Jeans ~ Intro to the Kinetic Theory of Gases,1967

AN INTRODUCTION TO

THE KINETIC THEORY

OF GASES

AN INTRODUCTION TOTHE KINETIC THEORY

OF GASES

BY

SIR JAMES JEANS

CAMBRIDGE

AT THE UNIVERSITY PRESS

1967

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www. Cambridge. orgInformation on this title: www.cambridge.org/9780521092326

© Cambridge University Press 1940

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 1940Reprinted 1946, 1948, 1952, 1959, 1960, 1962, 1967

Re-issued in this digitally printed version 2008

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-09232-6 paperback

PREFACE

I have intended that the present book shall provide such know-ledge of the Kinetic Theory as is required by the average seriousstudent of physics and physical chemistry. I hope it will alsogive the mathematical student the equipment he should havebefore undertaking the study of specialist monographs, such,for instance, as the recent books of Chapman and Cowling (TheMathematical Theory of Non-uniform Oases) and R. H. Fowler(Statistical Thermodynamics).

Inevitably the book covers a good deal of the same groundas my earlier book, The Dynamical Theory of Oases, but it iscovered in a simpler and more physical manner. Primarily Ihave kept before me the physicist's need for clearness and direct-ness of treatment rather than the mathematician's need forrigorous general proofs. This does not mean that many subjectswill not be found treated in the same way—and often in thesame words—in the two books; I have tried to retain all thatwas of physical interest in the old book, while discarding muchof which the interest was mainly mathematical.

I t is a pleasure to thank Professor E. N. da C. Andrade forreading my proofs, and suggesting many improvements whichhave greatly enhanced the value of the book. I am also greatlyindebted to W. F. Sedgwiek,* sometime of Trinity College,Cambridge, for checking all the numerical calculations in thelatest edition of my old book, and suggesting many improve-ments.

J. H. JEANSDORKINGJune 1940

[* W. F. Sedgwiek writes (1946): "As a rule I only checked one ortwo of the items in the tables. As regards these and the numericalresults given in the text, I did indeed as a rule agree, at least approxi-mately, with Jeans' figures, but in a few cases (see PhilosophicalMagazine, Sept. 1946, p. 651) my results differed substantially."

PUBLISHERS' NOTE.]

CONTENTS

Chap. I. Introduction

II. A Preliminary Survey 17

III. Pressure in a Gas 51

IV. Collisions and Maxwell's Law 103

V. The Free Path in a Gas 131

VI. Viscosity 156

VII. Conduction of Heat 185

VIII. Diffusion 198

IX. General Theory of a Gas not in a Steady State 225

X. General Statistical Mechanics and Thermo-dynamics 253

XI. Calorimetry and Molecular Structure 275

App. I. Maxwell's proof of the Law of Distribution

of Velocities 296

II. The#-theorem 297

III. The Normal Partition of Energy 299

IV. The Law of Distribution of Coordinates 301

V. Tables for Numerical Calculations 305

VI. Integrals involving Exponentials 306

Index of Subjects 307

Index of Names 309

Chapter I

INTRODUCTION

The Origins of the Theory

I. As soon as man began to think of abstract problems at all,it was only natural that speculations as to the nature and ultimatestructure of the material world should figure largely in his writingsand philosophies.

Among the earliest speculations which have survived are thoseof Thales of Miletus (about 640-547 B.C.), many of whose ideasmay well have been derived from still earlier legends of Egyptianorigin. He conjectured that the whole material universe consistedonly of water and of substances derived from water by physicaltransformation. Earth was produced by the condensation ofwater, and air by its rarefaction, while air when heated becamefire. About 500B.C. Heraclitus advanced the alternative viewthat earth, air, fire and water were not transformable one into theother, but constituted four distinct unalterable "elements", andthat all material substances were composed of these four elementsmixed in varying proportions—a sort of dim anticipation ofmodern chemical theory. At a somewhat later date, Leucippusand Democritus maintained that matter consisted of minute hardparticles moving as separate units in empty space, and thatthere were as many kinds of particles as there are differentsubstances.

Unhappily nothing now remains of the writings of eitherDemocritus or Leucippus; their opinions are known to us onlythrough second-hand accounts. From these we learn thatthey imagined their particles to be eternal and invisible,and so small that their size could not be diminished; hence thename arofAos—indivisible. The particles of any particular sub-stance, such as iron or water, were supposed to be all similar toone another, and every one of them carried in itself all the attri-butes of the substance. For instance, Democritus taught that theatoms of water, being smooth and round, are unable to hook on to

2 INTRODUCTION

each other, so that they roll over and over like small globes;on the other hand, the atoms of iron, being rough, jagged anduneven, cling together to form a solid body. The "atoms" ofthe Greeks corresponded of course to the molecules of modernchemistry.

Similar views were advocated by Epicurus (341-276 B.C.), butrejected by Aristotle. In a later age (A.D. 55) Lucretius advancedsubstantially identical ideas in his great poem De return natura.In this he claimed only to expound the views of Epicurus, mostof whose writings are now lost.

Lucretius explains very clearly that the atoms of all bodiesare in ceaseless motion, colliding and rebounding from oneanother. When the distances which the atoms cover betweensuccessive rebounds are small, the substance is in the solid state;when large, we have "thin air and bright sunshine". He furtherexplains that atoms must be very small, as can be seen eitherfrom the imperceptible wearing away of objects, or from theway in which our clothes can become damp without exhibitingvisible drops of moisture.*

There was little further discussion of the problem until themiddle of the seventeenth century, when Gassendif examinedsome of the physical consequences of the atomic view. He assumedhis atoms to be similar in substance, although different in size andform, to move in all directions through empty space, and to bedevoid of all qualities except absolute rigidity. With these simpleassumptions, Gassendi was able to explain a number of physicalphenomena, including the three states of matter and the transi-tions from one to another, in a way which difiered but little fromthat of the modern kinetic theory. He further saw that in anordinary gas, such as atmospheric air, the particles must be verywidely spaced, and he was, so far as we know, the first to conjecturethat the motion of these particles could account by itself for anumber of well-known physical phenomena, without the additionof separate ad hoc hypotheses. All this gives him a very specialclaim to be regarded as the father of the kinetic theory.

* See an essay by E. N. da C. Andrade, "The Scientific Significance ofLucretius", Munro's Lucretius, 4th edition (Bell, 1928).

f Syntagma Philosophicum, 1658, Lugduni.

INTRODUCTION 3Twenty years after Gassendi, Hooke advanced somewhat

similar ideas. He suggested that the elasticity of a gas resultedfrom the impact of hard independent particles on the substancewhich enclosed it, and even tried to explain Boyle's law on thisbasis.

Newton accepted these views as to"the atomic structure ofmatter, although he suggested a different explanation of Boyle'slaw*. He wrotef

" I t seems probable to me that God in the beginning formedmatter in solid, massy, hard, impenetrable, moveable particles. . . , and that these primary particles, being solids, are incom-parably harder than any porous bodies compounded of them;even so hard as never to wear or break in pieces."

Hooke was followed by Daniel Bernoulli, % who is often creditedwith many of the discoveries of Gassendi and Hooke. Bernoulli,again supposing that gas-pressure results from the impacts ofparticles on the boundary, was able to deduce Boyle's law for therelation between pressure and volume. In this investigation theparticles were supposed to be infinitesimal in size, but Bernoullifurther attempted to find a general relation between pressure andvolume when the particles were of finite size, although stillabsolutely hard and spherical.

After Bernoulli, there is little to record for almost a century.Then we find Herapath§ (1821), Waterston|| (1845), Joule^f (1848),Kronig** (1856), Clausius (1857) and Maxwell (1859) taking up thesubject in rapid succession.

* See § 50, below.f Opticks, Query 31 (this did not appear until the second edition of

the Opticks, 1718).X Daniel Bernoulli, Hydrodynamica, Argentoria, 1738: Sectio decima,

"De affectionibus atque motibus fluidorum elasticorum, praecipueautem aeris."

§ Annals of Philosophy (2), 1, p. 273.|| Phil. Trans. Roy. Soc. 183 (1892), p. 1. Waterston presented a long

paper to the Royal Society in 1845, but this contained many inaccuraciesand so was not published until Lord Rayleigh secured its publication,for what was then a purely historical interest, in 1892.

1f British Association Report, 1848, Part II, p. 21; Memoirs of theManchester Literary and Philosophical Society (2), 9, p. 107.

• • PoggendorffJ8 Annalen.

4 INTRODUCTION

In his first paper* Clausius calculated accurately the relationbetween the temperature, pressure and volume in a gas withmolecules of infinitesimal size; he also calculated the ratio of thetwo specific heats of a gas in which the molecules had no energyexcept that of their motion through space. In 1859, Clerk Max-well read a paper before the British Association at Aberdeen, f inwhich the famous Maxwellian law of distribution of velocitiesmade its first appearance, although the proof by which Maxwellattempted to establish it is now universally agreed to have beeninvalid. J In the hands of Clausius and Maxwell the theory de-veloped with great rapidity, so that to write its history from thistime on would be hardly less than to give an account of the subjectin its present form.

The Three States of Matter2. Most substances are capable of existing in three distinct

states, which we describe as solid, liquid and gaseous. The typica]example is water, with its three states of ice, water and steam.It is natural to conjecture that the three states of matter corre-spond to three different types or intensities of motion of the funda-mental particles of which the matter is composed, and it is notdifficult to see how the necessity for these three different statesmay arise.

We know that two bodies cannot occupy the same space; if wetry to make them do so, repulsive forces come into play, and keepthe two bodies apart. If matter consists of innumerable particles,these forces must be the aggregate of the forces from individualparticles. These particles can, then, exert forces on one another,and the forces are repulsive when the particles are pushed suf-ficiently close to one another. If we try to tear a solid body intopieces, another set of forces comes into play—the forces ofcohesion. These also indicate the existence of forces betweenindividual particles, but the force between two particles is nolonger one of repulsion; it is now one of attraction. The fact thata solid body, when in its natural state, resists both compression

* "Ueber die Art der Bewegung welche wir Warme nennen", Pogg.Ann. 100, p. 353.

t Phil. Mag. Jan. and July 1860; Collected Works, 1, p. 377.X See Appendix i, p. 296.

INTRODUCTION 5and dilatation, shews that the force between particles changesfrom one of repulsion at small distances to one of attraction atgreater distances, as was pointed out by Boscovitch in 1763.*

3. The Solid State. Somewhere between these two positionsthere must be a position of stable equilibrium in which twoparticles can rest in proximity without either attracting or re-pelling one another. If we imagine a great number of particlesplaced in such proximity, and so at rest in their positions ofequihbrium, we have the kinetic theory conception of a mass ofmatter in the solid state—a solid body. Modern X-ray techniquemakes it possible to study both the nature and arrangement ofthese particles. In solid bodies of crystalline structure, the" particles " are atoms and electrons, arranged in a regular three-dimensional pattern ;f in conductors the electrons are free tothread their way between the atoms.

When the particles which constitute a solid body oscillateabout their various positions of equilibrium, we say that the bodypossesses heat. The energy of these oscillatory motions is, in fact,the heat-energy of the body. As the oscillations become morevigorous, we say that the temperature of the body increases.

We may make a definite picture by supposing that the oscil-latory motions are first set up by rubbing two solid objects to-gether. We place the surfaces of the two bodies so close to oneanother that the particles near the surface of one exert per-ceptible forces on the particles near the surface of the other; wethen move the surfaces over one another, so that the forces justmentioned draw or push the surface particles from their positionsof equilibrium. At first, the only particles to be disturbed will bethose which are in the immediate neighbourhood of the partsactually rubbed, but gradually the motion of these parts willinduce motion in the adjoining regions, until ultimately themotion spreads over the whole mass. This motion represents heatwhich was, in the first instance, generated by friction, and thenspread by conduction through the whole mass.

• Theoria Philosophise Naturalis (Venice, 1763; English translation,Open Court, Chicago and London, 1922). See especially §§ 74 ff.

f See W. L. Bragg, Crystal Structure, and innumerable other booksand papers.

6 INTRODUCTION

As a second example, we may imagine that two solid bodies,both devoid of internal motion, impinge one upon the other—asfor instance a hammer upon a clock-bell. The first effect of thisimpact will be that trains of waves are set up in the two bodies,but after a sufficient time the wave character of the motion willbecome obliterated. Motion of some kind must, however, persistin order to account for the energy of the original motion. Thisoriginal motion will, in actual fact, be replaced by small vibratorymotions in which the particles oscillate about their positions ofequilibrium—according to the kinetic theory, by heat motion.In this way the kinetic energy of the original motion of the solidbodies is transformed into heat-energy.

4. The Liquid State. If a solid body acquires more heat, theenergy of its vibrations will increase, so that the excursions of itsparticles from their positions of equilibrium will become larger.If the body goes on acquiring more and more heat, some of theparticles will ultimately be endowed with so much kinetic energythat the forces from the other particles will no longer be able tohold them in position; they will then, to borrow an astronomicalterm, escape from their orbits, and move to other positions.When a considerable number of particles are doing this, theapplication of even a small force, provided it is continued for asufficient length of time, can cause the mass to change its shape;it does this by taking advantage time after time, as opportunityoccurs, of the weakness of the forces tending to retain individualparticles.

If still more heat is provided, a greater and greater number ofthe particles will move freely about; finally, when all the particlesare all doing this, the body has attained the state we describe asliquid.

So long as the body is in the solid state, the particles whichexecute vibrations will usually be either isolated electrons oratoms. In the liquid state it is comparatively rare for eitherelectrons or atoms to move as independent particles, because theforces binding these into molecules are usually too strong to beovercome by the heat-motion; thus the particles which moveindependently in a liquid are generally complete molecules.

Until recently it was supposed that the molecules of a liquid

INTRODUCTION 7

moved at random, and shewed a complete disorder in theirarrangement—in sharp contrast to the orderly arrangement ofthe particles in a solid. Recent investigations suggest the need formodifying this view. A molecule in a liquid is probably actedon all the time by about as many other molecules as it would beif in the solid state. The forces from these neighbouring moleculesimprison it in a cell from which it only rarely escapes. The maindifference between a solid and a liquid is not that betweencaptivity and freedom; it is only that the particle of a solid is heldin fetters all the time, while that of a liquid is held in fettersnearly all the time, living a life of comparative freedom only inthe very brief intervals between one term of imprisonment andthe next. Further X-ray technique has shewn that there is acertain degree of regularity and order in the arrangement of theliquid molecules in space.*

This knowledge is derived only from a statistical study of themolecules of a liquid; no known technique makes it possible to seethe wanderings of individual molecules. Perhaps this is not sur-prising, since even the largest of molecules are beyond the limitsof vision in the most powerful of microscopes. But if a number ofvery small solid particles—as, for instance, of gamboge or lyco-podium—are placed in suspension in a liquid, these particles areset into motion as the moving molecules of the liquid collide withthem and hit them about, now in this direction and now in that.These latter motions can be seen through a microscope, so thatthe solid particles act as indicators of the motions of the moleculesof the liquid, and so give a very convincing, even if indirect, proofof the truth of the kinetic theory conception of the liquid state.They are called Brownian movements, after the English botanist,Robert Brown.

For, in 1828 Brown had suspended grains of pollen in water, andexamined the mixture through a microscope. He found that thepollen grains were engaged in an agitated dance, which was to allappearances continuous and interminable. His first thought wasthat he had found evidence of some vital property in the pollen,

* See in particular, "Recent Theories of the Liquid State", N. F. Mottand R. W. Gumey, Physical Society Reports on Progress in Physics(C.U. Press), 5 (1939), p. 46.

8 INTRODUCTION

but he soon found that any small particles, no matter how non-vital, executed similar dances. The true explanation, that theparticles were acting merely as indicators of the molecular motionof the liquid in which they were immersed, was given by Delsauxin 1877 and again by Gouy in 1888. A full mathematicaltheory of the movements was developed by Einstein and vonSmoluchowski about 1905.*

In 1909 Perrin suspended particles of gamboge in a liquidof slightly lower density, and found that the heavy particlesdid not sink to the bottom of the lighter liquid; they were pre-vented from doing so by their own Brownian movements. If theliquid had been infinitely fine-grained, with molecules of in-finitesimal size and weight, every solid particle would have hadas many impacts from above as below; these impacts, coming ina continuous stream, would have just cancelled one another out,so that each particle would have been free to fall to the bottomunder its own weight. But when they were bombarded by mole-cules of finite size and weight, the solid particles were hit, now inone direction and now in another, and so could not lie inertly onthe bottom of the vessel. From the extent to which they failedto do this, Perrin was able to form an estimate of the weights ofthe molecules of the liquid (§ 16, below) and this agreed so well withother estimates that there could be but little doubt felt as to thetruth either of the kinetic theory of liquids, or of the associatedexplanation of the Brownian movements.

A molecule of a liquid which has escaped from its orbit in theway described on p. 6 may happen to come near to the surface ofthe liquid, in which case it may escape altogether from the attrac-tion of the other molecules, just as a projectile which is projectedfrom the earth's surface with sufficient velocity may escape fromthe earth altogether. When this happens the molecule leaves theliquid, and the liquid must continually diminish both in mass andvolume owing to the loss of such molecules, just as the earth'satmosphere continually diminishes owing to the escape of rapidlymoving molecules from its outer surface. Here we have the kinetictheory interpretation of the process of evaporation, the vapourbeing, of course, formed by the escaped molecules.

• See § 180, below.

INTRODUCTION 9

If the liquid is contained in a closed vessel, each escaping mole-cule must in time strike the side or top of the vessel; its path isnow diverted, and it may fall back again into the liquid after acertain number of impacts. In time a state may be reached suchthat as many molecules fall back in this way as escape by evapora-tion; we now have, according to the kinetic theory, a liquid inequilibrium with its own vapour.

5. The Gaseous State, On the other hand, it is possible for thewhole of the liquid to be transformed into vapour in this way,before a steady state is reached. Here we have the kinetic theorypicture of a gas—a crowd of molecules, each moving on its ownindependent path, entirely uncontrolled by forces from the othermolecules, although its path may be abruptly altered as regardsboth speed and direction, whenever it collides with anothermolecule or strikes the boundary of the containing vessel. Themolecules move so swiftly that even gravity has practically nocontrolling effect on their motions. An average molecule ofordinary air moves at about 500 metres a second, so that theparabola which it describes under gravity has a radius of curva-ture of about 25 kilometres at its vertex, and even more elsewhere.This is so large in comparison with the dimensions of any con-taining vessel that we may, without appreciable error, think ofthe molecules as moving in straight lines at uniform speeds,except when they encounter either other molecules or the walls ofthe containing vessel. This view of the nature of a gas explainswhy a gas spreads immediately throughout any empty space inwhich it is placed; there is no need to suppose, as was at one timedone, that this expansive property is evidence of repulsive forcesbetween the molecules (cf. § 50, below).

As with a liquid, so with a gas, there is no absolutely directevidence of the motions of individual molecules, but an indirectproof, at one remove only, is again provided by the Brownianmovements. For these occur in gases as well as in liquids; minuteparticles of smoke* and even tiny drops of oil floating in a gasmay be seen to be hit about by the impact of the molecules ofthe gas.

• Andrade and Parker, Proc. Roy. Soc. A, 159 (1937), p. 507.

10 INTRODUCTION

Recently E. Kappler* constructed a highly sensitive torsion-balance, in which the swinging arm was free to oscillate in analmost perfect vacuum. The swinging arm had a moment ofmomentum of 0*235 millionth of a gm. cm.2, and when it swungin a gas at pressure of only a few hundreds of a millimetre ofmercury, its period of oscillation was about 15 seconds. When theoscillations were recorded on a moving photographic film, it wasseen that they did not proceed with perfect regularity. The speedof motion of the arm experienced abrupt changes, and, as weshall see later (p. 129), there can be no doubtthat these were caused by the impacts ofsingle molecules of the gas.

Perhaps, however, the simplest evidence ofthe fundamental accuracy of the kinetic theoryconception of the gaseous state is to be foundin experiments of a type first performed byDunoyer.f He divided a cylindrical tube intothree compartments by means of two parti-tions perpendicular to the axis of the tube,these partitions being pierced in their centresby small holes, as in fig. 1. The tube was fixedvertically, and all the air pumped out. A smallpiece of sodium was then introduced into thelowest compartment, and heated to a sufficienttemperature to vaporise it. Molecules of sodiumare now shot off, and move in all directions.Most of them strike the walls of the lowest compartment of the tubeand form a deposit there, but a few escape through the hole in thefirst partition, and travel through the second compartment of thetube. These molecules do not collide with one another, since theirpaths all radiate from a point—the small hole through which theyhave entered the compartment; they travel like rays of lightissuing from a source at this point. Some travel into the upper-most compartment through the opening in the second partition,and when they strike the top of the tube, make a deposit there.

• Ann. d. Phys. 31 (1938), p. 377.t L. Dunoyer, Comptes Rendus, 152 (1911), p. 592, and Le Radium, 8

(1911), p. 142.

Fig. 1

INTRODUCTION 11

This is found to be an exact projection of the hole through whichthey have come. The experiment may be varied by removing theupper partition and suspending a small object in its place. This isfound to form a " shadow " on the upper surface of the tube; if thehole in the first diaphragm is of appreciable size, an umbra andpenumbra may even be discernible.

Stern* and Gerlach as well as a band of collaborators atHamburg, and many workers elsewhere, haveexpanded this simple experiment into theelaborate technique known as "molecularrays". The lower compartment of Dunoyer'stube is replaced by an electric oven or furnacein which a solid substance can be partlytransformed into vapour (see fig. 2). Some ofthe molecules of this vapour pass through anarrow slit in a diaphragm above the oven,and, as before, a small fraction of these passlater through a second parallel slit in a seconddiaphragm. These form a narrow beam ofmolecules all moving in the same plane (namelythe plane through the two parallel slits) and Fig. 2

this beam can be experimented on in various ways, some of whichare described later in the present book.

Mechanical Illustration of the Kinetic Theory of Gases

6. It is important to form as clear an idea as possible of theconception of the gaseous state on which the kinetic theory isbased, and this can best be done in terms of simple mechanicalillustrations.

We still know very little as to the structure or shape of actualmolecules, except that they are complicated structures of elec-trons and protons, obeying laws which are still only imperfectlyunravelled. At collisions they obey the laws of conservation ofmomentum and energy, at least to a very close approximation.They also behave like bodies of perfect elasticity. For, as weshall shortly see, the pressure of a gas gives a measure of the

* Zeitschr.f. Phys. 39 (1926), p. 751.

12 INTRODUCTION

total kinetic energy of motion of its molecules, and this is knownto stay unaltered through millions of millions of collisions.

Since, however, it is desirable to have as concrete a repre-sentation as possible before the mind, at least at the outset, weshall follow a procedure which is very usual in the developmentof the kinetic theory, and agree for the present to picture a mole-cule as a spherical body of unlimited elasticity and rigidity—tomake the picture quite definite, let us say a steel ball-bearing ora minute billiard ball. The justification for this procedure lies inits success; theory predicts, and experiment confirms, that anactual gas with highly complex molecules behaves, in manyrespects, like a much simpler imaginary gas in which the mole-cules are of the type just described. Indeed, one of the moststriking features of the kinetic theory is the extent to which it ispossible to predict the behaviour of a gas as a whole, while re-maining in almost complete ignorance of the behaviour and pro-perties of the molecules of which it is composed; the reason is thatmany of the properties of gases depend only on general dynamicalprinciples, such as the conservation of energy and momentum.It follows that many of the results of the theory are true for allkinds of molecules, and so would still be true even if the moleculesactually were billiard balls of infinitesimal size.

Since it is easier to think in two dimensions than in three, letus begin with a representation of molecular motions in twodimensions. As the molecules of the gas are to be represented bybilliard balls, let us represent the vessel in which the gas is con-tained by a large billiard table. The walls of the vessel will ofcourse be represented by the cushions of the table, and if thevessel is closed, the table must have no pockets. Finally, thematerials of the table must be of such ideal quality that a ball onceset in motion will collide an indefinite number of times with thecushions before being brought to rest by friction and other passiveforces. A great number of the properties of gases can be illustratedwith this imaginary apparatus.

If a large number of balls are taken and started from randompositions on the table with random velocities, the resulting stateof motion will give a representation of what is supposed to be thecondition of matter in its gaseous state. Each ball will be con-

INTRODUCTION 13

tinually colliding with the other balls, as well as with the cushionsof the table. The velocities of the balls will be of the most varyingkinds; a ball may be brought absolutely to rest at one instant,while at another, as the result of a succession of favourable col-lisions, it may possess a velocity far in excess of the average. Weshall, in due course, investigate how the velocities of such a col-lection of balls are distributed about the mean velocity, and shallfind that, no matter how the velocities are arranged at the outset,they will tend, after a sufficient number of collisions, to groupthemselves according to a law which is very similar to the well-known law of trial and error—the law which governs the groupingin position of shots fired at a target.

If the cushions of the table were not fixed in position, the con-tinued impacts of the balls would of course drive them back.Clearly, then, the force which the colliding balls exert on thecushions represents the pressure exerted by the gas on the walls ofthe containing vessel. Let us imagine a movable barrier placedinitially against one of the cushions, and capable of being movedparallel to this cushion. Moving this barrier forward is equivalentto decreasing the volume of the gas. If the barrier is moved for-wards while the motion of the billiard balls is in progress, theimpacts both on the movable barrier and on the three fixedcushions will become more frequent—because the balls have ashorter distance to travel, on the average, between successiveimpacts. Here we have a mechanical representation of the in-crease of pressure which accompanies a diminution of volume ina gas. To take a definite instance, let us suppose that the movablebarrier is moved half-way up the table. As the space occupied bythe moving balls is halved, the balls are distributed twice asdensely as before in the restricted space now available to them.Impacts will now be twice as frequent as before, so that the pres-sure on the barrier is doubled. Halving the space occupied by ourquasi-gas has doubled the pressure—an illustration of Boyle'slaw that the pressure varies inversely as the volume. This, how-ever, is only true if the billiard balls are of infinitesimal size; ifthey are of appreciable size, halving the space accessible to themmay more than double the pressure, since the extent to whichthey get in one another's way is more than doubled. At a later

14 INTRODUCTION

stage, we shall have to discuss how far the theoretical law con-necting the pressure and density in a gas constituted of moleculesof finite size is in agreement with that found by experiment for anactual gas.

Let us imagine the speed of each ball suddenly to be doubled.The motion proceeds exactly as before,* but at double speed—just as though we ran a cinematograph film through a projectorat double speed. At each point of the boundary there will betwice as many impacts per second as before, and the force of eachimpact will be twice as great. Thus the pressure at the point willbe four times as great. If we increase the speed of the balls in anyother ratio, the pressure will always be proportional to the squareof the speed, and so to the total kinetic energy of the balls. Thisillustrates the general law, which will be proved in due course,that the pressure of a gas is proportional to the kinetic energyof motion of its molecules. As we know, from the law of Charles,that the pressure is also proportional to the absolute temperatureof the gas, we see that the absolute temperature of a gas mustbe measured by the kinetic energy of motion of its molecules—aresult which will be strictly proved in due course (§ 12).

Still supposing the barrier on our billiard table to be placedhalf-way up the table, let us imagine that the part of the tablewhich is in front of the barrier is occupied by white balls movingwith high speeds, while the part behind it is similarly occupied byred balls moving with much smaller speeds. This corresponds todividing a vessel into two separate chambers, and filling one witha gas of one kind at a high temperature, and the other with a gasof a different kind at a lower temperature. Now let the barrieracross the billiard table suddenly be removed. Not only will thewhite balls immediately invade the part which was formerlyoccupied only by red balls, and vice versa, but also the rapidlymoving white balls will be continually losing energy by collisionwith the slower red balls, while the red of course gain energythrough impact with the white. After the motion has been inprogress for a sufficient time, the white and red balls will be

* In actual billiards the angles of course depend on the speed, but thisis a consequence of the imperfect elasticity of actual balls. With idealballs of perfect elasticity, the course of events would be as stated above.

INTRODUCTION 15

equally distributed over the whole of the table, and the averagevelocities of the balls of the two colours will be the same. Here wehave simple illustrations of the diffusion of gases, and of equalisa-tion of temperature. The actual events occurring in nature are,however, obviously more complex than those suggested by thisanalogy, for molecules of different gases differ by something morefundamental than mere colour.

Next let the barrier be replaced by one with a sloping face, sothat a ball which strikes against it with sufficient speed will runup this face and over the top. Such a ball will be imprisoned onthe other side, but one which strikes the slanting face at slowerspeed may run only part way up, and then come down again andmove as though it had been reflected from the barrier. The barrieris in effect a mechanism for sorting out the quick-moving from theslow-moving balls.

Let the space between the sloping barrier and the oppositecushion of the table be filled with balls moving at all possiblespeeds to represent the molecules of a liquid. A ball which runsagainst the sloping face of the barrier with sufficient speed to getover the top will represent a molecule which is lost to the liquidby evaporation. The total mass of the liquid is continuallydiminishing in this way; at the same time, since only the quick-moving molecules escape, the average kinetic energy of themolecules which remain in the liquid is continually being lowered.Thus, as some molecules of the liquid evaporate, the remainderof the liquid decreases in temperature. Here we have a simpleillustration of the cooling which accompanies evaporation—aprocess which explains the action of nearly all modern refriger-ating machinery.

7. One further question must be considered. No matter howelastic the billiard balls and table may be, the motion cannotcontinue indefinitely. In time its energy will be frittered away,partly perhaps by frictional forces such as air-resistance, andpartly by the vibrations set up in the balls by collisions. Theenergy dissipated by air-resistance becomes transformed intoenergy in the air; the energy dissipated by collision is transformedinto energy of internal vibrations of the billiard balls. What, then,does this represent in the gas, and how is it that a gas, if consti-

16 INTRODUCTION

tuted as we have supposed, does not in a very short time lose theenergy of translational motion of its molecules, and replace it byenergy of internal vibrations of these molecules, and of radiationtravelling through the surrounding space?

The difficulties raised by this and similar questions formed amost serious hindrance to the progress of the kinetic theory formany years. Maxwell drew attention to them, and Kelvin,Rayleigh and many others worried over them, but it was onlyafter the introduction of the quantum theory by Planck and hisfollowers in the early years of the present century, that it becamepossible to give anything like a satisfactory explanation. Thisexplanation is, in brief, that the analogy between billiard ballsand molecules fails as soon as we begin to consider questions ofinternal vibrations and the transfer of their energy to the sur-rounding space. The analogy, which has served us well for a longtime, breaks down at last. For the motion of billiard balls, as ofall objects on this scale of size, is governed by the well-knownNewtonian laws, whereas the internal motions of molecules, andtheir transfer of energy to the surrounding space in the form ofradiation, are now believed to be governed by an entirely differentsystem of laws. In the present book, we shall develop the kinetictheory as far as it can be developed without departing from theNewtonian laws, and shall notice the inadequacy of these laws incertain problems. The newer system of laws, constituting themodern theory of quantum mechanics, is beyond the scope of thepresent book.

Chapter II

A PRELIMINARY SURVEY

The Pressure in a Gas

8. The present chapter will contain a preliminary discussionof some of the principal problems of the kinetic theory. We shallnot go into great detail, or strive after great mathematicalaccuracy or rigour—this is reserved for the more intensive dis-cussions of later chapters. Our immediate object is merely to makethe reader familiar with the main concepts of the theory, and togive him a rapid bird's-eye view of the subject, so as to ensurehis not losing his bearings inthe more complete discussionswhich come later.

The main actors in thedrama of the kinetic theoryare rapidly-moving molecules;*the principal events in theirlives are collisions with oneanother and with the boundaryof a containing vessel. Letus consider the latter eventsfirst.

To begin with the verysimplest conditions, let us firstimagine that a single molecule is moving inside a cubical contain-ing vessel of edge Z, and colliding with the six faces time after time.We shall suppose that the molecule is infinitesimal in size, and alsoperfectly elastic, so that no energy will be lost on collision with theboundary. Thus the velocity of motion, which we shall denoteby c, will retain the same value throughout the whole motion.We shall also suppose that at each collision the molecule bouncesoff the wall of the vessel at the same angle as that at which itstruck. Thus we get a succession of paths such as a, 6, c, d9 e,...in fig. 3.

Fig. 3

18 A PRELIMINARY SURVEY

We can find the lengths of successive paths by a very simpleartifice. Let fig. 4 represent a honeycomb of cells, each of whichis a cube of edge I. Draw the first path a of the molecule in thefirst cell, exactly as it is described in the actual cubical vessel, andextend the line a indefinitely through the honeycomb of cells.Then it is easy to see that the intercepts b, c, d, e,... made by suc-cessive cells will be exactly equal to the paths 6, c, d, e,... of theactual molecule in the real vessel. If we imagine a point startingalong the path a at the instant at which the actual molecule startsin the closed vessel, and moving at thesame speed c as the real molecule, thenthe paths a, b, c, d, e,... in the honeycombwill be described at exactly the sametime as the paths a, 6, c, d, e,... in thereal vessel.

Let rectangular coordinates Ox, Oy,Oz be taken in the honeycomb, parallelto the three directions of the edges of thecube, and let A, /i, v be the directioncosines of the line abcde.... In the courseof unit time, we must suppose ourmoving point to describe a distance calong this line, so that it will travel adistance cA parallel to Ox, cju, parallel toOy and cv parallel to Oz. In travellingthe distance cA parallel to Ox, it will encounter faces of the honey-comb perpendicular to Ox at equal distances I apart. Each suchencounter will of course coincide with a collision between the realmolecule and a face perpendicular to Ox. The number of suchcollisions in unit time is therefore cXjl.

At each such collision, the momentum parallel to Ox is re-versed. If m is the mass of the molecule, this is of amount mcA,so that the total transfer of momentum, or "impact", betweenthe molecule and the face of the vessel is 2racA. The total of allsuch impacts between the moving molecule and the two facesperpendicular to Ox must be

, cA 2mc2A2

2racA x — = —z—.

b/

/

• /

/fc

//

Fig. 4

A PRELIMINARY SURVEY 19

There are similar impacts with the other pairs of faces, of totalamounts

2mc2a2 , 2racV

—r-and -T - -Thus the total of all the impacts exerted by the molecule on

all the six faces in unit time is

( A 2 + 2 + 2)

This, the total impact exerted in unit time, is also the totalpressure exerted on the six faces of the cube.

If there are a very great number of molecules inside thevessel, of masses mvm2,mz,... moving with different velocitiesCi, c2, c3..., and so small as never to collide with one another,the total impact they exert on all six faces is

2( 1 + l + zcl + . . . ) . (1)

If these molecules move in all directions at random, they willobviously exert equal pressures on the six faces of the cube. Apthe total area of the six faces is 6Z2, the pressure p per unit area iagiven by

1 2 2

The numerator in this fraction is twice the total kinetic energyof motion of all the molecules in the gas, while the denominator isthree times the volume of the gas. Thus we have

2 x kinetic enemypressure = Q _ . _ , (2)

and see that the pressure is equal to two-thirds of the kinetic energyper unit volume.

Since kinetic energies are additive, pressures must also beadditive. Thus Hie pressure exerted by a mixture of gases is thesum of the pressures exerted by the constituents of the mixtureseparately. This is Dalton's law.

If the volume of the vessel is allowed to change, while themolecules and their energy of motion are kept the same, we seethat the pressure varies inversely as the volume. This is Boyle's law.


If we write C2 for the average value of c2 throughout the gas,the average being taken by mass, we have

while m1 + m2 4-... = pv,

where v is the volume and p the density of the gas. From equation(2) the pressure is now given by

(3)

so that C2 = ^ . (4)P

With this formula we can calculate the speed of molecularmotion for a gas in any physical condition we please. Ordinaryair at room temperature and at a pressure of one atmosphere isfound to have a density of 1*293 grammes per litre. Insertingthese values of p and p in equation (4), we find that the mole-cular velocity is about 500 metres per second, which is roughlythe speed of a rifle bullet.

9. We have not yet shewn that the pressure has the same valueat all regions on the surface of the vessel, nor that its amountdepends only on the volume of the vessel, independently of itsshape—these facts are established in the next chapter (§33).Neither have we taken account of the collisions of the variousmolecules with one another, but it is easy to see that these do notaffect the result. For a collision between two molecules does notof itself produce any pressure on the boundary, and neither doesit affect the value of expression (2), since energy is conserved at acollision.

I t may at first seem strange that the pressure should depend onthe state of things throughout the whole of the gas in a vessel, andnot only on the state of things close to the boundary on which thepressure is exerted. One molecule, for instance, in the far interiorof the gas may be moving with immense velocity. How, it may beasked, can this molecule exert a correspondingly immense pres-sure on the boundary from which it is completely disconnected?The answer is, of course, that after a short time either the moleculewill itself strike the boundary or else, through the medium ofcollisions, will communicate its high energy to other molecules


which will strike the boundary in due course. For we must noticethat formula (2) does not give the pressure at a single instant oftime, but only the pressure averaged through one second, or ofcourse through some other comparatively long interval of time.

A more serious question is whether we were justified in assum-ing that the molecules bounce off the walls of the containing vesselat the same angle, and with the same speed, at which they metthis wall. The assumption was introduced solely for the sake ofsimplicity, and is probably not true (see § 34, below). But it is inno way essential to the proof. All that is necessary is that themolecules should, on the average, leave the walls with energy equalto that with which they arrive. The aggregate of the impactswhich the molecules exert on the walls before they are broughtto rest is then given by half of expression (1); the other halfrepresents the impacts which the walls exert on the moleculesto restore their motion. Then the total pressure is as before.

Even the supposition that the average energy of a molecule isunchanged by impact with the boundary may not of course bestrictly true. If a very hot gas is put into a very cold vessel, weknow that the vessel will acquire heat from the gas, and willcontinue to do so until the temperatures of the gas and vessel areequal. The transfer of heat from the gas to the vessel can only takeplace when molecules collide with the wall of the vessel, so thatthey do not, on the average, leave it with as much energy as theybrought to it. This does not have any great influence on the calcu-lation of the pressure, because the transfer of heat is so slow aprocess that very little energy can be transferred at a single col-lision. But it points the way to other questions of interest.

Equipartition of Energy

10. Let us consider in some detail the transfer of energy at thecollisions of two molecules, which will be supposed to be smoothhard spheres of perfect elasticity, and will ultimately be takento belong to the wall and the gas respectively. Let us take theline of impact at collision to be the axis Ox, this not necessarilycoinciding with our former Ox, which was perpendicular to oneof the walls.


Let the molecules have masses m, m\ and let their componentsof velocity before collision be

u, v9 w and u\ v', wr.Since the impact occurs along the axis Ox, the components of

velocity parallel to Oy and Oz are unaltered, so that we maysuppose the velocities after the collision to be

u,v,w and u',vryw'.

Gas WallFig. 5

The equations of energy and momentum now take the simpleforms

mu + m'u' =

or, by a slight transposition,m(u2 - u2) = - m'{u'2 - u'2)f (5)

m(u-u) = -m'(u'-u'). (6)

From these equations we obtain at once, by division of corre-sponding sides,

u + u = u' + u' (7)or v! — u = — (u'— u),

shewing that the relative velocity is simply reversed at collision,a necessary consequence of perfect elasticity.

Solving equations (6) and (7), we find that the velocities aftercollision are given by

')u = ( m - m > + 2 m V , (8)

' )^ ' = {m'-m)u' + 2mu. (9)


Now if m is the mass of the molecule of the wall, and m' that ofthe molecule of the gas, this collision results in the wall gainingenergy of amount

— u2) =

2mm' , , , w ,-j-^(mu -\-mu){u —u)

zmm r/ r f> o\ . / /\ ft-jrx [(m u * — m^J) + (m — m ) uu ].

When a collision takes place, the velocity u of the wallmolecule may be either positive or negative. Since this mole-cule has, on the whole, no continuous motion along the axisoix, and does not change its position except for small in-and-outoscillations, the average value of u must be zero. Thus if weaverage over a large number of collisions, the average value ofuu' is zero, and we find as the mean gain of energy to the wall

2 m m ' • , r

_ W^f{mu ~mu)'where mu2 denotes the mean value of mu2, and so on.

Thus the vessel gains in energy and so rises in temperature ifthe average value of m'u'2 is greater than the average value ofmu2, and conversely. If the vessel and the gas have the sametemperature, the vessel neither gains nor loses energy on theaverage, so that we have

m ^ = m/V~2. (10)11. Suppose next that two kinds of gas are mixed in the vessel,

the mass and velocity-components of the molecules of the secondkind being denoted by m", u", v"', w", and suppose further thatboth kinds of gas are at the same temperature as the vessel itself.Then there is no loss or gain of energy to the vessel through col-lisions with either kind of molecule, so that, in place of equation(10), we must have

m^V7"2 = mu2 = mfu/2. , (11)Since the molecules will be moving in all directions equally,

we also havemV2 = m'w'2i = mV2 , (12)

and there are similar equations for the second kind of molecules.


Equation (11) gives at once the relation

i (13)

Thus when two gases are mixed at the same temperature, theaverage kinetic energy of their molecules is the same. The lessmassive molecules must move, on the average, faster than themore massive, the difference of speed being such as to make theaverage kinetic energy the same in the two cases. This is a specialcase of a much wider theorem known as the equipartition ofenergy. The general theorem is of very wide application, de-pending as it does only on the general laws of dynamics. (See§211 below.)

An illustration, of a rather extreme kind, is provided by theBrownian movements. Andrade and Parker* find that theparticles of freshly-burned tobacco smoke have an averagediameter of about 1*6 x 10~6 cm., so that their mass must be ofthe order of 10~18 gm., or 20,000 times the mass of a molecule ofair. If the particles are suspended in air, their average speedwill be about a 140th of that of the air molecules—say 3 metresa second. The combination of ultra-slow speed with ultra-largesize makes it possible to study their motions in the microscope.Yet even here we do not see this actual motion, but only themotion resulting from a succession of free paths, each of whichis performed with this velocity on the average.

Another illustration of the same dynamical principle can befound in astronomy, being provided by the motion of the stars inspace. The stars move with very different speeds, but there isfound to be a correlation between their speeds and masses, thelighter stars moving the faster, and this correlation is such thatthe average kinetic energy of stars of any specified mass is(with certain limitations) equal to that of the stars of any othermass. Here, as so often in astronomy, the stars may be treated asmolecules of a gas, and the facts just stated seem to shew that thestars have been mixed long enough for all types of stars to haveattained the same "temperature".

• Proc. Roy. Soc. A, 159 (1937), p. 515.


Temperature and Thermodynamics12. Equation (13) has shewn that the average kinetic energy

of motion of a molecule depends only on the temperature, andit is important to discover the exact law of this dependence.

We must first define temperature. Many ways of measuringtemperature depend on the physical properties of particularsubstances. The ordinary thermometry, for instance, depends onthe physical properties of water, glass and mercury. If we usedalcohol or gas in our thermometer instead of mercury, we shouldobtain a slightly different temperature-scale. In fact, there areat least as many ways of measuring temperature as there aresubstances wherewith to measure it. But, outside all these,there is one more fundamental way which is independent of theproperties of any particular substance. Clearly this is the waywhich should be used in the kinetic theory, since this is concernedwith the properties of all substances equally.

According to thermodynamical theory, the addition of a smallamount of heat-energy dQ to a mechanical system at a tempera-ture 0 produces an increase dQ/f(d) in a physical quantity 8,which is defined to be the "entropy" of the system. Here/(0)is a function, so far unspecified, of the temperature 6, whichmay so far be measured in any way we please; a change inthe method of measuring 6 is compensated by a change in theform of the function / .

This expresses no property of nature or of matter so far. I tmerely defines S as the quantity obtained by successive alge-braic additions of the quantity dQjf(O), and so as

But the investigations of Carnot* shewed that there is a certainparticular form for/(0), for which the quantity S defined in thisway depends only on the physical state of the mechanicalsystem concerned. If we start from any physical state A, andafter a series of additions and subtractions of heat-energy bringthe system back again to its original state A, then S will also

• Reflexions sur la puissance motrice du feu (English translation,Macmillan (1890)), or Preston, Theory of Heat, Chapter vm, Section n.

2 JKT


return to its original value. Thus for such a cycle of changes, wemust have

This expresses Clausius' form of the second law of thermo-dynamics. The same thing may be expressed more mathema-tically by the statement that if

dSdQ

dsthen dS is a "perfect differential". These relations are true forall mechanical systems, and so do not depend on the physicalproperties of any particular substances except in so far as theseare involved in the measuring of 6.

Lord Kelvin pointed out in 1848* that the foregoing factsmake it possible to build up a thermometer scale which shall beindependent of the physical properties of all substances; weneed only define the new temperature T by the equation

The scale obtained in this way is known as the "absolute"scale. I t is indefinite only to the extent of a multiplying constant,since T might equally well be chosen to be any multiple of /(#).In practice, this constant is chosen so that the difference betweenthe freezing and boiling points of water (under the same con-ditions as for the centigrade scale) is equal to 100 degrees. Tothis extent, and this only, the "absolute" scale is related to aparticular substance. On this scale, the entropy 8 is defined by

We proceed to connect the results already obtained with thisabsolute scale of temperature.

Returning to our cubical vessel of volume lz, let us supposethat one of its faces can slide in and out, like a piston in a cylinder,so that we can change the volume of the vessel. At first let it beof volume P, as before, and contain N molecules of average

• W. Thomson, Proc, Camb. Phil. Soc. (1848) and Trans. Roy. Soc.Edinburgh (1854).


kinetic energy E, each molecule being a hard elastic sphere ofinfinitesimal size.

Now suppose that a small quantity of energy is added, in theform of heat, to the gas in the cylinder. Part of this may warm upthe gas, so that the average energy of each molecule is increasedfrom E to E 4- dE\ as there are N molecules, thisuses up an amount NdE of the energy. Anotherpart of the added energy may be used in ex-panding the volume v of the gas. If the movableface increases its distance from the opposite faceby dl, the volume will be increased from lz toI2(l + dl), an increase of volume dv equal to l2dl.As the gas exerts a pressure pi2 on this movableface, there must also be a force pi2 acting on themovable face from outside to hold it in position.When the gas expands to the extent justdescribed, this force is pushed back a distance dl, and the energyconsumed in doing this is pl2dl or pdv. The sum of these twoamounts of energy must be equal to the amount of heat-energyadded to this gas. Calling this dQ, we must have

dQ = NdE-{-pdv. (14)

On inserting the value already found for p in equation (2), this

Fig. 6

becomesdQ

,__= 2NE .NdE + ——dv. (15)

Thus the value of dS is given by_ dQ N , - 2NE

Since S depends only on E and v, we havea s N <M

We obtain the value ofdEdv

either by differentiating dS/dv

with respect to E} or by differentiating dS/dE with respect to v.Equating the two values so obtained, we find

3-2


This shews that EjT must be a constant, and gives us our firstinsight into the physical meaning of absolute temperature; wesee that it is simply proportional to the average kinetic energyof motion of a molecule. We may in fact write

where R is a constant which is usually called Boltzmann'sconstant. We have already seen (§11) that E has the same valuefor all gases at any assigned temperature, so that R is the samefor all gases; it is in fact a universal constant of nature. Its value,as we shall see later, is 1«379 x 10~16 in C.G.S. centigrade units.

We have thus found that the average kinetic energy of motionof each molecule is proportional to the absolute temperature,the exact relation being

(17)Since the motions of the molecules can have no preference for

particular directions in space, the mean values of u2, v2 and w2

must all be equal, so that relation (17) may be expressed in thef o r m nm2^^2 = mw2 = RT (18)

The Gas-laws

13. With these relations we can at once express the relationsbetween pressure, volume and temperature. In equation (2),namely

2 x kinetic energypressure = — 3 x volume

we may put the kinetic energy equal to NE, where N is the totalnumber of molecules and E is the average kinetic energy of each.The relation now becomes

If the N molecules consist of Nr molecules of one kind, N2 ofa second kind, and so on, so that N = Nx + N2+ ...,we have

N2RT N3RT


Thus the pressure exerted by a mixture of different gases isequal to the sum of the pressures which would be exerted by theconstituents separately, which again brings us to Dalton's law (§ 8).We have already noticed that when the temperature is kept con-stant, p varies inversely as v (Boyle's law); equation (19) shewsfurther that when the volume is kept constant, p varies as T, theabsolute thermodynamic temperature (Charles' law). Further,when p is kept constant, v varies as T\ if a mass of gas is kept atconstant pressure, its volume will change in exact proportionalitywith the absolute temperature. Thus the absolute temperatureis the temperature which would be read off a thermometer inwhich the expanding substance was a gas of the ideal kind wehave been considering, the molecules being hard spheres ofinfinitesimal size, exerting no force except at collisions. This,indeed, expresses the fundamental principle of gas-thermometry.It enables us to fix the zero-point of the absolute temperaturescale—the absolute zero of temperature. This is found to be

-273-2°C. (20)

At this temperature, E = 0, which means that all the moleculesare at rest; there is no molecular motion, so that the substancecannot be in the gaseous state. As the temperature is raisedabove this point, motion ensues, the average kinetic energy permolecule being always proportional to the temperature measuredfrom this zero-point.

If we warm up the gas inside an enclosure which is kept atconstant pressure, both p and v must remain constant, so that,from relation (19), NET remains constant. It is interesting tonotice that heating up the enclosure does not, as might at firstbe thought, increase the total quantity of heat inside it; as Tincreases N correspondingly decreases, so that all the heat weprovide goes elsewhere. For instance, when we light a fire in aroom, we do not increase the heat-content of the air of the room:this remains unchanged. By increasing the pressure, we drivesome molecules out of the room, and the original amount of heat,now being distributed over a smaller number of molecules, givesmore energy to each.


14. Formula (19), namelypv = NRT,

shews that the number of molecules in a gas of specified volume,temperature and pressure is the same for all gases, and so is inde-pendent of the nature of the gas.

This is commonly known as Avogadro's law, having been pro-posed by Avogadro in 1811 as a hypothesis to explain the factthat gases unite, both by weight and by volume, in simple pro-portions which are measured by the ratios of small integralnumbers.

Without the kinetic theory to help us, it would have been rathernatural to think that, if a gas had specially massive molecules, thestandard pressure of one atmosphere would be produced by asmaller number of molecules than in a gas with light molecules.The kinetic theory shews that at any specified temperaturemassive molecules move more slowly than light ones, so that inactual fact the pressure exerted per molecule is the same for all,namely RTjv. Thus the number of molecules per c.c. needed toproduce one atmosphere pressure at the standard temperature of0° C. is the same for all gases. This number, which is known asLoschmidt's number, will be denoted by NOi and its numericalevaluation is naturally of great importance for the kinetictheory of matter.

Closely associated with it is another number, which measuresthe number of molecules in a gramme-molecule of any substance—i.e. in a mass of the substance in which the number of grammesis equal to the molecular weight of the substance. This iscommonly known as Avogadro's number and will be denotedbyN,.

Evaluation of Avogadro's and Loschmidt's Numbers

15. These numbers can be evaluated in a great variety ofways. Virgo* has enumerated more than eighty differentexperimental determinations which had been made by 1933,and the number continually increases.

* Science Progress, 108 (1933), p. 634, or Loeb, Kinetic Theory of Gases(2nd edn, 1934), p. 408.


The numbers can be evaluated most accurately from electro-lytic phenomena. When a mass of any substance is broken upelectrolytically, a certain quantity of electricity is consumed, thisbeing required to provide the electronic charges on the ions.Experiment shews that 9648-9 electromagnetic units of chargeare required for every gramme-molecule of the substance.According to the most recent determinations, the electroniccharge e—i.e. the charge on a single electron or ion—is given by

e = 4-803 x 10-10 electrostatic units

== 1-602 x 10~20 electromagnetic units

to an accuracy of within 1 part in 1,000.* Thus the above-mentioned 9648-9 electromagnetic units are equal to

electronic units of charge. This, then, must be the number ofmolecules in a gramme-molecule of any substance, i.e. in a masswhich contains a number of grammes equal to the molecularweight of the substance. Since the molecular weight of hydrogenis 2-016, a gramme-molecule of hydrogen weighs 2-016 grammes,so that the number of molecules of hydrogen to a gramme is

The density of hydrogen at standard temperature and pressureis 0-00008987, so that a cubic centimetre of hydrogen at standardtemperature and pressure weighs 0-00008987 gramme. The num-ber of molecules it contains is accordingly

JVo = 0-00008987 x 2-987 x 1023 = 2-685 x 1019 (22)

This is Loschmidt's number.16. Another, although far less exact, determination of Nx can

be made from observations on the Brownian movements in theway already explained. Perrin, who first developed the method,obtained values uniformly larger than those given above. Later

* Birge, Nature, 137 (1926), p. 187; Phys. Rev. 241 (1937), p. 241;H. R. Robinson, Reports on Progress in Physics, 4, Physical Society(1938), p. 216; Nature, 23 July 1938, p. 159.


observations by H. Fletcher* gave the value Nt = 6*03 x 1023,with a probable error of about 2 per cent of the whole, but a stillmore recent determination by Pospisilf gives Nx = 6-22 x 1023.From experiments on the Brownian movements of his torsion-balance (§5), Kappler deduced as the value of Boltzmann'sconstant (see equation (60) below), R = 1-36 x 10~16. Combiningthis with the relation pv = NRT, we can deduce as the valueofN0,

& = 2-67 x 1019.

Kappler estimated the probable error of his determination ofR to be 3 per cent, so that there is a certain element of luck in theagreement between this and the true value of No.

17. The largeness of the numbers N± and No gives a measure ofthe fine-grainedness of the molecular structure of matter. Someconception of the degree of fine-grainedness implied may perhapsbe obtained from the following considerations.

The number of molecules in a drop of water one cubic millimetrein volume will be

3-34 xlO19,18-016 xlO3

whence we can calculate that if the water is allowed to evaporateat such a rate that a million molecules leave it every second, thetime required for the whole drop to evaporate will be 3*34 x 1013

seconds, which is more than a million years.Again, a man is known to breathe out about 400 c.c. of air at

each breath, so that a single breath of air must contain about1022 molecules. The whole atmosphere of the earth consists ofabout 1044 molecules. Thus one molecule bears the same relationto a breath of air as the latter does to the whole atmosphere of theearth. If we assume that the last breath of, say, Julius Caesarhas by now become thoroughly scattered through the atmosphere,then the chances are that each of us inhales one molecule of it withevery breath we take. A man's lungs hold about 2000 c.c. of air,so that the chances are that in the lungs of each of us there areabout five molecules from the last breath of Julius Caesar.

* Phys. Rev. 4 (1914), p. 440.t Ann. d. Phys. 83 (1927), p. 735.


The 2*685 x 1019 molecules in a cubic centimetre of ordinary airmove with an average speed of about 500 yards in a second. In asingle second, then, the molecules in a cubic centimetre of airtravel a total distance of 13 x 1021 yards, or 12 x 1018 kilometres,which is about 150,000 times the distance to Sirius.

18. Since a cubic centimetre of gas at 0° C. and at standardatmospheric pressure contains 2*685 x 1019 molecules, the averagedistance apart of adjacent molecules must be about

(2*685 xlO19)-* cm.,or 3*34 x 10~7 cm. If we pass 100 km. up in the atmosphere—broadly speaking to auroral heights—we come to pressures ofonly about a millionth of an atmosphere; here the average dis-tance of the molecules is only about 3 x 10~5 cm. Out in inter-stellar space the pressure is probably less by a further factor ofabout 10~15; here adjacent molecules are, on the average, a fewcentimetres apart.

Molecular Masses19. Asthere are 2*987 x 1023molecules in a gramme of hydrogen,

the mass of the hydrogen molecule must be 3*347 x 10~24 grammes,and that of the hydrogen atom 1*673 x 10~24 grammes.

The masses of other molecules will be in exact proportionto their molecular weights; that of oxygen, for instance, ofmolecular weight 32, is 53*12 x 10~24 grammes.

The Specific Heats of a Gas20. As we have seen, the heat-energy of a gas is the aggregate

of the energies of its separate molecules, and the temperatureof the gas gives a measure of this energy. Thus to raise thetemperature of a gas, the energy of its molecules must beincreased. The amount of heat or of energy needed to raise thetemperature of unit mass by 1° is called the specific heat of thegas; it may be measured either in heat- or in energy-units.

Equation (14) already obtained, namelydQ = NdE + pdv, (23)

tells us how much energy is required to produce any specifiedchange in the temperature and volume of a gas, and so providesa basis for a discussion of the specific heats of a gas.


Specific heat measurements are usually made under one of twoalternative conditions:

(1) At constant volume, i.e. in a closed vessel.

(2) At constant pressure, i.e. in a vessel which is open tothe air or other space at a fixed pressure.

To study specific heats measured at constant volume, we putdv = 0, and the last term in equation (23) is immediatelyobliterated; all the heat goes to increasing the energy of themolecules, and the equation becomes

The specific heat at constant volume, Cv, which is the heat re-quired to raise unit mass of the gas through a temperature changeof 1° C, is given by

°v~dT*

or again by Cv = N — ,

where N is now the number of molecules in a unit mass of the gas.Since Nm — 1, we may replace Nbyl/m. We have so far supposedthe heat measured in energy-units. One calorie or heat-unit isequal to J energy-units, where J is the mechanical equivalentof heat, given by

J = 4-184 xlO7,

so that if Cv is measured in heat-units, its value is

If the measurement of specific heat is made at constant pres-sure, the gas expands as its temperature is raised, and dv must nolonger be put equal to zero. The value of the pressure is given by

pv = NET,

so that when the pressure is kept constant, we obtain by differ-entiation A7DJ/77


and equation (20) assumes the form

dQ = NdE + NRdT,whence the specific heat at constant pressure, Cp> measured inheat-units, is given by

_ldQ_ 1 IdE

The difference and ratio (y) of the specific heats are seen to begiven by R

y = ri> = 1 + - ^ 1 _ . (27)' Cv dEjdT

If the molecules are hard elastic spheres, the value of E is f RT,and the value of Cp/Cv becomes If.

