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of Matter 

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Wiley 






This book provides a basis for the study of matter by 
describing its atomic and molecular nature and by 
showing how the macroscopic behaviour of matter 
can be related to its fundamental structure. These 
relationships are examined without recourse to 
quantum mechanics or thermodynamics, although the 
results of these theories are described where neces- 
sary. 

Many interesting problems, together with their solu- 
tions, are included to test a student's understanding 
and extend his interest. 



Contents 



1 The Study of The Properties of Matter 

2 Atoms, Molecules and the States of Matter 

3 Interatomic Potential Energies 



Energy, Temperature and The Boltzmann 
Distribution 



5 The Maxwell Speed Distribution and the 
Equipartition of Energy 

6 Transport Properties of Gases 

7 Liquids and Imperfect Gases 

8 Thermal Properties of Solids 

9 Defects in Solids; Liquids as Disordered Solids 
Solutions to Problems 

Reading List 
Index 



Properties of Matter 



The Manchester Physics Series 

General Editors 
F. MANDL : R. J..EJJ1IS0N : D. J. SANDIFORD 

Physics Department, Faculty of Science, 
University of Manchester 



This series is planned to include the following volumes: 

Properties of Matter: B. H. Flowers and E. Mendoza 



Electromagnetism: I. S. Grant and W. R. Phillips 



Atomic Physics: J. C. Willmott 



Optics: F. G. Smith and. J. H. Thomson 



Statistical Physics: F. Mandl 



Solid State Physics: H E Hall 



PROPERTIES 

OF 

MATTER 



B. H. Flowers, F.R.S. 

Langworthy Professor of Physics, 
University of Manchester 

E. Mendoza 

Professor of Physics, 

University College of North Wales, Bangor 




John Wiley & Sons Ltd. 

LONDON NEW YORK SYDNEY TORONTO 



Copyright © 1970 John Wiley & Sons Ltd. All 
Rights Reserved. No part of the publication may 
be reproduced, stored in a retrieval system, or 
transmitted, in any form or by any means, elec- 
tronic, mechanical, photocopying, recording or 
otherwise, without the prior written permission 
of the Copyright owner. 

Library of Congress Catalog Card No.70-118151 

ISBN 471 26497 Cloth bound 
ISBN 471 26498 9 Paper bound 



Set on Monophoto Filmsetter and printed by 
J. W. Arrowsmith Ltd., Bristol, England 



Editors' Preface to the 
Manchester Physics Series 



In devising physics syllabuses for undergraduate courses, the staff of 
Manchester University Physics Department have experienced great diffi- 
culty in finding suitable textbooks to recommend to students; many 
teachers at other universities apparently share this experience. Most books 
contain much more material than a student has time to assimilate and are 
so arranged that it is only rarely possible to select sections or chapters to 
define a self-contained, balanced syllabus. From this situation grew the 
idea of the Manchester Physics Series. 

The books of the Manchester Physics Series correspond to our lecture 
courses with about fifty per cent additional material. To achieve this we 
have been very selective in the choice of topics to be included. The emphasis 
is on the basic physics together with some instructive, stimulating and 
useful applications. Since the treatment of particular topics varies greatly 
between different universities, we have tried to organize the material so 
that it is possible to select courses of different length and difficulty and to 
emphasize different applications. For this purpose we have encouraged 
authors to use flow diagrams showing the logical connection of different 
chapters and to put some topics into starred sections or subsections. 
These cover more advanced and alternative material, and are not required 
for the understanding of later parts of each volume. 



vi Editors' preface to the Manchester Physics Series 

Since the books of the Manchester Physics Series were planned as an 
integrated course, the series gives a balanced account of those parts of 
physics which it treats. The level of sophistication varies : 'Properties of 
Matter' is for the first year, 'Solid State Physics' for the third. The other 
volumes are intermediate, allowing considerable flexibility in use. 'Elec- 
tricity and Magnetism' , 'Optics' and 'Atomic Physics' start from first year 
level and progress to material suitable for second or even third year 
courses. 'Statistical Physics' is suitable for second or third year. The books 
have been written in such a way that each volume is self-contained and 
can be used independently of the others. 

Although the series has been written for undergraduates at an English 
university, it is equally suitable for American university courses beyond 
the Freshman year. Each author's preface gives detailed information 
about the prerequisite material for his volume. 

In producing a series such as this, a policy decision must be made about 
units. After the widest possible consultations we decided, jointly with the 
authors and the publishers, to adopt SI units interpreted liberally, largely 
following the recommendation of the International Union of Pure and 
Applied Physics. Electric and magnetic qualities are expressed in SI units. 
(Other systems are explained in the volume on electricity and magnetism.) 
We did not outlaw physical units such as the electron-volt. Nor were we 
pedantic about factors of 10 (is 0.012 kg preferable to 12 g?), about 
abbreviations (while s or sec may not be equally acceptable to a computer, 
they should be to a scientist), and about similarly trivial matters. 

Preliminary editions of these books have been tried out at Manchester 
University (and in the case of 'Properties of Matter' also at Bangor 
University) and circulated widely to teachers at other universities, so that 
much feedback has been provided. We are extremely grateful to the many 
students and colleagues, at Manchester and elsewhere, who through 
criticisms, suggestions and stimulating discussions helped to improve the 
presentation and approach of the final version of these books. Our partic- 
ular thanks go to the authors, for all the work they have done, for the 
many new ideas they have contributed, and for discussing patiently, and 
frequently accepting, our many suggestions and requests. We would also 
like to thank the publishers, John Wiley and Sons, who have been most 
helpful in every way, including the financing of the preliminary editions. 

Physics Department F. Mandl 

Faculty of Science R. J. Ellison 

Manchester University D. J. Sandiford 



Preface 



A radical revision of the undergraduate physics syllabus of the Uni- 
versity of Manchester was undertaken in the year 1959. This exercise 
involved the participation of many members of the academic staff. It was 
eventually decided to base the whole of the syllabus upon two introductory 
first-year courses, one concentrating on the general properties of wave 
motions, the other based upon the statistical properties of matter con- 
sidered as a collection of interacting atoms and molecules. The latter 
course, consisting of about 34 50-minute lectures, was first given in the 
1959/60 academic session under the title 'Properties of Matter'; over the 
years it has developed into the present book. 

Our aim has been to show how the macroscopic quantities describing 
matter in bulk can be related to each other in terms of the microscopic 
properties of molecules and their interactions. This of course is the subject 
matter of statistical thermodynamics. To the purist this subject can only 
be tackled after a thorough grounding in advanced mechanics, thermo- 
dynamics and the quantum theory. But the spirit of inquiry amongst 
undergraduates, and the incentive to devote their time and energies to 
these rigorous pursuits can more readily be generated, it seemed to us, if 
they can first be made aware of what much of physics is about in a more 
rough and ready fashion. It is perhaps contrary to the present fashion, 
but we have omitted all quantum considerations from the foundations of 
this work, confining ourselves to a few passages here and there which are 
in the nature of 'see the next exciting instalment' when, indeed, quantum 



viii Preface 

theory is necessary rather than merely desirable in order to understand 
some macroscopic phenomenon. Similarly, we have excluded any discus- 
sion of the second law of thermodynamics and its consequences — at the 
risk, here and there, of doing violence to the distinction between internal 
energy (which we calculate) and free energy (which we do not). We hope 
that we have at least identified the points at which the distinction matters. 
However, we consequently have not always been able to avoid the phrase 
'It can be shown that . . .', although we have tried to avoid any implication 
of it except in peripheral matters. We comfort ourselves by suggesting that 
physics would be very dull unless there were always some things left 
outstanding in this way. 

More importantly, however, we have been forced to restrict severely 
the number of kinds of matter we were prepared to discuss. We have 
excluded all discussion of ionized plasmas, of polymers and of biological 
materials. Each of these, it seemed to us, requires a book to itself. We 
have touched, although briefly, on the engineering properties of materials 
limiting ourselves to a discussion of the strength of real solids — for our 
concern has been rather to show that these properties can in principle be 
related to the microscopic properties. Argon, gaseous, liquid and solid, 
figures ubiquitously. It is perhaps the simplest element from our point of 
view, the ideal element, about which much experimental information is 
available to us for our simple-minded analysis. Apart from that, we have 
mostly confined ourselves to gases, liquids and solids consisting of small 
molecules, and to simple ionic substances and metals. 

This is the way in which much of the study of the properties of matter 
developed historically. We hope that we have succeeded in bringing back 
some of the excitement of the original discoveries ; certainly we found it 
exciting to rediscover some of these ourselves. 

The course has been given in modified form in Manchester since 1959 
and in Bangor since 1965, by others as well as by ourselves. We are 
indebted to several of our colleagues who, as lecturers or tutors, have 
contributed much to its gradual development. We are particularly in- 
debted to Dr David Caroline, as well as to the editors of the Manchester 
Physics Series for their friendly but penetrating criticisms and suggestions. 
Most of all we are indebted to more than a thousand of our students whose 
own efforts to understand what we were trying to do has been our main 
encouragement and incentive. They are not, of course, responsible for the 
remaining imperfections in our book. 

B. H. Flowers 
Eric Mendoza 



List of Symbols 



A,B 


constants 


A 


area 


A 


activation energy 


CI) CIq 


atomic or molecular diameter 


a,b 


constants in van der Waals' equation 


a 


linear expansion coefficient ; Madelung constant 


P 


volume expansion coefficient 


y 


ratio of specific heats ; surface tension 


y G 


Griineisen constant 


c 


specific heat, usually with suffix: C p , C v 


c 


speed of molecule ; speed of light 


D 


diffusion coefficient 


d 


distance 


E 


energy 


e 


charge on electron 


s 


depth of interatomic potential well 


F 


force 


/() 


function 


& 


faraday 


r\ 


viscosity coefficient 


J 


flux of particles 


K 


bulk modulus 


°K 


degrees absolute 


k 


Boltzmann's constant 


K 


velocity coefficient of chemical reaction 


K 


thermal conductivity 


JT 


kinetic energy 


u 


latent heat at low temperatures 


$£ 


Lorentz number 


£ 


length 


A 


wavelength 


X 


mean free path 


M 


molecular weight 


m 


mass of atom 



x List of Symbols 

N Avogadro's number 

n number, number density 

rt coordination number 

Jf number per unit area 

v frequency 

v E Einstein frequency 

P pressure 

P[ ] probability function 

Px'PyPz momentum components 

p, q indices of interatomic potential energy 

r radial distance 

p density 

s strain 

a collision cross-section ; conductivity 

T temperature 

t time 

t characteristic time 

U x drift velocity 

V large volume 
V molar volume 
v small volume 
v x ,v y , v z velocity components 

V potential energy 
(D angular velocity 

tAt sections or subsections marked with a star may be omitted, 
if the reader so wishes, as they are not required later in the book 



Contents 



1 THE STUDY OF THE PROPERTIES OF MATTER 

1 . 1 The Study of the Properties of Matter .... 1 

1.2 Orders of Magnitude ....... 2 

1.3 Units and Systems of Units ...... 5 



2 ATOMS, MOLECULES AND THE STATES OF MATTER 

2.1 Atoms, Ions and Molecules ...... 8 

2.2 Gases, Liquids and Solids . . . .15 



3 INTERATOMIC POTENTIAL ENERGIES 

3.1 Molecular Dimensions . ...... 22 

3.2 Interactions between Electrically Neutral Atoms and 
Molecules ........ 23 

3.3 Binding Energy and Latent Heat . . . .31 

3.4 Surface Energy ........ 33 

3.5 Elastic Moduli 40 

3.6 Vibrations in Crystals : Simple Harmonic Motion . 44 

3.7 Metals 47 

3.8 Ionic Crystals ........ 53 

Problems ......... 57 



Contents 



ENERGY, TEMPERATURE AND THE BOLTZMANN 
DISTRIBUTION 



-I. Probability Functions 



4. 1 Heat and Energy 

4.2 Concepts of Probability Theory- 

4.3 Thermal Equilibrium ...... 

4.4 Boltzmann Distributions — I. A Gas of Independent 
Particles under Gravity ..... 

Appendix A 

A. 1 Dependence of the Probability Function on Energy and 

Temperature 
A. 2 Form of Probability Function .... 
A. 3 Extension to Macroscopic Systems 

Problems ........ 



62 

67 
74 

82 



90 
92 
94 
96 



THE MAXWELL SPEED DISTRIBUTION AND THE 
EQUIPARTITION OF ENERGY 

5 . 1 Velocity-Component Distribution -P[u J 

5.2 Speed Distribution P[c] 

5.3 The Equipartition of Energy 

5.4 Specific Heats C p and C v 

5.5 Activation Energies 
Problems . 



97 
106 
114 
117 
127 
131 



6 TRANSPORT PROPERTIES OF GASES 

6.1 Transport Processes ..... 

6.2 Solutions of the Diffusion Equation : the yjt Law 

6.3 Diffusion and the Random Walk Problem 

6.4 Distribution of Free Paths .... 

6.5 Calculation of Transport Coefficients 

6.6 Knudsen Gases ...... 

Appendix B 

B.l Diffusion Coefficient in Gases 

Problems ....... 



136 
145 
150 
153 
159 
169 

171 
174 



7 LIQUIDS AND IMPERFECT GASES 

7.1 Relations between Solid, Liquid and Gas 

7.2 The Approach to the Liquid State . 

7.3 Van der Waals' Equation 

7.4 Application to Gases .... 
"Ar7.5 Refinements to Van der Waals' Equation 

"Ar Starred sections or subsections may be omitted, if the reader so wishes, as they are not 
required later in the book. 



176 
185 
188 
193 
203 



Contents 



xin 



7.6 Critical Constants 208 

7.7 Fluctuation Phenomena . • .211 

7.8 Properties of Liquids on Estimated Van der Waals' Equa- 
tion 217 

Problems 222 



8 THERMAL PROPERTIES OF SOLIDS 

8.1 The External Forms of Crystals 

8.2 X-Ray Structure Analysis 

8.3 Amplitude of Atomic Vibrations in Solids 

8.4 Thermal Expansion and Anharmonicity . 

8.5 Thermal Conduction in Solids 

8.6 Electrons in Metals . 

Problems ..... • 



225 
233 
236 
238 
244 
254 
261 



9 DEFECTS IN SOLIDS: LIQUIDS AS DISORDERED SOLIDS 



9.1 Deformation of Solids . 

9.2 Brittle Materials . 

9.3 Deformation of Ductile Metals 
*9.4 Growth of Crystals 
*9.5 Point Defects 
^r 9.6 Diffusion in Solids 
*9.7 Diffusion in Liquids 

Problems .... 

Solutions to Problems 

Reading List 

Index .... 

Physical Constants and Conversion Factors 



263 
269 

274 
281 
286 
290 
297 
304 

306 
311 
313 
inside back cover 



CHAPTER 




The study of the properties of 
matter 



1.1 THE STUDY OF THE PROPERTIES OF MATTER 

Throughout the whole of the nineteenth century, one of the open ques- 
tions of science was whether matter was composed of atoms or not. 
Nobody had yet been able to perform experiments with single atoms, 
certainly no-One had ever seen one, and for a long time no-one knew even 
the order of magnitude of the sizes of atoms — whether their diameters 
were typically of order 10" 5 cm or 10" 50 cm. One method of attack was to 
try and correlate as many different properties of solids, liquids and gases 
as possible on the basis of simple postulates about the forces which atoms 
exerted on one another. The earliest attempt at describing these forces was 
made by Boscovitch in 1745. Sixty years later, a triumph was scored when 
Laplace, arguing from the fact that the rise of a liquid in a capillary tube 
was observed to be independent of the thickness of the wall of the tube, 
deduced that atomic forces must act only over short distances. He was 
able to deduce theoretically the form of the surface-tension law for 
liquids, that the force exerted by surface tension should be proportional 
to the length of a cut in the surface — and this was verified experimentally. 
Much later, in the 1860's and 70's, the transformation of gas into liquid 
was demonstrated for many substances when it became technically 
possible to produce high pressures and low temperatures. The similarities 



2 The study of the properties of matter Chap. 1 

and regularities in behaviour of several substances, predicted on the basis 
of crude atomic models, added plausibility to those models. Above all, 
the rough agreement between estimates of the sizes of atoms based on 
widely differing kinds of experiments (about eight completely different 
methods all gave atomic diameters of the order of 10~ 7 -10~ 8 cm) made 
the atomic hypothesis fairly secure by 1900. 

Thus the subject called -Properties of Matter' or 'Heat' was at one time 
an exciting one. Physicists measured surface tensions and latent heats and 
elasticities and tried to correlate them under an all-embracing atomic 
theory. But with the discovery of sub-atomic particles and the invention of 
counting devices which could detect single atoms or ions, the subject lost 
its urgency. By the early years of the twentieth century, no-one anywhere 
doubted that matter was atomic in structure and that atoms were of the 
order of 10 8 cm in diameter. Experimenters still measured surface ten- 
sions, latent heats and elasticities, but these had now become respectable, if 
routine, activities in their own right. Books came to be written entitled 
'Properties of Matter' which described highly sophisticated apparatus for 
measuring quantities of this kind, and gave elaborate calculations on the 
twisting of laminas and the bending of beams, but never mentioned the 
word 'atom'. 

It is the purpose of this book to try and recapture some of the spirit of 
the old approach. We will in fact start from statements about the shapes 
and sizes of atoms and the forces holding them together, and then show 
how the properties of solids, liquids and gases can be deduced. It is our 
purpose to show that, given the potential energy between two atoms of 
known atomic weight, it is possible to estimate the density of the solid 
and its specific heat, its thermal expansion and elasticity, the surface 
tension and latent heat and viscosity of the liquid, the diffusion constant 
and thermal conductivity and specific heat of the gas and the velocity of 
sound through it : they are all related properties of matter. 

1.2 ORDERS OF MAGNITUDE 

The estimates we shall make will rarely be exact ones. Since the object 
of our discussions will be mainly to show that we can identify the forces 
or mechanisms underlying certain phenomena, it will serve our purpose 
if we can show that using approximate methods we can get roughly the 
right answer. To improve on rough estimates usually demands a great 
increase in mathematical complexity, and it would achieve little if we 
risked obscuring the line of the argument by getting involved in com- 
plicated manipulations merely to add a few percent to the accuracy of the 
result. In many operations, it is by contrast extremely important to know 



1 .2 Orders of magnitude 



Hydrogen gas 




Fig. 1.1. A diagram drawn by John 
Dalton, in his epoch-making book A 
New System of Chemical Philosophy 
published in 1810. The gas is shown as 
a regular arrangement of atoms, which 
is quite wrong. Not till this idea was 
supplanted could any real progress be 
made. 




Fig. 1.2. An early idea of the effect of one atom on 
another — from a book published by Boscovitch in 1 745. 
Compare this diagram with Fig. 3.4. The oscillations 
in this graph were postulated to account for the 
structure of a gas as pictured above, but this remains 
an astonishingly penetrating attempt to explain the 
properties of matter in fundamental terms. 





4 The study of the properties of matter Chap. 1 

the exact values of certain parameters. Chemical engineers, for example, 
need to know thermal constants to five-figure accuracy in order to predict 
whether they can manufacture a certain product economically or not. 
Similarly, a bridge might collapse if a designer made a mistake by a factor 
of two in some of his data. But for our present purposes, we will be content 
if we can estimate that a thermal or chemical change may take place at 
some temperature of the order of a. few hundred degrees absolute; and we 
will regard it as satisfactory if, given the atomic constitution of both, we 
can predict from first principles that steel is a good deal stronger than 
butter. 

oo° 
<oooo 

Qto OO 
oo 

Fig. 1.3. Possibly the very first 'modern' drawing of the molecules in the 

solid, liquid and gas phases. A 'doodle' from one of Joule's notebooks, done 

while he was working out the implications of the conservation of energy and 

realizing that the old static picture of gases was wrong (1847). 

This emphasis on the importance of orders of magnitude must not be 
taken to disparage the crucial role of accuracy in experimental measure- 
ments nor to suggest that exact theories need not be pursued. Indeed the 
existence of new and unexpected phenomena is sometimes shown up 
when the discrepancy between observation and theory is quite small. 
Measurements of the specific heats of gases, for example, give results 
which can be predicted in order of magnitude by simple theories based on 
the laws of classical physics, but the persistent disagreement between 
precise measurements and exact classical theories was the first evidence 
that the laws of classical mechanics themselves were not applicable in all 
circumstances. Again, unexpected discrepancies between existing theories 
and measurements of the specific heats of solids at lower temperatures 
could similarly only be explained by using a quantum approach ; measure- 
ments on metals showed that electrons deviated sharply from the classical 
behaviour that had been expected. These important phenomena (which 
will all be discussed later in this book) would not have been discovered had 
physicists been content merely to make rough estimates. Nevertheless, if 
a rough estimate does give a result which agrees in order of magnitude 



1 .3 Units and systems of units 5 

with observation, this can usually be taken to mean that the correct 
mechanisms have at least been identified. This will be the main theme of 
this book. 




Fig. 1.4. A very early picture, one of the first to be drawn to 

scale, of a gas — the molecules of air in a volume 10~ 4 cm 

square by 10~ 8 cm thick— not long after Avogadro's 

number had been reliably estimated. Kelvin 1883. 



1.3 UNITS AND SYSTEMS OF UNITS 

When we make a statement about a physical quantity, like 'the mass of a 
proton is 1.66 x 10 ~ 24 g', the datum consists of three parts : the number of 
order unity, the power of 10 and the unit (1.66, 10~ 24 and the gram, 
respectively). Many students have a habit of remembering the first figures 
to high accuracy, but of forgetting the power of 10 or ignoring the units. 
In our collective memory we can recall students who have insisted that the 
sun is 3,000 miles from the Earth, that the diameter of an atom is 
109,737 cm, that Planck's constant is 6.6 x 10 23 unknown units, that 
gravitation is the cohesive force that holds solids together. Wildly wrong 
statements like these are sillier than saying that butter is a good material 
for building bridges. In quoting physical quantities, it is usually more 



6 The study of the properties of matter Chap. 1 

important to get the power of 10 and the units correct — to be sure of the 
order of magnitude — than it is to quote the digits at the beginning. But 
having got the figures correct, the units must be quoted; data without 
units are devoid of meaning. 

To be of any use, a system of units should be self-consistent and prefer- 
ably it should deal with numbers which are comprehensible to the human 
imagination. All well-known systems satisfy the first condition (S.I., c.g.s., 
even British units) but it is the second that it is difficult to fulfil. In almost 
any physical problem, one encounters numbers which are extraordinarily 
large or small by everyday standards; numbers like 10 23 or 10" 16 which 
occur often in physics, cannot easily be visualized. Further, any unit 
quantity which is suitable for one problem is often quite unsuited for 
another. A coulomb (C), for example, is a tiny thing compared with the 
charge on a gram ion (10 5 C) but enormous compared with the charge 
on a single ion or electron (10" 19 C). To avoid very large or small factors, 
it is natural to choose units which are large when we measure large things 
and small when we measure small ones. It is natural to measure atomic 
weights in grams, and the mass of single atoms in atomic mass units ; 
to insist on measuring all masses in kilograms, say, is merely perverse. 
Similarly the centimetre or the metre are suitable units of length for many 
common objects but single atoms are best measured in Angstrom units : 
lA = 10" 8 cm. The reader must, therefore, be prepared to change his 
units with the problem. 

It is a sad fact that different authors of research articles use different 
systems of units, where not only do the symbols stand for different mag- 
nitudes and dimensions, but different numerical constants also appear. 
In electromagnetic equations, statements of the same equation may or 
may not contain factors like c, 4n, e /i which arise from the units, and make 
them difficult to understand. In fact, few difficulties of this kind appear in 
this book : but the competent physicist must be prepared to be able to 
read papers written by authors who may be working in any system, and 
the student must be facile in all of them. 

1.3.1 Energy units 

The joule (J) is the common unit of energy. For some problems it is a 
suitable unit since the gas constant R (defined in section 4.4.2) is 8.31 J 
per degree and this is not a large number. For measuring molar binding 
energies or latent heats or heats of reaction, the kilojoule (10 3 J) is more 
appropriate. For measuring the corresponding energies of a single atom 
or molecule, however, which are roughly 10 24 times smaller, the joule is 
not appropriate. Nor is the common 'small' unit of energy, the erg, 
because it is only a factor of 10 7 times smaller than the joule and one 



1.3 Units and systems of units 7 

finds oneself dealing with awkward-sounding amounts of energy such 
as 10" n or 10" 14 erg. There does in fact exist an energy unit which is 
suitable for these purposes. It is the electron volt. This is the amount of 
energy acquired by an electronic charge when it falls through a potential 
difference of 1 volt: 1 electron volt (1 eV) = 1.60 x 10~ 12 erg = 1.60 x 
10" 19 J. (This relation can be calculated from the knowledge (a) that 
when 1 coulomb falls through 1 volt, 1 joule of energy is released, and (b) 
that the charge on an electron (section 2.1.2.) is 1.60 x 10~ 19 C). In these 
units molecular binding energies lie between 10 and 0.01 eV, and these 
numbers are easy to visualize and handle. 



CHAPTER 




Atoms, molecules and the 
states of matter 



2.1 ATOMS, IONS AND MOLECULES 

It is convenient for us to start with the statement that matter is com- 
posed of atoms. Atoms are not the fundamental units of nature because 
they themselves can be broken up into a few smaller, and in a sense more 
fundamental, units or elementary particles (electrons, neutrons and 
protons). But the conditions needed to make atoms disintegrate are 
rather extreme and are not normally met with (at any rate if we except 
radioactive substances whose nuclei disintegrate but which do not concern 
us particularly), so that from the present point of view we need go no further 
than to say that matter is built up of atoms. 

Under ordinary conditions, matter seems to be continuous. Given a 
small piece of any solid, for example, it is possible to cut it up into smaller 
fragments and to go on repeating this process ; there seems to be no limit 
to the fineness of subdivision, other than that set by the instruments 
available. But in fact (if we carry out the process by any method, under 
conditions of temperature and pressure which are not too extreme) there 
is a limit when we reach atomic dimensions. The illusion that matter is 
continuous is due to the extreme smallness of even the largest atoms, and 
to the very large numbers of them which are present even in a microscopic 
speck. 



2.1 Atoms, ions and molecules 9 

We shall be concerned mostly with the forces between atoms, and it is 
therefore necessary to describe their structure and the formation of ions 
and molecules, so that the origin of the forces which they exert on one 
another can be understood. 

2.1.1 Atomic number 

If a sample of a substance can be shown to consist of atoms all of one 
kind, that substance is said to be a chemical element. Three examples of 
elements are hydrogen (H), under normal conditions a gas which easily 
takes part in a number of chemical reactions with other elements ; helium 
(He), a gas which is chemically inert and hardly reacts at all with other 
elements ; and lithium (Li), a highly reactive metal. 

It is possible to find similarities and regularities among the physical 
and chemical properties of elements. Of the many characteristics which 
it is possible to select and use to arrange the elements in some sort of 
order, one has been found to have a special significance. When samples 
of the elements are bombarded with energetic electrons, X-rays are emitted, 
any element giving a spectrum containing many characteristic wave- 
lengths. Each of them, however, includes a recognizable group of four 
lines, called the /C-lines, whose wavelengths vary from element to element. 
The lines from hydrogen have the longest wavelength, those from helium 
the next longest, then lithium, and in this way it is possible to arrange the 
elements in order. On this basis, hydrogen is said to have atomic number 
1, helium 2, and so on. 

2.1.2 Structure of atoms 

An atom consists of a nucleus, consisting of neutrons and protons, which 
is extremely small (about 10 5 times smaller in diameter than the atom 
as a whole) but which contains almost all the mass. Around this nucleus 
is a cloud of electrons. This cloud is easy to visualize when there are many 
electrons ; but it is a fact, made comprehensible by quantum mechanics, 
that a single electron in an atom also behaves somewhat as if it were 
spread tenuously throughout a certain volume. Even though the electron 
is a point charge, it appears to an outside observer as if it were continuously 
spread out. The term 'electron cloud' is therefore appropriate even to an 
atom containing only a single electron. Each electron carries a negative 
electric charge, whose magnitude is a fundamental constant : 

e= 1.602 x 10" 19 C. 

It is found that the number of electrons inside any atom of an element is 
equal to the atomic number of that element. To maintain the electrical 



10 Atoms, molecules and the states of matter Chap. 2 

neutrality of each atom, the nucleus is positively charged, with a mag- 
nitude equal to that of all the electrons in the cloud outside it. Thus a 
hydrogen atom has one negatively charged electron surrounding a 
nucleus which has one unit of positive charge. Each helium atom has two 
electrons outside a nucleus carrying two units of charge, and so on. 

Between each electron and the nucleus there exist forces of attraction 
(Coulomb forces) which bind electrons to the nucleus — that is, they make 
it difficult for the electrons to escape. However, some electrons may be 
more tightly bound than others — they require greater energy to separate 
them from the nucleus — and at the same time the clouds are usually 
smaller in size. In helium and the other 'inert' or 'rare' gases such as 
neon and argon which resemble it, all the electrons are tightly bound 
and no further electrons can be added to the system if it is to remain 
stable. In lithium and the other alkali metals like sodium and potassium 
and the 'noble' metals copper, silver and gold, most of the electrons are 
tightly bound but there is one which is rather loosely bound, so that it 
does not take much energy to detach it from its atom. In hydrogen also, 
the single electron is only loosely bound. These elements are said to be 
monovalent and to contain one valence electron. Other elements have 
more than one loosely bound electron in each atom ; the alkaline-earth 
metals such as beryllium and magnesium each have two. In another group 
of elements, the halogens, which include the gases fluorine and chlorine, 
all the electrons are tightly bound but it is possible for another single 
electron to enter the existing cloud and become tightly bound too. Other 
atoms can accept more than one extra electron in this way ; oxygen for 
example can accept two, nitrogen three. Only the loosely bound electrons 
can enter into combinations in this way. The tightly bound ones remain 
undisturbed by ordinary chemical changes. 

2.1.3 Ions 

These are atoms which have lost or gained one or more electrons (while 
keeping their nuclei unchanged) so that they are no longer electrically 
neutral. An atom of hydrogen can lose its electron (which is of course 
negatively charged) so that it has an excess of one unit of positive charge. 
Its mass is very little different from that of a hydrogen atom since most of 
the mass resides in the nucleus which is unchanged. A chlorine ion is 
formed by adding one electron to the atom, oxygen can form two kinds 
of ions according as one or two electrons are added to the atom. The 
properties of ions are quite different from those of atoms not only because 
ions are charged but also because in ions all the electrons are tightly 
bound. The relative sizes of atoms of mercury, of a free ion and of the ions 
in metallic mercury are shown in Fig. 2.1. 



2.1 Atoms, ions and molecules 1 1 




[a) {b) 



Fig. 2. 1 . Scale drawings of (a) a mercury atom in the vapour at room temperature 

(4.4 A in diameter), and (b) a mercury ion (Hg ++ ) formed from an atom which has 

lost two loosely bound electrons (2.24 A in diameter), (c) The ions in the solid or 

liquid metal are a little bigger than this. 

Free ions can be formed in two ways. They can be produced by chemical 
action, when the initially neutral atom is very close to other atoms which 
contribute or take away electrons. They can also be produced when neutral 
atoms are bombarded by beams of particles such as energetic electrons, 
when the electrons inside the atoms are knocked out by what can be 
regarded as direct collisions. 

Any mass of matter in equilibrium must be electrically neutral, or very 
nearly so ; if any ions are present it is exceedingly probable that oppositely 
charged ions are to be found not far away. 

2.1.4 Molecules 

It is possible for the loosely bound electrons to be shared in various 
ways between different atoms, so as to bind those atoms together to 
form molecules. Some substances do so in the gaseous state but the 
molecules do not preserve their identity in the solid state; other sub- 
stances form molecules which exist in all three states, solid, liquid and gas. 
Hydrogen atoms find it favourable at ordinary temperatures to share their 
electrons so as to form molecules each containing two atoms. They can 
be pictured as two nuclei embedded in a cloud of electrons enveloping 
them both. The molecule forms a separate, stable entity, and at ordinary 
temperatures hydrogen gas consists almost entirely of such molecules. 
When cooled to very low temperatures so that the gas liquefies or solidifies, 
the molecules become quite tightly packed together but they still more or 
less retain their identity, though a given hydrogen atom may occasionally 
wander from molecule to molecule. Halogen gases also form diatomic 
molecules. Helium and the other rare gases are composed of molecules 
each containing a single atom (so that it is immaterial whether one calls 
these atoms or molecules). 



12 Atoms, molecules and the states of matter Chap. 2 

Metallic elements such as lithium exist as molecules only in the gas 
phase produced by heating the metal, though many of these molecules 
are ionized particularly if the temperature is high. Solid lithium metal 
however consists entirely of ions, each of which has lost one electron ; 
these valence electrons may be regarded as a gas which fills the space 
between the ions. Other metals have similar structures. The electron gas is 
mobile inside metals and this confers the electrical conductivity which is 
their characteristic. 

Similar processes of electron sharing can occur between atoms of 
different elements, to form molecules of compounds. For example, the 
water molecule is formed from one oxygen atom and two hydrogen 
atoms. It can be pictured as a rather large electron cloud (formed from 
those from the hydrogen atoms and the least tightly bound electrons 
from the oxygen atom) which envelops the oxygen nucleus and its most 
tightly bound electrons, somewhere deep inside near its centre, with the 
two hydrogen nuclei a little distance away. Steam at high temperatures 
consists predominantly of such molecules. In ice, the same molecules can 
also be distinguished, though the hydrogen atoms spend a proportion of 
their time wandering from molecule to molecule. Water also contains a 
proportion of molecules which have split up into ions, consisting of 
positively charged hydrogen ions and negatively charged hydroxyl ions 
(oxygen and hydrogen atoms with one extra electron). 



CI 

I Li 



# 



(a) ^^ ^^pWm? 

id) 

Fig. 2.2. (a) A molecule of lithium chloride which can exist in the gas. (b) In the solid 

each Li ion is surrounded by six CI ions — there are no distinguishable molecules. 

The electron clouds have been drawn with well defined surfaces. 

Another simple molecule is that of lithium chloride consisting of a 
lithium ion bonded to a chlorine ion. These molecules exist only in the 
gas phase, produced when the substance is vaporized, though even then 
a number of molecules break up into the constituent atoms or ions. In the 
liquid and solid, and when dissolved in water, any one ion cannot be 



2.1 Atoms, ions and molecules 13 

said to be definitely associated with any one other ion, so that no molecules 
exist any more. In the solid, any one lithium ion is surrounded by six 
chlorine ions, all symmetrically disposed around it, and each chlorine ion 
is similarly surrounded by six lithium ions. The electrons cannot be 
accelerated so the substance is ah insulator. In dilute solutions, every ion 
is separately covered by a layer of water molecules. In neither case can it 
be said that the compound exists in the form of molecules. 

Carbon can enter into the composition of a vast number of molecules of 
many varied shapes, sizes and properties and these constitute organic 
compounds. They include biological substances whose molecules are 
often of an astonishing complexity. Such molecules remain almost un- 
changed in solid, liquid and gas. The benzene molecule, for example, is a 
fiat ring of six carbon atoms each with a hydrogen atom close to it with 
electron clouds between, and more tenuous clouds above and below the 
ring. These molecules are present in benzene whether solid, liquid or 
vapour. 

2.1.5 Nuclear and chemical reactions 

Atoms and molecules are not indestructible, since the forces holding 
them together are not infinitely strong. Atoms can be converted into other 
atoms by nuclear reactions which alter the nuclei, molecules can be 
changed into other molecules by chemical reactions which affect the 
loosely-bound electrons and hence alter the groupings of the atoms. 
However, the forces (or amounts of energy) needed to make nuclei break 
up or combine are much greater than those required to promote chemical 
changes. This is made manifest by the temperature which must be attained 
before reactions of different kinds can be initiated — we shall see later that 
these temperatures on the absolute scale are a rough measure of the 
energies involved. Nuclear reactions, such as the combination of hydrogen 
nuclei to form helium, can only be made to take place at very high tem- 
peratures of the order of those found in the interior of stars, but many 
chemical reactions will proceed at ordinary temperatures, 10 3 or 10 4 times 
lower on the absolute scale. This topic is discussed again in section 4.4.3. 

2.1.6 Atomic masses and Avogadro's number 

Atomic and molecular masses can be determined to good accuracy by 
studying the masses of substances taking part in chemical reactions, and 
making suitable assumptions as to the nature of the reactions. Nowadays, 
atomic masses are measured to a high degree of precision by mass spectro- 
meters. The vapour is ionized by bombardment with a beam of electrons 
and the charged ions are deflected by electric and magnetic fields in such 
a way that the ratio of charge to mass can be determined. Since the charge 



14 Atoms, molecules and the states of matter Chap. 2 

must be equal to that of an electron or a multiple of it, the mass can be 
found. 

Observations with mass spectrometers showed that any element, as 
found naturally, contains atoms of different masses but having the same 
chemical properties. These are called isotopes of the element. It is the 
nuclei which differ from one another in mass although they all carry the 
same charge and the electron clouds round them are identical. 

To construct a scale of atomic masses, we take the mass of a given 
isotope as a standard and compare others with it. The mass of the most 
abundant isotope of carbon is conventionally taken as 12 atomic mass 
units, written 12a.m.u. Natural carbon is a mixture of isotopes with 
masses close to 12 and 13 a.m.u. and the average is 12.011 a.m.u. The mass 
of the average hydrogen molecule is 2.016 a.m.u ; it is a mixture of atoms 
of masses near 1 and 2 a.m.u. Oxygen has isotopes of masses close to 
16, 17 and 18 a.m.u. Thus the water molecule can have masses anywhere 
between 18 and 24 a.m.u. the value 18 being overwhelmingly the common- 
est. For many purposes, however, it is sufficient to quote average masses, 
rounded off to the nearest whole number. If the mass of a single molecule 
is M a.m.u. we define one gram molecule, often called one mole, of that 
substance to be M gm. The molecular weight of the substance however is 
usually denoted by M, without any units. 

It follows from this definition that the number of molecules contained 
in one mole of any substance is independent of its chemical composition 
or physical form. This number is called Avogadro's number and is given by 

N = (6.02257 + 0.00009) x 10 23 

which for most practical purposes can be taken as 

N = 6xl0 23 . 

This is the number of hydrogen molecules in 2.016 g of hydrogen, of 
carbon atoms in 12.011 g of carbon, of water molecules in 18.015 g of 
water or ice or steam, and so on. The value of the atomic mass unit in 
grams (the mass of an imaginary molecule of molecular weight equal to 1) 
is the reciprocal of N : 

1 a.m.u. = (1/N)g = 1.66xl0" 24 g. 

It is the enormous magnitude of Avogadro's number, which implies 
that the individual atoms are so extremely small, that gives the illusion 
that matter is continuous on the ordinary scale. 

The gram ionic weight can be defined in a similar way to the gram 
molecule. The total charge on a gram ion of any substance in which each 
ion carries a single electronic charge e is Ne units of charge. It is called 



2.2 Gases, liquids and solids 15 

the faraday and is equal to 

1#" = 1.60 x 10" 19 x 6.023 x 10 23 = 0.965 x 10 5 C. 

For most practical purposes 1 faraday can be taken as 10 5 C. 

So far, we have made several dogmatic statements about the atomic 
nature of matter, the masses of single atoms and the value of Avogadro's 
number. Of course, these facts are not self evident and historically they 
took a long time to prove and were bitterly disputed till surprisingly 
recent times. In the nineteenth century the proof that the atomic hypothesis 
was correct hinged on the observation that Avogadro's number could be 
measured in a large number of completely independent ways and in spite 
of the crudeness of the measurements, the result was always roughly the 
same. The earliest method involved a study of the rate of diffusion of gases 
combined with crude estimates of the volumes that the same gases would 
have occupied if they could have been liquefied or solidified. Other early 
methods depended on studies of the heat of formation of brass ; of the 
relation between the surface tension and latent heat of liquids ; of the colour 
of the sky and the absorption of starlight in the atmosphere ; of the charge 
on the electron and its relation to electrochemical changes ; other methods 
were also used. All gave values for N which at any rate had the same 
number of zeros after the first figure and in view of the extraordinary 
range of phenomena covered, this constituted convincing proof of the 
atomic hypothesis — though even as late as 1907 it was not universally 
accepted. Nowadays, it is possible to detect the effect of single atoms and 
by indirect methods to render them visible, and Avogadro's number is 
known to high accuracy. 

2.2 GASES, LIQUIDS AND SOLIDS 

As a useful, though not complete, classification it can be said that matter 
exists in three states, as gas, liquid or solid. This statement is justified by 
the fact that there exist many substances which can undergo sharp, easily 
identifiable, reproducible, and reversible transitions from one state to the 
other. Water is the classical example : its freezing and melting, boiling and 
condensation have been contemplated since the time of the ancient Greek 
scientists. There are obvious contrasts between the properties of ice, 
water and steam or water vapour which make their description as solid, 
liquid and gas quite unambiguous. Similarly, most metals are solid, they 
melt under well defined conditions of temperature and pressure to form 
liquids and boil at higher temperatures to produce gases. 

If all substances possessed such clear demarcations, it would be easy 
to define the different states of matter. But there are very many substances 



16 Atoms, molecules and the states of matter Chap. 2 

like glasses or glues which one normally thinks of as being solid but which 
do not melt at sharply denned temperatures ; when heated they gradually 
become plastic, till they become recognizably liquid. Other solids such as 
wood or stone are inhomogeneous and it is difficult to describe their 
structure in detail. 

We will therefore not attempt to present definitions of solids, liquids or 
gases ; there would be too many exceptions. Instead we shall describe the 
principal characteristics of these three states of matter in bulk, and relate 
them to their structure on the molecular scale. Later we shall describe 
some of the conditions under which any one state can take on some of the 
characteristics of the others. In the course of these rather brief summaries, 
we shall use terms like 'small' or 'large', 'quickly' or 'slowly' which are as 
yet a little vague. In later chapters, it will be shown how they can be 
precisely defined. 

2.2.1 Compressibility, rigidity, viscosity 

Among other properties, it will be useful to compare the compressibility 
and rigidity or viscosity of the different states of matter. If a given pressure 
acting on a substance produces a large relative change of volume, that 
substance is highly compressible. Rigidity is the ability to oppose or 
withstand forces directed towards changing the shape of a body, while 
keeping its volume constant ; this property refers to a purely static situ- 
ation. A related quantity is the viscosity, a measure of the resistance to 
changes of shape taking place at finite speeds. For example, a body moving 
through a medium has to keep pushing it aside to keep moving ; if the 
forces required to be exerted on the body are large, the medium has a 
high viscosity. 

2.2.2 Properties and structures of gases 

Gases have low densities, they are highly compressible over wide ranges 
of volume, they have no rigidity and low viscosities. The instantaneous 
structure of a small volume of gas is illustrated in Fig. 2.3. The molecules 
are usually a large distance apart compared with their diameter and there 
is no regularity in their arrangement in space. Given the positions of two 
or three molecules, it is not possible to predict where a further one will 
be found with any precision — the molecules are distributed at random 
throughout the whole volume. They are moving randomly with a mean 
velocity comparable with that of sound, of the order of 10 4 cm/s. Occasion- 
ally two or three of them may be found very close to one another so that 
their electron clouds overlap and they bind together. Such clusters are 
common at high pressures but they are usually short-lived. 



2.2 Gases, liquids and solids 



17 



The low density can be readily understood in terms of the compara- 
tively small number of molecules per unit volume, and the high compressi- 
bility follows from the fact that the average distance between molecules 
can be altered over wide limits. The lack of rigidity can be explained by the 
molecules being able to take up any configuration with equal ease. Further, 
the molecules can move long distances without encountering one another, 
so that there is little resistance to motion of any kind, which is the basis 
of the explanation of the low viscosity. 



/ifra 





wmm 






Fig. 2.3. Molecules in a volume of gas at room temperature, a cube of about 20A at 
a pressure of about 20 atmospheres (the molecules are pictured as simple spheres). 

2.2.3 Properties and structure of liquids 



Liquids have much higher densities than gases — comparing liquids with 
common gases under ordinary conditions the factor is of the order of 10 3 . 
Their compressibility is low. They have no rigidity but their viscosity is of 



18 Atoms, molecules and the states of matter Chap. 2 

the order of 10 2 times greater that that of ordinary gases. It is difficult to 
give a detailed picture of the structure of a liquid ; an attempt has been 
made in Fig. 2.4. The molecules are packed quite closely together so that 
each one is bonded to a number of neighbours ; in the illustration, that 
number is between 4 and 5. Given the position of one molecule, it is now 
possible to state how many molecules should be found in contact with it, 
which is a good deal more information than it is possible to give about the 
arrangement of the molecules in a gas. But still the pattern as a whole is a 
disordered one. The molecules are moving with just the same order of 
velocity as in a gas at the same temperature, though the motion is now 
partly in the form of rapid vibrations and partly translational. The 
configuration is therefore continually changing. 

This picture can be correlated with the macroscopic properties — the high 
density from the large number of molecules per unit volume, the lack of 
rigidity from the lack of order and the continual alteration of the arrange- 
ment. The comparatively close packing explains the low compressibility. 




Fig. 2.4. Molecules in a volume of liquid about 20A x 20A x 3A. 



2.2 Gases, liquids and solids 



19 



The fairly high viscosity arises from the fact that the molecules have to 
wriggle past one another in this irregular but closely packed arrangement, 
rather like people moving past one another in a dense crowd, where slow 
relative movements are easy but rapid ones are difficult. 

2.2.4 Properties and structure of solids 

Solids have practically the same densities and compressibilities as 
liquids. In addition they are rigid ; under the action of small forces they 
do not easily change their shape. 

An important property of those solids which have a well-defined 
melting-point is that if they are formed very slowly from the liquid state 
they are crystalline — that is, they form shapes bounded by plane faces with 




Fig. 2.5. Molecules in a volume of solid about 20A x 20A x 3A. They are close- 
packed, and the arrangement is highly regular. There is however a fault (dislocation) 
in the arrangement towards the left, which can be seen by holding the page horizontal 
and looking upwards along the rows. (Dislocations are discussed in detail in 

section 9.3.1). 



20 Atoms, molecules and the states of matter Chap. 2 

characteristic angles between them. Sometimes however the crystalline 
form is not obvious, especially if the solid is not produced under suitably 
controlled conditions ; the crystals may be too small to be seen so that the 
solid as a whole does not show the expected facets. Substances which do 
not melt sharply but show a gradual transition to the liquid when heated 
are said to be amorphous and show no trace of regularity of external shape. 

In crystalline solids, the molecules are arranged in regular three 
dimensional patterns or lattices, of which Fig. 2.5. is an example. If the 
crystal has been carefully prepared, the regular arrangement persists over 
distances of several thousand molecules in any direction before there is an 
irregularity, but if it has been subjected to strains or distortions the regular 
arrangement may be perfect and uninterrupted only over much shorter 
average distances. In metals the ions are closely packed together, so that 
the distance between the centre of an ion and that of one of its nearest 
neighbours is equal to the diameter of one ion, or something close to it. In 
other crystals, the packing together of the molecules may be relatively 
open, but even in a light solid such as ice the distance between the centres 
of any molecule and its near neighbours is only twice the diameter of a 
molecule. In solids, the molecules are again moving with the same order of 
magnitude of velocity as in gases or liquids, but the motion is confined to 
vibrations about their mean positions. 

Amorphous solids can be described as liquids of extremely high vis- 
cosity, which over long periods of time have been 'frozen' into one par- 
ticular configuration. Figure 2.4 could be taken to illustrate the arrange- 
ment of molecules in such a solid, with the great difference that the con- 
figuration hardly alters with time, the atoms hardly ever changing their 
relative positions, although they continue to vibrate. 

In terms of these structures, it is not surprising that solids have com- 
pressibilities and densities like those of liquids. The high rigidities can 
also be understood, for the molecules can only move with respect to one 
another with difficulty. In crystalline solids, changes of shape can only take 
place through molecules slipping into holes which exist at irregularities 
in the lattice and this is not easy. In amorphous solids, there are plenty of 
holes for the molecules to slip into in order to initiate a change of shape, 
but the bonds between neighbours cannot be easily broken by external 
forces acting on the solid as a whole. All solids therefore resist the action 
of external forces and this is just what we mean when we say they are 
rigid. 

2.2.5 Characteristic times 

There are some substances which are undeniably solids, yet which over 
a long period of time alter their shape under the action of only small 



2.2 Gases, liquids and solids 21 

forces. Glaciers, for example, although made of ice which is undoubtedly 
solid and crystalline, are found to be flowing slowly downhill if they are 
observed over a period of years. Lead is another well-known example, a 
metal of high density but comparative softness. Sheets of lead, sometimes 
used for covering the roofs of large buildings, will alter their shape over a 
period of decades, slowly creeping downwards under the action of their 
own weight. Under large pressures, a number of metals flow quite quickly 
and their ability to fill small cracks makes them suitable for use as washers 
or gaskets in situations where more fluid sealing compounds cannot be 
used. Indium and pure gold, as well as lead and other soft metals, can be 
used in this way. Thus, if we observe their behaviour for short times under 
the action of small forces, we class these metals as solids ; but if we study 
them under high pressures, or for times of the order of decades (10 8 -10 9 
seconds) if only small forces act on them, we say that they behave to some 
extent like liquids. 

By contrast, liquids and gases show resistance to bulk motion whose 
speed is comparable with the speed with which the molecules are moving. 
Water is well known to feel like a solid if one dives on to it instead of 
through it. If a gas is made to move with a speed comparable to the velocity 
of sound it can sustain very sharp changes of density ; instead of being 
uniform, it is divided into distinct regions of different temperatures and 
densities. The boundaries between them are called shock waves. Thus 
liquids or gases subjected to large forces for short periods of time exhibit 
some of the characteristics of solids. 

Descriptions of the properties of solids, liquids and gases should there- 
fore include estimates of the times over which it is necessary to extend the 
observations in order to decide whether a substance has rigidity or not, 
whether local variations of density can be sustained or not. Under ordinary 
conditions of temperature and pressure, lead is a solid if we are concerned 
with events taking place in times which are less than, say 10 7 seconds, but 
for experiments lasting more than, say 10 9 seconds, lead is a liquid. 
Similar but much shorter characteristic times can be defined for substances 
which are gases or liquids. In a similar way, when we say that a substance 
softens or melts over a certain range of temperature, we mean that below 
that range the characteristic time for flow under small pressures is very 
long, inside the range it decreases with rising temperature, and at higher 
temperatures it is small. 




CHAPTER 

Interatomic potential energies 



3.1 MOLECULAR DIMENSIONS 

A rough calculation of molecular dimensions can be made if the molar 
volume of a solid or liquid is known. This is given by 

V = M/p 

where M is the gram molecular weight and p the density in g/cm 3 .* 

For water, M = 18 g and p is about 1 g/cm 3 for both liquid and solid. 
V is therefore about 18 cm 3 . Now this is the volume occupied by 
N = 6 x 10 23 molecules. Thus the average volume occupied by a single 
molecule is 3 x 10" 23 cm 3 . If we regard this volume as a cube, its side 
must be about 3 x 10" 8 cm = 3 A. If it is a sphere or any other simple 
shape its linear dimensions will not differ much from this. The distance 
between molecules in water or ice must therefore be about 3 A. It has 
already been mentioned (section 2.2.4) that the lattice in ice and the 
packing in water are relatively open, so that this figure is greater than the 
diameter of a water molecule. 

In metals, the packing is usually very close, so that the diameter of an 
ion cannot differ much from (VJN) 1 ' 3 . For potassium, a light metal with 
large ions, M = 39 g, p = 0.86 g/cm 3 , so that V = 45.4 cm 3 and the 

* Many students firmly believe that the molar volume of all substances is 22.4 litres. This 
is indeed roughly true for gases under 'standard' conditions of temperature and pressure, 
°C and 1 atmosphere. It is NOT true for solids or liquids. 



3.2 Interactions between electrically neutral atoms and molecules 23 

diameter of an ion is about 4.2 A. For gold, one of the densest metals, 
M = 197 g, p = 19.3 g/cm 3 ; V = 10.2 cm 3 and the ionic diameter = 
2.6 A. Indeed the diameters of all monatomic ions or molecules and the 
mean diameters of the smaller polyatomic molecules are all between 
1.5 and about 5 A. From the fact that the densities of common gases are 
of the order of 10~ 3 g/cm 3 (of the order of grams per litre) it follows that 
the mean distance between molecules is of the order of a few times 10 A, 
which is much greater than the diameter of a molecule. This agrees with 
the situation pictured in Fig. 2.3. 



3.2 INTERACTIONS BETWEEN ELECTRICALLY NEUTRAL 
ATOMS AND MOLECULES 

In the previous chapter we described the different spatial arrangements 
which the molecules or ions could take up in solids, liquids and gases. 
These can be related in terms of the forces which the molecules or ions 
exert on one another as a function of their distance apart. In this section 
we will concentrate on electrically neutral atoms and molecules : metals 
will be dealt with in section 3.7, ions in section 3.8. 

We have to reconcile two apparently contradictory statements. First : 
liquids and solids are highly incompressible. From this it follows that 
when two molecules are squashed together so that they approach one 
another closely, they repel one another. Second: solids and liquids 
cohere — that is, their molecules tend to pull themselves close to one 
another. It takes force to stretch a solid, therefore molecules attract one 
another. We will denote the repulsive forces by F R and the attractive 
forces by F w . 

F R must be dominant when the distance between molecules is about 
1-2 A or less. F w must be dominant when the separation is about 2-3 A 
or greater. At some intermediate distance, say roughly 2 A, the two forces 
must be equal and opposite — repulsion balances attraction. (Obviously 
these figures must not be taken too literally. They are quoted as crudely 
typical.) We seek an algebraic representation for this. 

First we must have a sign convention for the direction of the force 
exerted by a molecule. Imagine the origin of coordinates to be taken at 
the centre of the molecule and a line drawn outwards towards another 
molecule on which it exerts a force. Call this the r-axis. A force which 
acts in the direction of r increasing — a force of repulsion, tending to 
separate the molecules — is reckoned positive (Fig. 3.1). A force of attrac- 
tion is, on the same convention, reckoned negative in sign. We will use this 
convention consistently. 



24 Interatomic potential energies Chap. 3 

Force exerted by molecule 



& 



Molecule 



r - axis 



Fig. 3.1. Convention for direc- 
tions of axes and forces. A 
repulsive force exerted by a 
molecule, in the direction of r 
increasing, is positive. 




Interatomic distance 



Interatomic distance 



Fig. 3.2. (a) Short range and long range forces ; both are repulsive, (b) Expo- 
nential fall-off. Again, a repulsion is shown. (Plotted in arbitrary units). 

All intermolecular forces, attractions and repulsions, become smaller 
as the separation increases. If we choose to represent this variation by a 
simple power law, this law might be 

Force = + (const.) 

choosing the + sign for a repulsion and the — sign for an attraction, a is 
some standard length and the index n is positive. Now if n is large, r~ n 
becomes rapidly smaller when r is increased and rapidly bigger when the 
distance is decreased. If n is small, however, then the force falls off com- 
paratively slowly at large distances though it also increases comparatively 
slowly at small distances, To take a specific example, Fig. 3.2(a), consider 
two forces proportional to (a/r) 1 and (a/r) 2 and called F x and F 2 respec- 
tively. The index 7 is for present purposes a large number and the 2 is a 
small number. At r — a, both forces are numerically equal. However at a 
small separation r = a/10, F x is 10 5 times bigger than F 2 ; but at a large 



3.2 Interactions between electrically neutral atoms and molecules 25 

distance r = 10a, F 2 is 10 5 times bigger than F : . A force like F x , dominant 
at short distances but negligible at large, is called a short range force. 
F 2 is a long range force. Besides simple power laws like (a/rf other forces 
are found in nature which vary like exp( — r/a) where a is some characteris- 
tic distance. Every time r is increased by a, this kind of force decreases by a 
factor e = 2.728, so that it falls to 1/20 of its value for every 3a (Fig. 3.2(b)). 
Thus exponentially varying forces are certainly short range. 

In this terminology, the repulsive forces between atoms are short range, 
the attractive contributions are rather longer range. The best expression 
for the total force has been shown theoretically to be of the type 

F = F R + F W 

= A.e~ rja -B(a/r)\ 

where the variable r is the distance between the centres of the atoms, a is 
some measure of the 'diameter' of an atom — not, of course, that it has a 
sharply defined surface like a billiard ball. 

The repulsive force is caused by the overlapping of the two electron 
clouds — this gives the exponential variation. The attractive force is 
called the van der Waals force. It arises from the distortion of the electron 
cloud of one molecule by the presence of the other. It exists even though 
the atoms are electrically neutral.* 

For many purposes it is an advantage to have an expression for the 
interatomic force which fits the true curve adequately but which has a 
simple analytical form. In fact, the exponential term can be replaced by 
one of the type (a/rf where n is 10 or 13 or some number like that, so that 

This is good enough for our purposes and we will adopt it. 

3.2.1 Potential energy 

Rather than deal with forces, it is more convenient to deal with the 
potential energy of two molecules with respect to each other. This is a 

* Many students firmly believe that the only force of attraction which can exist between 
electrically neutral atoms is the Newtonian gravitational attraction due to the masses. They 
assume that the attractive forces between atoms are gravitational. This assumption was 
made in the early nineteenth century by Dalton and other pioneers. Apart from the fact 
that the index n is wrong (2 for gravity, 7 for interatomic forces), the magnitude of the binding 
energy (see section 3.3) is a factor of about 10 3 ° times too small: 

1 ,000,000,000,000,000,000,000,000,000,000 

times too small. Indeed, gravitational forces can safely be forgotten in all problems which 
do not involve the Earth or bodies of comparable mass. 



26 Interatomic potential energies Chap. 3 

scalar quantity and therefore simpler to discuss than forces, which are 
vectors. 

With the sign convention of the last section — F positive if it acts in the 
direction of r increasing — the potential energy can be defined by the 
equation* 

Fir) = -t-^X (3-1) 

ar 

where F(r) means 'the force which depends on r' and i^{f) means 'the 
potential energy which depends on r\ The force is the gradient of the 
potential energy. Alternatively, 

T^W = ^(r )- [ F{r).dr, (3.2a) 

where r is a standard point and i^(r ) is the potential energy there. 
It is usually most convenient to take this constant ^(r ) as zero; if 
this is done then we have 



nr) = - f 



F(r) . dr. (3.2b) 



We are at liberty to choose the standard point r where we please. For 
many problems it is convenient to take r = oo. Then the potential 
energy at r is given by 

TT(r)= - f F(r).dr. (3.2c) 

J oo 

In some problems however it is convenient to define 'V to be zero at the 
origin, that is to take r = 0. The latitude in the absolute value of the 
potential energy causes no difficulty because we are usually concerned 
with measuring only changes of potential energy. 

As an example, consider the potential energy of two atoms derived from 
the force of repulsion 

A 

F = ?' 

where n > 1. Then 

f r a>_ A r 1 
1 r- " ^T 



TT = -A 

J r 



* Really the equations should be written in vector notation 
F(r) = - grad -jr(r) ; V{t) = ! F . dr 
but as we are dealing with central forces the simpler formulation is adequate. 



3.2 Interactions between electrically neutral atoms and molecules 27 

It is obviously convenient to take r = oo and to write 

nr)= A l 



n-1 r" _1 

A graph of this function resembles Fig. 3.3(a). The physical situation 
being described is that initially two atoms are an infinite distance apart 
and they are pushed together infinitely slowly so that finally they are 
separated by r. The potential energy is then increased by this amount. 
Whether one says that the energy of the second atom is increased with 
respect to the first, or vice versa, is irrelevant ; in fact it is better to talk 
about the potential energy of the whole system. The energy in fact resides 
in the field of force between the atoms. 

The position of stable static equilibrium of a system occurs when the 
forces are zero, and this is where the potential energy is a minimum. Left 
to itself, a system will always move so as to reduce its potential energy. 
It follows straightforwardly from the definitions that a repulsive force 
which decreases with distance always has a i^(r) curve of the type shown 
in Fig. 3.3(a) — or Fig. 3.3(b) which differs from it merely in the addition 
of an arbitrary constant — so that when the system moves to reduce 'V, 
it does so by increasing the separation r. Conversely, attractive forces 
have rising curves like Fig. 3.3(c). 

A system having a potential energy which is a function of displacement 
like Fig. 3.3(d) will tend to move into the position of minimum energy, 
where 

^ = °- 

This is the position marked r in the diagram. The system is then said to 
be in a potential 'well'. It is called a well because it looks like a hole in the 
ground. Work must be done, energy must be supplied, to get the system 
out of the well. The amount of energy needed to take the system far to the 
left would be e t ; to move it far to the right, e 2 . 

3.2.2 Interatomic potential energy 

The interactions between atoms can be represented by a graph of their 
potential energy as a function of the distance between their centres. 
Following the argument of section 3.2, it can be expected to be of the form 

where p is approximately 9 or 12, q is a smaller number and X and \i are 
constants. 



28 Interatomic potential energies Chap. 3 




Fig. 3.3. (a) i^(r) for a repulsive force, (b) The same situation with a shifted 
arbitrary zero for y. (c) ^"(r) for an attractive force, (d) A potential well. 



A very convenient form of this type of equation is 



r(r) = 




(3.3) 



This looks a good deal more complicated than the equation just above it 
but in essence it is the same, namely the sum of a l/r p term and a — 1/V 
term. A graph of this kind of function (for the special case of p = 12, 
q = 6) is shown in Fig. 3.4 and another example (p = 11, q = 1) is shown 
in Fig. 3.15(a). 

The reader should check that Eq. (3.3) has the following properties. 
The potential energy at r = oo, when the molecules are infinitely far apart, 
is zero. By putting dl^/dr = 0, it can be readily verified that the minimum 



3.2 Interactions between electrically neutral atoms and molecules 29 

value of the energy is — e, so that we can say that the depth of the well is e. 
This minimum occurs at r = a . This is the position of static equilibrium. 
In order to pull the atoms apart to infinity, the attractive forces would 
have to be overcome and this would require the expenditure of an amount 
of energy e. 




A/flL 



Fig. 3.4. Interatomic potential energy, Eq. (3.3), plotted 
for the important case of p — 12, q = 6 (the Lennard- 
Jones 6-12 potential). The potential energy has been 
plotted in units of s, the separation between centres in 
units of a . The curve crosses the axis at r = a (Eq. 3.5). 



One case which is specially useful for simple molecules is the Lennard- 
Jones 6-12 potential when p = 12, q = 6: it reduces to 



nr) 



(3.4) 



This is the function plotted in Fig. 3.4. There is a more symmetrical form 
of this equation, which introduces another parameter a : 



V(r) = 4e 



(3.5) 



This is exactly the same as Eq. (3.4), if we put a = Xjl ■ a — 1.12a. When 
the separation r between the centres is equal to a, the potential energy is 
zero ; thereafter, if the two atoms are squashed together a little more the 
potential energy rises steeply ; in other words, the force of repulsion in- 
creases greatly. If we regarded the atom as a kind of ball with a hard 



30 Interatomic potential energies Chap. 3 

surface, we would identify the diameter of the ball with the separation 
between centres at which the repulsion rises steeply. Fig. 3.4 would then 
be interpreted to mean that the diameter of one atom is a and that the 
position of static equilibrium occurs when the separation is a = 1.12a, 
so that atoms are 'nearly touching'. Since a and a are nearly equal to one 
another, both are good measures of the diameter of one atom. 

Two qualifications must be made about the use of this curve. The first 
is that it refers to the interaction between two molecules only. If a third 
molecule is in the vicinity, the force between the first two may be modified 
by the movement of the charges in the electron cloud. But we will assume 
that this effect is negligible. As a result, the total energy of an assembly of 
molecules will be taken to be the sum of the energies of every pair as given by 
this curve. The second qualification is that we have tacitly assumed that the 
molecules are spherical so that the energy is uniquely defined by their 
distance apart. If this is not so, then their relative orientations may be 
important. In general, however, we will limit discussion to simple molecules 
which do not depart too much from sphericity, where their separation r 
is easily defined and i^{r) does not depend on orientation. 

3.2.3 Nearest neighbour interactions 

When two atoms are in equilibrium and 'nearly touching', their 
potential energy is — £ (Fig. 3.4). If their separation is approximately 
doubled, their potential energy decreases to about — e/30, that is, by a 
large factor. Whereas two atoms which are nearest neighbours are bound 
together by an amount of energy e ('bound' in the sense that energy is 
required to separate them), two atoms which are next nearest neighbours 
are only very loosely bound to each other (Fig. 3.5(a)). This is another 
way of saying that the van der Waals forces are short range forces — 
though of course the repulsive forces are of even shorter range. 





Fig. 3.5. (a) Nearest and next-nearest neighbours. The 
potential energy of A and B is -e; that of AC is about 
30 times smaller, (b) An aid to counting nearest-neighbour 
interactions. Energy e is needed to 'cut' each pair apart. 



3.3 Binding energy and latent heat 31 

Provided the picture is not taken literally, it can be considered as if each 
atom were held to its nearest neighbours by some kind of bond which 
requires an amount of energy £ to cut it (Fig. 3.5(b)). Bonds between atoms 
which are further apart however are so weak that we can forget them. 
The purpose of this picture is to indicate the amount of energy that must be 
supplied in order to break up the structure. It must NOT be thought to imply 
that the electrons are more concentrated in certain regions. 

3.2.4 Potential energy dominant at low temperatures 

In the rest of this chapter we will show that it is possible to relate some 
of the macroscopic or large scale properties of solids and liquids to the 
potential energy between the atoms or molecules. 

We will make an important assumption, namely that the kinetic 
energy of the atoms or molecules is small compared with their potential 
energy. This is equivalent to saying that we will assume that the tempera- 
ture is low. At this stage, before we have described in detail what we mean 
by temperature, it is not possible to say precisely what is meant by a 'low' 
temperature ; in fact, a temperature which is low enough for our approxi- 
mation to hold for one substance may not be low enough for another. In 
quoting any data however, we will always take the precaution of referring 
to temperatures which are low enough for the substance concerned. The 
effect of this procedure will be to simplify our calculations. In order to 
estimate the energy of an assembly of a large number of atoms or molecules 
in a mass of liquid or solid at low temperatures we need only take into 
account the potential energy due to their interactions and we can neglect 
their kinetic energy. 

The result of the discussion of the last section, where we saw that the 
potential energy of such an assembly is dominated by the potential 
energy between nearest neighbours simplifies our calculations even further. 

3.3 BINDING ENERGY AND LATENT HEAT 

The energy required to change one mole of solid or liquid into gas at 
low pressure is called the binding energy. It is closely allied to quantities 
which can easily be measured experimentally, the latent heats of evapora- 
tion (liquid to gas) or sublimation (solid to gas). 

For precise calculations, there are difficulties when these quantities are 
compared, however. When we calculate binding energies, we usually take 
the pressure of the gas to be zero so that the separation between atoms is 
infinite. Experimentally, we usually take the pressure to be the vapour 
pressure at the temperature concerned. (For example, one might measure 
the latent heat to evaporate water at 100 °C to produce steam at 1 atmo- 
sphere.) The difference has to be allowed for, although it is small at low 



32 Interatomic potential energies Chap. 3 

temperatures. A more serious source of error is that latent heats of 
evaporation of liquids are functions of temperature — they decrease with 
increasing temperature and become zero at a high temperature above 
which the liquid cannot exist. (See for example Fig. 3.13(b) for the latent 
heat of liquid argon.) It is therefore meaningless to quote 'the latent heat 
of vaporization' of water or any other liquid as if it were a constant. 
However, we are restricting our discussion to low temperatures, and in 
that region latent heats tend towards limiting values : it is these which we 
consider. 

For rough estimates, we note that latent heats of melting, to convert 
solid to liquid, are small compared with latent heats of evaporation to 
convert liquid to vapour. For example, to convert ice to water at °C 
requires about 6x 10 10 erg/mol (that is, 340 J/g), to convert water to 
steam at a comparable temperature requires 45 x 10 10 erg/mol (2,500 J/g) 
— almost eight times as large. So we can approximate even further and, 
if no better data are available or only rough estimates are needed, we 
can say that the binding energy is not very different from the latent heat 
of evaporation. 

3.3.1 Estimation of £ from latent heat data 

On our approximation, the binding energy at low temperatures is equal to 
L — ex (number of pairs of nearest neighbours). 

It is useful to define the coordination number n, the number of nearest 
neighbours which surround a given atom or molecule. It can never 
exceed 12. For close-packed solids it can reach 12 ; for more open arrange- 
ments it is smaller, 6 or 10. In dense liquids, the coordination number is 
about 10. (In the two-dimensional pictures, Figs. 2.4 and 2.5, n is about 
4 or 5 and about 6 respectively.) 

For coordination number n, an assembly of N atoms has ?nN pairs of 
nearest neighbours ; the factor \ arises from the fact that each bond pictured 
in Fig. 3.5(b) links two atoms but it must only be counted once. Thus : 

L = ±nNe. (3.6) 

If we know L , this allows us to estimate e. At the same time, the smallness 
of the latent heat of melting can be understood, since the change of co- 
ordination number between solid and liquid is quite small (12 to 10 say), 
whereas between liquid and gas it is large (10 to zero). We can get consist- 
ent results for £ by using the latent heat of evaporation together with 
n = 10.* 

* Strictly our calculations refer to single atoms but they can be applied to molecules which 
do not depart too far from spherical shape. These include diatomic molecules like N 2 or 
molecules like CC1 4 which are roughly tetrahedral. Long chain molecules are ruled out. 



3.4 Surface energy 33 

For liquid nitrogen, consisting of diatomic molecules N 2 , the latent 
heat at low temperatures is about 210 J/g, the molecular weight is 28, 
so the molar latent heat of evaporation is 6 x 10 10 erg/mol. Then e is 
2 x 10~ 14 erg ~ 0.01 eV. This is the energy needed to separate two nitrogen 
molecules from one another. For carbon tetrachloride, CC1 4 , the latent 
heat is about 210 J/g, the molecular weight is 153 so that e is about 
10~ 13 erg or 0.05 eV. 

These are typical values. Most molecules have e of the order of 0.01 
to 0.1 eV. In Fig. 3.4, each division of the vertical (energy) axis is therefore 
of this order of magnitude ; each horizontal division represents a distance 
of a few Angstrom units. Compared with ionization energies, the energies 
required to remove an electron from the cloud surrounding typical atoms 
to convert them into ions, these are small amounts of energy. Ionization 
energies are commonly of the order of 1 to 10 eV, which is 100 to 1,000 
times as big. 



3.4 SURFACE ENERGY 

The surface energy of a solid or liquid is the amount of energy that is 
needed to create 1 cm 2 of new surface. The process can be pictured as 
follows. Imagine a column of solid or liquid of 1 cm 2 cross-section to be 
broken apart by some means, Fig. 3.6. Energy must be used in order to 
overcome the interactions between molecules on either side of the break. 
We will calculate this energy. Let there be JT molecules per cm 2 of cross 
section ; if the diameter of one molecule is a cm, then Jf is something 
like 1/a.o per cm 2 . 





Fig. 3.6. Creating new surfaces by cutting a solid or 
liquid column in two. 



34 



Interatomic potential energies Chap. 3 



After the break, each molecule in the surface is no longer surrounded 
by the full number n of nearest neighbours. Instead it has, on the average, 
only j\x neighbours, in the one hemisphere; therefore \nJf nearest 
neighbour interactions must be broken. This requires an amount of 
energy \ nJ^e : but it produces 2 cm 2 of new surface, 1 cm 2 each for the 
top and bottom halves of the column. The surface energy is therefore 
^wjVe erg/cm 2 . 

3.4.1 Surface tension 

Surface tension, a quantity which is easily measured experimentally 
and is allied to surface energy, is usually defined as the force exerted on a 
cut 1 cm long in the surface of a solid or liquid, a force which tends to 
close the cut. It will be denoted by y dyn/cm. Imagine a thin film of liquid, 
with upper and lower surfaces like a soap film, to be stretched across a 
wire frame, Fig. 3.7. One side of the frame is moveable. It is assumed that 
the film is many molecules thick. The force on the slider is 2yl dyn, where 
/ is its length. The factor 2 appears because there are two surfaces. If the 
slider is moved back a distance d, the work done is 2yld erg. We imagine 
this process to be done so slowly that heat can flow into the film so that 
any tendency to cool is counteracted and the temperature and y remain 
constant. Since the total area of surface is now increased by 2ld, the amount 
of energy supplied is y erg per unit area. The surface tension is therefore 
clearly related to the surface energy. They are not identical however, 
because of the heat energy flowing in during the process to keep the tem- 
perature constant. This is the same kind of difference as that between a 
binding energy and a latent heat. Again, however, if measurements are 
extrapolated to low temperatures there is little difference between the 
two, although for the roughest estimates it is not necessary to make even 
this correction. ((Fig. 3.13(d) shows the variation of surface tension with 
temperature for liquid argon and this is typical.) We will therefore write 



y = irvVe 



(3.7) 




Fig. 3.7. Stretching a liquid film to create new surface. 



3.4 Surface energy 35 

3.4.2 Estimation of £ from surface tension data 

It is clear that, with the considerable oversimplifications in our model 
of a liquid, we can relate the surface tension to the latent heat. Although 
the phenomena that we usually associate with these quantities are quite 
different, both are simply measures of the depth e of the potential well and 
of the sizes of molecules. In 1870 Kelvin used a similar analysis to estimate 
the size of water molecules from the known molar volume V , the heat of 
vaporization and the surface tension, effectively writing our equations in 
the form a = 2yV /L . This was one of the first methods for estimating 
Avogadro's number. 

