r REESE LIBRARY -n_n__rvrL--n_n, UNIVERSITY OF CALIFORNIA. A Deceived Accession No. / 6~ J J J . Class No. MOLECULES MOLECULAR THEORY MATTEE BY A. D. RISTEEN, S.B. BOSTON, U.S.A., AND LONDON PUBLISHED BY GINN & COMPANY 1896 lirlll COPYRIGHT, 1895 BY A. D. KISTEEN ALL RIGHTS RESERVED PREFACE. IN the multiplication of popular books on scientific subjects, the molecular theory of matter appears to have been strangely neglected. None of the works available to American readers pretend to give a complete, connected account of what is known of the constitution of matter, and the student who wishes to learn the present state of the molecular theory has to seek his information in the occasional articles that are scattered through the scientific journals. Dr. Watson's Kinetic Theory of Gases (a new edition of which has been recently published) is far too difficult for the undergraduates in our scientific schools and colleges ; Clausius's Kinetische Theorie der Gase (1889-91) has not yet been translated, nor has Meyer's Kinetische Theorie der Gase, so far as I am aware. Meyer's book is also out of print at present, although a new edition is in preparation. Lord Kelvin's delightful lecture on The Size of Atoms should be read by all students of physics, and it is now readily available, in the first volume of his Popular Lectures and Addresses. Crookes's classical papers on radiant matter should also be read; they are in the Proceedings of the Royal Society, beginning with the year 1874. The present volume is an attempt to elucidate the elements of the molecular theory of matter as it is held to-day. It is based on a lecture delivered on the 12th of last February, before the Washburn Engineering Society, of the Worcester Polytechnic Institute. In preparing the manuscript for the printer a considerable number of alterations have been made, and much new material has been added, though the form of presentation has been preserved. Special care has been takeni iv PREFACE. to exclude all matter except that which has an immediate and evident bearing on molecules; otherwise this book would have been a treatise on physics. For this reason many exceedingly interesting theories and discoveries have been omitted or dismissed with slight mention such, for example, as the investigations of Ostwald and others on solutions. It could hardly be expected that different men would agree on what should be passed over in this way, and it is likely that better judgment might have been "used in many places. It is also likely that there have been many oversights and accidental omissions. In some cases important theories have been passed over because it was believed that they could not be adequately discussed without introducing considerable digres- sions upon the phenomena they are intended to explain. Professor Ewing's theory of magnetism is an example of this, and Clausius's theory of electrolysis has been dismissed with a bare mention, for like reasons. Throughout this volume I have considered molecules in their physical aspect. There are numerous excellent works that discuss the chemical aspect of the molecular theory satisfactorily, of which the following may be particularly recommended: Meyer's Modern Theories of Chemistry, Hem- sen's Theoretical Chemistry, Ostwald's Outlines of General Chemistry, and Mendeleieff's Principles of Chemistry. A. D. RISTEEN. HARTFORD, CONN., September 1, 1894- . GENERAL CONSIDERATIONS. PAGE The Molecular Hypothesis S. 1 Dalton's Contribution 2 Similarity of Molecules -* . . 4 Hypothesis of Avogadro 5 Distinction between Molecules and Atoms . . . . . .6 Classification of Bodies 10 Molecular Constitution of Solids -T 12 Molecular Constitution of Liquids^ 12 Molecular Constitution of Gases f 14 Vri. THE KINETIC THEORY OF GASES. Preliminary Remarks . ... . . . . . . .14 Molecular Collisions and Free Path . . . . . . .15 The Cause of Gaseous Pressure . . . . . . .17 Molecules are Perfectly Elastic . . . . . . . .17 Velocities of Molecules Unequal 20 Statistical Method of Investigation 21 Fundamental Assumptions of the Original Kinetic Theory . . 22 Maxwell's Theorem .24 Illustrations of Maxwell's Theorem .26 I/- Determination of the Average Velocity of Translation of Hydrogen Molecules 29 ^Properties of Gaseous Mixtures 31 Degrees of Freedom 32 I/Generalized Theorems . . . . . . . . . .34 i 'Adaptation of the Foregoing Equations to the Generalized Kinetic Theory 37 ^Gaseous Pressure . . . 38 yi CONTENTS. PAGE Recalculation of the Average Molecular Velocity in Hydrogen . . 39 Pressure Produced by Several Sets of Molecules . . . .40 Avogadro's Law . 41 Boyle's Law . . . ... 41 l-Results of the Kinetic Theory compared with Results of Observation . 42 Temperature . . . 43 Absolute Zero . . . .46 Ratio of the Specific Heats of Gases 47 /Molecular Attraction in Gases 53 ^/Equations of Van der Waals and Clausius 56 I- Diffusion 58 Viscosity ' 59 Experimental Determination of /* 61 Kinetic Explanation of Viscosity 63 Free Path 66 High Vacua 68 The Radiometer 68 Crookes's Tubes . , 70 III. THE MOLECULAR THEORY OF LIQUIDS. Preliminary Remarks 74 *Free Evaporation . . * . 75 ^Cooling Effect of Evaporation 77 I Vapor Density 77 ^Vapor Pressure .79 Ebullition 80 Critical Points 82 Contraction and Compressibility 84 Surface Tension 86 Phenomena of Films . . . . . . . . .87 Other Surfa'ce Phenomena 89 Magnitude of the Surface Tension 90 Latent Heat of Vaporization 95 Investigation of the Work done in bringing a Molecule to the Surface 96 Numerical Estimation of the Work done in bringing a Molecule to the Surface , 98 CONTENTS. Vll IV. THE MOLECULAR THEORY OF SOLIDS. PAGE Condition of the Theory 102 Arrangement of the Molecules in Solids 103 Maxwell's Views concerning the Molecular Constitution of Solids . 106 Sublimation 108 Dissociation . . . . . 110 Solutions ... 112 Diffusion . . . ... 114 ^Osmotic Pressure . .117' Electrolysis 120 Saturation 121 Distillation ' ... 122 Supersaturation 123 Crystals . . . 124 Bounding Planes of Crystals . . . . . . . . 125 Molecular Structure of Crystals 127 V. MOLECULAR MAGNITUDES. Preliminary Remarks 133 The Electrical Method for Finding the Aggregate Volume of Mole- cules 133 Aggregate Volume of Molecules, from the Gas Equation . . . 134 Remarks on the Foregoing Results .135 Molecular Diameters by Clausius's Equation 138 Lord Kelvin's Electrical Method 139 Method by Camphor Movements . . . . . . . 142 The Surface Tension Method . . 143 Quincke's Determination of the Range of Molecular Attraction . 145 Other Methods of Investigation 146 Number of Molecules in a Unit Volume of Gas .... 148 Illustrations of Molecular Magnitudes 149 VI. THE CONSTITUTION OF MOLECULES. Preliminary Remarks 151 General Facts to be Explained . 152 Dulong and Petit' s Law 154 Viii CONTENTS. PAGE Front's Hypothesis 159 Periodic Law of Meyer and Mendeleieff 161 Elastic-Solid Theory of Light .164 Electro-Magnetic Theory of Light 168 Provisional Assumptions about the Constitution of Molecules . .171 Rankine's Hypothesis 173 Lord Kelvin's Vortex Theory 174 Dr. Burton's Strain-Figure Theory 180 Internal Vibration of Molecules 182 Gravitation.. . 190 Conclusion 196 APPENDIX. On the Integration of Certain Equations in the Text . . . 197 Rankine's Method for Calculating the Ratio of the Specific Heats of Gases 203 Plateau's " Liquide Glyc con " sidering that in it the particles of oxygen and hydrogen were united in pairs. Taking the hydrogen atom as the unit, it follows that the weight of the oxygen atom must be 8 ; for experiment shows that in a given mass of water ibhere is 8 times as much oxygen, by weight, as there is hydrogen. Carbon monoxide was represented by the symbol O0? an( i since for each unit of its oxygen (by weight) this gas con- tains f of a unit of carbon, it follows that the atomic weight of carbon is f of that of oxygen. Hence the weight of the .carbon atom is 6. Carbonic acid was represented by the symbol Q O* Ammonia, being the only known compound of hydrogen and nitrogen, was represented by the simple symbol 00 ; and as experiment shows that ammonia gas contains 4| times as much nitrogen as hydrogen, the atomic weight of nitrogen must be 4f. I have given you a general idea of the kind of reasoning Dalton used ; but in calculating the atomic weights I have made use of better experimental results than were available to him. A few of his own early determinations of the atomic weights are given in the following table : * ELEMENT. ATOMIC WEIGHT. Hydrogen - - 1.0 Nitrogen - - 4.2 Carbon - - - 4.3 Phosphorus 7.2 Oxygen - - - 5.5 * These were published in 1805. UNIVERSITY 4 THE MOLECULAR THEORY OF MATTER. Dalton's fundamental conception was correct, although the numbers that he used to express the atomic weights of the elements were erroneous, and so also were many of his formulae. We agree with him, however, in believing that the so-called " atomic weights " of substances are really the true relative iveiyhts of their atoms ; the weight of the hydrogen atom being taken as unity. Similarity of Molecules. Dalton assumed that all the molecules of any one substance are alike ; but I think this ought not to be admitted without some experimental evidence. Various methods for testing this assumption have been pro- posed, and while none of them are absolutely convincing, the general inference to be drawn from them is, that there is no sensible difference among the constituent particles of any given substance. I will tell you briefly of two of the methods of investigation that have been proposed. Graham's method consisted in passing pure hydrogen gas through a porous partition between two vessels. The first part of the gas that came through was collected and caused to pass through a second similar partition. The first portion that came through this partition was caused to pass through a third one, and so the process was continued until the hydrogen had passed through a considerable number of the partitions. The hydro- gen from the last operation was compared with the original hydrogen, and no difference between the two could be dis- tinguished. It was therefore considered that this gas, at least, is not a mixture of dissimilar particles ; because if it were, such a process as I have described could hardly fail to make some sort of a selection from among them, and the final gas would then be different from the primitive one. Another and a more convincing method of investigating the point in question, was tried by.Stas.* He determined the atomic weight of the same substance when prepared in different * Stas, Unter suchung en iiber die Gese+ze der chemischen Proporlionen. (Aronstein's translation.) HYPOTHESIS OF AVOGADRO. 5 ways, from different sources, and under different conditions of temperature ; and he found that the results were indis- tinguishable from one another. He also found the atomic weights of the elements to be the same, from whatever com- pound they were determined. His work was so accurate that it is not likely that a change in the atomic weight of more than the hundredth part of one per cent would escape detec- tion. Since there was no observable difference, we must con- clude that the atomic weights of his different samples were all sensibly alike ; and this indicates that the molecules of these substances were all alike, because otherwise we could reasonably expect a slight difference to be discernible when a substance was prepared from different sources, by different methods. From these and other investigations, we conclude that until some data tending to prove the contrary are pro- duced, we may reasonably proceed on the hypothesis that all the molecules of any given pure chemical substance are iden- tically alike. v Hypothesis of Avogadro. Soon after Dalton's theory had been announced, it was observed that there are simple volu- metric relations among gases when they combine. Thus it was noticed that 2 volumes of hydrogen combine with 1 volume of oxygen, to produce approximately 2 volumes of steam ; that 1 volume of hydrogen combines with 1 volume of chlorine to form 2 volumes of hydrochloric acid gas ; and so on. This being the fact, it was suggested by Avogadro in 1811, and independently by Ampere in 1813, that all gases, when under the same conditions of temperature and pressure,, contain the same number of molecules per unit of volume. This assumption also explains the observed fact that the densities \ of gases are proportional to their molecular weights ; for if * t#! and w 2 are the weights of the individual molecules of two gases, and -ZVj and N z are the numbers of the molecules in a unit volume of these gases, respectively, then the weights of a unit volume of the two gases are JV"i w 1 and N 2 w 2 , respec- 6 THE MOLECULAR THEORY OF MATTER. tively. Now the observed fact is, that these quantities are proportional to Wi and w 2 ; and hence we have the proportion from which it follows that N 1 = N 2 ' t that is, it follows that Avogadro's hypothesis is true. I do not know whether chemists at first received this hypothesis as the expression of an actual fact in nature, or merely as a sort of convenient working hypothesis. However this was, subsequent inves- tigation has made it increasingly probable, until now we must accept it as an established fact. Distinction between Molecules and Atoms. Molecules may be defined as the smallest parts into which a given sub- stance can be conceived to be divided, without changing its chemical character. An atom is not so easily defined. Up to this point, in fact, I have made no distinction between a molecule and an atom ; but if we are to accept Avogadro's hypothesis, it becomes necessary to make a distinction at once. For if you will think about it a moment, you will see that if one cubic inch of hydrogen, containing n molecules, combines with one cubic inch of chlorine, also containing n molecules, to produce two cubic inches of H 01, containing n molecules altogether, then the number of molecules in each 7? cubic inch of the H Cl gas is only ; whereas Avogadro's law requires us to assume the ' existence of n molecules in each cubic inch. It follows, therefore, that when the H and the Cl combine, their molecules do not simply unite in pairs. There is no way, in fact, to explain the observed facts, unless we assume that the molecules of H and Cl are both compound, and that when these gases combine, their molecules split in two, half a molecule of the one then uniting with half a mole- cule of the other, to produce a whole molecule of H 01. This is made plainer by the diagrams. Figs. 1 and 2 represent small and equal volumes of H and Cl, on the assumption that their molecules are simple ; and Fig. 4 represents the H Cl DISTINCTION BETWEEN MOLECULES AND ATOMS. 7 O FIG. 1. HYDROGEN MOLECULES. FIG. 2. CHLORINE MOLECULES. m . . . FIG. 3. THE FOREGOING GASES MIXED. FIG. 4. THE SAME GASES COMBINED (AVOGADRO'S LAW VIOLATED). 8 THE MOLECULAR THEORY OF MATTER. gas resulting from their combination. The space occupied by the H Cl gas is shown twice as large as that occupied by the component gases separately, because we know from experiment that when these component gases combine, the (^ eP <> * , %* * % / FIG. 5. HYDROGEN MOLECULES. FIG. 6. CHLORINE MOLECULES. &> FIG. 7. THE FOREGOING GASES MIXED. volume of the result is equal to double the volume of either one of the constituents. But you will see that the number of molecules per unit area is only half as great in the H Cl as it is in either the H or the Cl ; and this constitutes a violation of Avogadro's principle. Now if we conceive the molecules of H and Cl to be compound, as illustrated in Figs. 5 and 6 ? DISTINCTION BETWEEN MOLECULES AND ATOMS. 9 we shall have no such anomaly. Fig. 8 represents the result- ing H Cl, and you see that it fulfills Avogadro's law, as well as the observed fact of occupying two volumes. The com- pound nature of the molecules of the so-called elementary bodies is no mere logical figment ; there is direct experi- mental evidence of its truth. It is known, for example, that many substances in the nascent state, just being set free from Imeir compounds, are much more energetic in their chemical ^relations than they are under other circumstances ; and it is hard to explain this fact by any theory that assumes the molecules of such substances to be simple. On the other * 9

