilibrium with that metal, the pressure
outside 6 must be the correct one for equilibrium with it, and the pressures
outside the two metals must be related according to the Boltzmann
factor, to ensure equilibrium between the different parts of the gas. Thus
let the potential energy of a mole of electrons outside the metal a bo
Ea, and outside metal 6 be Eb. Then the Boltzmann factor leads to the
relation

P .(*»-*•).

IT = « "*T , (*-l)

•* o

where Pa, Pb are the pressures outside metals a and &. That is,

-a). (4.2)

Equation (4.2) is the condition of equilibrium of the gas. At the same
time, we must have the gas in equilibrium with each metal, which means
that Pa and Pb must be given by Eq. (2.1). Thus we have

SEC. 4] THERMIONIC EMISSION AND THE VOLT A EFFECT 469

The significance of Eq. (4.3) is clearer if we neglect the terms in

which are small though not entirely negligible. Then it is

Ea — Eh == La — Lb*

Neglecting the No(dCp/dN)'*, the L's are the latent heats. Thus we
have the important statement that the contact difference of potential
between two metals equals the difference of their latent heats, or approxi-
mately of their work functions. This relation is found to be verified
experimentally. The contact difference of potential can be found by
purely electrostatic experiments, and the work functions by thermionic
emission; the results obtained in these two quite different types of experi-
ment are in agreement. The small correction terms arising from the
No(dCp/dNYs lie almost within the errors of the experiments, so that we
hardly need consider them in our statement of the general theorem.
It is a characteristic of thermal equilibrium that it is the same, no
matter what agency or process brings it about. Thus in particular, two
metals in thermal equilibrium will take up a difference of potential equal
to the contact difference, so long as there is any agency whatever by
which charge can flow from one to the other. In the case we considered,
the electron gas formed the agency; it is a conductor and carried current
from one metal to the other. But we can readily imagine conditions
where the electron gas would not be an effective agency. For instance,
we may have the metals at room temperature. The density of the
electron gas at room temperature is so extremely small that for all prac-
tical purposes it has no electrons at all in it, and it would take an exces-
sively long time to transfer appreciable charge or produce equilibrium.
Nevertheless in time the equilibrium would be established and from
Eq. (4.3), since the temperature dependent terms are small, the contact
difference of potential at room temperature must be about the same as at
high temperature. But there are other much more effective ways of

470 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXVIII

transferring charge from one metal to another at room temperature : they
may be connected by a wire or other conductor. Thermodynamics now
requires that if this is done, the metals will automatically come to such
potentials that the points directly outside the two metals will differ by
the contact potential. This of course demands that the metals will
automatically become charged enough to produce these potentials in
outside space.

This mechanism furnishes the basis of one electrostatic method of
measuring contact potentials. The two metals whose difference of
potential is desired are made into plates, so that they can be brought
close together like the plates of a condenser. When they are close
together they are connected by a wire, so that they will set up the contact
difference of potential; then they are disconnected. The capacity of a
condenser is inversely proportional to the distance between the plates, so
that it is very large when the plates are close together, and it requires a
large charge to produce the required potential difference. Then the
plates, insulated from each other, are removed to a long distance apart.
Their capacity becomes much smaller and the charge, which of course
remains the same, raises them to a large potential difference, large enough
so that it can be readily measured with an electrostatic voltmeter.

We notice that in equilibrium, a metal with a low work function
becomes positively charged, one with a high work function negatively
charged, so as to set up the contact difference of potential. It is interest-
ing to notice the close similarity between this and the corresponding
situation of two atoms, one of low ionization potential and the other of
high ionization potential, in equilibrium, as we considered them in Chap.
XX, Sec. 4. If either one lacks an electron, we found that it would be the
electropositive one, the one with low ionization potential. Similarly
here we may consider the metals of low work function to be electropositive
ones, which lose electrons easily. We cannot push this analogy too far,
however, for while there is some parallelism between ionization potential
and work function, it is by no means very complete, so that we should not
arrange the metals in the same series of electropositive or negative charac-
ter by means of the work functions that we should from ionization
potentials.

There is an interesting graphical interpretation of our result that the
contact difference of potential between two metals equals the difference
of work function. In Fig. XXVIII-2, let us go down from the energy
Ea by the amount La, and down from the energy Eb by the amount L&.
Then from Eq, (4.3), the resulting energies, Ea — La, and Eb — L&, will
be approximately the same, so that the horizontal lines drawn at these
heights in the two metals, in Fig. XXVIII-2, will be at the same height
in these two, or any two, metals in equilibrium. Let us see what inter-

SEC. 4] THERMIONIC EMISSION AND THE VOLT A EFFECT 471

pretation we can give to these levels. The quantity La represents the
work done on an electron (or a mole of electrons, to be more precise), in
removing it from metal a at the absolute zero. We might adopt a very
crude picture of a metal: we might suppose it to be a region of constant
potential energy for electrons, in which the electrons simply formed a
perfect gas of high density. Then if the potential energy of an electron
jumped by the amount La in going out of the surface, we should under-
stand the interpretation of the work function. With this simple inter-
pretation, the equality of Ea — La and Eb — L& would mean that the two
metals set themselves so that the potential energy of an electron wavS the
same in each; if they were joined by a wire, there would be no jump in
potential in going from one metal to another. And if the electrons
satisfied Maxwell-Boltzmann statistics, the density of electrons within

ii
either metal would be greater by a factor eRT than the density outside.

Since L0 is much larger than RT at ordinary temperatures, this would
mean a very large density of electrons within the metal but, when one
calculates it, it is not unreasonably large. It would also mean that the
densities of free electrons should be the same within any two metals.
This would not be exactly the case, however, when we recalled the addi-
tional terms in Eq. (4.3), coming from the No(dCp/dNYs. These terms
can result in slight differences of potential within the metals and in
differences of electron densities.

The picture of a metal which we have just mentioned was elaborated
greatly some years ago, and was considered to be a reliable approximate
picture. One difficulty remained with it, however. The electrons
within the metal formed a perfect gas, and there was no reason why they
should not have the classical heat capacity of a perfect gas, f R per
mole for CV, independent of temperature. This would give a contribution
to the specific heat of a metal, in addition to that coming from atomic
vibration, and would increase the specific heat far beyond the value of
Dulong and Petit. Yet experimentally metals agree fairly accurately
with the law of Dulong and Petit, showing that the electrons can con-
tribute only a little, if anything, to their specific heat. This difficulty
showed that there was something fundamentally wrong with the simple
free electron picture of a metal, and that has proved to be the assumption
of the Maxwell-Boltzmann statistics for the electrons. In fact, electrons
have been found to satisfy the Fermi statistics. In the next chapter we
consider the application of this form of statistics to a detailed model of
the free electrons in a metal.

CHAPTER XXIX
THE ELECTRONIC STRUCTURE OF METALS

The electrons in a metal, like those in an atom, are governed by the
quantum theory, and a complete study of their motions is impossibly
difficult, on account of the enormous number of electrons in a finite piece
of metal, all exerting forces on each other. The only practicable approxi-
mation is similar to that used in Chap. XXI, Sees. 2 and 3, where we have
taken up the structure of atoms. There we replaced the force acting
on an electron, which actually depends on the positions of all other
electrons, by an averaged force, averaged over all the positions which
the other electrons take up during their motion. In the case of a metal,
then, we have a structure consisting of a great many positive nuclei,
arranged in a regular lattice structure, with electrons moving about
them. Each nucleus will be surrounded by a group of electrons forming
its inner, or x-ray, shells, just as we should find in individual atoms. The
remaining electrons, however, the valence electrons, will move very
differently from the way they would in separated atoms. The reason is
that the jiimensions of th^ orb^s qf th<* valence electrons in the fltoms
are of the same order of magnitude as the interat.nTrnn spring's fn
so that electrons of neighboring fl.tQ.mfl w^nH foTH fo

and profoundly affect their motion. Thus we must treat the problem as
if each valence electron moved in the field of the positive ions, consisting of
the inner shells and nuclei of all the atoms, and the averaged out field
of all the other valence electrons. Our problem of the electronic structure
of metals, then, is divided into two parts: first, we must find what this
field is like in which the electrons move; secondly, we must investigate
their motion in the field. Then we can build up a model of the whole
metal, treating each electron as if it moved in the field of which we have
spoken and remembering that on account of the Pauli exclusion principle,
or the Fermi statistics, no two electrons of the same spin can be found in
the same stationary state. First, we investigate the field inside a metal.
1. The Electric Field Within a Metal. — We have soen in Chap. XXI,
Sec. 2, that an electron in an isolated atom is acted on by a central force
on the average, equal to the attraction exerted by the nucleus, diminished
by a certain shielding effect on account of the other electrons. The
potential energy of the electron in such an atom was illustrated in Fig.
XXII-7. When such atoms are placed near each other, the potential

472

SBC. 1]

THE ELECTRONIC STRUCTURE OF METALS

473

energy of an electron at points between atoms decreases, as we saw in
Fig. XXII-7 (6). In a crystal of a metal, with a lattice of equally spaced
atoms, we have a similar situation, with a potential energy as shown in
Fig. XXIX-1. Here we show in (a) the potential in a single atom, as
in Fig. XXII-7 (a), and in (6) the corresponding thing for the whole
crystal. It will be seen that, at points well within the crystal, the
potential energy is a periodic function, reducing near each nucleus to the
value that we should have near the nucleus of an isolated atom but coming
to a maximum between each pair of nuclei. The field, or the force on an
electron, is given by the slope of the potential energy curve. We see that
it fluctuates violently from point to point, depending on where the
electron may be in an atom. The average field, however, is zero, if we
average over many atomic diameters. For this average field is found
from the difference of potential energy between widely separated points,
divided by the distance between, and we see that on account .of the
periodicity of the potential energy function, the difference of potential

Potent/at enery

FIG. XXIX-1. — Potential energy of an electron in (a) the central field representing an
atom; (6) a periodic field representing a crystalline solid.

at least between corresponding points near different atoms will be zero.
It is this average field that one speaks about in ordinary electrical prob-
lems, where we do not analyze things on a microscopic or atomic scale.
We see, then, that Fig. XXIX-1 corresponds to a metal in which no
current is flowing, so that by Ohm's law the electric field within the metal
is zero, or there is no difference of potential between different points. A
different figure would have to be drawn in the case of a current flow,
consisting of a curve like Fig. XXIX-1, superposed on a gradual change
of potential energy, representing the field within the metal. Such a curve
is shown in Fig. XXIX-2, though the over-all slope of the curve as drawn,
representing the applied field, is much greater than one would find in an
actual experimental case. We shall work at first with a metal in which
no current flows, so that the mean field is zero, as in Fig. XXIX-1.

As we go outside the metal, as Fig. XXIX-1 shows, the potential
energy of an electron rises and approaches an asymptotic value at infinity,
much as the potential energy of an electron outside an atom approaches
an asymptote at infinity. It is worth while looking a little in detail at
the nature of this asymptotic behavior. The force acting on a particular

474

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

electron somewhat outside a metal of course depends on the distribution
of the remaining valence electrons, which are still attached to the metal.
It is well known from electrostatics that a negative electron outside the
metal will induce a compensating positive surface charge on the surface
of the metal, as indicated in Fig. XXIX-3, where we show the electrical

FIG. XXIX-2. — Potential energy of an electron in a field representing a metal with a cuirent
flowing, and an average potential gradient.

lines of force running between the electron and the induced surface charge.
All the lines from the electron terminate on the induced charge; thus its
net amount must be just the charge of one electron. But this is what we
should expect. If the uncharged metal had N electrons, and one is
removed, there are N — 1 electrons left, and the positive surface charge
represents simply the averaged out deficiency left by the removed elec-

X /// l\Vx

V//i\\\>-

'!\

FIG. XXIX-3. — Lines of force between an electron outside a metallic surface and the
positive charge induced on the surface.

tron. Now it is easy to show that the lines of force in Fig. XXIX-3 are
just like those found from the electron of charge — e at a distance d from
the surface, and an equal and opposite charge +e at a distance d behind
the surface. This imaginary charge within the metal is called the electric
image of the electron. The dotted lines within the metal in Fig. XXIX-3
represent the imaginary continuation of the lines of force, which really
terminate on the surface of the metal, to the electric image. But if we

SBC. 21 THE ELECTRONIC STRUCTURE OF METALS 475

had the two charges +e and — e at a distance 2d from each other, the force
exerted by each on the other would be an attraction of magnitude e2/(2d)2.
This force is called the image force, and its potential, — e2/4d, is the
limiting value of the potential energy of the electron, at large distances
from the surface of the metal. It is this function which the curve of Fig.
XXIX-1 approaches asymptotically at large distances.

To solve the problem of the motion of an electron in a potential field
like that of Fig. XXIX-1 is a very difficult problem in quantum theory.
We shall describe its solution in a later section, but first we shall take up
an approximation to it, the free electron theory, which has enough
resemblance to the correct theory so that it can be used satisfactorily for
some purposes. In this approximation, it is assumed that the field acting
on the electrons in the metal is not only zero on the average, but zero
everywhere, or the potential energy is constant, equal perhaps to the
average value of the potential energy over the periodic potential of Fig.
XXIX-1. At the surf ace of the metal, — y
of course, the potential energy must
rise; as the simplest approximation we j

may assume that it rises discontinu- ., VVT__ . ~ ..- , A A. .

, „ . . i . , • i FIG. XXIX-4. — Simplified potential

OUSly from the Value which it has energy function for the free electron

within the metal to the asymptotic model of a metal,
value at infinity. That is, we replace the potential energy curve
of Fig. XXIX-1 by the simplified curve of Fig. XXIX-4. We shall
find in the next section that we can work out the motion of electrons
in such a potential energy completely, applying the results to the prop-
erties of metals.