Here we will use surface tensions to work out e for the same liquids as 
in section 3.3.1 and show that the results are comparable with the values 
deduced from latent heats. At its normal boiling point (77°K), liquid 
nitrogen has a density of 0.81 g/cm 3 and its surface tension is 8.7 dyn/cm. 
Its molecular weight is 28. The molar volume is therefore 35 cm 3 , and 
following the argument of section 3.1.2 the diameter of the molecule is 
3.9 A; hence Jf is 6.7 x 10 14 per cm 2 . If we take n = 10, £ must be 
0.5 x 10" 14 erg or 0.003 eV. This must be compared with 2 x 10" 14 erg 
(0.01 eV) which we deduced from latent heats. For carbon tetrachloride 
at room temperature the data are 1.6 g/cm 3 , 26 dyn/cm and molecular 
weight 153. The molar volume is 96 cm 3 , the diameter of a molecule 5.4 A, 
JT is 3.5 x 10 14 per cm 2 and if n = 10, e is 3 x 10" 14 erg or 0.02 eV. This 
must be compared with 10" 13 erg (0.05 eV) from the latent heat data. 
Both liquids therefore give figures which are consistent within a factor 4, 
and this must be considered good agreement in view of the crude handling 
of the data. 

* 3.4.3 The rise of liquids in capillary tubes 

One of the commonest methods of measuring the surface tension of a 
liquid is to measure its rise in a capillary tube of known radius. In this 
section we will discuss, in terms of interatomic potential energies, why 
many common liquids rise in a glass tube but mercury falls. 

When a glass tube is exposed to an atmosphere containing the vapour 
of a liquid, its surface is bombarded with molecules and some of these 
stick to the glass. The process is called adsorption. The whole surface 
quickly becomes covered with a layer one or two molecules thick. The 
molecules next to the glass may be attached very firmly: this can be 
deduced from the amount of energy (the heat of adsorption) which is 
observed to be given out when the surface is exposed in this way. 

* Starred sections or subsections may be omitted, if the reader so wishes, as they are not 
required later in the book. 



36 Interatomic potential energies Chap. 3 

But this tight binding usually does not extend very far because inter- 
molecular forces are of short range and the forces acting on a molecule 
outside the solid are determined by the nature of the outermost layers. 
After the surface of the glass has been covered with the first one or two 
layers of vapour molecules, further vapour molecules approaching the 
surface experience an attraction which is almost the same as if the entire 
tube were composed of these molecules. Its surface energy per unit area 
is practically equal to that of the liquid from which the vapour was 
produced. 

When a tube is first dipped into a liquid it has not risen in the tube 
and the surface is flat. Let us find the height to which it rises by calculating 
the change of potential energy when the tube is filled to an arbitrary height 
h and then let us write down the condition that the potential energy 
should be a minimum ; this determines the equilibrium value of h. Let the 
liquid have density p and surface tension y and let the radius of the tube 
be r. We can consider the tube to be filled in the following stages. We 
imagine a volume of liquid to be removed from the flat surface, just enough 
to fill the tube to the height h. It must have volume nr 2 h and surface 
area 2nrh (Fig. 3.8). To remove it from the rest of the liquid an amount of 
energy equal to (2nrh)y must be expended. Then we imagine this liquid 
to be changed to a cylindrical shape (which requires no change of surface 
area and hence no expenditure of energy). When it is raised vertically, its 
potential energy due to its weight is increased to {nr 2 hp)gh/2 since its 
mass is nr 2 hp and the height of its centre of mass is h/2. Finally the liquid 
can be imagined to be put inside the tube. A surface area of the inside of 
the tube equal to (2nrh) is covered and an equal area of the surface of the 
cylinder of liquid is also covered. Thus, since both surfaces have surface 
energy y per unit area, the surface energy is reduced by 4nrhy. 



surface of tube 
/"not covered 



correct amount 
of liquid removed 



surface of tube 
/not covered 



liquid raised into 
/'cylindrical shape 

surface of liquid 
'not covered 



surfaces 
/covered 



(a) it>) (c) 

Fig. 3.8. Energy changes when liquid rises in a capillary tube. 



3.4 Surface energy 37 

Thus the total increase of potential energy when the liquid rises to 
height h is 

U = 2nrhy + nr 2 h 2 pg — 4nrhy 

= nr 2 h 2 pg — 2nrhy. 

This equation shows that the reason the liquid rises in the tube is that 
the surface energy of the interior of the tube is reduced. 
The condition that U should be a minimum is that 



= 0, 



dU 

~dh 
that is 

7ir 2 hpg — 2nry = 

2y 
h = — . (3.8) 

rpg 

For carbon tetrachloride in a tube of 1 mm bore, h is equal to 6.5 mm 
since y = 26 dyn/cm and p — 1.6 g/cm 3 . 

This discussion should be valid for any liquid whose molecules are 
adsorbed on to the surface of the tube. For mercury in glass, however, 
conditions are very different. Under normal conditions mercury does not 
adhere to glass. Droplets of this liquid simply run off a glass surface. 

It should be noted however that under very special conditions mercury 
can be made to stick to glass, but that even then the adhesion is very weak. 
The effect is sometimes observed in McLeod gauges used to measure 
pressures in high vacuum systems. McLeod gauges have two limbs con- 
taining mercury, one open and the other closed. When the pressure is 
being measured in a system at extremely low pressure the mercury is 
pushed right up to the closed end of the tube and fills it completely. 
As part of the measuring procedure, the mercury in the other limb is 
then lowered and under normal conditions the mercury in the closed 
limb also falls, so that the two menisci keep at practically the same height 
as one another. But when conditions are exceptionally clean (the glass 
surfaces have been heated and the system has been evacuated for a con- 
siderable time), it is occasionally observed that the mercury in the closed 
limb does not fall but remains stuck to the glass. If the difference of 
heights is h, the mercury at the top of the closed limb is under a tension 
of pgh dyn/cm 2 , where p is the density of the mercury, and this tension 
must be resisted by the adhesion to the glass. As the level is further lowered, 
the mercury in the closed limb suddenly falls when the adhesion is broken. 



38 Interatomic potential energies Chap. 3 

Level differences up to about 10 cm are sometimes seen, corresponding 
to tensions of about 10 5 dyn/cm 2 . We can use this fact to estimate the 
energy required to separate 1 cm 2 of mercury from glass. Let us assume 
that once we have separated them by a distance of one or two atomic 
diameters, the force becomes very small. The work required to do this 
is equal to the tension times the distance, which is 10 5 dyn/cm 2 multiplied 
by 10" 8 cm, that is 10" 3 erg/cm 2 for a range of 1 A; for a range of 10 A 
it is 10" 2 erg/cm 2 . Interpreting this in terms of interatomic potential 
energies as we did in section 3.4, the depth of the well must be of the order 
of 10" 5 or 10" 6 eV (using Jf of the order of 10 15 per cm 2 and n about 10). 
This is a very small figure compared with quite weak van der Waals 
energies. Of course, the mercury glass bond always breaks at its weakest 
place and the average energy of adhesion must be rather larger than we 
have estimated. But allowing for this, it seems safe to assume that even 
under the most favourable conditions mercury adheres only weakly to 
glass. 

This means that when a glass surface is covered with mercury, the 
surface energy decreases at best by very little. Following the previous 
analysis, the condition for minimum total potential energy is that the 
liquid should be depressed inside a tube, which is what is observed. 

The important point about capillarity experiments is that they measure 
the interactions between molecules of the liquid and those of the surface 
of the tube. In order to measure the surface tension of the liquid alone, 
other methods have to be used. 

3.4.4 Speed of ripples over a liquid surface 

One interesting method of measuring the surface tension of a liquid is 
to find the speed of propagation of ripples across its surface. 

When a wave is travelling across a surface and the wave profile is 
sinusoidal, the area is greater than when the surface is plane. The surface 
energy is therefore increased (Fig. 3.9). This effect— which leads to a 
finite speed of propagation of the waves — is the one we are interested in. 
But at the same time, the weights of the parts of the wave which are 
displaced upwards and downwards also increase the potential energy of 
the system and this affects the speed of propagation too. It can be shown 
that for waves of wavelength A and frequency/, the speed defined by 

c = fA 

is given by 



3.4 Surface energy 



39 



where y is the surface tension of the liquid whose density is p, and g is the 
acceleration due to gravity. 




Fig. 3.9. Sinusoidal wave on a liquid. The perimeter ABCDE is longer than 

the undisturbed distance ACE (equal to the wavelength A) so that the surface 

energy is increased. 

It follows from the dependence of the two terms on A and A~ 1 respec- 
tively that when the wavelength is very great the second term under the 
square root sign is small ; the speed is then controlled by gravity alone 
and is not affected by surface tension. But when the wavelength is small, 
the gravity term becomes small and the speed is dominated by the surface 
tension term. For these short wavelength ripples, 



2ny 



so that they travel faster the shorter their wavelength. 

The method consists of generating ripples usually at audio frequencies, 
on the liquid in a tank. The frequency must be known and the wavelength 
is usually measured either by stroboscopic photography or by setting up 
a stationary wave pattern and measuring the distance between nodes. The 
liquid must be of sufficient depth for the bottom of the tank to have no 
influence on the waves. The disturbance does not in fact penetrate very 
deeply into the liquid ; the effective mass which takes part in the motion 
is only about one-tenth of a wavelength deep, to be precise A/4n. The 
amplitude of the disturbance at a depth of one wavelength is negligible 
and this gives a criterion for the depth of liquid to use. The analysis of 
the motion is set as a problem at the end of the chapter. It depends on the 
fact that if a system is disturbed and its potential energy is proportional 



40 



Interatomic potential energies Chap. 3 



to the square of the displacement, the motion is periodic in time. This 
topic is discussed in section 3.6. 



3.5 ELASTIC MODULI 

Elastic moduli are all defined by equations of the type 

(change of pressure) = (modulus) (fractional change of dimensions). 

For small changes of dimensions, usually less than 1 % or 0.1 %, the body 
regains its original shape and size when the forces are removed; the 
behaviour is said to be reversible (or elastic), and we will concentrate on 
this type of change. Furthermore, when the fractional changes of dimen- 
sions are extremely small — a factor 10 or so smaller than the limit where 
elastic behaviour ceases — the changes of dimensions are quite accurately 
proportional to the pressures. Thus over these very limited ranges, the 
elastic modulus is a constant for the material. (We will discuss the break- 
down of proportionality in section 3.7.1. and nonelastic behaviour in 
section 9.1.1.) 

Stretching and twisting by different geometrical arrangements of 
forces, and compression by uniform 'hydrostatic' pressures are different 
ways of producing deformations and correspondingly we define Young's 
modulus, the rigidity or shear modulus and the bulk modulus as in Fig. 
3.10. 



r 
l 

~"~r 

i_ 



1 




1 
1 


1 
1 
1 
1 
1 



(a) 



ib) 



(c) 



Fig. 3.10. Systems of forces and deformations denning elastic moduli, (a) 
linear tension producing extension, related by Young's modulus, (b) tan- 
gential forces producing an angle of shear, related by the rigidity and (c) 
hydrostatic pressure producing a change of volume, related by the bulk 

modulus. 



For many practical purposes, it is necessary to emphasize the differences 
between the three moduli but here we will concentrate on their similarities. 
For any one substance they are of the same order of magnitude. Usually 
the bulk modulus and Young's modulus are almost equal and the rigidity 



3.5 Elastic moduli 41 

modulus is a factor 2 or 3 smaller. This can be seen in the Table. But it 
must always be remembered that for liquids, the rigidity is zero (section 

2.2.3). 



Material Young's mod. Rigidity Bulk mod. 

dyn/cm 2 dyn/cm 2 dyn/cm 2 

Solid argon 7.0 xlO 9 3.0 xlO 9 6.0 xlO 9 

Sodium chloride 4.0 x 10 1 1 1.3 x 10 1 1 2.5 x 10 1 i 

Steel 2.0 xlO 12 0.8 xlO 12 1.8 xlO 12 



We will take the bulk modulus as the typical elastic parameter, because 
it is the easiest to calculate. 

The bulk modulus K of a material, the reciprocal of the compressibility, 
is defined by 



-- (S) 



(3.10) 



where V is the volume, which is decreased when a pressure P is exerted 
uniformly in all directions. Usually, it is assumed that the temperature is 
kept constant during the compression. K can be measured directly by 
exerting a known pressure and measuring the change of volume — a whole 
technology has grown up for producing enormous pressures without the 
substance leaking past the piston which compresses it. Usually the main 
source of error is due to the non-uniformity of the forces acting in different 
directions. Alternatively, the speed of propagation of sound waves through 
a material can be found. This depends on the compressibility (as mentioned 
in section 3.6.1.), though a number of corrections have to be applied if exact 
values of K are required. 

3.5.1 Bulk modulus and the 'Vif) curve 

A compressed body can do work if the pressure is released; thus a 
compressed body has potential energy. This energy is given by a term of 
the type (force) x (distance) or (pressure) x (volume) — this is shown 
explicitly for a gas in Fig. 4.2 but it holds for any body. Notice that the 
energy E increases when the volume decreases so that, 

d£ = -PdV. 



42 Interatomic potential energies Chap. 3 

Hence we can write 

This is an important relation. Pressure can usually be interpreted as an 
energy per unit volume, an energy density. However, one must be careful 
about this expression ; it assumes that no heat flows in during the process 
of compression, or in thermodynamic language the compression must be 
adiabatic. This conflicts with the usual definition of bulk modulus given 
above. But thermal effects become small at low temperatures, so once 
again our estimates will become better the lower the temperature. 
Substituting (3.11) in (3.10): 

ld 2 E 

K = y W. 

This expression is in macroscopic terms — that is, the variables are E, the 
energy of the whole block of material, and V its volume. We wish to 
express these in terms of 'Vif) the potential energy of a pair of molecules 
and r the separation between two molecules. We do this as follows. 

If we can express V in terms of r, we can express d 2 E/dV 2 in terms of r. 
In general, 

d£ _ dr d£ 
dK~cfF'~a>' 

d 2 £ d IdE dr\ d 2 E /dr\ 2 d£ dV 



dV 2 dV\dr dVj dr 2 \dV) dr dV 

This holds whatever the relation between Fand r. In particular, if we regard 
the molecules as little cubes. 

V = Nr 3 , 

where N is the number of molecules in the block and r the distance between 
two neighbours. Therefore 

dr 

Further, if deformations are small and we take only nearest neighbour 
interactions into account, we can say that to a good approximation r = a , 
where a is the separation for static equilibrium where the potential energy 
is a minimum, i.e. d£/dr = 0. Thus the second term on the right hand side 



3.5 Elastic moduli 



43 



of (3.12) is zero, and 



d 2 E 



dV 



K = 




9N 2 a*, 



'9Na . 



(3.13) 



Finally we must relate E, the energy of the block of material, to the energy 
of the individual molecules. As already emphasized, we will assume that 
their kinetic energy is small compared with their potential energy due to 
the intermolecular forces, which is equivalent to saying that we consider 
only low temperatures. We therefore relate E to *T(r). Taking only nearest 
neighbour interactions into account, we consider ^Nn nearest neighbour 
pairs; then 



E = iATnlT(r), 



and 



K = n 



d 2 TT(r) 
dr 2 



18a . 



(3.14) 



This expression holds whatever the form of f~(r). Let us assume the 6-12 
potential 



TT(r) = e 



?r->r 



Then 



whence 



d 2 -T(r) 
dr 2 



72s 
a 2 



(3.15) 



K = 



4ns 
al 



4iVne 8L f 



Nat 



V n 



(3.16) 



where V is the molar volume and L the binding energy per mole. This is 
a rather extraordinary relation. It predicts that, for solids which are bound 
together by van der Waals forces so that the 6-12 potential is a good 
description of the interatomic potential, the bulk modulus is 8 times the 
binding energy per unit volume. Implicitly, we are referring to low 
temperatures. This relation allows us to predict the order of magnitude of 



44 Interatomic potential energies Chap. 3 

any elastic constant, if we know the latent heat of evaporation or the 
surface tension. The factor 8 depends on the assumption of the 6-12 
potential but one expects a similar sort of factor for any molecular solid 
or liquid. 

3.5.2 Comparison of bulk modulus and latent heat data 

As in sections 3.3.1 and 3.4.2, we will use liquid nitrogen and carbon 
tetrachloride as typical molecular liquids. For liquid nitrogen, SL /V = 
(8 x 6 x 10 10 /35) erg/cm 3 = 1.4 x 10 10 erg/cm 3 . The measured bulk modu- 
lus of the solid at about the same temperature is 1.26 x 10 10 dyn/cm 2 . 
The agreement is excellent. For CC1 4 , at room temperature 8L /% = 
8 x 3.2 xlO 1 796 = 2.5 x 10 10 erg/cm 3 . The measured bulk modulus is 
1.1 x 10 10 dyn/cm 2 . The agreement within a factor of 2 must be considered 
good. 

3.6 VIBRATIONS IN CRYSTALS: SIMPLE HARMONIC 
MOTION 

We have already seen that molecules in a solid or liquid are vibrating 
about their mean positions (sections 2.2.3, 2.2.4). The purpose of the present 
discussion is to show how the frequency of vibration can be estimated, 
knowing the T^*(r) curve. 

Firstly we must establish some relations about simple harmonic 
motion. Consider a system subject to a restoring force directed towards 
an origin and proportional to the displacement x. With the sign convention 
of section 3.2, 

F = —ax. 

Such a system executes simple harmonic motion of frequency 

1 Ax 
2n\j m 
where m is the mass in motion. This well known result can be recast in 
terms of potential energies. We are at liberty to take the zero of potential, 
energy anywhere we wish. In this case we will choose it at x = 0, and 

V = |ax 2 . (3.17) 

The system is said to be in a parabolic potential well (Fig. 3.11(a)). 

We can define the curvature at any point of a curve as the reciprocal 

of the radius of curvature. It is shown in mathematics texts that for the 

curve y = f(x), 

d 2 y /dx 2 mR . 

curvature = -r. — . , , , , 2 ->3/2 • ^.io) 

{l+(dy/dx) 2 } 3/2 



3.6 Vibration in crystals : simple harmonic motion 



45 





Fig. 3.11. (a) Parabolic potential well, (b) A curved potential well which is 
roughly parabolic at the bottom. 

If the curve passes through a minimum then dy/dx is zero there and the 
curvature at the minimum is simply d 2 y/dx 2 . Thus for a parabolic potential 
well V = jtxx 2 , a is the curvature at the bottom of the well. 

Finally we can say that any reasonable curve which goes through a 
minimum is not very different from a parabola in the vicinity ^bf the 
minimum, Fig. 3.11(b). 

Gathering these results together, we can say that a system which has 
a i^(r) curve of the usual interatomic type will come to rest in static 
equilibrium at a separation a at the minimum. But if it is displaced 
slightly, it will undergo simple harmonic oscillations of frequency 



v = 



2n 



\d 2 y/dr\ =t 



m 



1/2 



(3.19) 



3.6.1 Einstein frequency 

Imagine now a solid in which all the molecules are fixed at their equi- 
librium positions in the perfect crystal lattice, except one which is free to 
vibrate. As a first approximation, dissect this one out of the lattice together 
with two neighbours, one on either side, and assume that the vibration 
takes place along the line joining them, Fig. 3.12. When the molecule 
moves to the left it goes nearer to one neighbour and further away from 
the other, To the approximation that the potential well near the minimum 
is a symmetrical parabola, the change of potential energy is twice that due 
to one neighbour. So, using the result of Eq. (3.15), 



d 2 1T 
dr 2 



144e 
a 2 



(2 neighbours in line). 



46 Interatomic potential energies Chap. 3 






Fig. 3.12. Linear vibrations of a molecule with 
two nearest neighbours. 

To a better approximation, imagine that we dissect out the one molecule 
surrounded by n nearest neighbours distributed uniformly over the 
surface of a sphere : these neighbours are now at all angles to the direction 
of vibration. The potential well is of depth ne and is a parabolic function of 
radial distance r. But to calculate the change of potential energy with 
displacement along a certain direction we have to average a factor cos 2 6 
over all directions. It emerges that 

—I = — ^ (n neighbours spherically disposed) 

2 1 "1 si* 1 



dr 2 J r=ao 3a 2 , 

like the previous expression but with the factor 3 coming from the aver 
aging. Thus the frequency 

1 /24ne 



Ve 2ti\I mal 



(3.20) 



where m is the mass of one molecule. It is called the Einstein frequency. 
This is a very rough estimate of the frequency of vibration, because of 
course all the molecules are vibrating at once and the potential energy 
of one molecule depends not only on its own position but on its neighbours'. 
Whereas this analysis implies that all molecules are vibrating at a single 
frequency, in a real solid many frequencies are present. But the order of 
magnitude is significant. It is no coincidence that the Einstein frequency 
and the bulk modulus both depend on (d 2 lT/dr 2 ) r=ao . The connection is 
that the speed of sound is given by 

//bulk modulus \ .. __ 

speed = / l->-^l) 

Y \ density / 

and using Eqs. (3.16) and (3.20) it can be verified that the Einstein frequency 
is the frequency of sound waves whose wavelength is about twice the 
intermolecular spacing. A full understanding of this result depends how- 
ever on a study of the propagation of waves through lattices of points 
rather than continuous media. 

3.6.2 Estimation of Einstein frequency 

From the data of 3.3. 1 and 3.4.2, the frequency of molecular vibrations in 
both liquid nitrogen and carbon tetrachloride is of the order of 10 12 c/s. 



3.7 Metals 47 

This has no immediate interest for us though later we shall see that it has 
important consequences for the thermal properties, notably the specific 
heat, of these substances (5.4.4). 

3.6.3 Experimental data for argon 

Argon is a rare gas which liquefies at about the same temperature at 
which air liquefies, 80° K. The molecule is a single spherical atom. Inter- 
atomic attractions are purely van der Waals forces. The solid has a close 
packed structure. It is therefore an 'ideal' molecular crystal and has been 
extensively studied down to very low temperatures. Experimental data 
are given in Fig. 3.13; suggestions for analysing them are given in a 
problem at the end of the chapter. 

One interesting use to which these data can be put is to verify experi- 
mentally that the 6-12 potential is a good representation. If we use the 
generalized p-q form of the interatomic potential energy, Eq. (3.3), then 
it can be verified that 

d 2 -T\ pqs 

~Tir\ = ~\ (3-22) 

dr lr=ao "o 

Thus comparing the bulk modulus with the binding energy L per unit 
volume, as in equation (3.16), the product pq can be measured. It will be 
found to be about 64, which is surprisingly close to 72. 

The sublimation energy has only been measured down to 70° K and 
this makes exact extrapolation to 0°K difficult. However, specific heats 
have been measured down to very low temperatures and we can then use 
the following energy cycle to find L . This uses quantities which have not 
yet been denned but is given here for completeness, (i) Start with the solid 
at T = 0°K and vaporize it; the energy required is L . (ii) Heat the gas 
to 83°K ; to present accuracy it is a perfect gas of specific heat f R, where 
R is the gas constant = 8.31 J/deg. (iii) Condense the gas to solid at 83°K— 
the energy released can be read off the graph, (iv) Cool to near 0°K— the 
energy extracted, deduced from specific heat measurements at these low 
temperatures, is 165 J/mol. The argon is now back in its initial condition 
and from the energy balance L can be calculated. 

3.7 METALS 

Metals consist of arrays of positive ions permeated by an atmosphere 
of free electrons (section 2.1.4). Each ion carries a positive charge and the 
electrostatic coulomb repulsions between the array of like charges would 
be enormous ; the electrons, all negatively charged, would also repel one 
another equally strongly. Thus at first sight we would not expect metal 



48 



Interatomic potential energies Chap. 3 



to cohere but to fly apart. But in fact any small region in the metal tends 
to be electrically neutral and the electrons tend to concentrate between 
the ions so that their negative charges cancel out or screen the effect of 
the positive ions on one another. 



10 



'o 



h. 
E 

o 
<->0 




50 



100 0|< 150 



(a) 



50 OK 80 
(c) 




Fig. 3.13. Data for Argon. Atomic weight = 40. Sources of data : (a) Densities 
— Dobbs and Jones, Rept. Prog. Phys. 20, 516 (1957); Mathias, Onnes and 
Crommelin, Leiden Comm. 131a (1912). (b) Latent heats— computed from den- 
sities and vapour pressure measurements of Clark, Din and Robb, Michels, 
Wassenaar and Zwietering, Physica 17, 876 (1951). (c) Compressibility — Dobbs 
and Jones, as above, (d) Surface tension — Stansfield, Proc. Phys. Soc. 72, 854 

(1958). 



3.7 Metals 49 

The problem of calculating the potential energy of an ion in the lattice 
is an extremely difficult one because the electrons are mobile and can 
redistribute themselves if the mean distance between ions is changed. 
Nevertheless we can make some general statements about the shape of 
the interionic potential energy curve. 

First, there must be a minimum in the curve because the metal coheres 
and energy equal in magnitude to a binding energy is needed to evaporate 
it. We also observe that a metal resists great compression and we interpret 
this to mean that when the ions themselves begin to overlap the potential 
energy increases very rapidly, in much the same way as in a molecular 
solid. Thus the i^(r) curve must resemble Fig. 3.4 near the minimum and 
at small values of r the curve must rise very steeply. As we have seen 
(section 3.2) we can use almost any rapidly increasing algebraic function 
of r to represent this. 

Next we must discuss the i^{r) curve on the other side of the minimum. 
A metal also resists being stretched or expanded so that the curve must 
rise in a similar sort of way. Now when we dealt with molecular solids we 
represented the potential energy of the van der Waals attractions between 
neutral molecules by an r -6 law; there are sound theoretical reasons for 
doing this. By contrast, there is no simple expression of this type to 
represent the subtle interplay of attractions and repulsions between the 
ions and the mobile electrons. Nevertheless the screening effect causes the 
interionic forces to be of short range. Quite arbitrarily we will therefore 
adopt an r~ 6 law for the potential energy at large r in metals also. The 
justification for this procedure is that it allows us to reach results which 
have the right order of magnitude so that, arguing in reverse, we can say 
that there must be a fairly strong resemblance between the real i^(r) 
curve in metals, and Fig. 3.4. We will therefore use the Lennard-Jones 
6-12 potential energy for metals also. But it must be clearly understood 
that for metals, in contrast to molecular crystals, it is only a crude approxi- 
mation having no theoretical justification. 

Some data for potassium and mercury are collected in the Table below. 
The coordination number in metals is always high, about 10 or 12. From 
the surface tension of the liquid (potassium at high temperature, mercury 
at room temperature) and also from the latent heat the depth e of the 
potential well can be calculated. The two estimates agree tolerably well. 
The value of e is comparable with that for molecular solids. 

The bulk modulus does not agree very well with the ratio SL /V 
(eight times the binding energy divided by the molar volume ; see section 
3.5.1). For potassium there is a discrepancy by a factor of 5 and for other 
metals it can be 10. The failure of this rather sensitive test shows that a 
6-12 potential is not a good representation of the potential for some 



50 Interatomic potential energies Chap. 3 



Potassium Mercury 



Atomic weight 39 200 

Density (g/cm 3 ) 0.86 14 

Binding energy L (erg/g atom) 1 1 x 10 1 1 7.8 x 10 1 l 

Surface tension of liquid (dyn/cm) 364 465 

Bulk modulus (dyn/cm 2 ) 0.4 x 10 1 1 2.7 x 10 1 ' 



Atomic volume (cm 3 ) 45 14 

Diameter of ion (A) 4.2 2.8 

efromL (eV) 0.19 0.16 

e from surface tension (eV) 0.14 0.09 

8L /K, Eq. (3.16) (erg/cm 3 ) 2 x 10 1 * 4.5 x 10 1 1 

Einstein frequency (c/s) ~ 10 ~ 10 



metals. If instead of 6 and 12 for the indices we use p and q, Eq. (3.3), then 
following through the calculation suggested in section 3.6.3 it is not 
difficult to show that the compressibility should be given by pqL /9V 
(SL /V is a particular case when pq = 72). Presumably the data for some 
metals mean that the product pq is sometimes a good deal smaller than 
72 ; the repulsions might vary more slowly than r" 12 or the attractive part 
of the interatomic potential might be of longer range than r~ b . 

3.7.1 Departures from Hooke's law 

In section 3.5.1 we limited the consideration of the elastic moduli of a 
solid to small departures from the position of minimum potential energy. 
The changes of dimensions and the pressures acting on the solid were 
both assumed to be small; under these conditions the deformation is 
proportional to the pressure and the substance is said to obey Hooke's 
law. Conditions like these are the ones usually encountered. 

Now it must be emphasized that a solid may be perfectly elastic (in 
the sense that the body regains its original shape and size when all the 
forces are removed) and yet it may not obey Hooke's law. Indeed strictly 
speaking for any finite deformation, Hooke's law cannot hold. If from 
experiment the deformation is proportional to the pressure, this merely 
means that the deformation has not been measured accurately enough. 

We can calculate the relation between pressure and deformation 
without imposing the condition that we are always near the minimum of 
the potential energy curve. Recapitulating some of the equations of 
section 3.5.1 : pressure P = -(dE/dV) where E = \N\iV and TT is the 
pair potential. Therefore P = -^Nn(di^/dV). Though it is not a very 
good approximation, we will use the 6-12 potential for metals. It is 



3.7 Metals 51 

convenient to write it in the generalized form of Eq. (3.3) : 



^HUt)"-W 



and then to rewrite it in terms of volumes. Putting V = Na% for the 
initial volume and V = Nr 3 for the volume under pressure : 

H(3H*)" 

Substituting this in the equation for the pressure 

'--HSl-Hffl-fiS) 

We can write this more elegantly by using the fact that the compressibility 
at small pressures is given by K = 4Nne/V . Therefore 

Further, it is common practice to use the fractional change of volume 
(V— v o)/K as a measure of the deformation; it is called the strain and is 
denoted by s. Thus 

v = TTS- ^ 

Substituting this and expanding by the binomial theorem for small s, 

P = -K(s-%s 2 +f s 3 ). (3.25) 

This relation between pressure and strain should be obeyed by any solid 
with a 6-12 potential — molecular solids like argon, as well as metals. But 
metals have been extensively studied and can be prepared as specimens 
capable of undergoing large deformations, and more data exist for them 
than for any other class of material, so we will concentrate on them. 

Our expression suggests plotting the ratio P/K as a function of strain s. 
This is reasonable : the definition of bulk modulus, extrapolated naively, 
implies that a pressure equal to K would reduce the volume by a factor e, 
so we can take K as a unit of pressure. The curve is plotted in Fig. 3.14. 
Positive pressures, which cause compression, are plotted downwards and 
negative pressures (that is, tensions) are plotted upwards. Hooke's law 
then appears as a straight line at 45°. As we shall see shortly, the small range 
of strains (between ±4 %) covered by the graph encompasses an extremely 
wide range of conditions, far outside any encountered in ordinary 
engineering. 



52 



Interatomic potential energies Chap. 3 




Fig. 3.14. Stress/strain curve predicted for a solid with a 6-12 potential (full 
line). Hooke's law is the dotted line at 45°. Positive strains, to the right, are 
extensions ; negative strains are compressions. Pressures plotted downwards ; 
tensions upwards ; both measured in units of K. Crosses are measurements of 
the elongation of an iron whisker (Brenner, J. Appl. Phys. 27, 1484 (1956)). 
Compression measurements on iron by Bridgman {Proc. Am. Acad. Arts. Sci. 
11, 187 (1949)) agree at s = and s = -0.017, marked by an open circle; at 
intermediate stresses Bridgman gives the coefficient of s 2 as 6.1 where we have 
4.5. Explosion waves show that for extremely large strains the points lie 
above our curve ; this is suggested by the arrow. (Al'tschuler et al, Soviet Phys. 
J.E.T.P.l, 606 {195S)). 

The trend of the curve is reasonable. When a solid is compressed, the 
repulsive forces come into play and a given pressure produces less strain 
than predicted by Hooke's law. The opposite holds for stretching. 

Results for iron are shown at a number of points. These represent a 
range of techniques which is probably as wide as can be imagined. On 



3.8 Ionic crystals 53 

the left the compression measurements are taken from experiments in 
which enormous pressures, of the order of 30,000 atmospheres, were 
generated hydraulically using thickwalled vessels with ingeniously 
designed pistons. These experiments showed that for small strains the 
coefficient of s 2 is 6.1, where our simple theory gives 4.5, so that the points 
lie below our curve ; but then they move upwards and at about s — — 0.017 
the points lie on the predicted curve. Other experiments have also been 
done at far higher pressures, right off the graph — experiments in which 
high explosive was detonated on the face of a thick iron plate and the speed 
of the shock wave determined. Measurements at s = — 0.2 and beyond 
show that the points lie above our predicted curve ; the arrow attempts 
to indicate this. This result may mean that the real repulsive force cannot 
be represented by a simple power law like r~ 12 . 

In contrast to these massive techniques, the other half of the curve 
represents experiments performed under a microscope. It is a fact that if 
iron or any other ordinary metal is strained beyond about 0.1%, it breaks. 
But it is possible to prepare 'whiskers', that is thin threads of the metal 
which — for reasons which will be described in section 9.4.2 — are by 
comparison immensely strong. The one used in these experiments was 
1.6 x 10 ~ 4 cm in diameter and a few millimetres long. It was stretched by 
applying a force equal to the weight of about 10 g and the extension 
measured. Of course, such an experiment measures Young's modulus, not 
the bulk modulus, but as we have stated in section 3.5, these are almost 
the same. 

Our simple theory gives deviations from Hooke's law of the correct 
sign, and correct magnitude within a factor of 2, which is satisfactory. 

3.8 IONIC CRYSTALS 

Lattices like those of sodium chloride and lithium chloride consist of 
arrays of ions, each positively or negatively charged (Fig. 2.2). The forces 
and potentials between two ions therefore consist now of three compo- 
nents : the repulsive part, the van der Waals part (exactly as for neutral 
atoms) and in addition the electrostatic attraction or repulsion. This is 
given by Coulomb's law. If each ion carries a charge of magnitude e the 
potential energy = + e 2 /4ne r (like charges) or — e 2 /4ns r (unlike 
charges). The energy is measured in joules if e is expressed in Coulombs, 
r in metres and e , the permittivity of free space is given by 4ne = 10~ 9 /9 
farads/metre. The signs follow the convention of Fig. 3.1, to express the 
direction of the attraction or repulsion. We summarize this as 

,^ e 2 f Like charges + 

' (^Coulomb = i' 



4ns r ) Unlike charges 



54 Interatomic potential energies Chap. 3 

Thus the total potential energy between two ions is 



nr) = ~5± 



r p r b 4ns r 

where p is about 10 and k and n should be of the same order of magnitude 
as for neutral atoms. 

We will now show that the r -6 term, the van der Waals attraction, is 
negligible compared with the coulomb potential and can be discarded. 
If the charge cloud were spherically symmetric, the Coulomb force and 
potential at an external point would be the same as if its charge were 
concentrated at the centre. This is a consequence of the inverse square law. 
Thus the potential energy between two ions cannot be very different from 
that between an ion and an electron the same distance away. But we know 
that to pull an electron out of an atom to make an ion, we have to do an 
amount of work (equal to the ionization energy) of the order of 10 eV, 
that is 10 -11 erg. Therefore, since the diameter of an atom is of the same 
order as the interionic spacing in crystals, the potential energy of two 
ions must be of this order of magnitude. The van der Waals energies are of 
the order of 0.1 eV, one hundred times smaller. 

In Fig. 3.15 we have attempted to plot the potential energy of two ions 
of like sign and unlike sign as a function of their separation. Also shown is 




Xb) 

Fig. 3.15. (a) The potential energy of two ions of unlike charge as a function of the 

distance r between their centres. The dashed curve is the potential energy of two 

neutral atoms, Fig. 3.4, plotted on the same scale, (b) The potential energy of two 

ions of like charge, which is repulsive at all distances. 



3.8 Ionic crystals 55 

the potential energy of two neutral atoms, to the same scale. This em- 
phasizes in an obvious way that van der Waals energies can be neglected 
for ions. We can therefore write for two ions : 

y/ ~, A e 2 f Like charges + 
^■^| Unlike charges- . <326) 

3.8.1 The binding energy of sodium chloride. The Madelung sum. 

Figure 3.15 also emphasizes another point : that the Coulomb potential 
is long range (section 3.2). It is therefore no longer justifiable to deal 
only with nearest neighbour interactions. The Coulomb potential due to 
an ion can be felt far into the lattice, and another method of calculating 
binding energies is needed. 

To start with, consider the Coulomb energy only of a line of ions, each of 
charge e alternately positive and negative as in Fig. 3.16, which has been 
dissected out of a lattice like that in Fig. 2.2. 

Let the interionic spacing be r. Then the energy of one ion due to its 
two neighbours (which necessarily have the opposite sign to it) is 
— 2e 2 /4ne r. The next nearest neighbours, distance 2r away and necessarily 
having the same sign as the ion considered, give potential energy 
4- 2e 2 /4ns 2r ; and so on. 




Fig. 3.16. A line of ions dissected out of an ionic lattice 

Thus the potential energy of the single ion in the infinitely long line is 

-2e 2 



4ns r 



(l-i+W+"0. 



Note that 



so that 



log e (l + x) = x-y+y 



l-i+i-i--- = log c 2 = 0.69. 



Therefore the potential energy of the single ion in the line is - l.3$e 2 /4ne r. 
This holds for any ion, positive or negative, anywhere in the line. 



56 Interatomic potential energies Chap. 3 

The constant which we have just worked out is called the Madelung 
constant a for a line of ions. The Madelung constant for the three- 
dimensional sodium chloride lattice has been calculated: it is 1.75, which 
is not very different from our 1.38. Indeed, all Madelung sums for simple 
lattices of ions of alternate sign are of the order of unity. They must be 
so, since the effect of a positive ion is to some extent cancelled out by the 
next nearest negative ion and so on. Thus although the Coulomb potential 
is long range, our calculations are greatly simplified by being able to say 
that the potential energy of an ion in a lattice is given by the energy of a 
nearest neighbour pair times the Madelung constant which is of the order 
of unity. 

So far we have dealt with the Coulomb part of the potential. When it 
comes to the short range repulsions, the k/r p term in which p is about 10, 
we may guess that only nearest neighbours need be counted. The repulsive 
potential energy of a single ion is therefore nX/r p , where n is the coordina- 
tion number. 

The potential energy of a pair of ions in the crystal can therefore be 
conveniently written 



ir = 



4ne o a 



!0 _ «0 

p\rj \r 



(3.27) 



This is of the p-q type of Eq. (3.3) with q = 1. By comparison, the depth 
of the well 

\ pj4ns a 

In one gram ion of sodium chloride, there are N Na + ions and N Cl~ 
ions, that is there are N pairs of ions. Thus when the interatomic spacing 
is r, the energy is NV per gram ion; when it is a , the energy is —Ne. 
This gives the binding energy of the substance. 

Now the index p of the repulsive potential is about 10. So to about 10% 
accuracy, we may say that the binding energy of an ionic crystal is equal 
to e 2 /4ne a the potential energy of one pair of adjacent ions times the 
Madelung constant times Avogadro's number. The binding energy is 
dominated by the Coulomb energy. 

The binding energy for sodium chloride has been determined as 
763 kJ/mol. The interionic spacing has been determined by X-ray analysis 
as 2.8 A that is 2.8 x 10 ~ 10 m. The charge e, on each ion, is one electron 
charge, 1.6 x 10" 19 C. Therefore 

Note 2 

= 860 k J. 



4ne a 



Problems 57 

If we diminish this by 10% to allow for the factor (1 - 1/p), the binding 
energy agrees extremely well. 

3.8.2 Elasticity of ionic crystals 

We can calculate the bulk modulus of an ionic crystal using the same 
method as for molecular crystals. The volume occupied by N pairs of 
positive and negative ions is 

V = 2Nal 
and equation (3.13) becomes 

{d 2 E/dr 2 ) r=ao = N(d 2 r/dr 2 ) r = ao 
lSNa 18Na 

where 'V is the pair potential energy. The easiest way to evaluate the 
second differential is to quote the result of Eq. (3.22), that it is equal to 
pqe/al, with q = 1. This leads straightforwardly to the result 

PL 



K = 



9V 



where L is the binding energy and V the molar volume. This is a special 
case of the relation pointed out in 3.6.3, namely that if the indices in the 
interatomic potential are p and q, the bulk modulus depends on the 
product (pq); here q — 1, so we can measure p directly. 

For sodium chloride, the molecular weight is 58.5 and the density 
2.18 g/cm 3 so that the molar volume V is 27 cm 3 . The binding energy 
is 7.6 x 10 12 erg/mol, and the measured bulk modulus at low temperature 
is 3.0 x 10 1 1 dyn/cm 2 . Substituting, p is found to be 9.4, which is reasonable. 

The speed of sound, calculated from the bulk modulus and the density, 
is about 4 x 10 5 cm/s. The Einstein frequency v E , corresponding to a wave- 
length of about twice the interionic spacing, is therefore about 5 x 10 13 c/s. 
Since the propagation of such a wave means that ions (charges) are moving 
it can lead to the absorption of electromagnetic waves of this frequency. 
This frequency lies in the infrared. 

PROBLEMS 

3.1. For gravitational forces, which obey an inverse square law, a sphere of large 
radius behaves as if it were a point located at the centre. In this question, the 
object is to link up the definition of potential energy V with that of gravitational 
potential energy as usually defined in elementary treatments. 

Write Newton's constant as G, the mass of the Earth M, the radius of the 
Earth a. 

(a) Calculate the force on a mass m at a distance r from the centre of the Earth 
(r > a). Take r radially outward, note that r can only be positive. Get the 
sign of this force correct. 



58 Interatomic potential energies Chap. 3 

(b) Calculate the potential energy, taking the value at r = oo to be zero. 

(c) Draw a graph of this function between r = a and r = oo. 

(d) Calculate the force on m at the surface of the Earth and by equating it to the 
weight mg, get an expression for g at the surface. 

(e) Calculate the potential energy at r = (a + h) where h is small compared with 
a, so that the square of h/a can be neglected. Show that the potential energy 
at (a+h) is greater than that at a by an amount mgh. Mark this increase 
clearly on your graph. 

3.2. Two small magnets are arranged as shown. The lower one is fixed, the upper 
one is restrained from moving horizontally, but is free to move vertically. 



^ 



77777777777777777 

The force between them is a repulsion of magnitude (2/x M 2 /h 4 ) where M is 
the magnetic moment of each, p, the permeability of vacuum and h is the 
distance apart of the two magnets. 

(a) Write down the potential energy due to the repulsion. 

(b) Draw a graph of this as a function of h. 

(c) Draw a graph of the potential energy of the upper magnet as a function of h, 
due to the Earth's gravity. 

(d) Draw a graph of the total potential energy as a function of h. 

(e) Calculate h for static equilibrium and indicate this point on the graph. 

(/) Using the methods of section 3.6, calculate the period of oscillation after 

the upper magnet is given a small displacement downwards and then 

released. 

3.3. A sinusoidal wave y = a sin 2%x/A, of wavelength A and amplitude a disturbs 

the surface of a liquid of density p and surface tension y. The problem is to 

calculate the frequency and speed of propagation; see section 3.4.4 and Fig. 3.9. 

(a) Calculate the potential energy of a whole wavelength, due to the up and 
down displacements of the weights of the two halves, as follows. Show that 
the displaced mass of the half wavelength between x = and x = A/2 is 
palA/n, where / is the width of the wavefront. Prove that its centre of gravity 
is at height na/S and hence that the potential energy of a whole wavelength 
is \pa 2 lAg. Note : J sin 2 a da — ^{a-sin a cos a). 

(b) Write down an integral expressing the length of perimeter of a whole 
wavelength of the sinewave. Assume a/A is small, expand the integrand by 
the binomial theorem and integrate it. Show that the increase of surface 
energy is n 2 a 2 ly/A. The total potential energy is the sum of this and (a); 
note the proportionality to a 2 . 

(c) Assume (from comparison with a complete analysis) that the penetration 
depth is effectively A/47i so that the mass in motion is effectively A 2 pl/4n. 



Problems 59 



Hence show that the frequency is 



v = 1 / A g 2n y 

A\J In pA ' 

(d) The (phase) velocity is vA. Sketch it as a function of A. 

(e) For waves with A = 1 mm on liquid argon at 100°K, show that surface 
tension contributes 30 times as much to the energy as the weight. Calculate 
the frequency and velocity. Design an experimental set-up to measure the 
surface tension. 

3.4. In a rough demonstration experiment, liquid nitrogen was contained in a small 
thin-walled, spherical glass dewar with a narrow neck. This was connected to 
a gas-meter so as to measure the volume of nitrogen boiled off. The meter and 
the gas passing through it were at room temperature, 20°C. While 25 g of liquid 
evaporated, the meter registered 0.76 cu ft of gas (1 cu ft = 28.3 litres). What is 
the molecular weight of nitrogen? The dewar was known to hold 122 cc up to 
a mark on the narrow neck. It was estimated that about 1 cc was always occupied 
by bubbles. The flask was weighed empty and full up to the mark ; the liquid 
weighed 98 g. What is the density? the molar volume? the diameter of a molecule? 
A capillary tube of internal diameter 0.55 mm was dipped into the liquid. The 
capillary rise was 8mm. What is the surface tension, and depth e of potential well? 

3.5. Calculate some of the atomic constants of argon using the data of Fig. 3.13; 
see section 3.6.3. 

(a) From the density, Fig. 3.13(a), calculate the molar volume V and hence the 
diameter a of an argon atom, assuming each atom to be a little cube. 

(b) If the details of the argon crystal lattice are taken into account, see section 
8.1.2, it can be shown that the molar volume is not Na% but Nal/y/l. 
Calculate a better value of a Q than in (a) above. 

(c) Estimate the depth e of the potential well from the latent heat data of 
Fig. 3.13(6), taking the heat of sublimation at 70°K to be a sufficiently good 
measure of L (Eq. (3.6)). 

(d) Extrapolate the heat of sublimation to T = 0°K using the energy cycle 
described in section 3.6.3 and calculate a better value of e than in (c) above. 

(e) Estimate e from the surface tension data of Fig. 3.13(d), extrapolating y to 
T = 0°K. 

(/) Estimate the bulk modulus K at T = 0°K from the compressibility graph, 
Fig. 3.13(c). Compare this value with 8L /% (Eq. (3.16)). 

(g) Check Eq. (3.22). (Refer if necessary to section 8.4.1.) Hence show that if the 
indices of the interatomic potential energy are p and q instead of 6 and 12, 
the bulk modulus is given by 

Use your values of K, L and V to estimate the value of pq. 
(h) Estimate the Einstein frequency v E (Eq. (3.20)). 

3.6. The molecules of a complicated organic molecule can be considered to be 
disc-shaped, of radius rem and thickness r/lOcm. When two molecules are 
close together face to face (like two pennies one on top of the other) an energy 
E is required to separate them; when they are placed end to end (like two 
pennies touching one another, edge to edge) an energy JE/30 is required. For 
these molecules, which are far from spherically symmetrical, it is no use quoting 
formulae for L or y (Eqs. (3.6), (3.7)) for spherical molecules ; it is necessary to 
imagine how these molecules can be packed together and to tackle the problem 
ab initio. 



60 Interatomic potential energies Chap. 3 

(a) Estimate the molar heat of evaporation. 

(b) Estimate the surface tension. 

(c) Near the surface of a solid or liquid, will the molecules tend to be oriented 
(0 randomly, {if} with their planes parallel to the surface, (Hi) with their planes 
normal to the surface? Give reasons. 

3.7. The crystal of sodium fluoride is a cubic structure in which alternate sites are 
occupied by Na + and F" ions, each ion having 6 nearest neighbours. The 
Madelung constant is 1.75 when referred to the smallest Na + -F~ separation, 
which will be denoted by a . The heat of formation of the crystal from its 
constituent ions is 900 kJ per mole of sodium fluoride. The density of crystalline 
sodium fluoride is 2.9 g cm -3 , and the atomic weights of sodium and fluorine 
are 23 and 19. Estimate a and Avogadro's number from the data. State clearly 
any approximations you are using in your calculations. 

3.8. The interaction between ions in sodium chloride can be described by their 
Coulomb interaction, plus a repulsive potential energy Aexp(-rfp) acting 
between nearest neighbours only. (This exponential term is used in place of the 
r" 12 term; A and p are constants). Obtain an expression for the lattice energy 
in terms of the nearest-neighbour separation a and the Madelung constant 
a for sodium chloride. Given a = 2.8 A, a = 1.75, and that the lattice energy 
is 763 kJ/mol, find p. 

3.9. In a medium of dielectric constant K, the potential energy of two charges e x 
and e 2 separated by distance r is e 1 e 2 /4n£ Kr. 

Water H 2 0, ethyl alcohol C 2 H 5 OH and ammonia NH 3 have dielectric 
constants of 80, 25 and 18 respectively at room temperature. These high values 
are due to the fact that the molecules are electric dipoles, with regions of + and 
— charge, which can be easily aligned by external fields ; they can crowd round 
a charged particle with their oppositely-charged ends all pointing towards it 
and so screen the particle from its neighbours. H 2 and NH 3 are compact mole- 
cules, C 2 H 5 OH is relatively large. 

(a) Use the data for the binding energy of sodium chloride, section 3.8.1, to 
calculate the binding energy of a single sodium chloride molecule. 

(b) Explain qualitatively the fact that the solubility of sodium chloride in 100 g 
of solvent is 37 g in water, 0.07 g in alcohol, 3 g in liquid ammonia ; the 
dissolved sodium chloride exists as Na + and Cl~~ ions. 

3.10. Calculate the Madelung constant for a line of dipoles, (a) all aligned in the same 
direction so that they repel one another with anr" 4 force, and (b) alternately 
parallel and antiparallel so that nearest neighbours attract, next nearest neigh- 
bours repel and so on. Evaluate the series numerically or by looking up Riemann 
zeta functions in Dwight's 'Mathematical Tables' or 'Tables of Integrals'. 

3.11. The molecular weight M, latent heat of evaporation L, and surface tension 
y at 20°C, of certain liquids are given in the table. Make a crude estimate of the 
molecular diameters in each case stating what approximations are made. 





M 


LJ/gm 


y dyn/cm 


Alcohol 


46 


856 


22 


Benzene 


78 


389 


29 


Mercury 


200 


272 


475 


Water 


18 


2,250 


73 



Problems 61 

3.12. (i) Prove that for a substance whose interatomic potential energy is given by 
Eq. (3.3), the relation between pressure P (applied hydrostatically) and 
volume v is 



where K is the bulk modulus at very small strains, v is the initial volume, 
and a and /? are equal to (p+3)/3 and (q + 3)/3 respectively. Hence show 
that for small strains s less than about 0.1 

a ,-E+p±* + ...\. 



(ii) For an ionic crystal, p is about 11 and q = 1. Compare this simple theory 
with measurements by Bridgman on the compression of sodium chloride. 



p 

Kg/cm 2 


s 


P 
Kg/cm 2 


s 


lxlO 4 

2 
3 


-0.038 
-0.068 
-0.093 


4xl0 4 
6 
8 
10 


-0.115 
-0.152 
-0.183 
-0.210 







One method of deducing the bulk modulus for very small strains is to note 
that on almost any theory P = —K(s — ys 2 ) where y is a constant, so that 
a graph of P/s against s should be a straight line from which K can be 
deduced. Use the measurements in the first table for this. 
(Hi) Convert the data of the next table into (P/K) against (v /v) raised to the 
appropriate powers, compare calculated and measured pressures and 
comment on the results. 
3.13. What conclusions can you draw about the natural vibrational frequencies of 
diamond, iron and lead from the following data (Y, M and p are respectively: 
Young's modulus in dyn/cm 2 , the atomic weight, and the density in g/cm 3 ) : 



M 



Diamond 8.4 xlO 12 12 3.5 

Iron 2.0 xlO 12 56 7.9 

Lead 0.18 xlO 12 208 11.4 



CHAPTER 




Energy, temperature and the 
Boltzmann distribution 



4.1 HEAT AND ENERGY 

We will take it for granted that heat is a form of energy. This statement 
is based on the experiments of Joule, the majority of which were of the 
same basic pattern. Weights held on strings could descend and so provide 
mechanical energy to drive a mechanism whose motion was resisted by 
friction of some kind and which grew hotter as it was driven. The change 
in mechanical energy was measured by the loss in potential energy of the 
weights, and the quantity of heat produced was measured in terms of the 
rise in temperature of the apparatus and its heat capacity. The mechanisms 
were very varied : a dynamo dissipating its energy in a resistance, a per- 
forated piston moving through viscous liquids, a conical bearing with 
friction between the rubbing surfaces, a system of paddles churning viscous 
liquids. In another investigation, air was compressed by a pump into a 
cylinder and the temperature rise was measured — here no frictional force 
was encountered during the compression of the gas, but work had to be 
done against the pressure it exerted. In all these experiments, the conver- 
sion factor relating the energy absorbed by the mechanism and the heat 
produced in it was the same within rough limits, ± 15 %. Since the mech- 
anisms were so diverse in type, it was unreasonable to suggest that this 
rough constancy could be a property of the substances or devices employed ; 
it could only be explained if heat and energy were physically identical. 



4.1 Heat and energy 63 

4.1.1 Ordered and random movements of molecules 

It has already been mentioned that the molecules of any substance are 
in ceaseless, rapid motion. The molecules of a gas move in straight 
trajectories till they collide, the molecules of a solid are in vibration about 
their mean positions, those in a liquid vibrate and also slip through the 
holes in the structure. These motions are random, in the sense that 
movement in one direction is just as likely as movement in any other, and 
also in the sense that any molecule changes its speed many times per 
second so that if we were to follow all the details of the motion we would 
find that the kinetic energy went irregularly through all possible values, 
from zero up to some large value. 

Consider a body which is big enough to contain a large number of 
molecules (though it might be of dimensions which are small on the 
ordinary scale) and let it be at rest. Then the total momentum of the 
molecules must be zero. If, however, we examined the momentum of one 
single molecule at any instant, we would almost certainly find it to be large. 
It is only by finding the vector sum of all the momenta or finding the 
average momentum of a very large number of molecules at any instant, or 
alternatively by finding the average momentum of one molecule over a 
long period of time, that we can come to any conclusion about the move- 
ment of the body as a whole. In the same way, we can consider a region of 
a body which is moving with a certain velocity. If we were to determine the 
instantaneous velocity of any one molecule we might well find that it was 
moving very fast in the opposite direction to the bulk motion, and we could 
deduce nothing about that bulk motion. But if we averaged the momentum 
of a large number of molecules in the region at one particular instant, or 
averaged the momentum of one molecule over a long time, we would detect 
a nett momentum, corresponding to the drifting of the body as a whole. 

It is important to point out that the statements just made are in fact a 
little too dogmatic. If we found the vector sum of the momenta of all the 
molecules in a body, it would be unlikely to be exactly zero. It would fluctu- 
ate about the value zero. For the averaging to have any physical signifi- 
cance, these fluctuations must be relatively small. We will see later (in section 
7.7.2) that this condition is satisfied if the number of particles in the assem- 
bly is large, and it will be assumed in the rest of this chapter that this is so. 

We are thus led to distinguish between the random movements of the 
molecules of a body (which add up vectorially, and hence average out 
to zero) and the movement of the body as a whole. The random move- 
ments are superimposed on the bulk or ordered movement, and the two 
can only be separated by averaging. An example of ordered motion is the 
macroscopic movement of any body such as a ball. A less obvious example 
is the flow of a liquid, either streamlined or turbulent, where a bulk velocity 



64 Energy, temperature and the Boltzmann distribution Chap. 4 

can be defined over each small region of the fluid. Another example is 
provided by waves of compression or rarefaction passing through any 
medium, or torsional waves through a solid, when the velocity and dis- 
placement due to the wave can be defined. The passage of an electro- 
magnetic wave through a transparent medium may also cause the atoms 
to vibrate in an ordered way. But in all these examples, the ordered move- 
ments have the random movements of the molecules superimposed on them. 

4.1.2 Temperature and random motion 

We can now extend the statement that heat is a form of energy and 
make it more precise, as follows : Temperature is a measure of the energy 
of the random motion of the molecules of a substance. (Temperature is 
not the only measure of this energy, but we will not pursue this fact here.) 
This statement is ultimately based on Joule's experiments. Later in this 
chapter we will show how the form of the relation can be deduced. 

A molecule in general possesses both kinetic and potential energy. 
Sometimes the kinetic energy dominates the situation, as in a gas of low 
density where the potential energy arising from close collisions can often 
be neglected. On the other hand, a solid owes its regularity of structure 
to the potential energy of interaction of an ion with its neighbours and as 
the kinetic energy merely leads to vibrations of the ions about their mean 
positions, the random variation of the potential energy is important. 

It must be emphasized that it is only the randomly varying contribution 
to the energy which is related to the temperature. Any ordered movements 
must be transformed away by a suitable choice of coordinate system. 
However, it is possible for ordered molecular movements to be converted 
into random ones. In fact, this is just what happens whenever mechanical 
energy is converted into heat. In Joule's experiments where liquids were 
stirred, the movements of the weights were converted by the mechanisms 
into bulk movements of the liquids — the surface waves, the eddies. But 
all the time, part of the energy of the ordered motion of the liquids was 
being converted or degraded by molecular collisions into random move- 
ments of the molecules and this caused the rise in temperature. In his 
experiments where gases were compressed by a piston, the ordered motion 
of the layer of gas being pushed back by the movement of the piston 
was transferred to a random movement of molecules throughout the gas 
so that the energy of the piston caused a rise in temperature. At the same 
time, the molecules of the piston were also heated. 

4.1.3 Degradation of ordered into random motion 

To see how this conversion or degradation is carried out, we consider 
first a simple example. Let a single molecule be projected into a layer of 



4.1 Heat and energy 



65 



gas (Fig. 4.1), with a velocity much higher than the random velocities of 
the molecules of the gas. In principle, we can imagine that this energy 
could be measured by some simple mechanical device, since only one 
particle is involved. The molecule must soon undergo a collision in which 
the laws of conservation of energy and momentum are obeyed. A propor- 
tion of its energy is transferred to the other molecule, both are deflected 
and the first continues at a slightly lower velocity. It undergoes a large 
number of such collisions, each time transferring some of its energy and 
deflecting the molecules of the gas till finally it emerges with a smaller 
kinetic energy than it started with ; in principle this loss of energy can be 
measured by mechanical means. Each molecule struck undergoes further 
collisions, at different glancing angles, with changes of direction each time, 
so that after quite a short time the excess of energy is carried to distant 
parts of the gas and is shared between all the molecules as extra kinetic 
energy of motion in all directions. The energy of the molecules is increased 
by an amount equal to the difference between E and £', but in macro- 
scopic terms we merely say that the temperature has increased. 



O 



O 



o 





o 


o 

o 
o 


o 




o 


o 


o 


\o 


o 




o 



o 

o 

o 

o 



Fig. 4.1. Passage of a single fast molecule of initial energy E through a layer 
of gas. It emerges with a lower energy E'. 



We can now consider a gas contained inside a cylinder fitted with a 
piston (Fig. 4.2). Let the area of cross section be A and the pressure of the 
gas P, and imagine the piston to be moved inwards a distance dx. This 
requires the expenditure of an amount of work PA • dx. But each molecule 



66 Energy, temperature and the Boltzmann distribution Chap. 4 

near the piston acquires an excess velocity and now plays the role of the 
single incident molecule in Fig. 4.1. Once again, the excess energy supplied 
by the moving piston is, after a large number of collisions, shared between 
all the molecules as a purely random movement, as a rise in temperature. 



Area/4 




Fig. 4.2. Gas contained in a cylinder with 
piston. When the piston moves the ordered 
motion of the molecules near it is ran- 
domized by collisions. 



If a solid is struck, the energy of deformation will at first probably be in 
the form of a compression wave travelling through it, so that it rings with 
a frequency of a few hundred or thousand cycles per second. This highly 
ordered motion can also be degraded into random motion, though the 
process is more complicated than in gases, but the final result is that the 
energy of the original disturbance is degraded into vibrations in all direc- 
tions and with all frequencies. Each vibrating molecule has both kinetic 
and potential energy at any instant (the potential energy between any pair 
of molecules being given by Fig. 3.4). Each of them separately varies 
randomly with time, and so does the sum of the two, but it is this total 
energy which we use to measure the temperature. If liquids are stirred, the 
molecules can'vibrate and also change their positions by slipping past one 
another. Again some of the energy of the ordered movement can be de- 
graded into increases of random molecular movement constituting rises 
in temperature. 

In all these examples, where work is done on a system and energy is 
apparently lost but is in fact divided up among a large number of particles, 
a thermometer would indicate a rise in temperature. Temperature is one 
of the measures of this randomized energy. 

4.1.4 Macroscopic variables and statistical specifications 

In practice, measuring instruments which we use for ordinary kinds of 
physical measurements are relatively massive, slowly-responding devices 
which are incapable of detecting the effect of individual molecules. Thus 
when we measure the pressure of a gas using an ordinary sort of pressure 
gauge, a mercury manometer or a diaphragm actuating a lever, the gauge 



4.2 Concepts of probability theory— I. Probability functions 67 

measures a time-average of the pressure. It does not register the individual 
impacts of the molecules, which occur with extremely high frequency. 
Similarly, when we measure the density of a gas we normally use a volume 
whose dimensions are very large compared with the distances between 
molecules. Figure 2.3, which pictures the molecules in a gas unevenly 
distributed throughout a volume, implies that the density fluctuates from 
place to place, but these fluctuations are never detected by a measurement 
using a large volume. Thermometers, similarly, measure time-averages of 
molecular energies. It is in fact possible to detect fluctuations in pressure, or 
density or energy under the right conditions using appropriate detectors, 
but for the moment we will restrict our considerations to ordinary instru- 
ments which measure average values of molecular quantities. 

The contrast between the complexity of the situation from the atomic 
point of view, and the simplicity of the large-scale measurements is a 
profound one. The one requires a knowledge of say 10 23 positions and 
velocities, the other a single dial reading. It is our task to relate these points 
of view and it is suggested at once that the elaborate specification of all 
the positions and velocities of the constituent molecules is not only 
impossibly complicated, but is actually irrelevant. 

We need only concern ourselves with those features of the assembly 
of molecules which allow us to calculate averages of molecular quantities. 
It turns out that the most we ever need to specify is the fraction of the total 
number of molecules to be found within certain limits of position or having 
speeds within a certain range. We need to know, for example, that 1 % 
of the molecules of a liquid of molecular weight 30 at 300°K have speeds 
between 10,000 and 10,320 cm/s ; it is not necessary to enumerate which 
molecules they are at any instant nor exactly where they are to be found. 
Specifications of this kind are all that is necessary to relate the macroscopic 
properties to the motions of the molecules. One often meets rather, similar 
statements about, say, the economic state of a nation. For many purposes, 
it is sufficient to enumerate what percentage of the population has earning- 
power within certain limits; the specification of the individuals does not 
matter. Statements of this kind are statistical, in the sense that they deal 
with percentages or probabilities but do not specify individuals. We 
must therefore study some of the concepts and theorems of probability 
theory. 



4.2 CONCEPTS OF PROBABILITY THEORY— I. PROBABILITY 
FUNCTIONS 

We will study a statistical problem which has nothing to do with physics. 
Consider a large population of people, several million in number, out of 



68 Energy, temperature and the Boltzmann distribution Chap. 4 

which we select a small sample of 100. Imagine that the height of each of 
these individuals has been measured. One good way of displaying the infor- 
mation is to plot a histogram which we construct as follows. We divide the 
total possible range of heights into convenient intervals or ranges (say 
2 cm wide), choosing them so that there is no ambiguity about classifying 
an individual whose height is at the end of a range. (For example, if the 
accuracy of measurement were .01 cm, we might choose ranges 160- 
161.99 cm, 162-163.99 cm and so on.) Then we count up the number of 
individuals in each range and plot them as a graph. The result might look 
like Fig. 4.3(a). The number in each range is drawn as a horizontal line 
covering the interval. From this diagram we can at once see that the 
population contains many members around 175 cm tall, and that very 
short and very tall members are rare. 

The histogram is not a smooth curve. Of course it consists of a series of 
steps because we are dealing with intervals of finite width — but sometimes 
the steps go up when they might be expected to go down. Intuitively one 
expects that the distribution of heights should be a smoothly varying 
function, but it is more or less obvious that the irregularities are present 
because the number of individuals in any range is small. For example, if 
the 100 people had by chance included only one more member in the 
range 166-168 cm and one less in the next lower range, then one of the 
dips would have disappeared. It will be shown in section 7.7.3 that in 
many circumstances, if the number of individuals in any class is most 
probably n, then quite probably the number might lie in the range between 
(n + Jn) and (n-Jn). For example, if the number of individuals might be 
expected to be 10, then (using different samples from the same population) 
we would probably find counts anywhere between about 7 and 13. If we 
expected 100 in any range, then counts between 90 and 110 would be 
common, and so on. The numbers are said to fluctuate between certain 
limits. Now it is obvious that as n becomes larger, Jn becomes larger too, 
but more slowly ; so the fluctuations become proportionally smaller, the 
larger the numbers are. For example, with 10 members in a range, the 
expected fluctuation is about ±30%; with 100 members it is ±10%; 
with 10 6 members it would be only ±0.1 % and so on. 

We will in future assume that the sample of the population which we 
take is very large— so large that the number in any range is itself so large 
that we can neglect fluctuations. We will then get a smoothly stepped 
histogram like Fig. 43(b), which refers to a sample of 10 6 people. 

The number in any range depends of course on the total number in the 
sample, being proportional to it. For many purposes (for example, com- 
paring samples of different size) this is inconvenient and it is better to refer 
to some standard number for the total sample. The best choice is 1. 



4.2 Concepts of probability theory — I. Probability functions 



69 



To convert Fig. 43(b) to this standard, all the ordinates of Fig. 4.3(b) 
have simply to be divided by 10 6 . If we take the ordinates of all the steps 
in Fig. 43(b) they add up to 10 6 ; in Fig. 4.3(c) they add up to 1. The histo- 
gram is said to have been normalized to a total population of 1. 



15 
| 10 

0) 

f 5 



'Ha) 



A 



■J* k 



160 170 180 190 
Height h 




0.15 



0.10 



0.05 



160 170 180 190 h 




160 170 180 1' 



160 170 180 190 h 



Fig. 4.3. (a) Typical histogram showing distribution of heights of 100 people. 
(b) Using scale at left : Expected distribution with 1 million people, (c) Using 
scale at right : Probability of a person's height being within 2 cm range of h. 
(d) Using scale at left : Probability of a person's height being within \ cm 
range of h. (e) Using scale at right : Same data, but referring to 1 cm ranges oih. 
(f) Probability function P[h], identical with (e) but using infinitesimally small 

steps. 



These ordinates are now called probabilities. From Fig. 43(b) we can 
read that if we take 10 6 individuals, there would probably be 80,000 whose 
height lay between 168 and 170 cm. But we can extend this statement by 
referring to Fig. 4.3(c) and saying that if out of the population we selected 
one single individual at random, the probability that his height lay between 
these limits would be 0.08. These two statements are equivalent to each 
other. 

This notion of probability is replete with philosophical difficulties 
which must be faced if a full understanding of many branches of physics is 



70 Energy, temperature and the Boltzmann distribution Chap. 4 

to be reached. But from our present point of view we can regard the 
probabilities of Fig. 4.3(c) as being merely a convenient shorthand for 
deriving the numbers of Fig. 4.3(b); the number in any range is the proba- 
bility of being within that range multiplied by the total number in the 
sample. 

We can now ask : what happens to the probability histograms if we 
reduce the size of the intervals — if for example we choose ranges of width 
\ cm instead of 2 cm. First, the steps now get narrower so that the histo- 
gram approximates better to a smooth curve. Secondly, the probabilities 
of lying within each range must now decrease, being proportional to the 
width of the ranges. (We must emphasize that the size of sample from which 
the histogram is derived must be so large that the numbers in any range 
still remain large or else the large fluctuations will reappear.) The curve 
for \ cm intervals is given in Fig. 4.3(d). But we can go further. It is possible 
to plot just the same results in a more useful way, so that the ordinates 
are independent of the width of the ranges. We can still do the counting 
and work out the data at \ cm intervals, but we work out the probability 
of finding an individual within ranges of unit width— conventionally, 
of width 1 cm. Here, one has to multiply the probabilities of Fig. 43(d) 
by 2, because the standard ranges are twice as wide as the one chosen in the 
counting. We then get Fig. 4.3(e). Of course we could get practically the 
same figures by taking Fig. 4.3(c) and dividing those ordinates by 2, because 
the standard range is only half as wide as the 2 cm ranges used there ; this 
would merely give a more crudely stepped histogram. Finally, we can 
take the limiting case of imagining the intervals to be infinitesimally wide 
but still calculating the probabilities for unit range of heights, Fig. 4.3(/). 
The only difference between this and Fig. 4.3(e) is that it is a smooth curve. 
It is called 'the probability function of h\ a phrase which is written P[h].* 
The equation for the curve of Fig. 4.3(/) is 

P[h] = 0.067905 exp[-(/i- 175) 2 /69.031]. (4.1) 

The value of this function at h = 185 is 0.01597 ; thus the probability that a 
member of the population has a height between 184.9 and 185.1 cm is 
P[h] dh = 0.003194. If there were 10,000 people in a sample, the number 
of people whose heights lay between these limits would most likely be 32. 
The properties of the probability function P[h] may be summarized 
as follows. The probability that an individual has a height within the range 
dh about h is P[h] dh. In a population of n people, the number with 

* This symbol must not be thought to imply the same analytical function of the variable 
every time. For example, we will later meet two probability functions P[u] and P[c] where 
u and c are respectively a velocity component and the total speed of a molecule ; the two are 
different in analytical form from one another. 



4.2 Concepts of probability theory — I. Probability functions 71 

heights within these limits is nP[h] dh. Finally, the fact that the graph has 
been normalized to a total population of 1 means that by convention 



f 

Jo 



P[h] dh = 1 (4.2) 



that is, the area under the graph is unity. In ordinary language, a proba- 
bility of unity is called certainty, and this equation says that it is certain 
that the height of any individual lies between zero and infinity, which is 
correct. 

One further point must be mentioned. In many situations where we 
consider the statistical distributions of the properties of molecules, we 
can take two quite different points of view. The first is similar to the one 
we have just considered. We can look at the assembly of molecules at 
any one instant and find the distribution of the relevant quantities (position 
or momentum coordinates) over them, in just the same way as we did for 
the heights in a population. But the second procedure has no analogue 
in the counting of people. It is to follow a single molecule for a long time ; 
its coordinates change continually, and we will find that they take on all 
possible values. For example, the speed of a molecule may be high or low, 
in any direction, at one time or another. We can then, in principle, find 
the fraction of the total time that the molecule spends in a given state. 
It is a fundamental assumption that these two kinds of distribution are 
identical. Thus if 1 % of the molecules of a certain liquid have speeds 
between 10,000 and 10,320 cm/sec (at any one instant), then one single 
molecule will, for 1 % of its time, be travelling with a speed between the 
same limits. When we come to take averages, we will in other words 
assume that sample averages are identical with time averages. 

4.2.1 Mean values 

The mean value of the height of an individual in a population is defined 
by 

, . , , total height of all members 

mean height h = — 

total number of members 

_ sum of terms : (height h) x (number with he ight h, dh) 
total number of members 



that is : 



C hP[K\ dh r 
J n = J [/l] dH (43a) 



72 Energy, temperature and the Boltzmann distribution Chap. 4 

where the integration must be carried out over all possible values of h. 
For the population of Fig. 4.3, the mean height is 

/•oo 

h = 0.067905 h exp[ - (h - 1 75) 2 /69.03 1] dh, 

J — 00 

where the range of h has to be taken from - oo to oo even though that is 
unrealistic in practice, the integrand being practically zero over most of 
the range. 

For finding mean values, both in this example and for the functions 
which will occur later in physical problems, the following integrals will 
be found useful : 



j V " 2d * = ^ J" 



V a 



» oo j 

xe~ a * 2 dx = — 

Jo 2a 


/» 00 

xe~ a * 2 dx = 

v — oo 


jVe—d*-^ 


j; b a- d,-i^ 


/•OO 1 

x 3 e" ax2 dx = x i 2 

Jo 2a 2 


/•oo 

x 3 e _ax2 dx = 

•/ — oo 



With these integrals it is not difficult to prove that the mean height in the 
above example is 175 cm. 

In general, the mean value of a quantity x is given by 



fxP[x]dx. (4.3b) 



In the same way we can also find the mean value of any power of x. 
Later it will be seen that mean square values of certain quantities are 
significant : 



= fx 2 P[x]dx. (4.3c) 



This is not in general identical with x 2 ; there is no reason why Jx 2 P[x] dx 
should be equal to (j"xP[x] dx) 2 . 



4.2 Concepts of probability theory — I. Probability functions 73 

4.2.2 Independent probabilities 

Consider now a second characteristic of each member of the same 
population which is quite independent of the first. As an example which 
we will assume to be independent of height, consider the marks m gained 
in any particular examination (and let us assume that m is a continuous 
variable). If h and m were not independent, we might find that the examina- 
tion score was proportional to the height of the examinee or some such 
relation which we will assume does not hold. We can again measure the 
probability that any individual scores between m and (m + dm). Let us 
write this 

P[m] dm 

where P[m] is the probability function of m, perhaps a different function 
from P[h\ 

Given these two characteristics h and m, we can ask what is the probabili- 
ty that a given individual has a height between h and (h + dh) and also 
scores between m and (m + dm) — a probability which we can write 

P[h, m] dh dm, 

where the notation P[h, m] means a function of the two independent 
variables h and m. We could find this from the data about the population 
by drawing h and m axes (Fig. 4.4) and representing each individual by a 
point with his coordinates. Then we could divide up the area into rec- 
tangular cells of size dh by dm, and we could count the number of points 
inside the appropriate cell ; finally we would divide this number by the 
total population n. In Fig. 4.4, only a few points have been drawn in the 
cell, but we must assume that the number is really so large that P can be 
considered as a continuous function of h and m. 

We can also relate P[h, m] to P[h] and P[m\ For if we choose any person 
at random from the population the probability that his height is within 
the desired range is P[h] dh. At the same time, the probability that his 
score is within the desired range is P[m] dm. Thus 

P[h, m] dh dm = P[h]P[m] dh dm. (4.4) 

This important equation expresses the fact that for any two character- 
istics which are entirely independent of one another, the probability of their 
happening together is the product of their separate probabilities. 

Familiar examples of this occur in games of chance. For example, a 
die has six faces which are equal except for the markings and the probability 
of throwing a given number, say a one, is therefore £. If two dice are 



74 



Energy, temperature and the Boltzmann distribution Chap. 4 



thrown, the numbers which turn up are independent of one another, so 
that the probability of throwing two ones is (£) 2 . 

