A KINETIC THEORY 
 
 OF 
 
 GASES AND LIQUIDS 
 
 BY 
 
 RICHARD D. KLEEMAN, 
 
 D.Sc. (Adelaide), B.A. (Cantab.) 
 
 Associate Professor of Physics, Union College; Consulting Physicist 
 
 and Physical Chemist; formerly 1851 Exhibition Scholar of the 
 
 University of Adelaide, Research Student of Emmanuel 
 
 College, Cambridge, Mackinnon Student, of the 
 
 Royal Society of London, and Clerk Maxwell 
 
 Student of the University of Cambridge 
 
 FIRST EDITION 
 
 NEW YORK 
 
 JOHN WILEY & SONS, INC, 
 
 LONDON: CHAPMAN & HALL, LIMITED 
 1920 
 
K :: 
 
 Copyright, 1920 
 
 BY 
 
 RICHARD D. KLEEMAN 
 
 BRAUNrtOHTH & CO. 
 
 BOOK MANUFACTURERS 
 
 BROOKLYN. N. Y. 
 
DEDICATED 
 
 TO MY PARENTS 
 
PREFACE 
 
 THE object of writing this book is to formulate a Kinetic 
 Theory of certain properties of matter, which shall apply 
 equally well to matter in any state. The desirability of 
 such a development need not be emphasized. The difficulty 
 hitherto experienced in applying the results obtained in the 
 case of the Kinetic Theory of Gases in the well-known form 
 to liquids and intermediary states of matter has been pri- 
 marily due to the difficulty of properly interpretating molec- 
 ular interaction. In the case of gases this difficulty is in 
 most part overcome by the introduction of the assumption 
 that a molecule consists of a perfectly elastic sphere not 
 surrounded by any field of force. But since such a state of 
 affairs does not exist, the results obtained in the case of gases 
 hold only in a general way, and the numerical constants 
 involved are therefore of an indefinite nature, while in the 
 case of dense gases and liquids this procedure does not lead 
 to anything that is of use in explaining the facts. 
 
 Instead of an atom, or molecule, consisting of a per- 
 fectly elastic sphere, it is more likely that each may be 
 regarded simply as a center of forces of attraction and 
 'repulsion. If the exact nature of the field of force sur- 
 rounding atoms and molecules were known, it would be a 
 definite mathematical problem to determine the resulting 
 properties of matter. But our knowledge in this connection 
 is at present not sufficiently extensive to permit a develop- 
 ment of the subject along these lines. But in whatever 
 
viii PREFACE 
 
 With the foregoing results as a basis, and the modified 
 definitions of the free paths of a molecule mentioned, the 
 foundation of a general Kinetic Theory can be laid which 
 applies to matter in any state, and which furnishes a num- 
 ber of important formulae. These formulas may be given 
 important extended forms containing quantities which are 
 arbitrary in so far as they satisfy the formulae as a whole. 
 Formulae may also be deduced along the same lines involving 
 instead of the molecular free path the projection of the mo- 
 tion of a molecule along a line. This projection and its 
 period are also arbitrary in so far as they satisfy the formulae 
 as a whole. Some of them find an interesting application 
 in connection with colloidal solutions. The foundation of 
 the subject may be said to be fairly complete, since it fur- 
 nishes the structure about which further advances may 
 be made so that the subject can be rendered more or less 
 complete. These advances will consist largely in expressing 
 the constants involved in the fundamental formulae in terms 
 of the constants of the molecular forces and the molecular 
 volume. If the formulae obtained in connection with vis- 
 cosity, conduction of heat, and diffusion, be applied to a 
 perfect gas, they assume the well-known forms, but the sym- 
 bols have somewhat different meanings. 
 
 The development presented is perfectly sound without 
 involving difficult mathematics. This has, in fact, been 
 one of the main objects kept in view. As a physicist I have 
 often failed to see the usefulness from a physical standpoint 
 of the extremely intricate mathematical investigations pur- 
 porting to work out to the utmost limit the results of certain 
 assumptions (usually in connection with molecular collision) , 
 and which have usually led to results whose usefulness seems 
 incommensurate with the labor involved, seeing that the 
 assumptions are usually not likely to be true. It is desirable 
 that there should be a simple and clearcut connection between 
 
PREFACE IX 
 
 theory and experiment, and the physical side of the sub- 
 ject has therefore been kept in prominence. 
 
 The development of a general Kinetic Theory of matter 
 will be of service in the study of chemical action, principally 
 in connection with the constant of mass-action and the reac- 
 tion velocity constants. On the whole the kinetic aspect 
 of the chemical interaction of molecules apart from the 
 thermodynamical aspect, which does not take into account 
 the individual nature of the different effects producing a 
 resultant whole, can only proceed along lines having such a 
 Kinetic Theory as a basis. Thus it will readily be seen 
 that the rapidity of a chemical reaction in a gaseous or 
 liquid mixture must be intimately connected with the num- 
 ber of molecules crossing a square centimeter per second 
 in the case of each constituent. The constant of mass-action 
 must evidently be intimately connected with the free dif- 
 fusion path of a molecule, etc. 
 
 The study of viscosity, conduction of heat, diffusion, 
 etc., has usually been confined to pure substances and to 
 mixtures whose constituents do not interact chemically. 
 It would evidently be of great interest to study these effects 
 in the case of substances partly dissociated, since the inter- 
 acting between molecules is then influenced by chemical 
 affinity. Further light may be thrown on the nature of this 
 property by the application of the formulae obtained. 
 
 I have previously published some investigations in 
 various scientific journals along the lines pointed out. Since 
 the development of a subject can only be gradual, it was 
 necessary to modify some of these results before incorporat- 
 ing them into this book. The other investigations in the 
 book which more or less complete the Kinetic Theory along 
 the lines mentioned I have not published previously. It 
 seemed that it would be better to present the subject as a 
 whole to the scientific public, since the various results are 
 
X PREFACE 
 
 intimately connected and could therefore not be presented 
 in detached parts without a good deal of reference to, and 
 recapitulation of, preceding parts being necessary, to the 
 inconvenience of the reader. 
 
 The book has also been brought up to date in matters 
 not connected with molecular collision, and has been treated 
 in a way so that the results are connected as directly as pos- 
 sible with the results of experiment. 
 
 The properties of matter treated in this book may be 
 said to depend mainly on the dynamical properties of mole- 
 cules modified in most cases by the molecular forces of attrac- 
 tion and repulsion. There is evidently, therefore, another 
 side to the subject of the properties of matter, namely that 
 which deals with those properties which depend in the main 
 on the molecular forces modified in some cases by the dynam- 
 ical properties of the molecules. Thus, for example, the 
 internal heat of evaporation and the intrinsic pressure 
 probably do not depend directly on molecular motion. 
 This part of the physico-chemical properties of matter will 
 be dealt with in a separate book under the title " Molecular 
 Forces. " The present book may serve together with the fore- 
 going as an introduction to the study of the purely ther- 
 modynamical aspect of material properties. 
 
 R. D. KLEEMAN. 
 UNION COLLEGE, SCHENECTADY. 
 March, 1919. 
 
CONTENTS 
 
 CHAPTER I 
 
 THE MOLECULAR CONSTANTS, AND THE DYNAMICAL PROPERTIES 
 OF A MOLECULE IN THE GASEOUS STATE 
 
 SEC. PAGE 
 
 1. A Brief Historical Summary of the Development of the Kinetic 
 Theory of Explaining the External Pressure and other Properties 
 
 of Gases 1 
 
 2. A Direct Experimental Proof that Matter consists of a large 
 number of Entities or Atoms 3 
 
 3. The Absolute Mass of an Atom determined from a Knowledge of 
 the Electric Charge e carried by an Electron 5 
 
 4. Indirect Experimental Evidence that the Molecules of Gases 
 and Liquids are in Rapid Motion 8 
 
 5. The Absolute Temperature; and the Equation of a Perfect Gas, 
 and of a Mixture of Gases 11 
 
 6. The Velocity of Translation of a Molecule in a Gas from 
 Dynamics 15 
 
 . 7. Maxwell's Law of Distribution of the Velocities of the Mole- 
 cules of a Gas 19 
 
 8. The Average Kinetic Energy Velocity of a Molecule, and the 
 Relation between the Kinetic Energy of a Molecule and its 
 
 Absolute Temperature 27 
 
 9. The Equipartition of Kinetic Energy between the Different 
 Molecules in a Mixture of Gases 31 
 
 10. The Number of Molecules per Cubic Cm. in a Gas 33 
 
 11. The Number of Molecules Crossing a Square Cm. in all Direc- 
 tions from one Side to the Other in a Gas 34 
 
 12. The First Law of Thermodynamics 37 
 
 y 13. The Specific Heats of Gases and Liquids 39 
 
 14. Evidence that Molecules and Atoms are surrounded by Fields 
 
 of Force 45 
 
 Molecular Interaction 48 
 
 xi 
 
4 
 
 xii CONTENTS 
 
 CHAPTER II 
 
 THE EFFECT OF THE MOLECULAR FORCES ON THE DYNAMICAL 
 PROPERTIES OF A MOLECULE IN A DENSE GAS OR LIQUID 
 
 SEC. PAGE 
 
 16. The Velocity of Translation of a Molecule in a Liquid or 
 Dense Gas when passing through a Point at which the Forces 
 
 of the Surrounding Molecules neutralize each other 51 
 
 17. The Total Average Velocity of Translation of a Molecule in a 
 Substance 54 
 
 18. The Number of Molecules crossing a Square Cm. in a Sub- 
 stance of any Density from one Side to the Other per Second . . 56 
 
 19. The Effect of Molecular Volume on p and n in the Case of an 
 Imperfect Gas 57 
 
 20. The Expansion Pressure exerted by the Molecules of a Sub- 
 stance of any Density 62 
 
 21. The Intrinsic Pressure; and the Equation of Equilibrium 
 
 of a Substance 69 
 
 22. Superior and Inferior Limits of n of a Substance 74 
 
 23. Superior and Inferior Limits of V t of a Substance 75 
 
 24. The Real and Apparent Volumes of a Molecule, and their 
 Superior Limits 77 
 
 25. Inferior Limits of n and Vt 80 
 
 26. The Equation of State of a Substance 81 
 
 27. The Conditions that the Equation of State has to satisfy 86 
 
 28. The Relation of Corresponding States 91 
 
 29. The Determination of the Quantities n, Vt, and 6, of a Sub- 
 stance 95 
 
 30. An Equation Connecting the Intrinsic Pressure, Specific Heat, 
 and other Quantities 103 
 
 31. The Mean Free Path of a Molecule under Given Conditions. . 106 
 
 CHAPTER III 
 
 QUANTITIES WHICH DEPEND DIRECTLY ON THE NATURE OF 
 MOLECULAR MOTION 
 
 32. The Coefficient of Viscosity of a Substance 110 
 
 33. The Viscosity Mean Momentum Transfer Distance of a Mole- 
 cule in a Substance 113 
 
 34. Formulae for the Viscosity in Terms of Other Quantities 116 
 
 35. Formulae for the Viscosity of Mixtures 142 
 
CONTENTS xiii 
 
 BEC. PAGE 
 
 36. The Coefficient of Conduction of Heat 147 
 
 37. The Mean Heat Transfer Distance of a Molecule in a Sub- 
 stance 148 
 
 38. Formulae for the Coefficient of Conduction of Heat in Terms 
 
 of Other Quantities 150 
 
 39. Formulae for the Coefficient of Conduction of Heat of a Mixture 
 
 of Substances 183 
 
 40. The Coefficient of Diffusion of a Substance 168 
 
 41. The Mean Diffusion Path of a Molecule in a Mixture . . 169 
 
 42. Formulae Expressing the Coefficient of Diffusion in Terms 
 
 of Other Quantities 170 
 
 43. Maxwell's Expression for the Coefficient of Diffusion of Gases 187 
 
 CHAPTER IV 
 
 MISCELLANEOUS APPLICATIONS, CONNECTIONS, AND EXTENSIONS 
 OF THE RESULTS OF THE PREVIOUS CHAPTERS 
 
 44. The Direct Observation of some of the Quantities depending 
 on the Nature of the Motion of a Molecule in a Substance, 
 and their Use 189 
 
 45. The Coefficient of Diffusion in Connection with Osmotic 
 Pressure and the Coefficient of Mobility 196 
 
 46. Partial Intrinsic Pressures 202 
 
 47. Conditions of the Equilibrium of a Heterogeneous Mixture 
 such as Two Phases in Contact 206 
 
 48. Osmotic Pressure expressed in Terms of the Kinetic Properties 
 
 of Molecules 208 
 
 49. The Velocity of Translation of Particles undergoing Brownian 
 Motion 210 
 
 50. The Osmotic Pressure of a Dilute Solution of Molecules and 
 their Velocity of Translation 213 
 
 51. A Direct Determination of N, the Number of Molecules in a 
 Gram Molecule 217 
 
 52. Formulae involving Stokes' Law 220 
 
 53. Extended Forms of the Diffusion Equations 222 
 
 54. Extended Forms of the Viscosity Equations 230 
 
 55. Extended Forms of the Heat Conduction Equations 238 
 
 56. The Constant Period Displacement Diffusion Equation and 
 
 its Applications 242 
 
 57. The Constant Displacement Diffusion Equation 252 
 
xiv CONTENTS 
 
 SEC. PAGE 
 
 58. The Constant Period and Constant Displacement Viscosity 
 Equations 255 
 
 59. The Constant Period and Constant Displacement Heat Con- 
 duction Equations 259 
 
 60. Another Method of Determining the Total Average Velocity of 
 Translation of a Colloidal Particle 262 
 
 61. The Distribution of the Molecular Velocities in a Substance 
 not Obeying the Gas Laws 265 
 
NOTE 
 
 IT will be useful to give a list of the most important 
 symbols used in a general way in this book, the kind of 
 molecule and phase to which a symbol refers in any given 
 case being indicated by the addition of suffixes and dashes 
 when necessary. 
 
 p = external pressure 
 P n = intrinsic pressure 
 P s = Osmotic pressure 
 
 v = volume of a gram molecule of a substance 
 d = external molecular volume connected with the con- 
 centration changes of a mixture 
 6 = apparent internal molecular volume connected with 
 
 molecular motion 
 p = density 
 
 T= absolute temperature 
 S = specific heat 
 n = number of molecules crossing a square cm. from one 
 
 side to the other per second 
 m = relative molecular weight 
 m a = absolute molecular weight 
 N = number of molecules in a gram molecule 
 N with suffixes = molecular concentration 
 V = average kinetic energy velocity of a molecule in the 
 
 gaseous state 
 
 V a = average velocity of a molecule in the gaseous state 
 V t total average velocity of a molecule in any state 
 77 = coefficient of viscosity 
 
 xv 
 
xvi NOTE 
 
 C = coefficient of conduction of heat 
 D = coefficient of diffusion 
 6 = rate of diffusion 
 
 Z n = mean momentum transfer distance 
 Z c = mean heat transfer distance 
 / 5 = mean diffusion path 
 $ = interference function 
 K = path factor 
 
 d = projection of the motion of a molecule, or its displace- 
 ment 
 t= period of displacement. 
 
A KINETIC THEORY 
 
 OF 
 
 GASES AND LIQUIDS 
 
 
 CHAPTER I 
 
 THE MOLECULAR CONSTANTS, AND THE DYNAMICAL 
 PROPERTIES OF A MOLECULE IN THE GASEOUS STATE 
 
 1. A Brief Historical Summary of the Develop- 
 ment of the Kinetic Theory of Explaining the External 
 Pressure and other Properties of Gases. 
 
 In order to explain the external pressure, or elasticity of a 
 gas, vapor, or liquid, it is necessary to introduce a theory, 
 since the mechanism giving rise to this property is not evident 
 to the eye. A purely mechanical explanation based on 
 the observed elasticity of solid materials, or of a mass of 
 fibrous matter, no doubt presented itself to the minds of the 
 early physicists. But since from the earliest times some 
 philosophers regarded matter as consisting of a number of 
 hard, indivisible, and similar parts, it was natural that it 
 should occur to some to explain the elasticity of a gas by the 
 motion and consequent change of momentum of these parts. 
 Thus Gassendi in the 17th century elaborated an atomic 
 theory of the properties of matter based upon the assump- 
 
2 
 
 . MOLECULAR CONSTANTS 
 
 tion that all the material phenomena can be referred to the 
 indestructible motion of atoms. He supposed that all atoms 
 are the same in substance, but different in size and form, and 
 that they move in all directions through space. A number 
 of processes, in particular the transition of matter from one 
 state to the other, were explained on this basis. Later these 
 ideas occurred independently to other investigators, who 
 elaborated them to a greater extent. Thus Daniel Bernoulli 
 in his Hydrodynamica, published in 1738, pointed out that 
 the elasticity of a gas may be explained by the impact of the 
 particles of which it was supposed to consist on the walls of 
 the containing vessel, and accordingly he deduced Boyle's 
 law for the relation between pressure and volume. Later 
 the subject was taken up by Herapath, Watertson, Joule, 
 Kronig, and with great success by Clausius. Some time 
 later Maxwell added some important contributions to the 
 Kinetic Theory. In the hands of Clausius and Maxwell it 
 developed with great rapidity and success. The subject 
 now attracted numbers of theoretical and experimental in- 
 vestigators who helped to perfect it theoretically, and by 
 testing the results experimentally demonstrated the sound- 
 ness of the underlying assumptions. Of the theoretical 
 investigators Boltzmann should be specially mentioned. 
 
 The endeavor by scientists in recent years to extend the 
 well-known mathematical investigations of the Kinetic 
 Theory of Gases to liquids and dense gases, supposing that 
 molecular interaction may be represented by the collision 
 of elastic spheres, has been almost barren of results. The 
 object of this book, as has already been pointed out in the 
 Preface, is to give the Theory such a form that these diffi- 
 culties are removed, and that it accordingly applies as con- 
 veniently to liquids as to gases. Van der Waals has already 
 rendered important service by means of his theory of con- 
 tinuity of state, in indicating the general nature of the relation 
 between the pressure, volume, and temperature of a sub- 
 
THE ATOMISTIC NATURE OF a RAYS 3 
 
 stance in any state, and of two states in equilibrium with each 
 other. 
 
 In this development a knowledge of the nature of the 
 relative distribution of matter in space is of foremost impor- 
 tance, and therefore claims first attention. 
 
 2. A Direct Experimental Proof that Matter 
 consists of a large Number of Entities or Atoms. 
 
 This was first furnished definitely by an experiment 
 devised by Rutherford and Geiger.* It was arranged that a 
 particles of radium were fired through a gas at low pressure, 
 exposed to an electric field. This gave rise to iomzation in 
 the gas which was measured in the usual way. The amount 
 of ionization obtained was considerably increased by in- 
 creasing the field to near sparking value. The velocity 
 given to the initial ions was then so large that they pro- 
 duced further ions by collision with neutral molecules. In 
 this way the small ionization produced by one a particle in 
 passing through the gas could be magnified several thousand 
 times. The sudden current through the gas, due to the 
 entrance of a single a particle in the detecting vessel, was by 
 this method increased sufficiently to give an easily measur- 
 able deflection to the needle of an ordinary electrometer. 
 Thus by limiting the number of a particles shot into the 
 vessel by means of a stop, a succession of throws of the gal- 
 
 * Proc. Roy. Soc. A. 81, p. 141 (1908); Phys. Zeit. 10, p. 1 (1909). 
 NOTE. There existed previously a good deal of indirect evidence 
 that matter consists of entities. Thus the law of constant proportion, 
 and others, in chemistry could very simply be explained if this were 
 the case. Also the formulae obtained for the viscosity, conduction of 
 heat, of a gas, deduced from the assumption that it consists of particles 
 of matter in motion, gave a general agreement with the facts. But 
 each of these results might also hold on mathematical grounds without 
 matter necessarily consisting of atoms or molecules. A definite proof 
 of the atomistic nature of matter was therefore desirable and of impor- 
 tance, and this was first furnished by the experiments quoted. 
 
4 THE MOLECULAR CONSTANTS 
 
 vanometer needle was obtained. This proved the atomistic 
 nature of the a radiation given off by radium. 
 
 Ramsay and Soddy* had previously shown that helium 
 was produced from radium emanation, and that it is there- 
 fore one of the products of the disintegration of radium. 
 It was suspected that helium might consist of a particles 
 which have lost their electric charge. This was proved by 
 Rutherford and Roydsf who showed that accumulated a 
 particles, quite independently of the matter from which 
 they were expelled, consist of helium. The radioactive 
 material was enclosed in a glass tube whose walls were so 
 thin that they were penetrated by the expelled a particles, 
 which were retained by a vessel containing the glass tube. 
 The matter collected in this way was tested spectroscopically 
 and otherwise, and found to consist of the gas helium. The 
 atomistic nature of one of the gases was thus proved; and 
 since different gases possess similar physical properties this 
 result is likely to hold for all of them. 
 
 Some interesting experiments by DunoyerJ may be men- 
 tioned which can only be. reasonably explained if matter 
 consists of entities which possess motion of translation. 
 A cylindrical tube was divided into three compartments 
 by means of two partitions perpendicular to the axis of the 
 tube, each partition being pierced centrally by a small hole 
 so as to form a diaphragm. The tube was fixed with the axis 
 vertical and a piece of some substance such as sodium, 
 which is solid at ordinary temperatures, placed at the bottom 
 of the lowest compartment. The tube was now exhausted 
 and the substance heated to a sufficient temperature to 
 vaporize it. The vaporized particles were projected in all 
 
 * Nature, 68, p. 246 (1903); Proc. Roy. Soc., A. 72, p. 204 (1903); 
 73, p. 346 (1904). 
 
 t Phil. Mag., 17, p. 281 (1909). 
 
 tComptes Rendus, 152 (1911), p. 592; Le Radium,Vlll (1911), p. 
 142. 
 
THE CHARGE ON THE ELECTRON 5 
 
 directions, some of which passed through the hole of the first 
 diaphragm at various angles. Of these some passed through 
 the hole in the second diaphragm forming a deposit on the 
 top of the tube. This deposit was found to coincide exactly 
 with the projection of the hole in the second diaphragm 
 formed by radii drawn from the hole in the first diaphragm. 
 A small obstacle placed in the path of the particles was found 
 to form a shadow on the upper surface of the tube. The 
 vaporized matter thus moved along straight lines, and there- 
 fore could conceivably consist -only of particles, i.e., atoms. 
 
 3. The Absolute Mass of an Atom is Most Ac- 
 curately Determined from a Knowledge of the Electric 
 Charge e Carried by an Electron. 
 
 For this reason, and that the accurate determination of 
 the value of e is important in itself, a number of different 
 principles and methods have been employed in recent years 
 to determine this quantity with the greatest possible ac- 
 curacy. 
 
 A determination of e from radioactive data was made by 
 Rutherford and Geiger.* The number of a particles passing 
 into a vessel of the kind mentioned in the previous Section 
 were directly counted, and also the total electrical charge 
 carried by the particles determined. It was thus found that 
 each particle carried a charge 9.3 X 10~ 10 units, and from 
 various evidence it was concluded that this was twice the 
 unit charge. Similar observations were carried out by 
 Regener,f who counted the particles by noting the scintilla- 
 tions that they produced on impinging on a diamond. The 
 value found by him was <? = 4.79X10~ 10 . 
 
 MillikanJ perfected a method in which minute drops of 
 
 * Proc. Roy. Soc., A 81, p. 162 (1908). 
 
 ^Ber. d. K. Preuss. Akad. d. Wiss., 38, p. 948 (1909). 
 
 % Phil. Mag., 19, p. 209 (1910); Science, 32, p. 436 (1910). 
 
6 THE MOLECULAR CONSTANTS 
 
 a non-volatile liquid are introduced between two parallel 
 and horizontal plates having a strong electric field between 
 them. The drops were obtained by spraying and became 
 electrically charged during the process. The movements 
 of individual drops under the action of gravity and the 
 electric field were observed by means of a microscope. By 
 adjusting the electric field to counteract gravity, it was 
 found possible to keep a charged drop suspended in nearly 
 the same position in the gas for an hour at a time. In that 
 case if X denotes the electric field, N'e, the charge on a drop 
 
 where r denotes the radius of the drop, g the gravitational 
 constant, and p the density of the material. The value of 
 r was obtained from observations of the rate of fall of the 
 drop under the action of gravity. According to Stokes' 
 equation the rate of fall V s is given by 
 
 where rj denotes the coefficient of viscosity of the gas through 
 which the drop moves. A number of experiments were made 
 to test the validity of Stokes' equation for drops of different 
 sizes, and a suitable correction was made from the observ- 
 ations. The value of N'e accordingly could be determined. 
 The experiments showed that each of these drops carried 
 a definite multiple of a small charge, which accordingly 
 is equal to e. The final result obtained gave a value* e = 4.77 
 X10- 10 E. S. U. 
 
 If e be the charge of electricity carried by the hydrogen 
 atom in electrolysis, and N the number of atoms in one 
 
 * The Electron, by R. A. Millikan. The University of Chicago 
 Science Series. 
 
THE ABSOLUTE MASS OF AN ATOM 7 
 
 gram molecule of hydrogen, it is known from experiments 
 on the electrolysis of solutions that 
 
 Now according to Rutherford and Geiger 
 e = 4.65X10- 10 E.S.U., 
 or e=1.55XlO- 20 E.M.U., 
 
 and accordingly 
 
 which is also the number of molecules in a gram molecule 
 of any substance, according to the definition of a gram 
 molecule. The absolute mass man of a hydrogen atom is 
 equal to 1/N, and hence 
 
 m a tf=1.60XlO- 24 grm. 
 
 The number of molecules in a cubic cm. of any gas at 
 standard pressure and temperature can now be shown 
 simply from density considerations to be equal to 2.78 X 10 19 . 
 Millikan's value of e gives 
 
 2V = 6.06X10 23 , 
 and 
 
 = 1.65 X10- 24 grin. 
 
 The remarkable result that the number of molecules 
 in a gas under standard conditions is a contant independent 
 of the nature of the gas, and the fact that a gas as a whole is 
 compressible, suggests that the molecules are not packed 
 closely together, but separated from each other by distances 
 much greater than their own dimensions, and that they 
 undergo niotion of translation so as to produce on the aver- 
 age an even distribution of matter in space, and give rise 
 to the external pressure of the gas, 
 
8 THE MOLECULAR CONSTANTS 
 
 4. Indirect Experimental Evidence that the Mole- 
 cules of Gases and Liquids are in Rapid Motion. 
 
 It will first be shown that there are definite theoretical 
 reasons that the molecules should be in motion. 
 
 Two ways only suggest themselves of explaining the 
 pressure exerted by a gas, viz.: (a) By forces of repulsion 
 between the molecules, (6) by a molecular motion of trans- 
 lation. It can be easily shown that the pressure cannot be 
 exerted according to (a) . The Joule-Thomson effect described 
 in Section 14 indicates that forces of attraction exist between 
 molecules which act over distances of the order of the dis- 
 tances of separation of the molecules of a gas. The positive 
 sign of the internal heat of evaporation of a liquid indicates 
 that this also holds for much smaller distances of separation 
 of the molecules. The pressure exerted by a gas cannot 
 therefore be produced by forces of repulsion; and the 
 molecules must therefore possess motion of translation to 
 account for the pressure. 
 
 It will be shown in Section 6 from dynamical considera- 
 tions that the magnitude of this motion decreases as the 
 mass of the molecule increases. Accordingly if the mole- 
 cules were sufficiently large to be visible to the eye, we 
 might expect that this motion could actually be observed. 
 It is evident that the motion will not be along a continuous 
 straight line, but zig-zag shaped, since the molecules will 
 collide with each other. This will also be the case with a 
 particle, or a conglomeration of thousands of atoms, con- 
 tained in a liquid or gas, for the particle would be struck on 
 account of its size by a large number of molecules at the 
 same instant, but the same number would not necessarily 
 strike it on each of two opposite sides, and thus the motion 
 of the particle would be rendered undulatory in character. 
 These conclusions are borne out in a striking way by experi- 
 ment. 
 
BROWNIAN MOTION 9 
 
 On observing by means of a microscope small particles 
 of matter suspended in a liquid, they are found to be under- 
 going rapid oscillatory motion. This was first studied by 
 Brown in 1827 in the case of pollen of plants suspended in 
 water, and hence has been called Brownian motion. The 
 motion is best observed by means of the ultra-microscope 
 introduced by Siedentopf and Zsigmondy.* In this appa- 
 ratus a parallel beam of sunlight, or arc-light, is passed 
 through the liquid under investigation at right angles to the 
 axis of the microscope. If the liquid is completely homo- 
 geneous no scattered light enters the microscope. On the 
 other hand if the liquid contains particles of matter these 
 appear, through scattering some of the light passing through 
 the liquid, as bright specks of light against a dark back- 
 ground. These specks of light undergo a rapid oscillatory 
 motion indicating the motion of the particles. Through 
 the contrast of the light and darkness a particle becomes 
 much more conspicuous in this arrangement than under 
 ordinary conditions, though a "proper view" of the particle 
 is entirely lost. It is necessary that the incident light be as 
 intense as possible, since the amount of scattered light 
 increases in proportion. Also the layer of liquid should be 
 as thin as possible in order to decrease the absorption of the 
 scattered light, and minimize the overlapping of the scat- 
 tered light from different particles. 
 
 The amplitude of the oscillation of a particle was found 
 to depend upon its size. For a diameter of l^u, or .001 mm., 
 of the particle, the amplitude is about equal to 1/z, while 
 for a diameter of 3^ the amplitude is practically zero. Par- 
 ticles 10-40/uM in diameter have amplitudes up to 20,u. 
 It appears from a discussion f of the ultra-microscope that 
 it is still possible to detect by this method a linear magni- 
 tude of 6X10~ 7 cm. = 6 juju, or about 1/100 of a light wave. 
 
 * Drud. Ann., 10, p. 1 (1903). 
 ftoco. tit., p. 14 (1903). 
 
10 THE MOLECULAR CONSTANTS 
 
 Brownian motion has also been observed in the case of 
 smoke particles in air. 
 
 Other designs of the ultra-microscope have been intro- 
 duced besides the foregoing, of which that by Cotton and 
 Monton* deserves special mention. In this arrangement 
 the beam of light is passed through the liquid in such a 
 manner that it is totally reflected from a cover glass placed 
 on top of the liquid. 
 
 Liquids containing small particles in suspension may be 
 prepared in various ways: by means of chemical reactions 
 involving precipitation, etc.; by sparking different metal 
 electrodes in a liquid, etc. < 
 
 Some well-known experiments by Crookes also show that 
 motion is associated with matter, and that an increase in 
 temperature gives rise to an increase in this motion. Thus 
 if a thin vane of light material is suspended vertically in a 
 tube partly exhausted and heat radiation is allowed to 
 fall upon it at right angles, it is deflected in the direction of 
 propagation of heat. This effect is strikingly shown by an 
 apparatus consisting of a number of vanes mounted on a 
 horizontal frame which can rotate round a vertical axis in 
 a partly exhausted tube, the planes of the vanes pass through 
 the axis of rotation and each alternate side is covered 
 with lampblack so as to make one side of each vane a better 
 absorber of heat than the other side. This apparatus is 
 known as Crookes' radiometer. An observer in the same 
 horizontal plane as the frame carrying the vanes is therefore 
 faced by blackened vanes on one side of the axis of rotation 
 and by unblackened vanes on the other side of the axis. 
 If heat radiation from a source in this plane is allowed to 
 fall on the arrangement it rotates rapidly in a direction as if 
 a greater pressure were applied to the blackened surfaces than 
 to the unblackened ones. This is caused by the temperature 
 of the vanes being raised above that of the gas through the 
 * Comptes Rendus, 136, p. 1657 (1903). 
 
DYNAMICAL EFFECTS OF MOLECULES 11 
 
 absorption of heat radiation, and hence heat being con- 
 veyed from the vanes to the gas by conduction. Therefore 
 if heat consists of kinetic energy, motion is given to the gas 
 in the immediate neighborhood of the vanes, which by 
 reaction gives rise to pressures acting upon them. Now 
 the absorption of heat by the blackened sides of the vanes 
 is greater than by the unblackened sides, and hence the rise 
 in temperature is greater in the former case, giving rise to a 
 greater transfer of kinetic energy to the gas, which results 
 in a greater pressure on the blackened sides than on the 
 unblackened ones. Therefore since the surface of a vane 
 rapidly loses the absorbed heat when not exposed to the 
 source a continual motion of the system results. 
 
 Further information as a guide to the development of a 
 kinetic theory of matter is obtained from a consideration 
 of the physical properties of a gas as a whole, which will be 
 described in the next Section. 
 
 5. The Absolute Temperature; and the Equation 
 of a Perfect Gas, and of a Mixture of Gases. 
 
 Experiments on the coefficient of expansion of a gas at 
 constant pressure have shown that it is practically inde- 
 pendent of the pressure, the temperature, and the nature 
 of the gas, and is approximately equal to .0036625. There- 
 fore if v a and v denote the initial and final volume of a gas 
 when its temperature is changed by t, we have 
 
 ...... (2) 
 
 If the volume of the gas is kept constant and the coefficient 
 of increase of pressure is measured, it is found to be inde- 
 pendent of the volume, temperature, and nature of the gas, 
 and practically the same as the foregoing coefficient, and 
 thus 
 
 p*po(H-.0036625t), ..... (3) 
 
12 THE MOLECULAR CONSTANTS 
 
 where p and po denote the final and initial pressures respec- 
 tively when the temperature is changed by t. 
 
 If we consider a mass of gas whose pressure is po at the 
 temperature C., and imagine the volume kept constant 
 while the temperature is lowered to 1, the pressure p 
 will be given by 
 
 If the cooling is continued to a temperature of 1/.0036625 
 or -273 C., then 
 
 p=po(l-l)=0, 
 
 i.e. the gas would exert no pressure on the walls of the con- 
 taining vessel at this temperature. According to the previous 
 Section this can only occur when the velocity of translation of 
 the molecules is zero. The temperature 273 C. has accord- 
 ingly been called the absolute zero. The temperature of a 
 substance may be measured from this zero and denoted by 
 T, in which case C. corresponds to 273 C., and in general 
 T = 273 +t. The foregoing equations then give 
 
 and 
 
 pccT, 
 
 while the experiments on the relation between p and v at 
 constant temperature give 
 
 "? 
 
 which is Boyle's law. Accordingly 
 
 pvccT, 
 or 
 
 pv = RiT, 
 
 where R\ is a constant. If v refers to a mass of M grams of 
 gas we may write R\ = RzM] experiment then shows that 
 
THE EQUATION OF A PERFECT GAS 13 
 
 R2 for different gases varies inversely as m the molecular 
 weight in terms of that of the hydrogen atom. Hence 
 finally the equation of a gas becomes 
 
 MET 
 
 where R is an absolute constant, and v refers to the volume 
 of a gas of mass M. If p denote the density of the gas it 
 follows that vp = M. The foregoing equation may also be 
 written 
 
 p RT , r RT 
 
 P=-=N C - W , ..... (5) 
 
 m a m Ar 
 
 where N denotes the number of molecules in a gram mole- 
 cule, N c the molecular concentration, and m a the absolute 
 molecular weight of a molecule. 
 
 In the case of a mixture of molecules e and r the total 
 pressure p according to equation (5) is given by 
 
 \r RT . , T Rl -* T R L 
 
 where p e and p r denote the partial pressures, N e and N r the 
 partial concentrations, and N er the total concentration of 
 the molecules. This equation is the same in form as equa- 
 tion (5), which therefore also holds for a mixture of any 
 number of constituents. 
 
 The value of R may be determined by means of the 
 foregoing equation on substituting the values of p, v, and T, 
 of a given gas. Since, however, in practice a gas does not 
 obey strictly equations (2) and (3), the value of the absolute 
 zero of temperature deduced from them in terms of any 
 arbitrary scale of temperature requires correction. The 
 thermodynamical equation (47) which involves the absolute 
 
14 THE MOLECULAR CONSTANTS 
 
 temperature is usually used for this purpose. If t denote 
 the temperature readings of any arbitrary thermometer 
 the equation may be written 
 
 which may be integrated in the form 
 
 fa 
 
 T_ r_y 
 T L (L 
 
 where the temperatures T and To on the absolute scale 
 correspond to t and to on the arbitrary scale. The quan- 
 tities on the right hand side of this equation can be directly 
 
 measured. The quantity ( ) is most conveniently de- 
 
 \ 6v/t 
 
 termined from experiments on the Joule-Thomson effect.* 
 A number of such determinations of To the absolute zero have 
 been carried out by a number of observers; in the Recueil 
 de Constantes Physiques, published under the auspices of the 
 Societe Franchise de Physique, the most probable value is 
 taken to be 
 
 To =- 273.09 C. 
 
 From values of p and v for oxygen Berthelot (Trav. 
 et Mem. Bur. Intl.) obtains 
 
 # = 8.315X10 7 . 
 
 Equation (4) is useful in helping to interpret the equa- 
 tion for the velocity of translation of a molecule deduced 
 from purely dynamical considerations in the next Section. 
 
 *For further information see Planck's Thermodynamics, pp. 12 
 131, and Partington's Thermodynamics, pp, 162-167, 
 
THE MOLECULAR MOMENTUM AND GAS PRESSURE 15 
 
 6. The Velocity of Translation of a Molecule in 
 a Gas from Dynamics. 
 
 Consider a single molecule of absolute mass m a moving 
 to and fro between two parallel walls at right angles to each 
 with a velocity V. The molecule on colliding with one of the 
 walls has its momentum changed by 2Vm a , and since this 
 occurs V/21 times per second, where I denotes the distance 
 between the walls, the change in momentum per second is 
 
 T7"O 
 
 equal to =-A Now according to dynamics 
 
 force X time = change in momentum, 
 
 and the force exerted by the molecule on the wall is there- 
 fore equal to the foregoing expression. Suppose now that 
 there are N c molecules per cubic cm. moving in all directions 
 between the walls. It will be convenient first to suppose 
 that these molecules consist of three streams each equal 
 to Nc/3 moving parallel to three axes at right angles to each 
 other. If one of these axes is taken at right angles to the 
 walls, WV c /3 molecules exert the foregoing pressure on each 
 square cm. of the walls. Therefore if p denotes this pres- 
 sure we have 
 
 N c m a V 2 V 2 
 
 where N c m a = p, the density of the gas. 
 
 Let us now suppose that the molecules move in all direc- 
 tions. Let us take a line at right angles to the walls and a 
 point on the line, and draw a sphere of radius r around it. 
 Molecules will pass through the point in all directions, which 
 is equivalent to supposing that n a molecules fall per square 
 cm. per second at right angles on the surface of the sphere. 
 Let us find an expression for the fraction of the molecules 
 passing through the point which move towards one of the 
 
16 THE MOLECULAR CONSTANTS 
 
 walls and make an angle 6 with it. On taking two lines 
 making angles 6 and 6+dO with the normal and rotating 
 these lines with the normal as axis, a belt whose area is 
 2wr sin 6-r-dQ is traced out on the sphere. The fraction in 
 question is therefore- equal to 
 
 2irr sin 6-r-dd-n a . Q -, Q 
 - = sin 8 - dd. 
 
 Thus of the total number of molecules N t between the walls 
 N t sin 6 dd molecules make an angle 6 with a normal to one 
 of the walls and move towards it. Each of these molecules 
 on striking the wall gives rise to a change in momentum 
 
 V cos 6 , . 
 equal to 2m a V cos 6 and this happens times per sec- 
 
 ond through rebounding from the two walls. The molecules 
 under consideration therefore exert a pressure equal to 
 
 ' wne-.de 
 
 I 
 
 upon the wall. The pressure P exerted on the whole wall by 
 the total number of molecules moving in all directions is 
 therefore given by 
 
 cos 2 0sin 6-dd 
 
 ~ m " i Jo' 
 
 = _m^ cos_ 
 
 31 
 
 If A a denotes the area of the wall, p the pressure per square 
 cm., and N c the number of molecules per cubic cm., we have 
 
 By means of these two equations the preceding equation 
 may be reduced to equation (6). It appears therefore that 
 
THE MOLECULAR VELOCITY OF TRANSLATION 17 
 
 we may represent the molecules, as far as the pressure they 
 exert is concerned, by three streams moving at right angles 
 to each other. 
 
 Since p and p can be directly measured, the velocity 
 of translation of a molecule is at once given by 
 
 (7) 
 
 If p is expressed in dynes and p in grams per cc. the velocity 
 V is expressed in cms. per second. 
 
 In illustration of this formula the values of V in cms. 
 per second for a few gases at standard temperature and 
 pressure are given by Table I. 
 
 TABLE I 
 
 Substance. 
 
 Formula. 
 
 F 9 ms - 
 sec. 
 
 Hydrogen 
 
 H 2 
 
 169 200 
 
 Helium 
 Methane . 
 
 He 
 CH 4 
 
 120,400 
 60060 
 
 Ethylene 
 
 C 2 H 4 
 
 45 420 
 
 Carbon dioxide 
 
 CO 2 
 
 36,250 
 
 Amyl propionate 
 
 
 20,030 
 
 C. tetrachloride . . .... 
 
 CC1 4 
 
 19390 
 
 Ethyl iodide 
 
 C 2 H 5 I 
 
 18820 
 
 Mercury 
 
 He 
 
 17000 
 
 
 
 
 Since in the foregoing investigation the velocity of a 
 molecule does not depend on the presence of another mole- 
 cule, V is independent of the volume of the gas. Therefore 
 according to equation (6) the pressure of a gas is propor- 
 tional to its density, or inversely proportional to its volume, 
 which agrees with the facts and is known as Boyle's law. 
 
18 
 
 THE MOLECULAR CONSTANTS 
 
 By means of equation (4) the foregoing equation may 
 be written 
 
 (8) 
 
 The velocity of translation of a molecule thus varies 
 inversely as the square root of its mass, and hence the 
 greater the mass the smaller the velocity. It is due to this 
 that it is possible to observe directly the motion of large 
 particles in a liquid, as described in Section 4. The velocity 
 is also proportional to the square root of the absolute tem- 
 perature, and hence is equal to zero at the absolute zero of 
 temperature. 
 
 FIG. 1. 
 
 In the foregoing investigation it has been tac^Jy assumed 
 that the apparent velocity of translation of a molecule is 
 not effected by the presence of other molecules. This, how- 
 ever, does not hold; the velocity is apparently increased by 
 the volumes of the molecules. Thus consider two molecules, 
 a and 6, which collide head on so that each retraces its path 
 as shown in Fig. 1. This amounts to the same thing as if 
 the molecules at the moment of collision were to be inter- 
 changed, the molecule a then taking the place of the. retreat- 
 ing molecule b, and the molecule b the place of the retreat- 
 ing molecule a. The velocity of each molecule is thus ap- 
 parently increased, since each passes instantaneously ovei 
 
THE MOLECULAR VOLUME AND GAS PRESSURE 19 
 
 a part of its path equal to the diameter of a molecule. Under 
 these conditions the walls are struck oftener per second by 
 each molecule, giving rise to a greater pressure than would 
 be obtained if each molecule had no volume or space asso- 
 ciated with it through which another molecule cannot pass. 
 The values of V given by equation (7) are therefore greater 
 than they should be. It is easy to see, however, that if the 
 diameter of a molecule is small in comparison with the 
 distance of separation of the molecules, as is the case with 
 a gas under ordinary conditions, the error introduced in 
 this way is small. It will be evident now that the deviation 
 of a gas from equation (7) is in part caused by the apparent 
 volume associated with each molecule. The remaining part 
 of the deviation is caused by the molecular forces, which 
 will be discussed in Sections 14, 15, and 21. 
 
 It has also been assumed that each molecule possesses the 
 same velocity. This does not hold in practice in a gas, since 
 the interaction between the molecules tends to continually 
 change their velocities. A theoretical investigation showing 
 that this is the case, and the law of distribution of velocities 
 obtained by Maxwell, will be given in the next Section. 
 But we may still express p by means of equation (6), where 
 V is now called the average kinetic energy velocity, a quan- 
 tity which will be discussed in Section 8. It is the velocity 
 which gives the average kinetic energy of a molecule. 
 
 7. Maxwell's Law of Distribution of the Veloci- 
 ties of the Molecules of a Gas. 
 
 It can be shown theoretically that the molecules in a 
 gas at any instant have not the same velocity; moreover, 
 the velocity of each molecule is continually changing in 
 magnitude. This is what we would expect on account of 
 the interaction between the molecules. According to Max- 
 well's law, which will be proved presently, the number of 
 
20 . THE MOLECULAR CONSTANTS 
 
 molecules N\ of N molecules which have a velocity lying 
 between V\ and V\+dV\ at any instant is given by 
 
 4A/Ti 2 ( Vl \ z 
 
 Nl ,*"g-e-(vj dV 1 ..... (9) 
 VwVp* 
 
 The most probable velocity is the value of V\ which gives 
 the foregoing expression a maximum value. Therefore if 
 the differential coefficient of the expression with respect to 
 FI is obtained and equated to zero we have 
 
 Vi = V, ....... (10) 
 
 Thus V p is the most probable velocity that occurs amongst 
 the gas molecules. 
 
 The probability of the occurrence of other velocities 
 decreases rapidly on increasing or decreasing the value of 
 FI from V p . That a molecule may have an infinitely small 
 velocity is of zero probability, and this also holds for an 
 infinitely large velocity. It appears that by far the larger 
 number of molecules have velocities differing but little 
 from the most probable velocity, and as a first approxima- 
 tion we may therefore suppose that the molecules have the 
 same velocity. 
 
 The most probable velocity V p must not be confounded 
 with the average velocity V a - The latter quantity is obtained 
 by adding up all possible velocities and dividing their num- 
 ber into the sum obtained. This corresponds to multiply- 
 ing the expression for NI from equation (9) by FI, integrat- 
 ing it between the limits Fi=0and FI= oo, and dividing 
 by N, giving 
 
 /* 
 ( 
 
 o 
 
 4F v \ x 2 e~** , C 
 
 ft T + J xe 
 
THE PERMANENCE OF OSMOTIC PRESSURE 21 
 
 on writing V\/V p =x. Thus the average velocity is greater 
 than the most probable. K 
 
 Maxwell's law can at once be strictly deduced if it can 
 be shown that the velocity components of a molecule along 
 three axes at right angles are independent of each other. 
 It will be recognized that this need not necessarily hold 
 without being proved, on account of the interaction between 
 the molecules. Maxwell's law has therefore been deduced 
 by a number of mathematicians in different ways from a 
 general investigation of molecular motion, but which all 
 involve one or more assumptions which are stated, or partly 
 hidden in the deductions. Mathematicians are therefore 
 not yet agreed that a perfect proof has been given, while 
 some maintain that the law cannot hold. The writer has 
 developed a thermodynamical proof that the velocity com- 
 ponents mentioned should be independent, and on this the 
 proof of Maxwell's law will be based in this book. 
 
 It can be shown from thermodynamics that osmotic 
 equilibrium is not of a temporary nature, but must be per- 
 manent. A number of thermodynamical formulae have been 
 deduced involving the isothermal separation of substances 
 by means of semipermeable membranes. Van't HofFs formula 
 for the heat of formation of a substance in the gaseous 
 state, for example, depends upon the properties of semi- 
 permeable membranes. The thermodynamical equation 
 (47) in Section 21 may evidently be applied to a system 
 involving semipermeable membranes. Now the isothermal 
 separation of substances must take place infinitely slowly 
 only, otherwise the process would not be isothermal. There- 
 fore the gases on the two sides of the semipermeable mem- 
 brane of a system must be in permanent equilibrium with 
 each other when the movable parts of the system are kept 
 in a fixed position, that is, in the limiting case when the 
 movable parts are kept in fixed positions the diffusion of 
 the different gaseous constituents from one side of the 
 
22 THE MOLECULAR CONSTANTS 
 
 membrane to the other must be infinitely small. If this were 
 not the case the formulae deduced would not apply in prac- 
 tice. Thus in the limiting case the diffusion of certain gases 
 through a given membrane may be infinitely small, while 
 other gases pass readily. Each of the constituents of a 
 gaseous mixture may thus exist on the two sides of a mem- 
 brane, when initially placed on one side, but the condition 
 of the system may be such that some of them exist on the 
 other side of the membrane in infinitely small amounts only, 
 which condition remains unaltered. 
 
 The partial pressure of each constituent of a mixture of 
 gases is the same as if the other gases were absent. Hence 
 the redistribution of the velocities of the molecules of a 
 gas that might take place on addition of another gas is such 
 that each molecule produces the same average pressure as 
 before, which is equivalent to each molecule having the 
 constant velocity V given by equation (7), which will be 
 called the pressure velocity in this Section. 
 
 Let us now consider a mixture of molecules a and b in 
 a chamber A B separated from a chamber A by a semi- 
 permeable membrane, which is permeable only to the 
 molecules a. A migration of the molecules a from the cham- 
 ber A through the membrane into the chamber AB, and in 
 the reverse direction is continually going on. This is shown 
 by the fact that on increasing the volume of the chamber 
 A molecules a at once pass into it from the chamber AB, and 
 that therefore there is a free passage for the molecules a 
 through the membrane which can only be interfered with 
 occasionally by molecules coming in the opposite direction. 
 The number of molecules passing in one direction per second 
 is evidently equal to the number of molecules passing in the 
 opposite direction, otherwise the number of molecules 
 on each side of the membrane would gradually change. 
 We have seen that the pressure velocity V of translation of 
 the molecules a is the same in the two chambers. The 
 
THE INDEPENDENCE OF MOLECULAR VELOCITIES 23 
 
 pressure velocity of each set of migrating molecules must 
 also be the same and equal to the foregoing velocity * other- 
 wise the pressure would gradually change in the two cham- 
 bers. Let N' denote the number of molecules migrating 
 from one chamber into the other when no molecules b are 
 in the chamber AB, and N" the number corresponding to 
 a given number of molecules b in the chamber. Thus the 
 addition of the molecules b to the chamber AB has the effect 
 of dividing the number of molecules N' into two parts, the 
 part N" which migrates and the part N' N" which does not, 
 the pressure molecular velocity being the same for each part 
 according to what has gone before. Now a set of molecules 
 of different velocities can be divided in one way only into 
 two parts satisfying the foregoing conditions, viz.: which 
 corresponds to the same distribution of velocities in each 
 part. For if the number of molecules N" is decreased by 
 n', the pressure velocity of these n' molecules must be equal 
 to V, since this holds for the molecules N" n f and N' 
 (N" n f ), which can only be realized in the foregoing way 
 unless each molecule a has the same velocity. \ Thus the 
 distribution of velocities between the molecules migrating 
 from the chamber A into the chamber AB is the same as 
 that of the molecules in the chamber A, and the distribution 
 amongst the molecules migrating from the chamber AB 
 into the chamber A is the same as that of the molecules in 
 the chamber AB. These two distributions can be shown to 
 be identical. If this were not the case the molecules passing 
 into the chamber A would give rise to a different distribu- 
 tion of velocities near the membrane than exists at other 
 parts of the chamber. This would effect the migration of 
 the molecules a from the chamber A into the chamber A B, 
 and thus disturb the equilibrium. * From this it follows 
 that the distribution of velocities between the molecules 
 a in the chamber A is the same as that in the chamber AB. 
 The addition of molecules b to the chamber AB, accord- 
 
24 THE MOLECULAR CONSTANTS 
 
 ing to the foregoing investigation, does not alter the dis- 
 tribution of the velocities between the molecules a from that 
 obtained wjien molecules a only are in the chamber. Thus 
 the velocity components at right angles to each other of 
 the molecules a are not changed by collision with the mole- 
 cules 6, and hence not changed by the collision of the mole- 
 cules a with each other, i.e., the velocity components are 
 independent of each other. This result will now be used 
 to prove Maxwell's law. 
 
 Let the rectangular components of the velocity V\ of a 
 molecule be denoted by a, 6, and c, in which case 
 
 Let the probability that the component velocity along the 
 x axis has a value lying between a and a-\-da be expressed 
 by the function /(a), similarly let /(&) and /(c) express the 
 probabilities that the component velocities along the y 
 and z axes lie between 6 and 6+d6, and between c and c+ 
 dc, respectively. The probability of the three components 
 occurring simultaneously is therefore /(a)-/(6)-/(c), since 
 the components are independent. The situation of the sys- 
 tem of coordinates in space is arbitrary, and therefore 
 
 /(a) /(&) /() = *i( 2 +6 2 +c 2 ), >/ 
 
 where <j>i is a definite function of Fi 2 . On differentiating 
 this equation, keeping Vi constant, and dividing the result- 
 ant equation by /(a) /(&) -/(c), we obtain 
 
 The differentiation of equation (12) under the same condi- 
 tion gives 
 
THE COMPONENT VELOCITY PROBABILITY 25 
 
 On multiplying this equation by the undetermined multi- 
 plier X, the foregoing two equations may be combined into 
 the equation 
 
 Since the changes da, db, and dc, are independent their 
 factors may be separately equated to zero, giving 
 
 
 and two similar equations involving b and c. The integra- 
 
 tion of these equations gives /(a) = C\e ^ , /(&) = C\e 2 , 
 
 -A c2 
 and/(c)=Ci6 2 , where Ci denotes an arbitrary constant 
 
 whose value is infinitely small, since the probability of a 
 certain case in an infinitely large number of possible cases 
 is infinitely small. We may therefore write Ci = C2-da, 
 
 and if besides we write ?> \fT/ we nave 
 
 -( Y 
 f(a) = C 2 e \v p l -da. 
 
