Equilibrium Statistical Mechanics [1st ed.] 0471031224, 978-0471031222

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Ritter Library Baldwin-Wallace College Berea,

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EQUILIBRIUM STATISTICAL

MECHANICS

EQUILIBRIUM STATISTICAL

MECHANICS FRANK

C.

^DREWS

Department of Chemistry University of Wisconsin

JOHN WILEY & SONS,

INC.

NEW YORK LONDON SYDNEY •

•

RITTER LIBRARY BALDWIN -WALLACE COLLEGE

Copyright

g

1963 by John Wiley

&

Sons, Inc.

All Rights Reserved

This book or any part thereof

must not be reproduced

in

any form

without the written permission of the publisher.

Library of Congress Catalog Card

Number: 63-22204

Printed in the United States of America

Chcarn

TO MY PARENTS

PREFACE

As times change, the demands on students of They must know more and more at

technology increase.

good

and

ever earher

and engineers have long recognized thermodynamics and the During the twentieth century, as more fundamental

stages of their education.

the need for

science

Scientists

training in such subjects as

properties of materials.

thinking has invaded these disciphnes, the study of the molecular structure

of matter has assumed greater emphasis compared to the macroscopic

For example, thermodynamics alone

studies.

is

seen as inadequate for

today's undergraduate or graduate student, unless he

is

at least

introduced

atoms and molecules which gives rise to the observable phenomena of thermodynamics. This, then, is the problem of statistical mechanics to rationalize the macroscopic behavior of matter through the knowledge that matter is composed of discrete particles. Rarely is there time in the busy curriculum to the mechanics of the

for a course in nothing but statistical mechanics.

Bits of

it

arise in courses

on mechanics, physical chemistry, thermodynamics, properties of materials, fluid mechanics, solid state electronics, and who knows how many others. Because of limitations in time, suitable books, and faculty training and interests, it is hard for students to come to grips with statistical mechanics to even the following extent: 1.

To understand what

method by which of

its

2.

it

statistical

proposes to do

this,

mechanics attempts to do, the

and the "truth" or "falsehood"

results.

To understand

the basic tools of statistical mechanics.

—

—

Like the

thermodynamics energy and entropy those of statistical mechanics are not easily grasped at first. But they are vital in comprehending the subject. 3. Using these tools, to study some of their simpler applications. This work often goes under such names as "the kinetic theory of matter" and "statistical thermodynamics." It gives insight into some tools of

vii

Preface

viii

of the "why's" of thermodynamics and the properties of materials. helps a person see other ways to use statistical mechanics in

It also

own thinking and work. 4. To do all the above within

his

points which

is

demolition

the

all

a framework of definitions and view-

directly capable

way out

of being extended without any

to the frontiers of statistical mechanical

research.

These four points represent

wonder

my

that reaching this goal has been hard for both students

no and

much agreement among

pro-

goal in writing this book.

teachers, because even today there

not

is

There

is

fessional statistical mechanicians (a singularly quarrelsome bunch)

many

of these points.

substituted others,

it

is

on

have discarded cherished approaches and because to me the substituted ones make more If I

in a way which seems clear. The methods of statistical mechanics are applied to a variety of problems. Such unity as the subject has comes from its dependence on the basic

sense.

I

have tried only to reach the goal

Throughout the entire book, the on understanding the fundamentals. The problems that are scattered throughout (with their hints and answers at the back) have been

principles, given in Part I of this book.

emphasis

is

picked to illustrate topics discussed

in the text, to present applications,

and to encourage independent thinking and understanding. They are a very important part of the book. The first twenty or twenty-one sections comprise the elementary part of the subject.

The amount of background assumed

for the reader

that usually

is

given by general chemistry, physics, and calculus courses, and by any

survey of thermodynamics, such as

chemistry and

mechanics

is

is

many undergraduate

given in undergraduate physical

engineering courses.

No quantum

needed, although such quantum mechanical conclusions as

energy levels and quantum states are introduced without apology.

Most

much quantum mechanics in their first year or two Others may just have to "buy" these results and see how

students run into this in college.

mechanics uses them.

statistical I

hope the reader enjoys

has appealed to so

many

this

book and

its

subject.

Statistical

mechanics

of the great minds of the past hundred years

Joule, Rayleigh, Clausius, Maxwell, Boltzmann, Kelvin, Gibbs, Planck, Einstein,

von Neumann, Dirac, Fermi, Debye

reader of this book should never

But that or even

is

all

—

^just

to

however, that he

name

is

a few!

The

reading a gospel.

true for any student of the sciences, as well as of the humanities,

life itself.

ideas and facts

from

feel,

those

we

We

must each make our own order out of the welter of

find ourselves in.

who have gone before to

In doing this

we have the heritage we are reaching,

guide us. For here

Preface

however

hesitantly,

toward that goal of

scientific

theory

to

ix

show

the

logical necessity of the behavior of the physical world from as simple a set

of basic assumptions as possible.

ACKNOWLEDGMENTS I

wish to thank the National Science Foundation for a grant which book. I also thank the many friends whose

facilitated the writing of this

comments helped C. F. Curtiss.

indebted to

J.

clarify

fuzzy points, in particular, R. B. Bird and

For valuable creative reviews of the manuscript E. Mayer and D. R. Herschbach.

Madison, Wisconsin September 1963

Frank

C.

I

am

Andrews

CONTENTS

PART I— BASIC THEORY 1.

2. 3.

4.

3

Thermodynamics versus Statistical Mechanics The Problem of Statistical Mechanics 6 Elements of Probability Theory 8

6.

The Ensemble in Statistical Mechanics The Isolated Equilibrium System 22

7.

System

8.

Exponential Reduced Probabilities 31 Energy and Entropy in Statistical Mechanics

5.

9.

PART

Introduction

I

in

Equilibrium with

10.

Entropy and the Third Law

11.

General Formulation of

GASES

II— IDEAL

Gas

12.

The

13.

Classical

14.

Classical

Gas

15.

Classical

Gas

16. 17. 18. 19.

20.

Heat Bath

25.

27 34

38

Statistical

Thermodynamics

47

Gas— Translational

Energy and Pressure

91

Summary Gas

of Ideal Boltzmann Gas in

Occupation

a Gravitational Field

Numbers— Effect

Metals—The Electron Gas Radiation The Photon Gas

—

42

45

101

106

of Statistics

23. Fermi-Dirac and Bose-Einstein Gases 24.

18

— Molecular Velocities 57 — Molecular Trajectories 65 73 Classical Gas — Equipartition of Energy 77 Function Quantum Gas — Partition Quantum Gas —Translational Energy and Pressure Quantum Gas — Internal Molecular Energies

21. Ideal 22.

Ideal

a

5

I

18

123

114

109

49

84

Contents

PART

TOPICS

III— SPECIAL 26. 27. 28.

—Semi-Classical Partition Function Equation of State 136 — —Van der Waals' Equation 142

Dense Fluids Dense Gases Dense Gases

Virial

29. Perfect Crystals

146

30.

Equilibrium Constants

31.

Gases

32.

Problems

33.

Fluctuations

— Complications in Statistical

34. Mechanical

APPENDICES

155 in

Partition Function

Thermodynamics

174

View

of

Thermodynamics

182

187

Answers and Hints to Problems

C. Definite Integrals

189

198

D. Glossary of Symbols, Values of Constants

INDEX

164

171

187

A. References B.

131

203

199

133

EQUILIBRIUM STATISTICAL

MECHANICS

PART

I

BASIC THEORY

INTRODUCTION

I

Most of

the physical

phenomena

studied by scientists and

engineers during the entire history of science are what are called macroscopic

phenomena. That

is,

they involve matter in bulk, say, a chunk of Nevertheless, during the past

metal, a beaker of hquid, or a tank of gas.

is composed made of atoms; and that it is these molecules, interacting among themselves according to some force laws or other, which in their enormous number give rise to the macroscopic, observable phenomena we see. It is hard to realize that this microscopic picture we all take for granted is really only a century old. True, it had been cropping up among philo-

hundred

years, scientists have

come

to beheve that all matter

of molecules

sophic writings for two thousand years, but

its

general acceptance by the

community came only in the latter half of the nineteenth century. Its acceptance arose from the same cause as that of most good theories from the scientist's desire to unify and make order out of the endless phenomena about him by showing that they all would follow logically if certain simple hypotheses were true. Certainly, one of the most successful of such creations of the scientific community is the molecular hypothesis, scientific

which has tied together so many previously unrelated theories and phenomena and led to so many remarkable new realms of study and technology.

Now, show

since the molecular theory

that a vast

each other

at

makes

it

should be possible to

number of molecules ratthng around and bumping

appropriate intervals would

phenomena we

sense,

see.

The motion of

all

into

indeed produce the macroscopic

these molecules should, of course,

known mechanical laws. Before 1900 the study was mechanics. The development of quantum mechanics

be describable by the limited to classical

during the period 1901 to 1926 brought greater precision to the description. The magnitude of the many-particle problem posed by macroscopic objects can be appreciated by realizing just

how many

molecules are

mole of any substance. If each molecule in a mole were the size of a small dried pea, there would be enough peas to cover the United States to a depth of several hundred feet. When we consider that

beUeved to be

in a

4

Basic

in either

Theory

quantum or

classical

mechanics

it

has proved impossible to solve

the equations which give the trajectories of just three mutually interacting bodies, the problem here, involving 6

x

10^^ bodies,

seems almost laughand z coordinates of each molecule a thousand to the page and a thousand per minute, it would take a million times as long to write them as the estimated age of the earth, and a freight train loaded with the pages would stretch to the sun and back about 40,000 times! How, then, might this problem be tackled? It has been a long, hard process even to get a language and a reasonable means of approach, and able.

In

fact, if

that process

we

is still

just wrote

going on.

down

the x, y,

What has developed

is

a mixture of mechani-

and probabilistic arguments called statistical mechanics. It is vitally important to understand what is mechanics and what is probability in order to interpret the results of this branch of science. Confusion between these arguments and the assumptions involved in them has resulted in a century of misconceptions and quarrels. Nevertheless, the first successes of statistical mechanics hastened the general acceptance of the molecular theory by the scientific community. Since that time, continuing successes have shed great light on countless physical phenomena. The physicists whose successes established the foundations of statistical mechanics in the scientific world were J. Clerk Maxwell of Cambridge and Ludwig Boltzmann of Vienna. These two did their major work between approximately 1860 and 1900. The entire field was carefully studied by J. Willard Gibbs of Yale, whose briUiant synthesis and creative exposition published in 1902 brought order and stability to the foundations laid by Maxwell and Boltzmann. The work has progressed with much debate and controversy to the present. Still, we must always be impressed by the great insight of Gibbs into the fundamentals of the subject, an insight prevented from complete success only by the fact that the full development of quantum mechanics was still twenty-five years in the cal

future.

2.

2

Thermodynamics Versus

Statistical

Mechanics

THERMODYNAMICS VERSUS STATISTICAL MECHANICS

It

is

thermodynamics and matter in bulk, that

important to understand the difference between statistical is,

mechanics.

Thermodynamics

with macroscopic objects.

It starts

deals with

with certain

from these and some other hypotheses and experimental facts derives relationships between the observed properties of substances. Thermodynamics does not try to explain through mechanics why a substance has the properties it does. The job of thermodynamics is to link together the many observable properties of macroscopic objects, so that knowledge of a few will permit calculation of many others. The "truth" of thermodynamics rests in its never faihng to give valid answers to problems involving macroscopic basic postulates called the "laws of thermodynamics," and

systems.*

thermodynamics takes no However, the causal "explanation" of the laws of thermodynamics mast he in the mechanics of the particles comprising the macroscopic system. One could present statistical mechanics in such a way as to "derive" the laws of thermodynamics. Using this approach, one might step back and exclaim, "Look, we have just derived the second law of thermodynamics!" No such approach is taken here. The laws of thermodynamics are valid and assumed known. Their formulation from statistical mechanics sheds hght on their meaning and ranges of vahdity. It does not make them any more "true." It is, however, gratifying that the assumptions on which statistical mechanics is grounded contain the thermodynamic laws as consequences. Of course, if it were otherwise, the assumptions would long ago have been Except perhaps

in discussing the third law,

account of the atomic and molecular structure of matter.

changed. Statistical mechanics may be viewed as the discipline which bridges the gap between the microscopic world of molecules, atoms, and fundamental particles and the macroscopic world of thermodynamics and the properties of materials. Given a picture of the microscopic world, statistical mechanics sets out to determine what macroscopic behavior to expect. Sometimes the macroscopic properties enable statistical mechanics to shed light on the microscopic picture causing them.

*

For further discussion, see Pippard, Chap.

1.

6

Basic Theory

The validity of the macroscopic results of statistical mechanics depends on the accuracy of the microscopic picture used. Primarily, it is mathematical difficulties that keep statistical mechanics from giving quantitative descriptions of most macroscopic phenomena. On the other hand, thermodynamics as a discipline is unaffected by any microscopic picture. In consequence, it is limited to relating the results of one set of measurements to another.

In practice

it is

important for a person to keep a clear picture

of which conclusions about his problem have thermodynamic validity

and which follow from statistical mechanics. For this reason, the author believes that thermodynamics and statistical mechanics should be studied as two separate disciplines. They may be studied concurrently, each shedding hght on the other; but one is a purely macroscopic study, the other a bridge between the macroscopic world and the microscopic.

3

The problem to which statistical mechanics addresses same for both equilibrium and non-equilibrium

basically the

itself is

conditions. is

THE PROBLEM OF STATISTICAL MECHANICS

One has

a system in the sense of thermodynamics:

that part of the physical wovld to which one directs his attention.

A system Further-

more, he has a microscopic picture of the system. This means knowledge of the mechanical behavior of the constituent particles, whatever they may be

—molecules,

atoms, ions, electrons, or photons.

This information

furnished by such microscopic disciplines as molecular chanics, spectroscopy, x-ray crystallography,

mental

is

quantum me-

and the physics of funda-

particles.

Certain macroscopic information

is

also

known about

the system as a

some macroscopic measurements. It may be a minimum; example, the volume, the number and type of particles, and the fact result of

for that

they are "in equilibrium at a given temperature" or "in equiUbrium with a given energy."

Or

it

may

characterizing a system that

equilibrium process.

information

is

be very extensive and detailed information is

about to evolve

in

some complicated non-

In the discussion that follows this macroscopic

called the initial information.

3.

The problem of

The Problem

of Statistical Mechanics

7

mechanics is to compute the probabilities of of other measurements one might choose to The computation involves finding how the initial

statistical

the various possible results

make on

the system.

information and the

known mechanical behavior of

particles affect various other

measurements.

might be made at the same time the the system

is

initial

the

constituent

These other measurements information

is

taken.