2. The Free Electron Model of a Metal. — In Fig. XXIX-4, there is a
volume, which we shall take to be V, in which the potential energy has
a constant value, which we choose to be zero. Outside this volume, the
potential energy is greater, by an amount Wa- An electron whose
energy is less than Wa is then free to wander freely through the volume
but cannot leave it, while an electron with energy greater than Wa can
travel anywhere, but it suffers a decrease of Wa in its kinetic energy on
leaving the volume. Electrons of the first sort form the picture furnished
by this model for the electrons bound in the metal, while those of the
second sort represent the fast electrons which can be emitted in thermionic
emission. Let us consider first the electrons with energy less than We.
These act exactly like molecules of a perfect gas, confined to the volume
V. We have already investigated such a perfect gas. We have found its
distribution of energy levels in Chap. IV, Sec. 1, and have applied the
Fermi-Dirac statistics to it in Chap. V, Sec. 5. Only one change must
be made in the formulas of that section to adapt them to our present
use. In developing the Fermi statistics, we assumed that only one mole-

476 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

cule could occupy each stationary state. With electrons, however, one
electron of each spin, or two in all, can occupy each state, so that the
allowed number of electrons per energy range is twice what we assumed
before. Then from Eq. (1.9) of Chap. IV, we find that if all energy levels
are filled, the number of electrons with energy less than € is

m)*eH (2.1)

and the number with energy in the range de is

At the absolute zero, then, if the gas contains N electrons, these will be
distributed in energy according to Eq. (2.2), up to a maximum energy
given by setting N(e) = N in Eq. (2.1). This maximum energy, which
was called €0o in Chap. V and which is usually called W% in the theory of
metals, is given, by analogy with Eq. (5.3), Chap. V, by

We have seen, in Eq. (5.5) of Chap. V, that such energies are of the order
of magnitude of several electron volts. In Table XXIX-1 we give values
of these energies for a series of metals, computed on the assumption that
the number of electrons equals the number of atoms, so that V/N is the
volume per atom. This can be computed easily from the lattice spacings
in Table XXVII-1, together with Eq. (3.1) of Chap. XIII, V/N = cr3,
where r is the lattice spacing, c a constant computed in Eqs. (2.6), (2.7),
and (2.8) of Chap. XXVII. We have no particular justification for the
assumption that the number of electrons equals the number of atoms.
For the alkali metals, where each atom furnishes one valence electron, the
assumption seems very plausible, and more elaborate methods, which
will be described later, justify it. For other metals, we should think at
first sight that the number of electrons per atom should be greater than
unity, since each atom has several valence electrons. On the other hand,
the more advanced theory shows in this case that the extra electrons do
not act very much like free electrons, and in some ways it is more reason-
able to take the number of free electrons to be less, rather than more, than
the number of atoms.

A number of properties of the electrons in a metal can be found from
our model. In particular, we can find the specific heat of the electrons
and can see in a qualitative way what their contribution to the equation
of state should be. For the heat capacity, Eq. (5.9), Chap. V, gives

(2-4)

SBC. 2] THE ELECTRONIC STRUCTURE OF METALS 477

TABLE XXIX-1. — FERMI ENERGY Wi OF METALS

Metal

Wi, volts

4.7

9.0

3.1

4.5

5.6

2.1

3.0

5.4

6.3

7.0

6.2

7.4

7.0

5.9

1.8

2.5

4.5

5.9

6.4

6.3

6.1

5.5

4.7

1.6

2.3

5.2

5.8

6.3

6.0

5.6

This is a heat capacity proportional to the temperature, and in Sec. 5,
Chap. V, we computed it for a particular case, showing that it amounted
to only about 1 per cent of the corresponding specific heat of free electrons
on the Boltzmann statistics, at room temperature. In Table XXIX-2
we show the value of the electronic specific heat at 300° abs., computed
from the values of Wi which we have already found, in calories per mole.
We verify the fact that this specific heat is small, and for ordinary pur-
poses it can be neglected, so that the specific heat of a metal can be found
from the Debye theory, considering only the atomic vibrations. At low
temperatures, however, Eq. (2.4) gives a specific heat varying as the first
power of the temperature, while Debye's theory, as given in Eq. (3.8),

478 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

TABLE XXIX-2.— ELECTRONIC SPECIFIC HEAT, FREE ELECTRON THEORY

Cv/T
calculated

Cv/T
observed

Cv (300°)
calculated

1 . 20 X 10~4

1 . 78 X 10~4

0.036

1.52

1.6

0.046

1 41

2.33 - 2.97

0.042

1.84

3 75

0.055

2.10

3.8

0.063

1.97

3.46

0.059

Pb ....

2.21

7 07

0 066

1.62

27.

0.049

1 13

17 4

0 034

1 38

0 041

1.40

16.1

0.042

Observations are taken from Jones and Mott, Proc. Roy. Soc., 162, 49 (1937). Calculations in
that paper assume different numbers of electrons per atom, instead of one per atom as we have done,
and secure somewhat better agreement with experiment. Specific heats are tabulated in calories per
mole.

Chap. XIV, shows that the specific heat of atomic vibrations varies as the
third power. At temperatures of a few dc»croou oK^in^ fko third
Of thejemperflf.iirp is sn mn^fr seller than the firs|f pnwpr fhof fhn
specific heat can be neglected, leaving only the heat capacity of the
electrons. The observed specific heat in this region is in fact proportional
to the temperature, and the constant of proportionality is roughly in
agreement with that predicted by Eq. (2.4). In Table XXIX-2 we tabu-
late the values of CV/T as observed, as well as those calculated by Eq.
(2.4) from the values of W i in Table XXIX-1. It is plain that the
agreement, while not exact, is good enough to indicate the essential cor-
rectness of our methods, for most of the metals, particularly for the
alkalies. For the transition metals, however, and in particular for the
ferromagnetic ones, the observed value of Cv/T is much greater than
the calculated one, a fact whose explanation we shall give later, in Sec. 6.
To understand the relation of the electrons to the equation of state
of the metal, we may consider the internal energy at the absolute zero as a

junction Of Volume. This quantify nf noiir^ ahrmlH Imvfl fl. mim'mnm

for the actual volume of the metal, rising as it is compressed or expandeH.
In the free electron model, the energy will depend on volume in two ways.
In the first r>lfl.fift. f.he lcinp.ti/^^ii^i!igirjH7^1_-flQj^^iiiJ-Qii vnlnrne. on account of
the fact, proved in Eq. (5.7), Chap. V, that the total kinetin energy is
\NWj, where Wj is proportional to V~H. as shown in Eq. (2.3^. Thus

iriiiofin ynpfgy T

+hft vnlijrnP.

varying

where r is foe distant between atoms, afoce V** is proportional to r,
This leads to a repulsive term in the energy. In the second place, the

SBC. 2]

THE ELECTRONIC STRUCTURE OF METALS

479

wi|l depefld on volnirift. on account of a

change of the quantity W* with volume. It requires a little more careful
analysis to see just what this change means, but it turns out that the
situation is as shown in Fig. XXIX-5. Here we show potential energy
curves like Fig. XXIX-1 for two different distances of separation. As
the distance decreases and the atoms overlap more, we see that the
potential energy of an electron between two atoms decreases. In fact, if
the total potential energy is the sum of Coulomb attractions to the various
atoms (a crude approximation, on account of the other electrons, which
complicate the situation), the potential energy at a given point of the
crystal would be a sum of terms — 1/d, where the d's are the distances to

FIG. XXIX-5. — Potential energy of an electron in a metallic crystal, for two differ-
ent distances of separation, illustrating the decrease of average potential energy with
decreasing lattice spacing.

the various atoms of the crystal. Then, since each of these d's is propor-
tional to r, the lattice spacing, we find that the potential energy at a point
between the atoms should decrease approximately like — 1/r. We can
take an average value over the periodic potential of Fig. XXIX-5 to
represent the zero of potential in the free electron picture. We see, in
other words, that this should not be chosen as being really zero potential,
but that it should be given a negative value, roughly proportional to
1/r. This value, then, is the mean potential energy of the electrons.
The net result is that we might expect to express the internal energy of
the crystal at the absolute zero as

(2.5)

where the first term represents the potential energy, the second the
kinetic energy, and where the constant B is given theoretically by

B -

(2.6)

480 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

using Eq. (2.3), and Eq. (3.1), Chap. XIII, for the definition of c. The
internal energy (2.5) is the general sort we expect, with a minimum and an
asymptotic value at infinity, not entirely unlike a Morse curve. As a
matter of fact, it is not very satisfactory for actually describing the
equation of state of a metal, on account of various approximations that
enter into it. First, the electrons of real metals are not free and their
kjpetifl energy is not given at all accurately by the free electron theory.
Secondly, the potential energy does not vary as 1/r, being really much
more complicated than this. And finally, the electrons do not really
move in an averaged out external field at all but are acted on by all the
other electrons in their instantaneous motions. For all these reasons, as
simple an expression as Eq. (2.5) is not adequate and is not actually as
satisfactory as the Morse function, which we have used in Chap. XXVII,
Sec. 2. In spite of this, it gives some insight into the mechanism of the
forces in the metal, describing them in rather different light from that
used in Chap. XXII, Sec. 4, where we treated them as a somewhat varied
form of homopolar bonds.

3. The Free Electron Model and Thermionic Emission. — We have
stated in the last section that electrons whose kinetic energy is greater
than Wa can escape from the volume V in Fig. XXIX-4, furnishing a
model for thermionic emission. Using this free electron picture, we can
easily calculate the rate of thermionic emission from a metal directly.
This of course must lead to the same result as the indirect method of
Chap. XXVIII, Sec. 3, for that is entirely justified thermodynamically,
but it may lead to greater insight into the mechanism of thermionic
emission. In the free electron theory, the electrons within the metal
form a perfect gasr and we can find foe* ^nmhp.r of electrons emerging_per
second by finding the number hitt.inpr fop mirfan.ft Inypy ^f t.l^ mftV1 frn™
the inside per second, and by multiplying hy (1 — r)1 where r is the
reflection coefficient, as in Chap. XXVIII. We must take account of one
fact here, however, which was absent in our calculation of the last chapter:
we must confine ourselves to electrons of energy greater than Wa, so that,
after passing the barrier, they will still have a positive kinetic energy and
a real velocity.

The situation of the barrier is different from what it would be with the
Boltzmann statistics, as can be seen most clearly from Fig. XXIX-6.
Here we have drawn energies, both inside and outside the metal, as in
Fig. XXIX-4. The zero of energy is taken to be at the bottom of the
picture. Then at the absolute zero there will be filled energy levels up to
the energy W% and empty levels above, the filled ones being shaded in
Fig. XXIX-6. We can now see that the potential energy of an electron
outside the metal, Wa, is related to Wi and to L0, the heat of vaporization

SEC. 3]

THE ELECTRONIC STRUCTURE OF METALS

481

of a mole of electrons at the absolute zero, discussed in the last chapter, by
the equation

(3.1)

For Lo represents the energy necessary to remove a mole of electrons from
the metal at the absolute zero, in equilibrium. For equilibrium, the
metal must be left in its lowest state, so that the removed electrons must
come from the top of the Fermi distribution, and they must have no
kinetic energy after they are removed from the metal. Thus each
electron is raised just through the energy LQ/N in the figure. In the
Fermi statistics, in other words, the work function represents the differ-
ence in energy between the top of the Fermi distribution and space outside
the metal. And the result of Sec. 4, Chap. XXVIII, that on account of
the contact potential the values of Ea — La and Eb — Lb were equal for two
metals at the absolute zero, means graphically that two metals will adjust

Fia. XXIX-6. — Occupied energy levels for the free electron model of a metal, at the
absolute zero, illustrating the relation between Wa, W%, and the thermionic work function
or latent heat of vaporization of electrons.

their potentials so that the top of the Fermi distribution is at the same
height in each metal, at the absolute zero. Let us find the small modifica-
tions in this occurring at higher temperatures.

The exact statement of Eq. (4.3), Chap. XXVIII, is that in any two
metals in equilibrium, the space outside the metals acquires such a
potential that if we subtract from it the amount

(3.2)

the result is the same for all metals. If it were not for the second term,
as we have just mentioned, this would take us down to the top of the
Fermi distribution for all metals. We shall how show that the last term
m£ans that really it will take us down to the level c0, rather than Wi or
eoo, as defined in Chap. V, Sees. 3 and 4, for all metals, so that any two
metals in equilibrium have their €0's at the same energy. To do this, let
us compute the second term of Eq. (3.2). We do not have CP\ but we
shall assume for the present that the metal has no thermal expansion,
so that CP = Cvt and we can use the result of Eq. (2.4). Substituting for
Wi from Eq. (2.3), this gives CV proportional to N*. Thus we have

482 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

dCv 1 Cv

Then, substituting in Eq. (3.2), and using Eq. (5.6), Chap. V, we have

Lo + tfo^ ^ - Lo + tf o(eoo - «o). (3.4)

But, referring to Fig. XXIX-6, we see that an energy lower by the amount
(3.4) than the energy outside the metal, is just at the level eo, verifying our
statement that this quantity is the same for all metals in equilibrium,
while the energies outside the metals, and the bottoms of the Fermi
distributions, are different for different metals.

Now we can proceed with our calculation of thermionic emission. We
wish to find how many electrons, with energy sufficient to surmount the

barrier of height Wa = Wi + -^> strike a square centimeter of the surface

J\l o

of the metal per second. Let the x axis be normal to the surface. Then
in going through the barrier, the x component of kinetic energy, pl/2m,
will be reduced by Wa, while the y and z components will be unchanged.
In other words, the electrons we are interested in are those whose x
component of momentum is greater than \/2mWa, while their y com-
ponents of momentum can be anything. In finding this number, we must
integrate over px, py, pz, rather than over the energy, for our limits of
integration depend on the p's. We ask first, then, how many electrons
with momentum in the range dpx dpv dpg will cross 1 sq. cm. per second
normal to the x axis. This, by the same methods used in the last chapter,
will be p9/m times the number of electrons per unit volume in the range
dpx dpv dp,. And in turn the number per unit volume in that range will
be the number of energy levels in that range, multiplied by the Fermi
factor

(p«/2m-«o)

e kT + 1

giving the fraction of those levels occupied by electrons, and divided by
the volume V of the gas to find the number of electrons per unit volume.
We must then find the number of energy levels in dpx dpv dp,. As in
Chap. IV, Sec. 1, this is (V/h*)dpx dpv dpz, except for the correction arising
from the electron spin. This doubles the number of allowed energy levels,

2V
leading to rdp» dpv dp, as the number of energy levels in dpx dpv dp,.

SBC. 3J THE ELECTRONIC STRUCTURE OF METALS 483

We are now prepared to find how many electrons, of energy sufficient
to cross the barrier, strike 1 sq. cm. of the boundary of the metal per
second. Using the statements made above, this is

J- >

The integral in Eq. (3.5) cannot be completely evaluated analytically.
But for the high energy of the electrons concerned, we are entirely justi-
fied in neglecting the term unity in the denominator of the Fermi function.
Then the calculation can be carried out at once, giving

mh\] - . ""'I _ „ ^ V V2SHT.

where in the integral over px we can introduce pi as a new variable to
simplify the integration. When we multiply by the factor (1 — r),
representing the fraction of electrons that penetrate the barrier, and when
we consider Eqs. (3.2) and (3.4), and Eq. (3.3) of Chap. XXVIII, we see
that this result agrees exactly with that found in Chap. XXVIII. As a
matter of fact, there is nothing in this simple model that would lead to
any reflection coefficient at all for electrons, so that we should really set
r = 0.