Further aspects of probability theory, notably fluctuations, are con- 
sidered in section 7.7. 






d/7 



height h 



Fig. 4.4. Counting the number of individuals whose height is between h 
and (h+dh) and score between m and (m + dm) in an examination. 



4.3 THERMAL EQUILIBRIUM 

We will in what follows consider systems which are in thermal equilib- 
rium ; we must try to describe what we mean by this. 

From the macroscopic point of view, or from the practical point of 
view, equilibrium is not too difficult to define even though it is seldom, 
if ever, strictly attained. The essential conditions are that the temperature 
must not change with time, however long the system is left, and that the 
temperature should be uniform throughout the system. 

These conditions imply other restrictions. It follows from our definition 
that there can be no flux or current of heat through the system — for this 
would produce nonuniformities of temperature. Thus a bar with heat 
fed into one end and extracted at the other may reach a steady state when 



4.3 Thermal equilibrium 75 

temperatures do not change with time, but this is not equilibrium because 
temperatures are not the same everywhere. It follows also from our 
definition that there cannot be any current or bulk movements of particles 
through the system, because these would give rise to differences of tem- 
perature. Thus a closed vessel partly filled with a liquid is not initially in 
thermal equilibrium, because the liquid must partly evaporate so that 
there must be a bulk movement of molecules upwards, driven by a minute 
temperature difference between liquid and vapour. Only when the space 
is saturated with vapour and the whole system has settled down to exactly 
uniform temperature will it be in equilibrium. It follows also from our 
definition that no system can be in equilibrium when it undergoes variable 
acceleration, or when dissipative forces are acting within it. 

The simple definition of equilibrium is therefore a restrictive one if 
strictly applied. The atmosphere of the Earth, for example, is not in 
equilibrium because molecules are escaping from the top of it and because 
it is absorbing radiation from the sun. Indeed our whole universe is 
able to function because it is not in equilibrium. But we might take a 
measurement on some property of a small part of the Earth's atmosphere, 
and if the changes which took place in it during the time occupied by the 
measurement were small enough, we could forget that the air was not in 
strict equilibrium. In a similar way, we can look again at the liquid 
evaporating into a closed space — strictly speaking, such a system only 
approaches equilibrium asymptotically and never actually attains it. 
But in practice we need often only wait a short time — minutes, say, or 
days — for the greater part of the change to have taken place. Subsequent 
changes are likely to be so slow that during the course of any experiment 
they are negligible. In such cases, equilibrium has for all practical purposes 
been reached. 

From the practical point of view, then, equilibrium is not too difficult 
an idea to describe. From the molecular point of view, however, all the 
simplicity of definition evaporates and the situation becomes one of the 
most elusive to describe. Because of fluctuations, the density in a small 
volume of gas inside a 'uniform' atmosphere continually changes so that 
on the atomic scale there are still random fluxes of particles. At the inter- 
face of a liquid and a vapour, seemingly quiescent and unchanging on the 
macroscopic scale, molecules are jumping out and returning all the time. 
A solid rod is continually expanding and contracting in length, imper- 
fections in the lattice are continually moving about. Let us for the moment 
then adopt a practical definition of thermal equilibrium and hope that 
by the time he has reached the end of this book the reader will have a 
deeper understanding of the meaning of thermal equilibrium at the 
molecular level. 



76 Energy, temperature and the Boltzmann distribution Chap. 4 

4.3.1 The Boltzmann law 

Let us now return to the problem of specifying the state of an assembly 
of a large number of molecules. We will assume that the assembly as a 
whole is at rest, but the molecules have random thermal motion. 

In classical mechanics, the state of a molecule must be specified by its 
coordinates of position, x, y and z, and its components of momentum 
p x , p y , p z parallel to the three axes. In the ordinary cartesian system, we 
might as well use velocities instead of momenta (they are related by simple 
equations of the type p x = mv x ). But since position coordinates and 
momenta are the conjugate variables used in the Hamiltonian dynamical 
equations which arc valid in all systems of coordinates, we will use 
momenta here. 

Let us now concentrate on one single molecule in an assembly in a 
container maintained at a temperature T. It does not matter whai: 
molecules are being examined, nor what the container is, as long as it 
functions as a thermostat which is set to maintain the temperature T 
constant. In fact, we can take an assembly of the same molecules — a 
portion of a solid embedded in the middle of a larger lump of the solid, 
for example — considering the whole mass acting as the thermostat for 
the smaller portion. It is one molecule in this assembly that we concentrate 
on. 

When a system is in thermal equilibrium, the coordinates and momenta 
of this one molecule are independent of one another. This means that 
wherever a molecule is located, its momentum may be of any magnitude 
in any direction. Any interdependence between the coordinates, such as 
the condition that fast molecules can only be found near the origin, is 
ruled out. Thus, six numbers or parameters are required to specify the 
state of one molecule. 

For simplicity, let us deal first with p x , the component of the momentum 
parallel to the x-axis, of the one molecule whose random thermal motion 
we are considering. 

The energy of this molecule is in general the sum of contributions from 
all the position coordinates and components of momentum, but the part 
of the energy contributed by p x is 

^(p x ) = ~, (4-5) 

2m 

where Jf stands for kinetic energy and the notation Jf(p x ) emphasizes the 
fact that Jf depends upon p x ; m is the mass of the particle and pl/2m 
might have been written \mv 2 x but we have chosen to use the more general 
formulation. 



4.3 Thermal equilibrium 77 

Then Boltzmann's law states that the probability that, in thermal 
equilibrium, the component of the momentum lies between p x and 
(Px + dp x )isgivenby 

Ac- jriPx)lkT dp x (4.6a) 

or in other words the probability function for this one component of 
momentum is 

Ac -*ip x )ikT t ( 46b ) 

This law is stated at this stage without derivation. The exponential 
factor itself is called the Boltzmann factor. 

In these expressions, A is a quantity which has to be chosen so that the 
integral of the probability over the whole possible range of p x is 1 — the 
probabilities have to be normalized. 

The quantity T is called the absolute temperature, and these equations 
define what we mean by absolute temperature. Later we will see that it is 
identical with the 'perfect gas scale' temperature (section 4.4. 1). 

The quantity k is a constant, called 'Boltzmann's constant', having the 
units J/degree, whose magnitude we will work out in section 4.4.1. 

This statement of Boltzmann's law is fairly general, but we will now 
write down an explicit example of a Boltzmann factor and use it to 
point out a number of features of Boltzmann's law. We choose cartesian 
coordinates, when the Boltzmann factor for the momentum component 
p x is 

Q -pH2mkT 

Let us see what the form of this factor implies — in particular we will show 
that it is consistent with the intuitive ideas of temperature already des- 
cribed. Firstly, it implies that the most probable value of p x (the x-com- 
ponent of momentum of the molecule in the assembly in equilibrium) 
is zero, and that large (positive and negative) values of p x are less probable. 
(The distribution follows the same kind of law as the distribution of heights 
of a population about their mean, the example of section 4.2.) Although 
for the molecule the mean value of p x is zero — this is obvious because the 
distribution is symmetrical about p x = 0, so that positive and negative 
values of p x are equally probable — the mean value of pi is not zero ; this 
in turn means that the contribution to the mean kinetic energy arising 
from p x (the mean value of Jt(p x ) or pl/lm) is not zero. In the next chapter, 
we work out this mean value and show that it is equal to jkT, where k 
is Boltzmann's constant. This is a very important result. It confirms the 
statement made in section 4.1.2 of this chapter that temperature is a 
measure of the mean kinetic energy of any molecule in an assembly in 



78 Energy, temperature and the Boltzmann distribution Chap. 4 

equilibrium ; we can say therefore that Boltzmann's law is consistent with 
our intuitive ideas. At the same time it is worth pointing out an elementary 
mathematical feature of the Boltzmann factor, namely that whereas we 
usually measure temperature in degrees, (kT) is a measure of the tem- 
perature in energy units, ergs or electron volts. The ratio Jf/ZcTis dimen- 
sionless — and since one can only take the exponential of a dimensionless 
number, it is not surprising that the kinetic energy and the temperature 
are associated in this particular way. Indeed, in Appendix A we demon- 
strate how the form of the Boltzmann factor can plausibly be derived from 
simple considerations based on Joule's experiments; we start with the 
fact that the probability function must be a function of (energy)/(tem- 
perature) where the temperature is measured in energy units — this is the 
only mathematically acceptable form it could possibly have. 

Having pointed out these aspects of Boltzmann's law, using a simple 
explicit form of the kinetic energy term, let us return to the more general 
formulation and extend its use. 

The Boltzmann law holds for each of the components of momentum 
p x , p y and p z independently. For example, the probability that the com- 
ponent of momentum parallel to y lies between the limits p y and 
(p y + dp y )is 

AQ-* (p * )lkT dp y 

Combining these two probabilities, we can say that the probability 
that the x and y components of momentum lie simultaneously between 
the limits p x , (p x + dp x ) and p y , (p y + dp y ) is 

A 2 e -X( Px )/kT dpx Q -xr( Py )/kT dp ^ 

Let us rearrange this expression. When exponentials are multiplied to- 
gether, we add the indices. So we can write the probability 

A 2 e-^^ + x ^ )] i kT dp x dp y . 

By extension, we can calculate the probability that the three components 
of momentum lie simultaneously within limits p x , (p x + dp x ) ; p y , (p y + dp y ) ; 
p 2 ,(p z + dp z ). This is 

A 3 e-* lkT dp x dp y dp z , (4.7) 

where Jf is the total kinetic energy of the molecule : 

X = X(p x ) + JT{p y ) + Jf(p z ) = ^(pl + P 2 y +Pl) = im(v 2 x + v 2 y + v 2 ). (4.8) 

We might have written Jf(p x ,p y ,p z ) to emphasize that the total kinetic 
energy depends on p x , p y and p z . 

The Boltzmann law also holds for the contributions to the energy 
which come from the position coordinates — these are always potential 



4.3 Thermal equilibrium 79 

energies. For example, the potential energy of a molecule undergoing 
simple harmonic motion is of the form 'Viz) = ^<xz 2 , where z is a dis- 
placement and the symbol 'Viz) emphasizes the fact that V depends on z ; 
the potential energy of a molecule in the Earth's gravitational field is of 
the form Viz) = mgz where g is the acceleration due to gravity and z is a 
vertical coordinate. These are two examples of the fact that energies which 
depend on position coordinates are potential energies. 

The Boltzmann law states that the probability that a molecule has a 
coordinate between z and (z + dz) is 

(constant)e- r(2)/kr dz, 

where 'Viz) is the contribution to the potential energy which depends 
on z. The constant is found by normalizing. For a harmonic oscillator, 
the probability is of the form (const)exp(-az 2 /2/cT)dz; for a particle in 
the Earth's gravitational field, it is (const)exp(-mgz//cT)dz. 

Extending the argument, the probability that the molecule is to be 
found between x, (x + dx), y, (v + dv), z, (z + dz) is 

(constant)e ~ ir/kT dx d v dz, (49) 

where V is the total potential energy — it might have been written 
V(x, v, z) meaning that it depends on x, y and z. Finally, the probability 
of finding the molecule in the state specified simultaneously by momenta 
and positions p x , ip x + dp x ) • • • z, (z + dz) is 

(constant) e~ E/kT dp x dp y dp z dx dy dz, (4.10a) 

where E is the sum of the kinetic and potential energies 

E = jf+ ir 

and the constant has again to be found by normalizing. Note that in all 
these expressions, the probability function has always been of the form 

e -£/*r 

where E is the energy which depends on the coordinates and momenta 
considered. 

So far we have dealt with the probability of finding a single molecule 
in a certain state specified by momenta p x , ip x + dp x ), ... and coordinates 
x, (x + dx), .... It is not difficult to extend the application of Boltzmann's 
law to two molecules and thence to a large number N. 

To do this, we first need a notation. Let us use p xl to denote the x- 
component of momentum of molecule 1, p x2 for the corresponding 
quantity for molecule 2, and so on. Then we can write down the Boltzmann 
factors for all the momentum and position coordinates, combine all the 



80 Energy, temperature and the Boltzmann distribution Chap. 4 

exponentials together by adding their indices. The result is that the 
probability of finding one molecule between limits p xl , (p xl + dp xl ), ..., 
z 1} (z 1 +dz 1 ) and simultaneously the second molecule between limits 
P X 2 , (Pxi + dp x2 ) ■■■z 2 ,(z 2 + dz 2 ) is 

(constant) Q' mT dp xl -dz x dp x2 • ■ ■ dz 2 , (4.10b) 

where E is now the sum of the kinetic and potential energies of the two 
molecules. In other words, the probability function for this state is 

(constant) Q~ mT . 

Similarly, the probability function for N molecules to be found with 
momentum near p xl ,...,p zN and coordinates near x 1 ...z N (6N para- 
meters in all) is 

(constant) e~ E/kT , 

where £ is the total energy of the whole assembly. This energy depends on 
the position and momentum coordinates of all the molecules, and it is 
(usually) an exceedingly complicated expression. Nevertheless, the form 
of the probability function is so simple that it is consistent with the need to 
describe a system in equilibrium by only a small number of parameters. 

4.3.2 Validity of Boltzmann factors 

The problem confronting physicists around the beginning of the cen- 
tury was whether matter really was composed of atoms and molecules, 
and the statistical methods whose main results we have just presented were 
evolved in order to tackle this question. To begin with, calculations 
were performed on exactly specified systems (such as gases, perfect and 
imperfect) but it was noted that terms of the type exp(-£//cT) kept 
appearing. Eventually it was realized that this kind of expression was not 
a characteristic of any one model of the way a particular lump of matter 
was constituted but was of the most general validity. 

In Appendix A an outline of an approach is given which emphasizes 
this generality of application. In it, one discusses the thermal equilibrium 
of a 'subsystem' within a 'system'. The result, that the Boltzmann factor 
gives the probability function for the subsystem to possess a given energy, 
is quite independent of what it is made of— provided only that it is in 
thermal equilibrium. It could be a collection of molecules of the same 
kind as those of the system, or different from them ; it could be a collection 
of large particles or it could be a single particle while the rest of the system 
could be a liquid. The subsystem could even be a large object (a metal bar, 
for example) 'immersed' in a 'gas' of similar but purely imaginary copies 
of the same object, exchanging energy with one another by some 



4.3 Thermal equilibrium 81 

unspecified mechanism whose only function was to ensure that the whole 
system was in thermal equilibrium. In order to relate the behaviour of the 
real object in these imaginary surroundings to its behaviour inside a real 
thermostat we need only assume that the behaviour of any body in thermal 
equilibrium under given conditions is independent of the mechanism used 
to bring it to that equilibrium. 

The final result of these discussions is that the Boltzmann factor is 
valid not only for a molecule (which we discuss in section 4.4 and later 
throughout this book) but for a large object such as a particle undergoing 
Brownian motion (which we discuss in section 4.4.2). 

We will therefore assume that Eq. (4.7) and Eq. (4.9) are both applicable 
to any object which is in thermal equilibrium and is subject to the laws 
of classical mechanics. For example, the probability that a particle of 
microscopic dimensions (large on the molecular scale), suspended in 
a liquid will be found within the range of coordinates z to (z + dz) is 
AQ\p{-Y'(z)/kT).dz where i^(z) is the contribution to the potential 
energy which depends on z. 

It is ironic that the intense scrutiny that the laws of statistical mechanics 
came under resulted in the establishment of their general validity. For 
this scrutiny was forced on physicists because certain results, notably 
predictions about specific heats at low temperatures, were in conflict 
with experiment. But the error lay not in the statistical methods but in the 
assumption that atoms were subject to classical mechanics. 

4.3.3 Kinetic and potential terms 

So far we have defined what we mean by a probability function and have 
quoted the form of the Boltzmann factor. This is the probability function 
for any system in thermal equilibrium at temperature T to have energy 
E. In turn, this will allow us to calculate mean values of molecular velocities 
and other parameters depending on them, and so to calculate the thermo- 
dynamic behaviour of many systems 

We always begin by writing down an expression for the energy E. For 
a collection of molecules this is immensely complicated in general, and to 
reduce it to manageable proportions we have to introduce simplifying 
assumptions. But in systems obeying the laws of classical mechanics, the 
Boltzmann factor can always be separated into a product of kinetic and 
potential energy terms for the molecules. The kinetic energy always con- 
sists only of quadratic terms of the type pl/2m (or jmv*). Whenever a 
molecule moves, whether it is moving in a straight line or in an orbit or 
oscillating about a fixed point, its kinetic energy can always be expressed 
by terms of this form. Therefore, when we write down a Boltzmann factor 
for a single molecule or for an assembly, the evaluation of the kinetic energy 



82 Energy, temperature and the Boltzmann distribution Chap. 4 

term, giving the distribution of velocities, does not depend on the physical 
nature of the assembly, on whether we are dealing with a solid, liquid or gas. 
When we come to calculate velocity distributions or the mean speeds or 
energies of molecules, this fact leads to some remarkable generalizations. 

The potential energy of the whole system is similarly the sum of energies 
of all the individual molecules. The contribution to this energy arising 
from fields of force such as gravitational, electromagnetic, electrostatic 
or magnetic fields, is in general a simple function of the coordinates and 
the Boltzmann factor breaks up into a product of simple terms. There is 
a further contribution due to interactions with other molecules, but 
since the distance apart of each molecule from every other molecule 
enters into the expression for this potential energy, it is usually an ex- 
tremely complicated function of the coordinates. It must be emphasized 
that in contrast to the kinetic energy term, the evaluation of the potential 
energy term depends very much on the physical nature of the assembly, 
on whether we are dealing with a solid, liquid or gas. 

The plan of the rest of this chapter is as follows. Since the Boltzmann 
factor incorporates the temperature, and we have stated that this is 
called the absolute temperature, we must relate it to other scales. We do 
this by considering (section 4.4) the equilibrium of an assembly of non- 
interacting particles which is an idealized model of a perfect gas and which 
bears some resemblance to many real gases. As a result we will be able to 
identify T and also calculate Boltzmann 's constant k. Underlying this 
discussion will be the need to show that Boltzmann's law gives self- 
consistent results. Then we will deal with another similar system, a 
suspension of tiny particles in a liquid (section 4.4.2). 



4.4 BOLTZMANN DISTRIBUTIONS— I. A GAS OF 
INDEPENDENT PARTICLES UNDER GRAVITY 

Consider a mass of gas, maintained by some means at a uniform tem- 
perature T. It is imagined to be in a very tall vessel — it will emerge later that 
the results are more interesting when the height is several kilometres 
and it is in the gravitational field of the Earth. The problem is to calculate 
the distribution of the gas molecules with height. 

We will not calculate the distribution of kinetic energies (that will be 
done in the next chapter) but we will deal only with potential energies ; 
the problem can be simplified so that we deal with a single potential 
energy term and the corresponding Boltzmann factor. 

Consider a single molecule. If we assume that the acceleration due to 
gravity g is constant with height (an approximation which is sufficient 
for our considerations) then the potential energy of each molecule of mass 



4.4 Boltzmann distributions — I. A gas of independent particles 83 

m due to its height z above an arbitrary zero (usually at the Earth's surface) 
is mgz. 

Each molecule also has potential energy due to the presence of neigh- 
bouring molecules. Most of the time, of course, the molecules are far 
apart and their interatomic potential energy is negligible. But when they 
collide with one another, their potential energy is, for a short time, by no 
means negligible. Now it is essential that there should be collisions 
between gas molecules because this is the mechanism whereby a gas 
reaches thermal equilibrium ; if it were not for these collisions the gas 
could never reach uniform temperature. But we will now make a further 
assumption — namely that the collisions are relatively rare so that 
averaged over a long time the interatomic potential energy is negligible. 
This can be achieved by ensuring that the gas is at low pressure. It is 
true that under these conditions, we might have to wait a very long time 
for the gas to reach equilibrium, but that does not concern us. We call 
such an assembly a gas of independent particles. 

Thus we reach a compromise : we choose an assembly of molecules 
where collisions do take place, even though the potential energy is large 
whenever they occur, because collisions are essential to enable thermal 
equilibrium to be reached. On the other hand, we choose an assembly 
where the collisions are relatively rare so that on the average the inter- 
atomic potential energy is negligible. Under these conditions, we can 
say that the only significant contribution to the potential energy of a 
molecule is that due to the Earth's gravity, namely mgz, where z is its 
height above an arbitrary zero. 

The probability of finding a molecule at a height between z and (z + dz) 
is given by the Boltzmann law : 

P[z] dz = Be- n ' )lkT dz = Be' mgz,kT dz. (4.11) 

We can evaluate the constant B by applying the condition that the 
molecule must be found somewhere between the zero of height and the 
top of the column — which for simplicity we will assume is at z = oo : 



f 

Jo 



P[z] dz = 1 (4.2) 



whence B = mg/kT. 
Hence the probability of finding the molecule between z and (z + dz) is 

P[z]dz = ^e-^ z/fcT dz. (4.12) 



84 Energy, temperature and the Boltzmann distribution Chap. 4 

It is worth remarking that had we chosen any other arbitrary zero for z, 
all that would have happened is that a constant factor would have ap- 
peared in our expression which would later have disappeared in the 
normalizing. 

(Though it does not concern us and the answer is obvious by common 
sense, we will digress to find the dependence on one of the horizontal 
coordinates, x. The potential energy does not depend on x : i^ix) = 0. 
Hence the probability of finding the molecule between x and (x + dx) 
contains the Boltzmann factor exp( — 0/kT) which is unity; and this 
probability is simply proportional to dx — in other words this implies 
that the molecule may be found anywhere in the xy plane with equal 
probability.) 

Returning to the z-variation, let us now consider a large number n 
of molecules. The number which we will find between z and (z + dz) is 

^e—^dz. 
kT 

Now the density of any substance is equal to the number of molecules 
per unit volume, multiplied by the mass of one molecule. Hence if p is 
the density at zero height (strictly, the limiting density between z = 
and dz where dz is small) then this last equation gives 

p(z) = p exp(-mgz/kT). (4.13) 

We can plot the variation of density with height, at any given tem- 
perature. The density falls off exponentially with height. A graph of 
exp( — z/z ) as a function of z is given in Fig. 4.5; it is identical in form 
with Fig. 3.2(b). When z is equal to zero it has the value unity. If we 
compare the function at two heights differing by z , say at z = h and 
(h + z ), we find that the value falls by a factor e = 2.717 ... ; z is called 
the scale height. At a height of 2z , the function falls by a factor of about 7 ; 
at 4z by a factor of about 50. 

Here, the scale height 

z = — ' (4.14) 

mg 

but we cannot yet evaluate this because we do not know the value of 
Boltzmann 's constant k. We will now find this. From this relation between 
density p and temperature T in a gravitational field, we can derive another 
between p and T as a function of pressure P. We do this by changing our 
approach completely — taking a macroscopic view of the same phenom- 
enon, regarding the gas now as a fluid having weight and capable of exert- 
ing a pressure without bothering about its molecular constitution and not 



4.4 Boltzmann distributions— I. A gas of independent particles 85 




Fig. 4.5. The exponential function exp(-z/z ). For an increase of z by an 
amount z , the function decreases by a factor e. 

concerning ourselves with how the gas exerts its pressure. We consider 
a tall vessel filled with the gas and deal with the equilibrium of a slice of 
it between the heights z, and (z + dz), Fig. 4.6. The pressure at z is greater 
than that higher up because of the weight of the gas above it ; this is true 
for a column of any material. Thus 

dP = — pg dz. 



I 

dz 



pressure (P + dP) 
pressure P 



Fig 4.6. Equilibrium of a gas in the Earth's 
gravitational field. 



86 Energy, temperature and the Boltzmann distribution Chap. 4 

The total pressure at height z is the integrated weight of all the gas above : 

/»00 /«00 

P = J dP= -g\ pdz, 

and using the relation (4.13) above, 

f 00 kT kTp 

P = -gPo exp(-mgz/kT)dz = — p [exp(-mgz/kT)]? = . 

J z m m 

It is convenient here to refer to a mole of the gas instead of a single 
molecule, by multiplying top and bottom by Avogadro's number N : 

NkTp 
M ' 

where M = Nm is the molecular weight. If V = M/p is the volume 
occupied by a mole of the gas, it follows that : 

PV= NkT. (4.15) 

This is a relation between pressure, volume and temperature for the gas 
of non-interacting particles which we have deduced from Eq. (4.12). It 
must be remembered that T is still defined by the Boltzmann law (4.11). 

4.4.1 Perfect gases 

It is well known that real gases obey an equation of just this type, under 
the correct conditions — namely that the pressure is sufficiently low. Air, 
for example, obeys this relation with accuracy at pressures of the order of 
a few atmospheres at ordinary temperature; at low temperatures the 
range of pressure over which the equation holds becomes very much 
smaller — but at all temperatures air, like any other gas, obeys this kind of 
relation in the limiting case of zero pressure. 

The temperature used in practice was originally the centigrade scale 
defined in terms of certain properties of water, and the equation was 
written 

PV= const(273 + T°C). 

It is convenient to shift the zero to - 273°C and to write the equation 

PV= RT (4.16) 

where T is known as the temperature on the perfect gas scale. 

We can identify (4.15) with (4.16) and draw the following conclusions. 
Firstly, a perfect gas behaves like a gas of non-interacting particles. 
Secondly, the temperature defined by the Boltzmann equation is indeed 



4.4 Boltzmann distributions — I. A gas of independent particles 87 

identical with that on the perfect gas scale ; it was this statement that we 
set out to prove. Temperature on the absolute scale is written T °K. 

Finally, Boltzmann 's constant k can now be found. As a result of 
measurements on a number of gases at low pressures, extrapolated back 
to what are called standard conditions of temperature and pressure, 
namely 1 atmosphere or 1.013 x 10 6 dyn/cm 2 , and 0°C or 273°K, it 
is found that V is then 2.24 x 10 4 cm 3 for all gases and that therefore R is 
8.31 J/mol.deg. for all substances. 

Since Nk = R, and N = 6x 10 23 , we have k = 1.38 x 10 -23 Jdeg -1 = 
0.86xlO _4 eVdeg- 1 . 

With these data we can now find the scale height of the Earth's atmos- 
phere, making a gross assumption which is quite untrue — that it is all 
at the same temperature. We use Eq. (4.14). For air, we can take 
m = 30 a.m.u. = 5 x 10" 23 gm. Let us take T = 300°K and g = 10 3 cm/s 2 . 
Then z = /cT/mg is about 8 kilometres. (We might equally well write 
z — RT/Mg, and take M — 30.) Thus if atmospheric pressure is taken 
to be 76 cm of mercury at the Earth's surface, it is 28 cm at 8 Km a height 
which is roughly the height of the world's highest mountains. At twice 
this height the pressure is about 10 cm. (Note that these heights are small 
compared with the Earth's radius which justifies our approximation that 
g is constant with height.) 

4.4.2 Examples of Boltzmann distributions— II. Brownian movement 

Consider a tiny solid particle immersed in a liquid. It exhibits Brownian 
motion, which means that it can be observed with a microscope to be in 
ceaseless, rapid motion. It moves randomly about in short discontinuous 
jumps, though if a higher magnification were used together with a higher 
speed of observation, each of those jumps would be seen to consist of a 
number of shorter jumps. These random movements are the result of the 
impacts of molecules of the liquid against the particle, and this is the 
mechanism for keeping the temperature of the particle constant on the 
average. 

We can write down the Boltzmann factor for one single particle, and 
hence calculate a quantity which is almost directly observable : the proba- 
bility of finding the particle between a height z and (z + dz) from the 
bottom of the vessel. To do this, we need to know its potential energy 
at a height z. It is 

-V(z) = m*gz 

where m* may be called the effective mass of the particle which takes into 
account the mass of liquid displaced : 

m* = v(p-p') 



88 Energy, temperature and the Boltzmann distribution Chap. 4 

where v is the volume of the particle, p and p' are the densities of solid and 
liquid respectively. (This result can be derived from the fact that the force 
on the particle in the direction of z increasing is — m*g.) 

We can therefore write: the probability of the particle being found 
between the limits of height z and (z + dz), at temperature T, is given by 

P[z] dz = P Q- m * gzlkT dz, 

where P is a normalizing factor which we will not evaluate. If we observed 
the particle wandering through the liquid over a long time, this expression 
would give the fraction of the total time it spent within these limits of 
height. Alternatively, if we had a large number of particles present (large 
enough for us to apply statistical methods but small enough for us to 
neglect any mutual attractions between particles), this expression would 
determine the average number to be found at any time between the given 
limits : 

ndz = n Q- m * gz/kT dz, (4.17) 

where n dz is the number between z = and dz. 

This is just the same form of variation as for molecules of a perfect gas, 
but now the scale height kT/m*g is in practice very small instead of being 
many kilometres because of the magnitude of m*. Typical numbers are 
given in a problem at the end of the chapter. The historical significance of 
this experiment is that it is possible to prepare particles (of a resin, gam- 
boge) whose sizes are great enough for their masses to be found, and yet 
small enough to undergo Brownian motion with a measurable scale 
height. Knowing m* (or more precisely, measuring v, p and p'\ the accelera- 
tion due to gravity and the temperature on the perfect gas scale, it was 
possible firstly to get a direct, almost visual, proof of the validity of the 
Boltzmann law and secondly (by measuring the scale height) to determine 
Boltzmann's constant. This was done by Perrin in the early years of this 
century. Fig. 4.7 is an adaptation of a photograph taken by focusing a 
microscope at different levels and then making a montage of them ; z 
was of the order of 10~ 3 cm. 

4.4.3 Characteristic temperatures 

By raising the temperature sufficiently, it is possible to break up any 
kind of bonding between particles. If at a low temperature the bonding 
together of the given particles lowers the energy of the assembly by an 
amount e per particle, then when there is a reasonably large probability 
that the energy of thermal agitation is also e, the interaction will be over- 
come and the bond broken. The nucleons forming a nucleus, the electrons 
and nucleus forming an atom, atoms combined together to form a 



4.4 Boltzmann distributions — I. A gas of independent particles 89 



molecule and molecules condensed to form a liquid or solid will all be 
knocked apart at sufficiently high temperatures. 
This will happen above a temperature T given approximately by 



kT 



(4.18) 



because the Boltzmann factor exp( — e/kT) is then of order unity (we justify 
this statement a little more precisely in section 5.5). If the temperature 
were 10 times lower the Boltzmann factor would have the value e -10 
which is very small, about 10 ~ 4 , and it would be improbable that sufficient 
thermal energy would be available. 



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z = *L 
o rng 



Scale Height 



Fig. 4.7. Distribution of resin particles in water as 

a function of height. Adapted from a photograph 

by Perrin (1910), actually a montage of several 

sections at different heights. 



This is a rough rule but it is remarkably powerful. For example, the 
binding energy of two molecules of many substances is of the order of 
10~ 14 to 10~ 13 erg. Correspondingly, each substance has a critical 
temperature above which the liquid phase can never form, usually of the 
order of 10" 14 /Jt to 10 _13 /fc degrees— that is, 10 2 to 10 3o K. Ordinary 
boiling and melting points are usually not very different in order of 
magnitude. In a different region of energy, atoms can be ionized at room 
temperature when they are bombarded with electrons whose energy is of 



90 



Energy, temperature and the Boltzmann distribution Chap. 4 



the order of 10 eV (that is, of the order of 10 -11 erg). We can therefore 
estimate that the atoms in a gas can be ionized by collisions with other 
gas atoms at temperatures of the order 10 _11 //c degrees, or roughly 
10 5o K. Indeed, gases do form ionized plasmas at such temperatures. 
Finally, in another and much higher energy range, nuclei can be dis- 
rupted by nuclear particles of about 1 MeV energy (around 10 " 6 erg). 
This means that one can expect nuclear reactions to take place in gases 
heated to about 10 9 or 10 10o K, which corresponds to conditions in the 
interiors of stars. 



APPENDIX A 

A.l Dependence of the probability function on energy and temperature 

In this Appendix we will attempt to make plausible the form of the 
probability function, quoted without proof in section 4.3.1. 

Consider a lump of matter, composed of molecules. Its temperature 
is maintained at a constant value T by some mechanism, a thermostat, 
which we will not consider. We call this lump of matter the system. We 
select a small part of the interior of the system which we call the 'sub- 
system', Fig. A.l. For example, the system might be a block of metal with 
a cavity containing gas : and we might select this gas to be the subsystem — 
or we might select part of the metal to be the subsystem. 



subsystem 




system 



Fig. A.l. A lump of matter (the system) maintained at 
temperature T; we select part of it, called the subsystem. 



We ask what is the probability that the molecules of the subsystem 
have particular positions and momenta. Since the system is in equilibrium, 



Appendix 91 

this probability will not change with time, even though the molecules are 
moving rapidly and randomly. 
If we had just one molecule in the subsystem, we could agree to write 

P[x, y, z ; p x ,p y ,p z ] dx dy dz dp x dp y dp z 

for the probability that the molecule was at a point whose coordinates 
lay in the range x to (x + dx), y to (y + dy), and z to (z + dz), while the 
components of momentum lay between p x and (p x + dp x ), p y and (p y + dp y ), 
p z and{p z + dp z ). 

If we had more molecules — and we usually have large numbers — then 
we have to write something more complicated. If there were n molecules, 
we would write 

P[x t . yi > 2 i ; Px, , P yi , P Zl ;x 2 ,y 2 ,z 2 ; p X2 ,p y2 ,p Z2 ;...x„,y„,z„; p Xn , p yn , p z J 
x dx! dy t dzj dp xl . . . dp yn dp Zn 

for the probability that molecule 1 is within a range dx! and dy x and dz x 
of the point (x l , y t , z t ) and moving with a momentum whose components 
are between limits p Xl and {p xi + dp Xi ) and so on ; while at the same time 
molecule 2 is at a point near (x 2 , y 2 , z 2 ) with momentum near (p X2 , p y2 , p Z2 ) ; 
and so on for all the molecules. 

This function P, which is denoted above to be a function of all these 
variables, could in general be very complicated. The fact that we are 
dealing with a system in equilibrium however leads to something much 
simpler. 

Since the subsystem is continually interchanging energy with the rest 
of the system by molecular collisions, it is natural to suppose that P might 
be a function simply of the energy E of the subsystem. (E itself depends 
on all the coordinates and momenta of all the molecules.) It also depends 
on the temperature T which characterizes the system as a whole. No 
other simple mechanical quantities can be thought of which could be 
relevant. (The total linear momentum and angular momentum of the 
system might be included but we rule them out by considering the system 
as a whole to be at rest.) 

Now the probability function is essentially dimensionless because of 
the meaning of probability. It must therefore be a function of the ratio 
between E and some other quantity which has the dimensions of energy. 
The only quantity we have at our disposal is T — which is a measure of 
energy, but whose dimensions are largely arbitrary since we can measure 
temperature in an arbitrary way. The simplest possibility is to say that 
P is a function of (E/kT) where k is a universal constant whose actual 



92 Energy, temperature and the Boltzmann distribution Chap. 4 

magnitude depends upon the units used to measure the temperature : 

P = f(E/kT). (A.1) 

This already looks a good deal simpler than our last expression for P, but 
of course E itself still depends on all the coordinates and momenta and is 
usually a very complicated expression. 

A.2 Form of probability function 

We can determine the form of the function/by considering the following 
special situation. Consider two independent systems both maintained at 
temperature T; for example, two cavities each containing gas, inside the 
same block of metal maintained at T. Let the energies of the systems be 
E x and E 2 respectively. The probability of finding system 1 in a given 
configuration is f(EJkT) and the probability of finding system 2 in 
another given configuration is f(E 2 /kT). But if now we consider the two 
systems together, the probability of finding both systems 1 and 2 in their 
given configurations at the same time must be 

f / £i + £ 2 

J \ kT 

since the energy of the two systems together is the sum of their separate 
energies. But since the two systems are independent, the probability must 
be the product of two separate probabilities : 

We will now show that this serves to determine the form of the function 
/. This can be seen intuitively by noting that since P (or/) is multiplicative 
whereas E is additive, the function must be of the kind 

log f(E/k T) = BE /k T+C (A.3) 

where B and C are constants not involving E. Alternatively, this result 
can be derived explicitly from (A.2) as follows. For simplicity write x 1 and 
x 2 for EJkT and E 2 /kT, and write /'(x) for df/dx where x is x t or x 2 or 
(x 1 +x 2 ). Carry out a partial differentiation of Eq. (A.2) all through 
with respect to x t keeping x 2 constant; then 

/'(*!+ x 2 ) = f\x x )f{x 2 ). 
Differentiating (A.2) with respect to x 2 keeping x t constant gives 
f'(x 1 +x 2 ) = f(x l )f'(x 2 ). 



Appendix 93 



Therefore 

f'(x 1 )f{x 2 ) = f{x,)f\x 2 ) 
or 

/'(*i) f'(x 2 ) 



fix,) f(x 2 ) 



= B 



where B is a constant, or more precisely B does not contain E t or E 2 
although it might depend on T. The solution of this equation is (A. 3) 
above. 
Therefore 

f(E/kT) = (constant) e BE/kT 

where the constant depends on temperature. Note that we lose no generality 
by writing B equal to + 1 or — 1 since we can absorb any other numerical 
factors in the constant k. Thus we have reduced the original complicated 
form of the expression for the probability of finding the assembly in the 
specified state to 

(constant) e BE/kT dx x dvi . . . dp zn 

where B is + 1 or — 1. 

Consider now the special case of one molecule in the assembly, which 
happens to be moving in the x-direction. The energy is simply pl/2m, 
independent of x. The molecule may therefore be found with equal prob- 
ability to have any x coordinate, but the probability of having momentum 
between p x and (p x + dp x ) is proportional to 

exp(Bp 2 x /2mkT)dp x . 

The top graph in Fig. A.2 shows this function for B = + 1. It would imply 
that it is almost certain that the molecule would have infinitely large 
momentum: this is not acceptable. The lower graph is for B = — 1, and 
implies that small momenta are more probable: this is reasonable. 
Therefore B = — 1. 

Thus we conclude that the probability of finding the assembly in the 
condition specified is 

(constant) e~ E/kT dx, dy x . . . dp z „. 

The value of the constant of proportionality is determined by the fact that 
the molecules must be found somewhere inside the accessible range of 
coordinates and momenta. Thus the integral over all variables must be 
equal to unity. 



94 Energy, temperature and the Boltzmann distribution Chap. 4 




Fig. A.2. Form of the functions exp(+p 2 /a 2 ) and exp(— p 2 /<x 2 ). 



By expressing the energy where possible as the sum of terms each 
depending on one coordinate or one component of momentum, the 
probability separates into simple terms each of the form of Eq. (4.6a). 

The above expression satisfies all our conditions. It allows the energy of 
independent systems to be additive but probabilities to be multiplicative. 
It includes the temperature in a way which agrees with intuitive ideas but 
at the same time it can serve as the definition of what we mean by tem- 
perature and in the text we call it the absolute temperature. 

A.3 Extension to macroscopic systems 

The language which we used in this discussion of the form of the 
probability function was a very general one : we spoke of systems and 
subsystems. Though in the text we concentrated on large assemblies of 
molecules, and though in deriving the form of the probability function we 
gave an example of a system consisting of gas molecules inside a cavity, 



Appendix 95 

there is no need to limit the discussion in this way. The Boltzmann law 
can be applied to any system in thermal equilibrium. 

For example, consider particles (each containing many molecules) 
suspended in a liquid and undergoing Brownian motion. The problem 
(which is discussed in section 4.4.2) is to calculate the distribution of the 
particles with height, taking into account the potential energy due to the 
Earth's gravity. We are not now interested in how the individual molecules 
behave inside each particle but rather in how the particle behaves as a 
whole ; we are interested in how the centre of mass of the whole particle 
moves, we do not enquire how the molecules move with respect to this 
centre of mass. 

We could begin by accepting the validity of the Boltzmann law for 
molecules and the distribution of their energies. With this as a starting 
point, we could then write down the energy of each molecule inside the 
particle in terms of its own position coordinates and momenta with 
respect to fixed axes. This expression would be an immensely complicated 
one. Then we could change the axes of reference to a set through the centre 
of mass of the particle and moving around with it : this would be a simple 
linear transformation in which for example the z-coordinate of the nth 
molecule would be written as z H = z'„ + h where z' n is the z-coordinate 
with respect to the new axes and h the height of the centre of gravity above 
the fixed origin. The expression for the energy would still be complicated. 
However, we would be able to group together a number of terms and 
separate out a term exp( — m*gh/kT) from the probability function, where 
m* is the effective mass of the whole particle. This is the sort of term we are 
interested in, since both h and m* refer to the whole particle, or in other 
words this term does not contain any molecular coordinates or momenta. 

But there is no need to go through this complicated procedure. There 
is no need to assume that the Boltzmann law is valid for molecules only, 
molecules which are members of large assemblies ; hence there is no need 
to begin each calculation at the molecular level. Our derivation of this 
law was valid for any system in thermal equilibrium. At one stage for 
example we considered a system of a block of metal with two cavities each 
filled with gas ; we might as well have considered a liquid containing two 
particles. A single particle can act as the subsystem and the Boltzmann 
law applies to it. Thus, we can select coordinates (such as h) which refer 
to one particle, calculate the corresponding contribution to the energy 
and hence write down the Boltzmann factor for this particle. This gives 
exactly the same answer as before, of course, but far more directly. 

Two remarks can be made here. First, this discussion was forced upon 
us because ordinary objects contain vast numbers of molecules and only 
statistical statements can be made about them. We have however ended 



96 Energy, temperature and the Boltzmann distribution Chap. 4 

by applying the results of this discussion to single objects and this might 
seem inconsistent. In fact it is not, because the mechanism for keeping its 
temperature constant is one of continuous bombardment — for example, 
collisions of a particle with molecules of the liquid in which it is suspended. 
This situation is so complex that again it cannot be followed in detail. 

Secondly, let us go back and examine a little more closely what is 
involved in accepting that the Boltzmann law is applicable to molecules. 
After all, each molecule itself has a structure and may contain many, even 
several hundred particles (electrons, nucleons). Thus if we accepted a 
strictly classical point of view it would not be sufficient to write down 3 
coordinates and 3 momenta for each molecule: many more would be 
needed. But our discussion has shown that, just as we need not write down 
the separate Boltzmann factors for every molecule inside a macroscopic 
particle but can write down the factor for the particle as a whole, so we 
need not begin by considering the constituent subatomic particles but can 
deal with the molecule as a whole. At least, then, our approach is self- 
consistent. (In addition, quantum mechanical considerations show that 
many types of motion, of the atoms inside a molecule or of the nucleons 
inside a nucleus which would be expected to occur if classical mechanics 
were universally valid, do not in fact take place at normal temperatures. 
The energy needed to excite them is much greater than kT where T is 
a normally accessible temperature; see section 4.4.3. This merely rein- 
forces the result for completely different reasons.) 

PROBLEMS 

4.1. In one of his experiments with resin particles suspended in water, Perrin used 
a microscope with a short depth of focus to count the number of particles 
in horizontal layers 6 microns apart (1 micron, jx = 10 -6 m = 10~ 4 cm). At 
17°C the numbers which he observed in the field of view were 305, 530, 940 and 
1880. He found the particles to have a radius of 0.52 /z and a density of 1.063 
gmcm -3 ; the density of water at 17°C is 0.999 gm cm -3 . Use Perrin's results 
to calculate Avogadro's number, given that the gas constant R is 8.314 J 
deg - * mol" * and g = 980 cm sec -2 . 
What would be the distribution with height for particles of density 0.935 



gmcm 



■39 



CHAPTER 




The Maxwell speed distribution 
and the equipartition of energy 



5.1 VELOCITY-COMPONENT DISTRIBUTION P[v x ] 

In Chapter 4, we saw that the probability of an assembly of n molecules 
possessing coordinates p xi ■ ■ • z n is given by 

(const)e- £/kT dp xl • • • dz„ (5.1a) 

where the x, y and z and the p x , p y and p z coordinates refer respectively to 
the positions and momenta of the individual molecules. Since the energy 
splits up into independent contributions, the potential energy ^depending 
only on the positions and the kinetic energy Jf only on the momenta, the 
probability can be expressed as the product of two entirely independent 
groups of terms : 

(const) e-*' kT dp xl ■ ■ • dp xn x e~ w dx t • • • dz„. (5.1b) 

We then considered an assembly of independent particles in which the 
potential energy due to the interactions between the molecules was taken 
to be negligible on the average even though collisions between molecules 
must occur in order to preserve thermal equilibrium. We did not consider 
the kinetic energy of the molecules, but by considering a special case of the 
assembly in the Earth's gravitational field, we showed that the quantity T 
appearing in the Boltzmann factor is identical with the thermodynamic 
temperature. 



98 Maxwell speed distribution and the equipartition of energy Chap. 5 

In this chapter we will concentrate on the kinetic energy rather than on 
the potential and deduce the probability that a molecule is in an assembly 
in thermal equilibrium at a given temperature and has a momentum 
component or total momentum or speed, within a certain range. 

The results which we will deduce are widely applicable. They hold for 
any state of matter, solid, liquid or gas, provided the laws of classical 
physics apply. 

To emphasize this point, we will first consider how the total potential 
energy of an assembly of N interacting molecules could be written down — 
interacting in the sense that each pair of atoms has a mutual potential 
energy. Considering the interaction of the ith molecule with the y'th for 
example, the potential energy depends on the distance r tj between them 
raised to the inverse 6th or 12th power (if they are neutral molecules) and 
to express these quantities in terms of the coordinates of the individual 
molecules is complicated. To extend this to all possible pairs of molecules 
gives a vast array of terms which includes many product terms and which 
cannot be arranged as a simple sum of terms each depending on a single 
coordinate. This is why in the previous chapter we limited the discussion 
to one of the few physical cases where there are on the average only 
negligible interactions — a perfect gas where the molecules are for most of 
the time at a great distance from one another. Without introducing gross 
simplifying assumptions, we cannot extend the discussion of the potential 
energy term to any interacting assembly such as a liquid or a solid or a 
dense gas. 

Even in strongly interacting assemblies, however, the total kinetic 
energy can always be written down as a sum of individual kinetic energies 
each depending on only one velocity component. For example, in a solid 
where the molecules are close together and mostly oscillate about their 
mean positions, the displacement of one molecule from its equilibrium 
position certainly causes its neighbours to move, either setting them 
oscillating at a single frequency or with a complicated frequency spectrum, 
or perhaps changing their places in the lattice. Nevertheless, if the instanta- 
neous velocity of one molecule is (v xl , v yl , v zl ) and that of a neighbour is 
( v x2> v yi-> v z2) tne energy of each is still the sum of terms of the type jmv xl or 
\mvl 2 , the kinetic energy therefore always separates out into the sum of 
simple terms and the corresponding Boltzmann factor always separates 
out into a product of single terms. 

Thus the momentum or velocity distribution which we will deduce is 
applicable to all physical systems. A solid and a liquid and a gas at high 
pressure and a gas at low pressure, all of the same molecular weight and 
at the same temperature have the same velocity distribution, even though 
the types of movements the molecules perform are quite different. In a 



5.1 Velocity-component distribution P[v x ] 99 

gas at low pressure the molecule moves in straight lines between relatively 
rare collisions, and their average interaction potential energy is negligible ; 
in a solid, the molecules are oscillating about their lattice points and have 
high potential energy. Nevertheless, if we considered a substance of given 
molecular weight in the gaseous and in the solid states at the same tempera- 
ture, the proportion of molecules whose velocity lay within a given range 
would be found to be just the same in the two states. The potential energies 
would be quite different, the distribution of kinetic energies would be 
identical. The discussion that follows is in no way limited to a gas — though 
we shall in fact apply the results to a gas and thereby gain insight into the 
way the pressure of a gas is produced. Later we shall apply the same 
results to solids and liquids. 

Consider an assembly of molecules each of mass m. It is convenient 
now to deal with velocities instead of momenta; let v x , v y and v z be the 
components of velocity parallel to x, y and z, so that p x = mv x , p y = mv y , 
p z = mv z . Since m is constant, the probability that the x-component of 
velocity lies between v x and (v x + dv x ) is 

PM dv x = A exp( - mv 2 x /2kT) dv x (5.2) 

where A is determined by the fact that the probability that v x must have 
some value between — oo and + oo is unity : 



/•oo 
J-, 



exp(-mvl/2kT)dv x = 1. 
We use one of the integrals listed in the Table on p. 72, namely 

exp(-ax 2 )dx = /-; 
J-oo V a 

whence A = (m/2nkT) 1/2 so that the probability distribution is 



m 



1/2 



P[Vx]dv * = [^f] 1 e ~ mv2x ' 2kTdv *-> (5-3) 

v x is the x-component of the velocity. A molecule travelling in any direc- 
tion has a component v x unless it happens to be moving exactly at right- 
angles to the x-axis, when v x is zero. P[v y ] dv y and P[v z ] dv z are identical 
in form. 

A graph of P[v x ] as a function of v x is given in Fig. 5.1 expressed in 
dimensionless form. It is a Gaussian distribution centred about the velocity- 
component v x = 0. The symmetry of the curve means that for every 
molecule travelling with a certain velocity in the + x direction, it is equally 
probable that another is travelling in the -x direction. Thus the average 
value of the total component of momentum is zero, as we would expect in 
a system at rest. 



100 Maxwell speed distribution and the equipartition of energy Chap. 5 




Fig. 5.1. Probability distribution P[>] of a component of 

molecular velocity in any direction called the x-axis. v = 

(mv 2 J2kTY 12 , a dimensionless measure of velocity. 



5.1.1 Experimental verification of the P[v x ] distribution for gases and solids 

It is possible to determine the velocity in any direction of a molecule in 
thermal equilibrium inside a piece of matter by observing the Doppler 
shift of a spectral line emitted by the molecule. When stationary, an atom 
emits a spectral line of a certain wavelength, A say, with the same in- 
tensity in all directions. An observer can measure this wavelength, the 
direction of observation being called the x-axis, taken as positive away 
from the observer. When the atom moves, in any direction, the observed 
wavelength is altered to a value A given by 



A-A ( 



(5.4) 



where v x is the velocity-component of the atom along the x-axis away from 
the observer and c is the speed of light. If the atom is moving at any angle to 
the line of observation, it is always the x-component of the velocity which is 
measured as a change of wavelength. Radiation can be received from all 
the atoms in the assembly, whether they are travelling towards the observer 
and give a shorter wavelength, or receding from him and give a longer 
wavelength, or moving at right-angles when their wavelength is unaltered. 
Thus the whole assembly of molecules containing these atoms produces a 
spread of wavelengths whose intensity at any wavelength is given by the 
number of molecules with the appropriate value of v x , that is, by the 



5.1 Velocity-component distribution P[v x ] 101 

probability distribution of v x : 

1(A) dA = J(A ) exp[ - mc 2 (A - A ) 2 /2A 2 kT] dA. (5.5) 

A spectral line which would, in the absence of thermal motion, be sharp is 
therefore broadened into one of gaussian shape whose width increases with 
temperature (Fig. 5.2(a)). 





(a) 



0.53 • 0.53 
6374.51 A 



Fig. 5.2. (a) Variation of intensity with wavelength of a spectral line broadened 

by the Doppler effect due to thermal motion, (b) Broadening of a line emitted 

by ionized iron atoms (M = 56) in the Sun's corona. Data from Dollfus, Compt. 

Rend. Acad. Sci. 236, 996 (1953). 



For observing this effect in gases, and for checking the validity of the 
P[v x ] distribution curve, optical emission lines in the visible region of 
wavelengths can be used. Even in gases however, the thermal motion of the 
molecules is not the only cause of the broadening of spectral lines. Every 
line has a 'natural width', a quantum effect, determined by the finite 
lifetime of the excited state of the atom which emits it ; many lines however 
have small natural widths. When molecules collide, the wavelength of the 
emitted line may be altered by the presence of the nearby molecule, but this 
effect can be reduced by working at very low pressures where collisions are 
infrequent. Finally, every optical instrument will broaden a monochro- 
matic line because of its finite resolving power, but this effect can easily 
be allowed for. Under favourable conditions, therefore, the Doppler 
broadening can be detected. The agreement with theory is always good. 



102 Maxwell speed distribution and the equipartition of energy Chap. 5 

Interesting applications of this effect have been made in astronomy, 
to determine the surface temperatures of stars. The Sun's corona, for 
example, is known to be a very tenuous gas of uniform high temperature. 
Among the spectral lines it emits is a red line (A = 6374 A), from highly 
ionized iron atoms which have lost nine electrons. This line has small 
natural width, but in the Sun's corona it has been observed to be a com- 
paratively broad line of Gaussian profile, 0.53 A from the centre of which 
the intensity falls to 1/e of its maximum value Fig. 5.2(b). The molecular 
weight of iron is 56 and with these data the temperature of the corona is 
2.1 x 10 6o K. This agrees with other independent estimates. 

Solids do not emit sharp visible spectral lines, so that this method of 
measurement cannot be extended to study the motion of atoms in solids. 
But nuclear radiations, gamma ray spectra, can under the right circum- 
stances be sharp enough. The complication here is that the radiation is so 
energetic that the nucleus recoils when it emits a gamma ray (an effect which 
is negligible with the lower-energy optical radiation) and there is a Doppler 
broadening due to recoil. However, radiation can be emitted with the 
recoil momentum being transmitted to the whole crystal instead of being 
taken up by the single nucleus — a quantum phenomenon called the 
Mossbauer effect — so that the recoil velocity is negligible, and the emission 
is sharply monochromatic. In principle, we could then study the Doppler 
broadening of such a line emitted from a crystal at a finite temperature ; 
it should resemble Fig. 5.2.(a). 



Cross 
section 



=fr^~ 




(a) 



(6) 



Energy (meV) 



Fig. 5.3. {a) Schematic layout of apparatus for measuring the distribution of 
the velocity-component in solids using the Mossbauer effect. A — device for 
moving the source rapidly backwards and forwards. B — cooled crystal 
emitting gamma rays. C — absorbing crystal at room temperature. D — counter 
which measures gamma rays passing through the absorbing crystal, (b) 
Absorption cross section per nucleus using iridium emitter and absorber, as a 
function of energy difference between emitted and absorbed radiation, 
allowing for recoil energy. The cross section falls by a factor e for 69 meV 
change of energy. From Visscher, Ann. Phys. 9, 194 (1960). 



5.1 Velocity-component distribution P[v x ] 103 

But there is a further complication because we cannot make a gamma- 
ray spectrometer to disperse different wavelengths and measure the profile 
of a line directly to the accuracy required. Instead we make use of the fact 
that if a given nucleus can decay from an excited state and emit radiation 
of energy E, then the same nucleus can absorb radiation of energy (E + E K ), 
where E R is the known energy of recoil, and thereby become excited. 
(Recoilless absorptions also occur but these can be disregarded.) We utilize 
this effect as follows (Fig. 5.3(a)). We have two crystals, one a source kept 
at low temperature so that the emitted gamma rays are monochromatic. 
It is mounted on a device which oscillates backwards and forwards at high 
speeds, comparable with thermal speeds of 10 4 or 10 5 cm/s. This motion 
creates a controllable Doppler shift of frequency of the radiation which 
then falls on a second crystal. This is the crystal the velocity distribution of 
whose atoms we wish to explore ; it is kept at a high temperature (say, 
room temperature). We measure the energy absorbed per second as a 
function of the frequency of the incident gamma rays which we can 
calculate from the known velocity of the source ; we must also allow for 
the effect of the recoil energy of the absorbing nuclei. 

The prediction is that the absorption should show a Gaussian variation 
with gamma ray energy, of width 2yJ(E R kT) where T is the temperature 
of the fixed crystal C and the factor -J(kT) comes from the Boltzmann 
distribution of thermal velocities. For iridium crystals with the absorber 
at 300°K, the curve is shown in Fig. 53(b). (The energy axis has been 
shifted to allow for the recoil energy and a spike due to recoilless absorp- 
tions has been deleted.) For iridium the gamma ray energy is 129 keV, so 
that E R = 0.046 meV. The essential point is the resemblance of the absorp- 
tion curve to the Boltzmann curve for velocity-component distribution, 
Fig. 5.1. 

5.1.2 The pressure of a perfect gas 

We have repeatedly emphasized that the P[v x ] distribution given by 
Eq. (5.3) is obeyed by assemblies of molecules in all states of rarefaction 
or condensation. The first application of this distribution law will, how- 
ever, be to a perfect gas and we will derive the PV = RT law again. 

Consider a gas contained in a vessel, one wall of which is plane, the vessel 
and gas being in thermal equilibrium. Let the y and z axes be drawn in 
the plane of the wall, the x axis normal to it. Molecules travel towards the 
wall, hit it and rebound. Thus each molecule suffers a change of momentum, 
the force on the wall being the rate of change of momentum. The impacts 
occur so frequently that this appears as a steady pressure (although 
refined measurements would show that it does fluctuate about its mean 
value) which we will now calculate. 



104 Maxwell speed distribution and the equipartition of energy Chap. 5 

We first make the rather unrealistic assumption that each molecule is 
reflected elastically at impact. In other words, if its initial velocity before 
impact is (v x ,v y ,v z ) then after impact it is ( — v x , v y , v z ); the normal 
component is reversed but the others are unchanged (Fig. 5.4(a)). This 
implies that the wall must be smooth on the atomic scale, which is of 
course impossible. At the end of this section we will show that this assump- 
tion is unnecessary. Since only molecules travelling towards the wall hit it, 
v x may have any value between and oo whereas v y and v z may have 
values from — oo to + oo. 





\a) 



(t) 



Fig. 5.4. (a) The impact of a molecule on a wall, assuming an elastic impact 

and specular reflection from a smooth surface, (b) More realistically, a molecule 

probably sticks to the wall for a finite time and is reemitted at random. 



The nett changes of momentum in the y and z directions from one impact 
are zero, but in the x direction there is a change of 2mv x per impact. 

Let us select out of the whole assembly those molecules with a velocity 
component between v x and (v x + dv x ). If there are a total of n molecules per 
cm 3 , there are nP[v x ] dv x molecules in this class. 

Any molecule at a distance equal to, or less than v x from the wall must 
hit it during one second. Therefore the number hitting an area A of wall in 
one second is equal to the number contained in a volume of area A and 
length v x . 

There are therefore nAv x P[v x ] dv x impacts per second on area A, each 
bringing a change of momentum 2mv x ; so the force on this area is 
2mnAvlP[v x ] dv x due to these molecules, since the force is the rate of 
change of momentum. Thus the contribution to the pressure is 
2mnvlP[v x ]dv x . 



5.1 Velocity-component distribution P[v x ] 105 

The total pressure from molecules of all velocities is found by integrating 
over all relevant values of v x : 

* go / m \ 1/2 f °° 

P = 2mn J^ v 2 x P[v x ] dv x = 2mnU^ J i£ e-"*'» r dt; x . 

(There should be no confusion between pressure P and probability func- 
tion P[ ].) This integral can be evaluated by writing 

mvl/lkT = a 2 ; , dv x = \ da, 

\ m I 

and by using one of the integrals on page 72 : 



I 



e-* 2 da = ^. 
o 4 



Then P = nkT, where n is the number of molecules per cm 3 . Let V be 
the volume of M grams of gas, containing N molecules ; then n = N/V. 
Thus 

PV= NkT= RT. 

This is the perfect gas law which we have already deduced by considering 
the potential energy (section 4.4).* 

We based this calculation on the assumption that the molecules are 
reflected specularly on impact, which implies among other things that the 
walls are smooth compared with molecular dimensions. It is much more 
realistic to assume that the individual impacts are as shown in Fig. 5.4(b) — 
a molecule sticks to the wall for a finite time and is reemitted later. In sec- 
tion 9.4.1 we show that 10" 8 s is a reasonable estimate of the 'sticking time' 
under certain conditions. While this is a short time in ordinary terms, it is 
long on the molecular scale; the atoms of the wall perform 1,000 or 
10,000 vibrations in that interval. When a molecule jumps off again there- 
fore, it does so at an angle and at a speed unrelated to the incident angle 
and speed. Let us recast the argument about the momentum change. 

Whatever the details of the impacts, we will nevertheless assume that the 
gas is in equilibrium. Then the number of molecules within any range of 
velocity must remain constant. Therefore if nAv x P[v x ] dv x molecules with 
velocity between v x and (v x + dv x ) are removed from the gas in each second 
by sticking to the area A, the same area must reemit the same number per 
second to preserve the equilibrium. Therefore, the overall momentum 
change is the same as we calculated before, even though any one individual 
molecule may be emitted with quite a different velocity from that at impact. 
This demonstrates that in order to derive the PV = RT law, we do not 

* This cannot pretend to be a new result — it is merely a confirmation of the perfect gas law 
by a different method. 



106 Maxwell speed distribution and the equipartition of energy Chap. 5 

have to assume any special form of impact at the walls, but merely that the 
assembly is in thermal equilibrium. 

5.2 SPEED DISTRIBUTION P[c] 

The previous discussion leads to the statement that the probability 
that any one molecule in any assembly (solid, liquid or gas) in thermal 
equilibrium has velocity components between v x and (v x + dv x ), v y and 
(v y + dv y ), v z and (v z + dv z ) is equal to the product of three factors: 

P[v x , v y , v z ] dv x dv y dv z = P[v x ] dv x P[v y ] dv y P[v z ] dv z 

l m \ 3/2 



\2nkTj 



e -M»i + v$ + vl)l2kT dVxdv ^ dVt 



= /_m_\ 3 e -«c*/2*r d d d (56) 

\2nkTj 

where we have written the total speed 

c = (v 2 x + v 2 + v 2 ) 1 ' 2 . (5.7) 

We will now find the probability that the total speed of a molecule 
lies between limits c and (c + dc). We will no longer be concerned with 
velocity components v x , v y and v z , or in other words all molecules moving 
with the same speed will be classed together, no matter which direction 
they are moving in. We have, in fact, to integrate over all angles. Thus we 
require an expression of the type P[c] dc. 

5.2.1 Transformation of coordinates 

When we were concerned with v x , v y and v z , the velocity of any particle 
was represented by a point within a framework of (v x , v v , v z ) axes, and the 
number of points within a parallelepipedal element of volume dv x dv y dv z 
was counted in a manner analogous to Fig. 4.4. In this new system of 
counting, the state of the assembly is represented by exactly the same set 
of points, but the speed c of a molecule is represented by the length of a 
radius vector from the origin and we no longer have a simply shaped 
element of volume. 

We can transform from dv x dv y dv z to dc using the following intuitive 
method. The range between c and (c + dc) is represented by a spherical 
shell bounded by spheres of radius c and (c + dc), Fig. 5.5. The required 
number of points, integrated over all directions, is therefore proportional 



5.2 Speed distribution P[c] 107 

to the volume of the shell which is 4nc 2 dc. We can therefore write 

' m \ 



3/2 



P[c]dc = 4n\ 



2nkT] 



c 2 e- mc2/2kT dc 



(5.8) 



for the probability of finding a molecule with speed between c and (c + dc) 
irrespective of the direction in which it is travelling. This is the Maxwell 
speed distribution, the result we set out to find. 




Fig. 5.5. Counting of representative points in a shell 
bounded by spheres of radius c, (c + dc). 



A more rigorous method of transforming from dv x dv y dv z to dc is as follows. 
The velocity of a molecule is uniquely specified by its speed c, the angle 9 it 
makes with an axis (which we may take without loss of generality to be coinci- 
dent with the z or v z axis) and another angle <f> which it makes with a plane 
through this axis (which we may take to be the xz or v x v z plane). This is a system 
of spherical polar coordinates, Fig. 5.6(a). 9 can vary from to n, cf> from to 
2n. When c is varied by dc, 9 by d9, <f> by d^>, a volume 

(dc)(c d9)(c sin 9 d</>) = c 2 sin 9 dc d9 d<f> 

is generated, Fig 5.6(6). Thus the probability that the velocity of the molecule 
lies between c and (c + dc), at angles between 9 and {9+d9\ (f) and ((f>+d(f)) is 

/ m \ 3 > 2 

P[c, 9, 0] dc d9 dtf> = — ~\ c 2 e - mc2/2fcr sin 9 dc d9 dd> 
\2nkTJ 

which is exactly equivalent to Eq. (5.6). P[c] does not contain 9 or <£, which 
merely expresses the fact that all directions are equally probable. If this 



108 



Maxwell speed distribution and the equipartition of energy Chap. 5 



expression is integrated over all possible directions, the little volume element 
becomes the spherical shell of Fig. 5.5. We then recover the Maxwell distribution 
(Eq. (5.8)) since 

f f sin0d0<ty*=4w. 

Jo Jo 




(b) 

Fig. 5.6. (a) Spherical polar coordinates, (b) Generation of volume element by 
variations dc, d6, d(f>. 



5.2 Speed distribution P[c] 109 

5.2.2 The Maxwell distribution 

The form of the Maxwell distribution is shown in Figs. 5.7(a) and 5.7(b). 
The curves are unsymmetrical (in contrast to the curve of Fig. 5.1 for the 
velocity-component) and pass through the origin and only positive values 
of speed have any meaning. Writing the ratio 

$mc 2 /kT = a 2 

so that a is a dimensionless quantity proportional to the speed, the 
distribution law becomes 

PO]dff = ^<7 2 e~ ff2 d<7 

and this function is plotted in Fig. 5.7(a). All masses and temperatures are 
represented by this one graph. The area under the curve is unity — this 
follows directly from the normalization of the individual components 
and expresses the fact that the speed of a molecule is certain to lie between 
zero and infinity. We can graph P[c] directly however, if we select any 
given mass of molecule and any given temperature. Two such curves, for 
M = 28 and T = 100°K and 1,000°K respectively are shown in Fig. 5.7(b). 
The areas under these graphs are also unity. When the temperature is 
raised, the maximum of the curve moves to higher values of speed, as is 
to be expected. At the same time, the spread of speeds increases, the curves 
becoming broader. 

Three characteristic values of the speed can be usefully defined — the 
most probable speed where P[c] goes through a maximum, the mean 
speed and the root-mean-square speed. They do not differ very greatly 
from one another. 

The most probable speed c m can be found by setting dP/dc = which 
gives 

c m = (— ) • (5.9) 

The mean speed c is found (following section 4.2.1) from 
f 00 2 l2kT\ 1/2 

Jo cp[c]dc= >M - U28c - l5l0) 

The mean value of c 2 , called the mean square speed c 2 (which is useful in 
finding the mean kinetic energy and is not equal to the square of the mean 
speed), is given by 

3kT 

c 2 P[c]dc = . (5.11a) 

m 



c = 



-f 

Jo 



110 Maxwell speed distribution and the equipartition of energy Chap. 5 



P[&\ 




3x10 - 



2x10 




1x10 



cm/s 



Fig. 5.7. (a) The Maxwell distribution of speeds expressed in terms of the 
dimensionless measure of speed a = (mc 2 /2kT) 112 . (b) The Maxwell 
distribution for M = 28, T= 100°K and T = 1,000°K. The unit of 
speed is 1 cm/s. The probability that the speed of a molecule lies 
between 20,000 and 20,001 cm/s is 3 x 10~ 5 at 100°K; in 28 g of this 
substance, there would be nearly 2 x 10 19 such molecules. 



5.2 Speed distribution P[c] 111 

The square root of this quantity is called the root-mean-square speed 
c 

— /3/cT\ 1/2 

c rms = (c 2 ) 1/2 =p^ = 1.225 c m . (5.11b) 

The value of the mean kinetic energy of all the molecules follows from the 
mean-square speed : 

mean kinetic energy = \mc 2 = \kT. (5.12) 

This result, Eq. (5.12), is a special case of a very general theorem, the 
equipartition of energy, which will be dealt with at length later in this 
chapter. In view of its importance, it will be derived again in another way. 
Instead of finding the mean value of c 2 directly, we may proceed by finding 
the mean values of v 2 and v 2 and v 2 . Since the components are independent 
variables, the mean values are additive : 

c 2 = vl + v^+v^. (5.7) 

In outline, the calculation is as follows. The mean value of v 2 is given by 

_ r °° / m \ 1/2 f 00 

K = J v 2 x P[v J dv x = \—A J v 2 x exp( - mvl/lkT) dv x . 

The same integral has been encountered in section 5.1.2 in the calculation 
of the pressure of a perfect gas, although the limits of integration are now 
— oo to oo, instead of to oo. The result is that 

uj = kT/m. (5.13) 

The same expression holds for the other two contributions, so that once 
again 

c I =3kT/m (5.11a) 

and the mean kinetic energy is \kT. 

5.2.3 Magnitude of the characteristic speeds 

The magnitude of these average speeds can be calculated once (kT/m) 1 ' 2 
is known, or (RT/M) 1/2 . Putting R = 8.31 Jrnol" 1 deg -1 , and referring 
to nitrogen (M = 28) and T = 0°C = 273°K, this factor is 2.8 x 10 4 cm/s, 
so that the r.m.s. speed is nearly 5 x 10 4 cm/s. Since M varies between 2 
and 200 from the lightest to the heaviest element, these speeds all lie in 
the range 10 4 -10 5 cm/s for the elements, whether solids, liquids or gases, 
at room temperature. 

Sound waves consist of ordered movements of molecules, in which 
energy and momentum are propagated through the medium from 



112 Maxwell speed distribution and the equipartition of energy Chap. 5 

molecule to molecule superimposed on the random movements of the 
molecules. The ordered motion in a solid, for example, might consist of 
sinusoidal vibrations of the molecules in the direction of propagation or 
transverse to it, and being a collective mode of motion it can be separated 
from the random thermal motion on which it is superposed. It is shown in 
standard texts on wave motion that in all substances the speed of propaga- 
tion of the sound is comparable with the characteristic speeds of the 
Maxwell distribution. The calculation for gases is given in section 5.4.2. 
In air, for example, at 300° K the mean speed is 470 m/s, the speed of 
sound which consists of longitudinal vibrations only, is 350 m/s. In copper 
at the same temperature, the mean speed is 316 m/s, while the speeds of 
longitudinal and transverse sound vibrations through an unbounded 
volume of the metal are 456 and 225 m/s respectively. 

5.2.4 Experimental verification of the P[c] distribution 

The Maxwell distribution has been verified experimentally for gases. 
The most direct methods depend on two techniques — the production of 
molecular beams and the measurement of their speed distribution using a 
time-of-flight or chopper method, analogous to Fizeau's method for 
measuring the speed of light. The experiments of Lammert (1929) are 
typical. Mercury was heated to 100°C in an oven which had a small hole 
in one side through which the vapour could escape as a molecular beam. 
The entire apparatus was under high vacuum. Inside the oven the mole- 
cules were practically in equilibrium at 100°C since the rate of loss of 
molecules was small, so the speeds were distributed according to the 
Maxwell law. Once a molecule escaped through the hole its speed was not 
likely to change since it probably never collided with another molecule. 
Thus though the beam travelled through a space which was not maintained 
at the same temperature, it was a sample of those molecules inside the 
oven whose direction of travel happened to lie in the direction of the beam. 
Let us call this the x-axis — then the method produced molecules whose 
total velocity vector (of magnitude c) was parallel to x. It was the c- 
distribution of the beam which was measured ; it was not possible to 
measure the velocity component v x of molecules not travelling parallel to 
x, for such molecules were simply not in the beam. Inside the oven, any 
molecule travelling towards the hole with speed c could escape in time dt 
if it were within a distance c dt of the hole. Hence the number of such 
molecules escaping per second is proportional to cP[c], which means that 
the distribution function of the speeds of the molecules in the beam was 
of the type c 3 exp(-mc 2 /2kT), not the Maxwell distribution though 
clearly derived from it. 

The speeds were found by passing the beam through a velocity selector 



5.2 Speed distribution P[c] 



113 



consisting of two discs (Fig. 5.8(a)), each having 50 narrow radial slits in 
it, rotating rapidly on a common axis parallel to the beam. The disc 
further from the oven was turned through a small angle 5 with respect to 
the first. A molecule passing through a slit in the first disc and travelling 
with a speed c took a time l/c to travel the distance / between discs. If 
the speed of rotation of the discs on their axis was co rad/s, the molecule 
met a slit in the second disc if 8 = col/c ; or if the angular width of each 
slit was 2y, molecules with speeds in the range col/(S + y) to col/(S — y) could 
get through both slits; molecules with speeds outside this range were 
stopped, co could be varied so as to select different ranges of speeds. In the 
edge of the discs were wider slits of total angular width equal to that of all 
the narrow slits together, but these were so wide that they passed molecules 



W 



w 







+ 2y 



= =•.=•= = 


=■ = =r ^= = 


V 






J 



l/C 

/ 

it 



& 



\a) 



r^'-H 



Fig. 5.8. (a) Slotted wheel with slits acting as a velocity selector. Schematic 
layout: W — wheels (seen edge on), rotating on common axis; O — oven; 
C — cold surface. All located inside a high vacuum. 



20% 



10% - 



I 


- 1- ! 






1 


- ,—l L^ 

1 1 1 1 1 1— -L 1 fc 



4 x10 4 cm/s 



(b) 



Fig. 5.8. (b) The percentage of the total intensity of a molecular 
beam within given limits of speed. Mercury vapour at 100°C. 
calculated from Maxwell distribution. ob- 
served. Data from Lammert, Z. Physik 56, 244 (1929). 



114 Maxwell Speed distribution and the equipartition of energy Chap. 5 

of all speeds. The two emergent beams, selected and unselected, fell on a 
surface cooled with liquid air which trapped the molecules, and their in- 
tensities were compared by finding the times needed to produce deposits 
which were just visible. In one experiment, / = 6 cm, <5 = 4.18°, 2y = 
Tiro rad and the discs rotated at 70 rev/s so that the range between 340 and 
390 m/s was selected. The beam through the wide slits was just visible 
after 4' 40", the other after 51' 45" ; thus 9.0 % of all the molecules in the 
beam had speeds in this range. The complete plot is shown in Fig. 5.8(b), 
which gives the observed intensities and that predicted for a beam with 
M = 200, T = 373° K. Other methods of measuring the intensity of the 
beam have been used. 