Relation (26) is obeyed by most gases throughout a substantialrange of physical conditions (Carnot's law). Also the ratio ofthe specific heats is equal to about If for the monatomic gasesmercury, helium, argon, etc., suggesting that the atoms of thesesubstances behave, in some respects at least, like the hardspherical balls which we have taken as our molecular model. Onthe other hand air and the permanent diatomic gases have avalue of Cp/Cv equal to about If under normal conditions, sug-gesting that their molecules are not adequately represented bythe hard spherical balls of our model. This is perhaps hardlysurprising, seeing that these molecules are known to consist oftwo distinct and separable atoms.

21. Complex molecules of this kind may rotate or have internalmotion of their parts, so that clearly we must suppose they canpossess energy other than the kinetic energy of their motion; letus suppose that on the average this energy is (5 times the averagekinetic energy of motion. The average total energy of a moleculeE is now given by

Carrying through the analysis as before, we find


££ (29)

in which J3 has been supposed not to vary with T.

The relation Cp-Cv = -*?-,

which is true for most gases, still holds, but in place of relation(27) we have

Thus the value of/? can be deduced from the observed value ofCp/Cv for any gas. This gives us some knowledge of the structureof the molecules of gases, which will be produced and discussed inits proper place.

Maxwell's Law22. So far we have been concerned only with the average

kinetic energy, and the average speeds of molecules; we have hadno occasion to think of the energies or speeds of individual mole-cules. And it is obvious that the speeds of motion of the variousmolecules cannot be all equal; even if they started equal, a fewcollisions would soon abolish their equality. For the solution ofmany problems, it is necessary to know how the velocities ofmotion are arranged round the mean, after so many collisionshave occurred that the gas has reached its final steady state inwhich the distribution of velocities is no longer changed bycollisions.

To study this problem, let us imagine that the molecules of agas still move in a closed vessel, but that they are under the in-fluence of some permanent field of force—gravitation will serveto fix a concrete picture in our minds, although it is more in-structive to imagine something more general. Under these con-ditions, the velocity of a molecule will change continuously as itmoves from point to point under the forces of gravity or the otherpermanent field of force, and will also change discontinuouslyeach time it collides with another molecule.

Let us now fix our attention on a small group of moleculeswhich occupies a small volume of space dx dy dz, defined by the


condition that the ^-coordinate lies within the small range fromx to x + dxy while the y- and ^-coordinates lie within similar rangesfrom y to y + dy, and from z to z + dz. We shall not concern our-selves with all the molecules inside the small element of volume,but only with those particular molecules of which the velocitycomponents are approximately u, v, w; to be precise we shallconfine our attention to molecules with velocities such that thew-component lies between u and u + du, while the v- and ^-com-ponents lie within similar small ranges from v to v + dv and w tow+dw. The number of molecules in this group will of course beproportional not only to the volume dxdydz, but also to the pro-duct of the small ranges dudvdw. Thus it is proportional to theproduct of differentials

dudvdw dxdydz.

It is also proportional to another factor of the nature of a"density", which specifies the number of molecules per unitrange lying within the range in question. This factor naturallydepends on the particular values of x, y, z and of u, v9 w. Let ussuppose that, whatever these values are, it is

f(u,v,w,x,y,z)y

so that the number of molecules in the group under considera-tion is

f(u,v,w,x,y,z) dudvdw dxdydz. (31)

As the motion of the gas proceeds, changes will occur in thevalues of x, y, z and u, v, w for individual molecules, and also inthe values of dx, dy, dz, du, dv and dw for the group as a whole.Let us suppose that after a small time dt, these quantities havebecome

x',y',z',u',v',w' and dx',dy'\dz*,du',dv',dw\

respectively. These are of course given by

x' = x + udt, etc. (32)

and ur = u + Xdt, etc., (33)

where X, Y, Z are the components of force per unit mass on amolecule at x> y, z.


If the gas is in a condition of steady motion, this group ofmolecules will exactly step into places which have just beenvacated by a second group of molecules, which occupied theseplaces at the beginning of the interval of time dt, and so will beequal in number to this second group. The number in the secondgroup is, as in formula (31),

f(u',v',w',x'yy',z')du' dv' dw' dx' dy' dz', (34)

so that if the gas is in a state of steady motion, expressions (34)and (31) must be equal.

The relation between u, v, w, x, y, z and u\ v', w', x\ y', z'is that expressed in equations (32) and (33). From these rela-tions, it is fairly easy to shew that the product of the sixdifferentials du' dv1 dw' dx' dy' dz' is precisely equal to theproduct du dv dw dx dy dz; we need not spend time over a detailedproof, since this would only constitute a very special case of ageneral theorem to be proved later (§206). The products ofdifferentials in formulae (31) and (34) being equal, we musthave the equation

f(u,v,w,x,y,z) = f(u',v',w',x',y',z') (35)

Thus/(w, v, w, x, y, z) is a quantity depending only on the valuesof u, v, w and of xy y, z, which does not change in value as a mole-cule follows out its natural motion without any collisions takingplace. One such quantity is of course the energy of the molecule,which we may denote by E; clearly then

f(u,vfw,x,y,z) = E (36)is a solution of equation (35). A more general solution is

f(u,v,w,x,y,z) = 0(E), (37)

where 0(E) is any function whatever of E, as for instance itssquare or its logarithm. Here E, the total energy of a movingmolecule, is given by

E = £m(w2 + v2 + w2)-f;\;, (38)

where x *s ^he potential energy of a molecule in the field of force,so that the forces acting on the molecule are given by


It is quite easy to shew* that this expresses the most generalsolution possible of equation (35), but we need not delay over thisproof, as a complete discussion of the whole problem will be givenlater (Chap. x).

23. We have now found that if the law of distributionis given by equation (37), where 0 represents any functionwhatever, the distribution of velocities will not be altered bythe natural motion of the gas, so long as no collisions takeplace..

In general, however, the law of distribution will be altered bycollisions, and the question arises whether there is any specialform for the function 0 such that collisions have no effect. If so,on inserting this form of 0 in equation (37), we shall obtain a lawof distribution which remains unaltered both by the naturalmotion of the gas and by the occurrence of collisions between itsmolecules. Such a law of distribution must of course represent atrue steady state.

As we shall see later (§ 189), a gas in which abundant collisionsare occurring behaves exactly like the fluid of hydrodynamicaltheory; at every point there is a pressure of the amount given byequation (3) or (19), namely

pRT

m

If the gas is in a steady state at a uniform temperature T,variations in this pressure hold the gas at rest against the forcesexerted by the external field. As we have supposed these forcesto be X, Y, Z per unit mass, the hydrostatic equations whichexpress this are

dp y

or, inserting the values of p and X,

RTdp __ _pdxm dx mdx*

* See, for example, Jeans, Astronomy and Cosmogony (2nd edn.),p. 364.


These three equations have the common integral, well knownin hydrodynamical theory,

BTlogp = — x + a constant,or

p = Be~*IRTy (39)

where B is a constant.If the steady state of the gas is not to be destroyed by collisions,

p must vary with the potential x ^ the way indicated by thisequation.

Nowp expresses the law of distribution in space alone, whereas/ expresses it for both space and velocities. If we are to find atrue steady state, it must be by extending the p given by formula(39) into a formula for/ by the addition of velocity terms in u, vand w. Since/? is a function of x only, while/must be a function of

the obvious extension of formula (39) is

f(u, v, w, x, y,z) = A Be RT ,

where A B is a new constant, being A times the former constant B.We shall see later that this is the usual law of distribution for

a gas in its steady state. I t is customary to write h for 1/2RT, sothat the formula becomes

f(u,v,w,x,y,z) = ABe-hn*u2+v2+w*>-2hx (40)

The two factors B and e~2hx express the law of distribution ofthe molecules in space in accordance with formula (39). Theremaining factors

must therefore express the law of distribution of velocity com-ponents for the molecules at any point of space.

24. This law was first discovered by Maxwell, and so is com-monly known as Maxwell's law. It occupies a very central positionin the kinetic theory, and will be fully discussed in a later chapter.

A more rigorous proof will also be given. The proof just givenfails entirely in mathematical rigour, but is of interest becauseit provides a physical interpretation of the exponential factor


which is the outstanding feature of Maxwell's law. We haveseen that this is exactly analogous to the exponential in thewell-known formula (39) which expresses the falling off ofdensity of a gas in a field of force. This latter formula shewsthat the chance of a molecule having high potential energy fallsoff exponentially with the amount of the potential energy: inthe same way Maxwell's law tells us that the chance of a moleculehaving high kinetic energy falls off exponentially with theamount of the kinetic energy. The two exponentials are ofexactly similar origin.

We can see still more clearly into the matter by supposingour gas to be the atmosphere, and the field of force to be thatof the earth's gravitation. We may think of a molecule of airwhich reaches the top of the atmosphere as getting there as theresult of a succession of lucky collisions with other molecules,each of which gives it enough potential energy to climb onerung higher in the ladder of the atmosphere. The chance of anymolecule experiencing a succession of n lucky hits is proportionalto a factor of the form e~an, where a is a constant. Thus we canpicture in a general way why the chance of a molecule havingpotential energy x *s proportional to e~Px, where ft is a newconstant. This gives a physical insight into the origin of theexponential factor in formula (39).

In the same way we may think of a molecule with an excep-tionally large energy E, as having acquired it by a successionof lucky collisions. If E is the average kinetic energy of amolecule, the number of lucky collisions needed to give kineticenergy E is proportional to E — E, and the chance of a moleculehaving experienced this number of lucky collisions is propor-tional to e-ME~®, which again is proportional to e~Ê.

The value of an exponential with negative index, such ase~x*, falls off very rapidly as x becomes large, but it does notbecome actually zero until x reaches the value ± oo. Applyingthis to formula (41), we see that there are molecules with allvalues of u, v, w right up to u, v, w = ± oo, but large values ofu,v9w are excessively rare. Still even the largest values are notprohibited by the formula.

To notice a simple consequence of this, we know that pro-


jectiles escape altogether from the earth's gravitational field iftheir velocity of projection exceeds about 11 kilometres persecond. In the earth's atmosphere, the average molecularvelocity is one of hundreds of metres per second. Thus molecularvelocities of 11 kilometres per second are excessively rare, but arenot non-existent. Of the molecules in the outermost layer ofthe atmosphere, a certain small proportion must always havevelocities in excess of this. These molecules in effect constituteprojectiles which will pass off into space never to return. For thisreason, the earth is continually losing its atmosphere, althoughat an excessively slow rate.

The molecules which are most likely to attain speeds of11 kilometres a second are those of smallest weight, for thesehave the highest average velocity. Thus hydrogen will be lostmore rapidly than nitrogen, and nitrogen more rapidly thanoxygen. A simple calculation shews that any hydrogen in theearth's atmosphere would be speedily lost.

The corresponding velocity for the moon is only about 2-4kilometres per second, so that when the moon had an atmosphere—as it must have had at some time in the past—the rate ofescape must have been comparatively rapid. This is why themoon no longer has an appreciable atmosphere. The atmosphereof the other planets and their satellites may be discussed in thesame way, with results which are found always to be in agreementwith observation.

The mean value of u2 + v2 -f w2, which we have denoted by C2,can be obtained from formula (41) by a simple integration (cf. §91,

below), and is found to be —=— or 3 — T. In a similar way, the' 2hm m

mean value of the velocity *J(u2 + v2 + w2)} which we shall denote by2

c, is found to be -7—=—r, which is 0-921 times C. so that C= l-086c.^(nhm)

25. If two gases are mixed, and the mixture has attained itssteady state, the temperature must be the same for both, so thath, which is equal to 1/2RT, must be the same for both. Thus thelaws of distribution for the two gases will be of the form

^ e-hm(u2-'rv2iw2)^ (42)


A' e-hmXu'*+v'2+w'*)9 (43)

where m' and u',v\w' denote the mass and components ofvelocity of a molecule of the second gas.

If we introduce new velocities u" ,v" ,w" defined by

this second law may be put in the formA' e-hm(u"i+v"2+w"*)

which is identical with (41) except for u"', v", w" replacing u, v, w.Thus the molecular velocities in the second gas are uniformly*](m/m') times those in the first gas. As a particular case of this,

(44)

so that the average molecular energy is the same in the two gases,the result already obtained in § 11.

The Free Path

26. We may next glance at some problems connected with thefree paths of the molecules—i.e. the distances they cover betweensuccessive collisions.

Let us fix our attention on any one molecule, which we maycall A9 and consider the possibility of its colliding with any oneof the other molecules B,C,D,.... If, for the present, we pictureeach molecule as a sphere of diameter cr, then a collision will occurwhenever the centre of A approaches to within a distance cr ofthe centre of B, C, D, or any other molecule, Thus we may imagineA extended to double its radius by a sort of atmosphere, andthere will be a collision whenever the centre of B,C,D,... passesinto this atmosphere (see fig. 7).

Now as A moves in space, we may think of it as pushing itsatmosphere in front of it, and for each unit distance that A moves,the atmosphere covers a new volume nor2 of space. If there arev molecules per unit volume, the chance that this volume shallenclose the centre of one of the molecules B,C,D,... is nvcr2.If the molecules B, C, Dt ... were all standing still in space,

4-2


the average distance that A would have to travel before it ex-perienced a collision would be

177W2*

This calculation reproduces the essential features of the free-path problem, but is generally in error numerically because ittreats the molecule A as being in motion while all the othermolecules stand still to await its coming. Thus it gives accuratelythe free path of a molecule moving with a speed infinitely greater

Fig. 7

than that of the other molecules (§114). I t also gives very ap-proximately the free path of an electron in a gas, because this,owing to its small mass, moves enormously faster than the mole-cules of the gas. It must however be noticed that cr no longerrepresents the diameter either of a molecule or of an electron,but the arithmetic mean of the two, and as the diameter of theelectron is very small, cr is very nearly the radius of the molecule.

In a variety of cases numerical adjustments are neededwhich will be investigated later (§§ 108 fi\). When all the mole-cules move with speeds which conform to Maxwell's law, theaverage free path is found to be

1 (45)


Viscosity

27. The free path in a gas which is not in a uniform steady stateis of special interest. A molecule which describes a free path PQof length A with ^-velocity u may be regarded as transportingx-momentum mu from P to Q. If the gas is in a steady state, wemay be sure that before long another molecule will transport anequal amount mu of momentum from Q to P, so that, on balance,there is no transfer of momentum.

Suppose, however, that the gas is not in a steady state, but isstreaming in parallel layers, with a velocitywhich varies from one layer to another. Tomake a definite problem, suppose that the gasis streaming at every point in a directionparallel to the axis Ox, that its velocity of

Q

\

streaming is zero in the plane z = 0, andgradually increases as we pass to positive valuesof z (fig. 8). Let the velocity of streaming atany point be denoted by u, which is of course ° \ ,the average value of u for all the molecules _,.at this point.

Again every molecule which describes a free path PQ withvelocity u transports momentum mu from P to Q. If the freepath crosses the plane 2 = 0, the molecule transports momentumacross the plane 2 = 0.

The molecules which cross this plane from the positive to thenegative side have, on the average, positive values of u, becausethey start from regions where the gas is streaming in the directionfor which u is positive, and so transport a positive amount ofmomentum across the plane 2 = 0 into regions where the momen-tum is negative. But the molecules which cross in the reversedirection start from regions in which u is prevailingly negative,and so on the average carry a negative amount of momentumacross the plane into regions where the momentum is positive.Thus both kinds of free path conspire to equalise the momentum,and so also the velocities, on the two sides of the plane. It isas though there were a force acting in the plane 2 = 0 tendingto reduce the differences of velocity on the two sides of this


plane. This provides the kinetic theory explanation of gas-viscosity.

A rough calculation will give an idea of its amount. Let atypical molecule describe a free path with momentum parallel toOx of amount mu, and let us suppose the projection of the freepath on the axis of z to be A'. Then the velocities of streamingdiffer at its two ends by an amount

duX~dz>

and if the momentum mu is that appropriate to P it is in-appropriate to Q by an amount

m A ' ^ . (46)az

If there are v molecules per unit volume, and we suppose eachto move with an average velocity c, the number of molecules whichcross unit area of the plane z = 0 in both directions in unit time isof the order of magnitude of vc\ multiplying this by expression(46), we find that the transfer of momentum per unit area perunit time across the plane z = 0 would be

^,du -^,duvc x raA -7- = OCA -7-

dz dz

if each free path had the same projection A' on the axis of z. Ifthe free paths are distributed equally in all directions in space andof average length A, we easily find that the average value of A'is iA. Inserting this value for A' into the formula just obtained,we find that the transfer of momentum is

The theory of viscosity tells us that the transfer is

du' & •

where 77 is the coefficient of viscosity. Thus

(47)


This simple calculation needs innumerable adjustments, butmore exact analysis to be given later shews that the actual valueof the coefficient of viscosity does not difFer greatly from that justfound.

Conduction of Heat

28. A moving molecule transports energy and mass as wellas momentum. A molecule which describes the free path PQ offig. 8 with energy E may be regarded as transporting energyE from P to Q. If the mean energy E at P is equal to that at Q,this transfer of energy will be equalised by another in the oppositedirection. But if the gas is arranged in layers of equal temperatureparallel to the plane of xy, the two mean energies will not beequal, and the first molecule carries, on the average, an excessof energy _

dEdz

to Q, while the second carries a deficiency of equal amount to P.Just as in our earlier discussion of momentum, we find that acrossunit area of any plane there is a transfer of energy equal to

fz , or *****. (48)

Now the theory of conduction of heat tells us that the transferof energy is

where k is the coefficient of conduction of heat and J is themechanical equivalent of heat. Comparing expressions (48) and(49), we find as the value of Jc,

7 vcXdE

We have found as the values of ij (the coefficient of viscosity)and Cv (the specific heat at constant volume)

V =

C = - -• JmdJmdT'


Using these relations, equation (50) may be put in the form

The three quantities Jc, rj and Cv which enter into this equationcan all be determined experimentally, and the accuracy withwhich they satisfy the equation provides a test of the truth of ourtheory. We shall see in Chapter vn that the equation is satisfiedwith very fair accuracy; the relatively small discrepancies aresuch as can be ascribed to the simplifying assumptions we havemade and to the fact that molecules have more complexity ofstructure than our theory has allowed for.

29. Finally, if we regard the moving molecules of a gas astransporters of mass, we arrive at a theory of gaseous diffusion,which will be discussed in due course.

Numerical value of the Free Path

30. Of the quantities which enter into the viscosity equation

V = ipc*>

we have seen how to calculate c, and TJ and p can be determinedexperimentally. Thus A can be deduced from this equation, or,better, from the more exact equations to be given later. We shallfind that the free path in ordinary air is about 6 x 10~6 cm. or a400,000th part of an inch. In hydrogen, under similar conditions,the free path is about 1'125 x 10~5 cm. The length of mean freepath, as is clear from formula (45), is independent of the tempera-ture or speed of molecular motion, and depends only on thedensity of the gas. If we double the number of molecules in ourgas, keeping its volume unaltered, each molecule has twice asmany molecules to collide with, so that at each instant thechance of collision is twice as great as before. Consequently themean free path is halved. In general the length of the mean freepath is inversely proportional to the number of molecules percubic centimetre of gas. If the pressure is reduced to half amillimetre of mercury, the gas has only 1:1520 of normalatmospheric density, so that the free path in air is about a tenthof a millimetre. In interstellar space, the density may be as lowas 10~24 gm. per c.c. and the free path is of the order of 1016 cm.,


or 100,000 million km. In internebular space, where the densityof gas may be as low as 10~29 gm. per c.c, the free path will be ofthe order of 1016 km., or about 1000 light-years.

Comparing the values just obtained for the length of the freepath with those previously given for the velocity of motion, we findthat the mean time of describing a free path is about 1-3 x 10~10

seconds in air under normal conditions, about 2 x 10~7 secondsin air at 0° C. at a pressure equal to that of half a millimetreof mercury, and about a thousand million years in internebularspace—out here a molecule travels for a long while before meetinganother!

As a consequence of these long free paths, molecules out ininterstellar space can exist in conditions which are impossibleunder the far more crowded conditions of laboratory samples,with the result that lines and bands are observed in nebularspectra which cannot be reproduced in the laboratory, althoughtheoretical calculations shew that they ought to occur in gasesat extremely low densities.

Molecular Diameters

31. Having estimated the free path from an experimentaldetermination of the coefficient of viscosity, we may proceedto estimate the molecular diameter from the relation

A. == ~/— •

We find (§ 152) that the diameter of a molecule of air is about3*75 x 10~8 cm., while that of a molecule of hydrogen is onlyabout 2-72 x 10"8 cm.

Thus in gas at atmospheric pressure the mean free path of amolecule is some hundreds of times its diameter (160 times forair, 400 times for hydrogen). When the pressure is reduced to halfa millimetre of mercury, the free path is hundreds of thousandstimes the diameter. It is generally legitimate to suppose, as afirst approximation, that the linear dimensions of molecules aresmall in comparison with their free paths.

More definite figures for the lengths of free paths and the sizeof molecules will be given later, but it is no easy matter to discuss


these questions with any precision; we cannot even define thesize of a molecule. The difficulty arises primarily from our ignor-ance of the shape and other properties of the molecule. If mole-cules were in actual fact elastic spheres, the question would besimple enough; the size of the molecule would be measured simplyby the diameter of the sphere. The molecules are not, however,elastic spheres; if they are assumed to be elastic spheres, experi-ment leads to discordant results for the diameters of these spheres,shewing that the original assumption is unjustifiable. Not onlyare the molecules not spherical in shape, but also they are sur-rounded by fields of force, and most experiments measure theextension of this field of force, rather than that of the moleculesthemselves.

These questions will be discussed more fully at a later stage(§§146ff.).

Chapter III

PRESSURE IN A GAS

CALCULATION OF PRESSURE IN AN IDEAL GAS

32. Leaving our general survey, we now turn to a more de-tailed study of various special problems. The present chapter willbe concerned with the calculation of the pressure in a gas.

This has already been calculated on the supposition that themolecules of the gas are hard spheres of infinitesimal size, whichexert no forces on one another except at collisions. We nowgive an alternative calculation, which still assumes the moleculesto be infinitesimal in size and exerting no forces except atcollisions, although not necessarily hard or spherical.

33. Let dS in fig. 9 represent a small area of the boundary ofa vessel enclosing the gas. Let therebe v molecules per unit volume of thegas, and let these be divided into classes,so that all the molecules in any oneclass have approximately the samevelocities, both as regards magnitudeand direction. Let vv v2,... be the num-bers of molecules in these classes, sothat vx + v2+... = v.

dS

Fig. 9Since the molecules of any one classall move in parallel directions, theymay be regarded as forming a shower of molecules of densityvt per unit volume, in which every molecule moves with thesame velocity. Some showers will be advancing towards thearea dS of the boundary, some receding from it. Let us supposethat the shower formed by molecules of the first class is ad-vancing.

The molecules of this shower which strike dS within an intervalof time dt will be those which, at the beginning of the interval,lie within a certain small cylinder inside the vessel (see fig. 9).

52 PRESSURE IN A GAS

This cylinder has dS for its base, and its height is uxdt, where uîs the velocity of molecules of the first shower in the directionnormal to dS, which we shall take to be the axis of x. Thus thevolume of the cylinder is u^tdS, and the number of moleculesof the first shower inside it is v-^dtdS.

Each of these molecules brings to dS momentum mux in adirection normal to the boundary. Thus in time dt the wholeshower brings momentum mv-û^dtdS, and all the showerswhich are advancing on dS bring an aggregate momentumZmv1v^dtdS9 where the summation is over all showers for whichu is positive.

The pressure between the boundary and the gas first reducesthis ^-momentum to zero, and then creates new momentum inthe opposite direction (u negative).

The pressure which, acting steadily for a time dt, will reducethe original momentum to zero is Zmv-^dS, where the sum-mation is again over all showers for which u is positive.

The pressure needed to create the new momentum afterimpact can be calculated in precisely the same way. Weenumerate the molecules which have impinged on dS within aninterval dt, and find that their momentum would be producedby a steady pressure Zmv^^dS, where the summation is nowover all showers for which u is negative.

Combining these two contributions, we find that the totalpressure pdS on the area dS is given by

pdS = Zmv&ldS, (52)

where the summation is over all the showers of molecules.The value of Zvxu\ is the sum of the values of u2 for all the

molecules in unit volume, and this is equal to vu2. Thus equation(52) may be written in the form

p = mvu2 — pu2. (53)

As we have seen (§ 12) that

rav? = mv2 = mw2 = \mC2 = RT9 (54)

equation (53) again assumes the new forms

= VRT. (55)

PRESSURE IN A GAS 53

If the gas consists of a mixture of molecules of different kinds,the summation of equation (52) must be extended to all the typesof molecules, and the final result, instead of equation (55), is

p = (v + v' + ...)RT. (56)

If a volume v of homogeneous gas contains N molecules in all,then (v + v' + ...)v = N, and equations (55) and (56) may becombined in the single equivalent equation

pv = NRT. (57)34. In making this calculation, it has not been necessary to

make any assumption as to the way in which momenta or velo-cities are distributed in the various showers after reflection fromthe boundary.

Certain experiments by Knudsen* have led him to the con-clusion that the directions of mole-cular motion after impact are notin any way related to the direc-tions of motion before impact, butthat the molecules start out afreshafter impact in random directionsfrom the boundary, so that thenumber which make an anglebetween 6 and d + dd with thenormal to the surface will be pro-portional to sin Odd. If this were lg*the true law, it is easily seen that when molecules are reflectedfrom a minute area of the inner surface of a sphere, the numberwhich would strike any area of the sphere would be propor-tional simply to the area. Knudsen has tested this by reflect-ing molecules from a small area of a glass sphere which waskept at room temperature, and allowing them, after reflection,to form a deposit on the remainder of the sphere which was keptat liquid air temperature (fig. 10). The law was found to be wellobeyed.

From the same supposition, Knudsen has deduced certainother properties which he findsf to be in good agreement with

* Ann. d. Phys. 48 (1915), p. I l l , and The Kinetic Theory of Gases(Methuen, 1934), pp. 26 ff. f 4nn. d. Phys. 48 (1915), p. 1113.


observations on the flow of gases. The assumption in question hasalso been tested by R. W. Wood,* who finds that it agrees veryclosely with experiment. The exact law of reflection, althoughimmaterial for the calculation of normal pressure, is of importancewhen tangential stresses have to be estimated, as in the flow ofgases through tubes (§§ 143, 144 below).

Langmuirf and others have supposed that when a gas is incontact with a solid, those molecules of the gas which collide withthe surface of the solid are either partially or wholly adsorbed bythe solid, and subsequently re-emitted by the surface in a mannerwhich does not depend on their history before adsorption tookplace; the molecule, so to speak, ends one life when it strikes theboundary and after an interval starts out again on a new existence.This supposition has obtained considerable experimental con-firmation and provides a rational explanation of Knudsen's law.A great deal of work has been done on the subject, but many com-plications and difficulties remain.}

35. As we have already noticed (§§ 13, 14), equation (57) pre-dicts all the well-known laws of gases—the laws of Avogadro,Dalton, Boyle and Charles. These laws have of course only beenshewn to be valid within the limits imposed by the assumptionsmade in proving them. The principal of these assumptions havebeen that the molecules (or other units by which the pressure isexerted) are so small that they may be treated as points in com-parison with the scale of length provided by intermoleculardistances, and that the forces between molecules are negligibleexcept at collisions. Thus the laws are best regarded as ideallaws, which can never be absolutely satisfied, but which aresatisfied very approximately in a gas of great rarity. Animaginary gas in which the molecules exert no forces except atcollisions, and have dimensions so small in comparison with theother distances involved that they may be regarded as points isspoken of as an "ideal" gas. The foregoing laws are always true

* Phil. Mag. 30 (1915), p. 300; see also Phil. Mag. 32 (1916), p. 364.t Phys. Rev. 8 (1916), p. 149; Journ. Amer. Chem. Soc. 40 (1918),

p. 1361; Trans. Faraday Soc. 17 (1921), Part III.t For a good summary, see Loeb, Kinetic Theory of Gases (2nd edn.,

1934), pp. 338 ff.


for an ideal gas; for real gases they will be true to within vary-ing degrees of closeness, the accuracy of the approximationdepending on the extent to which the gas approaches the stateof an ideal gas.

The method of evaluating the pressure which has been givenin § 33 in no way requires that the medium should be gaseous,although the resulting laws of Dalton, Boyle and Charles areusually thought of only in relation to gases. Clearly, however,these laws must apply to any substance with a degree of approxi-mation which will depend only on the nearness to the truth of theassumptions just referred to.

In point of fact the laws are found to be true (as they oughtto be) for the osmotic pressure of weak solutions. They arealso true for the pressure exerted by free electrons movingabout in the interstices of a conducting solid, and also for thepressure exerted by the "atmosphere" of electrons surroundinga hot solid. Each of these pressures p may be assumed to be givenby formula (55), where v is the number of free electrons per unitvolume.

Numerical Estimate, of Velocities

36. From equation (55) we at once obtain the relation

C* = ^ (58)

which, as we have already noticed, provides the means of esti-mating the molecular velocity of a gas, as soon as correspondingvalues of p andp have been determined observationally.

The density of oxygen at 0° and at the standard atmosphericpressure of 101323 x 106 dynes per sq. cm. is found to be0-0014290, whence we can calculate from equation (58) that

C = 461*2 metres per second.

Also we have seen (§ 24) that C = l«086 times c, the mean velocityof all molecules, so that

c = 425 metres per second.

We have also obtained the relation


where T is the temperature on the absolute scale, and R is a con-stant which is the same for all gases. Thus as between one kind ofmolecule and another, G varies as ra~*. Knowing the values of Cand c for oxygen at 0°, it is easy to calculate them for any othersubstance. It is also easy to calculate C and c for other tempera-tures since equation (59) shews that both vary as T*, the squareroot of the absolute temperature.

37. In this way the following table has been calculated:

Molecular Velocities at 0° C.

Gas (or othersubstance)

HydrogenHeliumWater-vapourNeonCarbon-monoxideNitrogenEthyleneNitric oxideOxygenArgonCarbon-dioxideNitrous oxideKryptonXenonMercury vapour

AirFree electron

Molecularweight(0 = 16)

2-0164-002

18-01620-1828-0028-0228-033001320039-944400440282-17

131-3200-6

IAy(H=l)

Rm

4127x104

2077 x104

462 x104

412 xlO4

297 x104

297 x104

297 x104

277x104

260 x104

208 x104

189x104

189 xlO4

101x 104

63 x 104

41-6 xlO4

[287 x 104]1-515 xlO1 1

C(cm. per sec.

at 0° C.)

1839 xlO2

1310x102

615x102

584 xlO2

493 x102

493 x102

493 xlO2

476x102

461x102

431x102

393 x102

393 x102

286 x102

227 x 102

185 xlO2

485 x102

1-114 xlO7

c(cm. per sec.

at 0° C.)

1694 xlO2

1207 x102

565 x102

538 x102

454 xlO2

454 x102

454 x102

438 x102

425 x102

380 x JO2

362 x102

362 x102

263 x102

209 x102

170 xlO2

447 x102

1-026 xlO7

We find that for oxygen Rjm — 259-6 x 104, while the valueof m is found, as in § 19, to be 53-12 x 10~24 grammes. Hence,by multiplication,

R = 1-379 xlO"16. (60)

This quantity is a universal constant, depending only on theparticular scale of temperature employed. It will be rememberedthat f R is the kinetic energy of translation of any moleculewhatever at a temperature of 1° absolute (cf. equation (17)). I tis often convenient to denote this by a single symbol a, so that

a = f R = 2-068 x 10~16. (61)


The kinetic energy of translation of a molecule (or free atomor electron) at a temperature of T degrees absolute is now <xT.

Other numerical values which are frequently of service are

RT0 = 3-767 x 10-14, aT0 = 5-651 x 10-14,

where To = 273-2° (centigrade), the temperature of melting ice(0° C.) on the absolute scale.

38. The order of magnitude of the molecular velocities justobtained might have been foreseen, without detailed calculation,in the following way.

Any disturbance in a gas will of course first produce an effecton the molecules in its immediate neighbourhood. When thesemolecules collide with those in the adjacent layers of gas, theeffect of the disturbance is carried on into that layer, and so onindefinitely. Thus the molecules of a gas act as carriers of anydisturbance, and the disturbance is propagated through the gaswith a velocity comparable with the mean velocity of motion ofthe molecules, just as, for instance, news which is carried by relaysof messengers spreads with a velocity comparable with the meanrate at which the messengers travel. The propagation of a dis-turbance in the gas is, however, nothing but the passage of a waveof sound, whence we see that the mean molecular velocity in anygas must be comparable with the velocity of sound in the samegas. Actually the velocity of sound a is given by the well-knownhydrodynamical formula

t

where y is the ratio of the two specific heats of the gas in question(cf. § 222 below). On replacing p by its value, %pC2, this equationbecomes

For diatomic gases at ordinary temperatures, y =» If, so that

a = 0-683C = 0-742c, (62)

shewing the actual relation between the velocity of sound and thevelocities G and c for these gases.

For instance, the table gives for air at 0° C, G = 485 metresper second, whence formula (62) leads to a «=• 331*3 metres per

3 JKT


second for the velocity of so and in air, which approximates veryclosely to the true value.

Effusion of Gases

39. The general order of magnitude of molecular velocities canalso be discovered experimentally by making a minute hole in thewall of a containing vessel and allowing the imprisoned gas tostream out; the velocity of efflux is nothing else than the velo-cities of the individual molecules. These would have been simplymolecular velocities inside the vessel had the hole not beenpresent. The hole in the vessel causes the molecular velocities topersist out into space, where they can be measured observa-tionally. In fig. 9 (p. 51) imagine that the element of surface dSis replaced by a minute trap-door, which is suddenly opened.The number of molecules which stream through the trap-doorin time dt is of course equal to the number which would haveimpinged on the element dS of the boundary had the trap-doorremained closed. If each molecule were moving directly towardsthe trap-door with a velocity c, the mean velocity of all themolecules of the gas, this number would be

vcdSdt,

so that the rate of efflux, measured in terms of mass per unittime per unit area of aperture, would be

mvc=pc. (63)

Actually half of the molecules are moving away from theaperture, so that this formula must be reduced by a factor of \,and those which are moving towards the aperture are movingin all directions at random, and therefore according to the lawsinddddcj). A simple integration shews that this reduces formula(63) by a further factor of J. When both of these considerationsare taken into account, formula (63) must be replaced by

\pc.

Thus the rate of efflux is the same as if the whole gas ofdensity p streamed out of the aperture with a uniform velocity \c.

PRESSURE IN A GAS

Using the value for c obtained in § 24, we find that the formulacan also be put in the form

RT(64)

shewing that the rates of efflux of different gases at the same densityand temperature vary inversely as the square roots of the molecularweights of the gases.

For gases at the same pressure and temperature, the densitiesare proportional to the molecular weights, so that the rate ofefflux varies as the square root of either. This of course followsmore directly from the equipartition of energy (§11).

40. In 1846 Graham* made some experiments to test this latterlaw, measuring the speeds of efflux of various gases comingthrough fine perforations in a brass plate. The results are shewnin the following table:

Efflux of Oases

HydrogenMarsh gasEthyleneNitrogenAirOxygenCarbon-dioxide

-y/( density)(air=l)

0-2630-7450-9850-9861-0001-0511-237

Rate of efflux(air=l)

0-2760-7530-9870-9861-0001-0531-203

Had the law been completely confirmed, the numbers in thesecond and third columns would have been identical; we shouldthen have had a direct experimental proof of the law of equi-partition of energy. Actually the numbers given in the tableseem rather to suggest that the law is not obeyed with any greataccuracy.

Knudsen found the reason for this in 1909;f it is simply thatas the molecules are not mere points of infinitesimal size, theycollide with one another and affect one another's motion

* Phil. Travs. Roy. Soc. 136 (1846), p. 573.t Ann. d. Phys. 28 (1909), p. 75.


in passing through the perforations. He performed experimentssimilar to those of Graham, but used a hole only 0-025 mm.diameter, cut in a platinum strip of only 0-0025 mm. thickness.He worked down to pressures as low as 0-01 mm. of mercury, atwhich gaseous colKsions are very rare indeed, and found that thetheoretical laws were well obeyed so long as the mean free pathin the issuing gas was at least ten times the diameter of thehole. As soon as the mean free path became less than this, rathermore gas came out than was predicted by the formula, and as thepressure was still further increased, the efflux of gas graduallypassed over to that predicted by the ordinary hydrodynamicalformula for the flow of fluid through a hole.

A number of experimenters have tried to determine themolecular weights of various gases by a use of formula (64), butKnudsen's results make it clear that accurate results cannot beexpected unless extreme precautions are taken.

Formula (64) is applicable only to a gas flowing out into a per-fect vacuum. If there is any gas on the farther side of the orifice,some of the molecules of the issuing gas will collide with themolecules of the external gas and be driven back, thus reducingthe rate of efflux. If, however, the external gas is of very lowdensity, there will be few collisions of this kind, and formula (64)will still give a good approximation to the rate of efflux.

Transpiration of Gases41. For experimental purposes, it is often better, instead of

using a single orifice or perforation,to use the large number of very smallorifices provided by the interstices ina plug of porous material, such as un-glazed earthenware or meerschaum.The phenomenon is then spoken ofas "transpiration" rather than "effu-sion".

Imagine a vessel of gas divided into two parts by such a porousplug (fig. 11). Transpiration or effusion will be going on from eachside of this plug to the other. If the two chambers into which thevessel is divided are denoted by A and JS, there will be some gas


from A crossing through the porous plug into B, and similarlysome from B crossing into A. If the pressures in both chambersare sufficiently low, both rates of transpiration may, as an approxi-mation, be supposed given by formula (64). If the gases in thetwo chambers are the same in all respects, the two rates of effusionwill of course be the same. If, however, one chamber is keptwarmer than the other, then the rates of effusion will not be thesame, and we have the phenomenon of thermal transpiration.

Let TA, TB be the temperatures of the two chambers, and letthe corresponding densities and pressures be pA, pB and pA, pB.If the temperature difference is permanently maintained, theflow of gas will continue until a state is attained in which the flowfrom A to B is exactly equal to that from B to A. Formula (64)shews that this state will be reached when

The ratio of the pressures pA, pB is accordingly given by

PB PBTB *TB'

Thus if the two chambers are kept unequally heated, a flowof gas will be set up, and will continue until the difference ofpressure between the two sides is established which is expressedby equation (66).

Osborne Reynolds* investigated this phenomenon in a seriesof experiments in which the two chambers were kept at tempera-tures of 8° C. and 100° C. When a steady state was attained, thepressures were measured, and it was found that, so long as thepressure was sufficiently low, equation (65) was satisfied withvery considerable accuracy. At higher pressures this equationfailed, as was to be expected.

If the chambers A and B in fig. 11 are not only connected bythe porous plug, but also by an external pipe, which keeps thepressures in A and B equal, then a steady state cannot be at-tained so long as the temperatures are kept permanently atdifferent temperatures TA, TB. Instead of this, there will be asteady flow of gas through the cycle formed by the chambers

* Phil. Trans. 170, II (1879), p. 727.


A, B and the pipe, a flow which is suggestive of and analogous tothat of a thermoelectric current.

Cohesion of Oases

42. An entirely different situation occurs if the chamber B infig. 11 contains no gas, while chamber A is filled with gas kept attemperature TA. This gas will flow through the plug or orifice intothe chamber B, and its temperature as it arrives in the chamberJS, say TB, could be measured by a thermometer placed in JS.

Suppose that the molecules of the gas had been held togetherby strong forces of cohesion, so that each molecule was attractedby the other molecules of the gas, oi at least by those in its im-mediate proximity. Then each molecule, while passing throughthe plug, would be under an attraction towards the molecules inthe chamber A, and as this attraction would reduce its velocity,the average velocity of molecules arriving in B would be lessthan the average velocity of molecules in A; the temperature TB

would be less than TA.Thus we see that an examination of the temperature of a gas

after transpiration or effusion will give us information as to theexistence or non-existence of forces of cohesion in a gas. Experi-ments of this general type had been devised and conducted byGay-Lussac in 1807 and Joule in 1845, although they had usedmerely a tube and a tap in place of the porous plug used by thelater experimenters. The more sensitive form of apparatus wasdevised by Lord Kelvin, who carried out a delicate and crucialset of experiments in collaboration with Joule.*

The earlier experiments had failed to detect any temperaturechange in the gas, shewing that the forces of cohesion in a gaswere at least small. In the more elaborate experiments of Jouleand Kelvin, slight falls of temperature were observed; for in-stance, in an experiment in which air passed by transpirationfrom a pressure of about four atmospheres to a pressure of oneatmosphere, the change of temperature observed was a fall of0*9° C. In general it was found that for air and many of the more

• The original papers will be found in the Phil, Trans. Roy. Soc.(143, p. 357; 144, p. 321; 150, p. 325 and 152, p. 579). See also LordKelvin's Collected Works, 1, p. 333.


permanent gases the cooling, although appreciable, was veryslight, while for hydrogen a slight heating was observed. Thischange of temperature was known as the Joule-Thomson effect.It did not, as was at first thought, establish the existence of aforce of cohesion in gases—or of negative cohesion in hydrogen.For as the gas passed through the porous plug—generally a wadof cotton-wool—the pressure fell, so that the gas actually goingthrough the plug was doing work in pushing forward, by itspressure, the less dense gas in front of it, while work of the samekind but greater in amount was done on it by the denser gasbehind. The net result was of course a gain of energy to the gastraversing the plug, so that even if this gas consisted of infini-tesimal billiard balls which exerted no cohesive forces on oneanother, it would still emerge with an increase of energy, andthis would necessarily appear as an increase of temperaturewhen equilibrium had been attained (cf. § 92 below).

When the observed results are corrected to allow for this, it isfound that the molecules of all gases, including hydrogen, losekinetic energy in passing through the plug. The loss for air isfound to be 0-051 calorie per gramme for each atmosphere dropof pressure; the corresponding figure for hydrogen is 0-06 calorieper gramme. It follows that the molecules of a gas may notproperly be treated as points exerting no forces on one another;they attract one another, so that there are forces of cohesion in agas, just as there are in a liquid or a solid.

CALCULATION OF PRESSURE IN A NON-IDEAL GAS

43. In §33 the pressure in a gas was calculated on the assump-tions that the molecules were infinitesimal in size, and that theyexerted no forces on one another except when they were actuallyin collision. We now see that neither of these assumptions is truefor an actual gas, and so must proceed to calculate the pressurefor a real gas in which the molecules are of finite size, and exertforces of cohesion on one another even when they are not incontact. Many such calculations have been made, the mostfamous being that of Van der Waals (1873) to which we now turn.

PRESSURE IN A GAS

Van der Waals* Equation

44. We shall face our difficulties separately, and so begin bysupposing that the molecules are of finite size, being spheres offinite diameter cr, but that these exert no forces on one anotherexcept when in actual contact.

Let there be N molecules A, B, 0, ... in a volume v, andimagine, as before, that a sphere of radius cr is drawn round thecentre of each of these mole-cules (fig. 12). Since the centresof two molecules cannot bewithin a distance less than orfrom each other, it is im-possible for the centre of mole-cule A to lie within any of thespheres of radius cr surround-ing the (N - 1 ) other moleculesB, C, D, .... Each of thesespheres has a volume f nor3, sothat their combined volume isf (JV ~ 1) TTO"3. Since N is a verygreat number, we may neglectthe difference between N — 1and N and write this totalvolume in the form ^Nncr*.

We shall proceed on thesupposition that this totalvolume is only a small frac-tion of the total volume v of Fi 12the gas. In this case we maydisregard the possibility of two of the spheres of radius or inter-secting, and so suppose that the space outside the spheresis of volume v—%Nnor*. This is, then, the space available for thecentre of molecule A. Thus the chance of finding the centre ofmolecule A inside a specified volume dv is nil if this volume liesinside one of the spheres, and otherwise is

( 6 7 )


Now let molecule A be moving with a velocity of componentsut v, w, the component u being, as before, perpendicular to acertain region dS of the boundary. The molecule A will hit thisregion dS of the boundary within a small interval of time dt if,at the beginning of this interval, its centre lies within a smallelement of volume udSdt, which is at a distance \(T from theboundary. Let us now identify this element of volume with thedv of the last paragraph.

We must now consider the two possibilities separately of thiselement of volume lying within one of the N — 1 spheres of radiuscr, and of its not so lying.

We can imagine each sphere divided into two hemispheres by aplane through its centre parallel to d8; we may describe these asthe nearer and farther hemispheres. Each point in the fartherhemisphere is farther from dS than the centre of the sphere, sothat no such point can ever be at a distance less than \<T from theboundary. Since the element of volume is only at a distance \<rfrom the boundary, it can never lie in one of the farther hemi-spheres, but all points in the nearer hemispheres are open to it.The total volume of the farther hemispheres is KiV—1)TTCT3,or, again disregarding the difference between JV— 1 and N, isf JVW8. Thus the chance of dv lying inside one of the spheres is

V '

while the chance of its not so lying is

<68)

In the former case there is no chance of a collision. In the lattercase the chance of a collision within the interval dt is, fromformula (67),

N*° (69)

The chance of a collision is found by multiplying together thetwo probabilities (68) and (69) and so is

Ndv v—b


where b = f NTKT*, (70)

and so is four times the total volume of all the molecules of thegas. Since b has already been supposed to be small in comparisonwith v, we may neglect (b/v)2, and write this in the form

Ndvv~^b'

We accordingly see that when b/v is small—i.e. when the aggre-gate volume of the molecules is small compared with the spaceoccupied by the gas—the effect of the finite size of the moleculesis adequately allowed for by replacing v by v — b> i.e. by re-ducing the volume of the vessel by four times the total volume ofN molecules. When this is done we have as the equation givingthe pressure, in place of equation (57),

p(v-b) = NRT. (71)

Since we are neglecting squares of b/v, this may be put in thealternative form

( ^ } (72)

which is frequently found useful.When b/v is not treated as a small quantity, the calculation of

the pressure presents a much more serious problem. Boltzmann*has carried the calculation as far as squares of (b/v)2, and findsthat the pressure is given by

but this is hardly found to agree better with observation than thesimpler equation (72).

45. A second correction is necessitated by the forces ofcohesion, the existence of which we noted in § 42. So long as amolecule is sufficiently remote from the surface, forces act on itwhich will vary continually, both in direction and magnitude,but which, when averaged over a sufficient interval of time, arelikely to cancel out, and leave an average resultant equal to zero.Thus this second correction will not affect the pressure of the gas

* Vorleswngen liber Gastheorie, 2, § 51.


at internal points; this is given by equation (71). But to calculatethe pressure of the gas at or near the boundary, these forces mustbe taken into account.

Let us fix our attention on a molecule which is as near to theboundary as it can go. If the force from each adjacent moleculeis resolved into components in and at right angles to the boundary „all directions in the boundary plane are equally likely for the firstcomponent, but the second component is invariably directed in-wards. Thus when we average over a sufficient length of time, theresultant force will be a force directed inwards at right angles tothe boundary.

Thus the effect of all the intermolecular forces is that of apermanent field of force acting upon each molecule at and nearthe surface. I t is this field of force which is often pictured asgiving rise to the phenomena of capillarity and surface-tension inliquids. It can be regarded as exerting a steady inward pull uponthe outermost layer of molecules of the gas. In magnitude thispull will be proportional jointly to the number of molecules perunit area in this layer, and to the intensity of the normal com-ponent of force. As each of these two factors is directly pro-portional to the density of the gas, the pull will be proportionalto the square of the density; let us suppose that it is of amountcp2 per unit area, where c is a constant depending only on thenature of the gas. Then the molecules, when they reach theboundary, are no longer deflected by impact alone, but by thejoint action of their impact with the boundary and of this pull.Thus their change of momentum may be supposed to be pro-duced by a total pressure p + cp2 per unit area, instead of by thesimple pressure p.

Hence equation (71) must be further amended by writing it inthe form

or again, replacing p by Nm/v, and putting cNhn2 = a,

v-b) = NET. (73)

This is Van der Waals' equation connecting #, v and T. So longas we confine our attention to a single mass of gas, a and 6 are


constants, but in general they depend on the amount of gas aswell as on its nature, a being proportional to the square and b tothe first power of the amount of gas.

Dieterici*8 Equation

46. Van der Waals' treatment of the forces of cohesion over-looks the fact that when cohesive forces exist, some moleculeswhich would have reached the boundary had there been no suchforces may never reach the boundary at all, being deflected bythe cohesion forces before their paths meet the boundary. Suchmolecules exert no pressure on the boundary, whereas Van derWaals' argument has supposed them to exert a negative pressure.Because of this, equation (73) admits of negative values for p,although an examination of the physical conditions shews thatthe true value of p must necessarily be positive.

This objection is of no weight so long as it is clearly recognisedthat equation (73) is true only to the first order, as regards devia-tions from Boyle's law, but the equation is often, and very use-fully, applied to cases where these deviations cannot properly beregarded as small. Because of this, Dieterici* has proposed analternative equation of state which is not open to this lastobjection. We have seen that the molecules which are close to theboundary of the gas act as though they were drawn back into thegas by a permanent field of force. Let x be the work done indrawing a molecule from the interior of the gas to any assignedposition near the boundary against these forces, then the densityp' at this point is, as in equation (39),

where p is the density in the interior, and so is what is ordinarilymeant by the density of the gas. We now take this point closeup to the boundary of the gas, and find that the pressure on theboundary p is given by

where p' and x a r e the density and potential at the boundaryof the gas.

• Wied. Ann. 65 (1898), p. 826 and 69 (1899), p. 685.


Thus the field of force reduces the pressure to a value which isless than that given by Boyle's law by a factor e~xIRT. If weassume that the pressure is similarly reduced when the moleculesare of finite size, we obtain the pressure at the boundary in theform

N R T vIRT

p „ e~xiRT.* v-b

The work x is clearly proportional to/?, to a first approximation,and is easily seen to be equal to a/Nv, where a is the a of Van derWaals, leading to the equation

NETp = tl^L e-aINRTvm (74)

This equation was first given by Dieterici in 1898, and is found, aswe shall see later, to fit the observed facts considerably betterthan the equation of state of Van der Waals.

The General Calculation of Pressure

47. Logically, neither of the two foregoing pressure calculationsis completely satisfactory, as can be seen in the following way.The correction a arises from forces which the molecules exert onone another when reasonably near to one another, i.e. when theircentres are at distances somewhat greater than or apart. Thecorrection 6 arises from forces which the molecules exert on oneanother when their centres are at exactly the distance cr apart.It cannot, however, be supposed that the forces acting on naturalmolecules can be divided into two distinct types in this way.The forces between two molecules must change continuously withthe distance, so that the a and 6 of Van der Waals' equation oughtto be different contributions from a more general correction, andso ought to be additive. This is not the case with the equationseither of Van der Waals nor of Dieterici.

There is, however, a wider calculation of pressure, first givenby Clausius* in 1870, to which this objection does not apply. Itproceeds by studying the motions of the molecules of a gas asthese move under the influence of perfectly general forces, these

* PhU. Mag. Aug. 1870.


including the forces at collision, the forces of cohesion betweenmolecules, and the forces of impact at the boundary. In this waythe pressure is found in terms of the general forces betweenmolecules. To this we now turn.

The Virial of Clausius

48. We suppose the motion of any molecule in a gas to begoverned by the Newtonian equations

d2xX = m-jp, etc., (75)

where X, F, Z are the components of the total force acting on themolecule. Multiplying these three equations by xy y9 z respectivelyand adding, we obtain

d I dx dm dz -VI-me*, (76)

where c, as before, is the velocity of the molecule.

As the motion of the gas proceeds, - (x2 + y2 + z2) continually

fluctuates in value, but there is no tendency to a steady increaseor decrease. If, then, we sum over all the molecules of the gas, thefirst term on the right of equation (76) disappears, and we have

\Zmc2 = -%2;(xX + yY + zZ). (77)

The expression on the right is known as the Virial of Clausius;we see that it is equal to the total kinetic energy of translation ofthe molecules.

Contributions to the virial are made by all the forces whichever act upon the molecules. We may divide these into

(1) The forces at collisions between molecules.(2) The forces of cohesion between molecules.(3) The forces of impact between molecules and the boundary

of the containing vesseL


Let us first examine the contribution made by (3); it is throughthis that the pressure is introduced into our analysis.

Let dS be any element of surface of the containing vessel, thecoordinates of its centre being x, y, z, and the direction-cosines ofits inward normal being Z, m, n. If p is the pressure of the gason this surface, the total of all theforces which this element of surfaceexerts on all the molecules of the gashas components

ipdS, mpdS, npdS.Hence the contribution of all theseforces to the virial is

(78)

l,m9n dS

da)

Fig. 13Let r be the distance of dS from

the origin, and 6 the angle between the normal to dS and theline joining the origin to dS (fig. 13). Then

Ix + my + nz = — r cos 6,

and expression (78) may be replaced by ^roosdpdS or \r%pdh),where do) is the element of solid angle which dS subtends at theorigin.

If p is assumed to have the same value at all points of thesurface, the total contribution to the virial made by all parts ofthe surface is

tpff>rzdo),

where the integral extends over the whole surface, and this isequal to j ^

where v is the whole volume of the containing vessel. Equation(77) now becomes

Z), (79)

where only intermolecular forces are now to be included in thelast term.

49. Suppose that the molecules are infinitesimal in size, butnot necessarily spherical in shape, and that they exert no forces


except when in actual contact. When a collision occurs, actionand reaction are equal and opposite, so that X, F, Z have equaland opposite values for the two molecules concerned. Since weare supposing the molecules to be infinitesimal in size, x, y, z havethe same value for the two molecules, so that the contributionwhich all the intermolecular forces, including the forces of col-lision, make to

is nil. Since we are supposing these to be the only forces in action,equation (79) becomes

pv = E\mc%,

which is the equation already obtained for the pressure in anideal gas.

Whatever the shape and size of the molecules, and whateverthe forces between them may be, the corrections to be appliedto this simple equation are all contained in the more generalequation (79), which may be written in the form

<pv = Z\mc* + \Z{xX + yY + zZ). (80)

50. This equation shews that the pressure may be regarded asmade up of two contributions, one coming from the kineticenergy of motion of the molecules, and the other from thepotential energy of their intermolecular forces, including ofcourse the forces of collision.

The essence of the kinetic theory view of pressure is that itstresses this first contribution; the second figures only as a smallcorrecting term. Newton had shewn* that if the pressure ofa gas was proportional to its density, this pressure could beaccounted for by repulsive forces between the particles of the gas;this view of course neglects the motion of the molecules and attri-butes the whole pressure to the second term in equation (80).We can, however, see that this view is untenable, for the followingreason.

Let us imagine the containing vessel, together with the gasinside it, expanded to n times its original linear dimensions. Thecoordinates x, y9 z are all increased n-fold, and as observation tells

* Principia, Prop, XXTTI, theorem xvin.


us that pv remains constant, equation (80) shews that X, Y,Zmust be decreased to 1/n times their original values. In otherwords, the intermolecular forces must vary inversely as thedistance, as indeed Newton shewed. This is, however, an im-possible law, since it would make the action of the distant partsof the gas preponderate over that of the contiguous parts, and sowould not give a pressure which would be constant for a givenvolume and temperature as we passed from one vessel to another,or even from one part to another of the surface of the same vessel.We therefore conclude that the pressure of a gas cannot beexplained by assuming repulsive forces between the molecules;it must arise mainly from the motion of the molecules.

51. We next proceed to calculate the contribution of the inter-molecular forces to the virial. Let us assume that the force be-tween two molecules is a repulsive force <f>(r), which depends onlyon the distance r they are apart. If the centres of the moleculesare at x, y, z, xf, y\ z', and if X, Y, Z, Xr

t Y\ Z' are the com-ponents of the forces acting on them, then

so that the contribution to ZxX made by the force between thesetwo particles is

xX + x'X' = ^(x-x')*.r v

The contribution to Z(xX + yY + zZ) is therefore

Thus equation (80) may be replaced by

pv = \Zmc*+\EZr<l>(r), (81)

where the summation extends over all pairs of molecules.52. Since the gas contains N molecules, the summation on the

right of equation (81) must be taken over %N(N— 1) pairs ofmolecules.

Let A9 B be the two molecules of any such pair. If the mole-cules were simply points exerting no forces on one another, the


chance of A and B being at a distance between r and r + dr fromone another would be

the numerator being the volume of the shell of thickness dr sur-rounding molecule A, and the denominator v being the wholespace possible for the centre of B. Thus the number of pairs ofmolecules having their centres within a distance r of one anotherwould be

^ W (83)

since it is obviously legitimate to neglect the difference betweenJV-landiV.