Or,

if

"in equilibrium" time will not even be a factor in the problem.

Or, the other measurements might be

made

at a later time after the system,

in this case non-equihbrium, has evolved for a while.

One can

only compute probabilities of various results of other measureFor most macroscopic measurements, however, the tremendous number of particles involved gives rise to a great simplification. The most probable result of a measurement proves to be overwhelmingly probable. Significant deviations from it have only vanishing probabilities. This means that once statistical mechanics has found this overwhelmingly likely result of a measurement, it may be safely predicted that this result is the one which will be found in the system. Therefore, the calculated values of properties like pressure and density given by statistical mechanics are usually these most probable values. This procedure is justified and

ments.

discussed in Sec. 33.

Whether or not a

particular result of a macroscopic

overwhelmingly probable, the problem of

statistical

give the relative probabihties of all possible results.

measurement

mechanics

Clearly,

it is

still is

is

to

necessary

to phrase statistical mechanics in the language of probability theory.

In order to do of

all

this,

one must understand the theory of probability,

quite in the abstract

may make

and divorced from physical content.

dull reading, but the points discussed should

in their physical context.

It is

first

This

come to life later know which

very important, however, to

parts of statistical mechanics arise from the physics and which parts

simply from the mathematical language of probability theory.

8

Basic

Theory

4

ELEMENTS OF PROBABILITY THEORY

The theory of probabiUty

a well-defined branch of

is

mathematics. Like any other branch of mathematics, it is a self-consistent way of defining and thinking about certain ideahzations. When it is used

—

of scientist, it becomes no more than a logical method of thinking examining the consequences of a set of assumptions. To the scientist mathematics is simply one of his logical tools broadly speaking, the logic of quantity. Whenever in science mathematical conclusions seem

by the

—

it just means that the assumptions must be modified so they no longer imply the objectionable conclusions.

unbelievable,

In this section statistical

we consider

mechanics

is

the language of probability theory, in which

expressed.

We

parallel the

formal discussion with

a simple example in smaller print, to lend substance to the formal presentation.

Probability theory treats the properties of a completely abstract ideali-

zation

we

call

an ensemble.

An ensemble

is

a collection of members,

each of which has certain characteristics. Depending on the problem, there may be just a few members, many members, or an infinite number

of members. Consider as an ensemble a certain hypothetical collection of cat has certain characteristics

—color,

sex, age,

number of

cats.

Each

teeth, weight,

Note: Color and sex are certain qualities that characterize each cat. and number of teeth are parameters that take only discrete values. Weight and length are continuous parameters characterizing length.

Age

(since last birthday)

the cats.

The probability

P

of a certain characteristic

is

defined by the following

equation

... P(characteristic)

=

number of members of ensemble with total

characteristic

—

number of members of ensemble (4-1)

If the characteristic of interest

is

labeled

/,

then Eq. 4-1

may

be rewritten

in the obvious notation

(4-2)

4.

The

probability of characteristic

is

/

Elements of Probability Theory

9

simply the fraction of members of

the ensemble that possess characteristic

/.

While Eq. 4-1 may agree with

about the likelihood of selecting a member at random from the ensemble and finding it to possess characteristic /, this is less important than the fact that the mathematical definition of probability our

is

intuitive ideas

given by Eq. 4-1 or 4-2.

The probability of a male cat is defined as the number of male cats ensemble divided by the total number of cats.

Most of

in the

the mathematical properties of probabilities follow directly

from Eq. 4-2 If characteristic is

/

appears

in

no members of the ensemble,

PC)

=

!1=P(/, +

P{i) plus P{j):

PO,.

(4-5)

n If no cat is both yellow and black, the characteristics of yellow and black color are mutually exclusive. In that case, the probability of a cat's being either yellow or black is the probability of a yellow one plus the probability of a black one.

If P{i)

is

the probability of characteristic

ability o^ not finding characteristic

P(not

/)

=

'^^^^^^

/,

then

1

—

/*(/) is

the prob-

/:

= '1^^^ =

1

-

P(i).

(4-6)

DISCRETE PARAMETERS For the moment, consider only a number;

that

is,

consider

characteristics that are represented

some parameter

that

takes on

by

discrete

Basic Theory

10

values.

Assume

that each

member

of the ensemble has some value of this

This will always be the case

parameter.

The allowed The probability

of probability theory

in the use

in statistical mechanics.

values of the /th characteristic are

given the symbol

that characteristic

li

is

/,.

/

has the value

simply

= -^.

P,{h)

(4-7)

n

Since no

member

of the ensemble can have two different values of the

parameter, the condition of mutual exclusiveness

is

satisfied automatically

for the validity of Eq. 4-5.

A

simple extension of Eq. 4-5

all possible values

\\

of

the

the following:

is

2P,(/,)

=

/,

=-= I^' n n

probability that satisfies Eq. 4-8

is

often called the normalization condition. cat

is

(4-8)

l.

/,

A

Each

The sum o/Pi(li) over

parameter yields unity:

characterized by

is

said to be normalized,

some unique value of

its

probability of either a 2 year old or a 3 year old cat

age.

is

the

and Eq.

4-8

Therefore, the

sum

of the in-

Furthermore, the sum of the probabilities over all the possible ages must yield unity, since it includes all cats in the ensemble. dividual probabilities.

Each member of

the ensemble is characterized by some value /^ of one were to compute the value of any function of /,, say, g{li), for each member of the ensemble and find the average of all these values, his result would be the ensemble average of g, which is written g. The formula for finding g from the probabilities P^ is easily obtained from the definition of the average of a set of numbers as the sum of the numbers

parameter

/.

If

divided by the

number of numbers: g

=

;7

=

/I

2

g(member)

all

members

of

ensemble

I

n[g(l,)]gil,)

all possil)le

values of 1

g

= lPi(iMhy

(4-9)

4.

Each cat has an age

Consider the function

/.

for the cats in the ensemble /^

uct over

all

possible ages

/-.

/-

II

The average value of /^

the "ensemble average of /^," written

is

one simply multiplies

calculate

Elements of Probability Theory

times P{l) for each

and sums

/

/-.

this

To

prod-

/.

Once

the ensemble average of g is found, the amount by which g(lj) from the mean may be computed for any value of ^ and given the symbol dg{lj) differs

:

-g=

gUi)

(4-10)

^g(li).

The ensemble average of the deviation from

the

mean

is

of course zero:

^ = I Pimg(h) - g] u

=

lPi(lMh)-glP.ili), li

6i which

=

g

li

-

=

g

(4-11)

0,

from

The mean positive mean negative deviation

from the definition of the average.

arises

deviation from the average exactly cancels the it.

one seeks a useful measure of the average deviation of g{l^ from its mean value, the signs of positive and negative deviations must be omitted If

somehow this

is

A

so they will not cancel.

simple and useful

way

to

accomphsh

to average the square of bg{l^)

'^'

= lPimg(ii)-gf

=

lPi(h)[g(lf-2g{h)g-\-f] li

= d? = The

2

-

-^

gil^)

other function

square of

its

g

-2

2g

+ 1

-2

g

,

7-f.

left-hand side of Eq. 4-12

In that case,

is

g

(4-12)

is

positive unless %(/,)

would have the same constant value g the average of

its

is

zero for

for all

/,.

all

/,.

For any

square will always be greater than the

average because, in averaging the squares, greater weight

given to the larger contributions.

Suppose the ensemble contained only four cats, aged 3, 6, 7, and 9. The ensemble average age is 25/4 or 6.25 years. The average deviation from the mean is (—3.25 — 0.25 + 0.75 4- 2.75)/4 = 0. The mean square deviation, however, is [(3.25)^ + (0.25)2 + (975)2 + (2.75)2]/4 = 4.69, which is identical to (32

+

The ensemble parameter

l^.

62 -h 72

+

92)/4

-

(6.25)2.

establishes a value of Pj(/J for each possible value of the

Viewed

in this

way, P^C/J

is

a.

function of the value of the

Basic Theory

12

parameter

/^

in the usual sense

of the word

a function of h

Pi(li) is

:

a unique value o/Pi(li)

for every allowed value ofh Suppose that each member of the ensemble

is

if

implied.

is completely described by s values of a set of s paramgiven by the which are different characteristics the values numbered are 2, ...,/,..., characteristics 1, 3, eters. If the Such an ensemble l^, I3, are l^, /,. parameters respective of the 4, jf,

.

.

.

.

,

.

.

,

not only defines the simple one-parameter probabilities of Eq.

The

also defines so-called joint probabilities.

4-7,

but

it

joint probability F-^ that

not only the parameter representing characteristic

is

/

/,

but also the

It is is defined as follows: parameter representing characteristic y is the fraction of members of the ensemble for which not only i has the /_,

value

/j-

but alsoy has the value

PijOr

/_,:

/;)

=

—

n(L and

/,)

,.

.^.

(4-13)

•

n

The

joint two-parameter probability P^j

the two parameters

The joint

/,

and

a function of the values of

is

/_,.

probability that a cat of age 5 has 18 teeth

is

the fraction of cats

ensemble which are both 5 years old and have 18 teeth. The structure of the ensemble establishes such a probability as a function of age and number of teeth for all possible values of the two parameters. in the

Consider the formal relationship between

/*,;(/,, Ij)

and

Pi(li).

How

would one obtain P^A) from P,,(/„ /,)? In P^/i) as given by Eq. 4-7, the ensemble for which n(li) of the numerator is the number of members of the parameter / has the value /,. It makes no difference what value parameter j has; all values of parameter/ contribute to nil,) so long as parameter / has the value /,. One may think of n{l,) as the sum of «(/, and /;) over all possible values of

/;:

n(/.)

Therefore, P^//)

may

= 2 "(/, and

be found from /*,//„

i'.a)

=

—=

=

Ij)

-I"aand/,), n h

n

Piild

(4-14)

/,).

(4-15)

lPuih,h)I,

Knowledge of Pi{l^, and,

P,//,,

/;)

as a function of

by the analogous equation, Pj{h)

=

/,

and

/,

is

thus enough to find

Pj{lj):

lPiAh,lj)-

(4-16)

4.

Elements of Probability Theory

13

The simple

probability that a cat is 5 years old is just the fraction of cats Finding this probability has nothing to do with the cat's teeth. If one likes, however, he may view the total number of 5 year old cats as the sum of the number of cats age 5 with teeth, plus the number age 5 with 1 tooth, plus the number age 5 with 2 teeth, .... The probability that a cat is age 5 is identical to the sum of the joint probability of age 5 and some number of teeth over all possible numbers of teeth.

of age

5.

The ideas of Eqs. 4-15 and 4-16 are much used. The probabilities P^ and Pj are said to be of lower order than joint probabilities like P^^ which

More information about

contain them.

knowing /, and Pj

and

the ensemble

is

provided by

than by knowing P^ as a function of separately. The process of obtaining a lower

P^j as a function

of

as a function of

/;

/^

/^

order probability from one of higher order, as

and 4-16, is and the lower order

Eqs. 4-15

in

often called reducing the higher order probability,

reduced probabilities. This is a usage of the word "reduced" different from that sometimes employed in referring to dimensionless variables, calling them "reduced variables." probabilities thus obtained are called

In the event that s different parameters completely describe each

of the ensemble, the ultimate in joint probabilities the

first

member

the probability that

l^ and the second has the value 4 and the and the 5th the value //.

parameter has the value

third has the value

/g

,,

^1.2,3.

The

is

.

.

and

.

.

.

,

.,.(/i, /2,

•

•

•

,

,, »(/i U = -^

by Eq. 4-17

joint probability given

probability for the ensemble.

It is

and

is

/,

and

.

.

.

and

/,)

(4-17)

.

n

sometimes called the complete

a function of all the s parameters

which describe a member of the ensemble.

If

it is

known

as a function of

the parameters, then one has complete knowledge of the ensemble.

That

he knows the relative magnitudes of the numerator of Eq. 4-17 for all possible combinations of values of the i' parameters. This enables him is,

to construct a replica of the ensemble completely, since the relative

of members of each type Quite evidently,

all

other probabilities relating to the ensemble

formed by reducing the complete probability. P^(l.)

= I2 /l

•

•

'•>

means

I ^1,2,3

may

be

In particular,

sUl^

/2,

•

•

,

/J.

(4-18)

'.

(fXCludillK

In words, Eq. 4-18

number

known.

is

;,)

that in counting

members of

the ensemble in

order to find the numerator of Eq. 4-7, one pays no attention to the values of any of the parameters other than /. if parameter / has the value /,, then that

member of the ensemble contributes may have.

values the other parameters

to the numerator, whatever

Basic Theory

14

Suppose that a cat were characterized by only its age and number of Then if one knew the joint probability as a function of age and number of teeth, he would have full knowledge of the ensemble. He would know the relative number of cats in the ensemble with every possible combination of age and number of teeth. He could find the simple probability of age by summing over all numbers of teeth and vice versa. Nothing else would have meaning in such an ensemble. teeth.

Characteristics

andy

/

are said to be explicitly correlated whenever the

structure of the function P^il^) varies with the value of

are uncorrelated, then, whenever

istics

/',(/;)

is

The character-

Ij.

independent of

Ij.

This

same function of /^ if figured from the complete ensemble as it is if figured from an ensemble composed of only those members with a particular value of ly That is, if / andy are uncorrelated,

means

that

/*,(/,) is

P^IQ

=

the

!![/l)

=

!!i/il!liy

n

In Prob. 4-2,

for all values of/,, and/,..

proved that Eq. 4-19 imphes the analogous equation

it is

p^g^^='^^'£i^, If characteristics

(4-19)

n{lj)

/

(4.20)

and j are uncorrelated, P^ and Pj are said

to be inde-

pendent probabilities. characteristics / and j may be independent even when P^A) ^^^ both depend on the value of some third parameter. In fact, even if andy have a common cause which implies coupling between the values

Two

Pj(lj) /

/;

and

/_,,

Eq. 4-19

is

satisfied so

long as explicit knowledge of

change the probability distribution for

As a that if/

of Pi

result of the definition of uncorrelated characteristics,

andy

Ij

does not

/,.

are uncorrelated, then the joint probability P,j

is

it

follows

the product

andP/ Pij(l,

and

/;)

=

ndi and

;7(/,)

/,)

n

n

n(h)

n(lj)

=

/i(/,

and

Ij)

n(l

P^{l^)-P^(l,).

(4-21)

n

In science, the joint probabiHties of independent characteristics are

made

to factor.

characteristic istics

is

All the dependence of the joint probability on each

contained in

its

own term

in the product.

If character-

are not independent, their correlation prevents factoring the joint

probability in such a characteristics.

way

as to separate

its

dependence on the various

4.