1 2 + ?v> + ...) M %m(v? + v? + ...)= A r ^ = ~2' u where it? is the quantity defined by equation (5), and M is the total mass of the gas. Hence we have %Mu 2 = 5163. (10) Now the weight of the hydrogen under consideration being .005592 of a pound, its mass will be .005592 + 32.2 = .0001737. Substituting this for M in equation (10), and solving for u, we have u = 7710, and F * = 7103 feet per second. You will please notice that it is not claimed that this is the average velocity of hydrogen molecules. All that can be said of it is, that it is their average velocity provided the assump- tions about them that I described to you a few moments ago * See equation (6). PROPERTIES OF GASEOUS MIXTURES. 31 correspond to the actual facts. I shall shortly give you a better determination of the velocity of the molecules of a gas one which is independent of the assumption of any particular form for the molecules. Properties of Gaseous Mixtures. If the kind of reasoning by which equation (1) was obtained is applied to a medium composed of a set of ^ spheres each with the mass m l9 a set of N 2 spheres each with the mass m 2 , and so on, the spheres in each set being exactly alike and very numerous, and every sphere being hard, smooth, small, and perfectly elastic, as before, we shall find that the different sets will mix with one^i another uniformly, and that the velocities in the spheres of! each set will be- distributed precisely as though the other sets* were not present. The most familiar example of a gaseous mixture, in nature, is air ; and one of the most striking things about air is its constancy of composition. In a chemical com- pound we should expect the proportions of the components to be constant ; but in a mere mechanical mixture of oxygen and nitrogen, for instance, we might naturally expect to find a material difference in composition in two samples, when one is taken, say, on the Himalaya Mountains, and the other on the shores of the Arctic Ocean. The fact that no such differ- ence in composition exists becomes particularly significant when we know that the kinetic theory shows that constancy of composition would necessarily result, if gases really are composed of small spherical molecules such as I have described. Another very important deduction, due, I believe, to Maxwell, is that in a mixture composed of several sets of elastic spheres, the average velocities in the different sets will not be equal ; i" the set in which the molecules are heaviest will have the smallest average velocity ; and, in general, the velocities will be such that the average kinetic energy of a molecule of one set will be precisely equal to the average kinetic energy of a molecule of any other set. This is one of the most remarkable propositions in the whole kinetic theory of gases. 32 THE MOLECULAR THEORY OF MATTER. Degrees of Freedom. I am going to tell you of some much more general theorems about gases, but before doing so I want to explain what is meant by the expression " degrees of freedom." A particle constrained to move in a given straight line is completely described when we have stated its distance from some fixed point in that line. A particle constrained to move in a given plane is completely described when we have stated its distance from two fixed, intersecting straight lines in that plane. A particle in space is completely described when we have stated its distance from three fixed, intersect- ing planes. In the first case we say that the particle has one degree of freedom, because it has only one coordinate ; in the second case we say it has two degrees of freedom, because it has two coordinates, either one of which. may vary inde- pendently of the other. In the third case we say that the particle has three degrees of freedom, because it has three coordinates, any one of which may vary, independently of the others. The Century Dictionary defines a degree of freedom as "an independent mode in which a body may be displaced." The number of degrees of freedom of a body in free space can never be less than three. It may be more than three, however. In fact, it must be more than three, if the body is anything more than a mere particle. Consider, for example, a finite straight line, of given length. One end of it, say A, has three degrees of freedom it can be anywhere in space. The other end, B, also has three coordinates, but between these coordinates and those of A there is an equation expressing the fact that the length of AB is constant. By means of this equation we could eliminate one of the six coordinates, leaving only five that are really inde- FIG. 17. A RIGID BODY IN SPACE. DEGREES OF FREEDOM. 33 pendent; and we therefore say that such a line as this has five degrees of freedom. A rigid body in space has six degrees of freedom : it becomes fixed when three of its points are given. Consider, for instance, the points A, B, and (7, in this sketch (Fig. 17). Each of these points has three coor- dinates, making nine coordinates in all ; and among these there are three equations, expressing the constancy of the distances AB, BC, and CA. By means of these three equa- tions we can eliminate three of the nine coordinates, leaving onty six that are really indepen- dent ; and hence we say that rigid bodies have six degrees of freedom. Bodies that are not rigid have more than six degrees of freedom ; the number of degrees that they possess being always determined by the number of independent FIG. is. -A SYSTEM OF JOINTED coordinates required to fix them. For example, a series of n jointed rods has (2n-}-3) degree's of freedom. We may perhaps get a clearer idea of the precise significance of the expression " degrees of freedom " by select- ing a mixed system of coordinates for defining the position of a body. Let us, for example, again consider a rigid body in space. The center of gravity of this body is free to move parallel to any of the three axes of reference. . (This is the same thing as saying that the center of gravity of the body is perfectly free, because any imaginable motion of it can be resolved into components parallel to x, y, and . See Fig. 19.) Moreover, the body is free to rotate about three axes, parallel respectively to x, y, and z. (All other rotations can be resolved into component rotations about these three axes. See Fig. 20.) Considering the three translations and the three rotations, we see that a rigid body in space has six degrees of freedom. Similarly, a geometrical figure con- strained to move in a plane has three degrees of freedom : its center of gravity can move parallel to either axis, and 34 THE MOLECULAR THEORY OF MATTER. the figure itself can rotate about an axis perpendicular to the plane in which it is constrained to move. If I have made, the meaning of the expression "degrees of freedom'' FIG. 19. A SPHEEE WITH THREE COMPONENT TRANSLATIONS. FIG. 20. A SPHERE WITH THREE COMPONENT ROTATIONS. clear to you, we are prepared to pass to the consideration of the more general theorems about molecules that I spoke of a moment ago. Generalized Theorems. In the earlier days of the kinetic theory of gases, molecules were assumed to be spherical, in order to avoid the tremendous mathematical difficulties that would arise if any other form were assumed. There are excellent reasons, however, for believing that most molecules are not spherical ; and mathematicians therefore turned their attention to the more general case in which no particular shape was assumed. You will understand that the analysis of this general problem is very difficult ; but a satisfactory amount of progress has been made with it, and I will tell you what has been discovered, thus far. Let us consider a medium composed of any number of sets of bodies, such that the bodies belonging to each set are exactly like one another, though a body belonging to one set may be totally unlike a body belonging to another set. Let these bodies have any GENERALIZED THEOREMS. 35 number of degrees of freedom (which number of degrees may be different in the different sets), and let them be acted on by parallel forces (such as gravity), or by forces tending towards fixed centers, or by internal forces (that is, forces acting within the individual bodies, between their parts). Let all the bodies be very small in comparison with the total space they occupy, so that the chance of their colliding three or more at a time is practically nothing. Moreover, let them be very numerous, and let them be perfectly elastic, and let them be smooth, so that when they collide the only force tending to make them rotate is that due to normal impact. Let them be set in motion among one another with any distribution of velocities ; and let them be hard, but not infinitely so, the force called into play during collision being very great, but not necessarily infinite (as it would be if the hardness were infinite) ; and let the duration of a collision be exceedingly short, yet not necessarily zero. These are the assumptions made by the kinetic theory of gases as it exists to-day. You will see that they are vastly more general than those I described to you in connection with the earlier investigations. The conclusions that have been drawn from them are as follows : (1) After a short time, the law of distribution of positions and velocities in each set of the generalized bodies (or molecules, as we shall call them henceforth) will be pre- cisely the same as it would be if all the other sets were absent ; so that each set behaves as a vacuum to all the rest, so far as the distribution of velocities, and the density of aggregation of the molecules in any given region, are con- cerned. (2) This law of distribution of the velocities in each set of molecules is the same as that given for spherical mole- cules in equation (1). (3) The average kinetic energy of translation of the molecules of any one set is equal to the average kinetic energy of translation of any other set. (4) The total kinetic energy of each set of molecules is divided up equally among the different degrees of freedom of that set. This last theorem is undoubtedly the most remarkable proposition 36 THE MOLECULAR THEORY OF MATTER. about molecules ever enunciated. It is due to Boltzmann, and it seems not to have met with, unqualified acceptance among mathematicians. Lord Kelvin, even, says that he " never felt it possible to believe in that theorem regarding the distribution of energy." My opinion is worth little, compared with his, yet it seems to me that there can be no doubt about the validity of the reasoning on which this theorem is based, provided internal vibrations are excluded from consideration. It has never seemed proper, to me, to consider a possible mode of vibration as a " degree of freedom " ; and I think that it is the extension of Boltz- mann's theorem to the vibrational energy of molecules that gives rise to the objections of Lord Kelvin and others. The theorem does not apply to single molecules, of course. Any molecule, selected at random, may nqt be rotating at all, or it may be rotating about some axis and yet have its center of gravity stationary, or it may be entirely motionless, or it may have component motions of translation parallel to all three axes, and a rotation which is the resultant of rotations about three axes, parallel to x, y, and 2, respectively ; and if the molecule is not a rigid body, it may have other motions also as many, in fact, as it has degrees of freedom. But Boltz- mann' s theorem asserts that if rectangular axes be drawn in a medium composed of a multitude of flying molecules, each with n degrees of freedom, the total kinetic energy in the medium will be so distributed that one nth of it will be due to the velocity-components that are parallel to x, one nth to the velocity-components that are parallel to ?/, one nth to those that are parallel to , one nib. to the rotation-components whose axes are parallel to x, one nth to the rotation-compo- nents that are parallel to y, and so on, one nth. of the total kinetic energy of the medium being due to the sum of the component motions in each degree of freedom. I think you will see, now, why the determination of the average velocity of hydrogen molecules that we made a little while ago, is unsatisfactory. We considered only the kinetic energy due THE GENEKALIZED KINETIC THEORY. 37 to the translation of the molecules ; and this amounted to assuming that the molecules are not set in rotation by their collisions. We shall see, later, that hydrogen molecules are set in rotation by their collisions ; and we shall find, in con- sequence, that the average velocity we deduced for them was too great. Adaptation of the Foregoing Equations to the Generalized Kinetic Theory. The extension of our conception of mole- cules, from spheres to smooth, elastic bodies of any form, does not involve any very radical changes in the formulae deduced from the consideration of the motion of the spheres. Thus equation (1) will still express the number of molecules whose velocities of translation lie between v and v + dv, and equa- tions (2) will also hold true, since they are derived from (1) by a process of reasoning which in no way involves the form of the molecules. But we can no longer consider the kinetic energy of the molecules to be all translational. In fact, Boltz- j mann's theorem tells us that the kinetic energy is divided up equably among the different degrees of freedom; and as translation involves only three degrees of freedom, it follows that the kinetic energy of translation, in a gas, is equal to - . k ; where n is the total number of degrees of freedom pos- sessed by a molecule of the gas, and k is the total kinetic energy in the gas. Hence, for k, in equation (3), we must O 7, write ; and we must make the same substitution in (4), which is derived from (3). Therefore (4) becomes (ii) and a = 2 . . 4 3M or nMo? , k a = Equation (6) remains unchanged, since k does not appear in it. UNIVERSITY 38 THE MOLECULAR THEORY OF MATTER. Gaseous Pressure. I am sure I have nearly exhausted your patience with mathematical formulae ; but there are one or two things more to which I want to call your attention, before passing on to matters involving less mathematics. By considering the molecules as projectiles striking against the walls of the containing vessel, we may find out what their mean velocity must be, in order to produce the observed pressure of the gas. The advantage of this method over the one I have already given you in discussing hydrogen gas is, that it does not require us to make any assumptions concerning the constitution of the molecules, except the very general one that their elasticity is perfect or, more correctly speaking, that their average "coefficient of restitution" is unity. With- out going through with the actual calculation (which would involve theorems in mechanics that you have probably not yet studied), let me say that if we confine our attention to one kind of gas that is, to a gas whose molecules are all alike, each having n degrees of freedom and if we assume that there are no forces except those due to collisions, then the expression for the pressure against a unit area of the containing vessel conies out ' where A is the absolute density of the gas, M is its total mass, and k is its total kinetic energy. Now if the velocities of translation of the individual molecules are v i9 v Z) v 8 ,..., and m is the mass of a single molecule, then the kinetic energy of translation of the system will be Substituting the value of the parenthesis as obtained from equation (5), the expression for the kinetic energy of trans- lation becomes - Nmu*, or - Mu z , z & AVERAGE MOLECULAR VELOCITY IN HYDROGEN. 