 Since the sum of the probability of an event happening and 
 the probability of it failing is equal to unity, we have for 
 all possible cases 
 
 ( 
 
 */-oo 
 
 C 2 I e W -da=l, 
 
 the integral expressing the sum of all the probabilities. 
 Similarly we have 
 
 ( 
 
 J-co 
 
 C 2 e <#>=!, 
 
26 THE MOLECULAR CONSTANTS 
 
 which on multiplying by the preceding equation becomes 
 
 /* r* _oH-6_2 
 C 2 2 e v p * -da-db=l, 
 
 J -&J-<x> 
 
 or 
 
 C 2 2 V P 2 ( ( e- (x * + ^dx-dy=l, 
 
 J &J-V) 
 
 if we write a/V p = x and b/V p = y. If x and y are now 
 regarded as coordinates in a system of rectangular coordinates 
 they may be transformed into polar coordinates by writing 
 
 x 2 +y 2 = r 2 and dx'dy = r'dr-d(}>, 
 which transforms the foregoing integral into 
 
 C 2 2 V 2 
 
 2 C C" e- r2 
 Jo Jo 
 
 and hence 
 
 C 2 = - 
 
 Accordingly 
 
 1 ( b Y 
 -da, /(6) = -*e-^-J . d b, 
 
 V p Vir 
 
 1 -(-^-Y 
 
 and the probability that the three components a, b, and c, 
 occur simultaneously is therefore 
 
 1 
 
 To obtain the probability of the occurrence of a certain 
 velocity Vi let us as before take a, 6, and c, as coordinates, 
 in which case 
 
THE DISTRIBUTION OF MOLECULAR VELOCITIES 27 
 
 and 
 
 da-db'dc=Vi 2 -dVi' smO-dft-dcf), 
 
 where denotes the angle between V\ and the c axis, and 
 the angle between the a axis and the projection of V\ on 
 the (a, 6) plane. The probability of a velocity V\ having a 
 definite direction is therefore 
 
 1 -(^lY 
 
 >/ FI -dVi- sin Q-d8'd<t>. 
 
 7T 
 
 The probability independent of any direction is obtained 
 on integrating with respect to 6 from to TT, and with respect 
 to $ from to 27r, which is taking into account all possible 
 directions. This can easily be shown to give 
 
 which expresses the probability that a molecule has a veloc- 
 ity lying between Vi and V \-\-dV\. The fraction of N 
 molecules possessing this velocity is then immediately given 
 by equation (9), which completes the proof. 
 
 It will easily be seen that the average velocity of trans- 
 lation of a molecule depends on the nature of the distribu- 
 tion of the velocities amongst the molecules. But the aver- 
 age kinetic energy of a molecule is independent of this 
 distribution, as will appear from the next Section. 
 
 8. The Average Kinetic Energy Velocity of a 
 Molecule, and the Relation between the Kinetic 
 Energy of a Molecule and its Absolute Temperature. 
 
 If Ni molecules in a cubic cm. of a gas have a velocity 
 Vi t and N2 molecules a velocity V2, etc., where 
 
 Ni+N 2 + . . N n =N c , 
 
28 THE MOLECULAR CONSTANTS 
 
 the total number of molecules in the cubic cm., the average 
 kinetic energy of a molecule is given by 
 
 , (NiV 1 *+N 2 V2 2 + . . . N n V n *\ 
 *(- Nl +N 2 +...N n - J' 
 
 where m a denotes the absolute mass of a molecule. This 
 expression for the kinetic energy may also be written 
 
 where V will be called the average kinetic energy velocity 
 of a molecule, and is given by 
 
 AW-f 
 \ tfi 
 
 +N 2 V 2 *+ . . . N n V n 2 
 
 . . .N n 
 
 This velocity is evidently not equal to the average velocity 
 V a , which is given by 
 
 _N 1 V 1 +N 2 V 2 + . . .N n V r 
 N l +N 2 + . . . N n 
 
 It can be shown that the velocity V is independent of the 
 distribution of the molecular velocities. Thus if pi denote 
 the partial pressure of the N\ molecules having a velocity 
 Vi, and p 2 the partial pressure of the N 2 molecules having 
 a velocity V 2 , etc., the total pressure p of the gas is given 
 by 
 
 But according to equation (6) 
 

 MOLECULAR ENERGY, TEMPERATURE, AND MASS 29 
 and 
 
 pn =^-X 2 , 
 
 and hence 
 
 This equation, which is same as equation (6), expresses V 
 in terms of p and m aj and it is therefore independent of the 
 distribution of molecular velocities. 
 
 The kinetic energy of a gram molecule of molecules in 
 the gaseous state therefore becomes 
 
 = T = yT= 1.247 Xl&T ergs, . . (13) 
 
 by the help of equation (8). The average kinetic energy 
 of a single molecule, which is 1/N, or 1/6.2X10 23 , the fore- 
 going value, is therefore equal to 
 
 (14) 
 
 The values of R and N used are given in Sections 5 and 3. 
 Thus the average kinetic energy of a molecule, whatever the 
 distribution of molecular velocities, is simply proportional 
 to the absolute temperature, and thus independent of molec- 
 ular mass. 
 
 It will be of interest to obtain the connection between 
 the average kinetic energy velocity V of a molecule and its 
 most probable velocity V p according to Maxwell's law. The 
 number of molecules of N c molecules in a cubic cm. whose 
 velocities lie between FI and V\+dV\ at any instant is 
 equal to 
 
 INcVi 2 (IiV , v 
 e-( Vp )-dV, 
 
30 THE MOLECULAR CONSTANTS 
 
 according to the preceding Section. The pressure these 
 molecules exert is obtained by multiplying this expression 
 
 by -~^li 2 , according to equation (6). Therefore on inte- 
 
 o 
 
 grating the foregoing product from Vi = Q to FI = QO we 
 obtain the total pressure p exerted by the molecules, that is 
 
 IWaNcV* C^ 4 
 
 - I x 4 e~ x 
 
 3V 7T JQ 
 
 ~ x -dx 
 
 3V7 
 
 since e x -dx=- , where p denotes the density. On 
 
 Jo * 
 
 comparing the foregoing value of p with that given by equa- 
 tion (6) we obtain 
 
 /Q\ 
 
 = (l) 
 
 or the average kinetic energy velocity is about 23% greater 
 than the most probable velocity. The relation between 
 V and V a is obtained from a comparison of the foregoing 
 equation and equation (11), which gives 
 
 8,, 
 
 Thus the average velocity is about 92% of the average 
 kinetic energy velocity. 
 
 In the case of a mixture of gases of molecules r and e the 
 total pressure p is given by 
 
 e 2 . 
 1 
 
THE EQUATION OF A GASEOUS MIXTURE 31 
 
 on applying equation (6)*, where p e and p r denote the par- 
 tial pressures, N e and N r the partial concentrations, m ae 
 and m a r the absolute molecular weights, and V e and V r the 
 average kinetic energy velocities of the molecules e and r 
 respectively. If the temperature of each set of molecules 
 is the same an application of equation (8) to the foregoing 
 equation gives 
 
 where N er denotes the total concentration of the molecules 
 per cubic en., and m e /m a e=nir/mar=N the number of mole- 
 cules in a gram molecule of a pure substance. , This equation, 
 which is the same as equation (5), agrees with the facts, 
 showing that when two gases are mixed they rapidly assume 
 the same temperature. This would arise in part through 
 both gases being in contact with the vessel whose tempera- 
 ture they would gradually assume, in other words, the 
 vessel usually being a conductor of heat would act as an 
 intermediary in adjusting the gases to the same temperature. 
 The question then arises, would an equalization of tempera- 
 ture take place through the interaction of the molecules 
 alone, and what is the distribution of velocities when the 
 two sets of molecules in some way or other have acquired 
 the same temperature? This will be discussed in the next 
 Section. 
 
 9. The Equipartition of Energy between the 
 Different Molecules in a Mixture of Gases. 
 
 When two sets of molecules of different kinds at different 
 
 temperatures are mixed, the average kinetic energy of each 
 
 molecule of each set, according to the Law of Equipartition 
 
 of Energy, eventually becomes the same through molec- 
 
 * We may do this since each set of molecules acts independently. 
 
32 THE MOLECULAR CONSTANTS 
 
 ular interaction, and the corresponding distribution of the 
 molecular velocities at any instant is the same as if the gases 
 were isolated. This law is usually proved, or attempts are 
 made to prove it, from purely dynamical considerations 
 in treatises on the Kinetic Theory of Gases. The analysis 
 is very intricate, and the suppositions introduced are not 
 unobjectionable. The subject has always attracted a good 
 deal of attention from mathematicians with the object of 
 putting the proof on a sounder basis. In the earlier develop- 
 ment of the subject the investigations of Maxwell and 
 Baltzmann are pre-eminent. 
 
 The dynamical proof of the law usually depends on the 
 assumption that the molecules consist of perfectly elastic 
 spheres not surrounded by fields of force, and that the 
 equipartition of energy is solely caused by molecular collision. 
 As a matter of fact in practice the equipartition would be 
 caused in other ways if not caused by molecular collision. 
 Thus we know from experience that a mass of a hot gas 
 located in a colder gas radiates heat to the latter till the 
 temperature is equalized. A well-known example of this is 
 the radiation manifested by the hot air rising from a fire, 
 or from a furnace. This shows that each individual mole- 
 cule in a gas is continually radiating heat energy whose 
 amount per second depends on its kinetic energy. There- 
 fore if two sets of molecules at different temperatures are 
 mixed heat will be radiated from one set of molecules to the 
 other till the average kinetic energy of each molecule is the 
 same, or the two sets of molecules have the same tempera- 
 ture, as would be the case if the two sets of molecules were 
 separate but adjacent to each other. It will follow then from 
 Section 7 that after temperature equilibrium has been 
 obtained the distribution of velocities in each set of mole- 
 cules is the same as if it were isolated. 
 
 It seems unnecessary and futile, therefore, to endeavor to 
 establish the Law of Equipartition of Energy on assumptions 
 
EQUIPARTITION OF ENERGY 33 
 
 relating to the interaction of molecules, when the law fol- 
 lows directly from the fact that a molecule is continually 
 radiating heat energy. Although equipartition of energy 
 would be brought about as indicated, it is very likely that it 
 would also be brought about by the collision of elastic spheres, 
 since it follows from mechanics that on the average two 
 colliding bodies of different kinetic energies have their 
 energies more evenly distributed after collision. In fact, 
 every kind of interaction between two molecules, whether 
 by " collision," or through the molecular forces of attrac- 
 tion and repulsion, has on the average the effect of redis- 
 tributing the kinetic energy in favor of the body possessing 
 the lesser energy. A definite mathematical proof that 
 equipartition of energy may take place along these lines is 
 difficult on account of having to deal with a large number 
 of molecules possessing different velocities, each of which 
 interacts some time with one or more molecules, and it is 
 therefore necessary to show that equilibrium in energy 
 partition exists after an infinite time beginning from an in- 
 definite state of affairs. 
 
 10. The Number of Molecules per Cubic Cm. in 
 a Gas. 
 
 According to equation (5) the number N c of mole- 
 cules per cubic cm. of a gas is given by 
 
 where N and R are given in Sections 3 and 5. Hence if 
 two different gases possess the same temperature and 
 pressure each contains the same number of molecules per 
 cubic cm. This result is known as Avogadro's Law. It 
 follows also that two gases at different temperatures con- 
 tain the same number of molecules per cubic cm. if the 
 ratio p/T has the same value for both. It will appear 
 
34 THE MOLECULAR CONSTANTS 
 
 from Section 5 that these results also hold for mixtures of 
 gases, in which case p denotes the sum of the partial 
 pressures. 
 
 The foregoing equation may also be written 
 
 , (16) 
 
 m m 
 
 according to equation (5), from which it follows that two 
 gases not necessarily at the same temperature contain the 
 same number of molecules per cubic cm. if they have the 
 same values for the ratio p/m. 
 
 Since the molecules possess motion of translation this 
 leads us to the consideration of another quantity. ^ 
 
 11. The Number of Molecules Crossing a Square 
 Cm. in all Directions from one Side to the Other in a 
 Gas. 
 
 This number is of importance, since it is one of the funda- 
 mental quantities occurring in the different formulae relating 
 to a gas. It can be expressed in terms of quantities which 
 can be measured directly, and hence its numerical value 
 obtained when desired. We will see in Section 29 that its 
 value can also be found in the case of a liquid. It will be 
 convenient first to obtain the number corresponding to the 
 supposition that the molecules move parallel to three axes 
 at right angles to each other. We have seen that this suppo- 
 sition may be made when dealing with the connection between 
 the velocity of translation of the molecules and the pres- 
 sure which they exert. Let no denote the number of mole- 
 cules crossing a plane one square cm. in area per second in 
 one direction situated at right angles to one of the foregoing 
 axes. It follows then that 
 
THE PRESSURE FACTORS OF A MOLECULE 35 
 
 where V a denotes the average velocity of a molecule, N c 
 the molecular concentration, and p the density of the gas. 
 To prove this we may suppose that we are dealing with a 
 single cubic cm. of gas whose faces are parallel to the axes 
 mentioned, in which case a molecule having a velocity V x 
 will cross it V x /2 times in one direction per second. There- 
 fore if there are Ni, N2, Ns . . . , molecules having respec- 
 tively the velocities Vi, 2, V$ . . . , we have 
 
 N e (N l Vi+NjV 2 +. .} N c V 
 
 61 N c j 6 
 
 Another expression for no whic,h is very useful may be 
 obtained. The pressure p in dynes exerted by a gas may be 
 written 
 
 (18) 
 
 where A represents an appropriate factor, which has other 
 important applications which will be found in Section 20. 
 The value of the quantity A may be determined by substi- 
 tuting in the foregoing equation successively for the quan- 
 tities p, no and V a , assuming that V a = V, from the equations 
 (17), (8), and (4), giving 
 
 (19) 
 
 where w a /m=1.6lXlO~ 24 , the absolute value of the mass 
 of a hydrogen atoin. 
 
 Let us now consider the case when the molecules are 
 moving in all directions. Let a point 6 be taken on a plane 
 abc in the gas, Fig. 2, arid a sphere of radius r described 
 round it. The molecules passing through the point 6 strike 
 the surface of the sphere at right angles, and let us therefore 
 
36 THE MOLECULAR CONSTANTS 
 
 suppose that their number corresponds to S molecules 
 entering the sphere per second per cm. 2 of its surface. The 
 number of molecules impinging on a circular belt of the 
 sphere of breadth r-dd, and radius r cos. 6 is therefore 2irr 2 S 
 cos dO. The force exerted by a molecule impinging on the 
 plane abc at an angle 6 is proportional to its component 
 velocity at right angles to the plane according to equations 
 (17) and (18), and thus according to equation (18) equal to 
 
 ^ A, or sin 0-A. Hence the molecules entering the 
 
 abc 
 
 FIG. 2. 
 
 upper hemisphere, which lie in the solid angle TT, on impinging 
 on the point b exert the force 
 
 ( 2 2Trr 2 SA sin 0- cos 6~de=Trr 2 SA. 
 Jo 
 
 If u denotes this number of molecules we have u = 2irr 2 S, 
 and the pressure exerted may therefore be written u-A/2. 
 Hence if n denotes the number of molecules crossing a square 
 cm. from one side to the other in all directions, the pressure p 
 of the gas is given by 
 
 . . ,'(20) 
 
CORRECTION FACTORS FROM MAXWELL'S LAW 37 
 
 This equation may be used to find n. The equation may 
 also be written in the form 
 
 (21) 
 
 by means of equations (19) and (4), where v denotes the 
 volume of a gram molecule of molecules. 
 
 In the foregoing investigation we have assumed that 
 V a =V, or the average velocity of translation of a molecule 
 is equal to the average kinetic energy velocity. But this is 
 not the case according to Section 8. According to Max- 
 
 /~8~ 
 well's law V a =+V=.922V. If this is accepted and taken 
 
 \O7T 
 
 into account the value of A is 5.521 XlQ~ 20 VTm, or 1.085 
 times that given by equation (19). In subsequent investi- 
 gations we shall, however, use the value of A given by 
 equation (19). The necessary change if desired can always 
 be readily made. 
 
 The value of n given by equation (21) is corrected accord- 
 ing to Maxwell's law by multiplying the right-hand side 
 
 /~8~ 
 of the equation by */*-i or by .922. 
 
 Since a molecule in motion represents a certain amount 
 of kinetic energy, general energy considerations will be of 
 interest and importance. 
 
 12. The First Law of Thermodynamics. 
 
 According to this law energy is indestructible. Since the 
 expenditure of energy on a substance is accompanied by a 
 change in temperature heat represents a form of energy. 
 The amount of change in temperature depends on the heat 
 capacity, or the specific heat, of the substance. By defini- 
 tion the heat capacity of water between the temperatures 
 
38 THE MOLECULAR CONSTANTS 
 
 14.5 C. and 15.5 C. is unity, and the amount of heat ab- 
 sorbed corresponding to this change in temperature is called 
 a calorie. Other definitions of the calorie have been pro- 
 posed, but the foregoing is mostly used. The units of heat 
 capacity and mechanical work are quite arbitrary, and there- 
 fore a factor according to the above law should exist which 
 would enable heat units to be converted into mechanical 
 units, and vice versa, that is, if W denotes the amount of 
 work in ergs expended to produce an amount of heat Q in 
 calories, the relation between the two quantities is expressed 
 by the equation 
 
 W = JQ, 
 
 where J denotes the factor in question. 
 
 The first law of thermodynamics was first enunciated by 
 Mayer in 1842, who obtained a value for J from the dif- 
 ference between the specific heats of air at constant pres- 
 sure and at constant volume, which is equal to R/J accord- 
 ing to the next Section. The value obtained is not very 
 accurate on account of the existence of the Joule-Thomson 
 effect according to which a certain amount of work is done 
 in overcoming molecular attraction during the expansion 
 of a gas. But it would have been possible to obtain a much 
 better value by this method by using the gas at a tempera- 
 ture and pressure for which the Joule-Thomson effect is 
 zero; the method has, however, now been superseded by 
 others. In 1843 Joule began a series of classical experi- 
 ments to test the law from different aspects, the best values 
 of J being obtained from the experiments on the agitation 
 of liquids. More recently J has been determined from the 
 heating produced by an electric current. In the Recuiel 
 de Constants Physiques (Gauthier-Villars, 1913) published 
 under the auspices of the Societe Frangaise de Physique, 
 its most probable value is taken to be 
 
 J=4.184X10 7 . 
 
THE TMOLECULAR KINETIC AND INTERNAL ENERGY 39 
 
 We may now consider more closely the nature of heat 
 capacity of substances. 
 
 13. The Specific Heats of Gases and Liquids. 
 
 The specific heat of a perfect gas at constant volume 
 consists of the sum of the changes in kinetic energy of 
 motion of translation and internal molecular energy per 
 degree change in temperature. If the specific heat is ex- 
 pressed in calories, the first term is equal to 3R/2J for a 
 gram molecule according to the preceding Section, and 
 
 equation (13); the second term may be written ( ^), 
 
 \ ol / v 
 
 where u a denotes the internal molecular energy per gram 
 molecule expressed in terms of calories. Hence if S' v and 
 S v denote the specific heats per gram molecule and per 
 gram respectively we have 
 
 .. ..... (22) 
 
 and i 
 
 (23) 
 
 If the pressure of the gas is kept constant during the 
 change of temperature additional heat is absorbed in doing 
 
 the external work ^(-^) expressed in calories, which is 
 J \51 IP 
 
 equal to R/J according to equation (4). Therefore if S' P 
 and S p denote the specific heats at constant pressure per 
 gram molecule and per gram respectively we have 
 
 *--+(!), ...... 
 
 and 
 
 S' P , ...... . . (25) 
 
40 
 
 where 
 that 
 
 and 
 
 THE MOLECULAR CONSTANTS 
 
 a perfect gas. It follows then 
 
 \~8TJ VS7V 
 
 R 
 :,= 7 , 
 
 S' C 3R dUg 
 
 (26) 
 
 (27) 
 
 In the special case that 
 becomes =1.666. 
 
 =0 the foregoing equation 
 
 It will be instructive to consider the ratio of the specific 
 heats for a number of gases at C. given in Table II. It 
 
 TABLE II 
 
 Substance. 
 
 Formula. 
 
 S 'P 
 
 SV 
 
 Mercury 
 
 Hg 
 
 1 666 
 
 Argon 
 
 A 
 
 1.667 
 
 Carbon monoxide 
 
 CO 
 
 1 403 
 
 Carbon dioxide 
 
 CO 2 
 
 1311 
 
 Ethelene 
 
 C 2 H 4 
 
 1.245 
 
 Propane 
 
 C 3 H 8 
 
 1.153 
 
 Methyl ether 
 
 C 2 H 6 O 
 
 1 107 
 
 Ethyl ether 
 
 C 4 Hi O 
 
 1 097 
 
 
 
 
 will be seen that for the mon-atomic gases argon and mercury 
 the ratio is practically the same as that given by the fore- 
 going equation, and that therefore for these gases ( ~ ) = 0, 
 
 i.e., the kinetic energy of motion of translation only changes 
 with change of temperature. This does not, however, hold 
 
INTERNAL ENERGY AND MOLECULAR COMPLEXITY 41 
 
 when the molecules of the gas consist of two or more atoms, 
 in a general way the deviation of this ratio from the value 
 1.666 increases with the complexity of the molecule. This 
 is what we would expect since a change in temperature 
 increases the violence of molecular collision through the 
 increase of velocity of translation, which would give rise to 
 an increase in the velocity of rotation of each molecule, 
 changes of the atomic configuration through collision and 
 increased velocity of rotation, etc. 
 
 It will be instructive to calculate the value of ( ^) 
 
 \l/9 
 in the case of one of the complex gases say ether. We have 
 
 5R (duA 
 2/+U7V, 
 
 and therefore 
 
 . QR 
 2J ' 
 
 fSua\ 17. 
 \8Tj v 2 
 
 and thus the heat absorbed in changing the molecular 
 internal energy is very much greater than the heat 3R/2J 
 absorbed in changing the kinetic energy of motion of trans- 
 lation, or in doing the external work R/J on carrying 
 
 out the heating at constant pressure. The values of ( -^ j 
 
 thus vary very much with the complexity of the molecule. 
 
 The ratio of the two specific heats of a gas, it may be 
 pointed out, can be determined with a much greater accuracy 
 than either of these quantities separately from measure- 
 ments of the velocity of sound V s which is given by the 
 equation 
 
42 THE MOLECULAR CONSTANTS 
 
 where p and p denote the pressure and density of the gas. 
 The value of V s for any given gas may be determined in the 
 laboratory by measuring the wave length X in a Kundt's 
 tube corresponding to a musical note of known frequency 
 n, these quantities being connected by the equation V s = n\. 
 It may be noted that since sound consists of the propagation 
 of waves of compression and rarefraction, the velocity with 
 which they travel is closely connected with the velocity of 
 translation of the molecules. 
 
 The specific heat S'a at constant volume per gram mole- 
 cule of a liquid, or gas not obeying Boyle's law, is expressed 
 by the equation 
 
 where U e denotes the heat absorbed in internal energy 
 changes on allowing the substances to expand at constant 
 temperature till its volume is infinite. This equation may 
 be proved by passing the substance through a cycle and 
 equating to zero the internal work done. Thus on expand- 
 ing the substance at constant temperature till its volume is 
 infinite, and then raising the temperature of the resultant 
 gas by 5T at constant volume, the internal work U e +S' v - 5T 
 is done. On compressing the substance to its original vol- 
 ume at constant temperature, and lowering its temperature 
 by dT at constant volume, the internal energy changes by 
 U e dUeS'vr8T. On equating the sum of the internal 
 energies to zero equation (28) is obtained. 
 
 If the equation of state (Section 26) of the substance 
 
 under consideration is known the value of I ^j can at 
 once be calculated. According to thermodynamics 
 
 A (6p\ 
 SV)T \STJ,- P ' 
 
SPECIFIC HEATS UNDER VARIOUS CONDITIONS 43 
 
 where U denotes the total internal energy of a substance of 
 volume v, temperature T, and pressure p. On multiplying 
 this equation by dv and integrating it between the limits 
 v and infinity we obtain 
 
 -u.-u.-r. = 
 
 The right hand side of this equation may be evaluated by 
 writing the equation of state in the form p = <j)(T, v) and sub- 
 stituting for p. 
 
 Similarly it can be shown that the specific heat S' P i 
 per gram molecule at constant pressure is given by 
 
 The internal specific heat S' ip i per gram molecule at constant 
 pressure is therefore given by 
 
 (30) 
 
 which is m times the specific heat S g per gram. 
 
 In the case that the molecules of a substance are partly 
 dissociated the expressions for the two specific heats for the 
 gaseous state will have to be modified. Consider a gram 
 molecule of molecules containing N molecules of which N s 
 are dissociated, each into q molecules. The kinetic energy 
 
 o r> 
 
 term has then to be replaced by the term 
 
 3R _ 
 2j'd 
 
 (\ 
 -^) has to be replaced by the term 
 51 /v 
 
 HN S \ 
 
44 THE MOLECULAR CONSTANTS 
 
 where H denotes the heat of formation of a gram molecule 
 of molecules. Equations (22) and (24) are accordingly re- 
 placed by the equation 
 
 8u a \ _Ns(8H\ _H(8N S \ 
 dTj, W\*TJ. N\WJ,' 
 
 and the equation obtained by substituting S' p , ( ^) , 
 
 \ ol I v 
 
 for S f c , (-TT) in the foregoing equation, and adding the 
 
 In the case of a mixture of substances it is useful to 
 define the quantity partial specific heat. Thus if S er denotes 
 the internal heat capacity at constant pressure of a mixture 
 of molecules e and r, it may be written 
 
 (32) 
 
 where S e denotes the internal specific heat associated with 
 the molecules e, and S r that associated with the molecules 
 r. This equation may also be written 
 
 S er = N'rS r +N' e S e , ..... (33) 
 
 where N' T and N' e denote respectively the numbers of mole- 
 cules r and e in the mixture, and s r and s e the specific heats 
 of a molecule r and of a molecule e respectively. The change 
 in heat capacity that the mixture undergoes on adding a small 
 number of molecules say of r, is expressed by differentiat- 
 ing the foregoing equation with respect to N' r , giving 
 

 THE PARTIAL SPECIFIC HEATS OF MIXTURES 45 
 
 Now the value of s r of a molecule r depends on the nature 
 and relative positions of the surrounding molecules, and 
 the same holds for s e . These conditions will evidently be 
 altered to a vanishing extent by the addition of a small 
 number of molecules r to the mixture, and therefore very 
 
 (\ / \ 
 
 -r^r) and (-TT^-) are equal to zero. 
 .ON r/P \ON r/P 
 
 The foregoing equation then gives 
 
 t), w 
 
 and thus s r can be determined by measuring (-^Trr) The 
 
 value of s e may now be determined from equation (33), 
 or in the same way as s r . In the latter case the values of 
 s r and s e obtained should make equation (33) vanish, and 
 in this way therefore the accuracy of the determinations 
 may be tested. 
 
 It may be mentioned here that the matter in Sections 
 30 and 39, has a bearing on specific heats. 
 
 The quantity U e in the foregoing equations has in prac- 
 tice a finite value depending on the temperature, unless 
 the substance behaves as a perfect gas, in which case the 
 quantity is zero. It appears, therefore, that on changing the 
 temperature of a substance not obeying the gas laws, a part 
 only of the heat energy absorbed is converted into kinetic 
 energy of translation of the molecules. The remaining part 
 is expended in other ways, most probably in overcoming 
 the attraction between the molecules, evidence of the 
 existence of which will now be considered. 
 
 14- Evidence that Molecules and Atoms are sur- 
 rounded by Fields of Force. 
 
 There is a good deal of evidence that molecules and 
 atoms are surrounded by strong fields of force which decrease 
 
46 THE MOLECULAR CONSTANTS 
 
 rapidly in intensity from the center outwards. The internal 
 heat of evaporation L of a liquid, for example, represents 
 almost entirely work done against these forces during the 
 separation of the molecules. We may write accordingly 
 
 where IJ\ U% denotes the change in potential energy of 
 the liquid during evaporation due to the molecular attrac- 
 tion, and HI u u the change in internal energy of the mole- 
 cules. The free kinetic energy of the molecules is not altered 
 during the process of separation according to Section 21, 
 and hence u\ u a represents the change in molecular energy 
 due to changes in atomic configuration and velocity of molec- 
 ular rotation. It is likely to be small in comparison with 
 U\ Us, the work done against molecular attraction. 
 
 When the volume of a dense gas not obeying Boyle's 
 law is increased at constant temperature, a larger amount 
 of heat is absorbed than corresponds to the external work 
 done. The explanation is the same as in the case of the 
 evaporation of a liquid. 
 
 The forces in question are appreciable over much greater 
 distances of separation of the attracting molecules than of 
 the order of the distance of separation in the liquid state, 
 i.e., 10 ~ 8 cm. This is shown by the Joule-Thomson effect. 
 On passing a stream of gas at a volume v per gram and 
 pressure p through a porous plug on the other side of which 
 the gas has a volume v f and pressure p f the temperature 
 of the stream emerging from the plug is not the same as that 
 entering it. If the gas does not obey Boyle's law p'v' is 
 not equal to p v, and a part of the temperature change is 
 accounted for by the external work done in the process, 
 which is equal to p'v' p v. This can evidently at once be 
 calculated. The other part of the temperature change is 
 accounted for by the work necessary to separate the molc- 
 < -i ilcs against their molecular forces during their passage 
 
THE JOULE-THOMSON EFFECT 47 
 
 I 
 
 I through the porous plug, which is done at the expense of 
 their kinetic or heat energy. The temperature change is 
 usually negative, but it may also be positive, depending on 
 the values of p, p', v, and v'. In the first case energy is 
 required to separate the molecules, and hence attraction is 
 the paramount force, while hi the second case work is done 
 on the molecules, and the paramount force is therefore 
 
 > repulsion. The change in internal molecular energy during 
 the process, represented by u\ u a , is probably negligible. 
 Thus we see that forces of attraction and repulsion exist 
 
 i between molecules whose relative magnitudes depend upon 
 
 | the distances of separation; and the resultant force may 
 therefore be represented by the sum of at least two terms, 
 one representing attraction and the other repulsion, the 
 value of the attraction term decreasing according to the 
 nature of the internal heat of evaporation of a liquid and 
 the foregoing experiments more rapidly with the distance of 
 separation of the molecules than the repulsion term. 
 
 At the absolute zero of temperature the molecules of a 
 substance have no kinetic energy of translation. It is fairly 
 certain that the volume of the substance under those condi- 
 tions is not zero. There exists therefore very probably 
 another repulsion term in the law of force counteracting 
 the attraction term for very small distances of separation 
 of the molecules, in which case the apparent volume of the 
 substance would not vanish at the absolute zero. 
 
 The considerations in the foregoing paragraph discard 
 the notion of molecular volume. It is important in this 
 connection to note that the volume associated with matter 
 manifests itself to us only by resistance to force; and mole- 
 cules may therefore be, and probably ought to be, simply 
 regarded as centers of force. The fact that the apparent 
 volume of a molecule depends on external conditions such 
 as the temperature. -etc., appears to show that this view 
 ought to be taken. (See Sections 19, and 24.) 
 
48 THE MOLECULAR CONSTANTS 
 
 It follows from the foregoing considerations that as a 
 first approximation the force of attraction between two 
 molecules may be represented by an expression consisting 
 of the sum of three terms, one of which has a positive and 
 the remaining two negative signs. The numerical value 
 of one of the negative terms decreases more rapidly with 
 the distance of separation of the molecules than the value 
 of the positive term, while the value of the other negative 
 term decreases less rapidly. The value of the last-mentioned 
 term will thus be predominant for large distances of separa- 
 tion of the molecules, giving rise to a repulsion between 
 them which shows itself as a heating effect in Joule-Thomson 
 experiments. The value of the other negative term will 
 be predominant for close distances of approach of the 
 molecules, giving rise to a repulsion between them which 
 shows itself as an apparent volume of the molecules. For 
 intermediate distances of separation of the molecules the 
 positive term will be predominant and the force one of 
 attraction. 
 
 It is obvious that the effect of the molecular forces on 
 the properties of a substance as a whole arises through 
 their effect on the relative motion of the molecules. This 
 leads us to consider the influence of one molecule upon the 
 motion of another in a substance. 
 
 15. Molecular Interaction. 
 
 In treatises on the Kinetic Theory of Gases molecules 
 are usually supposed to consist of perfectly elastic spheres 
 having no fields of force surrounding them. The interaction 
 of the molecules would then consist simply of collisions. 
 This is no doubt far from being the case in practice, accord- 
 ing to the previous Section. If molecules and atoms consist 
 simply of centers of force it would be hard, if not impossible, 
 to define what is meant by a collision. The mathematical 
 
THE COMPLEXITY OF MOLECULAR INTERACTION 49 
 
 investigations based on molecular collision may therefore 
 lead to results having little to do with the actual facts. 
 It appears therefore that the proper development of the 
 subject should be along lines which do not involve molec- 
 ular collision. 
 
 This will be emphasized by a consideration of the gen- 
 eral nature of the interaction of two molecules when the law 
 of force of interaction is of the nature described in the previous 
 Section. This is graphically shown in Fig. 3, which in a gen- 
 eral way shows the paths described by a molecule having a 
 high velocity in approaching a molecule at rest, correspond- 
 
 FIG. 3. 
 
 ing to different distances of the molecule at rest from the 
 line of prolongation of the initial direction of the moving mole- 
 cule. For small values of these distances the path of the 
 molecule will be mainly affected by the repulsion existing 
 on close approach; for greater distances the path is mainly 
 affected by forces of attraction, which deflects the moving 
 particle towards the one at rest; for still greater distances a 
 slight repulsion between the molecules tends to deflect the 
 moving molecule in the opposite direction. When both 
 molecules are moving their paths are similar to the foregoing, 
 the molecules then simultaneously approach or recede from 
 each other, or move in the same direction. 
 
 The nature of the path of a molecule, it will be seen, 
 
50 THE MOLECULAR CONSTANTS 
 
 depends considerably on the initial direction of motion of 
 the molecule, and in each case is complex. In practice 
 each path would probably be much more complex than indi- 
 cated by Fig. 3, which does not pretend to exhibit the 
 actual state of affairs that exists in practice, but has been 
 drawn to give the reader merely some idea of the complexity 
 of the motion of a molecule under the influence of another 
 molecule. It is evident, therefore, that the effect of the 
 molecular forces, in whatever connection, could scarcely 
 be represented by the effect of the collision of elastic spheres. 
 In the case of a liquid the paths of two molecules under 
 each other's influence would be affected by the vicinity of 
 other molecules, and would therefore be very much more 
 complicated than those indicated by Fig. 3. 
 
CHAPTER II 
 
 THE EFFECT OF THE MOLECULAR FORCES ON THE 
 DYNAMICAL PROPERTIES OF A MOLECULE IN A 
 DENSE GAS OR LIQUID 
 
 16. The Velocity of Translation of a Molecule 
 in a Liquid or Dense Gas when passing through a 
 Point at which the Forces of the Surrounding Mole- 
 cules neutralize each other. 
 
 It is often assumed, without any apparent reason, that 
 the average velocity of a molecule in a liquid or dense gas 
 is the same as that it would have in the gaseous state at the 
 same temperature. But this is not likely to hold on account 
 of the interaction of the molecules due to the existence of 
 molecular forces. Information in this matter is obtained 
 from considering what is meant by a thermometer indicating 
 the temperature of a gas in which it is placed.* From the 
 Kinetic Theory it follows that a thermometer placed in 
 a gas assumes its temperature through being bombarded 
 by the gas molecules. The temperature indicated does not 
 depend on the number of molecules falling per square cm. 
 per second on the surface of the thermometer bulb, but 
 only on the average kinetic energy of a molecule in the 
 gas according to Section 8. The velocity with which a mole- 
 cule actually strikes the bulk depends, however, on the 
 attraction exerted by its material. Therefore, if the bulb 
 is covered with a layer of material of much greater density 
 * R. D. Kleeman, Phil. Mag., July 1912, pp. 100-102. 
 51 
 
52 THE EFFECT OF THE MOLECULAR FORCES 
 
 than the bulb itself the velocity of impact may be very much 
 increased. But the temperature indicated by the ther- 
 mometer is not altered thereby, as we know from experiment. 
 Thus the temperature indicated corresponds to the kinetic 
 energy of a molecule when not, under the attraction of the 
 thermometer bulb, or the additional velocity given to the 
 molecule by the attraction of the bulb has no effect on the 
 temperature indicated. 
 
 Again, if a very dense solid were placed somewhere in 
 the gas, the molecules in its zone of attraction would pos- 
 sess a 'greater velocity of translation than outside the zone. 
 The temperature indicated by the thermometer would, how- 
 ever, not be altered thereby, but would correspond to the 
 kinetic energy of the motion of translation of the gas 
 molecules outside this zone, and the zone of attraction of 
 the thermometer bulb, i.e., to the kinetic energy when the 
 molecules are not under the action of any forces. 
 
 The same result will also hold in the case of a liquid, 
 or dense gas, in which case the temperature indicated by 
 the thermometer corresponds to the velocity each molecule 
 has when not under the action of a force, which occurs when 
 passing through a point where the forces of the surrounding 
 molecules neutralize each other. There are evidently num- 
 bers of such points in a liquid which change their positions 
 with the motion of the molecules. The velocity which a 
 molecule has when passing through such a point is not 
 necessarily always the same, but an average velocity may 
 be associated with it. When the molecule is not passing 
 through such a point hi a dense gas or liquid it is under the 
 action of forces, and its velocity on the average is then 
 greater (Section 17) than the foregoing average velocity. 
 Therefore the average velocity of a molecule when not 
 under the action of forces may be called its average minimum 
 velocity. 
 
 It is evident then that if we consider two masses of the 
 
THE AVERAGE MINIMUM MOLECULAR VELOCITY 53 
 
 same substance, one in the perfectly gaseous state and the 
 other in the liquid state, and they possess the same tem- 
 perature, the average kinetic energy of a gas molecule is 
 the same as the average minimum kinetic energy of a mole- 
 cule in the liquid state, in other words, the average kinetic 
 energy of a molecule when not under the action of a force in 
 a liquid or dense gas is the same as the average kinetic energy 
 it would have in the perfect gaseous state at the same tem- 
 perature. 
 
 We may also reason in a reverse manner. A thermometer 
 placed in a liquid indicates its temperature through being 
 bombarded by the liquid molecules. The velocity of impact 
 and the general nature of the bombardment may be changed 
 without changing the temperature indicated by changing 
 the nature of the thermometer bulb, or compressing the 
 liquid at constant temperature. The question then arises: 
 what is the connection between the temperature indicated 
 and the velocity of translation of a molecule under stated 
 conditions in the liquid? Now the velocity of a molecule 
 that means anything definite in a liquid corresponds to its 
 independent velocity, or the velocity when not under the 
 action of an external force, and we must therefore endeavor 
 to connect this velocity with the temperature indicated. 
 Now in the case of a gas a thermometer indicates the tem- 
 perature corresponding to the kinetic energy of a molecule 
 when not under the action of a force. There is no reason 
 whatever why this should not hold in the case of a liquid. 
 We conclude, therefore, that the average kinetic energy of a 
 molecule in a liquid under these conditions is the same as 
 in the gaseous state at the same temperature. 
 
 This result may be established in a somewhat different 
 way. Consider two chambers having a wall in common filled 
 with gases at the same temperature. Suppose that one-half 
 of the common wall adjacent to one of the gases is replaced 
 by a much denser material so that the molecules strike the 
 
54 THE EFFECT OF THE MOLECULAR FORCES 
 
 wall on that side with a greater velocity than on the other 
 side. This change in the nature of one side of the wall will 
 not give rise to a flow of heat from one side to the other, 
 since this would be contrary to the laws of thermodynamics. 
 The heat effect produced by the molecules on impinging 
 on the wall therefore does not depend on the velocity of 
 impact, but on the velocity when not under the action of the 
 attraction of the wall. The foregoing result follows then in 
 a similar way as before. 
 
 We will now turn our attention to the velocity of a mole- 
 cule at other points in a liquid than those considered. 
 
 17. The Total Average Velocity of Translation 
 of a Molecule in a Substance. 
 
 When a molecule is not passing through a point in a 
 dense substance at which the forces of the surrounding 
 molecules neutralize each other, its velocity is likely to 
 differ, on account of the effect of molecular attraction and 
 repulsion, from its velocity at such a point, which we have 
 seen in the previous Section is the same as that it would 
 have in the perfectly gaseous state at the same temperature. 
 The total average velocity of the molecule, corresponding to 
 the complete path described during an infinitely long time, 
 is therefore likely to differ from the velocity defined as 
 the average minimum velocity. When a substance is in the 
 perfectly gaseous state these two velocities are evidently 
 very approximately equal to one another, or denote approxi- 
 mately the same thing. But evidently this cannot be the 
 case when the density of the substance is such that the motion 
 of the molecule is constantly influenced by the attraction 
 and repulsion of the surrounding molecules. In fact, it 
 can be shown* that in the case of a liquid the total average 
 
 * R. D. Kleeman, Phil. Mag., July, 1912, pp. 101-103. 
 
THE TOTAL AVERAGE MOLECULAR VELOCITY 55 
 
 velocity is several times that of the average minimum 
 velocity. 
 
 Suppose that the volume of a gram of liquid is changed 
 by a small amount at a low temperature in contact with its 
 saturated vapor by changing the temperature. The energy 
 spent per molecule in overcoming the molecular attraction 
 is equal to m a -dL, where L denotes the internal heat of 
 evaporation in ergs per gram and m a the absolute molecular 
 weight of a molecule. The average force acting on a mole- 
 cule during the expansion is therefore -i-w a , where x denotes 
 
 dx 
 
 the distance of separation of the molecules. Now x = ( j , 
 where p denotes the density of the liquid, and the force may 
 
 therefore be written 3ra a 2/3 p 4/3 -r~- We may assume without 
 
 dp 
 
 any serious error that on the average the expenditure of 
 energy on a molecule during its motion of translation is 
 proportional to the distance traversed, and the force acting 
 equal to the foregoing value. Therefore, when a molecule 
 traverses a distance equal to half the distance of separation 
 of the molecules it may on the average gain or lose the 
 
 amount of energy -ra a p , which corresponds to a change 
 Z dp 
 
 in velocity in cms. equal to A/3p-r-. For example, in the 
 
 casfe of ether at C, P = .7362, aid g. 2.72*118X10* 
 
 dp .UU14 
 
 corresponding to a change of 10 C., which gives a change in 
 velocity equal to 1.69 X10 5 cms. /sec. of the molecule. The 
 average minimum velocity of a molecule, which corresponds 
 to that of a molecule in the gaseous state at the same tem- 
 perature, is equal to 3.03 XlO 4 cms. /sec. We see therefore 
 that the total average velocity of a molecule in a liquid may 
 be several times that corresponding to its temperature. 
 
56 THE EFFECT OF THE MOLECULAR FORCES 
 
 It will be convenient now to connect this velocity by a 
 general equation with a quantity which has already been 
 discussed in connection with gases. 
 
 18. The Number of Molecules crossing a Square 
 Cm. in a Substance of any Density from one Side to 
 the Other per Second. 
 
 If the molecules consisted in three equal streams moving 
 at right angles to each other, and a plane one square cm. 
 in area were taken at right angles to one of the streams, 
 N c V t (2 X 3) molecules would cross it in one direction per sec- 
 ond, and the same number in the opposite direction, where 
 N c denotes the number of molecules per cubic cm., and V t 
 the total average velocity. This expression may be obtained 
 and along the same lines as the similar expression in equa- 
 tion (17). If, however, the molecules move in all directions, 
 as is actually the case, the number of molecules n crossing 
 a square cm. from one side to the other in all directions is 
 double the above number, that is 
 
 N c V t 
 rc = ^-. ...... (35) 
 
 This may be proved by means of the geometrical considera- 
 tions used in Section 11. The result may, however, be at 
 once deduced from the results obtained in that Section. 
 Thus it was shown that the pressure of a gas may be written 
 
 A 
 
 where n denotes, if the molecules consist of three equal 
 streams at right angles, the number crossing a square cm. 
 taken at right angles to one of the streams, and n denotes 
 the number of molecules when they move in all directions, 
 as is the case in practice. Hence it follows that n = 
 
THE INTERNAL MOLECULAR VOLUME OF A GAS 57 
 
 the result just used, which obviously does not hold only for 
 a gas, but in general. 
 
 A molecule probably does not consist of a point in space, 
 but possesses a real or apparent molecular volume, which 
 influences its motion and consequently the properties of the 
 substance of which it forms part. This effect will be first 
 considered in connection with a gas. 
 
 19. The Effect of Molecular Volume on p and 
 n in the Case of an Imperfect Gas. 
 
 We have seen in Section 6 that the effect of the volume 
 of the molecules of a substance is to increase its external 
 pressure from that it would have if the molecules possessed 
 no volume. The effect of a molecular volume is evidently 
 to decrease the space available for molecular motion of 
 translation. It is therefore equivalent to supposing that 
 the molecules are devoid of volume, and that the volume 
 of the substance (as a whole) is decreased by an amount 
 b, as is shown by means of a diagram in Fig. 4. This quan- 
 tity is the apparent volume of the molecules connected with 
 molecular motion. The external pressure p in this case 
 might thus be written 
 
 in conformity with the equation p = RT/v of a perfect gas, 
 where v refers to a gram molecule of molecules. 
 
 It will not be difficult to see that this result will also 
 hold if the apparent volume of the molecules is caused by 
 molecular forces of repulsion, which do not permit an ap- 
 proach of two molecules beyond a certain distance. 
 
 If the molecules of a gas consist of a number of hard 
 elastic spheres not surrounded by fields of force, their vol- 
 ume would not interfere with their velocity of translation, 
 
53 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 which would simply correspond to the temperature of the 
 gas. This will be made clear by an inspection of Fig. 4 
 according to which a set of molecules possessing molecular 
 volume of the foregoing kind move in the space v with the 
 same velocity as if they were devoid of volume and moved 
 in the space v b. Equality of velocities is necessary in 
 the two cases since the external pressures are the same. 
 Since the molecular forces may give rise to an apparent 
 volume of the molecules as well as affect their velocity of 
 
 FIG. 4. 
 
 translation, the foregoing considerations suggest a defini- 
 tion of molecular volume which is quite definite and applies 
 to all states of matter. Thus the apparent molecular volume 
 of a substance may be defined as the quantity associated 
 with the molecules whose magnitude affects the external 
 pressure of the substance at constant -temperature and vol- 
 ume, but does not affect the average velocity of translation 
 of the molecules. The latter property is expressed by the 
 equation 
 

 THE MOLECULAR VOLUME AND VELOCITY 59 
 
 where V t denotes the average velocity of translation, and 
 b the apparent volume of the molecules. The velocity 
 V t is a function of the temperature and therefore 
 
 5VA /SVA / 56 \ /SFA _ /SVA 
 P/i \8b/T, v (dT/^ \5T)6.,~\8T)* t ,' 
 
 by the preceding equation. It is also a function of the 
 volume and therefore 
 
 /5VA /5FA /6\ /SFA = /*FA 
 V to /r \dbJ T ,v \dv/ T \5v J T ,b \dv J T t i>' 
 
 These equations express the relations between V t and 6 
 according to the above definition of b. 
 
 It should be carefully noted that 6 is a perfectly definite 
 mathematical quantity, though there might be some diffi- 
 culties in defining the exact nature and geometrical con- 
 figuration of the real, or apparent, volume of a single mole- 
 cule. 
 
 The apparent volume of a molecule, whatever its cause, 
 is likely to change little if at all with the density of the 
 substance. The value of 6 may therefore without the risk 
 of introducing any serious error be taken constant over small 
 changes in density. This is very useful in the determina- 
 tion of the values of 6 from simultaneous equations, as is 
 carried out in Section 29. 
 
 The value of 6 would vary with the temperature if caused 
 by forces of repulsion between the molecules. Thus two 
 molecules moving towards each other in a substance along 
 the same straight line continue approaching each other 
 until their kinetic energy of translation is completely trans- 
 formed into potential energy of repulsion, after which they 
 retrace their path. Hence a nearer approach must take 
 place with an increase of temperature, which corresponds 
 to an increase in kinetic energy, and a corresponding decrease 
 in the value of b would result. 
 
60 THE EFFECT OF THE MOLECULAR FORCES 
 
 It will be evident on reflection that the forces of attrac- 
 tion and repulsion between two molecules might be changed 
 at constant temperature and volume so that V t remains 
 the same, in which case a change in b may result, or so changed 
 that 6 remains the same, in which case a change in V t may 
 result. This shows that the definition of b given is admissible. 
 
 A change in temperature or density of a substance may 
 evidently result in a change of both b and V t . The fore- 
 going equations then express that whether or no b changes, 
 the changes in V t are the same as if b remained constant. 
 
 It can be shown that n, the number of molecules passing 
 through a square cm. in one direction per second, is inde- 
 pendent of the apparent volume of the molecules, since the 
 velocity of translation is also independent of this quantity. 
 Let us suppose that we are initially dealing with molecules 
 devoid of volume, and to which is then given an apparent 
 volume b. We may then suppose that the resultant mole- 
 cules possess no volume and occupy the volume v b of 
 the volume v for motion of translation. If N c denote the 
 number of molecules per cubic cm. when the gas occupies 
 the volume v, the number when it occupies the volume vb 
 is N c v/(vb). The number of molecules crossing a square 
 cm. per second is accordingly changed from n to nv/(v b), 
 since the velocity of translation remains unaltered, where 
 n corresponds to the molecules having no volume and 
 occupying as a whole the volume v. But in practice the 
 molecules with their apparent volumes would be distributed 
 throughout the volume v, instead of being separated from 
 their volumes as shown in Fig. 4, and the number of mole- 
 cules crossing a square cm. per second is therefore equal to 
 the foregoing expression reduced in the ratio of v b to v, 
 which makes it equal to n, the value when the molecules 
 have no volume. 
 
 This result can be very simply deduced from equation 
 (35). If the apparent volume b of the molecules is changed 
 

 THE MOLECULAR VOLUME AND GAS PRESSURE 61 
 
 without changing V t , by definition n remains constant 
 according to this equation, and it is thus independent of 
 the volume of the molecules. This method of obtaining 
 the result is perhaps not so instructive as the preceding 
 one. The result is mathematically expressed by the equation 
 
 () =- 
 
 \db/ T,V 
 
 By means of this equation and the Differential Calculus 
 it can be shown, similarly as in connection with the quan- 
 tity V t) that 
 
 5n\ /5n\ /dn\ S5n 
 
 and 
 
 These equations express the relations between n and 6 
 according to its definition. 
 
 If the molecules of a gas were devoid of volume and molec- 
 ular forces the laws of a perfect gas apply, and the external 
 pressure p of the gas may then be written 
 
 according to equation (20) in Section 11. 
 
 The acquisition of an apparent volume b by the mole- 
 cules does not change n, we have seen, but it changes the 
 external pressure in the ratio of v to v b. Hence the external 
 pressure of a gas under these conditions may be written 
 
 Since n is defined and used in connection with the pres- 
 sure produced by the molecules of a substance, the crossing 
 of a plane by a molecule, if the molecules are devoid of 
 volume, is obviously associated with the crossing of its 
 
62 THE EFFECT OF THE MOLECULAR FORCES 
 
 center of mass. This also holds when the molecules possess 
 volume, according to the method the subject has been 
 developed. 
 
 The external pressure of a gas is equal to the expansion 
 pressure due to the motion of translation of the molecules 
 tending to expand the gas. But in the case of a liquid or 
 dense gas this does not hold, due to the effect of the forces 
 of attraction and repulsion between the molecules, and the 
 expansion pressure of a dense substance is therefore a quan- 
 tity requiring special consideration. 
 