We have seen, in other words, that our free electron model, using the
Fermi statistics, leads to thermionic emission agreeing with our previous
deduction from thermodynamics. This is hardly remarkable, for any
model whatever, correctly worked out according to thennodynamic
principles, would have to do the same thing. But we have a somewhat
better physical understanding of the reason for the rapid increase in
emission with increasing temperature: only those electrons that happen
to have enough energy inside the metal to surmount the barrier can
escape; the number of such electrons increases very rapidly with increas-
ing temperature, the more rapidly the lower the work function is.

The model we have used is, of course, too simplified. A more accurate
model would likewise have to agree with our deduction of the previous
chapter, but it could differ in its final results from the free electron model in
having a different reflection coefficient and a different value of No(dCp/dN).
And the calculation would differ in that it is only with free electrons,
unperturbed by atoms, that we can find the number colliding with 1 sq.
cm. of surface per second as simply as we have done here. Then in wave
mechanics the reflection coefficient is not so simple as in the classical

484 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

theory of free particles. Finally, the change of work function with
temperature, coming from the term No(dCP/dN), is a decidedly more
complicated thing than we have assumed. On account of thermal
expansion, the volume of an actual metal changes with temperature.
Then the energy levels of the electrons are bound to change, bringing a
change explicitly with volume but actually with temperature, in the
height of the barrier, in €0, and so on, quite apart from anything we have
had to take up. All these complications make a direct calculation of
thermionic emission from a correct model a very difficult thing. And
they make the calculation of the preceding chapter all the more important,
for it is derived from straightforward thermodynamics and from the
properties of the electron gas in empty space, about which there can be
no doubt, and it does not depend on the nature of the metal at all.

4. The Free Electron Model and Electrical Conductivity. — By slight
extensions of the free electron theory, one can explain the electrical
conductivity of a metal. Let there be an external electric field E,
impressed on the metal. Then the electrons will no longer move with
iniform velocity in a straight line, as in a perfect gas. Instead, they will
be accelerated, the time rate of change of momentum of each electron
equaling the external force. If the field is applied at a certain instant,
this external field by itself would result in a net electric current, building
up proportionally to the time, if no other forces acted on the electrons.
We must assume in addition, however, that the electrons meet some sort
of resistance, proportional to their average or .drift velocity, which hence
is proportional to the current. When we consider this resistance, we find
that the current, instead of building up indefinitely, approaches an
asymptotic value, proportional to the external field, and hence obeys
Ohm's law. The time taken to reach this steady state is quite negligible
in comparison with ordinary times, so that we find that the current in
the conductor is proportional to the field. We can easily formulate this
argument mathematically and see how the electrical conductivity depends
on various properties of the electrons. The argument does not depend
on the Fermi statistics and follows equally well from Boltzmann statistics.

There are N/V electrons, each of charge --e, per unit volume in the
metal. These will have a certain distribution of velocities (the Maxwell
distribution or the Fermi distribution), of which we need only the prop-
erty that the mean velocity is zero, in the absence of an external field. In
the field E, the force on each charge will be — eE, so that, if px is the
component of momentum of an electron in the direction of the field (which
we take to be along the x axis), we shall have

SBC. 4] THE ELECTRONIC STRUCTURE OF METALS 486

But the velocity of an electron is its momentum divided by its mass, and
the electric current density u is the number of electrons per cubic centi-
meter, times the charge on each electron, times the velocity of each.
Thus we have

so that Eq. (4.1) can be rewritten

from which we can at once verify that the current increases proportion-
ally to the time. But now let us assume that there is an additional force
acting on the electrons, proportional to their velocity, of the nature of a
viscous resistance. Let the force acting on a single electron be — p»/r,
where r is a constant. Then Eq. (4.1) becomes

rfj? = -eE - &. (4.4)

The meaning of r is easily seen if we ask for a solution of Eq. (4.4) in the
case where the external field is zero. Then we have at once

_*
px = const, e r, (4.5)

so that r is the time in which the original momentum of an electron would
be reduced to l/e of its value on account of the friction. The quantity r
is sometimes called a relaxation time. Then we have

fa^N^E-*, (4.6)

of which the solution satisfying the initial condition of no current at
t = Ois

e (4.7)

which reduces to

«•-?£*-»*> (4-8)

at times large compared to the relaxation time, where by definition o is
the electrical conductivity, given on the free electron theory by

486 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

The conductivity increases as we see with the number of free electrons
available to carry the current and with the time in which each one can
he speeded up by the field before it reaches a stationary speed on account
of the resistance. It is obvious that Eq. (4.8), though it gives an explana-
tion of Ohm's law, does not lead to a calculation of the conductivity in
terms of known quantities, because though we have seen how to estimate
N/V, there is no way of estimating the relaxation time r. We can, of
course, reverse the argument, and from known conductivities and the
values of N/V, assumed in Table XXIX-1, find what values of relaxation
time would be required. These times are given in Table XXIX-3, from
which we see that they are very short, of the order of 10~14 sec.

TABLE XXIX-3. — RELAXATION TIMES FOR ELECTRICAL CONDUCTIVITY, FREE ELEC-
TRON THEORY

T, seconds

Li 0.89 X 10~14

Na 3.26

Mg 1.88

Al 2.20

K 4.07

Ca ... 3.31

Ti . . 1.98

Cr . ... .. . 1.62

Fe . 0.42

Co . . ... .... 0.52

Ni .. . . 0.56

Cu ... .. 2.50

Rb. . ... 2.70

Sr . . . 0.79

Mo. . ... 0.97

Pd. . . . 0.50

Ag .. . .... 3.96

Cd .... . 0.97

Cs . . 7.50

Ta . 0 41

W . . 1.00

Os. ... ... . . . 0.08

Pt . . . . . .... 0.48

Au 2.43

The electrical conductivities used in computing this table are either for 0 or 20°C.

There is nothing in the free electron theory to explain the existence of
the resisting force or the relaxation time. It is usually described as
coming from collisions between the electrons and the atoms, and thus
cannot properly be explained unless we take the atoms into account
specifically. We can see something about the mechanism of the collisions,
however, by considering the motion of the electrons in a momentum
space, similar to that of Chap. IV, Sec. 1, in which px, pv, pg are plotted as
variables, and each electron is represented by a point. As we saw in that

SBC. 4] THE ELECTRONIC STRUCTURE OF METALS 487

section, we may imagine a lattice of points in momentum space, one to a
volume h*/V, each point corresponding to an energy level, so that there
can be two electrons, one of each spin, at each point. According to the
Fermi distribution, at the absolute zero, the N points will be located at
the points within a sphere of radius p = \/2mWi about the origin, so
that the density of electrons within the sphere will be the maximum
allowable value, while none will be found outside the sphere. At higher
temperatures, the distribution will vary gradually, rather than discon-
tinuously, from the maximum density within the sphere to zero outside,
but the change will come within a very narrow region about the surface of
the sphere1. Now if there is an external impressed field, the momentum
of each electron will increase proportionally to the time. That is, each
of the points will move in the x direction (if the field is along x) with a
velocity given by Eq. (4.1). Since this velocity is the same for all points,
the whole sphere of points will drift along the x axis with uniform velocity,
so that after a time t it will be displaced a distance — eEt along the px
axis, since —eE = dpx/dt is the velocity of the points.

Now let us consider the effect of collisions on the distribution function.
The original distribution was in equilibrium, according to the Fermi
distribution, so that collisions will leave it unchanged, but the displaced
distribution is not. Collisions of electrons with lattice points correspond
to a sudden jump of a representative point from one region of momentum
space to another. There are at least two principles governing such jumps
that we can understand easily. In the first place, an electron cannot
jump to a stationary state which is already occupied, on account of the
exclusion principle. In the second place, if the collision of aiTelecTffon
and an atom is elastic, as we shall assume, it takes place with constant
energy or only slight dissipation of energy in the form of elastic vibrations
of the lattice, so that the representative point jumps from one location to
another at the same or slightly smaller distance from the origin, in the
momentum space. Then we can see in Fig. XXIX-7 that there are only
a few collisions possible. In this figure we show the undisplaced sphere,
the displaced sphere, and the crescents A and J5, in which A includes
points having more energy than Wi, representing electrons that have
been accelerated by the field, and B represents points of less energy than
Wi, from which electrons have been removed by action of the field. The
likely collisions are essentially those in which a point from 'crescent A
jumps to an unoccupied point in 5. These collisions have the effect of
disturbing the distribution, reducing the number of points with positive
momentum, -and increasing the number with negative momentum, verify-
ing our statement that the distribution was not in equilibrium, so that
collisions disturb it. We can now understand the general nature of the
motion of representative points, subject to the external field and to

488

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

collisions. They stream steadily in the direction of increasing x, on
account of the field, but as points enter the crescent A they are likely to
have collisions taking them back to B and starting the process all over
again. An equilibrium will be set up, in which the number of points
entering A per second will be equal to the number of collisions sending
points from A to B per second. These collisions prevent the sphere from
moving indefinitely far to the right; the farther it goes7~tEe"~gfeater the
crSce^"TBeco"mes,~the more collisions there are, so that equilibrium
corresponds io Ta finite displacement of the sphere, though the individual
points are "moving as we have just stated.

Undispl&ced
sphere

-Displaced
sphere

Fia. XXIX-7. — Diagram of occupied levels in momentum space, in free electron model
of a metal. The points of the displaced sphere (shaded) are occupied when the electrons
have been accelerated by an external field.

The current density is proportional to the mean momentum of the
electrons or to the displacement of the sphere from the center of the
momentum space. The time rate of change of current, or the velocity
of the center of the sphere, then depends on two things: on the velocity
of the individual points, which is proportional to the external field, and
on the number of points leaving A and entering B per second. The
collisions, represented by these points jumping from A to B, have the
effect of slowing down the motion of the sphere as a whole, though not of
its individual points. The number of collisions is proportional to the
number of points in A, and this is proportional to its area, which in turn
is proportional to the displacement of the sphere or to the current density.
Thus we see from our mechanism that we have essentially the two terms
in dux/dt given in Eq. (4.6), though the second term, proportional to the
current density, arises from collisions rather than from a frictional force.
It is then clear that if we can calculate the probability of collision we can

SBC. 5J THE ELECTRONIC STRUCTURE OF METALS 489

compute the conductivity, using Eq. (4.9). We cannot go further with
this, however, without considering the interaction of electrons and atoms
more in detail. We can only say that the probability of collision is
inversely proportional to the relaxation time ry since doubling the number
of collisions will halve the time required for dissipating the momentum, so
that the conductivity should be inversely proportional to the probability
of collision, or the specific resistance should be directly proportional to the
probability of collision, a very reasonable result.

5. Electrons in a Periodic Force Field. — In the three preceding sec-
tions, we have been dealing with the free electron approximation, in
which we assume that the electrons in a metal are not acted on by any
forces. Now we shall give a brief and qualitative discussion of the
changes brought about when we remember that the electrons are really
acted on by a periodic force field, as shown in Fig. XXIX-1. It is quite
impossible to understand these changes without knowing a few simple
facts about wave mechanics, and we shall proceed to give some simple
illustrations of the wave nature of the electron.

It has been shown, both theoretically and experimentally, that an
electron of momentum p has many of the properties of a wave of wave
length X = h/p, where h is Planck's constant. This is shown most
clearly in an experimental way by the phenomenon of electron diffraction,
in which a beam of electrons striking a crystal is diffracted much as a
beam of x-rays of the same wave length would be. Theoretically the
wave conception of electrons is shown most clearly in the explanation of
the quantum condition, the condition used in Chap. Ill, Sec. 3, to fix the
energy levels in the quantum theory. We shall illustrate this by the
problem in which we are particularly interested at present, the perfect
gas. Consider a particle moving under the action of no forces in a region
bounded by x = 0, x = X, y = 0, y = F, z = 0, z = Z, or in a volume
XYZ = V. Classically, if the particle starts out with components of
momentum px, pv, pz, it will suffer various reflections at the walls of the
region, traveling between collisions with the walls in a straight line with
constant velocity and momentum. At each reflection, the component of
momentum perpendicular to the wall will be changed in sign but not in
magnitude, while the other two components will be unchanged. When
we take account of all possible reflections, then, we shall find the particle
traveling equal fractions of the time with the eight possible momenta
given by the possible combinations of sign in ±px, ±pv, ±pz. Corre-
sponding to this, in wave mechanics, a particle of momentum p*, pv, pz
is associated with a plane wave of the form

sin

490 INTRODUCTfO^mfLCHtiMlCAL PHYSICS [CHAP. XXIX

analogous to Eq. (2.8), Chapter JHl, where we were discussing elastic
waves. The quantities Z/X, w/X, n/X, in Eq. (5.1), are given by the
relations

1 _ P* m _ Pv n _ P* /K 9\

x"T x"P x"T (6'2)

the vector form of our previous relation X = A/P- Now when we com-
bine the waves corresponding to the various combinations of ± signs, we
find a standing wave, just as we did in a similar case in Chap. XIV, Sec. 2.
It turns out that the wave must satisfy boundary conditions of reducing
to zero on the surfaces of the enclosure, at x = 0, x = X, etc. Then,
just as in Eqs. (2.11) and (2,12), Chap. XIV, we must have a wave
represented by the function

A . 0 . . . 2irmy . 2trnz ,_ ON

A sin 2irvt sm — — sin — r-2 sin — r— > (5.3)

A A A

where in order to satisfy the condition that the amplitude is zero at
x = X, etc., we must have

21X 2mY 2nZ ,e A.

-y- = **, -y- = *ir, — = *•> (5.4)

where 8Xt sv, s» are positive integers. Using Eq. (5.2), we can rewrite
these equations as

sxh syh szh , .

P* = 2X> Pv = 2Y' P* = 2Z' (6<5)

The conditions (5.5) determine the momentum, and hence the energy, of
the electron. But they are just the conditions that would be determined
by the quantum theory, as in Chap. Ill, Sec. 3. There we determined
energy levels by considering a phase space and by demanding that the
area of the curve enclosed by the path of the representative point in the
phase space be an integer times h. In Fig. XXIX-8 we show the x — px
projection of the phase space for the present case. As the particle travels
from x = 0 to x = X, its x component of momentum is px. At X there
is a collision with the wall and the momentum changes to — pxy again
reversing when the particle rejburns to x = 0. The area of the rectangular
path is then 2Xpx, so that we must have

2Xp, = a, P* = ^> (5.6)

where sx is an integer, with similar relations for the y and z components.
But Eqs. (5.6) are identical with Eqs. (5.5), showing that the wave con-
ception of the electron leads to the same quantum conditions that we
found earlier by quantizing the areas of the cells in the phase space.