Elements of Probability Theory

15

number of teeth

a cat had were completely independent of the cat's one would expect the probability for age and the probability for teeth to be independent of each other. The joint probabiUty would factor. Clearly, this is not the case as old cats are likely to have fewer teeth. Therefore, these characteristics are correlated, and the joint probability does not If the

age, then

factor.

CONTINUOUS PARAMETERS When

one or more characteristics of the members of the ensemble are

given by continuous parameters, the problem

is

Con-

slightly different.

may have any one of an infinite number of values, ensemble may be chosen to have an infinite number of members.

tinuous parameters

and the

This need not be alarming, even though Eq. 4-7 has infinity in the denominator. The feature of interest in the continuous case is not the probability that parameter bility that

lies

it

/

has some particular value

between two values

x^

and

x^ -f Ax^.

x^

but the proba-

This probability

is

simply defined as the fraction of members of the ensemble that have this parameter between x^ and x^ + Ax-. Even with n infinite, this fraction is well defined.

The number of members for which the value of the parameter lies x- and x^ }- Axj is a function of both .r, and the interval. If one

between

shrinks the interval

to an infinitesimal value dx-, the probability

A.r,

may

be used to define a probability density f-: ^(between

and

.r,

x,

+

dxA

= fM dx,.

u

The

probability density/,

at a given value

of

x^,

(4-22)

simply the density of members of the ensemble.

is

number of members.

divided by the total

It

is

defined formally as fi(Xi)

"/j(

=

between

x^

and

+

A.r,)'

(4-23)

n Ax.

Aj-,->OL

In terms of/,, the probability that parameter is

.r,

lim

/

lies

between

x-

and

x-

+

A.r,

given by the integral

P,(between

x^

The equations involving

.r,

+ Ax.) =

discrete parameters

f^(x/) dx/.

may be

(4-24)

taken over directly

summations are replaced by integrations. the probability density must be normalized to unity, as in

into the continuous case In particular,

and

if

the

Eq.4-8: J/.(a:,)^/.r,

=

1,

(4-25)

Basic Theory

16

where the

integral

over

is

all

average of any function of

x^. Also, the ensemble found by the equation analogous to 4-9:

allowed values of

x^ is

g=^gi^m^i)dx„

(4-26)

•J where again the integral

over

is

all

allowed values of

/

frequently convenient to normalize

It is

unity.

If

it

would have else, it is

x^.

to something other than

were properly called a "probability density," the normalization to be to unity. Therefore, when/^ is normalized to something

commonly

called a distribution function, perhaps because

it

shows how the values of the parameter are "distributed" among the

members of

general term embracing

book, any

may

"Distribution functions"

the ensemble.

be viewed as a

different normalizations employed.

all

In this

distribution functions used will be normalized to unity.

The number of

whose length

cats

between L and L

lies

+

ciL is

simply

the density of cats in the ensemble with length L (in units of cats/length) times dL. The probability that a cat has length between L and L + dL is

by the total number in the ensemble //. be called the probability density for length of cats /i,(Z.). Then /z,(L) dL is the probability a cat's length lies between L and L + dL. The probabilitv that a cat has length between any two values Z-i and L.2 is simply the integral the

number of such

The

density divided by n

cats divided

may

Pdi < L
li^ in terms of /^ 2, Express the ensemble average of h(lo, I3, 4) in terms eof an integral over /^ 2, Express this same average as an integral over e.

.

.

.

.

.

,

,

,

.

.

.

,

/2,3,5V'2> '3' 4)-

4-7. In the

paragraph following Eq. 4-16,

it is

stated that Pjjili, Precisely

more information than PjUi) and PjUj) do between them.

Ij)

contains

what is the additional information? Is there any circumstance under which knowing P, and Pj separately would furnish as much information as knowing f^j?

Basic

18

Theory

THE ENSEMBLE

5

IN

STATISTICAL MECHANICS

This section treats the

probabihty theory.

ensemble are

like,

In particular,

what

way

that statistical mechanics- uses

determines what the members of the

it

their characteristics are,

Not

represent these characteristics.

and what parameters

until Sees. 6 to 8 will consideration

be given to finding what function of the parameters the probabilities are.

The problem of statistical mechanics begins with a physical system about which some macroscopic information is known. The system is considered to be a mechanical object made up of its constituent particles. The known macroscopic information is far short of a complete mechanical specification of the state of the A^ particles. Yet, mechanics is useful only

when

applied to particles in a completely specified mechanical state.

Imagine how useful cussing a

game of

classical

mechanics would be, for example, in

billiards if

one knew only that the three

balls

dis-

were

"someplace on the biUiard table." It is this

very lack of detailed information that requires the addition of

probability theory to the mechanics. cal state the

N particles

probability.

are in,

we

Since

we do not know what mechani-

say that each possible state has

its

This probabihty distribution for the various states

function of the information that

is

known about

the system.

It is

own is

a

hoped

that the probabihties used will lead to conclusions which actually corre-

spond to the probabilities of various

results of

measurements on the

system.

Another way to express this is to say: One constructs an ensemble to Each member of the ensemble is a mental creation which is in a definite mechanical state. However, all members of the ensemble must reflect the known macroscopic information about the system. Since that much is known about the system, the probabihty of its not represent the system.

having those

known

ensemble must

properties

reflect that

is

zero.

Thus,

all

the

members of

the

information.

A clear distinction must be drawn between the system and a member of The system is the physical object about which we hope to make predictions. Members of the ensemble are only mental construc-

the ensemble.

which give substance to the use of probability theory. Of course, we do not need to construct, even mentally, an infinite number of members tions

The Ensemble

5.

in Statistical

with 10^^ particles in a fixed mechanical state.*

We

Mechanics

19

only seek the com-

plete probability as a mathematical function of the necessary parameters.

The ensemble and

members

its

are nevertheless useful to think about;

they give a tangible feeling to the various probabilities, which might

otherwise seem very abstract functions.

Choice of suitable parameters to characterize the members of the ensemble depends on the definition of a completely specified mechanical state for the

being

A'^

particles.

employed

or a mixture of both, which has acquired the

classical,

classical

In turn, this depends on the kind of mechanics

quantum,!

determining the microscopic picture:

in

name "semi-

mechanics."

CLASSICAL mechanics: In classical mechanics, an A^-particle state

and momentum of each

the position

are polyatomic molecules, the position

must be

given.

mechanical

If the system

particles, the

is

complete

set

member

of the ensemble

is

and the

momentum

a vector

is

are

6N

of

A'^

completely specified

when

If the particles

and momentum of each atom

considered to be composed of

particles

set

is

particle are given.

N classical

of parameters that characterizes a

the set of A^ position vectors that locates the vectors,

given by three numbers, say,

one for each particle. Since and ~ components, there

its x, y,

continuous parameters characterizing each

member

of a classical

ensemble.

The word "momentum" has been used despite the fact that

The reason

is

momenta and

here, rather than "velocity,"

velocities are often simply related.

that classical statistical mechanics

is

much

the variables used are the positions of the particles

simpler

when

and the momenta

Throughout physics, momenta appear as more fundamental mechanical variables than velocities. The 3N position components plus the 3A^ appropriate momentum

appropriate to these positions.

* Statistical

mechanics credits people with great hypothetical mental

agility.

Still, it

makes only reasonable demands. t Actually, this book never considers a consistent quantum mechanical treatment, which would be based on the density matrix. This is discussed by Tolman, Chap. IX. If required, its

Most of

study should prove

much

simpler after reading this book.

Appendix A, which cover either quantum mechanics or statistical mechanics, have chapters devoted to classical mechanics. The most thorough presentation is probably that of Tolman, Chaps. II and III, but it is J

difficult.

the larger references given in

Basic Theory

20

components which together form the complete classical

classical

set of parameters for the ensemble have been given the name of the phase space of the problem. Just as a point in ordinary space is the specification of

the three coordinates that locate

a point in phase space

it,

tion of the 67V coordinates that locate is

three-dimensional and phase space

the classical ensemble

it.

is

the specifica-

is

Thus, ordinary Cartesian space

member of

6A^-dimensional. Each

completely characterized by a single point in

is

Sometimes the 3A^ position components are referred to and the 3 A'^ momentum components as the momentum space. This makes the complete phase space the sum of configuration space plus momentum space. phase space.

separately as the configuration space of the problem

The concept of phase space where the

classical picture

is

is

discussed in greater detail in Sec. 13,

actually used.

QUANTUM MECHANICS* In quantum mechanics, a system whose volume is finite may exist in any one of an enormous number of discrete states. Since the allowed

quantum

states

arbitrary

way and

number

will

a

number given

to each state.

This single

completely specify the quantum state for the

Therefore, the only parameter needed to describe a

ensemble

in

some quantum

are discrete, they can in theory be ordered in

the

quantum mechanical

picture

is

particles.

A'^

member

this

single

of the discrete

number.

The mechanical behavior of particles in nature is in fact described by quantum mechanics, not classical. One might ask why classical mechanics with

its

6A^ different continuous parameters

The reasons

mechanics. discrete

parameter

is

are several.

is

ever used in statistical

Quantum mechanics

allowed quantum states are for an TV-particle system

its

single

is

usually impossibly

In classical mechanics, that part of the problem at least

difficult.

solved.

with

simple in principle, but the job of finding what the

Also, classical mechanics has the advantage that

visualize

N

momentum times the

particles,

each with

coordinates, than

quantum and

cases help one learn

how

it

its is

it

is

three position coordinates

an

is

easier to

and three

quantum state. Somebecome identical, and these

A^-particle

classical pictures

to choose whichever description

is

needed to

simplify the problem at hand. There are many satisfactory books on quantum mechanics, some of which are listed Appendix A. Also, most of the larger books listed there that deal with statistical mechanics have chapters devoted to quantum mechanics as well. *

in

5.

The Ensemble

Mechanics

in Statistical

SEMI-CLASSICAL MECHANICS One last description of the microscopic many cases, the positions and momenta of

picture

quately be described by classical mechanics.

themselves

The

cules).

may have

is

internal state of a molecule

done

often useful.

may

In

very ade-

However, the

particles

complicated structure (such as polyatomic mole-

adequately by classical mechanics.

What

is

the particles

A

may

almost never be described

quantum

description

in this case is to describe the centers

cules by classical positions

is

necessary.

of mass of the mole-

and momenta and to use quantum mechanics Such a mixed or hybrid

to describe the internal states of the molecules.

commonly been called semi-classical. The parameters needed to describe a member of a semi-classical ensemble are the following: ?)N continuous position components for the centers of mass of the A^ particles, ?>N continuous momentum components for the centers of mass of the A^ particles, and the set of discrete quantum numbers needed to fix the internal state of each of the A^ particles. If v different quantum numbers describe the internal state of each particle, description has

then a

member of

a semi-classical ensemble

by (6 + v)N parameters. The purely classical description

is

is

characterized completely

adequate only when one has no

interest in the internal states of the molecules.

Such a case could be the

study of a monatomic fluid hke an inert gas. If the atoms

are in their

all

lowest electronic state, the fact that the atoms actually have a structure of their

own may be

tronic excitation

neglected.

As

the temperature

is

increased and elec-

becomes more probable, the structure begins

active role in the physical properties.

Then, even gases

be treated semi-classically. In Parts

and

II

III, all

like

to play

an

argon must

three descriptions will

have occasion to be used.

DISCUSSION Just it

how

valid

is

the statistical mechanical

approach? It is likely that All measurements on

represents the ultimate in "scientific truth."

systems yield partial information.

If one were able to look at a system and learn the exact position, momentum, and internal state of each particle, then the ensemble would have to reflect that information. Each member of the ensemble would then have to be identical. An ensemble consisting of just one member would be adequate, and that member would be identical to the system. In that case it would be meaningless to

Basic Theory

22

introduce the concept of probability into the many-particle mechanics

There would be no need for

used to study the system.

statistical

me-

chanics.

However, complete information information obtainable, one

still

is

ne\er obtained; yet with the partial

wishes to predict as accurately as possible

the results of measurements. Statistical mechanics can give this prediction

and

give

correctly

it

—

it

consists in the distribution of probabilities of

various measurements on a system, which

information previously

known about

is,

naturally, a function of the

the system.

In the long run. as with any theory, the success of statistical mechanics

on its usefulness in practice. It has pro\ ed very useful in the instances where its mathematical difficulties have not been excessive. Its usefulness in the future in more difficult cases will depend on whether it can be visualized more and more simply, thus easing the mathematical difficulties. The rapid development of large computers will certainly help. Recent work by Alder and W'ainwright.* points to an interesting direction of in\estigation by actually following the mechanical motion of a large number of classical particles on a computer. rests

6

THE ISOLATED EQUILIBRIUM SYSTEM

This section treats the method whereby the macroscopic

known about

information

At

this point

it is

the system

is

incorporated into the ensemble.

necessary to particularize to only those systems

many

known

and useful topics from consideration, but treating non-equilibrium problems would quadruple the length of this book and lea\e the reader far from satisfied at the end. Non-equilibrium statistical mechanics is a current research field, one which has been fought o\er for about a centurv. It has a way to go to be "in equilibrium." This eliminates

fascinating

yet.

In Sec. * B. J. .\lder

5.

three different

and T.

33, 1439 (1960).

E.

commonly emploved microscopic

Wainurisht.

/.

Chem. Pins.,

11, 1208 (1957);

pictures

31, 459 (1959);

6.

The

System

Isolated Equilibrium

23

were discussed, along with the parameters used in each picture to characterize a member of the ensemble. The rest of Part I is based on a purely quantum description. Not only are the formal manipulations simpler, but also, after all, quantum mechanics is the correct description of a system of

and

sical

In Parts II and III, methods of passing to the clas-

particles.

semi-classical pictures

when appropriate

will

be examined in

detail.

THE EQUILIBRIUM CONDITION Systems tions. 1.

to be "in equihbrium" may be in one of several condimost commonly encountered are the following:

known

The

three

The system

a

is

V

volume

containing

N

particles of

known

type

completely isolated by insulating walls from the rest of the universe. particles of known type in 2. The system is a volume V containing

N

thermal contact with a heat reservoir or thermometer which

by the temperature

is

characterized

T.

3. The system is a volume V in thermal contact with a heat reservoir or thermometer which is characterized by the temperature T, and it is also open to the exchange of matter with a particle bath or reservoir which is characterized by the chemical potential fi.

In this section, condition microcanonical. is

In Sec.

called canonical.

ensemble In

all

is

called

is

1

7,

studied;

condition 2

is

the resulting ensemble

is

called

studied; the resulting ensemble

In Sec. 33, condition 3

is

studied;

the resulting

grand canonical.

cases, the system

is

known

to be "at equilibrium."

necessary to define the equilibrium condition carefully.

therefore

It is

One

associates

time independence with equilibrium and also a certain constancy in

macroscopic properties. However, as one examines any macroscopic "equilibrium" system more and more closely, he learns much specific information about it. Local variables, such as the density, will be found to fluctuate from place to place in the system. These fluctuations, caused

by the motion of the particles in the system, are constantly arising and being dissipated. So if one knows too much about a system that would otherwise be considered "in equilibrium," he destroys the time inde-

pendence associated with the equilibrium condition. The ensemble which correctly represents the equilibrium condition must simultaneously represent all possible fluctuations, each weighted according to

its

probability.

required of a system

known

Constancy

in

macroscopic properties

to be "in equilibrium."