 0. The Expansion Pressure exerted by the Mole- 
 cules of a Substance of any Density. 
 
 The pressure exerted by a gas in a closed vessel upon its 
 walls is, according to the Kinetic Theory of Gases, due to the 
 change in momentum the molecules undergo through col- 
 lision with the vessel's walls. The velocity of translation 
 of the molecules, the density and external pressure of the 
 gas, are then connected by equation (7) in Section 6. This 
 equation holds, however, strictly only if the walls of the 
 vessel do not exert any attraction upon the molecules. It 
 holds very approximately in the cases usually occurring in 
 practice, in which the size of the vessel is so large that the 
 greater mass of the gas is only slightly under the influence 
 of the attraction of the walls. But under certain conditions 
 this influence may be so large that the equation cited does 
 not hold. It will be of importance to investigate this effect 
 of molecular forces more closely, as it has a bearing on the 
 kinetic properties of liquids. Thus let AB and CD, in Fig. 
 5, represent two opposite walls of a vessel, and let the planes 
 A'B' and C'D' denote the boundaries of the zones in which 
 the attraction of the walls is of appreciable magnitude. 
 A molecule on entering one of the zones has its velocity 
 increased through the attraction of the material of tt$ 
 
THE EXPANSION PRESSURE EQUATION 
 
 63 
 
 adjacent wall. Thus the molecule would exert a pull upon 
 the wall during the whole time it is in the zone. Now it 
 follows from the dynamical equation Ft = m a V that the aver- 
 age pull per unit time exerted by the molecule during the 
 time it is in the zone, is balanced by the average thrust 
 acting in the opposite direction upon the wall during that 
 period, due to the reversal of the direction of the additional 
 momentum acquired by the molecule in the zone. The 
 force exerted by the molecule upon the wall during one 
 rebound is thus not affected by molecular attraction. The 
 
 B 
 
 FIG. 5. 
 
 total pressure exerted by the molecules of the vessel on the 
 wall would therefore depend only on the number impinging 
 per square cm. per second on the wall. This number is 
 equal to the number of molecules entering a zone per square 
 cm. per second. It evidently depends upon the velocity 
 of translation of the molecules outside the zones, and the 
 number of molecules per cubic cm. in that region. If the 
 thickness of each zone is small in comparison with the 
 diameter of the vessel, the average velocity of translation 
 of the molecules may be taken equal to their velocity out- 
 side their zones. 
 
 If, however, the thickness of the zone is comparable 
 with the diameter of the vessel this no longer holds. Thus 
 
64 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 suppose that the walls AB and CD are so close to one another 
 that the zones of attraction of the walls overlap as is shown 
 in Fig. 6. The attraction exerted by one of the walls would 
 be neutralized by that of the other wall in a plane EF which 
 lies midway between the walls. The velocity of a molecule 
 in that plane would therefore be the same as that it would 
 have between the planes A'B' and C'D' in Fig. 5. For 
 according to Section 16 the velocity of translation of a mole- 
 cule at a point where the various forces neutralize each 
 
 other, is, at constant tempera- 
 ture, the same as its velocity 
 when under the action of no 
 force. The velocity for all points 
 outside of the plane EF, would, 
 however, be considerably greater 
 than that in the plane. The 
 latter velocity would therefore 
 be considerably smaller than 
 the total average velocity. As 
 before the attraction of the 
 walls would not affect the force 
 exerted by a molecule corresponding to a single rebound. 
 But it would evidently affect the number of times a 
 molecule crosses from one wall to the other per second. 
 Therefore the pressure the molecules would exert upon the 
 walls AB and CD under the conditions shown in Fig. 6 
 would indirectly be increased by the molecular attraction 
 of the walls. The pressure per square cm. on each wall 
 would accordingly be equal to nA/2, where n denotes the 
 number of molecules crossing a square cm. per second of 
 the plane EF from one side to the other this quantity 
 being affected by the attraction of the walls on the mole- 
 cules, and A/2 denotes the force exerted by a single mole- 
 cule on rebounding from one of the walls, this quantity 
 having the same value as when the walls possess no attrac- 
 
 i 
 i 
 i 
 
 F 
 
 FIG. 6. 
 
THE EXPANSION PRESSURE EQUATION 65 
 
 tion, its value having been determined in Section 11. A 
 similar state of affairs exists in a liquid, or in a substance 
 which does not behave as a perfect gas, the attraction be- 
 tween the molecules taking the place of the attraction of 
 the foregoing walls. 
 
 These deductions will now be used to find an expression 
 for the expansion pressure P e of a substance tending to 
 expand it, due to the motion. of translation of the mole- 
 cules. The number of molecules n crossing a plane one 
 square cm. in area in all directions from one side to the 
 other, is equal to the number crossing in the opposite direc- 
 tion. We may therefore suppose, if the molecules possess 
 no volume, that each stream of molecules is reflected from 
 the foregoing plane and thus exerts a pressure upon it. 
 Hence the expansion pressure P e would be equal to the 
 number of molecules crossing the plane in one direction 
 multiplied by the factor A/2, which expresses the force 
 exerted by a single molecule, which factor is the same as 
 applies in the case of a perfect gas according to the pre- 
 ceding investigation, that is, under these conditions, 
 
 But if each molecule has an apparent volume it may exert 
 a pressure across the plane without crossing it, and the fore- 
 going formula has to be modified accordingly. We have 
 seen in Section 19 that the molecular volume does not affect 
 the value of n, but increases the expansion pressure in the 
 ratio of v to v b. Hence in general the expansion pressure 
 is given by 
 
 4 = n -^2.543 X W- 20 \/Tm, (38) 
 
 v-b 2 
 where v denotes the volume of a gram molecule, and b 
 
66 THE EFFECT OF THE MOLECULAR FORCES 
 
 the molecular volume, the value of A being given by 
 equation (19). This equation may be written 
 
 (39) 
 
 by means of equation (35), where V t denotes the total 
 average velocity of a molecule, and N c the molecular con- 
 
 centration. Since V= J , and ^ = m a j (Sec- 
 \ m 2 \ m 
 
 tions 6 and 11), the foregoing equation may be written by 
 the help of the equation vN c m a = m, 
 
 where V denotes the average kinetic energy velocity of a 
 molecule, and b is taken to refer to a gram molecule 01 
 molecules. 
 
 The foregoing equations are not corrected for the dis- 
 tribution of molecular velocities in the gaseous state, since 
 the function A in equation (38) , from which the other equa- 
 tions are derived, has not been thus corrected. If this cor- 
 rection is carried out according to Maxwell's law the right 
 
 hand side of each equation has to be multiplied by ( + 1 j, 
 
 or 1.085, according to Section 11. It may be noted that if 
 the kinetic energy velocity V of a molecule in the gaseous 
 state is eliminated from equation (40) by means of the 
 equation F a =.9227 given in Section 8, and the equation 
 is corrected for Maxwell's law, an equation is obtained in 
 which V is replaced by V a the total average velocity of a 
 molecule in the gaseous state. The equation in this form 
 is independent of the distribution of molecular velocities 
 when applied to the gaseous state, since then V t = V a . 
 If a substance behaved as a perfect gas at all densities 
 
THE EXPANSION PRESSURE OF MIXTURES 67 
 
 we would have V t =V a and 6 = 0. The expansion pressure 
 is then equal to the external pressure, according to equa- 
 tion (40) in the corrected form where V a takes the place of 
 V, and the gas equation. But since this is not the case 
 in practice, for Vt>V a according to Section 17, and b is 
 not zero according to Section 29, the expansion pressure 
 is greater than the pressure p = RT/v. Hence there must 
 exist a negative pressure acting in the opposite direction 
 to the external pressure to ensure equilibrium of the sub- 
 stance, since in practice the external pressure is usually 
 less than RT/v. This negative pressure is called the intrin- 
 sic pressure and is discussed in the next Section. 
 
 The foregoing results may be extended to a mixture of 
 molecules. Let us consider a mixture of two different mole- 
 cules e and r. Since the expansion pressure of a substance 
 is caused by the motion of translation of the molecules, 
 and each molecule produces pressure independently, the 
 total expansion pressure is equal to the sum of the expan- 
 sion pressures exerted by the molecules e and r. The expan- 
 sion pressure exerted by the molecules e is evidently equal to 
 
 V e 
 
 where n e denotes similarly as before the number of 
 molecules e crossing a square cm. from one side to the 
 other per second, m e denotes the relative molecular weight 
 of a molecule, v e may be taken to refer to the volume of the 
 mixture containing a gram molecule of molecules e, and 
 b' e denotes the apparent volume in the volume v c which 
 the molecules e and r appear to possess in obstructing the 
 motion of a molecule e. Similarly the expansion pressure 
 exerted by the molecules r is equal to 
 
 n r jT-2.543 X 10- 20 VTm r , 
 
68 THE EFFECT OF THE MOLECULAR FORCES 
 
 where the symbols, v r , n r , m r , and b' r , have meanings similar 
 to the symbols, v e , n e , m e , and b' e . The total expansion pres- 
 sure is therefore given by 
 
 (41) 
 
 and its value in the case of a mixture of any number of 
 different kinds of molecules is therefore given by 
 
 . . (42) 
 
 Similarly corresponding to equations (39) and (40) we have 
 
 . . . (43) 
 
 and 
 
 " ...... <> 
 
 for any number of constituents in the mixture. The fore- 
 going equations may be corrected for Maxwell's distri- 
 bution law similarly as before. 
 
 Approximate values of b' e and b' T in equation (41) may 
 be obtained from the values of b e and b r referring to the 
 constituents in the pure state. The value of b' e arises from 
 the interaction of molecules e with each other and with 
 the molecules r, while the value of b' r arises from the inter- 
 action of the molecules r with each other and with the mole- 
 cules e. It is evident, therefore, that if N r denotes the 
 concentration of the molecules r, and v e the volume of the 
 mixture containing a gram molecule of molecules e, we have 
 approximately 
 
 where N denotes the number of molecules in a gram mole- 
 cule of a pure substance. The first term on the right-hand 
 
THE MOLECULAR VOLUME OF MIXTURES 
 
 69 
 
 side of the equation expresses the apparent molecular 
 volume of the molecules e when a molecule e interacts with 
 them, while the second term expresses the apparent volume 
 of the molecules r when a molecule e interacts with them. 
 Similarly for b' r we have approximately 
 
 "N 
 
 I 
 
 where N e denotes the number of molecules e in a volume 
 of the mixture containing a gram molecule of molecules r. 
 
 The expansion pressure P e may now be connected with 
 other quantities. 
 
 21. The Intrinsic Pressure; and the Equation of 
 Equilibrium of a Substance. 
 
 On account of the attraction between the molecules of 
 a liquid, or dense gas, a negative pressure is associated 
 with it usually called the intrinsic pressure, which acts in 
 
 B 
 
 FIG. 7. 
 
 the opposite direction to the external pressure regarded 
 from an external point. The existence of this pressure 
 follows from the following consideration. Suppose a sub- 
 stance is cut into two parts A and B by an imaginary 
 plane ab as shown in Fig. 7. If attraction between the 
 molecules exists the parts A and B exert an attraction 
 
70 THE EFFECT OF THE MOLECULAR FORCES 
 
 upon each other which gives rise to a pressure between 
 them. This pressure per square cm. in the plane ab is 
 numerically equal to the intrinsic pressure, or negative 
 pressure which the parts A and B exert upon each other. 
 
 The expansion pressure P e exerted by the molecules in 
 crossing a square cm. in the plane ab is evidently balanced 
 by the intrinsic pressure, which will be denoted by P, 
 and the external pressure p, that is, 
 
 Pn+P = Pe ....... (45) 
 
 or 
 
 according to equation (38). The foregoing equation* 
 is the equation of equilibrium of a substance. The quan- 
 tity P n may be expressed very approximately in terms of 
 quantities which may be measured directly, as will now 
 be shown. 
 
 The average position of a molecule in a substance cor- 
 responds to a point about which the surrounding mole- 
 cules are symmetrically situated. The molecular forces 
 therefore neutralize each other at such a point. The aver- 
 age kinetic energy of a molecule at its average position is 
 therefore the same as that it has in the perfectly gaseous 
 state according to Section 16. Now we may suppose that 
 each molecule occupies its average position in a substance 
 at the same time for purposes of calculation, for the prop- 
 erties of a molecule in connection with its forming part of 
 a substance are average properties. It appears then that 
 if the volume of a substance is changed by 8v at constant 
 temperature, the average minimum or temperature kinetic 
 energy is not changed. Hence the heat 5U absorbed repre- 
 
 * Reconstructed form of an equation given by the writer in the 
 Phil. Mag., July, 1912, pp. 101-108. 
 
THE INTRINSIC PRESSURE EQUATION 71 
 
 sents, if np dissociation occurs as we will suppose, the 
 work dU a done against molecular attraction in separating 
 the molecules, and in changing their internal molecular 
 energy by 8u m . The change in energy du m may be caused 
 by a change in the atomic configuration of the molecules, 
 and a change in their velocities of rotation, etc. The latter 
 energy is very likely small in comparison with the former, 
 in which case we have 
 
 P n -5v=dU a =dU, 
 
 since work is done against the intrinsic pressure during 
 expansion. Now according to thermodynamics we have 
 
 where p and T denote the pressure and absolute tempera- 
 ture of the substance, and hence 
 
 '-T-* ; {48) 
 
 since by the help of the Differential Calculus 
 
 /dp\ fdv\ //dv\ l/5v\ l/dv\ 
 
 (ay.- - (*f),/ y r' a - w;, ^ = "fe)- 
 
 where a denotes the coefficient of expansion at constant 
 pressure, and the coefficient of compression at constant 
 temperature of the substance. These coefficients can be 
 measured directly and hence numerical values of P n be ob- 
 tained. The values obtained in this way are likely to be 
 very approximately correct, since the dissociation of the 
 molecules with the expansion of a substance is usually 
 negligible, and the change in internal molecular energy we 
 would expect to be small in comparison with the work 
 
72 THE EFFECT OF THE MOLECULAR FORCES 
 
 done against molecular attraction, i.e., against the intrinsic 
 pressure. 
 
 Table III gives the values of the intrinsic pressure at 
 different temperatures for a few liquids calculated* by 
 
 TABLE III 
 
 ETHER, (C 4 H 10 O). 
 
 
 
 
 
 TCL 
 
 
 tc 
 
 010 
 
 or 
 
 P 
 
 r n =-Q- 
 
 p'atmos. 
 
 
 
 
 
 atmos. 
 
 
 13.5 
 
 169 
 
 .001574 
 
 .7214 
 
 2669 
 
 232.3 
 
 25.4 
 
 190 
 
 .001632 
 
 .7077 
 
 2467 
 
 237.3 
 
 63 
 
 300 
 
 .001809 
 
 .6620 
 
 2026 
 
 250.0 
 
 78.5 
 
 367 
 
 .001892 
 
 .6421 
 
 1812 
 
 253.6 
 
 99 
 
 539 
 
 .001992 
 
 .6105 
 
 1375 
 
 255.2 
 
 BENZENE, (C 6 H 6 ). 
 
 15.4 
 
 87 
 
 .001215 
 
 .8840 
 
 4083 
 
 271.8 
 
 50.1 
 
 111 
 
 .001305 
 
 .8466 
 
 3796 
 
 291.6 
 
 78.8 
 
 126 
 
 .001379 
 
 .8145 
 
 3850 
 
 305.5 
 
 CHLOROFORM, (CHC1 3 ). 
 
 
 
 101 
 
 .001107 
 
 1.5264 
 
 2991 
 
 289.8 
 
 20 
 
 128 
 
 .001294 
 
 1.4885 
 
 2970 
 
 303.7 
 
 40 
 
 162 
 
 .001484 
 
 1.4503 
 
 2869 
 
 315.8 
 
 60 
 
 204 
 
 .001670 
 
 1.4108 
 
 2726 
 
 326.9 
 
 PENTANE, (C 5 Hi 2 ). 
 
 
 
 229 
 
 .001465 
 
 .6454 
 
 1747 
 
 203.5 
 
 20 
 
 318 
 
 .001589 
 
 .6262 
 
 1337 
 
 211.9 
 
 40 
 
 416 
 
 .001721 
 
 .6062 
 
 1294 
 
 219.2 
 
 60 
 
 486 
 
 .001830 
 
 .5850 
 
 1260 
 
 225.0 
 
 * R. D. Kleeman, Proc. Camb. Phil Soc., Vol. XVI, Pt. 6 (1912), 
 p. 545. 
 
THE INTRINSIC PRESSURE EQUATION 73 
 
 means of the foregoing formula, and for comparison the 
 
 r>/77 
 
 values of the external pressures p' given by p' ' which 
 
 Tfl 
 
 the substances would have if they behaved as perfect gases. 
 The values of the coefficient /3 in the Table refer to the 
 compression per atmosphere, and hence the values of P n are 
 expressed in terms of the same quantity. The values of p' 
 have therefore also been expressed in terms of this quantity 
 by dividing the value of R in the gas equation (Section 5) 
 by 10 6 . The values of a, /3, and p, were taken from Landolt 
 and Bernstein's Tables, the values of p being unnecessary, 
 
 since they are so small in comparison with those of T- 
 
 p 
 
 that they may be neglected. The values of P n are striking 
 on account of their magnitude, being much greater than 
 p', and illustrate the powerful attraction that exists between 
 molecules for close distances of approach. Since the attrac- 
 tion of a molecule probably increases with its mass, the 
 magnitude of the intrinsic pressure for constant volume 
 probably varies in a similar way. It would evidently de- 
 crease with the density p of the substance since the attrac- 
 tion between two molecules decreases with their distance of 
 separation. 
 
 The intrinsic pressure is evidently greater than the 
 external pressure, and this holds therefore also for the 
 expansion pressure P e according to equation (45), or P n >p 
 and P e >p. Also according to this equation P e >P n , and 
 since P n >p' according to the Table, we also have P e >p'. 
 The result that the expansion pressure P e of a liquid is 
 greater than the external pressure p' it would have if it 
 behaved as a perfect gas is caused according to Section 20 
 by the molecules possessing an apparent volume, and that 
 the total average velocity of a molecule is greater than the 
 velocity it would have in the perfectly gaseous state, or 
 that 6 >0 and V t >V a . 
 
74 THE EFFECT OF THE MOLECULAR FORCES 
 
 If the equation of state (Section 26) of a substance be 
 known the values of a and )8 may be calculated for different 
 temperatures and densities of a substance by means of the 
 equations defining these quantities, or better, the value of 
 
 ( -=, ) may be directly calculated and hence the correspond- 
 
 \0l/t 
 
 ing intrinsic pressure be obtained. 
 
 It is possible now to obtain superior and inferior limits 
 of the values of n and V t of a substance. 
 
 22. Superior and Inferior Limits of n of a 
 Substance. 
 
 It will appear from an inspection of equation (46) that 
 a superior limit of n, the number of molecules crossing a 
 square cm. from one side to the other per second, which 
 will be written n", is obtained by supposing that the appar- 
 ent volume of the molecules is zero, or 6 = 0, in which case 
 the equation becomes 
 
 = 2.543X10-^ 
 
 The quantity P n on the right hand side of this equation, 
 we have seen, can be calculated from quantities determined 
 by experiment. 
 
 If P n is zero, V t has the same value as if the substance 
 were in the perfectly gaseous state, since it is independent 
 of the apparent molecular volume (Section 19). Hence 
 according to equations (35) and (8), the value of n under 
 these conditions, which will be written n', is given by 
 
 (50, 
 
 This is an inferior limit of n since V t cannot have a smaller 
 value than corresponding to the perfectly gaseous state, 
 
THE INTRINSIC PRESSURE EQUATION 
 
 75 
 
 Table IV contains as an illustration the superior and 
 inferior limits of n for a few substances in the liquid state*. 
 The values of P n used in the calculations are given in the 
 Table III. 
 
 TABLE IV 
 
 ETHER 
 
 PENTANE 
 
 *C. 
 
 Sup. Lim. 
 of n ID- 2 *. 
 
 Inf. Lim. 
 of n 10- 2 5. 
 
 t C. 
 
 Sup. Lim. 
 of n ID- 2 *. 
 
 Inf. Lim. 
 of n 10-25. 
 
 13.5 
 
 78.5 
 
 71 
 44 
 
 6.24 
 6.16 
 
 
 60 
 
 49 
 
 32 
 
 5.23 
 5.68 
 
 CHLOROFORM 
 
 BENZENE 
 
 
 60 
 
 65 
 54 
 
 6.30 
 6.50 
 
 15.4 
 
 78.8 
 
 108 
 94 
 
 7.0 
 7.12 
 
 It will be seen that in the case of each substance one of 
 the limits is many times that of the other, the actual value 
 of n of course lying between the two. Since the values of 
 P n are large in comparison with the values of p' (the external 
 pressure a substance would have in the perfectly gaseous 
 state (Section 21)), and the former are the outcome of molec- 
 ular attraction which in proportion effects the velocities 
 of translation of the molecules, and therefore the values of 
 n, we would expect the actual values of n to lie nearer to 
 the superior than to the inferior limits obtained. 
 
 23. Superior and Inferior Limits of V t of a 
 Substance. 
 
 These limits are obtained by substituting in equation 
 (35) the superior and inferior limits of n obtained by the 
 
 * If the values of n are corrected according to Maxwell's law they 
 have to be divided by 1.085 according to Section 20. 
 
76 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 method given in the previous Section. Table V gives the 
 values obtained in this way for a few substances. For 
 reasons stated in the previous Section we would expect 
 that the actual total average velocity of a molecule in a 
 substance sliould lie nearer '/o the superior than to the inferior 
 limit. The inferior limit is the velocity when the substance 
 is in the perfectly gaseous state. 
 
 TABLE V 
 
 ETHER 
 
 PENTANE 
 
 tC. 
 
 Sup. Lim. 
 
 w^: 
 
 Inf. Lim. 
 
 - v > w ': 
 
 tc. 
 
 Sup. Lim. 
 
 of F, 10* 5' 
 
 1 sec. 
 
 Inf. Lim. 
 
 of F 104^ 
 1 sec. 
 
 13.5 
 
 35.7 
 
 3.10 
 
 
 
 26.4 
 
 2.83 
 
 CHLOROFORM 
 
 . BENZENE 
 
 
 
 24.6 
 
 2.38 
 
 15.4 
 
 46.0 
 
 2.98 
 
 The quantities V t and n do not depend on the quantity 
 b by definition according to Section 19. Therefore if P n 
 and p were also independent of 6, equation (49) would give 
 the actual value of n instead of a superior limit, while equa- 
 tion (35) would give the actual value of V t . But p obviously 
 depends on b (Section 19) ; it is, however, small in comparison 
 with P n in the cases considered. Therefore if b did not depend 
 on the molecular forces P n would be independent of 6, 
 and the superior limits of n and V t obtained would very 
 approximately represent the actual values of these quan- 
 tities. There is some indirect evidence, however, with 
 which we will get acquainted as we proceed, that b is the 
 outcome of molecular forces, as we would expect. 
 
 The mathematical definition of b was given in Section 
 19, but so far no information about its value and properties 
 
THE CAUSE OF INTERNAL MOLECULAR VOLUME 77 
 
 has been discussed. It will be convenient now to discuss 
 the quantity from a certain aspect. 
 
 24. The Real and Apparent Volumes of a 
 Molecule, and their Superior Limits. 
 
 The property of volume of a substance is known to us 
 only 'through the resistance to force: thus if two portions 
 of matter are pressed together there is a resistance to the 
 operation which need not be accounted for by actual con- 
 tact taking place whatever that may mean, but by an 
 approach of the molecules of the substances to such dis- 
 tances that the resultant of their forces of repulsion equals 
 the force with which the substances are pressed together. 
 It seems unsafe therefore, and little warranted by the facts, 
 to associate with an atom an impenetrable perfectly 
 elastic volume of constant magnitude. There is no doubt, 
 though, that it may be said that a volume is associated 
 with each molecule through which the center of another 
 molecule cannot pass, but whose magnitude depends upon 
 external conditions. If this volume is due to the existence 
 of forces of repulsion, as is highly probable, its magnitude 
 will depend upon the force exerted in approaching another 
 molecule. Thus it is evident that in the case of a gas this 
 volume will decrease with increase of temperature, for this 
 is attended by an increase in the kinetic energy of the mole- 
 cules, and therefore by a decrease in the minimum distance 
 of approach, the molecules approaching each other to a 
 distance at which their kinetic energy is completely or 
 partly converted into potential energy of repulsion. It 
 seems futile, therefore, to expect to obtain a constant value 
 for the quantity in question since it varies with the external 
 conditions, which often are not altogether known. The 
 best we can hope to obtain is the order of magnitude of the 
 quantity, which is likely to be always the same. 
 
78 THE EFFECT OF THE MOLECULAR FORCES 
 
 The apparent volume of a molecule, or of a gram mole- 
 cule of molecules, which was investigated in Section 19, 
 is, however, a perfectly definite mathematical quantity. 
 But its connection with the "real volume," whatever that 
 may mean, cannot be ascertained with certainty, if at all, 
 since the apparent volume would depend on the geometrical 
 configuration of the "real volume." Thus for example, 
 Van der Waals has shown that if the "real volume" of a 
 molecule consists of an impenetrable sphere, the apparent 
 volume is four times that of the "real volume." Now if 
 the apparent volume is caused by forces of repulsion between 
 the molecules, it is difficult, if not impossible, to define exactly 
 what is meant by the "real volume," and therefore to deter- 
 mine its magnitude. It will be evident on reflection that 
 the "real volume" of a molecule is not exactly the volume 
 through which another molecule will never pass. 
 
 The volume occupied by a molecule, or associated with 
 a molecule, at absolute zero, is of special interest in this 
 connection. Since at that temperature the molecules 
 have no motion of translation each will take up a position 
 so that the forces of attraction and repulsion acting upon 
 it balance each other, in other words, since P e and p are 
 zero P n is zero according to equation (45). The apparent 
 and "real volume" are evidently equal to each other under 
 these conditions. The apparent volume of a molecule at 
 the absolute zero is likely to be greater than at a higher 
 temperature, since the velocity of the molecules due to 
 their temperature is likely to make the molecules approach 
 closer to each other in opposition to their repulsion (Sec- 
 tion 15) than they would otherwise. This volume is thus 
 a superior limit of the apparent molecular volume for higher 
 temperatures than the absolute zero. An approximate 
 value of this volume at the absolute zero may be obtained 
 from considerations involving the coefficient of expansion, 
 or by means of Cailletet and Mathias' linear diameter 
 
THE MOLECULAR VOLUME AT THE ABS. ZERO 79 
 
 law. According to this law the densities pi and p2 of a liquid 
 and its saturated vapor respectively are given by the equa- 
 tion 
 
 where T denotes the absolute temperature, and a and c 
 denote constants depending only on the nature of the 
 liquid. These constants may be determined from values 
 of pi and P2 corresponding to two different temperatures 
 of the liquid. At the absolute zero, or T = 0, we accordingly 
 have pi = a, since p2 = and 7 7 = 0. As an illustration the 
 values of v of a gram molecule for a number of substances 
 at the absolute zero, or the values of ra/pi, and the cor- 
 responding values of V'Q and (Vo)^ for a single molecule, 
 are given in Table VI. It will be seen that the values of 
 
 TABLE VI 
 
 Substance. 
 
 V 
 c.c. 
 
 V 
 c.c. 
 
 i 
 
 v o 
 
 V io. 
 
 c.c. 
 
 cv>x 
 
 108. 
 
 cm. 
 
 Oxygen 
 
 20.8 
 
 49.2 
 
 2.37 
 
 33.5 
 
 3.22 
 
 Nitrogen 
 
 25.0 
 
 70 
 
 2.80 
 
 40.2 
 
 3.42 
 
 Carbon dioxide 
 Ether .... 
 
 25.5 
 71.7 
 
 96 
 280 
 
 3.77 
 3.91 
 
 41.1 
 115 
 
 3.44 
 
 4.87 
 
 Benzene 
 Carbon tetrachloride . . 
 Propyl acetate .... 
 
 70.6 
 72.2 
 
 86.2 
 
 256 
 276 
 345 
 
 3.63 
 
 3.82 
 4.00 
 
 114 
 
 116 
 139 
 
 4.84 
 
 4.88 
 5.18 
 
 
 
 
 
 
 
 V'Q increase with increase of molecular weight, as we would 
 expect. The values of (i/o)** probably give the order of mag- 
 nitude of the apparent diameter of a molecule under various 
 conditions. The Table also gives the values of the critical 
 volume v e , and the values of the ratio V C /VQ, which, it will 
 be seen, lie in the neighborhood of 4. The values of VQ 
 and v c were taken from a Table given in Nernst's Theo- 
 retische Chemie, 7th ed., p. 234. A superior limit of b is 
 thus v c /4. 
 
80 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 An important application of the foregoing results may 
 immediately be made. 
 
 25. Inferior Limits of n and V t . 
 
 The results of the preceding Section enable us to obtain 
 another inferior limit for n of a substance by means of 
 equation (46), on writing b = v c /4i, which is a superior limit 
 of 6. On substituting the inferior limit of n thus obtained 
 in equation (35) it gives an inferior limit of V t . Table VII 
 
 TABLE VII 
 
 cms. 
 CO 2 AT 40 C. F a = 4.165Xl0 4 - 
 
 Sec. 
 
 p in 
 atmos. 
 
 Inf. Lim. 
 of n 1Q26. 
 
 Inf. Lim. 
 
 of v t io< ?5- 
 1 sec. 
 
 V in 
 atmos. 
 
 Inf. Lim. 
 of n 10. 
 
 Inf. Lim. 
 
 of V t 10 C *L 
 sec. 
 
 70 
 
 3.88 
 
 4.60 
 
 94 
 
 14.4 
 
 5.94 
 
 80 
 
 5.76 
 
 4.79 
 
 100 
 
 16.4 
 
 6.24 
 
 85 
 
 8.01 
 
 5.06 
 
 112 
 
 18.7 
 
 6.63 
 
 contains the inferior limits for C02 at 40 C. calculated from 
 the values of P n +p obtained in Section 34 and given in 
 Table XIV. It will be seen that at constant temperature 
 the inferior limit of Vt, the total average molecular velocity, 
 gradually increases with increase of pressure or density of 
 the substance, and that its value corresponding to an external 
 pressure of 112 atmos. is about 50 per cent greater than the 
 values of V a , the total average velocity of a molecule of the 
 substance in the perfectly gaseous state at the same tempera- 
 ture*. This definitely shows that the total average molec- 
 
 * The values obtained for V t and V a have not been corrected for an 
 uneven distribution of molecular velocities. According to Maxwell's 
 Law of Distribution each value has to be divided by 1.085 according to 
 Sections 8, 11, and 20. 
 
DENSITY AND MOLECULAR VELOCITY 81 
 
 ular velocity at constant temperature in a substance not 
 obeying the laws of a perfect gas increases with the density, 
 and that it may be considerably larger than the velocity 
 when the substance is in the perfectly gaseous state. The 
 results thus fall into line with those of Sections 16 and 17, 
 and we will see later with those of Section 29. 
 
 From this it follows that n according to equation (35) 
 increases more rapidly with the density of a substance 
 than proportionally to it. 
 
 In the previous Sections we have mainly considered 
 individual properties of molecules. It will be of interest 
 and importance now to consider some properties of a sub- 
 stance as a whole which depend on the dynamical and 
 attractional properties of the molecules. 
 
 26. The Equation of State of a Substance. 
 
 The state of a given mass of a substance according to 
 experiment is completely defined by two variables. There- 
 fore each of the variables of a substance may be expressed 
 in terms of any two by means of an equation. Such a rela- 
 tion is called an equation of state of the substance, and 
 contains two independent variables. The most useful 
 equation is that connecting the variables pressure, volume, 
 and temperature. 
 
 One of the equations of state is 
 
 P e , ...... (51) 
 
 obtained in Section 21, where p, P n , and P e denote the 
 external, intrinsic, and expansion pressures respectfully 
 of the substance. This equation may also be written 
 
 = 2.534 X10- 20 \7X . . . (52) 
 
 according to equation (38), where n denotes the number of 
 molecules crossing a square cm. per second, and 6 the appar- 
 
82 THE EFFECT OF THE MOLECULAR FORCES 
 
 ent volume of the molecules. According to equation (40) 
 and subsequent remarks the preceding equation may also 
 be written 
 
 (53) 
 
 where V t denotes the total average velocity of a molecule 
 of a substance for any state, V a the average velocity in the 
 gaseous state, v the volume of a gram molecule, 6 the appar- 
 ent molecular volume, and R the gas constant. The fore- 
 going equations are fundamental in character, since they do 
 not depend upon any hypothesis. 
 
 If equation (52), (like (53)), is corrected according to 
 Maxwell's law of distribution of molecular velocities in the 
 gaseous state, the right-hand side of the equation has to 
 
 be multiplied by *-, or 1.085, according to Section 20. 
 
 The problem that remains to be solved in connection 
 with the foregoing equations is to find an expression for the 
 quantities, P e , n, V t , and 6, in terms of other quantities, 
 particularly in terms of the quantities p, v, and T, which 
 can be measured directly. This cannot be done without 
 the introduction of assumptions, and therefore yields results 
 which are likely to hold only approximately. 
 
 Van der Waals has proposed the equation of state 
 
 t & RT /*. *\ 
 
 p+- 2 = ...... (54) 
 
 where a and b are constants. On comparing this equation 
 with equation (53) we see that P n = a/v 2 , and V t =V a . 
 According to the latter equation the velocity of translation 
 of the molecules is not influenced by molecular attraction. 
 But this is inadmissible according to Sections 17, 20, and 
 25, nor does it hold approximately, for we will see in Section 
 
VAN DER WAALS' EQUATION OF STATE 83 
 
 29 that when a substance is in the liquid state the value of 
 V t may be more than four times the value of V a . Thus 
 the values of b calculated from Van der Waals' equation 
 would be too large. 
 
 Van der Waals deduced his equation from the supposi- 
 tions that the total average velocity of a molecule is not 
 affected by molecular attraction, or is the same as the 
 velocity in the gaseous state, and b represents the apparent 
 volume of a gram molecule of molecules as a whole due to 
 a true impenetrable spherical volume being associated 
 with each molecule. The value of b is then four times the 
 true volume occupied by the molecules. 
 
 The equation from Van der Waals as a whole, however, 
 fairly well represents the facts, that is, the values of each of 
 the quantities p, v, and T, may be obtained by means of 
 it given the corresponding values of the remaining two 
 quantities. The equation is particularly useful for such 
 determinations on account of its simplicity. 
 
 It can easily be shown that the intrinsic pressure term 
 a/v 2 in Van der Waals' equation corresponds to attraction 
 between two molecules varying inversely as the fourth power 
 of their distance of separation. Thus the molecules of a 
 substance may be divided into a number of parallel rows 
 separated from each other by a distance xi, the distance 
 of separation of the molecules. Therefore if A\/x s repre- 
 sents the law of molecular attraction, the attraction of one- 
 half of a row on the other half may evidently be written 
 A2/x s , where A 2 represents an appropriate constant. Since 
 l/x 2 rows pass through a square cm. situated at right angles 
 to the rows, the attraction of the molecules in one-half of a 
 cylinder of unit cross-section and infinite length on the 
 molecules in the other half is given by 
 
 A 2 
 
 x s+2> 
 
84 THE EFFECT OF THE MOLECULAR FORCES 
 
 and this is the intrinsic pressure. Now x\ ( ) = ( - ) , 
 
 \ P/ \ wi / 
 
 where p denotes the density, and v the volume of a gram 
 molecule of the substance, and thus the intrinsic pressure 
 may be written 
 
 4 
 
 A 3 p 3 , or - 
 
 s-f-2 
 where A 3 and A 4 are constants. Therefore when - = 2, 
 
 o 
 
 s = 4. 
 
 The effect of molecular attraction is, however, not so 
 simple that it may be expressed by a single term. According 
 to Section 14 it should contain at least two additional terms 
 having negative signs which express repulsion between the 
 molecules. The attraction terms are probably approxi- 
 mately represented by the foregoing single term involving 
 the inverse fourth power. The subject of molecular attrac- 
 tion and its bearing on the form of the intrinsic pressure 
 term is not of primary importance in connection with the 
 object of this book, and will therefore not be discussed 
 here any further but reserved for another place. For pur- 
 poses of calculating the values of one of the variables p, v, 
 and T, from given values of the other two variables most 
 of the empirical equations of state answer equally well. 
 An account of a number of them will be found in Winckel- 
 mann's Handbuch der Physik, IT. Edition, pp. 1135-1143. 
 
 Equation (53) may be given a form by means of which 
 the values of V t and P n of a substance may approximately 
 be calculated. The quantity V t may be taken equal to 
 the sum of two terms, one equal to V a , and the other equal 
 to Vi, which represents the effect of molecular attraction 
 on the value of V t . Now a molecule will pass over a distance 
 propprtjonal to x\ in passing out of the influence of one mole- 
 
A USEFUL FORM OF THE EQUATION OF STATE 85 
 
 cule and under the influence of another. Therefore if A\/x s 
 represents the law of molecular attraction the change of 
 kinetic energy of the molecule over that distance may be 
 
 written ~^ 1 -, and the velocity V\ is therefore approximately 
 
 given by 
 
 D 2 A? 
 
 where D$ is a constant. Equation (53) may accordingly 
 be written 
 
 A* Z) 3 RT . 
 
 ' ' ' ' (55) 
 
 o 
 
 where V a = \l7r\ according to Sections 6 and 8. 
 
 \OTT\ m 
 
 On applying this equation to three states of a substance not 
 differing much from one another at constant temperature 
 the quantities A*, Ds, and b, may be taken as constant and 
 determined from these equations. The values of V t and 
 P n corresponding to these three states may then at once be 
 calculated. Similarly the equation may be applied to another 
 set of three states and the corresponding values of V t , P n 
 and b, obtained, and so on. The values of V t and b obtained 
 in this way by means of equation (55) will obviously not 
 be so accurate as those obtained by means of equation (67) 
 in Section 29. The former equation can, however, be more 
 readily applied to the facts. The values of 6 obtained by 
 this equation should more accurately represent the facts 
 according to the definition of b in Section 19 than those 
 obtained by Van der Waals' equation, since Vt is not taken 
 equal to V a . If s is not known it may be determined by 
 applying equation (55) to four different states of the sub- 
 stance. 
 
86 THE EFFECT OF THE MOLECULAR FORCES 
 
 The quantities P e and P n in the equation of state must 
 very approximately satisfy the relation obtained on eliminat- 
 ing p from equations (51) and (48), which gives 
 
 It follows from this equation that if P n is a function of v 
 only P e is of the form T-4>(y), where (j>(v) is a function of v. 
 Van der Waals' equation of state satisfies the above condi- 
 tion, but of course it does not follow that therefore its form 
 is correct. It is fairly certain that P n is a function of T 
 as well as of v. For the intrinsic pressure of a substance, 
 since it depends upon molecular attraction, will depend upon 
 the change in atomic configuration of a molecule with change 
 in temperature. Since such a change undoubtedly occurs, 
 as is shown by the deviation of the ratio of the specific heats 
 considered in Section 13 from the value 1.66, the form of 
 the function for P e is not as simple as a linear function of T. 
 
 27. The Conditions that the Equation of State 
 has to satisfy. 
 
 If the pressure of a substance is plotted against its volume 
 at different temperatures, curves of the form shown in Fig. 
 8 are obtained. For certain temperatures each curve con- 
 sists of two parts, each of which has one of the terminal 
 points lying on a line (dotted in the figure) parallel to the 
 volume axis. Thus corresponding to the pressure indicated 
 by the dotted line the substance can only have either of 
 the two volumes corresponding to the abscissae of the 
 points at the extremities of the line. As the temperature 
 of the substance is increased the length of the dotted line 
 is decreased, and for a certain temperature, called the critical 
 temperature, it entirely disappears. 
 
THE CONTINUITY OF STATE 
 
 87 
 
 Van der Waals has proposed a theory to explain this 
 behavior of substances. According to it the two curves 
 corresponding to a given temperature are theoretically 
 parts of a single curve as shown in Fig. 9, the form of the 
 absent part being such as to render the curve continuous. 
 The fact that the intermediate part of the curve is not 
 realized in practice is explained by supposing that a sub- 
 
 FIG. 8. 
 
 stance in each of the states represented by it is in unstable 
 equilibrium, accordingly a disturbance from the outside 
 would make the substance change to one or both the stable 
 forms represented by the points a and 6 at the extremities 
 of this part of the curve. 
 
 Thus according to this theory there is no discontinuity 
 of state; and therefore the equation connecting p and v 
 
88 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 for the two stable portions of a curve will also represent the 
 unstable condition. 
 
 It will appear from an inspection of the curves in Fig. 9 
 that the point e, which corresponds to the critical point, 
 
 FIG. 9. 
 
 is a point of inflexion. The geometrical properties of the 
 point are expressed by the equations 
 
 and 
 
 ft] =0- 
 
 \ oV / T 
 
 to 2 
 
 (57) 
 (58) 
 
 which the equation of state has to satisfy at the critical 
 point. These equations express relations between the con- 
 stants of the equations of state and the critical constants. 
 
CONDITIONS INVOLVING THE ABSOLUTE ZERO 89 
 
 The foregoing equations, it may be mentioned, can also be 
 deduced from the Laws of Thermodynamics. 
 
 The equation of state must also possess the property 
 that it vanish if the substitutions p = 0, T = 0, and # = <, 
 are made, which correspond to the absolute zero of tem- 
 perature. This may indicate that a relation exists between 
 the coefficients of the equation, or that it has a certain form. 
 
 If the temperature and volume at constant pressure of 
 a substance are plotted against each other instead of the 
 pressure and volume at constant temperature, a set of 
 curves similar to those shown in Fig. 8 is obtained. A 
 portion of each curve cannot be realized in practice. The 
 theoretical curve representing this portion must for reasons 
 of continuity cut a volume ordinate in three points, similarly 
 as shown in Fig. 9. But this cannot hold at the absolute 
 zero, since the absolute temperature cannot have negative 
 values, and therefore no part of the curve can lie on the 
 negative side of the volume axis. The part of the curve 
 lying between the volumes VQ and infinity inclusive in that 
 case coincides with the volume axis. This condition is 
 expressed by the equation 
 
 which holds for v = infinity, v v , and inclusive values, 
 corresponding to p = 0, and T = Q. This probably indicates 
 that the equation of state must have a certain general form 
 which satisfies this condition. 
 
 Experiment shows that the variables pressure, volume, 
 and temperature, of a liquid in contact with its saturated 
 vapor, may each be expressed by an equation in terms of 
 one of the variables, and this holds also for the saturated 
 vapor. It can be shown that this is a consequence of Van 
 der Waals' theory of the continuity of state, and the Laws of 
 Thermodynamics. Thus it follows from thermodynamics 
 
90 THE EFFECT OF THE MOLECULAR FORCES 
 
 that the work done during an isothermal process between 
 given limits is independent of the path of the process. Hence 
 the work done in passing from the point a to the point b 
 in Fig. 9 is the same for the two paths indicated, one of which 
 corresponds to that obtained in practice with a substance, 
 while the other to that given by the equation of state. This 
 condition is expressed by the equation 
 
 -t>i) = I p-dv, .... (60) 
 
 where the suffixes 1 and 2 refer to the liquid and vaporous 
 states respectively. 
 
 On eliminating two of the variables pi, vi, V2, and T 
 from the foregoing equation and the two equations obtained 
 by applying the equation of state to the liquid and the 
 vaporous state of the substance, a number of equations are 
 obtained each of which contains two variables only. 
 
 Another set of equations each of which contains two varia- 
 bles only may be obtained as follows.* According to Clapey- 
 ron's equation the internal heat of evaporation L of a gram 
 molecule of a substance is given by 
 
 L=\T-j^-p\(v 2 -vi) (61) 
 
 Now according to equation (47) 
 
 and therefore Clapeyron's equation by the help of equa- 
 tion (60) may be written 
 
 dv = -j 7 j 1 (v2 vi) (62) 
 
 * R. D. Kleeman, Phil Mag., Sept., 1912, pp. 391-402. 
 
CONDITIONS INVOLVING THERMODYNAMICS 91 
 
 From this equation and the equation of state, a number 
 of equations may be obtained similarly, as before, each of 
 which contains two variables only. 
 
 Now if one of the variables is eliminated from two 
 equations taken from the foregoing two sets of equations 
 which contain the same two variables, an equation is ob- 
 tained containing one variable only. This equation must 
 evidently identically vanish with respect to this variable. 
 This requires that some of the constants in the equation 
 be equated to zero. Since these constants are functions 
 of the constants of the equation of state, the equations thus 
 obtained express relations between the latter constants. 
 The number of these equations evidently depends on the 
 form assumed for the equation of state. 
 
 It appears, therefore, that the equation of state has to 
 satisfy a number of conditions which govern its form and 
 the relation between its constants. Other equations of 
 condition besides those pointed out may of course exist, 
 which have not yet been discovered. 
 
 The equation of state possesses besides an important 
 property which indicates certain properties of its constants. 
 This property, which is discussed in the next Section, is 
 based upon the facts, but a theoretical reason may also be 
 found for it. 
 
 28. The Relation of Corresponding States. 
 
 The quantities p, v, and T, of a substance may be expressed 
 as fractions or as multiples n, r^ and rs, of their critical 
 values thus: p = rip c , v = r2V c , and T = r%T c . According to 
 the relation of corresponding states if the value of each of 
 two of the quantities n, r<i, and rs, for each of a number 
 of substances be taken the same, the remaining quantity 
 has the same value for each substance. In the case that 
 
92 THE EFFECT OF THE MOLECULAR FORCES 
 
 each substance consists of two phases in equilibrium, and 
 one of the foregoing quantities is taken the same for each 
 substance, each of the remaining two quantities is the 
 same for each substance. Substances under these condi- 
 tions are said to be at corresponding states, or obey the 
 relation of corresponding states. This relation was first 
 deduced by Van der Waals from his equation of state by 
 applying to it the conditions of the critical point expressed 
 by equations (57) and (58). The resultant equations 
 (one of which is the equation of state), on eliminating the 
 constants they contain, give an equation of the form 
 
 which expresses the first part of the relation in question. 
 The second part follows from the fact that the equations 
 of state of each of the phases of the two phases in equilib- 
 rium can be obtained from the foregoing equation and the 
 thermodynamical equation (60), giving three equations 
 each of which involves only two of the above ratios. 
 
 Meslin* has investigated the general conditions under 
 which the relation may be deduced. The equation of state 
 in its most general form is 
 
 lMP v, T, 0i, 02, . . )=0, 
 
 where gi, g% . . . , are constants depending only on the 
 nature of the substance. Suppose that the various equations 
 of condition applying to the equation of state are equal 
 in number to the foregoing constants. On eliminating 
 these constants an equation is obtained which is of the form 
 
 v, T,pc,v c , T c ) = 0. 
 * Comp.Jlend., 116, 135, 1893. 
 
CORRESPONDING STATE CONDITION EQUATIONS 93 
 
 This equation must be independent of the nature of the 
 units chosen and therefore be of the form 
 
 Therefore if fa, (or ^i), has the same fundamental form for 
 each substance the relation of corresponding states at once 
 follows. 
 
 The number of equations of condition applying to a 
 substance is not exactly known according to the previous 
 Section, but it is evidently greater than two, and the num- 
 ber of constants that must be associated with an equation 
 of state to obtain the relation of corresponding states is 
 therefore at present undetermined. If we are given the 
 relation of corresponding states, it obviously follows that 
 the number of the foregoing equations of condition is equal 
 to the number of constants in the equation of state. 
 
 The relation has been tested by a number of observers 
 of whom Young may be especially mentioned, and found 
 to agree fairly well with the facts. (An account of these 
 investigations will be found in Winckelmann's Handbuch 
 der Physik, 2d Edition, pp. 936-946). The deviations that 
 occur are probably entirely due to polymerization, which 
 renders a substance a mixture, and to which the relation- 
 ship is not likely to apply. 
 
 The relation may be extended to a number of other 
 quantities besides those mentioned which occur frequently 
 in this book. According to Section 21 
 
 On substituting for p, v, and T from the equations expressing 
 these quantities in terms of their critical values we obtain 
 
 r ** (63) 
 
94 THE EFFECT OF THE MOLECULAR FORCES 
 
 where r is a corresponding state quantity. At the critical 
 point the equation becomes 
 
 where r'$ is a constant, and thus the preceding equation 
 may be written 
 
 P n = r 5 P nc , ...... (64) 
 
 where r?> is a factor obeying the relation of corresponding 
 states, and P nc denotes the intrinsic pressure at the critical 
 point. Equation (63) can immediately be tested by the 
 facts. The intrinsic pressures of ether and benzene in the 
 liquid states at temperatures corresponding to 27 7 c /3 can 
 be calculated by the method of Section 21 and are found 
 to be 2756, and 3893 atmos. respectively. The ratio of 
 these pressures is .708, while the corresponding ratio of 
 the critical pressures is .733, which is practically equal to 
 the preceding ratio, as should be the case. 
 
 It can be shown in a similar way that the quantities 
 n" and V"t, which are superior limits of the quantities n 
 and V t , Sections (22) and (23), should also obey the rela- 
 tion of corresponding states. 
 
 If the quantity b in the equation (52) may be expressed 
 as a fraction obeying the relation of corresponding states 
 of the critical volume v c , or of the volume at the absolute 
 zero, which is highly probable, it can be deduced in a similar 
 way from this equation and equations (63) and (35) that 
 n and V t obey the relation of corresponding states. But 
 even if b did not possess this property, V t would obey the 
 foregoing relation approximately since b is usually small 
 in comparison with v. 
 
 It will be of importance now to discuss a method of ob- 
 taining the actual values of the foregoing quantities in any 
 given case. 
 
AN EXPRESSION FOR THE QUANTITY n 95 
 
 29. The Determination of the Quantities n, 
 V t , and b, of a Substance. 
 
 These quantities may most conveniently be determined 
 by means of the equation 
 
 20v ' (65) 
 
 obtained from equations (46) and (48) in Section 21. We 
 have seen that the quantity 6 is likely to vary little with 
 the density of a substance. Therefore over small regions 
 of densities it may be taken as constant. The quantity 
 n of a substance we would expect to increase more rapidly 
 than proportional to the density. If the substance behaved 
 as a perfect gas n would be proportional to the density, or 
 inversely proportional to the volume according to Section 
 11. For densities at which the substance does not obey 
 the law of a perfect gas the molecular attraction according 
 to Section 25, has the effect of increasing n above the value 
 corresponding to the density. We may therefore suppose 
 n to consist of the sum of a number of terms, the first expres- 
 sing the effect of the average minimum velocity (Section 16) 
 on the value of n, and the other terms the effect of the 
 increase in velocity produced by molecular attraction. 
 The first term corresponds to supposing that the substance 
 behaves as a perfect gas, and it therefore varies inversely 
 as v and may thus be written C\/v, where Ci is a constant 
 which may be determined from the substance in the gaseous 
 state. Of the remaining terms the first depends on the 
 chance of each molecule coming under the influence of 
 another molecule, which varies inversely as v 2 , and the 
 second term depends upon the chance of each molecule 
 coming under the influence of two molecules, which varies 
 
96 THE EFFECT OF THE MOLECULAR FORCES 
 
 inversely as v 3 , and so on. Thus the sum of these terms 
 may be written 
 
 and hence 
 
 + . . .C z /v*. . . (66) 
 
 The quantities 2, Cs, . . . C z are evidently functions of 
 v, for the chance of two molecules coming close together 
 will be influenced by the forces of attraction and repulsion 
 of the surrounding molecules, and hence 2 will depend on 
 the molecular concentration, or on v, similarly in general 
 the chance of any number of molecules coming close together 
 will be influenced by the molecular forces of the surrounding 
 molecules, and hence 2, C%, . . . C z will be functions of 
 v. But it will be evident on reflection that the variation of 
 n with v in the foregoing equation will in the main be ex- 
 pressed by the inverse powers of v, in other words, the 
 quantities 2, 3, . . . C z are functions of v which are not 
 sensitive to variations of v. They are obviously functions 
 of T since the chance of a number of molecules coming 
 close together will depend on their velocity of translation. 
 
 It follows from probability considerations that the terms 
 C2/v 2 , Cz/v 3 , . . . Cz/v z , decrease rapidly in magnitude in 
 the order they stand. For the chance of each molecule com- 
 ing close to another molecule and influencing its velocity 
 of translation is equal to a fraction pi, and the chance 
 of each pair of molecules coming close to a third molecule 
 and influencing its velocity of translation is evidently 
 therefore equal to pip2, where p2 is a fraction which is prob- 
 ably smaller than p\, etc., and accordingly 
 
 If the quantities C 2 , Cs, . . . C 2 be expanded in inverse 
 powers of v, an expression for n is obtained which may be 
 written 
 
AN EXPRESSION FOR THE QUANTITY n 97 
 
 .where 20, Cs a , - - - C za , are functions of T only. It will 
 be evident on reflection that since the first term of each 
 expansion will be large in comparison with the remaining 
 terms, the terms C^a/v 2 , Cz a /v 3 , . . . C 2a /v z decrease in 
 magnitude in the order they stand, and probably rapidly. 
 
 In using equation (66) we may as a first approximation, 
 from what has gone before, omit all the terms on the right- 
 hand side except the first two. Equation (65) then assumes 
 the form* 
 
 (67) 
 
 since Ci = A / = according to equation (21), where 
 m a \ o 
 
 A = 5.087 X 10 ~ 20 \/Tm according to equation (19), a denotes 
 the coefficient of expansion and /3 the coefficient of com- 
 pression per dyne at the absolute temperature T, v denotes 
 the volume of a gram molecule of the substance and b the 
 apparent volume occupied by the molecules, m a denotes 
 the absolute and m the relative molecular weight of a mole- 
 cule, and R the gas constant whose value is given in Section 
 5. If the distribution of molecular velocities is taken into 
 account the right-hand side of the foregoing equation accord- 
 ing to Maxwell's law has to be multiplied by 1.085 (Section 
 20). 
 
 The quantity 2 we have seen is a function which varies 
 little with v and therefore over a small region of densities 
 it may be taken as constant. Therefore on applying equa- 
 tion (67) at constant temperature to a substance at two 
 densities not differing much from each other, two equa- 
 tions are obtained from which 2 and 6 may be determined. 
 The values of n corresponding to these two densities may 
 then be obtained by equation (66) retaining two terms 
 only of the right-hand side. Similarly the equation may be 
 * Not previously published and used. 
 