Suppose, however, that molecules at a distance r repel oneanother with a force (j){r). Then we shall find, in § 228, that theprobability of finding two molecules at a distance r apart is lessthan the probability of finding the same two molecules at a dis-tance oo apart (oo here denoting any distance great enough for themolecules to be out of range of each other's action) by a factor

e-*% (84)

where x ^s ^ e work which must be done in bringing the twomolecules together from an infinite distance, so that

/•co

x = J r (r)dr.

Making this correction to expression (83), the number of pairsof molecules at a distance r apart will be

v

Multiplying this by ^r<p(r) and integrating over all values of r,

(85)

dxor, since <f>(r) = —-r,(XT


Replacing %2Jmc2 by its value NRT, equation (81) assumes theform

pv = NRT + ^ r ^* Zv Jo drJo

or pv = N11T(I + ^ \ , (86)

Replacing l/RT by its value 2h, and integrating by parts, wereadily find that B may be put in the alternative form

B = 2TTN r r2( 1 - e~2hx) dr. (88)Jo

53. If the molecules approximate closely to hard elastic spheres,the forces between molecules only come into play when r is verynearly equal to cr; for all other values of r,

dr

Thus we may replace r3 by crz in equation (87) and find

3BT h dr

ihRT

where b is the b of Van der Waals' equation.Equation (86) now becomes

(89)

which agrees with Van der Waals' equation (72).Clearly, then, equation (86) is a generalisation of Van der

Waals' equation, the general quantity JB, defined by equation(88), replacing the simple b of Van der Waals. The quantity B isgenerally known as the Second Virial Coefficient.

If forces of cohesion of the kind specified in §45 are alsosupposed to act, these forces will have a contribution to make tothe virial. To the first order of small quantities, we may, in calcu-

76 PRESSURE IK A GAS

lating ZEr<j)(r), ignore the effect of the forces of cohesion on thedistribution of density of the gas. The value of ZZr<f>(r) is there-fore obviously proportional simply to p2 per unit volume of thegas. Allowing for this addition to the virial, equation (89) mustbe replaced by

pv ** NRTI 1 + ?\ - cp2v9

where c is the same as the c of § 45, and is independent of thetemperature. Or again this last equation may be written

agreeing with Van der Waals' equation (73) as far as the firstorder of small quantities.

54. The quantity B is easily evaluated when the repulsiveforce <fi(r) varies as an inverse power of the distance, say(j){r) as /or*. We then have

^

2nN2 f °° // ( \so that rZ^(r) = ^ - J ^ e ^-lrt~l)dr

2hii\± ( 3 \-l) \ s-lj'

(2hv \s

It follows that B can be put in the form

B = f TTN(T\where cr8 is given by

This amounts to supposing that the molecules behave like elasticspheres, but that these have a diameter cr which depends on h,and so varies with the temperature; this merely represents thefact that at high temperatures the collisions are more violent, sothat the molecules penetrate farther into each other's fields offorce before being brought to relative rest.


If Bo and <70 are the values of B and <r at 0° C, the generalvalues at temperature T are given by

3

v

<r = c

55. The evaluation of B for more complicated laws of forcepresents a very intricate problem of mathematics. Keesom* hasstudied the problem when the molecules are regarded as rigidspheres of diameter cr surrounded by an attractive field of forceproportional to ju>r~8, and has evaluated B in the form of an infiniteseries

where n is the work done in separating two molecules which arein contact, and removing them to an infinite distance apart.Keesom has alsof evaluated B when the molecules are rigidspheres each containing an electric doublet at its centre, the valueagain being expressed as an infinite series.

J. E. Lennard- Jones % has discussed the problem when themolecules are supposed to repel according to the law of force§

and has obtained for B the value

){rJBi) F{y)>where y •» \

Mm-\)-S3 " * ' f

and ny)-yn-m\r(~^yYi

This contains the two simple formulae previously given asspecial cases.

• Comm. Phys. Lab. Leiden, Supp. 24 B (1912), p. 32, and Proc.Amsterdam Akad. 15 (1912), p. 1406. f Lx- ante-

% Proc. Roy. Soc. 106 A (1924), p. 463. § See § 149 below.


GAS THERMOMETRY

56. In spite of its imperfections, the equation of Van derWaals undoubtedly provides the most convenient basis for dis-cussing the behaviour of a gas over those ranges of pressure,density and temperature within which the deviation fromBoyle's law is small. We consider now some of the physical pro-perties of a gas predicted by the equation of Van der Waals.

Changes of Constant Volume

57. Let us first suppose the volume of the gas to be kept con-stant, and the temperature be raised from To to Tv

Ifp0, Piare ^n e pressures corresponding to these temperatures,Van der Waals' equation tells us that

(92)

^j (93)

By subtraction we get

This shews that the increase in pressure is proportional to theincrease in temperature; in fact px is given by the formula

where KP is a "pressure-coefficient" given by

NRK»-p~o¥=bY (94)

or, from equation (92), by

(£)£- (95)

In practical work, the initial temperature To is usually takento be 0° C, so that if 0 is the temperature on the centigrade scale

Pi = ( 0)


Equation (95) shews that Kp depends on the density but noton the temperature, so that for a given density of gas, the pressure-coefficient is independent of the temperature.

This law naturally is true only within the limits in which Vander Waals' equation is true. Regnault* proved that it was veryapproximately true under ordinary conditions by filling gasthermometers with different gases and shewing that they gaveidentical readings over a large range of temperature. More recentand more exact experiments have shewn, as might be expected,that the law is by no means absolutely exact or of universalvalidity. Full tables of values of Kp will be found in the Recueilde constantes physiques.f As a specimen may be given the fol-lowing values, obtained by Chappuis in 1903. J

Values of Kp

Temperature

0° to 20°0° to 40°0° to 100°

For nitrogen(po= 1001-9 mm. at 0° C.)

K9 = 0-00367540-00367520-0036744

For COa(£>0 = 998-5 mm. at 0° C.)

*„ = 0-0037335000372990-0037262

Callendar§ gives the following values for the pressure-coefficients (0° to 100° C. at initial pressure 1000 mm.) of threeof the more permanent gases:

Air 0-00367425,Nitrogen 0-00367466,Hydrogen 0-00366254,

while for neon and argon, Leduc^J has found the values:

Neon To = 5-47° C. to 2\ = 29-07° C, Kp = 0-003664,

Argon To = 11-95° C. to Tx = 31-87° C, Kp = 0-003669.

A gas in which the pressure obeys the Boyle-Charles lawpv = NRT exactly is described as a "perfect" gas. For such a

* Mem. de VAcad. 21, p. 180.t Pp. 234-40. The pressure-coefficient KP from range 0 to 0' is there

denoted by ffi. % Lx. p. 234.§ Phil. Mag. 5, p. 92. ^ Comptes Rendus, 164 (1917), p. 1003,


gas, a = 0, so that KP= or 0-003660. It will be noticed that

the pressure-coefficient of hydrogen approximates very nearlyto that of a perfect gas, shewing that the value of a is extremelysmall for hydrogen. Thus the value of Kp is almost independentof the volume of the gas. For this reason the Comite internationaledes poids et mesures decided on the constant volume hydrogenthermometer as standard gas-thermometer. The small valuewhich is known to exist for a is recognised in the stipulation ofthe committee that the volume at which the gas is used is to besuch that there is a pressure of 1000 mm. at 0° C. But nitrogengas-thermometers are also employed, and are found to possesscertain advantages.

Changes at Constant Pressure58. Let us next suppose that the pressure is kept at the constant

value p, and that changing the temperature from To to Tx is foundto change the volume from v0 to vv The two equations analogousto (92) and (93) of § 57 are

(96)

Neglecting the product ab which is a small quantity of thesecond order, we obtain, on subtraction,

We find from this that v1 is given by

V1 = V0{1 + KV(T1-T0)},

where KV is a "volume-coefficient" given byNR

On eliminating NR between this and equation (96), we obtain

))


This is more complicated than the formula for the pressure-coefficient (95) in that it depends both on the volume and thepressure.

For a perfect gas, a = 6 = 0, and the value of KV9 like that ofKpj becomes 0-003660. The following table, similar to that givenon p. 79, contains some observed values of KV.

Values of KV

Temp.

0°to 20°0° to 40°0° to 100°

For nitrogen(p = 1001-9 mm.)

*:„ = 0-00367700-00367500-0036732

For CO2(p - 998-5 mm.)

KV = 0-0037603000375360-0037410

For COa(p = 517-9 mm.)

Kv = 00037128000371000-0037073

Further values will be found in the Recueil de constantesphysiques, from which the above are taken.

The absolute scale of temperature can, as we have seen, bedefined without reference to any particular substance. Thismakes it a perfect scale for abstract problems and theoreticaldiscussions, but no thermometer can ever be constructed toread absolute temperatures directly. If any gas existed whichwas "perfect" in the sense explained above, a gas-thermometerusing this gas would give readings of absolute temperature. Inthe absence of such a perfect gas, the absolute scale can beapproximately realised by making the necessary small correc-tions to gas-thermometers in which ordinary gases are used.These corrections depend on the extent to which the valuesof KP and KV for the gases used differ from the theoretical value0-003660 for a perfect gas. Tables of such corrections will befound in books of physical constants.*

Evaluation of a and b

59. From an experimental evaluation of the "pressure-coefficient" Kp given by equation (95), the quantity a can beobtained at once, and when this is known, the value of b can beobtained from the observed value of the volume-coefficient,

• E.g. Kaye and Laby, Physical Constants, 8th edn (1936), p. 47.


For instance, using Callendar's value for KV for air at a pressureof 1000 mm. mercury, we have (equation (95)), with TQ = 273-2,

(1+^-0.003673,

while ^ = 0-003660.

At this pressure, therefore,

Si = pQT0 x 0-000013 = 0-0047 atmosphere pressure.

Thus for air at 1-3158 atmospheres pressure at the boundary,the forces of cohesion result in an apparent diminution of pressureof 0-0047 atmosphere, or about one-three hundredth of the whole,so that the pressure in the interior of the gas is 1-3205 atmo-spheres. This gives some idea of the magnitude of the forces ofcohesion.

At other pressures, a/v2 is proportional to l/v29 and so to the

square of the pressure. For instance, at a pressure of 1 atmo-sphere,

a~2 == 2649-5 in C.G.S. units = 0-00260 atmosphere.

When a has been determined in this way, we can determine bfrom the observed values of KV. This determination is of specialinterest, because from it we can calculate directly the value of or,the diameter of the molecule or of its sphere of molecular action.From the discussion of a great number of experiments byRegnault, Van der Waals deduced the following values for b:

Air 0-0026,Carbon-dioxide 0-0030,Hydrogen 0-00069.

These values refer to a mass of gas which occupies unit volumeat a pressure of 1000 mm. of mercury.

A more recent method of determining b depends on the measure-ment of the Joule-Thomson effect. Calculations by Rose-Innes*lead to the following values for b:

Air 1-62, Nitrogen 2-03, Hydrogen 10-73.

• Phil. Mag. 2 (1901), p. 130.


Referred to a cubic centimetre of gas at normal pressure,

Air 0-00209,Nitrogen 0-00255,Hydrogen 0-00096.

For helium, Kamerlingh Onnes* has determined the valuefor 6:

Helium 0-000432.

Values of Molecular radius \v

60. The value of b is, as in equation (70), equal to f 2VW3, andsince the values of 6 have been determined for a cubic cm. of gasat normal pressure, we may take N — 2-G9 x 1019 (§ 15), and sodetermine <x immediately.

The values of \v deduced from the best values of 6 are asfollows:

Gas

HydrogenHeliumNitrogenAirCarbon-dioxide

Value of b(1 o.e. of gas)

0-000960-0004320-002550-002090-00228

Observer

Rose-InnesKamerlingh OnnesRose-InnesRose-InnesVan der Waals

Value of \(r

1-27X10-8

0-99 x 10~8

1-78 x 10~8

1-66 x 10~8

1-71 x 10~8

ISOTHERMALS

61. One of the most instructive ways of representing the rela-tion between the pressure, volume and temperature of a gas isby drawing " isothermals'' or graphs shewing the relation betweenpressure and volume when the temperature is kept constant.There will of course be one isothermal corresponding to everypossible temperature, and if all the isothermals are imagineddrawn on a diagram in which the ordinates and abscissae re-present pressure and volume respectively, we shall have a com-plete representation of the relation in question.

* Cornm. Phys. Lab, Leiden, 102a (1907), p. 8,

84 PRESSURE A GAS

Isothermals of an Ideal Gas

62. For an ideal gas, the relation is expressed by the equation

pv = NRT. (100)

To represent this relation by means of isothermals, we take pand v as rectangular axes and draw the group of curves obtainedby assigning different constant values to T in equation (100). Thecurves all have equations of the form pv = const., and so are asystem of rectangular hyperbolas, lying as in fig. 14. These arethe isothermals of an ideal gas.

OFig. 14

Isothermals of a Gas obeying Van der Wools' Equation

63. Let us next examine what isothermals will correspond toVan der Waals' equation

NRT. .(101)

It will be noticed that if the system of curves shewn in fig. 14is pushed bodily through a distance b parallel to the axis of v,they will give the system of isothermals represented by theequation

p(v-b) = NRT, (102)

and on further drawing down every ordinate through a distance


a/v2 parallel to the axis of p, we shall obtain the system of iso-thermals represented by Van der Waals* equation (101).

The isothermals are found to lie as in fig. 15 in which the thickline AB is the curve p = —a/v2, while the line BCD is v = 6.

V F D

Fig. 15

The isothermals corresponding to high temperatures naturallylie exactly like those in the earlier fig. 14, the isothermals of anideal gas, but at lower temperatures divergences begin to appear.Below a certain temperature, the value ofp does not steadily in-crease as v decreases; on the contrary, after increasing for a time,the value of p reaches a maximum, then decreases to a minimum,after which it again increases.

These maxima and minima must of course occur at the pointsat which dp/dv = 0.

From equation (101), we may write the equation of the iso-thermals in the form

loglp-f ~ | oonst., (103)


and the points at which dpjdv = 0 are seen to be given by

__a(v-2b)P ^ ~ -

Clearly this value of p is positive for all values of v between 26and oo. Within this range it attains a maximum value, which isreadily found to be given by

f i (104)

The isothermals for values of T greater than that given by thisequation can have no points at which dp/dv = 0, and so are every-where convex to the axis of v.

64. The isothermals of a real gas will lie like those shewn infig. 15 so long only as the gas does not differ too much from anideal gas. The curves in fig. 15 will accordingly represent theisothermals of a real gas with accuracy in the regions far removedfrom both axes, but not near to these axes. We must inquire whatalterations must be made in these curves in order to represent theisothermals of a real gas.

The isothermal T = 0 is represented in fig. 15 by the brokenline made up of the curve A B and the vertical line BCD. The trueisothermal is, however, known with accuracy. If a gas is cooledto temperature T = 0 and is then compressed, the pressureremains zero until the molecules are actually in contact. Let thevolume in this state be denoted by v0. The pressure may now beincreased to any extent and the volume will still retain the samevalue v0, this being the smallest volume which can be occupiedby the molecules. Now %. being the smallest volume into whichN spheres each of diameter a can be comprefesed, is easily foundto be given by

while the value of 6 is (cf. equation (70))

b = P W 3 = 2-96v0. (105)

Thus the true isothermal T = 0, instead of being the curveABCD in fig. 15, consists of the two lines vE, EF. If we imagine

PRESSURE IN A GAS 87the curves in fig. 15 so distorted that the point B is made tocoincide with the point E, and the curve ABOD with the linesvEF, we shall obtain an idea of the run of the isothermals of areal gas. The curves will perhaps lie somewhat as in fig. 16, inwhich both the vertical and horizontal scales have been largelyincreased over those employed in fig. 15, but the vertical scale hasbeen increased much more than the horizontal.

Isothermals of a gas obeying Dieterici's Equation

65. The isothermals of a gas which is supposed to obeyDieterici's equation may be discussed in a similar manner.

v=v0Fig. 16

If we neglect the exponential factor, the isothermals are againgiven by equation (102), namely

and, as before, the curves may be obtained by shifting the iso-thermals of an ideal gas (fig. 14) through a distance b parallel tothe axis of v. If we now restore the exponential factor, we mustreduce the value of p at each point of each isothermal by a factore-aiNRTv^ a n ( j the resulting curves will be the isothermals required.They will again be found to lie as in fig. 16.


Again the network of isothermals for the lower temperaturescontain regions in which p and v are decreasing together. Theequation of an isothermal may be written in the form

logp + log (t;-&) + ^jrv = const.,

so that the points at which p has its maximum and minimumvalues on any isothermal are given by

dp ^ I a-f = 0 or v-b~ NRTv*f

so that ^ l

Between the values v = b and v = oo, we see that T is every-where positive. It attains a maximum value given by

* NBTTb ( 1 0 6 )

Continuity of the Liquid and Gaseous States

66. According to Van der Waals* equation, the temperaturedetermined by equation (104),

2lNEb9

has the special property that at all temperatures above thistemperature a decrease of volume is always accompanied by anincrease of pressure, but below it every isothermal containsstretches in which the pressure and volume decrease together.Every point on these latter stretches represents a collapsible orunstable state, since, if the gas is already yielding to pressure,each contraction increases the disparity between the resistanceof the gas and the force exerted on it—the gas is like an army inretreat which becomes more and more demoralised with everyyard it retreats.

The equation of Dieterici makes exactly similar predictionsexcept that the critical temperature, determined by equation(106), is given by

4NBb'


Let us fix our attention on any one collapsible state of the kindpredicted by both equations, say that represented by the pointX in fig. 16. On the same isothermal as X, there must clearly betwo other points Y, Z which represent states having the sametemperature and pressure as X. At each of these two pointsdp/dv is positive, so that the two states in question are bothstable; they ought therefore both to be known to observation.The point Z obviously represents the gaseous state; the point Ycorresponding to lesser volume is believed to represent the liquidstate.

With this interpretation it is at once clear that so long as a gasis kept at a temperature above that of the isothermal PXPP2, noamount of compression can force it into the liquid state. Thetemperature of the isothermal PXPP2 is called the " critical tem-perature " of the substance. We see that so long as a gas is keptabove the critical temperature, no pressure, however great, canliquefy it.

67. A gas which is below the critical temperature is usuallydescribed as a vapour. We therefore see that the line PP2 in fig. 16is the line of demarcation between the gaseous and vapour states,while PPX is the line of demarcation between the gaseous andliquid states.

It remains to examine the demarcation between the liquid andvapour states, which is at present represented by the unstableregion in which dpjdv is positive. If U is any point in this region,common experience tells us that there is a stable state in whichthe pressure and volume are those of the point U. What is thisstate?

Let us imagine a line drawn through U parallel to the axis ofv. Let this cut any isothermal in the points X, Y, Z, the two latterrepresenting stable states—liquid and vapour respectively. Asthese two states have the same pressure, it follows that a quantityof vapour in the state Z can rest in equilibrium with a quantityof liquid in state Y. By choosing these quantities in a suitableratio, the combination of the two will be represented by the pointU. Here, then, we have an interpretation of the physical meaningof the point U. As the vapour is compressed at the temperatureof the isothermal 8ZQXRY, the substance remains a vapour

4 JKT


until the point Z is reached. At this point condensation sets in;some of the substance is in the state Z but some also is in the stateF. As the ratio of these amounts changes, the representativepoint moves along the straight line ZXUY until, by the time thepoint Y is reached, the whole of the matter is in the liquid state.After this the substance, remaining wholly in the liquid state,moves through the series of changes represented by the pathYQ'N.

There is of course an element of arbitrariness in this, for insteadof describing the path SZUYN the substance might equally wellhave been supposed to describe the path SR'RYN, keeping atthe same temperature throughout; or any other path composedof two stable branches of an isothermal joined by a line of constantpressure. In other words, there is no unique relation between thepressure and temperature of evaporation or condensation. Thisis, however, in accordance with the known properties of matter;there are such things in nature as super-cooled vapours whichmay be represented by the range ZQ in fig. 16, and super-heatedliquids, which may be represented on the range YR.

Under normal conditions, however, when there are no compli-cations produced by surface-tensions, particles of dust, or otherimpurities, there must be a definite boiling point correspondingto every pressure, and the path of a substance from one stateto another, given the same external conditions, must be quitedefinite. So far we have not arrived at any such definiteness.

Maxwell* and Clausiusf both attempted to obtain definitepaths for a substance changing at a constant temperature. Theyreached the conclusion that if the line 8ZXYN in fig. 16 is torepresent the actual isothermal path from 8 to N, it must be sochosen that the areas ZQX, XRY are equal. For, imagine thesubstance starting from Z, and passing through the cycle ofchanges represented in fig. 16 by the path ZQXRYXZ, the firstpart of the path ZQXRY being along the curved isothermal, andthe second part YXZ along the straight line. Since this is aclosed cycle of changes, it follows from the second law of thermo-

dynamics that — = 0, where dQ is the total heat supplied to

• Collected Works, 2, p. 425. f Wied. Ann. 9 (1880), p. 337.


the substance in any small part of the cycle and the integral istaken round the whole closed path representing the cycle. Sincethe temperature is constant throughout the motion, this equation

becomes \dQ = 0, so that the integral work done on the gas

throughout the cycle is nil. As in § 12, this work is equal to \pdv

and therefore to the area, measured algebraically, of the curve infig. 16 which represents the cycle. Hence this area must vanish,which is the result already stated.

R

Fig. 17

68. The figure which is obtained from fig. 16 upon replacing thecurved parts of isothermals such as ZQXRT by the straight lineZXY is represented in fig. 17. This figure ought accordingly torepresent the main features of the observed systems of isothermalsof actual substances.

Comparison with Experiment

69. Fig. 18 shows the isothermals of crabon-dioxide as found inthe classical experiments of Andrews.* The figures on the lefthand denote pressure measured in atmospheres (the isothermals

• Phil. Trans. 159 (1869), p. 575 and 167 (1876), p. 421.


only being shewn for pressures above 47 atmospheres), whilethose on the right denote the temperatures centigrade of thevarious isothermals.

The isothermal corresponding to the temperature 31-1° is ofgreat interest, as being very near to the critical isothermal, thevalue of the critical temperature being given by Andrews as30-92° and by Keesom as 30-98°.On this isothermal, as on all thoseabove it, the substance remainsgaseous, no matter how great thepressure.

The next lower isothermal,corresponding to temperature21-5° C, shows a horizontal rangeat a pressure of about 60 atmo-spheres. As the representativepoint moves over this range,boiling or condensation is takingplace. Thus at a pressure ofabout 60 atmospheres the boilingpoint of carbon-dioxide is about21-5° C. The ratio of volumes inthe liquid and vapour states is Fig. 18

equal to the ratio of the two values of v at the extremities of thehorizontal range—a ratio of about one to three.

The lowest isothermal of all corresponds to a temperature of13-1° C. Here the inequality between the volumes of the liquidand the gas is greater than before. In fact an examination of thegeneral theoretical diagram given in fig. 17 shews that as the tem-perature decreases the inequality must become more and moremarked, so that in all substances the distinction between theliquid and gaseous states becomes continually more pronouncedas we recede from the critical temperature.

The Critical Point

70. The point represented by P in fig. 16 is generally describedas the "critical point". Just as there are points in England atwhich three counties meet, so here is a point where three states


meet—the liquid, the vapour and the gaseous. And, because ofthis, a substance at the critical point possesses certain remarkableproperties.

The values of the temperature, pressure and volume at thecritical point are usually denoted by Tc, pc and vc. The tempera-ture Tc is of course the critical temperature, above which lique-faction is impossible; pc and vc may be defined as the pressure andvolume at which liquefaction first begins when the substance isat a temperature just below Tc.

We have seen that the existence of this critical point is impliedin the equations of both Van der Waals and Dieterici. The twoequations make different predictions as to the position of thepoint, but this is not surprising, since both equations are onlytrue when the deviations from Boyle's law are small, and thecritical point is in a region in which these deviations are quitedefinitely not small. Thus both predictions as to the position ofthe critical point are unwarranted extrapolations. The predictionsare shewn in the following table:

Equation of

Van der Waals

Dieterici

NRT0

0-30 "b

0-25?6

Pc

0-037 ~62

0-034 «

3006

2-006

NRT,PeVo

2-67

3-69

The last column is of interest from the fact that NR Tc/pcvc is, onany theory, a pure number, having no physical dimensions at all.If Boyle's law still held at the critical point, this number wouldof course be equal to unity; we see how far the critical point isfrom the regions in which Boyle's law holds.

Actual observations of critical data are recorded in the tableon p. 94 overleaf.

These data shew that the properties of different gases varywidely, and that for most gases neither the equation of Van derWaals nor that of Dieterici comes particularly near to the truth.Generally speaking, however, the equation of Dieterici appears

PRESSURE IN A GAS

to be more accurate than that of Van der Waals, particularly forthe heavier and more complex gases.

Gas

HydrogenHel iumNitrogenOxygenNeonArgonX e n o nCOaHClCC14C6H«C6H6BrC6H5C1

Van der WaalsDieterioi

T

2-80—

1-501-46———

1-86

————

300200

NRTe

3-273-263-423-423-423-423-603-613-483-683-713-783-81

2-673-69

71. It is interesting to compare the two equations at placesother than the critical point.

According to the equation of Dieterici,

' pv(l-b/v)'

so that, on taking logarithms of both sides,

2dk = logl, a , (NRT\ . 62 b3

and F ^ = v l o g l ^ r ) + 6 + ^ + 3 ^ + - -According to Van der Waals,

whence we obtain

a , INRT

NRTv^2\NRT

PBESSTJRE IN A GAS 95

If we neglect powers and products of the small quantities aand 6, both equations predict that

(NRT\

\pv~)

should be constant along any isothermal. The following tablegives, in its last column, the value of this quantity along thecritical isothermal (T = 187-8° C.) of isopentane, as calculatedfrom the observations of Young. The values of p in the thirdcolumn are those given by Dieterici's equation (74) for the tem-perature 187 • 8° C. and the pressure recorded in the second column.*

Critical Isothermal of Isopentane

Critical temperature = 187-8° C. Critical volume = 4-266 c.c. per gramme.

Volume v9per gramme

2-42-52-62-8303-23-64-0

4-34-65678910121520304050608090100

Pressure p,m m . of mercury

4908040560349802894026460254902505025020

250102500024990248402440023710229302204020300179801484010950857070686001461441323750

Pressure p(calc.)

4273035810320902839026780260002542025320

25300253002524024880—

23400—

2159019850175401456010770850870255978460441273740

, /NRT\v log (6 \ pv )

1-2711-4861-6691-9382-1032-2052-3262-402

2-4472-4832-5202-5642-5772-5822-5762-5752-5682-5482-5642-5262-6242-6252-6522-6802-6372-680

* Ann. d. Phys. 5 (1901), p. 58.


( NRT\1 is approximately constant for all

pressures less than about 12 atmospheres, shewing the rangewithin which a first approximation holds. There is, however, verytolerably good agreement between the observed and calculatedvalues of p far beyond this range, indicating considerableaccuracy in Dieterici's equation.

Reduced Equation of State

72. The critical temperature, pressure and volume formnoteworthy milestones in the various states of a gas, and soprovide valuable units for the measurement of temperature,pressure and volume in general.

Let us introduce quantities t, p and t> defined by

x-h*-i-*-i (107)

so that t denotes the ratio of the temperature of any substance toits critical temperature, and so on. The quantities t, p} t) are calledthe reduced temperature, pressure and volume respectively.

According to Van der Waals' equation, we have (cf. equations

(104)> a 8a

so that the equation reduces to

K)H)-S» <•»>It will be noticed that this equation is the same for all gases,

since the quantities a, 6 which vary from one gas to another haveentirely disappeared. An equation, such as that of Van derWaals, which aims at expressing the relation between pressure,volume and temperature in a gas, is called an equation of state,or sometimes a characteristic equation or gas-equation. Equation(108) may be called the "reduced" equation of state of Van derWaals, and is the same for all gases.

On the other hand, according to Dieterici's equation, the valuesof t, p and t) are given by equations (106) and on substituting


these values for Tc, pe, vc in Dieterici's equation, we find that thecorresponding "reduced" equation of state is

1) = t e 2 ^ ~ ^ \ (109)

Corresponding States

73. If either of these equations could be regarded as absolutelytrue for all gases, it appears that when any two of the quantitiest, p, & are known, the third would also be given, and would be thesame for all gases. In other words, there would be a relation of theform p =/(t, fc>), in which the coefficients in / would be inde-pendent of the nature of the gas.

The same is true for any equation of state whatever, providedthat this contains only two quantities which depend on the par-ticular structure of the gas in question—say, for instance, thesame two as in the equation of Van der Waals, representing thesize of the molecules and the cohesion-factor. For if a and brepresent these two constants, the equation of state will be arelation between the five quantities

p, v9 T, a and 6.

The critical point is fixed by two more equations, these being(VfY\ (I If)

in fact -f- = 0 and -r s = 0> a n ( i these involve the critical quanti-dv dv2

ties Tc, pc and vc as well as a and 6. From the three equations justmentioned we can eliminate a and 6, and are left with a singleequation connecting T, p, v and Tc, pc, vc. Considerations ofphysical dimensions shew that it must be possible to put thisequation in the form

*>=/(t,»), (no)in which the coefficients i n / are independent of the nature of thegas. This is the result which has been already established for thespecial equations of Van der Waals and Dieterici.

Assuming that the gas-equation can be expressed in the form(110), two gases which have the same values oft, p and & are saidto be in "corresponding" states. Clearly for two gases to be in


corresponding states, it is sufficient for any two of the threequantities t, p and & to be the same for both.

74. It is sometimes asserted as a natural law, that when two ofthe quantities t, p and & are the same for two gases, then the thirdquantity will necessarily also be the same, and this supposed lawis called the "Law of Corresponding States". This law will, how-ever, only be true if the reduced equation of state can be put inthe form of equation (110), and this in turn demands that thenature of the gas shall be specified by only two physical constants,as for instance the a and 6 of Van der Waals. There is of course noquestion that the law is true as a first approximation, because theequations of both Van der Waals and Dieterici are true as a firstapproximation, but if the law were generally true, all the entriesin the table on p. 94 would be equal, which is obviously not thecase.

The law of corresponding states can be expressed in the formthat by contraction or expansion of the scales on which p and vare measured, the isothermals of all gases can be made exactly thesame. This same statement can be put in the alternative formthat graphs in which logp, log/> and logT are plotted againstone another will be the same for all gases.

O T H E R E Q U A T I O N S OF STATE

The Empirical Equation of State of Kamerlingh Onnes

75. Following Kamerlingh Onnes, let us introduce a quantityK defined for any gas by

Pcvc

a consideration of physical dimensions shewing that K must bea pure number, and let us further put

_ t)

With this notation, Van der Waals' equation (108) reduces to

PRESSURE IN A GAS

which can also be written in the formt _ 27

11 - 8t>.

tn+^-2 7 1

4 0 9 6 ^ 4 0 7 6 8 ^•...) (112)

As innumerable observers have found that an equation of thistype is not adequate to represent the various states of a gas,Kamerlingh Onnes proposed the more general empirical form

• — + - T + J T + - X + -?T » (113)v* v t 2 VA4 VAO VtO I ' v '

where S3, K, 2), . . . are themselves series of the form

SBt ' t 4 t 6 '

an expansion which contains no fewer than twenty-five adjustablecoefficients.

If the law of corresponding states were true, the coefficients inthe expansions (111), (112) ought to be the same for all gases.

Actually Kamerlingh Onnes* found that an equation of stateof the form (113) can be obtained which expresses with fairaccuracy the observations of Amagat on hydrogen, oxygen,nitrogen and C4H10O, also of Ramsay and Young on C4H10O andof Young on isopentane (C5H12). The coefficients in this equationare found to be those given in the following table:

io8b104c10* bio7e109f

l

117-796135-58066-023

-179-991142-348

2

-228-038-135-788

-19-968648-583

-547-249

3

-172-891295-908

-137157-490-683

508-536

4

-72-765160-94955-85197-940

-127-736

5

- 3 1 7 251109

- 2 7 1 2 24-582

12-210

• Encyc. d. Math. Wissenschaften, 5 (1912), 10, p. 729, or Comm. Phy$.Lab. Leiden, 11, Supp. 23(1912), p. 115.

100 PEBSSXJRE IN A GAS

The circumstance that the same equation of state is valid forall these six gases shews of course the wide applicability of the lawof corresponding states. On the other hand, the coefficients of theequation of state of argon and helium are found to differ very sub-stantially from those in the above table. Indeed KamerlinghOnnes found that the isothermals of helium* can be representedvery fairly between 100° C. and — 217° C. by the equation

which is simply Van der Waals' equation adjusted by the inclusionof Boltzmann's second-order correction (§44). It must howeverbe noticed that the value of Tc for heliuin is very low, about 5-3°abs., so that the range of temperatures studied by Prof. Kamer-lingh Onnes is about from t = 1 0 t o t = 70, and for these highvalues of t it is inevitable that Van der Waals' equation should inany case give a good approximation.

The Equation of State of Clausius

76. Various attempts have been made to improve Van derWaals' equation by the introduction of a few more adjustableconstants, which can be so chosen as to make the equation agreemore closely with experiment.

One of the best known of these improved equations is that ofClausius, namely

On putting c = 0, the equation becomes similar to that of Vander Waals, except that the a of Van der Waals' equation is re-placed by a'/T] in other words, instead of a being constant, it issupposed to vary inversely as the temperature. For some gasessuch an equation is found to fit the observations better than theequation of Van der Waals.

If, however, c is not put equal to zero, but is treated as anadjustable constant and selected to fit the observations, there isfound to be no tendency for c to vanish. The following table shews

* Comm. Phys. Lab. Leiden, 102a (1907).


the values of a', 6 and c which are found by Sarrau* to give thebest approximation to the observations of Amagat:

Gas

NitrogenOxygenEthyleneCarbon-dioxide

a '

0-44640-54752-6882-092

b

00013590-00089000009670-000866

0

0-0002630-0006860-0019190-000949

c/b

0190-751-98110

77. We have already seen that, if Van der Waals' equationwere true, the law of corresponding states would follow as anecessary consequence. The reason for this is that in Van derWaals' law there are only two constants, a and 6, which re-spectively provide the scales on which the pressure and volumecan be measured in reduced coordinates.

The equation of Clausius, on the other hand, contains threeseparate constants, a', b and c, and of these two, namely b and c,provide different scales on which the volume can be measured:these two scales only become identical if 6 and c stand in a constantratio to one another. Thus if the law of corresponding states weretrue, the ratio c/b would be the same for all gases. The last columnof the above table shews, however, that there is not even anapproximation to constancy in the values of c/b.

78. Clausius originally devised formula (114) in an effort tofind a formula which should fit the observations of Andrews oncarbon-dioxide. It was found that this formula, although fairlysuccessful in the case of carbon-dioxide, was not equally successfulwith other gases, and Clausius then suggested the more generalform

which contains five adjustable constants. For carbon-dioxide, itis found that n = 2 and a'" = 0, so that the equation reduces to(114), but for other gases n and a!" do not approximate to thesevalues. For instance Clausius finds that for ether n = T192, forwater-vapour n = 1-24, while to agree with observations onalcohol, n itself must be regarded as a function of the temperature

• Compte8 Rendus, 114 (1882), pp. 639, 718, 845.


and pressure, having values which vary from 1*087 at 0°C. to0-184 at 240° C.

Beatty and Bridgeman have suggested an equation of state,also containing five adjustable constants, of the form

They find that this represents at least fourteen gases, to withinan accuracy of one-half of one per cent, down to the critical point.

I t is, however, obvious that there can be no finality in any ofthese formulae; it is possible to go on extending them indefinitelywithout arriving at a fully satisfactory formula, as might indeedbe anticipated from the circumstance that they are purely em-pirical, and not founded on any satisfactory theoretical basis.

Chapter IV

COLLISIONS AND MAXWELL'S LAW

79. In § 8 we traced out the path of a single molecule underthe simplest conceivable conditions. It is only in very exceptionalcases that such a procedure is possible. Usually the kinetic theoryhas to rely on statistical calculations, based on the suppositionthat the number of molecules is very great indeed; the moleculesare not discussed as individuals, but in crowds. In doing this, thetheory encounters all the well-known difficulties of statisticalmethods. Indeed these confront us at the very outset, when weattempt to frame precise definitions of the simplest quantities,such as the density, pressure, etc. of a gas.

The Definition of Density

80. Let us fix our attention on any small space inside the gas,of volume Q. Molecules will be continually entering and leavingthis small space by crossing its boundary. Each time this happens,the number of molecules inside the space increases or decreases byunity. Thus the number of molecules inside the space will berepresented by a succession of consecutive numbers, such as

n, n+1, ra + 2, n+1, n, n + 1 , n, w—1, n, n-l, n — 2,n— 1, n, n + 1 , ny r&—1,....

This succession of numbers will fluctuate around their averagevalue, say n. Since the fluctuations arise from the passage ofmolecules over the boundary, they will be proportional to thearea of this boundary, and so to the square of the linear dimen-sions of the space, but the number n will be proportional to thevolume of the space, and so to the cube of its linear dimensions.As the volume is made larger, the fluctuations become more andmore insignificant in proportion to n.

In an actual gas the volume Q can usually be chosen of suchsize that it shall contain a very large number of molecules, andyet be very small compared with the scale on which the physical

104 COLLISIONS AND MAXWELL'S LAW

properties of the gas vary; a cubic millimetre is often a suitablevolume. The number of molecules contained in an element of thiskind, divided by the volume Q of the element, will give the mole-cular density of the gas, which is usually denoted by v. Actuallyv is a fluctuating quantity, but if the volume Q can be chosensufficiently large, the fluctuations are inappreciable. In such acase v may be treated as a steady quantity, and we may think ofit as the number of molecules per unit volume in a specified regionof the gas.

If each molecule is of mass m, and

p will measure the mass of all the molecules per unit volume; it isthe " density" of the gas.

The Law of Distribution of Velocities

81. We have already seen that the molecules of a gas do not allmove at the same speed; the components of velocity can have allvalues. For instance, when the gas has reached a steady state,they are distributed according to Maxwell's law (§ 23).

To discuss the velocities of a great number N of molecules, letus take any point as origin, and draw from it a number of separatelines, each line representing the velocity of one molecule, both inmagnitude and direction. The point at the extremity of any suchline will have as its coordinates u, v, w9 the components of velocityof the corresponding molecule. The distribution of points ob-tained in this way is of course the same as the distribution ofthe molecules would be in space if they all started together at apoint and each moved for unit time with its actual velocity.The number of points of which the coordinates lie between thelimits u and u + du, v and v + dv, w and w + dw may be denotedby rdudvdw, where r corresponds to the v above, being the"density " of points at the point u, v, w. The number of moleculesof which the velocities lie between u and u + du, v and v + dv,w and w + dw will now be T du dv dw. If N is the total number ofmolecules under consideration, it is often convenient to replacer by Nf. If it is further necessary to specify the point u, v, w atwhich/is measured, we may write f(u, v, w) instead of/. Because

COLLISIONS AND MAXWELL'S LAW 105

of the occurrences of collisions, f(u, v, w) is a fluctuating functionof u, v, wy but under suitable conditions the fluctuations will beinappreciable.

To avoid the continual repetition of these limits, let us describea molecule of which the components of velocity lie between u andu + du, v and v + dv, w and w + dw as a molecule of class A. Thenthe number of molecules of class A is

Nf(u, v, w) du dv dw.

Since the total number of molecules under consideration is N9

the probabiKty that any single molecule selected at random shallhave velocity-components lying between u and u + du9 v andv + dv9w and w + dw will be f(u> v, w) du dv dw.

The Assumption of Molecular Chaos

82. Let us fix our attention on any small volume of the gaswhich contains N molecules. We may imagine that we measurethe velocity of each molecule, represent it by a point u, v, w inour diagram, and so calculate f(u, v, w) for all values of u, v andw. Having done this, we may proceed to some other group of Nmolecules and again calculate/(w, v, w) for all values of u, v and w.The two sets of values of/may prove to be the same, in which casewe say that the two groups of molecules have the same distribu-tion of velocities; or obviously they may be different, as for in-stance if the two groups of molecules were taken from regions ofthe gas which were at different temperatures.

Let us now suppose that the second group of molecules ischosen from the same region of the gas as the first, but that it ischosen in a very special way. Let it consist of all the moleculeswhich lie within a certain small distance e of molecules which arethemselves moving at exceptionally rapid speeds. The questionarises as to whether this special way of choosing our group ofmolecules has any influence on the distribution of velocities. Itmight at first be thought, with some plausibility, that there wouldbe some such influence—a molecule near to a second and rapidlymoving molecule is more likely than others to have collided withthis second very energetic molecule quite recently, in which caseit might have acquired an undue amount of energy from it.


Boltzmann introduced into the kinetic theory the assumptionthat no such influence exists; he called this the "assumption ofmolecular chaos", but was unable to give any proof, or even anydiscussion, of its truth. We also shall make this assumption,ultimately justifying our procedure by a proof (given in AppendixIV), that the assumption is true. We shall make the assumptionin the form that the chance that a molecule of class A shall befound within any small element of volume dxdydz is

vf(u,v,w)dudvdwdxdydz9 (H5)

where / is the law of distribution of velocities for the group ofmolecules in a small volume Q which includes the smaller elementdxdydz. We assume this to be true for every small element ofvolume dxdydz, no matter how this is chosen.

The Changes produced by Collisions when theMolecules are Elastic Spheres

83. From the statistical point of view, the state of a gas is fullyknown when its density and the law of distribution of velocitiesare known at every point. As every collision between moleculeschanges the velocities of the two molecules concerned, the re-peated occurrence of collisions must produce continual changes inthe law of distribution of velocities. We shall first discuss thesechanges in a general way; ultimately we shall find that every gasmay, after a sufficient time, reach a steady state, i.e. a state whichis not altered by collisions, so that the density and law of distribu-tion of velocities remain statistically the same at every point ofthe gas throughout all time—just as a population can attain asteady state, so that births and deaths and the normal process ofgrowing old do not alter its statistical distribution by ages. Thissteady state is specified by Maxwell's law, which we have alreadyobtained in § 23.

To begin by discussing the problem in its simplest form, let ussuppose that the molecules of the gas are smooth rigid spheres,and that the physical conditions are the same at every point ofthe gas. Let us first imagine the gas to be in a state in which themolecular density and the law of distribution of velocitiesf(u,v,w) are the same at every point of space. Since there is


nothing to differentiate the various regions in space, this uni-formity of distribution must persist throughout all time, althoughthe actual form of the function/ may change with the time.

84. Let us first consider the simplest possible kind of collision,a direct head-on collision between two molecules which aremoving with velocities u and u' along a line which we may take tobe the axis oix. As in § 10, the total momentum along the axis ofx, namely m(u + u'), will remain the same after collision, while therelative velocity u' — u changes its sign, because of the perfectelasticity of the molecules. It follows that the molecules merelyexchange their velocities along the axis of x; if the velocitiesbefore collision are u, u', the velocities after collision are u\ urespectively.

Any additional velocities the molecules may have in the direc-tions of Oy and Oz will persist unchanged after the collision, since

Fig. 19

there is no force to change them. Thus if the molecules hadvelocities before collision

u, v, w and u', v', w\their velocities after collision will be

u', v9 w and u, v\ w'.

By considering a number of collisions with velocities very near,but not exactly equal to, the foregoing, we see that if the velocitiesbefore collision lie within the limits

u and u + du, v and v + dv, w and w + dwand u' and u' + du', v' and v' -f dv\ w' and w' -f dw\

then the velocities after collision will lie within the limits

vf a,ndu' + du', v and v + dv, w a,ndw + dwand u and u 4- du, vr and vr + dvr, wf and w' + dw'.


In discussing such a group of collisions, we may speak of theproduct of the six differentials du dv dw du'dv'dw' as the spreadof the velocities. We now see that the total spread of the sixcomponents before collision, namely du dv dw du' dv' dw'', isexactly equal to the total spread after collision, namelydu' dv dw du dv' dw'—a result we shall need later.

So far the collision has been supposed to take place with its lineof impact along the axis of x, the relative velocity being u' — u.In such a case we obtain the velocities after the collision by super-posing the relative velocity along the line of impact (taken withits appropriate sign) on to the velocities before impact. Now asthis rule makes no mention of the special direction (namely Ox)that we have chosen for the line of impact, it must be of universalapplication. So also must the other result we have obtained—that the total spread dudvdwdu' dv'dw' of the velocities beforecollision is equal to the total spread after collision.

85. Let us now remove the restriction that the line of impact isto be in the direction Ox, and consider a more general type ofcollision, which we shall describe as a collision of type a, and shalldefine by the three following conditions:

(i) One of the two colliding molecules is to be a molecule ofclass A, defined by the condition that the velocity-componentslie within the limits u and u -f du, v and v + dv, w and w -f dw.

(ii) The second colliding molecule is to be of class B, definedby the condition that the velocity-components lie within thelimits u' and u' + du', v' and v' + dv', w' and w' -h dw'.

(iii) The direction of the line of impact is to have direction-cosines 1,171,71, and is to lie within a specified small solidangle do).

If the molecules all have the same diameter or, a collision willoccur each time that the centres of any two molecules come withina distance or of one another. Let us now imagine a sphere ofradius or drawn round the centre of each molecule of class A. Wecan mark out on the surface of this sphere the solid angle do)specified in condition (iii), thus obtaining an area a2dco, in a direc-tion from the centre of the sphere of direction-cosines Z, m, n.Clearly a collision of type a will occur every time that the centreof a molecule of class B comes upon this area a2 dco.


Let the relative velocity of the two molecules at such a collisionbe F. In any small interval of time dt before such a collision occurs,the second molecule will move relative to the first, through adistance V dt in the direction of V. Hence at a time dt before thecollision, the centre of the second molecule must have lain on anarea cr2dco which can be obtained by moving our original areacr2 da> through a distance Vdt from its old position in the directionopposite to that of V. And if a collision is to take place some timewithin the interval dt, the centre of the second molecule must lieinside the cylinder which this element cr2 dco traces out as it makesthe motion just mentioned (see fig. 20).

Fig. 20

The base of this cylinder is a2 dco; its height is V dt x cos d, whered is the angle between the direction Z, m, n and that of V reversed.Hence its volume, which is the product of base and height, isVcr2cos6do)dt. The assumption of molecular chaos now comesinto play, and we suppose that the probability that a moleculeof class B shall lie within this small cylinder at the beginning ofthe interval dt is

vf(u',v'9w')du'dv'dw'Va2cos ddcodt (116)

This, then, must also be the probability that our single moleculeof class A shall experience a collision of type a within the intervalof time dt.


Each unit volume of the gas contains vf(u9 v9 w) dudvdw mole-cules of class A, and on multiplying this by expression (116) weobtain the total number of collisions of type a which occur in aunit volume of the gas in a time-interval dt. The number is foundto be

v2f(u,v,w)f(u',v\w') Fcr2cos0dudvdwdu'dv'dw'dtodt (117)

Each of these collisions results in one molecule leavingclass A.

86. Let us next consider a type of collision which results in amolecule entering class A. We shall describe this as a collision oftype fi, and shall define it by the three conditions that

(i) After the collision, one of the molecules is to be of class A.(ii) After the collision, the other molecule is to be of class B.(iii) The direction of the line of centres at impact is to satisfy

the same condition as for a collision of type a (p. 108).

In brief, the two molecules have velocities before collisionwhich are identical with those which the two molecules of a typea collision acquire after collision, and vice versa.

Let the velocities before collision be

u, v, w and u\ v\ w' with spreads du dv dw and du' dv' dw'.

Making the necessary alterations in formula (117), we see that thenumber of collisions of type /? which occurs per unit volume intime dt is

v2f(u, v, w)f(u'9 v'9 w') Vcr2 cos 6 du dv dw du' dv' dw' dco dt.(118)

The quantities v9 c2, cos 6, da> and dt stand unaltered, since

these have the same value for both types of collision. Further-more, the result obtained in § 84 shews that the total spread ofvelocities after collision is equal to the total spread before, so that

du dv dw du! dv' dw' = du dv dw du' dv' dw'

and formula (118) may be written in the form

v2f(u9 v, w)f(u'9 v', w') Vcr2 cos 6 du dv dw du' dv' dw' do) dt.(119)


The Condition for a Steady State

87. We may notice that expression (119), which specifies thenumber of collisions of type /?, is identical with expression (117),which specifies the number of collisions of type a, except thatf(u, v, w)f(u', v', w') replaces f(u, v, w)f(u', v', w').

If, then,/(w, v, w)f(u'y v\ W) >f(u, v, w)f(u', v\ w%

collisions of type ft will be more frequent than collisions of type a,and class A of molecules will gain by collisions of these two types.If

f(u,v,w)f(u',v',w') = /(w,t;,w)/(tt>',w'), (120)then collisions of types a and /? occur with equal frequency, sothat class A neither increases nor decreases its numbers as theresult of collisions of these two types. If equation (120) is true forall values of the velocity-components, then class A neither gainsnor loses as the result of collisions of any type whatsoever. Thesame is true of every other class of molecules, so that if equation(120) is true for all velocities, each class retains the same numberof molecules throughout all collisions—the gas is in a steady state.

It is, then, very important to examine whether equation (120)can be satisfied for all velocities. Since a collision changesu, vy w, u\ v\ w' into % v, w9 u'9 v\ w\ the equation merely ex-presses that/(^, vy w)f(u'y v' y w') remains unchanged by a collision,or again, taking logarithms, that

log/(tt,t%iiO + log/(<f>'>t©') (121)

remains unchanged at a collision.This expression is the sum of two contributions, one from each

molecule, and we immediately think of a number of quantities ofwhich the sum is conserved at a collision. To begin with there arethe energy %m(u2 + v2 + w2)y and the three components of angularmomentum (mu,mVymw)] these give four possible forms forlogf(u, V, w). A fifth is obtained by taking log/(u, v, w) equal to aconstant (conservation of number of molecules, i.e. mass), andit is obvious that there can be no others. For if any additionalform were possible for log/, there would be five equations givingU, Vy Wy V! y V'y V)' Ul tCHUS Of U, Vy Wy U'y V' y W', SO t h a t U, Vy Wy U\ V'y V)'


would be determined except for one unknown. There must, how-ever, be two unknowns, as the direction of the line of centres isunknown.

The general solution of equation (120) is therefore seen to be

logf(u, v, 10) = aLxm{u2 + va 4- w2) -f a2mu+azmv + aAmw -f a6,(122)

where av a2, a3, a4, a6 are constants. These may be replaoed bynew constants and the solution written in the form

K v, w) = o^mKu - u0)2 + (v- v0)

2 + (w - w

or, if we still further change the constants,f{u, v, w) = ê-^Wtt-^+^-^^+^-ô)*!, (123)

in which A,h,u0, vOi w0 are new arbitrary constants.If f(u,v,w) is of this form, the gas will be in a steady state,

i.e. the distribution of velocities will not be changed by collisions.This value of / is a slight generalisation of that previouslyobtained in § 23 (formula (41)).

88. Let us now examine what these various constants mean inphysical terms. As the index of the exponential in formula (123)is necessarily negative or zero for all values of u, v and w, wesee that f(u9 v, w) has its maximum value when the index is zero,i.e. when u = u0, v = v0, w = w0.

This shews that the velocities of most frequent occurrence axethose in the immediate neighbourhood of u0, vOi w0.

Further, since (u — u0) occurs only through its square, positivevalues of u — uQ occur exactly as frequently as the correspondingnegative values, so that the average value of u — u0 is zero. Thusthe average value of u is w0, so that u0, v0, wQ must be the com-ponents of velocity of the centre of gravity of the whole gas,while the quantities u — uQy v — vQ, w — w0, which figure in theexponential in equation (123), are simply the components of thevelocity of a molecule relative to the centre of gravity of the gas.

Let us next writeu-uo = U,

w — w0 = W,


so that u = uo + U, etc. We now see that the velocity of anymolecule may be regarded as made up of

(i) a velocity u0, vQ, w0 which is the same for all molecules, beingthe velocity of the centre of gravity of the gas, and

(ii) a velocity U, V, W relative to the centre of gravity of thegas, which of course is different for each molecule.

We may speak of the former velocity as the "mass-velocity"of the gas, and of the latter velocity as the "thermal" velocityof the molecule, since it arises from the thermal agitation of thegas. From formula (123), the law of distribution of thermalvelocities is

/(U, V, W) = ê-û2+v2+w2>. (124)

This brings us back to Maxwell's law, which we obtained in§ 23. We see that h is identical with the h of § 23, which wasthere found to have the value 1/2BT.

The right-hand member of this equation shews that theprobability that the thermal velocity of a molecule shall havecomponents which lie within the limits U and U+dU, V andV + dV, W and W + dW is

If we integrate this from

U = - o o to+00, V = - o o to+00 and W = ~ooto+oo,

we obtain the total probability that the molecule shall havevelocity-components lying somewhere within these limits. Butas every molecule must have thermal velocities which lie some-where within these limits, this total probability must be equal tounity, so that we must have

+ 00

cccA I I I e~~ m(u +v +wv(jj[j dVdW = 1n)

— 00

The value of the integral is easily found to be (n/hm)*(Appendix VI, p. 306), so that

A*


Any one of the N molecules has an amount

of thermal energy, so that the total thermal energy of the gas isobtained by summing the quantity over all the N molecules.

The number of molecules having a thermal velocity of whichthe components lie between the limits

U and U +dU, V and V + dV, W and W+dW

is ,

and the contribution which these molecules make to the totalthermal energy is

%mNA{U2 + V2 + W2) e-*m<u2+va+w2>dU dV dW.

Summing this over all values of U, V, W, we find the total thermalenergy of the N molecules to be

+ 00

\mNA f f f(U 2 + V2 + W2) c-*»«u«+v«+w«> dU dV dW.— 00

The value of the integral is easily found to be32

so that the total thermal energy is3N

V A5m5\mNA /TT1--J7-.

The average thermal energy of a molecule is accordinglyor fiJJ7, in agreement with formula (17).

Steady State in a Mixture of Gases89. A slightly more complex problem arises when the gas

consists of a mixture of molecules of two different kinds. Let ussuppose that the molecules of both kinds are hard smooth spheres;let mv (Tv vv /j refer to the first kind of molecule, while ra2, cr2, v2i /2

refer to the second kind. We proceed as before and obtain, inplace of formula (117),

Wfifr, v, w)M<u't v\ w1) V [fto-i + er2)]2

x cos 6 du dv dw du' dv' dw' da) dt


for the number of collisions of type a in which the second moleculeis supposed to be of the second kind.

If the gas is to be in a steady state, there must be no net loss orgain to molecules of class A through collisions with molecules ofthe second kind. The condition for this is found to be that (inplace of equation (120))

Mu,v,w)f2(u',v'w') = / 1 ( w , v , ^ ) / 2 ( t t » ' ) , (125)

and we see, just as before, that logfx and log/2 must representquantities which are conserved at collision. Again we find thatlog/i and log/2 may be molecular energies, or components ofmomentum, or constants, or a combination of all five, so thatfinally fx and f2 must be of the form

fx{u9vyw) =

f2{u,v,w) =

where h, u0, v0, w0 are the same in both formulae.The fact that h is the same for the two kinds of gases shews that

in the steady state the average kinetic energy of the two kindsof molecules is the same. This is a special case of the generaltheorem of equipartition of energy to which we have alreadyreferred (§11).

Remembering that the absolute temperature in a gas ismeasured by the average kinetic energy of its molecules, we seethat when two gases are mixed they will in time reach a steadystate in which the temperature is the same for both.

THE MAXWELLIAN DISTRIBUTION

90. We may notice first that the components U, V, W of thermalvelocity enter separately into Maxwell's law. To be more precise,if we write

(^)* (126)

then equation (124), which expresses Maxwell's law, can bewritten in the form

[<j>(U)dU][<f>(V)dV][<f>(\N)dWl

(127)


This shews that the probability of a molecule having a velocityof components U, V, W is the product of three independentprobabilities, one of which depends only on U, one only on V andthe third only on W. It follows that the probability of a moleculehaving any specified velocity in the direction of Ox is inde-pendent of the velocity-components of the molecule in otherdirections, being in fact determined by the formula

<f>{U)d\J =

To recapitulate, we have now found that equation (124) repre-sents a steady state in which the law of distribution of thermalvelocities (U, V, W) is given by the formula

,V, W) = / ^ V C - ^ U H V H W ^ e ( 1 2 8)

while the law of distribution for a single component is

(129)

Equation (128), which expresses Maxwell's law of distribution,was first given in his 1859 paper. The proof by which he sought toestablish the law is now generally agreed to have been unsound.It began by assuming, without any attempted justification, thatthe distributions of U, V, W were independent. This proof is re-produced in Appendix I (p. 296) on account of its historic interest.The proof given in the present chapter is a modification of oneoriginally given by Boltzmann and Lorentz, but this also pro-ceeded on the basis of an unjustified assumption, namely that ofmolecular chaos (§82). In respect of logical completeness, then,there is little to choose between the two proofs, but the proof justgiven has the merit of giving a vivid picture of the physical pro-cesses at work in a gas, besides providing a number of formulaewhich are otherwise useful.

The investigations of Boltzmann and Lorentz went furtherthan this. Equation (123) contains the most general solution ofequation (120), but it has not been shewn that it represents allpossible steady states. Boltzmann and Lorentz proved this; their


proof is given in Appendix II, and so will be assumed henceforth.Thus by giving different values to the constants h, u0, vOi w0 inequation (123) we obtain all the steady states which are possiblefor a gas. It is further proved in Appendix II that a gas which isleft to itself will gradually approach to one or other of these steadystates, and must finally attain it after sufficient time has elapsed.