39 . since M= Nm. Translation involves only three degrees of freedom; and hence, by Boltzmann's theorem, we have the proportion or 1 2 K = - nMu. 6 Substituting this value of k in (12), we have From this equation we can calculate the value of u when we know the pressure and density of a gas ; and having found u, we can calculate V by means of equation (6). Recalculation of the Average Molecular Velocity in Hydrogen. A cubic foot of hydrogen at 32 Fahr. and under atmospheric pressure (i.e.) 2117 pounds to the square foot) weighs .005592 of a pound. Hence its mass is ' , or .0001737. Substituting this for A and 2,117 for p, in (13), we have .0001737 Then, from equation (6), we have F = 6,047 X .9213 = 5,571 feet per second, which is the average velocity of hydrogen molecules, under the given conditions. If two gases have the same pressure, then, by (13), And as equation (6) shows that the average velocity of the molecules of a gas is proportional to u, it follows from (14) that in any two gases having the same pressure, the average 40 THE MOLECULAR THEORY OF MATTER. molecular velocities are inversely proportional to the square roots of the densities of the gases. This enables us to calcu- late the molecular velocities in other gases very readily, when the molecular velocity of any one gas is known. By the help of equation (14) and the velocity we have just obtained for hydrogen, I have calculated the following table of the average molecular velocities in several familiar gases. TABLE OF THE AVERAGE MOLECULAR VELOCITIES OF GASES, IN FEET PER SECOND, AT 32 FAHR., AND ATMOSPHERIC PRESSURE. GAS. DENSITY. (H = l) AVERAGE VELOCITY. Hydrogen 1.00 5,571 Oxygen 15.90 1,394 Nitrogen 14.01 1,488 Carbonic Oxide 13.96 1,491 Carbonic Acid 21.94 1,189 Pressure Produced by Several Sets of Molecules. I have told you that the mathematical investigation of the general- ized molecules described a few moments ago, shows that if there are several sets of such molecules flying about in the same field, the distribution of each set, and the distribution of velocities in each set, will be the same as though the other sets were not present. It follows from this that the pressure on the bounding walls, produced by all the sets together, will be equal to the sum of the pressures that the several sets would produce, if each existed in the same space alone. You/ will probably recognize this . as the equivalent of Dalton's law, which states that in a mixture of gases the resulting pressure is the sum of the partial pressures due to the several constituent gases. Another way of stating this law is, In a mixture of gases, each behaves like a vacuum to all the rest AVOGADRO'S LAW. 41 Avogadro's Law. Going back to equation (12), let ' confine our attention for the moment to a unit volume of gas. In this case M becomes identical with A ; for, by definition, the absolute density of a gas is its mass per unit volume. Hence (12) becomes ' Now if two gases have the same pressure, that is, if Pi= we have, from (15), From this equation it also follows that O KI O K% MI w 2 ' But we have seen that either member of this equation repre- sents the kinetic energy of translation of the molecules of the corresponding gas. Hence it follows that if two gases have the same pressure, they will have, per unit of volume, the same kinetic energy of translation. Now if k' represents the kinetic energy of translation per molecule; and N is the num- ber of molecules per unit volume, then we may express this last fact thus : If two gases have the same pressure, then So that if k\ = k' 2 , then NI = N 2 . Hence, finally, we may say that if any two gases have the same pressure, and the") same kinetic energy of translation per molecule, then these V gases will contain the same number of molecules per unit ) volume. If we read "temperature" in the place of "kinetic energy of translation per molecule," this statement becomes identical with Avogadro's law. Boyle's Law. Returning once more to equation (12), let us observe that the definition of "density" gives us, in all cases, the equation THE MOLECULAR THEORY OF MATTER. Substituting this value of A in (12), we have 2k M 2k (16) p = , or pv = -- nM v ' n We may, for convenience, transform this equation thus : Zi O K -. _ 7 . where k' is the kinetic energy of translation, per molecule, and N (which does not vary so long as we confine our atten- tion to some particular mass of gas) stands for the number of molecules in the gas under consideration. We are strongly reminded, by this equation, of Boyle's law, which states that the product, pv, is constant so long as the temperature of the gas does not vary ; and if we make the single assumption that the sense-impression that we call " temperature " is really our mode of perceiving molecular kinetic energy of translation, we shall find that the results of the kinetic theory of gases cor- respond very closely with the facts as actually observed. Results of the Kinetic Theory Compared with the Results of Observation. These correspondences may be exhibited as follows : RESULTS OF THE KINETIC THEORY. 1. When two or more sets of molecules are put into the same region of space, they diffuse into one another, until the molecules of each set become uniformly dis- tributed throughout this space. 2. The density of a medium composed of several sets of mole- cules is equal to the sum of the densities the individual sets would have, if each existed sepa- rately in a space equal to the given space. RESULTS OF OBSERVATION. 1. When two or more gases are put into the same vessel, they diffuse into one another, until each becomes uniformly distrib- uted throughout the vessel. 2. The density of a gaseous mixture is equal to the sum of the densities of its component gases. TEMPERATURE. 43 KESULTS OF THE KINETIC THEORY. 3. The pressure on the bound- aries, due to a medium composed of several sets of molecules, is equal to the sum of the partial pressures due to its constituent sets. 4. In a molecular mixture there is one physical quantity which is the same for every set of molecules ; and that is, the average kinetic energy of transla- tion, per molecule. 5. If two molecular aggregates exert the same pressure on their containing-walls, and have the same kinetic energy of translation, per molecule, then they will also contain the same number of molecules per unit of volume. 6. In any given mass of a molecular aggregate, the product of the pressure and volume is proportional to the average kinetic energy of translation, per molecule. RESULTS OF OBSERVATION. 3. The pressure on the con- taining-vessel, due to a gaseous mixture, is equal to the sum of the partial pressures due to the constituent gases (Dalton's law). 4. In a gaseous mixture there is one physical property which must be the same for each of the constituent gases; and that is, the temperature. 5. If two gases exert the same pressure on their containing-ves- sels, and have the same tempera- ture, then they will also contain the same number of molecules per unit of volume (Avogadro's hypothesis).* 6. In any given mass of gas, the product of the pressure and volume is proportional to the absolute temperature. (This in- cludes the laws of Boyle, Charles, and Gay Lussac.) Temperature. It is evident that there is a sufficient agreement between the properties of actual gases, and those of the ideal molecular medium we have considered, to make it very probable that gases have some such constitution as we have imagined the molecular medium to have. At all events we have found, as yet, no contradictions. ~You may not be ready to admit, however, that the assumption made * I have included Avogadro's hypothesis among the "observed prop- erties," because it was inferred from observation before it was deduced from a study of the motions of discrete elastic particles. 44 THE MOLECULAR THEORY OF MATTER. with regard to temperature is an admissible one. The dis- cussion of this point belongs properly to thermodynamics, but since it has a close bearing on molecular theories, you may allow me to say a few words about it. We all know what is meant when we say that one body is hotter or colder than another we refer to certain sensations that would be experienced if we should touch the bodies, or come very near to them ; but it is quite a different thing to devise a rigid scale that will enable us to measure differences in temperature in such a way that we can say that a difference of 10, for instance, on one part of the scale, is equivalent, in some sense, to a difference of 10 on any other part of it. In order to devise such a scale we shall have to fall back on some general principle or law; and our "temperature sense" does not furnish us any such principle at least, not with any that is sufficiently exact for scientific purposes. It is necessary, therefore, to seek for some such principle in the world out- side of ourselves, and to define "temperature" arbitrarily, so as to make it to conform to that principle ; always remember- ing, of course, that the scale of temperature finally selected must be such, that measures obtained by means of it shall not be perceptibly inconsistent with the crude observations we can make directly, by means of our temperature sense. The commonest form of thermometer consists of a glass vessel containing mercury. It is graduated by immersing it in the steam arising from boiling water, and in a mixture of ice and water, marking the points at which the mercury stands under these circumstances, and dividing the space between these marks into (say) 100 equal parts, which are called degrees. The scale so formed is perfectly arbitrary, since it involves whatever peculiarities of expansion mercury may have ; and these peculiarities cannot be investigated, without reasoning in a circle, unless we can find some kind of an absolute scale of temperatures which shall be independent of the properties of any particular substance. The mercury thermometer does not contradict our senses, it is true, but neither would ther- TEMPERATURE. 45 mometers made with a host of other liquids ; and yet the thermometers made with these other liquids would not agree among themselves, nor with the mercury thermometer. (Water is inadmissible as a thermometric fluid, for low tem- peratures at any rate, because near the freezing point it gives readings that contradict the direct evidence of the senses.) Now, I am not going to take you into the mazes of ther- mometry. I wanted to call your attention to the fact that " temperature " is not such a definite conception as one would be apt to imagine unless he had thought it over carefully, and that a thermometer scale is an arbitrary thing. I thought this might make it easier for you to admit that "temperature' 7 may be the sense-impression that corresponds to molecular kinetic energy of translation. Yet I should not want to leave you with the impression that heat measurement is not an exact science. Let me add, therefore, to what I have said, that Lord Kelvin has provided us with what he calls an absolute thermometric scale, which is quite independent of the properties of any particular body. He obtains this scale from thermo-dynamical considerations,* and although his definition of temperature is quite as arbitrary as any other definition of it, it is the only one yet proposed that rests on a thoroughly scientific basis. For the purpose of fixing the size of his degrees, he defines the difference in temperature between boiling water and melting ice to be 100; and he then finds that the temperature of melting ice, on his " abso- lute scale,' 7 is 273.1. Furthermore, he finds that the readings of the air thermometer are almost identical with those of his absolute scale, provided allowance is made for the difference of 273.1 Centigrade degrees that exists between the zero of the absolute scale and the zero of the ordinary Centigrade scale. I have made out a table, here, giving the corrections to the readings of several kinds of thermometers, to reduce them to their equivalents on Lord Kelvin's absolute scale. In all but the last column the readings are supposed to have. * See the article Heat, in the Encyclopaedia Britannica. 46 THE MOLECULAR THEORY OF MATTER. been previously corrected for calibration, expansion of the glass, error of the fixed points, etc., and the table gives only the correction that is made necessary by the imperfection of the thermometric fluid itself. The last column gives the corrections to be applied to a certain crown-glass, mercurial thermometer that was investigated by Regnault. The cor- rections in this column are far smaller than those in the preceding one, because they include the correction due to the expansion of the glass, and the glass-expansion and mercury- expansion corrections are of opposite sign. TABLE OF CORRECTIONS, FOR REDUCING THERMOMETER READINGS TO THE ABSOLUTE SCALE. HEADING OF THERMOMETER. (Centigrade Degrees.) THERMOMETRIC SUBSTANCE. AIR. (Constant Volume.) AIR. (Constant Pressure.) MERCURV. (Alone.) MERCURY AND CROWN GLASS. o.oo o.oo o.oo 20 - .03 - .04 + .20 40 - .04 - .05 + .29 60 - .04 - .05 + .30 80 - .02 - .03 + 0.20 100 .00 .00 .00 o.oo 120 + .03 + .03 -0.29 + .08 140 + .06 + .07 -0 .70 + .21 160 + .10 + .11 -1 .19 -1- .36 180 + .14 + .16 -1 .80 + .51 200 + .18 + .20 -2 .49 + .48 220 + .22 + .25 -3 .29 + .42 240 + .27 + .29 -4 .17 + .37 260 + .31 + .34 -5 .15 + .11 280 + .36 + .39 -6 .23 - .16 300 + .41 + .44 -7 .39 - .67 Absolute Zero. One of the most interesting things about the absolute scale is, that it has a zero usually called the absolute zero below which it appears to be impossible to RATIO OF THE SPECIFIC HEATS OF GASES. 47 cool things. You will note the bearing of this on the kinetic theory of gases ; for if temperature is really our mode of perceiving the translatory kinetic energy of molecules, then if we should gradually abstract from a gas its translatory kinetic energy, its temperature would seem to fall lower and lower, until finally, when we had abstracted all of this energy, the temperature would reach a point lower than which it could not go. There would no longer be any translatory kinetic energy to perceive ; and hence we should have reached an absolute zero, below which it is not thinkable that the gas could be cooled. I may say, in fact, that if we define temper- ature as Lord Kelvin defines it (and this is the only rational way yet proposed), and if we admit that the sense-impression that we call " temperature " is our mode of perceiving the kinetic energy of translation of molecules, then the kinetic theory of gases becomes even more remarkable than we have found it to be ; because we can then deduce from it all the fundamental equations of thermodynamics. ' Ratio of the Specific Heats of Gases. If a certain mass of gas, having a volume v at the pressure p , expands adiabatically that is, without either receiving or giving out heat as heat we know from thermodynamics that its pressure and volume are connected by the relation p v y=pvy (18) where y is the ratio of the specific heat of the gas at constant pressure to its specific at constant volume. Now if this given mass of gas is allowed to expand indefinitely, or until its volume becomes infinite, the total amount of work it can do is Work oo = fp . dv: (19) If we substitute in this equation the value of p as obtained from (18), and perform the integration, we have (20) 48 THE MOLECULAR THEORY OF MATTER. Now the work that a gas can do under such circumstances (assuming that there are no forces between the molecules) is equal to its total kinetic energy, k Q . Equating k to the second member of (20), we have Pv =(y-l)k , (21) and comparing this with equation (16) we see that y l = -, or y = l + ? (22) n n From this equation we can calculate the ratio of the specific heats of a gas, if we know n, the number of degrees of freedom of its molecules ; and conversely, we can calculate from it the number of degrees of freedom of the molecules, if we know the ratio of the specific heats of the gas. Now the smallest value that n can have is three ; for it must take at least three coordinates to fix the position of a body in space. The value n = 3 corresponds to the case in which the molecules are smooth spheres, incapable of being set in rotation by their mutual impacts. Mercury, for chemical reasons, is believed to contain only one atom in its molecule ; and hence it will be interesting to see whether the ratio y, calculated on the hypothesis of a smooth, spherical molecule, corresponds with the actual value of this ratio for mercury vapor. We find, for n = 3, y = If 1.666 ; and the ratio of the specific heats of mercury vapor, as determined experimentally by Kundt and Warburg, is 1.66. The agreement of this with the calculated value lends considerable plausibility to the supposition that the molecules of mercury are smooth and spherical, and, incidentally, to the whole kinetic theory. In the case of other gases the agreement is not so satisfactory. For example, the molecules of hydrogen, oxygen, nitrogen, and carbonic oxide, are believed, for chemical reasons, to consist of two atoms. We are led, naturally, to examine the results of the hypothesis that their molecules each consist of two smooth, spherical atoms, rigidly united by attractive forces or other- wise. The number of degrees of freedom that we have to EATIO OF THE SPECIFIC HEATS OF GASES. 