SBC. 5]

THE ELECTRONIC STRUCTURE OF METALS

491

The conditions (5.5) or (5.6) are not exactly the same that we used
earlier, in Eq. (1.6), Chap. IV, in considering the same problem. There
we found

Pv = --

nnh
P* - ->

(5.7)

where nxj Uy, n, were integers. The difference is that in Chap. IV we
were not considering the collisions with the wall and the reversal of
momentum produced by those collisions. Our present treatment is
correct, both according to ordinary quantum
theory and to wave mechanics, for a gas really
confined in a box. The results are just the
same, as far as the distribution of energy Px
levels is concerned. In a momentum space,
Eq. (5.5) gives a lattice of allowed momentum
values, each value corresponding to a volume
(h/2X)(h/2Y)(h/2Z) = A3/87 of momentum
space. On the other hand, Eq. (5.7) leads o
to a volume h?/V for each allowed value, so
that there are only an eighth as many mo-
menta in a given range in Eq. (5.7) as in Eq.
(5.5). But the integers in Eq. (5.7) can be
positive or negative, while those in Eq. (5.5) x
must be positive. Thus in Eq. (5.7) we have FlG. xxix-8.-Path of repre-

points in all eight OCtailts of momentum scntative point in momentum
i •! • TI /f f\ Ji n ^ space, for a particle reflected from

space, while in Eq. (5.5) they are confined ^alls at x = 0 and x - x.
to the first octant. The number of allowed

points within a sphere of given radius, corresponding to given energy,
is then the same according to either equation, leading in either case
to Eq. (1.9), Chap. IV, for the number of states with energy less than
«, which we have used as the basis of our treatment of the perfect gas.
While the two expressions (5.5) and (5.7), then, are equivalent as far
as the distribution of energy levels is concerned, the argument on
which Eq. (5.7) is based is more satisfactory for treating electrical
conductivity and the flow of electrons. If we consider electrons in a box,
there can be no net flow of current, for as many electrons will be traveling
in one direction as in the opposite direction, on account of reflection at the
boundaries. To get conductivity, it must be possible for more electrons
to travel one way than the other. This is most easily handled by neg-
lecting the walls of the box and by assuming that an electron continues
with the fixed momentum p, as in the derivation of Eq. (5.7). The
corresponding treatment according to wave mechanics proves to be to
use only one wave, a traveling wave like Eq. (5.1), but to apply the

492 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

boundary conditions that the wave must reduce to the same phase at
x = 0 and x = X, at y = 0 and y = F, and at z = 0 and z = Z. This
demands

where nx, nu, nz are integers, which reduces to the conditions (5.7). As in
Eq. (5.7), the integers n can be positive or negative, so that the corres-
ponding particles can be traveling in either direction.

We can now consider the effect of a periodic potential field on the
electrons. The relation of waves in a constant potential field to waves in
a periodic field is very much like that between the vibrations of a continu-
ous medium, treated in Chap. XIV, and vibrations of a weighted medium
or periodic set of mass points, discussed in Chap. XV. The calculation
which we made there of y2, the square of the frequency, is similar to that
which is made here of the energy of the electron. Thus, there, for the
continuous medium, we had Eq. (2.18), Chap. XIV, giving

while here we have

P2 A2 [YnA2 /nA2 /n.V~|

= 2^ = 2^L w + w + w J' (5ilo)

In Chaps. XIV and XV, we found that Eq. (5.9) did not really hold for a
solid composed of atoms. We found that there was instead a periodic
dependence of v2 on sx, sy, and sx, as illustrated in Fig. XV-1. This arose
because of a periodic dependence of the wave itself on the s's, as illustrated
in Fig. XV-2, and it was closely tied up with the fact that the shortest
wave that could be propagated in the crystal had a half wave length equal
to the interatomic distance, so that this wave corresponded to opposite
displacements of neighboring atoms. Similarly in the wave mechanics
of electrons in a periodic potential, there is a periodicity, and the shortest
wave is that which is in opposite phases at successive atoms. We cannot
give any sort of derivation of the behavior, without much more knowledge
of wave mechanics than we have developed here, but the analogy with the
mechanical vibrations is correct in most details. In place of the reciprocal
space which we had in Chap. XIV, Sec. 2, and Chap. XV, Sec. 2, we have
our momentum space, each wave function being represented by a point in
this space. The momentum space is divided into certain polyhedra, or
Brillouin zones, such that the energy repeats periodically in each zone
and each polyhedron contains as many stationary states as there are
atoms in the crystal. There is one difference between this case and that

SBC. 5]

THE ELECTRONIC STRUCTURE OF METALS

493

of elastic vibrations, however. In the case of vibrations, there were a
number of independent solutions of the equation giving v2 as a function of
5*, sy, s,; the number was equal to the number of atoms in the unit cell, in
the simple case of one-dimensional vibration which we took up there, but
in general it equals three times this number, when we consider both
longitudinal and transverse vibration. Here, however, the relation
giving E as a function of nx, ny, ng has an infinite, rather than a finite,

(a)

(b)

(c)

Fia. XXIX-9. — Energy as a function of momentum px, for an electron in a one-
dimensional lattice, (a) Almost free electron. (6) Same as (a), but plotted in the first
Brillouin zone, (c) More tightly bound electron.

number of solutions. These are often called energy bands. As in Chap.
XV, Sec. 2, we can handle these different bands in cither of two ways: in
the first place, we can confine our attention to the central Brillouin zone,
plotting E as a function of the n's for each energy band; or in the second
place, we can plot one band in one zone, another in another, in such a
way that they fit together in a reasonable way. The corresponding two
ways of plotting the elastic spectrum were shown in Fig. XV-4 (c).

Now we are prepared to understand the actual behavior of the elec-
tronic energy levels in a crystal, as a function of the momentum. In
Fig. XXIX-9 we plot curves of E vs. pxj for a one-dimensional model of a

494

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

crystal, in two cases. In (a) we show a case in which there is only slight
departure from constancy of the potential, so that we have almost the
free electron case. For free electrons, of course E = p2/2w, so that the
curve would be a parabola. The curve in (a) is plotted so as to show its
resemblance to a parabola, each energy band being plotted in a different
zone. It will be seen that for most energies and momenta the parabola is

FIG. XXIX-10. — Energy bands as a function of internuclear distance. The graph is
drawn for metallic sodium, showing the bands that go into the 2a, 3s, and 3p levels of the
atom at infinite separation. The 2s level is an x-ray level, and 3« is the valence electron
level. The energy gap which appears between 3s and 3;> in the figure, at distances of less
than 6 A, is really filled with bands from higher atomic levels.

a good approximation, but there is a discontinuity of energy near the
momenta at the edges of the zones. In (6) we plot the same curves
reduced to the central Brillouin zone. The case (c) shows a much greater
departure from free electrons. This is a case more like what is met in an
actual metal. We plot this only in the central zone; it is so far from
the case of free electrons that there is no sense trying to make the curves
resemble a parabola by plotting as in (a). Here there are a number of
bands of very low energy, with almost a constant energy throughout the

SBC. 61 THE ELECTRONIC STRUCTURE OF METALS 495

band. These correspond to the inner, x-ray energy levels of an electron
in a force field representing a single atom, and as we should expect from
the fact that the orbits of the corresponding electrons do not overlap at
all, these energy levels are in almost exact agreement with those of iso-
lated atoms. The higher bands, however, are fairly broad; that is, the
energy difference between the top and bottom of a band is considerable.
These correspond to the valence electron levels of the separated atoms and
are broadened on account of the perturbation of the valence electron of
one atom when it overlaps another.

It is very instructive to plot the energy bands in another way, as in
Fig. XXIX-10. Here we show the top and bottom of each band, as a
function of internuclear distance, as we vary the size of the crystal. This
shows very clearly the way in which the energy levels at infinite separation
go into sharp levels, as in the isolated atoms. On the other hand, as the
distance is decreased, the bands broaden, the broadening beginning at
about the interatomic distance where the orbits in question begin to
overlap, so that the valence electron levels are broadened at the normal
distance of separation in the metal, while the x-ray levels are not but
would be broadened if the crystal were compressed to a much smaller
lattice spacing. Fig. XXIX-10 is drawn for a three-dimensional lattice,
rather than a one-dimensional one as in Fig. XXIX-9. In the one-
dimensional case, there is an energy gap between each band and its
neighboring band, a gap in which there are no energy levels. On the
other hand, in three dimensions, this is not in general the case. With
the lower, widely separated bands there are gaps, but in the case of the
valence electrons the bands overlap and the gaps are filled up.

6. Energy Bands, Conductors, and Insulators. — In discussing atomic
structure in Chap. XXI, we first found out about the energy levels of an
electron in a central field, and then we built up a model of the atom by
assuming that the electrons of the atom were distributed among these
energy levels, the lower ones being filled with two electrons each, one of
each spin, and the upper ones being empty. Similarly here we have taken
up the energy levels of an electron in a periodic potential field, and next
we must ask what levels are actually occupied in the metal. We have
mentioned that each of the energy bands consists of N stationary states,
if there are N atoms in the metal. For the x-ray levels, these stationary
states all have almost exactly the same energy, while for the higher levels
they vary considerably in energy. In a crystal, then, each band can
accomodate 2N electrons, N of each spin, or two electrons per atom, one
of each spin. We first fill up the x-ray levels, having just the same
number of electrons per atom as in the isolated atoms and with just about
the same energy. The remaining electrons go into the valence electron
bands. It then becomes a question of great importance how many such

496 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

electrons there are, and how much they fill the bands up. To consider
this question, let us consider a series of elements, such as Ne, Na, Mg,
etc., each containing one more electron than the one before. If we
formed a crystal of Ne atoms, there would be ten electrons per atom, just
enough to fill the energy bands coming from the atomic Is, 2s, and 2p
electrons. These are all fairly narrow bands, not overlapping with others.
Thus all the electrons of a neon crystal would be in filled energy bands.
With sodium, however, there is one more electron per atom, and it must
go into a valence electron band. The band coming from the atomic 3.s
electron is broadened a good deal at the actual distance of separation of
sodium atoms, and in fact its E vs. p curve is a good deal like that for free
electrons. This band can hold two electrons per atom, but only one is
available, so that it is only half full. In the momentum space, if the
energy is approximately proportional to p2, as with free electrons, the
occupied levels will then fill an approximately spherical volume half as
large as a Brillouin zone, or just the same size that we should have for
free electrons with one free electron per atom. In magnesium, with two
valence electrons per atom, we might think at first sight that both would
go into the 3s band, filling it. But actually there is no gap between this
3s band and the one coming from the atomic 3p levels. Some of the
levels of the 3s band lie higher than the lowest ones of the 3p band. The
two valence electrons per atom, or 2N for the crystal, will go into the low-
est 2N states of the combined bands, meaning that some will go into the
3s band, some into the 3p, neither one being entirely filled. As more and
more electrons are added, the bands fill up more and more, but for the
characteristically metallic elements the levels all overlap, so that we never
have the situation of a certain number of filled levels, all the rest being
empty, with a gap between.

It can now be shown that if a crystal has all its electrons in filled
bands, it must be an insulator; conductivity comes essentially from
partly filled bands. To see this, we need to know the effect of an external
field on the energies of the electrons, in a periodic field. It turns out
that, just as in Sec. 4, the external field brings about a constant rate of
change of momentum, the points representing the various electrons drift-
ing with uniform velocity in the momentum space. The relation between
momentum and energy, however, is as we have found in this section.
Thus we may see, for example in Fig. XXIX-11, what will happen. At
the instant when the field is applied, the levels indicated by shading in (a)
are assumed to be occupied. This corresponds to a whole Brillouin
zone, or a whole band, being filled. As time goes on, the momenta all
increase, so that the occupied levels have shifted along to those shaded in
(6). But on account of periodicity, the levels that have been filled in
going from (a) to (b) are exactly equivalent to those vacated, so that

SBC. 6]

THE ELECTRONIC STRUCTURE OF METALS

497

there has been no net change of the electrons at all, and no resultant elec-
tric current has been set up. Furthermore, no change of distribution
within this band can be brought about by collisions, for thorn are no
empty levels to which electrons can jump. In other words, the electric
field has no effect on a filled band. This argument is not exactly correct;
it neglects the polarization effect, which the field of course can produce on
electrons in closed shells. But it is correct in showing the lack of con-
ductivity from filled bands. In a partly filled band, on the contrary, the
behavior is qualitatively, though not quantitatively, like that described in
Sec. 4. Only a part of a Briilouin zone is filled with electrons, so that

(a)

(b)

FIG. XXIX-11. — Occupied electronic levels in Brillouin zones, one-dimensional crystal
(a) At time of application of external field. (6) After lapse of time, illustrating that on
account of the periodicity the occupied levels really do not change with time.

when the filled region is shifted by the field, the levels that are filled are
different from those that are vacated, and there is a net change in the
distribution, resulting in a current.

Using the statements just made, we see that a crystal of neon, or
other material having only filled electron bands, will be an insulator, while
a metal, having partly filled bands, will be a conductor. One can set up
energy bands for chemical compounds, as well as for elements, and if the
compounds are held together by homopolar bonds, it turns out that the
energy bands are such that certain bands are entirely filled, the others
entirely empty, so that these materials are insulators. For metals, on
the other hand, even though the conductivity is explained essentially as
on the free electron theory, the wave mechanical picture can contribute
several important points to the theory. In the first place, the current
produced by a certain change of momentum, or by a given field acting
through a given time, is not the same as in the free electron theory. We

498 INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

can see this from the fact that a filled band of electrons produces no net
current, though it would according to the free electron theory. As a
matter of fact, as the number of electrons in a band increases, the current
produced by a given field at first increases proportionally to the number
of electrons, as if they were free, but as the band is more nearly filled the
current increases less rapidly than the number of electrons, then reaches
a maximum, and decreases to zero when the band is filled. As a result of
this effect, the current produced is less than if the electrons were really
free. We may define an effective number of free electrons, equal to the
number that would produce the same current as the electrons actually
present. For the alkali metals, this effective number of free electrons is
almost exactly equal to the actual number of valence electrons, but for
other metals it is much less, so that even though there are more valence
electrons than in the alkalies, the effective number of free electrons is less.
A second point in which the exact theory affects the conductivity is
in the matter of the collisions of the electrons with the atoms, resulting in
the time of relaxation which we have discussed in Sec. 4 in connection
with the resistance. We found in Eq. (4.9) that the specific conductivity
was proportional to the relaxation time, or the specific resistance propor-
tional to the probability of collision. Now in wave mechanics the picture
we form of the collision of an electron with an atom is the scattering of the
wave representing the electron, by the irregularity of potential represent-
ing the atom. But we really have scattering, not by a single atom, but
by a whole crystalline arrangement of atoms. If these are regularly
spaced, the electron wave will not be scattered, any more than a light
wave is scattered in passing through a crystal. It is only the deviations
from homogeneity, the irregularities in density, that produce scattering.
Thus if the lattice is perfectly regular, electrons will travel through it
undeviated. This is the case at the absolute zero of temperature. At
higher temperatures, however, there will be fluctuations of density, on
account of the temperature vibrations of the atoms. The amount of
scattering of the wave, or the probability of collision of the electron with
atoms, will be proportional to the mean square deviation of density from
the mean, or to the square of the amplitude of atomic vibration, which
in turn is proportional to the energy of the vibrating atoms, or to the
temperature. Thus we expect the probability of collision and the specific
resistance to be proportional to the temperature. Of course, it is a well-
known experimental fact that this is true, and this simple explanation of
the temperature variation of specific resistance is one of the most impor-
tant results of the wave mechanical theory of conductivity. More
elaborate methods make it possible to estimate the magnitude of the
scattering and of the resistance, and the agreement with experiment is
good.