The extent

to

is

not

which

24

Theory

Basic

constancy

is

expected should be calculable from the probability function,

however.

Time independence, on the other hand, The time independence

rium condition.

is

certainly part of the equilib-

rests

not

in

the macroscopic

properties of the physical system, but in the state of knowledge one has

about the system. With condition 1 above, he knows that the volume is constant, the number and type of particles are constant, and the energy

is

is constant. Furthermore, his own state of ignorance about any other features of the system is constant. The system may have got into its equilibrium condition from any of a number of previous conditions, but it has sat around long enough that it now is "at equilibrium." Information about its former condition is completely lost.

of the system

In statistical mechanics, a system the information one has about

it

is in

a condition of equilibrium when

has reached a time independent minimum.

This condition depends only on the material comprising the system and constraints (such as fixed volume, energy, temperature). It is independent of the system's history. In order to rule out so-called "steady states," one should also require that the immediate surroundings of the equilibrium system be in equilibrium too. its

THE MICROCANONICAL ENSEMBLE Consider an equilibrium system the type and

in condition

1

above.

number of particles and of the system volume

Knowledge of is

sufficient for

quantum mechanics in principle to determine all the many allowed A^particle quantum states.* The states can all be ordered, perhaps in order of increasing energy, and a number assigned to each. The only additional knowledge possessed is that the system is isolated and therefore has a mation, and

this

fixed energy ^'system-

It is

desired to build this infor-

information only, into the ensemble.

There must be no

spurious information buih into the ensemble accidentally.

One

faces this

enormous number of allowed quantum states, each of which must be assigned a probability. The only dynamical feature known about the system

is

ment on

its

energy

E^y^^^^^^

Therefore, he places the following require-

the complete probability at equihbrium:

The only dynamical

may depend is the energy of the state. The probability of the /th TV-particle quantum state may be denoted by Pyii). Thus, P v(0 is the fraction of the members of the ensemble that feature on which the probability of a state

* If

the classical or semi-classical picture were being used, this knowledge

determine

all

describe a

member

the allowed mechanical states,

of the ensemble.

and thus

all

would

still

the parameters needed to

6.

quantum

are in

state

£", is

Isolated Equilibrium

The above requirement may then be

/.

P.v(/)

where

The

the energy of the

=

System

written (6-1)

/>^.(£,),

N particles

in

quantum

state

/.

This requirement or hypothesis bears some discussion, since on

based

all

of equilibrium

it

is

Suppose the probabilities were assigned as functions of something

statistical

of the allowed quantum states other than the energy

25

mechanics.

— for example, the number of particles in a particular

That would mean that for some reason numbers of particles in that cubic centimeter.* However, without having measured the number of particles there, we have no right to bias the distribution in that way. And if we had made the measurement, we would have too much information for an "equilibrium" system and the time independence would cubic centimeter of the system.

we were

biasing the probability distribution for or against certain

be destroyed, as discussed above. Thus, the basic hypothesis of Eq. 6-1 seems to have been forced on us. Anything other than the energy on which the probabilities might be allowed to depend would not have been part of the macroscopic information available under these circumstances. Such additional information would have been spurious, thus having no place in the ensemble. Nevertheless, Eq. 6-1 may be viewed as an assumption if one wishes, or else as a consequence of the aims of statistical mechanics. An interesting consequence of Eq. 6-1 is the fact that all quantum states with the same energy have the same probability. In quantum mechanics, two different states are called degenerate if they represent systems with the same energy. Since Pv(0 is a function only of the value of E^, the probabilities of all degenerate

quantum

states are equal.

One might ask whether an ensemble

constructed in some way according would remain independent of time as each member evolved. The answer is that it would, and the proof is based on a rather general conclusion from quantum mechanics called the principle of detailed balancing.^ This to Eq. 6-1

of the ensemble in state A B going into state A. The transition is only allowed at all if energy is conserved, so the energy of A and B are equal. Therefore, at equilibrium the ensemble has the same number of members in both A and B. Since on the average the same number per unit time go from /I to 5 as go from B to A, the structure of the ensemble remains constant in time. principle says that the probability that a

will

*

go into

state

This type of biasing

theory."

B is

is

the

same

as that of

member one

in state

treated mathematically in the discipline called "information

In the opinion of the author, the

mathematics of information theory is less convincing than the simple physics of the problem. The interested reader may consult L. Brillouin, Science and Information T/ieory, Academic Press, New York, 2d edition, 1962. t

Tolman,

p. 521

;

Powell and Crasemann,

p. 420.

Basic Theory

26

The system being considered

in this section has measured energy, and, from the rest of the universe, it must keep that energy.* Each member of the ensemble must have this same energy, E^y^^^^^. Thus, PyiO is identically zero for states whose energy differs from -fi'sygtem'

since

isolated

it is

and PyiO

is

-P.v(^system) foi" the others

v(^system)

{i^i

=

i^system),

(£,

^

Esystem)

(6-2) I-*

I

Only the degenerate states with energy ^'system have probabilities, the same constant value for all such states.

A

non-vanishing

distribution of probabilities constructed according to Eq. 6-2 to

is what Gibbs called a microcanonical distribution. The ensemble that such a distribution represents is called a microcanonical ensemble. Gibbs did not say why

represent an equilibrium system of fixed energy

he chose

this

name, but the name has stuck. The "micro"

the m/f/-oscopically sharp energy requirement.

prefix connotes

The "canonical" connotes

the general acceptance of the ensemble described by Eq. 6-2 as representative

of the isolated equilibrium system.

The microcanonical distribution of probabilities ofN-particle quantum states, is a reflection of the following necessary and sufficient

as given in Eq. 6-2, conditions

The probability of an N-particle quantum state

1.

is

a function only of the

energy of that state.

The system has known

2.

total

energy

E,^,^^„j.

PROBLEM 6-1.

have

Suppose there are ^r quantum

total energy £'system-

What

is

states for

an

A^-particle

system which

the value of Py(i) for a particular

quantum

state? * Or very close to it. Actually, there is always an uncertainty in the energy of any system due to the Heisenberg principle. Broadening slightly the sharp energy requirement of this section changes the conclusions in no important way.

System

7.

SYSTEM

7

Equilibrium with a Heat Bath

in

27

EQUILIBRIUM

IN

WITH A HEAT BATH

isolated, the equilibrium

Suppose that instead of being system of interest

is

known

to be at a certain temperature because

it is

"in equilibrium with a heat bath characterized by that temperature."

This

is

the second possible equilibrium condition discussed at the begin-

ning of Sec.

6.

This section considers the question of what

is

implied in

the use of the following distribution of probabilities for this isothermal case:

=

P,.(/)

(7-1)

ae-^""'.

Gibbs called the distribution represented by Eq. 7-1 the canonical distribution, and ensembles constructed in this way canonical ensembles. An interpretation of Eq. 7-1 rests on the realization that the energy of each A^-particle quantum state

sum of

is

usually capable of being written as the kinetic energies of the individual

small energy contributions:

particles, potential energies of the particles relative to external fields,

internal energies,

energies

and perhaps

others.

=

E,

where the

e's

are the

many

This fact €,

+

intermolecular

molecules,

various

of the

e,

+

may €,

potential

be written

+ •-,

(7-2)

small contributions.

MATHEMATICAL IMPLICATIONS To prove: A

necessary condition for the use of Eq. 7-1

only dynamical feature on which the probability of a

be the energy of the

state.

(This condition

is

the

state

same

is

that the

may depend

as that given

by

Eq. 6-1.)

Proof: Simple examination of Eq. 7-1 shows the statement to be true.

The quantity a is determined by normalization, and the quantity /? shown in Sec. 9 to be l/kT, where T is the absolute temperature and k a constant. The only thing left in Eq. 7-1 is £,. Q.E.D.

To prove: A

necessary condition for the use of Eq. 7-1

is

factor, with a separate term for each contribution of Eq. 7-2.

is

is

that P\iEi)

Basic Theory

28

Proof: Inserting Eq. 7-2 into Eq. 7-1 yields

=

P^^,(i)

ae-"^'

To prove: The two

e-^''

e'^'^

Q.E.D.

.

(7-3)

necessary conditions just given are also sufficient

conditions for the use of Eq. 7-1.

two necessary

In other words, the

conditions are enough to force the canonical form, Eq. 7-1.

Proof: The

necessary condition

first

^.v(0

The second condition becomes

=

Py(E)

is

that

=

Py

^2

+

•

•)

=

^i(^i)

In P.vC^i

+

€2

+

•••)

=

In Pi{€,)

is

•

•

(7-4)

•)•

•

+

•

•

,

If

Eq. 7-6

this

(7-5)

•

In P^ie^) -f

7-5.

7-4,

•

is

•

•

(7-6)

.

differentiated

becomes d In P.v(£)

_

ajnP/e,) dej

dej

d In Pv(£)

dE

Be

dej

dE Since the left-hand side of Eq. 7-9 just a function of £,

for the differentiation.

d\n

P.jej)

(7-8) dej

a In P/e,) (7-9) Be,

in no way a peculiar function of ej made no difference in Eq. 7-7 which e

is

clearly

it

_ _ ~

d In Pv(£)

was chosen

+

A(^2)

•

simply the logarithm of Eq. €j, it

^2

Coupled with Eq.

factor.

+

Equation 7-6

is

+

Pyi^i

^.v(^i

with respect to

but

Eq. 6-1

is

The

result

would be the same. Thus

the right-hand side of Eq. 7-9, which at most could be a function of

must not even depend on

e^

and must be a constant. The

^

=

-'

BE If the left

we

and

=

result

constant.

is

e^,

simply

(7-10)

Bej

right sides of Eq. 7-10 are multiplied

by dE and integrated,

obtain

lnPv= -^^+ where the constant of Eq. 7-10 of Eq. 7-11

may

is

called

—

/?.

be called In a and Eq. 7-11

Pv(£)

=

fl^""^-

(7-11)

^,

The

may

integration constant

A

be rewritten

Q.E.D.

(7-12)

7.

System

in

Equilibrium with a Heat Bath

29

PHYSICAL IMPLICATIONS The

implications of the

first

condition for the vahdity of the canonical

ensemble, Eq. 7-4, are discussed in Sec. that

PyiE) must

factor, with a separate

6.

The only other condition

is

term giving a separate independent

probability for each contribution in the sum, Eq. 7-2.

This condition

is

seen to be met as soon as one has a picture of the

mechanical purpose of the heat bath. In Eq.

7-1

it is

clear that states with

any value of E^ have finite probabilities. There is no restriction whatever on the size of E^, as there is in the microcanonical ensemble. This is because one does not

know

The system might

the energy of the system.

indeed have any energy. All that

is

known

is

that the system

is

in equilib-

rium with a heat bath. This bath must be capable of giving up or receiving any finite amount of energy without appreciably changing. Therefore, one pictures the heat bath as an infinitely large* system at equihbrium. The last remaining question is why such a heat bath permits Py to be factored, as in Eq. 7-5. First we must ask what is present in the system to correlate explicitly the contributions in the various e's, and thus prevent

For example, what physical feature might correlate amount of kinetic energy of some particle with the amount of rotational energy of another particle way off in another part of the system? In fact, this factorization.

the

it is

the

interesting that whatever this physical feature e's

This

defined.

sum of

must correlate

is, it

all

with each other and in exactly the same way, however they are is

a consequence of the fact that

the various

symmetric

in the

Any

e's.

Py

function of the

is

a function only of the

sum must be completely

terms in the sum.

would be so?ne For example, if the system were isolated, its energy would be fixed. Thus, all the e's would have to add to give £'^j,.(^,,„. In that case, if one of the e's were an appreciable fraction of the total energy available, this would be felt as a limitation on Clearly, the only correlation of this type imaginable

requirement on the total energv of the system.

the energy available to

all

the other contributions.

heat bath eliminates this type of correlation. physical feature that could correlate the

The presence of

the

There being no other

e's explicitly,

Py must

factor,

and the canonical form follows. * Actually, is

it

makes

little

or even whether there

is

difference for macroscopic systems

one.

Due

to the

how

enormous number of

large the heat bath

particles contained in

macroscopic systems, the probability that the system energy departs appreciably from its This makes the is negligible, even if an infinite heat bath were present. canonical ensemble useful for calculations even when the heat bath is either small or non-existent. This is proved and discussed in Sec. 33. However, rigorous use of Eq. 7-1 clearly demands an infinite heat bath. average value

Basic Theory

30

The canonical distribution differs from the microcanonical in that the requirement of microscopically sharp total energy is removed. The system may have any energy. However, all A'^-particle quantum states with the same energy have the same probability. We might think of a canonical ensemble as constructed from a large number of microcanonical ensembles, one for each different possible value of E, weighted according to Eq. 7-1.

Both the microcanonical and the canonical ensembles represent a system with a fixed number of particles A'^. Gibbs called these petit is ensembles, in contrast with those in which the number of particles ensembles, they may be congrand and not fixed. The latter he called

N

sidered as

composed of a

large

number of

give the desired dependence of the

Grand ensembles

petit ensembles,

probability on

the

weighted to

number of particles.

are considered in Sec. 33.

In concluding the discussion of the canonical distribution, it must be emphasized that the exponential form, Eq. 7-1, still may represent corre-

between parts of the system; not explicit, but implicit. For example, consider two particles. Of the group of e's, one may be chosen to represent the potential energy of interaction between these particles. When they are close together, this is large and affects markedly the lations

probability of the configuration.

and may

probabilities

When

they are far apart, this

is

small

two particles are correlated, in that involving one of them depend on the position of the other

be neglected.

Clearly the

through terms in the energy involving both their positions simultaneously. This type of correlation is discussed further in Sees. 26 to 28.

summary, quantum states,

the

In

and

of probabilities

canonical distribution

as given in Eq. 7-1,

is

of N-particle

a reflection of the following necessary

sufficient conditions:

The probability of an U-particle quantun) state is a function only of the energy of that state. Note: It must be meaningful to talk about separate quantum states for the system. If the system is interacting too 1.

strongly with

its

surroundings for this to be so, then the theory breaks

down. 2.

The systen)

is

only an

infinitesimal part of a composite equilibrium

situation consisting of system plus heat bath.

bath

is

The only purpose of

the heat

to serve as an infinite source or sink of energy.