It has not been possible to measure the speed distribution of molecules 
in solids directly. X-ray and neutron diffraction measurements can give 
information about the distribution of the amplitudes of vibration of 
molecules about their lattice points, but it would be necessary to know the 
frequencies of the vibrations to convert these measurements into speeds. 
Evidence of a different kind comes from the passage through solids of slow 
neutrons which suffer large changes of momentum whenever they collide 
with atoms. After several such collisions the neutrons come into thermal 
equilibrium with the solid. The speed distribution of an emergent beam 
(which can be determined by a time-of-flight technique) is always found 
to be of the Maxwell type appropriate to the temperature of the solid. 
Only if the atoms in the solid had the same distribution (though corre- 
sponding to their heavier mass) would this result be found. 

We will see later that specific heat measurements give an insight into 
the velocity and speed distributions in solids and that these show that the 
Maxwell distribution only holds at sufficiently high temperatures. At low 
temperatures, the motion of the atoms is not described accurately enough 
by classical mechanics. Quantum mechanics has to be used instead and 
this leads to different results. 



5.3 THE EQUIPARTITION OF ENERGY 

When we write down the Boltzmann factor for a particle in an assembly 
in thermal equilibrium, we have to know its total energy E. So far, we have 
made use of the fact that E can be split up into kinetic and potential energies 
which depend on different and independent variables, and by separating 
those variables we were able to deduce some useful results. One of these 
was (Eq. (5.12)) that when the kinetic energy of one molecule in an assembly 
is \mv 2 x +\mv 2 y +\myl, then the total kinetic energy of all N molecules in 
thermal equilibrium at temperature T is f JVfeT, and the mean energy of 
one molecule is f/cT. Thus the mean energy does not depend on v x or v y 



5.3 The equipartition of energy 115 

or v z in any way but only on the temperature T. It is the purpose of this 
section to show that this result is a special case of a much more general 
theorem, which is one of the most important results of classical physics. 

It is useful to begin by noting that so far we have not written down expli- 
citly all the forms of energy which a molecule can possess. (We will assume 
that external fields of force, such as the Earth's gravity, or electric or 
magnetic fields are absent ; this restricts the discussion but the results are 
nevertheless of significance). The only form of energy which has been 
explicitly used in calculations has been the translational kinetic energy 
of the centre of mass, consisting of the three terms of the type \mv 2 
mentioned above. We will now consider the energy of angular rotation 
which every molecule of finite size must possess, and the energy of vibration 
due to % internal oscillations inside a molecule containing more than one 
atom, or due to the oscillations of a molecule about its equilibrium position 
inside a solid lattice. Our procedure should really be to consider what 
conjugate momenta and position coordinates are needed to write down 
the Hamiltonian expression for the energy. Here we will quote some 
results without proof and we will use velocities rather than momenta. 

First let us consider a rotator, that is a rigid body of arbitrary shape, 
rotating without any constraint. Then its energy of rotation can be written 

E r = \l^\ +$I 2 oj 2 2 +i/ 3 e»5 (5.14) 

where the J's are the principal moments of inertia and the co's are the 
angular velocities about the three mutually perpendicular principal axes 
of the body. This is the kinetic energy term. We will not consider any 
potential energy which depends on the angular orientation of the body. 
Thus, if a molecule of a perfect gas can be considered as a rigid rotator, 
the energy must have six terms in it, three of the type \mv 2 x and three of 
the type jlco 2 . Classically, every co can vary between — oo and +00. 

Now let us consider a linear simple-harmonic oscillator, that is a point 
particle which oscillates about an equilibrium position along a line or, 
for example, a pair of particles whose oscillation about their centre of 
mass is in one dimension only. Then two coordinates, a displacement x 
and velocity v x are needed to specify the instantaneous state of the 
oscillator. The energy is partly kinetic, partly potential. For the single 
particle : 

E = jmv x +j(xx 2 

where a is the restoring force per unit displacement. It follows that a 
particle capable of simple harmonic oscillations in three dimensions, such 
as a molecule in a solid lattice, has six terms in the expression for its energy 
of oscillation. It is, in principle, possible for all the velocities and all the 



116 Maxwell speed distribution and the equipartition of energy Chap. 5 

displacements to take any values between + oo and — go. (Notice that E 
for the one-dimensional oscillator could be written as ?ooco where x is 
the amplitude ; the sum of the two terms is a constant. But this expression 
for the energy is not appropriate for the present purpose, since it is 
required to write it in terms of those coordinates which are needed to 
specify the instantaneous state of the body completely ; for an oscillator 
in thermal equilibrium, the amplitude is not constant. In just the same way, 
the translational kinetic energy must be written in terms of v x , v y and v z 
and not merely of c.) 

Typical terms in these expressions are 



\mv 2 x , ?Ico 2 , 



i<xx 2 



and all of them are of the same type, a constant times the square of a 
coordinate. Such terms are called degrees of freedom* 

Because of the similarity in the form of these terms we can say at once 
that angular velocities, for example, are distributed in just the same way 
as translational velocities. The probability that co 1? a single component 
of the angular velocity of a molecule in thermal equilibrium, has a value 
between co x and (a> x + da> 1 ) is 

/ / \ 1/2 
P\ca l '\dxo 1 = U^T exvi-lMfi kT ) da >i 

an expression completely analogous to P[v x ] dv x (Eq. 5.3). Further it was 
proved that the mean value of \mv 2 x was \kT (which follows from Eq. (5.13)). 
Exactly the same result must hold for the mean value of any of the other 
terms in the energy. The mean value of \l<£>\ for an assembly in thermal 
equilibrium must be jkT. Exactly the same result must hold for the mean 
value of \l(o\ and \l(X)\. The mean value of \a.x 2 for an assembly of oscil- 
lators in thermal equilibrium must also be \kT. 

This is the theorem of the classical equipartition of energy. Every degree 
of freedom, that is, every quadratic term in the energy — translational, 
oscillatory or rotational — contributes y/cTto the mean energy. An assembly 
of N particles in thermal equilibrium, each with/ degrees of freedom has a 
mean energy per particle of \fkT, a total energy oijfNkT. 

The implications may be stated in another way. Consider an assembly 
of molecules, each one capable of several sorts of motion (oscillation about 

* It has become customary in statistical mechanics to use the phrase 'number of degrees of 
freedom' in this way, namely to mean the number of quadratic terms in the energy. The 
reader should be warned that this is in conflict with the more usual definition in ordinary 
mechanics which restricts the number of degrees of freedom to the number of kinetic energy 
terms. 



5.4 Specific heats C p and C v 117 

a lattice point together with rotation about one or more axes, for example : 
or translation throughout a volume together with one or more modes of 
internal oscillation of each molecule). Then according to the equipartition 
theorem, in thermal equilibrium each possible mode of motion will be 
excited and the amount of energy in each mode is predictable if the tem- 
perature is known. 



5.4 SPECIFIC HEATS C p AND C v 

One purpose of the next section is to describe experiments which provide 
the most direct tests of the validity of the theorem of the equipartition of 
energy and of the classical mechanics on which it is firmly based. The 
mean energy of each molecule in a mass of material cannot be measured 
in any direct experimental way, but specific heats can be measured and 
these are closely related to it. Our first task will be to define specific heats 
and to develop some relations between them, then to describe the experi- 
mental results. It will emerge that the theorem of equipartition of energy 
is not of universal validity, and that this is because classical mechanics 
cannot adequately describe atomic vibrations and oscillations under all 
conditions and must be replaced by quantum mechanics. Historically the 
failure of the equipartition theorem to predict the correct specific heats of 
gases was one of the first symptoms of the inadequacy of classical mech- 
anics ever to be observed. 

If dQ is the quantity of heat energy supplied to a standard mass of a 
substance under certain conditions and the temperature is thereby raised 
by dT, then the specific heat is defined as dQ/dT under those conditions. 
The most convenient standard mass is the mole and the appropriate units 
are J/mol deg. Grams can be used instead of gram mols, and it is occasion- 
ally useful to deal with the specific heat per unit volume of a substance. 

Practically all bodies when heated under conditions of constant 
pressure will expand. This expansion absorbs energy in two possible ways. 
First the external forces acting on the body are pushed back. At the same 
time, the mean distance between the atoms of the body increases and the 
potential energy of interaction between atoms is changed — under normal 
conditions it is increased. It is therefore useful to define two limiting sets of 
conditions under which specific heats can be measured — constant volume 
when the body is constrained by external forces not to expand and all the 
heat energy goes into raising the temperature, and constant pressure where 
some of the energy is absorbed by the process of expansion. The specific 
heat at constant volume is denoted by C v , the specific heat at constant 
pressure by C p . 



118 Maxwell speed distribution and the equipartition of energy Chap. 5 

5.4.1 The difference (C p - C v ) 

C v is the quantity which is more readily calculated theoretically for any 
assembly of atoms or molecules since potential energies of interaction 
between atoms are constant under conditions of constant volume. In fact 
if E is the mean energy of a standard mass of material, equal to the mean 
energy of one molecule multiplied by the number of molecules, then 

C p is the quantity most easily measured experimentally since laboratory 
work is usually carried out under conditions of constant pressure. The 
difference between the two quantities, (C p -C v ), is therefore of significance, 
to compare experimental measurements with theoretical predictions. 

For a perfect gas, (C p — C v ) is easy to calculate because the potential 
energy of interaction between the atoms is always zero. The energy E, 
which in thermodynamics is conventionally called the internal energy, 
therefore does not change when the gas expands at constant temperature. 
The extra energy absorbed by the expansion at constant pressure must be 
equivalent only to the work done in overcoming the external pressure. 
The energy required to produce a given temperature rise under conditions 
of constant pressure is equal to that needed to produce the same tem- 
perature rise when the volume is kept constant, plus the energy absorbed 
as work done against the external pressure. It has already been shown 
(4.1.3) that if a body expands in volume by dV against pressure p, the 
work done is p dV (writing the A dx of 4.1.3 as dV). Thus 

C p dT= C v dT+PdV 



or 



p 




C n -C=P—\ (5.16) 



It must be stressed that this holds only when the internal energy of the 
substance does not change with volume. For a perfect gas, PV = NkT, 
whence it follows that 

C p -C v = Nk = R. (5.17) 

Different gases can have appreciably different values of the molar specific 
heats C p and C„, but the difference (C p -C v ) for any one gas must always 
be equal to 8.31 J/deg, at all temperatures. 

For a substance whose internal energy varies with volume, the difference 
(C p - C v ) must also vary with volume, and must involve the compressibility 



5.4 Specific heats C p and C v 1 19 

and expansion coefficient. It is shown in standard thermodynamic texts 
that 





~ IdE \ 


dV 


c — c = 


P+\t-\ 


— 


V 


_ \dV) T _ 


\dT 



and using the second law of thermodynamics it may be shown that this 
can be written 

C p -C v = TP 2 KV (5.18) 

where ft is the volume coefficient of thermal expansion, K the isothermal 
bulk modulus : 

o 1 l dV \ 

'-vM, (519) 

K= - V (%) T < 310 > 

and V is the molar volume if C p and C v are molar specific heats. Inserting 
typical figures, (C p - C v ) for most liquids and solids at room temperature 
are of the order of R/3 and R/10 respectively. Though the volume expansion 
is very small, the pressures which would be needed to counteract it are 
very large so that their product is by no means negligible. For very hard 
substances such as diamond, (C p - C v ) is however very small, of the order 
offl/1,000. 

Experimental methods for measuring C p for solids and liquids are 
straightforward in principle. A known quantity of heat is introduced into 
the specimen, usually from an electric heater in good thermal contact, and 
the temperature rise measured, usually with a thermocouple or resistance 
thermometer. It is essential to reduce heat losses from the specimen as 
much as possible. The specimen is isolated in a high vacuum and the 
surroundings are heated separately so as to follow the temperature of the 
specimen as closely as possible. It is also essential to make sure that the 
temperature is uniform throughout the specimen before any readings are 
taken. From the measurements of C p and a knowledge of the expansion 
coefficient and compressibility, C v can be calculated. 

5.4.2 Ratio of specific heats C p /C v 

The methods just described are not very suitable for taking measure- 
ments on gases, particularly at the low densities required for them to 
behave like perfect gases. Their specific heat per unit volume is small, and 
the heat absorbed by the containing vessel may be comparatively large. 



120 Maxwell speed distribution and the equipartition of energy Chap. 5 

While it is therefore difficult to measure either C p or C v directly, it is 
however simple to measure their ratio 

7 = C p /C v . (5.20) 

A knowledge of y, together with the fact that (C p — C v ) = R, allows both 
C p and C v to be found : 

R y R 

c — c = — 

y — 1 v y—\ 

y is measured by studying adiabatic changes in a gas. 

Adiabatic changes are processes (such as changes of volume) which 
occur so slowly that the assembly of molecules never departs far from 
thermal equilibrium even though a finite time is required to reestablish 
the Boltzmann distribution disturbed by the change ; and at the same time 
no heat must flow into the body, d<2 = 0. The adiabatic compression of a 
gas must be a slow process in which heat cannot escape or enter and since 
work is done on the gas its temperature must rise. In practice, a compromise 
has to be struck between the need for slowness of change and the require- 
ment of thermal isolation, and it is usually necessary to accomplish any 
changes quite quickly. 

Sound waves through a gas are adiabatic. They consist of local alterna- 
tions of compression and rarefaction which produce small local alterations 
of temperature and these are not dissipated. It is shown in standard texts 
on wave motion that the speed of a sound wave is given by 

//bulk modulus \ 
Speed = 7( density ) < 3 ' 21) 

where the bulk modulus is — V(dP/dV) under adiabatic conditions. We 
now calculate this quantity for a perfect gas. 

Consider 1 mole of a gas which does work by changing its volume and 
into which heat also flows, so that the pressure, volume and temperature 
all change infinitesimally. Then 

dQ = C v dT+PdV (5.21) 

Now the relation PV = RT also holds, so that 

PdV+VdP = RdT 
and we can use this to eliminate dT: 

dQ = -^(PdV+VdP) + PdV. 
R 



5.4 Specific heats C p and C v 121 

Using the relation C p -C v = R, this becomes 

RdQ = C p P dV+C v VdP. 
If dQ = 0, the adiabatic bulk modulus 

~ v Itt] =< ^P = yP> (5-22) 



so that the speed of sound is given by 

Measurements of the speed of sound therefore give y. At the same time, 
the reason that the speed of sound is comparable with but not equal to 
the mean speed of the molecules, Eq. (5.10), can be appreciated, since they 
differ only by a factor y/{ny/S) which lies between 0.75 and 0.8 for most gases. 

The speed of sound can be measured from the transit time of a pulse of 
sound (such as a gunshot) between two points a known distance apart, or 
from simultaneous measurements of frequency and wavelength of a 
sinusoidal note. Ruchhardt's method is an interesting alternative way of 
finding the adiabatic bulk modulus yP directly. A resonator consisting of 
a vessel of a few litres' capacity and filled with the gas under examination 
is fitted with a vertical glass tube into which a steel ball fits closely, Fig. 5.9. 
The ball must be accurately spherical, the tube accurately cylindrical and 
the clearance between the two must be of the order of 0.001 cm so that if 
the ball is at the top of the tube it sinks only very slowly due to leakage of 
gas past it. If the ball is given a sudden displacement from its position of 
near equilibrium it oscillates, typically with a period comparable with one 
second. The restoring force is the adiabatic elasticity of the gas in the vessel. 
The system may be considered as a Helmholtz resonator with the ball 
acting as a heavy driving piston. 

If the displacement is x, the change of volume is Ax, where A is the 
cross sectional area of the tube. Then from the definition of elasticity, the 
change in pressure is 

dV yP 

dP = -yP- = -^A X 

where Kis the volume of the gas. The force on the ball is the pressure times 
the area 

yPA 2 

F = — — x 



122 



Maxwell speed distribution and the equipartition of energy Chap. 5 



which shows that the motion is simple harmonic, of period 

,. Vm 

x = 2n 



yPA' 



where m is the mass of the ball. Observation of the period therefore gives y. 



O-T— 



ft 



direction of 
displacement x 
and of positive 
forced 



— ' rest position 



(a) 



[b) 



Fig. 5.9. (a) Simplified version of Ruchhardt's apparatus, {b) Ball dis- 
placed from rest position. 



5.4.3 Results of specific heat measurements 

The previous sections have described the principles behind the methods 
of measuring specific heats and how C v can be deduced from them. For 
solids and liquids the most easily measured quantity is C p and then 
{C p — C v ) can be calculated knowing the expansion coefficient and com- 
pressibility. 

For gases at low pressures the most easily measured quantity is C p /C v 
and since (C p — C v ) always has the value R, C v can be calculated. Thus, 
since C v = (dE/dT) v we have a searching method for testing the truth of 
the law of equipartition of energy. 

We will deal with the results for solid elements first. The predicted molar 
specific heat for a solid having one atom (or ion or rigid molecule) at each 



5.4 Specific heats C p and C v 123 

lattice point, capable of oscillating harmonically in three dimensions is 

C * = N [£ffr T ) = 3R (524) 

which has a value near 25 J deg" 1 mol~ * at all temperatures. This rule is 
in fact obeyed by a large number of solid elements at room temperature. 
It was discovered empirically in the early nineteenth century by Dulong 
and Petit and was even used as a guide in determining atomic weights in 
the days when there was some ambiguity about them. But at low tempera- 
tures, the specific heat always falls below this predicted value. The graphs 
for all elements have the same form and can be superposed by merely 
altering the horizontal scales ; Fig. 5.10 shows the curves for three elements. 
At high temperatures, copper and lead have their expected specific heats 
of 3R. At 300° K, however, the value for copper begins to fall off, at 150°K 
it is decreasing rapidly with temperature, and at 50° K it is very small 
indeed. Argon does not begin to show a decrease until quite low tempera- 
tures are reached. Diamond follows the same sort of curve except that it 
would only approach its full value of 3R if the graph were extrapolated to 
almost 2,000° K. These results for low temperatures are in sharp conflict 
with the equipartition law. 

The same kind of discrepancy occurs for gases as well. Atoms have finite 
sizes so that they must each have three principal moments of inertia. An 
atom of a monatomic gas should therefore have three quadratic terms for 
its energy of rotation of the type ^hcoi as well as three terms of the type 
\mv\ for its translational energy. Thus there should be 6 degrees of free- 
dom per atom each contributing \kT to the energy, so that C v should 
be equal to %R and y = f = 1.33. In fact helium and other rare gases 
including argon, as well as mercury, all of which are monatomic, are found 
to have y quite close to 1.67 so that C v = f.R. Evidently three of the 
possible degrees of freedom are not excited. 

5.4.4 Quantum theory and the breakdown of equipartition 

These evident failures of the equipartition theorem mean that some of 
the ideas of classical mechanics itself are at fault. For solids, the explanation 
of the small specific heats at low temperatures hinges on the fact that the 
frequency of vibration of the atoms or molecules or ions about their lattice 
positions is very high, of the order of 10 12 or 10 13 cycles per second (see 
sections 3.6.2, 3.7 and 3.8.2 for estimates of the Einstein frequency). Now it 
is a result of quantum theory that the energy of an oscillator of frequency v 
can only take discrete values, jhv, § hv, f hv and so on, where h = 6.6 x 10" 27 
erg. s or 4 x 10~ 15 eV. s is Planck's constant. We no longer separate the 
energy into potential and kinetic contributions, but talk only of the total 
energy and this can only change by discrete amounts hv. 



124 



Maxwell speed distribution and the equipartition of energy Chap. 5 



3/? 



2/? 



Equipartition holds 







Copper 


/Lead 


/ / 






'/ 








100 200 

Temperature 
(a) 



<3* 



50 °K 



Temperature 
(6) 



300 K 



3/? 




Equipartition holds 








2/? 

R 
n 



















£ 



2/h/ 
A 



k 



Temperature 
(c) 



Fig. 5.10. (a) The specific heat C„ of lead and copper as a function of tempera- 
ture, (b) Specific heat data for solid argon. C v was derived from the measured 
C p using Eq. (5.18); this can be checked using K (the reciprocal of the com- 
pressibility) and V from Fig. 3.13 and /? derived from the density data : 



T°K 



20 



40 



60 



80 



jSdeg" 



0.4 



1.2 



1.5 1.8 xlO -3 



(c) Curves for C v for all substances can be superposed by altering the horizontal 

scale as shown; the Einstein frequency for copper and lead can then be 

estimated by comparing the two curves. 



5.4 Specific heats C p and C v 125 

In solid organic crystals or solidified rare gases or other soft substances, 
where v E is of the order of 10 12 cycles per second, the energy of a vibrating 
molecule can only increase in steps of about 7 x 10 ~ 15 erg or 0.004 eV. At 
very low temperatures (when kT is small compared with this quantity, so 
that the factor exp(-hv E /kT) representing the probability of a molecule 
being in the next higher state is small ; to be precise, when T is small com- 
pared with 40°K), the oscillators can vibrate only in their lowest state. The 
energy of oscillation is therefore hardly changed by changing the tem- 
perature, so the specific heat is very small. At high temperatures (say 
400° K in this case), the discreteness of the vibrational energy levels makes 
little difference to the mean amount that the oscillators can take up, and 
the classical law holds. In hard substances like ionic crystals or diamond, 
where the Einstein frequency is 10 or 50 times greater than in soft crystals, 
the specific heat is low even at room temperature. 

To explain the unexpectedly low specific heats of gases at ordinary 
temperatures we have to invoke the fact that the energy of a rotating body 
is quantized. It can only possess discrete values and can only change by 
discrete amounts. For monatomic helium molecules, the steps are of the 
order of 10 eV in size ; thus the minimum energy of rotation that a helium 
atom may possess is of this order. If its total average energy is very much 
less than this amount, it is highly improbable that it should be rotating 
at all. Since kT at room temperature is of the order of ^j eV, it is evident 
that rotation of the atoms in helium at room temperature does not take 
place. For monatomic mercury, the steps are of the order of 100 times 
smaller, but still the thermal motion is not sufficient to excite a significant 
number of mercury atoms into rotation. Thus they only possess three 
translational degrees of freedom and the ratio of specific heats is 1.67. 

We have just discussed monatomic gases ; there is disagreement with 
the classical equipartition theorem for diatomic molecules also. Many 
such molecules — like hydrogen, oxygen, chlorine — are dumbell shaped, as 
in Fig. 5.11. We would expect each molecule to be capable of translation 
(three energy terms of the type \mv\ where m is the mass of the molecule 



OO db -OOe 

(a) [b) (c) 

Fig. 5.11. Modes of rotation of a dumbell-shaped diatomic 

molecule. Specific heat measurements indicate that (a) and 

(b) occur, (c) does not. 



126 Maxwell speed distribution and the equipartition of energy Chap. 5 

and v x is the velocity component of its centre of mass), as well as rotation 
(three terms of the type jliCoj) and also vibration in and out along the 
line joining the two atoms as if they were connected by a spring (two terms, 
for potential and kinetic energies as for any harmonic oscillator). This 
gives a total of 3 + 3 + 2 = 8 possible degrees of freedom corresponding 
to y = ^ = 1.25. In fact, it is observed that many diatomic gases have y 
close to 1.4 (that is, j) at room temperature which seems to imply that 
only 5 degrees of freedom are excited at room temperature. Thus, the 
value for H 2 is 1.408; for N 2 it is 1.405, for NO 1.400 and for 2 1.396. 
Again, in order to explain these data we invoke the quantization of energy. 
The rotation (c) resembles that of a single atom and is again eliminated 
because the minimum energy of rotation is much larger than kT at 
ordinary temperatures. Of the 8 degrees of freedom, 7 remain ; evidently 
2 more must be eliminated. It is not obvious whether these are the two 
terms in the spring-like internal vibration, or the two rotations (a) and 
(b) in Fig. 5.11. An exact analysis in fact shows that the vibration is of 
very high frequency and it is the energy of that motion which is not equi- 
partitioned ; the two rotations (a) and {b) are excited. 

The quantization of energy is of course not invoked only to explain the 
specific heat data ; other phenomena are also explained at the same time. 
For gases, the most direct confirmation comes from their spectra, from 
the interpretation of which it can be confirmed that certain rotations and 
internal vibrations do not take place at ordinary temperatures. 

We have seen that at temperatures where kT is small compared with 
the energy steps, the value of C v is lower than predicted on classical equi- 
partition theory. We would however expect that at sufficiently high tem- 
peratures, more modes of rotation or vibration will be excited. An increase 
of C v and a decrease of y at high temperatures is therefore to be anticipated. 
This is indeed found experimentally — notably with hydrogen whose C v 
changes from jR below 50° K to %R at room temperature. This must mean 
that at 300° K, 5 degrees of freedom are excited but at 50° K only 3. Further, 
it must mean that two of the modes of rotation or internal vibration must 
have energy steps whose magnitude is comparable with kT where T is 
100°K or 200°K — say 0.01 eV. To take another example, chlorine is ob- 
served to have y equal to 1.355 at room temperature and this corresponds 
neither to 5 degrees of freedom (y = 1.4) nor to 6 degrees (y = 1.33). 
Presumably at room temperature 5 of its degrees are fully excited and 
another is partially excited and the value of y ought to decrease to 1.33 
at temperatures not too far above room temperature. 

Among polyatomic molecules, H 2 S has y = 1.340 which is close to the 
value f , corresponding to 6 degrees of freedom; this is taken to mean 
that the internal vibrations are not excited but all 3 of the rotational 



5.5 Activation energies 127 

modes occur. Larger polyatomic molecules have low values of y showing 
that some internal vibrations are excited, presumably because their 
frequencies are sufficiently low. 

Our detailed study of specific heats demonstrates the power as well as 
the limitations of the equipartition of energy theorem. We may restate this 
theorem as follows. In an assembly in thermal equilibrium at temperature 
T, each degree of freedom is excited and contributes jkT to the total 
energy provided that kT is large compared with any energy level spacings 
which an exact quantum-mechanical analysis shows must exist. If the 
equipartition theorem gives a result which is in conflict with experiment, 
then this gives some insight into the energy spacings of the modes of 
motion which are not excited. 

5.5 ACTIVATION ENERGIES 

We will now use the Maxwell speed distribution to derive a result of 
general applicability in chemistry as well as physics. It concerns activation 
energies. 

It frequently happens that a system can lose energy provided it can first 
overcome a barrier. For example, a molecule sitting on the surface of a 
solid in site X (Fig. 5.12(a)) may be able to find a site Y where its energy 
is lower. But in order to get there it has to jump or roll over an intervening 
molecule. Its interatomic potential energy as a function of position there- 
fore resembles that in Fig. 5.12(b). A , the height of the barrier which 
must be jumped, is called the activation energy. 

Chemical reactions are often characterized by activation energies. It 
often happens that two substances can be mixed together without reacting 
but if they are heated a reaction starts. This reaction may itself give out 
heat. A well-known example is provided by iron and sulphur which can 
exist as a mixture at room temperature without undergoing any change, 
but which when heated react together to form iron sulphide — the reaction, 
once started, giving out much heat. 

The interatomic potential energy as a function of the distance between 
the centres of two molecules must be of the type in Fig. 5.12(c). As they 
approach they repel one another (compare Fig. 3.3(a) and (b)) but if they 
can overcome the barrier of height A then they can stick together. This 
complex molecule is presumed to initiate or to be the product of the 
chemical reaction. 

We wish to calculate the probability that this takes place. Let us for 
simplicity assume that the reaction takes place in the gas phase — that is, 
the two reacting gases F and G are mixed together. Whenever a molecule 
of F approaches a molecule of G with a kinetic energy greater than A , 



128 Maxwell speed distribution and the equipartition of energy Chap. 5 

the two can react. Having fallen into the well, the complex molecule must 
then lose its excess energy but we will assume that this process can take 
place easily. We call this a reactive collision. 






X Y Position 

(b) 

'X 



Distance between centres 
(c) 

Fig. 5.12. (a) and (b) A molecule can lose energy by transferring from site X to site Y 
but needs activation energy A in order to do so. (c) Potential energy of two molecules 
which can undergo a chemical reaction. 



The problem of calculating the relative speeds of two molecules (and 
hence the kinetic energy which one imparts to the other on collision) is a 
complicated one — some of its aspects are dealt with in the next chapter. 
Let us therefore make two gross simplifying assumptions, firstly that all 
the molecules are stationary except for one F molecule which is moving 
with some arbitrary speed c, and further that whenever it meets a G 
molecule it hits it head on. The effect of these assumptions may (as usual) 
be expected to make the result incorrect only by a factor of order unity 
but to leave the form of the result correct — and it is the form which is 
important here. 

Let there be n G molecules per cm 3 of the species G. In one second, the 
one moving F molecule travels a distance c. If its area of cross-section is a 
(a quantity which is defined more exactly in the next chapter) it sweeps 
out a volume ac in one second. In this volume there are n G ac molecules 
of the other gas ; this is therefore the number of collisions it makes in one 
second with molecules of G (and this result remains true even if the path 
is not a straight one). The probability that one molecule is indeed moving 



5.5 Activation energies 129 

with speed between c and (c + dc) is given by P[c] dc, Eq. (5.8), so that if 
the one F molecule follows the Maxwell distribution it will probably 
encounter 

n G (TcP[c] dc 

molecules of G in one second at a speed between c and (c + dc). 

If now we imagine n F molecules of F per cm 3 , all of which we can treat 
in the same way, the number of such encounters is 

n F n G acP[c] dc 

in each cm 3 per second. 

Now let us put in the condition that the kinetic energy \mc 2 must be 
equal to A or exceed it, that is c ^ (2A /m) 1/2 . The number of reactive 
collisions per unit volume per second is 



/•OO 

n R = n F n G acP[c\ dc 

J /2A 
V m 

= 4™ F n G <x(^) ' J 



c 3 Q- mc2/2kT dc 



/ 2A 
V m 



after writing P[c] in full, and rearranging. 

When realistic modifications are made to allow for the facts that all F 
and G molecules are moving so that we must calculate the relative kinetic 
energies, and that not all collisions are head-on, the result happens to be 
identical with this expression except that a reduced mass m F m G /(m F + m G ) 
must be used in place of the m. 

We can substitute x = (mc 2 /2kT) as the variable. This gives 

A mC A 

dx = — — ■ dc 
kT 

IkTV 12 f°° 
n R = 2 3l2 n F n G <r\ — xe x dx. 

R \TimJ J Ao/kT 

This can be integrated straightforwardly (it is left for the student as an 
exercise in integration by parts) and the result is a sum of two terms of the 
form {kTy l/2 exp(-A /kT) and (/cT) 1/2 exp(-A /kT). The important 
point is that the result contains the factor exp(- A /kT). Because of the 
rapid variation of exp(X) with X, the variation of the exponential factor 
dominates the variation of the number of reactive collisions ; over the 
fairly narrow temperature interval usually encountered, the other factors, 



130 Maxwell speed distribution and the equipartition of energy Chap. 5 

T~ 1/2 and T 1/2 , vary only slowly by comparison and can be regarded as 
roughly constant. The presence of the factor exp(- AJkT) in the expres- 
sion for the number of molecules which can jump a barrier of height A 
is an important result. We have deduced it for a simplified model of a 
chemical reaction in the gas phase but it is valid for any system where 
the Maxwell distribution holds. In the system shown in Fig. 5.12(a) and (b) 
for example, the number of molecules which can jump over the barrier of 
height A is proportional to exp( — A /kT). 

If we can observe experimentally the number of activated molecules (or 
some macroscopic property which is directly proportional to this number) 
at different temperatures, then we can deduce the activation energy. The 
easiest way is to use a graphical method, plotting the number on a logarith- 
mic scale against 1/T. This gives a straight line whose gradient is — A /k. 
When measuring the gradient it is important to remember that an increase 
of a number by a factor 10 adds 2.303 to its natural logarithm. 

The rates of very many chemical reactions have been measured and 
analysed in this way to find the activation energy. Our simple theory 
corresponds, in chemists' language, to a bimolecular reaction ; they express 
the rate of chemical change in terms of a 'velocity coefficient' K, which is 
proportional to the constant connecting n R with the product n F n G in our 
notation. Curves of log K against 1/T and of log(n R /n F n G ) against 1/T 
therefore have the same gradient. 

To take one example, the reaction 

CH 3 I + C 2 H 5 ONa=CH 3 OC 2 H 3 + Nal 

in alcohol solution was one of the earliest to be measured accurately 
enough to give consistent results. The concentration of the reactants was 
determined to begin with, and by measuring the concentration of one of 
them after the lapse of several minutes the speed of reaction was measured 
and hence the velocity coefficient obtained. A graph of K on a logarithmic 
scale against 1/Tis shown in Fig. 5.13. Note that the reaction rate increases 
by a factor 40 while the temperature increases only by 10 % from 273°K 
to 303°K. The slope indicates that the activation energy is 8.1 x 10 4 J/mol 
or about 0.85 eV/molecule, an amount comparable with the ionization 
energy of sodium or iodine. 

If we allow ourselves to take a highly simplified view, this fact tells us a 
good deal about the molecules themselves and the complex which must 
be formed when the two reacting molecules are in the act of colliding. If 
the Na in the C 2 H 5 ONa were in the form of an ion Na + and the I were 
present as I" in the other reacting molecule, then we would expect an 
attraction, not a repulsion, between the two molecules. The bonds which 
link the Na and I atoms inside their respective molecules cannot therefore 



5.5 Activation energies 131 

be ionic but are in fact covalent. During the collision the electrons must 
be redistributed inside the molecules so as to ionize these atoms — which 
requires about 1 eV of energy — and sodium iodide can be formed. 



200 



100 - 



S 1 




3.7 



x10" 3 VT 



Fig. 5.13. Plot of velocity coefficient K (on logarithmic scale) against 1/Tfor 
chemical reaction in solution. The temperature varies from 303°K at the 
left to 273°K at the right. An increase of K by a factor 10 increases log e .K by 
2.303. Data from Hecht and Conrad, Z. Physik. Chem. 3, 450 (1889) reworked 
in Moelwyn-Hughes Kinetics of Reactions in Solution, Oxford 1947. 



PROBLEMS 

5.1 The length of the metre is defined in terms of the wavelength of an orange line 
in the spectrum of the krypton isotope of mass 86 a.m.u., a line which under 
certain conditions is very narrow. The wavelength is denned to be 6057.8022 A. 
With a lamp immersed in liquid nitrogen at 63°K, the width of the line for the 
intensity to fall by a factor e is 0.0037 A. (a) Estimate how much of this width 
is due to Doppler broadening, (b) What would be the width if the lamp were 
run at 1,000°K? 

5.2 (a) A spherical planet of mass M has some gas molecules near it. Write down 

an exact expression for the force on a gas molecule at a large distance r from 
the centre of the planet, and hence for the potential energy. (Use G for New- 
ton's universal constant of gravitation.) 



132 Maxwell speed distribution and the equipartition of energy Chap. 5 

(b) The planet and the molecules are in thermal equilibrium at temperature T. 
Write down the Boltzmann factor for a molecule at distance r from the 
centre of the planet. Get the sign of the exponent correct. 

(c) Write down the volume of a spherical shell bounded by radii r, (r+dr), and 
hence calculate the probability of finding the molecule between these limits. 

(d) What is the value of this expression when r is infinite? 

(e) This means that one of the following is true : 

(i) the density of the atmosphere decreases exponentially with r 
(ii) the density of the atmosphere cannot reach equilibrium 
(Hi) the density of the atmosphere is zero outside a certain radius 
(iv) the density of the atmosphere decreases as 1/r. 

5.3 Prove that the height of the centre of gravity of an atmosphere is equal to its 
scale height. (Assume a plane Earth, g = constant.) Write down an expression 
for the total energy (kinetic energy of the molecules plus potential energy due 
to the Earth's gravity) for one mole of an atmosphere of a monatomic perfect 
gas. Hence prove that its specific heat is f R. 

5.4 At ordinary temperatures, nitrogen tetroxide (N 2 4 ) is partially dissociated 
into nitrogen dioxide (N0 2 ) as follows : 

N 2 4 ^2N0 2 

0.90 g of liquid N 2 4 at 0°C are poured into an evacuated flask, of 250 cm 3 . 
When the temperature in the flask has risen to 270°C, the liquid has all vaporized 
and the pressure is 960 mm Hg. What percentage of the nitrogen tetroxide has 
dissociated? 

5.5 Calculate the escape velocity from the Earth's gravitational field. Calculate the 
r.m.s. velocity of helium atoms at room temperature. According to some 
theories, the Earth's atmosphere once contained a large percentage of helium. 
Explain the fact that helium is now a rare gas. (Assume that the Earth's tempera- 
ture has remained constant at all times at 300° K, molecular weight of helium = 
4.) 

5.6 This problem gives an interesting insight into the energy relations in adiabatic 
expansions. A piston moves with constant velocity and each molecule under- 
goes a kind of 'Doppler' change of velocity after reflection from it. 

(a) A molecule of mass m approaches a wall with velocity u x and is specularly 
reflected. The wall moves with velocity £, as shown. Which of the following 
expressions is correct for the velocity after reflection? 

-(u x -2£) -(u,-£) -(U. + S) -(u x +2£). 

Write down the kinetic energy after reflection, assuming that £ is very 
small compared with u x , so that £ 2 can be neglected. 



^ U x £. 

O * — ► £■ 

I 



Problems 133 

(b) Consider a molecule travelling in an arbitrary direction towards the wall, 
with C still in the x-direction. Following an argument like that in section 
5.1.2, write down the number of impacts on area A of the wall in a small 
time dt, undergone by molecules whose x-component of velocity is between 
u x , (u x +du); hence the change of kinetic energy of this class of molecule in 
this time. Hence find the change of kinetic energy of all molecules. Express 
the increase of volume dV in terms of the distance moved by the wall, etc. 

(c) Assume the gas is perfect and monatomic. During the small time, the volume 
is practically constant Write down the heat capacity of the gas in terms of 
the specific heat per molecule and the total number of molecules ; do not 
confuse number of molecules/cc with total number of molecules. Hence 
calculate the change of temperature dT. 

(d) Show that dT/T + 2 dV/lV = 0, and that this is consistent with the PV y law 
of adiabatic expansions. 

(e) Write down the analogous equation for a polyatomic perfect gas. 

(/) What modifications to the discussion are needed if the molecules are not 

assumed to be reflected specularly? 
(g) Describe briefly how the ordered motion of the molecules after reflection is 

degraded into thermal motion (i) in microscopic terms (ii) in macroscopic 

terms. 

5.7 Consider a magnetic dipole in a magnetic field. Using the coordinate system 
of Fig. 5.6(a) let the vertical axis represent the direction of the field, and the 
radius vector represent the dipole. From Fig. 5.6(b), it can be seen that for a 
sphere of unit radius constructed about the origin, an element of area generated 
by dd, d<j> is sin dd d(j>. 

The potential energy of a dipole of moment m at an angle 6 to a field H is 
— m fx H cos 6. 

For an assembly of N independent dipoles at temperature T : 
(i) Write down the probability that a dipole is oriented between 6, (6 + dd), </>, 

(ii) Normalize this expression by integrating over all possible values of 6 and </> 

and equating this probability to 1. 
(Hi) Write down the probability of a dipole being oriented between 6, (6+dd), 

irrespective of the angle $. 
(iv) Each dipole has a moment m cos 6 parallel to the field. Write down the 

contribution from those of AT dipoles which are oriented between d, (9+ dd). 
(v) Hence find the total magnetic moment from all N dipoles. Show that it 

tends to Nmln H/3kT when (m n H/kT) is small. 

(Note : |(e x + e " x ) = cosh x ; ^(e* - e " x ) = sinh x ; the limit of (coth x - 1/x) 

when x is small is x/3.) 

5.8 Crystals of sodium chloride show strong absorption of electromagnetic radia- 
tion at wavelengths of about 6xl0 -3 cm. Assuming this to be due to the 
vibrations of individual atoms, calculate (a) the frequency of the vibrations ; 
(b) the potential energy of a sodium atom as a function of its distance d from its 
equilibrium position, assuming the vibration to be simple harmonic; (c) the 
probability distribution of 5 at T = 400°K; (d) the r.m.s. value of S at 400° K. 
The atomic weight of sodium is 23. 

5.9 A galvanometer mirror is suspended on a thread, inside a box containing air. 
Its moment of inertia for torsional swinging is /. The torsion constant of the 
fibre is n. 6 is the angle of torsion, co the corresponding angular velocity. While 
swinging, the total energy (given by jla> 2 +jnd 2 ) is constant. 



134 Maxwell speed distribution and the equipartition of energy Chap. 5 

(a) How many degrees of freedom does the system have as regards this motion? 

(b) What is the total energy of random swinging of the mirror? 

(c) If /i = 10" 6 dyn cm, what is the r.m.s. value of the angle of deflection in 

radians? 

(d) Which of the following mechanisms produces these movements? 

(i) expansion and contraction of the mirror, varying J 
(ii) radial expansion and contraction of the fibre, varying n 
(Hi) thermal motion of screw dislocations in the fibre 
(iv) random collisions of air molecules with the fibre, interchanging spin 
(v) random collisions of air molecules with the mirror. 

(e) The box is evacuated so that there are very few molecules in it. The box 
remains at temperature T. Which of the following values does the r.m.s. 
deflection now have? 

(i) zero 

(ii) reduced in the ratio of the pressures 
(Hi) same as before 
(iv) increased in ratio of the pressures 
(v) infinite. 
(/) What is the mechanism which produces the movement now? 
5.10 The molecules of a substance are known to consist each of two atoms, rigidly 
fixed to one another, like a dumbell. The mass of each atom is 10 a.m.u. so that 
the molecular weight is 20 a.m.u. Each atom consists of a heavy but extremely 
small nucleus containing practically all the mass, surrounded by a larger 
spherical 'cloud' of electrons whose total mass is only about l/1000th that of the 
nucleus. 

(a) Estimate the order of magnitude of the moment of intertia of the molecule 
for rotation about the axis shown. 

(b) What is the mean kinetic energy in this mode of rotation of one molecule in 
the solid phase at room temperature (assuming it is not prevented from 
rotating in any way)? 

(c) What is the mean frequency of this rotation at room temperature? 

(d) What value of C v , the specific heat at constant volume, would be expected 
if the substance were solid at room temperature if the laws of classical 
physics were applicable? 




(e) What value of C v would be expected if the substance were a gas at room 

temperature if the laws of classical physics were applicable? 
(/) What value of y = C p /C v would you then expect to find? 
(g) What value of y would you expect to find in practice? 



Problems 135 

5.11 Hydrogen gas is contained in a thermally insulated cylinder with a moveable 
piston. If the pressure on the piston is suddenly reduced to 0.38 of its original 
value as a result the volume of the gas is immediately doubled, estimate the 
specific heat at constant volume per mole for hydrogen. 

5.12 The electrical conductivity a of a certain class of solid is predicted to vary 
according to the law 



r 

^ -A jkT 



T 

where C is a constant, k = 0.86 x 10 ~ 4 eV/deg and T is the temperature. A is 
an activation energy, the energy required for an elementary charge to be moved 
from its atomic site. 

Measurements of a (in arbitrary units) for ice as a function of Tare as follows : 

a 31 135 230 630 

T 200°K 220°K 230°K 250°K 

Plot a suitable straight-line graph and deduce A . 

Decide whether conduction in ice is dominated by (a) electron conduction 
with an energy gap A = 0.1 eV, {b) proton transport involving the breaking of 
a hydrogen bond, A = 0.25 eV, (c) transport of complex ions requiring 
simultaneous breaking of 4 hydrogen bonds, A = 1 eV. 




CHAPTER 

Transport properties of gases 



6.1 TRANSPORT PROCESSES 

So far we have concentrated on the properties of solids, liquids and gases 
which are in equilibrium. In this chapter we will deal with systems which 
are nearly but not quite in equilibrium — in which the density (or the tem- 
perature or the average momentum) of the molecules varies from place to 
place. Under these circumstances there is a tendency for the non-uniformi- 
ties to die away through the movement — the transport — of molecules down 
the gradient of concentration (or of their mean energy down the tem- 
perature gradient or their mean momentum down the velocity gradient). 
We will define certain transport coefficients and show how they can be 
estimated for gases. 

Although the systems we consider are non-uniform in some way and 
cannot therefore be in thermal equilibrium nor obey the Maxwell speed 
distribution exactly, we will always make the assumption that the depar- 
ture from equilibrium is only small. We will therefore assume that no 
error will be introduced if we take the speed distribution inside any region 
of the substance to be Maxwellian. 

6.1.1 Diffusion as a transport process 

Diffusion is the movement of molecules from a region where the con- 
centration is high to one where it is lower, so as to reduce concentration 
gradients. This process can take place in solids, liquids and gases (though 



6. 1 Transport processes 137 

this chapter will be mostly concerned with gases). Diffusion is quite 
independent of any bulk movements such as winds or convection currents 
or other kinds of disturbance brought about by differences of density or 
pressure or temperature (although in practice these often mask effects due 
to diffusion). 

One gas can diffuse through another when both densities are equal. For 
example, carbon monoxide and nitrogen both have the same molecular 
weight, 28, so that there is no tendency for one or other gas to rise or fall 
because of density differences; yet they diffuse through each other. 
Diffusion can also take place when a layer of the denser of two fluids is 
initially below a layer of the lighter so that the diffusion has to take place 
against gravity. Thus, if a layer of nitrogen is below a layer of hydrogen, a 
heavy stratum below a light one, then after a time it is possible to detect 
some hydrogen at the bottom and some nitrogen at the top, and after a 
very long time both layers will be practically uniform in concentration. 

Diffusion coefficients of gas a in gas /? can be measured with a suitable 
geometrical arrangement of two vessels with different initial concentra- 
tions together with some method of measuring those concentrations — a 
chemical method or mass spectroscopy for example. If the rates of change 
of concentration with time are plotted, the diffusion coefficient can be 
deduced ; the equations describing the process are given in section 6.2. 

It is also possible to measure coefficients of self-diffusion, of a gas a in 
gas a for example. This can be done by using two isotopes having the same 
shape and size of molecules, the same interaction potential and almost the 
same mass, but which are nevertheless detectably different — one isotope 
might be radioactive, the other not. One method is described in section 
6.2.1. Mass spectrometer methods can also be used. Applied to a solid, one 
is described in section 9.6.1. 

6.1.2 The diffusion equation 

We will begin by taking a macroscopic view of the phenomenon, that is, 
we will write down equations which involve such variables as concentra- 
tions or fluxes but will not specifically mention individual molecules. We 
define the concentration of a as the number of molecules n per unit volume, 
and we consider the simple case where n varies with one coordinate only 
which we call the x-axis. In Fig. 6.1, the concentration at all points in the 
plane x is n, at (x + dx) it is (n + dn). Then diffusion takes place down the 
concentration gradient, from high to low concentrations ; we are assuming 
that bulk disturbances are absent. We next define the flux J of particles 
as the number of particles on average crossing unit area per second in the 
direction of increasing x. Notice that both concentration and flux can be 



138 



Transport properties of gases Chap. 6 



measured in moles instead of numbers of molecules : this is equivalent to 
dividing all through our equations by Avogadro's number N. 

In general, the flux J may change with position x and may also change 
with time t . In other words, J may be a function of x and t so we write it 
as J(x, t). Of course, there are circumstances where J may be the same 
for all x, or where it is constant with time, but the most general situation 
is that J does depend on both. 

n n + d/7 



x x + dx 



Fig. 6.1. Coordinates used in the definition of 
diffusion. 



It is an experimental fact that, at any instant, the flux at any position x 
is proportional to the concentration gradient there : 



J(x, t) oc 



dn 
dx 



or 



J(x,t) = -D 



dn 
dx 



(6.1) 



where D is called the diffusion coefficient. This is known as Fick's law. 

By itself, Eq. (6.1) is adequate to describe 'steady-state' conditions 
where currents and concentrations do not change with time so that the 
flux can be written J(x). For example, if a tube of length / cm and constant 
cross-section A cm 2 has molecules continually introduced at one end and 
extracted at the other at the same rate, the concentration gradient becomes 

— An//, where An is the difference of concentration between the two ends. 
The number of particles crossing any plane in the tube per second is then 

— DA An// and this does not change with time. 

Consider, however, the much more general situation where initially a 
certain distribution of concentration is set up and then subsequently the 
molecules diffuse so as to try to reach a uniform concentration. Concentra- 
tions are, therefore, changing with time and particles must be accumulating 
in the region between x and (x + dx) or moving from it. Therefore, the 



6.1 Transport processes 139 

number crossing area A of the plane x is not equal to that crossing the 
same area at (x + dx). The flux entering this volume is 

The flux leaving the slice can be written J xn + dx where 



J xo +dx = J xo +\ — \dx+ • 



xo + dx 

dJ] 
dxJT' 



and we can neglect higher terms. The rate of movement of molecules 
from the slice is equal to the difference between the two values of A J, and 
also equal to the volume of the slice, A dx, times the rate of decrease of n : 

dJ * , dn , 
——Adx = — -^dx 
ox dt 

that is 

dJ dn 

dx dt' 

Combining this with equation (6.1) and eliminating J : 

dn 8 ( dn\ d 2 n 



(6.2) 



if we assume that D is a constant independent of the concentration. This 
is called the diffusion equation, and since n depends on x and t it could be 
written n(x, t)* 

Thus we have a system of three equations. (6.1) is an experimental law 
linking the flux at any point with the concentration gradient there. (6.2) is 
the continuity equation expressing the fact that molecules cannot dis- 
appear, and (6.3) combines these two equations. Eq. (6.1) is adequate for 
steady-state conditions, where conditions do not vary with time ; but for 
the general case (6.3) may be used. 

* If the process takes place in 3 dimensions, J is a vector whose components are (J x , J y , J z ) 
and Eqs. (6.1) and (6.2) become 

t • , . , . . ~/.<3" .d n dn\ 

J = iJ x +]J y + kJ 2 = -£> i— +j_ + k— = -Dgradn 

\ ox dy dzj 

dn dJ x dJ v dJ, 

-— = ^+— + — = div J 
dt ox dy oz 

where i, j, and k are unit vectors parallel to x, y and z. Eliminating J : 

8n *■ . ~ * ^ ^2 (d 2 n d 2 n d 2 n 

= _div -£>gradn )=EF7 2 n = D — -+-_+-_ 

dt \dx 2 dy 2 dz 2 



140 Transport properties of gases Chap. 6 

These are typical of transport equations — with the proviso that for 
energy and momentum diffusion, the coefficients in the three equations 
are not all identical as they are here. 

6.1.3 Heat conduction 

The conduction of heat is also a process of diffusion in which random 
thermal energy is transferred from a hotter region to a colder one without 
bulk movement of the molecules themselves. In a hot region of a solid body, 
the molecules have large amplitudes of vibration ; in a hot region of a gas 
they have extra kinetic energy. By a collision process, this energy is shared 
with and transferred to neighbouring molecules, so that the heat diffuses 
through the body though the molecules themselves do not migrate. The 
macroscopic equations describing conduction in one dimension x are, 
firstly, the experimental law for the heat flux 

Q = - K _ (6.4) 

(where Q is the heat flux across unit area, measured in W cm -2 , k is the 
thermal conductivity and T is the temperature) and, secondly, the con- 
tinuity equation 

dQ dT 

which expresses the conservation of energy in the form that the heat which 
is absorbed by a slice of a body goes into raising its temperature. C is the 
specific heat per unit mass, p the density so that Cp is the specific heat per 
unit volume. Combining these two equations to eliminate Q : 

dT l K \ d2T l(k * 

where (ic/Cp) is called the thermal diffusivity by analogy with Eq. (6.3). 
Eq. (6.4) by itself is adequate for steady-state conditions, as when for 
example heat is fed into one end of a bar and extracted at the other and all 
temperatures are constant with time, and Tcan be calculated as a function 
of x alone. But when conditions are not steady, and T varies with time as 
well as position, Eq. (6.6) describes the situation.* 

6.1.4 Measurement of thermal conductivity of gases 

To measure the thermal diffusivity, one has to arrange for temperatures 
to vary with time and to measure the speed of propagation of these 

* Many students are familiar with (6.4) and with the concept of thermal conductivity 
but have never met (6.6) and thermal diffusivity. In fact, transient heat flows are of great 
technical importance. 



6.1 Transport processes 141 

temperature changes. This is difficult with gases (which we are concen- 
trating on in this chapter), where convection currents may be set up. It is 
most convenient, therefore, to measure the thermal conductivity. Usually 
this is done by applying Eq. (6.4) directly, using the simplest geometrical 
arrangement. Commonly, the gas is enclosed between two concentric 
heavy, metal cylinders. Power is supplied electrically to the inner one ; the 
temperature difference is measured directly with a thermocouple. It is 
important to correct for the conduction of heat through the electrical 
leads which can be done by pumping all the gas out and measuring how 
much heat is still conducted across. It is also important to make sure that 
the gas does not set up a pattern of convection currents, which it can do 
rather unexpectedly at certain values of the gap width and pressure. This 
effect can be detected by using a different size of apparatus and also by 
checking that the power varies inversely as the gap. 

6.1.5 Viscosity 

For completeness, a third simple transport process — the diffusion of 
momentum by viscous forces — will be mentioned here, though only 
briefly. Viscous motion of fluids can be far more complicated than diffusion 
or heat conduction and we will be forced to consider only the steady-state 
equation. 

Consider a gas or liquid confined between two parallel plates (Fig. 6.2). 
Let the lower plate be stationary and the upper plate be moving in the 
direction shown, which we will calJL the x-direction. Molecules of fluid 
very near the plate will be dragged along with it and have a drift velocity, 
U x parallel to x, superposed on their thermal velocity. We will assume 
that U x is much less than the mean thermal speed or the speed of sound. 
Molecules of fluid near the stationary plate will, however, remain more or 
less with zero drift velocity. 

moving plate 

z 



~ UL 



'y 



stationary plate 

Fig. 6.2. Coordinates used in the definition of viscosity. 

Eventually a regime will be set up in which there is a continuous velocity 
gradient across the fluid from bottom to top. In this state, molecules will 
be continuously diffusing across the space between the plates and taking 



142 Transport properties of gases Chap. 6 

their drift momentum with them. Considering an area of a plane parallel 
to the xy plane in the fluid, molecules which diffuse across from above 
to below will carry more drift momentum than those which diffuse from 
underneath to on top. In other words, the more rapidly moving layer 
tends to drag a more slowly moving layer with it, because of this diffusion 
of momentum. 

In macroscopic terms, a shearing stress (force per unit area) is necessary 
to maintain this state of motion. The experimental law is 

dU x 

P xz = r\-~ (6.7) 

oz 

where P xz is the force per unit area in the x direction due to a gradient of 
U x in the z-direction and rj is called the coefficient of viscosity. Provided the 
direction of the force is clearly understood, it is not necessary to include a 
minus sign, as this depends on the convention for the choice of axes. 

We started by considering a fluid in Fig. 6.2, but Eq. (6.7) can be applied 
to solids because the right-hand side can be written dd/dt, where 6 is an 
angle of shear. It is difficult to imagine a solid subjected to a shear which 
goes on increasing with time, but it is quite common for solids to be sheared 
to and fro in an oscillatory fashion. Forces are then required to provide the 
accelerations, but in any case the viscosity gives rise to the dissipation of 
energy and the production of heat. It is usual to refer to this as due to the 
internal friction of solids. 

It is implied in Fig. 6.2 that dUJdz is a constant and that U x increases 
proportionally to z. This is so if the coefficient rj is a constant. For many 
liquids this holds, but there are notable exceptions where n varies with 
the velocity gradient or rate of shear so that the velocity profile is not linear. 
Blood, for example, flows with a much lower viscosity through narrow 
capillaries than measurements of the flow through wide tubes would 
indicate — which is fortunate because otherwise one's heart would have to 
generate several horsepower to maintain circulation. Other suspensions 
such as cement also have low viscosities when agitated. Oil paint is fluid 
when worked rapidly with a brush, but when laid on a vertical surface 
and sheared only by a small force due to its weight, it does not fall off. 
Such liquids are called thixotropic. Other liquids have opposite behaviour. 
Whenever we apply Eq. (6.7) to a fluid, therefore, it will be assumed that we 
are dealing with a gas or a 'Newtonian' liquid for which n is independent 
of the rate of shear. 

When we come to write down equations representing the motion of a 
fluid while it is not in a steady state but accelerating, we meet a situation 
which is much more complicated than the diffusion or heat conduction 
cases. For one thing, there are always mass-acceleration terms which have 



6.1 Transport processes 143 

no analogue in the other phenomena. For another, a kind of regime may 
be set up where the flow is not streamline as illustrated in Fig. 6.2 but 
turbulent, and vortices or eddies are present which add an element of 
randomness to the flow pattern. Whether or not it is set up depends on 
the ratio of the inertial to the viscous terms. We can, however, usefully 
adopt a mathematical representation of the simple situation of Fig. 6.2. 
We can imagine the liquid divided into layers, each one sliding over the 
one underneath it on imaginary rollers like long axle rods parallel to the 
y-axis. These rollers are not there in any real sense, but they can lead one 
to define a quantity called the vorticity which is always present in a flowing 
fluid even when no macroscopic vortices are present. (In a simple case 
like Fig. 6.2 the vorticity degenerates into the velocity gradient.) Now in 
the general case of an accelerating fluid with non-uniform velocity it is 
the vorticity which diffuses throughout the fluid, though the equation it 
obeys is not of a simple form. For obvious reasons we will not pursue this 
topic but will be content with the steady-state Eq. (6.7). 

6.1.6 Measurement of the viscosity of gases 

In his classic experiments to measure the viscosity of gases at low pres- 
sures, Maxwell used a torsion apparatus in which a number of circular 
glass discs were arranged to swing in between fixed ones (Fig. 6.3). He 
found the damping coefficient of the oscillations. If we neglect the energy 
loss in the torsion wire itself and assume that the discs would go on 
swinging for a very long time if all the gas were removed, we can calculate 
the damping as follows. 

Consider one surface of one plate, and select an annulus between radii 
r and (r + dr). Then (assuming streamline flow) the force on this annulus, 
whose area is 2nr dr, is 

dF = — — (2nr dr) 
a 

where the linear velocity is rco, co being the angular velocity, and d is the 
spacing between adjacent moving and stationary surfaces. The contribu- 
tion to the couple is the radius times the force : 

dG = 2^ r 3 dr 



and the total couple is 

f 

Jo 



27tna> C a , , nnoj . 



d J 2d 

where a is the radius of the disc. If there are n discs, each with two surfaces, 
there are 2n such contributions. The equation of motion of the system 



144 



Transport properties of gases Chap. 6 



when swinging freely is 



d 2 {nna*ri)dO n 
dr d dt 



where co — d9/dt, I is the moment of inertia, [i the torsion constant of 
the suspension. This is the equation of a damped oscillation. The time 
required for the amplitude to decrease by a factor e is 21 /B, where B is the 
coefficient of the second term in the equation. Thus rj can be determined. 



*■ 




Fixed surfaces 



Moving surfaces 




Fig. 6.3. Principle of the apparatus for the measurement of viscosity by the 
damping of torsional oscillations, (a) assembly of discs, (b) section of apparatus. 



In Maxwell's final apparatus, there were 3 swinging discs (n = 3) with 
d = 0.469 cm. / was determined as 7.33 x 10 4 g.cm 2 ; the radius a was 
effectively 13.1 cm, after allowing for the width of the suspension arrange- 
ment in the centre ; the period was 72.5 s. In one experiment with air at 
21°C, the (natural) logarithmic decrement was determined as 0.073, which 
meant that 13.7 swings were needed to damp the amplitude by a factor e. 
From these data, r\ = 2.47 x 10 ~ 4 g/cm s. A number of corrections were 
needed to allow for edge effects and for torsional damping in the suspending 



6.2 Solutions of the diffusion equation : the yjt law 145 

wire. This method has been used for measuring the viscosity of liquids as 
well as gases. 



6.2 SOLUTIONS OF THE DIFFUSION EQUATION : THE jt LAW 

It is worthwhile studying two solutions of the diffusion equation. The 
first corresponds to the following initial conditions : A semi-infinite prism 
of material has area of cross section A ; the length is along the x-axis and 
the ends are at x = and x = oo. On the face x = 0, N molecules are 
initially all concentrated in a thin layer and are subsequently allowed 
to diffuse into the material. We will denote the number at time t which are 
within a slice between x and (x + dx) by n(x, t)A dx. Then the appropriate 
solution of Eq. (6.3) shows that the concentration 

The function is shown in Fig. 6.4 for a number of values of the time. The 
following statements should be verified : (a) that the function does indeed 
satisfy the diffusion equation, which can be shown by direct substitution, 
and (b) that the total number of molecules remains constant and equals 
AT at any time t, which can be shown by integrating n(x, t)A dx from 



/ = V4D 




Fig. 6.4. Concentration as a function of x for different 
values of time. 



146 Transport properties of gases Chap. 6 

to oo, using an integral from the Table on p. 72. It is obvious, from the 
diagram, that the concentration always remains greatest near the starting 
place and falls off with increasing distance, and that the spread increases 
with time, which is all very reasonable. 

One very interesting aspect of the diffusion process can be deduced 
from this solution. On the microscopic scale, diffusion is of course a 
random process, and it is impossible to predict exactly how far one 
particular molecule will travel. But if we were to scale down the curves 
of Fig. 6.4 so as to refer to one molecule instead of N , these curves would 
then be the probability function P[x] for the nett distance travelled by a 
single molecule at any time (see section 4.2). 

We can, therefore, use the curves of Fig. 6.4 to calculate the mean nett 
distance travelled by a molecule at any time t. This is 

A f 00 
x(t) — — xn(x, t) dx. 



Using an integral from the Table on p. 72, we find 

x = -^{Dtf 12 . (6.9) 

V 71 

Thus the mean nett distance travelled is proportional to the square root 
of the time. This is perhaps an unexpected result : one is used to travelling 
twice as far in twice the time, but for the random process of diffusion this 
is not so. Of course, some molecules go much further than this, others 
less far, and it is the mean which we have calculated. Stated differently, our 
result shows that to diffuse a mean distance x, the time required is propor- 
tional to x 2 . This is an important characteristic of the diffusion process. 

Before leaving this problem, note that substituting T for n and taking 
D to signify the thermal diffusivity, we have the solution to the problem of 
a semi-infinite slab with a finite amount of heat generated on the surface 
and subsequently allowed to be conducted away. 

Another solution to the diffusion equation refers to the problem of a vessel 
of cross-section A with two layers (say of liquid or gas, so long as convec- 
tion is avoided) each of depth 1/2 and of initial concentration n molecules/ 
cm 3 and zero respectively, Fig. 6.5(a). Diffusion starts at zero time. It is 
obvious that after an infinite time the concentration throughout the vessel 
must be uniform and equal to n /2 molecules/cm 3 , Fig. 6.5(b). After a 
time t , the concentration as a function of distance along the vessel is given 
by Fig. 6.5(c). (The solution is a Fourier series.) 

Initially the concentration difference is n , but the mean concentration 
decreases in one half and increases in the other. The time interval required 



6.2 Solutions of the diffusion equation : the yjt law 147 



for the difference of concentration to decrease by a factor e is called the 
relaxation time x for the diffusion ; it is a natural unit of time to use for 
describing the process. It emerges from the analysis that 

t = l 2 /n 2 D. (6.10) 

Note that once again, a time is porportional to the square of a length. 

Area A 




1/2 



1/2 



05/7. 
o 

c 
o 
o 

o 



/ // = r 


//-a> 








1 


1 * 



(b) 



1 l 



(c) 



Position 



Container 
' Q 



Narrow 8 
' tube g 



Container 
P 



id) 




(e) 



Fig. 6.5. (a) Molecules of gas initially occupying lower half of vessel, upper 
half being filled with another gas (not shown) to avoid convection, (b) after 
an infinite time, (c) concentration as a function of position at different times, 
(d) 'lumped' volumes and a tube, (e) concentrations in P and Q as function of 

time. 

Without doing an exact analysis, some insight into this result can be 
gained from a crude model of the process. Imagine both halves of the vessel 
replaced by containers P and Q (Fig. 6.5(d)) one filled with the same total 
number of molecules as before and the other empty. Let these two be 
connected by a narrow tube which allows diffusion to take place. In the 



148 Transport properties of gases Chap. 6 

language of the electrical engineer we have replaced the distributed 
capacitance and conductance of Fig. 6.5(a) by lumped capacitances and 
a pure conductance in Fig. 6.5(d). (This language becomes even more 
appropriate if we translate the diffusion problem into the heat conduction 
problem, when heat capacities and thermal conductances are used.) 

To make the setups comparable, let us put the same total number of 
particles in both halves, namely n Al/2. Further, since the average distance 
that a molecule has to diffuse in Fig. 6.5(a) is something like 1/2, and it has 
to travel through an area of cross section A, let the narrow tube have the 
same ratio (area)/(l#ngth), namely 2A/1 ; the rate of diffusion will then be 
the same. We have : 

number of molecules leaving P per second 

= number entering Q per second 

= (flux J) x (area of narrow tube). 

To the approximation that we are dealing with "lumped" components we 
can use Eq. (6.1) so that 

J = D(conc'n in P — conc'n in Q)/(length of narrow tube). 

Therefore 

Al drip Al dn Q AD(n P — n Q ) 
~Y~di = ~2~~dT = 1/2 

where n P + n Q = n . The solution is 

»P = ^(l + e-n « Q = ^(l-e-</<) 



2 



where 



T 






Thus the concentrations in P and Q approach their final values exponen- 
tially, Fig. 6.5(e), with time constant l 2 /SD which is not very different from 
the l 2 /n 2 D quoted before. 

This square law can have quite startling effects. Diffusion coefficients 
of small molecules in liquids like water at ordinary temperatures are of 
the order of 10 ~ 3 cm 2 s _1 . Given a tube 1 cm long, concentrations will tend 
to equality in times of the order of 20 minutes. But the time for a 1 m tube 
would be reckoned in months and for a 10 m tube it is decades. A famous 
example of this is a very tall vertical tube, fixed to the wall of a lecture 



6.2 Solutions of the diffusion equation : the Jt law 149 

theatre in Glasgow University. Eighty years ago it was filled by Lord 
Kelvin, the lower half with blue-green copper sulphate solution and the 
upper half with water. It is still very far from uniform in concentration. 

6.2.1 Measurement of the diffusion coefficients of gases 

Diffusion coefficients of one gas through another (or with certain cor- 
rections, of a gas through itself), can be measured using similar arrange- 
ments. In the experiments of Mifflin and Bennet to measure the D of 
argon through argon at room temperature at very high pressures for 
example, two volumes V, each 36 cm 3 , were connected by a bar of length 
/ = 3.8 cm made of porous bronze (Fig. 6.6(a)). The pores were of average 
diameter 2x lCT 4 cm.* The total cross-sectional area of the pores was 
A = 0.36 cm 2 , about ^ of the area of the bar. One volume, which we will 
call Q, was filled with ordinary argon, the other, P, with argon at the same 
pressure containing a small concentration of 37 A. This is a radioactive 
isotope whose presence could be detected by the ionization it produced — 
each volume V was in the form of an ionization chamber in which the 
current was porportional to the concentration of 37 A. The half-life of the 
37 A was large compared with the time taken by the experiment. 

Using the same notation as in section 6.2, the equations are 



-V 



.drip 
dF 



V 



dn Q _ AD(n P — n Q ) 
~d7~ / 



Steel casing Porous bronze 




Ionization 
chamber P 

(o) 



Ionization 
chamber Q 




30 40 
Time in hours 



(6) 



Fig. 6.6. (a) Apparatus to measure diffusion of gases at high pressures, (b) 

Difference of currents in P and Q in units of 10 ~ 14 amp, plotted on log scale, as 

function of time on a linear scale. 

* It is shown later in this chapter that it is important that the diameter of the pores should 
be greater than the mean free path between collisions in order to measure D correctly. This 
condition was in fact satisfied here. 



150 Transport properties of gases Chap. 6 

so that (« P — n Q ) varies as exp( — tjx), where x = V 1/2 AD. A plot of log(i P — i Q ) 
against time was, therefore, a straight line of negative slope 2AD/VI, the 
i's being the currents. Figure 6.6(b) shows the results of one run. The 
currents are in units of 10~ 14 amp. The slope of this graph is 

(In 49 - lnl7)/40 hours" l = (3.89 - 2.77)/(40 x 3,600) 

= 0.78xl0 _5 s- 1 . 

Hence D = 1.48 x 10" 3 cm 2 s" *. This is the diffusion constant of 37 A in 
40 A. A small correction must be applied in order to calculate the self- 
diffusion constant of 40 A through 40 A; it will be mentioned here although 
it can only be understood in the light of the discussion of section 6.5. 
We assume that the interatomic potential energies of the two kinds of 
atom are the same but that their mean speeds c are different, because by 
the equipartition law the mean kinetic energy is the same for both. This 
gives a 2 % correction to c and hence to D. The results of these experiments 
are quoted in section 6.5.4. 

6.3 DIFFUSION AND THE RANDOM WALK PROBLEM 

We have so far dealt with diffusion in macroscopic terms with little 
reference to the paths followed by individual molecules. In fact, each mole- 
cule follows a random path, moving in more or less straight lines between 
collisions with other molecules, but travelling backwards almost as often 
as forwards (Fig. 6.7). It is the purpose of this section to show that the yjt 
law and other characteristics of diffusion are merely consequences of the 
'random walk' of each molecule. 




Fig. 6.7. A random path followed by a molecule, after Perrin. 

The distance that a molecule travels between collisions is called the/ree 
path. It may be of any length, in any direction. 

Let us, however, make a crude model of the random path by saying 
(a) that each free path is of the same length, and (b) that the molecules can 



6.3 Diffusion and the random walk problem 151 

only move parallel to the +xor — x direction — that we are dealing with 
a sort of 'one-dimensional gas'. It may be guessed that these simplifications 
allow us a considerable insight into the diffusion process although the 
value of the diffusivity so calculated is likely to be wrong ; even then, it is 
not likely to be wrong by a large factor such as 10 but rather by a factor 
like 2 or 3. 

The simplified problem is this. A molecule starts from the origin and 
moves a distance ± / along the x-axis ; having done so, it can then move a 
further distance ± /. Thus at the end of two such moves, it may have followed 
one of four possible sequences of +/ or — / moves: namely, ( + /+/) or 
( -I- / — /) or ( — /+/) or ( — /—/). All of these are equally likely to occur. The 
nett distance travelled may, therefore, be + 2/ (which may be achieved in 
only one way and, therefore, has probability 1/4) or (reached in two ways ; 
probability 1/2) or — 2/ (probability 1/4). At the end of three moves, there 
are eight possible sequences which may have been followed, all equally 
likely. The end point may be +3/(1 sequence only), or +/ (there are 3 
sequences with 2 positive-going moves and 1 negative-going), or — / (again 
three ways of achieving this) and — 3/ (one way only). Thus the probabilities 
are 1/8, 3/8, 3/8 and 1/8 respectively. Notice that the numbers 1, 2, 1 and 
1, 3, 3, 1 occur as coefficients in the expansions of (x + y) 2 and (x + y) 3 . Let 
us now generalize our results to sequences of N moves (Figure 6.8 shows 
two typical sequences of 10 moves, for illustration.) We can say that the 
total number of possible sequences is 2 N , and the number of ways of 
achieving 2 positive-going steps and {N — Z) negative-going ones is the 



+ 







Fig. 6.8. There are 2 10 different sequences of 10 moves, 
all equally probable. 10 !/7 !3 ! = 120 of these consist of 
7 positive-going and 3 negative-going moves; two of 
these are illustrated (slightly displaced in a vertical 
direction for clarity). 



152 Transport properties of gases Chap. 6 

coefficient of x z y N ~ L in the expansion of (x + y) N , namely 

N\ 
Z!(N-Z)T 

These coefficients are called the binomial coefficients. 

In slightly different terms, the probability that at the end of N moves 
the molecule will have travelled a nett distance x — SI (that is, S steps in 
the +x direction), by having made j(Af + S) positive-going steps and 
%{N — S) negative-going steps is 



We may assume that the molecule moves with speed c during its free path 
so that the time required to make the N steps is t = Nl/c. Thus Eq. (6.11) 
gives the probability that a molecule will have travelled a nett distance x 
in time t. 

We can now refer back to the problem of which Eq. (6.8) is the solution 
and alter it slightly to make it correspond exactly to the present one — 
namely by making it refer to 1 molecule instead of N and to diffusion in 
the ±x direction instead of +x only. We can interpret the solution to 
mean that the probablity that a molecule diffuses a distance between x 
and (x + dx) in time t is 

1 e-* 2/4Dt dx. (6.12) 



2(nDt) 



1/2' 



We can now show, by purely mathematical manipulations, that — unlikely 
though this might seem — this is identical with Eq. (6.11), in the limit when 
N tends to infinity, when t = Nl/c and x = SI. We need to make one 
additional intuitive statement (which can, however, be properly proved), 
that if N is very large, there is a great probability that S is small compared 
with N; this corresponds to the fact that the binomial coefficients are 
small at the beginning and end of the expansion and largest in the middle. 
The key is to use Stirling's formula for N ! when N is large :* 

log AT! = (N + ±)\ogN-N + log{27z) 112 . (6.13) 

* Natural logarithms to base e are meant, of course. For many purposes 

logiV! = JV log JV-JV 

is a good enough approximation, for large JV. But with all the terms present, it is remarkably 
accurate even for small JV. It gives 10! as 3.60 x 10 6 instead of 3.63 x 10 6 , for example. For 
JV = 10 10 it is very accurate indeed. 



6.4 Distribution of free paths 153 

Eq. (6.11) can be written 

log P = log Nl-logP^jl-logP^I 1-108 2" 



and after some tedious algebra this gives 

, D , /2\ 1/2 /N + S+ll / S\ lN-S+l\. I. S 

On the assumption that S/N is small we can write 
, m s \ s s2 

togll+ riv-^ + '- 



Then 



or 



2 \ 1/2 S 2 
l0gF = l0g U -2» 



which in terms of distance x and times t may be written 

1/ 2 \ 1/2 
P = - — - e-* 2/2cl, 2l. (6.14) 

2\7icltJ 

Comparing with Eq. (6.12), 2/ takes the place of dx (since in this simple 
model there can be particles only at points separated by 21). The two 
equations are identical in form and 

D = \c\. 

Thus the essential features of the diffusion process are reproduced by 
this simplified 'random walk' problem. A more accurate analysis 
(Appendix B) gives a coefficient in the expression for D of ^ instead of \\ as 
mentioned above, this kind of discrepancy is to be expected of the over- 
simplified one-dimensional model. 