49 consider in such a system is 5 (the freedom to rotate about the line of centers of the spheres is not considered, because as the spheres are assumed to be perfectly smooth the impacts of the molecules cannot set up rotations about this axis). When n = 5, we have y = 1 -f- f = 1.400. The accepted values of y for these gases, as given by experiment,* are presented in this table : TABLE or VALUES OF 7. GAS. EXPERIMENTAL y. CALCULATED y. OXV"G11 1.402 1 400 Nitrogen Hydrogen 1.411 1.412 1.400 1.400 Carbonic oxide .... 1.418 1.400 At first thought this seems like a very satisfactory agree- ment ; but it is not so. We have assumed, in deducing equa- tion (22), that the effects of the intermolecular forces are insensible ; but it can be shown that if they were sensible we should have to modify equation (22) so as to make it read where x is a small positive quantity, vanishing when the forces between the molecules are insensible. You will please note particularly that x is necessarily positive if the forces are attractive, so that the calculated values of y would be smaller, if we take these forces into account, than it would be if we neglected them and considered x to be zero ; whereas the observed fact is, that the values of y are larger than the computed value obtained by making x = 0. This constitutes an objection to the kinetic theory, which is worthy of serious consideration. Mathematicians have endeavored to account * These experimental results are from the Encyclopaedia Britannica, article Steam Engine. 50 THE MOLECULAR THEORY OF MATTER. for the observed discrepancy in various ways, but without any very distinguished success. You will find a suggestion made in Mr. Watson's little book on the kinetic theory of gases ; but as it involves some rather intricate considerations, and cannot be regarded at present as anything more than a suggestion, I shall not trouble you with it this evening. I have an idea that the true explanation will be found to involve the consideration of gaseous dissociation. It is known that many gases exhibit this phenomenon in a marked degree when their temperature is sufficiently high. According to what I have told you, raising the temperature of a gas is really the same thing as accelerating its molecules. When the mean speed of the molecules reaches a certain value, the shocks due to the molecular collisions become so great that the internal attractive forces existing within the molecules are no longer sufficient to hold them together. They break up, therefore, into simpler molecules, or perhaps into their constituent atoms. This phenomenon is known as dissociation. Owing to the great variety of velocities that exist within any given mass of gas, the dissociation does not take place suddenly, when the gas reaches a particular temperature. If the temperature of the gas be gradually raised, there will come a time when a considerable number of the molecules possess the velocity requisite for dissociation, although the great mass of them may still have velocities that are con- siderably below this critical value. Dissociation then com- mences. If the temperature of the gas be now kept constant the dissociation does not proceed until the molecules are all split apart, because many of the dissociated parts, coming together again at velocities less than the critical velocity, re-combine and produce new molecules like those of the original gas. You will see, therefore, that at any given temperature dissociation proceeds only until there is an equilibrium established between the molecules that are break- ing up, and those that are re-forming. If the temperature be now raised, the average velocity will come nearer to the KATIO OF THE SPECIFIC HEATS OF GASES. 51 critical value, and when equilibrium has been established at this new temperature, the number of molecules that exist in the dissociated condition at any given instant will be greater than before. If the temperature be high enough, the num- ber of dissociated molecules that happen to collide, during any given time, with velocities sufficiently small to allow of re-combination, may be so insignificant that we cannot recognize the presence of these re-combined molecules by any of the chemical or physical tests at our command. The gas is then said to be wholly dissociated. Now it seems probable that in any gas there must be some molecules in a state of dissociation, even though the temperature may be far below the critical value conventionally called the temperature of dissociation ; for we have seen that in any given mass of gas there are always some few molecules moving with extreme velocities velocities great enough to produce dissociation. We may therefore conceive hydrogen gas, for example, to be an aggregation of molecules, by far the greater number of which are diatomic with 5 degrees of freedom, but some of which, nevertheless, are monatomic with 3 degrees of freedom. The value of y for such a gas would be intermediate between the value calculated for n 5 and that calculated for n 3 ; or in other words, it would lie between 1.400 and 1.666, but far nearer the former value than the latter. It seems to me that the calculated and observed values of y can be reconciled in this way, but before we could prove this to be the fact we should have to make a rigid mathematical investigation of the theory of dissociation. If this be the true explanation, it is evident that the value of y must increase when the tempera- ture of the gas increases ; for at higher temperatures there would be a greater proportion of dissociated molecules present. I know of no experiments sufficiently accurate to test this point. There is, of course, a possibility that the observed values of y are incorrect, owing to the existence of some unrecognized source of error tending to give results uniformly too great ; but this would have to be proved to be the fact 52 THE MOLECULAR THEORY OF MATTER. before we could accept it as the true explanation of the diiference between calculation and observation. The ratio of the specific heats of gases is usually calculated from the velocity of sound as observed in the gases ; and as the neces- sary measurements are difficult to make, we find that with different data different results are obtained. Thus from Dulong's data for dry air we have y = 1.410, while from the data for air as given by Eegnault and Poisson, we find y = 1.401. So far as the gases in the table are concerned, it is worthy of note that a recent determination of y for hydro- gen (given by Clausius and quoted in Watts's Dictionary) gives y = 1.3852. I also find that if the value of y be calcu- lated by combining the specific heat at constant pressure as observed by Regnault, with the difference between the specific heats as calculated by E-ankine's method, we have, for oxygen, y = 1.398 ; using Regnault's results for the density of oxygen, Thomson's determination of the absolute zero, and Griffiths's value of the mechanical equivalent of heat.* In view of the differences that exist among the different experimental deter- minations of y, I think it would be unwise to conclude that the present discrepancy between the kinetic theory and the apparent facts may not be cleared up satisfactorily in the future. We must note, however, that since 3 is the smallest number of degrees of freedom that a free body in space can have, it follows from (22) that y If is the largest ratio of the specific heats that the kinetic theory is capable of explain- ing. If it can be shown that the ratio of the specific heats of any gas is greater than this, it looks as though the kinetic theory would have to go. You may be interested to know that for highly superheated steam the value of y is 1.30. This seems to correspond to a molecule with 6 degrees of freedom ; for with this value of n, equation (22) gives y 1.333. The observed value of y for carbonic acid gas is 1.263, which seems to indicate 7 degrees of freedom ; for * The same method gives for H, y = 1.408 ; and for N, y = 1.407. See Appendix. . MOLECULAR ATTRACTION IN GASES. 53 when n = 7 equation (22) gives us y = 1.286. These values of n, for steam and for C0 2 , seem to indicate that steam molecules are rigid bodies, and that the molecules of C0 2 may consist of two smooth bodies jointed together in some way ; but in speculating in this manner on the actual forms of the molecules we are going a good way beyond the limits of positive knowledge. Molecular Attraction in Gases. Thus far we have as- sumed that the molecules of a gas do not attract one another, but that the phenomena of gases result from the motions of the molecules, unrestrained except by their collisions with one another and with the walls of the containing-vessel. It appears, from the agreement of the results of this assumption with the observed facts, that the effects of the mutual attrac- tions that may exist between the individual molecules of a gas are small, on the whole. The forces, when they exist, may be great ; but since their effects are scarcely noticeable, we must conclude that under ordinary circumstances the sphere of sensible action of these forces is quite small in comparison with the length of the average free path of the molecules. We may investigate the attraction or repulsion that may exist between gaseous molecules, by allowing a given gas to expand 1 into a vacuum so that it shall do no external work. If there is an attraction between the molecules, the gas will be cooled; for some of its kinetic energy will be transformed into poten- tial energy. The experiment being performed, it is found that there is almost no change in temperature produced by simple expansion, when the gas does no external work. Very accurate experiments by Thomson and Joule, however, showed that there is a slight temperature change, though it is so small that it could readily escape observation. In order to avoid eddies and other such sources of error, Joule and Thomson caused the gases they experimented upon to flow from one vessel into the other through a porous plug.* The * Encyclopaedia Britannica, article Heat. 54 THE MOLECULAR THEORY OF MATTER. initial pressure, in their experiments, ranged from 1 up to 5 or 6 atmospheres, the final pressure being 1 atmosphere in every case ; and the cooling effect was found to be propor- tional to the difference between the initial and final pressures. In the case of air, the cooling effect was 0. 208 C. per atmosphere. With carbonic acid gas it was 1. 105 C. With hydrogen, on the other hand, there appeared to be a heating effect of .039 C. per atmosphere. (Owing to its peculiar behavior, Regnault called hydrogen un gaz plus que parfait "a more than perfect gas.") These experiments show that there are attractive forces between the molecules of air, and also between those of carbonic acid gas. In expanding, the molecules of these gases have become more widely separated, and energy that was before kinetic and therefore sensible as heat has become potential in overcoming the attractive forces. The behavior of hydrogen is anomalous. No other gas exhibits a heating effect when expanding into a vacuum. The discussion of this isolated phenomenon would take so much time that I shall not enter upon it this evening. We must note that the cooling effect observed by Joule and Thomson affords a rough indication that the attraction between gaseous molecules does not follow the inverse-square law with which we are so familiar in the laboratory and the observatory. It indicates, in fact, that the attraction varies something like the inverse fourth power of the distance. For let us assume that the law of attraction is where D is the mean distance between two neighboring mole- cules. Then if the gas expands until this mean distance becomes D 19 we may express the work done against the attractive -forces somewhat as follows : 'MOLECULAR ATTRACTION IN GASES. 55 Now, as D is a linear dimension in the gas, the volume of the gas will vary as the cube of D. Hence, we may write v = t)'D s , and v l = b A 3 - Substituting, in (24), the values of D 8 and D*, as given by this equation, we have Work = */l_l\ 3 \v vj But as the temperature of the gas is practically constant, we have Substituting, in (25), the values of v and v l , as given by this equation, we have, finally, x^> That is, the assumed law of attraction* indicates that when^a gas expands into a vacuum, the amount of work done in over- coming the internal attractive forces 'or, what is the same thing, the amount of kinetic energy that disappears and ceases to be sensible as heat is proportional to the difference between the initial and final pressures ; and this is the 1 observed fact. I am aware that equation (24) is only a crude expression for the amount of internal work, and that con- sequently the reasoning I have given you does not prove that the inverse fourth-power is the true law of variation of inter- molecular attraction ; but I think it is fair to say that the reasoning shows, at least, that when two molecules approach each other, the attraction between them increases faster than the inverse-square law would indicate. Although the reason- ing I have given you does not prove that equation (23) rep- resents the true law of molecular attraction, Mr. William Sutherland has recently shown, in a more rigorous manner, that this is very likely the fact.f I may, perhaps, venture *See equation (23). t See his recent articles in the Philosophical Magazine. 56 THE MOLECULAR THEORY OF MATTER. the suggestion (though it has never yet been proven) that the attraction existing between the molecules of a gas is due to the same ultimate cause as gravitational attraction (whatever that cause may be), and that the so-called Newtonian law of inverse squares is only an approximate expression for the relation between gravitation and distance, which is sensibly accurate only when the distance, D, greatly exceeds the dimensions of a molecule. If this conjecture be correct, the true law of gravitational attraction is probably a function capable of development in terms of descending powers of D, thus : f(V) = ^ + ^ + ^ + - (26) where the terms after the first have very small coefficients, so that they are insensible except when D is itself extremely small, and the reciprocals of its higher powers correspondingly large. Equations of Van der Waals and Clausius. The equation pv = Rr (27) being deducible from the kinetic theory only when the as- sumption is made that the effects of the attractive forces between the molecules are insensible, and it being admitted that even in the so-called permanent gases these forces are not absolutely insensible, it would naturally be expected that accurate observations would show that this equation is not perfectly fulfilled by any gas. And it is so. We can no longer regard this simple law as anything more than a very good first approximation to the truth. All gases exhibit variations from it, and these variations are quite marked at high pressures, when the molecules are crowded closely together. Many formulae have been proposed as "second approximations," and of these the most famous is undoubtedly that given by Van der Waals. over which there has been much controversy. It will be evident to you, I think, that EQUATIONS OF VAN DER WAALS AND CLAUSIUS. 