Sac. 6] THE ELECTRONIC STRUCTURE OF METALS 499

Further applications of the theory of energy bands can be made to
the interatomic forces in metals and other solids. We have already
spoken, in Sec. 2, of the relation of the free electron theory to the equation
of state. This relation can be made much more accurate and quantitative
by means of the energy band theory. In Fig. XXIX-10 we have shown
the way in which the energy bands vary with interatomic distance. In
an approximate way, we can find how the energy of the crystal varies with
lattice spacing by adding the energies of the various electrons it it, though
further corrections must be made to get accurate results. We see that
most atomic energy levels widen out as the atoms approach, their centers
of gravity staying roughly constant, but then rising as the atoms come
very close together. If, then, the crystal had only filled bands of elec-
trons, we should expect the energy to be roughly constant for large
interatomic distances but to rise for smaller distances. That is, there
would be no attractions between atoms, apart from Van dcr Waals forces,
which are neglected in this treatment, but at smaller distances there would
be repulsions, resulting in the impenetrability of atoms. This is what we
should expect in an inert gas like neon, for instance. But if we consider
a crystal like sodium, we have a band of valence electrons only half full,
the bottom half being occupied. This bottom half represents a band of
electrons which spreads out as the interatomic distance decreases (as
we found in Sec. 2 to be the case with free electrons), but whose center of
gravity first decreases, before its final increase at very small interatomic
distance. Thus the mean energy has a minimum; this is what is responsi-
ble for the binding of the metallic crystal. Calculations using this model
for the alkali metals give very good agreement with experiment. It is
clear from this example that the essential for metallic binding is a band
whose lower half is filled, while its upper half is empty. The broader the
band is and the more electrons it holds, the tighter the binding. Thus
in the transition groups of elements, we have noted in Chap. XXVII,
Sec. 2, that the binding increases in strength as we add more d electrons,
weakening again as the d shell is filled. These d electrons have a band
which is rather narrow but which is capable of holding ten electrons. The
first five go into the bottom half of the band, contributing to the binding,
while the last five go into the upper half, weakening the binding again.

We have just stated that, though the d band in the transition group
elements is rather narrow, resulting from the small overlapping of the
rather small d orbits, still it can contain ten electrons. This means that
the number of energy levels per unit range of energy must be very much
greater than it would be for free electrons. This has an interesting
application to the electronic specific heat of these metals. The general
expression for the specific heat of a system obeying the Fermi statistics
is given in Eq. (4.8), Chapter V,

500

INTRODUCTION TO CHEMICAL PHYSICS [CHAP. XXIX

= (**£\ LVr,

(6.1)

where dN/d,€ is the number of energy levels per unit range. Since this is
much larger for the d shell than for free electrons, we infer that the elec-
tronic specific heat should be unusually large for a transition metal. This
is observed experimentally, as we have already shown in the discussion of
Table XXIX-2.

The binding of valence crystals can also be explained from the stand-
point of energy bands. In Fig. XXIX-12 we show energy bands for
diamond, a typical crystal held by homopolar bonds. We see that the

FIG. XXIX-12. — Energy bands in diamond. The lower band is occupied, the upper one
unoccupied, illustrating the energy gap above the occupied levels.

bands look rather different from what they do in a metal, in that they are
divided into an upper and a lower half, with a gap between. There are
enough electrons in carbon to fill the bands coming from the Is atomic
level, and the lower band coming from the atomic 2s and 2p. Then there
is a wide gap between the top of the filled level and the first empty level,
showing that diamond must be an insulator. But now we notice that the
filled level dips down sharply as the interatomic distance decreases, before
commencing its rise. Thus diamond is strongly bound, as we know from
its high heat of vaporization and low compressibility. Similar bands
occur in other valence compounds, though they have not been extensively
investigated.

From the examples considered, we see that the effect of the periodic
potential on the motion of the electrons is essential to an understanding

Sac. 61 THE ELECTRONIC STRUCTURE OF METALS 501

of interatomic or intermolecular binding in crystals, as well as of the
electrical properties. We have been able to do no more in this chapter
than give a suggestion of the type of results to be obtained. To go
further, one must make a great deal more study than we have of the
principles of wave mechanics. The same remark might, in fact, be made
about a great many of the topics taken up in this book. A well-trained
chemical physicist should be an expert in the quantum theory and wave
mechanics, as well as in thermodynamics and statistical mechanics. Our
effort in this book, however, has been to show that one can really get
surprisingly far and can understand nature surprisingly well, with rela-
tively elementary facts about the fine structure of matter and the prin-
ciples governing its behavior.

PROBABLE VALUES OP THE GENERAL PHYSICAL

CONSTANTS

Velocity of light
Mechanical equivalent
of heat

Normal atmosphere
Gas constant

Ice point

Volume of 1 mole of

perfect gas, n.t.p.
Avogadro's number

Boltzmann's constant k

Planck's constant
Electronic charge

Electron volt

e = 2.998 X 1010 cm. per second

1 cal. = 4.185 X 107 ergs

1 kg.-cal. = 4.185 X 10l° ergs

1 atm. = 1.013 X 106 dynes square centimeter

R = 8.314 X 107 ergs per degree per mole

_ 8.314 X 107

"" 4.185 X 107

= 1.987 cal. per degree per mole

— .08205 1. atm. per degree per mole *
Temp. = 273.2° abs.

Volume = 0.08205 X 273.2 = 22.41 1.
NQ = 6.03 X 1023 molecules per mole
8.314 X 107

6.03 X 1023
= 1.379 X 10~lfl erg per degree
h = 6.61 X 10"27 erg-sec.
e - 4.80 X 10~10 e.s.u.

Mass of atom of unit

atomic weight
Mass of electron

^ 1.60 X 10~12 X 6.03 X 1028

4.185 X 1010
= 23.05 kg.-cal. per mole

Mass - 1.66 X 10"24 gm.
m - 9.10 X 10^8gm.

— isW X mass of unit atomic weight
Rydberg energy Rhc = 2.17 X 10"11 erg

= 13.56 electron volt
= 313 kg.-cal. per mole
Radius of first hydro-
gen orbit GO = 0.53 A.

The values for the large-scale constants are taken from the well-known tabulation
of Birge, Phys. Rev. Supplement , Vol. 1, 1929. The atomic and electronic constants
are taken or computed from the more recent values tabulated by Dunnington, Bull.
Am. Phys. Soc., 14 (1), 17 (1939).

508

SUGGESTED REFERENCES

The reader may very likely want to refer to other texts dealing with the same
general subject as the present one. There are of course a great many books, both old
and new, treating thermodynamics, statistical mechanics, or both. We may mention
first the classical treatises, L. Boltzmann's " Vorlesungen tiber Gastheorie," reprinted
by Barth in 1923, and Willard Gibbs's "Elementary Principles in Statistical Mechan-
ics," reprinted in his "Collected Works" by Longmans, Green and Company, in
1931. Somewhat more recent are two books by M. Planck: "Treatise on Thermo-
dynamics," published by Longmans, Green and Company, and "Heat Radiation,"
published by P. Blakiston's Sons and Company, the latter containing many of the
statistical ideas connected with the introduction of the theory of quanta. Another
standard text, dealing principally with phase equilibrium, is the "Lehrbuch der
Thermodynamik," by J. D. van der Waals. "The Dynamical Theory of Gases," by
J. H. Jeans, Cambridge University Press, is a standard text on kinetic theory; more
recent texts on the same subject are "Kinetic Theory of Gases," by Loeb, and a book
by the same title by E. H. Kennard, both published by McGraw-Hill Book Company,
Inc. "Kinetische Theorie der Warme," by K. Herzfeld, published by Vieweg, com-
bines the principles of statistical mechanics with application to matter, in somewhat
the same way as the present text. Among more recent texts of thermodynamics one
may mention "Thermodynamics," by E. Fermi, Prentice-Hall, Inc., 1937; "Textbook
of Thermodynamics," by P. Epstein, John Wiley & Sons, Inc., 1937; "Heat and
Thermodynamics," by J. K. Roberts, Blackie and Son, Ltd., 1933; "Modern Thermo-
dynamics by the Method of Willard Gibbs," by E. A. Guggenheim, Methuen and
Company, Ltd., 1933; and "Heat and Thermodynamics," by M. W. Zemansky,
McGraw-Hill Book Company, Inc., 1937. Recent texts on statistical mechanics
include the standard treatise, "Statistical Mechanics," by R. H. Fowler, Cambridge
University Press, 1929; (a revision and condensation of this work, in collaboration
with Guggenheim, is understood to be in preparation); "Statistical Mechanics with
Applications to Physics and Chemistry," by R. C. Tolman, Chemical Catalog Com-
pany, 1927; the successor to that volume, "Principles of Statistical Mechanics," by
Tolman, Oxford University Press, 1938; "Statistical Physics," by Landau and Lif-
schitz, Oxford University Press, 1938; and " Quantenstatistik und ihre Anwendungen
auf die Elektronentheorie der Metalle," by L. Brillouin, Springer, 1931. A number
of texts on physical chemistry and chemical thermodynamics bear closely on the topic
of this book. First of course is the well-known "Thermodynamics and the Free
Energy of Chemical Substances," by Lewis and Randall, McGraw-Hill Book Com-
pany, Inc., 1923. Others are "A Treatise on Physical Chemistry," by H. S. Taylor,
D. Van Nostrand Company, Inc., 1932; and "A System of Physical Chemistry," by
W. C. McC. Lewis, Longmans Green & Company, 1921. Closely related is "The New
Heat Theorem, its Foundation in Theory and Experiment," by Nernst, E. P. Dutton
Company, Inc., 1926, describing some of the early applications of quantum theory to
thermodynamics.

Thermodynamics and statistical mechanics, as well as the structure of matter, can
hardly be understood without a study of atomic and molecular structure, and of the
quantum theory which underlies them. Suggested texts in this general field are

505

506 INTRODUCTION TO CHEMICAL PHYSICS

"Introduction to Modern Physics," by F. K. Richtmyer, McGraw-Hill Book Com-
pany, Inc., "Atoms, Molecules and Quanta," by Ruark and Urey, McGraw-Hill Book
Company, Inc., "Introduction to Atomic Spectra," by H. E. White, McGraw-Hill
Book Company, Inc., "The Structure of Line Spectra," by Pauling and Goudsmit,
McGraw-Hill Book Company, Inc. For somewhat more mathematical treatments
of the same field, including wave mechanics, useful treatments are found in "Introduc-
tion to Quantum Mechanics," by Pauling and Wilson, McGraw-Hill Book Company,
Inc., "Elements of Quantum Mechanics," by S. Dushman, John Wiley & Sons, Inc.,
"Wave Mechanics, Elementary Theory," by J. Frenkel, Oxford University Press;
"The Fundamental Principles of Quantum Mechanics," by E. C. Kemble, McGraw-
Hill Book Company, Inc.; as well as the treatment in "Introduction to Theoretical
Physics," by Slater and Frank, McGraw-Hill Book Company, Inc.

Many texts deal with specific subjects taken up in one chapter or another of the
present book. One may mention "The Phvsics of High Pressure." by P. W. Bridg-
man, The Macmillan Company, 1931, and "The Thermodynamics of Electrical
Phenomena in Metals," also by Bridgman and published by The Macmillan Com-
pany; "Metallography," by C. H. Desch, Longmans, Green and Company, for a
treatment of phase equilibrium; "Measurement of Radiant Energy," by W. E.
Forsythe, McGraw-Hill Book Company, Inc., for black-body radiation; "Crystal
Chemistry," by C. W. Stillwell, McGraw-Hill Book Company, Inc., with more
detailed treatments of many of the points taken up in the Part III of the present
book; "Valence and the Structure of Atoms and Molecules," by G. N. Lewis, Chemical
Catalog Company, and "The Nature of the Chemical Bond," by L. Pauling, Cornell
University, 1939, for discussion of valence bonds; "The Crystalline State," by W. H.
and W. L. Bragg, The Macmillan Company, 1934, and "The Structure of Crystals,"
by R. W. G. Wyckoff, Chemical Catalog Company, for more detailed information
about crystal structure; "Photoelectric Phenomena," by Hughes and DuBridge,
McGraw-Hill Book Company, Inc., for treatment of electronic questions; "Properties
of Metals and Alloys," by Mott and Jones, Oxford University Press, "Elektronen-
theorie der Metalle," by H. Frohlich, Springer, and "The Theory of Metals," by A. H.
Wilson, Cambridge University Press, for the structure of metals.