A more complete proof of all this, which involves neitherMaxwell's assumption of the independence of U, V and W northe assumption of molecular chaos, is given in Appendix IV(p. 301).

Thermal Motions

91. Let us next suppose that the thermal velocity U, V, W ofa molecule is of amount c, and that its direction lies within a smallsolid angle da). Then U2 + V2-}- W2 = c2, and by a well-knowntransformation of coordinates

dUdVdW = c2dcdco.

The law of distribution now assumes the form

Ae-hmd2c2dcd(o.

The direction of U, V, W only enters this formula through thedifferential da), so that the chance of this direction lying withinany solid angle is exactly proportional to the magnitude of thesolid angle. In other words, all directions are equally likely, asindeed we should expect. For when the gas has no mass-motionthere is no reason why thermal velocities should favour onedirection more than another. When the gas has a mass-motion,the principle of relativity takes charge of the situation, and shewsthat the distribution of thermal velocities must be the same aswhen the gas has no mass-motion.

If we integrate over all possible directions for da), the law ofdistribution becomes

4:7rAe-him2c2dc. (130)

In fig. 21 the thin line shews the graph of the curve

y = e~*\

while the thick line is the graph of the curve

118 COLLISIONS AND MAXWELL S LAW

the factor 2 being introduced in this latter curve to make theareas of the two curves the same. The thin curve gives a graphicalrepresentation of the distribution of any single component(U, V or W) of molecular velocity while the thick gives the samefor the total molecular velocity c.

0-8

0-6

0*4

Q'2

'"-

j/

\

/

\

/

\

//f

\

1(

\\//

y(\

y

\

—

\

\

V

s s

\ V\

vs

\—.= am*

0 0-2 0-4 0-6 0-i 1-0 1-2 1-4 1-6 1-8 2-0 2-2 2*4

Fig. 21

On the thin curve the maximum ordinate occurs at y = 0,shewing that the most frequent value of U is U = 0. The valueof y falls off rapidly as x increases, shewing that large values of Uare rare. The average numerical value of U is

and so is represented by the ordinate of x = 0-5642 in fig. 21.Molecules with more than four times this velocity are representedby the area of that part of the thin curve which lies to the rightof the ordinate x = 2-257, and this is seen to be very small indeed.Thus it is very rare for any component of molecular velocity tohave as much as four times the average value.

The thick curve exhibits the distribution of c among the mole-cules of the gas and here the maximum ordinate is that of x = 1,so that the most frequent value of c is l/(Am)*. The graph shews


that values of c which are more than 2 | times the most frequentvalue of c are very rare. The average value of c, which we shalldenote by c, is

9

Jo /Jand so is just double the average value of any single coordinateU, V or W.

The mean value of c2 may be denoted by c2, but is more com-monly denoted by (72. Its value is readily found to be

C2 - — - —2hm hm'

so thatc = 0-921G and C = l-086c.

With C2 defined in this way, the average thermal energy of amolecule is ^mC2, and the thermal energy per unit volume is\mvG% or \pC2; it is the same as though the whole gas were movingforward through space with a velocity C, or l-086c.

Mass-Motion and Molecular-Motion

92. We have seen that in the most general "steady state"possible, the motion consists of a thermal motion compoundedwith a mass-motion. The mass-motion has velocity componentso» vo> wo> wkile the thermal motion has velocity components

u—u09v — vo,w-wOi which we have denoted (p. 112) by U, V, W.The kinetic energy per unit volume of the gas is

and since 27U = ZV = I W = 0,

we have

where p is the mass-density of the gas, given by p =» mv.


This shews that the total energy of a gas may be regardedas the sum of the energies of its mass-motion and its thermalmotion.

Let us suppose that a vessel containing gas, which has so farbeen moving with a velocity of which the components arew0, v0,

wo> is suddenly brought to a standstill. This will of coursedestroy the steady state of the gas, but after a sufficient time thegas will assume a new and different steady state. The mass-velocityof this steady state will obviously be nil, and the energy whollymolecular. The individual molecules have not been acted uponby any external forces except in their impacts with the containingvessel, and these leave their energy unchanged. The new mole-cular energy is therefore equal to the former total energy. Thisenables us to determine the new steady state. In the languageof the older physics, one would say that by suddenly stopping theforward motion of the gas the kinetic energy of this motion hadbeen transformed into heat. In the language of the kinetic theorywe say that the total kinetic energy has been redistributed, so asnow to be wholly molecular.

An interesting region of thought, although one outside thedomain of pure kinetic theory, is opened up by the considerationof the processes by which this new steady state is arrived at. Toexamine the simplest case, let us suppose the gas to be containedin a cubical box, and to have been moving originally in a directionperpendicular to one of the sides. The hydrodynamical theory ofsound is capable of tracing the motion of the gas throughout alltime, subject of course to the assumptions on which the theory isbased. The solution of the problem obtained from the hydro-dynamical standpoint is that the original motion of the gas isperpetuated in the form of plane waves of sound in the gas, thewave fronts all being perpendicular to the original direction ofmotion. This solution is obviously very different from that arrivedat by the kinetic theory. For instance, the solution of hydro-dynamics indicates that the original direction of motion remainsdifferentiated from other directions in space through all time,whereas the solution of the kinetic theory indicates that a stateis soon attained in which there is no differentiation betweendirections in spaoe.

COLLISIONS AND MAXWELL S LAW 121

The explanation of the divergence of the two solutions isnaturally to be looked for in the differences of the assumptionsmade. The conception of the perfect non-viscous fluid postulatedby hydrodynamics is an abstract ideal which is logically incon-sistent with the molecular constitution of matter postulated bythe kinetic theory. Indeed in a later part of the book we shallfind that molecular structure is inconsistent with non-viscosity;and shall be able to shew that the actual viscosity of gases issimply and fully accounted for by their molecular structure. Ifwe introduce viscosity terms into the hydrodynamical discussion,the energy of the original motion becomes "dissipated" by vis-cosity. On the kinetic theory view, this energy has been convertedinto molecular motion. In fact the kinetic theory enables us totrace as molecular motion energy which other theories arecontent to regard as lost from sight.

Fig. 22

93. Maxwell's law of distribution of velocities has an obvioussimilarity to the well-known law of trial and error. If a marksmanfires shots at a target, the result will look somewhat as in fig. 22.The marksman tries, in firing each shot, to make both the x andthe y coordinates as small as possible. Subject to certain assump-tions, which we need not discuss here, Gauss shewed that the two

5 J K T


efforts may be treated as independent. The chance that the occoordinate shall be between x and x + dx is

-e-*x2dx, (131)

where K is a constant which measures the skill of the marksman.If the marksman is skilled, small values of # are frequent, so thatK is large; the more skilled the marksman, the greater the valueof K, this becoming infinite for an ideal marksman of perfectskill. Formula (131) is similar to Maxwell's formula (126) for thechance of a velocity component between U and U+dU, exceptthat K replaces hm, which, as we have seen, is inversely pro-portional to C2 and so to the absolute temperature of the gas.Thus we may imagine Maxwell's formula arrived at by all themolecules aiming at zero for each component velocity. The totalof their errors of marksmanship provides the total thermal energyof the gas, so that a high temperature corresponds to bad marks-manship and conversely.

When a marksman aims at a two-dimensional target, as infig. 22, the law of distribution of his shots is

n

or, transforming to plane polar coordinates r, d, is

-e-^rdrdd.71

If the target is three-dimensional, the corresponding lawbecomes i K\\

i

or, in three-dimensional polar coordinates r, d, <f>,

M e

which is now identical with Maxwell's law (§91). Just as (infig. 22) the marksman aims at a radial distance r = 0, so here themolecules aim at a total velocity c = 0; the average velocity is

x u - / , i _ x T- IJ 1-1284not, however, c = 0, but as we have already seen c = -7—rj-.

(hm)*


94. Although Maxwell's law is comparatively simple in itself,its introduction into a physical problem often leads to compli-cated, and even intractable, mathematics. It is sometimes per-missible to avoid the introduction of Maxwell's law, and obtain arough approximation to the solution of a physical problem byassuming that all the molecules have exactly the same velocity,and in order that this velocity may be consistent with the actualvalues of the pressure and of the kinetic energy, this uniformvelocity must be supposed to be C.

Some idea of the amount of error involved in this approxima-tion may be obtained from a study of fig. 21. Since C2 = 3/(2hm),and hm is taken equal to unity in drawing the curves of fig. 21, theapproximation amounts to the assumption that the whole areaof the curve is collected close to the abscissa

x = j \ = 1-225.

The graph shews that the approximation is a very rough one.95. It is sometimes required to know how many of the mole-

cules of a gas have a speed greater than or less than a given speedc0. Out of a total of N molecules the number which have a speedin excess of c0 is, by formula (130),

and, on integrating by parts, this becomes

2N ( ^

In terms of the probability integral, or error function defined by

2 f °°erf a; = -j- e~x2dx,

this number becomes

Nlerfx + ^xe-**), (132)\ V77 /

where x = (hm)*c0 = J- Hj j .


From a table of values of erf a:, it is easy to calculate the numberof molecules either having velocity greater than any value c0, orwithin any range c0 to cx. The general run of the numbers to beexpected can often be sufficiently seen by an inspection of thecurves of fig. 21, but for exact calculations the table given inAppendix V will be required. The values of 1— erf x are givenin the fourth column of this table.

Experimental Verification of Maxwell's Law

96. The more general discussion given in Chap. X below willshew that Maxwell's law is not only applicable to the molecules ofgases, but also to those of liquids, to the atoms of solids, to thefree electrons in solids and even to the Brownian movements ofparticles suspended in a liquid or a gas. Although the law is ofsuch outstanding importance in many branches of physics, it isonly in recent years that experimental confirmation of its truthhas been obtained.

The first confirmation of the law was provided by the electronsescaping from a hot metallic filament. In the interior of a con-ducting filament, electrons are moving in all directions. Thosewhich reach the surface with a speed above a certain limit mayovercome the restraining forces at the surface and pass into outerspace, much as molecules escape from the surface of a liquid byevaporation. The speed with which any individual electron leavesthe surface provides a record of the speed with which it was ori-ginally travelling inside the metal. By putting various retardingpotentials on the escaping electrons, it is possible to find whatproportion of the electrons are travelling with speeds above asuccession of assigned limits, and so to plot out the law of distri-bution of velocities, both outside and inside the filament.

Experiments of this general type were originally performed by0. W. Richardson and F. C. Brown,* and later by Schottky,fSih Ling Ting % and J. H. Jones.§

• Phil. Mag. 16 (1908), pp. 353, 890; 18 (1909), p. 681.t Ann. d. Phys. 44 (1914), p. 1011.t Proc. Roy. Soc. 98 A (1921), p. 374.§ Proc. Roy. Soc. 102 A (1923), p. 734.


The original experiments of Richardson and Brown were con-cerned only with the electrons escaping from hot platinum, andshewed that Maxwell's law was obeyed with considerableaccuracy. Then Schottky obtained similar results with carbonand tungsten, except that all velocities appeared to be sub-stantially higher than could be reconciled with the temperatureof the filament. The work of Jones cleared up these difficulties,and finally Germer,* making an extensive series of experimentsat temperatures which ranged from 1440° abs. to 2475° abs.,shewed that the speeds of the electrons not only satisfied Max-well's law with accuracy, but also that their absolute magnitudesagreed exactly with the temperature of the hot filament.

97. Meanwhile, attacks were being made on the far moredifficult problem of testing the lawdirectly for molecular velocities. In1920, Sternf planned an arrange-ment, a development of that ofDunoyer (§5), by which he hopedto be able to measure molecularvelocities directly. Fig. 23 shews across-section of the apparatus. Atits centre is a platinum wire Wheavily coated with silver. Sur- s#

rounding this is a cylindrical drum D containing a narrow slit S,and still farther out another concentric drum PP'. The wholeapparatus can be rotated as a rigid body about W.

The complete apparatus is put inside an enclosure in whicha good vacuum can be obtained. If the wire W is now heated toa suitable temperature, atoms of silver are emitted in profusion,and if the gas-pressure is sufficiently low, so that the free pathis very great compared with the dimensions of the apparatus,some of these silver molecules will pass through the slit S andform a deposit on the outer drum PP'. If the apparatus is atrest, this deposit will of course be formed at a point P exactlyopposite 8, but if the whole apparatus is rapidly rotating, the

* Phys. Rev. 15 (1925), p. 795.t Zeitschr.f. Phys. 2 (1920), p. 49; 3 (1920), p. 417, and Phys. Zeitschr.

21 (1920), p. 582.

126 COLLISIONS AND MAXWELLS LAW

outer cylinder moves on through an appreciable distance whilethe molecules are crossing from the slit S, so that the deposit isformed farther back than if there were no rotation. Atoms whichmove at different speeds will of course form deposits at differentplaces, and by measuring the position and intensities of thevarious deposits, it is possible to calculate the distribution ofvelocities in the atoms by which the deposits were formed.Unfortunately the apparatus did not prove sensitive enough togive very convincing results.

98. In 1927 E. E. Hall* designeda new, and more sensitive, formof the same apparatus, which is shewn diagrammatically in fig. 24.

Fig. 24

An electrically heated oven represented at the bottom of thediagram contains a vapour. Molecules emerge through theopening 0, moving in all directions, and the slit S selects a beamof parallel moving molecules. These move in the direction OSat different speeds which ought, as we believe, to conform toMaxwell's law. Above the slit 8 is a rotating drum, with a slit inits side which, once in each revolution, comes directly above thefixed slit S. If the drum were rotating very slowly, moleculeswould, on each of these occasions, pass through the slit and forma deposit on the point of the drum opposite to the slit. If, how-ever, the drum rotates so fast that the speed of its surface is com-parable with that of the molecules, the drum has turned throughan appreciable angle before the molecule strikes its farther side,so that the point of deposition of the molecule is displaced by adistance which will be inversely proportional to the velocity of

• See Loeb, Kinetic Theory of Gases (2nd ed. 1934), pp. 132, 136.

COLLISIONS AND MAXWELLS LAW 127

the molecule. If the drum has a radius of 6 inches and rotates 100times per second, the speed of its surface is 314 feet per second,and a molecule which is travelling at 1000 feet per second willtake ^Q second to cross the drum. In this interval the surface ofthe drum will move

f i | feet or 7 | inches,

so that the point of deposition will be displaced this far from thepoint opposite the slit. In this way we obtain a sort of velocity-spectrum of the speeds of the molecules of the vapour.

10

Z 4

/

J\

\\\\\

1-0 2-0 3-0DisplacementFig. 25

I. F. Zartman* and C. C. Kof experimented with bismuthvapour in this way, but the velocity spectrum they obtained,shewn in fig. 25, does not entirely agree with that to be expectedfrom Maxwell's law. Ko finds, however, that his spectrum agreeswith that to be expected if the vapours consisted of a mixture ofmolecules of Bi, Bi2, and Bi3, in the proportion at 827° C. of44 :54 :2, the molecules of each kind obeying Maxwell's law. Inthis way a rather indirect proof of Maxwell's law is obtained.

* Phys. Rev. 37 (1931), p. 383.t Journal Franklin Institute, 217 (1934), p. 173,


99. More recently, V. W. Cohen and A. Ellett,* experimentingwith various alkali atoms in a similar manner, have obtained avery convincing confirmation of Maxwell's law.

The same experimenters! have also studied the velocities in abeam of potassium atoms which had been scattered by a crystalof magnesium oxide. They found that these conformed to Max-well's law, the absolute values being those appropriate to thetemperature of the crystal, independently of the origin of thebeam of atoms before striking the crystal. This suggests that theatoms were adsorbed by the crystal surface, and subsequentlyre-emitted from it by a process of evaporation, in accordance withthe suggestion of Maxwell and Langmuir (§34).

In both these last experiments, the atoms were sorted out,according to their velocities of motion, by passing them througha magnetic field; the atoms, being charged, had their paths curvedby the field, the radius of curvature being exactly proportionalto the speed of motion. A more complicated way of magneticsorting of atoms has been devised by Meissner and Scheffers,Jwho have verified Maxwell's law with great precision for atomsof lithium and potassium.

100. These magnetic methods are useful only for sorting outelectrically charged atoms; a magnetic field has no influence onmolecules or neutral atoms. In 1926 Stern§ proposed to sort outthese by a method rather like the toothed wheel method whichFizeau used to measure the velocity of light. In 1929 Lammert||constructed an apparatus on these lines and obtained a verysatisfactory confirmation of Maxwell's law for mercury atoms.

In fig. 26, 0 contains mercury vapour. A slit in this and asecond slit S limit the emerging mercury atoms to a parallelmoving beam. After passing through S, the atoms of the beamencounter in succession two wheels Wx, W2, both mounted on thesame shaft, and rapidly rotating. Each wheel has a slit in it, ofthe same width as the slit 8. Once in each revolution of the wheel

* Phys. Rev. 52 (1937), p. 502.f Ellett and Cohen, Phys. Rev. 52 (1937), p. 509.t Phys. Zeitschr. 34 (1933), p. 173.§ Zeitschr. f. Phys. 39 (1926), p. 751.|| Zeitschr. f. Phys. 56 (1929), p. 244.

COLLISIONS AND MAXWELLS LAW 129

Wl9 the whole beam of molecules passes through the slit in Wl9

and impinges on the wheel W2. The slit in W2 is not parallel to thatin Wx but is set to be at an angle of about 2 degrees behind it.Thus those molecules which travel exactly the distance betweenWx and W2, while the wheels are turning through 2 degrees, willpass through the slit in W2; molecules which travel at otherspeeds are either too late or too early to get through. By runningthe wheels at different speeds, it is possible to pick out the mole-cules which travel at any desired speed; these are allowed to forma deposit on a transparent screen, and from the density of thisdeposit the number of molecules can be estimated. In this wayLammert shewed that Maxwell's law was satisfied to a con-siderable degree of accuracy.

Fig. 26

Kappler obtained yet another confirmation of the law by theuse of the miniature torsion-balance already described in §5.When the arm is immersed in gas of very low density, the impactsof the separating molecules can be detected and analysedstatistically. The results obtained indicate that the velocities ofthe individual molecules conform to Maxwell's law.

101. A certain amount of confirmation of Maxwell's law is pro-vided by astrophysical observation. The atmosphere of a starconsists of a mixture of atoms of different kinds moving with allpossible velocities. If an atom of hydrogen is moving towards theearth with a speed v, and emitting the line Hoc, its light whenreceived on earth and resolved in a terrestrial spectroscopewill not be found in the normal position of the Ha line, but


will be seen to be displaced through a wave-length dX givenby

dX v

in accordance with the usual Doppler-effect. As the atoms havea very wide range of speeds, the line they produce is of substantialbreadth, and the distribution of intensity in this line will againprovide a sort of velocity spectrum of the moving atoms. Onceagain, however, this gives no direct proof of Maxwell's law, sincethe atoms which emit the light are not all at the same depth inthe star's atmosphere; the light we receive comes from places atdifferent temperatures, so that again the velocity spectrum doesnot come from a single Maxwellian distribution of velocities, butfrom a great number superposed. The observed distribution isfound to be that to be expected from a number of superposedMaxwellian distributions.

102. The objections just mentioned do not apply to a gasmade luminous in the laboratory, for here the temperature,pressure, etc. are the same throughout the gas. Ornstein andvan Wyck* have studied the velocity spectrum emitted byhelium gas at very low pressure, the gas being made luminousby passing an electric discharge through it. The observed velocityspectrum agreed well with that to be expected from a Maxwelliandistribution, but at a temperature some 50° C. above that of thegas in the tube. They thought the difference might be attributedto the general heating of the gas by the discharge.

* Zeitachr.f. Phys. 78 (1932), p. 734.

Chapter V

THE FREE PATH IN A GAS

103. We have already seen how viscosity and conduction ofheat can be explained in terms of the collisions of gas molecules,and of the free paths which the molecules describe between col-lisions. As a preliminary to a more detailed study of these andother phenomena, the present chapter deals with various generalproblems associated with the free path, the molecules again beingsupposed, for simplicity, to be elastic spheres.

Number of Collisions104. Let us begin by making a somewhat detailed study of the

collisions which occur in a gas when this is in a steady state, sothat the velocities are distributed according to Maxwell's law.It is clear that the occurrence of collisions cannot be affected byany mass-motion the gas may have as a whole, so that we maydisregard this, put u0 = v0 = w0 = 0, and suppose the law ofdistribution of velocities to be

(133)

where c2 = u2 -f v2 -f iv2.

In § 85 we calculated the frequency of collisions of variouskinds in a gas in which the molecules were similar spheres, all ofdiameter o*. We found the number of collisions of class a occurringper unit volume per unit time to be

v2f(u, v, w)f(uf, v',w') Vcr2 cos ddudv dwdu' dv' dw' do).

If we replace f(u, v, w) and f(u'yv',w') by the special valuesappropriate to the steady state, this number of collisions becomes

_/ tiin \21 I e-hmw+v*+u>*+u'*+tf*+**) Va2 cos 6 du dv dwdu' dv' dw' da.

(134)Here V is the relative velocity at collision, given by

132 THE FREE PATH IN A GAS

105. Expression (134) depends only on the velocities at col-lision, except for the factor cr2 cos 6 da). Clearly a2 da) is an elementof area on a sphere of radius cr, AsayP in fig. 27, and cr2 cos 6 do) isits projection Q on a plane ABat right angles to V. We may infact replace a2 cos ddco by dS,an element of area in this plane;we then see that the projectionis equally likely to be at allpoints inside a circle of radiuscr, as is of course otherwiseobvious.

In terms of the usual polar co-ordinates 6, <p, the factors which Fig. 27

involve the angles in expression (134) may be rewritten in theform

coaddco = sind cos 6 ddd<f> = Isin2dd(20)d<f> (135)

When collision occurs, the relative velocity along the line ofimpact is reversed, while that at right angles to this line persistsunchanged. Thus the relative velocity is bent through an angle20, and the polar coordinates of its direction after collision are20, <j). Formula (135) now shews that all directions are equallylikely for the relative velocity after collision, a result first givenby Maxwell.

106. Integrating expression (134) with respect either to dS orto 6 and <f>, we replace cr2 cos Odd) by ncr2, and find that thenumber of collisions at all angles of incidence is

___ \ e-h u*-H,Hw*+u>*+v>*+**) V<j2dudvdwdu'dv'dw'.

(136)

This is the number of collisions per unit time per unit volume ofgas in which the two molecules belong to what we have described(§ 85) as class A and class B respectively—that is to say, the firstmolecule has a velocity of which the components lie within thelimits u and u -f du, v and v -f dv, w and w + dw, while the second

THE FREE PATH IN A GAS 133

has a velocity of which the components lie within the limitsv! and u' + du\ v' and v' + dv\ vf and vf + dvf.

Let us fix our attention for the moment on the ^-componentsof velocity, namely u and u'. These enter formula (136) throughthe relative velocity F, and also through the factor

e-hm(u*+u'*)dudu'. (137)

We can plot values of u and vf in a plane diagram as shewn infig. 28. We can also plot them in the same diagram with the axesturned through an angle of 45°, thenew coordinates £, £' now beinggiven by Ji

yuf

£--L(«-«), go.-^(

Transforming to these new co-ordinates, we find that

Fig . 28

since u2 + u'2, the square of the distance from the origin, is equalto £2 + £'2; and dudu', an element of area in the plane, may bereplaced by d^dQ. Thusformula (136) for the number of collisionsbetween molecules of classes A and B becomes

hm( hm

(138)

in which rj = -j-(v' — v), etc. The reason for this transformation

of coordinates is seen as soon as we express the value of F in thenew coordinates, for we find that

F2 = (u' - uf + (vf -

so that F does not depend on ^',^',^'. Thus we may at onceintegrate formula (138) for all values of £', rf and £', obtaining

Ttv1 ^ \


This is not, as might at first be thought, the number of collisionsin which £, 7/, £ he within limits d£, dy, d£; it is twice that number,since each collision has been counted twice—once as a collisionbetween a molecule of class A and one of class B, and again as acollision between a molecule of class B and one of class A. Thusthe true number of collisions in which £, 7j, £ lie within assignedlimits d£drj d£ is half the foregoing, or

(139)

There is an obvious resemblance to Maxwell's law of distribu-tion of velocities. Again we see that all directions for £, ?/, £ areequally likely, and as the relative velocity V has componentsV2£, ^2TJ, V2£> it follows that all directions are equally likely for F,the relative velocity before collision. We have also just seen(§ 105) that, whatever the directions may be of the velocitiesand relative velocity of the molecules before collision, alldirections are equally likely for the relative velocity aftercollision.

107. Integrating formula (139) over all possible directions forF, we obtain, as in § 106,

(140)

as the total number of collisions in which the relative velocitylies between F and V + dV. If we finally integrate this from F = 0to F = oo, we obtain for the total number of collisions of allkinds

Since (§91) c = 2{n'hm)~ii this may be replaced by

These results will be required later.


Free Path

108. The free path is usually defined as the distance which amolecule travels between one collision and the next. We haveseen that the v molecules in a unit volume of gas experience

collisions per unit time. Since each collision terminates two freepaths, the total number of free paths is twice the foregoing ex-pression, or

Snv2(r2c.

The total length of all these free paths is of course the totaldistance travelled by the v molecules in unit time, and this is vc.Hence by division we find that the average length of a free path is

(142)V277W2

This is generally described as Maxwell's mean free path; aswe shall see later (§§ 117, 135), there are other ways of definingand evaluating the mean free path in a gas.

Free Path in a Mixture of Gases

109. I t is rather more difficult to calculate the free path in amixture of gases in which there are molecules of different sizes.

As before, the constants of the molecules of different types willbe distinguished by suffixes, those of the first type having a suffixunity (mv(rvv1,f1)9 and so on. We shall require a system ofsymbols to denote the distances apart at collision of the centresof two molecules of different kinds. Let these be S1V 812, $23>

e^c->8pQ being the distance of the centres of two molecules of typesp, q when in collision. Obviously

£12 = i(*i + <r2), Sn = <rlf etc.

As in § 89, we find for the number of collisions per unit timebetween molecules of types 1 and 2 and of classes A and Brespectively (as defined in § 85)

vivzf\{u> v> w)f?ku'> v'>w') Pî2 c o s 0 dudvdwdu' dv' dw'da)t


where V is the relative velocity, and dco the element of solid angleto within which the line of centres is limited.

Replacing /1?/2 by the values appropriate to the steady state,and carrying out the integration with respect to d<oy the numberof collisions per unit time is found to be

(143)

this expression being exactly analogous to expression (136)previously obtained.

Let the velocity-components u, v, w, u',v', w' now be replacedby new variables, u, v, w, a, /?, y given by

u — _J: *__ etc.; ot = u-u, etc., (144)

so that u, v, w are the components of velocity of the centre ofgravity of the two molecules, and a, /?, y are as usual the com-ponents of the relative velocity V. Writing

we find thatVYi

m1Jrm2

We readily find, by the method already used in § 106, thatdudu' — duda, so that expression (143) may be replaced by

miv%\ r~ e \V8l2dudvdw dad/3dy.

(145)

110. On integrating with respect to all possible directions inspace for the velocity c of the centre of gravity, we may replacedudvdw by 4nc2dc while similarly, integrating with respect toall possible directions for F, we may replace docdfidy by 4:7rV2dV.The number of collisions per unit volume per unit time for whichc, V lie within specified small ranges dcdV is therefore

J c2 V*dcdV.(146)


Integrating from c = 0 to c = oo, the number of collisions forwhich V lies between V and V + dV is found to be

p^h7l3 S\2e ™>+™>V F W , (147)

and again integrating this expression from V = 0 to V = 00, thetotal number of collisions per unit volume per unit time betweenmolecules of types 1 and 2 is found to be

ff-^-M (148)h \tnx m2j

This formula gives the number of free paths of the vx moleculesof the first type in unit volume, which are terminated per unittime by molecules of the second type. When the difference be-tween the two types of molecules is ignored, it reduces to twiceexpression (141) already found, the reason for the multiplyingfactor 2 being that already explained on p. 134.

111. If we divide expression (148) by vl9 we obtain the meanchance of collision per unit time for a molecule of type 1 with amolecule of type 2. Summing this over all types of molecule, thetotal mean chance of collision per unit time for a molecule oftype 1 is found to be

(149)

where the summation is taken over all types of molecule. Themean time interval between collisions is of course the reciprocalof this.

The total distance described by vx molecules of the first kindper unit time is

while the total number of free paths described by these v1

molecules is equal to vx times expression (149). By division, themean free path for molecules of the first type is found to be

(151)


When there is only one kind of gas present, this reduces toformula (140) already found.

If there are only two kinds of gas present, one with moleculesof far greater mass than the other (m2 > mx), the lighter moleculeswill move enormously more rapidly than the heavier because ofthe equipartition of energy. We may now put rn^Jm^ = 0, andformula (151) reduces to

A-i = —r=z

If the diameter of the lighter molecules is much smaller thanthat of the more massive, the first term in the denominator maybe neglected, and the free path is given by

1 4Ai = ^ ^ = ^ r (152)

This is the simple case already considered in § 26, our formula(152) agreeing with the formula there obtained. It gives the freepath of electrons threading their way through a gas.

D E P E N D E N C E OF F R E E P A T H ON V E L O C I T Y

112. This last calculation will have suggested that the freepath of a molecule may depend very appreciably on the velocityof the molecule; a molecule moving with exceptional speed mayexpect a free path of exceptional length, while obviously amolecule moving with zero velocity can expect only a free pathof zero length.

It is important to examine the exact correlation betweenthe speed of a molecule and its expectation of free path.

We shall suppose that the gas consists of a mixture of moleculesof different types, as in § 109.

Let us fix our attention on a molecule of the first type, movingwith velocity c. The chance of collision per unit time with amolecule of the second type having a specified velocity c! is equalto the probable number of molecules of this second kind in acylinder of base nS\2i and of height F, where V is the relativevelocity.


The second molecule is supposed to have a velocity c'. Let 6, <}>be angles determining the direction of this velocity, 6 measuringthe angle between its direction and that of c, and 0 being anazimuth. The number of molecules of the second kind per unitvolume for which c',d,<t> lie within small specified rangesdc',dd,d<f> is

yaftpV e-WVa sin

The result of integrating with respect to <fi is obtained byreplacing d<j> by 2n, and on multiplying this by nSl2V, we obtainfor the number of molecules of the second kind which lie withinthe cylinder of volume nS\2V and are such that c', 6 lie within arange dc', dd,

W2v2S\2 vWra | Ve-hm^cf2 sin ddddc' (153)

When c, c' are kept constant, the value of V varies only with 6,being given by

F2 = c2 + c'2-2cc'cos0,

whence we obtain by differentiation, still keeping c, c' constant,

VdV = cc' sin Odd.

Thus we can replace expression (153) by

J i (154)

and proceed to integrate with respect to F. The limits for F arec + c' and c ~c', so that

F W = |c(c2 + 3c'2) when c'>c

= f c'(c'2 + 3c2) when c' < c.

Thus the result of integrating expression (154) with respect to F is

when c'>c, fi>2£2

2^Trhzm\e-hm*c'2c'(c2 + 3c / 2)dc\ (155)

when c'<c, ±v2S212 <Jnhzml e~hm^2 — (c'2 + 3c2) dc' (156)

cIf we now integrate this quantity with respect to c' from c' = 0

to c' = oo (using the appropriate form according as c' is greater


or less than c), we obtain, as the aggregate chance per unit timeof a collision between a given molecule of the first type movingwith velocity c, and a molecule of the second type moving withvelocity c',

^ ^ [™c\c* + 3c'2

o cThe former of the two integrals inside the square bracket can

be evaluated directly. To integrate the second integral, wereplace hm2c'2 by y2. After continued integration with respect tot/2, we find as the sum of the two integrals in expression (157),

3 r return* " |

i c 7 P m l c 2 e ' h m z c 2 + ( 2 A m 2 ° 2 + l ) e~y*dy -(158)

If we introduce a function* i]r(x) defined by

ifr(x) = xe~xZ+(2x2+l) P\~y2dyt (159)Jo

expression (158) may be expressed in the form

-j===Jr(c<Jhm2),

and hence if we denote expression (157) by 012, its value is foundto be

»1 2 = s ifr(cvhm2). (160)

With this definition of (912 we see that when a molecule of thefirst kind is moving with a velocity c, the chance that it collideswith a molecule of the second kind in time dt is 012dt.

113. If we change the suffix 2 into 1 wherever it occurs, weobtain an expression <9n for the chance per unit time that amolecule of the first kind moving with velocity c shall collidewith another molecule of the same kind.

• The value of I e~yt dy cannot be expressed in simpler terms, so that.' o

t/r(x) as defined by equation (159) is already in its simplest form. Tablesfor the evaluation of ft{x) are given in Appendix v.


By addition, the total chance per unit time that a moleculeof the first kind moving with velocity c shall collide with amolecule of any kind is

I0l3 = 0n + 012 + 013 + .... (161)XS XX Xm XO \ f

In unit time the molecule we are considering describes a dis-tance c; hence the chance of collision per unit length of path is

u'(162)

The mean free path Ac, for molecules of the first kind movingwith velocity c, is accordingly

cAc = ~jyT~. (163)

When there is only one kind of gas, this assumes the form

* C kmC2 (164)

114. This formula expressing Ac as a function of c is, unhappily,too complex to convey much definite meaning to the mind, andwe are therefore compelled to fall back on numerical values. Thefollowing table, which is taken from Meyer's Kinetic Theory ojOases (p. 429), gives the ratio of Ac (equation (164)) to Maxwell'smean free path A (equation (142)) for different values of c, fromc = 0 to c = oo.

c/d

00-250-50-6270-8861-01-2531-5351-7722345600

hmc2

_

_234_

_

-

Ac/A

00-34450-64110-76470-96111-02571-13401-21271-25721-28781-35511-38031-39231-39891-4142

A/A,

002-91121-56041-31111-04070-97490-88190-82470-79540-77650-73800-72440-71820-71490-7071

We notice that for infinite velocity Ac

found.: V2A, as previously


Probability of a Free Path of given Length

115. It is of interest to find the probability that a moleculeshall describe a free path of any assigned length.

Let/(Z) denote the probability that a molecule moving with avelocity c shall describe a free path at least equal to I, After themolecule has described a distance I, the chance of collision withina further distance dl is, by formula (162), equal to dlj\c. Hencethe chance that a molecule shall describe a distance Z, and then afurther distance dl, without collision is

This must however be the same thing a,s f (I+ dl) or

Equating these expressions, we havedM _ JJQdl " V

of which the solution is(165)

the arbitrary constant of the integration being determined by thecondition that /(0) = 1. Thus a shower of N molecules, eachoriginally moving with the same velocity c, will be reduced innumber to

Ne-*l*c (166)

after travelling a distance I through the gas. Typical numericalvalues are given below, N being taken to be 100:

~= 0 0-01 002 0 1 0-2 0-25 0-33 0-50 1 2 3 4 4-16

Ne-l'*c= 100 99 98 90 82 78 72 61 37 14 5 2 1

The occurrence of the exponential in expression (166) shewsthat free paths which are many times greater than the mean freepath will be extremely rare. For instance, only one molecule in148 will describe a path as great as 5AC, only one in 22,027 apath as great as 10Ac, only one in 2«7 x 1043 a path as great as100Ac, and so on.

THE FREE PATH IN A GAS 143116. Formula (166) can be tested experimentally in various

ways, and each test provides a value of the free path Ag, fromwhich it is then possible to deduce the molecular diameter o\

When a shower of electrons or other swiftly moving particlesis shot through a gas, it is natural to assume a law of diminutionof the form

i = ioe-°x, (167)

where i0 is the initial current, i the current after a distance x,and a is a coefficient of extinction. Innumerable experimentershave found that a law of this type represents the facts. We seethat formula (167) is identical with (166), a being the reciprocalof the free path Ac.

In recent years, a number of experimenters* have testedformulae of this type for beams of electrons projected into gaseswith low velocities which are comparable with those which occurin the kinetic theory. In general the lengths of free path andmolecular diameters deduced from them agree well with thosecalculated in other ways. Here and there, however, difficultquestions arise. The most difficult is perhaps connected withelectrons of low velocity moving through the monatomic gases—helium, argon, neon, krypton and xenon. As a certain electronicspeed is approached, the mean free path undergoes a violentdecrease, as though the atomic diameter was greatly increased.For still lower velocities the free path is abnormally long, almostas though the atoms had become transparent to the electrons.The kinetic theory alone is not able to provide a satisfactoryexplanation of these phenomena; to obtain this we must availourselves of the methods of wave-mechanics. But until we cometo these quite low electronic velocities, experiment shews thatcollisions occur in the way predicted by pure kinetic theory.

Max Bornf has tested formula (167) for the free path of com-plete atoms with satisfactory results. A beam of silver atoms isprojected in the usual way from an electric furnace, so that each

* H. F. Mayer, Ann. d. Phys. 64 (1921), p. 451. C. Ramsauer, Ann. d.Phys. 64 (1921), p. 513. R. B. Brode, Phys. Rev. 25 (1925), p. 636.M. Rusch, Ann. d. Phys. 80 (1926), p. 707. R. Kollath, Phys. Zeitschr.31 (1930), p. 986.

t Phys. Zeitschr. 21 (1920), p. 578.


atom of the beam falls on one or other of four screens of glasscooled to liquid air temperature. These four screens are atdifferent distances x1,x29xz,zA. The number of molecules whichstrike the screens per unit area can be estimated from the densityof silver deposit on the glass, and are found to be approximatelyin the ratio

e-axt . e-ax2 : e-axz

In this way, Born found the free path of silver atoms movingthrough air at atmospheric pressure to be about 1-3 x 10~5 cm.More recently Bielz,* using a refinement of the same method, hasfound the free path of silver atoms moving through nitrogen atatmospheric pressure to be about 1-29 x 10~6 cm.

P. Knauerf has tested the formula for the scattering of abeam of complete molecules, shot both into the same gas as thebeam and also into mercury vapour. The predictions of thekinetic theory are confirmed at high velocities, but at lowvelocities it is found necessary to call in the wave-mechanicsto explain the observations.

117. On differentiating formula (165), we see that the prob-ability that a molecule moving with a velocity c shall describe afree path of length between I and I + dl is

e-^cf. (168)A*

The foregoing results apply only to molecules moving with agiven velocity c. If the velocities are distributed in accordancewith Maxwell's law, then the fraction of the whole number ofmolecules which at any given instant have described a distancegreater than I since their last collision will be

(169)

This function is not easy to calculate in any way. As the resultof a rough calculation by quadrature, I have found that throughthe range of values for I in which its value is appreciable, it does

• Zeitschr.f. Phys. 32 (1925), p. 81.t Zeitschr.f. Phys. 80 (1933), p. 80 and 90 (1934), p. 659.


not ever differ by more than about 1 per cent, from e~1#04^/A, whichis the value for molecules moving with velocity 1/VAra.

At any instant a fraction

of the whole number of molecules is moving with velocity c. Thesemolecules, on the average, are starting to describe distances c/0each before collision. Hence the average distance XT that all themolecules will travel before collision is given by

Jo © V 773

This integral can only be evaluated by quadrature. The evalua-tion has been performed by Tait* and Boltzmann,f who agree inassigning to it the value 0-677, leading to a value for XT whichis some 4 per cent less than the value of the free path calculatedin § 108.

It must be noticed that these two free paths are calculated indifferent ways. That calculated in § 108 is the mean of all thepaths described in unit time, that just calculated is the meanof all the paths being described at a given instant, or to be moreprecise, is the mean of all the distances described from a giveninstant to the next collision (cf. § 134 below). This latter iscommonly known as Tait's free path.

LAW OF D I S T R I B U T I O N OF V E L O C I T I E S I N COLLISION

118. In many physical problems, it is important to considerthe distribution of relative velocities, ratios of velocities, etc. inthe different collisions which occur. We attempt to obtain ex-pressions for various laws of distribution of this type.

In formulae (155) and (156) we obtained expressions for thechance per unit time that a molecule of type 1 moving with

• Edinburgh Trans. 33 (1886), p. 74.t Wien. Sitzungsber. 96 (1887), p. 905, of Gastheorie, 1, p. 73.


velocity c should collide with a molecule of type 2 moving with avelocity between c' and c' + dc'.

The number of molecules of type 1 per unit volume movingwith a velocity between c and c + dc is

Multiplying expressions (155) and (156) by this number weobtain, as the total number of collisions per unit time per unitvolume between molecules having specified velocities withinranges dc, dc\

when c' > c,

^ V > a SI2h*m\m\e-Kmi*+*hft*)c*c\c* + 3c'2) dcdc', (170)

when c' < c,

¥*V2 #12 A M m|c-îc2+w2c'2)cc'2(3c2 + c'2) dcdc> (171)

119. We proceed next to find the number of collisions in whichthe velocities c, c' stand in a given ratio to one another. Letc = KC', and let the variables in expressions (170) and (171) bechanged from c, c' to K, c!. Clearly the differential dcdc' becomesc'dtcdc'', and the two expressions become

when K> 1,

¥ ^ 2 #12 3 mi ™\ e-^2^^2+m2)/c(3A:2 + 1) dKc'6 dc', (172)

when K< 1,f6dcf (173)

On integrating these expressions with respect to c' fromc' = 0 to c' = oo, we obtain the number of collisions for which /c,the ratio of the velocities, lies within a given range dfc. Thenumbers are readily found to be

when/c>l, ^^2^121—jr~^\ 2 ^ ^ » (174)

when/c<l, ^ l ^ ^ l — r ^ l 7 2^—\i^- (175)


The total number of collisions per unit time per unit volume mayof course be derived by integrating this quantity from K = 0 toK = oo. It is found to be

(176)

which agrees, as it ought, with formula (148).120. The law of distribution of K in different collisions can be

obtained by dividing expressions (174) and (175) by the totalnumber of collisions (176). This law of distribution is found to be

w h e n i o l , 5 , ) J u / J , , ' A (177)

\, 5 m\m\when/c<l, - ; —

2(m + m

The law of distribution of values of K when the molecules aresimilar is obtained on taking m1 — m2. We must notice howeverthat if we simply put m1 = m2 in expressions (177) and (178) eachcollision is counted twice, once as having a ratio of velocities Kand once as having a ratio of velocities l/#c. It seems simplest todefine the value of K for a collision in this case as the ratio of thegreater to the smaller velocity, so that K is always greater thanunity, and we then obtain the law of distribution by puttingm1 = m2 in expression (177) and multiplying by two so that eachcollision shall only count once. The law of distribution is found to

^±})dKi

of which the value when integrated from K = 1 to K — oo is unity,as it ought to be.

P E R S I S T E N C E OF VELOCITY AFTER COLLISION

121. The next problem will be to examine the average effectof a collision as regards reversal or deflection of path. We shallfind that in general a collision does not necessarily reverse thevelocity in the original direction of motion, or even reduce it to


rest: there is a marked tendency for the original velocity to persistto some extent after collision. It is obviously of the utmostimportance to form an estimate of the extent to which thispersistence of velocity occurs.

Persistence of Velocity when the Molecules areSimilar Elastic Spheres

122. Let us again represent molecules by elastic spheres, andconsider two molecules of equal mass colliding with velocitiesc, c'. In fig. 29 let OP and OQ represent these velocities, and letR be the middle point of PQ. Then we can resolve the motion ofthe two molecules into

(i) a motion of the centre of mass of the two, the velocity ofthis motion being represented by OR, and

(ii) two equal and opposite velocities relative to the centre ofmass, these being represented by RP and RQ.

Imagine a plane RTS drawn through R parallel to the commontangent to the spheres at the moment of impact, and let P ' , Q'

Q Pf

be the images of P, Q in this plane. Then clearly RP' and RQf

represent the velocities relatively to the centre of gravity afterimpact, so that OP' and OQ' represent the actual velocities inspace.

We have already seen (§104) that all directions are equallylikely for RP', RQ', the velocities after collision, so that the"expectation" of the component of velocity of either moleculeafter impact in any direction is equal to the component of ORin that direction.


123. Let us now average over all possible directions for thevelocity of the second molecule, keeping the magnitude of thisvelocity constant. In fig. 30 let OP, OQ as before represent thevelocities of the two colliding molecules, and let R be the middlepoint of PQ, so that OR represents the velocity of the centre ofgravity of the two molecules. We have to average the componentsof the velocity OR over all positions of Q which lie on a spherehaving 0 for centre. It is at once obvious that the average com-ponent of OR in any direction perpendicular to OP is zero. Wehave, therefore, only to find the component in the direction OP,say ON. We must not suppose all directions for OQ to be equallylikely, for (cf. § 85) the probability of collision with any two

Fig. 30

velocities is proportional to the relative velocity. Thus theprobability of the angle POQ lying between 6 and d + dd is notsimply proportional to sin Odd, but is proportional to PQ sin Odd,for PQ represents the relative velocity. The average value of thecomponent ON is therefore

ON = ^ — . (180)PQa\n6dd

JoLet us now write

so that V2 = c2 + c'2-2cc'cos0. (181)


Then

ON = \{OP + OM) = |(c + c'cos0) = ~(3c2 + c '2-F2) .

(182)

By differentiation of relation (181), we have

VdV = cc' sin Odd,

so that equation (180) becomesc ' 2 - F2) V*dV _ 3c2+ c'2 J F W

V ~ Ac AcJV*dV(183)

the limits of integration being from F = c' ~c to F = c' + c.Performing the integration, we find that

w h e n o c ' , OJy- 1 0 e ( 8 6 i+

+ c > 1 ) , (184)

wheno<C, ^ ^

124. Since these expressions are positive for all values of cand c', we see that whatever the velocities of the two collidingmolecules may be, the "expectation" of the velocity of the firstmolecule after collision is definitely in the same direction as thevelocity before collision. Naturally the same also is true of thesecond molecule.

If we denote ON, the "expectation" of velocity after collisionof the first molecule in the direction of OP, by a, then the ratioctjc may be regarded as a measure of the persistence of thevelocity of the first molecule.

Formulae (184) and (185) give the values of a, and hence of thepersistence ajc. I t is at once seen that the values of oc/c dependonly on the ratio cjc', and not on the values of c and c' separately.If, as before (§119), we denote cjc' by AT, the values of the per-sistence are

oc 1 5K* + 1- _ _ _ _ _ , ( I 8 6 )


125. These expressions are too intricate to convey muchmeaning as they stand. The following table gives numerical valuesof the persistence a/c corresponding to different values of /c, theratio of velocities:

%co 4 2 l i 1 § i i 0c

- = 0-500 0-492 0-473 0-441 0-400 0-368 0-354 0-339 0-333c

- = 0 i | f 1 1£ 2 4 o oc

It now appears that the persistence is a fraction which variesfrom 33 | to 50 per cent, according to the ratio of the originalvelocities. From the values given, it is clear that we are likely toobtain fairly accurate results if we assume, for purposes of roughapproximation, that the persistence is always equal to 40 percent of the original velocity.

126. By averaging over all possible values of the ratio K, we canobtain an exact value for the mean persistence averaged over allcollisions.

Each collision involves two molecules of which the roles areentirely interchangeable. Let us agree to speak of the moleculeof which the initial velocity is the greater as the first molecule,so tha t c/c' or K is always greater than unity.

The persistences of the velocities of the two molecules involvedin any one collision are respectively

15AC4+1 , 3 + 5/c2

and10/C2(3AC2+1) 5 ( 3 K 2 + 1 ) '

the first of these expressions being given directly by formula (186),while the second is immediately obtained by writing l/#c for K inexpression (187).

The mean persistence of the two molecules concerned in thiscollision, being the mean of the two expressions just found, is

25** + 6*2+1( '


A few numerical values of this quantity are found to be:

/ c = l H H 2 3 4 oomean persistence = 0-400 0-401 0-404 0-413 0-415 0-416 0-417

The law of distribution of values of K in the different collisionswhich occur has been found in formula (179) to be

K. (189)

Multiplying together expressions (188) and (189) and inte-grating from K = 1 to K = oo, we find that the mean persistenceof all velocities after collision is

Thus the average value of the persistence is very nearly equalto f, the value when the molecules collide with exactly equalvelocities.

Persistence when Molecules have Different Masses

127. The calculations just given apply only when the moleculesare all similar. Let us now examine what value is to be expectedfor the persistence when the molecules have different masses andsizes, but are still supposed to be elastic spheres.

Consider a collision between two molecules of masses ml9 ra2;let their velocities be a, b respectively as before, and let then-relative velocity be F. Maxwell's result (§ 104) is still true thatall directions are equally likely for the velocities after impactrelative to the centre of gravity, and the expectation of anycomponent of velocity after collision is exactly that of thecommon centre of gravity.

We may accordingly proceed to average exactly as in § 123.But if OR in fig. 30 represents the velocity of the centre of gravity,R will no longer be the middle point of PQ; it will divide PQin such a way that mxRP = m2RQ. We then find, in place of


relations (182),

rnZ^ jn c'cos0) (190)

So long as c and c! are kept constant, we can average this exactlyas before. The first term, being constant, is not affected byaveraging, while the average value of the second term

(c + c cos 0)

is equal to — times the average of the term \(c-\-o' cos 6)

already found in § 123.Hence, if (a/c)e denotes the value of the persistence oc/c when

the two masses are equal, we have, in the general case in whichthe masses are unequal,

rn ^ / * \

This gives the persistence of the velocity c of the molecule ofmass %, the values of (<x/c)e being given by the table on p. 151.The persistence is of course a function of the two quantitieswi1/m2 and c'/c

128. If we assume as a rough approximation that the value of(a/c)e is equal to 0-400 regardless of the ratio of velocities /c, thenequation (191) reduces to the approximate formula

- = m i " " ^ 2 , (192)c m1 + m2

which of course depends only on mx\m2. This formula, however,must not be applied when the ratio mjm^ is either very large orvery small.

When mjm^ is very small, c will be large compared with c' inpractically all collisions, so that K is very great and the appro-priate value to assume for {ajc)e is \, this corresponding to K = oo.From equation (191) we now obtain the approximate formula

(193)m2 /

JKT


In the limit when m1 vanishes in comparison with m2, thepersistence vanishes, as of course it must; at a collision the lightmolecule simply bounces off the heavy molecule; all directionscan be seen to be equally likely by the method of § 104, and there-fore the persistence is nil.

At the opposite extreme, when — is very large, the appropriate772-2

values to assume are K = 0 and I -1 = J. The approximate

formula derived from equation (191) is now

a « V - | m , /m, \\77i2 /

In the limit when m2 vanishes, the persistence becomes equalto unity. This also can be seen directly: the heavy molecule merelyknocks the light molecule out of its way, and passes on with itsvelocity unaltered.

129. In place of these approximate formulae, it is quite feasibleto obtain a formula accurate for all values oim^m^ by averagingthe exact equation (191).

The result is

(195)

where /i2 = m^\mv On substituting this into equation (191), weobtain a formula giving the average persistence of velocity forany ratio of masses.

130. It is readily verified that when fi = 1 the value of thisaverage persistence is 0-406, as already found in § 126.

When fi is very small, formula (195) reduces to

I + A «_A3 15 35

and similarly when /i is large, the expansion is


From formulae (195) and (191), the following values can becalculated:

— = 0 TO I i 1 2 5 10 oom1 i o & ^

- =0-333 0-335 0-339 0-360 0-406 0-432 0-491 0-498 0-500We

- | = 1-000 0-879 0-779 0-573 0-406 0-243 0-152 0-086 0-000

These figures shew that the persistence is always positive butmay have any value whatever, according to the ratio of the massesof the molecules.

131. For laws of force between molecules different from thatbetween elastic spheres, the persistence of velocity will obviouslybe different from what it is for elastic spheres. Clearly, however,everything will depend on our definition of a collision. If we sup-pose that a very slight interaction is sufficient to constitute acollision, then the mean free path will be very short, while thepersistence will be nearly equal to unity. If, on the other hand,we require large forces to come into play before calling a meetingof two molecules a collision, then the free path will be long, butthe persistence will be small, or possibly even negative. In fact,the variations in the persistence of velocities just balance thearbitrariness of the standard we set up in defining a collision. Thisbeing so, it will be understood that the conception of persistenceof velocities is hardly suited for use in cases where a collision isnot a clearly defined event.

Chapter VI

VISCOSITY

132. At a collision between two molecules, energy, momentumand mass are all conserved. Energy, for instance, is neithercreated nor destroyed; a certain amount is transferred from oneof the colliding molecules to the other. Thus the moving moleculesmay be regarded as transporters of energy, which they may handon to other molecules when they collide with them. As the resultof a long chain of collisions, energy may be transported from aregion where the molecules have much energy to one where theyhave but little energy: studying such a chain of collisions we havein effect been studying the conduction of heat in a gas. If weexamine the transport of momentum we shall find that we havebeen studying the viscosity of a gas—the subject of the presentchapter. For viscosity represents a tendency for two con-tiguous layers of fluid to assume the same velocity, and this iseffected by a transport of momentum from one layer to theother. Finally if we examine the transfer of the molecules them-selves we study diffusion.

For the moment; we must study the transport of momentum.We think of the traversing of a free path of length A as the trans-port of a certain amount of momentum through a distance A.If the gas were in a steady state, every such transport would beexactly balanced by an equal and opposite transport in the re-verse direction, so that the net transport would always be nil.But if the gas is not in a steady state, there will be an unbalancedresidue, and this results in the phenomenon we wish to study.

Oeneral Equations of Viscosity

133. Consider a gas which is streaming in a direction parallelto the axis of x at every point as in fig. 31, the mass-velocity uQ

being different for different values of z, so that the streaming is inlayers parallel to the plane of xy.

VISCOSITY 157

°\= 210

Let us denote mu, the a-momentum of any molecule by fi9 andlet Jl denote the mean value of /i at any point of the gas. Then Jlis equal to mu0, and is a function of z only.

For definiteness let us suppose that z and Jl both increase as wemove upwards in fig. 31. Molecules will * z

be crossing any plane z = z0 in bothdirections, and transporting a certainamount of p, across this plane as theydo so. The amount of /i which any t^~~ \P Rparticular molecule transports acrossthis plane will of course depend onthe whole history of the moleculebefore meeting the plane. We shallhowever begin by supposing that itdepends only on the history of the jrig. 31molecule since its last collision; weshall assume that the average molecule carries the amount of /iappropriate to the point at which this last collision occurred.

134. Consider, then, a molecule which meets the plane z = zQ

at P, having previously come from a collision at Q. Let thevelocity components of the molecule be u9 v9 w9 and let the velocitybe regarded as consisting of two parts:

(i) a velocity u0, of components u0, 0, 0, equal to the mass-velocity of the gas at P ;

(ii) a velocity c, of components u — u0, v, w9 the molecular-velocity of the molecule relatively to the gas at P .

Let QP in fig. 31 represent the path of the molecule since itslast collision, and let RP represent the distance travelled by thegas at P in the same interval of time, owing to its mass-velocityuQ, 0, 0. Then QR will represent the path described by the mole-cule relative to the mass-motion of the surrounding gas. Let thelength QR be denoted by A,., and let this make an angle 6 with theaxis of z.

Since all directions of this molecular-velocity may be regardedas equally probable, the probability of 6 lying between 6 andd + dd is proportional to ainddd. The number of moleculesper unit volume having relative molecular-velocities for which

158 VISCOSITY

c and 6 lie within specified small ranges dc, dd may therefore betaken to be

$vf(c)sinddddc, (196)

where f(c) dc = 1, in order that the total number may be equalJo

to v.The number of molecules having a velocity satisfying these

conditions, which cross a unit area of the plane z = z0 in time dt,is equal to the number which at any instant occupy a cylinder ofbase unity in the plane z = z0 and of height c cos 6 dt; it is therefore

\vcf(c) cos 6 sin ddddcdt. (197)

We shall suppose that on the average these molecules carrythe amount of ^-momentum /i appropriate to the point Q, andnot that appropriate to the point P.

The z-co-ordinate of Q isZQ — AT COS d,

so that the average value of /i at Q is less than that at P by anamount

Hence the group of molecules under discussion carry, on theaverage, an amount of /i

(198)

in excess of that appropriate to the point P, where Ac is themean free path for a molecule moving with velocity c.

The total excess of fi which these molecules carry across theplane Z = ZQ is equal to the product of expressions (197) and(198), namely

— | Ac cos dvcf (c) cos dsinddddt.

On integrating the expression just found with respect to 6, weobtain the total transfer of momentum by all molecules withvelocities between c and c + dc, whatever their direction. Thelimits for 0 are 0 to TT, values of 6 from 0 to \TT covering moleculeswhich cross the plane from below, and values of 6 from \n to n

VISCOSITY 159

those which cross the plane from above. The result of this inte-gration is

a negative sign indicating that the transfer is from above tobelow. On further integrating from c = 0 to c = oo, we obtain forthe amount of momentum carried across unit area of the plane intime dt by all molecules,

cAcf(c)dcdt = 3 ( ~" I cAc, (199)

where cAc denotes the mean value of cAc averaged over all themolecules of the gas.

In the foregoing argument it might perhaps be thought thatAr should have been replaced by |AC instead of by Ac. For if QO(fig. 31) is the whole free path described before collision occurs,there is no reason why PO should be less than PQ, so that theprobable value of PQ might be thought to be Â.