PROBLEM of the information initially equilibrium conditions. Equations 6-2 and 7-1 give the probability distributions that represent the first 7-1.

At the beginning of

known about

Sec. 6, there

three different

is

a

list

commonly encountered

8.

two of

Exponential Reduced Probabilities

Discuss for each of these two

these.

enters the probabihty distribution.

was not

distribution that

8

initially

Is there

how

each

bit

of

initial

31

information

any information in the probability

known?

EXPONENTIAL REDUCED PROBABILITIES

In

many

quantum

cases, the probabilities for

We

the complete A^-particle system are not necessary.

states

of

are sometimes

how much of the total energy is present in one or two e that make up E. This section considers partially

interested only in

of the contributions

when

the question of

the reduced probability for a few energy contribu-

tions can legitimately be expressed as

P,(.,)

Equation e~^^'

by

The

8-1

the well-known

is

often called the

itself is

=

oie-P-^.

(8-1)

Bo/tzmann distribution of energies, and Boltzmann factor.

desire to look at just a few of the e's necessitates a

parameters used to describe the ensemble.

quantum

state

/,

we may

give the values of

than

is eg

and the

10^^ e's in the

third

7-1,

first

contribution

and .... Since there

is eg

sum, Eq.

our interest

is

in the very

probabihty describing the expected distribution of energy

One

these parts of the system. that

it

this condition * Often a

mere

is

to each

not

listing

e's

It is

nothing new to the analysis of

it

here.

It

must be emphasized is

in

few of is

as this joint probability, with physical particles,

discussed in Sec. 22.

does not uniquely determine the quantum state for the states may have the same set of e's. This introduces

quantum

this section, since

permits the formulation of Eq. 8-1 the system

/

For a system composed of identical

e.

trivial.

of the

system. Several different

much reduced

in just a

condition that must certainly be met

be meaningful to view state

meaning attached

in the

Then P\(Ej) may is e^ and may well be more

the e's.*

all

be thought of as the joint probability that the the second

change

Instead of simply listing the

a particular

in

a slight complication of the notation

complete analogy to the way we have formulated

that Eq. 8-1 gives the probability that a small part

quantum

state

which has energy

e^.

party have the same energy, each has the same probability, Eq.

/of

If several states for

8-1.

Basic Theory

32

For the moment, suppose the system

be represented by a canonical

to

ensemble, P,.(/)

=

This complete probabihty allowed values of

the

all

Pj(^j)

may

=

"

ae^"'''^''-

This

e^.

11---1

a

(8-2)

'. '

be reduced to Pj{€j) by

except

e's

=

a^-^^'

is

summing over

all

analogous to Eq. 4-18:

e-/'("+^--->.

(8-3)

€1,62. ... except €j

In order for Eq. 8-3 to simplify to Eq. 8-1, a second condition

The

sets

of values of

e's

be independent of the value of physical systems, but in is

discussed in Sec.

17.

€j.

many cases

When the

sum just gives new constant a:

the multiple yield the

P,(6,)

This condition

may

it

necessary:

=

is

generally not

A which may

22

ae-i^'^

=

ae-''^'

=

ae-^-^

.

.

•

•

.

met

in

be approximated. The problem

approximation of independence

a constant

Cj, 62,

F,(6,)

is

over which the summations are performed must

•

^"'"^"'^'

2

except

is

made,

be combined with a to

(8-4)

e,-

A,

(8-5)

(8-6)

With this result, the Boltzmann distribution, we can say that the particles "obey Boltzmann statistics." The last question we have on the validity of the Boltzmann distribution is whether one must start with a canonical ensemble for the system. Must the system be in equilibrium with a heat bath before the Boltzmann is valid ? This question can be easily answered (in the negative) back to the purpose of the heat bath, as discussed in Sec. 7. It was seen there to be simply a source or sink of energy which permits any value of the total energy to be possible. Suppose the system is isolated; then the microcanonical distribution properly describes it, and there is

distribution if

we

refer

the restriction ^1

For

Pj{€j) to differ

lead to party getting

ment of Eq.

8-7.

+

62

from Eq. its

+

8-1,

^gj.gfe^.

We

•

=

jEsystem.

(8-7)

however, the physical processes that

share of the energy must explicitly feel the require-

If part

J

is

only an infinitesimal portion of the total

system, the interesting values of

of

•

its

energy

would not expect part

y's

will

be infinitesimal fractions

energy to be influenced by

Eq. 8-7 under such circumstances. All the other parts of the system, with which part j is interacting, may be viewed as an effective "heat bath." This permits the reduced distribution function to be exponential, even

Exponential Reduced Probabilities

8.

though the system as a whole

is

We may

isolated.

33

incorporate this

Boltzmann distribution

physical conclusion into the theory by using the

for infinitesimal parts of even isolated systems, so long as the other

conditions for the validity of Eq. 8-1 are met.

With an

isolated system, very large values of e

probable than as given by Eq. 8-1 because of the

would be even

less

of the restriction

effect

Eq. 8-7 in eliminating vV-particle states from consideration.

In fact, e's

greater than ^'system rnust have zero probability, rather than the exponential. However, for systems containing an enormous number of

by the time

particles, e~^^ is already so small

e

becomes comparable to

^system that the difference between the exact probability and the exponential is

completely negligible.

In summary, the Boltzmann distribution of reduced probabilities for a portion

of the system, as given

in Eq. 8-1

is

,

a reflection of the following necessary and

sufficient conditions

The probability of an N-particle quantum state

1.

is

a function only of the

energy of that state.

The part

2.

j

of the system whose energy

being considered has physical

is

significance.

The amount of energy

3.

ej

present

of the total energy available to the

N

in

part

is

j

only an infinitesimal fraction

particles.

The sets of values of energies the other parts of the system may have are

4.

for all practical purposes independent of the value of

€j.

Consider a system for which the Boltzmann distribution

is

valid.

being so, the ratio of the probability of finding a given particle to that of finding

it

in state y

=

e-P^''-''\

In a very loose usage of language, Eq. 8-8 ratio of the populations of the

have (^ is

e^

—

Clearly,

shown

€_,

»

1//1

in Sec.

it is

is

level

is

is

(8-8)

commonly

This usage

states.

most unlikely

The "natural"

Sometimes several an energy

two

is

measure of energy

way

to say this

is

/

is /i"^.

which Since

that the "natural"

IjkT.

distinct particle states

called degenerate.

Its

have the same energy.

degeneracy

is

equal to the

of different particle states with that energy. Suppose level gi

said to give the

discussed further

for particles to be in states

unit or

9 to be l/AT, another

unit of temperature

/

is

Z(£l)

in Sec. 22.

This

in state

and leveiy a degeneracy

gj.

Then

/

Such number

had a degeneracy

the ratio of the probability of finding

Basic Theory

34

a given particle

energy

in

a state with energy

^^(^')

3

It is

e,

to that of finding

it

in a state

with

€j is

en.)

e

=1

important to realize the difference between what

Eq. 8-8 and what

is

is

expressed by

The introduction of

expressed by Eq. 8-9.

the de-

generacies g, often called the statistical weights, redirects one's concern

from quantum used

in this

to the other

states to

energy

book, although is

it is

levels.

in

The energy

easy, however, because the

to a degenerate level

number of

simply the degeneracy of the

is

level

language

is

not

many. Changing from one language states contributing

level.

PROBLEMS Discuss in your

8-1.

own words the reason why reduced probabilities for may be considered as exponentials of the energy.

isolated equilibrium systems 8-2. Discuss the difference

exp

9

(

between the following two mathematical expressions

— 2 eJkT]

and

^ exp

(

— eJkT).

ENERGY AND ENTROPY IN STATISTICAL MECHANICS

In this section the statistical mechanical analogs of the

energy and entropy of thermodynamics are found, giving a mechanical interpretation to

all

of thermodynamics.

The presentation

is

limited to

systems represented by a canonical ensemble, and the thermodynamic variables are found as averages over this ensemble.

These would be the

logical values to predict for a system in equilibrium with a heat bath.

may not how near

As

noted

in Sec. 3, the

ensemble average

always agree with what

found

in a system.

The problem of

the ensemble average the

is

9.

Energy and Entropy

thermodynamic variables are expected

to

in Statistical

lie is

Mechanics

35

Also,

treated in Sec. 33.

the use of canonical ensembles to represent even isolated systems discussed there. These problems need cause no concern until then.

The

quantum

probability of the /th A^-particle P^.(/)

where a

=

reciprocal of the

books, Z) and

t

is

=

1

(9-2)

le -pEi

normahzing constant

simply the

sum over

a

The sum

Q

thermodynamic it

given the symbol

all states

may

the Zustandssumme,

some

(9-3)

statistical

thermodynamics, because

be calculated from

and

(in

of the exponential:

as a function of the variables on which

quantities

Q

i

has a central role in

known

it is

is

2 "-"''

2 = 1=

called

(9-1)

the normalizing constant:

is

a

once

is

ae-f''^

i

The

state

is

it

it

depends,

directly.

for a while (Tolman) there

that the natural translation, sum-over-states, would

become

all

Planck

was hope

its

English

However, the name partition function, used by Darwin name. and Fowler for an analogous quantity, has become generally accepted for Q.

a function only of jj and of the set of available quantum corresponding energies. The set of states in turn is found in principle by using quantum mechanics and is a function of only the number and nature of the particles comprising the system and the conClearly,

states

and

straints.

^

is

their

The only

constraint on systems considered in this

book

is

the

volume; however, others could easily be incorporated. In summary, is a function of /5, of V, and of the number and nature of particles.

Q

In terms of the partition function, the probability of a state, Eq. 9-1,

becomes Py(i)

=

-

.

(9-4)

Q In terms of Q, the ensemble average of the energy of an TV-particle state

is

36

found

Basic

Theory

in the

manner of Eq.

4-9:

£=1 PyO)Ei = ^--z

9.

Use of

this in

Energy and Entropy

d{ln

One may view

+

Q

_„

E^)

=

^

(9-11)

dqrev.

of state, the entropy

S.

The

the differential

left-hand side of Eq. 9-11

ential of a function of state (that

number and kind of

the reversible heat possesses

makes ^q^^jT

integrating factor 1/T, which

is

the

of a function

indeed the

differ-

a function which depends only on the

is,

particles, the

must be proportional to

temperature.

37

Eq. 9-11 as a statistical mechanical expression of part of

the second law of thermodynamics

(i

Mechanics

Eq. 9-9 permits expressing the reversible heat in statistical

mechanics:

fore,

in Statistical

volume, and the temperature). There-

where Tis the absolute thermodynamic

l/T,

If the proportionality constant

is

called Ijk, then

(9-12)

Choice of ^ essentially fixes the size of the unit of T. This is conventionally done by reference to the properties of the "ideal gas." In both Sees. 13 and 18 this reference is made and the value of k, called Boltzmann's constant,

The

is

determined.

left-hand side of Eq. 9-11

may

be identified as Ijk times the

differ-

ential of the ensemble! average of the entropy:

dS If this

is

integrated

==

d{k\nQ + -

from zero temperature to

(9-13)

T,

an expression

is

obtained

for S:

S= The

klnQ + - +

(9-14)

81.

integration constant ^o does not depend either

states or

on

T.

It is

convenient to

set

discussed in greater detail in Sec. 10.

on the

^o equal to zero.

The entropy

set

of quantum

This practice

in statistical

is

mechanics

then becomes simply

S

= klnO + -

.

(9-15)

T

In this section, the statistical mechanical expressions for the energy

and entropy of a system have been obtained.

Also, the parameter

/5

of

previous sections has been proved proportional to the reciprocal of the

The overbar has been used to indicate that E and ensemble averages of the energy and entropy. However, they must be the same as the E and S of classical thermodynamics. While the

absolute temperature.

S are

Basic

38

Statistical

Theory

mechanical picture permits fluctuations in the measured

E and S

of physical systems, the picture of classical thermodynamics does not allow this. Therefore, the unique values of E and 5" in thermodynamics

must be the

E

and S of statistical mechanics. This question is discussed and 34. The overbars could be dropped at this time,

further in Sees. 33

but they are retained for

examined

in greater detail

10

clarity.

In the next section, the entropy

is

and the third law of thermodynamics considered.

ENTROPY AND THE THIRD

The entropy of a system

as

LAW

given by Eq.

9-14

is

undetermined to within an additive constant Sq. This constant is independent of both the temperature and the set of available iV-particle quantum states. Therefore, if some final condition for the system could

be reached by an imaginable reversible path from the

initial

condition of

would drop out in determining lS.S for the process. The imaginable path must be reversible, since the results of Sec. 9 are valid only for reversible changes; the system must be represented by a the system, then Sq

canonical ensemble during the entire process under consideration.

A5

calculating the entropy change

path could be imagined, choice of sets

Sq equal to zero. This

is

for

In

any process for which a reversible

5*0 is

immaterial.

For

simplicity,

one

by no means necessary; the Helmholtz and

Gibbs free energies could simply be redefined to ehminate S^. physical content of these equations

is

unchanged by neglecting

S^.

The This

applies even to the third law of thermodynamics, discussed below.*

A

more

intuitive

way of

writing

S may

be obtained from Eq. 9-14 or

9-15 using the following identities: In

2 = -In a,

lFv(0 =

l,

(10-1)

(10-2)

i

Ei= -kT\ne-^^\ *

For an

interesting discussion of the value of 5o, see Schrodinger, pp. 15-17.

(10-3)

Entropy and the Third Law

10.

One

finds, starting with

5=

39

Eq. 9-15,

/clnQ

+

E T 1

T

i

= -^1 PxiO In a - 2 ^.v(0 In e'"^' = -fc2^.v(01n(fl£'-^^0, /c

i

i

S=

-k^PyiO^nPyO),

(10-4)

i

S The entropy of

=

-/cirTP^.

the system

is

(10-5)

proportional to the ensemble average of the

logarithm of the probability of an A^-particle

state.

This average

is

a

closely one can pinpoint the state he would expect the A high degree of predictive ability regarding the system

measure of how system to be in. implies a low entropy and vice

versa.

this section

by some

The more quantum

This

to a system, the higher the entropy.

is

states available

illustrated in the

remainder of

specific considerations.

Suppose the system is known for certain to be in a particular quantum state, state y. Then, in the summation of Eq. 10-4, Pv(/) is zero for all states buty; Pyij) is unity, so In P-^ij) is zero. Therefore, S is zero (or else ^o)

quantum

whenever the system

is

definitely

known

to be in a particular

state.

might be possible under certain conditions to know the quantum a system is in. At the absolute zero of temperature, the only states allowed non-vanishing probabilities by the canonical ensemble are those It

state

with the lowest lying energy.

same probability 1/^", where

Each

^

state with this lowest energy has the

is

number of

the

states with the lowest energy (in other words, A'"

is

different A^-particle

the degeneracy of the

lowest energy level for the system). At absolute zero, the entropy

by Eq. 10-4 as

^(0°K)=

S(0 K)

Quantum mechanical to be unity,

given

-/cf—,ln

= -kin

^'

is

,.