In addition to these texts, various handbooks, tables, etc., are of great service.
Both the "Handbuch der Physik" and the "Handbuch der Experimentalphysik "
contain several volumes dealing with the general subjects treated in the present
book, treated both experimentally and theoretically. Some specific references are
made to them in the text. For numerical data, Landolt-Bornstein's "Physikalisch-
Chemische Tabellen" are invaluable, supplemented by the "International Critical
Tables." In the field of atomic spectra, "Atomic Energy States," by Bacher and
Goudsmit, McGraw-Hill Book Company, Inc., has very complete data, and the field
of crystal structure is covered by the "Strukturbericht," a supplement to the "Zeit-
schrift fur Kristallographie." Band spectra and molecular structure are included in
the volume of tables in " Molekulspektren und ihre Anwendungen auf Chemische
Probleme," by H. Sponer, Springer. Finally, though we have not listed many refer-
ences to original articles, the reader will do well to consult review articles in Chemical
Reviews and Reviews of Modern Physics, as well of course as becoming familiar with the
large literature in the Journal of Chemical Physics, Journal of the American Chemical
Society, as well as foreign periodicals.1

INDEX

A priori probability, 36, 38, 127
Absorption of radiation, 309-310, 322-333
Absorptivity, 309-310, 325-326
Acetic acid, data regarding melting point,
259

heat of vaporization, 434

structure of molecule, 427

Van der Waals constants, 408
Acetylene, structure of molecule, 428

Van der Waals constants, 408
Activation, energy of, 159-164, 257
Adiabatic processes, 13, 17-19
Aliphatic compounds, 420-428
Alkali halides, data regarding crystals,
381, 393, 395

data regarding melting point, 259

equation of state, 390-396

residual rays, 254-255

specific heat, 253
Alloys, 270-290, 458-459
Aluminum, crystal structure, 447

data regarding melting point, 259, 261

Debye temperature, 237

equation of state and crystal structure,
451, 454

molecular volume, 261

specific heat, 236

thermal expansion, 261
Ammonia, data regarding melting point,
259

formation of ammonium complex,
272-273

heat of vaporization, 414

valence structure of molecule, 401

Van der Waals constants, 408
Ammonium bromide, data regarding
crystals, 382

rotation of ions, 300

Ammonium chloride, data regarding
crystals, 381

rotation of ions, 293, 300
Ammonium iodide, data regarding crys-
tals, 382

Ammonium ion, 357, 378

Angular momentum, quantization, 40,

135, 339
Annealing, 282
Antimony, crystal structure, 447, 449

data regarding melting point, 259
Argon, atomic volume, 384

data regarding crystals, 416

data regarding melting point, 259

specific heat, 130

Van der Waals constants, 408
Aromatic compounds, 428-432
Arsenic, crystal structure, 447, 449
Assembly, canonical, 46—53
fluctuations in, 101-111

microcanonical, 46

in molecular phase space, 69-72

statistical, 32-35
Atomic number, 336
Atoms, structure, 321-351
Attractive forces between molecules,
130-133

and atomic theory, 352-376

in solutions, 271-277

and Van der Waals' equation, 182-184,

194-196

Avogadro's law, 60
Avogadro's number, 60-61, 128

Barium, crystal structure, 447

equation of state, 451
Barium chloride, data regarding melting

point, 259
Barium oxide, sulphate, selenide, tellu-

ride, data regarding crystals, 381
Barometer formula, 62-64
Benzene, data regarding melting point,

259

structure of molecule, 428-431

Van der Waals constants, 408

Beryllium, crystal structure, 447

equation of state, 451
Beryllium oxide, sulphide, selenide, tellu-
ride, data regarding crystals, 382

507

508

INTRODUCTION TO CHEMICAL PHYSICS

Binary systems, phase equilibrium, 270-

290

Bismuth, crystal structure, 447-449
data regarding melting point, 259
Black-body radiation, 307-320, 325-326

and Einstein-Bose statistics, 85
Body-centered cubic structure, and

metals, 445-447
and molecular vibrations, 232
and order-disorder, 293
Bohr, frequency condition, 323

hydrogen atom, 340
Boiling point, 166-169, 171-180

of chain compounds, 422
Boltzmann factor, and thermionic emis-
sion, 468

Boltzmann statistics, application to per-
fect gas, 115-129
fluctuations, 101-104, 109
and kinetic method, 86-96
relation to Fermi-Dirac and Einstein-
Bose statistics, 68-72
Boltzmann's constant, 33, 61
Boltzmann's H theorem, 90
Boltzmann's relation between probability

and entropy, 34
Boundary conditions, vibrating chain of

atoms, 244

vibrating solid, 227-228
Boyle's law, 30, 60

deviations from, 190-198
Brass, phase equilibrium, 270, 287-288
Bridgman, 24, 181, 201, 220
Brillouin zone, and electrons in metals,

493-501

and molecular vibrations, 233
Bromine, characteristic temperature, for

rotation, 136
for vibration, 142

heat of dissociation, interatomic dis-
tance, Morse constant, 132
and homopolar bonds, 400-408
and organic compounds, 425-426
Butane, heat of vaporization, 434
structure of molecule, 421
Van der Waals constants, 408

Cadmium, crystal structure, 447
data regarding melting point, 259, 261
Debye temperature, 237
equation of state and energy, 451, 454

Cadmium, molecular volume, 267

thermal expansion, 261
Cadmium oxide, sulphide, selenide, tel-

luride, data regarding crystals, 382
Caesium, compressibility, 202
crystal structure, 447
data regarding melting point, 259
equation of state and energy, 451, 454
Caesium bromide, fluoride, iodide, data

regarding crystals, 381
Caesium chloride, crystal structure, 378-

379, 381

Calcite stnicture, 397-398
Calcium, crystal structure, 447

equation of state and energy, 451, 454
phase equilibrium in alloys, 274
Calcium carbonate, structure, 397-398
Calcium chloride, data regarding melting

point, 259
Calcium oxide, sulphide, selenide, tel-

luride, data regarding crystals, 381
Caloric theory of heat, 4-6
Calorie, 6

numerical value, 8
Canonical assembly, 46-51

and Maxwell-Boltzmann distribution,

52-53

Carbon, and homopolar bonds, 400-407
and organic compounds, 420-434
(diamond), crystal structure, 379, 424
Debye temperature, 237
energy bands, 500
energy constants, 455
(graphite), crystal structure, 429
(molecule), characteristic temperature,

of rotation, 136
of vibration, 142

heat of dissociation, interatomic dis-
tance, Morse constant, 132
Carbon bisulphide, Van der Waals con-
stant, 408
Carbon dioxide, data regarding melting

point, 259

heat of vaporization, 414
triple point, 167

valence structure of molecule, 405
Van der Waals constants, 408, 411
vapor pressure and latent heat of

vaporization, 188-189
Carbon monoxide, characteristic tern*

perature, for rotation, 136
for vibration, 142

INDEX

509

Carbon monoxide, crystal structure, 417
data regarding melting point, 259
dissociation, 133

heat of dissociation, interatomic dis-
tance, Morse constant, 132
heat of vaporization, 414
Van der Waals constants, 408, 411
vibrational specific heat, 144—145
Carbon tetrachloride, boiling point, 426

data regarding melting point, 259
Carbonate ion, valence structure, 406
Carboxyl group, 427
Cells in phase space, 38-43
and Fermi-Dirac and Einstein-Bose

statistics, 65-85
and interatomic forces, 369
and kinetic method, 86-100
CH, characteristic temperature, for rota-
tion, 136

for vibration, 142

heat of dissociation, interatomic dis-
tance, Morse constant, 132
Change of phase, 23, 166-190, 2S&-269
Characteristic temperature, for atomic
vibrations in crystals, Debye, 235-
236

for ionization, 322, 335
for rotation, 136
for vibration, diatomic molecules, 141-

142

Charles's law, 60

Chemical constant, and chemical equilib-
rium, 155-156
of diatomic gas, 140
of monatomic gas, 118, 120, 128
and thermionic emission, 463-464
and vapor pressure, 178-180
Chemical equilibrium, 150-165
Chlorine, characteristic temperature, for

rotation, 136
for vibration, 142
data regarding melting point, 259
heat of dissociation, interatomic dis-
tance, Morse constant, 132
and homopolar bond, 400-408
and organic compounds, 425-426
Van der Waals constants, 408
Chloroform, boiling point, 426
heat of vaporization, 434
Van der Waals constants, 408
Chromium, crystal structure, 447
data regarding melting point, 259

Chromium, equation of state and crystal

structure, 451, 454
Clapeyron's equation, 174-181, 220
Clay, structure, 440
Cobalt, crystal structure, 447

data regarding melting point, 259

equation of state and energy, 451, 454
Cold-working, 457

Collisions, and chemical reactions, 150-
154, 158-165

effect on approach to equilibrium, 86-
92, 96-100

elastic, 327

and electrical resistance, 487-489

of electron and metal, 460-467

inelastic, 327

and radiation, 326-333

of second kind, 331
Complexion, 33-43

and Fermi-Dirac and Einstein-Bose

statistics, 65-85

Compressibility, of alkali halides, 392-
395

and entropy of melting, 261

and fluctuations, 110

of metals, 452-454

of solids, 200-205, 218-221

and thermal expansion, 238-240

and thermodynamics, 19

of water and ice, 170
Concentration, and chemical equilibrium,
153-154

and mixtures of gases, 121-124

and phase equilibrium in binary sys-
tems, 270-290
Conductivity, electrical, 456, 484-489,

495-501
Configuration, atomic, 342

table of, 346-347
Conservation of energy, 3-9
Conservative force, 3
Contact difference of potential, 467-471,

480-482
Copper, alloys with nickel, 458

crystal structure, 447

data regarding melting point, 259

Debye temperature, 237

energies of electrons in atom, 341

equation of state and energy, 451, 454

order-disorder in alloys, 293-304

phase equilibrium in alloys, 270, 279-
282, 287-288

510

INTRODUCTION TO CHEMICAL PHYSICS

Corresponding states, and Van der

Waals' equation, 187
Coulomb energy, 353, 361-363, 368-369
Covalent bond, 373-376, 400-407

and organic compounds, 420-434

and silicates, 435-443
Cristobalite structure, 441
Critical point, 166-169

and phase changes of second order, 291

and Van der Waals equation, 184-187
Crystals, electric field in, 473, 489-501

equation of state, 211-221

ionic, 375, 377-399

and liquid, 256-258

of metals, 444-451

molecular, 414-419
Cuprous chloride, bromide, iodide, data

regarding crystals, 382
Curie point, 292, 297-304
Cyclohexane, structure of molecule, 424

Pebye, ionic crystals, 390-393

specific heat of solids, 222-240, 245,

253-255
Decanc, heat of vaporization, 434

Van der Waals constants, 408
Degeneracy, in space quantization, 139,

339

Degradation of energy, 12
Degree of order, 294-304
Dependent variables, 17-18
Detailed balancing, 91

and radiation, 324

Diamond, crystal structure, 379, 424,
444-447

energy bands, 500

melting point, 449
Diatomic molecules, 130-145, 400-414

comparison with metals, 455
Dichlorbenzene, structure of molecule,

431
Dielectric constant, 365

and Van der Waals forces, 411
Diffusion, 12-13

Dimethylamine, structure of molecule,
427

Van der Waals constants, 408
Dipoles, 354-361

moment of HC1 and HBr, 404-405

moments of molecules, table, 408

Disorder, 10-12, 32-38, 43-46

in alloys, 293-304

Dispersion of elastic waves, 223, 234, 244
Dissociation, 152-154
Distribution function, 33-35, 46-51

in Fermi-Dirac and Einstein-Bose
statistics, 65-85

for fluctuations, 104-107

in Max well-Bo tzmann statistics, 52-

Double bonds, 401-402, 428, 431
Ductility, 456
Dulong and Petit's law, 213

deviations from, 222-240

Effective volume of molecules, and Van

der Waals' equation, 183, 195-196
Einstein, black-body radiation, 324-326
photoelectric effect, 316-320
specific heat functions, for gases, 144,

147

for solids, 214r-215, 253
Einstein-Bose statistics, 52, 65-85
and black-body radiation, 326
and fluctuations, 108-109
and kinetic method, 96-100
and perfect gas, 126
Elastic collisions, 327
Elastic vibrations of solids, and specific

heat, 222-255
Electron affinity, 338
Electron gas, and Fermi-Dirac statistics,

and metallic structure, 475-484
Electron volt, numerical value, 132—133,

318
Electrons and atomic structure, 337-351

and structure of metals, 472-501
Electrostatics, and field in metal, 472-

489

and interatomic forces, 353-367
and ionic crystals, 385-390
Emission of radiation, 309-310, 317-320,

322-333

Emissive power, 309-310, 325-326
Energy (see Conservation of energy;
Internal energy; Kinetic energy;
Potential energy)

Energy of activation, 159-164, 257
Energy bands in metals, 493-501

INDEX

511

Energy density, in radiation, 310-316,

324-326
Energy levels, 41-42

of atomic systems, 322, 338-344
Ensemble (see Assembly)
Enthalpy, and Joule-Thomson effect,

197-198

and latent heats, 175-178
and thermodynamics, 20-21
Entropy, of diatomic gas, 140
and equilibrium of phases, 170-173
in Fermi-Dirac and Einstein-Bose

statistics, 69-72
and fluctuations, 107
of fusion, 171-180, 258-269
and kinetic method, 89-91, 98-99
of mixture of gases, 121-123, 128-129
of perfect gas, 117-119, 127

Fermi-Dirac statistics, 78-79
and phase change of second order,

291-304

and phase equilibrium in binary sys-
tems, 272-290
of solids, 207-218

and statistical mechanics, 32-35, 43-51
and thermodynamics, 9-14, 17-18, 21
of vaporization, 171-180
Equation of state, of imperfect gases,

182-198

of ionic crystals, 385-396
of metals, 450-456, 479-480
of perfect gas, 58-61

Fermi-Dirac statistics, 82
and phase equilibrium, 169-170
of solids, 199-221
and thermodynamics, 16-18, 22-23,

2&-30

Equilibrium, between atoms and elec-
trons, 333-335

between liquid and solid, 266-269
between metals and electron gas, 463-

464
between phases, 166-181, 184-190

in binary systems, 270-290
chemical, 150-165
and radiation, 325

thermal, 13-15, 23, 37-38, 46-51, 96, 98
Equipartition of energy, and Maxwell's

distribution, 57-58
and specific heat, of polyatomic gases,

134, 144, 146
of solids, 213

Ethane, heat of vaporization, 434
hindered rotation, 147-148
structure of molecule, 402, 420-421
Van der Waals constants, 408

Ethyl alcohol, data regarding melting

point, 259

heat of vaporization, 434
Van der Waals constants, 408

Ethyl ether, heat of vaporization, 434
structure of molecule, 427
Van der Waals constants, 408

P^thylene, structure of molecule, 402, 428
Van der Waals constants, 408, 411

Eutectic, 284-285

Exchange, and interatomic forces, 367-
374

Excitation of atoms, 321-333, 343

Exclusion principle, 342

and interatomic forces, 369-372

Explosion, 158-159

External work, 3, 7-9, 17, 21-22
and statistical mechanics, 49

Face-centered cubic structure, descrip-
tion and figure, 415

in inert gases, 416

and metals, 445-447

and molecular vibrations, 232

and order-disorder, 293
Fermi-Dirac statistics, 52, 65-85

and atomic structure, 342

and exchange effect, 369

and fluctuations, 108-109

and kinetic method, 96-100

and metals, 471, 475-484

and perfect gas, 126
Ferromagnetism, 292-293
Fibers, silicate, 439

Field, electric, and interatomic forces,
35&-360, 366

in metal, 472-501
First law of thermodynamics, 7-8, 19

and statistics, 49-51
Fluctuations, 32. 101-111
Fluorite structure, 396-397
Forces between molecules, 130-133

interpretation from atomic theory,
352-376

in solids, 271-277

and Van der Waals' equation, 182-184,
194-196

512

INTRODUCTION TO CHEMICAL PHYSICS

Formic acid, structure of molecule, 427
Free electrons in metals, 475-489
Free energy, Gibbs, and chemical equili-
brium, 154-168
of diatomic gas, 140
and equilibrium of phases, 170-180
and melting, 265-269
of mixture of gases, 123-124
of perfect gas, 120
and phase changes of second order,