The fallacy in this reasoning becomes obvious on consideringthat after a molecule has left P, its chances of collision are exactlythe same whether it has just undergone collision at P or hascome undisturbed from Q. Hence PO = Ac, and therefore, by asimilar argument, PQ = Ac.

A simple example taken from Boltzmann's Vorlesungen* willperhaps elucidate the point further. In a series of throws with asix-faced die the average interval between two throws of unity isof course five throws. But starting from any instant the averagenumber of throws until a unit throw next occurs will be five, andsimilarly, working back from any instant, the average numberof throws since a unit throw occurred is also five.

135. We may putc\c = cl, (200)

where I is a new quantity, which is of course the mean free pathof a molecule, this mean being taken in a certain way. This is notthe same as any of the ways in which it was taken in the lastchapter, so that we do not obtain an accurate result by replacing

* Vol. l , p . 72.

160 VISCOSITY

I by any of the known values of the mean free path. At the sametime the mean values calculated in different ways will not greatlydiffer from one another, and as our present calculation is at bestone of approximation, we may be content for the moment tosuppose I to be identical with the mean free path, however calcu-lated. The extent of the error involved in this procedure will beexamined later.

136. We have shewn that the aggregate transfer of momentumper unit of time across a unit area of a plane parallel to the planeof xyia

d^. (201)

If we replace fi by its value mu, Jl becomes mu0, and the transferof momentum is seen to be

This transfer of momentum results of course in a viscous drag ofequal amount across the plane z = z0. Now a viscous fluid,moving with the same velocity as the gas at every point, wouldexert a viscous drag

' dx

per unit area across the plane z = z0, where TJ is the coefficient ofviscosity of the fluid. Thus the gas will behave exactly like aviscous fluid of viscosity ij given by

V = ipcl. (203)

137. We can now obtain some insight into the molecularmechanics of gaseous viscosity. Let us imagine two molecules,with velocities u, v9 w and — u, v, w, penetrating from a layer atwhich the mass-velocity is 0, 0, 0 to one at which it is uQy 0, 0. Bythe time the molecules have reached this second layer, we mustsuppose that their velocities are divided into two parts, namely,

u — uQyv,w and uOi 0,0

for the first, and — u — u0, v, w and uQ, 0,0

for the second. The first part in each case will represent molecular-

VISCOSITY 161

motion, and the second part will represent mass-motion. Nowin § 92, it was found that the total energy of the gas could beregarded as the sum of the energies of the molecular and mass-motions; indeed, the sum of the energies of the molecular-motionsof the two molecules now under discussion is easily seen to be

m(u2 + v2 + w2) + mu%.

The first term is equal to the energy of the molecular-motionof the two molecules at the start; the second term represents anincrease which must be regarded as gained at the expense of themass-motion of the gas. Thus we see that the phenomenon ofviscosity in gases consists essentially in the degradation of theenergy of mass-motion into energy of molecular-motion; thisexplains why it is accompanied by a rise of temperature in thegas.

Corrections when Molecules are assumed to be Elastic Spheres

138. From want of definite knowledge of molecular structure,two errors have been introduced into our calculations. In thefirst place we have neglected the persistence of velocities aftercollision, and in the second place we have ignored the differencebetween two different ways of estimating the mean free path.If the molecules are elastic spheres, it is possible to estimate theamount of error introduced by both these simplifications.

We may begin by an exact calculation of cAc, to replace theassumption of equation (200). The quantity required is clearly

^ = f°7(c)Accdc. (204)Jo

On substituting the value for Ac given by equation (164), andputting

we find- y roo4(^m)*c5c-7imcafflO4(Am)*c5e-*1ll^J 4 [

= — ,=rdC = , T J 7Jo 7Tva2iJf(c\hm) TTshmv(T2j o ¥\x)

-= =V20

where x = c \lhm = -= = .V20

162 VISCOSITY

Thus if I is defined by equation (200), we must take

(205)

The integral can only be evaluated by quadratures. Tait* andBoltzmannf agree in assigning to I a value equal to 1-051 timesMaxwell's mean free path calculated in § 108.

139. A more serious error has been introduced by neglectingthe persistence of velocities which was investigated in the lastchapter. When a molecule arrives at P after describing a path ofwhich the projection on the axis of z is £, with a velocity of whichthe component parallel to the axis of z is w, then, on tracing backthe motion, we know that as regards the previous path of eachmolecule the expectation of average velocity parallel to the axisof z is 6w, where 6 measures the persistence. The expectation ofthe projection of this path on the axis of z may therefore be takento be #£. Similarly, the expectation of the projection of each of thepaths previous to these may be taken to be 6%, and so on. Thusthe molecule must be supposed to have come, not from a distance£ measured along the axis of z, but from a distance

Z+OG+0*G+... = JL. (206)

We must not, however, assume that such a molecule on arrivingat the plane z = z0 has, on the average, a value of fi appropriate

rto the plane z = zo + -—^. For the molecule has not travelled a

rdistance _ n undisturbed, and at each collision a certain amount

1 — uof its excess of momentum will have been shared with the col-liding molecule. Of the various simple assumptions possible, themost obvious one to make is that at each collision the excess ofmomentum above that appropriate to the point at which thecollision takes place is halved, half going to the colliding moleculeand half remaining with the original molecule. Making this

• Collected Works, 2, pp. 152 and 178.t Wien. Sitzungsber. 84 (1881), p. 45.

VISCOSITY 163

assumption, it is clear that the excess of momentum to beexpected is not that due to having travelled undisturbed adistance equal to that given by expression (206), but a distance

Thus we must insert a factor 1/(1 — 6) in the integrand ofequation (205) before integration, the value of 6 being obtainedfrom the table on p. 151. As the result of a rough integrationby quadratures, I find

so that the viscosity coefficient is given by

= 0-461 ^ 1 ^ . (209)V 2 2

140. This formula, although undoubtedly better than formula(203), is still only an approximation. It might be possible to im-prove still further on the rough assumptions just made, and soobtain results still closer to the truth. This, however, seems un-necessary, since exact numerical results are obtainable by themathematical methods explained in Chap. IX below. There weshall see how Chapman, following Maxwell's method, has arrivedat the exact formula

^ = 0-499-7^-, (210)*J2ncr2

and Enskog has confirmed this, by an entirely different method.

VARIATION OF VISCOSITY WITH DENSITY

141. Equation (210) shews that rj is independent of the densityof the gas, when the molecules are elastic spheres. And it is clearthat whatever structure we assume for the molecules of the gas,I will, to a first approximation, vary inversely as the number ofmolecules per unit volume of the gas, so that formula (203)must give a value of rj which is independent of v. Increasingthe number of molecules in a given volume increases the

164 VISCOSITY

number of carriers, but decreases their efficiency as carrierspari passu; for on doubling the number of molecules per unitvolume, the free path is reduced by half so that the momentumof any carrier will only differ, on the average, by half as muchfrom the average momentum at the point of its next collision.

Thus we obtain Maxwell's law:

the coefficient of viscosity of a gas is independent of its density.

In spite of its apparent improbability, this law was predictedby Maxwell on purely theoretical grounds, and its subsequentexperimental confirmation has constituted one of the moststriking triumphs of the kinetic theory.

Some of the physical consequences of this law are interesting,and occasionally surprising. For instance, the well-known law ofStokes tells us that the final steady velocity v of a sphere fallingthrough a viscous fluid is given by

where a, M are the radius and mass of the sphere, and 3t0 the massof fluid displaced. Since TJ is, by Maxwell's law, independent ofthe density, it follows that, within the limits in which StokesJs lawis true, the final velocity of a sphere falling through air or anyother gas will be independent of the density of the gas, or morestrictly will depend on the density of the gas only through theterm M — Mo, which will differ only inappreciably from M. Thus,a small sphere will fall as rapidly through a dense gas as througha rare gas. Again the air-resistances experienced by a pendulumought to be independent of the density of the air, so that theoscillations of a pendulum ought to die away as rapidly in a raregas as in a dense gas, as was in fact found to be the case by Boyleas far back as 1660.*

To test the truth of the general law, Maxwellf fixed threeparallel and coaxal circular discs on a common axis, and thensuspended them by a torsion thread in such a way that the threemovable discs could oscillate between four parallel fixed discs.

* Thomson and Poynting, Properties of Matter, p. 218.f Phil. Trans. 156 (1866), p. 249, or Coll. Works, 2, p. 1.

VISCOSITY 165

As in Boyle's pendulum experiment, the oscillations were foundto die away at the same rate whether the air were dense or rare,up to pressures of one atmosphere.

142. At very much higher pressures at which the free path isbecoming comparable only with molecular diameters, Maxwell'slaw fails altogether as is only to be expected. For we havesupposed that a free path transports momentum through adistance A with a velocity c, but we have so far overlooked thatthe collision which ends the free path transports it through afurther distance v with infinite velocity. Thus the transport forfree path is through a distance of the form A + cr cos 9.

Enskog* has shewn that, because of this, the value of Tjjp willnot be constant, but will vary as

^ + 0-8000 + 0-7614-,

where b and v mean the same as in Van der Waals' equation.This factor attains a minimum value of 2-545 when v = 0-87266,so that the general value of y/p can be put in the form

£ +0-8000+ 0b

This formula is found to represent observed variations ofrjjp fairly well, especially at high densities. The table on p. 166gives the values of rj for nitrogen at 50° C, as observed byMichels and Gibson, f together with the values calculated fromEnskog's equation just given.

The minimum in rjjp is clearly marked. The value v = 0*87266,at which theory predicts that it ought to occur, is reached atabout 580 atmospheres.

Similar variations in rjjp for carbon-dioxide have been ob-served by Warburg and v. Babo,J the minimum value being

* K. Svenska Veten. Handb. 63 (1922), No. 4; see also Chapman andCowling, The Mathematical Theory of Non-uniform Gases (1939), p. 288.

t Proc. Roy. Soc. 134 A (1931), p. 307.% Wied. Ann. 17 (1882), p. 390, and Berlin. Sitzungsber. (1882), p. 509.

The numbers here given were subsequently corrected by Brillouin (Lecons8ur la viscosite des fluides, 1907), and our text has reference to thesecorrected figures.

166 VISCOSITY

reached at a pressure of 77-2 atmospheres, at which the densityis 0*450 gm. per cu. cm. Enskog has verified that these variationsalso are in accordance with his theoretical formula.

Viscosity of Nitrogen (50° C.) at high Pressures

Pressure(atmospheres)

15-3757-60

104-5212-4320-4430-2541-7630-47421854-1965-8

v/p(observed)

0-011790-0032740-0019280-0011480-0009520-0008870-0008660-0008590-0008700-0008890-000909

TJX 10 6

(observed)

191-3198-1208-8237-3273-7312-9350-9378-6416-3455-0491-3

7}X 108

(calculated)

181190205224266308348380418455492

A similar, although smaller, dependence of viscosity on densityhas also been observed in hydrogen at moderate pressures byKamerlingh Onnes, Dorsman and Weber.*

At the other limit of excessively small pressure, remarkabledepartures from Maxwell's law may occur through the free pathbecoming comparable with, or even greater than, the dimensionsof the vessel in which the experiment is conducted. If the mole-cule has not room to describe a free path equal to the theoreticalfree path assumed in § 140, the resulting formula obtained for theviscosity must obviously fail. If I cannot, from the arrangementof the apparatus, be greater than some value Zo, then rj (cf.equation (209)) cannot be greater than %pcl0, and so ought tovanish with p. This is found to be the case. As far back as 1881,Sir W. Crookesf measured the viscosities of gases at pressures ofonly a few thousandths of a millimetre of mercury, and obtainedvalues much smaller than those at higher pressures, which tendedto vanish altogether as the density of the gas vanished. Laterinvestigators have abundantly confirmed his conclusions.

* Comm. Phys. Lab. Leiden, 134a (1913). See also Winkelmann'sHandbuch der Physilc (lite aufl.), pp. 1399, 1406.

t Phil. Trans. 172 (1881), p. 387.

VISCOSITY 167

A case of special interest, which we shall now discuss, occurswhen gas flows through a tube of small or capillary cross-section.

The Flow of Gas through Tubes

143. When gas flows through a tube, the layer of gas incontact with the wall of the tube is usually held at rest by thetube, while the rate of flow increases as we pass inwards towardsthe centre of the tube.

Let us suppose the tube to be of circular cross-section,of radius R, and length L. Let the gas at any distance r fromthe axis be flowing along the tube with a velocity v.

The cylinder of gas of radius r coaxal with the tube willtit)

experience a viscous drag —y-r- per unit area over its whole

surface, and as this is of area 2nrL, the whole cylinder experi-ences a viscous drag of amount

o T d v

— znrLw — .1 dr

If the flow is steady, so that the gas moves without accelera-tion, this viscous drag must be exactly balanced by the differ-ence of the pressures at the two ends of the cylinder. This willbe nr2(p1—p2), where plf p2 are the pressures per unit area atthe ends of the tube, so that px—p2 is the pressure-differenceemployed to drive the gas through the tube. Thus

- 2nrL?i •£ = nr2 (px -p2).

Dividing throughout by — 27rrLi] and integrating, we obtain

where A is a constant of integration.If there is no slip between the gas and the walls of the tube,

v must be zero when r = R. Determining A from this condition,equation (211) becomes

# # ( 2 J 2 _ r a ) # (212)

168 VISCOSITY

The total volume V of gas which flows through the tube inunit time is

= {R

J 0

(213)

This is Poiseuille's formula for the flow of gas through a tube.Since the quantities F, L, R and px—p2 all admit of easy measure-ment, it provides a convenient method for measuring 7] ex-perimentally.

Experiment shews, however, that the flow of gas in tubes ofvery small diameter is often greater than that given by thisformula, as though the gas slipped at its contact with thewalls of the tube. In 1860 Helmholtz and Piotrowski* shewedthat a slip of this kind actually occurred in liquids, and laterKundt and Warburgf shewed the same for gases. A greatnumber of investigators, starting with Maxwell,J have dis-cussed the theory of this slip.

If such a slip occurs, let v0 be the velocity of the layer of gaswhich is in contact with the walls of the tube. This slip willproduce a viscous force on the gas which will be jointly pro-portional to v0 and to the area of surface, namely 2nRL, overwhich slip occurs. We may take the whole viscous drag on thegas to be 2nRLevOy where e is a constant. Equating this to theforce urging the whole gas through the tube,

so that VO-

The constant A in equation (211) must now be adjusted soas to give this value for v when r = R. We accordingly find, asthe general value of v,

and, instead of equation (213), for the flow per unit time,

(214)

• Wien. Sitzungsber. 40 (1860), p. 607.t Pogg. Annalen, 155 (1875), p. 337. J Coll. Works, 2, p. 7G3.

VISCOSITY 169

We see that the slip at the walls increases the flow by afraction fy/eR. This is generally known as the Kundt andWarburg correction. I t is unimportant so long as R is large incomparison with rjje, but when R is small compared with rj/e, ittakes control of the whole process, and results in the flow beingproportionate to R3 instead of to i?4.

144. The quantity rj/e, which measures the ratio of thefriction of the gas on itself to the friction between the gas andthe solid, is generally called the "coefficient of slip". Maxwell*gave much thought to the problem of its evaluation. He assumedin the first instance that of the molecules which impinged on asolid wall only a fraction 1 —/ are reflected back "specularly",i.e. at the same angle as that at which they struck the wall, theremaining fraction/leaving the wall at random angles which areindependent of the earlier motions of the molecule.

From a discussion of the experiments of Kundt and Warburg,Maxwell concluded that / must be about \. The experiments ofKnudsen described in § 34 seem, on the contrary, to indicatethat, under some circumstances at least, the correct value of/is not far from unity; most or all of the molecules start out inpurely random directions after impact.

Blankensteinf has directly measured the coefficients of slipbetween a number of gases and polished oxydized silver, andfinds for / the values 1-00, 1-00, 0-99 and 0-98 for helium,hydrogen, oxygen and air respectively. For reflection fromother solids the value of / may be substantially smaller. ThusMillikanJ gives the valuesAir or CO2 on machined brass, old shellac, or mercury/= 1-00Air on oil 0-895CO2 on oil 0-92Air on glass 0*89Air on fresh shellac O79

145. The Kundt and Warburg formula (214) is found toagree well with experiment so long as the pressure of the gas issufficiently high, but fails entirely, as might be anticipated,

* Coll. Works, 2, p. 703. See also Loeb, Kinetic Theory of Gases(2nd ed., 1934), pp. 285ff.

t Phys. Rev. 22 (1923), p. 582. % Phys. Rev. 21 (1923), p. 217.

170 VISCOSITY

for pressures so low that the free path is comparable with, orgreater than, the diameter of the tube. In this case it is re-markable that the rate of flow is independent of both theviscosity and the density of the gas. The appropriate analysishas been given by Knudsen.*

We may suppose that the velocity with which the gas flowsthrough the tube is small in comparison with the molecularvelocities, and that these latter conform to Maxwell's law for agas with a mass-velocity equal to the velocity of flow. Thetotal mass of the molecules which impinge on unit area of thewall of the tube in unit time is thus Jpc, as in § 39. The averagevelocity along the tube is now the same at all points of the gas,say u0. Thus the total momentum that the moving moleculesgive up to the walls of the tube in unit time is 2nRL x lpcu0.This, as before, must be equal to nR2(p1-pz)9 so that

The total mass of gas which flows through the tube in unittime is accordingly*)"

m & p * ( 2 1 5 )which, as already remarked, is independent of both TJ and p.Knudsen has verified that this formula agrees well with ob-servation. Comparing it with Poiseuille's formula, we noticethat the flow increases as the cube instead of as the fourthpower of R.

V A R I A T I O N OF V I S C O S I T Y WITH T E M P E R A T U R E

146. We turn next to examine how the viscosity of a gasvaries with its temperature.

Since c is proportional to the square root of the absolutetemperature, formula (210) shews that if the molecules were true

* Ann. d. Phya. 28 (1909), pp. 75, 999; 32 (1910), p. 809; 34 (1911),p. 593, and later papers. See also M. Knudsen, The Kinetic Theory ofGases (Methuen, 1934), pp. 21 ff.

t Knudsen gives a formula which is less than this (erroneously as itQ

seems to me) by a factor «-, but the experimental tests agree with thisoTT

formula better than with formula (215).

VISCOSITY 171

elastic spheres, the value of?/ would be proportional to the squareroot of the temperature.

As a matter of fact, 7} is found to vary a good deal more rapidlythan this as the temperature increases. The divergence betweenexperiment and the theoretical value obtained on the assumptionthat the molecules are elastic spheres is, however, one that couldhave been predicted. The assumption in question is, at best, onlyan approximation, and we must continually examine what devia-tions are to be expected from the results to which it leads.

The peculiarity of a system of elastic spheres is that the motionremains geometrically the same if the velocity of every sphere isincreased in the same ratio. If the molecules are surrounded byrepulsive fields of force, this is no longer the case; increasing thevelocities increases the degree to which the molecules penetrateinto each others fields of force at collision, and so has the resultof decreasing the effective sizes of the molecules.

Thus if the molecules of a gas are surrounded by fields of force,and we attempt to represent these molecules by elastic spheres,we must suppose the size of these spheres to vary with the tem-perature of the gas. The spheres are large at low temperatures,small at high.

It follows that in formula (210), r/ must be supposed to dependon the temperature both through the factor c in the numerator,and also through the factor a*2 in the denominator. Thus the valueof 7) will not vary simply as the square root of the temperature,but will vary more rapidly with the temperature than this.

From the way in which TJ is observed to vary with the tempera-ture, we can obtain some information as to the fields of forcesurrounding the molecules. For from equation (210) we cancalculate <r as a function of the temperature T. Now the value ofa at any specified temperature is, roughly speaking, the averagedistance of closest approach of the centres of two molecules incollision, so that the mutual potential energy of two molecules ata distance a is, on the average, equal to the kinetic energy of thevelocities along the line of centres before collision.

We readily find (formula (140)) that the average value of F2,the square of the relative velocity before collision, is § C2. Thus thesquare of the velocity of each molecule relatively to the centre of

172 VISCOSITY

gravity of the two colliding molecules will be, on the average,§ C2. The probability that this velocity is at an angle between 6and 6 + dd with the line of centres is 2 sin 6 cos 6 dd, so that theaverage square of the relative velocity along the line of centresFcosflis

28indcos*0dd--

The kinetic energy which has been destroyed by the inter-molecular field of force when the molecules are, on the average,at their point of closest approach at distance cr apart is therefore\mQ2 or RT. Thus the mutual potential energy of two moleculesat a distance cr apart will be RT, where T is the temperaturecorresponding to the value of cr in question. The force of repulsion

dTbetween two molecules at a distance cr is accordingly — R -=—.

If the law of force is jiv8, we must have

giving on integration

*. (216)

Here cr denotes the distance of closest approach of two mole-cules at an encounter, and when the orbits are at all curved, thisis not quite the same thing as the diameter of the sphereobtained by supposing the molecules to be elastic spheres.Thus equation (216) will give a value of cr which will differ bya numerical multiplier from the value obtained in § 54 by con-sidering the deviations from Boyle's law. This multiplier willof course vary for different values of s. It will reduce to unityfor elastic spheres, and will differ most from this for the smallestvalues of s.

In § 54 we found that molecules with a law of force fir~8 couldbe regarded as elastic spheres for the purpose of calculating thepressure, if <r were supposed given by

* - \iFnE-rA9-\ r i-r-r <2 1 7)

VISCOSITY 173

This agrees with (216) except for the numerical factor, and agreescompletely, as it ought, for elastic spheres (s = oo). We see thatmolecules which are point centres of force may be treated aselastic spheres, both as regards pressure and viscosity, but thespheres must be of different sizes in the two cases, except of coursewhen the molecules really are spheres.

Although formulae (216) and (217) become identical when 8 isinfinite, the divergence between them may be very considerablewhen s is small. The lowest value for s which can be supposedto occur for any gas is probably about s = 5 (cf. § 147, below),and when s = 5,

Thus for such a gas as carbon-dioxide, for which 5 = 5-6, we mayexpect a difference of as much as 50 per cent between the valuesof cr calculated from viscosity and Boyle's law.

In such a case as this, however, the calculation from Boyle'slaw fails because 6, which from equation (212) ought to vary asI7-*, is supposed, in evaluating b experimentally, to remainindependent of the temperature.

Whatever the value of the numerical multiplier may be, it1 -?-

appears that —^ will vary as T8*1, so that rj will vary as Tn, wheren = = * + d h # (218)

147. For some gases, rj is found as a matter of experiment tovary approximately as a power of T, being represented withvery tolerable accuracy by the formula

where 7]0 is of course the coefficient of viscosity at 0°C. Agood instance is helium; the agreement between the observedvalues of rj for this gas and the values given by formula (215)is exhibited in the table on p. 179 below.

Clearly the molecules of substances for which there is a goodagreement of this kind may be regarded as point centres of

174 VISCOSITY

force, repelling according to the law /j/r5, where s is given byequation (218).

If a gas is constituted of molecules which are not of this type,it will still be possible to represent the viscosity-coefficient bya formula of the type of (219), through any small range oftemperature we please. For the value of n is at our disposal,and may be chosen so as to give the right value to drj/dT, theslope of the viscosity-coefficient, through this small range. But thisvalue of n will not usually satisfy the experimental data throughanother range; for this some other value of n must be chosen.

The following table gives the values of n for a number ofgases through ranges which include 0° C, together with the

Values of n and sfor Certain Oases

Gas

HydrogenDeuteriumHeliumfNeonNitrogenCarbon-monoxideAirOxygenHydrochloric AcidArgonNitrous oxideCarbon-dioxideChlorine

Authority*

1234566547668

Value of n(observed)

0-6950-6990-6470-6570-7560-7580-7680-8141030-8230-890-93510

Value of s(calculated)

11-311-014-613-78-88-758-467-404-977-196155-65-0

• Authorities:1. Kamerlingh Onnes, Dorsman and Weber, Verslag. Kon. Akad. van

Wetenschappen, Amsterdam, 21 (1913), p. 1375.2. Van Cleave and Maas, Canadian Journ. of Research, 13B (1935),

p. 384.3. Kamerlingh Onnes and Weber, Verslag. Kon. Akad. van Weten-

schappen, Amsterdam, 21 (1913), p. 1385.4. Trautz and Binkele, Ann. d. Phys. 5 (1930), p. 561.5. Markowski, Ann. d. Phys. 14 (1904), p. 742.6. Values given by Chapman and Cowling (The Mathematical Theory

of Non-uniform Gases, Cambridge, 1939), calculated from dataof various experimenters.

7. Schultze, Ann. d. Phys. 5 (1901), p. 163 and 6 (1901), p. 301.8. Trautz and Winterkorn, Ann. d. Phys. 10 (1931), p. 522.

t See § 148, below.

VISCOSITY 175

values of s calculated from relation (218). An instance of thecloseness of agreement between formula (219) and observationwill be found below (§ 149).

If the molecules of a gas were in actual fact elastic spheres,the value of n would of course be zero, and s would be infinite.Molecules for which s is large are frequently described as " hard ",and those with smaller values of s as "soft". Our table shewsthat, generally speaking, the hardest molecules are those ofsimplest structure. The monatomic molecules helium and neoncome first, then diatomic molecules of simple structure—hydrogen and deuterium, followed by nitrogen, carbon-monoxideand oxygen. The softest molecules are those of chlorine andhydrochloric acid for which 8 is nearly five. This value seems toconstitute a sort of natural lower limit for s.

148. We next examine the absolute value of rj. We havealready seen that TJ is proportional to mc/cr2, or to ^JmRT/cr2.Using the value of cr given by equation (216), we find that thecoefficient of visoosity must be given by an equation of the form

V = A^l^Rf^RT(8fi

1)Yi, (220)

where A is a numerical constant.Chapman* has determined the value of this constant by very

complicated analysis. To a first approximation he found itsvalue to be ,

A *f , (221)( )

where /2(s) is a pure number depending only on s, its actualvalue being

I2(s) = 7TT sin2 d'd doc,Jo

where 6' and a have the meanings assigned to them in § 196below. This integral was introduced by Maxwell f in 1866. Hecalculated that when 5 = 5 its value is 72(5) = 1-3682, andshewed that when 5 = 5 formula (221) is exact.

* Phil. Trans. Roy. Soc. 211 A (1912), p. 433.t Coll. Works, 2, p. 42.

176 VISCOSITY

In a later paper*, Chapman carries the calculations to asecond approximation, and finds that the value of A given byequation (221) must be multiplied by a factor which increases con-tinuously from unity when s = 5 to 1-01485 when s = oo. Thus theerror in using approximation (221) for A is never more than about1£ per cent, and as this is smaller than experimental errors ofobservation, it is hardly worth carrying the approximationfurther. Before leaving this question, it may be remarked thatEnskogf has given the factor by which A must be multipliedwhen the molecules repel as the inverse 5th power of the distancein the form

+ 2 ( 5 -

This of course reduces to unity when $ = 5 and has the valuel2^ or 1-01485 when 8 = oo, thus agreeing with Chapman's valuejust quoted.

For hard spherical molecules (a = oo) these formulae lead tothe value of 7/ already given in formula (210).

Sutherland's molecular model

149. The foregoing theory has been concerned only withmolecules which attract or repel according to the law fir~*. I tcannot be supposed that any actual molecules are of so simplea type as this.

For, as we have already seen, the supposition that the molecularforce falls off as an inverse power of the distance leads to formula(216) which requires <x to vanish absolutely at very high tempera-tures. It seems more natural, and more in accordance withmodern knowledge of molecular structure, to suppose that amolecule possesses a hard kernel which is not penetrated by othermolecules no matter how violent the collision between themmay be.

• Phil. Trans. Roy. Soc. 216 A (1915), p. 279.t Kinetische Theorie der Vorgdnge in mdssig verdilnnten Oasen (Inaug.

Dissertation, Upsala, 1917). Enskog's analysis is given by Chapman andCowling, in a somewhat different form, in their book The MathematicalTheory of Non-uniform Gases (C.U.P., 1939).

VISCOSITY 177

As far back as 1893, Sutherland* had assumed that the effectivevalue of <r at temperature T is

where C, a^ are constants, cr^ being the value of cr when T = oo,and therefore being the diameter of the hard kernel of the mole-cule, while C is the temperature at which cr2 = 2<r§.

As the temperature varies, the coefficient of viscosity varies

as cl, and so as — , or again as ~—^. Thus if T/0 is the coefficient

of viscosity at 0°C. (T = 273*2), the general coefficient of vis-cosity 7j at temperature T will be given by

T \» (7 +273-2

which is Sutherland's formula for the viscosity at temperature T.For many gases this formula meets with very considerable

success in predicting the variation of viscosity with temperature.As an illustration may be given the following tables, taken froma paper by Breitenbach,f in which the observed and calculatedvalues of the viscosity are compared.

(*?0 =

Temperature

-21-2° C.15-099-3

182-43020

Temperature

-20-7°C.15-099-1

182-4302-0

Ethylene0-00009613, G = 225-9)

7} (observed)

0-00008911006127815301826

rj (calculated)

0-00008901012127815191833

Carbon-dioxide: 0-00013879, C = 239-7)

rj (observed)

0-00012941457186122212682

rj (calculated)

0-00012841462185722162686

• Phil. Mag. 36 (1893), p. 507. f d. Physik, 6 (1901), p. 168.

178 VISCOSITY

The following values for C have been found by differentobservers:Helium C = 80-3 (Schultze), 78-2 (Schmitt), 70 (Rankine).Neon C = 56 (Rankine).Argon C = 169-9 (Schultze), 174-6 (Schmitt), 142 (Rankine).Krypton C = 188 (Rankine).Xenon C = 252 (Rankine).Hydrogen C= 72-2 (Rayleigh), 71-7 (Breitenbach), 79 (Suther-

land), 83 (Schmitt).Nitrogen C = 102-7 (Trautz and Baumann), 118 (Smith).Carbon-monoxide C= 100 (Sutherland), 118 (Smith).Air C= 111-3 (Rayleigh), 119-4 (Breitenbach), 113

(Sutherland).Nitric oxide C = 128 (Trautz and Gabriel).Oxygen C = 138 (Markowski), 138 (Schmitt).Chlorine C = 325 (Rankine).Nitrous oxide C = 260 (Sutherland), 274 (Smith).Carbon-dioxide (7 = 239-7 (Breitenbach), 277 (Sutherland), 274

(Smith).Ethylene C = 225-9 (Breitenbach), 272 (Sutherland).Methyl chloride C = 454 (Breitenbach).

Authorities:Schultze, Ann. d. Phys. 5 (1901), p. 165, and 6 (1901), p. 310.Schmitt, Ann. d. Phys. 30 (1909), p. 398.Rankine, Proc. Roy. Soc. 84 A (1910), p. 188 and 86 A (1912), p. 162.Rayleigh, Proc. Roy. Soc. 66 A (1899), p. 68 and 67 A (1900), p. 137.Breitenbach, Ann. d. Phys. 5 (1901), p. 168.Sutherland, Phil Mag. 36 (1893), p. 507.Smith, Proc. Phys. Soc. 34 (1922), p. 155.Trautz and Baumann, Ann. d. Phys. 2 (1929), p. 733.Trautz and Gabriel, Ann. d. Phys. 11 (1931), p. 607.Markowski, Ann. d. Phys. 14 (1904), p. 742.

We should obviously expect the more permanent gases,which have low critical temperatures and low boiling points,to have low values for the Sutherland temperature C. Rankine*has noticed that most gases for which data are available havecritical temperatures equal to about 1*14 times G.

On the other hand, Kamerlingh Onnesf finds very definitelythat the viscosity of helium at low temperatures cannot be re-presented by Sutherland's formula with anything like the

* Proc. Roy. Soc. 86 A (1912), p. 166.t Kamerlingh Onnes and Sophus Weber, Comm. Phys. Lab. Leiden,

1346, p. 18.

VISCOSITY 179

accuracy given by the simpler formula (219). This is shewn in thefollowing table: the second column gives the values of y observedfor helium, the third column gives the values calculated fromformula (219) on taking T/0 = 0-0001887, n = 0-647, while thelast column gives values of TJ calculated by Sutherland's formula,taking G = 78-2.

Viscosity of Helium

Temperature

183-7° C.99-818-717-6

-22-8-60-9-70-0-78-5

-102-6-183-3-197-6-198-4-253-0-2581

TJ (observed)

0-000268123371980196717881587156415061392

0918608176081320349802946

/ rp \0-64771 ( I

H273-270-0002632

2309197019651783160315581515*1389

0918508213081550348902887

7) (calculated,Sutherland)

0-00026822345197919741771156315131460131707450628062101350092

* This entry, which was obviously wrong in the original table, hasbeen recalculated.

Very similar results have also been obtained for hydrogen byKamerlingh Onnes, Dorsman and Weber,f while a general failureof Sutherland's formula to represent viscosity at low tempera-tures has been noticed and discussed by Schmitt, Bestelmeyer,Vogel and others.^

More General Laws of Force

150. This has led to a study of the way in which viscosity woulddepend on temperature with more general laws for the forcesbetween molecules. In 1924 J. E. Lennard Jones§ introduced atentative law of force . .

t Comm. Phys. Lab. Leiden, 134a (1913).J For references see Chapman, Phil. Trans. 216 A (1915), p. 342.§ Proc. Boy. Soc. 106 A (1924), p. 441, and subsequent papers.

180 VISCOSITY

and examined the resulting dependence of viscosity on tempera-ture. This was soon discarded in favour of the more general law

J W - rn rm'

Hass6 and Cook* have calculated the resulting formula for thecoefficient of viscosity in the special case of n = 9, m = 5. Theyfound that, with a suitable choice of values for A and /i, theformula could be made to fit the experiments well for hydrogen,nitrogen and argon, but could not be made to fit for helium,carbon-dioxide and neon. The agreement of their formula withobservation for hydrogen and argon is shewn in the two tablesbelow.

Viscosity of Hydrogen

Abs. temp.

457-3373-6287-62730261-2255-3233-2212-9194-4170-289-6370-8720-04

7} X 10 7

(observed)

12121046877844821802760710670609-3392-2319-3

105-111

r/xlO7

(Hasse andCook)

12261060878846820806756708664603380320111

rjxlO7

(KamerlinghOnnes)

12071052875843816803757709666608389329137

Here the second column gives the best observed value of theviscosity coefficient, while the third column gives the valuecalculated by Hass6 and Cook. The last column gives the valuecalculated by Kamerlingh Onnes for a simple repulsive forcevarying as r"11*2, and it will be seen that the agreement withobservation is not enormously less good than with the morecomplex law of Hass6 and Cook.

• H. R. Hass6 and W. R. Cook, Proc. Roy. Soc. 125 A (1929), p . 196.

VISCOSITY

Viscosity of Argon

181

Abs. temp.

456-4372-8286-3272-9252-8232-9212-9194-3168-789-9

7} X 1 0 7

(observed)

324327512207211619871854169715751379735-6

?xlO 7

(Hass6 and Cook)

321227452216212919931855171115721373686-9

V I S C O S I T Y IN A M I X T U R E OF G A S E S

151. As the proportions of two kinds of gas in a mixture changefrom 1:0 to 0:1, the coefficient of viscosity of the mixture willof course also change, starting from the coefficient of viscosity T/Xof the first gas, and ending at the coefficient of viscosity of thesecond gas TJ2. But, as Graham* found in 1846, the change maynot be continuous, and for certain proportions of the mixture thecoefficient of viscosity of the mixture rj12 may have a value greaterthan either of the coefficients of viscosity r/l9 r\2 of the pure gases.

To a first rough approximation, the viscous transfer ofmomentum in a mixture of gases may be regarded as the sumof the transfers by the different kinds of molecules separately.Thus if />!,p2,/>3,... are the densities of different kinds ofgas, cl9c2,c3,... the average velocities of the molecules of thesedifferent kinds and \ v A2, A3,... the average free paths, we mayexpect the viscosity of the mixture to be given by

For a simple binary mixture, we should expect a coefficient ofviscosity TJ12 given by

or, substituting for A2 and A2 from formula (151),

7 l — • ^2— (222)

• Phil. Trans. Roy. Soc. 136 (1846), p. 573.

182 VISCOSITY

where ijv T/2 are the coefficients of viscosity of the two constituentswhen pure, and Av A2 are quantities which depend on the massesand diameters of the molecules of the two components.

When pJPi is zero, the value of T/12 given by this formulareduces of course to 7]x\ when />2/Pi *s small, it is

Pi \-^2Thus if IJ2>TJ1A1A29 a small admixture of the second gas in-

creases the coefficient of viscosity. If this inequality is satisfiedand rj2 is also less than r/l9 the value for r/12 must clearly rise to amaximum as pJPi is increased and then descend again to its finalvalue rj2.

A formula of this type was first given by Thiesen.* Schmitt,fand later Schroer,J have found that such a formula representsmany, or even most, of the experimental data on the viscosity ofbinary mixtures with very fair accuracy.

The exact theoretical investigation of viscosity in a mixtureof gases is very complicated. Formulae for the coefficientsof viscosity of mixture have been given by Maxwell, Kuenen,Chapman and Enskog, and in every case the theory predicts amaximum value for a certain ratio of the gases in accordance withobservation. Maxwell's investigation§ deals only with moleculesrepelling as the inverse fifth power of the distance; Kuenen|| dealswith elastic spheres, the formulae being corrected for the phe-nomenon of "persistence of velocity " explained in Chap, v; whileChapman^} and Enskog** discuss the viscosity of gas-mixture byfollowing the general methods which are explained in Chap, ixbelow.

Chapman's analysis shews that a formula of the type of (222)must necessarily fail to represent the facts with any completeness;

* Verhand. d. Deutsch. Phys. Gesell. 4 (1902), p . 238.t Ann. d. Physik, 30 (1909), p. 303.% Zeitschr. f. Phys. Ghent. 34 (1936), p. 161. § Coll. Papers, 2, p . 72.|| Proc. Konink. Ahad. Wetenschappen, Amsterdam, 16 (1914), p.

1162 and 17 (1915), p . 1068.If Phil. Trans. 216 A (1915), p. 279, and 217 A (1916), p. 115; Proc.

Roy. Soc. 93 A (1916), p. 1.*• Kinetische Theorie der Vorgdnge in mdssig verdiinnten Gasen, Inaug.

Dissertation, Upsala, 1917.

VISCOSITY 183

the necessary formula is found to be of the more generaltype

where al9 a12, a2 and b are new constants depending on the mole-cular masses, the law of force, and the temperature. Chapmanshewed that a formula of this type represented the observationsof Schmitt,f and later Trautz J and his collaborators have estab-lished further agreement between this formula and experiment.

Gas

MonatomicHeliumNeonArgonKryptonXenonMercury

DiatomicHydrogenCarbon-monoxideNitrogenAirNitric oxideOxygenHydrogen sulphideChlorine

PolyatomicWater vapourNitrous oxideCarbon-dioxideMethane (CH4)Benzene (C6H6)

Mol.wt.

4204084

131201

22828—30323371

1844441678

rj (observed)at 0° C.

0-0001880-0003120-0002100-00023300002110-000162

0-000085740-00016650-0001670-0001720-00017940-0001920-0001180-000122

0-0000870-00013660-0001370-000108 (20° C.)0-0000700

Authority*

111112

34226267

24222

\cr (calc.)in cm.

1-09 xlO-8

1-30 x 10-8

1-83 x 10-8

2-08 x 10-8

2-46 x 10-8

313 xlO-8

1-36 x 10-8

1-89 xlO-8

1-89 xlO-8

1-87 x 10-8

1-88 xlO-8

1-81 x 10-8

2-32 x 10"8

2-70 xlO-8

2-33 x 10-8

2-33 x 10-8

2-33 xlO-8

204 x 10"8

3-75 xlO"8

* Authorities:1. Rankine, Proc. Roy. Soc. 83 A (1910), p. 516; 84 A (1910), p. 181.2. Kaye and Laby, Physical Constants (8th edn., 1936).3. Breitenbach, Ann. d. Phys. 5 (1901), p. 166.4. C. J. Smith, Proc. Phys. Soc. 34 (1922), p. 155.5. Eucken, Phys. Zeitschr. 14 (1913), p. 324.6. Rankine and Smith, Phil. Mag. 42 (1921), pp. 601, 615.7. Rankine, Proc. Roy. Soc. 86A (1912), p. 162.

f L.c. ante.t Ann. d. Physik, 3 (1929), p. 409; 7 (1930), p. 409; 11 (1931), p. 606.

184 VISCOSITY

D E T E R M I N A T I O N OF S I Z E OF M O L E C U L E S

152. We have already, in §31, had an instance of the calcula-tion of molecular radii from the coefficients of viscosity. When thecoefficient of viscosity of any gas has been determined by experi-ment, it is possible to regard equation (210) as an equation for <r,and so obtain the molecular radius on the supposition that themolecules may be regarded as elastic spheres.

The table on p. 183 gives the coefficients of viscosity of variousgases, and the values of £<r, calculated from equation (210). Thesevalues are at least of the same order of magnitude as the valuescalculated from the deviations from Boyle's law in §60. Thereason why still better agreement cannot be expected will be clearfrom what has already been said in § 146.

Chapter VII

CONDUCTION OP HEAT

Elementary Theory

153. Chapter n contained a very incomplete discussion of theconduction of heat in a gas. We shall now attempt a more exact,although still imperfect, investigation of the problem.

The principle is that already explained. A molecule whichdescribes a free path of length I with velocity G and totalenergy E is regarded as transporting energy E through adistance I, so that on balance there is a transport of energyfrom regions in which E is large to regions in which E is small—i.e. from places of high to places of low temperature.

Let a gas be supposed arranged in layers of equal temperatureparallel to the plane of xy. Let E denote the mean energy of amolecule at any point in the gas, so that E will be a function of z.

Let us fix our attention on the molecules which cross a unit areaof the plane z = z0. Some molecules will cross this unit area afterhaving come a distance I from their last collision in a directionmaking an angle 0 with the axis of z. The last collision of thesemolecules must accordingly have taken place in the plane

z = z0 — I cos 6.

We shall, for the moment, make the simplifying assumptionthat the mean energy of these molecules is that appropriate to thisplane, and this may be taken to be

dEJ?-Zcos0^p, (223)

dzwhere E is evaluated at z = z0.

The number of molecules which cross the unit area in questionin a direction making an angle between 6 and d + dd with the axisof z per unit time is (cf. formula (197))

and if we assume that each of these has an average amount of7 JKT

186 CONDUCTION OF HEAT

energy given by formula (223), the total flow of energy across theunit area of the plane will be

p

lE-lcoadjÔiddd \cl

(224)

If E had been independent of z, this flow of energy would ofcourse have been nil, for as much would have crossed the plane inone direction as in the other. But if E increases with z, the mole-cules which cross the plane in the direction of z decreasing, carrymore energy than those crossing the plane in the reverse direc-tion, since they come from regions in which z is greater. Thusthere is a resulting flow of energy in the direction of z de-creasing.

If k is the coefficient of conduction of heat, the flow of heatacross unit area of the plane z = z0 in the direction of z increasing

dT dTis — k -=-, so that the flow of energy is — Jk -zr- > where J is the

oz dzmechanical equivalent of heat.

Equating this to expression (224),T1 dT

from which it follows that the value of k is

kl v ' d d E

From equation (24) we have the relation

v~ JmdT'

where Cv is the specific heat at constant volume, and again, fromequation (203), if r/ is the coefficient of viscosity,

rj — \vclm.

Using these relations, equation (225) becomes

k = rjCv. (226)

CONDUCTION OF HEAT 187

154. The flow of energy across the plane z = z0 is at the ratedT

of — Jk -=— per unit area in the direction of z increasing. Across

the plane z = z0 + dz, the corresponding rate of flow is

Consequently the slab of gas for which z lies between z0 andzQ + dz gains energy at a rate

d2TJkj^dz (227)

per unit area, which must increase the temperature of the slab.dT

To raise the temperature at a rate -=- requires energy per unitCut

area equal to

Equating this to expression (227), we obtaindT

which is the ordinary Fourier equation of conduction of heat.Introducing the value of k from equation (226), this becomes

,2281( 2 2 8 )

which is the special form of the equation of conduction appro-priate to the kinetic theory.

Exact Theory

155. Equation (226) could not in any case be exact and detailedanalysis shows it to be far from exact. There must clearly be anexact relation of the form

k = erjCvi

where e is a numerical multiplier, but the evaluation of e presentsa problem of great complexity. Many attempts have been madeto evaluate e by the approximate methods employed in the lastchapter, but none of them has met with much success. For the


special case of molecules repelling according to the inverse fifthpower of the distance, Maxwell gave an exact theory which ledto the value e — 2-5 (see § 199 below). For monatomic gasesChapman* has evaluated e by methods explained below (cf. §187).In his first paper he obtained e == 2-500 as a first approximationfor all laws of force of the form fir~s, this including of course thespecial case s = 5 studied by Maxwell. In his second paper hefinds that further approximations alter this value by less thanone percent of its value. The greatest error in the first approxima-tion is found to occur in the case of elastic spheres, for which thevalue of eis 2-522.

Enskogf has obtained identical results, and has further giventhe general formula for monatomic molecules repelling as theinverse sth power of the distance

€ = 2 .

( s - 5)2

5 +4(5-lJ7ll«§

which reduces to Maxwell's exact value e = f when s = 5.

E X P E R I M E N T A L V A L U E S

156. We proceed to examine the relation between k and rjwhich is found experimentally.

Monatomic Gases. The following table of recent determinationsoikjrjCv> the quantity we have denoted by e, is given by Enskog: %

Helium§ at 0° C, e = 2-40,„ -191-6° C., e=2-23,„ -252-1° C., e=2-02.

Argon§ at 0° C, e = 2-49,„ 182-5° C, e=2-57.

Neon|| at 10° C, e = 2-501.

* Phil. Trans. Roy. Soc. 211 A (1912), p. 433 and 216 A (1915), p.279.

t See footnote to p. 182. J L.c. p. 104.§ Eucken, Phys. Zeitschr. 12 (1911), p. 1101; 14 (1913), p. 324.|| Bannawitz, Ann. d. Phys. 48 (1915), p. 577.


Other investigators have found similar, although not identical,values. Thus Schwarze** in 1903 found e =* 2-507f f for heliumand e = 2-501 for argon, while Hercus and LabyJ% g*ve e = 2*31for helium and e = 2-47 for argon. Thus the theoretical law seemsto be confirmed to within the limits of experimental error exceptat low temperatures.

Other Gases. The following table contains, in its sixth column,some observed values of IC/T]CV for various gases. The valuesdo not, in general, approximate either to 2-5 or to any othervalue.

Values of k and of k/7)Cv

Gas

HydrogenHeliumCarbon-monoxideNitrogenEthyleneAirNitric oxideOxygenArgonCarbon-dioxideNitrous oxide

k{obs.)

0-00039700-00033000000054250-0000566000004070-0000566000005550-00005700-00003894O-OOOO33700000351

Authori ty*

11111111213

V (P. 183)

0-00008570-0001880-0001660-0001670-00009610-0001720-0001790-0001920-00021000001370-000137

2-420-746t0-177O-178|0-274 §0-172||0-16701560-0745H01560-148

k/vCv (obs.)

1-912-401-851-911-551-911-861-902-491-581-73

i(9y-5)

1-902-441-911-911-551-911-881-902-441-721-73

* Authorities:1. Eucken, Phys. Zeitschr. 14 (1913), p. 324.2. Schwarze, Ann. d. Phys. 11 (1903), p. 303.3. Value assumed by Eucken (I.e.). This is the mean of determinations by Winkelmann

and Wiillner.

t Determined by Vogel, and quoted by Eucken. The value of Cv for helium given by theformula (25) is, however, 0-767.

t Calculated from formula (25). Eucken takes Cv - 0177 , Pier gives Cv - 0 1 7 5 .§ The mean of values given by Winkelmann (Pogg. Ann. 159 (1876), p. 177) and Wullner

(Wied. Ann. 4 (1878), p. 321).|| Direct experimental value.U The value assumed by Eucken. Schwarze uses the value Cv *= 00740, based upon an

experimental determination of Cp by Dittenberger (Halle Diss. 1897). Pier gives Cv - 0-0746.The theoretical value given by formula (25) is 0-767.

An inspection of the values obtained shews, however, thathjrjGv is greatest for monatomic gases, and least for gases in whichthe molecules are of most complex structure (ethylene, carbon-

** Ann. d. Phys. 11 (1903), p. 303.ft Eucken (Phys. Zeitschr. 14, p. 328) state3 that Schwarze gives too

high a value for helium owing to miscalculation of the value Cv.JJ Proc. Roy. Soc. 95 A (1918), p. 190.


dioxide, etc.). In other words kjriCv is largest when the specificheat Cv originates most in internal motions of the molecules.This leads us to suspect that when the molecules are regarded ascarriers of energy, a distinction must be made between the energyof motion in space and the energy of internal motion. If we re-place Cv by its value as found in § 21, equation (226) becomes

(229)

We have seen that when the energy is wholly translational(fi = 0), the value of Jc given by this formula must be multipliedby (approximately) f. Eucken* has suggested that the simplerformula (229) may be accurate for the transport of internalenergy, there being (cf. §212, below) no correlation between thevelocity of the molecule and the amount of internal energycarried.

Combining these two contributions to the transport of energy,we arrive at the formula

R

(230)

the last two forms being obtained on substituting the valuesof Cv and y from equations (29) and (30). According to thisequation, e, or hjrjCv, ought to have a value J(9y —5) whichdepends only on y, the ratio of the specific heats. In the lastcolumn of the table on p. 189 the values of J(9y — 5) are given, andare seen to be in fair agreement with the observed values of k/rjCv.

CONDUCTION OF H E A T IN R A R E F I E D GAS

157. The theory of conduction of heat, like that of viscousflow, assumes a special form when the free path is larger than thedimensions of the apparatus. Molecules then transport heatacross the whole apparatus at a single bound, so that the wholeof the gas must be assumed to be at a uniform temperature;

• Phys. Zeitschr. 14 (1913), p. 324.


there is no longer a gradual temperature gradient. The appro-priate mathematical theory has been developed by Knudsen. *

We consider a rarefied gas enclosed in a vessel, and fix ourattention on a small area dS of the inner wall of the vessel.

If all the molecules of the gas moved with the same velocity c,the number of molecules impinging on the area dS in unit timewould be, as in § 39,

\vcdS.

Each of these would deliver up energy \mc2 to the wall, so thatthe total energy transferred from the gas to the element dSwould be

If the molecules do not all move with the same speed, we mustaverage c3 in this expression. If the speeds are distributedaccording to Maxwell's law, we find (cf. Appendix vi, p. 306)that the average value of c3 is 4/\Z(7r/&3m3), and the total transferof energy from gas to wall, per unit area and unit time, becomes

where p is the pressure in the gas.If the wall is at the same temperature as the gas, the net

transfer of energy between the wall and the gas must be nil, sothat the wall must yield back energy to the gas at a rate Jpc.Suppose however that the gas is at some temperature To

(absolute), while the wall is at some other temperature Tx.The simplest tentative assumption to make is that while the

molecules of the gas are in their condition of adsorption by thewall (§ 34), they acquire the mean energy which corrrespondsto the temperature of the wall, and subsequently leave the waDwith this mean energy. The molecules will now take away fromthe wall TJTQ times the amount of energy they took to it, sothat there will be a net transfer of energy (still measured

* Ann. d. Physik, 31 (1910), p. 205, 33 (1910), p. 1435 and laterpapers. See also M. Knudsen, The Kinetic Theory of Gases (Methuen,1934), p. 46, and Loeb, Kinetic Theory of Gases (2nd ed., 1934), pp.325ff.


per unit area per unit time) from the wall to the gas of

Thus the heat dissipated through contact with a rarefied gaswill be jointly proportional to the temperature difference Tx — TQ

and to the pressure p.158. Experiment confirms this law of proportionality, but

generally speaking, does not confirm the multiplying factor informula (231). The actual transfer of heat is substantially lessthan that predicted by formula (231), as though the interchangeof energy between the solid wall and the adsorbed molecules ofgas were far from complete, an explanation suggested byv. Smoluchowski* in 1898, and again by Soddy and Berryf in1910.

This led Knudsenf to introduce a quantity a which he de-scribed as a "coefficient of accommodation". He imagines, inbrief, that when a molecule is adsorbed by the wall, its energy ismot adjusted through the whole range of temperature differenceTx—TQi but only through a fraction a of this range. Formula(231) must now be replaced by

2 To a> ( 2 3 2 )

and it is immediately possible to determine a by experiments onthe dissipation of heat.

The value of a is found to depend very largely on the tempera-ture and other physical conditions, such as, in particular, thecleanness or otherwise of the solid surface. Almost all valuesfor a from 0 to 1 appear to be possible. Thus for CO2 in contactwith platinum heavily coated with platinum black, Knudsenfound a = 0-975; for helium in contact with tungsten, J. K.Roberts§ found that a was equal to 0-057 at 22° C, and fellsteadily to 0-025 as the temperature was lowered to — 194° C. A

• M. v. Smoluchowski, Wied. Ann. 64 (1898), p. 101.f Proc. Roy. Soc. 83 A (1910), p. 254 and 84 A (1911), p. 576.t 4nn.d.P%?. 34 (1911), p. 593, 36(1911), p. 871, and 6 (1930), p. 129.

For a full discussion of the subject see Loeb, Kinetic Theory of Gases(2nded., 1934), pp. 321ff.

§ Proc. Roy. Soc. 129 A (1930), p. 146 and 135 A (1932), p. 192.


study of the relation between a and the temperature suggestedthat a would be found to vanish at the absolute zero.

General dynamical considerations suggest that the value of ashould be substantially less than unity. In § 10 we consideredthe impact of gas molecules of mass m' on wall molecules ofmass m, both molecules being treated as hard elastic spheresobeying the Newtonian laws of motion. We found that theaverage impact resulted in the gas gaining energy of amount

2mm'

wu' • I ± 4 m m 'Inis is equal totimes the interchange of energy which would occur if the gasmolecules took up the temperature of the wall completely, andso corresponds to a "coefficient of accommodation" a given by

_ 4mm' __ /m —a== ( + 'Y^ \ +

This value of a is always less than unity, and is substantiallyless except when the masses m, m' are nearly equal. For morecomplicated molecules and more complicated impacts, the theo-retical value of a would no doubt be very different from thesimple value just found. Various attempts have been made tocalculate a "coefficient of accommodation" which shall agreewith experiment, and from these it seems to emerge quite clearlythat the problem is one for wave-mechanics, and so is outsidethe scope of the present book.* Jackson and Howarthf havefound a wave-mechanics formula which represents the observa-tions of Roberts mentioned above with very fair accuracy.

159. Our theoretical calculations have been based on thesupposition that the only energy which the impinging moleculecan transfer to the wall is its kinetic energy of motion \mc2. Wehave already seen (§21) that many molecules have other energybesides this, and the question arises as to what happens to this

* J. M. Jackson, Proc. Camb. Phil. Soc. 28 (1932), p. 136; C. Zener,Phys. Review, 37 (1931), p. 557 and 40 (1932), pp. 178, 335; Jackson andMott, Proc. Roy. Soc. 137 A (1932), p. 703.

f Proc. Roy. Soc. 142 A (1933), p. 447.


other energy when the molecule impinges on a solid. Knudsen*has devised ingenious methods for probing this question ex-perimentally and has reached the conclusion that the internalmolecular energy also has an accommodation coefficient, which,to within the limits of experimental error, is the same as theaccommodation coefficient for the kinetic energy of translation.Thus if the internal energy is /? times the kinetic energy ofrotation, the total transfer of energy is (1+/3) times that givenby formula (232).

CONDUCTION OF H E A T AND E L E C T R I C I T Y IN S O L I D S

Conduction of Heat

160. In 1900 Drudef propounded a theory of conduction ofheat in solids, according to which the process is exactly similar tothat in gases which we have just been considering, except thatthe carriers of the heat-energy are the free electrons in the metals.

According to the simplest form of this theory, the coefficientof conduction of heat in a solid will be given by equation (225),namely

in which all the quantities refer to the free electrons in the solid,so that v is the number of free electrons per unit volume, I is theiraverage free path as they thread their way through the solid, andso on. Taking E = § BT, this becomes

wlR (234)

Conduction of Electricity161. Drude's theory supposes that the free electrons also act

as carriers in the conduction of electricity. If there is an electricforce S in the direction of the axis of #, each electron will be actedon by a force Se, and so will gain momentum in the direction Oxat a rate Se per unit time. The time required to describe an average

• Ann. d. Phys. 34 (1911), p. 593.t Ann. d. Phys. 1 (1900), p. 566.


free path I, with average velocity c, will be l/c9 so that in describingsuch a free path, the electron will acquire an additional momentumin the direction of the axis of x equal to Sel/c.

Since the mass of the electron is very small compared with thatof the atom or molecule with which it collides, we must suppose(cf. § 130) that there is no persistence of velocities after collisions,so that an electron starts out from collision with a velocity forwhich all directions are equally likely, and, in describing its freepath, superposes on to this a velocity

Edme

parallel to the axis of x. It follows that at any instant the freeelectrons have an average velocity u0, parallel to the axis of x,given by

I Eel

Across unit area perpendicular to the axis of x, there will be aflow of electrons at the rate vu0 per unit time, and these will carrya current i given by

ISveH% = veu0 = - — — .0 2 me

The coefficient of electric conductivity cr is defined by the rela-tion i = crS, and is therefore equal to the coefficient of S in theabove equation. To the order of accuracy to which we are nowworking, c may be supposed to be the velocity of each electron,so that we may put \m{c)2 = § ET, and the conductivity is given

This is Drude's formula for electric conductivity.