=

k\nJV.

calculations for

and thus 5(0°K)

— (10-6)

some models of

to be zero.

A

crystals

show

generalization of this,

Basic Theory

40

5

that

^

as

r -> 0,

is

known

thermodynamics as Planck's statement

in

of the third law. In light of the discussion of Sq above, Planck's statement would be better replaced by the following: A perfect crystal is defined as

^

one for which

is

unity.

Then, the entropy change AS

is

zero for any

process at absolute zero for which a reversible path could be imagined, if

The mechanical formulation

the reactants and products are perfect crystals.

of the third law

The question not, but lots of

seen as a consequence of Eq. 9-14.

is

The answer, probably number of features involving

are there any perfect crystals?

is,

them come

A

close enough.

the entire macroscopic crystal could lead to values of

.>''

other than unity.

For example, crystals with permanent magnetic moments may be aligned in any one of several directions with the same energy. It is believed that, for most crystals completely in equihbrium at 0°K, ^V is very small. even as large as in the miUions or billions, 5(0°K) for However, for the crystal, as given by Eq. 10-6, is still of the order of k. In Part II, k is found to be about 3.3 X 10"^* cal/deg. Experiments measure entropies to at best 10"^ cal/deg. Clearly, J'^ must be of the order of lO^"'" before the residual entropy of a crystal at absolute zero becomes measurable. How might such an enormous value of ^K arise ? The degeneracy must not be so much a property of the crystal as a whole as a property of the

^

particles that

enormous

size

make of

A^,

As an example,

it

up.

the

In this way, advantage

number of particles

may

be taken of the

in the crystal.

consider crystalline carbon monoxide.

them

While they

CO molecules have enough energy that is immaterial direction they enter the lattice, CO or OC. After the

are freezing out, the

it

which formed and the temperature greatly reduced, the energy differences between a perfect and a random array may seem more important, but then it is too late for the CO molecules to turn around. They are frozen in. The resulting crystal with randomly oriented molecules may not have

to

crystal

in

is

lowest energy level quite so low as a

its

difference

is

not measurable.

If

an

more regular quantum

jV-particle

by giving the direction of orientation of the

N lattice points in the

crystal, then there

is

but the specified

molecule at each of the

are 2^^ different

monoxide

available to the crystal of carbon

CO

crystal,

state

quantum states A number

at absolute zero.

of this immensity leads to a measurable residual entropy:

S(0°K)

N

=

fc

In

2^

= Nk In 2.

Avogadro's number, this is 1.38 cal/mole-deg. This has actually been observed,* as follows: The entropy of gaseous CO at room temperature was calculated by the methods of Part II of this

If

*

J.

is

O. Clayton and

W.

F. Giauque, /.

Am. Chem. Soc,

54, 2610 (1932).

10.

book.

The

result

Entropy and the Third Law

41

was compared with the entropy found by integrating 0°K up to room tem-

the experimentally determined ^q^^^.jT from almost

The experimental value proved

perature.

by

to be less than the calculated

This suggests that

cal/mole-deg.

1.0

l.OA^/1.38

particles

entered the crystal as

units.

Some of the

A^ total

ordered in

entered the crystal;

others

random dimers when they

actually

may have been may have had

time to rearrange before the temperature got too low.

The above can be

CO.

as with

Another

(where the molecules states, all

A

generalized very easily.

One

degeneracy in any one of several ways. is

still

through

at

with the same low energy.

through

its

introduce a orientation,

having several states for rotation

its

can rotate

may

particle

is

almost 0°K), or several electronic

If the crystal contains

N^

particles

with particle degeneracy g^ and N^ particles of type 2 with particle degeneracy g^ and so on, then the residual entropy is

of type

1

^(0°K)

=

N^k

In gr

+

N^k\ng^

+

---

(10-7)

.

Examples of residual entropies other than that of CO are plentiful.* is a linear molecule and has residual entropy of

Nitrous oxide, N2O, A^^ In 2.

On

entropy of

the other hand, nitric oxide,

\Nk

In 2.

NO,

has experimental residual

This indicates that just half as

expected are orienting randorrily.

One assumes

many

that the

particles as

NO

molecules

dimerize before crystallizing out as either

NO ON The

crystal of

CH3D

ON °^

NO.

has residual entropy of

Nk

In 4, since there are four

which the deuterium may lie. Hydrogen presents a complicated case, because normal hydrogen is a mixture of three parts of socalled ortho hydrogen {g = 3) and one part of so-called para hydrogen {g = 1). Only the IN of the molecules that are para hydrogen contribute directions in

to the residual entropy

^Nk

In 3.

PROBLEM Consider the residual entropy of carbon monoxide. Inability to prequantum state at 0°K led to a measurable residual entropy. However, one of the possible quantum states is the completely ordered one. If, unknown to the experimenter, the system had happened to get into that state, what would the experiment have shown? Does this mean that more heat would have to be added to raise that system to the same final condition as the one in the text ? If so, where would the excess heat go ? Considered in this light,why should measurements of residual entropies be reproducible? 1

0-1.

dict the precise

*

An

extensive discussion

is

given by Fowler and Guggenheim, pp. 191-229.

42

Basic Theory

GENERAL FORMULATION OF

II

STATISTICAL

THERMODYNAMICS

The ensemble average of

the energy of a system

is

given in terms of the partition function in Eq. 9-5, and the entropy

One might

given by Eq. 9-15.

ask, then, whether

it is

is

possible to express

the ensemble averages of all thermodynamic variables in terms of the partition function. This would reduce the entire problem of statistical thermodynamics to the single one of calculating the partition function. The basic thermodynamic variables one would like to predict from statistical mechanics are the energy E, the enthalpy H, the Helmholtz free energy A, the Gibbs free energy G, the entropy S, the specific heats Cy and Cp, and the pressure P. With Eqs. 9-5 and 9-15, which already give E and S, the rest of these are easily found. The easiest starting point in statistical thermodynamics is the Helm-

holtz free energy:

A From

Eq. 9-15,

this is

= E-TS.

(11-1)

seen to be

A= E-Tk\nQ- E, A = - kT In This equation statistical

is

important because

it is

(11-2)

Q. a simple

way

to

remember the

mechanical expressions for the thermodynamic variables.

If

remembered, the rest can be derived in the following way: If a system consisting of N^ particles of type 1, A^2 particles of type 2, is caused to undergo infinitesimal reversible changes in temperature, volume, and numbers of particles, then the change in Helmholtz free energy is given by Eq. 11-2

is

.

The summation

is

over

the symbol dAjdN^ particles of type

/

is

all

types of constituent particles.

the change in A^^

-|-

.

.

The meaning of

caused by increasing the number of

1 The meaning of the symbol dN^ number of particles of type /.

from N^ to

the actual change in the

A

.

is

II.

General Formulation of Statistical Thermodynamics

This purely mathematical equation

dynamic counterpart, one of the

may

so-called

dA= -SdT-

be compared with Maxwell relations:

PdV+J^l^idN,.

its

43

thermo-

(11-4)

i

The symbol

//j is

the molecular chemical potential of the /th constituent.

from the molar chemical potential or partial molar free energy customarily used by chemists in being smaller by a factor of Avogadro's number. It is consistently used in this way throughout this book, because It differs

the emphasis in statistical mechanics

is on the particle nature of matter. Comparing Eq. 11-4 with Eq. 11-3 permits the following identifications, known in thermodynamics as some of the so-called Gibbs relations:

Coupled with the simple form of ^4, Eq. 11-2, Eqs. 11-5 give simple ways remember S, P, and fx^. In addition, Eq. 11-1 gives E\

to

E=A H and

G

+

TS;

(11-6)

are given by their definitions:

H = E + PV; G = H - TS = A + PV; and the heat

capacities are given

by

c;=(^)

(11-7) (1 1-8)

differentiation

;

Cp=l^) = (-)

.

(11-9)

\dTL,.,

dT/y^Xi

At this point we give a summary of the ensemble averages of the various thermodynamic variables* (see top of next page). These equations show why the partition function is so widely used to calculate thermodynamic variables from properties of the molecules. Whenever one has a system which is described by a canonical ensemble, if e~''^i

may

be conveniently

summed

over

all

states to yield

Q, then the

thermodynamic variables follow immediately. Needless to say, this has not solved the problem of equilibrium statistical mechanics. What it has done is to establish a framework, throwing the entire problem into the calculation of the one quantity Q as a function of the variables on which it depends. Of course, finding Q for systems other than ideal values of

*

all

Note, for any function of temperature ^'^(T),

d^^d^d£^ldj_ ddlT) _ _ _1_ dT

diiclT

kdjj

clT

^

kT^d[i'

44

Basic

Theory

1-10)

H=

o(d

kT-

\nQ dT

+ kTV

a In

6 dv

V,Ni

= -kTlnQ; dlnQ G = -kT\n Q + kTV

A

dV

1-11)

1-12) 1-13) )

^

'T,Ni

d\nQ 1-14)

S=k

d{T\nQ)

dT

= k\nQ + V,Si

-

1-15)

T 1-16)

1-17)

P=

ain(2 1-18)

fcTI

5^

/r,.v,

by no means easy, and in most cases it represents an unsolved problem. Much of the rest of this book is devoted to studying this problem gases

is

for various kinds of systems.

PART

II

IDEAL GASES

THE IDEAL GAS

12

Introductory presentations of thermodynamics always spend a great deal of time treating the properties of the so-called ideal gas. Although no real gases have exactly these properties, there are The properties of the ideal gas are still good reasons for treating them. sufficiently simple that they

do not add to the already serious problem

of understanding the thermodynamic concepts. Furthermore, the behavior of many real gases approximates that of an whose behavior does not are most naturally

ideal gas.

Those

real gases

treated by equations similar

to the ones for ideal gases, simply with corrections

added for the "degree of

non-ideality."

The

ideal gas

is

on the extrapolated

a hypothetical substance, based

behavior of real gases in the limit of vanishing densities.

Many studies how the

of the equations of state of real gases have been made, showing pressure P, the temperature T, and the density limit of vanishing density, the ratio

approach

1/^,

where

the

is

/?

PVjN

NjV

are related.

for all real gases

is

In the

found to

thermodynamic integrating factor for the

reversible heat, developed in Eq. 9-11

lim

PV ^-^ =

N

density-,

The

1

=

-

scale of temperature or size of a degree

at its triple point.

(12-1)

is

conventionally determined

side of Eq. 12-1 for a gas in equilibrium with

by studying the left-hand water

kT.

/?

Boltzmann's constant k

is

fixed

by the relation-

ship

lim density-,0

= — N

(273. 1600 deg)/c

for real gases in equiUbrium with water at

value of ^ found experimentally

Equations such as these or in terms of moles.

may

is

1.38

its

x

triple point.

47

The approximate

lO"^" erg/deg.

be written either

In the latter case,

(12-2)

in

terms of molecules

one employs the universal gas

48

ideal

Gases

constant R;

R= where

jVq is

Nok,

(12-3)

Avogadro's number, the number of molecules contained

in

a

mole.

Understanding the ideal gas

on an understanding of

in terms of molecular theory

real gases.

has led to the following picture of real gas molecules are objects which interact with each other.

20

must be based

Interpretation of experimental data

At

:

In general, molecules

fairly large distances,

two

13.

Translational Energy and Pressure

Real gases approach ideal behavior

As

the density

is

decreased, the

in the

10

A

fields

of force

is

enough

close

of each other) gets smaller.

Finally, in the limit of vanishing density, the

each others'

Hmit of vanishing density.

number of pairs of molecules

to interact appreciably (say, within

49

number of molecules within

completely negligible. The fact that molecules

interact with each other has

no bearing on the properties of a gas

in that

limit.

We therefore choose to define an ideal gas an one in which the molecular interactions through intermolecular forces are ignored.

molecular potential energy

is

Since the inter-

neglected, the entire energy of the system

is

contained in the kinetic and internal energies of the particles themselves.

Under the conditions

cited in Sec. 8 for the validity of exponential

to the

two properties of an

reduced

shown

in Sec. 13 to lead

ideal gas often cited in

thermodynamics

distribution functions, this mechanical picture

is

books: 1 The molar energy of a given gas and not of the volume:

2.

The thermal equation of

state

is

a function only of the temperature

is

PV =

NkT.

(12-5)

CLASSICAL GAS-

13

TRANSLATIONAL ENERGY AND PRESSURE

In Sees. 13 to

16

a classical mechanical picture

used to learn a surprising amount about ideal gases.

It is

mechanics actually governs molecular dynamics, and the

ment of

Sees. 17 to 19 duplicates

some of

Sees. 13 to 16.

the classical picture gives considerable physical insight.

more

closely the type of

argument the reader

will

is

quantum quantum treat-

true that

Nevertheless,

Also,

it

encounter

resembles if

he ever

50

Ideal

Gases

Studies non-equilibrium statistical mechanics, is

and physical

useless

where the partition function

insight vital.*

In this section, the classical ensemble for describing the translation of

molecules is

is

discussed and the resulting one-particle distribution function

described in

When

some

particle

is

found

The normalized distribution function is found. ensemble average of the translational energy of a

detail.

this is used, the

to be \kT.

The energy of the system

tion only of T, not of V, as

is

ensemble average of the pressure

is

Eq. 12-5 as equation of state

is

through

statistical

is

required by Eq. 12-4.

if

/5

seen to be a func-

Furthermore, the

also obtained, yielding precisely

equated to XjkT.

This

identifies,

mechanics, the absolute thermodynamic temperature

of Eq. 9-12 with the ideal gas temperature, and it identifies the constant k of Eq. 9-12 with Boltzmann's constant, obtained experimentally from Eq. 12-2.

THE CLASSICAL ONE-PARTICLE DISTRIBUTION FUNCTION In a classical ensemble, ignoring the internal structure of the molecules, the parameters describing one particle are the three position coordinates

and the three momentum coordinates. For brevity these are written as vectors r and p respectively.! Since these parameters are continuous, a one-particle distribution function of the form treated in Eqs. 4-22 to 4-29 is needed. The joint probability that some particle of interest has its x coordinate between x and x + dx, and has its ij coordinate between ?/ and y + dy, and has its z coordinate between z and z -\- dz, and has its momentum component in the x direction between p^. and p^ + dp^, and has its momentum component in the y direction between py and py + dpy, and has its momentum component in the z direction between p^ and p^ + dp^ is fi{x, y,

The

z,

p^, Py, p^)

dx dy dz dp^ dpy

dp,.