98 
 
 THE EFFECT OF THE MOLECULAR FORCES 
 
 applied to another pair of densities not differing much from 
 each other and from the othei pair of densities, and the 
 corresponding values of n calculated. The values of 2 
 and b obtained in the two cases are not necessarily the same, 
 though they will obviously differ very little from each other. 
 Thus by a repeated application of equations (67) and (66) 
 the values of n and b may be obtained for different densities 
 of a substance. The accuracy of the results will evidently 
 increase with the number of density values into which a 
 given region of densities is divided. On the whole the 
 method should give very good results. 
 
 On having obtained values of n of a substance, the cor- 
 responding values of V t , the total average velocity of a 
 molecule, may be obtained by means of the equation 
 
 3rc 
 
 Vt ~N c ' 
 given in Section 18. 
 
 As an example of the foregoing investigation the values 
 of n and V t have been calculated for C02 at C. correspond- 
 ing to the pressures 100 and 200 atmospheres, which reduce 
 the substance to the liquid state, the results being given in 
 Table VIII. 
 
 TABLE VIII 
 
 CO 2 AT C. 
 
 p in 
 atmos. 
 
 in atmos. 
 
 r in c.c. per 
 grm. mol. 
 
 n. 
 
 v cms. 
 
 1 
 
 100 
 200 
 
 1 
 
 2730 
 4586 
 
 45.03 
 
 42.90 
 
 3.68 X10 23 
 7.17X10 26 
 7.80 X10 26 
 
 3.96 X10 4 
 1.56X10 5 
 1.62X10 5 
 
 The values of P n which were used to obtain the values 
 of Ta/0 by equation (48), were deduced by the method of 
 Section 21 from Amagat's experimental results on the 
 
THE INCREASE OF V t WITH THE DENSITY 99 
 
 relation between v and p. Th0 Table contains the values 
 of n and V t corresponding to, a pressure of one atmosphere, 
 in which case the substance is in the gaseous state, 
 the values of the foregoing quantities in that case being 
 given by equations (21) and (35). It will be seen that 
 the value of V t is increased to about four times its original 
 value as the pressure of the substance is increased from 1 
 to 100 or 200 atmospheres. This falls into line with the 
 results of Sections/17, 21, and 25, according to which the 
 effect of molecular attraction on the molecular velocity 
 in a substance in the liquid state should be quite large. 
 If the moleculajf- forces had no effect on the magnitude of 
 the motion of i translation of the molecules, as is usually 
 supposed, the v^alue obtained for 2 should have been zero, 
 or at least veryi small. The result conclusively shows that 
 the factor V t /V a in equation (53) cannot be put equal to 
 unity, and that the right hand side of Van der Waals' equa- 
 tion of state must be modified accordingly. The values of 
 n and V t obtained may be corrected according to Max- 
 well's law by dividing them by 1.085. 
 
 The value of b, the apparent molecular volume of a 
 gram molecule of molecules at the pressure 100 and 200 
 atmospheres is equal to 12.2 in cc., and thus is about J of 
 the total volume of a gram molecule of the substance. It 
 is smaller than the volume at absolute zero, which according 
 to Section 24 is approximately J the volume v c at the 
 critical point, or equal to 25.5 cc. This difference is due, 
 as was explained in Sections 19 and 24, to the molecules 
 getting nearer to each other at higher temperatures than the 
 absolute zero through possessing kinetic energy of motion 
 of translation. 
 
 If the value of the right hand side of the equation (67) 
 is calculated for C02 at a temperature 0C. and pressure 
 of 50 atmos., using the foregoing values of the constant b 
 and C 2 , the value of 2562 atmos. is obtained. This agrees 
 
100 THE EFFECT OF THE MOLECULAR FORCES 
 
 well with the value of 2542 atmos. obtained directly for the 
 left-hand side of the equation. 
 
 This investigation may be extended to a mixture of 
 substances. In the case of a mixture of say molecules 
 e and r, we would have 
 
 r _ .. .. .. i 
 
 (68) 
 
 according to equations (41), (45), and (48), where n e denotes 
 the number of molecules e crossing a square cm. from one 
 side to the other in the mixture, and n r has a similar meaning 
 with respect to the molecules r. Now we may write 
 
 2r /n\ 
 
 > (69) 
 
 and 
 
 similarly as before, where C\ r and C\ e may be obtained from 
 the substances r and e isolated and in the gaseous state. 
 The resultant equation will then contain the four variables 
 C f 2r, C f 2e, V e and b' r , which may be determined by applying 
 the equation to the mixture at constant temperature cor- 
 responding to four densities not differing much from each 
 other. Similarly a mixture of any number of constituents 
 may be treated. 
 
 Another method may be used depending on variations 
 of the relative concentration of the constituents of the 
 mixture instead of variations of its density at constant 
 relative concentration. Thus for n e and n r we may in this 
 case write 
 
 nr=a r 'Nr+ar"N 2 r+a/"NrN e } 
 and , . "... (71) 
 
CALCULATION OF MOL^CULAVrME! .\ 101 
 
 where a/, a/', a/", a/, a/', and a/", are constants of which 
 a/ and a/ may be determined according to Section 11 
 from the constituents isolated and in the gaseous state, 
 and N e and N r denote the concentrations of the molecules 
 e and r respectively. Similarly for be and &/ we may write 
 
 and ,.... (72) 
 
 where k e , k r , and k er are constants. It will be recognized 
 that k e denotes the apparent volume due to the interaction 
 of a molecule e with the remaining molecules e, and k r has 
 a similar meaning, while N r v e kcr denotes the apparent 
 volume due to the interaction of a molecule e with the mole- 
 cules r, or vice versa, etc. On substituting from these equa- 
 tions in equation (68) an equation is obtained containing 
 seven variables, which may be determined by applying at 
 constant temperature the equation to seven different rela- 
 tive concentrations of the mixture differing little from each 
 other. 
 
 In the case of a dilute solution we may evidently write 
 
 n r =a r Nr } 
 and , (73) 
 
 where a r anu a e are constants. The foregoing process is 
 thus much simplified in this case. 
 
 The foregoing method may be simplified by using the 
 values of b e r and 6/ calculated from the values of b e and b r 
 of the constituents in the pure state; or we might make 
 the assumption that b r ' = b e ', which would probably intro- 
 duce no serious error. 
 
 In Section 20 a method was given for finding approxi- 
 mate values of be and b/ from the values of b e and b r of the 
 constituents in the pure state. Approximate values of 
 
102 : \f*I]S $FfitrrPF : T^E MOLECULAR FORCES 
 
 C f 2r and C f 2e may similarly be obtained, and hence approxi- 
 mate values of n r and n e be obtained from equations (69) 
 and (70). Let CZ T stand for 2 in equation (67) when the 
 substance consists of molecules r, and similarly let Cz e stand 
 for 2 when the substance consists of molecules e. The 
 quantity C2r/v r 2 then depends on the chance of each mole- 
 cule in a pure substance of volume V T per gram molecule 
 coming close to another molecule and changing its velocity 
 of translation. This dependence may be exhibited in a 
 certain way which is useful in dealing with mixtures. The 
 chance of each molecule to get close to another molecule 
 is proportional to the product of the concentrations of the 
 molecules, or proportional to N r N r . Since N T m a TVr=mr, 
 where m ar denotes the absolute and m r the relative molec- 
 
 ular weight of a molecule, we have N r =, where K denotes 
 
 V T 
 
 a constant. Thus the foregoing chance is proportional tc 
 l/v r 2 , and the resultant change in n r expressed by. C2r/v r 2 . 
 But if the gram molecule of molecules r contains mole- 
 cules e besides, a molecule r may have its velocity changed 
 through coming close to a molecule e. The chance of that 
 happening to each molecule r is proportional to the product 
 of the concentrations of the molecules r and e, or proportional 
 to NgNr* This chance according to the foregoing results is 
 proportional to l/v r v e , where v e denotes the volume of the 
 mixture containing a gram molecule of molecules e. The 
 resultant change in the value of n r through the addition of 
 
 molecules e is therefore approximately given by 
 
 \/C 2r C 
 
 where Cz e and Ci r refer to the molecules e and r in the pure 
 state. The term C f 2r/v r 2 in equation (69) consists of the 
 sum of the foregoing two changes in n r brought about by 
 molecular interaction, or 
 
 C f 2r _ C2r , V Cir 
 o o "T" 
 
 o o 
 V r 2 V r 2 V T V e 
 
THE INTERNAL ENERGY OF A SUBSTANCE 103 
 Similarly it can be shown that 
 
 nT 
 
 V e 2 V e V r V e 
 
 The values of n r and n e obtained from equations (69) and 
 (70) by the help of equations (74) and (75) are likely to be 
 fairly accurate. The accuracy may be increased by using 
 the third and higher power terms in the series for n r and 
 n e in the equations. 
 
 The deduction of equation (65) used in this Section 
 involves the use of the quantity intrinsic pressure. This 
 quantity is therefore one of importance, and it will there- 
 fore be of interest to obtain further relations between it 
 and other quantities. 
 
 30. An Equation Connecting the Intrinsic Pres- 
 sure, Specific Heat, and other Quantities. 
 
 If U denote the internal energy of a gram molecule of 
 a substance, we have according to the Differential Calculus 
 that 
 
 (*U\ = (8U\ (8v\ (5U 
 
 \8TJ P \~~ 
 
 The term on the left-hand side of the equation is the specific 
 heat per gram molecule at constant pressure and was written 
 Sipi in Section 13. For U we may write 
 
 (76) 
 
 fts in Section 13, where U a denotes the potential energy of 
 molecular attraction per gram molecule, u m the internal 
 
 molecular energy, and - the kinetic energy of the mole- 
 
 cules when situated at points at which the forces of the 
 surrounding molecules neutralize one another, this energy 
 
104 THE EFFECT OF THE MOLECULAR FORCES 
 
 being equal to the kinetic energy in the gaseous state accord- 
 
 ing to Section 16. According to Section 21, - ? = P B the 
 
 ov 
 
 intrinsic pressure, where v denotes the volume of a gram 
 
 molecule, and since -(-r^) =a the coefficient of expansion, 
 v\ol / P 
 
 the equation may be written 
 
 /p ,(8u m \ \ .{&U a \ ./du m \ 3R . 
 
 S iPl = P+- +- + + - (77) 
 
 The quantity l-r ) is verv probably negligible in 
 \ ov IT 
 
 comparison with P n , and may therefore be omitted from the 
 equation. The quantity (-r^) expresses the change in 
 
 potential energy of molecular attraction of a substance 
 with change in temperature at constant volume. It is very 
 likely not zero in the case of a substance consisting of com- 
 plex molecules, since according to Section 13 the relative 
 distribution of the atoms in a molecule is changed with 
 change in temperature, which would give rise to a change 
 in the forces of attraction of the molecules upon each other. 
 
 For the same reason the quantity ( ~] is not likely to 
 
 be zero in the case of a complex substance. 
 
 But if the molecules consist of atoms it is highly probable 
 
 that these quantities and the quantity (-r^) are zero. 
 
 \ dv )T 
 
 The equation then becomes* 
 
 (78) 
 
 and may thus be used to calculate P n in such a case. 
 * Not previously published. 
 
THE POLYMERIZATION OF SUBSTANCES 105 
 
 If the value of P n for a substance in the liquid state, 
 which is mon-atomic in the gaseous state, is calculated 
 by the foregoing equation, and it does not agree with that 
 given by equation (48), it shows that the substance is 
 polymerized in the liquid state. Thus, for example, equa- 
 tion (78) gives the value of 8.8 X10 4 atmos. for P n for Hg 
 at C., while equation (48) gives 1.25 X10 4 atmos. It 
 appears therefore that Hg is polymerized when in the liquid 
 state. There is a good deal of other evidence pointing to 
 the same conclusion. 
 
 Equation (78) may also give approximate values of P n 
 in the case of substances consisting of complex molecules, 
 since in most cases the differential quantities in equation 
 (77) are relatively so small that they may be neglected. 
 The difficulty of applying the equation to a substance arises 
 principally through being dependent on a knowledge of 
 
 o r> 
 
 the nature of the molecules, since the quantities and 
 
 v depend upon it. In the case of a partly polymerized sub- 
 stance a considerable error may therefore be introduced. 
 
 In the previous Sections we have considered various 
 properties of substances which do not depend directly on 
 the nature of the motion of a molecule and on its rapidity. 
 There are, however, some properties that do, and which 
 will be considered in the next Chapter. These properties 
 are directly connected with the recurring nature of the 
 path of a molecule, which exists according to the laws of 
 probability. This introduces the idea of a recurring path 
 of average length associated with a molecule, which will 
 be discussed in the next Section, the results being introductory 
 to the matter contained in the next Chapter. 
 
106 THE EFFECT OF THE MOLECULAR FORCES 
 
 31. The Mean Free Path of a Molecule under 
 Given Conditions. 
 
 A molecule in a substance has its velocity continually 
 changed in direction and magnitude through the influence 
 of the surrounding molecules. At certain points along its 
 path it may therefore satisfy certain conditions in respect 
 to the surrounding medium. The straight lines joining 
 these points will be called the molecular free paths correspond- 
 X ing to the given conditions .i These paths will obviously 
 not have the same length, from probability considerations, 
 but evidently the mean free path, the mean length of a large 
 number of paths, should have a definite magnitude. 
 
 It is of importance and interest in this connection to 
 calculate the probability that a molecule will pass over a 
 distance x without satisfying the conditions of the free path. 
 This was first done by Clausius in connection with the 
 mean free path of molecular collision. The investigation 
 will be given a form here which applies to any kind of a free 
 path. 
 
 The probability P x that a molecule will pass over the 
 distance x without satisfying the path conditions may be 
 written 
 
 and the probability that it will pass over the path x+ dx 
 is therefore 
 
 But this is also according to the rules of probability equal 
 to the product of 'the probability P x and the probability 
 Pai that the molecule will pass over the distance dx without 
 satisfying the path conditions, that is 
 
PROBABILITY OF PATH OF GIVEN LENGTH 107 
 
 Now if I denotes the mean free path of the molecule, the 
 probability of it satisfying the path conditions in passing 
 over the distance dx is dx/l, and the probability of it not 
 satisfying the path conditions is therefore 
 
 1-^=1* ; 
 
 I 
 
 Hence the preceding equation may be written 
 
 -f )-' 
 
 which on integrating gives 
 
 - -'+* 
 
 where C denotes an arbitrary constant. Since the probability 
 that the molecule will not satisfy the path conditions in 
 passing over the distance x = is unity, it follows that C= 1, 
 and hence 
 
 Px=e~T, ...... (79) 
 
 which expresses the probability P x that the molecule will 
 pass over the distance x without satisfying the free path 
 condition, in terms of the quantities x and the mean free 
 path I. 
 
 The number n x of 100 molecules passing over the distance 
 x without satisfying the mean free path conditions is given 
 as an illustration for a number of different values of x in 
 Table IX, where x is expressed in terms of the mean free 
 path I. It will be seen that a molecule rarely passes over 
 <* distance greater than its mean free path. 
 
 The probability that a molecule has a free path lying 
 between the lengths x and x+dx is evidently 
 
 P p _g ~~ ~[ . ^/ (80) 
 
 according to equation (79). 
 
108 THE EFFECT OF THE MOLECULAR FORCES 
 
 TABLE IX 
 
 n x . 
 
 X. 
 
 n x . 
 
 X. 
 
 n x . 
 
 X. 
 
 99 
 
 O.Oll 
 
 78 
 
 0.251 
 
 14 
 
 21 
 
 98 
 
 0.02Z 
 
 72 
 
 0.3331 
 
 5 
 
 31 
 
 90 
 
 0.1Z 
 
 61 
 
 0.51 
 
 2 
 
 41 
 
 82 
 
 0.21 
 
 37 
 
 1. 01 
 
 1 
 
 4.51 
 
 In Section 21 the number of molecules n crossing a plane 
 one square cm. in area in a substance was investigated. 
 In connection with the investigation of this Section it is 
 of interest to determine the probability that the path of 
 a molecule crossing the plane has a length lying between x 
 and x-\-dx. It is evident that on moving the plane from one 
 position to another parallel to itself its chance of cutting a 
 free path of length x will be proportional to the value of x. 
 Therefore, since a molecule in its migrations has different 
 consecutive paths the probability of the plane cutting one 
 of length x is equal to the ratio of x to the sum of all of the 
 different paths of the molecule divided by their number. 
 The latter quantity is equal to the mean free path I, and hence 
 the ratio is equal to x/L This probability must be multi- 
 plied by the probability that the molecule has a path lying 
 between x and x+dx which is given by equation (80). 
 Therefore the probability P c that the free path of a molecule 
 which lies between the lengths x and x+dx cuts the plane 
 
 is 
 
 (81) 
 
 It should be noted that the foregoing investigation is inde- 
 pendent of the nature of the conditions defining the free path 
 of a molecule and that therefore these conditions may be 
 given any form we please. 
 
 It will be of interest to compare the square of the aver- 
 
THE MEAN OF SQUARES OF MOLECULAR PATHS 109 
 
 age path I, or I 2 , with the mean of the squares of the differ- 
 ent paths. The latter quantity is given by 
 
 j^v U/J(/ LJ\, , 
 
 I 
 
 by the help of equation (80). Thus the latter quantity is 
 double the former. 
 
 We are in a position now to develop formulae for the 
 quantities viscosity, conduction of heat, and diffusion, 
 which depend directly on the nature of the motion of a 
 molecule, in terms of quantities referring to the nature of 
 this motion, and other quantities. In order that the formulae 
 may be given forms applying to different states of matter 
 I have introduced new definitions in connection with the 
 path of a molecule under different conditions, which make 
 no reference to molecular collision, and apply to all states 
 of matter. They lead to formulae involving quantities 
 usually not considered in treatises on the Kinetic Theory 
 of Gases. These give interesting information as to how the 
 various molecular properties give rise to, or modify, the 
 viscosity, conduction of heat, and diffusion, of a substance. 
 I have previously made an attempt to obtain formulae for 
 the foregoing properties of matter independent of the idea 
 of molecular collision*, and finally developed the results 
 given in this book, which, however, differ considerably 
 from the previous results, and are therefore more or less new. 
 
 * Phil. Mag., July, 1912, pp. 101-118. 
 
CHAPTER III 
 
 QUANTITIES WHICH DEPEND DIRECTLY ON THE NATURE 
 OF MOLECULAR MOTION 
 
 32. The Coefficient of Viscosity of a Substance. 
 
 A solid in contact with a liquid or gas at rest experiences 
 a resistance when displaced depending on the nature of the 
 gas or liquid, the resistance remaining constant when the 
 motion is uniform. This arises through matter being carried 
 along by the surface of the solid. This property of liquids 
 and gases is known as viscosity. It can be shown that 
 it primarily arises from the fact that the molecules of a 
 substance undergo a rapid motion of translation. Thus 
 suppose that n molecules per square cm. per second strike 
 the lower surface of the solid A shown in Fig. 10, which is 
 moving with a velocity V\ parallel to its surface. If the 
 average momentum parallel tg/ the surface of the solid 
 of the molecules which per second are about to collide with 
 the solid, is nV^rria, the momentum after collision is that 
 corresponding to the velocity of motion of the solid, or 
 equal to nVim a , if the colliding molecules assume the velocity 
 of the solid, and thus the solid imparts a momentum equal 
 to nm a (Vi 2) to the substance per second. According 
 to the equation Ft = m a V, this momentum is equal to the 
 force F that has to be applied to the solid per square cm. 
 of the surface in order to maintain uniform motion, or 
 F = nm a (Vi 2)- The layer of liquid in the immediate 
 vicinity of the surface is accordingly set in motion, and this 
 
 110 
 
VISCOUS RESISTANCE AND VELOCITY GRADIENT 111 
 
 motion is communicated to the next layer, and so on. The 
 whole liquid is thus set in motion along planes parallel to 
 the surface of the solid, the velocity of motion decreasing 
 with the distance from the surface. 
 
 It follows from the above equation that the force neces- 
 sary to maintain a velocity Vi of the solid depends upon the 
 value of FI Fg, which is proportional to the velocity 
 
 ^ 
 
 FIG. 10. 
 
 gradient of the liquid in the immediate vicinity of the 
 surface, and thus the foregoing equation may be written 
 
 = nm a K 
 
 dV 
 5x 
 
 (82) 
 
 where the distance x is measured at right angles to the 
 surface of the solid, and K denotes a constant. If the velocity 
 gradient is unity the corresponding force F is called the 
 coefficient of viscosity, and will be denoted by 77. Hence 
 if a plane moving with a velocity V\ is at a distance d from 
 a fixed plane, the force F that has to be applied per square 
 cm. of the moving plane may be written 
 
 which accordingly becomes equal to 77, when the velocity 
 gradient is unity. The magnitude of the viscosity may be 
 
112 THE NATURE OF MOLECULAR MOTION 
 
 smaller than indicated by the foregoing investigation 
 through the molecules on striking the moving solid not 
 acquiring exactly its velocity. This effect is known as 
 slipping of the fluid over the surface of contact of the solid. 
 Experiment has shown that there is no appreciable slipping 
 in the case of liquids, and this also holds in the case of gases, 
 except at very low pressures. This point will be further 
 discussed in Section 34. It will not be difficult to see that 
 if the moving solid exerted a repulsion upon the molecules 
 of the surrounding fluid this would have the effect of tend- 
 ing to turn back the molecules approaching it without 
 giving to them a motion parallel to that of the surface of 
 the solid. In that case slipping would occur. It is not 
 improbable that the double layer of electricity which exists 
 at the interface of a fluid and solid may give rise to some 
 slipping in this way. 
 
 The coefficient of viscosity of a substance is usually 
 obtained in practice by measuring the volume v\ of the sub- 
 stance which passes per second through a narrow tube of 
 radius r and length LI under the action of a pressure p. 
 If the substance escapes from the tube with a velocity which 
 is negligible in comparison with the velocity that would be 
 obtained with a tube of large radius, the value of T\ according 
 to Poiseuille is given by 
 
 If, however, the escaping liquid possesses an amount of 
 kinetic energy which cannot be neglected the term /-=r- has 
 
 to be subtracted from the right-hand side of the foregoing 
 equation where V denotes the velocity of the escaping 
 fluid and p its density. 
 
 The viscosity of a substance depends not only on the 
 velocity of translation of a molecule, but on the nature of 
 
THE TRANSFER OF MOMENTUM BY A MOLECULE 113 
 
 the motion, and it will therefore be necessary to define a 
 quantity connected with the path described by a molecule. 
 
 33. The Viscosity Mean Momentum Transfer 
 Distance of a Molecule in a Substance. 
 
 Consider a viscous medium in motion parallel to the plane 
 AB which is kept at rest, as shown in Fig. 11. Let abc 
 
 A B 
 
 FIG. 11. 
 
 denote the path of a molecule moving towards the plane 
 A B, and which therefore passes progressively into slower 
 moving portions of the substance. The molecule during 
 its course loses momentum, continually but in varying 
 amounts to the medium parallel to the direction it is moving, 
 through the interaction of the molecules due to their molec- 
 ular forces of attraction and repulsion and their molecular 
 volume. Similarly a molecule passing in the opposite 
 direction continually acquires momentum from the medium. 
 The medium evidently does not acquire or lose momentum 
 only at isolated points. We may suppose, however, that 
 
114 THE NATURE OF MOLECULAR MOTION 
 
 the effect is equivalent to a row of points being associated 
 with the molecular path, not necessarily lying on it, at 
 which only the medium acquires or loses momentum. Fur- 
 ther let us suppose that the momentum lost by the medium 
 at a point is equal to V^m a} where 2 denotes the velocity 
 of the medium at the point, and m a the molecular weight 
 of a molecule, this momentum being absorbed by the migra- 
 ting molecule, while the momentum gained by the medium 
 at the same point is equal to V\m a , where Vi denotes the 
 velocity of the medium at the preceding point, this momen- 
 tum being abstracted from the migrating molecule. The 
 distance between two consecutive points will be called the 
 momentum transfer distance corresponding to the path of 
 the migrating molecule. 
 
 The molecule thus appears to abstract the momentum 
 V\m a from a point in the medium and to transfer it to a 
 consecutive point, while at the latter point it abstracts 
 the momentum V2m a from the medium and transfers it to 
 a point consecutive to the latter, and so on. The positions 
 of these points, and their number per unit length of path, 
 are to a certain extent arbitrary, as will easily be recognized, 
 and it is therefore necessary to introduce conditions which 
 will render the lines joining them quite definite. One condi- 
 tion that the points obviously have to satisfy is that the 
 flow of momentum in the substance should be uniform 
 everywhere. But this condition alone is not sufficient. 
 Let us therefore impose the two additional conditions that 
 the number n of molecular paths crossing a square cm. 
 from one side to the other shall be equal to the number of 
 transfer distances passing through the square cm., and 
 that all directions of a transfer distance in space are equally 
 probable. We will see in the next Section that these con- 
 ditions completely determine the distribution of the points 
 in question. According to Section 31 the various transfer 
 distances corresponding to these points would not be of 
 
RESULTANT FORCE ON A MOVING MOLECULE 115 
 
 the same magnitude, but grouped about a mean transfer 
 distance l n according to a general law whose form was ob- 
 tained. 
 
 It should be noted that according to the definition of a 
 transfer distance the length of path of a molecule between 
 two points is equal to the sum of the corresponding momen- 
 tum transfer distances. The transfer distances will there- 
 fore be associated with the molecular path in the way 
 shown in Fig. 11. 
 
 A migrating molecule is evidently continually undei the 
 action of the resultant force arising from the forces of the 
 
 Molecular path 
 
 FIG. 12. 
 
 surrounding molecules. This resultant force is continually 
 changing in direction and magnitude. It will therefore pass 
 through a series of maxima and minima of different mag- 
 nitudes, which may graphically be illustrated by the curve 
 in Fig. 12. The largest maxima correspond to the closest 
 possible approach of two molecules, while the other maxima 
 correspond to various distances of approach, whose effect 
 is modified by the situation of the surrounding molecules. 
 Thus some of the maxima and minima will have very small 
 though finite values. This holds for a gas as well as for a 
 liquid. The theoretical points at which a molecule is sup- 
 posed to absorb momentum from the medium or transfer 
 
116 THE NATURE OF MOLECULAR MOTION 
 
 momentum to it evidently therefore do not necessarily 
 correspond to maxima or minima values of the resultant 
 force acting on the molecule, and besides on account of the 
 small values of some of the maxima and minima several 
 of these are likely to be situated between two such consecu- 
 tive points. These considerations show that if molecular 
 forces exist between molecules the momentum free path 
 cannot be defined with respect to molecular collision, but 
 must be defined along such lines as developed in this Section. 
 
 It will not be difficult to see that even if the molecules 
 consisted of hard elastic spheres not surrounded by fields 
 of force the average distance between two consecutive col- 
 lisions in a gas need not be equal to the length of the aver- 
 age momentum transfer distance as just defined. In fact 
 it can be shown that these distances are not equal to one 
 another. But the opposite is usually tacitly implied when 
 the mean free path of a molecule in a gas is defined according 
 to molecular collision and then used to obtain a formula 
 for the viscosity of a gas in the usual way. Of course, some 
 mathematicians have begun to realize this and endeavored 
 to determine the appropriate factor that has to be asso- 
 ciated with the mean free path in the formula. But this 
 cannot lead to anything definite because the interaction 
 between molecules is largely, if not altogether, due to the 
 existence of the molecular forces. 
 
 A general formula for the viscosity of a substance in the 
 gaseous, liquid, and intermediary states of matter, in terms 
 of the momentum transfer distance will now be developed 
 and applied to the facts. 
 
 34. Formulae for the Viscosity in Terms of 
 Other Quantities. 
 
 Let us consider as before a substance moving in planes 
 parallel to a plane AB which is at rest, as is shown in Fig. 
 
THE TRANSFER OF MOMENTUM ACROSS A PLANE 117 
 
 13. Suppose that n molecules per square cm. cross the 
 plane EF from one side to the other. The migration of 
 these molecules gives rise to a transference of momentum 
 across the plane from top to bottom which per square cm. 
 is numerically equal to the coefficient of viscosity if the 
 velocity gradient is equal to unity. The expression for this 
 momentum we will now proceed to find. Consider the 
 momentum transfer distances which cut the plane EF and 
 are inclined at the angle 6 with a perpendicular to the plane. 
 If we suppose for purposes of calculation that these distances 
 start from the same point which we will suppose lies in the 
 plane EF, it follows from the figure, which shows the sec- 
 Vl >^ *- 
 
 FIG. 13. 
 
 tion of a hemisphere of radius z made by a plane at right 
 angles to the plane EF, that the number of these distances 
 is equal to 
 
 where z denotes the length of one of these distances, r = z sin 6, 
 and this number is therefore equal to 
 
 nsme-dd. 
 
 If n z denote the number of the foregoing distances whose 
 lengths lie between z and z+dz, we have 
 
 n z 
 
 z 
 n sin 6 - dO --^e *, dz t 
 
118 THE NATURE OF MOLECULAR MOTION 
 
 by the help of equation (81), where Z, denotes the mean 
 momentum transfer distance. The molecules corresponding 
 to these distances abstract the momentum n 2 m a Vi from 
 the plane EF, which moves with the velocity Vi, and trans- 
 fers it to the plane CD which moves with the velocity V 2 . 
 From considerations of equilibrium it follows that an equal 
 number of molecules migrate in the opposite direction. 
 These abstract the momentum n z m a V2 from the plane CD 
 and transfer it to the plane EF. Thus a transference of 
 momentum across the plane takes place which is equal to 
 
 n,m a (V l -V 2 ). 
 
 On taking the velocity gradient equal to unity we have 
 Vi-V 2 = 
 
 2COS0 
 
 and the foregoing expression accordingly becomes 
 n z m a z cos 0. 
 
 To obtain the total momentum transferred per square cm. 
 across the plane EF, or the coefficient of viscosity 77, the 
 foregoing expression on substituting for n z , must be integrated 
 from to o with respect to the transfer distance z, and 
 from to 7T/2 with respect to the angle 0, giving 
 
 rf r 2 2 _L 
 
 i) = nm a I I cos sin j-^e l r, - d6 - dz. 
 Jo Jo l n 
 
 Since 
 
 ri 
 
 cos 0' 
 
 and 
 
 JT< 
 
 rv -?- 
 
 __/j I- . fly O7 
 
 j o e U"* **J 
 JO lr, 2 
 
A FORM OF THE VISCOSITY EQUATION 119 
 
 the foregoing equation becomes 
 
 r i =nm a l r ,, ....... (83) 
 
 where n denotes the number of molecules of absolute molec- 
 ular weight m a crossing a plane of one square cm. from one 
 side to the other per second, and l n denotes the mean momen- 
 tum transfer distance. 
 
 The foregoing equation in the form it stands is mainly 
 of use in determining the value of Z,, since 77 may be measured 
 directly, and n determined by the method described in 
 Section 29. 
 
 The quantity l n may be expressed in terms of other 
 quantities. Let us suppose that the substance is represented 
 by another substance possessing the same viscosity and 
 expansion pressure, but whose molecules possess no volume 
 of the kind represented by the symbol 6 (Section 19). This 
 is possible since the law of molecular attraction of the 
 representative substance is initially left arbitrary, and may 
 therefore be given a form that the foregoing conditions are 
 satisfied. The law may evidently involve as many arbitrary 
 constants as we please. The viscosity of the representative 
 substance is given by 
 
 and l\ have meanings corresponding to n and 
 l^ in equation (83). Since the representative substance 
 has the same expansion pressure as the original substance, 
 and the molecules of the former substance have no volume, 
 we have from equations (38) and (45) that 
 
 vb 
 where the quantities n, v, b, P n , p, and A, refer to the original 
 
120 THE NATURE OF MOLECULAR MOTION 
 
 substance, and the preceding equation may therefore be 
 written 
 
 ,. (84) 
 
 Each quantity in this equation, except l' n , may be determined 
 directly or indirectly by experiment, and the latter quantity 
 is therefore determined by the equation. Tfye fact that 
 the equation contains a variable which does not refer directly 
 to the original substance shows that it may be represented 
 by another substance under the conditions stated. 
 
 On comparing the foregoing equation with equation 
 (83) we see that 
 
 and thus /',,, the mean transfer distance in the representative 
 substance, is evidently an inferior limit of l r We will see 
 later that /, does not differ appreciably from l\ except 
 when the density of the substance corresponds to that of 
 the liquid state. 
 
 It is often of interest to compare the distance of separa- 
 tion d of the molecules in a substance with the quantity 
 
 Wi 
 Z',. We have immediately d = , where w\ is a constant. 
 
 For l r n we may write l\ = W2V, where W2 is a function of v 
 which becomes a constant when the substance is in the 
 gaseous state. The ratio of l\ to d is then given by 
 
 (86) 
 
 where ws is a function of v which becomes a constant when 
 the substance is in the gaseous state. 
 
 The quantity I' in equation (84) may be expressed in 
 
THE INTERFERENCE FUNCTION OF VISCOSITY 121 
 
 terms of quantities which may be taken to refer to the 
 original substance. If the molecules in the representative 
 substance were devoid of forces of attraction and repulsion 
 which extend beyond a small distance from each molecule, 
 Z',, would be inversely proportional to the chance of one 
 molecule encountering another, and thus inversely propor- 
 tional to the concentration of the molecules, or proportional 
 to the volume v, and thus we may write 
 
 under these conditions, where K^ is a constant at constant 
 temperature. The existing forces of attraction and repulsion 
 have the effect of modifying the interaction of two mole- 
 cules, in which case l' n would not vary according to the fore- 
 going equation. The effect of the molecular forces may be 
 expressed by introducing a factor into the right-hand side 
 of the foregoing equation, which is determined from the 
 following considerations. The chance of a migrating mole- 
 cule coming under the influence of another molecule, and 
 the pair of interacting molecules coming under the influence 
 of a third molecule, is proportional to the square of the 
 molecular concentration, or inversely proportional to y 2 ; 
 the chance of a migrating molecule coming under the influ- 
 ence of another molecule, and the pair of interacting mole- 
 cules coming under the influence of two molecules, is inversely 
 proportional to v 3 ; and so on. Thus the factor in question 
 consists of the series 
 
 (87) 
 
 where 0',/v 2 expresses the average effect of a single mole- 
 cule on a pair of interacting molecules, (fr'^/v 3 the effect of 
 a pair of molecules, and so on, while 3% denotes the sum of 
 the series involving v. The quantities <' n , <",,, . . . , are 
 
122 THE NATURE OF MOLECULAR MOTION 
 
 functions of v and T, which, however, probably vary little 
 with v. For I,, and l' n we have therefore the expressions 
 
 , - (88) 
 and 
 
 The interpretation of equation (88) is interesting. 
 When two molecules interact the process is modified by the 
 surrounding molecules, which effect may be called molec- 
 ular interference. The value of b depends on molec- 
 ular interference, but probably only to a small extent. 
 But the quantity <, depends entirely on it, and this quan- 
 tity may therefore be called the interference function in 
 viscosity. The effect of molecular interference on the 
 value of Z,, according to equation (88), is expressed by 
 the quantities b and <,,, but mainly by $,, in other words, 
 the effect of molecular interference on the value of l n 
 which is not included in the value of b is represented by 
 the interference function <,,, which may now be used 
 without referring to the representative substance. 
 
 Equation (83) referring to the original substance may 
 now be written in the forms 
 
 v 2 
 
 , ..... (90) 
 
 , . .. . (91) 
 
 and 
 
 ..... (92) 
 
 by means of equations (35), (38), (45), and ^88), remember- 
 ing that N c m a v = m, the molecular weight in terms of that 
 of the hydrogen atom. 
 

 THE QUANTITIES IN THE VISCOSITY EQUATION 123 
 
 The values of n, b, and V t , the number of molecules 
 crossing a square cm., the apparent molecular volume, 
 and the total average velocity, respectively, may be deter- 
 mined by the method described in Section 29, and the value 
 of the intrinsic pressure P n by the method described in 
 Section 21. The calculated values of n. b, and V t depend 
 on the assumed distribution of molecular velocities, * and may 
 be corrected according to Maxwell's law, if desired, in the 
 way described. The value of A in equation (92) not cor- 
 rected for the distribution of molecular velocities is given by 
 
 A = 5.087 X W~ 20 V^m 
 
 according to Section 11. If corrected according to Maxwell's 
 law the value has to be multiplied by */--, or 1.085. It 
 
 will appear, however, from Sub-section a that the calcu- 
 lated value of K^ similarly corrected involves the factor 1.085, 
 and that therefore the uncorrected values of /c, and A may 
 be used in equation (92). It follows then that in equations 
 (90) and (91) the uncorrected values of K n , n, b, and V t 
 may be used, since p is independent of molecular motion 
 (Section 8), P n is obviously so by nature, and therefore 
 the factor K n /A in equation (92) is represented in the pre- 
 ceding equations by factors involving * n, 6, and V t . 
 
 The value of K, for a given substance is immediately 
 determined by applying equation (92) to the substance in 
 the perfectly gaseous state, which corresponds to v = oo , 
 and therefore to P n = and <>,, = 0. It may be noted that 
 according to equations (85), (88), and (89) we have 1^ = 
 // = , when the original and representative substances 
 are in the perfectly gaseous state, and the value of K n is 
 therefore the same for both substances. It evidently de- 
 
 *This distribution refers to molecules in the perfectly gaseous 
 state. 
 
124 THE NATURE OF MOLECULAR MOTION 
 
 pends on the molecular forces and the molecular volume 
 of the molecules of the original substance. 
 
 It will be evident on reflection that the quantities 
 n, V t , 6, P n , and p are affected to a certain extent by the 
 influence of the molecules of the substance on two molecules 
 while they interact, or are affected by molecular interfer- 
 ence. The quantity <,,, or the interference function, there- 
 fore, represents the effect of molecular interference on the 
 viscosity of the substance which is not represented by the 
 foregoing quantities in equations (90), (91), and (92). 
 Thus we may now use these equations without referring to 
 the representative substance. 
 
 The series for the interference function <, which occurs 
 in the foregoing equations may be given a simpler form 
 which holds approximately. The probability of a migrat- 
 ing molecule coming under the influence of another mole- 
 cule, and the pair of interacting molecules coming under 
 the influence of y molecules, is much greater than the prob- 
 ability of their coming under the influence of y+1 molecules, 
 
 and thus the terms in this series stand in the order of mag- 
 
 ,' i // 
 
 nitude j>^-> - , and are likely to decrease in mag- 
 nitude very rapidly. As a first approximation we may 
 
 , / 
 therefore retain only the term -^-, or replace the series by 
 
 $' 
 the term . , where x would differ little from 2. 
 
 It is instructive to apply equation (91) to a hypothetical 
 substance whose molecules consist of perfectly elastic spheres 
 not surrounded by fields of force. The value of V t then 
 corresponds to that when the substance is in the gaseous 
 state at the same temperature according to Section 19, 
 and <,, represents the interference by actual contact of one 
 or more molecules with two molecules about to collide or 
 undergoing collision. Since in all cases $,, in the representa- 
 

 THE VISCOSITY OF A GAS AND ITS DENSITY 125 
 
 live substance represents an effect due to molecular attrac- 
 tion, the quantity will have the same sign in this particular 
 case as found in practice, namely positive according to Sub- 
 section (d) . The effect of the true molecular volume accord- 
 ing to equation (91) would thus be to increase the viscosity 
 by means of the positive apparent volume b, and the 
 molecular interference represented by <>,,, to both of which 
 it would give rise. 
 
 The number N t of times the mean transfer distance 
 associated with the path of a molecule is passed over per 
 second is a quantity of interest. Since the length of molec- 
 ular path between two points is equal to the sum of the 
 momentum transfer distances, we have immediately 
 
 by the help of equations (83) and (35), where V t denotes 
 the total average velocity of a molecule, and N c m a = p the 
 density of the substance. The value of V t may be deter- 
 mined by the method of Section 29 in the case of a> liquid, 
 or gas not obeying Boyle's law. 
 
 Applications of the foregoing equations will now be given. 
 
 (a) On applying equation (90) to the perfectly gaseous 
 state, which corresponds to v = co , it becomes 
 
 , . 
 (94 
 
 by the help of equation (21). It follows from this equation 
 that, since K, is constant at constant temperature, 77 is 
 independent of the density of the gas. This remarkable 
 result was first deduced by Maxwell,* and has been amply 
 confirmed by experiment. The deviations that have been 
 obtained are due in part to the conditions of the experi- 
 * This deduction depends on the method of molecular collision. 
 
126 THE NATURE OF MOLECULAR MOTION 
 
 ment not conforming to the conditions underlying the 
 deduction of the above equation. Thus for example, the 
 result is found to break down completely at extremely low 
 pressures. Now it will be evident from an examination 
 of the deduction of the foregoing equation that it will hold 
 only so long as Z, is small in comparison with the thickness 
 of the moving layer of gas. The equation would therefore 
 begin to break down when the value of l^ becomes com- 
 parable with the linear dimensions of the apparatus used 
 for measuring the viscosity. 
 
 There is, however, another cause operating giving rise 
 to a deviation of equation (94) from the facts at low pres- 
 sures. The effect of the slipping of the gas along the mov- 
 ing plane on the value of the viscosity increases as the 
 pressure decreases. Thus let V a denote the velocity of the 
 moving plane, V b the component velocity of a molecule 
 parallel to the plane before striking it, and V c the component 
 velocity of rebound parallel to the plane. Then if the mole- 
 cule undergoes slipping along the plane during rebound 
 V a >V c >Vb. The viscosities t\ and v\\ when there is no 
 slipping and slipping respectively are proportional to the 
 velocity gradients in the gas, and hence proportional to 
 V a and V c the corresponding velocities of the molecules 
 on rebound, which gives ij/rji = V a /V c . Now the difference 
 between V a and V b increases with increase of the mean 
 momentum transfer distance, or distance of the layer of 
 gas from which the molecule comes, and thus increases with 
 decrease of pressure. The difference between V a and V c 
 therefore also increases with decrease of pressure, which 
 increases the value of the ratio rj/rji according to the fore- 
 going equation. This corresponds to a decrease of 771, and 
 thus the effect of slipping becomes the greater the lower the 
 pressure. 
 
 The dynamical mechanism underlying the result that 
 the viscosity of a gas is independent of its density may be. 
 
GASEOUS VISCOSITY AND THE TEMPERATURE 127 
 
 illustrated by the following considerations. A molecule 
 transfers a certain amount of momentum to the gas at 
 the end of each transfer distance on migrating at right 
 angles to the motion of the gas in the direction of the decrease 
 of motion. If the concentration of the gas is halved the 
 length of each transfer distance is doubled, while the momen- 
 tum transferred at the end of each transfer distance is also 
 doubled since the velocity gradient of the gas remains 
 the same. Since a change in molecular concentration of a 
 gas does not alter the molecular velocities, the momentum 
 transferred per second by a molecule moving between two 
 parallel plates of material one of which is at rest while the 
 other moves parallel to itself is in the latter case double 
 that in the former. But since the number of molecules 
 per cubic cm. available for momentum transference in the 
 former case is half the number in the latter, the total momen- 
 tum transferred is in each case the same, or the viscosity 
 of the gas has not been altered by altering its density. 
 
 The quantity K^ is a function of the temperature. This 
 is shown by the calculated values of K, contained in Table 
 X for a number of gases at different temperatures. These 
 values are not corrected for Maxwell's law of distribution 
 of velocities since n given by equation (21) and substi- 
 tuted in equation (90) was not thus corrected. If this 
 correction is carried out the values in the Table have to be 
 
 multiplied by -J-, or 1.085. It will be seen that the values 
 
 of /c, increase with increase of temperature in the case of 
 each gas mentioned in the Table, and this was found to hold 
 for every other gas that was examined. This indicates, 
 since the mean momentum transfer distance /,, is equal to 
 KqV, that the distance over which a migrating molecule trans- 
 fers momentum in a gaseous medium is increased by an 
 increase of temperature. The reason probably is that the 
 greater the velocity of two molecules approaching and 
 
128 
 
 THE NATURE OF MOLECULAR MOTION 
 
 receding from each other the shorter is the time they are 
 under the influence of each other's attraction, and there- 
 fore the smaller is the change in momentum imparted to 
 each other. Large changes in momentum will therefore 
 take place only for close distances of approach, and these 
 will therefore take place less frequently along the path of 
 a molecule with increase of temperature. 
 
 TABLE X 
 
 MERCURY. m = 200.4 
 
 KRYPTON, m = 81.8 
 
 tC. 
 
 ,10'. 
 
 K, 10 . 
 
 tC. 
 
 r) 10'. 
 
 if, lO'o 
 
 300 
 
 380 
 
 5320 
 6560 
 
 2.98 
 3.94 
 
 
 100 
 
 2334 
 3063 
 
 2.97 
 3.33 
 
 ARGON. m = 39.9 
 
 HYDROGEN. m = 2 
 
 -183.2 
 
 183.3 
 
 735.6 
 2104 
 3243 
 
 2.62 
 3.83 
 4.57 
 
 -194.9 
 
 302 
 
 374.2 
 
 822 
 1392 
 
 5.69 
 6.69 
 7.81 
 
 ETHER, m = 74 
 
 ETHYL CHLORIDE, m = 64.5 
 
 
 100 
 212.5 
 
 689 
 967 
 1234 
 
 .92 
 1.11 
 1.24 
 
 
 157.3 
 240.6 
 
 935 
 1440 
 1714 
 
 1.34 
 1.64 
 1.79 
 
 CARBON DIOXIDE. m=44 
 
 ETH YLENE . m = 28 
 
 -21.5 
 100 
 302 
 
 1278 
 1972 
 2682 
 
 2.31 
 2.92 
 3.21 
 
 -21.5 
 99.25 
 302 
 
 891 
 1278 
 1826 
 
 2.02 
 2.38 
 2.73 
 
 M. ISOBUTYRATE. HI = 102 
 
 METHANE, m = 16 
 
 24 
 100 
 
 754 
 1122 
 
 .823 
 1.19 
 
 
 20 
 
 1040 
 1201 
 
 2.99 
 3.33 
 
THE CHARACTERISTIC CONSTANT IN VISCOSITY 129 
 
 The quantity K n may be interpreted in other ways. 
 17 represents the rate at which a plane at right angles to a 
 unit velocity gradient loses momentum per square cm. 
 per second. Since n molecules cross the plane per square 
 cm. per second in each direction, rj/n represents the momen- 
 tum conveyed by a molecule across the plane on crossing 
 it and recrossing it (sometime) in the opposite direction. 
 According to equation (90) this momentum is equal to 
 K^niaV in the case of a gas. Thus K, represents the momen- 
 tum conveyed per unit mass of the molecule at unit volume 
 of a gram molecule of the gas. Hence for substances of 
 equal K^ and the same molecular concentration the momen- 
 tum conveyed by a molecule is proportional to its mass. 
 
 A molecule crossing the plane gets a distance away 
 from it which on the average is proportional to the volume 
 v, * and the molecule therefore tal:e& a time proportional to 
 v/V a in crossing and recrossing the plane, where V a denotes 
 its average velocity. The amount of momentum conveyed 
 across the plane per second by the same molecule is there- 
 fore proportional to K,,m a y a , and thus independent of the 
 volume of the gas. 
 
 The quantity K^ according to equation (91) applied 
 to the gaseous state, is a measure of the total momentum 
 parallel to the motion of the gas conveyed per square 
 cm. per second across the plane per unit momentum of the 
 momentum of translation motion of a molecule. 
 
 A number of formulae of an empirical nature expressing 
 the variation of I,,, or K^V, with T at constant v, have been 
 given, but which need not concern us here.f It is useful 
 
 * This distance is evidently proportional to the chance of the path 
 of the molecule not undergoing a deflection per unit length, which is 
 proportional to I,,, and hence proportional to v, since I^^K^V. 
 
 t In these investigations the quantity l n is supposed to represent 
 molecular free path according to the method of collisions, which is 
 connected with the viscosity by the equation given at the end of this 
 Sub-section . 
 
130 
 
 THE NATURE OF MOLECULAR MOTION 
 
 to note, however, that l n and K n are roughly proportional 
 to the square root of the absolute temperature. 
 
 The viscosity coefficient of a gas appears to obey approxi- 
 mately the relation of corresponding states, and this holds 
 therefore also for the quantity K^ according to equation 
 (94). This is shown by Table XI, which gives the ratio 
 
 TABLE XI 
 
 Substance. 
 
 T c . 
 
 27V 
 
 nc 10 7 . 
 
 772C 10 7 . 
 
 roe 
 T ,c 
 
 Kr,c 10 10 . 
 
 C0 2 
 
 304.9 
 
 609.8 
 
 1625 
 
 2811 
 
 1.73 
 
 2.66 
 
 C 2 H t 
 
 282 
 
 564 
 
 1022 
 
 1802 
 
 1.76 
 
 2.19 
 
 A 
 
 152 
 
 304 
 
 1216 
 
 2298 
 
 1.89 
 
 2.97 
 
 N 2 O 
 
 318.4 
 
 637.8 
 
 1539 
 
 2885 
 
 1.88 
 
 2.41 
 
 H 2 
 
 32 
 
 64 
 
 272 
 
 645 
 
 2.36 
 
 6.46 
 
 of the viscosities corresponding to the temperatures T c 
 and 2T C for a few substances, where T c denotes the critical 
 temperature, which were interpolated from the viscosity 
 data given in Landolt and Bornstein's Tables, 4th edition. 
 The ratios, it will be seen, are approximately equal to each 
 other. The somewhat large deviation in the case of H2 is 
 probably due to the greater uncertainty attached to the data 
 in that case, and to a lack of sufficiently extensive data 
 for reliable interpolation. The deviation is, however, small 
 in comparison with the difference between the viscosity 
 of H2 and that of each of the other substances. The varia- 
 tion of K n with the temperature is therefore approximately 
 
 expressed by a function of the form c .^(- ) where K_ C 
 
 \TcJ 
 
 denotes the value of K,, at the critical temperature, and which 
 is thus a fundamental and characteristic quantity of a gas. 
 Its values for a few substances are given in Table XI, ob- 
 tained in the same way as the values given in Table X. 
 

 THE FREE PATH AND MOLECULAR SEPARATION 131 
 
 The value of /,, for a substance in the gaseous state at 
 standard temperature and pressure is obtained by multi- 
 plying the corresponding value of *,, by the volume of a 
 gram molecule of the substance under standard conditions, 
 which according to Avogadro's law has the same value for 
 all substances and is equal to 22,700 cc. 
 
 It is often of interest to compare the value of ^ with the 
 average distance of separation d of the molecules, which is 
 
 Im v\ l/i I v \ A 
 given by d = ( - ) = U- 23 1 , and the ratio of these 
 
 quantities is therefore given by 
 
 (95) 
 
 It will be found that on substituting for K n and v at standard 
 temperature and pressure that l n is considerably larger than 
 d, showing that except for distances of approach of two 
 molecules much less than d the effect of their interaction is 
 small. 
 
 If equation (91) is applied to the gaseous state, and l n 
 is written for K n v according to equation (88) applied to the 
 gaseous state, the equation assumes the form 
 
 IV .ml, 1 
 
 where = p the density of the gas. This equation is usually 
 
 obtained in treatises on the Kinetic Theory of Gases on 
 the supposition that each molecule has the same velocity 
 V t , where Z, is supposed to denote the mean free path between 
 consecutive collisions of a molecule. Since / cannot repre- 
 sent exactly this quantity the equation has sometimes 
 been modified by the introduction of an appropriate numer- 
 ical factor as pointed out in Section 33. Another factor is 
 introduced on taking into account the distribution of molec- 
 
132 
 
 THE NATURE OF MOLECULAR MOTION 
 
 ular velocities since this affects the chance of collision. 
 This represents the most that has been achieved in the way 
 of a Kinetic Theory of substances in connection with vis- 
 .cosity according to the idea of molecular collision. 
 
 (6) It will be of interest to consider the values of l\ 
 of some substances in the liquid state, since they are inferior 
 limits of \, from which they differ but little, and they can 
 usually be more easily obtained than the values of l n . Table 
 XII gives the values of Z', calculated by means of equa- 
 
 TABLE XII 
 
 ETHER 
 
 tc. 
 
 
 r 108 cm. 
 
 d 108 cm. 
 