Ratio of the two Conductivities

162. The Wiedemann-Franz law. By comparison of equations(234) and (235), we obtain


which is Drude's approximate formula for k/cr. From this equa-tion it appears that:

at a given temperature, the ratio of the electric and thermalconductivities must be the same for all substances.

This is the law of Wiedemann and Franz, announced by themas an empirical discovery in 1853.*

Various attempts have been made to obtain a more exactformula for k/cr. In 1911, N. Bohrf obtained the formula

* = ^(*n, (236)cr s~l\ej J

in which the electrons are supposed to be repelled from the atomsaccording to the law jir~s. According to this, the law of Wiede-mann and Franz is not strictly true, but k/cr varies only throughthe factor 2s/(s — 1) and as s varies from s = 5 to s = oo, this onlyvaries from 2-5 to 2.

The Law of Lorenz. From equation (236) it also follows that:the ratio of the thermal and electric conductivities must be pro-

portional to the absolute temperature.a law put forward on theoretical grounds by Lorenz in 1872.J

Comparison with Experiment163. For elastic spheres (s = oo) equation (236) reduces to

(237)

a formula originally given by Lorentz.§ Inserting numericalvalues, this equation becomes

k—- = 3-538 in electromagnetic units.

Extensive experiments to test this formula have been madeby Jager and Diesselhorst||, C. H. Lees f and many others. Ingeneral these confirmed the theoretical equations, except that

* Pogg. Ann. 89 (1853), p. 497.t Studier over Metallernes Elektrontheorie (Copenhagen, 1911).X Pogg. Ann. 147 (1872), p. 429 and Wied. Ann. 13 (1882), p. 422.§ The Theory of Electrons, p. 67 and note 29.|! Berlin. Sitzungsber. 38 (1899), p. 719, and Abhand. d. Phys.-Tech.

Reichsanstalt, 3 (1900), p. 369.If C. H. Lees, "The effects of low temperatures on the Thermal and

Electrical conductivities of certain approximately pure metals andalloys", Phil. Trans. 208 A (1908), p. 381,


the observed values of kjcr were always somewhat too high.Jager and Diesselhorst experimented at 18° C. and at 100° C.Putting T = 291-2 (i.e. 18° C.) in formula (235), and measuringk in work, instead of in heat, units (i.e. omitting the J in thedenominator), formula (235) gives

- == 4-31 x 1010 at 18° C.cr

The simpler formula of Drude leads to § times this value,namely

k- = 6-46xl010at 18° C.cr

The following are examples of the results obtained experi-mentally by Jager and Diesselhorst:

for three samples of copper, k/cr = 6-76, 6-65, 6-71 x 1010,for silver, k/tr = 6-86 x 1010,for two samples of gold, kjcr = 7-27, 7-09 x 1010.

As temperature coefficient of this ratio they find:

for two samples of copper, 0-39, 0*39 per cent,for silver, 0-37 per cent,for two samples of gold, 0-36, 0-37 per cent.

The theoretical value, as given by equation (237), is 0-366 percent. Lees finds that there is a greater divergence from formula(237) as the temperature of liquid air is approached.

It is more difficult to test the theoretical values for k and crseparately, since the formulae for these coefficients separatelycontain the quantities v and Z, for which it is difficult to form areliable numerical estimate. But such evidence as is availableshews quite definitely that the formulae for k and cr separatelydo not shew anything like so good an agreement with observationas that shewn by the formula for their ratio k/c

Indeed the phenomenon of electric super-conductivity shewsthat this must be the case. At helium temperatures the resistancemay be only a fraction 10"11 times the resistance at ordinarytemperatures, but it is impossible to believe that at these tem-peratures the free path can suddenly increase its length by afactor 1011.

Chapter VIII

DIFFUSION

E L E M E N T A R Y T H E O R I E S

164. The difficulties in the way of an exact mathematicaltreatment of diffusion are similar to those which occurred in theproblems of viscosity and heat conduction. Following the pro-cedure we adopted in discussing these earlier problems, we shallbegin by giving a simple, but mathematically inexact, treatmentof the question.

We imagine two gases diffusing through one another in adirection parallel to the axis of z, the motion being the sameat all points in a plane perpendicular to the axis of z. Thegases are accordingly arranged in layers perpendicular to thisaxis.

The simplest case arises when the molecules of the twogases are similar in mass and size—like the red and whitebilliard balls we discussed in § 6. In other cases differencesin the mass and size of the molecules tend, as the motionof the molecules proceeds, to set up differences of pressure inthe gas. The gas adjusts itself against these by a slow mass-motion, which will of course be along the axis of z at everypoint.

Let us denote the mass-velocity in the direction of z increasingby w0, and let the molecular densities of the two gases be vl9 v2.Then vv v2 and w0 are functions of z only.

We assume that, to the approximation required in the problem,the mass-velocity of the gas is small compared with its molecular-velocity, and we also assume that the proportions of the mixturedo not change appreciably within distances comparable with theaverage mean free path of a molecule. We shall also, to obtain arough first approximation, assume that Maxwell's law of dis-tribution of velocities obtains at every point, and that h is thesame for the two gases.

DIFFUSION 199

165. The number of molecules of the first kind, which cross theplane z = z0 per unit area per unit time in the direction of zincreasing, is now

fff > (238)in which the limits are from — oo to + oo as regards u and v, andfrom 0 to oo as regards w.

In accordance with the principles already explained, v mustbe evaluated at the point from which the molecules started aftertheir last collision. Those which move so as to make an angle 6with the axis of z may be supposed, on the average, to come froma point of which the z coordinate is z0 — A cos 0, and at this pointthe value of vx may be taken to be

( | ^ (239)

Inserting this value for vx into expression (238), this expressionbecomes the difference of two integrals. The first is expression(238) with vx taken outside the signs of integration and evaluatedat z = z0. The value of this integral is easily found to be

^ i U ( f e + ) 5 (240)

where cx denotes the mean molecular-velocity of all the moleculesof the first kind.

The second integral is

(241)

Owing to the presence of the multiplier Al - ^ I, this expression

is already a small quantity of the first order, so that in evaluatingit we may put w0 = 0. Replacing cos# by w/c, it becomes

in which the integral is taken over all values of u and v, and overall positive values of w.

200 DIFFUSION

This expression is easily evaluated by noticing that it has justhalf the value it would have if taken over all values of u, v, w,

/ di>i\ w2

and is therefore equal to 4A ~~= times the average value of —\oz) c

taken over all molecules, in a gas having no mass-motion. This1 14. 4-U- 1 f i l ~U2 + V2 + W2

average value is equal to one-third of the average ofc

or c, and is therefore lcv Hence expression (241) is equal to

Combining this with expression (240), we find as the totalvalue of expression (238),

in which all quantities are to be evaluated in the plane z = z0.This is the total flow of molecules across unit area of the plane

z = z0, in the direction of z increasing. The corresponding flow inthe opposite direction is

The rate of increase of the number of molecules of the first kindon the positive side of the plane z = z0, measured per unit timeper unit area, is the difference of these two expressions. Denotingthis quantity by Fl9 we have

A = ^ 0 - 3 A l - g ^ l -

Similarly, for the rate of increase of molecules of the second kind,

If the flow is to be steady, the total flow of molecules over everyplane must be zero, and this requires that

Further, the pressure must be constant throughout the gas, sothat we must have

^1+^2 = cons.,

DIFFUSION 201

whence, by differentiation with respect to zf

__ i i e ocz oz

EKminating w0 from these equations, we now obtain

The number of molecules of the first kind, in a layer of unitcross-section between the planes z = z0 and z = zo + dz, is vxdz\

the rate a t which this quantity increases is -~ dz, but is also found(tt

7) T1

to be — ^ d z , by calculating the flow across the two boundaryvZ

planes. Hence we have

dt " ~ d z 9

and on using the value of / \ just found, and neglecting smallquantities of the second order, this becomes

Equation (243) is the well-known equation of diffusion, ®12

being the coefficient of diffusion of the two gases. Hence thecoefficient of diffusion is given by formula (244). It is symmetricalas regards the physical properties of the two gases, but dependson the ratio vj^ in which they are mixed.

The foregoing analysis is essentially the same as that given byMeyer in his Kinetic Theory of Oases, and formula (244) isgenerally known as Meyer's formula* for the coefficient ofdiffusion.

* The actual value of £)12 given by Meyer (Kinetic Theory of Gases,p. 255, English trans.) is |7r times that given by formula (244). Meyer'sformula has, however, attempted to take into account a correction whichis here reserved for later discussion (§ 168). Meyer does not claim thatliis correction is exact.

202 DIFFUSION

Coefficient of Self-diffusion

166. Formula (244) becomes especially simple when themolecules of the two gases are approximately of equal size andweight. If we agree to neglect the differences in size and weight,we may take A and c to be the same for each gas, and so obtain

© = £Ac. (245)Comparing this with the corresponding approximate formula

(203) for the coefficient of viscosity

we obtain the relation 2) = - . (246)P

The quantity 2) obtained in this way may also be regarded asthe coefficient of self-diffusion or interdiffusivity of a single gas.It measures the rate at which selected molecules of a homogeneousgas diffuse into the remainder, as for instance red billiard ballsinto white billiard balls in our analogy of § 6, or (approximately)one isotope into another isotope of the same element. Hartek andSchmidt have made diffusion experiments with normal hydrogenand a mixture rich in para-hydrogen, while Boardman and Wild*have experimented with mixtures of gases—as for instancenitrogen and carbon-monoxide—in which the molecules are notonly of equal mass, but are believed to be of very similar structure(see §176).

Dependence on Proportions of Mixture

167. In the special case just considered, the value of 5) isindependent of the proportion of the mixture, but in the moregeneral case formula (244) shews ®12 ought to vary with theproportions of the mixture. In the limiting case in which vjv^ = 0,

__ x 2 ^12 = 3 l C l = ^ h

and there is a similar formula for the case of v2/v1 = 0, in which mx

and m2 are interchanged.

* Proc. Roy. Soc. 162 A (1937), p. 516.

DIFFUSION 203

Thus the coefficients of diffusion in these two cases stand in theratio ~

^"l-o _ Tl (9,47)

shewing that the value of 3) ought according to Meyer's formulato vary with the proportions of the mixture to the extent towhich m1 differs from m2. For example, for the diffusion ofH2—CO2, the extreme variation would be 22 to 1, for A—He itwould be 10 to 1, and so on.

We shall see later that the observed variation of 3)12 with vjîs nothing like as great as is predicted by this formula. For themoment we proceed to correct the formulae for persistence ofvelocities, and shall find that the corrected equations predict amuch smaller dependence of 3)12 on vjv^.

Correction to Meyer's Theory when the Moleculesare Elastic Spheres

168. We shall consider first the correction to be applied to thesimple formula for self-diffusion, namely

© = JAc (248)

= ^. (249)

Actually, two sources of error have been introduced into theseapproximate equations, the first arising from the assumptionthat A is the same for all velocities, and the second from neglectof the persistence of velocities.

As regards the first, A must simply be replaced by Ac and takenunder the sign of integration in expression (241). The upshot ofthis is, that instead of A in the final result, we have Z, where

\ce-h™2czdc ^—

/ .

00

e~hnu:2c2dco

This however is exactly the same as the I of the viscosity formula,so that this correction affects 3) and K exactly similarly, multi-plying each by 1*051, but does not affect equation (249).

204 DIFFUSION

169. There remains the correction for the persistence of velo-cities. We found in §139, that when a molecule arrives at theplane z = z0 in a given direction, the expectation of the distanceit has travelled in that direction is not A, but kX, where

Here d is the persistence of velocities at a collision between twomolecules of equal mass, of which the value was found in § 126to be 0-406. Thus the expectation of the molecule belonging tothe one gas or the other is not that appropriate to a distance Aback, but to a distance kX, and the effect of "persistence" istherefore to multiply the value of ® given in equation (249) bya factor k. Also, as we saw in § 139, the effect of persistence on thecoefficient of viscosity is to multiply the simple expression\Xcp by a factor 1/(1 - \6).

The values of ® and TJ, both corrected for persistence, areaccordingly

and the corrected form of equation (249) must be

** \-0 p

Putting 6 = 0-406, the value found in § 126, this becomes

© = 1-342?. (250)

It is of interest to examine into the origin of the difference be-tween the effect of persistence of velocities on diffusion on theone hand, and on viscosity and conduction of heat on the other.Diffusion, it will be seen, is a transport of a quality, while viscosityand heat-conduction are transports of quantities. The differencerests ultimately upon the circumstance that qualities remainunaltered by collisions, whereas quantities do not.

DIFFUSION 205

When the molecules are not of equal mass, it is more difficult toestimate the effect of persistence.

When the molecules were equal, the expectation of the distancea molecule had come was increased by persistence from A to

A (251)1 — C 7

When the molecules are of unequal masses, the persistence isdifferent at different collisions, and instead of expression (251),we shall have one of the form

X+p\+pq\+pqrA +..., (252)

where p, q, r,... are the different persistences at the various col-lisions. Suppose we are considering the motion of a molecule ofmass mx in a mixture of molecules of masses mv m2 mixed in theproportion vxjv2. Then of the quantities p,q,r,... a certain pro-portion, say /?, of the whole will have an average value d = 0-406,these representing collisions with other molecules of the firstkind, while the remainder, a proportion 1 — /? of the whole, willhave an average value which we shall denote by 612, this being thepersistence for a molecule of the first kind colliding with one ofthe second kind.

Let P denote expression (255), and let s denote fid+(l — y#)#i2>this being the expectation of each of the quantities p,q,r}

We haveP = X-\-p\+pqX+pqrX+ ...9

Ps — sX+psX +pqsA+...,

and hence, by subtraction,

P(l-s) = \ + (p-s)\+p(q-s)\+pq(r-s)\+....

Clearly the expectation of the right-hand side is A, for theexpectations of p — s, q — s, r — s,... are all zero. Hence the ex-pectation of P is

( 2 5 3 )

Accordingly the effect of persistence in this mixture of gases is toincrease A to a value of which the expectation is that on the right-hand side of equation (253).

206 D I F F U S I O N

170. In § 111 we found for the mean chance of collision per unittime for a molecule of the first kind, moving in a mixture of twokinds of gas,

In this the first term results from collisions with molecules of thefirst kind, and the second from collisions with molecules of thesecond kind. These two terms must therefore stand in the ratiofi: 1— /?, so that

fi _ 1 - P .. - i tmi

which again is equal to A1? the free path of a molecule of the firstkind, by equation (151).

Using this value for /?, we find for the value of expression (253),the free path of a molecule of the first kind increased by per-sistence,

p . . !( 1 - 0 ) 7 2 7 7 ^ 0 - ? + ( l - J ^

2 (254)and there is a corresponding quantity P2 for the second molecule.

On replacing A^ A2 in equation (244) by their enhanced values,as just found, we obtain as the form of Meyer's equation, aftercorrection for persistence of velocities,

In this formula the value of 6 is always 0-406, while thevalue of 012 depends, as was seen in § 128, on the ratio of the twomasses. It was there found that 012 was of the form

n _ m1-al2m2

m1

where a12 was a small positive number, depending on the ratio ofthe masses, but lying always between 0 and J, and equal to 0*188for equal masses.

DIFFUSION 207

When vx is small, we have, instead of the limiting form given in§167,

12 3vx + v2 3(l + a12)7r(v1 + v2)8l2Nnh\m1 mj(256)

When v2 is small, the limiting form is the same except that a12 isreplaced by a21. Thus the ratio of the extreme values of © asvxjv2 varies is

instead of the ratio m%:m1 found from Meyer's formula (247).Since the extreme values possible for a are 0 and J, it appears thatthe greatest range possible for ® is at most one of 4:3.

Thus, when persistence of velocities is taken into account,Meyer's formula yields values which do not vary greatly with theproportion v1: v2 of the mixture.*

The Stefan-Maxwell Theory

171. Equation (243) is of the same form as the well-knownequation of conduction of heat: it indicates a progress or spreadingout of the gas of the first kind, similar to the progress and spread-ing out of heat in a problem of conduction. The larger ® is, themore rapidly this progress takes place; % is largest when the freepaths are longest, and vice versa. Long free paths mean rapiddiffusion, as we should expect.

Now the formula for the mean path Ax in a mixture of two gaseswas found in § 111 to be

A ' ( 2 5 7 )

* This was pointed out in a valuable paper by Kuenen ("The diffusionof Gases according to O. E. Meyer", Supp. 28 to the Comm. Phys. Lab.Leiden, Jan. 1913). Kuenen took a uniformly equal to 0-188, its valuewhen m1 — m2, and assumed the number of collisions to be in the ratiov\: vivzt so that his result is different from mine, but the principle wasessentially the same. In a later paper by the same author (Comm. Phys,Lab. Leiden, Supp. 38) the mass difference was taken into account.

208 DIFFUSION

where S12 is the arithmetic mean of the diameters of the twokinds of molecules. The larger the denominator in this expression,the smaller A2 will be, and so the slower the process of diffusion.Both terms in the denominator of expression (257) accordinglycontribute something towards hindering the process of diffu-sion.

The second of these terms arises from collisions of the moleculesof the first kind with molecules of the second kind, and that thesecollisions should hinder diffusion is intelligible enough. But it isnot so clear how collisions of the molecules of the first kind withone another, represented by the first term in the denominator ofexpression (257), can hinder the process of diffusion. When mole-cules of the same kind collide, their average forward motionremains unaffected by the conservation of momentum, andit is not easy to see how the process of diffusion has beenhindered.

Stefan* and Maxwellf have accordingly suggested that col-lisions between molecules of the same kind should be entirelydisregarded; we then have effective free paths given by theequations

> A2 =

(258)

in place of equation (257). Using these values for the free paths,equation (244) becomes

m\c2-{-m\c1^12

\(— + — \, (259)mi\m1 m2

l

• Wien. Sitzungsber. 63 [2] (1871), p. 63, and 65 (1872), p.323.

t Coll. Scientific Papers, 1, p. 392, and 2, pp. 57 and 345. See alsoBoltzmann, Wien. Sitzungsber. 66 [2] (1872), p. 324, 78 (1878), p. 733,86 (1882), p. 63, and 88 (1883), p. 835. Also Vorlesungen iiber Gastheorie,1, p. 96.

DIFFUSION 209

or, in terms of the molecular-velocities,*

^ = dsi^1 (260)

If the two kinds of molecules are of equal mass and size, thisbecomes

l (261)

which may be contrasted with Meyer's uncorrected formula (245).Using Chapman's corrected formula (210) for the coefficient

of viscosity,97 = 0.499 pC

V277W2'

equation (261) assumes the form

® = 1-336^, (262)P

agreeing almost exactly with equation (250) which was obtainedon correcting Me}êr's formula for persistence of velocities.

When the molecules of the two gases are unequal in mass andsize, it is more difficult to compare equation (259) with equa-tion (255) which was obtained by correcting Meyer's formula forpersistence of velocities; there is the outstanding difference be-

* Meyer, using the value D12 already explained (see footnote to p. 201),obtains a value for D12 on Maxwell's theory equal to JTT times this,namely

and this same value is given by Maxwell (I.e. ante and Nature, 8 (1873),p. 298). On the other hand, Stefan (Wien. Sitzungsber. 68 (1872), p. 323),Langevin (Ann. de Chirn. et de Phys. [8], 5 (1905), p. 245), and Chapman inhis first paper (Phil. Trans. 211A (1912), p. 449) all arrived at the formula

which is only three-quarters of the value above. Chapman and Langevinboth extended their method to the general law of force /ir~*; their methodwas somewhat similar to that of Maxwell as given in Chap, ix (§ 201),but they assumed Maxwell's law of distribution to hold, so that theirresults were only exact for the case of a = 5, for which their result agreeswith Maxwell's formula (261).

210 DIFFUSION

tween the two, that Meyer's formula depends on the ratio vx\v%,whereas Maxwell's formula (259) does not. In the limiting caseof vx = 0, Meyer's formula reduces to formula (256), which isidentical with Maxwell's formula (259) divided by the factor(1 + a12), where a12 is the small number defined in § 128. Thus theformulae approximate closely; they naturally cannot agree com-pletely, since the one predicts slight variation with vjv^ while theother predicts none at all.

EXACT GENERAL FORMULAE

172. We have so far only obtained approximate formulae, andthis only for the very special case of molecules which may betreated as elastic spheres.

When the molecules are treated as point centres of forcerepelling according to the law fir~s, the methods of Chapmanand Enskog, which are explained in Chap, ix below, allow thevalue of ®12 to be calculated, by successive approximations, toany desired degree of accuracy.

A first approximation, arrived at by the assumption that thecomponents of velocity u, v, w relative to the velocity of mass-motion were distributed according to Maxwell's law, had beengiven by Langevin* in 1905; the same formula was given inde-pendently by Chapmanf in 1911. To a first approximation thevalue of ®12 is found to be given by

( '

where, in the notation of § 195 below,

J^s) — 4TT COS2 \d''a da,Ja=0

When s = 5, this is exact, and reduces to

*F K '

where Ax = 7^5) = 2-6595,

• Ann. de Chim. et de Phys. [8], 5 (1905), p. 245.t Phil. Trans. 211 A (1912), p. 433.

DIFFUSION 211

a formula which had been derived by Maxwell as far back as1866, by methods explained below (§201).

When s has a value other than 5, the exact value of ® 12 will beobtained on multiplying the above approximate value of ®12 bya numerical multiplier,

Self-Diffusion

173. Chapman has obtained the following table of the valuesof this multiplier from calculations carried as far as a secondapproximation for special values of s in the case in which the twokinds of molecules are exactly similar (self-diffusion).

5 = 5 5 = 9 8 = 17 5 = 00

Multiplier = 1-000 1-004 1-008 1-015

It appears that the multiplier increases steadily as s increasesfrom 5 upwards, and reaches its maximum of 1-015 when s = oo(elastic spheres). In the special case of elastic spheres, Chapmansupposes that the multiplying factor, to a third approximation,would have the value 1-017, and that this is the value to whichsuccessive approximations are converging, to an accuracy of onepart in a thousand. Assuming this, Chapman finds that theaccurate value of the coefficient of self-diffusion for elasticspheres* is

^ S (265)

= 1-200-. (266)

Pidduck,f following a method originated by Hilbert,f based onthe transformation of Boltzmann's characteristic equation (311),had previously arrived at the formula (265) for 5), except that thenumber in the numerator, calculated to three places of decimalsonly, was given as 0-151.

• Phil Trans. 217 A (1916), p. 172.t Proc. Lond. Math. Soc. 15 (1915), p. 89.t Math. Ann. 72 (1912), p. 562#

212 DIFFUSION

Diffusion in a Mixture174. Both Chapman* and Enskogf have shewn how to obtain,

by successive approximations, formulae for the general value ofS)12 in a mixture of gases. Chapman starts by taking the valuegiven by equation (263) as a first approximation. Denoting thisvalue by (®12)0, the general value of ®12 is put in the form

®i2 = ^ 4 ° > ......(267)

where e0 is a small quantity, to be evaluated by successive ap-proximations. As we have already seen, the value of e0 is zero forMaxwellian molecules repelling as the inverse fifth-power of thedistance. In other cases the formulae, even when carried only toa second approximation, are extremely complicated, and thereader who wishes to study them in detail is referred to theoriginal memoirs. J

The principal interest of these general formulae lies in theamount of dependence of ®12 on the proportion of the mixturewhich is predicted by them. We have already seen that theapproximate Maxwell-Stefan theory (§171) predicted that ®12

would be independent of this proportion; on the other hand, thetheory of Meyer predicted very great dependence when themolecules of the two kinds of gas were of very unequal mass,although this, it is true, was greatly reduced when "persistenceof velocities" was taken into account.

For the ratio of ®12 in the two extreme cases of vjv^ = 0 andvilv2 ~ °°> Enskog§ gives the formula (accurate to a secondapproximation) for the case in which the molecules may betreated as elastic spheres,

1 + mi22

1 12m?

This may be compared with the value m%\mx predicted by* Phil. Trans. 217 A (1916), p. 166.f L.c. ante (see footnote to p. 182).X Chapman, l.c. equations (13-07), (13-28) ff.; Enskog, l.c. equations

(168) ff. § L.c. p. 103.

DIFFUSION 213

Meyer's uncorrected formula (§ 167), and with our formula (§ 170)obtained by correcting Meyer's formula for persistence of velo-cities. To take a definite instance, it will be found that whenra^mg =10, the predicted values are as follows:

Meyer 10-000Meyer (corrected for persistence) 1*324Chapman-Enskog 1-072

COMPARISON WITH E X P E R I M E N T

175. I t will be convenient to consider first the experimentalevidence as to how far the coefficient of diffusion depends on theproportion in which the gases are mixed. A series of experiments*have been made at Halle to test this question, a summary ofwhich will be found in a paper by Lonius.f

Experiments were made on the pairs of gases H2—O2, H2—N2

and N2—O2 by Jackmann, on H2—O2 and H2—CO2 by Deutsch,and on He—A by Schmidt and Lonius. On every theory whichhas been considered, the greatest variation of ®12 with vjv2 oughtto occur when the ratio of the masses of the molecules differsmost from unity. The following tablej gives the values obtainedfor ® 12 with different values of vjv2 for the two pairs of gases forwhich this inequality of masses is greatest.

Pair of Gases(1,2 respectively)

H2—CO2

He—A

vi

»2

31J2-652-261-6610-4770-311

(observed)

0-213510-217740-227720-244180-249650-250400-254050-256260-26312

Observer

Deutschft

Lonius

Schmidt

Lonius»>

(calculated)(Chapman)

0-2120-2220-2260-2480-2500-2510-2540-2570-259

* R. Schmidt, Ann. d. Phys. 14 (1904), p. 801, and the followingInaug. Dissertations: R. Schmidt (1904), O. Jackmann (1906), R.Deutsch (1907), and Lonius (1909).

| Ann. d. Phys. 29 (1909), p. 664. See also Chapman, Phil. Trans.211 A (1912), p. 478. % Lonius, I.e. p . 676.

214 DIFFUSION

The last column gives the values calculated by Chapman fromhis theoretical formulae. In these calculations an absolute valueof ®12 is assumed such as to make the mean of the calculatedvalues of ®12 for each pair of gases equal to the mean of theobserved values.

The degree of variation in S)12 predicted by the formula ofChapman is at least of the same order of magnitude as thatactually observed. The superiority of Chapman's formulae overthe two others already discussed is shewn in the following table,which gives a comparison between the extreme values of ®12 forHe—A observed, and those predicted by these various formulae.(All values of S)12 are multiplied by a factor chosen so as to make^12 = 1 when vx = v2.)

2-651-000-311

(observed)

0-9611-0001-036

£)12 (calculated)

(Chapman)

0-9761-0001021

(Meyer;corrected)

0-9101-0001-110

(Meyer)

0-5481-0001-526

The foregoing discussion will have shewn that the actualvariation of ® 12 with the proportion of the mixture is, in any case,very slight. Consequently, throughout the remainder of thischapter we shall be content to disregard the dependence of ®ia

on the ratio vjv2'

Coefficient of Self-Diffusion

176. The formula which it is easiest to test numerically is thatfor self-diffusion, but the coefficient of self-diffusion of a gas intoitself is not a quantity which admits of direct experimentaldetermination.

A convenient plan, adopted by Lord Kelvin, * is to take a set ofthree gases for which the coefficients <2)12, 2)23? ®3i a r e known.All the quantities in formula (244) are then known with greataccuracy except only S12. Hence from the three values of

• Baltimore Lectures, p. 295.

DIFFUSION 215

®i2» ®23> ®3i w e c a n calculate S129 $23, Szl and so deduce valuesof <rv cr2, <rs. Instead of comparing these values with other de-terminations of <rv <r2 and cr3, Lord Kelvin inserted them intoformula (245) and so obtained the coefficients of self-diffusion ofthe three gases in question.

Lord Kelvin gives the following values of coefficients of inter-diffusivity of four gases, calculated from the experimental de-terminations of Loschmidt.

Pairs of gases D n

(12, 13, 23) 1-32(12, 14, 24) 1-35(13, 14, 34) 1-26

Pairs of gases 3)22(12, 13, 23) 0193(12, 14, 24) 0-190(23, 24, 34) 0-183

GasesHa —(1)O2 - ( 2 ) 'CO —(3)CO2—(4)

Mean

Pairs of gases(12,(13,(23,

13,14,24,

23)34)34)Mean

1-31

.0-169

.0175

.0-1780174

Mean

Pairs of gases(12,(13,(23,

14,14,24,

24)....34)....34)....Mean

0-189

..0106

..0111

..01090109

The agreement inter se of the values obtained by different setsof three gases gives a striking confirmation of the theory, exceptof course as regards the numerical multiplier which does not affectthe values obtained for ® u , 3)22, etc.

Somewhat similar although not entirely identical results wereobtained by Boardman and Wild,* particularly experimentingwith pairs of gases in which the molecules were nearly similar, sothat the coefficient of interdiffusion of a pair of gases could beidentified with the coefficient of self-diffusion of either. They ob-tained the following coefficients of self-diffusion, all at 15° C :

Hydrogen 1-43Nitrogen 0-203Carbon-monoxide 0*211Nitrous oxide 0-107

Carbon-dioxide 0-121

but their experiments did not yield completely consistent results.

• Proc. Roy. Soc. 162 A (1937), p. 611.

216 DIFFUSION

177. It remains to test the numerical multiplier. The calcula-tions of Chapman, Enskog and Pidduck combine in predictingthe relation (approximately)

® = 1-200^P

for elastic spheres, while Maxwell's theory given in Chap ixbelow predicts the relation (exactly)

© = 1-543^P

for molecules repelling according to the inverse fifth-power of thedistance.

In the following table, the second column gives the value as-sumed for 7] in the table of p. 183, the third column gives p, thefourth gives the value of 2) calculated from Loschmidt's experi-ments, and the fifth gives the value

Gas

HydrogenOxygenCarbon-monoxideCarbon-dioxide

V (P. 183)

0-00008570-00019260-00016650-000137

P

0-00008990-0014290-0012500-001977

D(p.215)

1-311-18901740-109

V

1-371-401-301-58

Within the probable error of the experiments, it appears that%plrj has values intermediate between the two values 1-200 and1*543 predicted by theory for elastic spheres and inverse fifth-power molecules. Not only is this so, but the values of ®p/?/ varybetween these limits in a manner which accords well with theknowledge we already have as to the laws of force (fir~8) in thedifferent gases concerned, as the following figures shew:

TheoryHydrogenCarbon -monoxideOxygenCarbon-dioxideTheory

Value of s(P- 174)

OO

1138-77-45-65 0

topV

1-2001-371-301-401-581-543

DIFFUSION 217

It is somewhat remarkable that the values oi%pjrj for hydrogenand carbon-monoxide, in which the molecules are comparatively"hard" (in the sense of § 147), approximate so closely to the value1-34 predicted both by Meyer's corrected theory (§ 169) and bythe Stefan-Maxwell theory (§171). This suggests that for ordinarynatural molecules the two simpler theories may represent theprocesses at work with as great accuracy as the more elaboratetheories of Chapman and Enskog so long as we remain in ignor-ance of the exact molecular structure which ought to be assumedin the latter.

Coefficient of Diffusion for Elastic Spheres

178. The following table gives the observed values of ®12 fora number of pairs of gases in which the molecules are com-paratively hard, having values of s greater than 7-4 in the table ofp. 174. The table gives also the values of S12 calculated from themby formula (260) (using values of c given on p. 56), and, in thelast column, the values of S12 calculated from the coefficient ofviscosity as on p. 183.

Gases

Hydrogen—Air—Oxygen

Oxygen—Air„ —Nitrogen

Carbon-monoxide—Hydrogen„ —Oxygen

(observed)

0-6610-679017750-1740-6420-183

(calc. from£12)

3-23 x 10-8

3183-693-743-283-65

(calc. fromviscosity)

3-23 x 10-8

3173-683-703-253-70

The agreement between the two sets of values of S12 is as goodas could reasonably be expected, providing a corresponding con-firmation of formula (260). When one or both of the two kinds ofmolecules involved is softer than those in the foregoing table theagreement is still good, although less striking than that foundabove, as is shewn in the table below.

8 JKT

218 DIFFUSION

Gases

Carbon-dioxide—Hydrogen„ —Air„ —Carbon-monoxide

Nitrous oxide—Hydrogen„ —Carbon-dioxide

Ethylene—Hydrogen„ —Carbon-monoxide

(observed)

0-5380-13801360-5350-09830-4860-101

(calc. from£12)

3-56 xlO-8

4034093-574-533-754-99

(calc. fromviscosity)

3-69 x lO"8

4-204-223-694-664-144-68

Thermal and Pressure Diffusion

179. In 1911 Enskog* predicted on theoretical grounds thatdiffusion must occur in any gas in which the temperature is notuniform throughout. The same is true when there is a pressuregradient, although this "pressure-diffusion" is of less practicalimportance than the "thermal diffusion" just mentioned. Chap-man| predicted the same phenomena independently in 1916, andDootsonf confirmed the existence of thermal diffusion in 1917.The discussion of these phenomena is postponed to the end ofthe next chapter.

RANDOM MOTIONS AND B R O W N I A N MOVEMENTS

180. In § 154 we obtained the equation of conduction of heatin a gas in the form

— = ^ — , (268)

while in § 165 we obtained as the equation of diffusion (equation(243))

^ 7 = ^ 1 2 ^ . (269)

Replacing ®12 by the value r/jp appropriate for self-diffusion,this becomes

dvi 7j 92*>1

dt p dz2 "(270)

• Phys. Zeitschr. 12 (1911), p. 533.f Phil.Trana.217A(1916),p. 164. % Phil.Mag.33(1917),p. 248.

DIFFUSION 219

We notice at once that this is of the same form as the equationof conduction of heat (268), so that heat is diffused through a gasat the same rate and according to the same laws as any selectedgroup of molecules. There is, of course, a reason for this. If therewere no collisions, both heat and molecules would proceedthrough the gas at the same speed, the speed of molecularmotion; when collisions impede progress, they affect the progressof both heat and molecules in the same way.

It will be understood that here, as throughout the remainderof this chapter, we are studying the phenomena in their broadoutlines only, and are not striving after numerical exactitude.The true value of ® 12 differs from the value rjjp we have assignedto it by a numerical factor, but we do not concern ourselves withthis in the present discussion.

The simplest solution of equation (269) is

A --£-

where A is any constant. This is the solution for a group ofmolecules which start from the plane z = 0 at the instant t = 0,and gradually diffuse through the gas. There are, of course, pre-cisely similar solutions to equations (268) and (270). We shallnow see how the same equations and solutions could have beenobtained directly from a detailed study of the molecular move-ments in the gas.

Random Motions

181. Let us consider the random motion of a single moleculeas it is hit about by collisions with other molecules. As we areconcerned with general principles rather than numerical exacti-tude, let us simplify the problem to the extreme limit by supposingthat all free paths are of the same length Z, and that all aredescribed parallel to the axis of z; we shall further suppose thatfor each free path the directions of z increasing and of z decreasingare equally likely.

Any molecule which has described a number N of free pathswill have advanced along the axis of z by a distance

±l±l±l± ... (to N terms).

220 DIFFUSION

Each of the signs in this series may be either + or —, so thatthere are 2^ possibilities in all. For some arrangements of signs,the molecule will have advanced a distance si along the axis of z.For this to be the case, there must be p positive and q negativesigns, where p, q are given by

p + q = N, p-q = s. (272)

The number of ways in which + and - signs can be arrangedto give this result is of course

N\p l q l 9

so that the chance P that the molecule shall have advanced adistance si after N free paths is

P = - ^ . (273)p! q!2N

This provides a solution to what is commonly known as theproblem of the " random walk ". A man takes a walk of N stepsof a yard each, and each step is as likely to be forward as back-ward; what is the chance that the end of his walk will find hims yards in front of his starting point? Obviously the answer isgiven by formula (273).

182. In the applications of this formula to kinetic theoryproblems, N9 p and q are all large numbers, so that the value of Pcan be simplified by using Stirling's approximation

to the value of n! when n is large. Using this relation, equation(273) takes the form

From equations (272) we find

DIFFUSION 221

so that P=ST 7 \7TF7 \3+i (274)

When N is made very large, the forward and backward pathsmust nearly cancel one another, so that in general s will be smallcompared with N, and p + \, q + \ may each be put equal to \N.Thus we may write

N)i-sY-(i = e2N

and P, as given by formula (274), assumes the approximatevalue

V nN(275)

This is the chance that after describing N free paths each oflength I, the molecule shall have advanced a distance si.

183. Let us now replace si by z and N by ctjl, where c is thespeed of molecular motion and t the time the motion has pro-gressed. Formula (275) becomes

^ ~ 2 " ' ( 276 )

which is in all essentials identical with the equation (271) alreadyfound. Equation (271) gives the density of molecules at adistance z from the plane z = z0, while equation (276) gives theprobability that any single molecule shall be found at thisdistance from z = z0. Clearly the values of P and vx must be thesame except for a multiplying constant.

For the exponentials to be the same, we must have

© = 1 ^ = ^, (277)

where rj has the uncorrected value obtained in equation (47).Thus the analysis just given would have enabled us to calculate% accurately except for the numerical adjustments required byvariations in the lengths of free path, persistence of velocities,and so forth. Its true importance is, however, that it presents

222 DIFFUSION

us with a vivid picture of the process of diffusion; we now seediffusion as a random walk.

When P is given by formula (276), the graph of P as a functionof z is an exponential curve like the thin curve in fig. 21 (p. 118).The maximum ordinate is always at the origin—no matter howlong diffusion goes on for, initial condensations are never entirelysmoothed out.

Using the formula of Appendix vi (p. 306), we readily findthat the average numerical value of z is (Zlctj-n)* or 0-798 Viet.Thus as diffusion proceeds, the average, or any other propor-tionate, value of s increases only as \lt; travelling at random, wemust take four times as many steps to travel two miles as totravel one.

Brownian Movements

184. The foregoing ideas are applicable to the Brownianmovements, although in a somewhat modified form. EachBrownian particle will perform a sort of random walk, but as itsmass is much greater than that of the particles it encounters,persistence of velocities becomes all important.

Following Einstein* and Smoluchowski,f we may treat allcollisions with other molecules as forming a statistical group,and regard the Brownian particle as ploughing its way througha viscous fluid. We accordingly suppose the particle to ex-perience a viscous drag in the direction opposite to its motion.We may suppose this viscous force to have components

dx dy dz~evdi' -ei}di' -67>dt>

where TJ is the coefficient of viscosity and e is a second coefficientwhich depends on the size and shape of the particle, as well as onthe physical properties both of the particle and of the fluidthrough which it moves. The equations of motion of the particleare now

d T n orX-eV^ = m-dT2,etc, (278)

* Ann. d. Physik, 17 (1905), p. 549 and 19 (1906), p. 371.t Bulletin Acad. Sci. Cracovie (1906), p. 577.

DIFFUSION 223

where X, Y, Z are the components of any extraneous forceswhich may act on the particle.

Exactly as with the virial equation of Clausius, we multiplythese three equations of motion by x, y, z, and add correspondingsides. This gives, after slight transformation,

}~(x2 + y2 + z2) = ^m^(x2 + y2 + z2)-mcK

Integrating through an interval of time t, and averaging overa number of molecules, we obtain

-\erjA{x2 + y2 + z2) = -m?t= -SRTt,

where A (x2 + y2 + z2) denotes the change in x2 + y2 + z2 in time t.Thus, finally,

A(x2 + y2 + z2) = *^t. (279)

Again, as in the random walk, the distance travelled in a time tis proportional to <\/t. It is also, naturally enough, proportionalto V^\ o r to c. Both these relations have been confirmed ex-perimentally by Perrin and many others.

If the Brownian particles are spheres of radius a, Stokes's lawgives the value of e as Gna, and equation (279) assumes the form

A(x2 + y2 + z2) = —U (280)

This equation has been used as a means of determining R,and hence Loschmidt's and Avogadro's numbers, with satis-factory results (cf. § 16, above).

185. So far we have discussed only the average value ofx2 + y2 + z2 for all the particles. The distribution of the individualparticles can be discussed in a similar manner, but is best foundby treating the motion as one of diffusion. As in §180, thedistribution of density must satisfy an equation of the form

+W w)' ( >where K is a constant, which is of the nature of a coefficient ofdiffusion.

224 DIFFUSION

Analogous to the solution (271) of our former equation, weobtain as a solution of equation (281),

v = je *K* , (282)

where A is a constant. For such a distribution of particles, wereadily find that the average value of x2 + y2 + z2 is 6Kt. Thesolution (282) is appropriate to a group of particles starting atthe origin at time t = 0, and gradually diffusing through space;for such a group we have already found the average value of

n f>rp

x2 + y2 + z2 to be 1 (equation (279)). Equating this to 6Kt,erj

we find

€71

This, then, is the value of K in the equation of diffusion (281),and with this value of K formula (282) will give the distributionin space of a group of particles which started at the origin atthe instant t = 0. On assigning this value to K in the moregeneral equation (281), we can trace the motion resulting fromany distribution of Brownian particles.

Chapter I X

GENERAL THEORY OF A GAS NOT INA STEADY STATE

186. The present chapter will deal with a variety of problemsin which the gas is not in a steady state, so that the velocities ofthe molecules are not distributed according to Maxwell's law.In particular we shall discuss the analogy between a gas and thefluid of hydrodynamical theory, and shall examine new methodsof dealing with the viscosity, heat conduction and diffusion ofa gas. The distribution of velocities will be supposed to be givenby the quite general law

f{u,v9w,x9ytz),

where the symbols have the same meaning as elsewhere in thebook, particularly in § 22.

Hydrodynamical Equation of Continuity

187. We first fix our attention on a small element of volumedx dy dz inside the gas, having its centre at £, TJ, £ and bounded bythe six planes parallel to the coordinate planes

The number of molecules of class A (defined as on p. 105) whichcross the plane x = £ — \dx into this element of volume in time dtwill be

vf{u> v>w>£> — \dx, rjy Q dydzdudvdwudt.

Similarly the number of molecules of class A which cross theplane x = E, + \dx out of the element of volume is given by

vf(u, v9wy£ + \dx, rj, £) dydzdudvdwudt.

By subtraction, we find that the element experiences a net lossof molecules of class A, caused by motion through the two facesperpendicular to the axis of x, of amount

w-[vf(u, v, w)]dxdydzdudvdwudtf

226 GENERAL THEORY OF A GAS

in which the differential coefficient is evaluated at £, TJ, £. The netloss of molecules of class A caused by motion through all six facesis therefore

u^- + v-^--\-Ww-\ [vf(u, v} w)]dudvdwdxdydzdt. ...(283)

On integrating over all values of u, v and w, we obtain the totalnumber of molecules which are lost to the element dxdydz intime dt. We may again write

uf(u, v, w)dudvdw = u0, etc.,

where uOi v0, w0 are the components of mass-velocity of the gas atx,y,z; this number is now seen to be

(284)

The number of molecules in the element at time t is, however,

vdxdydz, and that at time t + dt is tv + -j-dt\dxdydz. Thus the

net loss must be

Equating this to expression (284), we obtain

— -j-dtdxdydz.

l i l l - ° <2 8 5 )

If we multiply throughout by m, the mass of a molecule, andreplace vm by p, the density of the gas, this becomes

S+i^+i^+l*^ - ° ( 2 8 6 >This is the hydrodynamical equation of continuity, expressing

the permanence of the molecules of the gas—in other words, theconservation of mass.

The Hydrodynamical Equations of Motion

188. There is a somewhat similar equation expressing thatmomentum is conserved, or is changed only by the operation ofexternal forces.

NOT IN A STEADY STATE 227

Each molecule of class A carries ^-momentum of amountmu, so that the loss of a molecule of class A to the elementdxdydz involves a loss of -momentum equal to mu. The numberof molecules of class A which are lost to the element in time dt isgiven by expression (283), so that the loss of a-momentum fromthis cause is mu times expression (283) and the total loss is

mdxdydzdt \tu2^- + uv-^- + uw^-1 [vf(u, v, w)]dudvdw.JJJ\ ox cy dzf

This may be written in the form/ d — d — d — \

mdxdydzdt I-^-(vu2) + -^-(vuv) + w-(vuw) I, (287)

where u2 = ^2/(w, v, w)dudvdw,

uv = w/(w, v, w)dudvdw9 etc.,

so that u2, uv, etc. are the mean values of u2, uv, etc. at the pointA, y, z.

There is a further loss or gain of momentum from the action offorces on the molecules inside the element dxdydz. A force Xacting on a molecule in the direction of Ox causes a gain of x-momentum Xdt in a time dt. Combining the sum of these gainswith the loss given by expression (287), we find for the net in-crease of momentum inside the element dxdydz in time dt,

t, (288)

where Z denotes summation over all the molecules which wereinside the element dxdydz at the beginning of the interval dt.

The total ^-momentum inside the element dxdydz at time twas, however, mvuodxdydz. The gain in time dt is accordingly

-=- (vu0) mdxdydzdt,Cut

and on equating this to expression (288), we obtain-ft g 9 g ~i-j (vu0) + g- {vu2) + g- {vuv) + ^ (vuw) mdxdydz = EX.

(289)


These and the similar equations in y, z are the equations of motionof the gas, expressing that momentum is changed only by theaction of external forces.

189. In this equation, let us write

u = uo + U, etc.,

thus dividing the motion into the mass motion of the gas as awhole, and the thermal motions of the molecules. We haveu = uOi and U = 0, so that uv = uovo + U\/, and similarly for u2

and uw.If we multiply equation (285) throughout by u0, and subtract

corresponding sides from equation (289), the latter equationbecomes

d d

(290)

We proceed to examine more closely the system of forceswhich act upon those molecules of which the centres are insidethe element dxdydz—the system of forces which we have so farbeen content to denote by EX, EY, EZ.

These forces will arise in part from the action of an externalfield of force upon the molecules of the gas, and in part from theactions of the molecules upon one another.

If there is a field of external force of components S, H, Z perunit mass, the contribution to EX will be

Smv dxdydz. (291)

The remaining contribution of EX arises from the inter-molecular forces. As regards the forces between a pair of mole-cules, both of which are inside the element dxdydz, we see that,since action and reaction are equal and opposite, the total contri-bution to EX will be nil. We are left with forces between pairs ofmolecules such that one is inside and the other outside theelement dxdydz, that is to say, intermolecular forces which actacross the boundary of this element.


Let the sum of the components of all such forces across a planeperpendicular to the axis of x be iPxx>mxy>mxz P e r u m ^ area,and let us adopt a similar notation as regards forces acting acrossplanes perpendicular to the axes of y and z. Then the contributionto ZX from forces acting across the two planes x = £ ± \dx is

On adding similar contributions from the other planes, wefind that the ^-component of all the molecular forces which acton the element dxdydz can be put in the form

\ dxdx dy dzCombining this contribution to ZX with that already found in

expression (291), we find, as the whole value of ZX,

and there are, of course, similar equations giving the values ofZY and SZ.

Inserting this value for EX into equation (290), dividingthroughout by dxdydz and replacing mv by p, we obtain

fd d d d'

This may be compared with the standard hydrodynamicalequation,

where u0, v0, w0 is the velocity of the fluid at x, y, z, and Pxx, Pxy, Pxs

are the components of stress per unit area on unit face perpen-dicular to Ox.

We see at once that the gas may be treated as a hydrodynamicalfluid with stress components given by

,etc.J ( 2 M )


Gas in a Steady State

190. In the special case in which the gas is at rest and in asteady state, the molecular velocities will be distributed accord-ing to Maxwell's law at every point, so that

/>U2 = pu2 — vmu2 = -£=-,

as in § 88, andUV = VW = WU = 0.

Thus of the system of pressures specified by equations (292),that part which arises from molecular agitation reduces to asimple hydrostatical pressure of amount v/2h. Clearly the wsystem of pressures, arising from the intermolecular forces, mustalso reduce to a simple hydrostatical pressure m, and we thereforehave

*xx = *yy ~ *zz — 2Ji9

p = p = p =o^xy *xz *yz w #

The total pressure at the point x, y, z is therefore a simplehydrostatical pressure of amount Pgiven by

r, (293)

an equation which may be compared with the virial equation (80).

MaxwelVs Equations of Transfer

191. When we discussed the transfer of molecules across theboundary of a small volume dxdydz in § 187, we arrived at thehydrodynamical equation of continuity. When we discussed thetransfer of momentum in § 188, we arrived at the hydrodynamicalequations of motion. Let us now, following a procedure firstdeveloped by Maxwell,* consider the problem of the transfer ofany quantity Q which depends solely on the velocity of a mole-cule, as for instance u2 or uv.

• Phil. Trans. 157 (1386), p. 1, or Collected Works, 2, p. 26.


Let the mean value of Q at any point z, y, z be denoted by Q,so that

Q =11 \f(u, Vy w) Qdudvdw.

The number of molecules inside the small element of dxdydzat this point is vdxdydz, so that ZQ, the aggregate amount of Qinside the element, is given by

ZQ = vQdxdydz. (294)

There are various causes of change in ZQ. In the first place

some molecules will leave the element dxdydz, taking a certain

amount of Q with them. As in § 187, the total number of mole-

cules of class A lost to the element dxdydz in time dt is

u-^- + v-^- + Ww-\ {vf)dudvdw,

so that the total amount of Q lost by motion into and out of theelement is

dxdydzdt lu-^ + v-^ + w-^-] (vf) Qdudvdw.

If we writerr

\uQf(u, v, w)dudvdw = uQ, etc.,

so that uQ is the mean value of uQ averaged over all the moleculesin the neighbourhood of the point x, y, z, this loss of Q can be putin the form

~(v^) + (v^)] (295)

A second cause of change in ZQ is the action of external forceson the molecules. For any single molecule, the rate of change inQ from this cause is given by

dt du dt dv dt dw dt m \ du dv dw/'

where X, Y, Z are the components of the external force acting on


the molecule. Thus in time dt the total value of ZQ experiences anincrease

'3<3\"] ^ ^

where again the bar over a quantity indicates an average takenover all molecules.

Lastly, ZQ may be changed by collisions between molecules.If Q is any one of the quantities which have previously beendenoted by Xi>X2> ••• %5> namely, the mass, energy, and the threecomponents of momentum of a molecule, there is no such change,but if Q is any other function of the velocities such changes willoccur. In general let us denote the increase in ZQ which is causedin the element dxdydz by collisions in time dt by

dxdydzdtAQ. (297)

Then the total change in the aggregate value of Q for all themolecules in dxdydz is given by the sum of expressions (295),(296) and (297). The aggregate value of Q is however given byexpression (294). Equating the change in this to the sum of thethree contributions just calculated, we obtain

d . ^

This is the general equation of transfer of Q. If we put Q = m,the mass of a molecule, the equation reduces to (286), the hydro-dynamical equation of continuity, while if we put Q = mu, the^-momentum, the equation becomes identical with equation(289), the equation of motion of the gas.

If we multiply both sides of equation (285) by Q and subtractfrom corresponding sides of equation (298), this latter equationassumes the alternative form

4 -where Z denotes summation with respect to x, y and z. This givesa new, and more useful, form of the equation of transfer.


The difficulties of the equation are centred in the calcula-tion of AQ. Maxwell gave a general method for the calculationof AQ, which is explained in § 196 below, but found himself re-stricted by mathematical difficulties to the particular case inwhich the molecules attracted or repelled according to the lawof force /ir~5.

On identifying Q with various functions of u, v and w,Maxwell obtained equations which were suitable for the discussionof the problems of viscosity, conduction of heat, and diffusion.The numerical results he calculated have already been quotedin their appropriate places.

In 1915 Chapman, in a paper of great power, succeeded in ex-tending Maxwell's analysis, so as to apply to the general case ofmolecules repelling according to the law fir"8. Chapman's analysisis unfortunately too long to be given here even in outline, but hismain results have already been quoted and discussed. They in-clude of course Maxwellian molecules (s = 5) and elastic spheres(s = oo) as special cases.

Time of Relaxation

192. Maxwell's general equation of transfer (299) assumes itssimplest form when the gas has no mass-motion, so that

u0 = v0 = w0 = 0,

and when no external forces act, so that X — Y = Z = 0. If thedistribution of velocities is the same throughout the gas, theequation takes the form

expressing that the whole change in Q is caused by collisions.Taking Q = u2 — v2, in this equation, Maxwell calculated that

where r is a constant depending only on the structure of themolecules. Similarly, on taking Q = uv, he calculated that

AQ = —TV2UV9


where r is the same constant as before. Thus it appears that

u2 — v2 and uv both satisfy an equation of the form

of which the solution,

<t> = ft,*"**. ( 3 0 ° )shews that <j> decreases exponentially with the time at such a ratethat it is reduced to 1/e times its original value in a time l/rv.Maxwell calls this the "time of relaxation".

This time of relaxation measures the rate at which deviationsfrom Maxwell's law of distribution will subside. A glance atequations (292) shews that it must also measure the rate at whichinequalities of pressure must subside, as also the shear stressesPyx> Pxy> e t c -

Maxwell shewed that r is related to the coefficient of viscosity yby the relation

where p is the hydrostatic pressure, and this provides thesimplest means of evaluating r. Replacing p by its usual valuevBT, this relation becomes

D/7T

r = — . (301)V

To take a definite instance, the value of r/ for air at 0° C. is0-000172, and ET = 3-69 x 10~14, so that r = 2-15 x 10"10, andthe time of relaxation is

1 1 ,— = 7L—TT^ seconds.rv 6 x 109

It is, as we should expect, comparable with the time of describinga free path. Thus by the time that a few free paths have beendescribed by each molecule, the exponential on the right hand ofequation (300) has already become very small, so that the gas isalmost in the state specified by Maxwell's law.


The Lorentz-Enskog Analysis

193. In 1906, many years after the publication of Maxwell'swork, an investigation given by Lorentz* for a different purposesuggested a new line of attack on Maxwell's problem. This wasdeveloped and carried to a successful termination by Enskogf in1917.

When a gas is not in equilibrium or in a steady state, thedistribution of velocities will no longer be given by Maxwell'slaw. If, however, the law of distribution / i s known at the instantt, it is clearly possible in principle to follow out the motion of eachgroup of molecules, calculate the changes in/, and so obtain thelaw of distribution at the next instant t + dt, and similarly atevery subsequent instant. Thus the law of distribution is de-termined for all time when its value is given at any one instant.The function / must, then, satisfy an equation giving df/dt interms of / . We proceed to investigate the general form of thisequation, following a method given by Boltzmarm.J

194. Let the molecules be supposed to move in a permanentfield of force, such that a molecule at x, y, z is acted on by a force(X, F, Z) per unit mass. Then the equations of motion of amolecule, apart from collisions, are

du dv dw _dt~X' dt~ r > ~dt~L-

The number of molecules which at any instant t have velocitycomponents u, v, w within a small range dudvdw, and co-ordinates x, y, z within a small range dxdydz, will be taken to be

vf(u, v, w, x, y, z, t)dudvdwdxdydz (302)

Let these molecules pursue their natural motion for a time dt.At the end of this interval, if no collisions have taken place in themeantime, the u, v, w components of velocity of each moleculewill have increased respectively by amounts Xdt9 Ydt, Zdt,

* Theory of Electrons, Note 29, Teubner, Leipzig (1906) and DavidNutt, London (1909).

f Kinetische Theorie der V or gauge in mdssig verdiinnten Gasen (Inaug.Dissertation, Upsala, Almquist, 1917).

{ Vorlesungen iiber Gastheorie, 1, Chaps, n and in.


while the coordinates x, y, z will have increased respectively byamounts udt, vdt, wdt.

It now follows, as in § 22 above, that if no collisions occur,these molecules will have their velocity-components and space-coordinates lying within a small range dudvdwdxdydz sur-rounding the values u + Xdt, etc., and x + udt, etc. The numberof molecules within this range at the end of the interval dt is,however,

vf(u + Xdt, v+ Ydt, w + Zdt, x + udt, y + vdt, z + wdt, t + dt)x dudvdwdxdydz, (303)

and if no collisions occur, this expression must be exactly equalto expression (302). Expanding it as far as first powers of dt, andequating to expression (302), we obtain the relation

du dv dw dx dy dzj(304)

expressing the rate at which vf changes on account of the motionof the molecules, and the forces acting on them.

When collisions occur, these produce an additional change invf—let us say an increase at a rate

coll.

per unit time. On combining the two causes of change in (vf), wearrive at the general equation

cw c# 9w 9a; dy

+ r|(Hf)l (305)L ^ Jcoll.

This equation must be satisfied by vf under all circumstances.When the gas is in a steady state the right-hand member must ofcourse vanish.

No progress can be made with the development or solution

of this equation until the term -=- (vf) has been evaluated,L^ Jcoll.

and this unfortunately can only be effected to a very limited


extent. We saw in §§ 85, 86 that when the molecules are elasticspheres of diameter cr, this change is expressed by the equation

= j\\\\Aff' ~//') Va2 cos Odu'dv'dw'do).(306)

Let us now attempt to evaluate ^-(vf) when the mole-L^ Jcoll.

cules are regarded as point centres of force, attracting or re-pelling with a force which depends only on their distance apart.