6.4 DISTRIBUTION OF FREE PATHS 

The free paths of the molecules in a gas are not, of course, all equal in 
length. In fact it is most probable that the free path is short and quite 



154 Transport properties of gases Chap. 6 

improbable that it should be long. The average or mean free path will be 
denoted by X. The distribution of free paths follows the law : 



(probability of free path between x, x + dx) = e 



_ ~.-x/X 



dx 

T* 



(6.15) 



That the exponential form of this law is correct, is shown by the following 
argument. 

(Probability of collision between x, x + dx) 

= (probability of no collision in x) (probability of collision in dx) 

and identifying the factors in this and the identical Eq. (6.15), 

probability of no collision in x = e~ x/x (a), 

probability of collision in dx = dx/X (b). 




1 2 3 x/X 

Fig. 6.9. The function exp( — x/X) as a function of (x/X). 



But if (a) is correct, we would expect 

probability of no collision in dx = e~ dx ^ 
or 

probability of collision in dx = l—e~ dx/x 

= dx/X 



6.4 Distribution of free paths 155 

for small dx. But this is exactly what we found in (b) ; therefore, the law is 
self-consistent, and satisfies the conditions imposed by the random prob- 
ability of collisions. Further, it can be checked by integrating Eq. (6.15) 
that the probability that the free path lies between and oo is unity, as 
expected ; and the mean free path, given by 



f 



00 dx 



xe 

X 



does work out at X as required. Eq. (6.15), therefore, has all the desired 
properties. Figure 6.9 is a graph of exp( — x/X) as a function of x/X. It is 
identical in form to Fig. 3.2(b) and Fig. 4.5. 

6.4.1 Mean free path and collision cross-section 

Imagine all the molecules in a gas to be at rest, except one, which is 
moving with velocity v in a certain direction. 

This molecule will collide with another if the two centres get within 
a distance a of one another (where a is the diameter of one molecule, 
although we will see in section 6.4.3 that this quantity needs careful 
definition). In time t, the moving molecule travels a distance vt. Any other 
molecules that happen to be inside a cylinder of length vt and area na 2 
will collide with it. 

If there are n molecules/cm 3 , the moving molecule will, therefore, make 
nna 2 vt collisions. (Of course, the molecule would be deflected but we 
imagine the path to be straightened out.) 

Therefore, the mean distance between collisions is 

X = -i-x. (6.16) 

nna 

The quantity na 2 is called the collision cross-section of a molecule, denoted 
by a. It is equal to 4 times the geometrical cross-section of one molecule, 
Fig. 6.10. We can write 

X = —. (6.17a) 

na 

In a real gas, the molecules are not all stationary in this way, but are 
coming from all directions. This introduces a numerical factor : it is found 
that 

X = 4—- (6.17b) 

y/2 na 



156 Transport properties of gases Chap. 6 

Collision cross-section 
Molecule , 




Second molecule 



Fig. 6.10. The collision cross-section of a molecule of diameter a is na 2 . If the centre 
of a second molecule lies within a distance a of the centre of the first, a collision occurs. 



6.4.2 Estimates of mean free paths 

At 1 atmosphere pressure (that is, 760 mm of mercury), at 0°C, 1 mole 
of any ordinary gas occupies about 20 litres so that n = 6 x 10 23 /2 x 10 4 = 
3 x 10 19 per cm 3 . If the diameter of a molecule is 4 x 10" 8 cm, its area of 
cross-section is about 1.2 x 10" 15 cm 2 so the collision cross-section 
' = 5 x 10" 15 cm 2 . Thus X is of the order of 7 x 10" 6 cm, which is about 
200 diameters. 

The pressure of a gas is proportional to n ; hence, reducing the pressure 
by a certain factor increases X by the same factor. At 1/15 mm pressure, the 
mean free path in air is 1/15 mm— this is a useful datum for remembering 
the order of magnitude. At 10" 6 mrn pressure, the mean free path would 
be 5 m, which is bigger than most ordinary containers. 

6.4.3 The dynamics of collisions 

The exact calculation of the collision cross-section of the molecules of a 
gas is a major problem (even when purely classical laws are assumed as 
they will be here). If molecules were like billiard balls with a definite 
diameter, as we have tacitly assumed in the previous section, there would be 
no problem. There would be no doubt when two molecules collided and 
their trajectories were deflected. But with real interactomic potentials, 
two effects arise which have no counterparts in the billiard-ball model. 

Firstly, there is an effect due to the attractive forces ; this is important 
at low temperatures when the mean thermal energy kT is comparable 
with the depth e of the potential well. The forces cause the trajectories of 
two molecules to bend towards one another— the paths are correlated in 
the sense that the presence of one molecule influences the trajectory of the 
other. Thus, two molecules which in the absence of attractions might just 



6.4 Distribution of free paths 157 

miss one another, may just hit. Collisions are, therefore, more frequent ; 
the collision cross-section is increased and the mean free path decreased 
compared with a gas with no attractions. We will describe later, while 
dealing with van der Waals' equation, a crude method of allowing for this 
effect. 

Secondly, the repulsive forces (the r~ 12 term of the Lennard-Jones 
potential) can be interpreted to mean that molecules act rather like elastic 
spheres which can be compressed together. Two colliding head-on at high 
speed are momentarily pushed together so that the distance apart of their 
centres is smaller than when the two collide at low speed. We can, there- 
fore, not talk in a precise way about the diameter of a molecule which 
behaves like this. We can, however, take the distance of closest approach 
as a measure of the effective diameter. To see how this varies with the 
kinetic energy of either molecule, we will state some of the more important 
results of Newton's laws of motion applied to collisions and then see how 
these apply to molecular collisions. 

When two bodies collide, the centre of mass of the system is particularly 
important. It moves with unchanged velocity at all times during the 
collision. Thus, if the total mass M of the system is imagined to be concen- 
trated at the centre of mass, and this moves with velocity U G , then the 
quantity ^MU G is conserved — in other words, U G and jMU G are un- 
changed, before, during and after the collision (and this holds whether the 
collision is elastic or inelastic). From the point of view of an observer 
situated at the centre of mass and moving with it (i.e. in a frame of reference 
fixed with respect to the centre of mass), an elastic collision between equal 
spheres looks symmetrical. The two approach one another with equal and 
opposite velocities along parallel paths ; they are deflected, each with its 
speed unchanged and they go off again on parallel paths, Fig. 6.11(a). 

From the point of view of any other observer moving with constant 
velocity, but not situated at the centre of mass (for example, an observer 
who is stationary in the laboratory, Fig. 6.11(b)), the total kinetic energy of 
the system is equal to the kinetic energies of the two particles with respect 
to the centre of mass plus the term \MU G * 

During an elastic collision, some of the kinetic energy with respect to the 
centre of mass is momentarily transformed into potential energy of elastic 

* This can be checked by measurement of the vectors in Fig. 6.11, which is drawn to scale. 
There is a well-known paradox of two trains each of mass m approaching one another, each 
with speed v. A stationary observer sees their kinetic energy as 2(\mv 2 ). An observer on one 
train sees the speed of the other as (2v) and its energy as jm{2v) 2 , which is different. The 
question is, how much energy is dissipated during an inelastic head-on collision. There is 
no paradox if it is remembered that the centre of mass of the system (stationary with respect 
to the earth but moving with speed v with respect to either train) conserves its speed after the 
collision. With respect to the observer in the train, energy jMUq = j(2m)v 2 is conserved and 
not dissipated. 



158 Transport properties of gases Chap. 6 

deformation. For a head-on collision, all of this kinetic energy is stored 
as potential energy at the instant where the spheres are reversing their 
velocities ; but for a glancing collision the tangential velocity of one sphere 
past the other is never zero and the spheres are less deformed. 




U G 



(b) 

Fig. 6.11. Velocity vectors for an elastic collision of two spheres of equal mass. 
(a) In the centre-of-mass frame of reference, (b) in the laboratory frame of reference, 
where one sphere is 'chasing' the other. The velocity of the centre of mass is 

shown separately. 

To calculate the distance of closest approach between the centres of two 
molecules, we therefore have to know the speeds of the two, which may 
have any value from zero upwards (as given by the Maxwell distribution) 
and their relative directions and whether the collision is head-on or glanc- 
ing, Fig. 6.12(a). The mean distance of closest approach is governed by a 
complicated averaging process, and it appears that only a fraction (about 
5 or ^) of the mean kinetic energy goes into squeezing the atoms together. 
It is certain, however, that the collision cross-section gets smaller with 
increasing temperature because the colliding atoms on average 'climb' 
farther up the potential energy curve. The overall result of these two 
effects of the interatomic potentials is that the effective collision cross- 
section varies with temperature as shown in Fig. 6.12(b) — which is 
schematic and not to be taken too literally. 

It is convenient here to mention another result which is applicable to the 



6.5 Calculation of transport coefficients 



159 



detailed discussion of the transport coefficients. Imagine one sphere aimed 
at another with a given velocity, but hitting it somewhere at random so 
that in a large number of throws it bounces off in all possible directions. 
Then in the centre-of-mass frame of reference, all directions are equally 
probable. But in the laboratory frame, after the velocity of the centre of 
mass has been added to all the velocity vectors, there is a greater probability 
that the direction of the initial velocity will be favoured. Referring back 
to Fig. 4.1, for example, there is a greater probability that the molecule 
will get through the layer than that it will be reflected back out. This holds 
even if both molecules in a collision are moving and the effect is referred 
to as the persistence of velocities. 



I Head-on 
\collision 




b A 




AT~< 



Temperature 



id) 



Fig. 6.12. (a) Distance of closest approach, (b) Variation of collision 
cross-section with temperature. 



6.5 CALCULATION OF TRANSPORT COEFFICIENTS 

We will now calculate the steady-state transport coefficients — diffusion 
coefficient, coefficient of viscosity and thermal conductivity — for gases, 
using the concept of the mean free path which we have developed. 

It is important to notice that whenever there exists a concentration 
gradient or a velocity or temperature gradient in a gas, whenever any 
transport process is taking place, then the system is, strictly speaking, not 
in equilibrium. Conditions may be steady, but the speed distribution will 
not follow Maxwell's law. The exact solution of all problems of this sort 
then becomes difficult. Here we will assume that any departures from 
equilibrium and from the Maxwell speed distribution are small, that drift 
velocities are small compared with the velocity of sound, for example, and 
that any gradients are small. 



160 Transport properties of gases Chap. 6 

6.5.1 Diffusion coefficients 

We have already shown in section 6.3 that a solution of the diffusion 
equation 

dn ^d 2 n 

dt dx< 



= D ^2 (6-3) 



can be reproduced on the simple assumption that the molecules follow a 
random walk. 

The object of this section is to deduce the other equation 

/ = -D- ,6.1) 

relating the flux J (the number of particles crossing unit area per second) 
to the concentration gradient. We will assume steady-state conditions, so 
that J does not depend on time and can be written J(z) only. 

We will begin by calculating an important quantity for a gas in equili- 
brium — the number of molecular impacts which occur per second on a 
wall (or on an area inside the gas, the molecules coming from one side 
only). Let the area of the wall be A. 

Let the wall be in the xy plane. Then any molecule which has a com- 
ponent of velocity v z normal to it and is contained in a volume of cross- 
sectional area A and length v z , hits the wall within one second. Therefore 
the number of impacts per second from molecules whose z-component of 
velocity lies between v z and (v z + dv z ) is 

nAv z P[v z ] dv z 

where n is the total number of molecules per unit volume and P[v z ] is the 
probability function of v z . (This result was written down in section 5.1.2.) 
Therefore the total number of impacts from all molecules moving 
towards the wall (that is, v z going from to oo) is 



nA 



/•co I m \ 1/2 f °° I kT \ 1/2 



using one of the integrals quoted on page 72 and the expression for 
the probability function from Eq. (5.3). We can rewrite this in more 
compact form by quoting another result, the expression for the mean speed 
c, namely 

2 l2kT\ l/2 

V 71 ! m / 



6.5 Calculation of transport coefficients 161 

Hence it follows that the number of molecular impacts per second on an 
area A of a wall exposed to a gas is 

impacts/s = \nAc. (6.18) 

This result only assumes that there is equilibrium so that the Boltzmann 
distribution is followed. 

Let us now select a small area dS normal to z in the gas — it can be 
imagined as a kind of little picture-frame suspended in the gas— and let 
us calculate the numbers of particles going through it per second, arriving 
from the +z and -z directions, Fig. 6.13. In equilibrium they would be 
equal, but in the presence of a small concentration gradient there is a nett 
flux. We will assume that near-equilibrium conditions hold. 



concentration n + Xgj 



oreads 



concentration n 



concentration n - X^ 



Fig. 6.13. A small area dS in a gas with molecules 
arriving from the +z and — z directions. 

We will now make an approximation, which leads ultimately only to an 
error by a numerical factor of order unity. It is, that every molecule which 
passes through dS made its last collision with another molecule in the 
plane parallel to dS at a distance of one mean free path X from it. This is 
not correct : on the average, each molecule makes its last collision a radial 
distance X from dS, but not a distance X in the z direction because many 
molecules travel obliquely to dS ; the average distance in the z direction 
is something smaller than X. Nevertheless we will make this assumption. 

Let the concentration in the plane of dS be n molecules per cm 3 . 

The molecules coming from the + z direction come from a region where 
the concentration is (n + Xdn/dz) molecules per cm 3 ; those arriving from 
below come from a region with (n-Xdn/dz) molecules per cm 3 . Hence 
more molecules come from above than below, and the nett flux of mole- 
cules per unit area is, in the direction of increasing z, 

1 / ,dn\ 1 / ,dn\ l..dn 



162 Transport properties of gases Chap. 6 

which is of the same form as Eq. (6.1) expressing Fick's law. Comparing 
this with Eq. (6.1), 

D = \ck. (6.19a) 

This is the same expression as we arrived at by considering the linear 
random walk* — and the error is the same. When the distribution of free 
paths is properly taken into account, together with the fact that the 
molecules can pass through d5 from all angles, the result is 

D = %cL (6.19b) 

This exact expression is deduced in Appendix B. 

Before comparing this result with experimental data, we will calculate 
the coefficients of viscosity and thermal conductivity for gases. 

6.5.2 Viscosity coefficient 

In the arrangement of Fig. 6.2, when the top plate is moving at constant 
velocity parallel to x and the gas is in a steady state, molecules continually 
diffuse across the gap and carry their x-drift momentum with them. Each 
molecule has drift momentum mU x , where U x is the velocity in the x 
direction and is a function of z, the distance from the lower plate. 

Again we image a small area dS inside the gas, and calculate the nett 
flux of drift momentum through it. Since a force is a rate of change of 
momentum, the nett flux is 



Force/area =\n~c 



mU + A d{mU > 



dz 



1 

-nc 

4 



mU -A- x 



dz 



where we have written m for the mass of one molecule, U for the drift 
velocity in the plane of dS and we are assuming that the concentration n 
is constant. Hence 

Force/area - i-l^" 



dz 



This is identical in form with Eq. (6.7). We may justifiably alter the \ to 
j, and write the coefficient of viscosity 

rj = \nmlc. (6.20) 

6.5.3 Thermal conductivity coefficient 

Exactly the same methods can be used to calculate the conduction of 
thermal energy across an area normal to a temperature gradient. The mean 

* Presumably we identify the step-length I with the mean free path X. 



6.5 Calculation of transport coefficients 163 

translational kinetic energy of molecules which collide in a region where 
the temperature T is given by 

|m? = \kT (5.12) 

and the nett rate of transport of energy across unit area in the direction of 
z increasing is 

Q = -nc\ [ -kT + X^- r c^kT -X\J } j 

1 -,3.dr 

2 2 dz 

This is identical in form with Eq. (6.4). If we again alter the i to | to com- 
pensate for our crude averaging, 

k = \nkkc. 

This is correct for a monatomic gas for which the specific heat per molecule 
is f/c. Remembering that n is the number of molecules/cm 3 we can write 
f k as nCJN, where C v is the specific heat per mole, (f R J mol~ i deg~ x for a 
monatomic gas) or alternatively as C' v which means the specific heat per 
unit volume (in J cm -3 deg -1 ). For polyatomic gases, rotational energy 
is transported together with translational energy, so that in general 

k = ^C v Xc=^C v Xc. (6.21) 

In words, the thermal conductivity of a gas is the specific heat per unit 
volume, times the mean free path, times the mean speed (which is nearly 
equal to the speed of sound), times a numerical factor of order unity. 

6.5.4 Comparison with experiment 

We have deduced expressions for the three transport coefficients for 
gases : 

D = ^c, (6.19b) 

y\ = ^nmXc, (6.20) 



where 



k = i~C v Ac, (6.21) 



1 1 
A = ^-— (6.17b) 

y/2 no 



164 Transport properties of gases Chap. 6 

We can compare these with the results of experiments. As may be expected, 
the qualitative agreement is good but the actual numbers are only correct 
within a factor 2 or 3. 

First, a remarkable fact can be deduced if we write the viscosity as 



1 



m 



rj = 



3V2 a 



(6.22) 



Here, m and a are both constants characteristic of a given gas, and c 
depends only on the temperature. Thus the viscosity of a gas should be 
independent of pressure. If the number of molecules/cm 3 is reduced, the 
mean free path increases and their product remains constant. This predic- 
tion was made by Maxwell and he undertook the experiments described 
in 6.1.6 to prove it. Modern measurements for argon show (Fig. 6.14(a)) 
that at 40°C it holds between 0.01 atmosphere and about 50 atmospheres 
pressure. The constancy is astonishing when one realizes that the left-hand 
side of the graph is a rough vacuum and the right-hand represent a gas at 
high pressure that needs a steel vessel to contain it.* The simple "billiard- 
ball" model seems to be adequate over this range. At extremely high 
pressures it breaks down, of course, Fig. 6.14(6). This is not unexpected — at 



IS) 

I 3 



9 2 





0.001 0.01 0.1 1 10 100 

pressure (atm) 
(a) 



F- I 


i 




w 10 






E 

^ a 
o> 8 


. 




O 
^ 6 


- 




4 


- 




2 











_i 1 1 — »~ 



500 



1000 1500 
pressure (atm) 

ib) 



Fig. 6.14. The viscosity of argon as function of pressure (a) at low and 

moderate pressure (T = 313°K) and (b) up to high pressures (T = 298°K). 

Dotted line calculated for a = 22 x 10" 16 cm 2 . Data from Michels, Botzen 

and Schuurman, Physica 20, 1141 (1954). 

* In a well-known demonstration experiment (the 'guinea and feather' experiment invented 
by Boyle) a light object, which descends only very gently when falling in air, is seen to drop 
much more rapidly through a rough vacuum. Yet the viscosity of the air is the same in the 
two cases. The paradox can be solved after reading section 9.7.1. 



6.5 Calculation of transport coefficients 



165 



1,500 atmospheres and room temperature, gaseous argon is denser than 
water and the mean free path is less than an atomic diameter. 
Similarly we can write the thermal conductivity 



1 C 



K — 



3^2 No 



(6.23) 



where the quantities N, a, C v are all constants for a given gas and c depends 
on temperature ; hence k should be independent of the pressure. Figure 
6.15(a) shows that this is so over the same wide range of pressures when rj 
is constant — although at extremely high pressures it breaks down again, 
Fig. 6.15(6). This constancy at ordinary pressures is one of the most 
unexpected predictions of kinetic theory. 



a> -z 
-o •-> 

E 

1 

b 1 





0.0010.01 0.1 



1 10 100 
pressure (atm) 




500 



\a) 



1000 1500 
pressure (atm) 

(£) 



Fig. 6.15. The thermal conductivity of argon as function of pressure (a) at 
low and moderate pressures (T = 313°K) and (b) up to high pressures 
(T = 398°K). Dotted line calculated for a = 8.8 x l(T 16 cm 2 . Data from 
Waelbroeck and Zuckerbrodt, J. Chem. Phys. 28, 523 (1958), and Michels, 
Botzen, Friedman and Sengers, Physica 22, 212 (1956). 

The fact that both r\ and k are proportional to c means that they should 
vary as y/T, the square root of the absolute temperature. Figure 6.16 
shows that this is roughly true. The decrease of collision cross-section at 
high temperature, described in section 6.4.3, is evident. It is perhaps un- 
expected that a gas should become more viscous when it is hotter ; one's 
ordinary experience is limited to liquids, which show the opposite be- 
haviour. 

By contrast to rj and k, the self-diffusion coefficient D of a gas is directly 
proportional to X by itself, and hence should be inversely proportional 
to the density p: the product Dp should be constant at all pressures. 
Figure 6.17 shows that this is roughly so in argon up to moderate pressures 
— at 300 atmospheres the density is 0.44 g/cm 3 . 



166 Transport properties of gases Chap. 6 




500 1000 1500 2000 K 
temperature 

(a) 



500 



1000 1500"K 
temperature 



(b) 



Fig. 6.16. (a) The viscosity and (b) the thermal conductivity of argon at 1 
atmosphere pressure as functions of temperature. Dotted curves calculated 
for a = 22 and 8.8 x 10" 16 cm 2 respectively. Data from Vasilesco, Ann Phys. 
(Paris) 20, 292 (1945), (Fig. 10); Kannuluik and Carman, Proc. Phys. Soc. 
65B, 701 (1952) and Schafer and Reiter, Naturwissenschaften 43, 296 (1956). 

Qualitatively, then, the theory holds very well over a wide range of 
temperatures and pressures. In terms of absolute values, the fit is less good. 
To get the curve of Fig. 6.16(a) for the viscosity, the cross-section a must 
be chosen as 22 x 10" 16 cm 2 . For the thermal conductivity, Fig. 6.16(b), 
the effective a = 8.8 x 1(T 16 cm 2 . For self-diffusion, Fig. 6.17, a = 
16 x 10~ 16 cm 2 . These give the diameter of a single molecule as 2.6 A, 
1.7 A and 2.25 A respectively. These are certainly of the same order of 
magnitude as the diameter 3.35 A, deduced from the density of the solid 
at absolute zero, Fig. 3.13(a); the discrepancy is in the expected direction. 



$ 



0.010 
0.008 
0.006 
0004 
0.002 



D in cm 2 /s 
p in mol/l 



100 



200 300 

pressure (atm) 

Fig. 6.17. Self-diffusion coefficient multiplied by density 

for argon as function of pressure (T = 323°K). Dotted 

line calculated for a = 16 x 10 -16 cm 2 . Data from Mifflin 

and Bennett, J. Chem. Phys. 29, 975 (1959). 



6.5 Calculation of transport coefficients 167 

It is perhaps not surprising that the transport processes in the gas and 
the density of the solid do not give exactly the same values for the molecular 
diameter. But the lack of self-consistence between the collision cross- 
sections themselves is worth trying to explain. One approach is as follows. 
In calculating the number of molecular impacts on an area of wall, we 
assumed implicitly that all directions of motion were equally likely. It is 
this step which is not strictly correct It is true for a gas in equilibrium, 
but during transport there is some sort of drift in a special direction. In 
diffusion, for example, there is a nett drift of molecules in the -z direction. 
Because of the persistence of velocity of the centre of mass of a colliding 
pair (see the footnote to section 6.4.3), more molecules tend to travel 
parallel to - z after collision than in other directions ; all directions are 
not equally likely before collision and so they are not equally likely after 
collision. Of course, this non-uniformity is small, but it is the transfer of 
molecules in just this -z direction that we are calculating. Similar con- 
siderations apply to heat conduction. But when we deal with viscous flow, 
we are concerned with x-rriomentum being transported in the - z direction 
(Fig. 6.2). It is the x-direction which is favoured before and after collisions 
inside dV, and the number passing through dS in Fig. 6.13 is altered. When 
the very difficult averaging processes are carried through, much of the 
discrepancy disappears. 

6.5.5 Effusion 

Suppose we have a tiny area dS which is part of a wall of a vessel con- 
taining gas. In Fig. 6.13, the xy plane can now be thought of as a wall, with 
the gas above it. Using Eq. (6.18), which states that when there are n 
molecules per cm 3 the number of impacts on unit area is \ric per second, 

No. molecules hitting area dS per second = \ric dS 

Pc 
4 kT 

, P 2 (2RT\ 112 

= i *r>M dS(6 - 24) 

where n is the number per cm 3 and c the mean speed, and we have used 
the relation P = nkT and Eq. (5.10) for c. 

Consider now a thin membrane with a hole cut in it, whose diameter 
is comparable with the thickness of the membrane. Let there be gas at a 
certain pressure on one side of this partition, and a lower pressure on the 
other. Gas must diffuse through the hole but we must distinguish two sets of 
conditions. If the diameter is large (more precisely, if it is large compared 



168 Transport properties of gases Chap. 6 

with the mean free path between collisions), any molecule suffers many 
collisions while going through the hole, and the description of the diffusion 
process as a random walk (superimposed on the bulk flow through the 
hole) is applicable. But if instead the dimensions are small compared to 
the mean free path— conditions which can be achieved using a tiny hole 
with gas at low pressure— then a typical molecule suffers its last collision 
some distance in front of the membrane and then goes straight through 
without further collision. In fact the number going through per second 
from one side to the other is equal to the number of impacts per second 
on an area equal to that of the hole. If n represents the difference of 
number-densities between the two sides, and P the difference of pressure, 
then the expressions given just about give the nett rate of transport of 
molecules from one side to the other. 

The process of diffusion through a small hole is called 'effusion'. It is 
important to notice that the rate of effusion at a given temperature is 
proportional to (1/M) 1/2 . A light gas, therefore, effuses more rapidly 
through a tiny hole than a heavy one. 

This fact has been used as the basis of a process of great technological 
importance to separate gases of different molecular weights. In particular, 
it can be used for enriching rare isotopes found in gases consisting of 
mixtures of isotopes. 

Imagine two compartments separated by a membrane with many small 
holes of suitable dimensions (Fig. 6.18). One compartment contains gas 
at a relatively high pressure— though it must be low enough to produce 
a sufficiently long mean free path— and the other side is continuously 
pumped to maintain a very much lower steady pressure. The 'partial 
pressures' exerted by the gases individually will be denoted by P lh , P 2h 
on the high pressure side and P u , P 2l on the low pressure side. 

Now the pressure of gas 1 on the low pressure side is proportional to 
the number of molecules effusing into it per second ; similarly for gas 2. 
Hence 

1 P lh /2RT\ 1/2 
P u _ 2jnkf\M, ) 
P21 1 P U [2RTV 12 
Ijn kT\ M 2 J 

(where we have written P lh in place of (P lh -P u ) for the pressure causing 
gas 1 to effuse through, and similarly for gas 2). Hence 

Pu (M 2 \ ll2 P lh 



(6.25) 
Pu Wi Pih 



6.6 Knudsen gases 



169 



This equation says that the fractional concentration of gas 1 on the low 
pressure side is a factor (M 2 /M l ) 112 times as great as on the high pressure 
side. There is an enrichment of the lighter gas on the low pressure side. 



high pressure 



low pressure 




Fig. 6.18. Effusion of gas mixture through a 
membrane. 

This is the basis of one method for enriching the rare isotope 235 U. 
Natural uranium is mostly 238 U with 0.7 % of the lighter isotope. From the 
metal, the gas uranium hexafluoride UF 6 can be produced. Its molecular 
weight is about 350, so the two isotopes give molecules differing by about 
1 % in mass. It follows that the concentration of the lighter isotope is 
increased by a factor 1.005— that is, from 0.7% to 0.7035 %— by a single 
passage through a membrane. The process can be made regenerative or 
many membranes can be used in cascade^ so that useful concentrations of 
the rare isotope can be obtained. 



6.6 KNUDSEN GASES 

It is not difficult to see why the viscosity and thermal conductivity of 
gases cease to be independent of pressure when the pressure is low (Figs. 
6.14(a) and 6.15(a)). It has already been pointed out (in section 6.4.2) that 
at extremely low pressures the mean free path X between collisions becomes 
very long. Already at 10~ 2 mm (easily attainable with a rotary pump) X 
is about 1 mm for a typical gas; at 10" 4 mm it is 10 cm. Ordinary pieces 
of apparatus commonly have dimensions of the order of millimetres or 
centimetres so that at low pressures the calculated mean free path may be 
larger than the apparatus. What this means in practice is that a molecule 
can go from one side of the apparatus to the other without making any 
collisions at all ; the mean distance it travels is dictated by the size of the 
apparatus and not by the properties of the gas. 

Gases at such low pressures are called 'Knudsen gases' after the scientist 
who first investigated them systematically. They are said to exhibit 
'molecular flow' instead of viscous flow. 



170 Transport properties of gases Chap. 6 

6.6.1 Viscous forces in a Knudsen gas 

As an example of Knudsen-type behaviour, consider the force between 
a moving and a stationary plate immersed in a low-pressure gas (Fig. 6.19 ; 
in contrast with Fig. 6.2, there is no velocity gradient in the medium 
between the plates). Consider a molecule which has struck the stationary 
surface and remained long enough to come to rest. It jumps off — and for 
simplicity we will imagine it to be emitted normally, like molecule P in 
the diagram. It then travels with its thermal speed all the way across the 
gap without colliding, until it hits the other surface. After this, one of two 
things might happen. It may stick for a long time and eventually be emitted 
in some direction at random, or it might act as if it were specularly reflected. 

moving plate 

/ 

Q/ \R 



stationary plate 
Fig. 6.19. Viscous drag with molecular flow. 

The experimental evidence suggests that the molecules mostly stick and 
for simplicity we will assume that they all do so. Each such molecule is 
travelling with x-component of relative velocity — U with respect to the 
moving plate and therefore transfers momentum — mil directly to it. The 
same overall result holds for molecules which travel obliquely across the 
gap. Molecules like Q in the diagram bring some thermal momentum to 
the right but this is cancelled out by molecules like R, of which there are 
an equal number. The net result is that momentum — mil is transferred 
from every molecule. 

The total rate of transfer of momentum — the force exerted by one plate 
on the other — is therefore equal to the number of molecules striking the 
moving plate per second, multiplied by mil. 

The number striking unit area per second is \nc. The derivation in 
section 6.5.1 is independent of considerations of mean free path; the result 
is true as long as there is equilibrium, or near-equilibrium. Therefore the 
rate of transfer of momentum (the force on area A) is ^AnmcU* 



* We could arrive at practically the same result if we started from the equations 

AU 

F = r] t] = jnmcl 

a 

and put the mean free path X equal to the separation d. 



Appendix B 171 

The drag is therefore proportional to the speed of the moving plate but 
does not depend on the separation (provided of course that this is small). 
Two oscillating disc arrangements, with different spacings between the 
discs (Fig. 6.3) would have just the same damping; a coefficient of viscosity 
based on Eq. (6.7) cannot be defined. At the same time, the dependence 
of the viscous drag on n alone, and not on the product (nk), means that 
the force is proportional to the pressure instead of being independent of it. 
This behaviour is beginning to be shown at the left of Fig. 6.14(a). 

For the flow of gas down a long circular tube, the analysis for viscous 
flow shows that the mass of gas transported per second is proportional 
to (j> A /r] where r\ is the viscosity coefficient and (f) the diameter of the tube. 
In the molecular flow region, where the effective mean free path is dictated 
by the diameter, the mass per second is proportional to 3 . 

Similar results hold for the other transport properties. Gases become 
thermal insulators, for example, at very low densities. An ordinary 
vacuum flask (dewar vessel) has double walls enclosing space evacuated to 
a sufficiently high vacuum for the mean free path to be limited by the 
spacing. Another interesting way of providing thermal insulation is to 
fill the interspace with a fine powder (derived from silica, and cheap to 
produce) which is in the form of tiny thin-walled hollow spheres, loosely 
packed. If the space is only roughly evacuated, the mean free path can be 
limited by the diameter of the spheres or the spaces between them, 
dimensions much smaller than the spacing between the walls of the vessel ; 
the thermal conduction falls below the value expected if there were no 
powder. Powder-packed vessels are much stronger mechanically than 
dewar vessels ; tank wagons can be insulated in this way. 

APPENDIX B 

B.l. Diffusion coefficient in gases 

In this Appendix we calculate the diffusion coefficient in a gas taking 
into account the distribution of free paths — that is, not making the approxi- 
mation which was made in section 6.5.1. 

Imagine a small area dS, normal to the z-axis (Fig. B.l), located some- 
where inside a large volume of gas. Consider a small volume dV located 
at distance r at an angle 9 to z. We will first calculate the number of 
molecules which pass through dS per second, having made their last 
collision inside dV. 

If there are n molecules/cm 3 in the neighbourhood of dV, there are ndV 
molecules inside this little volume. 

Each one, on average, undergoes one collision every time it travels a 
distance k ; that is, once in every (k/c) seconds. 



172 Transport properties of gases Chap. 6 




Fig. B.l. Coordinate system for calculating transport coefficients. 

Therefore, the number of molecules which suffer collisions inside dV in 
one second is (nc/X) dV. 

Each one of these molecules then goes off in some direction, and we 
assume that all directions are equally likely. The fraction which start off 
in the direction of dS is 

solid angle subtended by dS _ dS cos 

4tt 4?tr 2 

Notice that by writing the solid angle in this way, we are implicitly counting 
molecules that come from below as negative in number. This is because 
when 9 is greater than 90°, cos 6 is negative. We are eventually, therefore, 
going to calculate the nett flux of molecules going through dS from above 
to below— that is, the number going downwards minus the number 
going upwards, the nett rate of diffusion downwards. (If we wanted to 
calculate the total number of molecules going through dS irrespective of 
direction we would have to take |cos 6\ instead of cos in the expression 
for the solid angle.) 

Of this fraction that start off in the direction of dS, some will suffer a 
collision on the way. The fraction passing straight through dS will be the 
fraction whose mean free path is equal to or exceeds r. Now the probability 
of a free path between r and (r + dr) is 



exp 



r\dr 
IT 



Appendix B 173 

from Eq. (6.15), so the probability that it exceeds r, i.e., that it lies between 
r and oo, is 



1 f 00 



, >r ,/A dr = e" r/A . 

A 



Collecting these results, and using spherical polar coordinates as in Fig. 
5.6(6) so that 

dV= r 2 sin dr dd d(f), 

we calculate that the number of molecules which collide inside dV and 
then travel all the way to dS and go through it in one second is 

COS C YiC 

dS— - T -e~ r/x dV = dS—- d</3 cos 6 sin 6 dd e" r/A dr 
4nr 2 X 4nX 

where we are counting molecules coming from below as negative. It is 
convenient now to write cos 6 = \x, so that dfi = sin 6 d6. The number is 

nc 
dS— -d</3^d/ie~ r/A dr. (B.l) 

471/1 

If we integrate over all space — all values of r, 6 and <f> — we take account of 
all molecules passing through dS, wherever their last collision takes place. 
It is obvious, however, that only values of r comparable with the mean free 
path A are important, because the factor exp( — r/X) means that there are a 
negligible number of molecules whose last collision took place far from dS. 

The n which appears in this expression is the number of molecules per 
cm 3 inside dV. Let us now assume that there is a concentration gradient 
in the direction of z — that is, that n depends on z. 

If n(0) is the value of n at z = 0, the value of n at z is 

™ l dn \ ! il 82n \ 
n(z)^ ( ) + z(-) o+ -z^^) o + ... 

and higher terms are negligible because only small values of r are important. 
Writing z = r cos 6 = rfi, 

dn\ 1 - ^ld 2 n 



„(z) = „(o, +r/i ^j o+ - r V(^) o + 

We can substitute this in (B.l) and integrate to get the nett number of 
molecules travelling through dS from above to below per second. To cover 
all possible locations of dV, r goes from to oo, </3 from to 2n and from 
to n (fi from 1 to - 1). If we divide through by dS, we get the nett flux 



174 Transport properties of gases Chap. 6 

J of molecules per unit area per second from above to below. Arranging 
the terms : 



Ank 






/»2ti pec /»1 

« d(j) e~ r/x dr /xd/i 
Jo Jo J - 1 

d</> re" rM dr ^ 2 dji 

0^0 ♦'O J -I 



l/^n 



Let us first concentrate on the integrals with respect to p. The first and third 
give even powers of pL and when limits are put in, they give zero. Only the 
middle term survives : 



J. 



1 t i 2 d f i = ^fi 3 ]l l =i 

- i 



In this surviving term, the r integral 

re-^dr = X 2 



f 



and the 4> integral gives In. 

Hence the nett flux from above to below is 

4nX\dzJ \3/ 3 \dzj 

This is a flux of molecules downwards, in the direction of z decreasing. To 
put this in exactly the same form as Eq. (6.1), we have to calculate the flux 
in the direction of z increasing. We must, therefore, change the sign of J. 
The equation is then identical with (6.1) and 

D = Uc- (6- 19b) 

We have performed the averaging over all directions and all lengths of 
free path much more precisely than in section 6.5.1, and the overall result 
is to justify the factor j rather than \. 

PROBLEMS 

6.1. A cylindrical dewar vessel, containing water at 0°C, stands in a room where the 
temperature is 17°C. The glass walls are silvered to reduce heat input; the outer 
diameter of the inner wall is 10 cm and the inner diameter of the outer wall is 
10.6 cm, the space between being filled with nitrogen gas at 1 cm pressure, 
(a) Estimate the thermal conductivity of nitrogen gas. 



Problems 175 

(b) Calculate approximately the heat influx per cm height of the flask due to 
heat conduction. 

(c) Estimate the value (in mm Hg) to which the pressure must be reduced before 
the heat influx begins to fall off. 

(d) Deduce an approximate expression for the thermal conductivity of a 
Knudsen gas (compare footnote, section 6.6.1) and estimate the pressure at 
which the heat influx falls to ^th of its original value. 

6.2. A test-tube contains a liquid whose level falls slowly by evaporation. If the 
liquid is very volatile, the rate of evaporation is limited by the rate at which 
vapour molecules can diffuse through the air molecules. Assume that (i) the rate 
of fall of level is so slow that conditions are practically 'steady state' conditions 
(see section 6.1.1); (n) the number of vapour molecules/cm 3 is n v at the liquid 
surface and zero at the top end of the tube which is open to an infinite atmosphere. 

(a) Write down the steady state equation for the number of molecules diffusing 
across any plane per second. Hence, the mass per second crossing any plane. 

(b) Write down a simple equation giving the rate of loss of mass of the liquid 
in terms of the rate of fall of the liquid level, dh/dt . 

(c) Hence show that the distance h of the level below the open end is propor- 
tional tO y/t. 

(d) In a narrow tube (1 mm diameter), initially full of ether, the level falls by 
about 1 cm in 30 minutes, at room temperature. Deduce D for ether through 
air. Make a crude estimate of the diameter of an ether molecule. (Saturated 
vapour pressure of ether at room temperature = 40 cm ; it obeys PV = RT 
roughly. Molecular weight = 74. Density of liquid = 0.7g/cm 3 .) 

6.3. The pressure of a gas in a thermionic vacuum tube must be such that the electron 
mean free path is substantially larger than the linear dimensions. If the collision 
cross-section of an electron with a molecule is of the order of the geometrical 
cross-section of the molecule, estimate the pressure required in mm of mercury. 
Estimate also the maximum size of pin-hole that may be tolerated in the envelope 
if the tube is to last for at least one year. Rate of flow (g/s) through tube of 
diameter D, length /, between pressure P and a much smaller pressure is 

(nD*/256rjl)(M/RT)P 2 for viscous flow and ( > /27t/6)(Af//?7 , ) 1/2 (D 3 /0^ for 
Knudsen flow. 

6.4. A gas at a low pressure and at a temperature T is contained in a vessel from 
which it effuses through a small hole whose dimensions are small compared 
with the mean free path. Show that the number of molecules with speeds between 
c and (c + dc) leaving the vessel per second is GcP[c] dc (where G is a geometrical 
factor which need not be evaluated. See section 5.2.4.) Hence show that the mean 
kinetic energy of the molecules leaving the vessel is 2kT. This is greater than the 
energy, 3kT/2, of the molecules inside the vessel. Is the equipartition law 
violated? Explain this result qualitatively. (It is useful to ask whether the beam 
is in thermal equilibrium.) 

Note 



f 

Jo 



dx = —5. 
a -5 



CHAPTER 




Liquids and imperfect gases 



7.1 RELATIONS BETWEEN SOLID, LIQUID AND GAS 

Transitions between the solid, liquid and gaseous states of any one 
substance — solidification, melting, evaporation, sublimation and so on — 
can be brought about by varying the temperature T, pressure P and 
volume V. In any experiment to study these changes, the substance must 
(in principle) be placed inside a cylinder as in Fig. 4.2, so that the pressure 
acting and the volume occupied can be altered and measured. There must 
also be a thermostat for controlling the temperature. 

The results can be displayed in several ways. In Fig. 7.1, the axes are 
pressure and temperature, in Fig. 7.2 pressure and volume. The lines are 
called phase boundary lines and represent the conditions when transitions 
take place. The ranges of the variables where the solid, liquid and gas 
phases can exist are shown as areas, on both diagrams. At the pressure 
and temperature represented by TP, called the triple point (a point on 
Fig. 7.1, a line on Fig. 7.2), all three phases can exist together. 

In Fig. 7.1, a vertical line represents the course of an experiment at 
constant temperature. Three such lines are shown at low, medium and high 
temperature, labelled a, p and y respectively. The same lines are shown on 
Fig. 7.2 ; they are called isotherms. 

Let us follow the isotherm a, from low to high pressure — that is, moving 
upwards in Fig. 7.1 or across Fig. 7.2 starting from the low pressure, large 
volume region at the bottom right. At first, the substance behaves more or 



7.1 Relations between solid, liquid and gas 



177 



less like a perfect gas, obeying PV = constant, so that the isotherm in 
Fig. 7.2 is part of a rectangular hyperbola. However, when the curve 
reaches the phase-boundary line, the gas suddenly begins to condense to a 
solid, which is of course much denser. If the piston which applies the pres- 
sure is moved so as to decrease the volume, there is no rise of pressure but 
more and more gas solidifies so as to keep the pressure constant. The 
isotherm in Fig. 7.2 therefore turns horizontal. When the substance inside 
the cylinder has all solidified, it becomes relatively hard to compress and 
the isotherm turns almost vertical. 




Temperature 

Fig. 7.1. Pressure-temperature diagram, sometimes 
called the phase diagram showing the relations between 
the solid, liquid and gas phases of the same substance. 
Full lines are phase-boundary lines, broken lines are 
isotherms. C is the critical point, TP the triple point. 
The boundary between solid and gas is called the 
sublimation curve, between liquid and gas the vapour 
pressure curve, and between solid and liquid the 
melting curve. 



178 



Liquids and imperfect gases Chap. 7 




Volume 



Fig. 7.2. Pressure-volume diagram for the same sub- 
stance as in Fig. 7.1. Full lines are phase boundary 
lines, broken lines are isotherms. C is the critical point, 
the triple point is the horizontal full line. 

The isotherm p is of a different kind. As the pressure is raised, the gas 
condenses to a liquid. In Fig. 7.1, the lower branch of the curve is crossed. 
In Fig. 7.2 the isotherm turns horizontal. When the gas has all been 
liquefied and becomes relatively incompressible, the isotherm in Fig. 7.2 
turns steeply upwards. At some high pressure, the upper branch of the 
curve in Fig. 7.1 is reached, and the liquid begins to solidify, that is, to 
freeze. There is a second horizontal part in the isotherm of Fig. 7.2, 
corresponding to the contraction in volume from liquid to solid. When 
solidification is complete, the isotherm rises steeply again. Of course this 
or any other isotherm can be followed in the reverse direction ; then the 
solid would melt to a liquid, the liquid would boil to form the gas. 

The high temperature isotherm y is different again. The gas can be 
compressed to very high density (comparable with that of the solid) before 
it condenses and when it does so, it goes straight to the solid. 



7.1 Relations between solid, liquid and gas 179 

One isotherm, between (3 and y, has a special importance. It is the one 
going through the point C, called the critical point. C is at the abrupt end 
of the lower branch in Fig. 7.1, and the top of the phase boundary curve in 
Fig. 7.2. This critical temperature is the highest temperature at which the 
liquid can exist. 

7.1.1 Data for argon 

The isotherms for argon are plotted in Fig. 7.3. Both coordinates are 
logarithmic, since this allows great changes of conditions to be represented 




Molar volume ■ 



Fig. 7.3. P, Fdiagram for argon. Pressure and volume axes are on logarithmic 
scales. In this diagram, full lines are isotherms, broken lines are phase boundary 
lines. Sources of data : 600° isotherm : Lecocq, J. Rech. Centre Nat. Rech. Sci. 
p. 55 (1960) ; 400° isotherm : Michels, Wijker and Wijker, Physica 15, 627 (1949) 
200° and 1 50° isotherms : Michels, Levelt and de Graaff, Physica 24, 659 (1958) , 
100° isotherm : Holborn and Otto, Z. Physik 33, 1 (1925). Liquid and vapour 
densities : Mathias et al., Leiden Comm. 131a (1912), and Michels, Wijker and 
Wijker, as above. Vapour pressures : Clark, Din et al., Physica 17, 876 (1951). 
Solid density : Dobbs and Jones, Rept. Prog. Phys. 20, 516 (1957). 



180 Liquids and imperfect gases Chap. 7 

in one diagram. A pressure at the bottom of the graph (0.1 atmospheres or 
7.6 cm mercury) is a partial vacuum; at the top, 1,000 atmospheres is an 
extremely high pressure. On this log-log scale, a perfect gas isotherm 
becomes a straight line at 45°. 

The triple point is at 83.3°K, the critical temperature is 150.9°K. The 
80° isotherm is of the type a, the 100° isotherm resembles p. The isotherm 
labelled 150° is actually just 0.2° below the critical temperature, which 
is why it just misses the point C. The other isotherms, at 200° and above, 
are of the type y. At 200° K and 1,000 atmospheres, the molar volume 
occupied by the gas is about 34 cm 3 and its density 1.2 gm/cm 3 , which is 
comparable with the solid density of 1.6 gm/cm 3 . This isotherm does not 
meet the solid-phase boundary line till a pressure of 6,000 atmospheres, 
above the top of the diagram, is reached. 

7.1.2 Metastable states 

In this section we will consider the processes of boiling, condensation 
and freezing. The P, V diagram refers strictly to equilibrium conditions, 
where in the horizontal sections of the isotherms the coexisting phases 
must be at exactly equal temperatures. In a liquid-vapour transition for 
example, this implies that all the evaporation must take place slowly at the 
surface. But in practice the heat input into a liquid may greatly exceed the 
energy that can be carried away per second in this manner and when a 
liquid boils at a finite rate, bubbles arise in it. In fact, their appearance is 
popularly supposed to be essential to the process of boiling. We will see 
however that this means that the liquid must be hotter than the vapour 
and — more important — it implies that the liquid can exist under conditions 
not shown in Fig. 7.2. 

It is not easy to produce a bubble. Let us begin by considering one which 
arises in the middle of the liquid, beginning as a small hole of atomic size 
and then growing bigger. We can show that if such a hole or incipient 
bubble is too small it will collapse again ; only if its radius exceeds a certain 
critical value will it grow. The argument is as follows. A spherical hole of 
radius r has surface area Anr 2 and energy 4nr 2 y, where y is the surface 
energy in erg/cm 2 . Therefore, the energy per unit volume is 4wr 1 yj%wr' = 
3y/r erg/cm 3 . By arguments which have been sketched in section 3.5.1 
and will be reiterated in section 7.3, we can identify this with the pressure 
inside the bubble— actually a more precise analysis gives 2y/r where y 
means the surface tension, in dyn/cm. This estimate is not accurate for a 
hole of a few atomic diameters in size where only a few bonds have to 
be broken but it is correct in order of magnitude even then — and the 
dependence on 1/r shows that enormous pressures must exist in order to 
create the hole initially. These can only be produced if the liquid is super- 



7.1 Relations between solid, liquid and gas 181 

heated above its expected boiling temperature. Let us assume that local 
superheating raises the vapour pressure by AP. Then bubbles of radius 
less than 2y/AP cannot be sustained and will collapse, larger bubbles can 
grow. Let us assume that we have argon at around 100°K, but that the 
conditions of heating are such that locally the temperature might reach 
103°K — a moderate but not untypical degree of superheating under 
ordinary conditions. The vapour pressure curve gives AP equal to 1 atmo- 
sphere, 10 6 dyn/cm 2 . The critical radius is then 2,000 A, corresponding to a 
relatively large bubble occupying the volume of a million atoms of the 
liquid. Big bubbles of this kind can only be produced from smaller ones 
requiring greater superheating, so that it is obvious that in ordinary 
boiling liquids, the bubbles must be started in some other way. 

In fact most bubbles form or nucleate at solid surfaces in contact with the 
liquid. The vessel which holds the liquid is likely to be hotter than else- 
where, so that bubbles would in any case be expected to form on its walls. 
Thin flat layers of vapour are produced and these can collect together and 
balloon upwards as bubbles. If the surface is rough, less superheating is 
needed before the bubbles can swell up in this way, and then detach them- 
selves, though it is not at all obvious why this is so. Pieces of solid inside 
the liquid can also act as nucleation centres — glass seems to discharge a 
continuous stream of tiny particles into water so that there is never any 
shortage of them. Once a bubble does form, the liquid cools itself by 
evaporating into it and the temperature falls locally towards the equilibrium 
value. 

Under exceedingly clean conditions however, the only way the bubbling 
can start is in the body of the liquid so that great superheating can occur. 
One can achieve this by using chemically polished tubes of newly drawn 
glass. More simply, it is possible to suspend liquids in oil of the same 
density but having a higher boiling point — water, for example, can con- 
veniently be suspended in oil of cloves. Each liquid has a fairly well- 
defined temperature 7^ of maximum superheating. 10° below 7^,, it will 
last for hours ; 5° below T m it can be held for a few seconds ; at T m it explodes 
immediately. It will be seen later (section 7.8.1) that T m is quite close to the 
critical temperature T c . 

Thus superheated liquids can be kept for long periods and studied at 
unexpectedly high temperatures. These are called metastable conditions. 
Instead of following the expected isotherm GADF in Fig. 7.4, comparable 
with the isotherm /? in Fig. 7.2, isotherms like GAB can be produced. 
(AB does not represent the course of a superheating experiment but is a 
plot of the properties of the metastable liquid at constant temperature.) 

Similarly, gases may be made to exist at unexpectedly low temperatures 
or high pressures where the P, V diagram would indicate that they should 



182 Liquids and imperfect gases Chap. 7 

liquefy. They are called supersaturated vapours, and they exist because 
droplets have to exceed a certain critical size before they will grow; 
smaller droplets evaporate again. Condensation takes place most easily 
on rough solid surfaces or around small solid particles. In the absence of 
these, supercooled vapours can be kept for long periods and isotherms like 
DE of Fig. 7.4 can be plotted. 




Fig. 7.4. Part of a P, Fisotherm for liquid-gas 
transition, showing metastable states. FDAG 
is similar to part of the isotherm /? of Fig. 7.2. 
AB: superheated liquid. X: liquid under 
tension. DE : supersaturated vapour. 



Finally, liquids can be supercooled below their expected freezing points 
before they begin to solidify. Here the role of nuclei is to provide surfaces 
with the same crystal spacing as the solid so that this can grow by the 
addition of atoms to it one at a time instead of many atoms having to 
arrange themselves into a regular array all at once. 



7.1 Relations between solid, liquid and gas 183 

7.1.3 The tensile strength of liquids 

The tensile strength of a substance is the tension (measured in units like 
dyn/cm 2 ) which must be applied in order to break it. For a liquid, the 
stretching and breaking are represented by ABX of Fig. 7.4 : a liquid under 
tension is, of course, in a metastable state. 

The tensile strength of a metal or other solid specimen can be measured 
by gripping its ends in some way and then pulling them apart, but this 
method obviously cannot be used with a liquid. Instead, the liquid must 
be put in some sort of tube so that when tension is applied in some way 
to the liquid and a bubble appears in it and it breaks, one can usually not 
be sure whether the break occurred inside the liquid or between the 
liquid and the walls. In other words, either the tensile strength of the 
liquid itself or the adhesion of liquid to wall may have been measured — 
presumably the smaller of the two. 

It is observed that the tensile strength of liquids is much less than that 
predicted by theory. Part of the discrepancy may be due to the presence 
of tiny cracks in the walls of the tube which harbour minute bubbles of gas 
and prevent the liquid from entering — the very kind of nuclei which are 
postulated to account for ordinary boiling— and these constitute points of 
weak adhesion to the walls. It is indeed observed that if high pressure is 
applied for a short time to a liquid just before the tensile test begins, the 
observed strength is usually enhanced. This is presumably because the 
high pressure makes the bubbles dissolve in the liquid so that the molecules 
diffuse away, and do not come out again when the pressure is released. 

One simple method of demonstrating the breaking of liquids is to use a 
hypodermic syringe to suck up, say, water through the needle. If the 
plunger is pulled rapidly so that the liquid rushes in, then where it leaves the 
narrow needle and enters the wide barrel the liquid must be under tension : 
bubbles can be seen to be created there. The liquid is said to cavitate. 
Cavitation is responsible for a good deal of noise in plumbing systems, and 
for great losses of energy from ships' propellors. Under controlled con- 
ditions, cavitation can be produced inside the body of a liquid, far from 
any solid surfaces, by passing sound waves of high amplitude through it. 
The breaks take place during the rarefactions. Unfortunately, this is not a 
good way of measuring the tensile strength of a liquid because of the large 
changes of temperature which also occur. 

The best and most direct measurements have been taken with apparatus 
of the kind shown in Fig. 7.5(a), though without elaborate techniques for 
cleaning and degassing the inside of the tube the results are often a factor 
10 lower than accepted values. The Z-shaped tube is made of glass and is 
open at both ends. It is kept in a horizontal plane by fixing it to a horizontal 



184 Liquids and imperfect gases Chap. 7 

disc which is itself mounted at the end of a vertical shaft of a variable-speed 
motor. The tube is filled with liquid using a syringe, making sure that there 
is enough to extend round the bends into the arms. Then the tube is 
rotated. The liquid (surprisingly) is not thrown out but remains stably 
inside the tube. (This is because the centrifugal force on the liquid at A of 
Fig. 1.5{b) is less than at B because the distance OA is less than OB.) 



__ B 





(a) 



Fig. 7.5. (a) Tube for measuring the tensile strength of a 
liquid by centrifugal breaking, (b) Pressures at A, A', B, B' 
are all equal to atmospheric ; hence if the liquid is originally 
unsymmetrical, as shown, it must move back to a sym- 
metrical position. 



Solid impurities should of course be centrifuged outwards, and bubbles, 
similarly, collect at the centre. At low speeds, while the nett pressure is still 
positive, small air bubbles sometimes collect and do not grow. It is best to 
stop the rotation and coax them out of the tube and to hope that some of 
the bigger nuclei have been got rid of. 

At a distance r from the centre, a slice of thickness dr has mass (pot dr) 
where p is the density, a the area of cross section ; the centrifugal force on it 
is co 2 par dr, where co is the angular velocity in rad/s. Each slice of the 
liquid exerts an outward force on the slice next to it and the total force is 
greatest at the centre. Integrating between the two open ends, the tension 
(per unit area) is co 2 prl, where r is the radial distance to the free meniscus. 
The (positive) pressure of the atmosphere is added to this. The speed of 
rotation must be increased slowly till the liquid breaks, an event which can 
be easily seen because the liquid at the centre is always visible even when the 
tube is rotating fast. It is convenient to use a flashing stroboscope for 



7.2 The approach to the liquid state 185 

illumination, however, because this allows both r and the frequency of 
rotation to be measured easily. 

With ordinary tap water and no special cleaning of the tube, the liquid 
breaks when the frequency is about 5,000-10,000 rev/min, with r about 
2-3 cm. This corresponds to a tensile strength of 3-10 atmospheres. With 
elaborate cleaning, values of the order of 10 times higher are found. The 
tensile strength decreases with temperature. 

At room temperature, carbon tetrachloride has a tensile strength of 
276 atmospheres, mercury 425 atmospheres; for liquid argon at 85°K the 
figure is 12 atmospheres. 

7.2 THE APPROACH TO THE LIQUID STATE 

It can be seen that the range of temperature over which the liquid can 
exist at all is a very narrow one. It is bounded at the lower end by the 
temperature of the triple point and at the upper end by the temperature 
of the point C, the critical temperature. Typically, this is only a 2 : 1 range 
in absolute temperature. By contrast, the gas can certainly exist at all 
temperatures, and there is no evidence to suggest that the solid cannot 
exist at any arbitrarily high temperature provided the pressure is high 
enough. This distinguishes the liquid state sharply from the solid and gas — 
it requires explanation. 

An assembly of atoms in a rarefied gas is simple to treat mathematically, 
because each atom is effectively isolated and moves independently of all 
the others. At the other end of the scale, a perfectly regular solid can also be 
treated by comparatively simple mathematical methods — at any rate, if the 
amplitudes of vibration are not large — because all atoms are in identical, 
highly symmetrical environments. By contrast, the disordered array of 
atoms in a liquid is difficult to describe mathematically. Each atom has a 
large potential energy, comparable with that in a solid, due to its inter- 
action with a number of neighbours. But its environment is continually 
changing with time, and an atom is neither completely caged in by its 
neighbours as in a solid, nor perfectly free to move as in a gas. 

Now liquids can be derived from solids by melting, and this suggests 
the following approach. We can consider a single atom inside an otherwise 
regular solid to be displaced from its lattice position, Fig. 7.6(a). This 
produces a small region where there is disorder, a region of high density 
near a hole in the lattice. When we write down the energy of such a con- 
figuration, the kinetic term is of course unchanged, and the potential term 
— the difficult one to calculate because it depends on the distances between 
pairs of atoms — is also not too complicated because only a few pairs of 
atoms are concerned. Thus we can deal with a single displaced atom. Then 



186 



Liquids and imperfect gases Chap. 7 



we can imagine many such disordered regions to be produced. Fig. 1.6(b). 
As long as they are far enough away from one another, each region will be 
independent of all the others and the energy required to produce n of them 
will be just n times the energy to produce a single one so that it can still be 
calculated. When the degree of disorder becomes too great, however, the 
problem becomes intractable. But we can in this way produce a first 
approximation to a liquid — or at any rate, to a solid which is showing 
signs of melting. Certainly, though such an assembly with a small number 
of displaced atoms is really quite far from being a liquid, Fig. 7.6(c), it can 
be expected to indicate relations to look for in a real liquid. 



oooooo 
oooooo 
oooooo 

OCKpOO 

oooooo 



(a) 






icf) 

Fig. 7.6. (a) One atom out of place in an otherwise regular solid. (Compared 
with Fig. 2.5 the coordination number is low.) (b) Several atoms are displaced 
but the disordered regions are still more or less independent, (c) Atoms in a 
liquid, (d) A compressed gas at high density with several small clusters of 
atoms. Both (£>) and {d) begin to resemble (c). 



Alternatively, we can concentrate on the fact that liquids can be derived 
from gases by condensation. We can, therefore, begin with a gas at very 
low pressure and imagine the atoms brought closer and closer together so 
that they spend progressively more of their time near to one or more 



7.2 The approach to the liquid state 187 

neighbours and the potential energy of the clusters of atoms cannot be 
neglected. The gas is then said to be imperfect. Again, only an assembly 
with a small number of small clusters of atoms can be dealt with simply, 
but even this is a rough approximation to the onset of condensation and 
the formation of the liquid, Fig. 1.6(d). 

In Chapter 9 we will be concerned with imperfect solids, the onset of 
melting, and liquids in so far as they resemble solids. In this chapter, we 
will deal with imperfect gases, the beginning of condensation, and liquids 
as derived from gases. 

7.2.1 Laws of corresponding states 

We have emphasized the difficulties of writing down the potential 
energy term in liquids and have stressed that the procedure of regarding 
them as derived from imperfect gases cannot be pushed too far. We then 
meet a paradox. The simplest considerations allow us to derive an equation 
— van der Waals' equation of state — whose basis is particularly crude, but 
which is found to predict many properties of liquids themselves with 
surprising accuracy. By rights, van der Waals' equation should only 
succeed in describing the behaviour of gases which are still far from 
condensing ; yet it is capable of predicting, in order of magnitude, such 
properties as the latent heat of the liquid. 

There is no doubt that this surprising power is in part due to the 
felicitous analytical form of the equation, which is simple but adaptable. 
But the important fact is that once the interaction energy e between a 
pair of atoms is known, many of the properties of all the phases can be 
estimated, as we have already seen. In van der Waals' equation, we 
express energies not in terms of e itself but in terms of the critical tempera- 
ture T c (or rather kT c ), which is directly related to ^. The pressure and 
volume at the critical point C are also taken as standard parameters. 

Using the simple approach of Chapter 3, we were able to estimate the 
equilibrium properties of substances only at very low temperatures. Van 
der Waals' equation, however, suggests that we can legitimately compare 
the properties of two substances at their critical points — or at some given 
fraction of their critical temperatures, or under corresponding pressures or 
volumes. Conditions of this sort are called corresponding states. 

The real significance of van der Waals' equation is therefore that it 
predicts certain laws of corresponding states. As a real description of phase 
changes, it fails badly — it does not predict the existence of the solid, for 
example — and even to make it refer to liquids is to use a wild and un- 
justifiable extrapolation. Nevertheless, more accurate theories of con- 
densation are extremely difficult to construct, and do not lead much 
further. With these caveats, van der Waals' equation will now be derived. 



188 Liquids and imperfect gases Chap. 7 

7.3 VAN DER WAALS' EQUATION 

The object of the section is to develop a simple theory of imperfect gases 
and their relation to liquids. This means that we will try to explain as much 
as possible of Fig. 7.7, which has been dissected out of Fig. 7.2. 




Fig. 7.7. P, V isotherms for gas and liquid, extracted 
from Fig. 7.2. 

For an imperfect gas, we have to take some account of the potential 
energy of the atoms due to their interactions. These are of the form given 
by Fig. 7.8(a) which is identical with Fig. 3.4. r is the distance between the 
centres of two atoms. 

Since our theory will perforce only be crude, we will simplify the 
algebra by substituting a rough approximation for this curve — the square- 
well potential of Fig. 7.8(b). This has the form 

Potential energy i^(r) = oo 



r < a 

= — s a < r < oca 
= aa < r. 



(7.1) 



a is a number greater than 1 which presumably ought to be chosen so that 
the volume integral of the potential energy, 



f 

v a 



V{r)Anr 2 dr 



7.3 Van der Waals 1 equation 189 




n 
























A 


B C 












r 






7 


aa 





Fig. 7.8. (a) Interatomic potential energy as function of distance between 

the centres of two atoms ; identical with Fig. 3.4. {b) Square-well potential 

energy, an approximation to (a), adequate for many purposes. A, B and C 

correspond to Fig. 7.9. 





£ 



'c 
(c) 



/A B| C 



(of) 



Fig. 7.9. Interactions of molecules (each of diameter a) having square-well poten- 
tials, (a) r A is just greater than a and the system has energy -£. (b) r B is just less than 
aa and the energy is again -e. (c) r c is just greater than aa and the energy is zero. 
(d) Shows the centre of the second atom in relation to the first; the interaction volume 
v, lies between the two dashed outlines. 



190 Liquids and imperfect gases Chap. 7 

has the same value for the 6-12 and the square well potentials.* In this way 
one obtains 

a 3 -l=f i.e. a = 1.54. 
The potential implies that the atoms are incompressible spheres of 
diameter a ; the centre of two spheres cannot therefore approach more 
closely than a. At distances between a and aa, the potential energy is — e, 
as shown in Fig. 7.9(a) and (b). At distances greater than oca, there is no 
interaction, Fig. 7.9(c). Another way of describing this is to imagine each 
atom surrounded by an 'interaction volume', bounded by spheres of 
radii a and aa, as in Fig. 1.9(d), so that if the centre of another atom lies 
inside this volume the energy is — e. The size of this interaction volume is 

v . = f7ta 3 (a 3 -l)- (7.2) 

If we write v for the volume of a single molecule, then we have 



and 

v { = 8(a 3 -l)i; . (7.3) 

Now the problem is to calculate the mean energy of attraction of the 
assembly of molecules. To do this exactly, we should write down Boltz- 
mann factors representing the probability of different values of the 
energy. But this becomes impossibly complicated, so we will adopt a 
simple approximate procedure. Consider two atoms located somewhere 
inside a volume V. Assuming that all positions are equally probable 
(which is not strictly true at such short distances), the probability that 
one lies within the interaction-volume of the other is (vJV), and the po- 
tential energy of the pair is, on average, —{v i /V)s. If there are N atoms 
randomly distributed in this volume, we can select jN(N—1)~jN 2 
pairs of atoms. 

Therefore the potential energy of the assembly is, on average : 

ir= ~\ N2 [v) e - {1A) 

So the total energy, kinetic plus potential energy, for N molecules of the 
imperfect gas is 

jf + r = l N kT- l -N 2 \^\z. (7.5) 

In a perfect gas, the second term is zero. 

* We identify a with the a of Eq. (3.5). 



7.3 Van der Waals' equation 191 

Now we have already mentioned, in section 3.5.1, that an energy 
density or an energy per unit volume is equivalent to a pressure, although 
if we use the kinetic plus potential energy this relation is only true for 
adiabatic changes when no heat flows into the system. But for rough 
calculations, the error introduced is not too serious. For example, for a 
perfect gas the energy is all kinetic and has the value \NkT so that the 
rough rule would give P = \NkT/V, or in other words PV = \RT, 
instead of RT. Here, for the imperfect gas, we are calculating the effect of a 
small additional potential energy term, so it will serve our purpose to get 
the major term in the pressure correct (by omitting the factor f ) and use the 
rough rule for the extra term ; this is likely to introduce a numerical error 
but to leave the form of the expression correct. Then, deliberately leaving 
the denominator a little vague, 

p = RT-mVj/V)* (7 . 6) 

volume 

In this way we have accounted roughly for the attractive forces between the 
atoms. If we seek a dynamical interpretation of the reduction in pressure, it 
is that the atoms spend more time near one another than they would if 
there were no attractive forces, and this reduces the number of impacts on 

the walls. 

There is a second effect due to the interatomic potential energy, this 
time the repulsions which give the atoms their finite size. The volume 
available to the atoms to move freely about in is not V but something 
smaller. We should therefore, write the denominator on the right hand 
side as (V-b), not V, where b is a volume presumably of the same order 
of magnitude as the volume of all the molecules. 

It is not easy to decide the value of b exactly. One extreme argument is to 
imagine all the molecules gathered together in one lump— of solid or 
liquid— except for one single one, which would then obviously find a 
volume Nv not available for moving about in, where v is the volume of a 
single atom, f?r(a/2) 3 . Another extreme argument is to say that the centres 
of two gas molecules cannot approach more closely than a, so that a 
volume Sv is excluded around each molecule, making b eight times bigger 
than before. But this can be rejected as an overestimate because for a 
grazing collision between two molecules the trajectories are hardly 
deviated so that the centres come within just the same distance of one 
another as they would have done if the atoms were points ; there are many 
more glancing than head-on collisions. Presumably, then, b lies some- 
where between Nv and SNv . The most detailed analyses, taking 
proper account of collision dynamics show that 4Nv is the best estimate 
ofb. 



192 Liquids and imperfect gases Ch. 7 

It is convenient to write a in place of^Nhfi. It follows immediately from 
the connection between v it the interaction volume, and v , the volume of a 
molecule (Eq. 7.3) that 

a = N(ix 3 - l)be (7.7) 

where a is the measure of the range of the interatomic forces as shown in 
Fig. 7.8(b), and s is the depth of the potential well. We have also seen that 
(a 3 - 1) is about 8/3 so that a/b « 2.7 Ne. We have also seen, in section 
3.3.1, that the molar binding energy of the condensed phase at low 
temperatures is L = ^Nne (where n is the coordination number, about 
8 or 10). Hence roughly, 

a/b a £L . (7.8) 

Eq. (7.6) now reads 

„ RT-a/V 

p = -r=r (7 - 9) 

RT o_ 

since the second term is already a small correction; 

so that l P +JLy V -b) = RT. (7.10) 

This is van der Waals' equation of state. Eq. (7.9) differs from (7.10) in 
one important respect: (7.9) is quadratic in V, (7.10) is cubic, and this 
latter form is essential to the ability of van der Waals' equation to interpret 
many phenomena. 

We can plot P, Fisotherms by assigning a value to Tand calculating P as 
a function of V as in Fig. 7.10(a). These isotherms are of two kinds. At 
high temperatures, P decreases monotonically with V while at low 
temperatures the curves have maxima and minima, as expected of a cubic 
equation. We identify these two regions as above and below the critical 
temperature. It is obvious that the S-shaped curves are unphysical, but it 
is tempting to identify parts of them as representing the supersaturated 
vapour and the superheated liquid, and to draw horizontal lines across to 
represent the equilibrium mixtures, Fig. 7.10(b). No good reason for doing 
this can be adduced from the mode of derivation — the S-shapes are merely 
a result of imagining only pairs of atoms being near to one another — but if 
this procedure is accepted, then plausible (though not rigorous) arguments 
suggest that the line should be drawn so as to make the area under it equal 
to that under the curve. In other words, the areas of the two loops should 
be equal. 



7.4 Application to gases 193 





Fig. 7.10. (a) van der Waals' isotherms, (b) Horizontal line drawn so as to 
equalize the areas of the loops. 



7.4 APPLICATION TO GASES 

We have stressed that van der Waals' equation should be valid only when 
the density is not too great. We will therefore discuss its application to 
gases at low and moderate densities where the equation is applicable and 
at high densities where, as expected, it breaks down. We can regard a and b 
in the equation as constants whose values can be chosen to fit experimental 
data It will emerge that values of a and b can be chosen which allow 
several gas phenomena to be correlated. But when we compare these 
values of a and b with those expected from independent estimates of the 
sizes of molecules and the depth of the potential well, the agreement is in 
order of magnitude only— in particular, the value of b differs by about a 
factor 3 from that expected from the solid density. 

Finally we will consider briefly how the theory could be improved. 

7.4.1 The second virial coefficient B{T) 

One of the most powerful methods of displaying the way a real gas 
deviates from a perfect gas in its behaviour is to plot the ratio PV/RT as a 
function of increasing pressure or decreasing volume. There are theoretical 
reasons for preferring 1/Fas the variable, and curves for argon are shown in 
Fig. 7.11. Each curve refers to a fixed temperature. They are called virial 
plots. 



194 Liquids and imperfect gases Chap. 7 

Most of this graph refers to small volumes— this is the effect of using 
1/Kas the variable. When V is infinite, 1/Fis zero. When V is 100 cm 3 , 
which is quite small compared with normal conditions, 1/Fis only one- 
third the way along the axis in Fig. 7.1 1. The molar volume of liquid argon 
is about 30 cm 3 so that all the curves must be asymptotic to l/V = 0.033. 
At large volumes, that is when l/V tends to zero, PV/RT = 1 always, so that 
all the curves go through one point on the vertical axis. 




VV cm 1 

Fig. 7.11. Virial plots for argon. Sources of data as for Fig. 7.3. 



Many reasonable curves y = f{x) can be represented by a polynomial, 

y = a + /foe + yx 2 -\ . Here, each virial curve can similarly be represented 

by 

PV , B C 

RT V V 2 * 

The coefficients B,C ■•• are called the second, third and higher virial 
coefficients. They depend only on the temperature so that they should be 



7.4 Application to gases 195 

written B{T), C(T) and so on — in general, 

PV B(T) C(T) 

For the polynomial v = a + fix + yx 2 • • • , the gradient of the curve when 
x is small is equal to /? (since higher terms in the expression for the gradient 
are negligible). Thus the gradients of the virial plots when 1/7 is small, the 
initial gradients of the curves, are equal to B(T). A graph of B as a function 
of T for argon is given in Fig. 7.12. At low temperatures, when the virial 
curves start downward, B is negative. At temperatures near 410°K for 
argon, the virial curve starts horizontally, so that B = 0. Around this 
temperature, over a considerable range of pressures, the gas obeys the 
perfect gas law PV = constant (Boyle's law) with accuracy, whereas at 
other temperatures it deviates significantly at much smaller pressures. This 
is therefore called the Boyle temperature, denoted by T B . At high tempera- 
tures where the curves start upwards, B is positive and tends to a constant 
value. 

B(T) can be determined from experiments while PV/RT does not deviate 
too much from unity — under conditions, in fact, when van der Waals' 
equation should be valid. We will, therefore, rearrange (7.10) as a poly- 
nomial in the form of Eq. (7.11): 



P = 



RT a 

V-b V 2 



so 



PV = (l_ h \~ 1 a 



RT \ Vj RTV 

.2 



1 .t. « \1 & 

= 1 + \ h -Rf)v + V- 2+ '-- (7 - 12 > 



and comparing with Eq. (7.11) 



a 



B(T) = h ~Rf- (713) 

Van der Waals' equation therefore predicts that at very high tem- 
peratures B(T) tends asymptotically to b, while at low temperatures 
B(T) becomes large and negative, following a rectangular hyperbola. 
The Boyle temperature T B is evidently a/Rb. 

The detailed course of B(T) as calculated agrees quite well with experi- 
ment, as shown by the dashed curve of Fig. 7.12. This has been drawn with 



196 Liquids and imperfect gases Chap. 7 

b = 42cm 3 and a = 1.42 x 10 12 erg cm 3 /mol— values which give an 
adequate fit over the whole range and also give a/Rb close to the observed 
value, 410°K, of the Boyle temperature.^ 




Fig. 7.12. Second virial coefficient B{T) for argon. Dashed curve : van der 

Waals' curve calculated with a = 1.42 x 10 12 erg cm 3 /mol, b = 42 cm : 

B = [42 -(1 71 x 10 4 )/T] cm 3 . Data from Lecocq, J. Rech. Centre Nat. 

Reck Sci., p. 55 (1960). 

We can now apply a stringent test of the whole theory as constructed so 
far, by testing the prediction that 

o/b « \L ( 7 - 8 ) 

where L is the binding energy at low temperatures. Here a/b = 3.4 x 
10 10 erg/mol, compared with L = 7 x 10 10 erg/mol given in Fig. 3.13(b). 
The agreement is remarkable. 