 
 
 
 
 
 
 d ' 
 
 13.5 
 
 .001779 
 
 2.06 
 
 5.49 
 
 .376 
 
 25.4 
 
 .001649 
 
 2.08 
 
 5.52 
 
 .377 
 
 63 
 
 .001338 
 
 2.21 
 
 5.65 
 
 .392 
 
 78.5 
 
 .001241 
 
 2.35 
 
 5.70 
 
 .412 
 
 99 
 
 .001133 
 
 2.91 
 
 5.80 
 
 .502 
 
 BENZENE 
 
 15.4 
 
 .004387 
 
 3.25 
 
 5.22 
 
 .623 
 
 50.1 
 
 .003641 
 
 3.07 
 
 5.29 
 
 .581 
 
 78.8 
 
 .003000 
 
 3.60 
 
 5.36 
 
 .486 
 
 CHLOROFORM 
 
 
 
 .003827 
 
 3.05 
 
 5.01 
 
 .607 
 
 20 
 
 .003419 
 
 2.84 
 
 5.06 
 
 .562 
 
 40 
 
 .003073 
 
 2.73 
 
 5.10 
 
 .535 
 
 60 
 
 .002791 
 
 2.68 
 
 5.15 
 
 .520 
 
 tion (84) for a number of substances in the liquid state at 
 different temperatures, and for comparison the corresponding 
 values of d the average distance of separation of the mole- 
 
INFERIOR LIMITS OF THE FREE PATH 
 
 133 
 
 cules. The value of A used in the equation was not cor- 
 rected for the distribution of molecular velocities. If cor- 
 rected according to Maxwell's law the values of l\ in the 
 
 Table have to be multiplied by , or 1.085. Both the 
 
 quantities I \ and d have values of the order of magnitude 
 10 ~ 8 cm. The latter values are greater than the former, 
 as is indicated by the values of the ratio I'^/d. The values 
 of P n (which must be reduced to dynes) used in the cal- 
 culations are contained in Table III, while the values of 
 f\ were calculated by means of the empirical formula of 
 Thorpe and Rodgers.* 
 
 Table XIII gives the values of Z', for CO 2 at 40 C. 
 under high pressures in the gaseous state, and the correspond- 
 ing values of d. According to equation (85), taking the 
 
 TABLE XIII 
 
 CO 2 AT 40 C. 
 
 p in 
 atmos. 
 
 1'^ 108 cm . 
 
 d 108 cm. 
 
 p in atmos. 
 
 l' n 108 cm. 
 
 d 108 cm. 
 
 70 
 
 6.52 
 
 7.34 
 
 94 
 
 2.84 
 
 5.16 
 
 80 
 
 4.55 
 
 6.52 
 
 100 
 
 2.83 
 
 5.03 
 
 85 
 
 3.82 
 
 5.95 
 
 112 
 
 2.81 
 
 4.90 
 
 value of b for CO2 obtained in Section 29, the value of I',, 
 is smaller than the value of I,, for the greatest density by 
 about 16 per cent. Thus ^ and I' \ do not differ much from 
 each other. The values of P n +p and 17 used in these calcu- 
 lations are given by Table XIV. 
 
 (c) The number of times N t that the mean momentum 
 transfer distance of a molecule in a substance is passed over 
 per second, given by equation (93), is a quantity of inter- 
 est. This number for C02 in the liquid state at C. under 
 * Phil. Trans., A., 1894, p. 1. 
 
134 THE NATURE OF MOLECULAR MOTION 
 
 a pressure of 100 atmos. is 6X10 12 , where the value of V t 
 was obtained from Table VIII, ?? = . 000925, and p=.87. 
 The number obtained is mainly of interest on account of 
 its great magnitude. It is interesting to note that this 
 number is smaller than the number of times per second the 
 resultant force acting on a molecule due to the surrounding 
 molecules passes through a maximum or minimum. 
 
 (d) Let us next apply equation (92) to some of the 
 facts to obtain values of <,,. Table XIV contains values of 
 
 TABLE XIV 
 
 CO 2 AT 40 C. * = 2.578 X 10- 10 
 
 P in 
 
 P n +P 
 
 v per 
 
 r, 10 
 
 *n 
 
 4.03X103 
 
 atmos. 
 
 in atmos. 
 
 grm. mol. 
 
 
 
 V 2-12 
 
 70 
 
 128.7 
 
 245.7 
 
 200 
 
 
 
 80 
 
 200.9 
 
 172.2 
 
 218 
 
 .027 . 
 
 .073 
 
 85 
 
 295.9 
 
 130.7 
 
 269 
 
 .132 
 
 .132 
 
 94 
 
 611.2 
 
 85.35 
 
 414 
 
 .292 
 
 .325 
 
 100 
 
 719.3 
 
 78.96 
 
 483 
 
 .385 
 
 .385 
 
 112 
 
 854.3 
 
 73.24 
 
 571 
 
 .486 
 
 .448 
 
 <,, for C02 at different pressures at a temperature of 40 C., 
 at which the gas does not assume the liquid state however 
 great the pressure. The values of P n +p used where cal- 
 culated by means of Van der Waals' equation of state (Sec- 
 tion 26) writing for b the value 42.8 cc. per gram molecule, 
 which results from the conditions expressed by equations 
 (57) and (58). The value corresponding to a pressure of 
 70 atmos. thus obtained is very nearly equal to that given 
 in Table XVI, which was obtained by a different method. 
 The values obtained for P n +p are therefore likely to be at 
 least approximately correct. The values of 77 used are those 
 obtained by Phillips*. Equation (94) gives the value of 
 * Proc. Roy. Soc., A., Vol. LXXXVII, pp. 56-57. 
 
INFERIOR LIMITS OF THE FREE PATH 
 
 133 
 
 cules. The value of A used in the equation was not cor- 
 rected for the distribution of molecular velocities. If cor- 
 rected according to Maxwell's law the values of I',, in the 
 
 Table have to be multiplied by 
 
 , 
 
 or 1.085. Both the 
 
 quantities l\ and d have values of the order of magnitude 
 10 ~ 8 cm. The latter values are greater than the former, 
 as is indicated by the values of the ratio I'Jd. The values 
 of P n (which must be reduced to dynes) used in the cal- 
 culations are contained in Table III, while the values of 
 77 were calculated by means of the empirical formula of 
 Thorpe and Rodgers.* 
 
 Table XIII gives the values of ' for C0 2 at 40 C. 
 under high pressures in the gaseous state, and the correspond- 
 ing values of d. According to equation (85), taking the 
 
 TABLE XIII 
 
 CO 2 AT 40 C. 
 
 p in 
 atmos. 
 
 l'^ 108 cm . 
 
 d 108 cm. 
 
 p in atmos. 
 
 Vq 10 8 cm. 
 
 d 108 cm. 
 
 70 
 
 6.52 
 
 7.34 
 
 94 
 
 2.84 
 
 5.16 
 
 80 
 
 4.55 
 
 6.52 
 
 100 
 
 2.83 
 
 5.03 
 
 85 
 
 3.82 
 
 5.95 
 
 112 
 
 2.81 
 
 4.90 
 
 value of 6 for C02 obtained in Section 29, the value of l\ 
 is smaller than the value of l^ for the greatest density by 
 about 16 per cent. Thus ^ and V ^ do not differ much from 
 each other. The values of P n +p and 17 used in these calcu- 
 lations are given by Table XIV. 
 
 (c) The number of times Nt that the mean momentum 
 transfer distance of a molecule in a substance is passed over 
 per second, given by equation (93), is a quantity of inter- 
 est. This number for CO 2 in the liquid state at C. under 
 * Phil. Trans., A., 1894, p. 1. 
 
136 THE NATURE OF MOLECULAR MOTION 
 
 stant. The foregoing considerations indicate the fundamental 
 reasons for the magnitude of the value of the viscosity of a 
 substance, and the reason why its value increases rapidly 
 with the density when the substance does not behave as a 
 perfect gas. 
 
 Interesting information may now also be obtained on 
 the effect of molecular interference on the motion of a 
 molecule. 
 
 The positive nature of <!>,, indicates, according to equa- 
 tion (89), that the effect of molecular interference is to 
 increase l' n from the value it would have if it varied pro- 
 portionally to v, which corresponds to the absence of molec- 
 ular interference. The same remark applies to /,, which is 
 given by equation (88), since an increase of b would be 
 attended by an increase of molecular interference. This 
 signifies that the chance of momentum being transferred 
 by a migrating molecule to another molecule when moving 
 along the velocity gradient of the substance is reduced by 
 the vicinity of other molecules through the interaction of 
 their forces of attraction and repulsion. But since the excess 
 of momentum of the migrating molecule must eventually 
 be transferred to the medium the act of transference when 
 molecular forces exist is less frequent, but when it occurs 
 more momentum is transferred, than would be the case in 
 the absence of molecular forces. In the former case when 
 a molecule transfers momentum to the medium the differ- 
 ence between its velocity and that of the medium in the 
 direction the medium is moving is greater than in the latter 
 case, and hence we would expect that the amount of momen- 
 tum transferred would be greater. 
 
 Assuming that the series <, may approximately be rep- 
 resented by the term 0,/w* the values of </>, and x cor- 
 responding to the data in Table XIV were calculated from 
 the values of $,, corresponding to the pressures 85 and 
 100 atmos., giving the values 4.03 X 10 3 and 2.12 respectively. 
 
VALUES OF THE INTERFERENCE FUNCTION 137 
 
 The value of x does not differ much from 2 as was pre- 
 dicted. The last column of the Table contains the values 
 of <t>r,/v x for different pressures, calculated by means of the 
 foregoing values of 0, and x. They agree fairly well with 
 the values of $>,, in the preceding column. A better agree- 
 ment would evidently have been obtained if <, had been 
 taken a function of v which decreases with increase of v. 
 
 Table XV gives the values of $, at different temperatures 
 for a few liquids (whose density depends of course on the 
 
 TABLE XV 
 
 ETHER 
 
 tC. 
 
 Gas 
 r, 10'. 
 
 KTI lo 1 ". 
 
 *,. 
 
 0, io 4 . 
 
 13.5 
 
 723 
 
 .9612 
 
 1.09 
 
 1.148 
 
 99 
 
 942 
 
 1.074 
 
 1.235 
 
 1.857 
 
 CHLOROFORM 
 
 o 
 
 959 
 
 1.028 
 
 2.783 
 
 1.707 
 
 60 
 
 1 
 
 1163 
 
 1.129 
 
 1.802 
 
 2.011 
 
 BENZENE 
 
 15.4 
 
 755 
 
 .974 
 
 2.772 
 
 2.158 
 
 78.8 
 
 1079 
 
 1.263 
 
 1.154 
 
 1.058 
 
 temperature), calculated by means of equation (89), using 
 the values of /' contained in Table XII. The values of 
 KT, were calculated by means of equation (94) using the 
 values of 77 corresponding to the gaseous state given in the 
 second column of the Table. It will be seen that the values 
 of $,, in some cases increase with increase of temperature, 
 
138 THE NATURE OF MOLECULAR MOTION 
 
 which, it should be noted, is attended by an increase of 
 the volume, while in the other cases the values decrease. 
 Thus the effect of an increase of the temperature on the 
 value of $ appears to be in the opposite direction to the 
 effect of an increase of the volume. If the quantity ^ 
 behaves similarly for all substances, as we might expect, 
 the foregoing results would indicate that at constant tem- 
 perature it would decrease with increase of volume, as the 
 preceding results have already shown, and increase with 
 increase of temperature at constant volume. This indicates 
 that the chance of transfer of momentum by a migrating 
 molecule to a molecule of the surrounding medium through 
 the existence of molecular forces is decreased by an increase 
 of temperature. 
 
 Table XV also contains the values of </>,, in the approxi- 
 mate expression of ^/v x for $, putting x = 2. The values 
 of 77 for different volumes at constant temperature of the 
 substances mentioned in the Table might now be approxi- 
 mately calculated by means of equation (92) on calcula- 
 ting the values of P n +p by means of an empirical equation 
 of state according to Section 21. 
 
 Further investigation in connection with the experi- 
 mental values of 77 of liquids must proceed mainly along the 
 line of comparing the values of the characteristic quantity 
 $ of different substances at various temperatures and vol- 
 umes. This might furnish some information as to how this 
 quantity depends on the molecular weight besides on the 
 volume and temperature, and incidentally furnish further 
 information in connection with the molecular forces. 
 
 (e) It is interesting to apply equation (91) to a substance 
 not in the perfectly gaseous state at volumes for which 6 
 is small in comparison with i>, and <!>,, small in comparison 
 with unity, in which case the equation becomes 
 
 (96) 
 
EFFECT OF DENSITY ON MOLECULAR VELOCITY 139 
 
 Corresponding to the smallest value of v for which these 
 conditions hold rj usually differs considerably from that 
 applying to the substance in the perfectly gaseous state, 
 and this therefore also holds for V t and n. The value of 
 V t may therefore be calculated with fair accuracy by means 
 of the foregoing equation over a region at the beginning of 
 which a substance begins to deviate from the gas laws. 
 The corresponding values of P n +p, and hence of P n , may 
 be obtained from equation (92). Table XVI contains a 
 
 TABLE XVI 
 
 CO 2 AT 40 C. Kr, = 2.578 X 1.085 X 10~ 10 
 
 p in 
 atmos. 
 
 77x106. 
 
 per 
 grm. mol. 
 
 P n +P 
 
 in atmos. 
 
 P n 
 atmos. 
 
 V t / V a- 
 
 1 
 
 157 25,550 
 
 1 
 
 
 1.000 
 
 40 
 
 176 
 
 589.8 
 
 48.59 
 
 8.6 
 
 1.121 
 
 60 
 
 187 
 
 328.7 
 
 92.70 
 
 32.7 
 
 1.191 
 
 70 
 
 200 
 
 245.7 
 
 132.5 
 
 62.5 
 
 1.274 
 
 set of calculations carried out for CCb at 40 C. over a range 
 of volumes for which the foregoing conditions hold, since 
 
 4 03 X 10 3 
 approximately 6=12, Section 29, and $,,= - ' 
 
 .12 
 
 accord- 
 
 ing to Table XIV. The value of K^ in equation (96) was 
 determined by applying equation (94) to C02 in the gaseous 
 state at a pressure of one atmos. and temperature 40 C., 
 and correcting the value obtained according to Maxwell's 
 law. It will be seen that P n and the ratio V t /V a , where V a 
 denotes the average velocity of translation of a molecule 
 of CO2 in the gaseous state, which is given by equation 
 (8) and the equation F a =.922F, gradually increase with 
 increase of pressure. Thus the more the molecules come 
 under each other's influence the greater the total average 
 
140 THE NATURE OF MOLECULAR MOTION 
 
 velocity above that in the gaseous state, which falls into 
 line with what has been obtained before. It is of impor- 
 tance to notice, however, that the foregoing deduction of the 
 result does not depend on the results of Sections 16 and 17. 
 
 (/) On using the term ^/v 2 for the series $,, in equation 
 (90), which holds approximately according to Sub-section 
 d, and substituting for n the expression given by equation 
 (66) retaining only the first two terms, we obtain the equa- 
 tion 
 
 This equation contains the three unknowns b, 2, and </> r 
 They may be determined by applying the equation at con- 
 stant temperature to a substance at three densities not 
 differing much from each other. The values of n and V t 
 may then at once be obtained similarly as in Section 29. 
 
 It may be noted, however, that it is preferable to deter- 
 mine the quantities 6, 2, and < if possible, without using 
 simultaneous equations, or using as few as possible, as this 
 gives more reliable results. For the variables of a set of 
 simultaneous equations may usually be varied over a 
 considerable range and yet approximately satisfy the equa- 
 tions, and hence slight errors in the constants of the equa- 
 tions (furnished by experiment) may considerably affect 
 the values obtained for the variables. 
 
 (g) In Sub-section (a) of this Section it was shown that 
 the values of T? for substances in the gaseous state obey 
 the relation of corresponding states. This is also found to 
 hold when the substances are not in the gaseous state,* 
 as is shown by the approximate constancy of the ratio of 
 171 to 772 corresponding to the temperatures T c /2 and 47V7, 
 shown by Table XVII for a number of liquids. Since the 
 quantities P^+p, v, T, and K, in equation (92) obey this 
 * R. D. Kleeman, Proc. Camb. Phil Soc., Vol. XVI, Ft. 7, p. 633. 
 
PROPERTIES OF THE INTERFERENCE FUNCTION 141 
 TABLE XVII 
 
 ETHELENE-BROMIDE (C 4 H 4 Br 2 ) 
 
 ETHELENE CHLORIDE (C 4 H4C1 2 ) 
 
 *c. 
 
 ni and 1)2. 
 
 /*. 
 
 <c. 
 
 171 and /2. 
 
 r)i/r)2. 
 
 18.4 
 60 
 
 .01767 
 .009985 
 
 1.770 
 
 7.7 
 
 47.8 
 
 .01000 
 .005903 
 
 1.693 
 
 ETHYL PROPIONATE (CoHi O 2 ) 
 
 BUTYRIC ACID (C 4 H 8 O 2 ) 
 
 
 
 38.6 
 
 .007040 
 .003864 
 
 1.822 
 
 32.5 
 76.1 
 
 .01255 
 .007016 
 
 1.777 
 
 THIOPHEN (C 4 H 4 S) 
 
 BENZENE (C 6 H 6 ) 
 
 22.1 
 64.3 
 
 .00644 
 .00408 
 
 1.579 
 
 3.8 
 43.3 
 
 .008680 
 .004740 
 
 1.831 
 
 ETHYL BENZENE (C 8 Hi ) 
 
 PROPIONIC ACID (C 3 H 6 O 2 ) 
 
 36.7 
 80.9 
 
 .00550 
 .00357 
 
 1.540 
 
 33.4 
 
 77.2 
 
 .009169 
 .005605 
 
 1.635 
 
 OCTANE (C 8 Hi 8 ) 
 
 CARBON TETRACHLORIDE (CC1 4 ) 
 
 11.6 
 52.2 
 
 .006060 
 .003783 
 
 1.603 
 
 5 
 44.7 
 
 .0124 
 .00696 
 
 1.781 
 
 relation (Section 28 and Sub-section (a) of this Section), 
 it follows from this equation that the factor 1 +<>,, also 
 approximately obeys the relation of corresponding states. 
 
 We may therefore write l+^ ^[~^j ), where T c denotes 
 
 \ * PI 
 
 the critical temperature, and p c the critical density of the 
 substance. 
 
142 THE NATURE OF MOLECULAR MOTION 
 
 35. Formula for the Viscosity of Mixtures. 
 
 It will be evident from an examination of the investiga- 
 tion in the previous Section that the effect of each molecule 
 of a substance on its viscosity is additive in character. 
 Therefore in the case of a mixture the effect of the different 
 sets of molecules is additive. It will easily be seen there- 
 fore that in the case of a mixture of molecules e and r the 
 viscosity is given by 
 
 ..... (98) 
 
 where l^ r denotes the mean momentum transfer distance 
 of a molecule r of absolute mass m ar , n r the number of mole- 
 cules r crossing a square cm. from one side to the other per 
 second, and the remaining symbols have similar meanings 
 with respect to the molecules e. For l nT and l,, e we may write 
 
 ), .... (99) 
 
 Vr (/ r-LVer 
 
 and 
 
 ' 
 
 similarly as in the previous Section, where v r denotes the 
 volume of the mixture containing a gram molecule of mole- 
 cules r, b' T the apparent volume of the molecules r and e 
 in the volume v r with respect to the motion of a molecule r 
 (Section 20), K\ T a characteristic constant of the molecules 
 r when the mixture is in the gaseous state, N er the total 
 concentration of the molecules e and r, N the number of 
 molecules in a gram molecule of a pure substance, ^>, r the 
 effect of the molecules of the mixture on a molecule r inter- 
 acting with a molecule r or e, or the interference function 
 of the mixture with respect to the molecules r, and the 
 remaining quantities have similar meanings with respect to 
 

 VISCOSITY EQUATION FOR GASEOUS MIXTURES 143 
 
 the molecules e. It will be recognized that in deducing the 
 foregoing two equations along similar lines as equation 
 (88), we have written 
 
 N N 
 
 l 'v =K 'v inrCl+S,,, and Z' ne =*' !+<*>,), 
 
 -iV er IV e r 
 
 or replaced volume by concentration, which in the case 
 of mixtures is more convenient. It should be noted that in 
 
 N 
 the case of a pure substance say e, we have v= . It 
 
 is also more convenient to regard <!>', and <$',,,. as a series 
 of powers of the concentration instead of the volume. As 
 a first approximation we may write 
 
 and 
 
 similarly as in the previous Section. It will be easy to 
 see that the various quantities involved are functions of the 
 ratio of the number of molecules r to e, as well as of their 
 nature. 
 
 On applying equation (98) to the gaseous state after 
 substituting from equations (99) and (100), it becomes 
 
 Vv-M., (101) 
 
 by the help of equation (21), and since 
 
 v e N .N AT 
 
 = -, and + = Ncr, 
 
 V r V e V r 
 
 where N denotes the number of molecules in a gram 
 molecule, and m r and m e the molecular weights of a 
 
144 THE NATURE OF THE MOLECULAR MOTION 
 
 molecule r and e respectively in terms of that of the hydro- 
 gen atom. This equation expresses a relation between the 
 quantities K.\ T and K'^, which is of use in determining them. 
 Approximate values of K'^ and K\ C may be obtained 
 from the values of K^ and K^ corresponding to the con- 
 stituents separated from each other and in the gaseous state. 
 In the case of a pure substance r in the gaseous state 
 
 N 
 I,,, = a,,,. v r = K, f . Thus l^ varies inversely as the probability 
 
 of a molecule r coming under the influence of another mole- 
 cule r under given conditions, which is proportional to N r , 
 and hence l/K, r is the probability factor of N r . In the case 
 of a mixture of molecules r and e in the gaseous state 
 the transfer distance l^ r varies inversely as the proba- 
 bility of a molecule r coming under the influence of a mole- 
 cule r or e. This probability consists of the sum of the 
 probabilities of a molecule r coming under the influence 
 of another molecule r, and of coming under the influence 
 of a molecule e, since these processes are independent. 
 
 N 1 
 Now the first probability is equal to , where K, r refers 
 
 J\ Kyr 
 
 to the molecules r in the pure state, since the two proba- 
 bilities are independent and we may therefore suppose the 
 molecules e absent. The second probability is equal to 
 
 AT 1 1 
 
 -^ , where is the probability factor in this case. Hence 
 N K X K X 
 
 we have 
 
 Ner 
 
 : AT 
 
 or 
 
 _1_ fA^ ATeM 
 
 , 
 
 ( 102 ) 
 
CHARACTERISTIC FUNCTION APPROXIMATION 145 
 
 The factor I/K X may be expressed as a mean of the factors 
 I/K^ and I/K^ referring to the substances r and e in the 
 pure state. Thus we may write 
 
 or we may write 
 
 1=1)1+11 
 
 K X 2\Kr, r Kr,e\' 
 
 which would hold approximately. Thus an approximate 
 value of K'^ could be obtained from equation (102) in terms 
 of quantities referring to the constituents of the gaseous 
 mixture in the pure state. Similarly an approximate expres- 
 sion for K'^ may be obtained. The accuracy of the values 
 obtained may be checked by means of equation (101). 
 
 Approximate values of $' v and 0',^ may be calculated 
 from the values of nr and </>,,<, referring to the isolated 
 constituents of the mixture. In the case of a pure sub- 
 stance r the term Q^N^/N 2 is a measure of the probability 
 of a molecule r coming under the influence of another mole- 
 cule r and the pair coming under the influence of a third 
 molecule r. In the case of a mixture of molecules r and e 
 the term 4>' w N 2 er /N 21 is correspondingly a measure of the 
 probability of a molecule r coming under the influence of 
 another molecule r and the pair coming under the influence 
 of a molecule r or e, or a molecule r coming under the influ- 
 ence of a molecule e and the pair coming under the influence 
 of a molecule r or e. These four probabilities may be taken 
 as approximately independent of each other, and the total 
 probability therefore equal to their sum. We may there- 
 fore write 
 
 , . . . 
 
 ^ X ~ " ~ ~* ' 
 
146 THE NATURE OF MOLECULAR MOTION 
 
 where <t> nr refers to molecules r in the pure state, and $ x , 
 <f> v , and </> 2 , are appropriate probability factors. These 
 factors may approximately be expressed in terms of quan- 
 tities referring to the molecules r and e in the pure state. 
 Thus we may write 
 
 approximately; or we may write 
 
 approximately. Similarly an approximate expression for 
 d, \ e may be obtained. 
 
 The quantities just considered may be determined di- 
 rectly by the following method. In conformity with 
 equation (102) we may write 
 
 v 
 and 
 
 On substituting for K'^ and ^ e from these equations in 
 equation (101) and applying it to a gaseous mixture at 
 three different relative concentrations at constant tem- 
 perature three simultaneous equations are obtained from 
 which the constants a r , a re , and a e may be determined. 
 
INTERFERENCE FUNCTION APPROXIMATION 147 
 
 Similarly in conformity with equation (103) we may 
 write 
 
 and 
 
 On substituting for 4)\ r and tf^ in equation (98) transformed 
 by means of subsequent equations as indicated, and apply- 
 ing it to four different relative concentrations of the mixture 
 at constant temperature four simultaneous equations will 
 be obtained from which the constants b r , b rr e, b ree , and b e , 
 may be determined. The values of b' T and b' e involved, 
 it may be pointed out, may be obtained by the help of 
 Section 29. 
 
 Knowing the values of the foregoing seven constants 
 and the values of b' e and b' r , the viscosity may be calcu- 
 lated for any density and relative concentration of the 
 constituents. 
 
 Similarly a mixture of more constituents than two may 
 be treated. 
 
 36. The Coefficient of Conduction of Heat. 
 
 When heat flows from one part of a substance to another 
 at a different temperature without a bodily transference 
 of matter taking place the heat is said to be propagated 
 by conduction. The heat energy is then transferred from one 
 molecule to another in the direction of the flow of heat 
 through their interaction by means of their forces of attrac- 
 tion and repulsion. The coefficient of conduction of heat is 
 usually denned in connection with the flow of heat across 
 a slab of material of thickness d and infinite extent whose 
 surfaces are kept at the different temperatures ti and fa, 
 
148 THE NATURE OF MOLECULAR MOTION 
 
 where ti>t 2 say. The quantity of heat Q transferred per 
 second per square cm. of each surface is defined by 
 
 (104) 
 
 where C is a constant which is called the coefficient of con- 
 
 duction of heat, and -W-^ is called the temperature gradient 
 d 
 
 of the flow of heat. When the gradient is equal to unity 
 it follows from the equation that the coefficient C is equal 
 to the quantity of heat transferred from one side of the 
 slab to the othe.r per second per square crn. of surface. 
 The lines of flow of heat in the foregoing arrangement 
 would obviously be everywhere perpendicular to each sur- 
 face of the slab. In practice with a slab of finite dimensions 
 this is realized only near the central portion, and the amount 
 of heat transferred is therefore measured for this portion 
 only, the rest of the slab acting as a guard ring arrangement. 
 As in the case of viscosity, the flow of heat in a substance 
 is directly connected with the nature of the motion of a 
 molecule, and it will therefore be necessary to define a 
 quantity connected with the path described by a molecule 
 in migrating from one portion of the substance to another. 
 
 37. The Mean Heat Transfer Distance of a 
 Molecule in a Substance. 
 
 Consider a substance which is at a higher temperature 
 in the plane AB, Fig. 14, than in the parallel plane CD, 
 and which has a uniform temperature gradient between 
 the planes. Let abc denote the path of a molecule migrat- 
 ing progressively into layers at lower temperatures. The 
 molecule loses heat energy in continually varying amounts 
 to the medium at the expense of its kinetic energy (Section 
 16), and the change in its potential energy of attraction 
 
THE TRANSFERENCE OF HEAT BY A MOLECULE 149 
 
 brought about by passing progressively into denser layers 
 of the substance. We may suppose that the medium acquires 
 and loses energy at certain points only near the path of the 
 migrating molecule, which energy is abstracted from, or 
 transferred to, the migrating molecule as the case may be. 
 The amount of energy lost at a point will be taken equal 
 to 7 7 2>Sm, where T^ denotes the absolute temperature at the 
 point, and S m the internal specific heat at constant pressure 
 of the molecule, while the energy acquired at the point will 
 be taken equal to T\S m , where T\ denotes the absolute 
 temperature at the preceding point. The internal specific 
 heat S m of a molecule is equal to the internal specific heat 
 
 FIG. 14. 
 
 at constant pressure of a gram of molecules (Section 13) 
 divided by the number of molecules it contains. The line 
 joining two consecutive points will be called a heat transfer 
 distance in the substance, since the molecule abstracts the 
 quantity of heat TiS m at one extremity of the distance and 
 transfers it to the other extremity, while at the latter point 
 it abstracts the quantity of heat T 2 S m and transfers it to 
 the other extremity of the adjacent distance, etc. Let us 
 suppose, similarly as in the case of viscosity, that the posi- 
 tions of the points associated with the path of a migrating 
 molecule are so selected that the number of heat transfer 
 
150 THE NATURE OF MOLECULAR MOTION 
 
 distances, and the number of molecular paths, cutting a 
 plane of one square cm., are equal to one another, and that 
 all directions of a transfer distance in space are equally 
 probable. These two conditions determine the positions 
 and lengths of the transfer distances. These distances are 
 not equal to each other, but are grouped about a mean 
 distance l c according to the distribution law of Clausius 
 given in Section 31. 
 
 The mean heat transfer distance of a molecule is evi- 
 dently not equal to the mean momentum transfer distance 
 in viscosity, for one refers to the transfer of momentum 
 and the other to energy under different conditions. Experi- 
 mental evidence will be considered later showing that that 
 is so. In treatises on the Kinetic Theory of Gases both dis- 
 tances are usually assumed to be equal to the average dis- 
 tance between two consecutive collisions of a molecule. 
 
 We will now deduce formulae involving the mean heat 
 transfer distance as defined in this Section. 
 
 38. Formulae for the Coefficient of Conduction 
 of Heat in Terms of Other Quantities. 
 
 Let us suppose that the temperature of a substance is 
 higher in the plane AB, Fig. 15, than in the parallel plane 
 CD. Suppose that n molecules cross the plane EF from 
 one side to the other. This migration of molecules gives 
 rise to a transference of heat across the plane from top to 
 bottom which per square cm. is numerically equal to the 
 coefficient of conduction of heat if the temperature gradient 
 is unity. This amount of heat may be expressed in terms 
 of other quantities as follows. Consider the heat transfer 
 distances which cut the plane EF and are inclined at an 
 angle 6 to a perpendicular to the plane. For purposes of 
 calculation we may suppose that the upper extremities of 
 these distances are located at the same point which may be 
 
THE TRANSFERENCE OF HEAT ACROSS A PLANE 151 
 
 taken to lie in the plane EF. It follows then from the figure, 
 which shows a section of a hemisphere of radius z having the 
 point mentioned as center, that the number of these dis- 
 tances is equal to 
 
 l ~~" 
 
 where z denotes the length of one of these distances, r 
 z sin 6, and the number is thus equal to n sin 6 dB. 
 
 FIG. 15. 
 
 If n^ denote the number of the f oregoinj distances whose 
 lengths lie between z and z+dz, we have 
 
 n z = n sin 6 - d0 j-^ 
 
 z -L 
 
 dz, 
 
 by the help of equation (81). Each of the molecules cor- 
 responding to the foregoing distances abstract the energy 
 T\Sm from the plane EF which is at the temperature TI, 
 and transfers it to the plane GH which is at the temperature 
 TV An equal number of molecules inclined at the angle 
 6 to a line at right angles to the plane EF move in the oppo- 
 site direction. Each of these molecules abstracts the energy 
 T2S m from the plane GH and transfers it to the plane EF. 
 Thus on the whole these two sets of molecules transfer the 
 
152 THE NATURE OF MOLECULAR MOTION 
 
 energy n z (TiS m T2S m ) across the plane EF. On taking 
 the temperature gradient equal to unity we have 
 
 - 
 
 ' 
 
 z cos e 
 
 and the preceding expression for the energy transfer becomes 
 n z S m z cos 6. 
 
 The total energy transferred per square cm. across the 
 plane EF is obtained on substituting for n z in the foregoing 
 expression and integrating it from to oo with respect to 
 the transfer distance z, and from to 7r/2 with respect to 
 the angle 6, which gives 
 
 fir- 
 
 = nS m 
 
 JQ Jo 
 
 Z 2 l c 
 
 cos 0sin 0,-^e -dd-dz, 
 
 where C denotes the coefficient of conduction of heat. The 
 integral in this equation can be shown to be equal to l c 
 (Section 34), which reduces the equation to 
 
 (105) 
 
 where n denotes the number of molecules crossing a square 
 cm. from one side to the other, l c the mean heat transfer 
 distance, and S m the internal specific heat at constant pres- 
 sure per molecule. If S ff denote the internal specific heat 
 per gram at constant pressure of the substance we have 
 
 S m =S g m a , ...... (106) 
 
 where m a denotes the absolute molecular weight of a mole- 
 cule, and the preceding equation may be written 
 
 C = nm a S a l c ....... (107) 
 
 By means of this equation the value of l c for any state of 
 
AN INFERIOR LIMIT OF THE FREE PATH 153 
 
 matter may be calculated, since C and S g may be measured 
 directly and n determined indirectly according to Section 29. 
 The quantity l c may be expressed in terms of other 
 quantities similarly as the quantity Z, in Section 34, by 
 representing the substance by another substance possessing 
 the same coefficient of conduction of heat, internal specific 
 heat at constant pressure, and expansion pressure, but 
 whose molecules possess no apparent volume 6. The coeffi- 
 cient of conduction of the representative substance is evi- 
 dently given by 
 
 C = n'maS ff l'c, 
 
 where n'and l' c referring to the representative substance 
 have meanings similar to n and l c in equation (107). Since 
 the representative substance has the same expansion pres- 
 sure as the original substance, and the molecules of the 
 former substance have no volume, we have according to 
 equations (38) and (45) that 
 
 v-b A/2' 
 
 where the quantities n, 6, P n , p, v, and A refer to the original 
 substance. The preceding equation may therefore be writ- 
 ten 
 
 C= -jnmaSil'c = "m a / c . . '. (108) 
 v o A. 
 
 This equation determines l' c since each of the other quan- 
 tities it contains may be determined from the original 
 substance. 
 
 The relation between l c and l' c according to equations 
 (107) and (108) is given by 
 
 (109) 
 
 and thus V c is an inferior limit of l c . 
 
154 THE NATURE OF MOLECULAR MOTION 
 
 Equation (108) may be given another form along the 
 same lines as equation (84) in Section 34. If the mole- 
 cules in the representative substance were devoid of molec- 
 ular forces which extend beyond a small distance from each 
 molecule, the quantity l c would vary inversely as the chance 
 of one molecule meeting another, or be proportional to the 
 volume v, and we may write 
 
 l'c=KcV, 
 
 where K C is a constant at constant temperature. The exist- 
 ence of molecular forces extending some distance from each 
 molecule gives rise to the interaction of two molecules not 
 being independent of the surrounding molecules, and hence 
 l' c not being proportional to v. This effect may be expressed 
 by a factor introduced into the right-hand side of the fore- 
 going equation, which is obtained in the same way as a 
 similar factor in Section 34. Since the chance of a mole- 
 cule interacting with another molecule and the pair coming 
 under the influence of r molecules, is proportional to 1/V +1 , 
 the factor in question may be written 
 
 The quantities <' c , 4>" c , . . . , which are functions of T 
 and v, insensitive to v, are not identical, it should be noted, 
 with the quantities <',,, </>" . . . , in Section 34, because 
 the quantities /, and l c have somewhat different meanings, 
 but they will probably not differ much from each other. 
 The quantities l c and V c are therefore given by the equations 
 
 and 
 
 l' c = Kc v(l+3> c ) ........ (112) 
 
VARIOUS FORMULA FOR HEAT CONDUCTION 155 
 
 Equation (111) may be interpreted similarly as equa- 
 tion (88). The interaction of two molecules is modified 
 by the surrounding molecules, an effect which was called 
 molecular interference. The value of the apparent molec- 
 ular volume b in the equation depends to some extent 
 (which is probably small) on molecular interference, while 
 $ c depends wholly on it. The latter quantity may therefore 
 be called the interference function in heat conduction. 
 In value it probably differs not much from that of the 
 interference function $,, in viscosity. Therefore according to 
 equation (111) the effect of molecular interference on l c which 
 is not represented by b (a quantity which also occurs in 
 equation (88)), is represented by the function 3> c , which 
 may now be used without referring to the representative 
 substance. 
 
 Equation (107) may therefore be written in the forms 
 
 (113) 
 
 I, .... (114) 
 
 o\v uj I 
 
 and 
 
 '<), (115) 
 
 by the help of equations (35) and (46), and remembering 
 that N c m a v = m. The values of n, b, and V t , the number of 
 molecules crossing a square cm., the apparent molecular 
 volume in cubic cms. per gram molecule, and the total 
 average velocity in cms. respectively, may be determined 
 by the method described in Section 29. The value of the 
 intrinsic pressure P, which must be expressed in dynes 
 similarly as the pressure p, may be determined by the 
 method described in Section 21. The internal specific 
 heat S g per gram at constant pressure, and the conduction 
 
156 THE NATURE OF MOLECULAR MOTION 
 
 of heat C per second across a square cm. at right angles to 
 unit heat gradient, must be expressed in terms of the same 
 heat units, which for convenience may be taken the calorie, 
 or the mechanical heat unit the erg. The calculated values 
 of n, 6, and V t depend on the distribution of velocities as- 
 sumed for the molecules; they may be corrected according 
 to Maxwell's law, if desired, in the way described. The 
 value of A in equation (115) not corrected for the distri- 
 bution of molecular velocities is given by 
 
 A =5.087X10 ~ 20 VTm t 
 according to Section 11. If corrected according to Maxwell's 
 
 /o~ 
 
 law the value has to be multiplied by \--, or 1.085. It 
 
 will appear, however, from Sub-section a that the calculated 
 value of K C similarly corrected involves the factor 1.085 and 
 that therefore the uncorrected values of A and K C may be 
 used in equation (115). Since p is independent of the 
 molecular distribution of velocities (Section 8), and P n is 
 obviously so by nature, the dependence of the factor K C /A 
 in equation (115) on this distribution is represented in 
 equations (114) and (113) by factors involving the quan- 
 tities n, K c , b, and V t . The uncorrected values of these 
 quantities may therefore be used in these equations. It 
 may be noted here that the quantity, S g is probably very 
 approximately, if not altogether, independent of the dis- 
 tribution of molecular velocities. This follows according 
 to Section 13 from the fact that it depends mainly on the 
 change in kinetic energy and change in potential energy of 
 attraction of a gram of substance during a degree change of 
 temperature, both changes being independent of the dis- 
 tribution of molecular velocities. 
 
 The value of K C in these equations for a given substance 
 is determined by applying equation (113) to the substance 
 
THE HEAT CONDUCTION EQUATION FOR A GAS 157 
 
 in the perfectly gaseous state, which corresponds to v= oo , 
 in which case the equation becomes 
 
 (116) 
 
 by means of equation (21), where S a is now equal to the 
 specific heat at constant volume. The quantity K C , it should 
 be noted, is not identical with the quantity K n . It is a char- 
 acteristic constant depending on the molecular forces and 
 volume of the substance. 
 
 The values of the quantities n, V t , 6, P n , p, and S e are 
 affected to a certain extent by the influence of the mole- 
 cules of the substance on two molecules while interacting, 
 or influenced by molecular interference. The quantity 3> c 
 therefore represents the portion of a similar effect on the 
 heat conductivity of a substance which is not represented 
 by the foregoing quantities in equations (113), (114), and 
 (115). These equations may now be used without referring 
 to the representative substance. 
 
 It follows similarly as in Section 34 in connection with 
 the quantity <$,,, that the terms of the series for 3> c decrease 
 rapidly in magnitude. As a first approximation we may 
 therefore retain only the term <f>' c /v 2 , or replace the series 
 by the term <f> c /v x , where x would differ little from 2. It 
 will readily be recognized that the quantities <f>' c , <f>" e , . . . , 
 in the series for < c are functions of T and v, which are not 
 very sensitive to v. 
 
 From equations (107) and (83) we have 
 
 which gives the ratio of the mean heat transfer distance 
 associated with a molecule of a substance to the mean 
 momentum transfer distance. This would furnish some 
 
158 THE NATURE OF MOLECULAR MOTION 
 
 information about the difference in the effect of molecular 
 interference on the chances of a migrating molecule losing 
 momentum or kinetic energy to a medium under the con- 
 ditions of viscosity and heat conduction respectively, since 
 these chances are respectively proportional to I//, and 
 l/lc. 
 
 On substituting for l c and l v in the foregoing equation 
 from equations (111) and (88), it becomes after rearranging 
 
 = ^4 (118) 
 
 The quantities K n and K C in this equation may be determined 
 from the viscosity and heat conduction of the substance in 
 the gaseous state, while the quantities C, S g , and 77 (which 
 refer to any state) may be determined directly. The equa- 
 tion may thus be used to compare the quantities 3> c and 
 <f> r Since K C and K^ do not depend on molecular interference 
 according to their definitions, and S g also does not from 
 its nature, while both 3> c and <, depend entirely on it, it 
 follows from equation (118) that an increase of <I> C rela- 
 tive to <!>,, is attended by an increase of C relative to 
 77. ' Thus this equation gives some information on the relative 
 effects of molecular interference on the heat conductivity 
 and viscosity of a substance. 
 
 A few applications of the foregoing investigation will 
 now be given. 
 
 (a) Equation (116) gives the coefficient of heat conduc- 
 tion of a gas. Since K C is independent of the density of the 
 gas, it follows from the equation that this also holds for the 
 quantity C. Experiment has demonstrated the truth of this 
 result. It would of course cease to hold when the dimen- 
 sions of the vessel in which conduction takes place are 
 comparable with the length of the mean heat transfer dis- 
 tance, according to the conditions on which the deduction 
 of the equation is based. 
 
THE EFFECT OF TEMPERATURE ON CONDUCTION 159 
 
 The dynamical mechanism underlying this result may 
 be illustrated by the following considerations. A molecule 
 in a gas moving in the direction of the flow of heat transfers 
 a certain amount of heat to the medium at the end of each 
 transfer distance. If the concentration of the gas is halved 
 the length of each transfer distance is doubled, while the 
 heat transferred at the end of each transfer distance is also 
 doubled since the temperature gradient remains the same. 
 Since a change in molecular concentration of a gas does not 
 alter the molecular velocities, the heat transferred per second 
 by a molecule moving between two walls at different tem- 
 peratures in the latter case is double that in the former. 
 But since the number of molecules per cubic cm. available 
 for heat transference in the latter case is half the number 
 available in the former case, the total heat transferred is the 
 same in each case. 
 
 The value of K C is found to increase with increase of tem- 
 perature, as is the case with the similar quantity K,, con- 
 nected with the viscosity of the substance. This is shown 
 for a few gases by Table XVIII. The reason is undoubtedly 
 the same as that holding in the case of viscosity, namely 
 that the greater the velocity of a molecule the shorter the 
 time it is under the influence of another molecule, and the 
 smaller its chance of transferring heat energy, this chance 
 being measured by l/l c , or I/K C . 
 
 The values of K C in the Table are not corrected for the 
 distribution of molecular velocities since equation (116) 
 was derived from equation (113) on substituting for n 
 the expression given by equation (21). If they are cor- 
 rected according to Maxwell's law each value has to be 
 
 /Q 
 
 multiplied by -\-jj-, or 1.085, according to Section 11. The 
 
 same remarks apply to the values of the quantity K, in the 
 Table. 
 
160 THE NATURE OF MOLECULAR MOTION 
 TABLE XVIII 
 
 C 2 H 4 . TO = 44 
 
 iC. 
 
 S g cal. per gm. 
 
 C10 3 cal./cm.2 sec. KC 10' 
 
 ic, lO'o 
 
 *</",, 
 
 
 100 
 
 .3245 
 .3403 
 
 .0395 2.64 
 .0506 2.77 
 
 2.08 
 2.38 
 
 1.27 
 1.16 
 
 CO 2 . TO = 28 
 
 
 100 
 
 .1541 
 .1724 
 
 .0307 
 .0506 
 
 3.45 
 4.36 
 
 2.42 
 2.92 
 
 1.43 
 1.49 
 
 H 2 . TO = 2 
 
 -150 
 
 
 2.41 
 2.41 
 
 .1175 
 .3270 
 
 5.91 
 11.04 
 
 5.86 
 6.69 
 
 1.00 
 1.65 
 
 A. TO =39.9 
 
 
 
 .07379 
 
 .03894 
 
 9.61 
 
 3.83 
 
 2.51 
 
 Hg. TO = 200.4 
 
 203 
 
 .01476 
 
 .01846 
 
 7.70 
 
 1.78 
 
 4.33 
 
 On applying equations (118) and (117) to the gaseous 
 state it follows that 
 
 since ^, = 0, and 4> c =0, under these conditions. The cal- 
 culation of the ratio K C /K V for different gases shows that it 
 
PROPERTIES OF THE CHARACTERISTIC FUNCTION 161 
 
 is not equal to unity, or the quantities K C and K,, are not 
 equal to each other, and that the value of the ratio depends 
 on the nature of the gas and its temperature, as is shown 
 by Table XVIII. We would expect that these quantities 
 should not be equal to each other since they do not mean 
 exactly the same thing. For 1/7 C , or l//c c , is a measure of 
 the chance of heat energy being transferred by a migrating 
 molecule to its medium under the conditions of heat conduc- 
 tion, while 1/Z,,, or I/*,, is a measure of the chance of mo- 
 mentum being transferred to the medium under the con- 
 ditions of viscosity. The former chance is smaller than the 
 latter in gases, since according to the Table KC/K,, or 
 
 Y is greater than unity. 
 
 tr, 
 
 It will be of interest to give a more definite and direct 
 physical significance of K C , and compare it with that obtained 
 for K n in Sub-section (a) of Section 34. The quantity C 
 represents the amount of energy which flows per second 
 across each square cm. of a plane in a substance situated 
 at right angles to a unit temperature gradient. Since n 
 molecules cross the plane per square cm. per second in each 
 direction, C/n represents the energy conveyed by a mole- 
 cule across the plane on crossing it and recrossing it (some- 
 time) in the opposite direction. According to equations 
 (105) and (111) this energy is equal to K c S m v in the case of 
 a gas, where the value of S m is independent of the pressure 
 of the gas. Thus K C represents the energy conveyed by 
 a molecule per unit of its specific heat at unit volume of a 
 gram molecule of the gas. The quantity K,,, we have seen, 
 represents the momentum conveyed by a molecule across a 
 plane at right angles to unit velocity gradient per unit mass 
 of the molecule at unit volume of a gram molecule of the gas. 
 
 It can be shown similarly as in Section 34 that a mole- 
 cule takes a time proportional to v/V in crossing and re- 
 crossing a plane. The amount of energy conveyed by a 
 
162 
 
 THE NATURE OF MOLECULAR MOTION 
 
 molecule per second across a plane under the foregoing con- 
 ditions is therefore proportional to K C 8 m V, and thus inde- 
 pendent of the volume of the gas. 
 
 (6) Equation (117) may immediately be applied to 
 liquids to obtain the values of the ratio ljli\. Table XIX 
 
 TABLE XIX 
 
 Substance. 
 
 t C. 
 
 S g cal. per gm. 
 
 r> 
 
 C cal./cm. 2 sec. 
 
 yi, 
 
 E. bromide .... 
 
 15 
 
 .2135 
 
 .004212 
 
 .0 3 247 
 
 .275 
 
 E. iodide 
 
 15 
 
 .1641 
 
 .006231 
 
 .0 3 222 
 
 .217 
 
 Chloroform .... 
 
 15 
 
 .237 
 
 .006019 
 
 .0 3 288 
 
 .202 
 
 M. alcohol. . . . 
 
 15 
 
 .5868 
 
 .006429 
 
 .0 3 495 
 
 .131 
 
 C. tetrachloride 
 
 
 
 .2010 
 
 .01351 
 
 .0 3 2664 
 
 .0981 
 
 C. disulphide . . 
 
 15 
 
 .242 
 
 .003905 
 
 .0 3 343 
 
 .363 
 
 Benzene 
 
 10 
 
 .4066 
 
 .007631 
 
 .0 3 333 
 
 .107 
 
 contains the values for a number of substances. It will 
 be seen that they depend on the nature of the substances, 
 and that they are smaller than unity. We have just seen 
 that when the substances are in the gaseous state this ratio 
 is greater than unity. This difference is due to the effect 
 of the attraction of the molecules of a substance on a migrat- 
 ing molecule increasing with the density of the substance. 
 It indicates that an increase of molecular interference in- 
 creases to a greater extent the chance of a molecule trans- 
 ferring heat energy than of transferring momentum under 
 the conditions of heat conduction and viscosity respectively, 
 since these chances are measured by l// c and 1/Z,,. 
 
 According to the foregoing result, and the result that 
 KC/K, is greater than unity, it follows from equation (118) 
 
 and (117) that 
 
 is less than unity, and that therefore 
 
 <$,, ><<;, or the interference function of viscosity is greater 
 than the interference function of heat conduction. Since 
 each of these quantities is zero for matter in the gaseous 
 
DETERMINATION OF INTERFERENCE FUNCTION 163 
 
 state, an increase in the molecular interference as brought 
 about by an increase of the density of the matter already 
 in a dense state would therefore increase <,, to a greater 
 extent than 3> c . Therefore according to equation (118) 
 and the considerations following it the value of 17 is increased 
 to a greater extent by the increase of molecular interference 
 brought about by an increase 01 density of the matter than 
 the value of C. 
 
 (c) Equation (115) may be used to obtain the value of 
 < c for a liquid, and hence the corresponding value of <f>' c 
 may be deduced. The value of P n +p required for the cal- 
 culations may be obtained from the coefficients of expansion 
 and compression of the liquid according to Section 21. 
 It may also be obtained by the method of Section 46. The 
 value of K C may be determined by means of equation (116), 
 which applies to the substance in the gaseous state. At 
 present the data are not sufficient to permit such a calcula- 
 tion being carried out for a substance. 
 
 (d) If the values of C and S g be known for three differ- 
 ent densities of a substance not differing much from each 
 other, and the value of K C be calculated by means of equa- 
 tion (116), the values of b and </>' c may be determined by 
 means of a modified form of equation (113) obtained on 
 substituting for 3> c the approximate expression <A' C A 2 , and 
 for n the expression given by equation (66) retaining only 
 the first two terms. The equation will contain the three 
 constants, b, 2, and <// c , which may be determined by apply- 
 ing the equation to the substance at three different densi- 
 ties. This operation also determines incidentally the quan- 
 tities n and V t similarly as in Section 29. 
 
 39. Formulce for the Coefficient of Conduction 
 of Pleat of a Mixture of Substances. 
 
 It will be evident from a consideration of the investiga- 
 tion in the previous Section that in the case of a mixture 
 
164 THE NATURE OF MOLECULAR MOTION 
 
 the heat conductivity is equal to the sum of the effective 
 conductivities of the constituents. Hence in the case of a 
 mixture of molecules r and e we may according to equation 
 (105), write for the coefficient of conductivity 
 
 C = n r S m rlcr+n e S me l ce , .... (120) 
 
 where S mr and S me denote the partial molecular specific 
 heats of a molecule r and e respectively (Section 13), l cr 
 denotes the mean heat transfer distance of a molecule r, 
 Ice that of a molecule e, and n r and n e have the usual meanings. 
 Similarly as in Section 35 we may write 
 
 and 
 
 t- -V ^r (!+*'), . . . . (122) 
 
 l/e t/ e IV er 
 
 where the symbols have similar meanings, that is, v r denotes 
 the volume of the mixture containing a gram molecule 
 of molecules r, N& the total concentration of the molecules 
 e and r r , b' r the apparent molecular volume of the mixture 
 which appears to obstruct the motion of the molecules r, 
 K'CT the characteristic factor of the transfer path of a mole- 
 cule r when the mixture is in the gaseous state, and && 
 the heat conductivity interference function with respect 
 to a migrating molecule r, while the other symbols have 
 similar meanings. As a first approximation we may write 
 as before 
 
 fy 
 
 and 
 
 */ ^ ^ 
 
THE CONDUCTIVITY OF GASEOUS MIXTURES 165 
 
 where N denotes the number of molecules in a gram mole- 
 cule of a pure substance, and <$>' and 0' cc are functions of 
 T and Ner but which are insensitive to variations in Ner- 
 On substituting from equations (122) and (121) for la 
 and l ce in equation (120) and applying it to the gaseous 
 state, it becomes 
 
 c= '^ +MS '' (123) 
 
 by the help of equation (21), and since 
 
 V e N N ,, S mr , S me 
 
 = -, -- h = N er , bar = - , and Sft = - , 
 
 V r V e V r m a r Wlae 
 
 where Sgr and S ge denote the partial specific heats per gram 
 of the molecules r and e respectively, the ratio of the 
 number of molecules r to the number of molecules e whose 
 relative molecular weights referred to the hydrogen atom 
 are m r and m e , and N denotes the number of molecules in 
 a gram molecule of a pure substance. When the mixture 
 is in the perfectly gaseous state the values of Sgr and S ge are 
 the same as for the pure substances in the gaseous state at 
 constant volume. Equation (123) gives a relation between 
 the quantities *' and *'> which is of use in their determina- 
 tion. 
 
 The various quantities contained in the foregoing equa- 
 tions are evidently functions of , the ratio of the constitu- 
 ents. With the exception of the quantities v r , v e , b' e , and 
 6' r , they are not identical with the quantities contained in 
 the equations of Section 35. 
 