(13-1)

subscript on/i refers to the fact that knowledge of/i as a function of

the variables on which

it

depends gives

full

information about a single

Usually, /i is called the one-particle distribution function, sometimes the singlet distribution function.

particle.

*

Certain aspects of this classical mechanical study are frequently referred to as the of gases, but different people use the phrase with different meanings.

kinetic theory

Because of this lack of agreement, the phrase l I

_l_

^-^ (i-a)-' (ri)j A-:z-_

(21-4)

p(ri)

/ii(ri) is

often

more

interested in pressures than in

are easily related to each other through a modification of

the ideal gas law,

P = -kT,

(21-5)

V which was derived

in Eq.

varying gravitational slightly.

The

field,

13-21 for a field-free system.

the derivation of Eq. 13-21

tiny region of Fig. 13-3

must be

If there

is

a

must be modified

truly infinitesimal.

Then,

Gas

21. Ideal

in

a Gravitational Field

over so small a region, the change in gravitational potential

107

may

be

This changes the derivation between Eqs. 13-19 and 13-21

neglected.

only in replacing l/V, the spatial part of/^, by

=

A(r, p)

/7i(r):

„,(r)(-^yV^-'^^'".

(21-6)

\277m/

This replaces the equation of state, Eq.

13-21,

by the more general

expression,

P=

Nn^{r)kT

=

p{x)kT.

If the gravitational potential arises only

height h

from the

earth, then i^ij) at a

is

-Tir)

The

(21-7)

acceleration of gravity

=

mgh.

(21-8)

g has the value 980 cm/sec^

at sea level.

Use

of this with Eqs. 21-4 and 21-7 yields ^("2)

_

P(h)

^("2)

_

^-mg(h2-hi)/kT

_

g-Ma(h2-hi)/RT^

piK) (21-9)

Equation 21-9 gives the ratio of the pressures or densities at two different heights in the earth's gravitational field for an ideal gas of molecular weight M, each molecule of which weighs m. The system must be at equilibrium; therefore the temperature must be the same at both heights. Also, the acceleration of gravity heights.

The

g must vary

negligibly between the

two

density or pressure becomes smaller the higher one goes.

For gases with greater molecular weights, the drop-off than for lighter gases. barometric equation.

is more abrupt Various forms of Eq. 21-9 are often called the

The equation may be derived from purely thermodynamic arguments, which ignore the particle nature of the gas

completely.

The theory of

ideal gases has

been applied, interestingly enough, to

dilute suspensions of colloidal particles in liquids.

Colloidal particles are

huge by ordinary molecular standards, from about 10 A to 10,000 A. If they are suspended in a liquid, the constant bombardment by molecules of the liquid prevents their settling out under the influence of gravity. In dilute concentration, the colloidal particles prising a gas.

The

glected, except for

Of gas

is

liquid

its

is

may

be treated as com-

medium and is neon the mass of the colloidal particles.

treated as a continuous

buoyancy

effect

course, real colloidal particles interact with each other;

not ideal.

It is

thus the

observed, however, that as any particular property

Ideal

108

Gases

of a colloidal suspension

plotted against concentration, the extrapolated

is

value of this property at zero concentration takes the value if

it

would have

the gas were ideal.

For

densities of colloidal particles, Eq. 21-9

is

useful in the low density

limit,

p(h^)

where

m*

is

=

p(ft,)e— *^"'^-''i'/^^,

(21-10)

the particle weight corrected for buoyancy, that

is,

the

mass

of the particle minus the weight of the displaced liquid medium. Many studies of colloidal suspensions have been made using Eq. 21-10, but a major drawback

is

the relative smallness of the earth's gravitational

field.

During the period 1923 to 1925, in order to increase the gravitational much as several hundred thousand g, T. Svedberg developed the ultracentrifuge.t As is known from elementary physics, if a particle of mass m* is spun oj revolutions per second at a distance x from the axis field to as

of revolution, the centrifugal energy

-Teem This

may

=

is

-lm*oj'x\

be quickly verified by noting that

Fcent

=

Or

cent

=

it

tn

(21-11)

yields a centrifugal force *

2_

OJ X,

i ia (21-12)

/")i

OX which

is

probably more familiar to the reader.

In this case, instead of

Eq. 21-10, one has p{x,)

=

(21-13)

p{x,)e--'-'^^'-^^''"''^,

or In p{x,)

One may

thus find

= -

m* from

m*o/(x,^

-

X,')

^

the slope of the curve

j^ ^^^^^

made by

^21-14)

plotting In p(x)

against x^. The ultracentrifuge permits fairly accurate determination of the molecular weight of reasonably small molecules, using Eq. 21-14.

PROBLEMS Suppose you are going up in an elevator and your ears pop at the about 100 meters above the ground. What pressure change caused your ears to pop? Note: For small x, e~^ '^ \ — x. The average molecular weight of air is about 29. The temperature could be 300°K. 21-1.

thirtieth floor,

t This remarkable instrument and its application to a variety of problems are discussed by T. Svedberg and K. O. Pedersen, The Ultracentrifuge, Oxford University Press,

London, 1940.

22.

Occupation Numbers

— Effect

of Statistics

109

21-2. Consider that at the earth's surface, air is composed mostly of oxygen and nitrogen with average molecular weight of 29. About one part in a thousand, however, is hydrogen with molecular weight of 2. If the air is at about — 10°C to all heights, at what height does the concentration of hydrogen equal the concentration of oxygen and nitrogen put together ? At 800 km, what is the ratio of the concentration of hydrogen to that of the other species ? Assume that

Eq. 21-14 21-3.

is still

valid.

Below what height

gravitational field ?

will

f =

Let

be found just half the gas molecules in the earth's let T be constant at 0°C. The average

mgh and

molecular weight of air is 29. 21-4. Find the density of an ideal gas at some arbitrary point in a cylinder of radius R and length /. The cylinder contains TV molecules, each weighing m grams, and rotates about its axis with angular velocity co. Neglect the external gravitational field. 21-5. Experiments with colloids based on equations as simple as Eq. 21-9 provided some of the first direct evidence for the existence of molecules, one of the earliest tests of the statistical mechanics of macroscopic substances, and an accurate determination of Avogadro's number. J. Perrin and coworkers [Ann. chim. phys., 18, 5 (1909); Compt. rend., 152, 1380 (1911)] and others determined A^o from Eq. 21-9. Boltzmann's constant was replaced by RJNq. The gas constant R was known from work with gases. Their experiments are described by Glasstone [S. Glasstone, Textbook of Physical Chemistry (Macmillan and Co., London, 2d ed., 1953), p. 257]. They suspended granules of gamboge, a bright yellow gum resin, of density 1.194 in water at 25 °C. Their granules had average radii of 0.368 x 10^* cm. They found p(/?2)/ K^'i) = lO"-*^^ for /?2 — //i of only 0.01 mm. What value of Avogadro's number did they determine ?

OCCUPATION NUMBERS-

22

EFFECT OF STATISTICS

In this section the discussion returns to the subject

of the

first

state for

part of Sec.

an ideal gas

available particle state

can be

in

a state;

this section the

for

1

7.

is

It

was

stated there that an A^-particle

determined

if

the

number of

quantum

particles in

each

For fermions, at most only one particle bosons, any number can be in the same state. In is

given.

ensemble averages

A',

are found for both fermions

of the number of particles

and bosons.

in particle

These are really the ensemble averages of the occupation numbers, but most authors call state

/

Ideal

110

Gases

them simply the occupation numbers of the particle states. They correspond to what one would predict for the number of particles in state / in a system composed of fermions or bosons. For systems in which the ratio r of Eq. 17-9 is decreased toward unity from something very large, the properties change in important ways due to the fact that the particles are either fermions or bosons. The second and fourth conditions given in Sec. 8 for the validity of the Boltzmann distribution are not fulfilled. The gas is still ideal by the definition of no longer simply proportional to e"'^^'. Consequently, the ideal gas law no longer is the equation of state for the gas. The occupation numbers found for fermions and bosons must become proportional to e~'^^' only when the ratio r becomes much greater than unity. A system in equilibrium with a heat bath is represented by an ensemble Sec. 12, but

Ni

is

quantum

defining the probability of theyth TV-particle

P^.(j)

state:

= i--.

(22-1)

Qn This equation states are

is

well defined only

allowed

is

when knowledge of what

A^-particle

incorporated in the partition function:

1 e-^^'^^'"'"'. Q.x= JV'-particle

(22-2)

states

FERMI-DIRAC STATISTICS For an

ideal gas of fermions, each allowed particle state will

either zero or

one particle

The occupation number particle state

/

for fermions

/

and dividing by the

number

for fermions, N^^j^,

_

/

sum being over

occupied.

the

same

as the probability that

members of the ensemble with a particle in total number of members. The occupation is

therefore -^£'v(state)

^i.FD

the

is

be occupied. This probability and the occupation number

are found by counting the state

have

N different states will always be occupied.

in it;

= I ^v(state) = 2

all A^-particle

If the notation

Qy

—z

,

(22-3)

quantum states which have particle state and 2y° be used for the sums over all

N-particle states having particle state

i

occupied or unoccupied respectively.

Occupation Numbers

22.

— Effect

of Statistics

III

then Eq. 22-3 becomes

W,

=

,,^

^=

^'''

(22-4)

^

(22-5)

Equation 22-4 takes advantage of the states

may

be viewed as the

those with

it

occupied

is

In Eq. 22-5, the

unoccupied.

are divided by

Qy

In Eq. 22-6, the fact

.

recognized by factoring

every term.

The remainder

is

The be

/

ratio in the

e~''^'

unoccupied, so

/

/

if

it is

A^

is

enormous, the

ratio

is

absolute zero. Therefore, the ratio

of particle state

occurs in

it

all (A'^

is

/:

N or N —

^

1)-

particle

particles

1

high enough that the A^ lowest

This

is

true even for small N,

essentially independent of

may

—

given the symbol

not change noticeably for

this ratio will

the temperature

lying particle states are not solidly filled. if

ocfw/7/Wplus

denominator of Eq. 22-6 depends on choice of

The value of

in that state.

/

out of Qx', since sum over

only through the requirement that none of the

different choices of

but

particle state

numerator and denominator that Qy' has particle state /

equivalent to the

particle states having particle state

state

fact that the set of all A^-particle

sum of those with

/

even at

be taken as independent of choice

^

F

=

^ ^^^

,

(22-7)

and Eq. 22-6 becomes simply

— The value of

Fe-"''"'

1

'f^D

the ratio

p-l^,,f,T

F may

_^

1

,

_^ p^-..a-T

(22-8)

be obtained by normalization:

1^,,™=^=!^.,-,^, although evaluation of such a summation

is

(22-9)

rarely easy.

BOSE-EINSTEIN STATISTICS For an will

number of particles in state / quantum states. The occupation

ideal gas of bosons, every possible

be represented by

some

TV-particle

112

Ideal

number state

Gases

for state

i

is

the ensemble average of the

number of

particles in

i:

Kbe =

1

1

•

P.vCstate)

+

2

^

•

+

2

3-

^.vCstate)

iV'-particle states (2 particles in i)

JV-particle states (1 particle in i)

P,v(state)

+

•

•

2al'

+

.

.

•

(22-10)

JV-particle states (3 particles in i)

=

1

.

?a:'

Qn

+

2

•

2^"

+

3

•

Qn

(22-11)

. .

Qn

The primed notation is an obvious extension, the number of primes being This expression may be put into more the number of particles in state /.

convenient form by recognizing the fact that the set of states

two

may

be viewed as the

particles in

/,

sum

all A^-particle

of those with one particle in

/,

those with

those with three, etc

jr

_ e.v' + 2e.v- +

3g.v"'

+

In almost any conceivable situation, the terms in the series in numerator

and denominator of Eq. 22-12 get smaller as the series progress. This is more particles are fixed in state /, there are fewer others which may be in the other states. Thus there are fewer A^-particle states over which to sum. The reader may verify that Eq. 22-12 is identical to because, as

(2y(l+2^+3^ +

-

(22-13)

simply by multiplying Eq. 22-13 out. series

of the denominator, e~^^'

may

In the numerator and the second

be canceled in each term:

iV,i.BE

y.v-i

v.\-\

(22-14)

Occupation Numbers

22.

— Effect

of Statistics

113

Except for the difference between ^ y and Qj^-i, the series in the parenand may be canceled. This difference might be

theses are identical

noticeable in

sums

is

some of

when

the last terms, but

A^

large the effect

is

on the

negligible:

'^i.BE

=

„

fr „ - Qn

.

(22-15)

, '

Qn' ^i.BE

=

1 .

{QN'IQ%-^)e'"

An argument

-

(22-16)

1

what led to Eq. 22-7 can be made here. Only when the temperature is so low that all the particles are crowded into a handful of lowest energy levels could the ratio in the denominator depend on choice of state /. Thus, the ratio This resembles Eq. 22-6.

may

similar to

be written

B=

e"v-i .,

Qn' and Eq. 22-16

is

JT

^

Q N-l Qn

(22-17)

114

Ideal

Gases

FERMI-DIRAC

23

AND BOSE-EINSTEIN GASES

In this section, the statistical mechanical expressions for the various thermodynamic properties of ideal fermion or boson gases are found by a method analogous to that of Sec. 9. The statistical thermodynamics is here based on occupation numbers, in contrast to the use of the partition function in Sees. 17 to 20 for the Boltzmann gas. The Boltzmann limit of these results is of course identical to the results of Sees. 17 to 20. The partition function approach has the advantage of

being valid in

all cases, in particular, for

systems other than ideal gases.

However, within the partition function framework

it is

hard to incorporate

quantum state for fermions or bosons. On the other occupation number approach directly incorporates the correct of a quantum state, but it is limited to ideal gases. One must

the definition of a

hand, the definition

choose the approach best suited to the problem at hand. Occupation numbers for both fermions and bosons may be considered together by writing Eqs. 22-8 and 22-18 thus:

N,

We

D

have chosen

The value of Boltzmann

D

as

—^

The upper

(23-1)

.

compromise notation

will of course

limit.

=

for the parameters

F

and B.

be different in the two cases, except in the

sign applies for fermions, the lower sign for

bosons.

may

be used to express the normalization condition, Eqs. 22-9 and 22-19, and also the ensemble average of the energy of the This joint notation

system

^=

I^.iV.

=I ± .

1

(23-3)

1-,...De-

Since either one or the other of Eqs. 23-2 and 23-3

is

sufficient to fix

the two equations together should permit an interpretation of

of thermodynamic quantities.

D

D,

in terms

Fermi-Dirac and Bose-Einstein Gases

23.

D

1

15

In this case,

thermodynamic variables of the system is accommethod similar to that used in Sec. 9. the place of the logarithm of the partition function is taken by

the function

T,

Relating

to the

plished for fermions and bosons by a

defined thus "F

= ±2 In (1 ±

(28-4)

De-^'').

i

this

is

the obvious replacement for In

Sec. 9, In

Q

is

That

and the number of Sec. 9,

Had

when

TV also

11-14 for verify.

fl

Q

be shown

will

later.