57 the "volume" entering into our expressions should not be, strictly speaking, the actual space occupied by the gas, but rather the space in which the molecules are free to move ; that is, the empty space within the enclosure, or the space not actually filled by the molecules themselves. We may there- fore use the expression (v b) in the place of v in the fore- going formula, v still signifying the apparent volume of the gas, while (v b) is the empty part of it. The gas equation will then read (28) This formula is an improvement on the older one, but we have not yet taken account of the attractive forces between the molecules. If we should compress a gas, we should be assisted by this internal attraction ; and therefore the actual external pressure required to reduce the volume by a given amount would be less than that calculated by either (27) or (28). This much, I think, is admitted by all physicists ; but there is a considerable disagreement among them as to the actual algebraic form of the correction thus called for. Van der Waals considered it to be expressible by a term of the form ^ ; and his equation is, therefore, Kf i , *~ i , -~ /29^ (v b) v* Clausius proposed, in the place of 2 , the expression so that his equation is : However interesting Van der Waals's equation may be, it certainly does not represent the facts of observation satis- 58 THE MOLECULAR THEORY OF MATTER. factorily, unless we cease to regard a and b as constants. Clausius's equation, on the other hand, is found to represent the results of Andre ws's extensive experiments on C0 2 , with great fidelity. The values of the constants, in (30), for C0 2 are as follows : R= .003683, b = . 000843, c = 2.0935, ft = .000977. The pressures are to be reckoned in atmospheres, the temper- ature is to be measured on the absolute Centigrade scale, and the unit of volume is the volume of the gas itself at the freezing-point, and at atmospheric pressure. Diffusion. There are two properties of gases to which I must refer before passing to the consideration of liquids. The first of these is diffusion. It has often been remarked, by persons not conversant with the kinetic theory, that if the molecules of gases are moving at the rate of thousands of feet per second, it is hard to understand why one gas does not diffuse through another with correspondingly great rapidity ; so that if a bottle of some strong-smelling gas, like ammonia, were opened in one part of a room we could smell it in another part without any sensible lapse of time, even if the air of the room were, apparently, at perfect rest. The answer to this is, that although the molecules of ammonia vapor do have high velocities, they cannot travel in uninter- rupted straight lines across the room, like projectiles, because there are enormous numbers of air molecules in the way, with which they collide thousands of millions of times in a second. In a rough way, the case may be likened to an army of swift runners trying to pass, at the top of their speed, across a field thickly set with posts. Of course this is a very crude comparison, but it may serve to fix the idea. The runners would be merely turned aside, first one way and then another; but the ammonia molecules, rebounding from the air molecules that they strike, are actually turned about, and in VISCOSITY. 59 such diverse directions, that in any given region there are almost as many of them returning towards the bottle as there are going away from it. They are forced to describe zig-zag paths which are so very crooked that by the time a given ammonia molecule has reached a point actually ten feet dis- tant from the bottle, it has traveled, in all, an enormously greater distance probably quite a number of miles. I shall not give you the mathematical theory of diffusion, because it is very much involved. Furthermore, I do not consider that any very valuable information has yet been obtained from it. I say this with all due respect to the men who have worked over this theory so patiently. What they have learned about diffusion has been found out by great labor, and it is gratifying to know that the results of this labor accord well, in general, with similar results obtained by other methods. There is no doubt that very important facts will be learned, some day, from the study of diffusion, and I should not want to discourage any of you who may feel disposed to enter the lists and give battle to the formulae involved. Viscosity. The other property of gases to which I referred a moment ago is viscosity. It is by no means an obvious property, and its effects are quite small so small, in fact, that they are difficult to measure with precision. You are probably better acquainted with the viscosity of liquids than you are with the corresponding property of gases. Molasses and maple syrup are familiar examples of viscous liquids. You will understand that if two planes were submerged in molasses, and one of them was caused to move in its own plane, parallel to the other one and near it, there would be a certain drag exerted on the motionless plane, tending to pull it in the direction in which the first plane is moving. This " drag " would also be experienced by the fixed plane if any other liquid were substituted for the molasses ; and it is found that the same effect is observable, though in a muck 60 THE MOLECULAR THEORY OF MATTER. smaller degree, even when the experiment is performed with gases instead of with liquids. This property of fluids, in virtue of which they can transmit motion from one moving plane to another one parallel to it, is called viscosity. In many respects viscosity is analogous to friction; and physi- cists (particularly among the Germans) often call it the "internal friction" of the fluid. In Fig. 21, A represents the moving plane, and B the fixed one. If the fluid between them be conceived to be divided into exceedingly thin layers, as suggested in the diagram, we may conceive the resistance - 1 1 1 I I 1 -p 1 1 1 ] 1 1 1 1 1 B FIG. 21. DIAGRAM ILLUSTRATING VISCOSITY. to the motion of A to be due to the friction of these layers upon one another. If F is the force required to overcome this friction or, which is the same thing, if F is the force required to sustain the motion of A, it may be shown* that *- where S is the area of the plane A, V is its velocity, ,Z> is the perpendicular distance between A and B, and ft is a coefficient peculiar to the fluid experimented with, and called its coefficient of viscosity. * See Encyclopedia Britannica, article Hydro-mechanics, part III. EXPERIMENTAL DETERMINATION OF /a. 61 Experimental Determination of /*. The foregoing equa- tion may be written in the form DF sv' (31) Hence the coefficient of viscosity of a fluid can be determined, if we can measure the quantities D, F, S, and V. The only one of these that presents the least difficulty is F, which, for gases, is so small that its determination is ex- tremely difficult. Various methods have been tried, however, and owing to the skill and patience of the experimenters, the results are fairly satisfac- tory. Meyer determined F by a study of the action of air on vibrating pen- dulums. He worked with three pendulums, and considered that the most trustworthy result was that obtained from the shortest one of the three. The value of F thus de- duced by him gave FIG. 22. DIAGRAM ILLUSTRATING MAXWELL'S APPARATUS. fi = . 000184, the units being the centi- meter, gramme, and second. Another method, tried with success by Maxwell (and Meyer also), consisted in causing a circular disk to oscillate in its own plane, between two 62 THE MOLECULAR THEORY OF MATTER. others that were parallel to it. The way in which the oscil- lations died out gave a means of calculating F, and then, by substitution in (31), the value of /x was obtained. In the actual experiment, Maxwell used several disks fixed to a com- mon axis, instead of only a single one. Fig. 22 is a diagram illustrating the principle of his apparatus.* Maxwell's final result for the coefficient of viscosity of dry air was P = .0001878 (1 + .00365 1). f Meyer's result was /* = . 000190(1 + .00250- In both of these expressions t is the temperature, on the Centigrade scale, counted from the freezing point of water in the usual way. Maxwell's expression, as quoted above, implies that /* varies directly as the absolute temperature. Meyer found, by experiment, that it varies in proportion to the .77 power of the absolute temperature. That is, it varies proportionally to where = 273, or the absolute temperature of the freezing point on the Centigrade scale. The best value of the coefficient of viscosity of dry air at present obtainable is considered by KnottJ to be fl = . 000185(1 +.0028^. (32) Professor Knott also gives a table of the relative values of the coefficients of viscosity of other gases, as determined by Graham, Maxwell, Meyer, Kundt and Warburg, and Crookes. * For an engraving of the actual apparatus, see the Philosophical Transactions of the Eoyal Society of London, for 1866. t In all that is said in this volume about viscosity, the fundamental units are understood to be the centimeter, gramme, and second. \ Encyclopedia Britannica, article Pneumatics. KINETIC EXPLANATION OF VISCOSITY. 63 From this table we may perhaps infer the following relative and absolute values of the coefficients of viscosity of these gases, at the freezing point of water : TABLE OF COEFFICIENTS OF VISCOSITY OF GASES. (lAS COEFFICIENI c OF VISCOSITY. Relative. Absolute. Air 1 000 M = .000185 Oxygen Nitrogen 1.109 .972 .000205 .000180 Hydrogen .484 .000090 Carbonic Oxide .... Carbonic Acid .... .970 .855 .000179 .000158 Kinetic Explanation of Viscosity. Eeturning to our con- ception of a gas as an aggregation of swiftly moving mole- cules, let us see if it is not adequate to explain the viscosity that experiment has shown these bodies to possess. Referring back to Fig. 21, let us conceive the imaginary layers of gas there represented to be of such a thickness that the average "free path " of the molecules of the gas will be just sufficient to allow these molecules to cross one layer in the interval between two successive collisions. Now when the molecules in the upper layer strike against A, they receive from it a component motion in the direction of the arrow. Rebound- ing from Ay they cross the first layer and collide with the molecules of the second layer, communicating to them also a component motion in the direction of the arrow. The molecules in the second layer carry this component over to those in the third layer, and so on until the molecules in the last layer tend to communicate a component motion, in the direction of the arrow, to the plane B. This is the general nature of the kinetic explanation of viscosity in gases, though you will understand that in the rigorous mathematical 64 THE MOLECULAR THEORY OF MATTER. treatment of the subject it is necessary to take account of many things that I have not mentioned; such, for instance, as the law of distribution of velocities among the molecules. Several mathematicians, following out the general idea I have given you, have deduced from the kinetic theory of gases, expressions for the coefficient of viscosity. One of the most recent of these expressions is that of Clausius, which is (33) > where A is the density of the gas at C. and atmospheric pressure, as compared with air under the same conditions, T is the absolute temperature of the gas, T is the absolute temperature of the freezing point of water (273 C.), and X is the average "free path" of the molecules of the gas, at atmospheric pressure and at C. While it is true that this formula does not accurately represent the experimental facts, inasmuch as the exponent of the absolute temperature is known to be about .77 instead of -j-, the error introduced in this way is practically insensible for temperatures near the freezing point.* There is one very important thing I would like you to notice about this formula. It is, that the formula indicates that the coefficient of viscosity of a gas remains unchanged, no matter how much we rarefy the gas or compress it; for X and A are constants, since they stand for the free path and density at atmospheric pressure and at C. That * Mr. Sutherland has shown very clearly that the difference between theory and experiment, here mentioned, disappears if we take account of the attractive forces existing between the molecules of the gas. Assum- ing that these forces are proportional to the inverse fourth-power of the distance, he finds that Mo where MO is the coefficient of viscosity at the freezing point and M is the corresponding coefficient at the absolute temperature T. The value of . KINETIC EXPLANATION OF the viscosity of a gas does not depend density, lias been amply verified by experiment. Numerous observers have found that air at a pressure of only a few millimeters of mercury has substantially the same viscosity as air under the normal pressure of 760 millimeters. It is hardly necessary to say that this constitutes a most remark- able verification of the kinetic theory. Of course this state- ment about viscosity being independent of absolute density must not be pressed to extremes, or we should be saying that a vacuum has just as much viscosity as common air ; which would be absurd. A vacuum cannot have viscosity. This does not involve an error in Clausius's reasoning, however, for, as he says himself, his formula was deduced on the assumption that the free path is so small that terms involv- ing higher powers of it than the first can be neglected. There is no doubt about this being the fact under ordinary conditions ; but when the gas is rarefied the molecules have much more free space to move in. They do not collide so often, and hence their average free path becomes greater. As the exhaustion proceeds, there comes a time when the terms involving powers of the average free path higher than the first cannot be neglected ; and when that point is reached, the formula no longer has any pretensions to accuracy. the constant C, for C0 2 , is 277. The following table exhibits the agree- ment of this formula with Holman's experiments on COa : TABLE OF VALUES OF THE RATIO FOR C02. Mo TEMP. OBSERVED RATIO. CALCULATED KATIO. TEMP. OBSERVED KATIO. CALCULATED KATIO. 18 C. 1.068 1.066 119.4 C. 1.415 1.414 41 1.146 1.148 142 1.484 1.490 59 1.213 1.211 158 1.537 1.541 79. 5 1.285 1.280 181 1.619 1.614 100.2 1.351 1.351 224 1.747 1.746 66 THE MOLECULAR THEORY OF MATTER. Free Path. Clausius's formula is of no particular value for computing the coefficient of viscosity of gases, because it involves the free path, A, which is much harder to determine than the viscosity is ; but when the coefficient of viscosity has once been determined by experiment, Clausius's formula becomes of the most extreme importance for the inverse problem of determining the average free path. Thus, con- fining our attention to the case in which T = T , we may write the formula in the form 20.23 (34) Applying this equation to carbonic acid gas, for which A = 1.529 and /A = .000158, we have .000158 20.23 = .000 0063 cm. Similar calculations for the other gases whose coefficients of viscosity are in the table I gave you a few moments ago, give the following results : TABLE OF AVERAGE FREE PATHS AT ATMOSPHERIC PRESSURE AND C. DENSITY. AVERAGE I REE PATH. (Air = l.) Centimeters. Inches. Air* . . . 