 The values of K' ' and /. may be approximately calcu- 
 lated from the values of K CT and K ce referring to the sub- 
 stances isolated and in the perfectly gaseous state, by means 
 of a formula similar to equation (102) in Section 35. The 
 values thus obtained may be checked by means of equa- 
 
166 "THE NATURE OF MOLECULAR MOTION 
 
 .tion (123). Similarly the values of <f>' cr and 4>' C e may approxi- 
 mately be calculated by a formula similar to equation 
 (103). The foregoing quantities may also be determined 
 directly by the method given in the Section mentioned. 
 
 (a) It is instructive to apply equation (120) to the con- 
 duction of heat in metals. Since the molecules of a metal 
 are more or less in relatively rigid positions the heat energy 
 must in the main be transferred by carriers which are able 
 to move about more freely than the molecules. It has 
 been supposed that these carriers consist of electrons in the 
 free state which arise through a spontaneous dissociation 
 of the molecules into electrons and positively charged mole- 
 cules. Equilibrium exists when the number of electrons 
 produced in this way per second is equal to the number 
 neutralized through combination with oppositely charged 
 parent molecules. The state of equilibrium may therefore 
 be expressed by the equation of mass-action. 
 
 where N n denotes the number of neutral molecules per 
 cubic cm., N e the number of electrons or positively charged 
 molecules per cubic cm., and K denotes the constant of mass- 
 action. This constant is a function of the temperature, 
 density, and nature, of the metal. 
 
 If the molecules of the metal are designated by r and 
 the electrons by e in equation (120), we have n r =0, since 
 the molecules are more or less in relatively fixed positions, 
 and the equation becomes 
 
 C = n e S me lce ...... (124) 
 
 The value of l ce is probably governed by the spacing of the 
 molecules of the metal, since the size of an electron is very 
 small in comparison with that of an atom. As a first approxi- 
 mation we may therefore take l ce equal to the distance of 
 
THE MIGRATION OF ELECTRONS IN A METAL 167 
 
 separation of the atoms in a metal, which is usually of the 
 order 10 ~ 8 cm. The partial specific heat S me of an electron 
 probably differs little from \ m a F 2 = 2.012X10- 16 7 7 ergs, 
 its specific heat at constant volume in the gaseous state. 
 Whatever its value, it cannot be greater than the specific 
 heat per atom of the metal, which is equal to twice the 
 specific heat it would have in the gaseous state. We may 
 therefore with a fair degree of certainty calculate by means 
 of equation (124) the order of magnitude of n ey the num- 
 ber of electrons crossing a square cm. per second from one 
 side to the other in a metal. Thus for example in the case 
 of copper we have 
 
 C = .7198 cal. S me = 4.808X10- 24 caL, 
 and 
 
 \ 3 = 2.25XlO- 8 cm., 
 which gives 
 
 n e = 9.15X!0 30 . 
 
 It is of interest to compare this number with the number of 
 molecules crossing a square cm. from one side to the other 
 per second in a liquid or dense gas. Thus in the case of 
 CO2 at 40 C. under a pressure of 200 atmos. this number is 
 equal to 7.8X10 26 according to Table VIII. The density 
 of the C02 under these conditions is about equal to unity. 
 The value of l ce according to the foregoing considerations 
 would increase with increase of temperature in the case 
 of the pure metals, since they expand with increase of tem- 
 perature, and the increase may probably be taken approxi- 
 mately equal to the increase in the distance of separation 
 of the atoms. S me is very likely practically independent of 
 the temperature. Therefore for the cases that C decreases 
 with increase of temperature this very likely also holds for 
 n e according to equation (124). Some of the metals falling 
 
168 
 
 THE NATURE OF MOLECULAR MOTION 
 
 into this category are: lead, cadmium, iron, silver, bis- 
 muth, zinc, and tin. 
 
 Other investigations having a bearing on the electrons 
 in a metal will be found in Section 42. 
 
 40. The Coefficient of Diffusion of a Substance. 
 
 If the constituents of a mixture of substances are not 
 evenly distributed, diffusion from one part to another 
 will take place till equilibrium is obtained. The primary 
 
 Distance 
 FIG. 16. : 
 
 cause of diffusion is the motion of translation of the mole- 
 cules. This will at once appear from the consideration of 
 a mixture whose constituents r and e are distributed accord- 
 ing to the curves shown in Fig. 16, the concentration being 
 supposed uniform in planes parallel to the concentration 
 axes. Molecules are projected from each element of volume 
 of the mixture. The portion of the mixture between the 
 planes ab and cd will therefore receive more molecules r 
 from the portion of the mixture between the planes cd and 
 ef, on account of the concentration gradient of the mole- 
 cules r, than vice versa. Thus on the whole a migration 
 of molecules r in the direction of decrease of concentration 
 would take place. Similarly it follows that the molecules 
 
THE COEFFICIENT OF DIFFUSION 169 
 
 e on the whole migrate in the direction of decrease of con- 
 centration of the molecules e, or in the opposite direction 
 to the molecules r. If d r denote the number of molecules r 
 which on the whole are transported across a square cm., 
 
 and c denotes the concentration gradient, we have 
 8x 
 
 8 r = D r ., ...... (125) 
 
 where D T denotes the coefficient of diffusion of the molecules 
 r. When the concentration gradient is unity D r = 8 r . It 
 is evident that the coefficient of diffusion should depend on 
 the density of the mixture, the ratio of the masses of the 
 constituents, and the concentration gradient, at the point 
 the coefficient is measured. It is found, however, to be 
 approximately independent of the concentration gradient 
 over a considerable range of gradients. 
 
 41. The Mean Diffusion Path of a Molecule in 
 a Mixture. 
 
 The path of a molecule in a mixture in migrating from 
 one place to another is undulatory in character on account 
 of the interaction between the molecules due to the existence 
 of molecular forces and molecular volume. It differs the 
 more from a number of straight lines of various lengths 
 joined consecutively together at various angles, the greater 
 the density of the substance. The effect of this path on the 
 rapidity of the diffusion of a molecule from one place to 
 another may be represented by another path along which 
 the molecule is supposed to move, consisting of straight 
 lines joined together and lying near the path, and inci- 
 dentally intersecting it, as shown by Fig. 14. In supposing 
 that a molecule passes along its representative path it is 
 obvious of course that in general the molecule at any in- 
 stant does not occupy its actual position. But this does 
 
170 THE NATURE OF MOLECULAR MOTION 
 
 not matter when we are dealing with the behavior of a large 
 number of molecules as a whole. The straight lines consti- 
 tuting the representative path will be called the diffusion 
 free paths of the molecule. The average length of these 
 paths is left arbitrary to a certain extent by their defini- 
 tion, and we will therefore impose the conditions that the 
 sum of the representative free paths between any two points 
 a considerable distance apart is equal in length to the cor- 
 responding actual path, and that each direction of a free 
 path of given length is equally probable. These conditions 
 completely determine the magnitude of the mean free path, 
 as will appear from the next Section. These paths are not 
 equal in magnitude but are grouped about the mean free 
 path l d according to the law given in Section 31. 
 
 42. Formulae Expressing the Coefficient of Dif- 
 fusion in Terms of Other Quantities. 
 
 Let us consider a heterogeneous mixture of molecules 
 r and e in which the molecules are uniformly distributed 
 in planes parallel to the plane AB in Fig. 15, but non-uni- 
 formly distributed at right angles to the plane so that a 
 diffusion of molecules r towards the parallel plane CD takes 
 place, while a diffusion of molecules e takes place in the 
 opposite direction. We may suppose, to simplify the rea- 
 soning, that the molecular paths which cut the plane EF 
 in migrating towards the plane CD, begin their journey in 
 the former plane at the same point. If n r denote the total 
 number of molecules r crossing the plane EF per square 
 cm. from one side to the other, which is equal to the num- 
 ber of times the plane is cut by the corresponding free paths, 
 the number whose paths make an angle 8 with a perpendic- 
 ular to the plane is similarly, as in Section 34, given by 
 
 n r sin 6 - d6. 
 
MOLECULAR MIGRATION ACROSS A PLANE 171 
 
 The number of the foregoing molecules whose paths lie 
 between z and z-\-dz is equal to 
 
 Z _L 
 n r sin e-dd-^-e I 8r -dz, 
 
 * 6r 
 
 according to Section 31, where l sr denotes the mean dif- 
 fusion path of the molecules r. This number of molecules, 
 and the corresponding number of molecules moving in the 
 opposite direction, which we may suppose begin their journey 
 in the plane GH in Fig. 15, are respectively proportional 
 to the molecular concentrations of the molecules r in the 
 planes EF and GH. If N r denotes the concentration in 
 the plane EF, the concentration in the plane GH is evidently 
 
 . dNr , dN r ,, 
 
 N r z cos 0- , where : is the concentration gradient 
 dx dx 
 
 measured in the direction of increase of concentration. 
 Therefore the loss in molecules r in the plane EF is equal to 
 the difference in the foregoing concentrations divided by 
 NT, and multiplied by the preceding number of molecules, 
 which gives for the loss 
 
 1 dN r Z 2 ~Tfo 
 
 TT? r~ r w r sin 8 cos B^e r -dz- dd. 
 N r dx Pi, 
 
 The total loss of molecules r in the plane is obtained by 
 integrating the foregoing expression from to oo with respect 
 to the free path z, and from to 7r/2 with respect to the angle 
 6. On referring to similar integrals in Sections 34 and 38 
 it will be evident that the integral in question is equal to 
 
 rirhr dN r 
 Nr dx' 
 
 Similarly it can be shown that the gain of molecules e in the 
 plane EF is equal to 
 
 Tvfife' 1 
 
172 THE NATURE OF MOLECULAR MOTION 
 
 where the concentration gradient of the molecules e is meas- 
 ured in the direction of their increase of concentration, 
 and thus in the opposite direction of the gradient of the 
 molecules r. The total loss of molecules r and e in the plane 
 EF is therefore equal to 
 
 ~W r ~~dx' ~N7~dx' 
 
 which may be written M T M e . When the molecular paths 
 are in their original positions the foregoing number repre- 
 sents, on the whole, the gain in molecules on the lower 
 side of the plane EF. But the space occupied by these 
 molecules is not zero. A portion of the mixture must there- 
 fore be transported bodily across the plane in the oppo- 
 site direction to make room for the foregoing molecules. 
 Let P v denote the volume of this portion of the mixture. 
 The total number of molecules r transported on the whole 
 across the plane per square cm. in the direction of decrease 
 of concentrtaion gradient, which is equal to the diffusion 
 6 r , is accordingly given by 
 
 Hrhr dNr P,N r 
 
 Nr dx Ne+Nr 
 
 Let # r and & e denote the external molecular volumes of a 
 molecule r and of a molecule e respectively in the mixture, 
 i.e., the decrease in the volume of the mixture when a mole- 
 cule r or e at constant pressure is removed. These quanti- 
 ties are evidently connected by the equation 
 
 The quantity P v is therefore given by the equation 
 
THE GENERAL DIFFUSION EQUATION 173 
 
 or 
 
 '+ 
 
 On substituting this expression for P v in the preceding dif- 
 fusion equation, and then substituting the expressions for 
 M e and M r , the equation becomes 
 
 . _ e rhrNe dN T n e l &e N T dN e 
 
 ~N r &r + Ne# t \ N r dx :""* N e dx 
 
 which is a fundamental form of the diffusion equation. 
 The rate of diffusion of the molecules e, which takes place 
 in the opposite direction to the molecules r, is obtained on 
 interchanging the suffixes r and e in the foregoing equation. 
 It will readily be seen that the rates of diffusion for the 
 two different kinds of molecules may be written 5 r = K& e , and 
 d e = K& r , and hence the ratio of the rates of diffusion is 
 given by 
 
 H ........ (127) 
 
 In the case of a liquid mixture the quantities &, and # c are 
 not equal to each other. They may easily be measured in 
 practice since 
 
 8v dv 
 
 where v denotes the volume of the mixture. 
 
 In the case of a gaseous mixture d e = $ r , and N e +N r =* 
 C er a constant since the pressure is everywhere the same. 
 Equation (126) in that case becomes 
 
 r sre , e r T n OQ >. 
 
 -~ ~~ 
 
 If the concentration of one of the constituents of the 
 mixture is small in comparison with that of the other, say 
 
174 THE NATURE OF MOLECULAR MOTION 
 
 of the molecules r in comparison with the molecules e, 
 equation (126) becomes 
 
 The rate of diffusion is then independent of $ r and $ e > 
 
 The forms of the foregoing equations may be modified 
 by considering similarly as in the four previous Sections 
 a representative mixture which has the same rate of diffusion, 
 expansion pressure, and molecular volumes $ r and $ e , but 
 whose molecules do not possess an apparent molecular 
 volume 6 whose defining property according to Section 19 
 is that a change in its value produces a change in the external 
 pressure without changing the total average velocity. It 
 should be carefully noted that the quantities b and d do not 
 mean the same thing, one may be zero when the other is 
 not. The former quantity represents the obstruction the 
 molecules present to each other's motion, while the latter 
 quantity represents the change in the external volume of 
 a mixture on adding a molecule. It will be convenient 
 now to replace the volume v by the concentration N er 
 similarly as in Sections 35 and 39. The molecular free paths 
 of the molecules r and e may then be expressed similarly 
 as before by the equations 
 
 v r NK' 
 
 and 
 
 V r 
 
 \ 
 ) 
 
 . . . . (130) 
 
 where v r denotes the volume of the mixture containing a 
 gram molecule of molecules r, Ner the total concentration 
 of the molecules r and e, b' r the apparent molecular volume 
 of the mixture which appears to obstruct the motion of the 
 molecules r, K.' & the characteristic factor of the diffusion 
 
THE DIFFUSION EQUATION FOR A GAS 175 
 
 free path of a molecule r when the mixture is in the gaseous 
 state, and <' 5r the diffusion interference function with respect 
 to a migrating molecule r, while the other symbols have 
 similar meanings. As a first approximation we may write 
 similarly as before 
 
 where N denotes the number of molecules in a gram mole- 
 cule of a pure substance, and </>' 8r and </>' Se are functions of 
 T and Ner, but which are insensitive to variations in N er . 
 On applying equations (130) to the gaseous state they 
 become 
 
 I =K' 
 
 and 
 
 N 
 
 Substituting for l dr and I 8e from these equations in equation 
 (128) and obtaining expressions for n r and n e by means of 
 equation (21) the equation becomes 
 
 N RT{N e K' 8 r NrK' 8e \dN r 
 
 r ~ *f2~\lo~\ 7=~i -- T^r^T"' 
 N 2 er \ 3 \\/m r Vm e \ dx 
 
 since N e -}~Nr = Ner, v r m a rNr = m r , and v e maeNe = m e , where 
 mar and m ae denote the absolute and m r and m e the relative 
 molecular 'weights of the molecules r and e respectively. 
 This equation applies to the gaseous state and gives the 
 relation between the characteristic quantities K r dr and 
 K' Se which are functions of the temperature and the nature 
 of the molecules. 
 
176 THE NATURE OF MOLECULAR MOTION 
 
 Since l Sr =K' Sr N/N r when the mixture is in the gaseous 
 state, it will be evident from an inspection of Fig. 14, which 
 shows the relation between the free paths of a molecule 
 and its actual path, that l/l Sn or !/*'&. is a measure of the 
 chance of the resultant force on a migrating molecule passing 
 through a maximum or minimum and changing its direction. 
 
 When the mixture is in the gaseous state and the con- 
 centration of one set of molecules is relatively small, another 
 interesting significance may be attached to the quantity 
 K' 8r . Equation (129) then applies, and it indicates that for 
 
 the same concentration gradient per molecule, or f = 
 
 constant, the foregoing quantity is proportional to d r /n r . 
 Therefore, since n r denotes the number of molecules crossing 
 per second per square cm. a plane situated at right angles 
 to the direction of diffusion, and d r denotes the rate the 
 molecules diffuse across the plane, the quantity K' Sr is a 
 measure of the chance of a molecule which has crossed the 
 plane from one side to the other, to remain on the latter 
 side. 
 
 The quantities & Sr and $' Se in the foregoing equations 
 express the effect of the molecular interference of the mixture 
 on two interacting molecules in changing the value of 5 r , 
 which is not expressed by the other quantities which, are 
 similarly affected. 
 
 The values of n r , n e , b' r , and b' e in the foregoing equations 
 may be found by the methods of Section 29, or by those 
 given in Section 46. The values thus obtained may be cor- 
 rected according to Maxwell's distribution of molecular 
 velocities in the way described. The values of the other 
 quantities which cannot be measured directly may be 
 determined by methods which will now be described. 
 
 In the case of a dilute solution of molecules r in e the 
 value of l dr is immediately given by equation (129) since 
 8 r may be measured directly. If the coefficient of diffusion 
 
CHARACTERISTIC FUNCTION APPROXIMATION 177 
 
 for the same kind of mixture in the gaseous state were 
 measured, or otherwise were known, the value of K' ST would 
 be given by equation (133) to which equation (129) can 
 be reduced. The value of <' Sr could then be obtained from 
 equations (130). 
 
 In the case of a mixture in which one of the constituents 
 is not small in comparison with the other the quantities 
 K! Sr and K' 8e may be determined by means of equation (131). 
 Thus we may write 
 
 1 ai .l 
 
 K'ST N er 
 
 and 
 
 similarly as in Section 35, and substitute from these equa- 
 tions for the foregoing quantities in equation (131). The 
 three constants a r , a e , and a re , at constant temperature 
 may be determined by applying the resultant equation 
 to three diffusing gaseous mixtures of different relative 
 concentrations, which furnishes three simultaneous equa- 
 tions. The values of K' Sr and K' de may then be calculated 
 for any relative concentration of the molecules by means 
 of the foregoing two equations. 
 
 An interesting special case of such calculations corresponds 
 to NeQ. The resultant value of K' Sr may be used to calcu- 
 late the coefficient of diffusion of a molecule r in a gas of 
 the same kind (a quantity which cannot be measured 
 directly) by means of equation (131) putting JV e =0. 
 
 The values of K.' Sr and K' Se obtained by the foregoing 
 method would not be corrected for the distribution of 
 molecular velocities. If this correction is carried out accord- 
 ing to Maxwell's law each value has to be multiplied by 
 1.085. 
 
178 THE NATURE OF MOLECULAR MOTION 
 
 The quantities <j>' 5r and </>' 8e may now be determined by 
 means of equation (126) on substituting for <t>' 5r , <J>' ar , 
 I 5n and I 8e in the equation from the equations defining these 
 quantities, and for 0' Sr and cf> f Se from the equations 
 
 /V 
 
 and 
 
 deduced similarly as similar equations in Section 35. An 
 equation is obtained containing the four constants b e , b r , 
 bree, and brre at constant temperature. Therefore on apply- 
 ing the equation to four dense diffusing mixtures of dif- 
 ferent relative concentrations of the molecules r and e four 
 simultaneous equations will be obtained from which these 
 constants can be determined. The values of 0' 5r and </>' 8e 
 may then be calculated for any relative concentration of 
 the molecules r and e in the mixture by means of the fore- 
 going equations. An interesting special case of such a 
 calculation is that corresponding to N e = 0. 
 
 The number of times N s a molecule passes over its 
 mean free diffusion path l s , which is inversely proportional 
 to the number of times the force on a molecule passes 
 through a maximum and changes its direction of motion, 
 is given by 
 
 tf,-f', ...... (132) 
 
 ^5 
 
 where V t denotes the total average velocity of a molecule. 
 This equation follows from the fact that the representative 
 path of a molecule is equal in length to the actual path. 
 
 (a) As an application of the foregoing investigation let 
 us consider the diffusion of molecules r and e in the gaseous 
 state into each other. If the concentration of the mole- 
 
DIFFUSION IN DILUTE MIXTURE OF GASES 179 
 
 cules r is small in comparison with that of the molecules 
 e the coefficient of diffusion D r of the molecules r is accord- 
 ing to equations (125), (129), and (130), given by 
 
 Now according to equations (35) and (8) 
 
 ~3 3"\"^7' 
 and the foregoing equation may therefore be written 
 
 (134) 
 
 The coefficient of diffusion is therefore inversely propor- 
 tional to the total molecular concentration, since K.' Sr is 
 independent of the volume of the mixture. This result is 
 borne out by experiment. 
 
 The foregoing equation may also be written 
 
 . . . (135) 
 
 m r 
 
 since ]V er = 7.46Xl0 15 p/T r according to equation (15), 
 N = 6.2X10 23 , and R = 8.315 X10 7 , where p denotes the 
 external pressure. This equation was used to calculate 
 the values of K S given in Table XX for the inter-diffusion 
 of a number of different gases, where for convenience and 
 simplicity K S is now written for K f dn the values of D used cor- 
 responding to p = 760 cm. of mercury. The directions of 
 the diffusion to which the coefficients in the Table refer 
 are indicated by arrow heads. 
 
 It will be seen on inspecting the table that K S increases 
 with increase of temperature for the same two diffusing 
 gases. The chance of the resultant force on a migrating 
 
180 
 
 THE NATURE OF MOLECULAR MOTION 
 
 molecule per unit length of path passing through a maximum 
 and changing in direction is thus decreased by an increase 
 of temperature. Actually this means that some of the 
 smaller maxima or bends of the molecular path are more or 
 less smoothed out by an increase of temperature. This 
 would happen because the time any pair of interacting mole- 
 cules are under each other's influence is decreased by an 
 increase of temperature, since this increases the molecular 
 velocities, and hence the amount of deflection each mole- 
 cule undergoes is decreased. 
 
 TABLE XX 
 
 t C. 
 
 D 
 
 * 5 10' 
 
 D 
 
 Kg 1010 
 
 
 H 2 0-+C0 2 
 
 H 2 0->H 2 
 
 
 92.4 
 
 .132 
 .2384 
 
 2.84 
 3.31 
 
 .687 
 1.179 
 
 14.8 
 16.3 
 
 
 CH 4 O-CO 2 
 
 CH 4 O H 2 
 
 
 
 49.6 
 
 .0880 
 .1234 
 
 2.52 
 2.75 
 
 .5001 
 .6738 
 
 14.3 
 15.0 
 
 
 C 6 H 6 ^C0 2 
 
 C 6 H 6 -H 2 
 
 
 45 
 
 .0527 
 .0715 
 
 2.36 
 2.54 
 
 .294 
 .3993 
 
 13.1 
 14.2 
 
 
 C 9 H 18 O 2 -CO 2 
 
 C 9 H 18 2 -H 2 
 
 
 
 97.8 
 
 .0305 
 .0568 
 
 1.94 
 2.29 
 
 .1724 
 .3177 
 
 11.0 
 12.8 
 
PROPERTIES OF THE CHARACTERISTIC FUNCTION 181 
 
 It appears also that for the same medium the value of 
 KS decreases with an increase of molecular weight of the 
 diffusing molecule. This indicates that the chance of the 
 resultant force on a migrating molecule per unit length 
 of its path passing through a maximum and changing in 
 direction is increased by an increase in the molecular weight 
 of the molecule. The relative change in K S , it will be 
 noticed, is considerably smaller than the relative change in 
 the molecular weight. Thus an increase in molecular 
 weight acts in the opposite direction to an increase in tem- 
 perature on the value of K,,, as we might expect. It is 
 evident that an increase in molecular weight increases the 
 time the interacting molecules are under each other's influ- 
 ence, since this is attended by a decrease in their velocities, 
 and the force that they exert upon each other is also in- 
 creased. The values in the Table are not corrected for 
 Maxwell's law of distribution of molecular velocities the 
 correction corresponding to the introduction of the factor 
 1.085. 
 
 If the quantities n e and n r in equation (128), which applies 
 to the gaseous state, are expressed in terms of the correspond- 
 ing molecular velocities, the equation assumes the form 
 usually given in treatises on the Kinetic Theory of Gases. 
 The quantities l sr and l se are supposed to refer, however, 
 to the free paths of molecular collision of the molecules r 
 and e respectively, and therefore have not the same funda- 
 mental meaning given to them in this book. 
 
 (6) An interesting application of equation (129) may 
 be made in connection with the electrons in a metal. We 
 may write N e = K e n e , where N e denotes the number of 
 free electrons per cubic cm. in a metal, and n e the number 
 crossing a square cm. from one side to the other in all direc- 
 tions per second. A change in N e evidently produces a 
 change in n e through the increase in concentration of the 
 electrons and the change in their interaction upon each other. 
 
182 THE NATURE OF MOLECULAR MOTION 
 
 It will not be difficult to see that the change in n e from the 
 former .cause is large in comparison with that due to the 
 latter when the change in concentration is small. We may 
 therefore consider K e a constant for small changes of N e . 
 Equation (129) applied to the diffusion of electrons in a 
 metal may therefore be written 
 
 . dn e 
 
 Since n e is equal to the partial expansion pressure P e of 
 
 the electrons according to Section 20 the foregoing eauation 
 may be written 
 
 where 7-^ is evidently the force acting on a cubic cm. of 
 
 diffusing electrons. We may therefore suppose that the 
 electrons are uniformly distributed and this force produced 
 by an electric field X, so that 
 
 dPe 
 
 -r-, 
 
 ax 
 
 where e denotes the electric charge on an electron, and the 
 equation therefore given the form 
 
 2l 8e XeN e 
 
 8 e = - I . 
 
 The current / produced by the electric field is given by 
 
 where k denotes the electric conductivity of the material; 
 and the foregoing two equations therefore give 
 
 2. 
 
THE CONCENTRATION OF ELECTRONS IN METALS 183 
 
 where m e denotes the mass of an electron relative to the 
 hydrogen atom, and A = 5.Q87XlQ~ 20 V~Tm e . This equa- 
 tion may be used to calculate the number of electrons per 
 cubic cm. of a metal. 
 
 The value of the free diffusion path l Se of an electron in 
 a metal is probably in the main governed by the presence 
 of the atoms, and as a first approximation may therefore 
 be taken equal to the distance of separation of the atoms. 
 Approximate values of N e may thus be obtained. Thus 
 for example in the case of copper at C. & = 7.7X10~ 4 
 E.M.U., and since m=.001, e=1.6X10~ 20 E.M.U., T = 
 273, and l de = 2.25XlO~ 8 approximately, we obtain 
 
 N e = 2 X 10 24 approximately. 
 
 Equation (136) gives definite information about the 
 variation of N e with the temperature. Since k for the pure 
 metals varies approximately inversely as the absolute 
 temperature, it follows from the equation that 
 
 Therefore, since l Se can only increase with increase of tem- 
 perature, the value of N e decreases with increase of tem- 
 perature in the case of the pure metals. This result may 
 of course not hold in the case of alloys whose conductivity 
 usually increases with increase of temperature. 
 
 Experiment shows that the conductivity of a metal 
 is greatly affected by small amounts of impurities. These 
 should not alter the value of l se to an appreciable extent. 
 The change in conductivity must therefore be due according 
 to equation (136) to a change in N e induced by the impuri- 
 ties. This is probably brought about by the impurities 
 affecting the rate of dissociation of the atoms of the metal, 
 and therefore the corresponding constant of mass-action. 
 
184 
 
 THE NATURE OF MOLECULAR MOTION 
 
 From equations (136) and (124) we obtain 
 
 C 2.543 X W- 20 n e lceS m eVTme 
 k ~ N e l 8e e 2 
 
 This equation gives the value of the ratio 
 
 (137) 
 
 * 
 
 the quantities C and k can be measured directly, and the 
 remaining quantities are known constants. The ratio C/k 
 has approximately the same value for all pure metals at 
 the same temperature, and is approximately proportional 
 to the absolute temperature T. This is shown by Table 
 XXI, which was taken from Richardson's Electron Theory 
 
 TABLE XXI 
 
 Material. 
 
 Values of 
 C/k at 18 C. 
 
 Temp. Coef. 
 of Ratio. 
 
 Copper (pure) 
 
 6.65 X10 10 
 
 3.9X10" 3 
 
 Silver (pure) 
 
 6.86 X10 10 
 
 3.7X10" 3 
 
 Gold (pure) 
 Nickel (pure) ... 
 
 7.27 X10 10 
 6.99 X10 10 
 
 3.6X1Q- 3 
 3.9X10" 3 
 
 Zinc (pure) 
 
 7.05 X10 10 
 
 38X10" 3 
 
 Cadmium (pure) 
 
 7.06 X10 10 
 
 3 7X10~ 3 
 
 Lead (pure) 
 
 7.15X10 10 
 
 4.0X10" 3 
 
 Tin (pure) . ... 
 
 7.35 X10 10 
 
 3.4X10" 3 
 
 Aluminium 
 
 636X10 10 
 
 43X10" 3 
 
 Platinum (pure) 
 
 7.53X10 lft 
 
 4.6X10" 3 
 
 Palladium 
 
 7.54 X10 10 
 
 4.6X10" 3 
 
 Iron 
 
 8.02 X10 10 
 
 4.3X10" 3 
 
 Bismuth 
 
 9.46 X10 10 
 
 1.5X10" 3 
 
 
 
 
 of Matter. Since // is very probably independent of 
 the temperature, and this is also likely to hold approxi- 
 mately for S me , the partial specific heat of the electrons, it 
 follows that approximately 
 
APPROXIMATE CONDUCTIVITY EXPRESSION 185 
 
 Another expression for the electric conductivity of a 
 metal, which is well known, may be obtained as follows: 
 Consider a metal in which the electrons are under the 
 action of an electric field of intensity X. Let I denote the 
 length of the actual path of an electron at the end of which 
 it possesses the same amount of kinetic energy as at the 
 beginning, being the path over which all the energy imparted 
 to the electron by the electric field is imparted to the sur- 
 rounding electrons and molecules of the metal. It will 
 be noticed that again we define molecular path without the 
 introduction of molecular collision, as this is more satisfac- 
 tory. If V t as usual denotes the total average velocity 
 (Section 17) of an electron, the time t it takes to traverse 
 the distance I is given by t = l/V t . The effect of the electric 
 field is to give an acceleration equal to Xe/m ae to the electron, 
 whose absolute mass is m a e. If we suppose that this is 
 unimpeded along the path of the electron, it will have 
 
 Xet 
 a component velocity equal to - - at the end of the path. 
 
 m a e 
 
 This causes a drift of the electrons which is approximately 
 
 Xet 2 
 equal to - during the time t. The average velocity of 
 
 the drift is therefore "- , or - . The electric current 
 2m ae 2m a eV t 
 
 in the metal is therefore given by 
 
 and accordingly the electric conductivity k by 
 
 NJl _N?<?1 
 
 "'"oZ; Tr~aZ^.^~> .... V 100 ^ 
 
 by the help of equation (35). 
 
... 186 THE NATURE OF MOLECULAR MOTION 
 
 If the supposition is made that I = l c , and that the electrons 
 behave as if they were in the perfectly gaseous state,* the 
 ratio C/k formed from equations (138) and (105) is pro- 
 portional to the absolute temperature T, and its value 
 agrees approximately with that found by experiment. It 
 is hardly likely, however, that the suppositions made are 
 true, since the theoretical ratio shows great deviations 
 from the experimental when the metals contain small 
 amounts of impurities, which should not affect the supposi- 
 tions made. Moreover, it is obvious that in general I can- 
 not be equal to l c . What we can say with certainty only 
 is that the theoretical ratio of C/k has a factor of T which 
 for pure metals has a value corresponding to the electrons 
 behaving as if they were in the perfectly gaseous state, 
 according to the results of experiment. 
 
 From equations (138) and (136) on eliminating k we 
 have 
 
 I 2.545X10- 20 \/rm^ 
 
 l ~ 
 
 This equation identically vanishes if we assume that I = 2l 5e , 
 and that V t is given by equation (8), which corresponds 
 to the electrons behaving as if they were in the perfectly 
 gaseous state. Thus the latter assumption, and the assump- 
 tion that l = lce = 2lse, satisfy equations (124), (136), and 
 (138). 
 
 Accordingly the assumption that l ce = 2l Se) and that the 
 electrons bthave as if they were in the gaseous state, give 
 a value for the theoretical ratio of C/k expressed by equation 
 (137) which agrees with the facts. But the assumptions are 
 not likely to hold for the same reasons as stated previously. 
 
 * Which corresponds to n being given by equation (21), S me being 
 equal to 4.808 XlO~ 24 cal., and taking into account that vN e mae = m e 
 = .001. 
 
INVERSE FIFTH POWER LAW OF ATTRACTION 187 
 
 The expression for the ratio C/k obtained from equa- 
 tions (124) and (138) applied to the gaseous state is the 
 one usually given in connection with electric conductivity, 
 but the meanings usually attached to I and l ce are not 
 exactly the same as those given in this book. The ex- 
 pression for the ratio given by equation (137) is mathe- 
 matically more fundamental, as will easily be recognized. 
 
 The diffusion of gases may also be treated from another 
 aspect, which involves the attraction of the molecules upon 
 each other. 
 
 43. Maxwell's Expression for the Coefficient of 
 Diffusion of Gases. 
 
 If a molecule may be regarded simply as a center of 
 forces of attraction and repulsion the coefficient of diffusion 
 of a gas into another gas is a function of the law of force 
 between the molecules. The mathematics involved in 
 finding the function for any given law of force is, however, 
 of a very complicated character, and does not yield expres- 
 sions of any practical use except in one case. Maxwell* 
 has shown that if the attraction between two molecules 
 separated by a distance x may be represented by the expres- 
 sion B/x 5 , where B denotes a constant depending on the 
 nature of the molecules, the coefficient of diffusion D T of 
 a gas r into a gas e is given by 
 
 Pe 
 
 )r A f B 
 
 Pc^lc 
 
 (139) 
 
 where p r , p r , p e , and p e denote the partial densities and 
 pressures respectively of the molecules r and e whose molec- 
 ular weights are m f and m e , P = pr+Pe } and A c denotes a 
 * " Dynamical Theory of Gases " Collected Papers, Vol. II, p. 36. 
 
.188 THE NATURE OF MOLECULAR MOTION 
 
 numerical constant. The attraction between two molecules 
 may probably over a certain region be approximately 
 represented by a single term of the above form according 
 to Section 26. Actually the force between two molecules 
 according to Section 14 is expressed by a number of terms 
 which is probably greater than three. The equation has 
 been applied to gases by the writer * to calculate their 
 relative coefficients of diffusion, in connection with investi- 
 gations of the nature of the forces of molecular interaction. 
 The discussion of the results is therefore reserved for 
 another place. 
 
 In the previous Sections the diffusion, viscosity, conduc- 
 tion of heat, was investigated without any reference to the 
 exact nature of the law of force of molecular interaction. 
 The free paths introduced have not the direct physical 
 significance associated with the paths defined by the old 
 method of molecular collision. But we have seen that 
 some other important physical significance can be attached 
 to each path, and to the more important component factor 
 K about which really the interest of each kind of path cen- 
 ters. This procedure, there can be no doubt, is more funda- 
 mental than that involving molecular collision, and more- 
 over it is mathematically quite sound, besides being very 
 much simpler. The constants involved in the various 
 expressions obtained in this way are obviously functions of 
 the law of molecular attraction and repulsion. The results 
 obtained will be used in the next chapter to interpret 
 various aspects of Brownian motion, and the diffusion and 
 mobility of particles. The molecular free paths will also 
 be given extended forms along the lines worked out in the 
 previous Sections, by the aid of which useful and interesting 
 information may be obtained about molecular motion under 
 various conditions, 
 
 *Phil Mag., May, pp. 783-809, 1910. 
 
CHAPTER IV 
 
 MISCELLANEOUS APPLICATIONS, CONNECTIONS, AND 
 EXTENSIONS OF THE RESULTS OF THE PREVIOUS 
 CHAPTERS 
 
 44- The Direct Observation of some of the Quan- 
 tities depending on the Nature of the Motion of a 
 Molecule in a Substance, and their Use. 
 
 In Section 4 we have seen that the motion of transla- 
 tion of a colloidal particle in a liquid may be of such a 
 magnitude, depending on the size of the particle, that it 
 can conveniently be observed directly by means of the 
 ultra microscope. The motion is oscillatory in nature, 
 and takes place in haphazard directions, giving rise to a 
 migration of the particle which is also oscillatory and hap- 
 hazard in character. If the solution is given a motion at 
 right angles to the microscope by passing a stream of the 
 solution through the containing vessel, the motion of each 
 particle as it appears to the eye is the resultant of the motion 
 of the particle and that of the liquid, and thus the particle 
 appears to traverse an undulatory or wavy curve. This 
 method of observation was introduced by Svedberg*, 
 who accordingly speaks of the average amplitude A\ and 
 wave-length X of the apparent path of the particle. Fig. 
 17 shows as an illustration some curves that he obtained 
 in this way. The left-hand side of the figure shows cases 
 
 *Zs. f. Eledroch., 12, 1906, pp. 853-909; Zs. f. Phys, Chem., 71, 
 1910 ; p. 571. 
 
 189 
 
190 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 of motion of particles as they would appear to the eye if 
 the fluid were at rest, while the right-hand side shows the 
 curves they would trace out in the plane of observation 
 on giving a parallel motion to the fluid. If the actual motion 
 of a particle in a liquid at rest were along a number of straight 
 lines joined together consecutively at different angles into 
 a continuous line, and this motion were compounded with 
 a uniform motion in a given direction, the motion of the 
 
 particle projected on to a plane would appear as a curve of 
 a- similar character, namely consisting of straight lines 
 joined together at various angles. This is not observed 
 in practice, however, and it follows therefore that the motion 
 of a particle is not suddenly changed in direction at cer- 
 tain points only along < its path, but continually so more 
 or less, showing that it is continually under the influence 
 of the surrounding molecules. 
 
THE PROJECTION OF A PATH ON TO AN AXIS 191 
 
 If a point is chosen in the vicinity of each bend of the 
 actual path of the particle, and each consecutive pair of 
 points joined by a straight line subject to the conditions 
 that the sum of the lengths of the lines between two points 
 a considerable distance apart is equal in length to the cor- 
 responding path, and each direction of a line in space is 
 equally probable, each line corresponds to a diffusion free 
 path of the particle according to Section 41. 
 
 The projection of a diffusion path on to a plane can 
 be shown to reduce it on the average in the ratio 7r/4, since 
 it may point in any direction. Thus if l s denotes the length 
 of the mean diffusion path of the particles making an angle 
 with a line at right angles to the plane of observation, 
 and no particles pass through each point per second, the 
 length of the projection of these paths on to the plane is 
 equal to 
 
 /""T /*V 
 
 I 2?r/gsin 6-ls'dd- . 19 'l$smd = ^ sin 2 0-dd 
 Jo 47iV 2 Jo 
 
 'S^O I n l TT^Oy '5^0 I O n ls\ 
 
 = -jr- sin cos + =r Is s - sin 2 dB 
 * L Jo ^ I Jo 
 
 "T^;.:2jl Sin2 ^' 
 
 and thus is equal to ^pZ*, or J 5 for a single path. The 
 
 average projection p of a diffusion path on to an axis in the 
 plane of observation can now be shown to be given by 
 
 4 
 
 
 or 
 
192 MISCELLANEOUS APPLICATIONS CONNECTIONS 
 
 where <t> denotes the angle the projection -l s makes with 
 
 a line at right angles to the foregoing axis. 
 
 The projection p is evidently somewhat greater than 
 twice the average amplitude A\ of the curve traced out in 
 the plane of observation by the projected motion of a particle 
 when the solution is given a motion at right angles to the 
 axis of projection of p. It is evident that the closer the 
 actual path of the particle resembles a number of straight 
 lines joined together at various angles the smaller is this 
 difference. Its magnitude is impossible to determine theo- 
 retically, but it is probably safe to say that it is not likely 
 to be greater than 10 per cent, probably it is often much less. 
 Thus as a first approximation we may take l s equal to 4A \. 
 
 It is possible, however, to determine the exact average 
 value of the mean diffusion path l d from curves of the 
 nature shown in Fig. 17. The average wave length X of 
 *the curve is evidently equal to the average length of the 
 projection of the diffusion path on to an axis in the plane 
 of observation parallel to the direction of the motion given 
 to the solution. The magnitude of this projection, it should 
 be noticed, depends on the magnitude of the motion of the 
 solution. If n m denote the number of maxima in the curve 
 passed over by the particle in one second, and V 8 the velocity 
 given to the solution, we have the relation 
 
 Each of the quantities in the foregoing equation may be 
 measured directly. The length of path l p traced out in the 
 plane of observation in one second may also be obtained 
 by direct measurement from the curve. This length l p 
 is equal to the length of the curve that would be traced 
 out in the plane of observation if the particle passed over 
 
DIRECT DETERMINATION OF DIFFUSION PATH 193 
 
 its diffusion path according to its definition instead of over 
 its actual path. The value of l p /n m is thus equal to the 
 length of the average projection of the diffusion path on 
 to the plane of observation under the conditions of the 
 experiment. Therefore, since the projection of the dif- 
 fusion path on to an axis parallel to the motion of the fluid 
 in the plane of observation is X, it follows from geometrical 
 considerations that the projection of the path on to an axis 
 in the plane of observation at right angles to the motion 
 of the solution is 
 
 or 
 
 since n m = T,/X. This projection is independent of the motion 
 given to the solution, and is therefore equal to p the pro- 
 jection corresponding to the solution at rest. Therefore, 
 since we have previously obtained that p = l s /2, we have 
 
 (140) 
 
 Thus we see that it is possible to determine directly the 
 average diffusion path of a colloidal particle as defined 
 in Section 41. 
 
 It is possible therefore to calculate the coefficient of 
 diffusion and the total average velocity of such a particle. 
 According to equation (129) the coefficient of diffusion 
 D r of particles r in a substance consisting of particles e 
 is given by 
 
 '' N r ' 
 
 where n r denotes the number of particles r crossing a square 
 cm. from one side to the other per second, and N r the con- 
 
194 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 centration of the particles. Now according to equation 
 (35). 
 
 _VtrNr 
 Mr o > 
 
 while directly 
 
 where t r denotes the average period of the diffusion path 
 l sr of a particle r, and V tr its total average velocity. The 
 diffusion equation may therefore be written 
 
 D,=g. ...... (141) 
 
 The period t r is also given by 
 
 and may thus be determined from curves of the nature 
 shown in Fig. 17. Equation (141) may therefore be used 
 to determine the coefficient of diffusion of colloidal particles. 
 Since I have not previously published the definition 
 of the free diffusion path used, and the method of deter- 
 mining it directly in the case of a colloidal particle, no 
 values of it have yet been determined. But the average 
 values of the amplitudes A i (Fig. 17) of the curve described 
 by a particle in the plane of observation are, however,' 
 available, which, we have seen, are approximately equal 
 to l 6 /4. Svedberg has measured the average amplitude 
 AI, and average period t, of platinum particles in various 
 solvents. Using these values I have calculated the co- 
 efficients of diffusion of the particles, which will be found 
 in Table XXII. These values are probably nearer 'to the 
 truth than could be obtained by direct measurement. 
 The last column in the Table gives the product Drj, where 
 
EQUATION FOR TOTAL AVERAGE VELOCITY 195 
 
 ?7 denotes the viscosity of the solvent. It will be seen that 
 it is very approximately constant, and the coefficient of 
 diffusion of a particle thus varies inversely as the viscosity 
 of the medium. This is what we would expect to hold in 
 the case of particles considerably larger than a molecule. 
 
 TABLE XXII 
 
 Solvent. 
 
 Radius 
 in 
 cms. X 
 
 105. 
 
 Temp. 
 Cent. 
 
 Time t 
 in 
 seconds. 
 
 Viscos- 
 ity ij. 
 
 4 A, in 
 cms. X 
 
 105. 
 
 DX10' 
 cm. 2 
 sec. 
 
 Di) 10" 
 
 Acetone 
 E Acetate 
 
 0.25 
 0.25 
 
 18 
 19 
 
 0.032 
 0.028 
 
 .0023 
 .0046 
 
 14.2 
 9.4 
 
 2.10 
 1.05 
 
 4.83 
 483 
 
 Amyl acetate . . 
 Water 
 Propyl alcohol . 
 
 0.25 
 0.25 
 0.25 
 
 18 
 20 
 20 
 
 0.026 
 0.013 
 0.009 
 
 .0059 
 .0102 
 .0226 
 
 8.0 
 4.3 
 2.4 
 
 .82 
 .47 
 .21 
 
 4.84 
 4.80 
 4.75 
 
 An expression for the total average velocity of a colloidal 
 particle is obtained from the two preceding equations giv- 
 ing the period of the diffusion path, and equation (140), 
 which gives 
 
 V tr = l -^ s = 2\/l p 2 -V s 2 . (142) 
 
 A 
 
 Since values of l p are not available this equation cannot 
 yet be used to calculate values -of V tr . We may, how- 
 ever, use the approximate average value of 4Ai for 1 ST , 
 which gives to the equation the form 
 
 This equation would hold exactly if the actual path of a 
 particle were along a number of straight lines joined to- 
 gether at various angles. It has been obtained by Sved- 
 berg on this supposition, and used to calculate the veloci- 
 ties of particles. For the platinum particles in various 
 
196 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 solvents to which Table XXII refers he obtained the veloci- 
 ties given in Table XXVI. It will be seen that the velocities 
 do not differ much from one another, and therefore do not 
 seem to depend much on the nature of the solvent since the 
 particles have the same diameter. The values obtained 
 are, however, not exact, as explained before. 
 
 We have seen in Section 42 and this Section that the 
 coefficient of diffusion depends directly on the total average 
 velocity of a molecule and the nature of its motion. Since 
 this also holds for other quantities, the coefficients may also 
 be expressed in terms of them. This applies to the quan- 
 tities osmotic pressure and coefficient of mobility, and 
 their relation to the coefficient of diffusion will therefore 
 now be investigated. 
 
 45. The Coefficient of Diffusion in Connection 
 with Osmotic Pressure and the Coefficient of Molec- 
 ular Mobility. 
 
 Consider a heterogeneous solution of molecules e and r, 
 in which case diffusion of molecules from one place to 
 another takes place until the molecules are uniformly dis- 
 tributed. If a semipermeable membrane impervious to 
 say molecules r, were placed at right angles to a stream of 
 diffusing molecules r, they would exert a pressure in the 
 direction of their migration upon the membrane. . Since 
 the concentration of the molecules r is different on the 
 two sides of the membrane, this pressure is the difference 
 between the osmotic pressures of the molecules acting on the 
 two sides of the membrane. This pressure is therefore 
 the force acting upon the molecules tending to move them 
 to places of lower concentration, which manifests itself as 
 a pressure by reaction on the membrane placed in the 
 path of the molecules to prevent their migration. In the 
 absence of the membrane this force is spent in overcoming 
 
CONNECTION OF DIFFUSION AND MOBILITY 197 
 
 the viscous friction exerted by the mixture on the migrat- 
 ing molecules. We may therefore look upon each cubic cm. 
 of molecules r as being under a force equal to the difference 
 in the osmotic pressures of the opposite faces of the cubic 
 cm., which force is spent in giving motion to the molecules 
 , against the viscous friction of the medium. This idea of 
 looking upon diffusion is mainly due to Nernst. 
 
 In general if 5 r denotes the number of molecules diffusing 
 across a square cm. per second we have 
 
 8r = NrV' r , 
 
 where N r denotes the concentration of the molecules r, 
 and V'r the velocity with which each molecule r on the aver- 
 age is moving in the direction of decrease of concentration 
 gradient. Now according to the foregoing considerations we 
 may write 
 
 _M r dP sr 
 T N r dx ' 
 
 where M r denotes the coefficient of mobility of a molecule, 
 or its velocity under the action of unit force, and dP sr /dx 
 denotes the osmotic pressure gradient. Hence the preceding 
 equation may be written 
 
 (143) 
 
 Similarly in the case of the molecules e diffusing in the 
 opposite direction we have 
 
 (144) 
 
 From the preceding equations and equation (127) 
 we have, 
 
 f-.Uf,, .... (145) 
 
198 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 an equation which expresses the relation between the 
 osmotic pressure gradients of the molecules r and c, their 
 coefficients of mobility, and their molecular volumes. 
 
 In the case of a dilute solution say of molecules r in 
 e, the osmotic pressure obeys the gas laws, that is we may 
 write 
 
 m ar 
 
 where m r and m ar denote the relative and absolute molecular 
 weights of a molecule r, and the ratio m r /m a r is therefore 
 equal to the number N of molecules in a gram molecule 
 which according to Section 3 is equal to 6.2X10 23 . Hence 
 
 dx ~~N~"dx' 
 and equation (143) becomes 
 
 This gives for the coefficient of diffusion 
 
 ...... (147) 
 
 Thus if the value of either of the two quantities D r and M r 
 be known that of the other quantity may be immediately 
 calculated. Table XXIII gives the values of M, or the 
 mobility under unit force, of a number of different mole- 
 cules in different liquid and gaseous media, calculated 
 from the known values of D. Table XXIV gives the cal- 
 culated values of D for some electrically charged mole- 
 cules whose mobilities per unit force are known. They 
 agree fairly well with the values obtained by experiment. 
 
COEFFICIENTS OF MOBILITY AND DIFFUSION 199 
 
 TABLE XXIII 
 
 
 Gaseous 
 
 Gaseous 
 
 
 
 medium 
 
 medium 
 
 
 Molecules 
 
 of CO 2 at 
 760 cm. 
 
 of H 2 at 
 760 cm. 
 
 Liquid medium of water. 
 
 for which 
 D is given 
 in Table 
 
 pressure 
 and O C. 
 
 pressure 
 and O C. 
 
 
 XX. 
 
 
 
 
 I 
 
 
 M 10 - 12 
 
 M 10-12 
 
 Molecules. 
 
 Temp. 
 C. 
 
 , -, 2 
 
 M 10-9. 
 
 H 2 O 
 
 * 3.6 
 
 18.8 
 
 Hydrogen. 
 
 10 
 
 3.75 
 
 1.14 
 
 CH 4 
 
 2.4 
 
 13.7 
 
 f Nitrous 
 I oxide. 
 
 }l4 
 
 .63 
 
 .19 
 
 C 6 H 6 
 C 9 H 18 2 
 
 1.99 
 .83 
 
 8.02 
 4.70 
 
 f Cane 
 I Sugar. 
 
 }l2 
 
 .284 
 
 .087 
 
 TABLE XXIV 
 
 Nature of 
 gas in 
 which the 
 ions are ' 
 
 Values of M for 
 + and ions 
 calculated from 
 experimental 
 results obtained 
 by Wellisch.* 
 
 Values of D for 
 + and ions 
 calculated by 
 means of equa- 
 tion (147). 
 
 Values of D 
 obtained di- 
 rectly by ex- 
 periment by 
 Townsend.f 
 
 produced. 
 
 
 
 
 
 M- 10 12 
 
 M+ lO 12 
 
 D-. 
 
 D + . 
 
 D- 
 
 D + . 
 
 H 2 
 
 5.13 
 
 4.32 
 
 .200 
 
 .169 
 
 .19C 
 
 .123 
 
 O 2 
 
 1.16 
 
 .88 
 
 .0453 
 
 .0343 
 
 .0396 
 
 .025 
 
 C0 2 
 
 .55 
 
 .52 
 
 .0214 
 
 .0201 
 
 .026 
 
 .023 
 
 *Phil. Trans.; A, Vol. CXCIII, p. 129 (1900) 
 t Ibid., A, Vol. CCIX, p. 269 (1909). 
 
 On equating the coefficient of diffusion given by equa- 
 tion (146) with that given by equation (129), we have 
 
 Since n r = 
 
 V tr Nr 
 
 (Section 18), and V tr t r = l Sr , where t r denotes 
 
200 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 the time the molecule passes over the path I 8r with the velocity 
 V tr (Section 17), the foregoing equation may be written 
 
 7 2 
 
 I'dr 
 
 T 
 
 RT 
 
 r N' 
 
 (148) 
 
 This equation may be used to calculate the mobility under 
 unit force of particles of a size that undergo Brownian 
 motion. The values of I 8r and t r , we have seen in the pre- 
 vious Section, can be determined by direct observation. 
 Table XXV gives the mobility under unit force of particles 
 
 TABLE XXV 
 
 PLATINUM PARTICLES 
 
 OF AVERAGE RADIUS 
 
 .25X10~ 5 CM. 
 
 Medium. 
 
 Viscosity 
 
 M r w-*^ 
 
 sec. 
 
 Acetone . 
 
 .0023 
 
 5.36 
 
 E acetate 
 
 .0046 
 
 2.68 
 
 Amyl acetate 
 
 .0059 
 
 2.10 
 
 Water 
 Propyl alcohol 
 
 .0102 
 .0226 
 
 1.20 
 .54 
 
 suspended in different kinds of solutions calculated from 
 data contained in Table XXII, using for l Sr the approximate 
 value 4Ai. 
 
 An expression may be found for the force F on a colloidal 
 particle moving with the observed velocity V c under unit 
 electric field. This force cannot be calculated from elec- 
 trical data without the introduction of assumptions. A 
 colloidal particle as a whole is not electrically charged, but 
 is supposed to be surrounded by two parallel layers of 
 electricity of equal magnitude but of opposite sign. These 
 layers get distorted when an electric field is applied to the 
 solution, and the tendency of the particle to move so as to 
 
FORCE ON PARTICLE IN AN ELECTRIC FIELD 201 
 
 readjust this tends to give it a continual motion. The force 
 F acting on the particle is immediately given by 
 
 V c =FM r , 
 which may be written 
 
 (149) 
 
 by means of equation (148). Each quantity on the right- 
 hand side of this equation may be determined directly and 
 hence F calculated. 
 