The absolute value of b is not very good, however. Since we have seen 
that b should be about 4 times the volume of the molecules it would imply 
a molar volume of about 10 cm 3 for the solid instead of 26 cm 3 . A plausible 



7.4 Application to gases 197 

reason for believing that the value of B at high temperatures does never- 
theless depend on the volume of the molecules is provided by substances 
such as helium where the a term is small. Following the arguments of 
section 6.4.3, the value of b for a real gas would be expected to decrease at 
high temperatures because the molecules are not really hard spheres. In 
argon however, any small decrease of the b term in the expression 
(b-a/RT) is swamped by the change in the a/RT term. But in helium 
where a is small, a small decrease in B(T) at high temperatures is evident 
in the measurements — although the absolute value of b remains too 
small, even then. 

7.4.2 Specific heats of imperfect gases 

The specific heat C v of a gas obeying van der Waals' equation is the same 
as if the interaction terms a and b were removed and the gas became perfect. 
This is because during a heating at constant volume, the mean distance 
between molecules (and hence the potential energy) remains unchanged. 
Alternativejy we can argue that since 

C.-[§) f (5,5) 

(where we have written E for the mean value of the total kinetic plus 
potential energy) and 

- 3 a 

E = 2 RT ~V (7 - 5) 

for one mole of an imperfect monatomic gas, C v = f R as for a perfect 
monatomic gas. 

But if a gas is heated and expands to keep the pressure constant, a 
quantity of heat C p dTmust be supplied to raise the temperature by dT— 
not only to increase the kinetic energy of the molecules and to supply the 
work done, but also to increase their interatomic potential energy. For a 
monatomic gas : 

= ^RdT+-^dV+PdV. (7.14a) 

In general, for a gas whose specific heat at constant volume is C v , 

C p dT= C v dT+ (P + ^2) dV. (7.14b) 



C p dT=d\-RT--)+PdV 



198 Liquids and imperfect gases Chap. 7 

Here, dTand dFmust be related because the pressure must be constant. 
We can find this relation as follows. Since 

small variations of P, V and T must obey 



k + 172 d(V-b) + (V-b)d\P + 1 ^\ = RdT 



a \ ... / „ . _a 

V- 

P-^- 2 - 7 0jdV+(V-b)dP = RdT. 

We can neglect the term in ab/V 3 . In the present case, P is held constant, 
dP = 0, so that 

P — % )dV= RdT. 
V ) 

This is the special relation required to substitute in (7.14b) above. After 
simplifying, we get 

P + a/V 2 

C p- C »- R p_ a/V 2 

= R 1 + 



RVTj 

to sufficient accuracy. Thus for monatomic gas obeying van der Waals' 
equation 

5 2a C_ 5 4 a 

C„ = |R; C p = -R +Vf ; 7 = ^ = 5 + 5 ^^- (7-15) 

This means that the ratio y of specific heats is no longer a constant even 
if the classical equipartition of energy holds, y should increase as the 
density and temperature are decreased, and this variation allows us to 
measure a again. 

The ratio y was introduced into the discussion of perfect gases (section 
5.4.2) because their adiabatic elasticity is equal to yP, and the speed of 
sound is equal to ^{yRT/M); this allows y to be measured easily, which in 
turn allows both C p and C v to be calculated. For imperfect gases, however, 

* The same result can be deduced with more sophistication using the relation stated in 
section 5.4.1 : 

C„-C v = [P + (dE/dV) T ](dV/dT) p . 



7.4 Application to gases 



199 



all these results require modification. For gases obeying van der Waals' 
equation, it may be shown that the adiabatic elasticity is equal to yP 2 V/RT 
(which reduces to yP if PV = RT, as expected) and the speed of sound is 



c. = 



PV yRT 



RT\ M 



(7.16) 



Thus y can still be calculated from the measured speed of sound but with 
an additional factor which can be read off Fig. 7.11. For example, for 
argon at 200°K, at a pressure such that V = 200 cm 3 and l/V = 
0.005 cm -3 , Fig. 7.11 gives PV/RT equal to 0.79. The speed of sound is 
equal to 256 cm/s. From these data, y = 2.52 — quite a different value 
from the 5/3 = 1.667 found at low densities. 



200 



.200° K 



H* 



100 



0O1 



002 1/1/ 




1000 °K 



(o) 



(d) 



Fig. 7.13. (a) Ratio of specific heats y for argon, as a function of 1/Fat 200°K. 
When V = 200 cm 3 , l/V = 0.005 cm" 3 , y = 2.52. Hence the initial gradient 
of this graph is (2.52-1.67)/0.005 = 170 cm 3 . Data from Michels, Levelt 
and Wolkers, Physica 24, 769 (1958). (b) Initial gradient of this type of 
graph, as a function of temperature. Dashed curve : %a/RT; a = 1.42 x 10 12 
ergcm 3 /mol. Data for gradients taken from Nat. Bur. Std. (U.S.) Circ, 

564 (1955). 



A graph of y as a function of l/V at 200° K is given for argon in Fig. 
7.13(a). y begins by increasing linearly, as van der Waals' equation predicts, 
but then the curve turns downwards again when the density becomes large. 
The situation is analogous to that in the virial plots — van der Waals' 
equation predicts only the initial gradient. Proceeding as we did before, 
we can plot this quantity over a wide temperature range, and compare it 
with the predicted 4a/3RT. The two curves are shown in Fig. 7.13(b), using 
the same value of a equal to 1.42 x 10 12 erg cm 3 per mole as in Fig. 7.12 
for the van der Waals' curve. The agreement is good. Thus even the crude 



200 



Liquids and imperfect gases Chap. 7 



representation of interatomic forces provided by van der Waals' equation 
is enough to account qualitatively for the specific heat variations. 

7.4.3 Free expansion of gases 

In section 5.4.2 we discussed the temperature change when a perfect 
gas (with a = and b = 0) performs work by pushing a piston back — an 
adiabatic expansion with the performance of external work. If however a 
perfect gas undergoes an expansion without performing any external 
work, no energy is expended and there is no temperature change. This is 
called a free expansion and we will describe how it can be performed in 
principle although measurements of this kind are rarely performed in 
practice. 

Imagine a vessel constructed of rigid material, thermally insulating and 
of negligible heat capacity (Fig. 7.14). Inside, it is divided into two com- 
partments one of which contains a gas under pressure, while the other is 
empty. The wall dividing the compartments is then broken and gas flows 
so as to equalize the pressure throughout. Then, while the gas is flowing, 
one compartment gets hot because the gas there is being compressed more 
or less adiabatically, while the other side gets cold because it is doing the 
compressing. Imagine then that the two halves later reach equilibrium by 
exchanging heat with one another, and come to equal temperatures. If the 
gas is perfect, this final temperature will be exactly the same as the initial 
temperature. 





Fig. 7.14. Free expansion of a gas inside a rigid, insulated vessel. 

During the whole process, the gas is isolated from its surroundings by 
the rigid vessel so that energy neither enters nor leaves. The total internal 
energy, kinetic plus potential, of the gas must be constant. Let us therefore 
consider one mole of an imperfect monatomic gas, whose energy is given 
by the van der Waals' expression 



*-!"•-£ 



(7.5) 



7.4 Application to gases 201 

where V is the volume it occupies. This is conserved. If the subscript i 
denotes the initial and f the final conditions, then 

\ RT '-vrl RT '-y' ■ ,7 - 17) 

For 1 mole of argon expanding from 1 litre to 2 litres, that is from about 
20 to 10 atmospheres, the change of temperature would be 4° (taking a 
to be 10 12 ergcm 3 /mol). This is a large change but in practice the heat 
capacity of the vessel — necessarily thick walled — would decrease its 
magnitude and we will attempt no comparisons with the meagre experi- 
mental data. 

7.4.4 Joule-Thomson coefficient 

The Joule-Thomson porous plug experiment is a much more sensitive 
method of measuring the change of internal energy of a gas with pressure. 
It is a continuous process (as opposed to a "one-shot" process like the 
one just described) so that the temperatures ultimately reached by different 
parts of the apparatus do not depend on their heat capacities. 

In principle, Fig. 7.15, a gas is maintained at pressure P t (by an external 
compressor) and is brought to a known temperature T x . It is forced 
through a device which can maintain a pressure difference, and does not 
allow heat to be conducted across it. In the original experiments, carried 
out in 1852, this was a silk handkerchief. Nowadays, a cotton-wool plug 
or a porous ceramic plug is used ; often, just a long length of narrow-bore 
tubing with a small hole in the end. The gas then emerges into a space 



m 

mm 

pressure /° pressure P z 

temperature T A temperature T z 

Fig. 7.15. The flow of a gas from high to low pressure 
through a porous plug. 

* The general expression for the temperature change in a free expansion is 
C v (dT/dV) E = P- T(dP/dT) v 
which for a gas obeying van der Waals' equation is -a/V 2 . This gives 

C„dr= -(a/V 2 )dV 
in agreement with the above expression. 



202 Liquids and imperfect gases Chap. 7 

which is maintained at another known pressure P 2 . The temperature T 2 
at the exit side is measured. The gas is continuously forced through the 
plug, and measurements are made only when the parameters are steady. 

In such a process, the gas is certainly not isolated from its surroundings, 
so that the internal energies of 1 mole of the gas on one side and the other 
are not equal. However, we can still find a quantity which is conserved. 
First, we note that in the steady state there is no nett interchange of heat 
with the walls. Secondly, we can assume that the velocity of bulk move- 
ment of the gas is so small that its bulk kinetic energy can be neglected. 
Finally, we concentrate on the balance between the internal energy of the 
gas and the work performed on it or by it. 

Let 1 mole of the gas occupy volume V r on the entrance side. The work 
done on it to force it through the plug is jP . dV where P is constant at 
P t and V is changed from V^ to zero ; that is, (P 1 V t ). On the other side, the 
same gas performs work P 2 V 2 on the pump. The nett amount of work must 
come from the internal energy, so that 

E, +P.V, = E 2 + P 2 V 2 . 

In thermodynamics, E + PV is called the enthalpy, and it is this quantity 
which is conserved here. 

Expanding van der Waals' equation and neglecting the very small 
term in ab/V 2 : 

PV= RT-- + bP. 

We also have, for a monatomic gas, 

E = \ RT ~V (75) 

So 

5 2a 
E + PV = -RT + bP. 

We can now calculate the temperature change dT accompanying a small 
change of pressure dP. First we can, with little loss of accuracy, substitute 
PV = RT in the small a/ V term: 

E + PV = -RT — — + bP. 
2 RT 

Since this is conserved, 

d(E + PV) = 0. 



7.5 Refinements to van der Waals' equation 203 

This gives 

l RdT+ **dT-^dP+bdP = Q 
2 RT 2 RT 

dT 2a/RT-b 

- for a J-T process = & + 2«I7K7*- (7.19a) 

Writing 2a/VTin place of 2aP/RT 2 , the denominator is seen to be C p for 
the monatomic gas, Eq. (7.15). Generalizing, one obtains for the Joule- 
Thomson effect in any gas obeying van der Waals' equation : 

dT 2a/RT-b ^^ 

— for a J-T process = — -. (7.19b) 

dP F C p 

This is called the differential Joule-Thomson coefficient. Both dTand dP 
represent increases. In a real experiment, dP is always negative. Therefore, 
if (2a/ RT— b) is positive, the drop of pressure will cool the gas. Evidently 
this should occur if the temperature is low, because then the first term is 
large. Conversely, a heating should occur at high temperatures. The 
changeover from heating to cooling, at a temperature called the inversion 
temperature T h occurs when T = 2a/ Rb. Van der Waals' equation predicts 
that the inversion temperature should be twice the Boyle temperature T B . 
Agreement with these predictions is surprisingly good. Figure 7.16 
shows the observed differential Joule-Thomson coefficients for argon 
(extrapolated to zero pressures where C p is accurately 5R/2). The observed 
inversion temperature T t = 785°K; thus the ratio TJT B = 785/410 = 1.92, 
close to the predicted value 2. The dashed curve is that predicted from van 
der Waals' equation with a = 1.42 x 10 12 ergcm 3 /mol, b = 42 cm 3 , the 
same values used for the virial coefficients and y curves, Figs. 7.12 and 
7.13. 



* 7.5 REFINEMENTS TO VAN DER WAALS' EQUATION 

The problem of deducing an accurate equation of state, valid at all 
densities, is basically one of writing down the potential energy of an 
enormous number of interacting molecules. Van der Waals' equation 
takes only the first step in this direction. In the following sections we will 
consider some important effects which have not been taken into account, 
and some ways in which van der Waals' equation might be refined. 

* The general expression is (dT/dP) H = {T(dV/dT) p - V}/C p where H stands for enthalpy. 
For a gas obeying a virial equation, this reduces to { TB'(T) — B(T)}/C P to a first approxima- 
tion. 



204 Liquids and imperfect gases Chap. 7 




Fig. 7.16. The differential Joule-Thomson coefficient for argon as a 
function of temperature. Since 1 atmosphere = 1.01 x 10 6 dyn/cm 2 , 1°K/ 
atm ~ 10~ 6 °K/dyn.cm" 2 . Dashed curve : van der Waals' curve calculated 
with a = 1.42 x 10 12 erg cm 3 /mol, b = 42 cm 3 : Joule-Thomson coefficient 

= ( .203)°K/atm. Continuous curve: experimental results. Sources 



of data : as for Fig. 7.13(a), {b). 



7.5.1 Dimers 

When two atoms in a gas are a small distance apart comparable with 
their diameters, they have an appreciable (negative) potential energy. 
In an ordinary encounter between two atoms, they approach one another 
and then fly past ; it is only while they are close together that they have 
potential energy comparable with - &. The pairs of atoms which we have 
considered so far are not bound permanently together in any way ; they 
are merely pairs of atoms which happen for a short time (a fraction vJV 
of the total) to be in one anothers' vicinity. 

But the existence of the minimum in the T^"(r) curve means that it is 
possible to form pairs of atoms which are loosely bound together. These 
are called dimers. It can be shown theoretically that the commonest form 
of dimer is not two atoms statically stuck to one another, or oscillating 
as if joined by a spring; instead, they rotate round one another — in orbit 
round one another, like a double star. The two molecules 'touch' one 



7.5 Refinements to van der Waals' equation 205 

another and the mean frequency of rotation is about 10 11 rev/s. 

To form such a pair out of two isolated atoms requires rather special 
circumstances. As has been mentioned, two atoms which come together 
will usually fly apart again ; but if by chance there is a third atom in the 
vicinity at the right moment which can take away enough kinetic energy, 
then the pair can be left in a bound orbit. This is called a three-body 
encounter. Equally well, another collision with a sufficiently energetic 
atom can knock them apart again. The energy required to disrupt a dimer 
is practically equal to e if they are orbiting only very slowly, but it is 
reduced if they are orbiting fast. The 'centrifugal potential' L 2 /2I has to 
be added to the interatomic potential energy, where L is the angular 
momentum and / the moment of inertia of the system. There is therefore a 
limiting angular momentum above which the potential well is filled up and 
the dimer cannot exist at all. 

During the whole of its existence, a dimer makes an appreciable con- 
tribution to the potential energy of the gas. At high temperatures, dimers 
are likely to be knocked apart again after a short time, but at low tempera- 
tures when they are longer lived, their energy can dominate the second 
virial coefficient B(T). 

Dimers have been detected and their masses measured experimentally. 
This was first done by Leckenby and Robbins in 1965. The basic idea was 
to produce a narrow beam of gas atoms, bombard it with electrons so as 
to produce ions and then to analyse the masses present in the beam by 
passing it through a mass spectrometer. The biggest technical difficulty 
was to produce the narrow beam inside the high vacuum required for 
the mass spectrometer to function. This was done by allowing the gas to 
escape through a tiny hole from a reservoir where the pressure was of the 
order of 1 mm of mercury ; the thickness of the diaphragm and the dia- 
meter of the hole in it were both of the order of 10" 4 cm, comparable with 
the mean free path of the molecules in the gas. Under these conditions, the 
beam effused through without change of temperature or mean energy, 
a true sample of the molecules inside the reservoir. 

With argon (atomic weight 40) molecules of mass 80 were detected. 
They were shown unambiguously to have originated inside the reservoir 
and not spuriously as the result of any process inside the mass spectro- 
meter. In the gas at room temperature at 10 cm pressure the dimer 
concentration was found to be 1 in 10 4 , in agreement with theoretical 
estimates. Of course these experiments can only be conducted with low 
pressures in the reservoir ; dimer concentrations are greatly increased at 
high pressures. 

One unsuspected fact was revealed in these experiments. It has been 
mentioned in section 5.4.4 that specific heat measurements indicate that 



206 Liquids and imperfect gases Chap. 7 

monatomic molecules in gases at ordinary temperatures do not rotate 
about their centres, whereas polyatomic molecules (such as N 2 or C0 2 ) 
do rotate. Now a dimer of N 2 or C0 2 has three sorts of rotation going 
on inside it — the spinning of each molecule about its centre and the 
orbiting of the molecules round one another. Within this system, angular 
momentum must be conserved. If therefore one of the molecules stops 
spinning for any reason, the orbiting must speed up — and this will pro- 
bably cause the pair to fly apart. In monatomic gases however, this effect 
cannot occur. The result is that polyatomic gases contain fewer dimers 
than monatomic gases. 

7.5.2 Higher clusters 

The pairs of molecules which are not bound together but nevertheless 
possess some potential energy because they happen to be in one anothers' 
vicinity for a short time are called clusters of two. Bigger clusters are of 
course possible, clusters of three or more. It was thought for a long time 
that three molecules meeting at a point would be so rare an event that its 
probability could be neglected: but this is not true at high densities. 
Taking into account the potential energy of the higher clusters will 
obviously alter the equation of state. How it will do so can be guessed from 
the following line of argument. Let us first consider a gas of hard-sphere 
molecules, but with no attractive forces : that is, e is zero but a is not zero, 
or in other words a is zero but b is not zero. Then the equation of state 
becomes 

P(V-b) = RT. 
Expanding this as a virial equation 

PV _ V b b 2 b 3 

RT~ V-b ~ 1 + v + v* + V* + """ 

In other words, for the hard-sphere model, the second virial coefficient is 
b, the third b 2 and so on. However, we have seen that when we put in the 
attractive forces and consider clusters of two, the second virial coefficient 
becomes (b-a/RT) instead of b, while the others are left unchanged. We 
may therefore guess that if we consider clusters of 3 the third virial co- 
efficient will be modified and so on. The higher clusters are therefore 
important because they allow the higher virial coefficients to be calculated 
properly. 

We will now outline how the clusters of 3 could be dealt with, using the 
same rough methods as we did for the clusters of 2. The argument will 
not be followed through to the end because the approximations are too 
crude; the purpose of this calculation is to indicate some interesting 



7.5 Refinements to van der Waals' equation 207 

features of the argument which have their analogies in more sophisticated 
treatments. 

Consider a volume V containing a single molecule and a cluster of 2 
molecules. The interaction volume of the cluster is shown in Fig. 7.17. 
If the third molecule enters either of the volumes v A , it is near only one 
neighbour and loses energy e only. But if it comes within the volume u B , 
it interacts with both of them and loses energy 2s. Thus there are two 
distinct types of clusters of 3. 




Fig. 7.17. Interaction volume of a cluster of 2. 

Assuming that the distribution is still random, the probability that the 
third molecule finds itself within one or other of the volumes v A is 2vJV; 
within v B , the probability is v B /V. Thus the average extra energy lost by 
the cluster of 2 because of the presence of the third molecule is 

2v A v B 

E f- 2s — 

V V 

which can be written 2vJV because 

2v A + v B = 2v t . 

We must now put in the probability that the clusters of 2 was formed 
originally, and the fact that in an assembly of N molecules there are 
N(JV— 1)(N — 2)/3! « N 3 /3\ ways of selecting three of them. The average 
potential energy of the assembly is thus 

2! \Vj 3! \VJ 

We can then write down the energy density and get a small pressure term 
in V~ 3 as well as the V~ 2 term of van der Waals' equation. Expanded as a 
virial equation, the third virial coefficient can be picked out. It depends 
on temperature but is still of order b 2 . 



208 Liquids and imperfect gases Chap. 7 

The problem of computing the energy of all the possible types of 
cluster becomes rapidly more complex with the size of cluster, if more 
realistic potentials are used in place of the square well. Calculations of 
this type have nevertheless been pursued because it was hoped that the 
S-shaped curves of Fig. 7.10(a) would be eliminated if all clusters could be 
included. In fact, dimers and higher bound groups of molecules probably 
play a dominant role in the process of condensation. They have been 
detected by the same kind of experiment as that described in section 7.5.1, 
but allowing the gas to enter the vacuum through a comparatively wide 
nozzle so that it cooled to a low temperature by expansion. Dimers, 
trimers and all degrees of association were found up to 40-molecule 
aggregates, the upper limit of the instrument. Some of the biggest of these 
might almost be thought of as small droplets. 

7.6 CRITICAL CONSTANTS 

Having emphasized the inadequacies of van der Waals' equation and 
the fact that it cannot be expected to be valid beyond moderate densities, 
we will nevertheless apply it to high densities. Whereas practically all the 
former results could be deduced equally well from Eq. (7.9) instead of 
Eq. (7.10), the cubic form of van der Waals' equation is now essential. 

We have already noted that the isotherms of a van der Waals gas have 
an S-shaped form below a certain temperature, which we identify with 
the critical temperature. 

We can select the critical isotherm by first finding the locus of maxima 
and minima, and then finding the maximum of this curve. 

Starting from van der Waals' equation 



RT ji_ 

V^-b~V : 



P = 7T-T-772> ( 7 -10) 



we find the equation of the curve on which all turning points lie by differ- 
entiating with respect to volume, keeping the temperature constant, 



dP\ RT 2a 

+ TTT 



dVj T (V-b) 2 V 3 

= at the turning points. 

We can get a more useful expression by using van der Waals' equation to 
eliminate RT. We get 

2a _ RT _ P + a/V 2 
V*~ (V-b) 2 ~ (V-b) 



7.6 Critical constants 



209 



or 



P = a 



(V-2b) 

K 3 



(7.20) 



as the equation of the locus of maxima and minima, Fig. 7.18. The maxi- 
mum of this curve is given by equating the gradient to zero : 

— = ~(3b-V) = 0. 



ii i 

ii i 

ii i 

n i 

11 i 

ii 
ii 
ii 
H 
ii 
ii 
ii 
u 
n 
i! 
i 




Fig. 7.18. Full curve: locus of maxima and minima 

of van der Waals' isotherms, Eq. (7.20). Isotherms 

also appear in Fig. 7.10(a). The isotherm labelled T m 

is referred to in section 7.8.1. 



Thus at the critical point : 

V c = 3b. (7.21a) 

Substituting this value back in the equation for the locus of maxima and 
minima, 

a 



Pr.= 



21b 1 



(7.21b) 



210 Liquids and imperfect gases Chap. 7 

and putting both these in van der Waals' equation, 

It has already been shown, when a and b were originally defined, that 
we expect (from Eq. (7.8)) : 

alb = -Ne. 
1 3 

This gives 

c 81* 
= 0.79 e/k. 

Roughly speaking therefore, the critical temperature T c occurs when 

kT c ~ e (7.22) 

that is, when kT becomes comparable to the energy of interaction of two 
molecules. When the thermal energy exceeds this, the gas cannot liquefy. 
We could proceed to compare the critical constants of a number of gases 
with the values of a and b which we derive from the Joule-Thomson 
coefficient or from knowledge of the radii of the molecules and the form of 
the square-well potential. But it is more realistic to use van der Waals' 
equation to provide laws of corresponding states — for example, to 
compare liquids at their critical points. We can write for the van der 
Waals gas 

RT C 8a 27b 2 1 _ 8 

YJ C ~ 21Rb ~a~ 3b ~ 3' (? " 23) 

This relation is obeyed remarkably well. For argon, T c = 150°K,P C = 48.3 
atmospheres = 49 x 10 7 dyn/cm 2 , V c = 74.6 cm 3 , so that this ratio is 
3.41. This is within 30% of the predicted value, 2.67. Data for other gases 
are: 





T 

•*c 

°K 


atm 


cm 3 


RT C 


Nitrogen 


126 


33.5 


90 


3.43 


Carbon dioxide 


304 


73 


94 


3.63 


Water 


647 


218 


56 


4.35 



7.7 Fluctuation phenomena 211 

7.7 FLUCTUATION PHENOMENA 

It is convenient at this stage to introduce an important aspect of 
statistical theory, namely the fluctuations that occur in any statistical 
quantity. It is a concept which is applicable to any branch of statistics 
and it could have been introduced in section 4.2 where statistical ideas 
were discussed. However, the optical phenomena which occur at the 
critical points of liquids are among the most striking manifestations of 
density fluctuations in fluids. We will therefore concentrate on phenomena 
near the critical point and then extend the discussion to fluctuations in 
general. 

7.7.1 Critical point phenomena : critical opalescence 

The region of the critical point has been much studied and some peculiar 
effects seem to occur there. It seems that the temperatures at which the 
isotherm becomes horizontal, at which the meniscus disappears and at 
which the properties of the two phases become identical may all be 
different. Thus, over a narrow interval of temperature the fluid can exist 
in a tube as two layers of different density even though there is no sharply 
defined meniscus separating them. At the same time, it is known that the 
properties of the liquid and vapour in the region of the critical point are 
strongly influenced by minute traces of impurity and it is difficult to 
decide whether the observations are highly significant for a full under- 
standing of the process of condensation or whether they are in some sense 
spurious. 

One phenomenon however certainly uncovers some interesting 
physical ideas. The appearance of a normally colourless fluid at its critical 
point is remarkable. Illuminated by a beam of light, it looks diffuse and 
shimmering and intensely white, so that one instinctively thinks that a 
cloud fills the whole space. If the temperature is raised or lowered by as 
little as a fraction of a degree away from the critical point, the whiteness 
disappears and the gas or liquid appears colourless again as one thinks 
it should. This phenomenon is called 'critical opalescence'. 

The process undergone by the light in its passage through the fluid, 
going in as a beam but coming out diffusely in all directions, is called 
scattering. Now it can be shown that a large block of a perfectly regular 
crystalline solid at absolute zero scatters no light at all — it would be 
invisible except for reflections from the surface — provided that the wave- 
length of the light is much greater than the interatomic spacing, a condition 
which is always satisfied. The fact that the fluid scatters so strongly near 
the critical point indicates that it is in some way far from homogeneous. 
We will be able to show (in section 7.7.3) that this is indeed so, and that its 



212 Liquids and imperfect gases Chap. 7 

density varies appreciably from point to point at the critical point. We 
cannot however, give a satisfactory account of the optical effects, for the 
following reasons. It can be shown that when the density of a medium 
varies from point to point in an entirely random way, the variation being 
appreciable over distances small compared with the wavelength, then 
short wavelengths are scattered more than long ones and as a result the 
medium looks blue. The blue of the sky, for example, originates in the 
light scattered by the great thickness of air above one's head ; this in itself 
is sufficient proof that the air is composed of a completely random 
arrangement of small molecules. Evidently, a medium where the variations 
of density are random but particularly large would scatter blue light 
with great intensity. But in a highly compressed gas or a liquid the mole- 
cules are not arranged completely at random. There are regions where 
several molecules are nearly close-packed, whose arrangement over a 
short distance is fairly regular. It is this fact that causes the scattered light 
near the critical point to be white rather than blue. Measurements of the 
intensity at any angle can indeed give information about the scale of 
distance over which the molecules are ordered — which bears a distant 
relation to the average size of the clusters. Experiments show that at the 
critical pressure but 10 ~ 3 deg above the critical temperature, this 'correla- 
tion length' is about 1000 atomic diameters; to deg away it is about 
10 diameters. We cannot follow these arguments, however, but will 
content ourselves by showing that the variations in density which occur 
near the critical point are particularly large, without attempting to 
estimate the intensity of the scattered light or to predict its colour. 

7.7.2 Concepts of probability theory — II . fluctuations 

The variations of density from point to point which occur in every 
system but which are very great in a liquid at its critical point are a 
fluctuation phenomenon. We will now consider fluctuations in general. 
It must be emphasized that they occur in all systems and that they are not 
due to gross effects like unequal heating or nonuniform external pressures : 
they occur in systems in thermal equilibrium. Their origin lies in certain 
aspects of statistical theory which we have so far ignored. 

In section 4.2 we called attention to the fact that when we deal statistic- 
ally with small numbers of people, the characteristics of a single individual 
can quite upset the shape of a histogram. Thus, Fig. 4.3(a) which refers to 
a small sample of 100 people with at most 15 in any range, is not a regularly 
stepped histogram ; Fig. 4.3(b) which differs from it merely in the much 
larger numbers in any range, is regular. Such deviations from the most 
probable value occur in all types of statistical phenomena. We will, 
however, concentrate on physical examples. 



7.7 Fluctuation phenomena 213 

The most probable situation inside a gas or liquid in thermal equilib- 
rium is that (in the absence of external fields) its density is uniform at all 
points, on the average. But instantaneously this is not so, as can be seen 
by referring to the pictures of the molecular arrangements in gases, 
liquids and solids, Figs. 2.3, 2.4, 2.5. Liquids under ordinary conditions 
contain 'holes' whose size is comparable with 2 A cubed, though there are 
many regions where the packing is close ; and the pattern changes all the 
time. Obviously, then, the density can fluctuate over a wide range if we 
concentrate on small volumes. This can be seen by making a mask of 
size corresponding to 3 A square, and laying it down over Fig. 2.4 : the 
number of molecules encompassed varies from about 1 (near the hole in the 
lower right-hand corner) to about 2 wherever the packing is close. This is 
a 100% variation. If on the other hand we deal with large volumes, the 
fluctuations get relatively smaller; a mask corresponding to 10 A square 
encloses between about 20 and 24 molecules in Fig. 2.4, only a 20% 
variation. It is indeed a general principle that the smaller the average 
number of molecules enclosed contained in any arbitrarily selected 
volume, the proportionally larger are the fluctuations of density. 

Usually, in ordinary laboratory experiments, we cannot detect the effect 
of density fluctuations in gases or liquids, because we deal with large 
numbers of molecules, and most instruments cannot respond to the 
fluctuations. However, the blueness of the sky and the critical opalescence 
of fluids do detect them. In addition, other quantities can also fluctuate. 
For example, the movement of a small particle undergoing Brownian 
motion in a liquid (section 4.4.2) is determined by the mean momenta of the 
molecules within a small volume of liquid, comparable with the volume of 
the particle — and its jerky movement shows that this quantity fluctuates. 
In other words, the Brownian motion detects the fact that though the mean 
momentum crossing any plane in the interior of a stationary fluid averages 
out to zero over a long time, it departs from zero at any instant ; it fluctuates 
about the value of zero. In an analogous sort of way, the pressure exerted 
by a gas on a very small area of wall fluctuates about its mean value and 
this can also be detected — not using ordinary sluggish pressure gauges 
but rapidly-responding ones. 

7.7.3 Fluctuations of volume in an elastic system 

We can most simply calculate the probability of finding a certain density 
near a given point in any fluid by fixing attention on a given number of 
molecules and finding what volume they occupy. Consider therefore a 
fixed point X and select the n molecules which at any instant are to be 
found nearest to it. On the average they must occupy a volume v such 
that n/v is equal to the mean number-density averaged over the whole 



214 Liquids and imperfect gases Chap. 7 

volume. But at any instant they may occupy a volume v, greater or less 
than v ; of course, the number-density n/v is less probable than n/v .* 

We may calculate the difference in probability of these two states by 
calculating the difference of energy between them and then using the 
Boltzmann factor. 

Consider therefore a volume v of fluid in equilibrium at a certain 
pressure and alter its volume to v by expanding or contracting it, keeping 
the temperature constant. Following an argument similar to that in 3.5.1, 
we define 

(isothermal) bulk modulus K = - W^ (3.10) 

dP\ 



= - Vo{ Tv 



T 



nearly, if the change of volume is not too great. Therefore when the 
volume has increased from v to v the extra pressure acting is 

The energy required to increase the volume further by dv is 

so that the total energy required is 

AE = 5 f V {v - Vo )dv = U^^- = l -Kv s 2 (7.24) 



vo 



where s is the fractional change of volume, (v — v )/v , Eq. (3.24). This 
means that the energy is increased whenever it deviates from its equilibrium 
value, because the squared term must be positive. This is reasonable : the 
fluid pressure resists any change from the equilibrium value. 
The ratio 

probability of volume v = e _ Af/kT = e - (K *o/2iW (7.25) 

probability of volume v 

and this gives the probability of a volume fluctuation of magnitude s. 

* It does not matter if any one of the original n molecules diffuses away from the vicinity 
of X and its place is taken by another because (according to both classical and quantum 
mechanics but contrary to intuition) molecules are indistinguishable from one another. 
We are justified in thinking that j; or v is always occupied by the same molecules. 



7.7 Fluctuation phenomena 215 

We could proceed to work out the mean-square volume fluctuation 
? from the correctly normalized probability by averaging s 2 . But we can 
avoid all this by noting that the energy A£ is a squared term of exactly 
the same form as the \I(o 2 or \mv\ terms in the kinetic energy of a single 
molecule. Using the same terminology as before (section 5.3), the com- 
pressibility of a volume v of any substance confers on it one degree of 
freedom. However large or small v is, however many molecules it contains, 
its fluctuation of volume confers one single degree of freedom. 

Therefore the mean-square volume fluctuation is given by 

^Kv 7 = ^kT 
that is 



s 2 = 



v — v\ kT 



v / Kv 



(7.26) 



It is obvious that if v is small, the r.m.s. value of s is large and this agrees 
with the statement that fluctuations are largest in small volumes. 

Eq. (7.26) says that the smaller the bulk modulus K, the greater 
the fluctuations. The reason is simply that little energy is then required 
to cause a change of volume so that large changes of volume become 
probable. Now, the isothermal bulk modulus K is proportional to the 
slope of the PV isotherm and at the critical point this is zero. Eq. (7.26) 
could then indicate infinite fluctuations. Actually these cannot occur 
because higher derivatives of the slope have to be taken into account — but 
the fact remains that fluctuations of volume (or density) are large and 
light is strongly scattered. As explained above, the analysis of the optical 
effects will not be taken further. 

It is worth noting that results analogous to Eq. (7.26) for other elastic 
systems can be written down at once. For example, the length of a rod 
fluctuates because the atoms in it are in motion. If Y is Young's modulus, 
its increase of potential energy 

A£ = ^Yl s 2 
where / is its mean length, s the fractional change of length A/// . Hence 

? = fi- <7 - 27 > 

For a rod 1 m long with Y = 10 11 dyn/cm 2 , the r.m.s. fluctuation of 
length is of order 10 -11 cm at room temperature, which conforms with 
ordinary experience by being negligible. 



216 Liquids and imperfect gases Chap. 7 

7.7.4 Fluctuation in a perfect gas 

An important result will be derived as a particular case of Eq. (7.26). 
Consider the fluctuations of a volume containing n molecules of a perfect 
gas. Since it obeys PV = RT, 

K = - V (%) T = P (Z28) 

— the isothermal bulk modulus is equal to the pressure. Thus 

-y kT 

s = -^r- 
Pv 

where v is the mean value of the volume. Now n/v — N/V , where N is 
Avogadro's number and V is the molar volume at pressure P and tempera- 
ture T. This gives 

s = -L (7.29) 

(This value of s is the r.m.s. fluctuation.) This means that n molecules are 
contained in a volume which fluctuates between the values 

1+-H and v \l — r 

in the sense that these limits define the r.m.s. fluctuation. We can express 
this slightly differently by calculating the number of molecules contained 
in a. fixed volume v . If that number on average is n, it fluctuates between 
probable limits 

nlH — t\ and "I 1 — r ' 

that is, between n±y/n. For example, a gas under standard conditions 
contains 3 x 10 19 molecules in 1 cm 3 on average ; in fact the number has a 
r.m.s. fluctuation of roughly 5 x 10 9 . Expressed differently, this is a fluctua- 
tion of order 1 in 10 10 which is negligible. This is because the number of 
molecules is so large. But in a volume of 100 A cubed, there are only 30 
molecules on average, and the number therefore fluctuates between about 
25 and 35, which is roughly a 15% variation. Once again, the smaller 
the number, the relatively larger the fluctuations. 

This yjn law for a gas of independent particles has been derived in a 
rather roundabout way. It could have have been deduced more directly 
from first principles by expressing the fact that the probability of finding 
a molecule inside v is independent of where all the other molecules are. 



7.8 Properties of liquids estimated on van der Waals' equation 217 

Many statistical systems obey this kind of law. The problem of counting 
the heights of members of a population, referred to at the beginning of 
section 7.7.2, is typical ; numerical examples of the ^Jn rule have been 
given in section 4.2. 

7.8 PROPERTIES OF LIQUIDS ESTIMATED ON VAN DER 
WAALS' EQUATION 

Extending the application of van der Waals' equation even below the 
critical point into the liquid region is of course quite unwarranted. But 
it must be remembered that, following the discussion of section 7.2.1, 
useful laws of corresponding states can be written down. With this in 
mind we can estimate the tensile strength of a liquid using van der Waals' 
equation. We will also discuss certain aspects of the boiling of liquids. 

7.8.1 Tensile strength and superheating of liquids 

We have seen in Fig. 7.10(a) that van der Waals' equation gives isotherms 
at low temperatures which go below the P = axis, corresponding to 
states of tension. The minimum value of P on any isotherm corresponds 
to the tensile strength at that temperature. 

We have already calculated this value as 

a(V-2b) ,„„ rtX 

P = -^yT 2 ( 7 - 20 ) 

(with V < 3b to make sure we are at a minimum). The volume of the 
condensed phase is roughly equal to b, so the tensile strength at low 
temperatures should be roughly 

P = -£ (7-30) 

which is equal to — 27P C , where P c is the critical pressure. For liquid argon 
this is badly wrong. We have already quoted (section 7.1.3) that liquid 
argon has a tensile strength of 12 atmospheres ; van der Waals' equation 
predicts 1,300 atmospheres. 

We can interpret this equation by referring back to the relation a/b « ?L 
[Eq. (7.8)], where L is the molar binding energy at low temperatures. 
Our expression for the tensile strength is therefore equal to the energy 
required to vaporize a mole of liquid divided by the molar volume. Since 
pressure or tension is an energy density, we must be implying that when 
the liquid breaks, it half vaporizes. This is not the mechanism at all, 
and the van der Waals' approach is unrealistic. 



218 Liquids and imperfect gases Chap. 7 

By contrast, van der Waals' equation predicts with surprising accuracy 
the maximum temperature of superheating, possibly because now the 
liquid does all vaporize. This temperature was called T m and was defined 
in section 7.1.2. Referring to the complete family of isotherms, Fig. 7.10(a), 
we can say that if the branch corresponding to superheated liquid goes 
below the P = axis, then in any experiment at this temperature con- 
ducted at low (approximately zero) external pressure, the liquid can be 
contained. But if the minimum of the isotherm lies above the F-axis, then 
the liquid cannot exist under low external pressure. T m is therefore the 
temperature of that isotherm whose minimum first touches the P = 
axis. This isotherm is labelled T m in Fig. 7.18. Putting P = in Eq. (7.20), 
the condition is that V — 2b, and this in turn gives 

T m = ^T c . (7.31) 

In other words, van der Waals' equation predicts that liquids can be 
superheated, under small external pressures, to 0.85 of their critical 
temperatures. 

In Fig. 7.19, T m is plotted against T Q for a number of liquids. Data for 
some of the points are given here : 



Sulphur dioxide 323°K 430°K 

Ether 416°K 466°K 

Alcohol 477°K 516°K 

Water 543°K 647°K 



The line of gradient 0.85 passes quite well through the points and those for 
a number of other liquids. 

7.8.2 Vapour pressure of liquids 

In principle we can calculate the vapour pressure of a liquid once we 
know where the horizontal part of an isotherm is to be drawn so as to 
equalize the areas of the loops, as shown in Fig. 7.10(b). Unfortunately 
it is not possible to do this analytically because the cubic form of the 
equation makes it impossible to express the integrals in closed form. 

However, we may guess that the probability that a molecule can 
escape into the gas phase will contain the factor exp(-L /#T), and that 
therefore this factor will appear in the equation for the vapour pressure. 



7.8 Properties of liquids estimated on van der Waals' equation 



219 



o 

600 



500 



400 



300 



200 



100 




oj- 







100 



200 300 400 500 600 



700 
T 



Fig. 7.19. Plot of maximum temperature of superheating T m against 

critical temperature T c for several liquids. Data from Kennick, Gilbert 

and Wisner, J. Phys. Chem. 28, 1297 (1924). 

Thermodynamic and statistical arguments show that a better equation 
is of the form 

P = (const.) x T n e' LolRT (7.32) 

where n is equal to the difference of specific heats of vapour and liquid (in 
units of R), and is therefore a small number. It is quite reasonable that this 
factor T" should enter, because the latent heat varies with temperature. 

Over small temperature ranges, the exponential factor varies much 
more rapidly than the T" term and this allows a quick estimate of L 
to be made if vapour pressures are roughly known. Comparing vapour 
pressures at T x and T 2 , and taking only the exponential term into account, 

~ R\T 2 TA r\ Tl 



In 



(7.33) 



where T av is a mean between 7\ and T 2 . Water, for example, has a vapour 
pressure of about 2 cm at room temperature, that is about 300° K, while 
at its boiling point 373° K its vapour pressure is atmospheric, 76 cm. 
Simple arithmetic gives L « 5,700 R, close to the correct value of 



220 Liquids and imperfect gases Chap. 7 

4 x 10 4 J/mol, since R = 8.3 J/mol deg. Alternatively, since L for most 
common liquids which boil at ordinary temperatures is always of the same 
order of magnitude, the same data show that the vapour pressure of 
common liquids roughly doubles itself for a rise of temperature around 
10°K at ordinary temperatures. 

7.8.3 Supersaturated vapours 

We can now return to the nucleation of liquid droplets in vapours, and 
discuss how they grow, and why vapours can be supercooled. 

Consider a droplet of radius r. Compared with the same number of 
molecules in the interior of a large mass of liquid, the molecules of the 
droplet have greater energy. This is because some of them are at the surface 
and do not have their full coordination number or, in macroscopic terms, 
the surface tension gives the droplet a surface energy. Or, to state just the 
same result differently again, a pressure 2y/r is exerted on the molecules 
in the droplet (where r is its radius, y the surface tension). The distance 
between molecules in the droplet is slightly decreased because of this. 
Since pressure is an energy density, the extra energy is 2yV/r where V 
is the volume of the droplet (where, as usual, this is only rough but is 
good enough for order-of-magnitude estimates). Thus the activation 
energy for vaporization, the latent heat (which is proportional to the 
energy required to separate a pair of atoms) is reduced. Consequently 
the vapour pressure of the droplet, compared with that of a large mass of 
the liquid with a plane surface, is increased. Instead of a term exp( — L /R T) 
in the vapour pressure, we have exp( — L + 2yV/r)/R T where L , R and V 
must all refer to N molecules, that is V must be the molar volume. Thus 
compared with a mass of liquid with a plane surface (r = oo), the vapour 
pressure is increased by the factor exp(2y V/rRT) = exp(2yv/rkT), where 
v is the volume of a single molecule, k is Boltzmann's constant, r is the 
radius of the droplet. It is easy to see why the vapour pressure increases 
compared with a plane surface. A molecule just inside a curved surface 
finds it easier to escape because it does not have to break n/2 bonds (see 
section 3.4) but some smaller number, which is reduced all the more as the 
surface becomes more strongly curved. 

Consider now a vapour at pressure P in equilibrium with a liquid with 
a flat surface. This means, of course, that the liquid is just boiling or the 
vapour just condensing. The rate of arrival of molecules, ^nc/s.cm 2 , 
must be equal to the rate of escape ; therefore since n is proportional to 
the vapour pressure, both rate of arrival and rate of escape must be 
proportional to the vapour pressure. 

Next consider a droplet inside a vapour maintained at a certain pressure 
P, greater than P so that the vapour would be called supersaturated. Let 



7.8 Properties of liquids estimated on van der Waals' equation 221 

us calculate the radius of droplet which is in equilibrium. Compared with 
a plane surface, the rate of evaporation is increased by the factor 
exp(2y v/rkT), while the increased vapour density means that the rate of 
arrival of molecules is increased by the factor P/P . Thus for equilibrium 

— = G 2yvlrkT ( 7>34 ) 

Thus inside a supersaturated vapour at pressure P, a droplet of radius r c 
given by 

2yv (7.35) 

is in equilibrium. If (P/P ) is less than unity, r c would be negative which 
means that no droplet is in equilibrium. If P = P the surface must be 
plane. If (P/P ) = 1.1, ln(P/P ) « 0.1 and r c is of order 10" 7 cm so that the 
equilibrium droplet contains the order of 100 atoms. 

But we can show that this equilibrium is an unstable one. Imagine a 
vapour held at a pressure P but the radius of the droplet to be slightly 
decreased from its equilibrium value. Then the number of molecules 
condensing per second remains the same but the number evaporating 
increases. This tends to make the droplet shrink even further. Conversely, 
a droplet whose radius slightly exceeds r c must grow even bigger. 




P/R >1 




radius r 

Fig. 7.20. Energy of a droplet of radius r as a function of r. The energy 

of the same molecules inside a large mass of liquid, with no surface, is 

taken as zero. 



222 Liquids and imperfect gases Chap. 7 

In all the equilibria we have discussed so far, the condition that the 
forces are zero has been interpreted to mean that the potential energy is a 
minimum (section 3.2.1). Such a system, displaced, will return to equilib- 
rium. Here however the potential energy of the droplet expressed as a 
function of its radius must go through a maximum. There is certainly 
equilibrium when the radius has exactly the required value corresponding 
to the given pressure P, but displaced from this condition (i.e. slightly 
increased or decreased in radius) the droplet will evaporate or expand 
further and not return to equilibrium. 

Figure 7.20 expresses this. Each curve refers to a given degree of super- 
saturation, a given {P/P ). It expresses the surprising fact that, at a given 
temperature with a given pressure of vapour, the molecules condensed as a 
small droplet can actually have a higher energy than when they are in the 
vapour. The molecules inside the droplet have indeed each lost their 
energy of condensation (not the full e per pair but something less because 
of the finite temperature) ; but when the droplet is small there are, pro- 
portionally, so many molecules at the surface that the droplet as a whole 
actually has more energy than in the corresponding vapour. Only when 
the droplet is quite large does the process of condensation reduce the 
energy of the assembly. 

Thus there is an activation energy for droplets to form, and this at once 
explains why vapours must be supersaturated before they can condense. 
The surface energy of the critical droplet is Anrly but the activation energy 
(by thermodynamic arguments which we cannot reproduce here) is equal 
to one-third of this, 

A = Wh- (7.36) 

Thus the probability of forming a droplet contains the factor 
exp(— jnrly/kT). Expressed explicitly in terms of (P/P ) this is a com- 
plicated function, but it increases dramatically with (P/P ). For (P/P ) = 1.1, 
the activation energy is of order 10 - 12 ergs or 1 eV, and the Boltzmann 
factor is exp( — 40) at room temperature. If (P/P ) is 1.2, r c is reduced 
by a factor 2, the activation energy by a factor 4 and the Boltzmann factor 
becomes exp(— 10) which is a factor 10 10 times larger than before. This 
example, though incomplete, demonstrates the interplay of the different 
factors. 

PROBLEMS 

7.1. You are given that the critical temperature of hydrogen is 33°K. 

(a) Write down a relation between the inversion temperature and the critical 
temperature for a gas obeying van der Waals' equation. What is the inversion 
temperature of hydrogen? 



Problems 223 

(b) Show that at room temperature hydrogen gets hotter when it undergoes a 
Joule-Thomson expansion. j<,,„+„ 

(c) In the 1840's, Regnault found that at room temperature and moderate 
pressures, PV for hydrogen was greater than RT He called ^t a more than 
perfect gas' because all other gases known at the time had PV less than RT 
Explain this observation. 

7.2. Find the critical constants for Dieterici's equation 

P{y-b) = RTexpi^a/RTV) 

where a and b are constants and all other symbols have their usual meaning. 

What is the second virial coefficient and the Boyle temperature for Dietenci s 
equation? 

7.3. Methyl chloride CH 3 C1 has a critical temperature of 416°K. Tte hqiudhas 
density = 1 g/cm 3 at room temperature. The molecular weight is 50.5. The second 
virial coefficient is 



239 255 311 366 422 450 °K 



-764 -637 -401 -265 -184 -155 cm 3 /mol 



Data from J. S. Rowlinson, Trans. Faraday Soc. 45, 974 (1 949). 

(a) Estimate b and the diameter of a molecule from the molar volume of the 

(b) IfTan der Waals' equation held, what function of Tplotted against B would 
give a straight line graph? Plot such a graph and though it gives a pronounced 
curve, strike a reasonable straight line and estimate a. Hence estimate e. 

(c) Estimate £ from the critical data. 

Id) The molecule has a CI" ion at one end and a concentration of positive charge 
at the other. It therefore acts like an electric dipole ; the dipole moment n o\ 
charges +e and -e separated by distance / is el. In addition to the 6-12 
potential energies, two molecules therefore have an additional dipole- 
dipole interaction. This may be attractive or repulsive depending on relative 
orientation but on the whole it is attractive of order of magnitude n /Am ? r 
(J if a is in Coulomb metres and r is in metres). This swamps the r attraction 
so we are left with a 3-12 potential energy. Sketch the shape of the Y (r) 
curve. Estimate fi. . 

le) Compare it with the moment of charges ± e (where e is the electronic charge 
section 2.1.2) separated by 1 A. Sketch the charge distribution in the methyl 
chloride molecule. 
7 4 One mole of a van der Waals' gas is kept in a vessel at its critical volume but its 

temperature T is greater than T c . Show that its isothermal bulk modulus is 

3R(T- T)/4b Hence show that the mean square volume fluctuation is given by 



S" = 



9N T-T 



1 5 A random walker takes N steps in the +x or -x direction (see section 6.3). He 
takes one step every x seconds and each is of length /. What is (a) the most 



224 Liquids and imperfect gases Chap. 7 

probable nett distance travelled? (b) the likely fluctuation from this? (c) Verify 
the yjt law for his diffusion and write down an approximate expression for his 
diffusion coefficient. 
7.6. A very sensitive spring balance consists of a quartz spring suspended from a 
fixed support. The spring constant is a, i.e. the restoring force of the spring is 
- ax if the spring is stretched by an amount x. The balance is at a temperature T 
in a location where the acceleration due to gravity is g. 

(a) If a very small object of mass M is suspended from the spring, what is the 
mean resultant elongati on x of t he spring? 

(b) What is the magnitude ((x-x) 2 ) 1/2 of the thermal fluctuations of the object 
about its equilibrium position? 

(c) It becomes impractica ble to m easure the mass of an object when the fluctua- 
tions are so large that ((x - x) 2 ) 1/2 = x. What is the minimum mass M which 
can be measured with this balance? 




CHAPTER 

Thermal properties of solids 



8.1 THE EXTERNAL FORMS OF CRYSTALS 

To the physicist, intent on understanding the forces which bind matter 
together, the ideal form of solid is the crystal. There, the molecules are 
regularly arranged, so that the environment of any one of them is well 
denned and the problem of computing the energy of the assembly is 
comparatively simple. (This enthusiasm for crystalline solids is not shared 
by engineers, who ask for materials which are mechanically strong. 
Typically, they use materials which are deliberately made impure and 
contain more than one phase, and if they are crystalline at all, the crystals 
are very small.) 

The inner regularity of the molecular arrangement in crystals is made 
manifest by the regularity of their external forms. There is an obvious 
tendency for crystals to be bounded by plane faces. Any one substance 
usually forms crystals of a particular habit (that is, general shape), 
whether cubes or needles or hexagonal prisms or flat plates. No two 
crystals of the same substance are identical in shape ; certain facets in 
one specimen, compared with another, may be enlarged at the expense 
of others. But the angles between corresponding faces are remarkably 
reproducible. 

In Fig. 8. 1 for example, one cannot help feeling that the two crystals 
(a) and (b) have an underlying identity and that the different relative sizes 
of the faces is somehow accidental and unimportant. We can express this 



226 



Thermal properties of solids Chap. 8 



as follows. We construct normals to each face and then let each normal be 
moved parallel to itself so that it passes through the centre of a sphere. 
Each normal intersects the sphere in a point and the arrangement of these 
points on the sphere expresses the angular relationship of the faces.* 
The two crystals (a) and (b) give identical patterns, independent of the 
relative sizes of the faces, as shown in Fig. 8.1(c), except of course that by 
chance certain facets might be absent from one. 



N. 




X,>! 




(a) 



{b 





c) 



id) 



Fig. 8. 1 . (a) and (b) : Crystals with faces of different sizes but the same angular relation- 
ships. Normals to each face are shown. Three cube faces and four octahedral faces 
are visible, (c) : When the normals are translated to pass through the centre of a 
sphere, (a) and (b) give identical patterns, (d) : Faces near an octahedral face. 



In the very simple crystal forms shown, there are only two different 
kinds of face — those which are all at 90° to one another and would form a 
cube if the oblique faces were absent, and the oblique faces themselves 
which if the cube faces were absent would form an octahedron. We shall 
refer to these as 'cube faces' and 'octahedral faces'. Faces at other angles 

* It is possible to project these points on to a plane surface. Such projections are easier to 
handle than the 3-dimensional spherical patterns. 



8.1 The external forms of crystals 227 

are possible and indeed frequent, and some which might occur near an 
octahedral corner are illustrated in Fig. 8.1(d). But we shall not pursue 
this topic because most of the facts about the growth of crystals can be 
typified by referring only to the simplest types of face. 

It is observed that many edges of a crystal may be parallel to one 
another. Rotation of the crystal about an axis parallel to these edges 
therefore brings successive faces into parallelism. Such faces constitute 
a zone and the axis is called a zone axis. 

8.1.1 Optical goniometry 

The angles between the faces of any crystal can be measured using a 
reflecting goniometer. This can be constructed from a spectrometer or 
any other optical instrument which provides a parallel beam of white 
light from a collimator, and a telescope focused for parallel light, with a 
crosswire. Telescope and collimator are placed roughly at right-angles to 
one another and then clamped and left fixed. At the centre of the system 
the crystal is mounted, carefully aligned with a zone axis vertical. Each 
face can act like a tiny mirror, and when a face is set so that it reflects 
light into the telescope, an image of the slit of the collimator can be 
seen. If the crystal is turned about a vertical axis so that another face 
throws up an image in the same way, then the angle turned through by the 
crystal is equal to the angle between the faces. 

The construction of a typical goniometer head, with a crystal mounted 
on it, is shown in Fig. 8.2. The crystal is stuck on to a metal point so as to 
be near the optical centre of the system. The head must have sufficient 
adjustments* to allow the crystal to be accurately aligned. Two horizontal 
screws at right-angles are necessary to bring it to the centre of the instru- 
ment ; two concentric circular scales at right-angles are needed to align 



Telescope 




Fig. 8.2. A crystal holder for an optical goniometer. 
* Instrument makers call them 'degrees of freedom'. 



228 Thermal properties of solids Chap. 8 

certain edges (and a selected zone axis) of the crystal vertically. The precise 
adjustments require considerable patience but eventually it is possible 
to take measurements of the interfacial angles of a whole zone of perhaps 
a dozen faces. Then the crystal has to be remounted to take measurements 
on another suitably chosen zone, so that eventually all the angles can be 
measured and related. 

It is advisable to use a white light source, because internal refractions 
and reflections can give spurious images. With white light these are 
coloured because of dispersion, and are easily distinguishable from direct 
reflections from the faces. With a monochromatic source, this would not 
be so. 

With care, measurements accurate to a few minutes of arc can be taken, 
and the extraordinary symmetry of crystalline form can be revealed, 
using crystals little bigger than a pinhead. Optical goniometry is one of the 
most rewarding of practical exercises. 

8.1.2 Molecular arrangements; unit cells 

We will now show how the faces of a crystal can be related to the arrange- 
ment of the molecules inside it. We will in fact study solid argon : other 
molecular crystals and several metals have similar structures. It will 
emerge that the crystals are highly symmetrical ; cubes and other similar 
shapes are common. It must be understood that in nature such very high 
symmetry is not at all frequent. Of all the tens of thousands of crystals 
which have been catalogued, only about 5 % have this high symmetry. 
Nevertheless, many principles are illustrated by considering these sub- 
stances, and low symmetries will only be briefly mentioned. 

Solid argon crystallizes into a dense lattice (called 'cubic close packed' 
or 'face centred cubic') where each atom in the interior touches 12 nearest 
neighbours. Figure 8.3 shows the arrangement; this picture shows a 
brick-shaped external form with one corner removed. 

The most obvious feature of this or any other lattice is its repetitiveness 
or periodicity. To be precise, there is a unit of repetition of the three- 
dimensional pattern called the unit cell. For ease of illustration however, 
we will for the moment concentrate on two-dimensional patterns and 
their unit cells. 

Figure 8.4(a) shows a two-dimensional lattice of atoms. This pattern 
is not related to Fig. 8.3; it has been chosen only as an example of a two- 
dimensional lattice. A regular network, all of whose cells are identical, 
has been drawn on the lattice. These are unit cells for the lattice. Each 
contains just one atom. The shape of the cells is such that they can be 
packed together to cover the area entirely. Further, one unit cell can be 
translated parallel to one of its edges through a distance equal to the 



8. 1 The external forms of crystals 



229 



length of that edge and the new position of the atom noted. If the process 
is repeated parallel to both edge directions, the complete lattice can be 
built up. (Note that multiples of this unit cell could be used in the same way 
but one usually chooses the simplest unit.) 




Fig. 8.3. Atoms in solid argon (cubic close-packed structure). Some have been 
removed from a corner so as to form an octahedral face. 

There is in fact no unique way of choosing the unit cell for any lattice. 
Two other possible ones are shown in Figs. 8.4(6) and (c). They each still 
contain only one atom and indeed each has the same area as that in Fig. 
8.4(a). All o.ther possible simple unit cells for this lattice have the same 
characteristics. 

These ideas can readily be extended to three-dimensional lattices. The 
unit cell may contain several atoms or molecules : for example, the unit 



oooo 



oooo 



oooo 



oooo 



oeeQ 
oeee 




(a) 



(6) 



(c) 



Fig. 8.4 (a), (b) and (c). A two-dimensional lattice, not related to the argon 
lattice, showing possible two-dimensional unit cells. 



230 



Thermal properties of solids Chap. 8 



cell of the argon structure shown in Fig. 8.3 contains 4 atoms. Of the 
infinity of possible unit cells, one is usually preferred because its symmetry 
is the same as that of the crystal itself. For the argon crystal, this is a 
cubic cell with atoms at each vertex and in the centre of each face. 

Another obvious feature of the lattice shown in Fig. 8.3, as for any 
3-dimensional lattice, is that the atoms are ordered in planes. Further, 
in this cubic lattice each horizontal plane (for example, the topmost 
one) is exactly equivalent to the vertical planes which outline the shape 
and these in fact are parallel to the cube faces of Fig. 8.1(a). Within these 
planes, the arrangement of atoms is comparatively open, as seen on the 
left-hand face of Fig. 8.3. One of these cube faces has been dissected out in 
Fig. 8.5(a). Each atom has 4 neighbours within the plane and it touches 
another 4 in each of the adjacent parallel planes. If we call the diameter 
of one atom a , then there is one atom inside a square of area a% in these 
planes. 

In Fig. 8.3, some corner atoms have been removed to leave an oblique 
octahedral face, exactly corresponding to the octahedral faces of Fig. 8.1. 
Such a surface would appear flat on the macroscopic scale and this 
demonstrates how plane faces can develop on crystals and why the angular 
relations are maintained. 





(a) (d) 

Fig. 8.5 (a) A cube plane and (b) an octahedral plane of atoms. 



Within this octahedral plane (dissected out in Fig. 8.5(b)), the atoms are 
close packed. Each atom has 6 neighbours within the plane and touches 
another 3 in each of the adjacent planes. There is one atom in a rhombus 
whose angles are 60° and 120° and whose side is a — whose area, that 
is, is {J?>l2)al = 0.866ag. 



8.1 The external forms of crystals 231 

8.1.3 Surface energies of crystal faces 

Let us now calculate the surface energy of a crystal face, in much the 
same way as we calculated the surface tension of a liquid in section 3.4. 
Consider first a cube face. Each atom has only 8 neighbours (4 in its 
plane and 4 in the next plane) instead of 12. To make a new surface 
we have to cut 4 bonds per atom, each demanding energy e/2 (energy e 
is needed to cut one bond but two surfaces are formed). There are \ja\ 
atoms in 1 cm 2 so that the surface energy of this face is le/al erg/cm 2 . 
Similarly, the surface energy of an octahedral face is y/3s/al = \.11&ja\ 
erg/cm 2 because each atom has to have three bonds cut and there are 
2/y/3a.o atoms/cm 2 . The surface energy of the cube faces is therefore 
higher than that of the octahedral faces. 

Imagine now a crystal growing with plane faces. Ideally, it ought to 
grow so as to minimize its surface energy. This means that the octahedral 
faces should be prominent, the cube faces small. In different words, the 
cube faces will tend to grow fastest outwards so that they eliminate 
themselves. This paradoxical statement is illustrated rather sketchily in 
Fig. 8.6. The shapes are meant to indicate the form of the crystal at suc- 
cessive times. The cube faces grow fastest outwards but get progressively 
smaller in area, leaving the octahedral faces predominant, as required. 




Fig. 8.6. Stages in the growth of a 
crystal. 

In the growth of large crystals, surface energies seem in practice to 
play little part, though they are important for small ones. In any case, 
small temperature inhomogeneities which may be caused by the release 
of energy during the act of crystallization itself, have a profound effect 



232 Thermal properties of solids Chap. 8 

on growth. We will consider some details of the mechanism of growth 
in section 9.4.1. 

Another consequence follows from the different closeness of packing 
within different planes. Densely packed planes must be further apart 
(because the total number of atoms in a crystal must be the same, which- 
ever way we count them). Now many crystals can be cleaved by being 
struck sharply in certain directions using a heavy blade. If this is well 
done, the faces of the two halves are bright and plane. Planes of easy 
cleavage should — according to these simplified considerations — be those 
of large spacing, which automatically have small surface energies as we 
have shown, and are the ones which occur in natural crystals. Frequently 
this is so. 

8.1.4 Single-crystal specimens 

Unless they are specially prepared, solid specimens do not usually 
consist of single crystals, with their atomic planes parallel to one another 
in all directions. Rather they are polycrystalline with different orientations 
in different regions so that they can be considered to consist of numbers 
of small crystals of arbitrary shape all packed together, separated by grain 
boundaries. Certain properties of solids, notably thermal conductivities 
at low temperature, vary markedly if grain boundaries are present, and 
to investigate them it is necessary to convert polycrystalline specimens into 
single crystals. 

A single crystal may not have visible plane faces. It might be, say, 
in the form of a rod of circular cross-section. However, very often the sur- 
face has a shimmering appearance because it consists of little steps parallel 
to certain crystallographic directions which catch the light, and is not a 
curved surface at all. But whatever the external shape, the planes of atoms 
must all be parallel to one another throughout the specimen. 

In order to grow single crystals, one always begins with a 'seed' crystal, 
as perfect as possible. Using this as nucleus, single crystals can be grown 
provided the crystallization proceeds very slowly. In extreme cases, 
crystals a few millimetres in dimension may take months to form. 
Temperatures must be controlled within fine limits over these times and 
all surfaces must be clean so that no other centres of nucleation are present. 

If the solid is soluble in water, the seed can be put into a saturated 
solution from which the water is allowed to evaporate slowly. Molecules 
crystallize most easily on seeds and the procedure is mainly a matter of 
common sense— removing other nuclei, keeping the temperature constant 
and so on. 

Growing crystals by cooling the molten liquid is also possible, and is 
nowadays much more generally used. One method, capable of great 



8.2 X-ray structure analysis 233 

variation of detail, is to fill a tube with liquid at a temperature just above 
the melting point. The tube has a pointed end, and cooling the point 
causes the liquid to solidify there. In the restricted space, there is probably 
only room for one nucleus to grow (or else the seed can be put there), 
so that there is a tendency for a single crystal to form. Subsequently, a 
carefully controlled temperature gradient is maintained so that the crystal 
grows along progressively. The alkali halides, like NaCl and LiF, can 
be grown in this way. They melt at temperatures of the order of 1,000°K. 
Their vapour pressures are not negligible and the biggest technical 
problem is to stop them distilling away and depositing on cooler parts 
of the apparatus. In another temperature range, solid argon rods have 
been grown by just the same method ; this is described in some detail 
in section 8.5.1. 

Another method is 'crystal pulling', where the seed is dipped into the 
molten liquid and is gradually raised into a cooler part of the apparatus, 
solidifying and pulling a single crystal rod after it. Finally, in zone melting, 
the starting point is a polycrystalline rod. By some method of localized 
heating (such as a narrow beam of radiation, or induction heating), a 
narrow transverse section is melted. If conditions are right, the surface 
tension holds the liquid in place so that it does not run away. It is arranged 
that this melted zone can pass along the crystal, from one end to the other. 
If the seed is at the starting end, the whole rod eventually becomes 
crystalline. 

8.2 X-RAY STRUCTURE ANALYSIS 

The atomic arrangements which have been described have all been 
elucidated by X-ray diffraction experiments. When a beam of radiation 
falls on a periodic structure, a grating or a crystal, it is diffracted through 
large angles if the wavelength A is comparable with the repetition length d 
of the structure. Thus with crystals we are limited to waves whose A is 
a few Angstrom units. Any type of waves can be used for diffraction 
experiments ; electrons or other particles of the same wavelength would 
act in the same way. But there are experimental complications when 
charged particles are used, and also with neutrons which have a magnetic 
moment. In any case, the principles governing the diffraction are exactly 
the same for all types of waves so we will concentrate on X-rays. 

When a parallel beam of X-rays of given wavelength falls on a small 
crystal, most of it travels straight through, but some of the energy is 
diffracted into a number of beams (Fig. 8.7(a)). These can be recorded 
on film, or detected by counters and their intensities and angular distribu- 
tion measured. The scattering is done by the electrons in the atoms of the 



234 



Thermal properties of solids Chap. 8 



lattice (the nuclei are so heavy that they are not affected by the X-rays and 
do not scatter them). The X-rays therefore 'see' spheres whose size is 
comparable with their own wavelength, and the scattering power of an 
atom therefore depends on its size as well as the number of electrons in it. 
All the diffracted beams from a given crystal are part of the one family. 
The angles at which the beams occur depend on the periodicity of the 
lattice, that is, on the size of the unit cell, while the intensities of the beams 
depend on the arrangement within the unit cell. The full family of diffracted 
beams, expressed as an appropriate function of angles, is called the Fourier 
transform of the electron density in the crystal. 




(b) 

Fig. 8.7. (a) A beam of X-rays (covering a wide range of wavelengths) incident 
on a crystal emerges as a family of diffracted beams, (b) Bragg reflection from 

a plane. 



Although it is wrong in principle to take any one diffracted ray and to 
treat it in isolation from the rest of the family, nevertheless this was done 
in the early days of X-ray crystallography and has persisted because it is 
useful in interpreting diffraction patterns. As the method gives quantitative 
results about angles of diffraction using only simple mathematics, we will 
give the main result. 



8.2 X-ray structure analysis 235 

We isolate a set of parallel planes of atoms. In general, these are inclined 
at some angle to the cube faces but, within the planes, the centres of the 
atoms lie on some regular pattern, like those of Fig. 8.5. There are many 
directions in a plane which pass through the centres of atoms, and they 
do so at regular intervals. Two such lines of atoms (drawn for simplicity 
as dots) are shown in Fig. 8.7(b), an upper and a lower plane, separated by 
a distance d. The spacing along the lines is not equal to the distance 
between planes, and the atoms in the lower plane are not necessarily 
exactly below the upper ones but are displaced to one side. The third row 
is similarly disposed with respect to the second, and so on. 

Imagine now a beam of X-rays of wavelength A incident on an infinite 
plane of atoms. It can be shown that it diffracts radiation with a strong 
maximum of intensity in the direction of specular reflection, so that the 
glancing angle of reflection is equal to the glancing angle of incidence, 
just as if it were a continuous reflecting plane. We will only consider the 
intensity in this direction, and we can speak of the X-rays being reflected 
from the planes of atoms. 

Let us consider interference between radiation scattered by the top 
plane of atoms and the next plane. The atoms P and Q of Fig. 8.7(b) are 
typical and we know that if X-rays scattered from P reinforce those 
scattered from Q (because the phase relation between them is correct) 
then all other similar pairs of atoms throughout all the planes will also 
reinforce and the reflected wave will be of finite intensity. 

The condition for constructive interference is 

path difference XQ + Q Y = n A 

where n is an integer, 1, 2, etc. We must therefore calculate the distances 
XQ and QY. Call the angle PQX equal to a. Then angle PQY is equal to 
(18O°-a-20). We have (from APXQ) 

XQ = PQ cos a 

and (from APYQ) 

YQ = PQcos(18O°-a-20) 

= -PQcos(20 + a). 

Therefore 

XQ + YQ = PQ{cosa-cos(20+a)} 

which by a trigonometrical identity is equal to 2PQ sin sin(0 + a). If we 
drop a perpendicular from P to the lower plane, we also have 

d = PQsin(0 + oc). 



236 Thermal properties of solids Chap. 8 

Hence the condition for the presence of a diffracted beam is 

2dsind = nA. (8.1) 

Notice that d is the glancing angle with respect to the plane, but neither the 
spacing within the plane nor the lateral displacement of one set of atoms 
with respect to the other enters the expression ; only the distance d between 
parallel planes enters. 

Whereas a single plane of atoms reflects X-rays of any wavelength 
specularly, a stack of atomic planes only reflects a finite intensity at this 
angle if, in addition, the condition (8.1) is satisfied. Other wavelengths 
are cancelled out by interference. Eq. (8.1) is called the Bragg law. 

We can now take a single spot of the X-ray diffraction pattern and regard 
it as having been produced by a Bragg reflection from a stack of parallel 
planes of atoms, and this enables us to find the spacing d in terms of the 
wavelength A *. For example, with X-rays of wavelength 1.541 A (pro- 
duced from a copper target) one particular reflection from an argon crystal 
was observed to be produced at a glancing angle of 16.52°. Thus 

2xdx sin(16.52°) = n x 1.541 A, 

whence d = 2.71 A if n = 1, 5.42 A if n = 2, and so on. By itself, a single 
reflection does not allow n to be decided, nor is it possible to say which 
plane did the reflecting. These can only be decided by correlating all the 
reflections from the whole pattern — intensities as well as angles — and 
working out just what the unit cell is. (With substances which produce 
well-shaped crystals, the process is simplified if the reflections can be 
correlated with the external form.) A check on the value of n comes from 
comparing the atomic diameter deduced from other methods with that 
deduced from the unit cell ; an error of a factor of 2 is easily detected. 

In this way it can be decided that n = 1 for this reflection. Thus the side 
of the unit cell, from Fig. 8.5(a) twice the distance between neighbouring 
planes, is 5.42 A, and the diameter of an argon atom is 5.42/^2 = 3.83 A. 
Having used other estimates of the diameter to decide which multiple of 
this figure to use, the X-ray measurement is by far the most accurate. 

8.3. AMPLITUDE OF ATOMIC VIBRATIONS IN SOLIDS 

Consider a system in a parabolic potential well, whose equation (referred 
to axes through an origin at the bottom of the well) is 'V = jax 2 . If the 
system is given energy E it oscillates with an amplitude x given by 

* X-ray wavelengths were originally measured in absolute units using ruled diffraction 
gratings. Though comparatively coarse in spacing, they were held at very small glancing 
angles of incidence, so that the diffraction occurred. 



8.3 Amplitudes of atomic vibrations in solids 



237 



E — j<xxq. A geometrical construction for finding x is shown in Fig. 8.8. 
A line drawn at height E above the minimum intercepts the parabola at 
± x , because at x = x all the energy is potential energy. 

Now consider an atom inside a solid. We will assume that it has n 
nearest neighbours, bound to it by a potential of the 6-12 type. We have 
already seen (when we estimated the Einstein frequency of vibration 
in section 3.6.1, Eq. (3.20)) that the atom 'sees' a potential well, due to all 
its neighbours, which is nearly parabolic and has a curvature near the 
minimum given approximately by 24ne/al, where a is the separation 
between atoms. 




Fig. 8.8. Amplitude of vibration with 
energy £ in a parabolic potential well 



If we now extrapolate this assuming that the well is accurately parabolic, 
its equation referred to axes through the minimum is 



TT(x) = 



12ne 

2 

at 



(8.2) 



where x is the displacement from the minimum. (This result follows from 
the fact that the parabola jax 2 has curvature a at the minimum.) 

With these preliminaries, we can now estimate the greatest amplitude 
of vibration which the atom in a solid can have under ordinary conditions 
— namely, at the melting point. When the solid is at absolute zero, the 
atom is (according to the ideas of classical physics) at the bottom of the 
well. When it is at temperature T and the atom has a mean energy kT, 
it oscillates within the well — though not, of course, with a strict periodicity 
because it interchanges energy all the time with its surroundings. Now 
we know that the melting temperature (at ordinary pressures) is not very 



238 Thermal properties of solids Chap. 8 

different from the temperature of the triple point — and this in turn is 
always roughly half the critical temperature (80° K and 150°K for argon). 
We already know that the mean thermal energy of an atom at its critical 
temperature is about e, so that at the melting point it is about ^s. Thus 
the amplitude is given by 



12„e^j *-. ,8.3) 

that is, (x /a ) « 0.07, if n = 10. Thus we can say that in any substance 
with a 6-12 potential, the amplitude at the melting point is less than 1/10 
of the interatomic separation, which is quite small. 

For metals, the ratio of triple point temperature to e/k is nearer 1 or 3 
than 2- For potassium, for example, the melting point is 335°K, and from 
the data of section 3.7, e/k is about 0.16 eV so that the ratio is about 0.2. 
For mercury using data from the same table and a melting point of 234° K, 
the ratio is about 0.15. But the amplitude at melting is not greatly different 
from before. 