We fix our attention on an encounter between two molecules,moving, before the encounter, with velocities u, v, w and u\ v\ w',and so with a relative velocity F, given by

F 2 = (uf -u)2 + (v' -v)2+(w' -w)2 (307)In fig. 32 let 0 represent the centre of the first molecule movingin some direction QO with a velocityu, v, w, and let MNP represent the pathdescribed relatively to 0 by the secondmolecule before the encounter begins.When the second molecule comes towithin such a distance of 0 that theaction between the two molecules be-comes appreciable, it will be deflectedfrom its original rectilinear path MNP,and will describe a curved orbit suchas MNS, this orbit being of course inthe plane MNPO.

Let ROP be a plane through 0perpendicular to MN, and let MNmeet this plane in a point P. Let the polar coordinates of Pin the plane ROP be p, e, the point 0 being taken as origin, andany line RO as initial line. Clearly p is the perpendicular fromthe first molecule 0 on to MN, the relative path of the secondmolecule before encounter.

Let us calculate the frequency of collisions in which the secondmolecule has a velocity u\ v\ w' whose components lie within asmall specified range du'dv'dw', while its path before the en-counter is such that p, e lie within a small range dp, de. For such

Fig. 32


a collision the line MP must meet the plane ROP within a smallelement of area pdpde. Thus the number of such collisionswithin an interval dt will be equal to the number of moleculeswhich lie within a small volume pdpdeVdt, and have velocitieswithin the specified range du'dv'dw'. This number is

vf{u\ v\ w')du'dv'dw'pdpdeVdt (308)The number of molecules per unit volume having velocities

between u and u + du, v and v + dv, w and w + dw isvf(u, v, w)dudvdw,

so that the total number per unit volume of collisions of the kindwe now have under consideration is

v2f(u, v, w)f(u', vr, w')dudvdwdu'dv'dw'pdpdeVdt (309)The type of collisions now under consideration is similar to

that which we called type a in §85, and expression (309) isobviously a generalisation of our former expression (117). Pro-ceeding precisely as in § 86, we obtain, just as in formula (306),

(310)This, then, is the required generalisation of equation (306).

It clearly reduces to this latter equation for elastic spheres, thefactor a2cosddo) of equation (306) being exactly the factorpdpde of equation (310).

On substituting this value into equation (305), we obtain asthe characteristic equation which must be satisfied by/,

Lvf) = -[X^-+ Y^ + Z J^ + U A + « 1 + wf\(vf)dt [_ du dv dw dx dy dzj J'

+ f f f \[v\Tf'-Sr) Vdu'dv'div'pdpde (311)

For a mixture of gases, in which the different kinds of moleculesare distinguished by the suffixes 1,2,..., we obtain in a similarway a series of equations such as

v2(fj2-fj'2) Vdu'dv'dw'pdpde (312)


195. The condition for a steady state is that the right-handmembers of equations (311) and (312) shall be equal to zero.

If we try puttingf — Ae-hmKu-UoP-Hv-VoP-Ww-w^^ (313)

the integrals on the right of equations (311) and (312) vanish,so that this value of/ provides a solution when X = Y = Z = 0,and y/is independent ofx, y, z. It is of course the solution alreadyfound in §87.

On substitutingf AHi**^ (3X4)

we again obtain a solution, provided

X- dX Y_ dX 7_ tydx dy dz

Thus (314) is a solution when x *s the potential of the forcesacting on the molecule, the result obtained in § 23.

If, however, u0, v0, w0, h and v vary from point to point,formulae (313) and (314) do not provide a solution, for on substi-tuting them into the right-hand member of equation (311) wefind that the second term vanishes, while the first does not.

The case in which u0, v0J w0, h and v vary only slightlyfrom point to point is especially important for the study ofviscosity, conduction of heat and diffusion. In this case we mayassume, as an appropriate solution

where 0 is a small quantity of the first order, andf = Ae-hm[{u-uQ¥Mv-Vo)2+{w-wQnt (316)

Since / = f0 is a solution when u0, v0, wQ) h and v do not varyfrom point to point, equation (315) must necessarily provide asolution when these quantities vary to the first order of smallquantities.

The right-hand member of equation (312) contains the termv\^2/1/2 of which the value, by equation (315), is

Here/01 denotes the value of/0 for a molecule of the first kind,


and so on, and the product 0t0'2 *s omitted as being of the second

order of small quantities. Similarly

W i / i = 1 2/01/02(1 + ^1 + ^2) (318)

The conservation of energy and momenta at an encounter givethe relation

foJ'o*=foifo2, (319)

so that v1v2(fj2-fj'i) = Wu/i^+ff;-*!-*;).On substituting the tentative solution (315) into equation (312),/may be replaced by/0 everywhere except in the integrals; for ifwe retained terms in 0 in the remaining parts of the equation, weshould be including terms of the second order of small quantities.Equation (312) accordingly reduces to

(320)where

'2" 01 ~ 0 i ) Vdu'dv'dw'PdPde>

(321)

an equation in which every term is of the first order of smallquantities.

On dividing out by vxf01 and replacing /01 by its value fromequation (316), this equation becomes

/a ^ a v a _ a a a a\ot on ovCW 0W C7# Cl/ 02/

x {log vxA- hm[(u - w0)2 + (v - v0)

2 + (^ - ô)2]} = J-(322)

The solution (315) is indeterminate in the sense that changesin uQy v0, w0, vA and h are not independent of changes in 0. Forinstance, the total momentum in the direction Ox is readilyfound to be

+ m \ vfo&ududvdw (323)

and the same change can be made in this either by changing u orby changing 0.


Let us agree that u0, v0, w0, vA and h are to have the samephysical interpretation in solution (315) as they have in the steadystate solution for which 0 = 0, e.g. the velocity of mass-motionof the gas is to have components u0> v0, w0, and so on. This makesthe value of 0 quite definite for any assigned physical conditions.But for this condition to be satisfied, the second term in ex-pression (323) must vanish, with similar equations in v and w,and 0 must also satisfy the equations

tihvfo0dudvdw = O, (324)

= 0 (325)

With 0 restricted in this way, the equation of continuity isagain given by equation (285), so that

in which small quantities of the second order, such as u0 - areox

neglected. We now multiply both sides of this equation byfhmc2, add to the corresponding sides of equation (322), andfind that the latter equation reduces to

(1 + f hmc2) j log v + j t {| log h - hm[(u - u0)*

+ {v - v0)2 + (ti? -1^0

2]} - 2hm[(u -uo)X + (v- v0) Y + (w- w0) Z]

dy

dh dh Bh\ T

d-x+v^+wTzJ

= I' <3 2 7>where / is given by equation (321).

It will be remembered that this equation is only accuratewhen 0 satisfies five relations, expressed by equations (323),(324) and (325). The solutions in 0 will however be additive, since


the equations are linear; five solutions which contribute nothingto either side are

0 = 1 , mu, mVy mw, me2,

so that to any solution for 0 which satisfies equation (327) maybe added terms of the form

0 = B+ Cmu + Dmv + Emw + Fmc2,

and the constants B, C, D, E, F may be adjusted so as to satisfythe five necessary conditions.

Law of Force /ir~*

196. The next step must be to calculate / , as given by equation(321), and we can only do this by assuming definite laws for theinteraction between molecules at collisions. We therefore supposethat the molecules are centres of force repelling according to thelaw/^r-*.*

If two molecules of masses mv ra2 at a distance r apart exert arepulsive force of this amount, then it is easily shewn that themotion of either relative to the other is that of a particle of unitmass moving about a fixed centre of force, the potential energywhen the two are at a distance r apart being

As usual, the differential equation of the orbit is

2r* \\dd) } ~where r, 6 are now polar coordinates in the plane of the relativeorbit, h is the angular momentum pV, and C = |F 2 , where V,the relative velocity before the encounter begins, is identical withthe usual "velocity at infinity".

This equation has the integral

dr

h2 m1mi(8-l)K2'

in which the addition of a constant of integration is avoided by

* The method of § 196 is that of Maxwell, Collected Works, 2, p. 36.


taking the direction of the asymptote to the orbit to be the initialline 6 = 0.

Replacing h by pV, and further writing 7/ for pjr, so that rj is apure number, this becomes

drj0 =

where

(328)

(329)

The apses of the orbit are given by dr/dd = 0, and therefore by

5 - 1 WA simple graphical treatment shews that, when s is greater than

unity, this equation can only have one real root. Call this root9/0, then the angle, say 60i between the asymptote and the apsidaldistance will be given by equation (328) on taking the upper limitto be 7j0. The angle between the asymptotes, say 6\ is equal totwice this, and so is given by

After the encounter, the velocities par-allel and perpendicular to the initial lineare of course — V cos 6' and — V sin 6''.

For any value of s, there will naturallybe a doubly infinite series of possible orbitscorresponding to different values of p andF. Except for a difference of linear scale,however, these may be reduced to a singlyinfinite system corresponding to the varia-

2

tion of a or pV8'1. Fig. 33* shews somemembers of this singly infinite system forthe law of force fijrb. Fig. 33

• This figure is given by Maxwell, Collected Works, 2, p. 42. I am in-debted to the Cambridge University Press for the use of the original block.


In fig. 32 of § 194, the angle e was measured from an arbitrarilychosen line OR. Let us now suppose this to be the intersection ofthe plane POR with a plane through 0 containing the directionof NP and the axis of x, as in fig. 34.

In fig. 35, let OR, OX be the directions of this line OR and ofthe axis of x. Let OG be the direction of V, the relative velocitybefore collision, so that 0R> OX, OG all lie in one plane. Let theselines be supposed each of unit length, so that the points GXRlie on a sphere of unit radius about 0 as centre.

'A

Fig. 34

Let OY, OZ be unit lines giving the directions of the axes ofy and z, and let OG' give the direction of the relative velocity afterthe encounter. Then GOG' is the plane of the orbit, which is theplane NPO in fig. 34. Thus the angle RGGf is the e of § 194, whilethe angle GOG' is 6'.

Now suppose that, as usual, the velocities before collision areu, vy w and u\ v', w', so that the relative velocity V is given by

V2 = (uf - u)2 + (v' - v)2 + (w' - w)2,

and the velocities after collision are u, v, w and u', v',w'.From the spherical triangle G'GX,

cos G'X = cos GX cos G G' + sin G X sin GG' cos e,

in which we have cos G'X = — —=z—, cos GX = — r—, so that

u~-u' = (u' — u)cosd' - (u'-u)2 sin 6' cose (330)


Denoting the angle XOY by co2 and XOZ by (oz, we have, in asimilar way,

v-'v' = (vf - v) cos 0' + VF2 - (t/ - v)2 sin d' cos (e - (o2),

w — w' = (w' —w) cos d' + VF2 — (w' — w)2 sin 0' cos (e — a)3).

To determine CJ29 we notice that in the triangle OXY, XY = \nand XOY = OJ2; thus

(uf - tO (v' - v) + V[F2 - (uf -1^)2] [F2 - (v' - v)2] cos w2 = 0,

K - M) (w' - ti?) + V[F2 - (u' - w)2] [F2 - (wf - w;)2] cos co3 = 0.

These equations together with three equations of momentum,such as

m^ + mû' = rrtyû-\-m2u\ (331)

determine the velocities after collision.Eliminating u' from equations (330) and (331), we obtain

u = u + m% [2(uf - u) cos2 \d' + V F 2 - ^ - ^ ) 2 sin (9'cos e],

(332)

and there are of course similar equations giving v and w.

Solutions for 0

197. We are now in a position to proceed with the evaluationof / and the discussion of equation (327). From equation (329)we have

2 4_

pdpde = [(mx + ra2)/i/m1m2]8^XV

so that / = ZV2 [(mx + m2) H>\m\

where J stands for the quintuple integral

J = [[ i f [(&1 + 02-&1-02)adzdeV^1f'O2du'dv'dw'.

(333)

In this integral, it will be remembered that 01 is a function,as yet undetermined, of u, v and w; 0'2 is the similar function of


u', v\ wr for a molecule of the second kind, while &v 0'2 havecorresponding meanings in terms of the velocities after collision.Our problem is not to evaluate expression (333) for given valuesof 0, but to find values of 0 such that after integration expression(333) shall be equal to a certain algebraic function containingterms of degrees 0, 1, 2 and 3 in u, v and w, namely that on theleft of equation (327).

We see at once the simplification introduced by supposing,with Maxwell, that s = 5. In this special case, the powsr of Vdisappears entirely from the integral J .

In the more general case, as discussed by Lorentz* and Enskog, twe consider a tentative solution for equation (327) of the form

0 = ufaic).

With this value for 0, let the value of J as defined by equation(333) be denoted by

J = J(u),

where J(u), in addition to depending on u, depends on the variousconstants of the gas, ml9 ra2, h, etc. Similarly, let 0 = vcjy^c)lead to a value J = J(v) and let (j> = wtfi^c) lead to a valueJ = J(w).

Then a solution0 = {lu + mv + nw)^^) (334)

will clearly lead toJ = M(u) + mJ(v) + nJ(w). (335)

Now let Z, m, n be regarded as direction-cosines, so thatlu + mv + nw will be the component velocity along a certaindirection (I, m, n)\ it follows that the solution (334) will lead to avalue of J given by

J = J {lu + mv + nw), (336)

where J(lu + mv + nw) is the same function of lu + mv + nw asJ(u) is of u. Comparing (335) and (336), we have

J(lu + mv + nw) — U(u) — mJ(v) — nJ(w) = 0,

* Vortrdge iiber die Kinetische Theorie der Materie und der Elehtrizitdt(Leipzig, 1914), p. 185.

f Kinetische Theorie der V or gauge in mdssig verdiinnten Gasen (Inaug.Dissertation, Upsala, 1917).


whence it can be shewn that J(u) must be of the form

J(u) = uxM, etc.

Thus we have shewn that a solution

leads to a value of J of the form

By a somewhat similar method, Enskog shews* that a solution

leads to a value of Jof the form

while a solution & = uv<f>2(c)

leads to J =

198. Thus the solution of the general steady-state equationwill be

0 = fa(337)

where xjrl9 ^ 4 have the forms

and 2> ^3 a r e obtained from ^x by changing x into y, zrespectively, u into v, w and X into Y, Z.

Here ^(c), ^2(c) a r e functions of c and the constants of the gas

only. As we have already seen, they cannot be evaluated in finiteterms. Their expansion for a series of powers of c2 has been con-sidered by Enskog,f who has also calculated numerical results. J

We are now in a position to apply the analysis of the presentchapter to the exact solution of physical problems.

* Enskog, I.e. pp. 39, 40. t ^-c« Chaps, n and in.% Lx. Chap. iv.


Viscosity

199. In discussing viscosity, the gas may be supposed to be atthe same temperature throughout, so that

dh dh dhdx dy dz 9

and at uniform pressure throughout, so that

^v _ ^v _ ^v _ ndx dy dz

We may also suppose that no external forces are acting on thegas so that

X = Y = Z = 0.

As a consequence of this, r/rv i/r2, i/r3 are all equal to zero, andthe solution (337) reduces to

The law of distribution now becomes (cf. equation (315))

The pressures can now be calculated from equations (292).Neglecting the intermolecular pressures mxx, etc., these are foundto be of the form

<339»

where rj can only be calculated after ^(c) has been calculated.These are, however, precisely the equations of motion of a viscousfluid having a coefficient of viscosity r/.* Thus the pressures givenby equations (338) and (339) will be exactly accounted for byregarding the gas as a viscous fluid having a coefficient ofviscosity t\.

The value of rj has been calculated by Enskog,f who found thevalues already quoted in Chap. vi.

* Lamb, Hydrodynamics, p. 512. f L.c. pp. 88 ff.


Conduction of Heat

200. Consider next a gas which is in a steady state, and devoidof mass-motion, but which is not at a uniform temperature. Forsimplicity suppose that the gas is arranged in parallel strata ofequal temperature, so that the temperature is a function of zonly, and that no external forces act, so that X = Y = Z = 0.

In the general solution (337) of the steady state equation, wenow have ^ 2

= ^3 = ^4 — ®> s o

_ , 1 dp / 9 3 \ dh . . xr * I *> 3a; \ 2A/ 3o;j r i v 7

The relation between ^- and r - must be found from thedx dx

condition that there shall be no mass-motion, i.e. the process ofconduction must not be complicated by the addition of con-vection. To satisfy this condition, we must have

= 0.

On substituting for x}r1 in this equation, and carrying out the

integrations, we obtain a linear relation between ^- and x-.& dx dx

Using this relation, 0 is given as a multiple of ^-, or of

2hT2 dx'and from this it is easy to calculate the coefficient of conductionof heat, by the methods already used in § 199. The result is thatgiven in § 155.

Diffusion

201. To discuss the phenomenon of diffusion, we may in thefirst place suppose the gas to be at a uniform temperaturethroughout, and further take u0, v0, w0 all constant and X, Y, Zall zero. A sufficient solution of equation (337) is now seen to be

, , udv . . x0 ^ ^ { )


corresponding to diffusion in the direction of the axis of x. Theflow of molecules parallel to the axis of x, measured per unit areaper unit time, is

M — e-hmc \u ^r<f>i(c) \dudvdw.JJJ \n J \ vdx™ 7

The coefficient of diffusion is the coefficient of —dvjdx in thisexpression, and is found to have the values already quoted in§172.

Maxwell's treatment of the diffusion problem was based uponhis general equation of transfer (299), which assumes the form

^(^)-e^(^) = AQ, (340)

for steady motion parallel to the axis ofx. On putting Q = u andassuming the inverse fifth-power law, this equation becomes

^ = »vm 2 / -~ -AJUfio — um), (341)2hm1dx 1 2 2V m1m2(m1 + m) 1V 02 01/> v '1

where Ax is the constant introduced and defined in §172, anda comparison with the general equation of diffusion enabledMaxwell to determine the coefficient of diffusion.

The success of the method depended on the assumption of thelaw of the inverse fifth-power. Under this law An was found tobe proportional to u02 — u01 and the equation of diffusion followedat once. Under any other law Au is not proportional to u02 — u01;the value of A Q depends on the law of distribution of velocitieswhatever value is given to Q, and the equation of diffusion nolonger follows on putting Q = u.

To obtain the equation of diffusion from equation (340) in thegeneral case, it is necessary to assume Q to be equal to u multi-plied by a series of powers of c. The resulting equation* is foundto be of the same general type as (341), except for the important

* For details see Chapman, Phil. Trans. 217 A (1917), pp. 124, 181.The analysis given by Chapman in this paper is invalidated by an errorof algebra, as was pointed out by Enskog (Arkiv f. Mat. Astron. undFyaik, 16 (1921)), but the numerical consequences are not serious. Fora corrected discussion see Chapman and Hainsworth, Phil. Mag. 48(1924), p. 593.


difference that the left-hand member includes terms in dT/dxand dp/dx in addition to the term in dv/dx.

If T and^ do not vary with x, these additional terms disappear.But their presence in the general equation indicates that aprocess of diffusion is necessarily going on in any gas in which Tand p vary from point to point, exception being made of thespecial case in which the molecules repel according to the exactinverse fifth-power of the distance. These phenomena were firstpredicted on purely theoretical grounds by Enskog* in 1917;shortly afterwards they were predicted independently byChapman in the paper already quoted.

Thermal- and Pressure-diffusion

202. The phenomenon of "pressure-diffusion" originates inthe terms in dp/dx, but does not appear to possess any greatimportance physically. That of "thermal-diffusion", whichoriginates in the terms in dT/dx, is of importance becausenumerically its magnitude is comparable with that of ordinarydiffusion. Let us imagine that we have a tube or cylinder,originally filled with a uniform mixture of two gases, and let thetwo ends be kept permanently at different temperatures. As theresult of thermal diffusion, currents will be set up in the tube, themolecules of the heavier gas tending to diffuse in the direction ofdecreasing temperature and vice versa. There is a limit to theinequality of composition of the mixture which can be establishedby this means, for the inequality brings into play ordinary dif-fusion which acts in the opposite direction and tends to restoreuniformity of composition. Thus a steady state will ultimatelybe reached in which the proportion of the mixture will varygradually as we pass along the tube.

This predicted variation in the proportion of the mixture wasfirst confirmed experimentally by F. W. Dootson.f In a typicalexperiment, a tube with a bulb at each end was filled with amixture of hydrogen and carbon-dioxide in approximately equalproportions. One bulb was then kept for four hours at a steady

* Kinetische Theorie der Vorgdnge in mdssig verdiinnten Gasen (Inaug.Dissertation, Upsala, 1917).

t Phil. Mag. 33 (1917), p. 248.

252 THEORY OF A GAS NOT IN A STEADY STATE

temperature of 230° C, the other being kept water-cooled at10° C. At the end of the four hours samples were drawn off fromthe two bulbs and analysed, with the following results:

Hot bulb (230° C): 44-9 per cent H2; 55-1 per cent CO2,Cold bulb (10° C): 41-3 per cent H2; 58-7 per cent CO2.

The effect detected here is of the sign and order of magnitudepredicted by theory. In actual amount it is rather less than halfthe amount predicted on the supposition that the moleculesbehave like elastic spheres. The theoretical effect for elasticspheres is, however, greater than that for any other type ofmolecule, and it will be remembered that it vanishes altogetherfor molecules repelling according to the Maxwellian law jar~5.Thus the effect detected by experiment was about what was tobe expected, and as later and more accurate experiments* havegiven very similar results, the theory may be regarded as beingverified.f

The steady state phenomenon we have been considering is ofinterest in that it depends greatly upon the law of force betweenmolecules. It seems possible that it may in time lead to powerfulmethods for the investigation of molecular fields of force.

Chapman has shown how it can be used for the separation ofgases which have the same molecular weight (e.g. C2H4 and N2),and it has recently proved useful for the separation of iso-topes.

* Ibbs, Proc. Roy. Soc. 99 A (1921), p. 385, and 107 A (1925), p. 470;Elliott and Masson, Proc. Roy. Soc. 108 A (1925), p. 378.

t A fuller discussion of both the theory and the experimental verifica-tion of thermal diffusion will be found in Chapman and Cowling, TheMathematical Theory of Non-uniform Gases (C. U. Press, 1939), pp. 252-258.

Chapter X

GENERAL STATISTICAL MECHANICSAND THERMODYNAMICS

203. So far our molecules have been treated either as elasticspheres, exerting no forces on one another except when in actualcollision, or else as point centres of force, attracting or repellingaccording to comparatively simple laws. The time has now cometo discard all such restrictions, and treat the question in a moregeneral way, regarding the molecules as general mechanicalstructures, which may be as complicated as we please, consistingof any number of parts, capable of any kind of internal motionand exerting upon one another forces of any type.

Degrees of Freedom

204. The total number of independent quantities which areneeded to specify the configuration of any mechanical system iscalled the number of degrees of freedom of the system. Thisnumber does not depend on the motions, but on the capacitiesfor motion, of the various parts of the system; it is thereforerelated to the geometrical or kinematical, and not to themechanical, properties of the system.

For example, if a point is free to move in space, its position canbe specified by three quantities, as for instance x, y, z, the rect-angular coordinates of the point, so that a point which is free tomove in space has three degrees of freedom. A rigid body whichis free to move in space has six degrees of freedom, for the positionof the body can only be fully fixed when six quantities are known,as for instance x, y, z the coordinates of the centre of gravity of thebody, and three angles to determine the orientation of the body.Similarly, a pair of compasses has seven degrees of freedom, tworigid arms connected by a "universal joint" has nine, and so on.

Numbers of degrees of freedom are additive in the sense thatwhen a complex system is made up of a number of simplersystems, the number of degrees of freedom of the complex system

254 GENERAL STATISTICAL MECHANICS

is equal to the sum of the numbers of degrees of freedom of theconstituent systems. We see this at once on noticing that a know-ledge of the configuration of the complex system is exactlyequivalent to a knowledge of the configurations of all the con-stituent systems.

If atoms are regarded as points, each atom has three degreesof freedom. A diatomic molecule must therefore have six degreesof freedom. These can be counted up in a variety of different ways,but the total must always come to six. For instance, we mighttake the six degrees of freedom to consist of the three degrees offreedom of the centre of gravity of the molecule to move in space,the two degrees of freedom of the line joining the two atoms tochange its direction in space, and the one degree of freedom arisingfrom the possibility of the two atoms changing their distanceapart. Alternatively, we may suppose each atom to have its ownthree degrees of freedom as a point, so that the diatomic moleculeagain has six degrees of freedom.

In general, if atoms are regarded as points, a molecule composedof n atoms will have 3n degrees of freedom, but if atoms areregarded as rigid bodies capable of rotational as well as trans-lational motion, such a molecule will have 6n degrees of freedom.If electrons are regarded as points, a cluster of n-electrons willhave 3n degrees of freedom. If molecules are treated as points, agas consisting of N molecules will have SN degrees of freedom,while if each molecule has n degrees of freedom, the gas will havenN degrees of freedom.

The Motion of a General Dynamical System

205. A dynamical system which has n degrees of freedom needsn coordinates to specify its configuration. Let us call these

9v fe ••• <ln> (342)

and let us denote their rates of increase by

4v &. ... 4n- (343)

The configuration and motion of the system at any instant arecompletely known when the values of the above 2n quantitiesare known, and classical mechanics tells us that, if these quanti-

AND THERMODYNAMICS 255

ties are known at any instant, it is possible to trace out the motionof the system throughout all future time, or until it experiencessome disturbance from outside. The energy of the system dependsonly on these 2n quantities; let us denote it by E. It is convenientto introduce quantities known as momenta, which we denote by

Pv P* - Pn

and define by the equationsdE ^

pa = Ws, etc.

Ordinarily E will contain only squares and products of thevelocities ql9q29 ••• qn, so that the momenta are Linear functionsof these velocities.

For instance, if the dynamical system is simply a movingparticle of mass m, its coordinates may be supposed to be x, y, z>its velocities are x, y, z, its energy is

E = £ra(i'2 + 2/2 + z2),

so that the momenta aredE . ^—— = mx, etc.dx

In this case the momenta reduce to the ordinary momenta of amoving mass, mu> mv, mw.

A simplification is introduced on replacing the 2n quantities(342) and (343) by the 2n quantities

2v ?2> — ?». Pv P* ••• Pn- (344)

When the values of these quantities are known at any instant,the configuration and velocities of the system are known at thatinstant, so that the classical mechanics tells us that the motionof the system can be traced throughout all time.

The equations which determine this motion are the well-knownHamiltonian equations

dpa_ dE. dq,_dEdt ~ ~Ws' dt ~ dPa'

in which it is essential that E is expressed as a function of theq's and p's.


The Conception of a Generalised Space

206. Twice already we have found it convenient to representthe three components of velocity u, v, w of a molecule by a pointin space of coordinates u, v, w. The same artifice is found useful onmany occasions. And, although the space of nature possessesonly three dimensions, we need not limit our use of the artifice tospaces of three dimensions, for the spaces we use exist only in ourimagination. If the configuration and motion of a dynamicalsystem are specified by the 2n quantities (344), we can imaginethese quantities represented as the coordinates of a space of 2ndimensions. Such a space is called a "phase-space".

Each point in this space represents one set of values of the 2ncoordinates (344), and so represents one state of the system underdiscussion. As the natural motion of the system proceeds, thepoint moves in the phase-space, and its motion records the historyof the system. The rates at which the point moves in the differentdirections of the phase-space—i.e. the components of velocity ofthe point—are given by equations (345). The ratios of these com-ponents give the direction of motion at the point, and we see thatthis depends only on the coordinates of the point in the phase-space. Thus the moving points move along curves in the phase-space, the positions of which do not change with the time. Thesecurves are generally known as "trajectories".

We can imagine all such trajectories mapped out in the phase-space, and the 27i-dimensional chart so obtained will enable us tofollow out the motion of our dynamical system, starting from anyinitial state that we please. It is as though the currents in theAtlantic Ocean were always the same at each point of the Ocean.A chart could then be drawn, like the usual Admiralty chart,which would shew the motion of the surface-water at each pointof the Ocean. Such a chart would enable us to trace the motionof each particle of surface-water throughout its whole life in theOcean.

The problem before us is not, however, to study the motion ofour system when it starts from any particular set of initial con-ditions; we wish rather to find statistical properties which shallbe true for its motions, no matter from what particular state it


starts. We therefore imagine the whole of the 2w-dimensionalspace filled with moving points each describing its own curve, asdetermined by equations (345). We shall suppose these points tobe so thickly scattered in the space that they may be regarded asforming a continuous fluid; our object is now to make a statisticalstudy of the particles of this fluid. Let us first consider the motionof the points which originally occupy a small rectangular elementof volume in the phase-space extending from px to px + dply p2 top2-¥dp2i etc., and so of content dv equal to dpl9 dp2i ---dqn.

Let us follow these points in their motion through a smallinterval of time dt. Those which originally occupy the face px ofthe small element of volume are moving perpendicular to theface at a rate pv while those in the opposite face Pi + dpx are

moving at a rate p± + -^ dpv Thus these two layers of particles

are moving apart at a rate iJ-^dpv and by the end of the time dtcpx

their distance apart will have increased from dpx to

There is a similar increase in the distance apart of the slabs ofparticles which formed each other pair of faces, so that after aninterval dt the points under consideration will occupy a volume

Multiplying out and neglecting squares of dt, this becomes

idpt (346)

The values of pv qv etc. are of course given by equations (345).From these we at once find that

dpx oqt

and similarly for every other pair of coordinates, so that thecoefficient of dt in expression (346) is equal to zero, and the ex-pression itself reduces to dpx, dp2,... dqx, dq2 f.... In other words,


the points under consideration occupy exactly the same volumeat the end of the interval dt as they occupied at the beginning.Thus their density does not change in any small interval dt, andso must remain unaltered through all time.

In brief, there is no tendency for the representative points tocrowd into any special region or regions in the phase-space,a result which was first enunciated by Liouville, and is generallyknown as Liouville's theorem.

207. Imagine now that any dynamical system is found in-variably to possess a certain property (e.g. maximum entropy)after being left to itself for a sufficient time. This might a prioribe for either of two reasons: either that the points in the phase-space tend to crowd into those regions of the space for whichthe property is true, or else that the property is true for the wholeof the space. Liouville's theorem excludes the first alternative,so that the second must be the true one. We must now examinethis in some detail.

Normal Properties and the Normal State208. Let P denote any property—as for instance maximum

entropy—which the system under consideration may or may notpossess. Let us, as before, represent all possible states of thesystem by a collection of points filling the appropriate phase-space. Let the states in which the system possesses the propertyP occupy a volume Wx of the space, while those in which thesystem does not possess the property P occupy the remainingvolume W — W± of the phase-space. If we choose coordinates andmomenta for the system at random, we are in effect choosing apoint at random in the phase-space to represent the system. Thechance that the system shall possess the property P is ac-cordingly WJW.

Let us next examine what is the probability that a system initi-ally selected at random shall have the property P after followingout its natural motion for a time t. Imagine the phase-space to befilled with a cloud of representative points, so close together thatthey may be regarded as forming a continuous fluid, and let thesepoints be distributed initially so that the density of this fluid isuniform. Each of these points has an equal chance of representing


the system selected initially at random. Let this cloud of pointsmove for the time t, in accordance with the equations of motionof the system. Liouville's theorem shews that at the end of thistime, the fluid will still be of uniform density. Thus after any timet the number of points which represent systems possessing theproperty P will be a fraction WJ W of the whole, and therefore thesame fraction measures the probability that the system shallpossess the property P after time t.

It follows that if the system is found always to possess theproperty P after a sufficient time has elapsed, this can only bebecause the ratio WJiW — Wx) is infinite.

A property P which is such that the ratio WJ( W — Wj) is infinitewill be called a " normal property" of the system, while a systemwhich possesses all the normal properties of which it is capablewill be said to be in the "normal state".

T H E NORMAL STATE

209. So long as a system has only a finite number of coordinates,the ratio WJiW-Wj) corresponding to any property P willnecessarily have a finite value. But in a gas, the number ofdegrees of freedom is so large that it may be treated as infinite,so that it is not surprising that this ratio should become infiniteor zero, since Wx and W — W± will in general be functions of thenumber of degrees of freedom.

The various properties of the system will, in general, changewith the time, some of them perhaps slowly, some more quickly,some with extreme rapidity. Let us suppose that a property Px

may in general be expected to change in a time comparable withtv a property P2 in a time comparable with t2i and so on. After atime t which is very large compared with all of the quantitiestvt2,..., the system will have had ample time to change all itsproperties. In a statistical sense, the influence of the initial con-ditions will have disappeared; if the representative point startedin peculiar regions of the generalised space, in which any normalproperty does not hold, it will have had time to move away fromthese. The system may therefore be expected to possess all thenormal properties of which it is capable, and therefore to be inthe normal state.


Complications may arise through the system possessing pro-perties which are not capable of change at all, or for which thetime of change is infinite or nearly so.

For instance, a system which is perfectly self-contained andsubject to no external influence cannot change its angularmomentum, so that the property of having any particular angularmomentum is one which cannot be acquired or lost with the lapseof time. In the generalised space, one particular value of theangular momentum, namely zero, is far commoner than anyother, so that for the property of having zero angular momentumthe value of WJ( W — Wx) is infinite. Yet the having of zero angularmomentum, although a normal property of the system, is not tobe regarded as an essential of the normal state; it is not requiredby the definition of the normal state, since it is not a propertywhich the system is capable of acquiring.

On the other hand, if conditions are such that the system canchange its angular momentum, then the property of having thenormal value for its angular momentum must be regarded as oneof the properties of the normal state. For instance, a gas enclosedin a fixed closed vessel can change its angular momentum, andit is easily seen that the possessing of zero angular momentum isone of the normal properties of such a system. Hence in the finalstate of the system we must expect the angular momentum of thegas to be zero.

One property which can never be changed in a conservativesystem is that of having a certain value for the energy. Thus indiscussing the normal state, we need only consider systems havinga specified amount of energy. In the same way, if the system hasother quantities or properties which are invariable, account mustbe taken of this invariability in specifying the normal state. Thevarious complications which may arise in this way are somewhatdifficult to discuss in general terms, but are not difficult to treat inparticular cases, as the various examples which occur in thepresent book will shew.

The Normal Partition of Energy210. We shall first consider the normal properties associated

with the partition of energy.


Let the 2n coordinates of position and velocity (344) now bedenoted by 0V 82i... 62n, no distinction in notation being madebetween the two, and let us suppose that the energy E can bedivided into separate and distinct parts EvE2,... such that Et

depends only on one group of coordinates, say dv62, ...68; E2

depends only on another group, distinct from the former,@8+iy s+2» • • • @s+t> a n d s o on. Also let it be supposed that thenumber (s, t,...) of coordinates in each of these groups is so greatthat it may be treated as infinite. Such a case would for instanceoccur if we had oxygen and nitrogen mixed in a closed vessel.The total energy can be regarded as the sum

energy of oxygen•f energy of nitrogen+ energy of containing vessel.

ThenE = E1 + E2 + ... = f1(dv 0 a , ... e8)+f2(O8+1, 0M, ...0^) + .-.-Let us define the property P as being possessed by the system

when the energy is divided between the various groups of co-ordinates in such a way thatEx lies within a small range e extending from Ex to Ex + dEv

E2 „ „ „ „ „ E2 to E2 + dE2, etc.Then the property P holds throughout that part of the phase-space for which

#2> • • • 08) K e s between E1 and Ex + dEv (347)

* 0«+2> • • • ds+t) ti^ between E2 and E2 + dE2, (348)and so on.

The volume W1} throughout which the property holds, may bewritten in the form

where the integration is taken throughout the region defined bythe conditions (347), (348), etc. This again may be written in theform of the product

Wl = ( j j \ ) ( j \ j(349)


where the first integral is taken within the limits specified by(347), the second integral within the limits (348), and so on. Thefirst integral can depend only on E1 and dEl9 and so must be ofthe form F1(E1)dEv Similarly the second must be of the formF2(E2) dE2i and so on. Thus W± must be of the form

Wx = F1(E1)F2(E2)...dE1dE2... = FÊJFiiEJ ...e*.(350)

We shall not attempt to evaluate Wx directly, but shall attack thesimpler problem of finding for what particular values of Ev E2,...W± has its maximum value.

If we slightly change the partition of energy, by altering thevalues of EvE2i... to E1 + 8EVE2 + 8E2,..., but keeping theranges dEv dE2i... unaltered and each equal to e, the change inlog Wx is given by

8logW1 = 8logF1(E1) + 8logF2(E2) + ...dlogFM dlogF2(E2)

= —d%~ SEl + —dET~ * ( }

These changes SEV 8E2)... cannot be anything we please; theirsum must be zero, because the total energy Ex + E2 + ... must notchange its value. Thus we must have

SE1 = -SE2-SE3-8Et- ...,and equation (351) becomes

, (dlogF3(E3)"*( dEz dEx

If W± is to have its maximum value, this must vanish for allvalues of 8E2, 8EZ, ..., so that we must have

dlogFÊJ __ dlogi^2(^2) _dlog^(#3)dK " dK = dK '

x a o

The solution of these equations, together with

will give the most probable partition of energy for a system ofassigned total energy E.


Equipartition of Energy

211. Next suppose that part at least of the system we have hadunder discussion consists of the molecules of a gas, these being allsimilar and n in number. Part of the energy of this system willbe the energy of the translatory motion of these n molecules;let us identify this with Ev so that

where m is the mass of each molecule, and uv vv wx\ u2, v2, w2\...are the velocities of individual molecules. We can identifymuvmvlf mwvmu2,mv2, mw2, etc. with the dvd2,...ds of equa-tion (349), so that

F(E1)dE1 = [[[...d01d6t...d0ai (353)

where the integral is taken over all values of dvd2,... 08 for whichEx lies within the specified range of extent dEv

If the range for Ex were from 0 to some assigned value E'l9 theintegral could easily be evaluated, for the integration would thenextend throughout the interior of the boundary

u\ + v\+... K

or 61 + 01+ ... + #f = 2mE'l9which is a sphere in s-dimensional space. The integration merelyevaluates the volume of this sphere, and since the radius of thissphere is (2ra£")*, its volume must be C(2mE')l8, where C is aconstant of which the value need not concern us.*

On differentiating this in respect to E[, we find that the valueof the integral taken over all values which make Ex lie within arange dEx is

E

This is precisely the quantity we have called F(E1) dEv so that,on carrying out the differentiation,

dEL Ex *

* Actually the value of C is 7VT^-TT •


We have already identified s with the number of degrees offreedom of the n molecules to move in space, so that s = Zn. Ifn is a very large number, the difference between | s and | s — 1 maybe neglected, and we may write

dlogFjEJ __ fr _ In _ 3

where ex denotes the mean energy of motion fyn(u2 + v2 + w2) ofa single molecule, of which the value is known to beThus equation (352) assumes the form

dlogF(E1)_dlogF(E2) _ 1dEx dE2

Actual molecules may possess energy other than that of thismotion in space—as for instance energy of rotation or of vibra-tions of their internal structure. Often the energies of suchmotions may be expressed as a sum of squares of the coordinates.

In general, if different parts EvE2,... of the whole energyconsist of 8, t,... squared terms, we may evaluate the mostprobable values of Ev E2 in precisely the same way as above, andfind that, whatever the physical interpretation of the squaredterms may be, the most probable partition of energy is given, asregards those parts of the energy for which the energy function isquadratic, by the equations

E1 = \sRT, E2 = \tRT, etc., (356)

where s, t,... are the number of coordinates concerned in thequadratic functions El9E2,

It is proved in Appendix in that, as far at least as these partsof the energy are concerned, the partition of energy expressed byequations (356) is not only the most likely partition, but alsoexpresses a "normal" partition in the sense of §208; that is tosay, this partition is infinitely more probable than any other.

We have accordingly shewn that those parts of a system, sayEv E2i..., in which the energy is of quadratic type, will necessarilytend to the partition of energy specified by equations (356). These


equations express the Theorem of Equipartition of Energy in itsmost general form:

The energy to be expected for any part of the total energy which canbe expressed as a sum of squares is of amount %RT for everysquared term in this part of the energy.

The Normal Distribution of Coordinates

212. Not only will there be a normal partition of energy, butthere will also be a normal way for the separate coordinates to bearranged so as to give this particular energy. For the simple caseof a gas composed of molecules which behave like hard elasticspheres, this has been shewn to be Maxwell's law of distributionwhich we obtained in the form

f(u, v, w)dudvdw = Ce-hrtu*+v*^*)+a*mu+a*Tnv+a*mw+a*.(357)

In the present case let us suppose that in addition to itscoordinates and velocities in space x, y, z and u, v, w9 the moleculecontains internal motions expressed by additional coordinates$1* <f>2> <f>z> • • • • Let u s suppose that the energy of a molecule is ev

this of course being conserved at collisions, and that in additionto this there are other quantities e2, e3, e4,..., which are conservedat collisions. Then it is shewn in Appendix iv that the normal lawof distribution of coordinates is

<f>2 (358)

If the only quantity which remains constant is the energy, thisreduces to

Ce-2he^dxdydzdudvdwd^>1d(f>2...9 (359)

where 2h replaces ocx so as to agree with the notation already used,2h being equal to l/RT. If certain of the coordinates, say<f>v <f>2> - • • &> enter into the energy e only through their squares, sothat the value of e is of the form

where <P does not involve <f>v <f>2, ~- <f>s, but only <f>8+1, &+then the law of distribution may be written in the form


This shews that there is no correlation between the distributionsof (f>v <j)2,... <+>8. The law of distribution of any single coordinate,say <f>v is of the form

^ , (360)

the constant being determined from the condition that the inte-gral, taken from <f> = — oo to 0 = -j- oo, shall be equal to unity.

The mean value of the contribution from <p± to the energy is

of course -j or iRT, as before, and we have4/&

expressing the theorem of equipartition of energy in a new form.

THERMODYNAMICS

213. Let us suppose that our gas, containing vessel, etc. arein their most probable state, specified by the equations

dlogFÊJ _ dlogF2(E2) _ 1~dEx " dE2 ~"RT'

Let the gas now be heated from outside, and suppose that, afterthe gas has again attained its most probable state, we find thatElf E2,... have been increased to Ex + dEv E2 + dE2,....

With Wx defined by equation (350), let us putP = log Wx = log [FÊJ F2(E2)...] + a constant (362)

Then the heating of the gas changes P by an amountdlogFÊJj^ ^dlogFÊJj^

Using relation (361) this becomesv 1

RT'where dQ is the total amount of energy that has been added to thegas. Thus ,Q

f = SdP.


The second law of thermodynamics tells us (§ 12) that dQ/Tmust be a perfect differential; we now see that it is the differentialof RP where P is defined by equation (362), or again of -Rlog Wv

According to thermodynamical theory dQjT is equal to dS,where S is the entropy. We accordingly see that

S = RP + a constant = R log Wx + a constant.

We now see that the entropy 8 is the same thing as P, exceptfor a multiplying constant R, and an additive constant which isof no consequence because we are only interested in changes in S.It is also the same thing as log Wv the logarithm of the probabilitythat the energy of the system under discussion shall be dividedaccording to the partition

^V^2> • • • •

We have already seen that the most probable state is one inwhich Wx is a maximum; we may now put this in the form that itis one in which the entropy is a maximum.

To get a clearer idea of the physical meaning of entropy, let ussuppose that a system consists of only two parts, of energiesEVE2. Let the total energy E = E± + E2 remain unaltered, butlet a quantity of energy dQ pass from the first part Ex to the secondpart E2, so that the partition of energy is changed from Ev E2 to

Let the two parts of the system be at different temperaturesTv T2. Then the subtraction of energy dQ from the first part of thesystem reduces the total entropy by dQ/Tv while the addition ofenergy dQ to the second part of the system increases the totalentropy by dQ/T2. The net gain to the entropy is accordingly

If the process is one which occurs in the ordinary course ofnature, in which heat-energy can flow only from a hot body to acooler one, Tx is necessarily greater than T2. Hence in a naturalchange dS is positive—the entropy necessarily increases, and afinal steady state is only reached when the entropy has attainedthe maximum value which is possible for it.


Since 8 is equal to RlogWl9 where Wx measures the probabilityof a state, this is equivalent to saying that in a natural motionof a system the probability Wx continually increases, and thesystem only reaches a final steady state when the probabilityWx has reached its maximum value.

Entropy and Probability

214. So far we have defined and discussed entropy as somethingassociated with a temperature T, but temperature has no meaningexcept for a system, or part of a system, which is in a steady state.Entropy has in fact only been defined for a system in a steadystate.

The results just obtained make it possible to remove thisrestriction. When parts of the system have specific temperaturesattached to them, we have seen that

8 = Rlog Wx + a constant. (363)

But Wx has a definite meaning with reference to any partitionof energy in the system, so that we may define a more genera]entropy 8 by equation (363).

Here W1 is the volume of that part of the generalised spacewhich represents systems having the partition in question. Thisvolume measures, in a sense, the probability that this specialpartition shall prevail, so that the entropy is proportional to thelogarithm of the probability.

An increase of entropy is thus associated with an increase ofprobability. The law that the entropy always increases meansthat a system passes always from a less likely to a more likelystate. The law that the entropy is a maximum in the steady statemeans simply that the steady state is the state of maximumprobability.

We have so far thought of the steady state and the final stateas the same thing, but we now see that strictly speaking the twoare not identical. The value of W, the volume or probability,which corresponds to the steady state is far larger than that whichcorresponds to any other state. If the number of molecules in thegas were infinite, it would be infinitely larger. With the numberof molecules in the gas enormously large, it is enormously larger,


but not infinitely so. If Wx is the probability of the state ofmaximum entropy, and W2 that of any other specified state,WJW2 will be very large but not absolutely infinite.

Thus if a gas starts in the second state there is a very largeprobability that, after a sufficient time, it will be found in thesteady state. There is also a finite, although very small, prob-ability that if it starts in the steady state it will be found after atime t in the second state. Thus the increase of entropy is only aprobability, not a certainty, although the more we increase thenumber of molecules in the gas, the nearer this probability comesto unity*

If the system is allowed to undergo its natural changes for aninfinite time, it is almost a certainty that at some time it will passthrough the second state. This must be so unless the trajectoriesin the generalised space all avoid the region of volume W2, andany such avoidance would be contrary to Liouville's theorem.

We may, then, say that an increase of entropy is not a matterof certainty, but only of very high probability. And if the systemcontinues in existence for long enough, it is certain that at sometime a decrease of entropy will occur.

215. When applied to concrete instances, these results seem atfirst sight somewhat startling. To borrow an illustration fromLord Kelvin, if we have a bar of iron initially at uniform tem-perature, and subject neither to external disturbance nor to lossof energy, it is infinitely probable that, given sufficient time, thetemperature of one half will at some time differ by a finite amountfrom that of the other half. Or again, if we place a vessel full ofwater over a fire, it is only probable, and not certain, that thewater will boil instead of freeze. And moreover, if we attemptto boil the water a sufficient number of times, it is infinitelyprobable that the water will, on some occasions, freeze instead ofboil. The freezing of the water, in this case, does not in any wayimply a contravention of the laws of nature: the occurrence ismerely what is commonly described as a "coincidence", exactlysimilar in kind to that which has taken place when the dealer ina game of bridge finds that he has all the cards of one suit in hisown hand.

The analogy of the distribution of a pack of cards will help us


to see further into the problem presented by the entropy of a gas.In dealing cards, it is just as likely that the dealer will have thethirteen spades as that he will have any other thirteen cards thatwe like to specify. The occurrence of a hand composed of thirteenspades might, however, be justly regarded as a " coincidence",whereas the occurrence of any specified hand in which the cardswere more thoroughly mixed, could not reasonably be so re-garded. The explanation is that there are comparatively few waysin which a hand which is all spades can be dealt, but a greatnumber in which a mixed hand can be dealt.

A similar remark applies to the result of putting cold waterover a hot fire. There are comparatively few ways in which thefire can get hotter, and the water colder, but a great many ways inwhich the fire can impart heat to the water—a proposition whichbecomes obvious on looking at it in terms of the phase-space.Speaking loosely, it is just as likely that the water will freeze asthat it will boil in any specified way. There are, however, so manyways in which the water can boil, all these ways being indis-tinguishable to us, that we can say that it is practically certainthat the water will boil.

The increase of entropy, then, simply means the passage froma more easily distinguishable state to a less easily distinguishablestate, or, in terms of the generalised space, from a less probableto a more probable configuration.

216. In general, the energy of a dynamical system is the sumof two parts—kinetic and potential. We see from equations (362)and (363) that the entropy consists of two parts, the formerdepending on the energy of the molecules of the gas, and thelatter on their positions. So far we have considered only varia-tions in the first term, resulting from inequalities in the tempera-ture of the gas. Similar remarks could, however, be made aboutthe variations of the second term, these denoting inequalities inthe density of the gas. A single illustration, suggested by WillardGibbs,* will, perhaps, make clear what is meant.

If we put red and blue ink together in a vessel, and stir themup, common experience tells us that, if the inks differ initially innothing more than colour, the result of stirring is a uniform violet

• Elementary Principles of Statistical Mechanics, p. 144.


ink. Here we have the passage from a more easily distinguishableto a less easily distinguishable arrangement of coloured inks. If,however, we start by stirring a uniform violet ink composed of amixture of red and blue inks, then it is possible, although notprobable, that the effect of the stirring will be to separate the inksof different colour, so that one half of the vessel is occupied solelyby red, and the other solely by blue ink. And from the dynamicalstandpoint it is no less probable that this should occur, than thatwe should be able to start stirring inks which were separatedinitially as regards colour.

217. With reference to this subject, some well-known remarksof Maxwell* are of interest. He says: " One of the best establishedfacts in thermodynamics is that it is impossible in a system en-closed in an envelope which permits neither change of volumenor passage of heat, and in which both the temperature and thepressure are everywhere the same, to produce any inequality oftemperature or of pressure without the expenditure of work.This is the second law of thermodynamics, and it is undoubtedlytrue so long as we can deal with bodies only in mass and have nopower of perceiving or handling the separate molecules of whichthey are made up. But if we conceive a being whose faculties areso sharpened that he can follow every molecule in its course, sucha being, whose attributes are still as essentially finite as our own,would be able to do what is at present impossible to us. For wehave seen that the molecules in a vessel full of air at uniformtemperature are moving with velocities by no means uniformthough the mean velocity of any great number of them, arbitrarilyselected, is almost exactly uniform. Now let us suppose that sucha vessel is divided into two portions A and B9 by a division inwhich there is a small hole, and that a being, who can see theindividual molecules, opens and closes this hole, so as to allowonly the swifter molecules to pass from A to J5, and only theslower ones to pass from B to A. He will thus, without expendi-ture of work, raise the temperature of B and lower that of A, incontradiction to the second law of thermodynamics."

Thus Maxwell's sorting demon could effect in a very short timewhat would probably take a very long time to come about if left

* Theory of Heal, p. 328.


to the play of chance. There would, however, be nothing contraryto natural laws in the one case any more than in the other.

F L U C T U A T I O N S

218. Formula (358) or (359) shews that when no externalforces act on a gas, the normal state is one in which the density isuniform throughout. In this state, and no other, the entropy isa maximum.

Nevertheless, we saw in § 80 how the number of molecules inany small volume of the gas must continually fluctuate asmolecules enter or leave the volume. The entropy, then, cannever be permanently equal to its maximum value; there mustbe continual fluctuation below and up to this value, even whena gas is in a completely steady state. The phenomena we mentionedin §215, as for instance the freezing of a kettle of water put overa fire, result from fluctuations of this kind.

To study these fluctuations in detail, let us fix our attentionon a small element of volume Q in a gas of N molecules, and letQ be one 5th part of the whole volume of the gas. In the state ofmaximum entropy, the volume Q will of course contain one sthpart of all the molecules, namely, N/s, which we shall denote by n.We proceed to examine how the actual number will fluctuateabout its mean value n.

Since all positions in the gas are equally likely for each molecule(§ 82), the chance that any particular molecule A shall be inside Q

at any specified instant is - ; the chance of its being outside is3

1 — . Thus the chance that a group of p selected moleculess

A9B,C,... shall all be inside, while the remaining N—p moleculesare all outside, is /np . ,N_p

t) (•-.-) •Such a group of p molecules can be selected out of the N

molecules of the gas in

N\ = N(N- 1) (N- 2)... (N-p+1)p\(N-p)\~ pi


ways, so that the whole chance Q that there shall be p moleculesinside Q, and the remaining (N — p) outside, is

Q=N(N-l)(N-2)...(N-p+:

If Q is only a small fraction of the whole volume of the gas, 8 willbe large, and p small compared with N. In these circumstances,we may replace N(N— l)...(iV—p+1) by Np, and the value ofQ becomes

p\

(364)

This is the probability that the volume O shall contain pmolecules in place of its average number n. Ifp is itself a largenumber, although still small compared with N, we can expressthis probability in a simpler form by using Stirling's approxi-mation for p!, namely

Equation (364) now becomes

^ ( ) P (365)

Let us now put p = n + 5, so that S is the excess of moleculesin Q caused by fluctuations about the mean. Taking logarithmsof both sides, equation (365) becomes

log Q = log J2np- (n + 8)logll + - j + $

£2= log v 2nn — — + terms of higher order in S,

so that 8%j t o m (366)


This gives the chance that the number of molecules in the volumeQ shall differ by 8 from the average value n. Or, if we divide upthe whole gas into "cells" of volume Q, it gives the proportionof the whole number in which there are n + 8 molecules.

From the integrals given in Appendix vi (p. 306), we readilyfind that the mean numerical value of 8 is 0*798 //i, while themean value of 82 is exactly n. Thus as n increases, 8 increases as

219. An average cubic millimetre of air at ordinary pressurecontains 2-7 x 1016 molecules, so that the average fluctuation isabout 1*3 x 108 molecules—hundreds of millions of molecules inexcess or defect which, however, form only 0-0000005 per centof the whole. At a cathode-ray tube pressure of 0-001 mm., acubic millimetre contains only about 4 x 1010 molecules, so thatthe average fluctuation is about 200,000 molecules and there maybe appreciable variations of density over lengths comparablewith a millimetre. A gas which is near to its critical point isabnormally sensitive to such density variations, which nowshew themselves optically, giving rise to a characteristic opale-scence. From measurements on this phenomenon also, estimatesof Loschmidt's number have been made, ranging from6 x 1023 to 7-7 x 1023.* While these cannot claim a high degree ofaccuracy, they suffice to shew that the explanation in terms offluctuations is correct.

• S. E. Virgo, Science Progress. 108 (1933), p. 634.

Chapter XI

CALORIMETRY AND MOLECULAR STRUCTURE

Specific Heats

220. A brief account of the problem of calorimetry has alreadybeen given in §§ 20, 21 (p. 33). We proceed now to study theproblem in more detail.

Calorimetry

Let us suppose that the pressure of a gas is known in termsof its volume and temperature, so that

P=f(v,T). (367)

As in § 12, the equation of energy is

dQ = NdE + pdv. (368)

Specific heat at constant volume. If the volume v is kept constantwhile the heat dQ is absorbed, we may put dv = 0, and the equa-tion of energy becomes

dQ = NdE.

If Cv is the specific heat at constant volume—i.e. the heat re-quired to raise the temperature of the gas by one degree—thenthe amount of heat required to raise the mass Nm through atemperature-difference dT will be CvNmdT. Changing to unitsof energy, the number of ergs needed to raise the temperature bydT will be

JCvNmdT,where J is the mechanical equivalent of heat. This must then bethe value of dQ, or of NdE. Hence we have, as the value of Cv,

__±dEv~~ JmdT'

Specific heat at constant pressure. If the pressure is kept con-stant while the heat dQ is absorbed, both v and T change, so thatwe must not put dv = 0 in equation (368). There is a change ofvolume dv which is related to the change of temperature dT

276 CALORIMETRY AND MOLECULAR STRUCTURE

through the restriction that the value of p, as given by equation(367), shall remain unaltered. In fact we have the generalformula

*-(£)*•(!)*-.so that when dp = 0,

cpjcv

Thus when the heat dQ is absorbed at constant pressure(dp = 0), the equation of energy becomes

If Cp is the specific heat at constant pressure, this is equal toJCpNmdT, so that

Specific Heats of a Perfect Gas

221. If the gas is perfect, p = NBT/v, whence we readily findthat

cp/dT _ v _NRdpjdv "~ T~~ p '

and the formulae for the specific heats become

G - 1 dE

the formulae already obtained in § 20.

Molecular Structure

222. Experiment shews that for many gases Cp and Gv areapproximately independent of the temperature through a con-siderable range of temperatures and pressures in which the gasbehaves approximately as a perfect gas. This, as is shewn by areference to equation (369), must mean that dE/dT is a constant,

CALORIMETRY AND MOLECULAR STRUCTURE 277

and therefore that the mean energy of a molecule of the gas standsin a constant ratio to the translational energy. As in § 21, let usdenote this ratio by (1 -f/?), so that /? is the ratio of internal totranslational energy. Then

E = (l + fi)$RT, (371)

so that —_ = fi2(l+/?). (372)(tJ.

Equations (369) and (370) now become

Cp = [! + §(!+/?)] A , (374)

whence, on division,

( 3 7 5 )

The quantities dE/dT and /? (the two being connected by rela-tion (372)) can only be evaluated when the internal structure ofthe molecule is known. We have not sufficient knowledge of thisinternal structure to evaluate these quantities directly, but theirvalues can to some extent be determined from a comparison ofthe specific heat formulae and the experimentally determinedvalues of the specific heats, and the values obtained in this wayprovide a basis for the discussion of the structure of molecules.