1.0000 .000 00914 .000 0036 Oxv"en 1 1056 .000 00964 .000 0038 Nitrogen 9713 .000 00903 .000 0036 Hydrogen .06926 .000 01690 .000 0067 Carbonic Oxide .... Carbonic Acid .... .9545 1.5290 .000 00906 .000 00631 .000 0036 .000 0025 I think these figures will make it plain why one gas does not diffuse through another with a speed comparable with the velocity of translation of its molecules. Consider the case of * Calculated as a simple gas. FREE PATH. 67 hydrogen, for example. We found the average velocity of translation of its molecules to be 5,571 feet per second ; and as the average free path of its molecules, between collisions, is .000 0067 of an inch, you will see that, on an average, each of its molecules experiences : ^#=10,000,000,000* collisions with its neighbors every second. When one gas is diffusing through another one, the calculation is not so simple ; because in that case there are two kinds of molecules to be considered, and we must take account of the collisions of each molecule with those of its own set, and with those of the other set. Still, I think you will be quite ready to admit, from the calculation I have given you in the case of hydrogen, that the number of collisions per second experienced by each of the molecules of a diffusing gas would be so great that the explanation of the slowness of diffusion that I offered you a few moments ago is quite defensible. It may interest you to know the average number of collisions per second experienced by a molecule in some other gases. I have therefore cal- culated this table, by the same method used in the case of hydrogen, and from the data concerning the free paths and the molecular velocities that have already been placed before you. TABLE OF THE AVERAGE NUMBER or COLLISIONS PER SECOND EXPE- RIENCED BY A MOLECULE OF VARIOUS GASES, AT ATMOSPHERIC PRESSURE AND C. GAS. COLLISIONS PER SECOND. Oxygen 4,410 000,000 Nitrogen . 5 021 000 000 Hydrogen 10 040 000 000 Carbonic Oxide . . . . Carbonic Acid .... 5,014,000,000 5,741,000,000 * In round numbers. 68 THE MOLECULAR THEORY OF MATTER. High Vacua. The average free path of the molecules of a given gas is independent of the temperature, but it varies when the density of the gas is made to vary. For the average distance a molecule will travel, between successive collisions, will obviously be less when there are many molecules in a unit volume than it will be when there are but few of them. In fact, mathematical analysis shows that the length of the average free path is exactly proportional to the reciprocal of the density of the gas. It follows from this that we can make the average free path as great as we please, by dimin- ishing the density of the gas sufficiently. If, therefore, we should diminish the density by the aid of a good modern air pump until it were only (say) a millionth of its normal value at atmospheric pressure, we should thereby increase the average free path of the molecules to one million times the lengths I have given you in the table.* Thus in the case of C0 2 the average free path would become about two inches and a half, and in the case of hydrogen it would even become six inches and three-quarters. It would be natural to think that at such high exhaustions the residual gas would exhibit no phenomena at all that it would be indistinguishable from a vacuum. But this is not the case. We must remember that the contents of a vessel exhausted to this degree is called a vacuum "by courtesy only." There are still many millions of molecules left in it ; and as their mean free paths are now measurable in inches, the medium exhibits entirely new prop- erties, some of which I shall try to show you. The Radiometer. I have here a small piece of appara- tus (Fig. 23), kindly loaned for the evening by Professor Kimball, from the physical laboratory of the Institute. You will see that it consists of a glass bulb in which is a sort of wind-mill, mounted very delicately upon a pivot. The vanes of this wind-mill are blackened on one side, while on the other side they are bright. I will hold the bulb near the gas jet. * On page 66. THE RADIOMETER. 69 You see that the vanes are now flying around very rapidly, bright side foremost. I hold it still nearer the gas jet, and the vanes revolve with such speed that they cannot be sep- arately distinguished. This wonderful little instrument is called the radiometer; and I am going to try to make it clear to you why the vanes revolve. The ultimate phenomena on which the motion depends are somewhat com- plicated, and I shall only attempt to give you a general idea of them.* In the first place, the blackened sides of the vanes absorb more of the radiant energy from the gas jet than the bright sides do, and hence they become warmer. This difference in temperature is essential to the work- ing of the instrument ; and hence it is important to make the vanes of some fairly good non-conductor, such as mica, in order that the tempera- tures of their opposite surfaces may not become equalized by conduction. The difference in temperature be- tween the black and bright surfaces is probably small, but for brevity we may speak of these surfaces as the "hot side" and the "cold side," respectively. The molecules on the hot side of a vane are vibrating more energetically than they are on the cold side and hence they communicate heavier blows to such gas-molecules as chance to collide with them. Now, since action and reaction are equal, it follows that the gas-molecules react more powerfully on the hot (or black) surfaces than they do * For a more satisfactory discussion, see Maxwell's article in the Phil- osophical Transactions for 1879. FIG. 23. THE RADIOMETER. 70 THE MOLECULAR THEORY OF MATTER. on the cold ones, and hence there is a tendency to drive the vanes around as you saw them go, bright side foremost. This tendency is not sufficient to cause the vanes to revolve when the bulb is filled with air of ordinary density, however, because the average free path of the air-molecules is so extremely small, and the number of molecular collisions per second so enormously great, that the accelerated molecules that fly off from the hot sides of the vanes beat back the molecules of air in the immediate neighborhood of the vane sufficiently to cause a slight rarefaction of the air in front of the hot side. This rarefaction tends to make the vanes revolve black side foremost. When the air in the bulb is of ordinary density, these two opposing tendencies appear to be sensibly balanced, and no motion results. When the air in the bulb is moderately exhausted, the effect due to local rare- faction in front of the hot side preponderates, and the vanes slowly revolve black side foremost; but when the exhaustion is pushed to such a degree that the free path of the residual molecules becomes as great as the distance from the vanes to the glass bulb, the accelerated molecules no longer beat back their neighbors ; there is no local rarefaction ; the only remaining cause of motion is the reaction of the black sur- faces against the accelerated gas-molecules; and the vanes therefore revolve bright side foremost. Crookes's Tubes. The radiometer depends for its action on the presence of the walls of the glass bulb, and the larger the instrument is, the more perfect must the exhaustion of the bulb be ; for the essential thing about the instrument is, as I have explained, that the average free path of the mole- cules of the residual gas must be great enough to permit these molecules to strike against the walls of the bulb, after rebounding from the vanes, instead of against one another. I have here three other pieces of apparatus, the operation of which does not depend upon the dimensions of the containing tube. I am enabled to show you these tubes through the CKOOKES'S TUBES. 71 courtesy of Messrs. Queen & Co., of Philadelphia, to whom they belong, and who have very kindly loaned them for this occasion. In each of these tubes the degree of exhaustion is so great that the average free path of the residual mole- cules is several inches long. I will first show you this tube (Fig. 24), in which the vanes of the little fly are set obliquely, and are not blackened at all. As I connect the terminals of the battery with the ends of the loop of wire below the fly, you notice that this wire becomes hot. The gas-molecules that collide with the hot wire are driven off in all directions, at greatly increased speed, by the vigorous blows they receive from the mole- cules of the wire, which are now vibrating very energetically. Those that strike against the vanes of the fly impinge on them obliquely and cause them to revolve, as you see they are now doing. I will next connect the loop and the upper elec- trode with the respective terminals of the induction coil. You see that this also causes the fly to rotate. Before explaining why the induction coil causes the vanes to move, I will show you another tube (Fig. 25), which is very simple in construction, but very beautiful and instructive in operation.* It consists of two electrodes sealed into the tube, one of which is concave, or cup-shaped. As I throw the coil into action, you notice the hazy double cone of purplish light, the vertex of which is at the center of cur- * In exhibiting this tube and the next one, the lights in the lecture room were turned down. FIG. 24. A CROOKES'S TUBE WITH OBLIQUE VANES. 72 THE MOLECULAR THEORY OF MATTER. vature of the concave electrode ; and where the cone spreads out so as to intersect the outer tube, the glass shines with a beautiful golden fluorescence. Now the explanation of this phenomenon is, that as the molecules of residual gas come in contact with the concave electrode they receive a charge of electricity themselves, and are energetically repelled in a direction normal to the surface of the electrode. Flying away from the electrode, they necessarily pass close to its center of curvature, and being crowded together at that point, they brush against one another sufficiently to give rise, in some manner, to the purplish glow that you see. After they FIG. 25. A CROOKES'S TUBE WITH CONCAVE ELECTBODE. have passed the vertex of the cone of light, they diverge once more ; and when, continuing to move in straight lines, they come in collision with the sides of the tube, they excite a fluorescence in the glass which dies away towards the remote end of the tube, as the flying stream of molecules strikes more and more obliquely. I think you will see, now, why the induction coil caused the vanes of the other tube (Fig. 24) to revolve. The gas-molecules were projected from the elec- trode by electrical repulsion, and the effect was the same as when the wire loop was directly heated by the battery. There is one very strange thing about these high vacuum tubes, which I think no one has yet satisfactorily explained. CROOKES'S TUBES. 73 You will notice that when I reverse the direction of the cur- rent through the coil, the appearance of the tube (Fig. 25) is entirely changed. There is no longer any sign of the double cone of flying molecules. It appears that in the phenomena of electric repulsion in these tubes, it is the negative electrode exclusively that is concerned. It seems to make little differ- ence what part of the tube the positive electrode is in. I will now show you the remaining tube (Fig. 26), and I think you will agree with me that it is exceedingly beautiful when in operation. It consists of a horizontal tube containing a pair of parallel glass rails, along which a sort of little paddle- wheel can roll. The electrodes, you will notice, are on a level with the uppermost vanes of the wheel. I will now FIG. 26. A CROOKES'S TUBE WITH ROLLING WHEEL. start the coil in action, and you see the whole tube glorious with light and color. The molecular stream from the nega- tive electrode, beating on the upper vanes, causes the wheel to revolve and roll along the track toward the other end of the tube ; but just before it gets there I reverse the coil, and you see the wheel come to a stand-still and then begin to revolve in the opposite direction until it returns to its starting point. We can make it travel back and forth as many times as we please, by merely reversing the coil when the wheel nears the end of its course. You will notice, at the ends of the tube and along the bottom of it, what appears to be the shadow of the glass rails. It is not a true shadow, however. The gorgeous fluorescence that you see elsewhere is caused, as in the last tube I showed you, by the impact of the molecules 74 THE MOLECULAR THEORY OF MATTER. against the sides of the tube; and the dark places do not shine, simply because they are shielded from the molecules that are streaming away in straight lines from the negative electrode. To prove this to you, I move this small magnet about in the neighborhood of the tube, and you see the dark lines shifting from place to place, as they would not do if they were true shadows. The streams of electrified mole- cules, behaving in a certain sense like electric currents, are deflected by the magnet ; and different parts of the tube are shielded as I change the magnet into different positions, so that the pseudo-shadows appear to move about. Before leav- ing this interesting subject, let me say that these wonderful mechanical phenomena in high-vacuum tubes were discovered by Mr. William Crookes, whose researches in this department of physics have earned him a lasting renown. He has given us an experimental demonstration of the kinetic theory of gases, and in his apparatus we can almost see the molecules as they fly about. III. THE MOLECULAR THEORY OF LIQUIDS. Preliminary Remarks. I have already given you the kinetic definition of a liquid.* You will remember that liquids resemble gases in one respect, which is, that their molecules can move freely about among one another. They differ from gases, however, in having a much smaller average molecular velocity, and in having their molecules so close together that they are always well within the sphere of one another's attractive influence. Except when great accuracy is required, we found it possible to ignore the intermolecular forces in gases. This makes the molecular theory of these bodies comparatively simple. In liquids, however, no such simplification is possible, for the molecular forces are no longer insensible. Their effects are everywhere visible, and in discussing liquids we can never cease to consider them. * Page 12. FREE EVAPORATION. 75 This fact, and the more lamentable one that we do not yet certainly know the form of the molecular force-function (that is, the law of variation of the force with distance), render the study of liquids exceedingly difficult. Thus it is that although questions of the greatest moment are arising con- tinually in this field of molecular physics, to most of them we can make no answer at present. No mathematician has yet worked out the kinetic theory of liquids to anything like the extent to which the corresponding theory of gases has been pushed; and for this reason what I shall say about liquids will necessarily be of a fragmentary character. Free Evaporation. There can be no doubt that the mole- cules composing liquids have as great a variety of velocities as those in gases ; for there must be almost innumerable collisions among them, and even if there were an absolute equality of velocities at any given instant, the collisions would necessarily destroy this equality at once. Doubtless there is some law of distribution of velocities in liquids, corresponding to Maxwell's law in gases ; * but the form of this law has not yet been discovered. Admitting the fact that the velocities of the molecules are unequal, let us con- sider what would happen at a free surface of the liquid, assuming for the moment that above this free surface there is a boundless vacuum. A particle well within the liquid is attracted, on the whole, equally in all directions. A particle at the surface, however, is attracted only dowmvard. You will see, therefore, that when a molecule, in the course of its wanderings, comes to the surface, whether it will escape from the liquid or not depends upon the magnitude of the vertical component of its motion. If this vertical component is sufficient to carry the molecule beyond the range of sensible attraction of the liquid, the molecule will pass away indefi- nitely into the space above. On the other hand, if the vertical component of its motion is not sufficient to carry the mole- * See equation (1). 