296-304
and phase equilibrium in binary

systems, 270, 278-290
of solids, 205-211
and thermionic emission, 463-464
and thermodynamics, 22-23
and Van der Waals' equation, 184-

189
Helmholtz, and Fermi-Dirac and Ein-

stein-Bose statistics, 73, 79, 82
and melting, 265-269
of perfect gas, 119, 126
and second virial coefficient, 193-194
of solids, 205-211, 216-218
and statistical mechanics, 50-51
and thermodynamics, 21-22
Free expansion of gas, 30, 196-198
Freezing (see Melting)
Frequency of oscillation, diatomic mole-
cule, 141

molecular solid, 241-255
solid, 213-240
Friction, 3
Fusion, 23, 166-169, 171-176, 256-269

Gallium, crystal structure, 447

data regarding melting point, 259
Gas constant, 33

numerical values, 60
Gases, and equilibrium with other phases,

166-180

imperfect, and Van der Waals' equa-
tion, 182-198
perfect, 17, 30

chemical equilibrium in, 150-165
and Maxwell-Boltzmann distribu-
tion, 53-64
polyatomic, 130-149
thermodynamic and statistical treat-
ment, 115-129

Gases, perfect, translational energy levels
in quantum theory, 54-55

Van der Waals constants, 408
Gauss error curve, 106
Germanium, crystal structure, 444, 447-
449

melting point, 449
Gibbs, 32, 44, 107

Gibbs free energy (see Free energy, Gibbs)
Gibbs's paradox, 129
Glass, 256-258

structure, 442

variability of composition, 273
Gliding, 457
Gold, crystal structure, 447

data regarding melting point, 259

Debye temperature, 237

equation of state and energy, 451, 454

order-disorder in alloys, 293-294
Graphite structure, 429
Gravity, 4
Grtineisen, thermal expansion, 217-221,

238-240

of ionic crystals, 392-394
of metals, 451-456

H theorem, 90

Hafnium, crystal structure, 447

equation of state, 451
Halogens, characteristic temperature, for

rotation, 136
for vibration, 142
data regarding melting point, 259
heat of dissociation, interatomic dis-
tance, Morse constant, 132
and homopolar bonds, 400-408
and organic compounds, 425-426
Heat absorption, 7-9, 12-13, 20
and statistical mechanics, 49
Heat, of dissociation, diatomic mole-
cules, table, 132
latent (see Latent heat, of fusion;

Latent heat, of vaporization)
of reaction, 156-158

and equilibrium of ions and electrons,

334

specific (see Specific heat)
Heat capacity (see Specific heat)
Heat engine, 13, 172
Heat flow, 12-13
Heitler-London method, 367-368

INDEX

513

Helium, specific heat, 130

Van der Waals constants, 408
Helmholtz free energy (see Free energy,

Helmholtz)

Hexagonal close-packed structure, de-
scription and figure, 417
and metals, 445-447
and molecular vibration, 232
Hexane, structure of molecule, 423
Hindered rotation, 147-149, 417-418
Homopolar valence attraction, 373-376,

400-407

and organic compounds, 420-434
and silicates, 435-443
Hydrogen, characteristic temperature,

for rotation, 136
for vibration, 142
combination with oxygen to form

water, 151-164

data regarding melting point, 259
heat of dissociation, interatomic dis-
tance, Morse constant, 132
heat of vaporization, 414
and homopolar bonds, 400-408
interatomic potential, 371
and organic compounds, 420-434
specific heat, 137-138
Van der Waals constants, 408, 411
Hydrogen bromide, data regarding melt-
ing point, 259
valence structure, 404-405
Van der Waals constants, 408
Hydrogen chloride, characteristic tem-
perature, for rotation, 136
for vibration, 142

crystal structure and hindered rota-
tion, 417

data regarding melting point, 259
dipole moment, 358
heat of dissociation, interatomic dis-
tance, Morse constant, 132
heat of vaporization, 414
valence structure, 404-405
Van der Waals constants, 408
Hydrogen sulphide, valence structure,

405
Van der Waals constants, 408

Ice, crystal structure, 412, 418-419
polymorphic forms, 167-170
structure, 260

Image force, 461, 474-475
Impenetrability of matter, 130
Imperfect gases, and phase equilibrium,

166-170

and Van der Waals' equation, 182-198
Independent variables, 17-18
Indium, crystal structure, 447

data regarding melting point, 259
Induced emission, 325
Inelastic collisions, 327
Inert gases, crystals, 416
and periodic table, 345-350
volumes of atoms, 384
Insulators, and energy bands, 495—501
Integrals independent of path, 8, 13
Interatomic distances, in crystals, and
formulas for thermodynamic quan-
tities, 212-213

in crystals of inert gases, 416
in diatomic molecules, table, 132
in ionic* crystals, table, 381-382
in metals, 447

in organic compounds, 420-434
Interatomic forces, 130-133
interpretation from atomic theory,

352-376

in ionic crystals, 385-390
in metals, 451-456
in organic compounds, 433
and second virial coefficient, 191-196
and Van der Waals' equation, 182-184
and vibrations of atoms in crystals,

211-240
Interference of light, and quantum

theory, 319-320

Intermolecular forces, in gases, 410-414
Internal energy, 6-9, 17
at absolute zero, 179
and melting, 258-269
of mixture of gases, 123
of perfect gas, Boltzmann statistics,

117

Fermi-Dirac statistics, 77-78, 81-82
and phase change of second order, 295,

301

of solids, 205-220
of solutions, 275-277
Internal pressure, 182-184
Iodine, characteristic temperature, for

rotation, 136

for vibration, 142

crystal structure, 418

514

INTRODUCTION TO CHEMICAL PHYSICS

Iodine, dissociation, 133

heat of dissociation, interatomic dis-
tance, Morse constant, 132

and homopolar bond, 400-408

and organic compounds, 426
Ionic crystals, 375, 377-399
Ionic radii, 382-385
lonization, of atoms, 321-335
lonization potential, 322, 334, 343

table of, 348

Ions, and atomic structure, 321-335, 337-
338, 351

forces between, 357-358

formation in solution, 272-274, 290
Iridium, crystal structure, 447
Iron, crystal structure, 447

data regarding melting point, 259, 261

Debye temperature, 237

equation of state and crystal structure,
451, 454

molecular volume, 261

thermal expansion, 261
Irreversible process, 11-13, 16

and kinetic approach to equilibrium,
86-92, 96-98

and statistical mechanics, 43-46
Isomers, 423
Isothermal processes, 19
Isothermals, of solid, 200

and Van der Waals' equation, 184^186
Isotopes, 336-337

Joule, 5

Joule-Thomson effect, 196-198

Joule's law, 30, 115

Kinetic energy, and exchange effect, 369
and Maxwoll-Boltzmann law, 60
of polyatomic molecules, 134, 144
Kinetic theory, 15, 86-100
and chemical reactions, 151-154, 158-

165

and radiation, 324-333
and thermionic emission, 465-471,

480-484

Kirchhoff's law, 309-310
Krypton, atomic volume, 384
data regarding crystals, 416

Krypton, specific heat, 130
Van der Waals constants, 408

Lanthanum, crystal structure, 447

equation of state, 451
Latent heat, of evaporation of electrons,

464, 469-470, 481-484
of fusion, 171-180, 258-269
of vaporization, 171-180, 258-260
of metals, 452-454
of organic compounds, table, 434
table of, 414

and Van der Waals' equation, 189
Lattice energies, alkali halides, 395-396
Lattice spacings, ionic crystals, 381-382

metals, 447

Lead, crystal structure, 447
data regarding melting, 259
Debye temperature, 237
equation of state, 451
Lead bromide, chloride, iodide, data

regarding melting point, 259
Lewis, G. N., and homopolar bond,

400-408

Linear oscillator, and black-body radia-
tion, 314r-316

and equipartition of energy, 58
quantum theory, 39-40, 42
and vibration of atoms and molecules

in crystals, 211-240
and vibration of diatomic molecules,

140-149

Liouville's theorem, 37-38, 44-46, 88
Liquefaction of gases, 198
Liquids, 166-174, 256-269
Liquidus, 281

Lithium, compressibility, 202
crystal structure, 447
equation of state and energy, 451, 454-

455
molecule, characteristic temperature,

for rotation, 136
for vibration, 142
dissociation, 133
heat of dissociation, interatomic

distance, Morse constant, i32
Lithium fluoride, chloride, bromide,
iodide, data regarding crystals, 381,
393, 395

INDEX

615

Lithium nitrate, data regarding melting

point, 259
Longitudinal waves in solids, 222-240

Madelung, electrostatic energy of crys-
tals, 385-388

Magnesium, crystal structure, 447
data regarding melting point, 259,

261

equation of state and energy, 451, 454
molecular volume, 261
phase equilibrium in alloys, 274,

287-288

thermal expansion, 261
Magnesium oxide, sulphide, selenide,

data regarding crystals, 381
Manganese, crystal structure, 447-448
data regarding melting point, 259
equation of state, 451
Mass action law, 151-158

applied to atomic processes, 334r-335
Maxwell-Boltzmaim distribution, 52-64
and activation, 159
and Fermi-Dirac and Einstein-Bose

statistics, 74, 84
and fluctuations, 101-104
and mean moment of rotating dipole,

361,364

Maxwell's demon, 45-46
Maxwell's distribution of velocities,

55-58

in arcs, 333

and kinetic methods, 91-96
Maxwell's relations, 26
Mean free path, 328
Mechanical equivalent of heat, 5, 8
Mechanical work (see External work)
Melting, 23, 166-169, 171-176, 256-269

of alloys, 278-290
Melting points, of chain compounds, 422

ionic crystals, table, 381-382
Mercuric bromide, data regarding melt-
ing point, 259
Mercuric iodide, data regarding melting

point, 259
Mercuric sulphide, selenide, telluride,

data regarding crystals, 382
Mercury, crystal structure, 447
data regarding melting point, 259

Metallic bond, 374-376, 451-464
Metals, 376, 444-470, 472-501
Metasilicate ion, structure, 437
Metastable equilibrium, 163
Methane, boiling point, 426
data regarding melting point, 259
heat of vaporization, 434
valence structure of molecule, 401
Van dcr Waals constants, 408, 411
Methyl alcohol, data regarding melting

point, 259

heat of vaporization, 434
structure of molecule, 427
Van der Waals constants, 408
Methyl chloride, boiling point, 426
heat of vaporization, 434
structure of molecule, 425
Van der Waals constants, 408
Methyl ether, Van der Waals constants,

408
Methylamine, structure of molecule, 427

Van dcr Waals constants, 408
Mica, structure, 439
Microcanonical assembly, 46
Microscopic properties, 32-35
Microscopic reversibility, 88, 331
Mixture of gases, 120-124

and chemical equilibrium, 154-158
Molecular orbitals, 368
Molecular phase space, and Fermi-Dirac

and Einstein-Bose statistics, 65-86
and Maxwell-Boltzmann distribution,

52-64
Molecular volume, of liquids, table, 408

table of, 261

Molecules, sizes of, 409-410
in valence compounds, 375-376

(See also Diatomic molecules; Poly-
atomic molecules)

Molybdenum, crystal structure, 447
Debye temperature, 237
equation of state and crystal Btnicture,

451, 454
Moment of inertia, diatomic molecule,

134r-135

Momentum, 36, 38-43
and approach to equilibrium, 87
of radiation, 311

Monatomic gases, and Maxwell-Boltz-
mann distribution, 53-64

516

INTRODUCTION TO CHEMICAL PHYSICS

Morse curves, 133
and metals, 452
Multiplets, 344

Naphthalene, heat of vaporization, 434
structure of crystal, 432
structure of molecule, 430
Van der Waals constants, 408
Neon, atomic volume, 384
data regarding crystals, 416
specific heat, 130
Van. der Waals constants, 408
Newton's second law of motion, 3
NH, characteristic temperature, for

rotation, 136
for vibration, 142

heat of dissociation, interatomic dis-
tance, Morse constant, 132
Nickel, alloys with copper, 458
crystal structure, 447
data regarding melting point, 259
equation of state and energy, 451, 454
phase equilibrium in alloys, 274, 279-

282
Nitrate ion, 357

valence structure, 406
Nitric oxide, characteristic temperature

for rotation, 136
for vibration, 142
data regarding melting point, 259
heat of dissociation, interatomic dis-
tance, Morse constant, 132
specific heat, 137
valence structure, 403
Van der Waals constants, 408
Nitrogen, characteristic temperature, for

rotation, 136
for vibration, 142
crystal structure, 417
data regarding melting point, 259
dissociation, 133

heat of dissociation, interatomic dis-
tance, Morse constant, 132
heat of vaporization, 414
and homopolar bonds, 400-408
Van der Waals constants, 408
Nitrous oxide, Van der Waals constants,

408

Nonconservative force, 3
Normal modes of vibration, 215, 222-255
Nucleus, atomic, 336

Octane, Van der Waals constants, 408
OH, characteristic temperature, for rota-
tion, 136

for vibration, 142

heat of dissociation, interatomic dis-
tance, Morse constant, 132
Ohm's law, 484-489
Opalescense, 111

Order-disorder transition, 293-304
Organic compounds, 376, 420-434
Orthosilicate radical, structure, 435
Osmium, crystal structure, 447
Overtones, 215, 222-255
Oxygon, characteristic temperature, for

rotation, 136
for vibration, 142
combination with hydrogen to form

water, 151-164

data regarding melting point, 259
heat of dissociation, interatomic dis-
tance, Morse constant, 132
heat of vaporization, 414
and homopolar bonds, 400-408
Van der Waals constants, 408