In

a function of temperature, the constraints on the system, particles, that

of ^, V, and

is,

a reversible change

was made

In the procedure of

A'^.

in In Q,

N

was kept

fixed.

been allowed to vary in Eq. 9-6, then the equivalent of Eq. would have been found directly, as the reader may quickly

The function

T also depends

on

/5

and V and has //-dependence

in the parameter D.

In place of Eq. 9-6 for d In Q, the value of

changes in

/S,

V (or

the

— ^^

JT =

i

1

D

and

e's),

-f

y

(or

—

T

N and the

Je, 4-

altered

by

explicit

— ^D T ±

db De-'''^

is

e's):

1

(23-5)

'

De-^^' (23-6)

± 2 -^^ T t 1 ± De-"''

^^'

dT^ -Ed^- (^dE±-dD.

(23-7)

D

The

identifications in passing

23-2 and 23-3.

Two

from Eq. 23-6 to 23-7 were based on Eqs. employed here, instead of just Eq. 9-8:

identities are

d(E^)

d(N The quantity dE as

is

in

In

D)

Eq. 23-7

is

= Ed^ + = - JD + D

^dE, In

D

(23-8)

dN.

(23-9)

the negative of the reversible work,

discussed following Eq. 9-7.

—dw,

Employing Eqs. 23-8 and 23-9 permits

turning Eq. 23-7 into the desired form:

d^ = fidEdE This

is

d{Eft)

= kTdCV +

E(i

+ (ib\v + d(N In D) - \n D dN, - N In Z)) - r^ + ATln D dN.

compared with the general thermodynamic equation

for

(23-10)

dE

in a

1

Gases

Ideal

16

reversible change, including a change in

dE^TdS where

/x is

b\v

number of particles:

+

dN,

IX

(23-1 1)

Comparison of Eqs. 23-10 and 23-11

the chemical potential.

permits writing a statistical mechanical expression for both the chemical potential

and the entropy:

=

p

kT\n

D = -kTln

(23-12) Q.v-i

The chemical

potential thus appears to be a rather peculiar average which

more particle to the ideal gas. The energy same as that for the energy of a particle; in particular, have the same zero. If Eq. 23-12 is solved for D,

expresses the effect of adding one scale for

the

/Z is

these scales

D = The entropy

(23-13)

e''^"^.

obtained in a similar way:

is

dS The same argument

=

k diW

was used

as

S

=

/cT

+

El^

-

N\n

D).

(23-14)

from Eq. 9-13

in going

+ - - Nk In

D.

to 9-15 gives

(23-15)

T

By using Eq.

23-13, Eq. 23-4 for

T may be rewritten:

"*= ±Iln(l ±e (il-ei)ikT^ Similarly, the

(23-16)

two equivalent ways of rewriting the occupation numbers,

Eq. 23-1, are Afi-€i)lkT N,.

= M.-fi)I^T _^

1

1

^

Aii-eO/kT

(23-17)

it is worth noting a few thermodynamic from T. Passing from Eq. 23-5 to 23-7

Before concluding this section, results obtainable in general

proved that

(23-18)

It also

established that /5

dw

=

j^P

dV =

J^ i

— Se,

d€,

=

— dV

dV.

(23-19)

Fermi-Dirac and Bose-Einstein Gases

23.

Comparing

117

the second and fourth terms of Eq. 23-20 shows

(23-20)

(23-21)

It is

an easy matter for the reader to establish that

l=_^^T/a^ = |K proportional to

(23-22)

This

indeed the volume

in the case that

e^

dependence of

for the translational energy of free particles in a cubical

e^

is

F~^'^.

is

box, as shown by Eq. 18-6. Therefore, for an ideal monatomic gas.

PV= for both fermions It is

and bosons,

iE

(23-23)

as well as in the

interesting to note the difference in

E

Boltzmann

limit.

(or pressure, since they are

proportional) caused by the explicit effect of Fermi-Dirac or BoseEinstein statistics at low temperatures.

Consider

first

the normalization

condition 2iV;.

For fermions, each term it

does for bosons.

=

iV

in the

=2

A^^

.

±1

summation has a

(23-24)

•

larger

denominator than /7 must be

Therefore, in order to offset this effect,

larger for fermions than for bosons.

shows that

.._.-.L Aci-fi)/kT

However, examination of Eq. 23-17

becomes damped only for

particle energies larger than

This occurs at higher energies for fermions than for bosons;

fi.

therefore

the ensemble average of the system energy,

£=

_ I^.^.

,

=Z ,

is

Jfi-(i)/kT

;,,-..»>.

(23-25)

greater for fermions than for bosons.

Considering Table 17-1, we may think of kT as is not surprising. a measure of the upper bound, below which the energies of the occupied states must lie. As the temperature is lowered, the particles are squeezed This

and fewer states with energy below kT. Fermions are one particle per quantum state, while any number of bosons may be in the same state. Thus, in order to accommodate all the particles, fermions must go into higher energy levels than bosons; since into the fewer

restricted to only

Ideal

118

Gases

bosons may, if they wish, go into a single quantum state. In the extreme r = 0°K, the A^ fermions occupy the A'^ states of lowest energy while all A^ bosons are in the single state of lowest energy. The energy all

case of

(thus, the pressure) of a fermion gas

be

if

the particles were bosons.

is

These

"repulsion of the fermions due to

consequently higher than

it

would

results are often said to reflect a

quantum mechanical exchange forces^'' quantum mechanical con-

or an "attraction of the bosons leading to a densation.'"

This terminology

these phrases have for

is

unfortunate because of the connotations

most people.

The only reason it is harder to treat fermions or bosons correctly than Boltzmann statistics is that the summation of Eq. 23-16 to find so much more difficult to perform than the summation of the simple

T

to use is

exponential in Boltzmann

statistics.

However,

in the next

two

sections,

special cases of particular interest are briefly treated.

PROBLEMS 23-1.

Prove Eq. 23-23.

23-2. Consideration of the

requirement on the sign of

24

fi

form of Eq. 23-17

is

sufficient to

prove what

for bosons ?

METALS-THE ELECTRON GAS

One may

use the statistical mechanics of the ideal

Fermi-Dirac gas to gain considerable insight into the properties of metals.

done by introducing a greatly oversimplified model of the metallic The metal is treated as if it were composed of the outer valence electrons of its atoms and of the resulting positive ions. The positive ions form a lattice through which the electrons are completely free to move. The plus and minus charges of the ions and electrons are assumed to neutralize each other with only negligible forces resulting on the electrons. Such a model is certainly crude, but as a first approximation to the mechanical structure of metals, its simplicity and predictive successes make it well worth studying.

This

is

structure.

Metals—The Electron Gas

24.

As mentioned electron gas

Whether or not the depends on the

in Sec. 17, electrons are fermions.

may

be described by Boltzmann

ratio r of Eq. 17-9.

This ratio

statistics

calculated in Sec. 18 for structureless

is

The conclusion, Eq.

particles in a box.

119

would be correct for elecone of two states simply

18-10,

may

trons except for the fact that each one

be

in

by virtue of its spin.* For electrons, therefore, given by Eq. 18-10.

r is

simply twice the value

at which r equals unity is called the characteristic The temperature degeneration temperature of and is interesting to calculate for an electron

gas:

=

r

In

(24-1)

1,

(2MkQp V

e =

^

"

^'

^'P

(24-3)

.

2Mk

For metals, a reasonable estimate of p is obtained by assuming that each metal atom furnishes one valence electron and that a molar volume is perhaps 6 cm^. There would then be about 10^^ valence electrons per cubic centimeter of the metal.

The molecular weight of an

electron

is

only about

1/1840; therefore,

_ ~

3-V^(1.054x

2(1840)-^ g-mole'Hl -38

=

29,700 deg

X

10-^^erg-sec)^(6.02

about

or

3

x

x

10^=^

mole-^)(10^'^cm-y^

lO"'' erg-deg-')

10* deg.

(24-4)

Thus, a metal would have to be heated to almost a million degrees before Boltzmann statistics would yield a good approximation to the partition function of the electron gas.

Room

temperatures

T are

so

much

less

than

tion

number

be zero

This

is

large.

*

The

if e^

=

spin

references.

be treated

,«.-,L

(24-5)

.

+

1

exceeds the chemical potential at absolute zero,

because the exponential

On

may

zero, the occupa-

for fermions,

^will

that they

At absolute

as minor perturbations from absolute zero.

the other hand,

quantum number

is

in the

denominator of Eq. 24-5

if e, is less

either

+2

or

than

—\;

/Zq,

see

is

/7o.

infinitely

the exponential vanishes

any of the quantum mechanics

ideal

120

and Ni

Gases

unity.

is

Therefore, at absolute zero,

Ar=l, -7^

=

Ni This case

electrons completely

to them, which are

0,

(24-6) //q.

complete degeneration of the electron

called

is

e,.

.

fill

all

those below

It

//q.

has become

jUq

the Fermi limiting energy, or the Fermi level.

/A.

2

j

ni=0

(29-5)

.

to be simply

In (1

-

(29-6)

e-^">/^^).

}=i

Eq. 29-6 and the equations of Sec.

11, all

of a perfect crystal could be calculated

if

thermodynamic prop-

only the distribution of

The bonds holding together a most chemical bonds. As can be seen from Table 19-1 the forces of chemical bonds are so strong that vibrational frequencies are high enough to make lico = kT at temperatures of, say, 3000°K in a typical case. The weaker forces in a crystal typically give

the normal frequencies w^ were known. crystal lattice are less strong than ,

frequencies about ten times lower; thus liw

The

first

= kT at

about 300°K.

choice of frequency "distribution" considered here

proposed as an approximate treatment by Einstein 0).

=

co^,

a constant for

all

is

that

in 1907:

(29-7)

/.

and only 29-7 Equation would seem temperatures. low qualitatively reasonable at vibraenough that all high temperatures for respectable approximation a one tional modes are excited, because just how they came to be excited Einstein's theory proves to be all right at high temperatures

after another as the temperature increased

portance.

—would then

The use of an average frequency

would not be too unreasonable.

A

—

be of

as representative of

physical

little

them

imall

model could be given by

29. Perfect Crystals

mode

imagining each normal

about

its

to be the vibration of one of the

Each

equilibrium configuration.

Then

normal frequencies would be

same

in all

identical,

but

not a very sound model.

this is

The 29-7

all

N particles

particle should experience a

similar environmental potential energy, approximately the

three directions.

149

results of Einstein's theory are certainly simple,

is

used to define a characteristic temperature hcoj,

=

however.

If Eq.

©j^,

kQj,,

(29-8)

then Eq. 29-6 becomes

\nQ=- ^^^ 2T

3N

In (1

-

(29-9)

e'^^'^).

All thermodynamic quantities of interest could be calculated from Eq. 29-9.

In particular, the specific heat

interesting to

is

compare with

experiment ,2/5 Q E=kTH-^^^\ In

\

= £=

kT'

dT

^^ + kT'3N —^

p-^^

to0^-f-|^^;

The heat capacity of a universal function of Q^/r. crystal to crystal.

(29-10)

Jv,N,

crystal,

(29-11)

according to Einstein's theory,

The parameter

Experimentally,

it

Q^

will,

a

true that plots of heat capacity

is

versus temperature do have similar shapes for

may

is

of course, vary from

monatomic

crystals

often be superimposed by proper scaling of the temperature.

and At

high temperatures, 0^/ris small and Eq. 29-12 becomes

C^^3Nk,

r»0£.

(29-13)

This has long been recognized experimentally as the law of Dulong and Petit.

This

is

by no means an astonishing

result, since

limit of the contribution to heat capacity of

as

was discussed

Cy

is

in Sees. 16

and

19.

sensitive to the fraction of the

A'^

it is

just the classical

simple harmonic oscillators,

However, at low temperatures, where normal modes which are excited, the

limit of Eq. 29-12,

Ty

-^ 3iV/c(^ V^*^^,

r«

0^-,

(29-14)

ISO

Special Topics

approaches zero too rapidly for decreasing T because of the damping of the exponential. This is to be expected for a model which gives the same characteristic temperature to all normal modes. It is very difficult to excite

any of them

low, because very

Then, as

T

at temperatures little

Cy would

increases to O^,

Such behavior

is

much below 0^. Thus, Cy would be

heat could raise the temperature considerably. rise

rapidly and level off at 2)Nk.

not observed physically.

Clearly, Einstein's choice of a single frequency, Eq. 29-7

oversimplification.

Experimentally,

approaches zero as T^.

It is

it is

known

that for

most

is

a gross

crystals,

Cy

therefore reasonable to seek a distribution of

normal frequencies which makes Cy approach zero as T^ for low temperatures and approach the Dulong and Petit value 2>Nk for high temperatures. A physically reasonable interpolation between these limits was developed by Debye in 1912. It has proved very useful in correlating theory with experiment.

The Debye theory asks

the question,

what are the normal modes of is, have long wave length ?

vibration which have low frequencies, that

The answer crystal

is,

they should be directly analogous to the vibrations in a

by which sound waves are transmitted.

identical,

If they are

assumed to be

then the frequency distribution of these vibrational modes

It was shown there that if waves propagate within a volume V at speed c, the number of quantum states with frequencies between m and co + dw is

has already been found in Eqs. 25-4 to 25-15.

For vibrations in a lattice there are three possible polarizations, two transverse waves at right angles to each other and one longitudinal (in contrast to sound waves, which have only the longitudinal). So for lattice vibrations in a crystal of volume V, the number of quantum states or normal modes of vibration with frequencies between co and m -\- do) is

'^(^ + VCj trans

277

The sum

in the parenthesis

is

^

C2^trans

+ J-)

a property of the crystal and

in terms of an average speed of sound in the crystal

h^

C

(29-16)

Ciong/

-T— + -T— + -h Ciong Ci, trans

C2^trans

c,

is

often written

defined by

(29-17)

29. Perfect

In this notation, the

V whose

of volume

Crystals

number of independent normal modes

characteristic frequencies

lie

between

oj

151

in a crystal

and

oj

+

doj

is

3Vco^dco (29-18) Itt'c'

Once Eq. 29-18 Eq.

function,

adopted for the frequency distribution, the partition immediately gives the thermodynamic variables. item of importance remains to cut oflF the total number of is

29-6,

—

Only one last normal modes

enough degrees of freedom normal modes. The usual choice is the set of 3N modes of the same form as the 3A^ sound waves of longest wave length which may be propagated through the crystal. Thus, any frequency is Clearly, there are only

at 3A^.

in the crystal to allow

prohibited that

modes with

oj

is