76 THE MOLECULAK THEORY OF MATTER. 3 cule beyond the range of sensible attraction of the liquid, it will rise into the vacuous space only a short distance, its upward velocity growing less and less under the influence of the downward attractive forces until it vanishes altogether, after which the molecule will begin to fall back again, and it will finally plunge once more into the liquid. At the surface of a liquid, therefore, the molecules are continually describ- ing paths something like those indicated in this diagram (Fig. 27), where the dotted line represents the limit of sen- FIG. 27. DIAGRAM OF A LIQUID SURFACE. sible molecular attraction. Most of the molecules that start upward, fall back into the liquid ; and the escape of such of them as are moving fast enough to overcome the attraction of the liquid, constitutes the phenomenon that we call free evap- oration. I said that of those molecules that leave the liquid, the majority fall back again ; and perhaps I ought to explain how we know this to be the fact. If the converse were true that is, if the majority of them at once flew off into space we should have to conclude that the average molecular velocity in liquids is just about great enough to overcome the attraction of the liquid for a molecule about to leave it ; and the enormous latent heat of vaporization of liquids proves that this cannot be so. For example, we have to add a vast amount of energy to a pound of water before it will pass into steam ; and this shows that under ordinary circumstances the COOLING EFFECT OF EVAPORATION. 77 average kinetic energy of translation of a water-molecule is far too small to overcome the molecular attractive forces. Hence it follows that of the many molecules that come to the surface of a liquid in a given time, very few will per- manently escape, because very few have velocities sufficiently in excess of the average to enable them to pass away, directly against the attractive force of the liquid. Cooling Effect of Evaporation. We know that the tem- perature of a gas is proportional to the average kinetic energy of translation of the molecules of the gas. It is not so certain that this is the case with liquids, for these bodies are consti- tuted so differently that we can hardly assume them to act on oiir senses in precisely the same manner. Nevertheless, it is probable that there is some analogous relation between the kinetic energy of a liquid and the temperature of the liquid ; so that although the two may not be strictly propor- tional, we may fairly assume, I think, that- one of them is what mathematicians would call a continuous, one-valued, increasing function of the other. This being admitted, it is easy to see why evaporation cools a liquid. For it is plain that when a liquid is evaporating, it loses only those mole- cules which have a speed considerably greater than the average, the slower moving ones, as I have explained, being retained by the attraction of the liquid. This is equivalent to saying that the molecules that do fly off will carry away with them more than their equable share of the kinetic energy of the liquid. Hence while evaporation is going on, the average kinetic energy per molecule, in the- mother liquid, is continually growing less ; and this means; that the temperature of the liquid is falling. Vapor Density. Thus far we have spoken only of the phenomena of evaporation in a boundless vacuum, and we now come to the consideration of evaporation in a closed vessel. We will suppose that at the outset this vessel is 78 THE MOLECULAR THEORY OF MATTER. absolutely empty, and that at a certain instant a small quantity of water is admitted to it. During the first instant following the introduction of the water, the phenomena are precisely the same as we have seen them to be in the case of the boundless vacuum. The swiftest molecules fly off as before ; but they can no longer pass away indefinitely into space. They are now retained by the vessel, in which they will accumulate, constituting a gas or vapor whose density will go on increasing until a certain limit is reached. You will readily see that the molecules composing this vapor will travel in every direction, precisely as they do in other gaseous bodies. Many of them, therefore, will 'plunge back into the liquid again, and become an integral part of it once more. Now the number of molecules that leave the mother liquid in a given time will be quite independent of the density of the vapor overhead ; but the number that fly back into it again, in a given time, will be greater, the greater the density of the vapor. At the beginning of the evaporation the vapor will be rare, and the number of molecules that fly off in any given time will greatly exceed the number that return during that time. The density of the vapor will therefore increase. After a certain interval (an exceedingly short interval, meas- ured by ordinary standards), the density of the vapor will become so great that tftfe number of molecules plunging back into the liquid in a given time will become sensibly equal to the number that fly off from it in the same time. When this adjustment becomes perfect, the density of the vapor will no longer increase. It is then said to be "saturated." I would like to fix it clearly in your minds that a saturated vapor is one in which the number of molecules that plunge back into the mother liquid in any given time, is precisely equal to the number of molecules that rise out of the liquid, in the same time, and enter the vapor. You will see that any cause that tends to disturb this equality will also tend to alter the density of the vapor. For example, if we raise the tempera- ture of the system we shall destroy the equality in question ; VAPOR PRESSURE. 79 for we shall accelerate all the molecules, and hence more molecules will plunge from the vapor into the liquid in a given time than before, and more molecules will also come to the surface of the liquid from the interior. Furthermore, of the increased number of molecules that emerge from the interior of the liquid, a larger proportion than before will have velocities exceeding the critical velocity that a molecule must have in order to escape from the attraction of its fellows. Hence, on the whole, the density of the vapor will increase, approaching a new limit at which the number of in-coming and out-going molecules will again become equal. We see, therefore, that for any given vapor in contact with its liquid there is a definite density corresponding to each temperature. If we knew enough about the physics of liquids and vapors, we could express the relation between temperature and vapor density by means of a rational equation ; but unfortunately we are still very far indeed from possessing this knowledge. Our reasoning shows that the density of a saturated vapor in no wise depends upon the size or shape of the containing vessel; and this is known to be the fact. It is also inde- pendent of the area of the free surface of the liquid, though this, perhaps, may not be so evident. If the free surface be doubled (for example), we shall thereby double the number of both the out-going and the in-coming molecules ; hence the equality between the two will not be disturbed, and this shows that the vapor density does not depend upon the extent of the free surface of the liquid. Vapor Pressure. The pressure exerted by a vapor depends (1) on the average speed of the molecules composing the vapor, and (2) on the number of these molecules that strike against a unit area of the containing vessel, per second. The number of molecules that strike against a given area in a given time will depend on the number of molecules in a unit volume of the vapor that is, on the density of the vapor and on the average molecular velocity. Hence (since the average molec- 80 THE MOLECULAR THEORY OF MATTER. ular velocity is a function of the temperature) we may say that the pressure exerted by a vapor on the walls of the con- taining vessel will depend on (1) the density of the vapor, and (2) its temperature. This much is true of all gaseous bodies, as I have already explained to you while speaking of gases ; but in the case of a saturated vapor in contact with its liquid, we have just seen that the density of the vapor is itself a function of the temperature. Hence we conclude that the pressure exerted by such a vapor against the vessel containing it depends, ultimately, only upon the temperature of the vapor and its liquid. This fact has long been known from experiment, and many attempts have been made to find an equation which should represent the relation between the pressure and temperature of saturated vapors. Of the many equations that have been proposed, B/ankine's is probably as good as any. He found that the relation in question could be represented with remarkable accuracy by an equation of the form a-- where p is the pressure, t the absolute temperature, and a, (3, and y are constant quantities, to be determined for each liquid by experiment. 1 * Ebullition. Although we saw that the density of a satu- rated vapor is not, strictly speaking, a function of the area of the free surface of the liquid, it must nevertheless be borne in mind that the process of evaporation is of such a nature that it cannot take place unless there is some free surface. For it consists in the escape of certain molecules froin a free surface ; and where there is no such surface, obviously there can be no evaporation. It is not essential that the free sur- face should be at the top of the liquid. For example, when * For the values of a, /S, and y for various liquids, see Rankine's Mis- cellaneous Scientific Papers (London, Charles Griffin & Company, 1881). EBULLITION. 81 water is heated, the air that it holds in solution is deposited in small bubbles on the walls of the containing vessel ; and evaporation may take place across the surface of these bubbles, the bubbles increasing in size as they fill with steam, until presently they rise to the surface and break. If water be freed from such dissolved air, by protracted boiling or other- wise, it may be made to behave in a remarkable manner. Thus Dufour found that if drops of water so prepared are submerged in a mixture of oil of cloves and linseed oil (of specific gravity 1.000), they can be heated far above the boil- ing point, although exposed only to atmospheric pressure. Such drops have no free surface, and therefore no evapora- tion can take place from them. Dufour heated large drops of water, in this way, up to 248 Fahr., and he succeeded in heating smaller ones as high as 352 Fahr. (Under ordinary circumstances water could not be heated to 352 unless it were subjected to a pressure of at least 139 pounds, absolute, to the square inch.) Wheft these drops came in contact with the thermometer, or with the containing vessel, they passed instantly into steam, with a hissing sound. It seems probable that the explanation of such phenomena as these is, that when the liquid is heated gradually and uniformly, without any free surface, its molecules are accelerated throughout the mass, but in such a uniform manner that the momentum is nowhere sufficient to tear them apart against the attractive forces that exist among them. When the velocities become so great that in parts of the drop they are on the point of tearing the molecules apart, the least disturbance from with- out, such as a shock or a vibration or the contact of some foreign substance, may precipitate the disruption ; and after a free surface has once been formed, even though it may be exceedingly small, the drop will be almost instantly dissipated by evaporation across this surface. Ebullition differs from simple evaporation in the formation of such free surfaces in the interior of the liquid, or along the bottom and sides of the containing vessel, across which surfaces evaporation occurs 82 THE MOLECULAR THEORY OF MATTER. precisely as we have described it in connection with the upper, horizontal surface, or in connection with the air-bubbles sepa- rated from solution by heat. Ebullition appears to occur only when heat is supplied to the liquid faster than it can diffuse to the surface layers, or be carried there by convection. The cause of the formation of free surfaces in the interior of a boiling liquid is not well understood yet, and in fact it may be said, in general, that we have still much to learn concern- ing the phenomena of ebullition, both by the making of new experiments, and by interpreting those that have been made already.* Critical Points. Before leaving the subject of vapors and evaporation, I want to call your attention to a peculiar fact which was discovered experimentally, I believe, before it was explained theoretically ; though its theoretical explanation is quite simple. I refer to the fact that there is a temperature for every gas called its critical temperature such that the gas cannot be liquefied by pressure alone when its tem- perature exceeds this critical value, no matter how great the applied pressure may be. In speaking of the theory of evap- oration, I called your attention to the fact that a molecule cannot leave its liquid unless its upward velocity exceeds a certain limit. ISTow if molecules attract one another, there must be a similar proposition true of molecules that collide with one another in a vapor or a gas. Let me illustrate. A stone thrown upward by the hand does not travel far before the attractive force of the earth upon it overcomes its momen- tum, and causes it to fall back again. By using a rifle we can project a ball far higher into the air, but still it is only a matter of time when the momentum will be overcome, and the ball will fall back again just as the stone did. With a good modern cannon we can throw a projectile several miles * In connection with this point, the reader is recommended to consult the section on "Evaporation and Ebullition " in Preston's Theory of Heat (Macmillan & Co., 1894). CRITICAL POINTS. 83 into the air, and still it falls back. But it is conceivable that we might project one with such speed that it would leave the earth forever. Such a result could be realized without giving the projectile an infinite velocity. I have no doubt that Professor Alden will show you, or has shown you, in your course in mechanics, that if the retarding action of the air be omitted from consideration, an initial speed of 36,650 feet per second would be quite sufficient. Now, with this much premised, let us imagine two molecules of a gas in contact ; and let us suppose that a sudden impulse is given to one of them, to drive it away from the other. If the impulse is small enough, the disturbed molecule will only travel a short distance, and will then fall back to its original position; but we may give it such a speed that the attractive force of the fixed molecule will fail to bring it back, and in this case it will travel onward indefinitely. NOAV, just as in the case of the cannon-ball and the earth, there must be some intermediate velocity that will be just sufficient to separate the two molecules under consideration. We may call this the critical velocity ; and we may say, that if the molecules of a gas are moving about so that, on an average, when two of them collide they have a relative velocity greater than this critical value, the gas in question cannot be liquefied by pressure alone j for even if its molecules were forced almost into absolute contact with one another, their velocities would be sufficient to separate them again indefinitely, as soon as the pressure was removed. From this, and from the rela- tion between temperature and molecular velocity in gases, it follows that for every gas there is a temperature above which the gas cannot be liquefied. The critical temperatures of the so-called permanent gases are very low ; and that is why they resisted liquefaction until the condition necessary to success in the experiment became known. The following table gives the critical temperatures of certain of the more familiar gases and liquids : 84 THE MOLECULAR THEORY OF MATTER. TABLE OF CRITICAL TEMPERATURES. CHEMICAL CRITICAL TE MPERATURE. FORMULA. Centigrade. Fahrenheit. Hydrogen* . . -. . . Nitrogen . Oxv^en H N o - 229 - 124 - 118 - 380 - 191 180 Marsh gas CH 4 100 (?) 