For ionic solids, the calculation can be followed through by noting that 
an ion 'sees' a potential whose curvature is about 4ne/al if the neighbours 
are spherically disposed about it, as we assumed in the discussion of the 
Einstein frequency, sections 3.6. 1 and 3.8.2. For sodium chloride, e ~ 9 e V 
from the data of section 3.8.1, the coordination number is 6, the melting 
point is 1,073°K. Hence the amplitude at melting is one-thirtieth of the 
interatomic spacing. Looked at from this point of view, NaCl melts more 
easily than molecular solids or metals. This is caused by the presence of 
two sizes of ion in the lattice — small positive ions — and negative ions 
of about twice their radius. The small ions can slip easily between the 
large ones, more easily than in lattices where all the members are the 
same size. Thus though ionic solids like sodium chloride melt at high 
absolute temperatures, these are really quite low on this proportional 
scale. 

We return to the onset of melting and to some premelting phenomena, 
considered from a rather different point of view, in section 9.5.2. 



8.4 THERMAL EXPANSION AND ANHARMONICITY 

It is well known that solids expand when heated. The coefficient of linear 
thermal expansion is defined by 

1 / dl \ 



8.4 Thermal expansion and anharmonicity 



239 



where the subscript P denotes an expansion at constant pressure, usually 
atmospheric pressure. The coefficient of volume expansion /? is defined 
by a similar equation with volume V in place of linear dimension / 
and since V = Z 3 , it follows that /? = 3a. Typically a is of order 10 ~ 5 per 
degree for hard solids, 10 -3 per degree for soft ones, at ordinary tem- 
peratures. We will now relate these quantities to the V{r) curve. 

It must be realised at once that if the forces binding one atom to another 
were purely harmonic — if the potential well between a pair of atoms were 
exactly parabolic, even for large amplitudes of vibration — then the mean 
separation of two atoms would always be the same, whatever the ampli- 
tude, Fig. 8.9(a). (In the same way, the mean position of a simple pendulum 
remains fixed, whatever its amplitude of swing.) Thus, a solid bound by 
purely harmonic interatomic forces would not expand with temperature. 

The origin of the finite expansion coefficient must lie in the asymmetry 
of the i^(r) curve, which expresses the fact that two atoms can more easily 
be pulled apart from one another than pushed together. We can see this 
graphically, Fig. 8.9(b), by drawing horizontal lines across the well, 
representing different mean energies and hence different temperatures. 
With increasing energy, the mean separation tends towards greater 
separation, and the solid must expand. 

Systems in non-parabolic potential wells are said to execute anharmonic 
motion. 





(a) 



ib) 



Fig. 8.9. (a) The mean displacement of a simple harmonic motion is always 

zero, whatever the amplitude, (b) With an unsymmetrical potential energy 

curve the mean displacement increases with amplitude. 



8.4.1 Thermal expansion coefficient 

To calculate the magnitude of a, the thermal expansion coefficient, we 
will first find a convenient approximate equation for the interatomic 



240 Thermal properties of solids Chap. 8 

potential energy curve near the minimum, referred to axes through the 
minimum. If a is the value of r at the minimum and therefore the separa- 
tion of a pair of atoms at T = 0, we will denote {r-a ) by x, the displace- 
ment from the minimum. Then we will find the limits of x between which 
a pair of atoms vibrate when their energy is given, and since the positive 
swing is greater than the negative one, the mean value x is greater than 
zero. x/a is the total linear expansion between any pair of atoms at temper- 
ature T, and differentiating with respect to T gives the linear expansion 
coefficient. This procedure is not exact; it does not properly take into 
account the effect of the atoms at the side of any pair, nor is the mid-point 
of the swing an exact measure of the mean position ; but it is not far wrong. 

First we will seek an equation for the interatomic potential energy near 
the minimum — something better than the parabolic approximation which 
we used previously (page 45). The anharmonic terms, the extra ones not 
proportional to x 2 , will be the ones responsible for the thermal expansion. 

We start by quoting Taylor's theorem. If we have a function -V whose 
value at r = a is i^(a ), then the value at some other point is 

ldV\ 1 ld 2 i^\ 



i . .Jd 3 r 



+ ^-<">> 3 M + - (8 - 5) 

where we use the subscript to denote that the term must be evaluated at 
r = a . This expansion is true for any reasonable curve. Now let the point 
a be a minimum so that (d^/dr) = 0. Let us write ArV for the increase 
of potential energy compared with the minimum value, and x for (r — a ). 
Then 

.. x 2 /d 2 ^\ x 3 /d 3 ^\ 
W = A-rA +77ho- +■■■ (8-6) 



2!\ dr 2 / 3!\ dr /0 

The relation between the V, r coordinates and the i r , x coordinates is 
made clear in Fig. 8.10. 

This holds near the minimum of any curve. Let us concentrate on the 
6-12 potential which is appropriate for molecular solids like argon and 
with less accuracy (see section 3.7) for metals. Starting with Eq. (3.4) : 



V = lie 



d-T _ _12e 
dr a 



a \ 13 \Oq 
r 



(3.4) 
= when r = a 



8.4 Thermal expansion and anharmonicity 



241 



d 2 -r 

dr 2 


12e 


[»(?)"-'(?)'] 


72e v. 
= — r- when r = a 


dr 3 


L 

a 


2e 
T 




"l3xl4^)' 5 -7 


x8 — 
\ H _ 


1,512c 

al 



(3.15) 



when r = a . 



i\W 




iUV 




Fig. 8.10. Change of coordinates to an origin through the minimum. 

Thus, using the minimum as origin, as in Fig. 8.10, the interatomic potential 
energy is given by 

2 



M^ = 36e 



a 



-252e 



W ' 



(8.7) 



Since we have neglected higher terms and the coefficients are increasing, 
this approximation is only good up to about (x/a ) = 0.1. But, as we have 
seen, this is adequate even up to the melting point. The effect of the 
anharmonic x 3 term is clearly seen in Fig. 8.11. 




A{x/aJ 



,/ ^A(x/o Y - B{x/af 



(x/a) 



Fig. 8.1 1. A curve of the type A(x/a ) 2 - B(x/a ) 3 , 
for small (x/a Q ). 



242 Thermal properties of solids Chap. 8 

For simplicity, we will write this equation 



Go \«0 



Let us now choose a value of the total (kinetic plus potential) energy E t 
We find the limits of swing in the usual way, solving 



x' 2 



and we can do this by successive approximation. First, we neglect the B 
term entirely. This gives 

Next, we can use these rough values to calculate the small extra term and 
hence solve the equation more exactly. We write 



x \ 2 I x \ 2 l x \ I x \ 2 



E x =A\—\ -B\— I , , , 

\a J \ a ol \«o/ \a 

so that approximately 

£1 



^- x 
A-B 



A + BiEJA) 1 ' 2 

where the minus sign is associated with the positive square root and vice 
versa. Taking the square root : 

These are the second approximations to the solutions, good enough for 
present purposes. The mean of the two values is found by adding and 
dividing by 2 : 

* i/£i\ 1/2 /B£ 1/2 \ = be 1 

a -2\AJ U 3/2 / 2A 2 ' 
Putting in the values of A and B and simplifying : 

- = ^-£i. (8-8) 

a 72e 



8.4 Thermal expansion and anharmonicity 243 

This gives the total mean expansion, when the energy of the oscillating 
atom along the x-axis is E t . The coefficient of linear expansion is the 
differential coefficient of this with respect to temperature 

Now we have seen that when the total energy of N atoms (oscillating in 3 
dimensions) is E, the molar specific heat C p is given by 

C = ^ 
p dT 

for any experiment which takes place at constant pressure (including 
the expansion of a solid). Here, E x is the energy of a single atom oscillating 
in one dimension only, so E Y = E/3N. Therefore 

"^ (8 ' 10) 

At high temperatures, in the equipartition region, C p is not very different 
from 3Nk energy units/deg so a ~ ^ • k/e. For argon, e/k ~ 120 degrees, 
whence a ~ 1CT 3 per degree which is the right order for a solid. For 
metals, for which as we have seen the 6-12 potential can be used for rough 
estimates, e/k is of order 3,000 degrees, and a ~ 10" 5 per degree, again of 
the right order. Thus the identification of the mechanism causing thermal 
expansion is correct. 

*k 8.4.2 The Griineisen relation 

There is no need to restrict the discussion to high temperatures. The 
relation (8.10) shows that the expansion coefficient is proportional to the 
specific heat — which implies that a falls off at low temperatures and this 
is in fact observed. The expression indeed predicts that a/C p is a constant 
for any substance at all temperatures. As it stands, however, the ratio 
incorporates e, the interaction energy of a pair of atoms, whereas a and 
C p are macroscopic properties. It is more convenient to use one of the 
relations deduced in Chapter 3 to eliminate e in favour of a macroscopic 
quantity, and the one usually chosen is the bulk modulus (because it 
leads eventually to a dimensionless ratio, a pure number) ; 

4JVne 
K = -— (3.16) 

y o 

where n is the coordination number and V the molar volume. It must be 
noted that, strictly speaking, this K refers to adiabatic conditions; we 



244 Thermal properties of solids Chap. 8 

will write it K ad , although previously we did not emphasize the difference 
between it and the isothermal bulk modulus K T because we limited the 
discussion to low temperatures where differences vanish. 

Finally, having eliminated e and having incorporated both a molar 
volume and an elastic modulus which refers to volume changes, it is 
reasonable to refer to the volume coefficient of thermal expansion /? 
which is equal to 3a. In these terms our expression is 

This ratio should be a constant, independent of temperature, called 
Griineisen's constant y G for the solid. 

We may now invoke a thermodynamic result, namely that the ratio 
Kad/Cp, that is the adiabatic bulk modulus over the specific heat at 
constant pressure, is identical with K T /C V , the isothermal bulk modulus 
over the specific heat at constant volume. Thus we can write 

/ G s~i f-i \ ■ ) 

This is a surprising relation; one would hardly expect the thermal 
expansion coefficient to be related to the specific heat. In 1908, when 
Griineisen first announced his empirical law (in the form <x/C„ = constant 
for any metal) he was unable to account for it. In its complete form, 
Griineisen's equation relates changes of pressure, volume and temperature 
and is often referred to as an equation of state for solids. 

The absolute value of y G which we have calculated is roughly correct. 
We have already given sufficient data for computing it for solid argon — 
K T as its reciprocal the isothermal compressibility in Fig. 3.13(c), C v and 
the expansion coefficient in Fig. 5.10(b). Experimentally, therefore, y G 
has the value 2.8 all the way from 20° to about 60° K, falling to about 2.4 
at the melting point, 80° K. The simple theory presented here predicts 
about 4.5 for a close packed crystal structure with n = 12. 

For metals, the Griineisen y G is usually about 1.4. Ionic crystals usually 
have y G between about 1.5 and 2; the method of calculation for simple 
ionic crystals is outlined in a problem at the end of the chapter. 

8.5 THERMAL CONDUCTION IN SOLIDS 

The problem of describing the mechanism of the conduction of heat in 
solids is one of extreme difficulty. Here we can only try to give a brief 
sketch of some of the physical phenomena. 



= D— 2 (6.3) 



(6.6) 



8.5 Thermal expansion in solids 245 

We have seen, in sections 6.1.2, 6.1.3 of the chapter on transport pro- 
cesses, that the equations describing the diffusion of molecules and those 
describing the conduction of heat are formally very similar : 

dn d 2 n 

ct ox 

and 

dT _ I k \d 2 T 
dt \C v pJ dx 

Here, n is the concentration of a substance, measured in mol/cm 3 , diffusing 
with time t in a direction x, D is the diffusion coefficient ; T is the tempera- 
ture, (k/C v p) is called the thermal diffusivity. These equations were derived 
by general arguments and are valid for all states of matter. The symbols 
represent macroscopic properties of the substance and not the properties 
of the individual molecules. 

The formal analogy between the two equations justifies our using the 
expression that heat diffuses into a body. But we are justified in going 
further than that. We can look at the process on the atomic scale and 
bear in mind our success in describing diffusion in gases in terms of the 
random walk executed by each molecule, in terms of the mean free path. 
We can therefore try to bring together the concepts of the thermal motion 
of the molecules of a solid and the ideas of the random walk and the mean 
free path. 

Imagine one small region of a solid to be heated. The additional thermal 
energy must diffuse to distant regions. Molecules have large amplitudes 
of vibration there and oscillate violently about their mean positions. Now 
because of the interactions between molecules in the lattice, this must 
set their neighbours into oscillation and the result is that a wave dis- 
turbance is propagated outwards. It travels with the speed of sound. 
(The only difference between this kind of disturbance and an audible 
sound wave travelling through the solid is that the typical frequency, the 
Einstein frequency, is 10 10 times greater for the thermal vibrations.) 

Now a wave disturbance of this kind can have its direction of energy 
flow altered — that is, it can be scattered, just as a light beam can be 
scattered — by a number of processes. For example, it can meet the bound- 
ary of the solid when it will be internally reflected or it can meet an im- 
perfection of some kind such as a region of high or low density or an 
impurity atom of different mass from the bulk. In any case, the energy in 
the disturbance cannot travel unimpeded through the lattice. Every so 
often it will have its direction of propagation altered. The average distance 
that a disturbance travels in a solid before it is scattered is analogous to 



246 Thermal properties of solids Chap. 8 

the average distance travelled by a molecule in a gas, namely the mean 
free path. We shall denote this average distance in a solid by /. It is a 
property of the wave motion and has nothing to do with the distance 
moved by an individual molecule. 

Let us take our analogy between diffusion and thermal conduction 
one stage further. Let us say that the diffusion of heat is caused by the 
scattering in random directions, the random walk, of the disturbance 
carrying the additional thermal energy. Then quoting a result deduced 
in section 6.5.1, for the diffusion coefficient of a gas of molecules: 

D = \cX (6.19b) 

where c is the mean speed of the molecules (practically the speed of sound) 
and X is the mean free path. Therefore we expect that for the thermal 
diffusivity of a solid 

~^ = jc s l (8.13) 

CvP 

where c s is the speed of sound, X the mean distance travelled by a wave 
disturbance before being deviated, and the factor \ is perhaps a little 
arbitrary. 

As we have presented it here, we have perforce left the description of 
the flow of energy rather vague. When it is described precisely, in quantum 
terms so that the energy can be considered to have particle-like aspects, 
the analogy becomes exact and the equation for thermal diffusivity can be 
rigorously justified. 

This relation predicts that the thermal conductivity of a solid should 
be given by 

K = iC v pc s l. (8.14) 

In this expression, all the quantities are known — C v the molar specific 
heat, p the density in mol/cm 3 , c s the speed of sound — except a the 'mean 
free path'. 

We can therefore compare this equation with experimental measure- 
ments and use it to deduce the 'mean free path' ; then we can see what the 
results mean in atomic terms. It will emerge that in a given specimen of 
material at any given temperature there are several mechanisms for 
scattering the flow of energy and all of these operate at once although with 
varying effectiveness at different temperatures. Under any given set of 
conditions, the mechanism which causes the strongest scattering is the one 
which limits the mean free path and determines the thermal conductivity. 



8.5 Thermal conduction in solids 247 

8.5.1 Measurements on solid argon 

The techniques of making specimens of solid argon and of measuring 
their thermal conductivity are sufficiently unusual to be worth describing. 
The difficulty is that solid argon exists (under ordinary pressures) only 
below 83°K, the triple point, which is just above the normal boiling point 
of liquid nitrogen, 77° K. A rod of solid argon has to be grown, consisting 
of large crystals if possible (because these turn out to be the most interesting 
specimens to study) ; then heat has to be put into one end of the rod and the 
temperature gradient measured. All these manipulations have to be done 
inside Dewar vessels at liquid nitrogen temperatures and below, in an 
argon atmosphere out of contact with air. Then conductivity measure- 
ments must be made down to a few degrees absolute, using liquid helium 
as the refrigerant. The techniques used by Berne, Boato and de Paz will 
be described. 

The rod was grown inside a pointed glass tube G (Fig. 8.12) as described 
in section 8.1.4. This tube was surrounded by a copper sheath S, so 
arranged that its absolute temperature could be accurately controlled 
while in addition a small temperature gradient could be superimposed. 
Small electrical heating elements Hj and H 2 were wound for this purpose 
on the copper sheath, one at either end, while its lower end dipped into a 
bath of liquid nitrogen. The level of this bath was kept constant within ± 1 
mm, by an automatic topping-up arrangement. The power to H t was 
adjusted to keep the temperature of the lower end of the sheath just above 
the triple point, while H 2 caused the upper end of the sheath to be a few 
degrees hotter. Thus when pure argon gas was let into the apparatus, it 
liquefied at the bottom of the glass tube. Next, the power in H t — elec- 
tronically controlled — was gradually reduced so that the temperature fell 
steadily, at about 0.1° per hour. As a result, the temperature at a certain 
level in the glass tube fell to the triple point and this level travelled gradually 
upwards — inside the tube the argon solidified up to this level, covered with 
a thin layer of liquid. The rods, examined after the experiment, were found 
to consist of comparatively large crystals between 1 mm and 4 mm in size. 

Into the top of the tube there hung a long thin plastic rod R with a bead 
on the end, and the argon crystals grew round this and enclosed it. When 
the argon rod was long enough, 6 cm long, no more gas was introduced ; 
instead, the surrounding argon atmosphere was slowly pumped away. 
The argon rod began to evaporate a little from its surface and by gently 
pulling on the plastic rod (from outside the apparatus) the argon rod 
could therefore be detached from its glass mould and pulled into the 
upper part of the apparatus. Next, the liquid nitrogen was pumped 
away and liquid helium (4°K) syphoned in. At this very low temperature 
the rod became considerably harder, capable of withstanding the hazards 
of the next operation. 



248 



Thermal properties of solids Chap. 8 



Fig. 8.12. Apparatus for measuring the 
thermal conductivity of a crystalline solid 
argon rod A. The rod is shown raised into the 
upper part of the apparatus, clamped to the 
copper block B and with its heater H 3 and gas 
thermometers T t and T 2 all attached. G is the 
pointed glass tube in which the rod was 
grown, S the copper sheath, extended at the 
bottom and dipping into liquid refrigerant. 
H x and H 2 are the heaters for controlling 
the temperature while growing the rod. R is 
the plastic rod for raising the argon rod out 
of its mould. The whole apparatus is enclosed 
in a Dewar vessel. Details have been con- 
siderably simplified and no connecting tubes 
or electrical leads are shown. 




8.5 Thermal conduction in solids 249 

Under vacuum the top end of the rod was clamped to a copper block B 
and a heater H 3 attached to the other end, with two thermometers T* and 
T 2 at points in between. All these attachments were ready in position, 
with spring-loaded copper clamps held open by nylon strings. The argon 
rod was slid into position and the strings cut with a blade controlled from 
outside, so that the clamps closed. If the springs were too strong, the argon 
rod broke ; if they were too loose the thermal contact was poor and the 
temperatures were measured incorrectly. Of 50 experiments which were 
started, 12 were carried out to completion, and of these 4 gave results 
which were self-consistent and judged to be significant. 

T x and T 2 were helium-filled gas thermometers connected to a sensitive 
differential pressure gauge, so that the temperature difference (a few 
tenths of a degree) could be measured directly. The temperature of the 
copper block at the cold end of the specimen could be varied between 
3°K and 15°K, being surrounded either by liquid helium or 'warm' 
helium gas. The power fed into the specimen was of the order of milli- 
watts or tens of milliwatts, and the thermometers took several minutes to 
come to equilibrium. 

8.5.2 Impurity scattering of energy flow 

The results of measurements of the thermal conductivity of argon on 
four specimens are given in Fig. 8.13(a). The conductivity is plotted 
vertically and the temperature horizontally, both on logarithmic scales. 
At the left of the curves, the lowest measurements were taken near 3°K; 
the triple-point, 83°K, is towards the right. At the bottom of the curves, 
the lowest conductivity measured, 0.004 watt/cm.deg., is comparable with 
that of stone or glass at room temperature which are normally regarded 
as heat insulators. At the top, the best conductivity, 0.6 watt/cm.deg., is 
better than that of aluminium at room temperature, which is certainly 
regarded as a good conductor of heat. The 30:1 range of temperature 
pictured in this graph encompasses a 150 : 1 range of thermal conductivity 
and only logarithmic plots can display these wide variations. 

Above about 10°K, the curves are pretty well coincident for all speci- 
mens. Therefore, in this 'high-temperature' region, the dominant mechan- 
ism for scattering the energy flow must be one which does not depend 
on some variable quality of the specimens (such as their shape or size 
or purity) but must depend on the bulk properties of solid argon itself. 
We will return to this region later. 

Below 10° K however, different specimens have different conductivities, 
as evidenced by the displacement of the curves parallel to one another. 
The experimenters in fact noted that the specimens looked different from 
one another. For example, the best-conducting specimen was made from 



250 



Thermal properties of solids Chap. 8 



spectroscopically pure argon and was fairly transparent. The next one 
was grown from less pure gas, although it still looked clear; the worst 
specimens (actually prepared by a slightly different method from that 
described) were 'quite cloudy and opaque'. Since the transparency of a 
crystal is an indication of its perfection (see the discussion on the scattering 
of light, section 7.7.1), we deduce that the mean free path for diffusion 
of the thermal energy is limited, below 10°K, by imperfections in the 
crystals. Indeed, we expect wave motions going through regions of 
imperfection to be deviated. The imperfections may be grain boundaries, 
or small regions of different density caused by strains, or of different 
composition caused by impurities. 



E 
o 

>10 



<J* 1 



10' 



10' 



1CT 




1d 2 



10 



100 °K 1 
Temperature 



10 



100 °K 
Temperature 



(a) 



(£>) 



Fig. 8.13. (a) The thermal conductivity of four specimens of solid argon 
as a function of temperature, on logarithmic scales. Data from Berne, 
Boato and de Paz, Nuovo Cimento 46B, 182 (1966). (b) Specific heat C v 
plotted on similar logarithmic scales and expressed as specific heat per 
cm 3 . Data from Fig. 5.10(b). 



An analogous situation has already been encountered, in section 6.6.1 
when discussing thermal insulation with silica powder. In a real gas at 
low pressures, the mean free path of the molecules would normally be 
large and the thermal conductivity would have a certain value. But if 
we fill the space with a fine powder, the mean free path is limited by the 
distance between the grains ; the conductivity is reduced — and so, presum- 
ably, is the diffusion coefficient. In imperfect crystals, the imperfections 



8.5 Thermal conduction in solids 251 

(some of which also scatter light) similarly limit the mean free path of the 
energy flow. 

In any given specimen, these imperfections should not change in any 
way with temperature so that X should be constant. The density p and 
the speed of sound are also constant ; hence the thermal conductivity 
should be proportional to the specific heat C v . In Fig. 8.13(6), C v has 
been plotted on logarithmic scales— the data of Fig. 5.10 replotted and 
converted into specific heat per cm 3 using a molar volume of 22.6 cm 3 . 
Now if the conductivity k is proportional to C v and bpth are functions 
of T, then log k and log C v plotted against T (or log T) should be parallel 
curves. It can be seen that below 10° K this is indeed so. This then allows 
us to measure X the mean free path. The speed of sound c s is about 10 5 
cm/s. At 5°K, C v is equal to 0.0164 J/cm 3 .deg. For the best specimen, the 
thermal conductivity is equal to 0.55 watt/cm.deg, and for worst 0.015 
watt/cm.deg. From these data, X is equal to 10~ 3 cm and 3 x 10~ 5 cm 
respectively. These are reasonable figures ; for the transparent specimens, 
X is many times the wavelength of light, for the cloudy ones it is comparable 
with the wavelength, as we would expect. 

8.5.3 Long mean free paths : the Knudsen region 

One's belief in the validity of the energy-flow model is reinforced by the 
fact that it is possible to prepare crystals (not of argon but of other sub- 
stances) which are practically free from impurities, so that at low tempera- 
tures where the energy density is small, the mean free paths become 
extremely long, comparable with the dimensions of the crystal itself. 
The energy wave-trains then behave Hke a Knudsen gas. This means that 
when they diffuse down a rod they collide only with the surface of the 
specimen ; there is nothing else to collide with. 

Two stringent conditions must be fulfilled before this behaviour can be 
observed — one concerning the surface of the specimens and the other 
concerning their physical and chemical purity. 

In discussing the molecular flow of gases down tubes, it was not possible 
to predict whether the molecules stick to the surface and are reemitted at 
a random angle ('diffuse reflection') or whether they are reflected specularly. 
Here, we meet an analogous problem. If the surface of the crystal is smooth 
on the atomic scale, the energy flow will be reflected specularly and the 
flux of energy down the rod will be unaltered by the collision ; on the other 
hand, if it is desired to make the energy flow undergo a random walk, 
it must be reflected diffusely and the surface must be rough on the atomic 
scale. 

When a crystal is to be prepared having a long enough mean free 
path to observe Knudsen-type behaviour, an unusual degree of purity 



252 Thermal properties of solids Chap. 8 

is demanded. This is because the energy flow can be scattered by any 
departure from perfect regularity in the lattice. Now a foreign atom, of 
different mass from the rest, can act as a scattering centre. (In the same 
way, a wave passing down a stretched string is partly reflected at a knot 
or any small section of different density from the rest.) But any chemical 
element as found naturally consists of a mixture of isotopes, atoms of 
different masses. To prepare crystals which are pure enough for the 
present purpose, the elements therefore have to be pure isotopes. 

The most convincing measurements have been made on lithium 
fluoride, LiF, of which large single-crystal ingots can be grown, using 
the pointed-tube technique (section 8.1.4). Lithium as it occurs naturally 
is predominantly 7 Li with about 7.5% of 6 Li; fluorine is pure 19 F. The 
lighter 6 Li was therefore extracted before the LiF was prepared (and a 
later set of experiments showed that the thermal conductivity was 
thereby increased). Several specimens were cut from the same big crystal, 
in the form of square-section rods of different sizes. Finally, the surfaces 
were sand-blasted in an attempt to make them rough on the atomic 
scale. This treatment did not so much roughen the surface as produce 
a thin layer of damaged crystal structure just below the surface and a 
detailed analysis of the measurements showed that the most of the 
reflection was diffuse. 

The results are shown in Fig. 8. 14(a), together with specific heat measure- 
ments in (b), both on log-log scales. Below 20°K, the curves are parallel 
showing that k is proportional to C v , but the conductivity is larger the 
bigger the cross-section of the specimens. Though the curves look as if 
they are close together, they in fact show that k for the 7 mm specimen is 
between 5 and 10 times that for the 1 mm specimen. Note the enormous 
value of k at 20° K — about 200 times better than that of copper at room 
temperature, an astonishing fact when one considers that LiF is an 
insulator, since one is used to the fact that they are worse conductors of 
heat than metals. 

The speed of sound is 5 x 10 5 cm/s. Using all the data, it can be cal- 
culated that the mean free path in the 7 mm x 7 mm specimen is about 
3 mm ; in the 1 mm x 1 mm rod it is about 0.6 mm. Thus the mean free path 
is certainly of the same order as the dimensions, as predicted by the 
Knudsen-type theory. 

8.5.4 High temperature behaviour 

Above about 10°K for solid argon or 20° K for lithium fluoride, the 
thermal conductivity decreases with rising temperature. At high tempera- 
tures in the equipartition region, the graphs on log/log scales tend to 
become straight lines at 45° showing that kccI/T. Since C v is constant, this 



8.5 Thermal conduction in solids 253 

implies that the mean free path Xccl/T. As remarked previously, the fact 
that impurities or the mode of preparation of the specimens have no 
effect on this part of the curve— all graphs for different specimens of any 
substance are coincident— shows that the mean free path is dictated by 
some property of the lattice itself. 



10 



10 



10 




10 



10 



!io 3 



10 



10 100 °K 

Temperature 
(a) 



^6 5 



10 100 

Temperature 



Fig. 8.14. (a) The thermal conductivity of four extremely pure LiF rods 
as a function of temperature on logarithmic scale. The rods were nearly 
square in cross section of side approximately 7 mm (specimen a), 4 mm 
(specimen b), 2 mm (specimen c) and 1mm (specimen d). Data from 
Thacher, Phys. Rev. 156, 975 (1967). (b) Specific heat per unit volume 
below 20° K, the low temperature tail of the curve, far below the equi- 
partition region. Data from Scales, Phys. Rev. 112, 49 (1958). 



It is now accepted that this is the anharmonicity of the lattice vibrations. 
When two wave disturbances pass through one another, they create new 
frequencies which were not originally present in either. This process 
has no analogy under ordinary conditions for light waves, for example ; 
if it did, then a red beam passing through a green beam might generate 
ultraviolet and this does not occur. The generation of new frequencies 
from two wave motions is called 'frequency mixing' and electronic 
engineers are familiar with related effects using nonlinear circuit elements ; 
because of anharmonicity, a solid is similarly said to be a nonlinear 



254 Thermal properties of solids Chap. 8 

medium to the passage of sound waves. When a wave passes over another 
and generates a third frequency, this can undergo a sort of Bragg reflection 
with the nett effect that the direction of energy flow is altered. It is plausible 
that the effect should become more important as the amplitude of the 
atomic motions is increased — that is, as the temperature is raised. This is 
why the thermal conductivity decreases as the temperature increases. 

8.6 ELECTRONS IN METALS 

Metals can be described as lattices of ions, permeated by a kind of 
gas of electrons. We have already described (sections 2.1.2, 2.1.4) how 
some electrons inside atoms are tightly bound, while others are only 
loosely bound. It is the loosely bound ones which can wander through the 
lattice. They can be accelerated by electric fields and their energy can be 
increased by raising the temperature. 

In many metals such as copper, silver or sodium, the number of free 
electrons is about one per atom. In a molar volume of metal, therefore, 
there are about 6 x 10 23 free electrons. We shall call this number '1 mole 
of electrons'. 

The most obvious property of metals is that they are good electrical 
conductors. But not only do metals conduct electricity, they are also much 
better conductors of heat than insulators. For example, at room tempera- 
ture, sodium chloride which is an electrical insulator has a thermal 
conductivity of 0.06 watt/cm.deg, while metallic copper has k = 0.92 
watt/cm.deg. Among liquids, alcohol at room temperature and sodium 
at 200° C have thermal conductivities of 0.0017 and 0.82 watt/cm.deg 
respectively ; again the metal has the much higher thermal conductivity. 
Further, the thermal conductivity among metals increases roughly in 
proportion to the electrical conductivity a. For example, copper and zinc 
have electrical conductivities of 0.63 x 10~ 6 and 0.16 x 10~ 6 (ohm cm)' l 
respectively, that is in the ratio 3.8:1. Their thermal conductivities are 
respectively 0.92 and 0.265 watt/cm.deg, which are in the ratio 3.5:1. 
This rough proportionality holds for all metals. This fact was discovered 
experimentally by Wiedemann and Franz in 1853, who stated that 
k/o = constant for many metals at room temperature. In 1881, the 
Danish physicist Lorentz made measurements between 0°C and 100°C 
and was able to restate the law with another factor present — in the form 

\=^> (8-14) 

G 1 

a constant for all metals for all temperatures. It is called the Lorentz 
ratio or the Lorentz constant. 



8.6 Electrons in metals 255 

Typical values of k and a for a number of metals at 20° C (293° K) are 
given in the table. The conductivities cover a seven-fold range but the 
Lorentz ratio agree within 10% of one another— although some metals 
which are poor conductors do not agree so well. 

a k k/oT, 

Metal (ohm cm) -1 watt/cm.deg T=293°K 



Tin 


0.087 x 10 6 


0.64 


2.51 xlO -8 


Zinc 


0.174 


1.18 


2.32 


Aluminium 


0.354 


2.36 


2.28 


Copper 


0.591 


3.96 


2.28 



If we study one metal, copper, over a wide range of temperatures, we 
find that the Lorentz ratio remains fairly constant within 20% or so 
from 1,000°K down to 100°K. Below that temperature it changes quite 
rapidly, however (Fig. 8.15). 

From the proportionality of electrical to thermal conductivity over a 
wide range of temperature, we can surmise that the electrons are respon- 
sible not only for the transport of electricity, but also for heat conduction— 
and that this mode of heat transport is far more effective than other 
modes of heat transport through the lattice of a metal. 

There is one further experimental fact which intimately concerns these 
electrons in metals. Surprisingly, their specific heat is extremely small. 
If they behaved like a classical gas, and if 1 mole of metal contained about 
1 mole of mobile or conduction electrons, then they would be expected to 
contribute about f R to the measured molar specific heat of the metal. 
Copper for example would be expected to have a specific heat of fK = 25 
J/moldeg from the lattice and fR = 12.5J/moldeg from the electrons, 
a total of 37.5 J/moldeg. But at ordinary temperatures the observed 
value is not very different from that expected for the lattice alone so that 
all that can be said is that the contribution from the electrons is small. 
At extremely low temperatures however, below about 2°K, the lattice 
specific heat is expected to be vanishingly small yet the measured specific 
heat is not small and varies proportionally to T, Fig. 8.16(a). It can be 
represented by 

C v = 7.5 x 10 ~ 4 T J/moldeg 

for copper. This is interpreted to be the contribution due to the electrons. 
If we extrapolate it still using the linear law up to room temperature, it 
gives only a small contribution of 1 % to the total instead of the expected 
33% and this at least is self-consistent, (Fig. 8.16(b)). It follows inevitably 
that electrons in a metal do not behave like a classical gas. 



256 



Thermal properties of solids Chap. 8 



i 


i 


X. 


b 


10 










\ o 


en 




T3 


\ x 


F , 


\ 


^b 


■ V 


o 


F 


3 


o 




\ 




o 




-C 


n 


E 
i 1 ^~ 



500 1000 

Temperature 

(a) 







'O 




500 1000 

Temperature 



500 1000 °K 

Temperature 
(c) 

Fig. 8.15. Thermal conductivity, electrical conductivity and the Lorentz 
ratio of copper as functions of temperature. 



So far, we have assembled some experimental facts about metals. The 
plan for the following section will be to show how some of the properties 
of the electron gas can be deduced by harmonizing the Wiedemann- 
Franz law with the unexpected observation about the specific heat. The 
result will be in the form of an equation for the energy of the electron gas 
as a function of temperature — which is certainly non-classical in form 
and can only be understood in terms of quantum mechanics. Historically, 
the problem was not approached in this way, but the electrons were as- 
sumed to behave like a classical gas ; it was found possible to explain the 
electrical and thermal conductivities of metals but their specific heats 
remained mysterious. Because of its historical interest, we will reproduce 
this calculation also. 



8.6 Electrons in metals 



257 




0.0003 



: ^0.0002 



<o" 



1 2 3 °K 100 200 300 °K 

Temperature Temperature 

(a) (b) 

Fig 8.16. (a) Low temperature measurements, below 2°K, of the specific heat of 
copper. The predicted contribution from the lattice is shown as a shaded area, 
the linear part is interpreted to be the contribution from the electrons, (b) Measure- 
ments up to room temperature. The lower line is the extrapolated contribution 

from the electrons. 



* 8.6.1 Thermal and electrical conductivities of metals 

We calculate the thermal conductivity of a metal by imagining the 
lattice not to exist, but the space occupied by the metal to be filled with 
the electron 'gas'. This transports heat like an ordinary gas, although with 
the appropriate values of mean speed and specific heat. The electrons 
must be presumed to have a finite mean free path. Just what mechanism 
is responsible for altering the trajectories of the electrons need not be 
specified ; it might be collisions between electrons (a plausible suggestion 
but not a correct one) or collisions with imperfections or impurities or 
other departures from perfect regularity in the lattice such as thermal 
vibrations. Then 



k = %C' v cA 



(8.15) 



where c is the mean speed of the electrons, X the mean free path and C' v is 
the specific heat per cm 3 of electron 'gas'. If there are n electrons/cm 3 
inside the metal, then 

1 n o~ -, 

k=--C v cX 

where C v is the specific heat of 1 mole of electrons. 

Next we will calculate the electrical conductivity of the same 'gas'. 
Imagine the metal in the form of a rod or wire of cross-sectional area 



258 Thermal properties of solids Chap. 8 

A and length /. Let a voltage V act between the ends. Then the electric 
field is V//, and if the charge on the electron is e, the force on it is e\/l. 
This causes the electron to accelerate, with an acceleration eV/lm, where 
m is the mass of the electron. 

If the mean free path is A and the mean speed is c, then the mean time 
between collisions is A/c. We will assume that after a collision, the electron 
is brought to rest. During its mean free time it accelerates and reaches 
a final velocity equal to the acceleration multiplied by the time. It is 
brought to rest by the next collision and then the process repeats itself. 
Thus the mean velocity with which the electron drifts along the wire is 

_ _ 1 VeA 
2 mcl ' 

If there are n electrons per cm 3 , the number drifting across any plane in 
the wire is vn A, so the charge transported per second is 

_ A 1 \e 2 Xn A 

Current = vn Ae = — . 

2 mcl 

Ohm's law states 

A\ 
current = (conductivity a) — — 

so that we have derived Ohm's law and 

1 n e 2 X 

o = - -^r-. (8.16) 

2 mc 

Let us compare this expression with that for the thermal conductivity. 
Both contain the mean free path A, which is not surprising for transport 
processes with the same carriers. If we form the ratio, A cancels out : 

k ^ Un /N)C v cX = 2 C v mc 2 

o ^{n e 2 /mc)X 3 Ne 2 ' { } 

Now let us rewrite this expression in slightly more general terms, \rn~c 1 
is the mean kinetic energy of an electron — and we can assume that it is 
not very different from \mc 2 (notice the different averaging). Let us 
denote the mean energy of N electrons by E : 

E = \Nmc^ 
The specific heat C v is equal to dE/dT. Therefore 

k _ 4 E dE/dT 

a ~ 3 (Ne) 2 ' 



8.6 Electrons in metals 259 

Experimentally, k/<jT = Z£ the Lorentz constant. Hence, 

EdE/dT= l(Ne) 2 ^T 
so that 

E 2 = E 2 +l(Ne) 2 £eT 2 

where the term Eq is an arbitrary additive constant. We can take the 
square root of both sides, making the assumption (which we will justify 
shortly) that Eq is much larger than the T 2 term under our conditions. 
Then 



E = E 



\ 3 (Ne) 2 J? 
S E 



1/2 



= £o 3^ r2+ (g]8) 

to h 

This is the variation of energy of the electron gas with temperature which 
we deduce from the observation that the Lorentz ratio is a constant. 
Let us see what specific heat it predicts. We write 

_dE _3(Ne) 2 J? 
v dT 8 E 

It can be seen at once that at any rate this is of the correct form, in the sense 
that it agrees with observations at very low temperatures. If we put in 
numbers, we can find the one unknown E . 

Putting C v = 0.002 T J/moldeg deduced from measurements at low 
temperatures, Ne = !F the faraday equal to 10 5 coulombs, the Lorentz 
constant equal to 2.5 x 10 ~ 8 watt ohm/deg 2 , then E = 50,000 J/mol — an 
enormous value equivalent to 1 eV for each electron. (At least our assump- 
tion that El > the term in T 2 is justified and our method of taking the 
square root is self-consistent.) 

Thus the discussion of the Wiedemann-Franz law and the small 
specific heat of the electron gas leads us to conclude that even at absolute 
zero the electrons have enormous energies, the £ term. Being a constant, 
E does not show up in the specific heat measurements which only 
measure dE/dT: all these detect is the coefficient of the small T 2 term in 
the energy, and at ordinary temperatures the energy of the electrons 
changes by very little. 

These facts can only be explained in terms of quantum mechanics — 
in particular the uncertainty relation and the exclusion principle — which 



260 Thermal properties of solids Chap. 8 

we will not attempt to do. All we have achieved, while suggesting a mechan- 
ism for the conduction of electricity and heat in metals, is to point out yet 
another system to which the Maxwell distribution and the law of equi- 
partition of energy are not applicable. For completeness, we will mention 
that the specific heat of the electron gas as a function of temperature 
has been calculated quantum-mechanically and is shown in Fig. 8.17. 
It does eventually reach the equipartition value of |R expected of a 
monatomic gas, but only at the impossibly high temperature of 10 4 °K 
— when the metal would have ceased to exist. 




10 



Fig. 8.17. Predicted specific heat of 1 mole of electron 
gas. 



8.6.2 The classical calculation 

We have already explained that the approach which we have just 
adopted was not taken by the pioneers in the subject. It was at the very 
end of the nineteenth century that electrons ('atoms of electricity') were 
discovered in ionized gases and their elementary charge roughly measured 
by J. J. -Thomson. Within a few years, Lorentz and Drude were proposing 
that electrons formed a gas inside metals, in much the same way as we have 
done. But it was natural in those days, before the advent of even the most 
rudimentary form of quantum theory, to assume that the electrons really 
did behave like an ordinary gas obeying the laws of classical physics. 

The ratio of conductivities can be written 



k 4 C V E 
a 



3 {Nef 



Problems 261 

on either theory. If now we assume the classical equipartition laws : 

C v = $Nk, E = ^NkT, (5.12) 

then we find 

This is a very precise prediction — and it almost agrees with experiment. 
Putting R the gas constant equal to 8.31 J/mol deg and J* the faraday to 
10 5 coulomb (or k, Boltzmann's constant, equal to 1.38 x 1CT 23 J/deg, e, 
the electron charge, to 1.6 x 10~ 19 C) the Lorentz number is predicted 
to be about 2.1 x 10 ~ 8 watt ohm/deg 2 which is remarkably good. But of 
course the specific heat is completely incorrect and it was because of 
this that the whole theory had eventually to be demolished and rebuilt in 
quantum terms — a process which took more than 20 years. 

PROBLEMS 

8.1. An imaginary element of atomic weight 80 has density 1.2 g/cm 3 . Its crystal 
structure is known to be simple cubic and an experiment is carried out to 
determine its lattice spacing accurately. Using Cu Ka radiation (A = 1.54 A), 
a reflection from a cube plane is observed at 6 = 73° 45'. What value of n 
must be used in Bragg's equation for this reflection and what is the lattice 
spacing? At what angles 6 would reflections from octahedral planes be expected? 

8.2. An X-ray reflection from a crystal occurs at 6 = 45° when the crystal is main- 
tained at 0°C. When it is heated to 100°C, 9 decreases by 3.42 minutes of arc. 
What is the expansion coefficient of the substance? How is the argument affected 
if the crystal is not cubic in symmetry? 

8.3. Calculate the expansion coefficient and Griineisen constant for an ionic crystal. 
Start with the pair potential of Eq. (3.27). 

(a) Using Taylor's theorem, show that the equation near the minimum is 

A f- = JP-A ag2 l x \ 2 (P-1)(P + 4 ) «e 2 (*\ 3 
\ 2! /47te ao\«o/ 3! 47r£ ao\ao/ 

(Here a is the Madelung constant, not the expansion coefficient.) 

(b) Show that in the equipartition region, the linear expansion coefficient is 
equal to 

(p + 4\/4m: a \ 



p— lj\ ae 2 

where k is Boltzmann's constant. Estimate it for KC1 for which a = 3.1 A 
and Madelung constant = 1.75. 
(c) Use expressions for the compressibility and molar volume taken from 
section 3.8.2. Hence show that the Griineisen constant is (p + 4)/9. 



262 Thermal properties of solids Chap. 8 

(d) Compare this with the experimental measurements on KCL (Data from 
G. K. White, Phil. Mag. 6, 1425 (1961)). 



T 
°K 


deg ' 


J/mol. deg. 


V 
cm 3 


dyn/cm 2 


30 
65 

283 


9.90 x 10" 6 
51.6 
111.3 


8.31 
28.6 

48.7 


36.7 

36.75 

37.4 


1.95 xlO 11 

1.93 

1.69 



CHAPTER 




Defects in solids: Liquids as 
disordered solids 



9.1 DEFORMATION OF SOLIDS 

In section 3.7.1 we calculated the deformation produced in a solid by 
the application of pressures or tensions to it. We will now extend the 
discussion to deformations which are so great that the solid breaks. 

The conventional nomenclature is to call the fractional deformation 
the strain (as already mentioned) and the force per unit area the stress. 

Previously, we considered only hydrostatic stresses, that is pressures 
or tensions acting uniformly over the whole surface of the body (like those 
which exist inside liquids, which cannot support shear strains). In these 
the atoms in the body become uniformly squeezed or separated from 
their neighbours in all directions equally. Now in practice, one is very 
often concerned not with these hydrostatic forces but with tensions or 
pressures acting along a line, or else with twisting forces. However, 
we have noted (in section 3.5) that the order of magnitude of the bulk 
modulus — the initial slope of the stress/strain curve — is of the same 
order of magnitude as the other elastic moduli so we will continue to deal 
only with uniform hydrostatic stresses. 

We showed in section 3.7.1 that the strain s induced by a stress P is 
given approximately by 

P= -K(s-h 2 + ---) (3-25) 



264 Defects in solids : Liquids as disordered solids Chap. 9 

assuming that higher terms can be neglected for the small values of s 
we deal with in practice. (A positive stress is a pressure which produces 
compression, a positive strain is an increase of volume ; whence the minus 
sign in front of the leading term.) If we plot this function up to large values 
of s, beyond the range shown in Fig. 3.15, P should go through a maximum. 
In practice, this result can only mean that there is a maximum tension 
which the solid can withstand, and when this tension is exceeded the solid 
breaks. 

Therefore the tensile strength of a material should be given by the con- 
dition 

£-* 

that is, 

-K(l-9s) = 

s m * 10% 

where s m is the tensile strength. 

Our simple theory predicts that a body should be able to withstand a 
10% strain. Though we have derived this result only for hydrostatic ten- 
sions, the same result should hold for stretching or twisting strengths : 
a wire ought ideally to be able to be stretched by 10% in length before 
breaking, a rod should be capable of being sheared through ^ radian 
before shearing off. 

9.1.1 Ductile, brittle and plastic materials 

The behaviour of real metals is in strong contrast to this result. Some 
typical stress/strain graphs are shown in Fig. 9.1. In (a), the body is 
strained by only a small amount— say 0.001 % to 0.01 %. Then, when the 
stresses are removed, the body returns more or less exactly to its original 
shape and size. However, there exists a strain (typically of order 0.01 % to 
0.1%) called the elastic limit, marked E on Fig. 9.1(b); if the body is 
strained beyond E, and the forces then removed, it does not return to its 
original shape and size but remains permanently distorted. Subsequent 
stressings follow a complicated pattern, but when the strain is of the 
order 0.1 % to 1 %, the metal breaks. This means that metals are a factor 
of 10 or 100 times weaker than our simple theory predicts. 

Materials which behave in this way are called ductile. This refers to 
their property of distorting permanently before breaking. 

Metals can be prepared as single crystals and these are often very soft 
and ductile. They can be deformed by squeezing in one's fingers and 
in extreme cases will even flow under their own weight. If pulled, they can 
be extended to many times their own length— the strain can reach 10 or 




265 



10 to 10 Strain 
(c) 

Fig. 9.1. Stress (tension) strain curves for (a) ductile material in the 

elastic range, (b) ductile material strained beyond the elastic limit E, 

(c) brittle material. The curves are not to scale. 



266 Defects in solids : Liquids as disordered solids Chap. 7 

20 — though all these deformations are inelastic in the sense that they are 
permanent and the metal does not regain its original shape when the 
stresses are removed. These deformations are again quite different from 
those predicted in section 9.1. 

Some metals, however, like cast iron and many non-metals like glass 
or stone are brittle. Under small tensions they distort but then quite 
suddenly they break. There is no previous warning in the form of perma- 
nent stretching as with ductile metals and the halves or fragments can be 
fitted together afterwards to reconstruct the original shape. The stress/ 
strain graph is shown in Fig. 9.1(c), on greatly enlarged scales. This refers 
to tension: many brittle materials can, however, withstand quite large 
compressions. Houses and bridges can be built of brick and stone provided 
those materials are not called upon to withstand tension. 

Plastic materials (polythene, nylon, etc.) are different again in their 
behaviour. The molecules of many of them are long chains which may be 
crumpled or coiled and when the substance is stretched the chains may 
straighten ; later they may slip slowly over one another under the action 
of a constant force, as the weak bonds between the molecules break and 
form again. As a result, many plastics can suffer elongations of several 
times their original length without breaking. We will not however study 
these materials but will concentrate on substances built of simple mole- 
cules. 

9.1.2 Friction of metals 

The empirical laws of friction are well known. When a body rests on a 
horizontal plane, the horizontal force required to move it at constant low 
speed is proportional to the mass of the body but independent of the area 
of contact, Fig. 9.2(a) and (b). A brick-shaped object requires the same 
force to move it whether it is resting on a face of large area or one of small 
area. Over a wide range of conditions, the frictional force is almost 
independent of the speed of relative movement. 

We will now show that though friction is a force which acts at the surface 
of a body, it can be related to the bulk properties of the substance. 

It is known that when one body rests on another, the area of real contact 
between the surfaces is extremely small. Though each body has a macro- 
scopic area of several square centimetres, say, the area where atoms of 
one metal actually touch atoms of the other is only a tiny fraction of this. 
(The experimental proof of this statement came originally from measure- 
ments of the electrical resistance between two pieces of metal placed in 
contact under only small forces.) The reason is that no metal surface is 
plane on the atomic scale. There are always large bumps or asperities 
even on surfaces which appear to be smooth— at least, if one uses the 



9.1 Deformation of solids 267 



-F 



(o) 



(b) 




(c) 



Weight Mg 




(d) 



Force F ■ 




Fig. 9.2. (a), (b) The force required to move a given object at low speeds against 
friction is independent of the area of contact with the plane, (c) Points of real 
contact between two bodies, (d) Simplified model of contacts. Metal flows 
until the pressure due to the weight is equal to the yield strength of the metal. 
(e) Shearing of contacts by horizontal force. 



268 Defects in solids : Liquids as disordered solids Chap. 9 

word large' to mean large on the atomic scale. When two such surfaces 
are brought together, it is probable that only at points like P and Q (Fig. 
9.2(c)) will the two really touch. 

If this does occur, then the weight of the upper body is borne on a 
very small area of contact so that the pressure, the weight per unit area, 
is very large. As a result, the metal in these regions is crushed beyond its 
yield point and it flows. The two bodies may even weld together there. 

Let us call the maximum stress that the metal can bear in compression 
without flowing the yield strength, and let us denote it by S Y . 

Then the contacts will go on flowing and increasing their cross-sectional 
areas till the weight Mg of the body can be supported ; then movement 
will stop. Let the total area of real contact be A when this happens (Fig. 
9.2(d)). Then A is determined by 

^=S Y , (9.1) 

A 

Typically, S Y is of the order of 10 10 dyn/cm 2 for a metal. If the mass of 
the body is 100 g, Mg is 10 5 dyn and A is of the order of 10" 5 cm 2 — 
though a body weighing 100 g is likely to have dimensions of a few centi- 
metres. 

Now imagine a horizontal force F to act on the upper body. The 
contacts experience a shearing stress, Fig. 9.2(e), equal to this force 
divided by the area tangential to it, that is F/A. They can withstand this 
deformation until the stress reaches a limiting value called the shear 
strength, denoted by S s , when the contact snaps. In a real situation there 
are of course many contacts which break at different times and reform 
elsewhere, but the force required to move the body is given by 

- = S s . (9.2) 

A 

Eliminating A from these two equations, 

— = ^ = fi (9.3) 

Mg Sy 

where \i is called the coefficient of static friction, the ratio of the horizontal 
force acting on the body to its weight. Thus we have shown that \i is 
independent of the size or shape of the bodies and is equal to the ratio 
of the shear strength to the yield strength. Roughly, this is equal to the 
ratio of the rigidity modulus to Young's modulus, and this from the table 
in section 3.5 is roughly \. Indeed it is observed that the coefficient of 
friction between metals is of this order. 



9.2 Brittle materials 269 

In addition to the welding and snapping of the regions of contact, 
the coefficient of friction is affected in practice by a number of other effects 
which are of technical importance. For example, the ploughing of one 
asperity through another as the body moves sideways can be important. 
Lubricants and films of oxide on the surface can have a profound effect 
on friction. Further, when two different metals are in contact, the fact that 
one metal melts more easily than the other can alter the details of the 
mechanism. We will content ourselves however with having related the 
phenomena at the surface with the properties of the bulk material. 

9.2 BRITTLE MATERIALS 

Glasses and ceramics (like pottery and bricks, which are mostly metallic 
oxides) are brittle materials. They have disordered, non-crystalline 
structures: Fig. 9.3 is an attempt to picture some neighbouring atoms 
in such a solid. The interatomic bonds are highly directional in nature, 
being of the electron-sharing type (section 2.1.4). The coordination 
number of each atom is small and the structure more or less rigid. As 
mentioned previously (section 2.2.4), amorphous solids can be pictured 
as liquids 'frozen' into a particular disordered configuration which does 
not change with time. For a molecular solid, like solid argon in a non- 
crystalline state, the molecules are arranged as in Fig. 2.4 ; the coordination 
number is high, the bonds directed at all angles. Amorphous regions in 
metals have similar structures. Fig. 9.3 and Fig. 2.4 resemble one another 




Fig. 9.3. Atomic structure of an oxide glass. The atoms or ions have been drawn rather 
small for clarity. The lines joining them represent directional covalent bonds. 



270 Defects in solids : Liquids as disordered solids Chap. 9 

in that both are disordered, but they differ in that one lattice is much more 
rigid than the other. 

To explain their stress/strain curves, it has been proposed that brittle 
materials contain cracks, which may be of microscopic or even atomic 
dimensions, and which may either be wholly inside the material or else 
originate on the surface. Such imperfections weaken the material and it 
breaks when one crack suddenly spreads. A familiar example of such 
behaviour is the way one can break glass in a controlled way by first 
scoring the surface with a diamond and then bending the glass so as to 
open the crack. 

We can simulate the stresses round a crack of known shape but of 
large size, and study them using an optical method. Materials like glass 
or plastics when deformed exhibit double refraction. This means that the 
refractive index for visible light is different for light polarized parallel to 
the direction of the stretching, from that for light polarized at right-angles 
to it. Using polarizing plates and a quarter-wave plate in a way which is 
described in standard texts on optics, the strain in a photoelastic material 
can actually be seen and measured through the intensity of light it trans- 
mits. In particular, lines of constant strain can be made visible as dark 
lines, interference fringes, on an illuminated background. 

Cut 



Tension 




Fig. 9.4. (a) Stress pattern round the end of a cut in a slab of photoelastic material. 
(b) End of the cut showing semicircular end of radius p. 



Figure 9.4(a) shows a photograph of part of a slab of plastic (Columbia 
resin, CR 39) 6 mm thick, 19 mm wide and 7 cm long being stressed 
horizontally with a tension of several hundreds of kilograms. A parallel- 
sided cut, 0.4 mm wide and 1 1 mm long was made in it, transversely, with 
the ends carefully machined to be semicircular Fig. 9.4(b). The photograph 
concentrates on the end of the cut and it can be seen that the dark lines all 
crowd in there. 



9.2 Brittle materials 271 

In some preliminary experiments with a specimen of plastic of the same 
thickness but with no cut in it, the optical components were set so that 
the specimen was uniformly dark when there was no tension on it. When 
the tension was increased, the field lightened gradually and then went 
dark again; this occurred with a tension of 1,200 kg/cm 2 . Subsequently, 
the field went dark again every time the tension was increased by about 
this amount. Now each fringe on the pattern represents a line of constant 
strain : we can therefore say that if Hooke's law holds the tension at all 
points along one fringe differs from that at points in the next fringe by 
1,200 kg/cm 2 . The fact that all the dark lines crowd together therefore 
means that the tension varies very rapidly from point to point, and that it 
is a maximum at the end of the cut. 

From the intensity at the top and bottom of the specimen, the applied 
tension was estimated to be about 600 kg/cm 2 . There are 7 fringes between 
these regions and the end of the cut, so that at that point it is 8,400 kg/cm 2 
higher. Hence the tension at the end of the cut is about 15 times as great 
as the tension far from it. 

The theory of these stress/strain patterns in irregularly shaped bodies 
is complex : we will however quote the result that if a transverse cut is of 
total length / and the radius of the end is p, the stress in the neighbourhood 
of the end of the cut is enhanced by a factor 

Here, this is 2 x (11/0.2) 1/2 = 15, in good agreement with the measure- 
ments on the photoelastic specimen. 

It is obvious, then, that a crack inside a solid with a small radius p at 
the end — that is, a sharp pointed crack — concentrates the stress at the 
end. Whereas the tension far from the crack may be quite small, well 
within the tensile strength of the material, the stress at the point may 
become too great and the material will fail there. This occurs when the 
stress s is given by 

s m = s.2l l - (9.4) 

where s m is the expected breaking stress, in the absence of cracks. As the 
crack extends, / becomes larger and p remains the same or gets smaller, 
so that it continues to grow. 

There are two points to note. The first is that cuts or cracks which are 
parallel to the line of the tension have no effect as stress concentrators. 
This is the reason for the great strength of freshly drawn fibres of quartz 
or glass — they can withstand elongations of several percent and their 



272 Defects in solids : Liquids as disordered solids Chap. 9 

Young's moduli are higher than that of steel so that their breaking stresses 
are higher than that of steel. They owe this property to the fact that when 
they are prepared by pulling a filament of melted but highly viscous liquid 
just above the melting point, all cracks become pulled out also, but parallel 
to the length of the fibre. Until the surface becomes pitted by chemical 
action through exposure to the atmosphere, they retain their strength. 

The second point is that only the ratio of the length of the crack to its 
width enters into the stress ratio. A crack 4 A wide and 110 A long would 
be just as effective as the one in the specimen of Fig. 9.4. Now most 
ceramics are found to be inhomogeneous in their crystal structure when 
examined by X-ray diffraction. It is quite possible therefore that in a 
structure like Fig. 9.3 with rigid, directed bonds there may be misfitting 
atomic planes with few bonds crossing them. These behave like long 
narrow cracks, and can therefore act as stress concentrators. Glass on the 
other hand usually seems to break because of defects on the surface — 
scratches or particles of dust picked up during manufacture. The evidence 
for this comes from the study of the fractured surfaces. There is always 
a small area with a very smooth surface — so smooth that no roughness 
can be seen even under a microscope. But surrounding this small area, 
the surface is covered with small ridges. These different areas are thought 
to be caused firstly by the rapid tearing of the glass and secondly a forking 
of the crack as it accelerates. When a rod is deliberately notched and then 
bent and broken, so that there is no doubt where the break originated, the 
mirror-like surface is always found near the origin and the ridges radiate 
roughly from the same point. Thus we have a good method for locating 
the origin of any crack. All cracks seem to start from the surface, not from 
inside. 

The great strength of brittle materials when they are compressed is 
explained by the closing of the cracks and notches. Glass can be toughened 
using this effect, by arranging that the surface layers of a sheet of molten 
glass cool more quickly than the inside ; the inside of the sheet is the last 
to solidify. This has the effect of compressing the outer layers, that is, the 
atoms in the outer layers are closer together than those in the inner layers. 
Therefore when the surfaces are stressed, these tensions must first overcome 
the locked-in compressive stresses before any cracks can begin to open 
so the glass is stronger than when prepared normally. Concrete can 
similarly be prestressed by allowing it to harden under great pressure. 

9.2. 1 Dynamic behaviour of brittle materials 

It is good common sense to argue that if a number of cracks begin to 
spread through a solid, only one of them can 'win', and that once the 
solid has broken the stress is immediately relieved and the other cracks 



9.2 Brittle materials 



273 



must stop growing. In other words, a solid ought to break only at one 
place at once. It would be a gross and improbable coincidence if there were 
two cracks of exactly the same rate of growth so that a solid broke at 
two places at once. 

Yet if one bends a glass rod (say 25 cm long and 3 mm diameter) rapidly 
but firmly so that it breaks, it may fly into several pieces, three or four or 
more (Fig. 9.5.). This is seemingly impossible behaviour. But it is not 
difficult to demonstrate that the cracks occur one after the other and not 
simultaneously, so that the commonsense argument is not incorrect. 
Similar things happen when a rod is broken by straight tension. 



' : %TiMmif i fflMiiiiimiiiiil^F "*"* 







Photos by Colin King 



Fig. 9.5. Flash photographs of two different glass rods breaking under the action of 
shear — the rods were bowed upwards. 



The key to these phenomena is the presence of stress waves which are 
propagated along the rod. In a rod broken by bending, they start as a 
flexural pulse which whips back and forth along the rod. This is a compli- 
cated motion in which short wavelengths travel fastest and an initially 
compact pulse straggles ; later, the shortest wavelengths can start bouncing 
around from side to side inside the rod. Or if a rod is broken by pulling, 
then after the break the surfaces spring back and a compression pulse is 
propagated along each half. When it meets the far end it is reflected. 
If that end is free to move, being undamped, then the pulse is reflected 
as a tension pulse. For a short time, while the incident compression pulse 
is passing over the reflected tension pulse, the stresses cancel, but later 



274 Defects in solids : Liquids as disordered solids Chap. 9 

the tension appears — and the rod may break again, a little away from 
the end. 

9.2.2 Two-component materials 

Fibre-glass is a typical two-component material. It consists of glass 
fibres (which in the absence of surface flaws have great tensile strength) 
embedded in a matrix of soft, non-brittle plastic. Consider the fibres all to 
be parallel and the tension to be applied in their direction. Then practically 
all the stress is borne by the fibres. Some may crack across, but because 
they are separated from each other a crack cannot spread from one to 
another. If the matrix material adheres strongly to the fibres, then one 
broken fibre will be held in place through the strength of its neighbours. 
The tensile strength of the material is therefore extremely high. 

Its shear strength is small, however, because the fibres can be easily 
bent, so that aligned fibre glass is not a good structural material. However, 
if the fibres are tangled up and are aligned in all directions, the strength 
for all kinds of stress can be quite high. 

Wood is another common two-component material — cellulose fibres 
in a lignin matrix. Again, it is extremely strong in tension but its shear 
strength is low. This can be improved by laminating it to form ply- 
wood, in which successive layers have their fibres oriented in different 
directions. 

9.3 DEFORMATION OF DUCTILE METALS 

In section 9.1.1 we defined the elastic limit of a substance, and stated 
that soft metals prepared as single crystals can be strained beyond their 
elastic limits by the action of quite small forces. Cadmium notably can 
be pulled to 10 or 20 times its original length. 

When the surface of a single-crystal rod is examined after it has been 
pulled beyond its elastic limit, it is seen that its surface is covered with 
fine lines. These show that the material has divided itself into bands which 
have slipped or glided with respect to one another, Fig. 9.6. The direction 
of easy glide are close-packed crystallographic planes. Each section has 
slithered sideways, pulled over by the component of the tension parallel 
to the glide direction, but as a result the specimen as a whole has become 
longer. The elongation has taken place because each section undergoes 
shear with respect to the neighbouring ones. 

Each unslipped section has an almost perfect crystal structure: the 
deformation is located inside regions which are only a few atomic planes 
thick. The width of the unslipped regions may be anything upwards from 
a few thousand atomic spacings and the steps on the surface may be a few 



9.3 Deformation of ductile metals 



275 



hundred or thousand atomic spacings high, so that the structure is usually 
visible under a microscope. 

Thus the process of deformation in a ductile metal is quite different 
from that which we considered in section 9.1 where we worked out the 
tensile strength of a material — on the assumption that the deformation 
was elastic and would return to zero when the stresses were removed. 
There, we assumed that the strain was homogeneous, that the distance 
between every atom and its nearest neighbours increased by the same 
amount. But it appears that, beyond the elastic limit, the strain is far 
from homogeneous : large movements take place in relatively few planes. 




Fig. 9.6. Stretching of a single-crystal rod under tension 

by slipping in sections along direction of easy glide. The 

sheared rod is slightly longer than it was originally. 



9.3.1 Dislocation lines 

It is useful to study the strains in a metal where the slip has travelled 
only part way across the thickness. Fig. 9.7(a) shows this diagrammatically. 
The whole block, originally rectangular, is now distorted and out of true. 
Part of the left-hand front face has been pushed in a short distance, and 
an incipient step is visible on this face. With this deformation, part of the 
whole block has also slipped and the distortion of the atomic planes is 
visible on the right-hand face. The boundary between the slipped and the 
unslipped region is marked by a curve inside the block, shown by the 
shading on the slip plane. This boundary is called a dislocation line. 

The strain, the displacement of an atom relative to its neighbours, is 
greater near the dislocation line. But elsewhere, the crystal is almost 
perfect. 

It is these relative displacements in the vicinity of the dislocation line 
which we will have to study in some detail. Now when the dislocation line 
is curved, as in the diagram, the strains are difficult to describe. But two 
basic types of straight dislocation lines can be distinguished, and any 
curved line can be regarded as a combination of these. 



276 



Defects in solids : Liquids as disordered solids Chap. 9 




°o o o o 

; 1 1 \ I \ i \ 

aOOCxWOO 

OtyOO 

p6 6 6 6 6 

R 

id) 



O--Q---9--Q-0 



0--<>-c--6 

i ! ! ■ ■' 

a-6--6--6 




c) 



Fig. 9.7. (a) A strained cube with a slipped region bounded by the shaded 
arc. At A a screw dislocation emerges, at B an edge dislocation. The deforma- 
tion is in the direction RP. (6) Looking down on a screw dislocation ; A, P, 
Q, R correspond to A, P, Q, R in the cube above. In a perfect lattice the 
atoms would lie on separate lines, but here they lie on a helical surface, (c) 
Looking along an edge dislocation, (d) Small-angle grain boundary ; D and D' 
are edge dislocations. 



9.3 Deformation of ductile metals 277 

In the diagram, the slip is in the direction of RP. When the dis- 
location is a straight line parallel to the slip, as it is near A, it is called a 
screw dislocation. When it is perpendicular, as it is near B, it is called an 
edge dislocation. 

The arrangement of atoms near to a screw dislocation line is shown in 
Fig. 9.7(b). What were originally parallel planes of atoms have now be- 
come distorted into a helical surface, like that generated by a propellor 
rotating slowly as it moves forward. Starting on one plane at the top of 
the diagram and moving from atom to atom to the bottom and then up 
again, one arrives without discontinuity at the next plane ; in fact, there is 
strictly speaking only one single sheet of atoms, instead of parallel planes. 
The left-hand face, with the step, in Fig. 9.7(a) is the outermost layer of 
this sheet. 

The atoms near an edge dislocation are shown in Fig. 9.7(c). In the upper 
half, there are five columns of atoms, in the lower half only four. Similar 
patterns are visible on the right-hand face of Fig. 9.7(a), and also in Fig. 2.5. 

Dislocations may be produced by any irregularity during the growth 
of a crystal. They occur at grain boundaries, the junctions between 
crystals which do not fit perfectly because there is an angle between the 
sets of crystallographic planes. The resulting misfit can be described as 
a row of edge dislocations, regularly spaced, as in Fig. 9.1(d). Dislocations 
may also be clustered round any misfitting inclusions or foreign atoms 
inside a lattice. 

Dislocations can be seen most easily in specimens made in the form of 
very thin films, say 1,000 A thick, examined in an electron microscope 
capable of magnifying by a factor of the order of 10 5 . Dislocation lines 
which run more or less vertically through the thickness are visible. Points 
like A and B in Fig. 9.7(a) where dislocations emerge at surfaces of crystals 
are chemically active, because of the high energy of atoms with the wrong 
coordination. Thus they are vulnerable to chemical attack by appropriate 
reagents. Little holes called etch pits appear at odd places on the surface. 
Another chemical method for seeing dislocations is the 'decoration' 
of the lines inside transparent crystals like silver bromide and silver 
chloride. These substances are used in photographic plates because, under 
the action of light, free silver is formed which is opaque. Not surprisingly, 
it is formed preferentially on the dislocation lines so that these can be 
seen under a microscope. 

In a crystal of metal prepared under ordinary conditions, there are 
typically about 10 7 dislocations in any square centimetre of cross section. 
This sounds an enormous number, but expressed rather differently the 
same datum shows that such a crystal is highly regular. For if the inter- 
atomic spacing is 2 A, there are 5 x 10 7 atoms in 1 cm of length; 10 7 



278 Defects in solids : Liquids as disordered solids Chap. 9 

dislocations per cm 2 means one dislocation every 3 x 10 -4 cm, that is one 
every 15,000 atoms. For comparison, it may be noted that in the most 
perfect of single crystals which can be prepared the density is around 10 3 
dislocations/cm 2 . At the other end of the scale in heavily damaged metals 
where the crystal structure has been broken up there might be 10 12 
dislocations/cm 2 , one every 50 atoms. Even in the most disordered 
crystals, the assumption that most of the atoms are in their regular lattice 
sites is a valid one for many purposes. 

9.3.2 Movement of dislocations 

Having described what dislocations are, we can now show how they 
can produce slip in a crystal under the influence of small forces — tensions 
which are small compared with the elastic moduli. 

The essential process is the movement of a dislocation through the 
lattice. We will concentrate on edge dislocations. Similar arguments 
apply to screw dislocations but it will be left as an exercise for the reader 
to construct these. 

Imagine a block sheared as in Fig. 9.8. A dislocation is created on the 
left, runs from left to right and emerges from the block. The net result 
is that the top half of the crystal has sheared over. By itself, this process 
causes only a very small slip, only one atomic spacing high. But it is the 
basic process which is responsible for the inelastic extension of metals 
under small forces. Given 10 7 dislocations per cm 2 , a movement of each 
one of them would add a 2 A step to one side, and if all of them moved 
a 1 cm cube would shear over a distance of 2 mm, a 20 % strain. 

We can now look at Fig. 9.6 with much more understanding than before. 
One would guess at first sight that each slip band has slipped bodily 
across, like a penny in a pile which has been pushed askew. But now we 
can see that each section has not been moved like a rigid body — dis- 
locations have moved across instead — and when we say that a dislocation 
has moved, Fig. 9.9 shows that all we mean is that all the atoms in the 
slip plane have moved a small fraction of an atomic spacing, one after the 
other. 

It is not difficult to see qualitatively that dislocations, once created, 
can move through a lattice very easily. Not only are the individual 
displacements of the atoms very small, which implies small activation 
energies, but when one atom gains potential energy (by moving a little 
further from its neighbours) another loses an almost equal amount (by 
moving closer). Therefore during the movement the potential energy of 
the system hardly changes. 

Quantitatively, however, the mobility depends critically on the geometry 
of the lattice. The dislocations which we have pictured have all been narrow 



9.3 Deformation of ductile metals 



279 



ones, in the sense that there are only two or three atoms in a line which are 
badly, displaced from their regular positions. The number w of such 
atoms in a line is a useful measure of the width. But it is possible to produce 
wide dislocations, as in Fig. 9.9(b). Here there are many more displaced 
atoms in a line but the change of position of each one when the dislocation 
moves is much smaller than before so that wide dislocations are more 
mobile than narrow ones. A detailed analysis shows that if s d is the stress 
required to move a dislocation and K is one of the elastic moduli, then in 
order of magnitude 



K 6 * 



(9.5) 



e 2 * is equal to 500 so that the width co of the dislocation should have a 
dominant effect on the yield strength of a material. 




Fig. 9.8. Shearing of a block by the movement of a dislocation across it. 



Wide dislocations are favoured by large spacings between atomic 
planes ; such planes themselves are close-packed crystallographic planes. 
Hence one would expect that dislocations can move most easily in close 



280 Defects in solids : Liquids as disordered solids Chap. 9 

packed crystallographic planes. This result was observed experimentally 
in metals and has already been mentioned in section 9.3. 

Wide dislocations are also most easily formed in substances like metals 
where the bonds are non-directional. Indeed, very pure metals are ex- 
tremely soft. However, another effect enters as soon as a metal has been 
repeatedly deformed: it is observed that its hardness increases. This is 
called work hardening. It is interpreted to mean that the metal has become 
full of dislocation lines — and it is difficult for one line to pass through 
another. The lines repel one another because of the stresses round them ; 
thus the movement of dislocations is impeded, and the metal hardens. 
Ordinary metals also owe some of their hardness to another effect, the 
presence of impurities which can also impede the motion of dislocations. 




(a) 



OCCCOOCOD 
GCCODOCCCO 



U>) 



(c) 



id) 



Fig. 9.9. (a) and {b) : Dislocation moves one atomic space, (c) Relative positions 

of atoms in top line in (a) and (b), showing that each atom moves only a small 

distance, (d) Same diagram for a wider dislocation : co is about 8 whereas co 

is about 5 or 6 in (c). 

Hard metals like steel owe their combination of ductility and hardness to 
the judicious introduction of other elements such as carbon, and of 
several types of crystal modifications dispersed throughout the material. 
Narrow dislocations are characteristic of crystalline substances with 
covalent bonds, which are rigid and difficult to distort. Covalent materials 



9.4 Growth of crystals 281 

in any case have high elastic moduli, but in addition we can now see why 
they are not ductile. Diamond, for example, is the hardest substance 
known. 

* 9.4 GROWTH OF CRYSTALS 

We will now discuss the rate of growth of crystals and to simplify matters 
we will consider growth from the vapour phase. The classic studies were 
done on iodine crystals; iodine has a vapour pressure of 0.1 mm at room 
temperature. The rates of growth were typically about 0.1 mm per hour 
when the vapour pressure was 1.05 times the saturated vapour pressure — 
that is, when the vapour was 5 % supersaturated. In this section we will 
outline a simple theory and demonstrate that it is wrong by many orders 
of magnitude. We will assume that the initiation of a new layer of molecules 
on a plane crystal face is much like the nucleation of a droplet of liquid in 
a supersaturated vapour (see section 7.8.3). We will also assume that the 
crystal face is perfectly plane to begin with. If a single molecule arrives 
from the vapour and sticks to the plane, the number of bonds which it 
forms is only a small fraction of its full coordination number, about \. 
So this lone molecule probably evaporates again after a short but finite 
time. If, however, there is an 'island' already formed on the face (a two- 
dimensional droplet, as it were), the newly arrived molecule might lodge 
at the edge of it and then its coordination number approaches n/2 and 
there is a much greater chance of it sticking permanently ; the island acts 
as a nucleus, Fig. 9.10. Let us for simplicity assume that the island is 
circular. Then we can treat it like a cylindrical droplet of finite radius and 




Fig. 9.10. The formation of an 'island' from the aggregation 

of newly-arrived molecules landing anywhere on the 

crystal face. 