As an example of this procedure, we may examine the caseof air which for the moment, as frequently in the kinetic theory,may be thought of as consisting of similar molecules.

For y, the ratio of the specific heats, under a pressure of 1atmosphere, the following values have been obtained by Kochand others:*

Values of y for air at 760 mm. pressure:0=-79-3°C., 7 = 1-405,6 = 0° C, y = 1-404,0= 100° C.. 7=1-403,6 = 500° C, 7=1-399,0 = 900° C., 7 = l-39.f

* Kaye and Laby's Physical Constants (1936).t Value given by Kalahne.


These numbers shew that at this pressure y is almost inde-pendent of the temperature, and approximately equal to If.Equation (375), namely

( 3 7 6 )

now shews that /? = f.At higher pressures the value of 7 is by no means constant, as is

shewn by the following observations by Koch.*

Values ofy for Air at High Pressures

p = 1 atmos.OK

100 „200 „

0 = -79-3°C.

7 = 1-4051-572-212-33

d = 0° C.

7 = 1-4041-471-661-85

223. Generally speaking, it is found that for monatomic gases,and for the more permanent diatomic gases, there exists a rangeof the kind we have found for air, within which the specific heatsremain approximately constant. But in the case of more complexgases, it frequently happens that no such range appears to exist.

The table below gives observed values of 7 for a number of themore common gases of both types. The third column containsvalues of /? calculated from formula (376), but it will be re-membered that these values have no obvious physical meaningexcept within the range in which the specific heats remainapproximately steady.

The figures in this table, in conjunction with a large mass ofother experimental evidence, shew that the value of /? is approxi-mately equal to zero (7 = If) for the monatomic gases—mercury,krypton, argon and helium. It is nearly equal to §(7= If)throughout the steady range for a number of diatomic gases—hydrogen, oxygen, carbon-monoxide and others. When we passto temperatures below the steady range, /? is found to decreasewith great rapidity.

• Ann. d. Physik, 26 (1908), p . 551.


Gas

Monatomic gasesHelium at 18° C.Helium at -180°Argon at 0°Neon at 19°Krypton at 19°Xenon at 19°Mercury vapour at 310°

Diatomic gasesHydrogen at 16°Hydrogen at - 76°Hydrogen at -181°Nitrogen at - 20°Nitrogen at -161°Oxygen at 20°Oxygen at - 76°Oxygen at -181°Nitric oxideChlorineBromineIodineHydrochloric acid (HC1)Bromine iodide (Br I)Chlorine iodide (Cl I)

Polyatomic gasesWater vapourCarbon-dioxide

Nitrogen peroxide(NO2 at 150°)

Nitrous oxide (NaO)

r

1-631-6671-6671-642

J]

]]

L-689L-666L-666

L-407L-4531-5971-4001-4681-3991-4161-4471-3941-3331-2931-2931-401-331-317

1-3051-300

1-31

1-324

0-060-00000004

-0-040-000-00

0-640-470120-670-420-670-600-490-691-001-271-270-671-001-10

1-191-22

115

1-06

0-190-000-00013

- 0 1 30-000-00

1-901-410-372-001-272011-811-482-083-03-83-82-03 03-3

3-563-67

3-45

318

Observer

Behn and GeigerScheel and HeuseNiemeyerRamsay

Kundt and Warburg

Scheel and Heuse

MassonStrecker

MakowerLummer and

PringsheimNatanson

Leduc

The fourth column gives the value of 3/?, and we notice atendency for these values to cluster round integral values, exceptat low temperatures. These values are usually 0 for monatomicgases, 2 and 3 or 4 for diatomic gases; values near to 3/? = 1 areconspicuously absent. General dynamical theory gives someindication of what this may mean.

224. The energy of a molecule will consist always of threesquared terms representing the kinetic energy of motion, towhich may be added any number of other terms representingenergy of rotation, of internal vibration, etc.

If these latter terms are n in number, the average value of each


is \RT, which is also the average value of each squared term inthe kinetic energy of motion. Thus

\nET nfi- iRT 3'

so that n = 3/?. This is the number tabulated in the fourthcolumn on p. 279.

This shews why 3/?, which represents the number of degreesof freedom, ought always to be integral. The difficulty now is tounderstand how it is possible for n to have any other than integralvalues.

Let us examine different kinds of gas in turn.

Monatomic Gases225. The six monatomic gases in the table—mercury, xenon,

krypton, argon, neon and helium—all have very approximatelyy = If, J3 = 0, and n = 0. For these gases, then, there is noappreciable amount of molecular energy except that of transla-tion. This seems to indicate that the molecules of these gases mustbehave at collision like hard spherical bodies. If they did not doso, an appreciable fraction of the molecular energy of translation*would be transformed into some other form of energy at eachcollision. In these gases the molecule is of course identical withthe atom.

Although the atoms of these substances behave like hardspherical bodies at collision, there is abundant evidence that theyhave a highly complicated internal structure. The helium atomfor instance is known to consist of three parts—a positive nucleus,which is identical with the a-particle of radioactivity, and twonegative electrons. The helium atom made up in this way must,as a matter of geometry, have six degrees of freedom in additionto its three degrees of freedom of motion in space.

The explanation of why the specific heats of such an atom couldbe accurately obtained by taking /? = 0 in formulae (373) and(374) presented for many years a problem of the utmost gravity.It is now generally accepted that no satisfactory explanation canbe given in terms of the classical system of dynamics. In recentyears the new system quantum-dynamics has provided an ex-

CAL0RI3V1ETRY AND MOLECULAR STRUCTURE 281

planation not only of the behaviour of the monatomic moleculebut of many other problems of specific heats in addition. Thisis, however, outside the range of the present book.

After n = 0, the next value theoretically possible would ben = 1, giving y = If. No gas is known for which n and y havethese values, even approximately. This provides additional con-firmation of the truth of the kinetic theory, since no molecularsystem could conceivably have its energy expressible as the sumof four squared terms; either a rotation or a vibration wouldnecessarily introduce at least two squared terms into the energy,beyond the three terms representing motion in space. Thus amolecule for which n = 1 is an impossibility, and the value y = Ifcannot be expected to occur for any type of gas.

Diatomic Gases

226. After n = 0, then, the next value which is theoreticallypossible is n = 2, giving y = If. The table shews that n and yhave very approximately these values for hydrogen, nitrogen andoxygen at ordinary temperatures. This indicates two squaredterms in the energy which may either represent a vibration ofthe two atoms relative to one another along the line joiningthem, or a possibility of a rotation. We must notice that ingeneral the energy of rotation of a rigid body will be representedby three squared terms, but the energy of rotation of a bodysymmetrical about an axis may be represented by only two, norotation about the axis of symmetry being set up by collisions.

Whatever is the true physical origin of these two terms, thetable on p. 279 shews that their energy falls off rapidly as thetemperature falls, particularly in the case of hydrogen. Euckenand others* have found experimentally that as the absolute zeroof temperature is approached, the molecules of diatomic gasestend to lose all energy except that of translation, and so behavelike the molecules of monatomic gases, with the values /? = 0 and7 == If.

* Eucken, Sitzungsber. Berlin Alcad. d. Wissensch. 6 (1912), p. 141;Sckeel and Heuse, Ann. d. Phys. 40 (1913), p. 473; Schreiner, Zeits. / .Phys. Chem. 112 (1924), p. 1.

IO JKT

282 CALOEIMETRY AND MOLECULAR STRUCTURE

More complex Gases

221. It is difficult to discover any law or regularity in the valuesof n and y for more complex gases. Various attempts have beenmade to connect the values of n and y with the number of atomsin the molecule, but it is now clear that no general law can beexpected to relate y with the number of atoms, independentlyof the nature of the atoms. For instance, Capstick* found thefollowing values for the methane derivatives:

MethaneMethyl chlorideMethylene chlorideChloroform

CH4CH3CICH2C12

CHCL

r1-3131-2791-2191-154

n + 36-47-29 0

1 3 0Carbon tetrachloride CC14 1-130 15-4

We see that the introduction of the series of chlorine atomsincreases n very perceptibly at every step, without increasing thenumber of atoms in the molecule. Capstick found a similar lawfor paraffin derivatives; the second chlorine atom introducedinto the molecule produced a large change, although the firstmay or may not have done so.

A similar result was obtained by Strecker,f who found thathydrochloric, hydrobromic, hydriodic acids all have approxi-mately the same values as hydrogen, namely

y = l - 4 , n + 3 = 5,

while for chlorine, bromine and iodine, the values are approxi-mately

Chlorine y = 1-333 n + 3 = 6,

Bromine, Iodine y = 1-293 n + 3 = 6-8.

Similarly for the iodides of bromine and chlorine,Bromine iodide y = 1-33 n + 3 = 6,Chlorine iodide y = 1-317 rc + 3 = 6-3.

These figures shew that one halogen can be put in the placeof hydrogen without producing any difference in the values of y

• Phil. Trans. 186 (1895), p. 564; 185 (1894), p. 1.f Wiedemann's Annalen, 13 (1881), p. 20 and 17 (1882), p. 85.

OALORIMETRY AND MOLECULAR STRUCTURE 283

and n, but the substitution of the second halogen atom causes amarked increase in n.

MOLECULAR AGGREGATION

228. The discussion of the physical properties of gases given inthis and the preceding chapters has been based upon the supposi-tion that a gas can be regarded as a collection of separatedynamical systems, namely molecules, each of which retains itsidentity through all time. As a close to this discussion, we mayexamine what changes are to be expected if the supposition isregarded as an approximation to the truth only, and not as beingwholly true. We shall first consider what complications are intro-duced by the possibilities of molecular aggregation, leaving thediscussion of the converse process of dissociation until later.

We again simplify the problem by regarding molecules aspoint-centres of force, acting on one another with a force de-pending only on their distance apart. The chance of finding a freemolecule of class A inside an element of volume dxdydz is now

Aerhmc*dudvdw dxdydz, (377)

while the chance of finding two molecules of classes A and B inadjacent elements dxdydz and dx'dy'dz' is

Ah-h^c2^-2h*dudvdwdxdydzdu'dv'dw'dx'dy'dz',

where W is the mutual potential energy of the two molecules.If we replace the element dx' dy' dzr by a spherical shell of radii

r and r + dr surrounding the centre of the first molecule, this lastexpression becomes

W being a function of r. If we use the transformations of § 109,

u = %{u + u'), etc., a = u'—u, etc.,and write

we can transform the foregoing expression into

Sdy4,7rr2dr.(378)


The first factor after the A2 expresses the law of distributionof translational velocities for a double molecule. I t is of coursethe same as if each double molecule were a permanent structure ofmass 2m. The remaining factors express the distribution of thosecoordinates which may be regarded as internal to the doublemolecule.

229. So long as the motion of a double molecule is undisturbedby collisions, c2 remains constant, so that the energy equationshews that \rnV2 + 2W remains constant. The orbits which thecomponent molecules can describe about their common centre ofgravity fall into two classes, according as they pass to infinity ornot. Analytically these two classes are differentiated by the signof \mV2 + 2W. Double molecules for which \mV2 + 2W is positiveconsist of two molecules which have approached one another fromoutside each other's sphere of action, and which after passingonce within a certain minimum distance of each other, will againrecede out of each other's sphere of influence. On the other hand,double molecules for which \mV2 + 2W is negative consist of twomolecules describing orbits about one another, these orbits beingentirely within the two spheres of action, and this motion con^tinues except in so far as it is interrupted by collisions with othermolecules. Clearly double molecules of the first kind are simplypairs of molecules in ordinary collision. In discussing molecularaggregation we must confine our attention to double moleculesof the second kind, i.e. those for which \mV2 + 2W is negative.Such double molecules cannot be produced solely by the meetingof two single molecules. It is necessary that while the singlemolecules are in collision something should happen to change themotion—in fact to change the sign of %mV2 + 2xF. This mightbe effected by collision with a third molecule, or possibly if\mV2 + 2W were small at the beginning of an encounter, sufficientenergy might be dissipated by radiation for \mY2 + 2¥/to becomenegative before the termination of the encounter. We may dis-regard the consideration of this second possibility for the present,with the remark that if this were the primary cause of aggrega-tion, the equations with which we have been working would notbe valid, since they rest upon the assumption of conservation ofenergy for the molecules alone.


Integrating expression (377) over all values of u, v and w, wefind that vv the molecular density of uncombined molecules, is

Similarly, by integration of expression (378), we find forthe molecular density of double molecules,

(380)

in which the integration extends over all values of V and r forwhich \mV2 + 2\jr is negative.

The total number of constituent molecules per unit volume is

= i J l + 4 Ne-M

(381)

so that if q denotes the fraction of the whole mass which is free,

= 7=—TT? ' <382)TT?l2} \e

Eliminating A from equations (379), (380), etc., we obtain aseries of relations of the form

etc.

where $5, ^ , . . . are functions of the temperature only. Equationsof this type have formed the basis of practically every theory olaggregation and dissociation.*

* Compare, for instance, Boltzmann's Theory, Wied. Ann. 32, p. 39,or Vorlesungen iiber Gastheorie, 2, § 63; Natanson's Theory, Wied. Ann.38, p. 288, or Winkelmann's Handbuch d. Physik, 3, p. 725, or the theoryof J. J. Thomson, Phil. Mag. [5] 18 (1884), p. 233. These theories arebased on widely different physical assumptions, but all lead to equationsof the same general form as (403). The difference of the physical assump-tions made shews itself in the different forms for the functions <j>(T)y etc.


To study the variation of aggregation with temperature aknowledge of the exact form of the functions <f>(T), i/r(T), etc. isnecessary, but it is possible to examine the dependence of aggre-gation on density without this knowledge.

Dependence of Aggregation on Density

230. For a number of substances, it is probable that no greaterdegree of aggregation need be considered than that implied in theformation of double molecules. For such substances

v = vx + 2v2.

Neglecting the Van der Waals corrections, the pressure isgiven by (equation (5G))

p = RT(vx + v2) = \RT(v + vx) = \RvT(\+q), (384)

where q is introduced from equation (382). Thus it appears thatg, the fraction of the whole mass which is free, can be readilyobtained from readings of pressure and temperature.

The following table gives the values of 1 — q calculated in thisway from the observations of Natanson* on the density ofperoxide of nitrogen:

Aggregation of NO,

Temp.Value of 1 — q = -

p = 115 mm. I p = 250 mm. p = 580 mm. p = 760 mm.

0=-12-6°C.0 =d =0 =6 =

6 =

0°21°49-7°73-7°99-8°

151-4°

0-9190-837

0-25300840031

0-901

0-3700-1490050

0-8240-5500-2630093

Inappreciable0117

Here the single molecule is NO2, the double molecule is N2O4, andmore complex structures are supposed to occur only in negligibleamounts. The value of 1 —q is 2v2/v, and so measures the propor-tion by mass which occurs in the form N2O4.

Becueil de constantes physiques, p. 168,


Equation (383) predicts that the ratio of v2 to v\ ought to bethe same for all readings at the same temperature. We have fromthis equation

so that 1 Q - 1 j = (^j [2Vl<t>(T)] = 2itf (T).

Substituting the value of v given by equation (384), we obtain

ffl,

shewing that the ratio — ought to be the same for all readings

at the same temperature.The following table, calculated from the observations in the

table of p. 286, will shew to what extent this prediction of theoryis borne out by experiment:

Aggregation of NO2

Temp.

0 = 49-7° C.0=73-7°0=99-8°

Value of ^fa = »pq2 RT

p = 115mm.

0-6890-1670-056

p = 250 mm.

0-60801520043

p = 580 mm.

0-6800-1450-037

p = 760 mm.

0-037

Dependence of Aggregation on Temperature

231. The degree to which the aggregation varies with tempera-ture depends on the functions <f>{T), ^(T)y etc. introduced inequations (383). From equations (379) and (380), we find

(385)

At the highest temperatures (h very small) the value ofwill be clearly insignificant, but the presence of the exponentialin formula (385) suggests that after <j){T) has once become


appreciable, it must be expected to increase rapidly with fallingtemperature.

Our knowledge of the structure of matter is not sufficient toenable us to evaluate <fi(T), as given by equation (385), withprecision. Progress can only be made by the introduction ofsimple hypotheses as to the interaction of molecules, which mayprove to lead to results near to the truth.

Boltzmann* imagined that potential energy exists between twomolecules only when the centre of the second lies within a smalland clearly defined region which is of course fixed relative to thefirst, and that when the second molecule has its centre within this"sensitive region", the potential energy has always the samevalue W. This region does not necessarily consist of a sphericalshell, but if co denotes its total volume, equation (385) may bewritten in the form

where o) replaces the integral An r2drf which has represented the

extent of the "sensitive region" in our analysis. If we replace\hmV2 by x2, this equation may be expressed in the form

(386)

The upper limit of integration is determined by the conditionthat \mV2 + 2Wshall vanish, and is therefore given by £2 = — 2hW,the value of W being necessarily negative. If we put —W~ Eft,so that /? is positive, the value of £2 is 2hBfi or jJjT.

For some substances IFmay be so large that a good approxima-tion can be obtained by taking the integral in equation (386) be-tween the limits x = 0 to x = oo. The integration is now readilyeffected, and we find

in which W is negative. The degree of dissociation is then given by

Vorlesungen iiber Gastheorie, 2, Chap. vi.


which is Boltzmann's formula for molecular aggregation anddissociation.

An almost exactly identical treatment of the problem wasgiven by Willard Gibbs.*

The following table contains the densities of peroxide ofnitrogen observed at various temperatures by Deville andTroost,f the pressure being one atmosphere throughout, and alsothe values calculated from equation (387).

Aggregation of NO2

Temp.°C.

183-215401350121-5111-3100-190-0

Density(observed)

1-571-581-601-621-651-681-72

Density(calc.)

1-5921-5971-6071-6221-6411-6761-728

Temp.°C.

80-670-060-249-639-835-426-7

Density(observed)

1-801-922-082-272-462-532-65

Density(calc.)

1-8011-9202-0672-2562-4432-5242-676

Continuity of Liquid and Gaseous States

232. At very high temperatures, the series (381) reduces to itsfirst term, so that q = 1, and there are no molecules in permanentcombination.

At lower temperatures h is greater, so that not only is A greater,but the exponential e~h[^mV2+2W\ in which it will be rememberedthat the index is always positive, is also greater. The relativeimportance of the later terms of the series (381) is thereforegreater. Finally, we reach values of the temperature for which hhas so great a value that the series (381) becomes divergent.According to our analysis the molecules tend to form into clustersat this point, each containing an infinitely great number of mole-cules, and, ultimately, into one big cluster absorbing all themolecules. By the time this stage is reached the analysis hasceased to apply, as the assumption that the molecular clusters aresmall, made in § 230, is now invalidated. It is, however, easy to

* Trans. Connecticut Acad. 3 (1875), p . 108 and (1877), p . 343;Silliman Journal, 18 (1879), p . 277. Also Coll. Works, 1, pp. 55 and 372.

t Comptes Rendus, 64 (1867), p . 237.


give a physical interpretation of the point now reached: obviouslyit is the point at which liquefaction begins, and the collection ofmolecular clusters is a saturated vapour.

The series (381) may be regarded as a power series in ascendingpowers of A. Thus for a given value of h, say hQ, there is a singlevalue of A, say Ao, such that the series is convergent for all valuesof A less than Ao and is divergent for all values of A greater thanAo. In other words, corresponding to a given temperature, thereis a definite density at which the substance liquefies. This of courseis the vapour-density corresponding to this temperature. As hincreases, A will clearly decrease, and conversely, so that anincrease of pressure is accompanied by a rise in the boiling-pointof the substance.

Since A depends on v, the relation between correspondingvalues h0, Ao which has just been obtained may be expressed inthe form

/(*>, T) = 0, (388)

expressing the relation between v and T at the boiling-point of aliquid.

The Critical Point

233. For very small values of h, vx and v become identical, sothat the series (381) cannot become divergent. Thus for very highvalues of T equation (385) can have no root corresponding to aphysically possible state. If Tc is the lowest value of T for whichequation (385) has a root, then Tc will be a temperature abovewhich liquefaction cannot possibly set in, no matter how great thedensity of the gas; in other words, Tc is the critical temperature.

Ordinary algebraic theory tells us that there must be twocoincident values of v given by equation (385) to correspond tothe critical temperature Tc, agreeing with what is already knownas to the slope of the isothermals at the critical point.

Pressure, Density and Temperature

234. It will now be clear that when a gas or vapour is at a tem-perature which is only slightly greater than its boiling-point atthe pressure in question, it cannot be regarded as consisting of

CALOBIMETRY AND MOLECULAR STRUCTURE 291

single molecules, but must be supposed to consist partly of singlemolecules and partly of clusters of two, three or more molecules.If m is the mass of a single molecule, and if vv v2i vz,... have thesame meaning as before, the density is given by

In calculating the pressure, each type of cluster must betreated as a separate kind of gas, exerting its own partial pressure.We accordingly obtain for the pressure, as in § 230,

From a comparison of this equation with equation (383),remembering that vl9 v^... are functions of T and/), it is clear thatneither Boyle's law, Charles' law nor Avogadro's law will besatisfied with any accuracy.

235. The observed deviations from the laws obeyed by a per-fect gas must of course be attributed partly to aggregation, as hasjust been explained, and partly to the causes which have alreadybeen discussed in Chap. in. The two sets of causes are not, how-ever, altogether independent; so that it is not sufficient to con-sider the effects separately, and then add. The state of the questionis, perhaps, best regarded as follows.

The effect of the forces of cohesion is too complex for an exactmathematical treatment to be possible. We have therefore, inChap, n i and the present chapter, examined their effect with thehelp of two separate simplifying assumptions. In Chap, m,following Van der Waals, we regarded the gas as a single molecularcluster containing an infinite number of molecules; and in re-placing the whole system of the forces of cohesion by a permanentaverage force, we virtually neglected the effect of any formationsof small clusters inside the large cluster. In the present chapter,on the other hand, we have been concerned solely with the forma-tion of small clusters, and have disregarded the large clusteraltogether. The former treatment, because it failed to take accountof the formation of small clusters, led to the erroneous result(equation (71) and § 45) that the internal pressure is proportional


to the temperature; the treatment of the present chapter, be-cause it fails to consider the clustering of the gas as a whole, hasled to the erroneous conclusion that the internal pressure isidentical with the boundary pressure. The situation may then besummed up by saying that the treatment of Chap, in considersonly the tendency to mass-clustering, while that of the presentchapter considers only the tendency to molecular-clustering.

So long as the deviations from the behaviour of a perfect gasare small, the two tendencies may be considered separately, andthe total deviation regarded as the sum of the two deviationscaused by these tendencies separately. On the other hand, as weapproach the critical point the phenomena of mass-clustering andmolecular-clustering merge into one another, ultimately becom-ing identical at the critical point. The two effects are no longeradditive, for each has become identical with the whole effect.

It must be borne in mind that we have only found an exactmathematical treatment of either effect to be possible by makingthe assumption that the effect itself is small. In other words, sofar as our results apply, the effects are additive. It may be noticedthat the deviations from the laws of a perfect gas, which weradiscussed in Chap, in, fell off proportionally to IJT and 1/T2,whereas the deviations discussed in the present chapter varymuch more rapidly with the temperature.

Calorimetry

236. The formulae which have been obtained for the specificheats will be affected to a greater or lesser degree by the possi-bilities of molecular aggregation. For in raising the temperatureof the gas work is done not only in increasing the energy of thevarious molecules, but also in separating a number of moleculesfrom one another's attractions. This latter work will involve anaddition to the values of Gp and Cv such as was not contemplatedin the earlier analysis of §§ 220-223. We should therefore expectthe values of Cp and Cv to be in excess of the values obtained fromour earlier formulae, throughout all regions of pressure and tem-perature in which molecular aggregation can come into play. Forinstance, the specific heats of nitrogen peroxide have been


studied by Bertkelot and Ogier,* who give the following valuesfor Op:

From 27°„ 27°„ 27°„ 27°„ 27°

tototototo

67° C , C100°,150°,200°,300°,

\ = 1-62,1-46,1-115,0-85,0-64.

The excess in the values of Cp at the low temperatures may bereasonably attributed to the work required to separate moleculesof N2O4 into pairs of molecules of NO2.

Steam provides a further illustration of a somewhat differentnature. Wet steam is steam in which large molecular clustersoccur, dry steam is steam in which the molecules are all separate,and our quantity q measures what engineers speak of as the dry-ness of wet steam. For the value of y for wet (saturated) steam,Rankine and Zeuner give respectively the values 1-0625, 1-0646.The value for dry steam is about 1-30. If we used the formula

9

7=1 3(1+/?)

for the calculation of n, we should come to the conclusionthat 3 + 3/? had the value 32 for wet steam, and 6-6 for drysteam.

The large value of 3/? in the former case is fully in keeping withthe existence of large clusters of molecules, so large that each hasabout 32 degrees of freedom.

D I S S O C I A T I O N

237. A treatment similar to the foregoing may be applied tothe problem of dissociation. The former molecules must be re-placed by atoms, and the former clusters of molecules by singlemolecules.

Let us consider a gas in which the complete molecules are eachcomposed of two atoms, distinguished by the suffixes 1,2. As in

* Bull. Soc. Chimie [2], 37 (1882), p. 434; Comptes Rendus, 93 (1882),p. 916; Ann. d. Chim. et de Phys. [5], 30 (1883), p . 382; Recueil deconstantes pJiysiques, p . 108.


equations (399) and (400) the laws of distribution of dissociatedatoms and complete molecules are

r2 =r12 = ,

where W is the potential energy of the two atoms forming themolecule. The analysis will be simplified, and the theory suf-ficiently illustrated, by regarding the atoms as point centres offorce, of masses m1, m% respectively. Thus we obtain as the laws ofdistribution of velocities for the dissociated atoms

Ae-hmic*dudvdw)Be~hm^dudvdw]9 ( ]

and for the complete molecules

v2hWi+m* dadfldy4nr2dr.

(390)This latter law is arrived at in the same way as the law (378),except that the scheme of transformation of velocities must nowbe taken to be

mu + mu' ,<x, = u-u, etc.,ml + ra2

this being the transformation already used in § 109.238. Although the mathematical analysis is similar to that of

the aggregation problem, there is an important difference in thephysical conditions. The law of distribution (390) is limited tovalues of the variable such that

is negative; as soon as this quantity becomes positive the moleculesplits up into its component atoms. Now in the case of molecularaggregation, the attraction between complete molecules is notgreat, so that W is a small negative quantity, and the range ofvalues for V is correspondingly small. In the case of chemicaldissociation W is a large negative quantity, and the range for Vis practically unlimited.


An estimate of the value of W can be formed from the amountof heat evolved when chemical combination takes place. Forinstance, when 2 grammes of hydrogen combine with 16 grammesof oxygen to form 18 grammes of water, the amount of heat de-veloped according to Thomsen's determination is 68,376 units—sufficient to raise the temperature of the whole mass of water by3600° C. The value of V necessary for dissociation to occur istherefore comparable with the mean value of V at 3600° C, andthese high values of V will be very rare in a gas at ordinary tem-peratures. The exclusion from the law of distribution (410) of highvalues of V will therefore have but little effect either on the law ofdistribution or on the energy represented by the internal degreesof freedom, and we may, without serious error, regard the law ofdistribution as holding for all values of V.

In such a case, it appears that the molecule may be treatedexactly as an ordinary diatomic molecule, supposed incapable ofdissociation, but possessing six degrees of freedom, three trans-lational degrees represented by the differentials dudvdw, andthree internal degrees represented by the differentials dadfidy.

Since there are six degrees of freedom, the value of y will be aslow as 1J if potential energy is neglected, and will be even less ifpotential energy be taken into account. We have, however, seenthat for diatomic molecules y is fairly uniformly equal to If, andthis shews that the ordinary diatomic molecule must not betreated as consisting of two atoms describing orbits in the waywe have imagined.

We are here brought back to the difficulties which have alreadybeen encountered in § 225 in connection with the specific heats ofgases. The solution of these difficulties is not provided by the oldclassical dynamics but by the new quantum dynamics. Weaccordingly leave the question at this stage; it passes out of thescope of the kinetic theory, and so beyond the range of the presentbook.

Appendix I

MAXWELL'S PROOF OF THE LAW OF DISTRIBUTIONOF VELOCITIES

As already mentioned, Maxwell's original proof of the law of distribu-tion of velocities was given in a paper he communicated to the BritishAssociation in 1859. Except for a slight change of notation, the form inwhich it was given is as follows.*

"Let N be the whole number of particles. Let u, v, w be the com-ponents of the velocity of each particle in three rectangular directions,and let the number of particles for which u lies between u and u + du beNf(u) du, where f(u) is a function of u to be determined.

"The number of particles for which v lies between v and v + dv will beNf(v) dv, and the number for which w lies between w and w + dw will beNf(w) dw, where/ always stands for the same function.

"Now the existence of the velocity u does not in any way affect that ofthe velocities v or w, since these are all at right angles to each other andindependent, so that the number of particles whose velocity lies betweenu and u + du, and also between v and v + dv and also between w andw + dw is

Nf{u)f(v)f(w)dudvdw.If we suppose the N particles to start from the origin at the same instant,then this will be the number in the element of volume dudvdw after unitof time, and the number referred to unit of volume will be

NJ(u)f(v)f(w).

"But the directions of the coordinates are perfectly arbitrary, andtherefore this number must depend on the distance from the origin alone,that is

f(u)f(v)f(w) =Solving this functional equation, we find

f(u) = CeAu\ <j){u2 + v2 + w2) = C*eA(ut+vt+tviK"

This proof is now generally admitted to be unsatisfactory, because itassumes the three velocity components to be independent. The velocitiesdo not, however, enter independently into the dynamical equations ofcollisions between molecules, so that until the contrary has been proved,we should expect to find correlation between these velocities.

• J. C. Maxwell, Collected Works, 1, p. 380.

Appendix IT

THE ^-THEOREM

In Chap, iv (§ 87) we discovered certain steady states for a gas, but itwas not proved that a gas would necessarily attain to one of these steadystates after a sufficient time. The following proof of this was originallygiven by Boltzmann and Lorentz.

With the notation already used in Chap, rv, let H be defined by

H = / / Iflogfdudvdw, (a)

in which the integration extends over all possible values of u, v, w. ThusH is a pure quantity and not a function of u, v, w; it depends solely uponthe law of distribution of velocities and therefore remains unchanged solong as this law remains unchanged. Hence a necessary condition forasteady state is given by dH/dt = 0. We proceed to evaluate dH/dt in thegeneral case.

After an interval dt, the value of/ log/ corresponding to any specifiedvalues of u9 v, w will of course have changed into

or, what is the same thing, into

Hence the increase in H, which may be written —=- dt, will be given by

* {///<1 + I°gft I dudvdw^ dt, (6)

or, substituting the value of df/dt from equation (306),

-?- = v [[[[[Iff{l + \ogf) (]]'-fJ')V<r2 cos 0 dudvdwdu'dv'dw'dco.

(c)Since H depends on molecules of all classes (A, B, C, ...) equally,

formula (c) necessarily remains true if we interchange the roles of mole-cules of classes A and B. We then have the same equation, except that,as the first factor inside the integral, 1 + log/' replaces 1 + log/. Addingtogether these two values for dH/dtf we obtain

c o s ° dudvdwdu'dv'dw'du.

(d)

298 APPENDIX II

This equation expresses dH/dt as the sum of a number of contributions,one from every possible class of collision. The typical class of collision istaken to be class a, in which

u, v, w, u\ v', v/become changed into

u, v, w, u\ v', w'.

If we use the same equation, but take as the typical collision one ofclass ft, in which

u, v, w, u', v', w'become changed into

u, v, w, u', v', w\

we obtain, as a still different form for dH/dt,

(e)As in § 86, we may replace the product of the first six differentials on theright hand of this equation by dudvdwdu'dv'dw', and if we add thismodified value of dH/dt to that given by equation (d), we obtain

') (#' -//')Vor2 c o s ° dudvdwdu'dv'dw'du.

Now (logff — log//') Is positive or negative according asjjf' is greateror is less than / / ' and is therefore always of the sign opposite to that ofJJ'—ff. Hence the product

(logjsr-iog/f) (//'-//'),if not zero, is necessarily negative. Since V cos 0, the relative velocityalong the line of centres before impact, is necessarily positive for everytype of collision, it follows that the integrand of equation (/) is alwayseither negative or zero. Hence equation (/) shews that dH/dt is eithernegative or zero.

Hence as the motion of the gas progresses, H continually decreasesuntil it reaches a final state in which dH/dt = 0. We have seen that everycontribution to dH/dt on the right of equation (/) is negative or zero.Thus in the final steady state, every contribution is zero. In other words,we must have

for every type of collision, bringing us back exactly to the solutions wediscussed in § 87.

Appendix III

THE NORMAL PARTITION OF ENERGY

In §§ 210, 211 we found that the most probable partition of energy(Ev E2, ESf ...) was given by the equations (355)

d log FX(EX) _ d log F2(E2) _ d log FZ(EZ) __ 1dE1 "" dE2 " dEz ~ RT w

where Ex + E2 + Ez + ... = E, the total energy of the system.In the most general partition of energy, let us write, as in § 211,

P = log FX(EX) + log F2(E2) + log Ft(E%) + . . . .For any partition of energy Et + elf Et + e2t Ei + ez, ..., adjacent to the

most probable, the value of P is

If the total energy remains the same, we have Eex = 0, so that thesecond term on the right vanishes, and the equation reduces to

P - P

where Po is the value of P for the most probable partition of energy.In the special case in which E19E2f... consist of s,t,.,. squared

terms, this assumes the form

P—P — •• * fc

It has already been seen that the only stationary value of P is givenby e± = e2 = ... = 0. This makes P = Po, and an inspection of the righthand of equation (d) shews that this value is a true maximum.

As we recede from the value ex — e2 = ... = 0, it is clear that P — Pobecomes finite as soon as ex becomes comparable with *Js.RTt e2 withJt.RT, and so on. For such values of ev e2, ... the first term on the rightof equation (d) is infinitely greater than any of the succeeding terms, andthe value of P — Po reduces to

e2p p 57 _ _ _ L _ _ (*)

For values of elf e29 ... greater than these P — Po becomes equal to — 00From equations (350) and (362), the general value of Wx is

while the whole value of Wx + W2 will be

W1+W2=[f...epdeldez.... (g)

300 APPENDIX III

Thus the whole value of the integral (g) comes from a small rangeof values surrounding the values e1 = e2=. . . = 0; i.e. the values ofEv E2t ... given by equations (a). Thus the integral (g) reduces to theright-hand member of equation (/), the small range dEx being comparablewith Js.RT, the small range dE2 being comparable with jt.RT, and soon. These small ranges are of course small in comparison with the wholevalues of E19 E2, ...; thus dE1 is comparable with EJ^Js, dE2 with E2/Jt,and so on.

With such values for the small ranges dEl9 dE2, ..., the value ofWl + Wt given by equation (g) becomes identical with the value of W1given by equation (/). Thus we have WJW2 infinite, shewing that thepartition of energy now under consideration is a normal property of thesystem.

The proof that the remaining parts of the system, if any, in whichthe energy is not of this type, will necessarily tend to the partition ofenergy given by equations (e) is more difficult, since the sign of the termson the right of equation (c) must necessarily be a matter of uncertaintyso long as the form of the energy-function remains unspecified. It is,however, clear that the arrangement of the loci Ex = cons., E2 = cons.,etc., in the phase-space must, in every case, be of the same general typeas that in the simple case just considered, from which we may infer thatP—Po must, in the more general case also, be of negative sign. It againfollows that WJW2 must be infinite, so that the most probable partitionof energy as expressed by equations (a) is now seen to be a normalproperty of the system.

We accordingly see that every system must pass to a final state inwhich W19 and therefore also the entropy, is a maximum. In this waywe obtain an analytical proof of the second law of thermodynamics,which may now be regarded as being on a mathematical, instead of on apurely empirical basis.

Appendix IV

THE LAW OF DISTRIBUTION OF COORDINATES

We fix our attention on a part of a dynamical structure, this partconsisting of N similar units, which we may think of as molecules fordefiniteness, each unit possessing p degrees of freedom, and thereforehaving its state specified by 2p quantities </>v <f>2, • • • ^2J» these being co-ordinates of position and their corresponding momenta, as in § 205.

Imagine a generalised space of 2p dimensions constructed, having

A* <f>2> ••• 0 2 *as orthogonal coordinates. Then the state of any molecule of the systemcan be represented by a single point in this space, namely the pointwhose coordinates are equal to the coordinates <fiv <j)2, ... <f>2P specifyingthe state (i.e. velocity and positional coordinates) of the molecule. Thestates of all the molecules can be represented by a collection of points inthis space, one point for each molecule. We want to find the law accordingto which these points are distributed in the space.

Let 7 denote the density of points in this space—the quantity which weare trying to find—so that

Td<j)1d(l>2...d<j>2p (a)

will be the number of points (or molecules) such that <px lies between(f>x a n d <fix + d<f>lf <j>2 b e t w e e n <fi2 a n d <j>2 + d<p2, a n d s o o n .

Let us now suppose the whole of this space divided up into n smallrectangular elements of volume, each of equal size a), and let these beidentified by numbers 1, 2, 3, Let us fix our attention on a specialdistribution of points, which is such that the number of points inelements 1, 2, 3, ... are respectively av a2, o3, Let any distribution ofpoints giving these particular numbers ax, a2, a3, ... be spoken of as adistribution of class A. Similarly any distribution of points givinganother set of numbers bv b2, bZi ... may be spoken of as a distributionof class B, and so on.

Each point in the original generalised space will correspond, as in § 206,to a complete distribution of points in the space now under consideration.The distribution corresponding to some of these original points will bea distribution of points of class A, corresponding to others it will be adistribution of class B, and so on. We proceed to evaluate the volume,say WA, of the original generalised space which is such that the pointsin it represent systems for which the distribution of coordinates is ofclass A.

This volume is readily seen to be given by

)

302 APPENDIX IV

In this expression the first factor on the right hand represents thenumber of ways in which it is possible to distribute the N points repre-senting the N different molecules, between the n different elements,subject only to the condition of the final arrangement being of type A.The remaining factor, say A, given by

A = o)Njjj...dx1dx*dx3... (c)represents the volume of the generalised space which corresponds toeach one of these arrangements, Xi> #2» Xs> • • • being coordinates of partsof the system other than the N molecules under consideration. If wewrite

and use a similar notation for a system of class B, etc., then the volumesWA9 WB, ... are given by

WA = \0Af WB = A0*, etc. (e)

According to the well-known theorem of Stirling, the value of p\ whenp is very large approximates to the value

On taking logarithms of both sides, this becomesLt log p! = | log 277 + (p -f J) log p —p*P-QD

Taking logarithms of both sides of equation (d), and using this valuefor log £?!,

log 6A = log N\ - i log as\

= (N + i) log N- \ ( n - 1) log 27T~rn (as + J) log as.j-i

If we put1

this gives as the value of 6A

Since WA = A^, etc., it is clear that WA is proportional to e~NKa.The most probable partition of energy is obviously obtained by

making WA a maximum, and therefore K a minimum, for different valuesof Op a2, .... For the variation of Ka, as given by equation (/), we find

The variations Sc^, Sa2, ... are not independent. They are necessarilyconnected by two relations, and in some cases by more. Of the tworelations which are certain, the first expresses that the total number of

APPENDIX IV 303

molecules remains equal to the prescribed number N, and is thereforeexpressed by the equation

'if 8as = 0, (i)s = l

while the second relation expresses that the total energy of the Nmolecules is equal to the allotted amount Ev Let ex denote the energyassociated with a molecule represented by a point in cell 1, so that e2 isa function of <f>v <j>t9 ... fi2j>, the coordinates of the first cell. Let e2 be theenergy associated with a molecule represented by a point in cell 2, andso on. Then the total energy of the JV molecules, when the distributionof coordinates is of class A, will clearly be a1e1 + aze2+ ..., so that wemust have

and on variation of this we obtain the relation

Any other relations there may be will be derived from equations ofsimilar type, expressing that the total of some quantity /i summed overall the molecules will have an assigned value. The integral equation willbe of the form

Z ji5as = M,5 = 1

and the corresponding relation between the quantities Sc^, Sa7, ...will be

Z?V5*as = 0. (k)5 = 1

In many problems there will be six equations of this type, the differentji's representing three components of linear momentum, and three com-ponents of angular momentum. We may, however, be content to takeone relation as typical of all, and shall suppose it to be given by theequations just written down.

Following a well-known procedure, we now multiply equations (i),0), (k) by undetermined multipliers p, q, r, and add corresponding mem-bers of these equations and equation (h). We obtain

SK = £ J1 + 2a- + log -jf + P + qes + r A | $as,

and the maximum value of K is now given by the equations

l + ^ + logn^+p + qe5 + r/is = Of (5=1, 2, . . . n).

Since a19 a2, ... are all supposed to be large quantities, the term ^-~may be neglected, and we obtain

V?h - e-(l

304 APPENDIX IV

If r is the quantity defined in expression (a), the value of a9 will beTO), where r refers to the «th cell. Equation (I) becomes

NT = — e-(1+p>e-(«c+v*>.

n<o

which is true for all values of <f>v <f>2, ... <j>Zv9 since equation (I) was truefor every cell. Changing the constants, this equation may be rewritten

in which the one typical quantity /i is now replaced by the actual seriesof quantities filt fiz,.... Using this value for r, the law of distributionof coordinates (cf. expression (a)) is seen to be given by

From the result obtained in Appendix in, it follows that this distri-bution of co-ordinates is infinitely more probable than any other, and soexpresses a normal state of the system.

If there is no potential energy, there will be three co-ordinates x, y, zfixing the position of the molecule in space, and these will not occurexcept through the differential dxdydz. Thus the chance of finding amolecule in a small element of volume dxdydz, however chosen, will beproportional simply to dxdydz. This provides the justification for theassumption of molecular chaos introduced in § 82.

Appendix V

TABLES FOR NUMERICAL CALCULATIONS

The following tables will be found of use for the various numericalcalculations which are likely to be needed in connection with kinetictheory problems. The values of ft{x) are from a table by Tait in the paperalready referred to (p. 145).

X

0-10-20-30-40-5

0-60-70-80-91-0

111-21-31-41-5

1-61-71-81-920

212-22-32-42-5

2-62-72-82-930

X2

0010-040-090160-25

0-360-490-640-811-00

1-211-441-691-962-25

2-562-893-243-614-00

4-414-845-295-766-25

6-767-297-848-419-00

0-990050-960800-913930-852140-77880

0-697680-612630-527290-444860-36788

0-298200-236930-184520-140860-10540

0-077300-055580-039160-027050-01832

0-012150-007910-005040-003150-00193

0-001160-000680-000390-000220-00012

0-112460-222700-328630-428390-52050

0-603860-677800-742100-796910-84270

0-880210-910310-934010-952290-96611

0-976350-983790-989090-992790-99532

0-997020-998140-998860-999310-99959

0-999760-999870-999920-999960-99998

rlr(x)Denned by

equation (159),p. 140

0-200660-405310-617840-842001-08132

1-339071-618191-921322-250722-60835

2-995823-414483-865384-349394-86713

5-419116-005706-627157-283667-97536

8-702349-4646710-2623611-0954711-96402

12-8679813-8073414-7822515-7925516-83830

Appendix VI

INTEGRALS INVOLVING EXPONENTIALS

In making Kinetic Theory calculations, we are frequently confrontedwith integrals of the type

une~Au2du, (i)/ •

where n is a whole member. Such an integral can be evaluated in finiteterms when n is odd, and can be made to depend on the integral

-*u*du, (ii)

when n is even. In each case the reduction is most quickly performed bysuccessive integrations by parts with respect to u2. Tables for theevaluation of the integral (ii) have already been given in Appendix v.

When the limits of integration are from u = 0 to u = oo, the resultsof integration are expressed by the formulae

The following cases of the general formulae are of such frequentoccurrence that it may be useful to give the results separately:

Jo 2 V hm Jo 2hm

[~e-h™zu*du = ] \/~*, [°°e-h

Jo 4 V hzmz Jo

= | /~f- [<°e-h8 V h5ms Jo

Jo16 V h7m7

Each integral can be obtained by differentiating the one immediatelyabove it with respect to hm. In this way the system can be extendedindefinitely.

INDEX OF SUBJECTS

a-coefficient of Van der Waals, 67, 81Absolute scale of temperature, 26Absolute zero of temperature, 29Accommodation, coefficient of, 192Adsorption of molecules by a solid, 54,

191Aggregation, molecular, 283Air, ratio of specific heats of, 57, 278Assumption of molecular chaos, 105,304Atmospheres of planets, 42Atomic dissociation, 293Atomic theory, 2Avogadro's law, 30Avogadro's number, 30, 31

^-coefficient of Van der Wapls, 66, 82Boiling point, theory of, 190Boundary, effect of, on law of distribu-

tion, 64, 68density at, influenced by cohesion,

67, 292Boyle's law, 13, 19, 53

deviations from, 63, 291Brownian movements, 7,24,31,218,222

Calorimetry, 33, Chap, xi (p. 273); seealso Specific heats

Capillary tubes, flow of gas in, 167Carbon-dioxide, isothermals of, 91, 92Carnot's law (Cp -Cv) , 35

(thermodynamics), 25Chaos, molecular, 105, 304Characteristic opalescence at critical

point, 274Charles' law, 29, 54Chemical affinity, 259Clausius, equation of state, 100

virial, 70Coefficient of slip at a solid, 169Cohesion, 4, 62, 66, 82, 291Collisions, between elastic spheres, 21,

107, 131between molecules with law fir~*f 76,

172, 233, 242in a gas, dynamics of, 22, 106, 132,

237, 242in a gas, law of distribution of re-

lative velocities in, 134in a gas, law of distribution of

velocities in, 145

Collisions, in a gas, number of, 131in outer space, 33, 49

Conduction, of heat in a gas, 47, Chap.VH (p. 185), 249

of heat in a solid, 194of electricity, 194

Conductivities, ratio of thermal andelectrical, 196

Continuity, hydrodynamical equationof, 225

of liquid and gaseous states, 88, 289Correlation, between velocity and posi-

tional coordinates: absence ofcorrelation assumed in statisticalmethod, 105; justification, 304

between components of velocity:absence assumed by Maxwell,116, 296; justification, 304

Corresponding states, law of, 97, 99Critical point, 92, 96, 274, 290Critical pressure, 93Critical temperature, 93Critical volume, 93, 94

Dalton's law of pressure, 19, 29Degrees of freedom, 253, 280

of a molecule, 254, 280Demon, sorting, of Maxwell, 271Density, exact definition of, 103

fluctuations of, 103, 272Diameter of molecules, see SizeDiatomic gases, 279, 281, 293Dieterici, equation of state of, 68, 87,

94, 95, 96Diffusion of gases, 15, 198, 222, Chap.

Tin (p. 198), 249numerical values, 213

Diffusion, pressure, 218, 251Diffusion, thermal, 218, 251Dissipation of planetary atmospheres,

42Dissociation, 293Distribution, laws of, see LawDoppler effect, 130Dynamical system, general motion of,

254Dynamics of collisions, see Collisions

Effusion of gases, 58Electricity, conduction in solids, 194

308 I N D E X OF S U B J E C T S

Electrons, charge on, 131in a gas, 44, 138in a solid, 55, 194

Energy, distribution of, 299Entropy, 25, 267, 268, 300Equalisation of temperature, 15, 267,

270Equations of state, 98Equipartition of energy, 21, 43, 263Evaporation, 8, 15, 90

Final state of a gas, 259, 268Freedom, degrees of, 253, 280Free electrons, in a gas, 44, 138, 143

in a solid, 194, 197Free path, 43, Chap, v (p. 131)

calculation of, 43, 135numerical values, 48of electrons, 44, 138, 143, 197

Gas-thermometry, 78 ff.Gaseous medium, statistical mechanics

of a, 260 ff.Gaseous state, 9, 289Generalised space, 256

//-theorem, 297Heat, conduction of, 47, Chap, vn

(p. 185), 249mechanical equivalent of, 34mechanical nature of, 5, 14, 120specific, see Specific heats

Hydrodynamical equations of a gas, 226

Ideal gas, 54Internal energy of atoms and molecules,

35, 279Isothermals, 83 ff.

(experimental) of CO2, 91(experimental) of isopentane, 95

Joule-Thomson effect, 63

Law of distribution, definition of, 37,104

of coordinates, 265, 301of relative velocities in collisions, 134of velocities, 4, 36, 104, 239, 205, 301of velocities in collisions, 145

Law of force between molecules: fir~*,175, 188, 210, 233

pur-*, 76, 172numerical values of s, 174

Liouville's theorem, 258Liquid, superheated, 90

Liquid state, 6, 88, 89, 289Loss of planetary atmospheres, 42Low temperature phenomena: con-

duction of electricity, 197specific heats of gases, 279

Mass, and molecular motions, 119of molecule, 33

Mass-motion, equations of, 165, 167Maxwell's law, 4, 36, 40, 115, 121

experimental verification of, 124Mean free path, see Free pathMechanical equivalent of heat, 34Metals, theory of conduction in, 194Mixture of gases: diffusion, 208, 212

law of distribution, 114persistence of velocities, 152pressure, 19viscosity, 181

Molecular aggregation, 283Molecular chaos, 105, 304Molecular rays, 11Molecules: mass, 33

number per cu. cm., 31size, 49, 83, 183structure, 35, 276velocities, 20, 55, 56

Momenta, generalised, 255Monatomio gases, 188, 280

Normal properties, 258Normal state, 259Number of molecules per cu. cm., 30, 31

Osmotic pressure, 55

Partition of energy, 260, 266, 299Pendulum, decay of oscillations, 164Perfect gas, 79Persistence of velocities, 147, 162Planetary atmospheres, 42Pressure: in a gas, 13, 17, 28, Chap, in

(p. 51), 229, 230physical interpretation of, 3, 13, 17,

72, 290Pressure-coefficient, 78Pressure-diffusion, 251Probability and Thermodynamics, 268

Radius of molecules, see SizeRandom motion, 218Random walk, 220Rarefied gases, conduction of heat in,

191viscosity in, 168, 170

INDEX OF SUBJECTS 309

Reduced equation of state, 96Reflection of molecules from a solid,

21, 52, 169Relaxation, time of, 233Rotating gas, equilibrium of, 260Rotational energy of molecules and

atoms, 135, 277

Self-diffusion of a gas, 202, 211, 214Size of molecules, determinations of,

from ^-coefficient, 83from electron beams, 143from viscosity, 183

Solid state, 4Solids, conduction of heat and electricity

in, 194Sorting demon of Maxwell, 271Sound, propagation and velocity of,

57Specific heats, of a gas, 33, 275

at low temperatures, 279ratio of, 35, 36, 57, 279

Specular reflection, 169States of matter, 4, 88, 291Statistical mechanics, 253Statistical methods, 103Steady state in a gas, 111, 114, 230Stirling's theorem, 220Stokes's law, 164Stresses in a gas, 229Structure of molecules, 35, 279Supercooled vapour, 90, 290Superheated liquid, 90Sutherland's formulae, 177

Temperature, 14, 25absolute zero of, 29equalisation of, 15, 267, 269thermodynamical, 26

Thermal diffusion, 251Thermal effusion, 58Thermal motions of molecules, 117Thermodynamics, 25, 266

second law of, 26, 267, 300Thermometry, gas, 78Time of relaxation, 233Torsion balance of Kappler, 10, 32, 129Transfer, equations of, 230Transpiration of gases, 60Trial and error, law of, 121Tubes, flow of a gas through, 167

Van der Waals' equation, 64, 75, 88, 93Vapour, 9, 89

supercooled, 90, 290Velocity, persistence of, 147, 162

of molecules (numerical values), 56of sound, 57

Virial, 70the second coefficient, 75, 77

Viscosity, 45, Chap, vi (p. 156), 248at low pressures, 166in a mixture of gases, 181variation with density, 164variation with temperature, 70

Volume-coefficient, 80

Wiedemann-Franz law, 196

Zero, absolute, of temperature, 29

INDEX OF NAMESAmagat, 101Andrade, 2Andrade and Parker, 9, 24Andrews, 91Aristotle, 2Avogadro, 30

Bannawitz, 188Baumann, 178Beatty and Bridgeman, 102Behn and Geiger, 279Bernoulli, D., 3Berry, 192Berthelot and Ogier, 293

Bielz, 144Binkele, 174Birge, 31Blankenstein, 169Boardman, 202, 215Bohr, N., 196Boltzmann, 28, 66, 145, 159, 162, 285,

288, 297Born, M., 143Boscovitch, 5Boyle, R., 13, 19, 165Bragg, W. L., 5Breitenbach, 177, 178, 183Brillouin, 166

310 I N D E X OF N A M E S

Brode, R. B., 143Brown, F. C., 127Brown, Robert, 7

Catlendar, 79Capstick, 282Carnot, 25Chapman, S., 175, 176, 179, 182, 183,

188, 209, 210, 211, 212, 213, 218,250

Chapman, S. and Cowling, 165, 174, 252Chapman, S. and Hainsworth, 250Chappuis, 79Charles, 29Clausius, 3, 4, 60, 90, 100Cohen, V. W., 128Cook, 180, 181Crookes, Sir W., 166

Dalton, 29Democritus, 1Deutsch, 213Deville and Trost, 289Diesselhorst, 197Dieterici, 68 ff.Dootson, F. W., 218, 251Dorsman, 160, 174, 179Drude, 194Dunoyer, L., 10

Einstein, A., 8, 222EUett, 128Elliott and Masson, 250Enskog, D., 165, 166, 176, 182, 188,

212, 235, 246ff.Epicurus, 2Eucken, 183, 188, 189, 190, 281

Fletcher, H., 32Frank, 196

Gabriel, 178Gassendi, 2, 3Gauss, 121Gay-Lussac, 62Gerlach, 11Germer, 125Gibbs, Wiilard, 270, 289Gibson, 165Graham, 59, 181

Hall, E. E., 126Hartek, 202Hasse, 180, 181Helmholtz, H. von, 168

Heraclitus, 1Herapath, 3Hercus and Laby, 189Hilbert, 211Hooke, 3Howarth, 193

Ibbs, 252

Jackmann, O., 213Jackson, J. M., 193Jager, 196Jones, J. H., 124Joule, 3, 62

Kalahne, 277Kamerlingh Onnes, 83, 98, 99, 100, 166,

174, 178, 179, 180Kappler, E., 10, 32, 129Kaye and Laby, 183, 277Keesom, 77, 92Kelvin, Lord, 26, 62, 214Knauer, F., 144Knudsen, 53, 54, 59, 60, 169, 170, 191,

192, 194Ko, C. C, 127Koch, 278Kollath, R., 143Kuenen, 182, 207Kundt and Warburg, 168, 169, 279

Lammert, 128Langevin, 209, 210Langmuir, 54, 128Leduc, 79, 279Lees, C. H., 196Lennard-Jones, 77, 179Leucippu8, 1Liouville, 258Loeb, JO, 54, 126, 191, 193Lonius, 213Lorentz, H. A., 196, 235, 246, 297Lorenz, 196Loschmidt, 30, 215Lucretius, 2Lummer and Pringsheim, 279

Maas, 174Makower, 279Markowski, 174, 178Masson, 250, 279Maxwell, J. Clerk, 3, 4, 40, 90, 128, 164,

165, 168, 169, 182, 188, 209, 211,242, 271, 299

Mayer, H. F., 143

I N D E X OF N A M E S 311

Meissner, 128Meyer, 0 . E., 141, 201, 209Michels and Gibson, 1C5MUlikan, R. A., 169Mott, 193Mott and Gurney, 7

Natanson, 279, 285, 286Newton, Sir Isaac, 8, 72Niemeyer, 279

Onnes, Kamerlingh, 83, 98, 99, 100,166, 174, 178, 179, 180

Ornstein, 130

Perrin, J., 8, 31, 223Pidduck, 211Piotrowski, 168Poiseuille, 168Pospisil, 32Poynting, J. H., 165

Ramsauer, C, 143Ramsay, Sir W., 99, 279Rankine, 178, 183Rayleigh (3rd Baron), 3, 178Regnault, 79Reynolds, Osborne, 61Richardson, Sir O. W., 124Roberts, J. K., 192Robinson, H. A., 31Rusch, M., 143

Sarrau, 101Scheel and Heuse, 279, 281Scheffers, 128Schmidt, R., 202, 213Schmitt, 178, 179, 182, 183Schottky, 124

Schroer, 182Schultze, 174Schwarze, 189Sih Ling Ting, 124Smith, C. J., 178, 183Smoluckowski, 8, 192, 222Soddy, 192Stefan, 207, 209Stem, 11, 125, 128Stokes, Sir G. G., 164Strecker, 279, 282Sutherland, 177, 178

Tait, P. G., 145, 162, 305Thales of Miletus, 1Thiessen, 182Thomson, Sir J. J., 165, 285Thomson, Sir W. (Lord Kelvin), 26,

62, 214Trautz, 174, 178, 183

Van Cleave, 174Van der Waals, 63, 68Van Wyck, 130Virgo, 30, 274Vogel, 179Von Babo, 166

Warburg, 166, 168, 169, 279Waterston, 3Weber, 166, 174, 179Wiedemann, 196Wild, 202, 215Wood, R. W., 54

Young, S., 95, 99

Zartmann, 127Zener, 193

Documents

Jeans ~ Intro to the Kinetic Theory of Gases,1967