Palladium, crystal structure, 447

equation of state, 451
Partial derivatives, 16, 18-20, 23-30
Partial pressure, 120-121

and chemical equilibrium, 151-158
Partition function, and fluctuations, 106

and melting, 265-269

for perfect gas, 125-128

for rotation, diatomic molecule, 138-
140

and second virial coefficient, 191-196

for solids, 213-218

and statistical mechanics, 50-51

for vibration, diatomic molecule, 143
Pauli exclusion principle, 342

and exchange effect, 369

and metals, 475-476
Perfect gas, 17, 30

chemical equilibrium in, 150-165

and Maxwell-Boltzmann distribution,
53-64

polyatomic, 130-149

INDEX

517

Perfect gas, thermodynamic and statis-
tical treatment, 115-129
translational energy levels in quantum

theory, 54-55

Periodic potential in metal, 473, 489-501
Periodic table, 344-351
Permutations, 67

Phase changes of second order, 291-304
Phase diagram, 166-168

binary systems, 281-287
Phase equilibrium, 166-170

binary system, 270-290
Phase space, 36-43
and Maxwell-Boltzmann distribution,

52-64

Phosphine, valence structure of mole-
cule, 405

Van dcr Waals constants, 408
Phosphonium ion, 378
Photoelectric effect, 316-320
Photon, 316-320, 323-326
Planck, black-body radiation, 307-320,

325-326

probability and entropy, 34
Planck's constant, 39
Platinum, crystal structure, 447
data regarding melting point, 259
Debyc temperature, 237
equation of state and energy, 451, 454
Poisson's ratio and velocity of elastic

waves, 238, 240
Polarization, and interatomic forces,

363-367, 398-399, 410-414
of light, 308

Polyatomic gases, 130-149
internal coordinates, 124
Polymorphic phases, 167-170, 180-181,

22O-221, 448

Potassium, compressibility, 202
crystal structure, 447
data regarding melting point, 259, 261
Debye temperature, 237
equation of state and energy, 451, 454-

455

molecular volume, 261
molecule, characteristic temperature,

for rotation, 136
for vibration, 142
dissociation, 133

heat of dissociation, interatomic dis-
tance, Morse constant, 132
thermal expansion, 261

Potassium bromide, data regarding crys-
tals, 381, 393, 395

data regarding melting point, 259, 261
molecular volume, 261
thermal expansion, 261
Potassium chloride, data regarding crys-
tals, 381, 393, 395

data regarding melting point, 259, 261
Debye temperature, 391
molecular volume, 261
thermal expansion, 261
Potassium dichromate, data regarding

melting point, 259

Potassium fluoride, data regarding crys-
tals, 381, 393, 395
data regarding melting point, 259
Potassium hydroxide, data regarding

melting point, 259

Potassium iodide, data regarding crys-
tals, 381, 393, 395

Potassium nitrate, data regarding melt-
ing point, 259

Potential, electrostatic, 353-367
Potential energy, 3
of interatomic forces, 131-133, 191-

196, 352-376
and Maxwell-Boltzmann distribution,

54, 62-64
Pressure, 17

and chemical equilibrium, 151-158
and equation of state of ionic crystals,

392-396

and equation of state of metals,450-456
and equation of state of solids, 199-221
and equilibrium of phases, 166-181
of imperfect gases, and Van der Waals'

equation, 182-198

of mixtures of gases, and partial pres-
sure, 120-121, 128
of perfect gases, 58-60

Fermi- Dirac statistics, 79
of radiation, 311
Probability, a priori, 36, 38, 127
and Maxwell-Boltzmann distribution,

thermodynamic, 34
of transition, 324-333
Propane, heat of vaporization, 434
structure of molecule, 421
Van der Waals constants, 408
Pyrometer, 312-313

518

INTRODUCTION TO CHEMICAL PHYSICS

Quadrupole moment, 356-357
Quantum defect, 340-341
Quantum theory, 36, 38-43, 45-46

and atomic structure, 339-344

and equation of state and specific heat
of solids, 215-221, 234-240

and identical particles, 129

and kinetic method, 96-100

and liquids, 265

and radiation, 314-320

and specific heat of polyatomic gases,
138-149

and structure of metals, 489-501

and translational energy levels, 54H>5
Quasi-ergodic motion, 38
Quenching, 181

Radiation, 307-326
Radii, of atoms, 342
table of, 349

of ions, 382-385
Ramsauer effect, 330
Randomness, 9-12, 32-35, 43-46

and melting, 262-264
Rate of reaction, 150-154, 158-165
Ray leigh- Jeans law, 314
Reciprocal space, 230, 245
Recombination, 165
Reduced mass, 135, 141
Reflection of electrons by metals, 462, 465
Relaxation time, in electrical conduc-
tivity, 485-486

Representative point, 36-38, 65-66
Repulsive forces, between atoms, and
exclusion principle, 369-372

between ions, ionic crystal, 388-390

between molecules, 130-133

and Van der Waals' equation, 182-184,

194-196

Residual rays, 254r-255
Resistance, electrical, 484-489, 498-501
Resonance potential, 322
Reststrahlen, 254-255
Reversible processes, 10-13, 16

and statistical mechanics, 44-51
Rheology, 256-258
Rhodium, crystal structure, 447
Richardson equation, thermionic emis-
sion, 466

Rigid sphere atomic model, 131
Rotator, diatomic molecule, 134-140

and equipartition of energy, 58

in quantum theory, 40, 42
Rubidium, compressibility, 202

crystal structure, 447

data regarding melting point, 259

equation of state and energy, 451, 454
Rubidium fluoride, chloride, bromide,
iodide, data regarding crystals, 381,
393, 395
Rumford, 5

Ruthenium, crystal structure, 447
Rydberg number, 340

Saha, 335

Saturation of valence, 374-376

Second law of thermodynamics, 12-14, 16

and statistics, 19, 49-51
Second order, phase change, 291-304
Second virial coefficient, 190-196
Secondary emission of electrons, 461-462
Selenium, crystal structure, 444, 447, 450

data regarding melting point, 259
Shielding constant, 340-342
Shot effect, 108
Silica gel, 438
Silicates, 435-443
Silicon, crystal structure, 444, 447

and homopolar bonds, 400

melting point, 449

Silicon hydride, valence structure of
molecule, 405

Van der Waals constants, 408
Silver, crystal structure, 447

data regarding melting point, 259, 261

Debye temperature, 237

equation of state and energy, 451, 454

molecular volume, 261

thermal expansion, 261
Silver bromide, data regarding crystals,
381

data regarding melting point, 259-261

molecular volume, 261

thermal expansion, 261
Silver chloride, data regarding crystals,
381

data regarding melting point, 259, 261

molecular volume, 261

thermal expansion, 261

INDEX

519

Silver fluoride, data regarding crystals,

381
Silver nitrate, data regarding melting

point, 259
Simple cubic structure, and atomic

vibrations, 232
Sodium, crystal structure, 447

data regarding melting point, 259, 261
Debye temperature, 237
electronic energy bands, 494
equation of state and energy, 451, 454-

455
and thermodynamic functions, 200--

211

molecular volume, 261
molecule, characteristic temperature,

for rotation, 136
for vibration, 142
dissociation, 133

heat of dissociation, interatomic dis-
tance, Morse constant, 132
thermal expansion, 261
Sodium acetate, structure of molecule,

428
Sodium bromide, iodide, data regarding

crystals, 381, 393, 395
Sodium chloride, crystal structure, 378,

381

data regarding crystals, 381, 393, 395
data regarding melting point, 259, 261
Debye temperature, 391
molecular volume, 261
thermal expansion, 261
water solution, 285-287
Sodium fluoride, data regarding crystals,

381, 393, 395

data regarding melting point, 259
Sodium hydroxide, data regarding melt-
ing point, 259
Sodium nitrate, data regarding melting

point, 259

Sodium perchlorate, data retarding melt-
ing point, 259
Solids, binary systems, 270-290

equation of state and specific heat,

199-255

equilibrium with other phases, 166-181
ionic substances, 377-399
melting, 256-269
metals, 444-501
Solidus, 281
Solutions, 270-290

Space quantization, 139, 339
Specific heat, 17-20, 22-23
of compounds, 241-255
difference between Cp and CV, 90
electronic, 322
and fluctuations, 107-108
of free electrons, 471, 476-479, 49&-500
internal, of gas, 117
of ionic crystals, 390-393
of liquids, 262-265
of monatomic perfect gases, 61-62

Fermi-Dirac statistics, 78-79
and phase changes of second order,

291-304

of polyatomic gases, 136-149
of solids, 203-205, 213-214, 222-255
and statistical mechanics, 51
jind temperature variation of latent

heat, 177-178
variation with pressure and volume,

116

Spectrum, infrared, of crystals, 254-255
optical, and Kirchhoff's law, 300-310
of vibrational frequencies, solids, 225-

255
Spin, electronic, 339

and interatomic forces, 369-374
and metals, 476
Spontaneous emission, 324
Standing waves, 226-231, 242-252

of electrons in metals, 489-501
Stationary states, of electrons in atoms,

321-323, 338-344
of oscillator and rotator, 41-42
Statistical mechanics, 14, 32-85

applied to black body radiation, 307-

320
applied to chemical equilibrium, 154-

158
applied to equation of state of solids,

211-221
applied to equilibrium between phases

and vapor pressure, 178-180
applied to melting, 265-269
applied to perfect gas, 124-129
applied to polyatomic gases, 138-140,

142-145
applied to second virial coefficient,

190-196

applied to solubility and phase equilib-
rium, 270-304
Stefan-Boltzmann law, 307-313

520

INTRODUCTION TO CHEMICAL PHYSICS

Stirling's theorem, 70-72

Streamline flow of representative points,

37,94

Stresses and strains, 17, 199
Strontium, crystal structure, 447

equation of state, 451
Strontium oxide, sulphide, selenide, tellu-

ride, data regarding crystals, 381
Sugar, phase equilibrium in solution, 270
Sulphate ion, 357, 398
Sulphur dioxide, valence structure of

molecule, 406

Van der Waals constants, 408
Supercooling, 181, 256-258, 262-264

Tantalum, crystal structure, 447

equation of state, 451
Tellurium, crystal structure, 444, 447,

450
Temperature, 9, 12-14, 17

bath, and canonical assembly, 46-47

and fluctuations, 101
of inversion, Joule-Thomson effect, 1 98
and kinetic method, 96
Temperature-entropy diagram, 172
Terms, spectroscopic, 323
Thallium, crystal structure, 447

data regarding melting point, 259
Thallium bromide, data regarding melt-
ing point, 259

Thallium chloride, data regarding crys-
tals, 382

data regarding melting point, 259
Thallium iodide, data regarding crystals,

QQO
OoZ

Thermal equilibrium, 37-38, 46-51

and kinetic method, 96, 98
Thermal expansion, 19, 200-220, 238-
240, 261

of ionic crystals, 392-394

of metals, 450-456
Thermal pressure, 217-218
Thermionic emission, 460-471, 480-484
Thermodynamic formulas, 16, 23-30

table of, 27-29
Thermodynamic probability, 34

and Fermi-Dirac and Einstein-Bose

statistics, 69-72

Thermodynamic scale of temperature,
30-31

Thermodynamics, 14, 16-31
applied to black-body radiation, 307-

320
applied to chemical equilibrium, 154-

158
applied to equation of state of solids,

199-211
applied to equilibrium, between atoms

and electrons, 333-335
between metal and gas, 463-464
between phases, 174-178
applied to perfect gas, 115-124
applied to solubility and phase equilib-
rium, 270-304
applied to Van der Waals' equation,

18^-189

Threshold, photoelectric, 318
Tin, crystal structure, 444, 447-449
data regarding melting point, 259
Debye temperature, 237
Titanium, crystal structure, 447
Transition probability, 42, 88

of atoms, 322-333
Transverse waves, in electromagnetic

radiation, 313-314
in solids, 222-240
Traveling waves, 226-228
Triethylamine, Van der Waals constants,

408
Trimethylamine, structure of molecule,

427

Van der Waals constants, 408
Triple point, 166-167, 171-172, 181

and eutectic, 285
Tungsten, crystal structure, 447

equation of state and energy, 451, 454

Uncertainty principle, 40-41
Undetermined multipliers, 48

Valence forces, 130

explanations from atomic theory, 371-

376
Vanadium, crystal structure, 447

equation of state, 451
Van der Waals constants, 182-198

for molecular substances, 407-414
Van der Waals' equation, 182-198, 210

and molecular substances, 407-414

INDEX

521

Van der Waals forces, 182-198

and atomic structure, 356-374

and inert gases, 385

and molecular substances, 407-414

and solutions, 273
Van't Hoff's equation, 154-158
Vapor pressure, 166-169, 174-180

and Van der Waals' equation, 188-189
Vaporization, 23

entropy of, 171-180

heat of, 171-180

of metals, 452-^54

of organic compounds, table, 434

table of, 414

and Van der Waals7 equation, 189,

258-260
Velocity, of elastic waves, 227-240

of light, 308
Vibration, of diatomic molecules, 140-149

and light waves, 313-314

of molecules and atoms in crystals,

211-255

Vibrational degrees of freedom, 146
Virial, 190-396
Viscosity, 12, 13, 257
Volt, electron, 132-133, 318
Volta effect, 467-471
Volume, 17

and chemical equilbrium, 151-158

of imperfect gases, and Van der Waals'
equation, 182-198

molecular, table of, 261
of liquids, table, 408

of perfect gas, 58-61

and phase equilibrium, 168-178

of solids, 199-221

Water, crystal structure, 260, 418-419
data regarding melting point, 259
dissociation into hydrogen and oxygen,

151-164

entropy and free energy, 172-173
equilibrium between phases, 166-169
heat of vaporization, 414

Water, and solubility, 270-275, 285-290
valence structure of molecule, 401
Van der Waals constants, 408, 412-414
vapor pressure and latent heat of

vaporization, 188-189
vibrational specific heat, 146-147
Water glass, 438
Wave mechanics, 41, 307
and electrons in periodic potential,

489-501

and radiation, 323
Waves, elastic, in continuous media,

227-234
electromagnetic, and light, 308, 313-

320

in molecular media, 241-252
Work, 3, 7-9, 17, 21-22

and statistical mechanics, 49
Work function, thermionic and photo-
electric, 317-318, 464, 469-470,
480-484
Wurtzite, crystal structure, 379-380, 382

Xenon, atomic volume, 384
data regarding crystals, 416
specific heat, 130
Van dor Waals constants, 408
X-ray diffraction, liquids, 256
X-ray levels in atoms, 344

Zinc, crystal structure, 447
data regarding melting point, 259
Debye temperature, 237
energy, 454

order-disorder in alloys, 293-304
phase equilibrium in alloys, 270,

287-288
Zinc oxide, sulphide, selenide, telluride,

data regarding crystals, 382
Zincblende, crystal structure, 379-381,

382

Zirconium, crystal structure, 447
equation of state, 451

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