 Experiment shows that the observed velocity V c of a 
 colloidal particle varies inversely as the viscosity of the 
 solution, or as 1/r/, but is independent of the mass of the 
 particle. Hence for different solvents the force F is inversely 
 proportional to 77, and proportional to t r /l 2 dT) or inversely 
 proportional to the mobility, according to equation (148). 
 Since the mobility is proportional to the coefficient of 
 diffusion D according to equation (147), and Drj is constant 
 according to Table XXII, it follows that the force acting 
 on a colloidal particle in a solution under unit electric field 
 is approximately independent of all conditions at constant 
 temperature. 
 
 It will be of interest to determine the absolute value of 
 F in a special case. Thus Bredig found that V c in the 
 case of platinum particles suspended in water had a value 
 of about .00025 cm. per second for an electric field of one 
 volt per cm. According to Table XXII for platinum 
 particles of radius .25X10" 5 cm. suspended in water, t r = 
 .013 sec., 4Ai = .000043 cm. corresponding to T = 293, 
 while # = 8.315X10 7 , and N = 6.2X10 23 . Assuming that 
 = Z 5r , which holds approximately, we obtain that 
 
 F = 2XlO- 10 dyne 
 
. 202 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 for a platinum particle under the foregoing conditions. 
 This is the force that would act on the particle if it possessed 
 an electric charge equal approximately to lOOe. 
 
 The formulae for the diffusion and viscosity given in Sec- 
 tions 42 and 35 involve the quantities n r and n e , the 
 number of molecules r and e crossing respectively a square 
 cm. from one side to the other per second. These quan- 
 tities may be expressed in terms of quantities which have 
 not been denned previously. 
 
 46. Partial Intrinsic Pressures.* 
 
 In connection with the intrinsic pressures of a mixture 
 it will be convenient to introduce the quantity partial in- 
 trinsic pressure, whereby expressions for various quantities 
 are furnished which are of theoretical interest and often of 
 practical use. Let us consider a mixture of molecules 
 r and e cut into two parts by an imaginary plane ab as 
 shown in Fig. 7. Let P n n denote the attraction which 
 the molecules r in the portion B exert on the molecules 
 r in a cylinder of unit cross-section and infinite length 
 standing in the portion A with one of its bases on the plane 
 ab, and let P ng 2 have a similar meaning in reference to the 
 molecules e. Let P n re denote the attraction of the mole- 
 cules r in the portion B on the molecules e in the cylinder, 
 which is also equal to the attraction of the molecules e 
 in this portion on the molecules r in the cylinder. The 
 intrinsic pressure of the mixture in the plane ab, which is 
 the sum of the foregoing forces of attraction, is therefore 
 given by 
 
 , .... (150) 
 
 where the quantities on the right-hand side of the equation 
 
 may be called the partial intrinsic pressures of the mixture. 
 
 * Matter published for the first time. 
 
PARTIAL EXPANSION PRESSURES 203 
 
 The molecules r in one of the portions of the mixture 
 are thus attracted by the other portion as a whole by a 
 force which gives rise to the pressure P n n+P n re per cm. 2 
 in the plane ab. This force acting on the molecules r can 
 be sustained only by their expansion pressure, in other 
 words, no part of the force can be sustained by the expan- 
 sion pressure of the molecules e in the plane ab. For suppose 
 the part I T of the intrinsic pressure P n n+P nr e is sustained 
 by the part X e of the expansion pressure of the molecules e. 
 The molecules r may then be said to be under a component 
 force IT, and the molecules e under a component force 
 X e , acting in the opposite directions, which react upon 
 each other through the medium of the interaction of the mole- 
 cules r and e of the mixture. But if two forces act upon 
 two different sets of molecules in this way they would tend 
 to be set in motion in opposite directions, and would react 
 upon each other only through the viscous resistance they 
 would exert upon each other. The two sets of molecules 
 would therefore gradually get separated through diffusing 
 through each other. But this does not take place since 
 the mixture is in equilibrium. The forces in question are 
 therefore equal to zero, or the intrinsic pressure P n n+Pnre 
 associated with the molecules r is sustained by a part of 
 their expansion pressure. 
 
 A quantity similar to the partial intrinsic pressure may 
 be denned in connection with the external pressure of a 
 mixture. This pressure is exerted in part by each set of 
 molecules, and in the case of a mixture of molecules e and 
 r the total external pressure p may therefore be written 
 
 (151) 
 
 where p r and p e denote respectively the partial external 
 pressures of the molecules r and c. It is obvious that the 
 partial external pressure p T represents a part of the expan- 
 
204 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 sion pressure of the molecules r, and a similar remark applies 
 to the partial external pressure of the molecules e. It 
 follows then from Section 20 that if n r denote the number 
 of molecules r crossing a square cm. in the mixture from one 
 side to the other, and n e has a similar meaning with respect 
 to the molecules e, we have 
 
 Pr ~ nr* nre = n T 
 
 = 2.1 
 
 V r 
 
 and 
 
 V e A e 
 
 e 
 Vg ~"~ j 
 
 = 2.543 X lO- 20 ^ V VT^ e . (153) 
 
 The partial intrinsic pressures cannot yet be expressed 
 similarly as the quantity P n in terms of other quantities 
 which can be directly measured. Approximate numerical 
 values may, however, be obtained from the simultaneous 
 equations (150), (152), and (153), by neglecting the partial 
 external pressures in comparison with the partial intrinsic 
 pressures, which may usually be done, and calculating the 
 quantities n r , n e , b' r , b' e , and P n , by the method of Sections 
 21 and 29. 
 
 Approximate values may also be obtained by the fol- 
 lowing method. In the case of a pure substance the intrinsic 
 pressure according to Section 26 may as a first approxima- 
 tion be taken proportional to the square of the density p 
 of the substance. Applying this result to the kind of mixture 
 under consideration the quantities P nr * and P ne2 may evi- 
 dently be written 
 
 P nf , = JBVr, . . o . . . (154) 
 and 
 
 ..... (155) 
 
THE INTRINSIC PRESSURE AND LATENT HEAT 203 
 
 where p r and p e denote the partial densities of the molecules 
 r and e in the mixture, and B T and B e denote approximate 
 constants. As a first approximation the quantity P nT e is 
 then given by 
 
 ..... (156) 
 
 The values of the quantities B r and B e may be obtained 
 from the internal heats of evaporation L r and L e of gram 
 molecules of the substances in the pure state. Thus accord- 
 ing to Section 21 we have 
 
 L r = 
 
 where pi and p2 denote the densities of the pure substance 
 r in the liquid and vaporous states, and a similar equation 
 may be obtained for L e . These equations express B r and 
 B e in terms of L r and L e . 
 
 The values for the partial intrinsic pressures obtained 
 by this or the preceding method may be tested by sub- 
 stituting them in equation (150), and determining P n 
 (the intrinsic pressure of the mixture as a whole) by the 
 method described in Section 21. If an agreement is obtained 
 we may be fairly sure that the values of the partial intrinsic 
 pressures obtained are very approximately correct. 
 
 If the latter method is used to determine the partial 
 intrinsic pressures, we may use equations (152) and (153) 
 to determine n e and n r . We are thus furnished with another 
 method of determining the latter quantities, which is com- 
 paratively simpler to use than that described in Section 29. 
 
 The latter method of obtaining the partial intrinsic 
 pressures may evidently also be used to find the intrinsic 
 pressure of a pure substance. 
 
 In the case of a mixture of more substances than two 
 the corresponding partial intrinsic pressures will not be 
 
206 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 difficult to define, and may be obtained in a similar way as 
 described. 
 
 The partial intrinsic pressures of a heterogeneous mixture 
 may also be connected with its osmotic pressures. Before 
 discussing the connection it will be necessary to consider 
 some general conditions of equilibrium. 
 
 47. Conditions of the Equilibrium of a Hetero- 
 geneous Mixture such as Two Phases in Contact. 
 
 The relative concentration of the constituents of two 
 phases of a mixture is usually different and gradually changes 
 from one to the other in the transition layer which exists 
 at the boundary of the phases. In the case of a mixture 
 of molecules r and e, for example, equations (152) and (153) 
 must hold for both phases and the transition layer, and are 
 therefore two of the equations of equilibrium. 
 
 The partial external pressures p r and p e in these equa- 
 tions must have the same values everywhere, otherwise 
 diffusion from one portion to the other would take place, and 
 the system would not be in equilibrium. This follows at 
 once from considering two layers of liquid in which the 
 partial external pressures have the values p r , p e , and p' r , 
 p' e , respectively. Since the total pressure is the same 
 everywhere 
 
 Pe + Pr = p'e + P'r, 
 
 and hence 
 
 From the latter equation it follows that there is an excess 
 of pressure of the molecules r in one direction which is 
 balanced by an excess of pressure of the molecules e in the 
 opposite direction. But this would give rise to a diffusion 
 of the two sets of molecules through each other, reasoning 
 
EFFECT OF INTRINSIC PRESSURE GRADIENT 207 
 
 along the same lines as in the previous Section, and thus 
 disturb the equilibrium. Hence we must have 
 
 Pr = p'r and p e =p'e ..... (157) 
 
 If n r denotes the number of molecules r crossing a square 
 cm. from one side to the other, n' r the number crossing in 
 the opposite direction, and n e and n' e have similar meanings 
 with respect to the molecules e, we must also have 
 
 n' e = n e } 
 and L ...... (158) 
 
 
 otherwise diffusion from one place to the other would take 
 place. 
 
 The foregoing equations are of special interest in con- 
 nection with the transition layer, since this has molecular 
 concentration gradients the tendency of which is to give 
 to n r , the molecules crossing a square cm. in one direction, 
 a value different from n' r , the number crossing in the opposite 
 direction, and to give also different values to n e and n' e . 
 But there evidently exists a force in the layer at right angles 
 to it acting in the direction of increase of density, due to 
 the existence of molecular forces of attraction. This force 
 is equal to the intrinsic pressure gradient. It has the effect 
 of decreasing the velocity of the molecules moving in the 
 direction of decrease of density and increasing the velocity 
 of the molecules moving in the opposite direction. The 
 magnitude of this modification of the velocity of a molecule 
 depends on its nature, since the various molecules do not 
 possess the same forces of attraction. Thus the effect of 
 the intrinsic pressure gradient on the number of molecules 
 crossing a square cm. from one side to the other per second 
 is opposite to that of the concentration gradient, and there- 
 fore n r and n' r , and n e and n' e , may be rendered equal, as is 
 
.- 
 
 208 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 the case in practice. Besides, since the effect of the intrinsic 
 pressure gradient on a set of molecules depends on their 
 nature, the concentration gradient would not necessarily 
 be the same for each set of molecules corresponding to the 
 equilibrium conditions (152) and (153). Thus we see why 
 the relative concentration of the ingredients of a mixture 
 differs in the vaporous phase from that in the liquid phase. 
 It will also be evident that, since the attraction of a molecule 
 increases with its mass, the relative concentration of the 
 heavier molecule is likely to be smaller in the vaporous 
 phase than in the liquid phase. This is exemplified in prac- 
 tice, though it need not, and does not, hold in every case. 
 
 Similar conditions apply to a mixture of more than two 
 constituents. 
 
 48. Osmotic Pressure expressed in Terms of 
 the Kinetic Properties of Molecules.* 
 
 If the different portions of a mixture of substances are 
 not in equilibrium with each other, which would manifest 
 itself by a redistribution of the ingredients by diffusion, 
 the equations of equilibrium given in the foregoing Section 
 will not hold. Evidently equations (157) will not hold since 
 the vapor pressure of the molecules r depends on their 
 
 concentrations. A force equal to , where x is measured 
 
 ox 
 
 along a line of decrease of concentration, will therefore act 
 on each cubic cm. of molecules r. Equation (152) may hold 
 at certain parts of the mixture, though in general this would 
 not be the case. There is therefore an additional force 
 equal to 
 
 Matter published for the first time. 
 
THE NATURE OF OSMOTIC PRESSURE 209 
 
 acting on the molecules r per cubic cm. The total force F r 
 acting on the molecules r per cubic cm. is therefore given by 
 
 p _i_ \-^L VT $Hi-L-^L HrVr dfr'r 
 
 t l 2 *-*V .ifeT.2 (Vr-b'r) 2 ^ 
 
 <Arnrf 1 V r 
 
 T 2 
 
 '. . (159) 
 
 This force is equal to the osmotic pressure acting on the 
 molecules r. It gives rise to a diffusion of molecules equal 
 to n r n'r'y and equations (158) therefore also do not hold. 
 It is likely that the upper sign of the first term of the right- 
 hand side of the foregoing equation always holds in practice. 
 Similarly the force F r acting on the molecules e per 
 cubic cm. giving rise to a diffusion of molecules e in the oppo- 
 site direction to the molecules r, is given by 
 
 \Ae V e faeAe U e V e db'e 
 
 2 V- 
 
 ,A e n e / 1 v e \ 8v e 
 
 2 \v e -b' e ( Ve -b' e ) 2 /5x 
 
 . . (160) 
 
 Since n r depends on the motion of translation of the 
 molecules and molecular attraction, the foregoing equations 
 express osmotic pressure in terms of molecular motion, 
 molecular attraction, and the apparent molecular volume. 
 It will be evident on reflection that osmotic pressure must 
 arise through these properties of matter, and the equations 
 are therefore fundamental in character. They are, however, 
 of little use in practice to calculate osmotic pressure, since 
 we have no means yet of determining experimentally how 
 
210 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 the quantities n r , b' r , P nr2 , and P ner vary in a heterogeneous 
 mixture from one part to another. 
 
 Osmotic pressure as exhibited in connection with semi- 
 permeable membranes will be discussed in connection with 
 another subject. 
 
 Since the quantities n e and n T in the foregoing and 
 other formulae depend on the total average velocities of the 
 molecules e and r, it will be of importance to see if some 
 direct information can be obtained about the values of these 
 velocities. 
 
 49. A Method of Obtaining the Velocity of Trans- 
 lation of Particles undergoing Brownian Motion. 
 
 The path of such a particle we have seen in Section 44, 
 is zigzag in shape. The average velocity of translation of 
 the particle may therefore, according to this Section, be 
 determined from observations of the average period t of the 
 free path l s by means of the equation 
 
 (161) 
 
 Table XXVI contains the approximate velocities of plati- 
 num particles in different solvents determined in this way 
 from observations of the amplitudes A\ contained in Table 
 XXII made by Svedberg and described in Section 44, 
 which we have seen are approximately equal to Z a /4. It is 
 usually supposed that the average velocity of a colloidal par- 
 ticle is the same it would have in the perfectly gaseous state. 
 But on calculating the average velocity V a under these con- 
 ditions, which is given by V a = -922F and equation (8), it is 
 found that it does not agree with that observed, as will 
 appear from Table XXVI, which gives the velocities of 
 platinum particles obtained in these two ways. Now this 
 is what we would expect from the results of Sections 16 and 
 
THE MOTION OF COLLOIDAL PARTICLES 
 
 211 
 
 29. But it has yet to be shown why the actual velocities 
 might be smaller in the case of a colloidal particle than 
 corresponding to the gaseous state, when in the case of a 
 single molecule in a liquid we have seen it has a greater 
 value. It appeared from Section 44 that a colloidal particle 
 is continually under the influence of the surrounding mole- 
 cules in varying degrees, and continually bombarded from 
 all sides by molecules, whose resultant effect is not zero 
 
 TABLE XXVI 
 
 PLATINUM PARTICLES < 
 
 V a FOR GAS 
 
 3F AVERAGE RADIUS .25X10" 5 CM. 
 CM. 
 
 SEC. 
 
 Medium. 
 
 Viscosity. 
 
 V t 10 -551: 
 
 sec. 
 
 Acetone 
 
 .0023 
 .0046 
 .0059 
 .0102 
 .0226 
 
 444 
 336 
 308 
 324 
 266 
 
 E acetate 
 
 Amyl acetate 
 
 Water 
 
 Propyl alcohol 
 
 but continually changes in direction and magnitude, giving 
 the particle the well-known zigzag motion. Therefore 
 when the particle begins to move in any new direction its 
 motion is at once impeded by the viscous friction of the 
 surrounding medium, due to the particle being constantly 
 more or less under the influence of the surrounding mole- 
 cules. Hence if the initial velocity is that corresponding 
 to the gaseous state, or greater, it may be slowed down to 
 a much smaller value before the particle receives a new 
 impetus, and the total average velocity is therefore not 
 likely to be the same as in the gaseous state, but may be 
 much less. Section 60 also deals with this point. 
 
212 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 It is often assumed that the velocity of a colloidal 
 particle obtained from observations of its path does not 
 represent the true velocity, since the amplitude of the 
 oscillatory motion may occasionally be so small that it 
 cannot be observed by means of the ultra-microscope, 
 and what in truth is a wavy curve of small amplitude and 
 wave lengths would be taken for a smooth line. This is 
 no doubt partly true. But it is highly improbable that 
 only T V of the path of the particle should be possible of 
 being observed by the ultra-microscope. Otherwise the 
 oscillatory motion of the particle would largely correspond 
 to amplitudes whose magnitude could not be determined 
 by the microscope, on which are impressed amplitudes of 
 very much greater magnitude. This does not seem to be 
 in harmony with the distribution of the amplitudes according 
 to Clausius' law given in Section 31. For this reason, and 
 that the particle must always be subject to viscous fric- 
 tion in its motion, we conclude that the velocity need not 
 be that corresponding to the gaseous state, but is probably 
 much less. 
 
 Some experiments by Exner, who showed that particles 
 of diameters 1.3ju, .Q/z, and .4ju, have the velocities 2.7, 
 3.3, and 3.8, respectively, point to the same conclusion. 
 For the masses of the particles are proportional to the 
 cubes of their diameters, and their velocities in the gaseous 
 state therefore proportional to the f power of their 
 diameters. But this is evidently not realized. 
 
 If Vi denote the velocity of a particle at any instant 
 in passing over the distance 51, and R the viscous resistance 
 of the medium, we have 
 
 where m a denotes the mass of the particle. The resistance 
 R is proportional to Vi, or equal to K^V\^ where K\ denotes 
 
THE MOTION OF A PARTICLE UNDER IMPETUS 213 
 
 a constant, and hence the preceding equation on substitut- 
 ing for R and integrating gives 
 
 l=~(V^V e ), 
 
 K2 
 
 where V t denotes the initial velocity of the particle when 
 it changes its direction, V e the velocity when it is about 
 to receive a new impetus, and I the length of path between 
 the corresponding points. 
 
 It follows therefore that the rate of change of V\, or the 
 negative acceleration, is constant when the particle is not 
 propelled by the molecules of the medium. 
 
 The value of Kz is probably very different from the value 
 of the coefficient of mobility referring to the average velocity 
 of a particle parallel to a given direction considered over a 
 time that gives a constant velocity under the action of an 
 external force. The latter constant has been considered in 
 Section 45 and is the factor of V c in equation (149). 
 
 The result of this Section is of interest in connection 
 with the osmotic pressure of dilute solutions considered in 
 the next Section. 
 
 50. The Osmotic Pressure of a Dilute Solution 
 of Molecules, and their Velocity of Translation.* 
 
 Van't Hoff has shown that it follows from thermodynam- 
 ics that the osmotic pressure of a dilute solution obeys the 
 gas equation, that is, we may write 
 
 where N denotes the number of molecules in a gram mole- 
 * Not previously published. 
 
214 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 cule of a substance, and N r the concentration of the solute 
 molecules r in the solvent e. This result has given rise to 
 the erroneous statement often made that the velocity of 
 translation of each solute molecule is the same as that it 
 would have in the perfectly gaseous state. It is obvious 
 that if this be true for the solute molecules it must also 
 be true for the molecules of a pure liquid. But it can be 
 shown in many ways that this deduction cannot hold, though 
 Van't Hoff's law be true. 
 
 If the velocity of translation of a molecule is always the 
 same as in the gaseous state the expansion pressure of a 
 
 substance would be given by the expression -, according 
 
 to equation (40) in Section 20. Now the external pressure 
 of a substance is not equal to this expression, and therefore 
 forces must exist between the molecules which give rise 
 to an intrinsic pressure which, together with the external 
 pressure, balances the expansion pressure, as expressed by 
 equation (45). But if the molecules exert forces of attrac- 
 tion and repulsion upon each other these will have the effect 
 of changing the velocity of a molecule from what it would 
 have in the gaseous state as shown in Section 17. According 
 to Sections 25 and 29 this total average velocity of a mole- 
 cule in a liquid is many times what it would have in the 
 gaseous state. Another method used in Section 34 gave the 
 same result. 
 
 It can also be shown that this result is a consequence of 
 Section 48, which gives us besides other information. Let 
 us suppose that the vapor pressure of the solute r of a solu- 
 tion is zero, as is the case of a salt dissolved in water. Since 
 the molecules of the solute in the solution according to the 
 definition of a dilute solution are so far apart that their 
 influence upon each other is practically zero, we have P nT 2 
 
 = 0. According to Van't Hoff's law Fr = = ?~ along 
 
 dx N dx 
 
VELOCITY OF PARTICLES IN DILUTE SOLUTIONS 215 
 
 a concentration gradient in the mixture. Equation (159) 
 therefore in this case becomes 
 
 RT5N r A r v r 8n, A r n r v r dVr 
 
 N dX 2 V T -b' r 5Z 2 (Vr-b'rY dx 
 
 r dP 
 
 . 
 
 Now if the velocity of translation of a molecule r were the 
 same as in the gaseous state UT would be given by equation 
 
 (21) on adding suffixes r to the symbols, and ^ would 
 
 Ol\ r 
 
 be given by equations (162) and (20), in which case 
 
 6N r NA/ 
 
 dN r 
 
 Therefore on dividing equation (163) by - , and taking 
 
 uX 
 
 into account that N T Vrm ar = m r , it may be written 
 
 RT _ V r RT A r 
 " 
 
 8P nfe RTv r f I V r 
 
 ^ dN T N r \V r -b' r (v r -b' r ) 
 
 But the right-hand side of this equation is not identically 
 equal to the left-hand side, for - - is not approxi- 
 
 VT 
 
 mately equal to unity, since the value of &/ according to 
 Section 20 is determined by the volume of the molecules 
 e as well as that of the molecules r in a gram molecule of 
 molecules r, and the value of b' r is therefore not small in 
 
 comparison with v r ; for the same reason ^ is not zero; 
 
 oN r 
 
 ?r> 
 
 and - -^ is not zero since molecular forces exist. Thus 
 dN r 
 
216 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 in general the velocity of translation of the molecules r would 
 not .be equal to that in the gaseous state. 
 
 We may also note the following considerations in this 
 connection. If we suppose that the velocity of the solute 
 molecules in a dilute solution is the same as that which 
 they would have in the perfectly gaseous state their external 
 
 r>/7T 
 
 pressure is given by - n-> Now the apparent volume 
 
 V r r 
 
 b' r of the molecules of the mixture, or the apparent volume 
 which the molecules e and r appear to possess in obstructing 
 the motion of the molecules r, is not small in comparison 
 with v r . Its value is approximately equal to v r b e /v e , where 
 b e refers to the solvent e in the pure state. According to 
 the above supposition Van der Waals' equation holds, and 
 if applied to the solvent e in the pure state we have 
 
 RT 
 
 (Section 26) where P n , the intrinsic pressure, is large in com- 
 parison with the external pressure p, and large (Section 
 21) in comparison with the pressure p' the substance would 
 have if it behaved as a perfect gas. It follows therefore 
 from the equation that b e /v e is not small in comparison with 
 
 r>/77 
 
 unity. Hence on substituting v r b e /v e for b' r in - j^ it 
 
 V r r 
 r>/T7 
 
 becomes r- , , ., and thus the pressure exerted by the 
 
 solute molecules is many times that which they would have 
 in the perfectly gaseous state. This pressure is in part 
 balanced by the intrinsic pressure of the mixture. We 
 cannot, therefore, with any show of reason, say that because 
 the osmotic pressure of the solute molecules obeys the gas 
 laws their velocity of translation is the same as in the 
 gaseous state. 
 
THE CONSTANTS OF A DILUTE SOLUTION 217 
 
 The values of w, and P nre of a dilute solution are evidently 
 proportional to N T , while b' r is proportional to v r , and we 
 may therefore write 
 
 n r = aiNr, b'r = a 2 v r , and P nTe = 
 
 where ai, a 2 , and a 3 are constants. Equation (163) may 
 therefore be written 
 
 2RT(l-a 2 ) 2(1-02)03 
 - ---, . (164) 
 
 on taking into account that 
 
 which gives a relation between the foregoing constants. 
 
 A method of determining directly the number of mole- 
 cules in a gram molecule which is based on equation (162) 
 will now be described. 
 
 51 . A Direct Determination of N, the Number of 
 Molecules in a Gram Molecule. 
 
 Perrin* has determined directly the value of N by a 
 method of counting the number of particles in a dilute 
 colloidal solution. The gravitational attraction of the 
 particles tends to deposit them on to the bottom of the 
 vessel containing the solution. The concentration gradient 
 caused thereby gives rise to an osmotic pressure acting in 
 the opposite direction to the gravitational attraction. 
 Equilibrium exists when these forces balance one another, 
 which corresponds to an increase in the concentration of 
 the solution with increase of distance from the surface. 
 
 * C. R., 147 (1908), p. 530; 147 (1908), p. 594; Ann. Chim. Phys., 
 8, 18 (1909), pp. 5-174; Bull Soc. Fr. Phys., 3 (1909), p. 155; Zs. /. 
 Electroch., 15 (1909), p. 269. 
 
( 218 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 Since the osmotic pressure obeys the gas laws the distribu- 
 tion of the particles is similar to that of the molecules of a 
 gas under the action of gravity. The differential equation 
 of equilibrium is 
 
 F-dh=5P s , 
 
 where P s denotes the osmotic pressure at a distance h from 
 the surface of the solution, and F the force due to gravity 
 acting on the particles in a cubic cm. of the solution. If 
 v a denote the volume of a particle, p a its density, p the density 
 of the liquid, we have 
 
 F = v a (p a -p)gNc, 
 
 where N c denotes the concentration of the particles. For 
 P s we have 
 
 P= RT N 
 
 N c> 
 
 according to equation (162), and hence the equation of 
 equilibrium becomes 
 
 V a (pa p)gN c ' 8k = -JT^- 5N C , 
 
 which on integration gives 
 
 \os, . . (165) 
 
 where N' c and N" c denote the concentrations corresponding 
 to the distances h\ and Ji2 from the surface of the solution. 
 The values of N' c and N" c corresponding to given values of 
 h\ and hz were found by counting the particles in different 
 planes in a solution placed under the ultra-microscope. 
 Solutions of gamboge and mastic were used. 
 
 The densities of the particles in the solutions were 
 determined by two methods. In one method it was taken 
 the same as that of the substance in the undivided state, 
 
THE CONSTANT N FROM VAN'T HOFF'S LAW 219 
 
 while in the other a known volume of the solution was 
 evaporated and the residue weighed. The two methods 
 gave concordant results. 
 
 The volume of a particle was obtained, in one of the three 
 methods used, by counting the number of particles in a 
 given volume of emulsion and, knowing their weight and 
 density, the volume was immediately obtained. 
 
 In the second method the emulsion was slightly acidu- 
 lated, which has the effect of making the particles stick 
 together in little strings which adhere to the vessel's walls. 
 On measuring the length of a string and counting the num- 
 ber of particles in it, their radii could be determined. 
 
 The third method depended on an application of Stokes' 
 law to the rate of fall of the particles under the action of 
 gravity. 
 
 These methods gave concordant results. Thus in one 
 case the three methods gave for the same particle the diam- 
 eters .46/x, .455ju, and .45ju. Particles of mastic and gamboge 
 in a solution thus appear to be spherical in shape, and 
 their motion obeys Stokes' law. 
 
 These measurements determine the various quan- 
 tities in equation (165) except N, which is therefore expressed 
 in terms of these quantities. In this way Perrin obtained 
 
 N = 7.05X10 23 , 
 
 which agrees as well as can be expected with the value 
 given in Section 3. 
 
 It is very important to notice that this investigation 
 does not depend upon the velocity of translation of the 
 particles being the same as if they were in the perfectly 
 gaseous state. It depends merely upon the osmotic pressure 
 obeying the gas equation, which according to Section 50 
 may be the case independent of the velocity of translation 
 of the particles. This point is of importance because these 
 
220 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 experiments are occasionally cited as proving, on account 
 
 nfTI 
 
 of the occurrence of the factor -^- in equation (165), that 
 
 the colloidal particles in a solution have the same velocity 
 as they would have in the gaseous state. 
 
 52. A Free Path Formula involving Stokes' Law. 
 
 According to Stokes' law if a spherical particle of radius 
 r is under the action of a force F in a medium of viscosity 
 rj and density p, and there is no slipping (Section 34), 
 the velocity V c with which the particle moves is given by 
 
 F.- 
 
 Since V c = MF, where M denotes the coefficient of mobility, 
 the equation may be written 
 
 M = ^-. . . . (166) 
 
 671T77 
 
 On substituting this expression for M in equation (148) 
 it becomes* 
 
 h 2 RT 1 
 
 (167) 
 
 Since the value of l s differs little from four times the aver- 
 age amplitude A\ of a particle observed in practice (Sec- 
 tion 44) the equation should agree approximately with the 
 facts provided Stokes' law holds. If for example the values 
 of 4A i and t (Section 44) observed by Svedberg for platinum 
 particles are substituted in the left-hand side of the equation 
 for l s and t it does not agree even approximately with the 
 values of the right-hand side, as is shown in Table XXVII. 
 This may be caused by the values of kA\ differing more 
 * Not previously published. 
 
AN EQUATION OF MOTION OF PARTICLES 221 
 
 from the values of l s than is apparent. The values of 4Ai 
 are probably otherwise unobjectionable since the coefficients 
 of diffusion of platinum particles in different solvents cal- 
 culated from them (Section 44) give very approximately 
 Dry = constant. It is more likely that the motion of plati- 
 num particles in a solution obtained by the sparking of 
 platinum electrodes in a solvent does not obey Stokes' law. 
 This is perhaps not surprising since these particles are not 
 likely to be spherical in shape as required by this law, 
 but more likely consist of flakes, since they are produced 
 by portions of the electrodes being torn off through the 
 electrical discharge. 
 
 The particles of gamboge used by Perrin in the investi- 
 gation described in the previous Section are more likely 
 to be spherical in shape since they were prepared by rub- 
 bing gamboge in distilled water giving a yellow solution 
 containing particles of various sizes whose corners would 
 more or less be dissolved off. Thus the motion of such 
 particles might obey Stokes' law, as Perrin has directly 
 observed. Values of l s and t for such particles are, however, 
 not available to test equation (167). 
 
 But whatever the geometrical configuration of the 
 particle the velocity V c is likely to vary inversely as r/, or 
 
 (168) 
 
 where K c is a constant depending on the nature of the 
 particle; and accordingly equation (148), since V C = MF, 
 may be written 
 
 1 8 2 _3RT Kc nfiq x 
 
 T nr rj- 
 
 This equation appears to agree in a general way with the 
 facts. It might be used to calculate K c , which has been 
 carried out for platinum particles in Table XXVII, using 
 
222 MISCELLANEOUS APPLICATIONS, CONNECTIONS . 
 
 the data contained in Table XXII. The values obtained 
 are practically constant, as we would expect that they should 
 be if the mass of the particles is kept the same. 
 
 TABLE XXVII 
 
 PLATINUM PARTICLES OF RADIUS .25X10 CMS. 
 
 Solvent.' 
 
 107 's 2 /< 
 
 107 RT l 
 
 D ~N~2^, 
 
 K c 10- 
 
 Acetone 
 
 6.45 
 2.51 
 2.46 
 1.42 
 .640 
 
 10.9 
 5.43 
 4.24 
 2.45 
 1.11 
 
 3.688 
 2.934 
 3.696 
 3.693 
 3.681 
 
 E. acetate 
 Amyl acetate 
 
 Water 
 
 Propyl alcohol 
 
 
 Some interesting and important extensions of the vis- 
 cosity, conduction of heat, and diffusion formula? will 
 now be given which involve an extended meaning of the 
 molecular paths under various conditions. Following 
 these a number of important formulae related to the fore- 
 going will be developed involving the projection of the 
 motion of a molecule along an axis, instead of. the molecular 
 path. By means of these formulae further information 
 about the motion of a molecule and other of its properties 
 may be obtained. The foregoing developments are new, 
 and have not been published previously. 
 
 53. Extended Forms of the Diffusion Equations., 
 
 The most important property of the mean diffusion 
 path of a molecule defined in Section 41 which is contained 
 in its definition is, that the sum of the diffusion paths be- 
 tween two points is equal to the corresponding length of 
 the actual path of the molecule. This property enables us 
 to express the number of diffusion paths n crossing a plane 
 
AN EXTENDED TREATMENT OF DIFFUSION 223 
 
 one square cm. in area, a quantity which occurs in the 
 diffusion equations, in terms of quantities which can be 
 determined directly or indirectly. We may, however, give 
 other definitions to the diffusion path, which, however, do 
 not enable us to express in a simple way the corresponding 
 value of n in terms of other quantities. Thus we may take 
 a number of points on the path of a molecule subject to any 
 condition we please, join consecutive points by straight 
 lines, and suppose that the molecule in its migration passes 
 along these straight lines instead of along its actual path. 
 The points, it should be noted, need not even lie on the 
 actual path of the molecule. The straight lines thus obtained 
 constitute, as usual, the representative diffusion paths of a 
 molecule under the stated conditions. 
 
 Suppose for example that the points are taken on the 
 path of a molecule, and are so selected that the diffusion 
 paths are grouped about a mean path l' dr of any chosen 
 length according to Clausius 7 distribution law given in 
 Section 31, and that any direction of a path in space is 
 equally probable. It follows then from the investigation 
 in Section 42 that the expression for the coefficient of dif- 
 fusion of a dilute solution of molecules r in e is the same in 
 form as the coefficient given by equation (129), or 
 
 D, = ~jf, (170) 
 
 where I' Sr has replaced I 8r , and n' T the number of representa- 
 tive paths cutting a square cm. in one direction has replaced 
 n r . We may give I' Sr any value we please above a limit 
 determined later, and determine the corresponding value of 
 n'r from the equation. Since ri r is a measure of the chance 
 of a molecule crossing a plane one square cm. in area in 
 moving along its representative path, it follows from the 
 foregoing equation that this chance is inversely proportional 
 to l' 8r . 
 
.224 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 The quantity n' T in equation (170) may be expressed in 
 terms of other quantities which are of interest. Thus if 
 V' Sr denote the average velocity a molecule would have 
 if it passed over its representative path, instead of over its 
 actual path, it can be shown that 
 
 _ 
 
 along the same lines as equation (35) was obtained. We 
 also have directly that 
 
 7> 
 TTt v fir 
 
 where t 1 ' ir denotes the average time it would take the mole- 
 cule to pass over the average diffusion path l' &r , if it passed 
 over its representative paths. It should be noted that V' 8r 
 is not equal to the total average velocity V tr of the molecule 
 unless I'sr is equal to l Sr . By means of the foregoing two 
 equations equation (170) may be written in the forms 
 
 ^-^V* < m > 
 
 and 
 
 In these equations we may give l' dr any value we please 
 above a limit which will be obtained presently and determine 
 from them the corresponding values of V f Sr and t' Sr . 
 
 Since the path of a particle is undulatory, and the 
 points locating the diffusion paths lie upon the actual path 
 of the particle, it is evident that the smaller l' Sr is taken the 
 nearer is th - representative path equal to the actual path, and 
 the nearer V' Sr approaches V tr the total average velocity of the 
 molecule. Therefore V' dr cannot be smaller than V tr , and l' Sr 
 therefore according to equation (171) not smaller than 
 
A MOLECULAR PATH RELATION 225 
 
 Wr/Vtr- This limiting value of l' Sr corresponds to the value 
 of the diffusion path l Sr defined in Section 41, since the 
 representative and actual molecular paths are the same in 
 length. But the points locating the path l Sr cannot, and do 
 not, lie on the actual path. It follows therefore that under 
 the foregoing conditions l' Sr can have values only which are 
 somewhat larger than the value of l Sr . The paths are evi- 
 dently of interest only under these conditions, namely in that 
 their points of location lie on the actual path of the molecule. 
 
 If a molecule in a substance moved along a straight line 
 with a constant velocity, the period t' Sr would be proportional 
 to l' Sr , instead of proportional to (l' Sr ) 2 as indicated by 
 equation (172). It follows therefore that a molecule pur- 
 sues a zigzag course in a substance. It is interesting to 
 illustrate equation (172) by means of a diagram in this 
 connection. Thus let abed in Fig. 18 denote the actual path 
 of a molecule and ac and ad two selected mean diffusion 
 paths of which ad has double the length of ac. Now it 
 follows from equation (172) that if the molecule traveled 
 over its representative paths, the time taken in passing 
 over the path ad is 2 2 or 4 times the time taken in passing 
 over the path ac. And since the molecule takes the same 
 time in passing over its actual path as over its representa- 
 tive path it follows from the figure that the molecule on 
 the average takes times in the ratio of 1 to 4 in passing 
 over the actual paths abc and abed, where ad = 2ac. 
 
 It is obvious that we may substitute n' r , n' e , l f Sr , and 
 l' 6e for n r , n e , l Se , and l de in the general diffusion equation 
 (126), where the former symbols have meanings of the 
 nature just considered. The equation may be given forms 
 involving V' Sr , V' Se , t' Sr , and t' Se similarly as just shown. 
 
 The points on the actual path of a molecule which indi- 
 cate the location of the diffusion paths may be selected in 
 a different way than the foregoing, which is simpler and 
 physically more definite. Thus we may select the points 
 
226 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 
 
 so that each line joining two consecutive points is equal to 
 the same length instead of being grouped about a mean 
 path according to Clausius' law, and that any direction of 
 a line in space is equally probable. The deduction of the 
 diffusion equation is then simplified since Clausius' prob- 
 ability factor need not be introduced, and therefore no 
 integration with respect to z is necessary. The integral 
 with respect to 6 (Section 42) is the same as before, and 
 
 FIG. 18. 
 
 introduces the factor J into the right-hand side of the 
 diffusion equation, which otherwise would be cancelled 
 by the foregoing integral. The coefficient of diffusion in 
 this case for a dilute solution is therefore given by 
 
 _ 1 n" r l" S r 
 
 N r 
 
 where l"^ denotes the selected diffusion path, and n" r 
 the number of diffusion paths passing through an area of 
 
A GENERAL EXTENDED DIFFUSION EQUATION 227 
 
 one square cm. in one direction per second. The foregoing 
 equation may be written in the forms 
 
 V" 1" 
 D r = *p* (174) 
 
 and 
 
 DT= l~t^ (175) 
 
 similarly as before, where V" Sr denotes the average velocity 
 the molecule would have if it passed along its diffusion 
 paths, and t" 8r the average time it would take to pass over 
 a single diffusion path. It can be shown similarly as before 
 that l" 8r can have values only somewhat greater than the 
 average diffusion path I 8r . 
 
 The general diffusion equation (corresponding to equa- 
 tion (126)) may be written in this case 
 
 (l"&r) 2 N e dN r (l" Se ) 2 N r dNe 
 c*n j~ G*rr j 
 
 where the meaning of the different symbols is evident from 
 what has gone before. 
 
 The points on the path of a molecule locating the dif- 
 fusion paths may be selected in a third way which is of 
 interest. Thus the period the molecule takes to pass from 
 one point to the next may be taken the same. Let us suppose 
 in this connection that of N r molecules r per cubic cm. 
 n ri have a diffusion path l\ and n r2 a diffusion path /2, and 
 so on, corresponding to the period t'" sr . We may consider 
 each of these sets of molecules separately and apply the 
 preceding result to obtain expressions for the diffusion. 
 On taking into account that the concentration gradient 
 
 corresponding to the molecules n fl is ~ -^, and that 
 
 J\ f ax 
 
 n dN 
 corresponding to the molecules n r% is -- - r ^, and so on, it 
 
228 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 follows at once that the coefficient of diffusion of a dilute 
 solution is given by 
 
 - - ._, 
 '~6T'7,r ~1VT ~\-~T^' ' 
 
 where (l" f Sr ) 2 denotes the mean of the squares of the dif- 
 fusion paths. The corresponding general form of the dif- 
 fusion equation may now be written down without any 
 difficulty. 
 
 It will readily be recognized now that the points locating 
 the diffusion paths may be selected in other ways, in fact 
 according to any law we please. The foregoing three ways 
 are, however, the only ones of special interest and importance. 
 
 An interesting and important feature of the foregoing 
 equations is the fact that either the period or the diffusion 
 path may be given any value we please. We may therefore 
 immediately calculate the diffusion path corresponding to a 
 given period, or vice versa. These values furnish interesting 
 information about molecular motion. A set of such cal- 
 culations has been carried out in the next Section in con- 
 nection with similar equations involving viscosity. 
 
 The dependence of the diffusion path, or the period, 
 in equation (175) on more fundamental quantities may be 
 obtained by equating the coefficient of diffusion given by 
 the equation and that given by equation (129), and substi- 
 tuting for I 5r from equations (130), which gives 
 
 
 This equation may also be written 
 
 by means of equation (35). Thus for a selected constant 
 value of l"sr the period varies inversely as the total average 
 velocity V tr of a molecule r, as we might expect.. The 
 
RELATIONS OF THE PATH PERIOD 229 
 
 existence of the apparent volume b' r of the molecules r 
 and e has the effect of decreasing the period from what it 
 otherwise would be. This could not be recognized directly. 
 The existence of molecular interference has the same effect 
 since & Sr is positive. Also an increase in K' ST , which is 
 proportional to the diffusion path in the gaseous state at 
 standard pressure, tends to decrease the period. 
 
 The period may be expressed in terms of the partial 
 intrinsic pressures and other quantities on eliminating n r 
 from equation (178) by means of equation (152). 
 
 The variation of l" Sr for a given selected constant period 
 t" Sr is obtained by solving the foregoing equations with 
 respect to I" &. 
 
 Equations similar to the foregoing may be obtained 
 corresponding to a mixture which is not dilute. Thus on 
 equating the expressions for 5 r given by equations (126) 
 and (176), eliminating l dr and I 8e by means of equations 
 
 (130), and equating the factors of -^ and =-? separately 
 
 clx dx 
 
 to zero, which holds since the equation is an identity, we 
 obtain in the case of the molecules r an equation which is 
 the same as equation (178) having the factor 
 
 introduced into the right-hand side. Thus we see that in 
 the case of a solution of molecules r in e which is not dilute, 
 an increase in the number of molecules r relative to the mole- 
 cules e increases the period t" 8r . The existence of the external 
 molecular volume ft e of the molecules e decreases the period 
 from what it otherwise would be, while the external molecular 
 volume &T of the molecules r increases it. 
 
 The periods corresponding to the selected paths, or vice 
 versa, may approximately be calculated from equations 
 (178) and (179) on determining the approximate values 
 
,230 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 of the quantities on their right-hand sides by means of the 
 method described in Section 42. 
 
 54- Extended Forms of the Viscosity Equations. 
 
 The equations for the viscosity given in Sections 34 and 
 35 may be given extended forms similarly as the diffusion 
 equations in the previous Section. Points may be selected 
 on the actual path of each molecule and these joined, and 
 the supposition made that momentum is given to the sub- 
 stance parallel to its motion, or abstracted from the sub- 
 stance, at these points only by the molecule. The line 
 joining two consecutive points may be called a momentum 
 transfer distance under the stated conditions. The points 
 of location of the transfer distances may be selected 
 according to any given law; there are three ways only, 
 however, which are of interest, and which were used in con- 
 nection with diffusion. Thus we may suppose that the 
 transfer distances are grouped about their mean according 
 to Clausius' law; or are equal to each other; or correspond 
 to the same period for the molecule to pass from one point 
 to the consecutive point, the points in each case being so 
 selected that any direction of a line of given length is equally 
 probable. 
 
 In the first case the form of the viscosity equation re- 
 mains the same as that of equation (83), and for a pure 
 substance r may be written 
 
 (180) 
 
 where /' \ r denotes the mean momentum transfer distance 
 in this case, and n' r the number of representative paths 
 crossing a square cm. from one side to the other per second. 
 The latter quantity is given by 
 
 3 ' 
 
EXTENDED GENERAL VISCOSITY EQUATION 231 
 
 where V nr denotes the velocity the molecule would have if 
 it moved along its representative path, this quantity being 
 
 directly given by 
 
 V 
 
 v --JE 
 
 *ir. f > 
 f r,r 
 
 where t'^ denotes the period of the average transfer distance. 
 Equation (180) may therefore be written in the more impor- 
 tant form 
 
 (181) 
 
 The equation for the viscosity of a mixture of molecules 
 r and e is accordingly given by 
 
 N r m ar (I'^.a. , e 
 
 --- ~~ ' (182) 
 
 If the transfer distances are taken equal to each other, 
 it can be shown similarly as in the previous Section that 
 the corresponding viscosity equations are the same in form 
 as equations (181) and (182) having the factor \ introduced 
 into each right-hand side, that is, we may write 
 
 r, = ^y, (183) 
 
 and 
 
 _N r m ar (l"r, r } 2 , 
 r J~ a =Fr r 
 
 TV ' TV 
 
 where l'\ r denotes the selected constant momentum transfer 
 distance of a molecule r, t"^ r the mean of its periods, while 
 the other symbols referring to the molecules e have similar 
 meanings. 
 
 If the average period for a pure substance in equation 
 (183) is taken equal to the period in the case of equation 
 (175) applied to the diffusion of a molecule in molecules of 
 
332 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 the same kind, and this period eliminated from the two 
 equations, we obtain the equation 
 
 If the nature of the motion of a molecule were not altered 
 by a shearing motion given to the substance, l" v would be 
 equal to l" Sr , which corresponds to the normal motion of a 
 molecule. The foregoing equation, however, indicates that 
 this does not hold, as we might expect, and expresses the 
 effect of shearing motion on the value of l" v for unit 
 velocity gradient in the substance. 
 
 Since the viscosity of a gas is independent of its pressure, 
 or the molecular concentration, it follows from equation 
 (183) that the value of the average period t"^ for a constant 
 value of I" ^ is proportional to the molecular concentration 
 N T . This shows that the molecules in their encounters 
 deflect each other from their paths, and therefore the actual 
 path of a molecule between two points is greater than the 
 straight line joining them. We may therefore say that the 
 period a molecule takes to traverse its curved and zigzag 
 path between two points a constant distance apart in a 
 gas whose pressure is varied, is proportional to its chance of 
 encountering another molecule. 
 
 The viscosity of a liquid is greater than that correspond- 
 ing to the gaseous state, and hence according to equation 
 (183) the average period t rr v f r a constant value of V' v 
 in the case of a liquid, is smaller than proportional to N r , or 
 smaller than the chance of the molecule encountering an- 
 other molecule along its path. This would be the case if 
 the average velocity of a molecule in the liquid state is 
 greater than in the gaseous, which would decrease the time 
 it takes the molecule to pass over its path. This fits in with 
 results obtained previously. . . 
 
VALUES OF PERIODS FOR A GIVEN PATH 
 
 233 
 
 It is of interest to obtain by means of equation (183) 
 for different substances the average periods t" nT that a mole- 
 cule takes to pass from one point to another one mm. apart, 
 or corresponding to /",= .! cm. Table XXVIII gives the 
 values of the periods in the case of C02 at different pres- 
 
 TABLE XXVIII 
 
 VALUES OF t" T CORRESPONDING TO l" 1)T = .l CM. 
 
 C02 AT 40 C 
 
 P in 
 atmos. 
 
 Volume 
 of a gram 
 molecule. 
 
 r, 10>. 
 
 N r lQ2i. 
 
 l "r,r in 
 seconds. 
 
 70 
 
 245.7 
 
 200 
 
 2.52 
 
 1.49 
 
 80 
 
 172.2 
 
 218 
 
 3.61 
 
 1.95 
 
 85 
 
 130.7 
 
 269 
 
 4.74 
 
 2.08 
 
 94 
 
 85.35 
 
 414 
 
 7.26 
 
 2.08 
 
 100 
 
 78.96 
 
 483 
 
 7.86 
 
 1.92 
 
 112 
 
 73.24 
 
 571 
 
 8.48 
 
 1.75 
 
 ETHER 
 
 t C. 
 
 p- 
 
 ,,ia>. 
 
 Ar r io 
 
 *'V in 
 
 seconds. 
 
 13.5 
 
 .7214 
 
 1779 
 
 6.05 
 
 .67 
 
 99 
 
 .6421 
 
 1133 
 
 5.39 
 
 .94 
 
 CHLOROFORM 
 
 
 
 1.5264 
 
 3827 
 
 7.93 
 
 .66 
 
 60 
 
 1.4108 
 
 2791 
 
 7.33 
 
 .84 
 
 BENZENE 
 
 15.4 
 
 .8840 
 
 4387 
 
 7.38 
 
 .36 
 
 78.8 
 
 .8145 
 
 3000 
 
 6.65 
 
 .45 
 
234 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 sures at the temperature 40 C., and of three liquids at 
 different temperatures. The period of a molecule of CC>2 
 when it does not obey the gas laws evidently does not 
 depend much on the molecular concentration, the effect 
 of the increase of concentration being probably more or 
 less balanced by the increase in molecular velocity. The 
 order of the period is two seconds. In the case of liquids 
 the period is increased with an increase of temperature, 
 as we would expect, since the concentration is thereby 
 decreased which decreases the molecular velocity. Its value 
 appears to decrease with an increase in the molecular weight 
 of the liquid, and is of the order of one second. 
 
 In the case of a gas r at standard temperature and pres- 
 sure the period is under the foregoing conditions given by 
 the equation 
 
 -77- 4.5X10- 7 m r 
 
 t r,r=~ , .... (186) 
 
 where m T denotes the relative molecular weight, and r? the 
 viscosity which in this case is independent of the pressure. 
 An idea of the dependence of the period on the nature of 
 the gas may be obtained from an inspection of Table X 
 which contains values of TJ and m for a number of different 
 gases. 
 
 If the period is kept constant it can be shown similarly 
 as in the previous Section that the viscosity in the case of 
 a pure substance r is given by 
 
 N r m ar (I gr) 
 
 6 t' 
 
 where (Z"V) 2 denotes the mean of the squares of the transfer 
 distances corresponding to the constant period '", and an 
 equation consisting of similar terms may be obtained for a 
 mixture of substances. 
 
THE PATH PERIOD AND OTHER QUANTITIES 235 
 
 1 
 
 It can be shown in the same way as in the previous Sec- 
 tion that the smallest admissible values of l\ n l'\ n and 
 I"V are somewhat larger than the value of l^ defined in 
 Section 33. 
 
 The dependence of the extended transfer distances and 
 periods on the nature rf molecular interaction is obtained 
 on equating the expression for the viscosity obtained in 
 this Section with those obtained in Section 34. Thus in 
 the case that the transfer distance is kept a constant length 
 we obtain from equation (183) and equations (90), (91), 
 and (92), the equations 
 
 ,, 
 
 ' 
 
 . 
 
 C\Q(\\ 
 
 ' 
 
 on adding the suffix r to the symbols of the latter equations 
 in order to bring the notation into line with the former 
 equation, and taking into account that VrN r m a r = mr, where 
 
 A T = 5.087X10 
 
 It will be seen from these equations that the apparent 
 molecular volume 6 r , and the interference function $^ r 
 (which is positive), decreases the period from what it other- 
 wise would be. Thus the existence of molecular forces of 
 repulsion, which prevents the molecules approaching each 
 other within any degree of closeness, and the existence of 
 interference of the molecules of the substance with two 
 interacting molecules, have the effect of decreasing the 
 period. An increase in the total average velocity V tr has 
 the effect of decreasing the period, as is also directly evident. 
 
236 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 In the case of a substance not obeying the gas laws an in- 
 crease in velocity at constant temperature, it may be noted, 
 may be brought about by an increase in the density of the 
 substance (Sections 17 and 29). An increase in K^, which de- 
 creases with an increase of the molecular mass, has the effect 
 of decreasing the period, and this also holds for the volume 
 V T of a gram molecule. An increase in the molecular forces 
 of attraction would give rise to an increase in the intrinsic 
 pressure P nr , and in the number n r of molecules crossing a 
 square cm. per second, and thus give rise to a decrease of 
 the period. Equation (189) of the three equations (188), 
 (189), and (190), indicates best the dependence of the 
 period on the temperature at constant volume. It decreases 
 with an increase of temperature, since V tr , , r , and ^ 
 increase with an increase of temperature at constant volume 
 (Sections 16, 17, and 34), while b r is probably approximately 
 independent of the temperature. 
 
 The dependence of l'\ T on the fundamental properties 
 of a substance for a constant period is obtained on solving 
 the foregoing equations with respect to l n v . It is evident 
 that the effect of the changes in the quantities considered 
 on the quantity l'\ r is opposite in direction, and less (through 
 the extraction of square roots on solving the equations as 
 indicated) than in the case of the period. 
 
 In order to obtain the dependence of the periods on 
 other quantities in the case of a mixture of molecules r 
 and e, each of the terms on the right-hand side of equation 
 (184) is equated with one of the two terms to which it cor- 
 responds on the right-hand side of equation (98), and the 
 resulting equation transformed by means of equations (99) 
 and (100). Equations similar in form to equations (188), 
 (189), and (190) will be obtained in this way. 
 