282 Defects in solids : Liquids as disordered solids Chap. 9 

only one molecule high. Using exactly the same methods as in section 
7.8.3 for spherical droplets we can calculate the critical radius for a 
given degree of supersaturation of the vapour, the radius of island which 
can grow instead of evaporating. We do this by balancing the rate of 
evaporation from the curved surface against the rate of arrival of new 
molecules which might have landed anywhere on the crystal face and 
diffused to the island. (This last is different from the corresponding 
conditions during the formation of a liquid droplet). Finally we can 
calculate the activation energy A for forming such an island ; the prob- 
ability of a critical island forming contains the Boltzmann factor 
exp( — A /kT). The steps are as follows. Inside a cylindrical droplet of 
any length but of radius r, the excess pressure is y/r. Its vapour pressure 
is therefore increased over that of a plane surface by the factor 

P 

_ p yvlrkT 

Po~ 

where v is the volume of one molecule, y the surface tension, k Boltzmann's 
constant ; compare Eq. (7.34). The critical radius is 

yv 



c kT\n(P/P Y 

compare Eq. (7.35). The activation energy A for forming such an island 
is (following precise thermodynamic arguments) equal to one-half of the 
surface energy of the edge : 

A = nr c dy 

(compare and contrast Eq. (7.36)) where d is the diameter of a molecule 
(the height of the cylinder). 

Let us simplify these expressions by writing y = j J/ "m (see section 3.4) 
where A" = \ja% . The volume v of one molecule is al • Then : 

A - n ( U£ ) 2 / QA v 

Ao ~ l6kTln(P/P ) ( * 6) 

for an island of critical radius to be nucleated in a supersaturated vapour 
of pressure P. 

Let us now put in numbers. For a molecular crystal, let us take n = 10, 
£ = 0.1 eV and for room temperature kT = ^ eV. ; whence 



9.4 Growth of crystals 283 

Suppose the area of face is 1 mm square, 1CT 2 cm 2 . At a vapour pressure 
of 0.1 mm, the number of molecules arriving at unit area, \nc, is 10 18 per 
second. Further, the critical radius is always small and a new nucleus 
might be formed anywhere on the crystal face. Thus the number of 
critical nuclei formed per second is equal to the number of molecules 
arriving per second on the whole face multiplied by the probability 
that one of them has enough energy to form the nucleus : 

10 16 exp(-/4 //cT) = 10 16 exp[-400/ln(P/P )]. 

Consider a crystal growing at 0.1 mm per hour, which (for a = 3 A) is 
3 x 10 6 layers per hour or 1,000 layers per second. For this 

10 3 = 10 16 exp[-400/ln(P/P )]. 

This tells us the supersaturation required; it gives ln(P/P ) = 13, which 
means that the vapour pressure should be e 13 or 5x 10 5 times super- 
saturated. Experiments gave 5 % supersaturation for this rate of growth. 
Thus the theory is badly wrong. 

The weak point of this theory is that after an island has grown and 
covered the whole crystal face, it produces a perfect plane ; a finite time, 
very long on the molecular time-scale, has to elapse before another 
fluctuation occurs which is big enough to form another island. In order 
to explain the observed rate of growth, we need to postulate some sort of 
step against which newly arrived molecules can stick themselves, but a 
step which remains in existence even after the face has been covered 
with a new layer. 

9.4.1 Growth spirals 

The point of emergence of a screw dislocation at a crystal face provides 
just such a step — Fig. 9.11. We can imagine atoms arriving more or less 
uniformly all along the line and the crystal can grow. But the step does not 
advance as a straight line. Near the centre of the screw there are some sites 
where the step is not of the full depth, and the atoms lodging there soon 
convert the end of the straight ledge into a tight spiral. This spiral then 
grows upwards like a ziggurat, and the crystal grows as the turns of the 
spiral sweep round over the crystal surface — and they continue to advance 
however much the crystal grows. This is ultimately a consequence of the 
fact that there is only one sheet of atoms in a crystal containing a single 
screw dislocation. 

Spiral markings can be found on many crystals under a microscope, 
Fig. 9.11(e). The faces of an average crystal have many screw dislocations 
emerging on them, and no point of a face is more than a few hundred 
atomic distances from a spiral step. This fact obviously increases the 



284 Defects in solids : Liquids as disordered solids Chap. 9 

chance of permanently capturing a molecule from the vapour. But at 
first sight, it might appear that such a molecule would have to land right 
on the step, or within one atomic distance or thereabouts from the step, 
in order to be captured. But this is not so : we can show that a lone molecule, 
landing far from a step, sticks for a comparatively long time and diffuses 
an appreciable distance over the surface before being thrown off again. 
In its wandering it can meet a step and stick there, so that the probability 
of capture is greatly enhanced. 

We show this as follows. For an isolated molecule on a flat plane, the 
number of bonds it forms is about |rt where n is its full coordination 
number. The probability of its evaporating therefore contains the factor 
exp(-^ne//cT). 




Fig. 9.11. First four pictures show the early stages in the growth of a spiral 

step on a crystal face round the point of emergence of a screw dislocation. 

(e) shows the appearance of a growth spiral in the electron microscope : the 

steps are typically 100 A to 1,000 A across. 



9.4 Growth of crystals 285 

Now the molecules on which it is sitting are vibrating at the Einstein 
frequency (see section 3.6.1), which we call v E . One would guess that there 
is a maximum probability of the lone molecule being thrown off once 
per vibration. Therefore the chance per second that the molecule evapo- 
rates is v E exp( — Ine/kT). With reasonable numbers for a molecular solid, 
the 'sticking time' is about 10~ 8 s. The molecule can jump to a neigh- 
bouring site without evaporating, and the energy needed to do this is 
very small indeed compared with e. One can imagine the molecule sitting 
in a dimple on the surface and the activation energy required to make it 
jump to a neighbouring dimple is obviously very small. Since the cor- 
responding Boltzmann factor is almost equal to unity, it probably jumps 
once every vibration, so that its jump frequency is v E . Therefore it performs 
exp(ine/kT) jumps before evaporating. Each jump moves a distance a , 
but it performs a random walk. The net distance it goes in s steps is there- 
fore only a Js (see section 6.3). Hence it diffuses an average distance 

a exp(fne//cT) 

(where the effect of taking the square root of the exponential is to alter the 
I to |). For ns = 1 eV, kT = ^ eV, this is about 150 atomic diameters. 

This is comparable with the distance between the spiral steps which are 
to be found on the average crystal face. Therefore we can say that practi- 
cally every molecule that arrives from the vapour is eventually captured. 
Of course the crystal also loses molecules by evaporation. The number 
lost per cm 2 in one second is (\nc) evaluated at P , the equilibrium vapour 
pressure (because this must be equal to the rate of arrival of molecules, 
in equilibrium). If, however, the crystal is exposed to vapour at super- 
saturated pressure P, the rate of arrival of molecules is (i«c) evaluated 
at P. Therefore the net rate of gain of molecules is 



1 


~P ~ 


-nc 


1 


4 


Po 



per cm 2 in one second, where the first factor is about 10 1 8 for iodine vapour 
at room temperature. Putting in numbers, the supersaturation required 
for a 1 mm square crystal face to advance at a rate of 0.1 mm per hour is 
1 % — in good agreement with the 5 % observed. 

This detailed study of the mechanism of crystal growth is not incom- 
patible with the account given in section 8.1.3. If screw dislocations inside 
a growing crystal are oriented in all directions at random, then the growth 
will proceed as in Fig. 8.6. 



286 Defects in solids : Liquids as disordered solids Chap. 9 

* 9.4.2 Single-crystal whiskers 

Under the right conditions, crystals can grow in the form of tiny 
whiskers, typically a few thousand atoms in diameter and anything up 
to millimetres in length. NaCl and KC1 can, for example, be grown out 
of the sides of big crystals by exposing them to the vapour at tempera- 
tures of the order of 1,000° K. X-ray examination shows that these 
filaments are single crystals. It is thought that they are produced when 
the outer turns of the growth spirals on the surface of the big crystal 
are stopped from advancing because foreign impurity atoms have lodged 
there. This allows only the tight central turns of the spiral to grow up- 
wards and the result is a filament. Single-crystal whiskers are extra- 
ordinarily strong. For example, the specimen of iron capable of with- 
standing a 4% elongation which was elastic and reversible (see section 
3.7.1) was a single-crystal whisker. This strength is in marked contrast 
to the ductility of large single crystals of metals. But we expect a 
whisker to have only one single screw dislocation running down the 
middle of it. Such a line defect does not weaken a lattice when tension is 
applied parallel to it — in much the same way as a crack does not weaken 
a specimen of brittle material if it is parallel to the stress (section 9.2). 
The observed strength of single-crystal whiskers therefore confirms the 
proposed explanation for their growth. It should be mentioned for 
completeness that other mechanisms of growth are possible — some 
whiskers actually seem to grow from the base, pushing their way upwards. 

* 9.5 POINT DEFECTS 

We have described dislocations in solids, but there exists a much 
simpler kind of imperfection, the point defect. This is the single atom (or 
molecule or ion) which has been displaced from its lattice position. If it 
has lodged in a nearby site, wedged in between a near set of neighbours 
which have to move a little out of the way in order to accommodate it, 
it becomes an interstitial atom; it leaves a vacancy behind, Fig. 9.12. 
Vacancies can exist by themselves without any corresponding interstitial 
atom : in that case we can regard the displaced atom as having been 
removed to the surface of the crystal. 

Vacancies are important because their presence greatly enhances the 
speed of diffusion of atoms in solids, and this is because the activation 
energy for one to jump one atomic space is not large. We can follow a line 
of argument suggested in section 7.2 and regard liquids as resembling 
highly disordered solids containing many vacancies and interstitial atoms ; 
we can then gain new insights into diffusion mechanisms in liquids and 
thence into other transport processes. 



9.5 Point defects 287 

oooooooo 

O xQ o o o o 0,0 

o o y o o ""o 6 
oooooooo 

Fig. 9.12. Atoms removed to an interstitial position and 
to the surface, leaving vacancies behind. 

9.5.1 Concentration of vacancies 

First we will estimate the number of vacancies in a solid as a function 
of temperature. Atoms can jump out of place as a result of thermal agita- 
tion so that the displacing of an atom is thermally activated. Let us 
compare the energy of a perfect lattice with that of a lattice containing 
one single vacancy, with the displaced atom removed to the outer surface. 
We can imagine the atom to be cut out of the lattice by breaking n bonds 
(n the coordination number). This requires energy ne. The atoms around 
the vacancy then relax a little towards one another and this reduces their 
energy. Finally, putting the one atom back on the surface reduces the 
energy a little further still. The net result is that the energy of the whole 
crystal is increased by something like ins. For solid argon this is about 
0.02 eV, for sodium chloride it is about 3 eV, a much larger value because 
of the long range forces. 

Let the increase of energy of the crystal due to the presence of one 
vacancy be denoted by AE. The Boltzmann factors for the crystal with 
one vacancy and for the perfect crystal can be written down ; the ratio of 
probabilities of the two states is exp( — AE/kT). 

Thus for a solid in equilibrium the fraction of vacancies is 
exp(- AE/kT). This function increases with temperature and for 
a solid reaches its greatest value when the solid melts. We there- 
fore expect some warning of the onset of melting to be given in 
the last 10° or 100° before the solid melts, in the form of large numbers of 
vacancies. In section 8.3 we noted that the amplitude of atomic vibrations 
in a solid at the melting point was surprisingly small — one-tenth of the 
interatomic spacing in molecular solids, one-thirtieth in ionic solids. We 
will now be able to show that under these same conditions, vacancies are 
produced in large numbers. The smallness of the amplitude at melting 
therefore appears a little less mysterious, because we can accept that a 
lattice which is full of holes can collapse more easily than a perfect one. 



288 Defects in solids : Liquids as disordered solids Chap. 9 

There are indeed a number of 'premelting' phenomena which can be 
observed in many substances. But the fact that we can explain these 
does not imply that we can explain the phenomenon of melting itself. 
The sharpness of the melting point, the suddenness of the onset of the 
transition, cannot be explained in simple terms. All that we can show is 
that because of thermal equilibrium a substance can be quite inhomoge- 
neous on the microscopic scale. 

9.5.2 Premelting phenomena 

For rough calculations on molecular substances and metals near their 
melting point let us take the melting temperature 7} to be half the critical 
temperature T c : 

K If = 2^ l c = 2 £ 

(see section 8.3). Then AE/kT f must be about 4 for a close packed lattice 
with n = 12. Now exp( — 4) = j$ ; thus at the melting point we would 
expect about 2 % of the atoms to be displaced. This means that about 
1 atom in 4 linearly is out of place and really it represents a highly dis- 
ordered lattice. Vacancies are therefore very close to one another and the 
energy required to produce a new one is thereby reduced. So our estimate 
is, if anything, likely to be too low. Further, we have not taken the expan- 
sion of the lattice into account ; this would make it easier for atoms to 
slip through their neighbours. In other words, the activation energy 
should decrease with temperature. Again, this means that we are under- 
estimating the number of vacancies. Since we are dealing with exponentials, 
we may be wrong by orders of magnitude. In fact it will emerge that, 
numerically, any simple theory of this sort is not very useful. 

In thermal equilibrium there must be vacancies present. Perhaps 
unexpectedly, a perfect crystal cannot exist at a finite tempera- 
ture. Vacancies in turn affect the specific heat. In other words, if heat 
energy is put into a body, not all of it goes into raising the temperature 
but some goes into creating defects. The specific heat is thereby increased. 
We will calculate this with our simple theory. 

If there are N atoms, N exp( — AE/kT) are displaced and each one has 
an extra energy A£ in addition to the 3/cT it has (if it is a monatomic 
molecule) in the perfect lattice. So the mean energy is 

E = 3NkT+AE . N Q -^ kT . (9.7) 

The specific heat is given by 

Cv = of = lNk + Nk(^\\-^ kT . (9.8) 



9.5 Point defects 



289 



At the melting temperature, (AE/kT) is about 4, so that the extra term is 
about j R which is appreciable. We have just seen that this is likely to be 
an underestimate.* 

Some such rise, of about \ R, is present in the curve for argon, Fig. 5.10(6), 
but it is difficult to disentangle it from the variation at low temperatures 
due to quantization of the lattice energy. A comparable rise, of about $R, 
is present in the specific heat of krypton (melting point 115°K) which is 
clearly separated from the fall at low temperatures; but the difficulty 
with this substance is that the expansion coefficient is not known with 
sufficient accuracy to allow C v to be calculated. Exact comparisons are 
therefore not possible for either of these two simple substances. Potassium 
shows a similar rise before it melts at 335° K, Fig. 9.13. 



AR 




~0 100 200 300 °K 

Fig. 9.1 3. Specific heat C v of potassium as function of temperature. 



For ionic solids, we have already mentioned (section 8.3) when 
discussing NaCl that small positive ions can probably slip more 
easily through a vibrating lattice than larger negative ions; in other 
words, their activation energy should be small. The concentration of 
vacancies is of course very sensitive to the exact value of the activation 
energy and as a result, extreme variations in concentration occur from 
substance to substance. In NaCl, it is very small. In AgBr, however, it 



* The term is of the form x 2 . e x and this goes through a maximum when x 
decrease of A£ with temperature alters this to an increasing function. 



The 



290 Defects in solids : Liquids as disordered solids Chap. 9 

is about 2% at the melting point, although the crystal structure is the 
same as for NaCl and the ratio of radii is similar. However in AgBr the 
bonds are not wholly ionic in character but partly covalent; perhaps 
this decreases the activation energy significantly. The effect of the presence 
of 2% of vacancies is very marked. The rise in specific heat is large, 
the value just below the melting point being almost three times the 
expected equipartition value for the perfect lattice. 

Another premelting phenomenon will also be mentioned. The 
volume becomes greater when vacancies are formed. If there is no relaxa- 
tion of the atoms round a vacancy, each one adds a volume v to the volume 
of the crystal. The total increase of volume due to this effect is therefore 

AV= Nve- AE/kT . 

Nv is equal to the molar volume of the substance. This expansion must 
be added to that due to the anharmonicity. The extra term in the expansion 
coefficient is 



dT K kT 2 



Let us put AE/kT equal to about 4 at the melting point. For solid argon, 
7} is about 80° K and since exp( —4) is roughly 0.02, the extra term in the 
coefficient is about 1 x 10 ~ 3 . Now it is observed experimentally that the 
expansion coefficient at the melting point is 1.8 x 10~ 3 so that our simple 
theory implies that over 50 % of the measured effect is due to the vacancy 
contribution. But we have already seen (in section 8.4.2) that the Griineisen 
ratio (whose constancy implies that only the anharmonicity of lattice 
vibrations contributes to the expansion) decreases only by about 15 % as 
the melting point is approached in solid argon. Thus our value for the 
vacancy contribution must be an overestimate. In turn, this can only 
mean that the extra volume created by a vacancy is much less than the 
volume of one atom. It appears in fact that the neighbours relax a good 
deal and only about \v is added per vacancy as the melting point is 
approached. 

* 9.6 DIFFUSION IN SOLIDS 

We usually consider every atom in a solid to be bound to its own lattice 
site and never to move from it. But this is not so ; diffusion can take place 
in solids, albeit very slowly. An atom in a perfect lattice can change its 
position by jumping to an interstitial position and thence to a neighbour- 
ing interstitial position and so on. Each jump requires a high activation 



9.6 Diffusion in solids 291 

energy and so this process is comparatively rare. However, in an imperfect 
lattice where there are defects, diffusion is much easier. An atom can jump 
into a vacancy much more easily than into an interstitial position : the 
process of squeezing past its neighbours into an already existing hole 
requires rather less energy than that needed to create the vacancy initially 
— not much less, because of the relaxation of the atoms round the vacancy. 
After the vacancy has been occupied, another is left behind, Fig. 9.14, 
and the process can happen again. This mechanism must dominate the 
diffusion process and is the only one we will consider. 

ooooo ooooo 
ooo- o oo oo 
ooooo ooooo 

Fig. 9.14. The movement of a vacancy caused by an atom jumping into it. 

The probability of an atom at A being able to jump into a neighbouring 
site B is equal to 

(probability of a vacancy existing at B) 

x (probability of jump into vacancy). 

The first of these two factors we have already calculated as exp( — A£//cT) 
where A£ is the energy required to create the vacancy. 

The second factor we can calculate by noting the resemblance to the 
situation shown in Fig. 5.12(a) and (b), namely that in order to jump into 
a hole, an atom has to squeeze past an intervening position of higher 
potential energy. We need to know this activation energy. Let us call it 
A£j, which it would be reasonable to expect to be about half of A£. The 
Boltzmann factor is exp( — AE } /kT). 

The lattice is vibrating at about the Einstein frequency v £ , and one 
would expect an atom to have a maximum chance of jumping once per 
cycle. So the number of jumps per second is 

v e -A£/fcT Q -AEj/kT 

In a time t it makes 

y e -(A£ + A£j)/kT t 

jumps. Each one moves the atom a distance a . But the path is a random 



292 Defects in solids : Liquids as disordered solids Chap. 9 

walk and the net distance moved in s jumps is a ^/s. Hence 

distance moved = a (v E e- (A£+A£j)/fcT . t) 1 ' 2 . (9.9) 

Now when we discussed the connection between the random walk problem 
and diffusion, in sections 6.2 and 6.3, we showed that the mean distance 
travelled in a medium of diffusion coefficient D in time t is, neglecting 
numerical constants, (Dt) 1 ' 2 . We can therefore say that the coefficient 
of self-diffusion in a solid is given by 

Dxa 2 v E e- {AE+AE ' ),kT . (9.10) 

Our theory therefore predicts that diffusion should become more rapid 
as the temperature is raised. This is in complete contrast to diffusion in 
gases, which becomes more rapid when the temperature is lowered. We 
will see that this prediction agrees with observation and that the form of 
the variation with temperature is correct — but quantitatively, agreement 
is not good. 

9.6.1 Comparison with experiment 

In section 6.2 we described two solutions of the diffusion equation which 
are the bases of all measurements of coefficients of self-diffusion. One was 
used in a set of experiments to measure D for copper. The idea was to 
follow the diffusion of the radioactive isotope 64 Cu through a single crystal 
of 63 Cu. The isotope was converted into copper nitrate and then electro- 
deposited on to a flat surface of the single crystal as a layer about 50 A 
thick. 64 Cu, the only available radioactive isotope, has a half-life of only 
13 hours. This meant that the temperature had to be raised so that it 
diffused a measurable distance in a time not too long compared with 13 
hours. The specimen was therefore kept at as high a temperature as 
possible, in the region of 1,000°K, not very far below the melting point, 
where D is of the order of 10" J1 cm 2 sec" l so that in 1 day (10 5 s), the 
mean depth of diffusion y/(Dt) was about 10" 3 cm. To find out how far 
the isotope had actually travelled, the specimen was put in a precision 
grinding machine and extremely thin layers ground off the surface. The 
accuracy of this operation was about 10 ~ 4 cm. The amount of radioactive 
tracer in the metal taken from each layer was determined by counting the 
beta-ray activity. Knowing the depth of the layer below the original 
surface and allowing for the decay of radioactivity with time, D could be 
calculated. A plot of log D against 1/T is a straight line, as predicted, 
Fig. 9.15. 

Measurements were also made on solid argon by following the progress 
of the stable isotope 36 A through large crystals of 40 A. The isotope was 
introduced into the vapour — at the temperature chosen, the vapour 



9.6 Diffusion in solids 



293 



pressure is large— and some atoms entered the solid and diffused inwards. 
Thus the concentration of 36 A in the vapour decreased as time went on, 
and this was followed by continuously analyzing the vapour with a mass 
spectrometer. In a typical run, the vapour pressure was several centimetres, 
the initial concentration of 36 A was 12% and it decreased to about half 
in an hour. 



10 8 



10" 



10' 



10 



10" 



1300 1200 



1100 



1000 



900 °K 




0.0008 



0.0009 



0001 



00011 



VT 



Fig. 9.15. Plot of log D against 1/Tfor copper. The temperature is marked 
along the top. Data from Kuper and others, Phys. Rev. 96, 1224 (1954). 



Diffusion can take place rapidly along grain boundaries, since these are 
liable to be regions with large numbers of defects. Diffusion through 
polycrystalline specimens is therefore likely to be spuriously large, and 
in all these experiments the specimens were examined to make sure that 
they were single crystals. 

For metals, the experimental result is indeed of the form 

D = D Q- AolkT (9.11) 



294 Defects in solids : Liquids as disordered solids Chap. 9 

where the constant D and the activation energy A are typically of the 
orders of magnitude l-10cm' 2 /sec and 1 eV respectively. (Copper, for 
example, from Fig. 9.15 has D = 0.5 cm 2 /s, A — 2.0 eV). These magni- 
tudes give D ~ e" 40 = 10 -17 cm 2 /s at room temperature, which means 
diffusion through a mean distance of 1 mm in 10 15 seconds, 30 million 
years, at room temperature. 

For comparison, our expression gives a^v^ for the constant, and we 
can take a , the lattice spacing, to be 2.5 A and v E , the Einstein frequency, 
to be of the order of 10 13 cycles per second. Hence we would predict 
D ~ 6 x 10 ~ 3 cm 2 /sec, which is 100 times too small. For the activation 
energy, we would expect about 3e or 0.4 eV, which is a factor 2 or 3 too 
small. (This has a catastrophic effect on the value of the exponential term.) 

The experiments on argon reveal even worse discrepancies. They give 
D ~ 4 cm 2 sec" l , A ~ 0.11 eV. For an atomic spacing of 2 A and an 
Einstein frequency of 10 12 cycles per second we would expect D to be a 
factor 10 4 smaller and an activation energy 3 times smaller than that 
observed. It is however possible to advance plausible reasons for these 
discrepancies. The activation energy can be expected to decrease if the 
mean distance between atoms is increased ; this happens as the temperature 
is raised. We ought therefore to get somewhat better agreement with 
observation if we compared our predictions with measurements at 
constant density — measurements, that is, at high pressures which counter- 
act the thermal expansion. We return to this point in section 9.7.3. For 
solids, this would require the application of extremely high pressures 
and no measurements have been made. But in the absence of suitable 
measurements we can postulate some sort of decrease of the activation 
energy with temperature. A decrease by a factor 2 by the time the melting 
point is reached would indeed greatly reduce the discrepancies by orders 
of magnitude. However, we will not pursue this but will be content that 
the form of the expression for the diffusion coefficient is correct. 

9.6.2 Diffusion and electrical conduction in ionic crystals 

A most interesting related phenomenon is that ionic crystals, usually 
considered to be insulators, can conduct electricity because of the diffusion 
of ions under the influence of an electric field. (In metals the same phe- 
nomenon occurs but it contributes a negligible current compared with 
that carried by the electrons : in ionic substances the movement of the ions 
is the only possible process.) 

The potential energy of a vacancy or an interstitial ion as it moves 
through the lattice is shown in Fig. 9. 16(a). The situation is analogous to that 
in Fig. 9.14 except that we expect the changes of energy to be higher than 
those in a lattice of neutral atoms because of the electrostatic attractions 



9.6 Diffusion in solids 295 

and repulsions. When an electric field acts across the crystal, the energy 
of the ion is changed. If a voltage V acts between the faces of a slab of 
thickness /, the electric field is V// and the potential energy of a charge e 
coulombs changes by V ea /l when it moves a distance a . If V// is expressed 
in volts/metre and a is expressed in metres then the energy is measured 
in joules. The potential energy of an ion or a vacancy as a function of 
position is then shown in Fig. 9.16(6). 

i i AVeaJ ,.. „ 



Fig. 9.16. (a) Potential energy of a vacancy or interstitial ion at different 
positions in a lattice, (b) The same in the presence of a potential gradient. 

If the vacancy is in a potential well, the height of the barrier it must 
overcome in order to jump 'downhill' in the direction of the field is 
(AE j — ^Vea /l) where a is the interatomic spacing. To j ump in the opposite 
direction against the field, the activation energy is (AEj+^Veao/l). 
Therefore the probabilities of jumping in the two directions are respectively 
proportional to 

exp -(AE } -^Vea /l)kT and exp -(AE } +^\ea /l)/kT 

which makes it more probable that the ion will move in the direction 
of the field than against it. In other words, an electric current i flows, pro- 
portional to the difference between the two probabilities : 

i oc exp(-AE } /kT)[exp$Vea /lkT)-exp(-±Yea /lkT)]. (9.12) 

Now \ea /l is very small compared with kT under normal conditions. 
For example, if \/l is large, a kilovolt per centimetre, then putting 
e~l(T 19 C, a ~2A, Yea /l~ 10" 24 J, whereas /cT~lCr 21 J at 
room temperature. Since e* is equal to (1 +x) when x is small, the current 

ix -j^f e ~ AEjlkT - (9-13) 

To find a complete expression for the current, we should find the total 
number of ions (or vacancies) jumping in the direction of the field per 
second, and then multiply by the charge on the ion. The important results 
can however be seen at once : that the current is proportional to the voltage 



296 



Defects in solids : Liquids as disordered solids Chap. 9 



gradient (Ohm's law holds) and that the conductivity (which is the ratio 
of the current to the voltage) contains the same Boltzmann factor as the 
diffusion coefficient. 

It should be remarked that both positive and negative interstitial ions 
and also positive and negative vacancies can contribute to the conductivity, 
but it is to be expected that the mechanism with the smallest activation 
energy is likely to dominate the conductivity. 

We can therefore find the activation energy for diffusion of ions by 
two quite independent methods. We can use tracer methods to determine 
the diffusion coefficient, or we can find the electrical conductivity as a 
function of temperature. A plot of logD against 1/T and a plot of 
(log cr + log T) against 1/Tshould be straight lines with the same gradient. 
In practice, the conductivity of many ionic solids changes by a factor 10 6 
when the temperature changes by a factor 2 ; in other words, if we use logs 
to the base 10, log a changes by 6 while log Tonly changes by 0.3. Thus it is 
accurate enough to plot log a (by itself) against 1/T over a restricted tem- 
perature range and to take the mean gradient of that graph as the activa- 
tion energy for electrical conduction. 

Curves for NaCl are shown in Fig. 9.17. The resistivity is of 
the order of 10 3 ohm cm near the melting point, 10 6 ohm cm at 800° K. 
To measure high resistivities of this kind accurately, a high voltage is 
applied as a pulse and the current is measured as quickly as possible. 

In the diagram, only high temperature observations have been plotted, 
above about 800° K (1/Tless than about 0.0013). Below that temperature 
the behaviour changes and both the conductivity and the diffusion 
coefficient become dependent on the grain size and the presence of 



~ E 10° 
o 

**■ 

o -v 

-MO 



M.P 




^10 3 



itf 



10" 



'10° 



0001 0.0011 0.0012 0.0013 1/7" 



MP 



0.001 ooon 0.0012 0.0013 vr 



1000 900 800 °K 



1000 900 800 °K 



Fig. 9.17. Diffusion coefficient and electrical conductivity of NaCl as functions 

of temperature, (log 10 -l/T plots). Data from Mapother, Crooks and Maurer, 

J. Chem. Phys. 18, 1231 (1950). 



9.7 Diffusion in liquids 297 

impurities. Further, it is thought that positive and negative vacancies 
can migrate together, acting as an electrically neutral pair whose move- 
ment contributes to the diffusion process but not to the conductivity. 
Concentrating, then, on the high temperature region, we see that the lines 
are almost parallel. The activation energy is nearly 2 eV per ion, and we 
get a consistent picture of both the conduction and the diffusion if we 
assume that the small Na + ions are the most mobile and dominate both 
processes. 

* 9.7 DIFFUSION IN LIQUIDS 

We will now apply this theory, developed to account for transport 
through disordered solids, to liquids. This follows the plan outlined in 
section 7.2, Fig. 7.6. 

If this approach is correct, we would expect the coefficient of self- 
diffusion through liquids to be of the same form as that for solids, 
D = D Qxp(-A /kT). Of course, whenever we find it convenient we can 
regard liquids as derived from dense gases. This approach would lead us 
to expect D to be proportional to T, which is quite different behaviour. 
It is in fact reasonable to expect that when the density of the liquid is 
low, some features of its behaviour might be 'gas-like', but at high densities 
some features might be 'solid-like'. In the meantime, let us see whether 
the solid-like model of a liquid ever applies at all. 

We would expect the activation energy AE } for an atom to jump into a 
vacancy in a liquid to be much smaller than in the corresponding solid, 
because there is a 10 % expansion in volume between liquid and solid ; 
small changes of interatomic distance have a large effect on the activation 
energy. We would also expect AE, the energy of creation of a vacancy, 
to be smaller than in the solid. 

The diffusion coefficient of liquid argon (radioactive 37 A through 40 A) 
has been measured using yet another procedure, appropriate to liquids, 
based on section 6.2. A narrow capillary was made, closed at both ends 
by needle valves. The top one was opened and pure liquid 40 A was con- 
densed in; the valve was closed again. Then liquid argon containing a 
small percentage of 37 A was condensed outside. The bottom valve was 
opened for a known time, during which some 37 A diffused into the capil- 
lary. Convection, which might have disturbed the results grossly, was 
discouraged because the capillary was very narrow. Afterwards, the 
liquid in it was pumped away and the total amount of 37 A determined 
from its radioactivity. Knowing the time and the concentration in the 
surrounding liquid, D could be calculated. The measurements give 

D = 6.1 x 1(T 4 exp(-0.027//cT) 



298 



Defects in solids : Liquids as disordered solids Chap. 9 



where (kT) is to be measured in electron volts. The activation energy is 
indeed about 5 times smaller than in the solid and the form of the 
expression justifies our regarding a liquid as resembling a highly dis- 
ordered solid in some of its properties. We will now construct a theory of 
the viscosity of liquids, and we will do this by showing that there is a 
connection between the viscosity n and the diffusion coefficient D. Since 
we already know D, this relation allows us to estimate n. 

* 9.7.1 Stokes' law 

As a preliminary study, interesting for its own sake, we will estimate the 
force on a sphere of radius r which moves at constant velocity through a 
liquid of viscosity n. We can concentrate on a few little specks of liquid and 
follow their displacements as they go past the sphere. These define stream 
lines and they represent the way the liquid flows past the sphere, or the 
way the sphere pushes the liquid apart in order to pass through it. Fig. 9.18 
shows some of these. It implies that the flow pattern is symmetrical and 
that at some distance away from the sphere the liquid is hardly disturbed. 
This distance is called the penetration depth. We will make the assumption 
that the penetration depth is the same order as the radius of the sphere, 
which is reasonable and is what the diagram implies. We saw in section 
6.1.5 that the pressure on a moving object is given by (Eq. (6.7)) 

pressure = rj x velocity gradient. 
Here, the velocity gradient is of the order of magnitude v/r. 




Fig. 9.18. Stream lines round a sphere moving 
through a liquid. 



9.7 Diffusion in liquids 299 

Also implied in Fig. 9.18 is that the flow is non-turbulent. In the language 
of section 6.1.5, the kinetic energy of the mass in motion— the inertial 
energy— must be small compared with the energy dissipated against 
viscosity. This means we have to compare terms of the type \pv 2 (kinetic 
energy per unit volume, p being the density and v the velocity of the fluid 
in motion) with the viscous pressure nv/r (which is also an energy per unit 
volume). Thus pv 2 must be less than nv/r to make sure that the flow is not 
turbulent ; in other words, v must be less than rjr/p. For a sphere of radius 
10 cm moving through water for which n = 10~ 2 gm/cms, v should not 
exceed 0.1 cm/s; a spherical particle of radius 10" 3 cm can travel at 10 
cm/s and, extrapolating, a sphere of atomic dimensions can travel at 
10 6 cm/s and the flow will still be non-turbulent.* With these assumptions, 
the pressure on the sphere is nv/r. The surface area of the sphere is Anr 2 . 
Therefore the force on the sphere should be roughly Annrv. 

An exact analysis with proper regard to the flow pattern shows that 

force = 6nnrv (9.14) 

This is called Stokes' law. 

9.7.2 Einstein's relation between viscosity and diffusion 

We establish the relation between viscosity and diffusion by studying 
the Brownian motion of an assembly of particles. For clarity, we will 
consider the special situation of the fluid in the field of the earth's gravity, 
although it is not in fact necessary to restrict the calculation in this way. 
Further, for the sake of illustration we will first consider a gas and having 
shown that the results agree with previous calculations we will extend the 
method to liquids. We begin, then, by considering a gas in a gravitational 
field, as we did in section 4.4. There, we established the fact that the 
concentration n at a height z is given by 

n = noQ -m^T ( 413) 

where n is the concentration at zero height and m is the mass of one 
molecule. The new idea is that we will regard the equilibrium as being 
produced by two opposing motions. First, we have the tendency of each 
atom to fall downwards. Secondly, opposing this, we have the tendency 
of each atom to diffuse along the concentration gradient (which means a 
tendency to move upwards, because n decreases with height). We express 
this balance as follows. Let us take z to be positive upwards, and con- 
sistent with this let us take the drift velocity v z of the atom to be positive 

* The ratio vrp/rj where v is the limiting velocity for the onset of turbulence is called the 
Reynolds number. It is about 1 for a sphere. It is much smaller for a cylinder because the 
penetration depth is much greater. 



300 Defects in solids : Liquids as disordered solids Chap. 9 

upwards ; we will also take all fluxes of atoms (numbers crossing 1 cm 2 
in 1 second) as positive upwards. Then the net flux is given by 

J = -nv z -D—. (9.15) 

The first term on the right hand side is the number of atoms per unit 
area falling downwards under gravity — they are accelerating of course, 
but are brought to rest at intervals, and v z is their mean velocity. The 
second term is the rate of diffusion down the concentration gradient. 

For equilibrium, J = 0. 

Let us next evaluate v z , the mean drift velocity. If X is the mean free path 
between collisions, the mean free time between collisions is (X/c) av which 
is nearly equal to X/c, where c is the mean speed. In this time, every atom 
acquires an extra downward velocity equal to (acceleration) x (time), that 
is gX/c. Let us assume that after every collision, each atom is stationary. 
This is rarely likely to be true, because of the persistence of velocities (sec- 
tion 6.4.3) but it only introduces an error by a factor of order unity. With 
this assumption, the mean velocity downwards is half the terminal velocity, 
that is gX/2c. Further, let us put in the expression for n. This allows us 
to evaluate the nv z term completely. Finally, by differentiating we evaluate 
the dn/dz term. We find 



0= -~n e- mgz/kT + D^-n e- mgzlkT 
2c kT 



which gives 



D = 2m-c- < 9 " 16 > 



If we put kT = \mc 2 (not paying too much attention to the distinction 
between mean and r.m.s. speeds), we get 

D = ±Xc. 

The more exact expression arrived at in Appendix B is ^Xc. This is practi- 
cally our result and the discrepancy is due to our crude averaging. This 
digression has therefore justified the idea of regarding the equilibrium of 
the gas as a balance between a drift velocity in one direction and a diffusion 
in the opposite direction down a concentration gradient. 

Let us now return to the real subject of this section and use the same 
method for particles suspended in a liquid, undergoing Brownian motion — 
resin particles in water for example. We have already studied this system in 



9.7 Diffusion in liquids 301 

section 4.4.2, when we concluded that the concentration is given by 
exactly the same law : 

n = n exp(-m*gz/kT) (4.17) 

with m* the effective mass, corrected for the buoyancy of the surrounding 

liquid. 

Again in equilibrium we have a balance between drift downwards and 

diffusion upwards : 

= _„„ X -D^. (9.17) 

dz 

Let us assume here that the force on the particle is given by Stokes' law. 
Then the particle (of radius r) reaches a terminal velocity when the viscous 
force (6nnrv z ) is equal to the weight (m*g) : that is, 

m*g 



6nnr 
Then our equation reads 



= -^n e-^ T + D^n e-'"^ 
onrjr k 1 



which gives 



kT 
D = — . (9-18) 

dnnr 

This is the diffusion coefficient of the particles of radius r through a 
medium of viscosity n at temperature T. Finally, let us assume that the same 
approach is valid when the particle is an atom of the liquid itself. As we 
have presented it, this is a rather controversial step to take. There are two 
points which we have to consider. First, are we justified in applying Stokes' 
law for the force on the moving atom? In deriving this law we clearly 
regarded the sphere as immersed in a continuum, a medium without any 
atomic structure, in which it generated stream lines. This hardly seems 
justified if the sphere itself is of atomic size. But the mean free time between 
collisions is certainly very short— in contrast to the situation in a gas— so 
that a continuum is not too bad an approximation. Further, we have seen 
that Stokes' law implies that the penetration depth of the disturbance 
caused by the passage of the sphere is between 1 and 2 atomic radii : this 
is likely to be roughly correct, so that perhaps Stokes' law can be applied. 
Secondly, are we justified in talking of the effective mass m* of an atom 
inside a liquid of identical atoms? An incompressible liquid in a gravita- 
tional field has the same density at all heights (that is, the scale height is 



302 Defects in solids : Liquids as disordered solids Chap. 9 

infinite) and this is equivalent to saying that m* is zero. Admittedly, m* 
does not appear in the final relation, but it is always dangerous to cancel 
both sides of an equation by a quantity equal to zero. Perhaps the best 
justification of this step is to say that real liquids are compressible so that 
there is a small change of density with height and m*, though very small, 
is not zero. 

We say then that the relation 

kT 
6nrjr ^ ' ' 

where r is the radius of an atom, relates the diffusion coefficient and the 
viscosity of a liquid. This is called Einstein's relation between the two 
quantities. 

^k 9.7.3 Viscosity of liquids 

Using the expression for the variation of diffusion coefficient with 
temperature 

D = D e~ Ao/kT (9.11) 

where A is the activation energy per atom, Einstein's relation gives 

kT 

We can test this relation by comparing with experiment. It predicts 
that the viscosity decreases as the temperature is raised, in contrast 
to the behaviour of gases. 

Among the most interesting measurements of the viscosity of liquid 
argon are those shown in Fig. 9.19. They were performed in a very simple 
way. The liquid was contained in a metal tube, through which a cylindrical 
weight could fall very slowly, the clearance between tube and cylinder 
being only small. The cylinder had a magnetic core inside it and its 
position could be detected magnetically. The time required to fall a known 
distance was determined. This was calibrated by timing the fall of the 
cylinder using liquids of known viscosity. The apparatus was robust and, 
as we shall see, was used up to very high pressures. 

On the diagram, the two vertical dashed lines represent the triple point 
(83° K) and the critical temperature (150°K). The region between them 
represents the range of temperature in which the liquid can exist. 

The full curve AB represents the viscosity of argon in equilibrium with 
its vapour pressure. At A, the liquid is at a low pressure, has a density 
of 1.37 g/cm 3 , at B it is under 48 atmospheres and its density is 0.70 g/cm 3 . 



9.7 Diffusion in liquids 



303 




100 



200 300 °K 

Temperature 



Fig. 9.19. Viscosity of liquid argon as a function of temperature. Line AB: 
measurements on the liquid under its saturated vapour pressure, from the 
triple point to the critical point. Dashed line: calculated from D using 
Einstein's relation, Eq. (9.18). Line AC: measurements at constant density, 
1.37 g/cm 3 — the part to the right of the critical point refers to the gas. The 
descending curve shows liquid-like behaviour. Line BD; measurements on 
gas at constant density, p = 0.70 g/cm 3 . Data from Zhdanova, Soviet Phys. 
J.E.T.P. 4, 749(1957). 

The dashed curve close to it is calculated using the experimental values 
of D (section 9.7) and Einstein's relation, assuming an atomic radius of 
2 A (or, what comes to the same thing, a rather smaller radius together 
with a larger number than 671 in the expression for the force on a moving 
atom). The agreement is good, and it shows that Einstein's relation is 
valid within a factor close to unity. The upper curve AC represents 
measurements at constant density equal to 1.37 g/cm 3 (the density at the 
triple point temperature). For each measurement, the temperature was 
first raised to a chosen value and then the pressure was increased until 
the density reached the value 1.37 g/cm 3 ; the viscosity was then deter- 
mined. Thus for all points along this curve, the mean distance between 



304 Defects in solids : Liquids as disordered solids Chap. 9 

atoms was the same. This is the kind of experiment, mentioned in section 
9.6.1, which might be expected to give results more consistent with 
theoretical estimates. The activation energy calculated from this curve* 
is a factor 2 or smaller than that from the curve AB where the density 
varies — this is certainly a change in the right direction. The measure- 
ments at this density, 1.37 g/cm 3 were continued above the critical 
temperature ; thus the part of the curve AC to the right of the dashed line 
at 150°K represents measurements on the gas. The pressure needed at 
250°K to keep the density at 1.37 g/cm 3 is about 2,000 atmospheres. 
There is no discontinuity in viscosity when the liquid changes to the very 
dense gas — which means that we have a gas in which the transport of 
momentum proceeds by the same mechanism as in a liquid. This must 
mean that in these circumstances the atoms progress by a series of short 
jumps into holes in the assembly, and not by a series of long free flights 
terminated by collisions. 

By contrast, the line BD is the viscosity of the gas held at the lower 
constant density of 0.70 g/cm 3 , the density at the critical point. The 
viscosity now increases with temperature, which we might call gas-like 
behaviour although y\ is not proportional to ^/T as in a well-behaved gas. 
This must mean that the atoms now spend a considerable fraction of their 
time in free flight between encounters. 

Measurements were also taken of the viscosity of the liquid at small 
densities, just above 0.70 g/cm 3 . The curves, above BD and almost parallel 
to it, have not been shown. They also tend upwards which means that in 
the liquid near the critical point there are sufficient large holes in the 
structure for the motion of the molecules to be gas-like. 

PROBLEMS 

9.1. (a) Estimate the extra energy per unit length (in erg/cm) of an edge dislocation 
in a metal due to the low coordination and increased interionic spacing 
of some of the ions. 

(6) A specimen of metal has 10 9 dislocations per cm 2 of cross section. How 
much energy is stored in the dislocations per cm 3 ? Will this extra energy 
appear as a reproducible contribution to the specific heat, like that due to 
vacancies — in other words, is the number of dislocations a unique, function 
of the temperature? If not, why not? 

(c) Work hardening is due to the presence of large numbers of dislocations 
tangled together and repelling one another, so that large stresses must be 
applied before they can move. The extra stress (dyn/cm 2 ) must be of the same 
order as the energy density (erg/cm 3 ) due to the dislocations. Estimate the 
concentration of dislocations at which significant work hardening takes 
place in a typical metal at a strain of 1 %. 

* Determined by plotting log(>j/r) against l/T. 



Problems 305 

9.2. X-ray examination of a small region of a crystal of cubic symmetry and lattice 
parameter 4 A shows that it consists of two slightly misaligned crystals. The 
angle between the planes is 4o°- What pattern of etch pits would be expected 
along the grain boundary? See Fig. 9.7(d). 

9.3. The mobility q of a charged particle (such as a vacancy or an interstitial ion in 
an ionic solid) is defined by the equation 

mean drift velocity 
electric force Ee acting on charge 

where E is the electric field and e is the charge on the particle. The problem is to 
show that there is a relation between q and the diffusion coefficient D, using a 
method analogous to that in section 9.7.2. 

(a) Consider a steady electric field E acting across a slab of material, considered 
to be a continuum without atomic structure. A positive charged particle 
inside the slab is attracted to the negative face and vice versa. What is its 
potential energy at a distance h from the face compared with the value at the 
face? In equilibrium, what is the probability of finding the particle between 
h and (h + dh)l Show that the charge is distributed exponentially with h, 
with a scale length equal to kT/Ee. Estimate this length at room temperature 
in a field of 1 volt/cm for e equal in magnitude to the electronic charge, 
remembering that kT at room temperature is 1/40 eV. 

(b) Does this result conflict with the predictions of elementary electrostatic 
theory about charge distributions? 

(c) Set up a differential equation expressing the equilibrium as a balance 
between diffusion down the concentration gradient and a steady drift down 
the electric field. Apply this equation to the system discussed in (a). Deduce 
Einstein's relation : q = D/kT. 



306 Solutions to problems 

SOLUTIONS TO PROBLEMS 



Chapter 3 



3.1 (a) 

3.2 (a) 



GMm 

2 HqM 2 

3 h* 



(b) 



GMm 



(d) See Fig. P.2 (e) 



(c) See Fig. P. 1 (d) 

2fi M 2 \ 1/4 



GM 



mg 



CO 2tt 



,5/4 



3.3 (d) See Fig. P.3. 



2 5/4 /^ M 2 \ 1/4 _ / 2 /^ 



fc 



a{a+h) 




Fig. P.l 



Fig. P.2 



Fig. P.3 



(*) 



surface tension energy 47t 2 y 



, where y ~ 10 dyn/cm, p ~ 1.4 g/cm 3 , 



gravitational energy A z pg 
g ~ 10 3 cm/s 2 ; 210 c/s ; 21 cm/s. 

3.4 28 ; 0.81 g/cm 3 ; 34.5 cm 3 ; ~ 3.9 A ; 8.8 dyn/cm ; 0.0034 eV 

3.5 (a) 28.5 cm 3 , 3.55 A (b) 3.95 A (c) 0.0134 eV (note : n = 12) 
{d) 0.01 14 eV (e) 0.0124 eV 

(/) 2.5 x 10 10 dyn/cm 2 , compared with ~2.2x 10 10 erg/cm 3 or 1.7xl0 10 erg/ 

cm 3 according to value of L . 
(g) 83 or 66 (h) ~l.lxl0 12 c/s 

3.6 Assume close-packed discs in planes, n = 6, e = £/30 within planes ; n = 2 , 

e — between planes. 

(a) L = l.lNEpermol 

{b) If discs parallel to surface, JT = {l/ljlr 2 ) (see Fig. P.4; each triangle con- 
tains \ disc and has area V3r 2 ) and y ~ 0.1£/r 2 . If perpendicular to surface, 
jV = l/(2r x r/10), and y = 0.25 E/r z . Surface tension somewhere between 
these two. 




Fig. P.4 

(c) Tendency for parallel orientation because lower surface energy. 
3.7 a = 2.32 A N = 5.8 x 10 23 assuming r" 10 repulsions. 



Solutions to problems 



307 



3.8 Binding energy = 



Nae 2 






0.37 A 



3.9 (a) 4.55 eV 

(b) If solvents were structureless media, binding energy would be reduced by 
factor 80, 25, 18 respectively. But to screen one ion from another, solvent 
molecules must crowd round ions and size is important. 

3.10 (a) 2.404 (b) 1.803 

3.11 12A;9A;3.3A;7A 

3.12 K = 22.1 xlO 4 kg/cm 2 . 



Comparison with 



1 (M 14/3 
10 I V\ 



>V 4/3 
v 



shown in Fig. P. 5. Theory should be quite accurate. 




PfK 



Fig. P. 5 



3.13 _L[6a y/m] 1/2 gives 3.4 x 10 13 , 0.86 x 10 13 and 1.56 x 10 12 c/s. 

2n 



Chapter 4 

4.1 Effective mass 3.8 x 10" 14 g, scale height 10.0 fi. N = 6.47 x 10 23 . 
Inverted concentration gradient with same scale height. 



Chapter 5 

5.1 (a) 0.0022 A 

(b) 0.0104 A if other causes of broadening are independent of temperature. 

, x GMm GMm „, / GMm \ 

5.2 (a) --,-; -— (6)exp +w 



308 Solutions to problems 

(c) A4nr 2 exp(GMm/rkT) dr, where A is a normalizing factor. 

(d) exp factor -*■ 1, so that whole expression -» oo when r -*■ oo. 

(e) (ii) ; (c) cannot be normalized. 

5.3 Centre of gravity rises by R/Mg for 1° increase of temperature so that gravi- 
tational potential energy increases by R. 

5.4 31 % assuming that at 1 atm, both N 2 4 and N0 2 act like perfect gases. 

5.5 11.2 km/sec; 1.37 km/sec. 

High-speed 'tail' of Maxwell distribution allows more He atoms than 2 or N 2 
atoms to escape ; see problem 2 above, part (e). 

5.6 (a) (u x — 2£) ; \mu 2 x - ImuJ, 

(b) nAu x di • P[u x ] du x ; nAu x dx • P[u x ] du x ( - ImuJ,) ; 
-2nmdV J£ u 2 x P[u x ] du x = nkTdV 

(c) iknV;dT= -dVkTftknV 

(d) TV 213 = const., becomes PV 513 = const, since PV = RT 

(e) dT= -dVkT/C v V where C v is specific heat per molecule. 
(/) See discussion in section 5.1.2. 

5.7 (0 A sin 6 dd d</> exp(-((iH/kT) cos 0), A a normalizing factor. 
(ii) Integral over gives lit. Integral over 6 gives 

2sinhx , fiH x I 
where x — -r=, whence A = 



x 



kT' 4n sinhx 



(Hi) — r-: — exp( - x cos 6) ■ sin 9 dd 
2 sinhx 

x 

(iv) Nu cos 6 ^— — — exp( - x cos 6) d( - cos 6) 
2 sinhx 

(v) Integral over 6 gives iv>[coth x- 1/x] 

5.8 (a) 5 x 10 12 c/s. (b) 1.89 x 10~ 3 <5 2 J, where 8 in A 

(c) p[S] dd = 0.304 exp(- 34.2<5 2 ) d^ where 8 in A 

(d) 0.12 A 

5.9 (a) 2 (b) kT (c) 2.9 x 10" 4 rad (d) (v) (e) (Hi) 

(f) Bombardment by photons of thermal radiation. 
5.10(a) 6 x KT 39 gem 2 (b) \kT (c) 3 x 10 12 c/s 

(d) 6 vibrational + 3 rotational degrees of freedom ; f R 

(e) 3 translational + 3 rotational degrees of freedom ; f R 

(/)f 

(g) Rotations (c) take place, rotation about dumbell axis does not ; 5 degrees of 

freedom ; y near ^. 

5.11 1.4 

5.12 0.3 eV (b) 



Chapter 6 

6.1 (a) C v - 20.8 J/mol deg, a ~ 35 x 10" 16 cm 2 , c ~ 4.5 x 10 4 cm/sec ; 

k; ~ 1 x 10 ~ 4 watt/cm deg 
(b) 0.175 watt (c) 2xKT 3 m 
(d) k = %n(CJN)cd where d is spacing ; 2 x 10 ~ 4 m 

6.2 (a) -(DA/h)mn v 



Solutions to problems 



309 



(/,) _ Pl a dh/dt g/s where A is area of tube and p L is density of liquid 



(c) h 2 = 



2Dn„m 



Pl 



t 



(d) 0.11 cm 2 /s. Assume collisions with air molecules are much more frequent 

than with other ether molecules; then X = l/n A a E , where subscript A means 

air, E ether; D EA = %Xc E ; whence a E = 10" 15 cm 2 . 

6.3 Assume only collisions with air molecules are important ; X = l/n A <r where n A 

= number of air molecules/cm 3 . Take a = 10 x 10" 16 cm 2 . For small thermionic 



tube, dimension ~ 1 cm, volume 
10 3 cm 3 



1 cm 3 ; for large transmitting tube, dimen- 
Then P ~ 0.02 mm, 0.002 mm and area of 



sion ~ 10 cm. volume 
hole ~ 10" n cm 2 , 10" 12 cm 2 respectively (Knudsen flow). 
6.4 Maxwell distribution inside vessel. All molecules within cone of height c and 
solid angle determined by shape and size of hole will leave vessel in 1 sec; 
whence factors c and G respectively. 

G[*\mc 2 cP[c]dc 
Mean kinetic energy = ° G ^ cP[c]dc • 

Denominator is equivalent to inc. 

Expression GcP[c] dc for number escaping per sec shows that fast molecules 

predominate and distribution is non-Maxwellian. 



Chapter 7 

7.1 (a) T i = 2 iT c ;220°K 

(c) T B = $Ti = 110°K, so that at room temperature PV > RT. 

7.2 V. = 2b ■ T. = ^; P. = ^ ~ 3^; BiT) = (fc-^LJ ; % = ^ 

7.3 (a) 200 cm 3 ; 4.4 A 

(b) B plotted against 1/T; B = 280 -(2.18 x 10 5 )/Tis a rough fit— see Fig. P.6. ; 
a ~ 1.83 x 10 6 J cm 3 /mol ; using a/b ~ 2.7Ne, e ~ 0.025 eV. 




Fig. P.6 



310 Solutions of problems 

(c) 0.036 eV 

W "- = (^-4^]^^= — -l-^MxlO-Co, 

(f) 1.6xlO" 20 Cm; +5 electron charge 2 A apart, or larger charges nearer 
together. 

7.5 (a) zero (b) r.m.s. distance = ^/N/ (c) D = ?(/ 2 /t) 

7.6 (a) x = (b) — (c) -* 

a V a z 



Chapter 8 

8.1 6; 4.82 A; 16.1°, 33.6°, 56.1° 

8.2 Bragg's law gives dd/d — — d0/tan0 for small changes of d and 0; a = 10~ 5 . 
Expansion coefficient varies with direction. 

8.3 (b) 16xl0 _6 ,forp= 11 

(d) y = 0.85 at 30° K, 1.28 at 65°K, 1.45 at 283°K compared with 1.67 for p = 1 1. 



Chapter 9 

9.1 (a) ~ 10" 4 erg/cm 

(6) 10 5 erg/cm 3 . No, number depends on mechanical and thermal treatment of 

specimen, 
(c) Elastic energy = (^/2)s 2 ~ 10 8 erg/cm 3 ; 10 12 dislocations/cm 2 

9.2 DD 1 (Fig. 9.7(d)) is given by a/DD 1 = sin where 6 is angle between grains, 

a is lattice parameter. Line of etch pits 4460 A apart. 

9.3 (a) Veh/l ; Fe//feTexp( - Veh/lkT) dh, if / > h ; ^ cm. 

(6) Yes. Elementary electrostatics neglects the diffusion of electrons, because 

electric charge is treated as a continuous fluid. 
(c) Ddn/dh + qn(Ve/t) — 0, where n = no. of charged particles/cm 3 ; this gives 

n oc exp[ — q(Veh/lD)]. 



Reading List 



The following list contains a few suggestions for complementary and further 
reading. 

GENERAL 

R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, 

Vols. 1 and 2, Addison-Wesley, Reading, Mass., 1963. 
T. L. Hill, Lectures on Matter and Equilibrium, Benjamin, New York, 1966. 
F. Mandl, Statistical Physics (Manchester Physics Series), to be published by John 

Wiley, London. 
F. Reif, Statistical Physics (Berkeley Physics Course, Vol. 5), McGraw-Hill, New 

York, 1967. (Mainly relevant to chapters 4-9 of this book.) 
F. O. Rice and E. Teller, The Structure of Matter, Science Editions, New York, 1966. 

D. Tabor, Gases, Liquids and Solids, Penguin Books, Harmondsworth, Middlesex, 
1969. 

M. W. Zemansky, Heat and Thermodynamics, 5th ed., McGraw-Hill, 1968. 

CHAPTER 3 

A. H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964. 
C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966. 

CHAPTER 4 

R. D. Present, Kinetic Theory of Gases, McGraw-Hill, New York, 1958. 

E. Schrodinger, Statistical Thermodynamics, Cambridge University Press, Cam- 
bridge, 1964. 

H. D. Young, Statistical Treatment of Experimental Data, McGraw-Hill, New York, 
1962. 



312 Reading list 

CHAPTERS 5, 6 and 7 

R. D. Present, Kinetic Theory of Gases, McGraw-Hill, New York, 1958. 

CHAPTER 8 

B. E. Chalmers, Principles of Solidification, Wiley, New York, 1964. 

C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966. 

F. C. Phillips, An Introduction to Crystallography, 3rd ed., Longmans, London, 1966. 

CHAPTER 9 

F. P. Bowden and D. Tabor, Friction and Lubrication, Methuen, London, 1967. 

B. E. Chalmers, Principles of Solidification, Wiley, New York, 1964. 

A. H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964. 
J. Frenkel, Kinetic Theory of Liquids, Dover, New York, 1955. 

C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966. 

The following films deal with experiments which are described in this book. 
During the course of each film, measurements can be taken from the screen so that 
students can write down their own readings and later work out their own results. 
A booklet is available giving essential numerical and other data about each piece of 
apparatus. 

The series is called 'experiment'. 
No. 5 The Determination of Boltzmann's Constant (Chapter 4) 

No. 3 C p /C v for Helium, Nitrogen and Carbon Dioxide (Chapter 5) 

No. 4 The Effect of Pressure on the Thermal Conductivity 

of a Gas (Chapter 6) 

Nos. 1 and 2 pV Isotherms of Carbon Dioxide (Chapter 7) 

Films (16 mm sound colour, 15 minutes each) can be purchased from Granada 
International Productions Ltd., 36 Golden Square, London Wl, or hired from The 
British Film Institute Distribution Library, 42/43 Lower Marsh, London SE1. 



ndex 



References such as 304(9.1) are to Problems, where information is given which is not in 
the text. 



Activation energy 127, 222, 296 
Adiabatic changes 120 
Adsorption 35 

Amorphous solids 16, 20, 269 
Anharmonicity 239, 253 
Argon (Ar) 

critical constants 180, 210, 238 

density 48 

latent heats 48 

p-V-T diagram 179 

triple point 180, 238 
Argon gas 

Boyle temperature 195 

C p /C v 123, 199 

diffusion 166 

dimers 205 

inversion temperature 203 

Joule-Thomson coefficient 204 

2nd virial coefficient 196 

thermal conductivity 165 

virial plots 194 

viscosity 164, 166, 304 
Argon liquid 

diffusion 297 



Argon liquid — continued 
surface tension 48 
tensile strength 185 
viscosity 302 

Argon solid 

crystal structure 228, 236 
diffusion 292, 294 
elastic moduli 41, 48 
Griineisen constant 244 
melting 289 
single crystals 233, 247 
specific heats 124, 289 
thermal conductivity 247 
thermal expansion 124, 243 

Atomic mass 13 
units 14 

Atomic number 9 

Avogadro's number 14 
early measurements 15, 35 

Binding energy 6, 31 
ionic crystals 55 
metals 50 
molecular crystals 32 



314 



Index 



Boiling 180 

Boltzmann factor 77, 80, 90, 94 

separability 78, 81,97 
Boltzmann 's constant 87 
Boyle temperature 195, 203 
Bragg reflections 234, 252 
Brittle materials 266, 272 
Brownian movement 87, 213, 299 
Bubbles 180 
Bulk modulus 40, 41, 50, 57, 215, 244, 263 

and latent heat 43 

and speed of sound 41, 46, 120 

gases 16 

ionic solids 57 

metals 50 

molecular solids 43, 48 



Capillary rise 1, 35 
Carbon compounds 13 
Carbon dioxide (C0 2 ) 

critical constants 210 

dimers 206 
Carbon tetrachloride (CC1 4 ) 

bulk modulus 44 

Einstein frequency 46 

latent heat 33 

surface tension 35 

tensile strength 185 
Ceramics 269 
Characteristic speeds 109 
Characteristic temperatures i 
Characteristic times 21, 147 
Chemical reactions 13, 127 
Cleavage, crystal 232 
Clusters 187, 207 
Collision 

cross section 155 

dynamics 156 

three body 205 
Compressibility 16, 41 

see bulk modulus 
Coordination number 18 
Copper (Cu) 

diffusion 292, 294 

electrons in 254 

Lorentz ratio 255 

specific heat 123, 255, 257 
Corresponding states 187 
Coulomb forces 10, 53 



Covalent bonds, see electron sharing ; 

see diamond, glass, silver bromide 
Cracks 270 

Critical constants 179, 208 
Critical opalescence 210 
Critical temperature 89, 179, 210, 238 
Crystal 

cleavage 232 

faces 225 

growth 231, 281 

habit 225 
• structure 20, 228 

unit cell 228 

surface energy 231 

zone 227 
Crystals 

single 232, 247, 264, 281 

whiskers 286 

Degree of freedom 116 
Density 

gases 16 

liquids 17 

solids 19 
Diameter of molecules 22, 29, 155, 157, 

190 
Diamond (C) specific heat 123 
Diffusion 136, 138, 145 

and random walk 150 

gases 149, 161, 166, 171, 175 (6.2), 299 

liquids 297, 299 

of heat 245 

solids 290 
Dimers 204 
Dislocation 

edge 277 

lines 275 

screw 277 

width 279 
Doppler 

broadening 101 

shift 102 
Droplets, condensation on 220 
Ductile materials 264, 274 

Effusion 167 

Einstein frequency 46, 47, 50, 57, 245 

and specific heat 123 
Einstein relation 299 
Elastic limit 264 



Index 



315 



Elastic moduli 40 
Electrical conductivity 

ionic crystals 294 

metals 257 
Electron 

charge 9 

cloud 9 

diffraction 233 

'gas' in metals 10, 12, 254, 257, 260 

sharing 10, 11 

volt 7 
Element 9 
Energy 

density 42, 191,220,299 

equipartition 114, 116, 127 

units 6 

internal (£) 118, 197, 243, 258, 288 
Enthalpy 202 

Equilibrium, thermal 74, 159 
Equipartition of energy 114, 116, 127 
Expansion coefficient see thermal ex- 
pansion 
Expansion of gases 

adiabatic 121, 132 (5.6) 

free 200 

Joule-Thomson 201 



Faraday (unit) 15 
Fibre glass 274 
Fick's law 138, 162 
Fluctuations 63, 68, 75, 211 

elastic solid 213 

perfect gas 216 
Force 

Coulomb 10 

gravitational 25 

interatomic 23 

short and long range 24 

and potential energy 26 
Fourier transform 234 
Free expansion of gases 200 
Free path 153 

mean 155, 156, 169,246 
Frequency 

Einstein 46 

harmonic motion 46 

mixing 253 
Friction 266 

internal 142 



Gas constant (R) 87 

diffusion 149, 161, 166, 171, 299 

free expansion 200 

imperfect 187, 188 

Joule-Thomson process 201 

Knudsen 169 

Maxwell distribution 112 

specific heats 118, 123, 197 

speed of sound 112, 120, 199 

structure 16, 17, 23, 185, 212 

thermal conductivity 140, 162, 165, 
171 

viscosity 16, 143, 162, 164, 170, 304 
Glass 

brittleness 269, 273 

fibres 271 

melting 16 

structure 269 
Goniometer 227 
Grain boundaries 277, 293 
Gram molecule 14 
Growth spirals 283 
Griineisen constant 244, 261 (8.3) 

Habit, crystal 225 
Hamiltonian equations 76 
Harmonic 

motion 44 

oscillator 115, 123 
Histogram 68 
Hooke's law 50 

departures from 50 

Ice 20, 21, 135 (5.12) 

Impacts, molecular on wall 105, 160, 285 

Internal friction 142 

Interstitial atom 286 

Inversion temperature 203 

Ionic substance 12, 13, 53 

binding energy 55 

diffusion 294 

Einstein frequency 57 

elasticity 57 

electrical conductivity 294 

Griineisen constant 261 (8.3) 

melting 238 

see lithium chloride, lithium fluoride, 
silver bromide, sodium chloride 



316 



Index 



Ionization 

by collision 90 

energy 33, 54 
Ions 10,11 

Iron (Fe) stress/strain curve 52 
Isotherms 176 
Isotope 14 

enrichment 169 

Joule 4, 62 
(unit) 6 
Joule-Thomson process 201 

Kelvin 5, 35, 149 
Knudsen gas 169, 251 

Latent heat 6 

evaporation 31, 192, 196, 219 

melting, solidification 31 

sublimation 31 
Lattice, crystal 20, 228 
Lead (Pb) 

flow of metal 21 

specific heat 123 
Lennard-Jones potential 29 

used for metals 49 
Liquid 

adhesion to solids 35 

boiling 178, 180 

capillary rise 35 

compressibility 17 

condensation 178, 182, 220 

diffusion 297 

droplets 208, 220 

freezing 178 

latent heats 31, 192, 196 

range of existence 185 

rigidity 17 

structure 18, 185 

surface tension 34 

tensile strength 183, 217 

vapour pressure 177, 218 

viscosity 16, 19, 142, 298, 302 

see argon, carbon tetrachloride, nitro- 
gen 
Lithium chloride (LiCl) structure 12 
Lithium fluoride (LiF) 

single crystals 233 

thermal conductivity 253 
Lorentz ratio 254, 260 



Macroscopic variables 66 

Madelung constant 56 

Mass spectrometer 13, 205 

Maxwell 143, 164 

Maxwell distribution 107, 112, 260 

Mean free path 155, 156, 169, 246 

Mean molecular speeds 109 

and speed of sound 111 
Mean values 71 
Melting 16, 19, 177 

amplitude of vibration 238, 287 
Mercury (Hg) 11, 50 

adhesion to glass 37 

capillary fall 38 

latent heat 50 

Maxwell distribution 113 

melting point 238 

surface tension 50 

tensile strength 185 
Metals 10, 12, 48, 254, 260 

binding energy 50 

bulk modulus 41, 50 

ductility 264, 274 

Einstein frequency 50 

electrical conductivity 257 

electrons in 10, 12, 48, 254, 260 

friction 266 
internal 142 

Griineisen constant 244 

Lennard-Jones potential 49 

Lorentz ratio 254 

single crystals 264 

specific heats 123, 289 

surface tension 50 

thermal conductivity 257 

thermal expansion 243 

whiskers 286 

work hardening 280 

see copper, iron, lead, mercury, potas- 
sium, steel 
Metastable states 180 
Mobility 305 (9.3) 
Molar volume 22 
Mole see gram molecule 
Molecular flow 169 
Molecular weight 14 
Molecules 11 
Mossbauer effect 102 

Nearest neighbour interactions 30 



Index 



317 



Nitrogen (N) liquid 59 (3.4) 

binding energy 33 

bulk modulus 44 

dimers 206 

Einstein frequency 46 

surface tension 35 
Nucleus 9 
Nuclear reactions 13, 90 

Ordered molecular motion 63 

degradation 64, 66 
Orders of magnitude 5 

Penetration depth 298 
Perfect gas 86, 103 

pressure fluctuations 216 

temperature scale 77, 86 
Perrin 88, 150 

Persistence of velocities 159, 167 
Phase boundary lines 176 
Phase diagram 177 
Planck's constant 123 
Plasticity 266 
Point defects 286 
Porous plus experiment 201 
Potassium (K) 

bulk modulus 50 

latent heat 50 

melting 238, 289 

specific heat 289 

surface tension 50 
Potential 

centrifugal 205 

energy 25, 31 

intermolecular 27 
interionic 56 
Lennard-Jones 29, 49 
square well 188 

well 27 

anharmonic 239 
parabolic 46, 237, 239 
square 188 
Premelting phenomena 288, 290 
Pressure 

as energy density 42, 191, 220, 299 

perfect gas 86, 103, 191 

imperfect gas 188 
Probability 69, 70 

function 70 

independent 73 



Quantum theory 9, 123, 259 

Random 

molecular motion 63 

walk 150 
Reynolds number 299 
Rigidity 16, 40 

gases 16, 21 

liquids 17, 21 

solids 19, 21, 41 
Ripples 38 
Rotator 

energy 115, 125 
Riichhardt's method 121 

Sample average 71 
Scale height 

atmosphere 84, 87 

suspended particles 88 
Scattering 

energy flow in solids 249, 251 

light 211 

X-rays 233 
Shock waves 21, 53 
Silver bromide (AgBr) 

specific heat 290 
Simple harmonic motion 

see harmonic motion 
Size effects in thermal conduction 253 
Sky, blue of 212 
Sodium chloride (NaCl) 

binding energy 55 

bulk modulus 41, 57, 61 (3.12) 

diffusion 296 

Einstein frequency 57 

elastic moduli 41 

electrical conductivity 296 

melting 238, 289 

single crystals 233 

solubility 60 (3.9) 
Solids 

brittle 266, 272 

compressibility 19, 20, 40 

density 19, 20 

ductility 264, 274 

flow 21 

plastic 266 

rigidity 19, 20, 40 

structure 20, 185 

Young's modulus 40 



318 



Index 



Sound, speed of 16, 41, 46, 120 

imperfect gases 199 

and molecular speeds 112 
Specific heat 117, 197 

(Cp-CJ 118, 119, 198 

C p /C„ 119, 121, 198 

solid near melting point 288 
Spectral lines, broadening 101 
Stars 13, 90 

Statistical specifications 67 
Steady state 138, 159 
Steel 41, 280 
Stokes' law 298 
Strain 51, 263 
Streamline flow 142, 299 
Stress 263 

concentration 270 
Sun, surface temperature 102 
Superheating 181, 217 
Supersaturation 182, 220, 281 
Surface 

of crystal 231 

energy 33 

ripples 38 

tension 34, 35 
Temperature 

absolute 77 

Boyle 195, 203 

characteristic 88 

critical 179, 210 

inversion 203 

maximum superheating 218 

melting 177, 238 

perfect gas scale 77 

and random motion 64 

triple point 176 
Tensile strength 

liquids 183, 217 

solids 264 
Thermal conductivity 140 

effect of impurities 249 

gases 140, 162, 165, 171 

metals 257 

size effects 251 

solids 244 
Thermal diffusivity 140 
Thermal equilibrium 

see equilibrium, thermal 
Thermal expansion 1 19, 238 
Thixotropic liquids 142 



Time average 71 
Transport coefficients 136 
Triple point 176, 238 
Turbulent flow 142, 299 
Two component materials 274 

Unit cell 228 

Uranium hexafluoride (UF 6 ) 169 

Vacancy 286 

and diffusion 291 
Van der Waals 

equation 188 

refinements to 203 

forces 25 
Vapour 

pressure 218 

supersaturation 182, 220 
Velocity coefficient of chemical reaction 

130 
Velocity-component distribution 98, 103 
Virial coefficients 193 

2nd coefficient 195, 205 

higher coefficients 206 
Viscosity 16, 141 

gases 16, 143, 162, 164, 170, 304 

liquids 16, 19, 142, 298, 302 

solids — see internal friction 
Vorticity 143 

Water 12, 22 

critical constants 210 
maximum superheating 218 
tensile strength 185 

Whiskers, single crystal 286 

Wiedemann-Franz law 254 

Wood 16, 274 

Work hardening 280, 304 (9.1) 

X-rays 
characteristic 9 
diffraction 233 
structure analysis 233 

Yield strength 268, 279 
Young's modulus 40 
and fluctuations 215 

Zone 

crystal 225 
melting 233 



Physical constants and conversion factors 



Physical constants 

Avogadro's number 

Boltzmann's constant 

Gas constant 

Molar volume of perfect 

gas at STP 
Planck's constant 

Elementary charge 

Electron rest mass 

Proton rest mass 

Speed of light 

Conversion factors 
1 angstrom (A) 

1 electron volt (eV) 

kT = 1 eV 

1 atmosphere (atm) 



Value 

6.02 x 10 23 mol" 1 
1.38 x 10" 23 J deg" 1 
8.62 x 10" 5 eVdeg" 1 
8.31 J deg" 1 mor 1 

22.4 x 10 3 cm 3 mol" 1 

6.626 x 10" 34 J sec 

1.60 x 10 ~ 19 coulomb 

9.11 x 10" 28 gm 

1.67 x 10" 24 gm 

2.998 x 10 8 msec" 1 



10 



cm 



1.60 x 10" 19 J 
96.5 kJ mol" 1 

when 7" = 1.16 x 10*deg K 

1.01 x 10 6 dyn cm" 2 
1.01 x 10 5 Nm" 2 



Symbol 
N 
k 
R 

h 
e 



The Manchester Physics Series will be a collection of textbooks suitable for 
an undergraduate degree course in Physics, 

Each book will have been Individually developed, through a preliminary 
edition, to provide a reliable, self-contained text for an up-to-date course. 
Each book can be used independently of the other books in the series, while 
the organisation and scope of each book allows considerable flexibility in the 
selection and arrangement of different courses. 



The Manchester Physics Series 

General Editors 

F. Mandl; R.J. Ellison; D. J. Sandiford, 

Physics Departments Faculty of Science, 

University of Manchester 

Properties of Matter: B. H. Flowers and E. Mendoza 

Electromagnetism; I. S. Grant and W. R. Phillips 

Atomic Physics: J, C. WMImott 

Optics: F. G. Smith and J. H. Thomson 

Statistical Physics: F. Mandl 



Solid State Physics: H. E, Hall 



JOHN WILEY & SONS LTD London - New York ■ Sydney * Toronto 

ISBN 471 26497