 Approximate values of the periods corresponding to 
 given transfer distances, or vice versa, in the case of a 
 mixture, may be obtained from these equations on obtain- 
 
THE PARTITION OF MOMENTUM TRANSFERENCE 237 
 
 ing approximate values of the other quantities they contain 
 in the way described in Section 35. 
 
 In connection with the foregoing investigation the fol- 
 lowing remarks may help to clear up any difficulties encoun- 
 tered. If we take a plane parallel to the motion of the 
 substance corresponding to the velocity Vi, it follows from 
 considerations of equilibrium that the algebraical momentum 
 per molecule parallel to the plane on the average is equal to 
 Vim a . Therefore on considering two planes corresponding 
 to the velocities V\ and 2 of the substance, it follows that 
 the molecules crossing each plane in the same direction will 
 each have on the average its momentum parallel to the 
 planes changed from V\m a to Vznia on passing from one 
 plane to the other. This will, of course, not hold for each 
 molecule considered independently. These considerations 
 show that we may suppose that momentum is given by a 
 molecule to the medium and abstracted from it in these 
 planes only, which may be taken any distance apart. This 
 may be illustrated by considering a number of similar par- 
 allel planes corresponding to the velocities FI, Vi, . . . V e of 
 the substance. The momentum transferred from the first to 
 the last plane by a molecule is equal to ( (Vi F 2 ) + (2 Fa) 
 + . . . (V e -iVe)}m a , or equal to (Fi F e )m a , and thus the in- 
 termediate planes have no effect on the amount of momentum 
 transferred from the first of the planes to the other. It 
 follows, therefore, that a molecule crossing one of the fore- 
 going planes may, or may not, be taken to change its vis- 
 cosity momentum in crossing it, just as we please. Also a 
 molecule may recross a number of times each of two consec- 
 utive planes between the two points lying on the planes at 
 which only it is supposed to change its viscosity momentum. 
 The foregoing planes may therefore be taken to indicate 
 the location of the various points defining the molecular 
 paths used in the previous investigation ; and hence it follows 
 that these paths are permissible to use. 
 
238 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 It should be pointed out, however, that the representa- 
 tive molecular path probably cannot assume all numerically 
 possible values between two given limits, or that it is prob- 
 ably a recurring discontinuous function. It can easily be 
 shown, for example, that the values of the path I for a part 
 which is a straight line do not fit in with the relation l 2 /t = 
 constant. But the total region of discontinuity, if not zero, 
 is probably small in comparison with the remaining region. 
 
 55. Extended Forms of the Heat Conduction 
 Equations. 
 
 Extended heat conduction equations may be obtained 
 along the same lines as the extended viscosity and diffusion 
 equations in the preceding two Sections. We may suppose 
 as before that the heat transfer distances are defined by 
 points distributed according to any given law on the paths 
 of the molecules. The transfer distances are of interest only, 
 however, when the points are distributed according to one 
 of the three ways already used. 
 
 If the transfer distances are grouped about a mean 
 according to Clausius' law the heat conductivity equation 
 for a pure substance r may immediately be written 
 
 (191) 
 
 according to Section 38, where V CT denotes the average 
 heat transfer distance which may have any value above a 
 certain limit to be determined presently, n' T denotes the 
 number of transfer distances crossing a square cm. in one 
 direction per second, S or denotes the specific heat at con- 
 stant pressure per gram and m ar the absolute molecular 
 weight. If V cr denotes the average velocity a molecule would 
 have if it passed along its successive transfer distances, and t er 
 the time it would take to pass over the average transfer 
 
THE CONSTANT PATH CONDUCTION EQUATIONS 239 
 
 distance (which is the same as the time taken to pass over 
 the actual path), we have 
 
 Hence equation (191) may be written in the form 
 
 3 t' cr > 
 
 which is more important. The heat conduction equation 
 for a mixture of molecules r and e may now immediately 
 be written down. 
 
 If the distance between each pair of consecutive points 
 is taken the same and equal to l" C r, and t" cr is the average 
 period of the transfer distance, it can be shown along the 
 same lines of reasoning as contained in the preceding two 
 Sections that 
 
 for a pure substance, and 
 
 _ N r m ar S or (l" cr ) 2 . N e m ae S oe (I"*}* . 
 
 ~<r "FT ~s~ ~W ' 
 
 for a mixture of molecules r and e. 
 
 On the other hand if the period t'" T is kept the same it 
 can be shown along similar lines as before that 
 
 ~ j-argT \l / CT) /lOP^ 
 
 ~6 PT' ..... 
 
 in the case of a pure substance r, where (l" f cr ) 2 denotes the 
 mean of the squares of the corresponding heat transfer dis- 
 
240 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 tances. An equation consisting of the sum of similar terms 
 holds for the coefficient of conductivity in the case of a 
 mixture. 
 
 It can be shown along the same lines as in the preceding 
 two Sections that the least admissible values of l' C r, l"cr, 
 and T 77 ^. are somewhat larger than the value of l c as defined 
 in Section 38. 
 
 If the period in equation (193) is taken equal to the 
 period in equation (183) and it is eliminated from the 
 equations, the equation 
 
 is obtained. This equation gives some information about 
 the effect of a shearing motion of a substance, and of a flow 
 of heat, on the nature of the motion of a 'molecule. For 
 the quantity l r \ T refers to the motion of a molecule in a 
 substance undergoing a shearing motion corresponding to 
 a unit velocity gradient, and the quantity I" CT to the motion 
 of a molecule in a substance through which a flow of heat 
 takes place corresponding to a unit heat gradient. If the 
 molecular motion were not affected by these conditions 
 l"cr would be equal to l"^. But this is evidently not the 
 case according to the foregoing equation, since the term 
 under the radical is not equal to unity. According to Sec- 
 tion 38 the right-hand side of the equation is larger than 
 unity when the substance is in the liquid state, and smaller 
 than unity when in the gaseous state. Therefore on the 
 average the line joining two points on the path of a molecule 
 in a liquid for a constant period is increased when a velocity 
 gradient exists in the substance in comparison with the 
 magnitude of the line when a heat gradient exists, and the 
 opposite holds in the case of a gas. 
 
 The fact that the internal molecular energy of a complex 
 molecule is several times greater than the free kinetic energy 
 
A GENERAL FORM OF TRANSFER FORMULAE 241 
 
 of translation (Section 13), gives rise to a transference of 
 energy to the medium by the molecule in moving along a 
 heat gradient, or vice versa, which is considerably greater 
 than the change in kinetic energy. This will have the 
 effect of changing the nature of the path of the molecule 
 considerably from what it would be if the molecule possessed 
 no energy apart from its kinetic energy of translation. 
 The distortion of the molecular path by a heat gradient is 
 probably in the main caused in this way. 
 
 If the periods are taken equal in equations (193) and 
 (175) and eliminated we obtain the equation 
 
 l" 8r 
 
 W*-\ 
 
 ( . 
 
 The quantity l" Sr may be taken to correspond to the diffusion 
 of a molecule in a substance of the same kind instead of in 
 a mixture in which molecules r possess a concentration 
 gradient. The motion of a molecule in the former case is 
 normal. It would be interesting to test the equation under 
 these conditions. 
 
 The heat transfer distance and its period may be connected 
 with more fundamental quantities in the same way as similar 
 quantities in the preceding two Sections in the case of 
 viscosity and diffusion. 
 
 It will be seen that the diffusion, viscosity, and heat 
 conduction expressions given in this and the preceding 
 two Sections are each of the form l 2 /t multiplied by a factor 
 which depends on the conditions under which the molecules 
 are undergoing motion of translation. Each of the quan- 
 tities I and t may be taken to a certain extent as arbitrary, 
 but not at the same time, since one determines the other. 
 
 A set of equations for the diffusion, viscosity, and heat 
 conduction of substances will now be developed, which are 
 similar to the equations obtained in this and the preceding 
 
242 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 * 
 
 two Sections, and which involve the projection of the motion 
 of a molecule along a line. 
 
 56. The Constant Period Displacement Dif- 
 fusion Equation and its Applications. 
 
 Let us take an imaginary plane for reference at right 
 angles to the diffusion in a heterogeneous mixture and 
 determine the number of molecules of one kind that drift 
 across it per square cm. per second. The motion of the 
 molecules will be expressed in terms of their displacement 
 at right angles to the foregoing plane for a selected period 
 t. It will be evident that in each element of volume half 
 of the molecules during the time t undergo a displacement 
 in one direction, while the other half undergoes a displace- 
 ment in the opposite direction. Suppose that n rv n T2J n T3 , 
 . . . , molecules r whose displacements are respectively 
 dr t1 d T2) dr 3J . . . , during the period t r cross the plane per 
 square cm. in the direction of their diffusion say from 
 top to bottom. The molecules of displacement d Tl come 
 from a cylinder of height d T} and unit cross-section stand- 
 ing on the plane, while the molecules of displacement d rz 
 come from a cylinder of height d T2 standing on the plane, 
 etc. Therefore if n' fl , n' T2 , n' Tz . . . , molecules r of dis- 
 placements d Tl , d T dr 3 , . . . , respectively come from each 
 cubic cm. of the mixture we have 
 
 n ri = n' ri d ri , n Tz =n f r 2 d r etc. 
 We also have 
 
 2+rc' r2 + . . .)=N r , . . . (198) 
 
 since each molecule undergoes a displacement during the 
 period t r , one-half of the molecules undergoing a displace- 
 ment in one direction and the other half undergoing a dis- 
 placement in the other direction, where N T denotes the con- 
 
MOLECULAR DISPLACEMENT ACROSS A PLANE 243 
 
 centration of the molecules r. The total number of molecules 
 r which cross the plane in the direction of diffusion, or 
 from top to bottom, during the time t r is therefore equal to 
 
 But molecules r also cross the plane in the opposite 
 direction. Similarly as before the molecules of displacement 
 d ri come from a cylinder of height d fl and unit cross-section 
 standing on the plane, and the molecules of displacement 
 d Tz come from a cylinder of height d r2 standing on the plane, 
 etc. We may suppose, to simplify matters, that the molecules 
 r of displacement d fl which cross the plane of reference in 
 the direction of the diffusion, or from top to bottom, come 
 from a plane at a distance d ri /2 from the plane of reference, 
 while the molecules of displacement d Tl crossing in the 
 opposite direction come from a plane at the distance d r J2 
 on the opposite side of the plane of reference. The numbers 
 of these two sets of molecules are proportional to the con- 
 centrations of the molecules at the starting planes. If the 
 concentration at the upper starting plane is N r , that at the 
 
 . , 7 , dN r i dN T . Al _ 
 
 lower plane is N T d Tl - r - , where - T is the concentration 
 dx dx 
 
 gradient of the molecules r measured in the direction of 
 increase of concentration. We may therefore write KN r 
 
 and K( NT dr. r^ ) for the numbers of these two sets for 
 \ dx / 
 
 migrating molecules, where K denotes an appropriate 
 factor. Since n ri = KN r these numbers may also be written 
 
 n ri d ri dN r 
 ,, and m ^ -fc, 
 
 or 
 
 , , n' n d? n dN r 
 n' ri d fl and n' n d n -- -* 
 
,244 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 since n Tl = n' ri d ri . Therefore the excess of molecules r of 
 displacement d Tl crossing the plane of reference in the direc- 
 tion of diffusion over the molecules r of displacement d ri 
 crossing in the opposite direction in the time t r , is equal to 
 
 n' n d? n dN r 
 N r dx' 
 
 Similarly it can be shown that the excess of molecules r 
 of displacement d T2 crossing the plane is equal to 
 
 N r dx' 
 
 and so on. Therefore the total excess of molecules r that 
 cross the plane in one second due to the displacements they 
 undergo, is equal to 
 
 (2n' n d? n +2n' r # n + . . . ) ldN r = d? r dN r 
 2(2n' T1 +2n' r2 + . . . ) t r dx 2t r dx ' 
 
 by the help of equation (198), where 
 
 (2n' n d 2 n +2n' r2 d 2 n+ . . .)_^- 
 (2n' n +2n' n + . . . ) 
 
 the mean of the squares of the displacements. Similarly 
 it can be shown that the excess of molecules e that cross 
 the plane in the opposite direction per second due to the 
 displacements they undergo is equal to 
 
 QdN, 
 
 2t e dx ' 
 
 where d 2 e denotes the mean of the squares of the displace- 
 
 dN 
 ments corresponding to the period t e , and -^ the con- 
 
 CtOu 
 
 centration gradient of the molecules e measured in the 
 
DISPLACEMENT DIFFUSION EQUATION 245 
 
 direction of increase of concentration, and thus in the 
 opposite direction that the -gradient of the molecules r is 
 measured. Thus on the whole there is a gain of molecules 
 r and e per second below the plane equal to 
 
 2t r dx 2t e dx' 
 
 But the space occupied by these molecules is not zero. 
 A portion of the mixture must therefore be transported 
 bodily across the plane to make room for the foregoing 
 molecules. This may be taken into account in exactly the 
 same way as in Section 42. It will appear from that investi- 
 gation that the total migration 5 r of molecules r across the 
 plane per second is accordingly given by 
 
 2t r dx 2t e dx 
 
 where $ r and d e denote the external molecular volumes of 
 gram molecules of molecules r and e in the mixture. The 
 diffusion of the molecules e is obtained by interchanging 
 the suffixes r and e in the foregoing equation. 
 
 In the case of a dilute solution of molecules r in e the 
 foregoing equation becomes 
 
 d? r dNr 
 
 The coefficient of diffusion D r in this case is therefore given 
 by _ 
 
 A=J (201) 
 
 Thus if the molecular displacements in a mixture for a given 
 period could be measured the diffusion could be calculated. 
 
^246 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 This is of course impossible in the case of molecules, but 
 can be carried out in the case of Brownian particles. Exam- 
 ples of such investigations will be considered further on. 
 
 The foregoing equations are limited according to the 
 following considerations: Suppose that the displacements 
 of the molecules of a cubic cm. of matter were observed for 
 several consecutive periods each equal to t r . The value of 
 d 2 T for each set of observations should be the same, since the 
 equations are independent of the time of beginning of a 
 set of observations. Now this is the less likely to be realized 
 the smaller the value of t r . Hence there is a limiting value 
 of t r for lower values of which the equations do not hold. 
 This limiting value can definitely be determined as will 
 now be shown. If the path of each molecule consisted of a 
 number of straight lines joined at various angles, and their 
 lengths were grouped about a mean length according to 
 Clausius' probability law given in Section 31, while each 
 inclination of a line in space were equally probable, each 
 line or molecular free path would be independent of the 
 preceding path. The corresponding displacements, or pro- 
 jections of the free paths on to an axis, would therefore also 
 be independent of each other. But this would evidently not 
 hold for displacements corresponding to a fraction of a free 
 path. Now the average of the displacements d r , which is 
 given by 
 
 and the average free path l dr are connected by the equation 
 
 4d~r = kr (202) 
 
 according to Section 44. Hence it follows that a period 
 t r is inadmissible if the corresponding average displacement 
 has a smaller value than given by the foregoing equation. 
 
THE INFERIOR LIMIT OF THE DISPLACEMENT 247 
 
 This would be indicated in practice by the values of t r and 
 d? r not fitting in with equations (200) and (199). But in 
 practice the molecules do not move along a series of lines 
 joined together, especially in a liquid. We may, however, 
 (Sections 41 and 42) find a series of rectilinear paths lying 
 close to the actual path equal to it in length, which indicate 
 the average directions of different portions of the actual 
 path as shown in Fig. 14, and whose lengths are grouped 
 about a mean length according to Clausius' law, and whose 
 directions in space are unrestricted. The parallel planes 
 passing through the points of connection of these paths 
 divide the actual path into a number of parts which we may 
 take as being independent of each other as the correspond- 
 ing rectilinear paths. The smallest period t r admissible 
 in equations (199) and (200) would then be that correspond- 
 ing to the average displacement which satisfies equation 
 (202), where the symbol I 8r now denotes the average dif- 
 fusion path as defined and used in Sections 41 and 42. 
 Smaller values of t r and the corresponding values of d? r 
 will therefore not satisfy equations (199) and (200). 
 
 This result may be obtained in a somewhat different 
 way. Let us suppose as before that a molecule moves along 
 a series of rectilinear paths. Consider the displacement 
 corresponding to a portion of a rectilinear path for a period 
 , which is much smaller than the period of the path. The 
 corresponding displacement d\ is then proportional to t\, 
 or equal to tf! lt where C, is a constant. . The value of 
 d 2 r/t r in equation (200) under these conditions is therefore 
 equal to t T C2, where 2 is a constant. But this is obviously 
 impossible, and the period t r can therefore be given values 
 only equal to or greater than the period of the average path. 
 Since the actual path of a molecule may be on the average 
 represented by such a series of rectilinear paths, the same 
 result holds in reference to the period of the average rep- 
 resentative path. The deviation of the ratio from satis- 
 
248 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 .. 
 
 fying equation (200) on decreasing t r would, however, be 
 more gradual in this case. 
 
 If t'r denote the smallest admissible period that satisfies 
 equation (200) and d' r the corresponding average of the 
 displacements, the total average velocity V tr of a molecule 
 r is given by 
 
 (203) 
 
 by the help of equation (202), since t' T is then the period 
 of the average diffusion path l dr , and the representative 
 and actual path of a molecule are equal in length (Section 
 41). This equation may be used to find the velocity of a 
 colloidal particle, as will be described in Section 60. The 
 number of times n r per second that a square cm. is crossed 
 by molecules r in one direction is given by 
 
 _N r V tr _4N r d r r 
 
 Wr -3- -gjT-, - (204) 
 
 according to equations (203) and (35). 
 
 The foregoing remarks, and equations (203) and (204), 
 apply also to a mixture of molecules r in e which is not a 
 dilute solution. 
 
 The largeness of the value that may be taken for the 
 period t r is limited somewhat by the fact that in the deduc- 
 tion of equation (199) it is assumed that the number of 
 molecules undergoing a displacement d at each of two 
 equal elements of volume in a heterogeneous mixture at a 
 distance d apart is proportional to the concentration of the 
 molecules. This will of course not hold when the difference 
 between the concentrations in the two volumes is large. 
 This difference evidently depends on the concentration 
 gradient of the mixture. The admissible values of the 
 displacement and period therefore depend on the concen- 
 
THE MOBILITY IN TERMS OF DISPLACEMENTS 249 
 
 tration gradient, and increase with decrease of the gradient. 
 The same remarks apply to the limits of l' Sr in Section 53. 
 
 On equating the coefficients of diffusion given by equa- 
 tions (201) and (175), and eliminating (l n br ) 2 /t" &T from the 
 resultant equation by means of equation (178), or (179), 
 we obtain equations similar in form to the latter equations, 
 which express d 2 r /t r in terms of fundamental quantities. 
 The same remarks apply to these equations as were made 
 in connection with equations (178) and (179). 
 
 Equation (201), which applies to a dilute solution, has 
 previously been deduced * by Einstein by a different method, 
 without obtaining, however, its exact limitations. The 
 method that I have developed is more general and funda- 
 mental since it gives diffusion formulae for any relative 
 concentration of the constituents of the mixture without 
 referring to its state, and gives the exact limitations of the 
 formulae. It should be noted that no restrictions whatever 
 are made in the investigation in connection with the molec- 
 ular velocities. By means of this method other diffusion 
 formulae, and formulae for the viscosity and conduction of 
 heat, can be obtained in terms of molecular displacements. 
 These are given in subsequent Sections. 
 
 An application of the foregoing results to the Brownian 
 motion of colloidal particles will now be given. 
 
 (a) On equating the values of the coefficients of dif- 
 fusion given by equations (201) and (147) the equation 
 
 (205) 
 
 is obtained, which expresses the coefficient of mobility 
 of a particle, or the velocity under the action of unit force, 
 in terms of other quantities. If the particle is spherical in 
 shape and its size is so large that its mobility obeys Stokes 7 
 
 * Ann. der Phys., 19, pp. 371-381 (1906). 
 
.250 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 law, the coefficient of mobility given by the foregoing 
 equatiori and equation (166) may be equated giving 
 
 N , (206) 
 
 where r denotes the radius of the particle, and 77 the vis- 
 cosity of the medium. This equation has been applied to 
 the Brownian motion of colloidal particles of different 
 substances in different solvents, the values of _d r for a selected 
 value of t r being directly observed by means of the ultra- 
 microscope as described in Section 44, or recorded photo- 
 graphically by means of an appropriate arrangement. 
 Some of the earlier observations did not fit in well with the 
 equation. This was probably due to some extent that the 
 period was taken below the least admissible value, the 
 limitations of the equation not being known. Later experi- 
 ments happen to have been carried out for larger periods 
 and consequently gave a better agreement. It is also 
 hardly likely that the colloidal particles obtained by spark- 
 ing between various metal electrodes placed in a liquid 
 should in all cases be exactly, or approximately, spherical 
 in shape as required by the equation. We would expect 
 that the geometrical shape of the particles should depend 
 a good deal on the conditions of the sparking. A good 
 confirmation of Einstein's equation was recently obtained 
 by Nordlund,* who used particles obtained by sparking 
 mercury in a liquid. These particles are bound to be spherical 
 in shape since they would be in the liquid state, and hence 
 Stokes' law would hold. Nordlund determined N, the 
 number of molecules in a gram molecule, by means of the 
 equation, and obtained the value 
 
 5.91 X10 23 , 
 * Zs. f. Phy. Chem., 87, pp. 40-62 (1914). 
 
RATIO SQUARE OF DISPLACEMENT TO PERIOD 251 
 
 which is in close agreement with the values obtained by 
 other methods. 
 
 Einstein's formula has been exhaustedly tested by 
 Millikan* in the case of particles suspended in a gas, and 
 found to be in complete agreement with the facts. 
 
 According to equation (201) the value of is constant 
 
 t r 
 
 for different values of d r and t r . This is necessary to test 
 before using equation (206). Table XXIX shows the 
 
 TABLE XXIX 
 
 Periods. 
 
 Z d2 T . 
 
 4 
 
 u 
 
 1.039 
 
 1.039 
 
 2t 
 
 2.458 
 
 1.229 
 
 3t 
 
 3.957 
 
 1.319 
 
 4t 
 
 5.264 
 
 1.416 
 
 5t 
 
 6.685 
 
 1.337 
 
 61 
 
 7.963 
 
 1.327 
 
 Wt 
 
 13.528 
 
 1.353 
 
 agreement obtained by Nordlund in the investigations just 
 described. It will be seen that for periods of the displace- 
 
 ment d r from 3 to Wt inclusive Z - is practically constant 
 
 t r 
 
 where Z is a constant. The values for the periods It and 2t 
 do not fit in well with the others, and they are therefore 
 probably inadmissible values along the lines previously 
 explained. (See also Section 60.) The result is of interest 
 and importance since it is evidence of the soundness of the 
 deduction of equation (199), and incidentally of the deduc- 
 tion of the equations in Sections 57 and 58, and of the 
 
 * The Electron, by R. A. Millikan, the University of Chicago Science 
 
 Series. 
 
252 MISCELLANEOUS APPLICATIONS, CONNECTIONS - 
 
 equations in Sections 53, 54, and 55, each of which involves 
 an indeterminate linear magnitude connected with the 
 actual path of a molecule, and its corresponding indeter- 
 minate period. 
 
 57. The Constant Displacement Diffusion Equa- 
 tion. 
 
 Instead of considering the displacement of the mole- 
 cules for a selected period /, we may consider the different 
 periods of the molecules corresponding to a selected displace- 
 ment d. This case may be treated along similar lines as 
 that in the previous Section. Suppose that n' Tl molecules 
 r per cubic cm. have a period t ri in undergoing a displace- 
 ment d r in one direction at right angles to the plane of 
 reference, and that n' T2 molecules have a period t ra etc. 
 Each of these sets of molecules may be treated independ- 
 ently along the same lines as the investigation in the pre- 
 ceding Section. It follows immediately then from the 
 previous Section that the gain per second in molecules r 
 having the period t ri below the plane of reference is equal 
 to 
 
 n' n d? r dNr 
 
 dx> 
 
 and the gain in molecules r having a period t T2 is equal to 
 
 and so on. The total gain in molecules r is therefore 
 
 (i 
 
 ~ + ~ + ' * ' W'S"* 2 r ~S? > 
 
 where 
 
 ~~\ r / T" [TF; 
 Ti ^rj I -iVf 
 
A DISPLACEMENT DIFFUSION EQUATION 253 
 
 the mean of the reciprocals of the periods. Similarly it 
 can be shown that the gain in molecules e below the plane 
 of reference is 
 
 2 dx' 
 
 where d e is a selected displacement and t e ~ 1 the mean of 
 the reciprocals of the corresponding periods. The gain in 
 molecules r and e due to their displacements is therefore 
 
 ~ l d 2 r dN r t e ~ 
 
 2 dx 2 dx' 
 
 But the volume of these molecules is not zero, and if this is 
 taken into account similarly as in the previous Section we 
 obtain that the diffusion d r of the molecules r is given by 
 
 * _ 
 
 8r ~ 
 
 2 NN re dx e er dx 
 
 where & e and # r denote the external molecular volumes of 
 gram molecules of molecules e and r respectively in the 
 mixture. 
 
 In the case of a dilute solution of molecules r in mole- 
 cules e the foregoing equation becomes 
 
 fr T 
 
 and hence the coefficient of diffusion is given by 
 
 D r = '^f (209) 
 
 It can be shown along the same lines as in the previous 
 Section that the smallest admissible value of d r in equation 
 (209) corresponds to its value in the equation 
 
 4d r =l 5r , (210) 
 
254 MICSELLANEOUS APPLICATIONS, CONNECTIONS 
 
 where I 8r denotes the average diffusion path according to 
 Section 41. If d' T denotes the smallest admissible value of 
 d r and t' T the average of the corresponding periods, the 
 total average velocity V tr of a molecule r is given by 
 
 F, r =^= 4 | r , ..... (211) 
 
 t r t T 
 
 by means of the foregoing equation, since the average dif- 
 fusion path is equal to the corresponding actual path. The 
 number of times n r a square cm. is crossed per second by 
 molecules r in one direction is given by 
 
 according to equations (211) and (35). The foregoing 
 remarks and equations also hold when the mixture is not a 
 dilute solution of molecules r in e. 
 
 On equating the expressions for D r given by equations 
 (209) and (147) we obtain the equation 
 
 (213) 
 
 which may be used to determine the coefficient of mobility 
 M r of a colloidal particle. This equation and equation 
 (166) give the equation 
 
 RT 1 
 
 ..... (214) 
 
 which depends on Stokes' law, and may be used in the 
 same way as equation (206). 
 
 It will now be recognized that an expression for the 
 diffusion may be obtained along the same lines corresponding 
 to any law of selection of the displacements, or of the periods. 
 
DISPLACEMENT OF MOMENTUM 255 
 
 58. The Constant Period and Constant Displace- 
 ment Viscosity Equations. 
 
 The viscosity of a substance also may be expressed in 
 terms of molecular displacements at right angles to a plane. 
 Let us first obtain a formula for the viscosity in terms of 
 displacements corresponding to a constant period. 
 
 (a) Let the plane of reference be taken parallel to the 
 motion of the substance, and let us suppose that the velocity 
 gradient is unity, and for convenience of reference from top 
 to bottom. We will adopt the same notation as in Section 
 56, and suppose that we are dealing with a pure substance 
 of molecules r instead of with a solution of molecules r 
 in molecules e. Let us suppose that a molecule at the begin- 
 ning of a displacement (which takes place at right angles 
 to the motion of the medium) abstracts the momentum 
 V\m a from the medium, where V\ denotes the velocity of 
 the medium in a plane parallel to its motion passing through 
 the molecule whose absolute mass is m ar . This momentum, 
 we will suppose, is transferred to the medium at the end 
 of the displacement. Now it follows from considerations of 
 equilibrium that the displacement of a molecule in one 
 direction from one plane to another, both of which are 
 parallel to the motion of the medium, is accompanied by the 
 displacement of another molecule from one plane to the 
 other in the opposite direction. Therefore if V\ and 2 
 denote the velocities of the medium at two planes taken a 
 distance d from each other, the momentum transferred 
 from one plane to the other by a molecule and its associate 
 molecule is (V\ V2)m a , or dm a , since the velocity gradient 
 of the medium is unity, or (V\ Vz)/d=\. Since n ri mole- 
 cules of displacement d ri cross the plane of reference per 
 square cm. in each direction during the period t r , or n' Tl d ri 
 molecules, where n' Tl denotes the number of molecules in 
 a cubic cm. undergoing a displacement d rt in one direction, 
 
256 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 it follows that these molecules transfer the momentum 
 n f ri d 2 ri niar across the plane. Similarly it follows that the 
 molecules undergoing a displacement d ri transfer the mo- 
 mentum n'rf- r ^n ar during the period t r across the plane 
 per square cm. and so on. The total momentum trans- 
 ferred per second per square cm. of the plane, which is the 
 coefficient of viscosity r?, is therefore given by 
 
 _ N r m ar 1 2n' n d 2 n + 2ri r 4 2 r*+ . . . " 
 
 or 
 
 '-^r* (215) 
 
 where d 2 r denotes the mean of the squares of the displace- 
 ments of the molecules of a cubic cm. during the period t r . 
 
 It will not be difficult to see that in the case of a mixture 
 of molecules r and e the coefficient of viscosity is given by 
 the equation 
 
 _ rar , N e d 2 e m ae /0 ! >. 
 
 ~ ~~~~ 
 
 where d 2 e denotes the mean of the squares of the displace- 
 ments of the molecules e in a cubic cm. during the period t e . 
 It can be shown in a similar way as in Section 56 in con- 
 nection with diffusion that the smallest admissible value 
 of d 2 r in equations (215) and (216) corresponds to the 
 value of the mean of the displacements d r which satisfies 
 the equation. 
 
 43", = ^, ....... (217) 
 
 where l^ denotes the mean momentum transfer distance 
 of a molecule r as defined in Section 33. 
 
COMPARISON OF DIFFERENT DISPLACEMENTS 257 
 
 If d' T denotes the smallest admissible mean of the dis- 
 placements and t' r the corresponding period, the total aver- 
 age velocity is given by 
 
 (218) 
 
 while the number of molecules n r crossing a square cm. 
 in one direction is given by 
 
 obtained along the same lines as similar equations in Sec- 
 tion 56. 
 
 If equation (201) is supposed to apply to the diffusion 
 of a molecule in a substance of the same kind and the period 
 is taken the same as the period in equation (215) and 
 eliminated from the equations we obtain the equation 
 
 = =T^v7- (220) 
 
 d 2 Sr D T N r m ar ' 
 
 where the suffixes 8 and 77 have been added to the displace- 
 ments to indicate what each refers to. According to this 
 equation the ratio on the left-hand side is not in general 
 equal to unity. Thus the displacement of a molecule along 
 an axis is affected by a shearing motion of the substance at 
 right angles to the axis. 
 
 In the case of a dilute solution of molecules e in r the 
 value of d? r will very approximately be the same as for the 
 solvent in the pure state, and may accordingly be obtained 
 for any selected period t r from equation (215). The value 
 of d?~ e may then be obtained for any selected period t e from 
 equation (216). It would furnish interesting information 
 to. compare this value of d 2 e with that of a pure substance 
 
258 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 of molecules e, or if the solute consists of colloidal particles, 
 to compare the value with that observed. 
 
 If rj is eliminated from equation (215) by means of one 
 of the equations (90), (91), and (92), an equation is obtained 
 which expresses the ratio d 2 7 /t r in terms of more fundamental 
 quantities, the equation being similar to one of the equa- 
 tions (188), (189), and (190). 
 
 If we write t e = t r and d 2 e d 2 T in equation (216) it gives 
 for any selected value of t r a value of d 2 r which in a sense is 
 a mean value of d 2 e and d 2 T . It would be interesting to com- 
 pare this value with that of the substance r in the pure state. 
 
 The values of d 2 r and d 2 e in equation (216) may be 
 determined for any selected values of t r and t e by equating 
 each of the two terms on the right-hand side of the equa- 
 tion with the corresponding term on the right-hand side 
 of equation (98), and calculating the other quantities in- 
 volved in the way described in Section 35. 
 
 It can easily be proved along the same lines as the 
 preceding investigation that if the periods of the molecules 
 in a cubic cm. for a selected displacement d r are observed, 
 the coefficient of viscosity is given by 
 
 77 = V r d 2 r t^m arj (221) 
 
 where t r ~ l denotes the mean of the reciprocals of the periods 
 corresponding to the displacement d r and is given by 
 
 n 2nf_ n 
 
 ' ' 
 
 (2ri n +2n' n + . . . ) 
 In the case of a mixture of molecules r and e we have 
 
 The smallest admissible value of d r in equation (221) 
 satisfies the equation 
 
 U T = l v , . . , . , , (223) 
 
DISPLACEMENT OF HEAT ENERGY 259 
 
 which follows similarly as before. The formulae for the 
 total average velocity and the number of molecules r crossing 
 per second a square cm. from one side to the other in terms 
 of the smallest admissible displacement d' r and the cor- 
 responding mean of the periods t' r are the same in form as 
 equations (218) and (219). Similar relations apply in the 
 case of the molecules in a mixture of molecules r and e. 
 
 59. The Constant Period and Constant Dis- 
 placement Heat Conduction Equations. 
 
 We will now find an expression for the coefficient of 
 conduction of heat in terms of molecular displacements, 
 using the same notation as before, and taking the plane of 
 reference at right angles to the flow of heat, which we will 
 suppose takes place from top to bottom. A formula involv- 
 ing the displacements for a selected constant period will 
 be first obtained. Let us suppose that a molecule r at the 
 beginning of a displacement, (which takes place parallel 
 to the flow of heat at right angles to the plane of reference) 
 abstracts the energy TiS mr from the medium and transfers 
 it to the medium at the end of the displacement, where 
 TI denotes the absolute temperature of the medium at the 
 beginning of the displacement, and S mr the corresponding 
 internal specific heat at constant pressure per molecule. 
 Since no matter is transferred from one plane to another, 
 each of which is at right angles to the flow of heat, it fol- 
 lows that the displacement of the foregoing molecule is 
 accompanied by the displacement of another molecule in 
 the opposite direction, both displacements lying between 
 the same two parallel planes. Therefore if TI and TI are 
 the absolute temperatures in descending order of magni- 
 tude of the medium in these planes the energy T\S mr T2S mr 
 is transferred to the lower plane by a pair of correspond- 
 ing molecules during the period t r . If the distance between 
 
-260 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 the planes is d and the heat gradient is equal to unity 
 Ti T2/d=l, and the amount of heat transferred may be 
 written dS mr . Since n ri molecules of displacement d ri cross 
 the plane of reference in both directions during the period 
 t r , which number (Section 56) is equal to n' ri d ri , it follows 
 that these molecules transfer the energy n f ri d 2 Tl S^ T per 
 square cm. across the plane. Similarly it follows that 
 the molecules of displacement d zr transfer the energy 
 n' rt d 2 rj$mr per square cm. across the plane, and so on. 
 The total energy transferred per square cm. per second 
 across the plane, or the heat conductivity C, is therefore 
 given by 
 
 ri _N r S m l2n f n d? n +2n' r J? n + . . 
 
 " 2t r ( N T 
 
 or 
 
 where d 2 r denotes the mean of the squares of the displace- 
 ments of the molecules in a cubic cm. If S or denote the in- 
 ternal specific heat at constant pressure per gram we have 
 Smr= Syr^ar, where m ar denotes the absolute molecular 
 weight of a molecule r, and the foregoing equation may 
 therefore be written 
 
 d 2 r . 
 
 ...... ( } 
 
 In the case of a mixture of molecules r and e we have 
 
 r _ N r m ar S gr d 2 r , N e m af S oe d 2 e 
 
 ~~" ~"~" 
 
 It can similarly be shown that if the displacement d r 
 is kept constant we have 
 
 C = Nrm 2 rSar d 2 T t^, (227) 
 
THE ADMISSIBLE VALUES OF DISPLACEMENTS 261 
 and 
 
 where t. 1 denotes the mean of the reciprocals of the periods 
 of the molecules in a cubic cm. corresponding to the dis- 
 placement d Tl and t e ~ l has a similar meaning. 
 
 The least admissible value of the average displacement 
 d T in the foregoing equations can be shown, similarly as 
 in the preceding Sections, to be the value which satisfies 
 the equation 
 
 4d r = l cr , 
 
 where l cr denotes the heat transfer distance as defined in 
 Section 37. The magnitude that may be given to the value 
 of d r is limited somewhat by the fact that in the deduction 
 of the equations we have assumed that the specific heat 
 along the heat gradient at two points is independent of their 
 distance apart. But this cannot hold unless the heat 
 gradient is taken infinitely small. The formulae for the 
 heat conductivity may be given forms taking the variation 
 of S mr and N r along the heat gradient into account, but 
 on account of their complexity they possess no particular 
 interest or importance. 
 
 If the periods are taken the same in equations (225) 
 and (215) and eliminated we obtain the equation 
 
 (229) 
 
 where the suffixes c and r? have been added to the dis- 
 placements to indicate to what they refer. The right-hand 
 side of the equation is evidently in general not equal to 
 unity, and hence the nature of the motion of a molecule is 
 influenced in different ways by a heat gradient and a shear- 
 ing motion in a substance. It should be noted that cP^ 
 
262 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 refers to unit heat gradient, and d 2 ^ to unit velocity gradient 
 of the substance. According to Section 38 the right-hand 
 side of the foregoing equation is greater than unity when 
 the substance is in the gaseous state, and smaller than 
 unity when in the liquid state. Thus the displacement 
 of a molecule for a given period in a heat gradient is rela- 
 tively greater than in a velocity gradient of a substance 
 when it is gaseous, while the opposite holds when it is in the 
 liquid state. This appears to indicate that the path of a 
 particle in a gas is rendered relatively more undulatory 
 in character by a velocity gradient than by a heat gradient, 
 while the opposite holds in the case of a liquid. 
 
 The displacements and periods may be expressed in 
 terms of more fundamental quantities similarly as in the 
 previous Section. The results may be used to find approx- 
 imate values of the former quantities in the case of a mixture 
 by the help of Section 38. 
 
 60. Another Method of Determining the Total 
 Average Velocity of Translation of a Colloidal 
 Particle. 
 
 We have seen in Section 56 that equation (201) holds 
 only when the period t T is equal to, or larger, than that 
 corresponding to the average displacement which satisfies 
 equation (202). The smallest value of t r admissible may 
 be determined by using successively values of t r in decreas- 
 ing order of magnitude and observing the corresponding 
 values of the displacements, till values are obtained which 
 do not satisfy equation (201), or equation (206) if Stokes' 
 law holds. The value of t T corresponding to the transition 
 point is the smallest admissible value. On substituting 
 this value and the value of the corresponding average dis- 
 placement in equation (203) the value obtained for V tr 
 is the total average velocity of the particle. 
 
TEST OF ADMISSIBILITY OF DISPLACEMENTS 263 
 
 Since d r on the average is proportional to t r when its 
 value is inadmissible according to the Section cited, it fol- 
 lows from equations (201) and (206) that in that case 
 
 (A) 
 
 and 
 
 & r< RT 
 
 37T7Y 
 
 Thus as the value of the period is decreased a point is ulti- 
 mately reached when the ratio d 2 r /t r ceases to be constant, 
 and decreases proportionally to t r for further decreases 
 in l r . Therefore on plotting d 2 T /t r against t rj and drawing a 
 mean straight line through the points which indicate that 
 the ratio is constant, and a mean straight line through the 
 other points, the intersection of the two lines gives the 
 period t' r and displacement d' r which on being substituted 
 in equation (203) give the total average velocity of the 
 colloidal particle. 
 
 No experiments have of course yet been carried out with 
 the ob'ect of determining the total average velocity of a 
 coll; i lal ^article in this way. It seems probable, however, 
 that Nordlund in testing the constancy of d 2 r /t r in the 
 experiments described in Section 56 advanced into a region 
 in which the values of d 2 r and t, are inadmissible in equation 
 (206). Thus an inspection of Table XXIX shows that the 
 ratio in one set of experiments, though constant for periods 
 of 3 and 101 inclusive where t= 1.481 sec., decreases con- 
 siderably for smaller periods. If this deviation is genuine, 
 as it seems to be, it will be possible to determine from 
 Nordlund's experiments whether the velocity of a colloidal 
 particle is the same as if it were in the perfectly gaseous 
 state. The radius of the mercury particle corresponding 
 to the values in the Table cited was 2.66 X10~ 5 cm. The 
 
264 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 velocity that it would have if it behaved as if it were in the 
 perfectly gaseous state is 3.33X10" 1 cm. /sec. The average 
 value of d 2 r obtained by experiment corresponding to 
 periods which were multiples of 1.481 sec. was 1.813X10" 8 
 cm., and t r is therefore equal to 1.481 sec. The average 
 displacement is probably very approximately given by the 
 square root of d? r , and hence the apparent velocity of the 
 particle according to equation (203) is 
 
 3.64XlO- 4 cm./sec. 
 
 If this represents approximately the total average velocity 
 of the particle, as is rendered probable on account of the 
 deviations mentioned, it follows that this velocity is about 
 1/1000 of that the particle would have in the perfectly 
 gaseous state. This result fits in with those of Section 49. 
 Some experiments by Henri* should be mentioned in 
 this connection. He obtained cinematograph photographs 
 of the Brownian motion of colloidal particles of rubber of 
 IM radius in water. The displacements observed, which 
 corresponded to a period of ^V of a second, were four times 
 smaller than those calculated from Einstein's equation. 
 This deviation might have been caused by the period being 
 smaller than the least admissible value of the period, since 
 the deviations are in the right direction. It is true that 
 Henri tested whether W T /t T is constant, though this point 
 was not specially investigated, and found the ratio approxi- 
 mately constant. But the deviation of this ratio from a 
 constant when it occurs is probably very gradual, and 
 may therefore be approximately constant in the inadmissible 
 region for values lying between limits of considerable mag- 
 nitude. It is difficult to see how in Henri's experiments 
 an error could have come in to produce a deviation of the 
 magnitude mentioned. 
 
 * Compt. Rend., 146, 1024-1026, (1908). 
 
MOLECULAR VELOCITY AND ITS PROJECTION 265 
 
 It is highly desirable that experiments be carried out 
 with the object of evaluating the ratio d 2 r /t r of colloidal 
 particles for as small periods as possible in order to obtain 
 in one or more cases with certainty the smallest period that 
 will satisfy equation (206), and to calculate from that the 
 total average velocity of the particle concerned. It is 
 probably not accidental that in all cases that have come 
 under my notice the above ratio when not fitting in with 
 equation (206) fitted in with the inequality (A) which 
 corresponds to periods not admissible in equation (206). 
 
 61. The Distribution of the Molecular Velocities 
 in a Substance not Obeying the Gas Laws. 
 
 It does not seem possible as yet to obtain any definite 
 theoretical information about this distribution. The proof 
 of Maxwell's law given in Section 7 applies only to gases, 
 and the law need not therefore hold for a substance not 
 obeying the gas laws. It is very probable, however, that 
 the distribution of molecular velocities obeys Maxwell's 
 law in all cases. This may be tested directly for colloidal 
 particles in solution. Thus according to Section 7 the 
 probability that the velocity of a molecule has values lying 
 between Vi and V \-\-dV \ and is 
 
 This expression may be converted into one involving dis- 
 placements (Section 56) as follows: The projection of the 
 path of a particle between two points on to an axis is the 
 same as if the points were joined by a straight line. Now 
 if the path of each particle were divided into parts corre- 
 sponding to the same period t r , these joined by straight 
 lines, and the particle supposed to move along them, the law 
 
266 MISCELLANEOUS APPLICATIONS, CONNECTIONS 
 
 of distribution of velocities would probably not be altered 
 if t r is below a certain limit, where FI and V p would now 
 refer to the new conditions. Instead of directly transform- 
 ing the foregoing expression into one involving displace- 
 ments, it will be more convenient to derive the required 
 expression from one used in the deduction of the foregoing 
 probability expression. Thus we have seen in Section 7 
 that the probability that a molecule has a component 
 velocity lying between a and a + da is 
 
 The quantity V p in the expression may be expressed in terms 
 of the average d r of the displacements corresponding to the 
 period t r . According to Section 7 we have 
 
 V 
 
 17 _ o v P 
 
 v t 4^-, 
 
 Vir 
 
 where V t , the total average velocity of a molecule in any 
 state, now takes the place of F a , the average velocity in the 
 gaseous state. On applying equation 2p = l s in Section 44 
 to this case by considering the different molecular velocities 
 individually and adding up the results, we obtain 
 
 The foregoing two equations give the equation V p = 2 VV d r /t r , 
 which expresses V p in terms of d r . Therefore on substi- 
 tuting this expression for V p , and writing d r /t r for a, in the 
 foregoing probability expression, and considering a numb^ 
 of displacements N d , we obtain 
 
VELOCITY DISTRIBUTION AND DENSITY 267 
 
 for the number of the displacements N d which have values 
 lying between d r and d-\-d(d r ) corresponding to the period t r . 
 The values of the observed displacements appear to agree 
 with the foregoing law of distribution, and Maxwell's law 
 for the distribution of molecular velocities holds therefore, at 
 least approximately for colloidal particles. 
 
 The most probable displacement, which is obtained by 
 dividing the foregoing expression by Nd, differentiating it 
 with respect to d r , and equating the result to zero, is equal to 
 zero. The reason for this is not difficult to see. Consider 
 the molecular paths of the same length per period t r passing 
 in all directions through any given point. It is evident then 
 that the number of molecules associated with a given pro- 
 jection increases with a decrease in its value. It is of impor- 
 tance to point out that the average displacement obtained 
 by direct observation will therefore tend to be rather too 
 large than too small. The total average velocity of a col- 
 loidal deduced according to Section 56 will therefore tend 
 to be too large rather than too small, a point which is of 
 importance in determining whether this velocity is the same 
 as that corresponding to the gaseous state. 
 
 The value of the most probable velocity V p in Maxwell's 
 distribution law will depend, according to what has gone 
 before, on the solvent in which the particles are suspended 
 besides on their mass, and in the case of a molecule the value 
 of V p will increase with increase of density of the substance 
 at constant temperature, since we have seen that its total 
 average velocity increases with the density. 
 
SUBJECT INDEX 
 
 Absolute temperature: Deduction of, 12; correction of scale of, 14; 
 
 zero of, 14. 
 
 Amplitude: Of motion of colloidal particle, 9, 189, 190. 
 Atom: Absolute mass of hydrogen, 7. 
 Attraction: Forces of, surrounding molecules and atoms, 46, 47. 
 
 Brownian motion: Of particles in liquids, 9, 189, 190, 217-219, 250, 
 251; of particles in gases, 10, 251. 
 
 Calorie: Definition of, 38. 
 
 Colloidal particles: Preparation of, 10. 
 
 Conduction of heat: Coefficient of, 148; characteristic function of, 
 154, 159, 161; displacement formulae for, 259-262; extended forms 
 of formulae for, 238-241; formula for, of pure gas, 157; formula 
 for, of gaseous mixture, 165; general formulae for, of pure sub- 
 stance, 155; general formulae for, of mixtures, 164; interference 
 function of, 154, 158, 162, 163; mean transfer distance in, 148- 
 150; nature of 147; of gaseous mixtures, 165; of liquids, 162, 163; 
 of pure gases, 158, 159; of solids, 166, 167, 184-186. 
 
 Crookes' radiometer: Action of, 10. 
 
 Diffusion: Cause of, 168; characteristic function of, 174, 177, 179- 
 181; coefficient of, 169; connection of coefficient of, with mobility, 
 198; displacement formulae for, 242-249, 252-254; formula for, of 
 gases, 179; general formula for, 173; interference function of , 174, 
 175, 178; Maxwell's expression for, of gases, 187; mean path of 
 colloidal particle, 192, 193; tables of various coefficients of, 180, 199. 
 
 Displacement and its period: Constancy of ratio of square of first 
 to second, 251; table of values of, 251. 
 
 Effect: Joule-Thomson, 46, 47. 
 
 Electric conductivity: Formulae for, 182-186; ratio of, to heat con- 
 ductivity, 184-186. 
 
 269 
 
270 SUBJECT INDEX 
 
 
 
 Electrons: Charge carried by 7; concentration of, in metals, 182, 183; 
 
 dissociation of atoms into, 166; number of, crossing a square cm. 
 
 of a' substance per second, 167; variation of concentration of, 
 
 with change of temperature, 1&3. 
 Energy: Equipartition of molecular kinetic, in gaseous mixture, 31; 
 
 internal molecular, 41, 103; kinetic, per molecule of a gas, 29; 
 
 potential, of attraction, 103; total internal, of a substance, 103. 
 
 Factor: For converting calories into ergs, 38. 
 
 Gases: Equation of pure, 12; equation of mixture of, 13; equation 
 of, from dynamical considerations, 29, 30; value of R in equation 
 of, 14. 
 
 Heat: Latent of evaporation, 46, 205. 
 
 Heat gradient: Effect of, on molecular path, 240, 241. 
 
 Interaction : Of molecules and atoms, 48-50. 
 
 Intrinsic pressure : Cause of 69; comparison with other quantities, 
 
 73; formulae for, 70, 71, 104; from equation of state, 84; partial 
 
 intrinsic, 202-206; tables of values of, 72, 139. 
 
 Linear diameter law, 79. 
 
 Matter: Atomic nature of, 3-5. 
 
 Maxwell's law of distribution of molecular velocities: Consideration 
 of, 20-27. 
 
 Molecules: Formulae for number of, per c.c. in a gas, 33, 34; nature of 
 motion of, 225, 226; number of, per gram molecule, from charge 
 on the electron and electrolysis, 7; from observed displacements 
 and Stokes' law, 250; from counting and Van't Hoff's law, 217-219. 
 
 Mobility: Connection of coefficient of, with osmotic pressure, 197; 
 coefficient of, expressed in terms of molecular displacements, 249- 
 254; of a colloidal particle in an electric field, 200, 201; of a 
 colloidal particle in terms of its path and period, 199, 200; table 
 of coefficients of, 199. 
 
 Number of molecules n crossing a square cm. in one direction: De- 
 termination of, by an equation of equilibrium, 95-105; formula 
 for, corresponding to any state, 56; in a gas, 34-37; inferior and 
 superior limits of, 74, 75; inferior limit, of 80; probability series 
 for, 96. 
 
SUBJECT INDEX 271 
 
 Osmotic pressure: Cause of, in heterogeneous mixtures, 208, 209; 
 connection of, with molecular motion in a dilute solution, 213-217; 
 constants of, in heterogeneous dilute solution, 217; in connection 
 with diffusion, 197; permanent nature of, 21. 
 
 Path satisfying given conditions: Mean and mean of squares of, 109; 
 
 probability of, lying between given limits, 107; probability of, 
 
 cutting a plane, 108. 
 Polymerization: Of mercury, 105. 
 Pressure: External, given by equation of state, 81-86. 
 Pressure of expansion: Average, per single molecule in any state, 
 
 35-37; nature of, in a substance in any state, 62-64; of mixtures, 
 
 67, 68; total, in terms of other quantities, 65, 66. 
 
 Repulsion: Forces of, surrounding atoms and molecules, 47. 
 
 Shearing motion: In viscosity, 110, 111; effect of, on path of particle, 
 
 232. 
 
 Sound: Formula for velocity of, 41. 
 Specific heats: Of gases, 39, 40; of liquids and dense gases, 42-44, 
 
 103, 104; ratio of the two, of gases, 40, 41. 
 State, corresponding: Nature of, 91-94. 
 State, equations of: Conditions that they have to satisfy, at critical 
 
 point, 88; at absolute zero, 89; thermodynamical, 90, 91; various 
 
 forms, 81-86. 
 Stokes' law: Formulae involving, 220, 250; nature of, 220. 
 
 Thermodynamics: First law of, 37, 38. 
 
 Volume, external molecular: Definition of, 172, 173. 
 
 Volume, internal molecular, or b: Cause of, 78, 79; determination of, 
 95-105; mathematical definition of, 58; properties of, 58, 59, 61; 
 superior limit of, 78, 79. 
 
 Volume, real and apparent : Discussion of, 77, 78. 
 
 Velocity of a molecule: Average kinetic energy, in a gas, 28; average, 
 in a gas, 20; determination of total average, 95-103, 139; inferior 
 limit of total average, 80; most probable, in a gas, 20; variation 
 of, with time in a gas, 20; variation of, with time in a substance in 
 any state, 265-267; when not under the action of a force in a gas 
 or liquid, 52, 53. 
 
 Velocity of colloidal particle: Determination of, 195, 210, 262-265. 
 
272 SUBJECT INDEX 
 
 . 
 
 Viscosity: Cause of, 110, 111; characteristic function of, 121, 128, 
 129, 130, 144, 145; definition of coefficient of, 111; definition of 
 momentum transfer distance of, 113-116; displacement formula) 
 for, 255-258; extended formulae for, 230-238; formula for, of a 
 gas, 125, 126, 131; general formulae for, 122, 142, 143; interfer- 
 ence function of, 121, 134, 138, 140, 141, 145-147; measurement 
 of coefficient of, 112; various factors influencing it, 135-136. 
 
r 
 
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