148 I?} Carbonic acid ..... Nitrous oxide ..... Ammonia gas . . ... Chlorine ....... Sulphur dioxide .... Ether * . . C0 2 N 2 O NH 3 Cl S0 2 (C 2 H 5 ) 2 + 31 + 36 + 131 + 141 + 156 + 194 + 88 + 97 + 268 + 286 + 313 + 381 Alcohol Chloroform Carbon disulphide . . . Benzene C 2 H 6 CHC1 3 CS 2 C 6 H 6 + 235 -1- 260 + 272 + 281 + 455 -t- 500 + 522 + 538 Acetic acid C 2 H 4 O 2 + 322 -1- 612 Water . H90 + 365 + 689 Contraction and Compressibility. The well-known resist- ance of liquids to compression might seem to indicate that the molecules of these bodies are nearly in contact with one another; and the assumption that they are in contact has sometimes been made, for the purpose of deducing their diameters in accordance with a theorem of Clausius, to which I shall presently refer. The fact that liquids con- tract when cooled indicates, on the other hand., that their molecules are not in contact. I think I can tell you -how these two things are to be reconciled. I think the difficulty of compressing liquids arises from the fact that their mole- cules are describing curved paths t with considerable veloci- ties. They would fly off tangentially to these paths if it * Calculated by Mr. Sutherland. t See Fig. 10. CONTRACTION AND COMPRESSIBILITY. 85 were not for the inter-inolecular attractive forces that deter- mine the curvature. Normally, therefore, there is a sort of balance between the attractive forces and the centrifugal tendency due to the velocity of the molecules and to the curvature of their paths. The centrifugal tendency developed by liquid molecules may be quite considerable, because it is proportional to the square of the velocity of translation, and to the reciprocal of the radius of curvature of the path. The radius of curvature of the path must be exceedingly small, because it is probably of the same order of magnitude as the distances between the molecules. Its reciprocal, therefore, will be correspondingly large, and hence the liquid molecules may have a .considerable centrifugal tendency, even though their velocity of translation may be far less than it is in gases. Under normal conditions, as I have said, the attrac- tive forces and the inertia effects are balanced, and the liquid remains at a constant volume. When we cool the liquid we diminish its kinetic energy that is, we make its molecules go slower. This lessens the centrifugal tendency, the attrac- tive forces preponderate, and the molecules approach one another; that is, the liquid contracts. When, instead of cooling the liquid, we compress it, the phenomena are not so simple. In reducing the bulk by compression we cause the molecules to approach one another. This lessens their poten- tial energy, and therefore increases their kinetic energy; hence compression increases the average molecular speed. If all the other circumstances of the molecular motion remained the same as before, this increase of speed would increase the centrifugal tendencies of the molecules, and cause the liquid to resist compression. But in compressing a liquid we do not simply accelerate its molecules. In bringing them nearer one another we increase the attractive forces between them, and we undoubtedly alter the radii of curvature of their paths ; and to predict the actual behavior of the liquid we should have to take all these things into consideration. The mathematical treatment of this problem is quite difficult, and 86 THE MOLECULAR THEORY OF MATTER. I cannot positively say what result it would yield. Never- theless I am satisfied, in my own mind, that in forcing the molecules of a liquid closer together we increase the inertia effects far more than we increase the attractive forces, and that that is why liquids resist compression so powerfully. Surface Tension. The existence of molecular forces in liquids may be readily shown by experiment, and some of the experiments that have been devised for this purpose are exceedingly beautiful and suggestive ; but before describing any of them I wish to call your attention to a fact which enables us to discuss molecular forces more conveniently than we otherwise could. Consider, for a moment, a drop of liquid, of which this (Fig. 28) is an imaginary sectional view. Particles in the interior of the liquid are attracted equally in every direction ; but particles on the surface are attracted only inward, in a direction perpendicular to the sur- face of the drop. If we write the mathe- matical equations that would represent FIG. 28. DIAGRAM IL- LUSTRATING SURFACE the behavior of a drop when acted upon FORCES IN A DROP OF b y forces of this kind, we shall find that LIQUID. . . these equations are precisely the same as they would be in a similar drop whose parts do not attract one another, but which is enveloped by a water- tight skin, or membrane, in a state of uniform tension. The tension that such an imaginary membrane would have to have, to produce the observed phenomena of drops and other small portions of liquid, is called the surface tension of the liquid. It should be carefully noted that we do not assert that any such membrane actually exists. " Surface tension " is only a convenient conception that enables us to foresee, more clearly, the effects produced under given conditions by the molecular forces existing in liquid masses. I think I need not discuss PHENOMENA OF FILMS. 87 this conception further, because it is explained and illustrated in all good works on physics. It may be well, however, to call attention to the fact that the ideal surface membrane of liquids differs from all material membranes inasmuch as its tension does not increase when the surface of the liquid is extended in any way. When the surface of a liquid increases, it does so by the exposure of particles that were previously in the interior, and not by the stretching of the old surface, in the sense in which the word " stretch' 7 is ordinarily used. This point seems obvious enough, but I find that unless particular attention is called to it, students are apt to get erroneous ideas about surface tension. Phenomena of Films. Kegarding liquids as mobile bodies enclosed in contractile membranes in a state of uniform ten- sion, it will be plain, I think, that the most notable effects of molecular attraction will be observed when the mass of the liquid is small in proportion to the surface it exposes. Thus the phenomena of soap-bubbles, and of other forms of liquid films, are very striking. A simple and beautiful experiment illustrating surface tension can be performed with a piece of wire, a bit of sewing silk, and a solution of soap, suitable for blowing bubbles. 3 * The wire is bent into a ring, about 2-J- inches in diameter, and a soap film is formed across it by simply immersing it in the soap solution and withdrawing it again. To make the film adhere, it may be necessary to roughen the ring with a file. A piece of sewing silk, about three inches long, is tied together at the ends, so as to form a closed loop. The silk loop is then wetted with the soap solution, and laid gently on the film. (See Fig. 29.) It will lie indifferently in any position if the film be kept perfectly horizontal, its indifference being due to the fact that the forces acting on it are balanced in all directions. If the film be broken inside the silk loop, by the contact of a dry * Plateau's liquide glycerique is the best thing for blowing bubbles. See Appendix. 88 THE MOLECULAR THEORY OF MATTER. bit of wood or a hot needle, the loop instantly flies out into a perfect circle (Fig. 30), demonstrating the existence of the "surface tension" in a most interesting manner. (The same FIGS. 29 and 30. ILLUSTRATING THE SILK-LOOP EXPERIMENT. experiment may be performed with the ring and film vertical, if the silk loop be supported as shown in Fig. 31.) When the circular silk ring is floating on the soap film, it is instruc- tive to tap it lightly on the inside with a lead-pencil or other convenient small article. It springs away from the touch of the pencil as though it were a ring of tempered steel escap- ing from the blow of a hammer. This is because the tap with the pencil deforms it slightly, and the tension of the film im- mediately restoring it to the circular form causes it to react against the pencil very smartly, and to bound away from it with surprising quickness. An end- less variety of beautiful experiments can be tried with soap films with extremely simple apparatus; but we cannot dwell upon them longer this evening. The one experiment that FIG. 31. MODIFICATION OF THE SILK-LOOP EXPERIMENT. OTHER SURFACE PHENOMENA. 89 I have described will be sufficient to show the existence of the molecular forces satisfactorily. Other Surface Phenomena. The surface tension of a liquid may be altered by a variety of methods. Thus if a small sliver of cork be carefully wetted with alcohol along half of its length on one side, and be then thrown upon water, it will revolve, because the surface tension of dilute alcohol is less than the surface tension of pure water, and the cork is FIG. 32. A METHOD FOB OBTAINING A CLEAN WATER-SURFACE. therefore pulled more in one direction than the other. If a similar fragment of cork be wetted with alcohol or ether along one entire end or side, and thrown upon water, it will be pulled across the surface of the water bodily, for the same reason. Particles floating on the surface of clean water appear to be repelled by a drop of alcohol or ether, brought near to them on the end of a glass rod or the tip of a finger. This is because some of the vapor of the alcohol or ether con- denses on the surface of the water and alters its tension in the vicinity of the drop. Small particles of camphor floating 90 THE MOLECULAR THEORY OF MATTER. on clean water exhibit vigorous movements. Camphor is slightly soluble in water, and its solution has a lower surface tension than pure water. The camphor particles do not dis- solve with equal rapidity on all sides, and hence there is a resultant pull exerted on them, in the direction in which the water contains the least camphor in solution. For success in this experiment it is essential to have a perfectly clean water surface, as the least trace of oily matter deadens the move- ments remarkably. A good way to secure such a surface is shown in Fig. 32. An inverted glass funnel is connected with the faucet by a rubber tube, and the water is allowed to run freely for some time. The faucet is then turned off until only the least possible amount of water comes through it. A piece of camphor is next scraped clean, and a few of the last scrapings are allowed to fall on the water in the funnel. If the funnel was originally clean, they will spin about in a lively manner. We shall have occasion to refer to this experiment again, when we come to consider the size of molecules. Magnitude of the Surface Tension. If we could find out what the strain is on the silk thread shown in Fig. 30, we could calculate the intensity of the surface tension by the same formula that is used for calculating the bursting pressure of thin, hollow cylinders. It would be possible to devise an experiment that would give this strain, but it would hardly be worth while, because the surface tension of liquids can be determined by other methods more conveniently and with far greater accuracy. In fact, it can be determined directly, by means of an apparatus like that shown in Fig. 33, where S is a soap film, and W is a light wire, which can move up and down without sensible friction. By putting small weights in the pan we can readily discover what the supporting power of the film is. This result is to be divided by the length of that part of the wire W which is wetted by the film, and the quotient is the supporting power of the film, per unit length MAGNITUDE OF THE SURFACE TENSION. 91 of W. To find the surface tension we have to divide the supporting power thus found by 2, because the film has two surfaces. Experiments of this sort are interesting enough, but they are of small value, because they can only give us the surface tension of soap solutions, or of other liquids from which bubbles can be blown. Different methods have to be used, therefore, to determine the surface tension of pure water and the great majority of liquids. Many such methods have been pro- posed. One of them is based on the investigation of the curved surface of a liquid, where it touches the vessel Containing it. You are all FIG. 33. APPARATUS FOB MEASURING familiar with the appearance of the water curve, and I think you would be ready to admit that the experimental investigation of its shape would FIG. 34. THE WATER-CURVE; FROM HAGEN'S MEASURES. be very difficult. The difficulties have been overcome, how- ever, by numerous observers ; and I have plotted for you, 92 THE MOLECULAR THEORY OF MATTER. from Hagen's data, the form assumed by a water surface in the vicinity of a flat, vertical plate of polished brass (Fig. 34).* The dotted line shows the level of the water at an infinite distance from the plate, it being an asymptote to the water- curve. It can easily be shown that if y is the height, above the dotted line, of any point, P, on the water-curve, and a is the angle that the tangent at P makes with the dotted line, then ' = 2\^- sin|, (35) W where S is the pull exerted by the water surface on a line one unit long, and w is the weight of a unit volume of water. This equation is derived from the fact that the curvature of the surface at any point must be just sufficient to enable the superficial tension to support a column of water of height y. You will see that if we measure a and y at any point, and substitute their values in the equation I have just given, we can calculate S (iv being a known quantity). The principal objection to this method is the difficulty of determining the coordinates of the water-curve accurately. A very simple method of finding the surface tension consists in observing the height, h, to which a liquid will rise in a capillary tube of radius r. The surface tension, S, is then given by the equation f s ^ WTh 2 COS a where w is the weight of a unit volume of the liquid, and a is the angle which the surface of the liquid makes with the tube at the point where the two come into actual contact. With a clean glass tube the value of a for water is 0, and for mercury it is 128 52'. The capillary-tube method gives * The scale of Fig. 34 is such, that every dimension in the cut is ten times the corresponding dimension in nature. t For the derivation of this equation see Maxwell's article on Capillary Action in the Encyclopaedia Britannica. MAGNITUDE OF THE SURFACE TENSION. 93 fairly satisfactory results, but the angle a varies considerably with the degree of cleanliness of the surface of the glass tube, and it would be desirable to have a good method that would be free from any such objection. I believe that Lord Kelvin's method meets every requirement in this respect, and although I have not tried it, and do not know what results FIG. 35. LORD KELVIN'S APPARATUS FOR DETERMINING SURFACE TENSION. FIG. 36. NOZZLE OF LORD KEL- VIN'S APPARATUS. have been obtained from it by others, I am going to tell you, briefly, what his plan is.* His apparatus consists of two vessels, A and B (Fig. 35), connected by a flexible syphon in such a manner that the level of the liquid in B can be varied at will by raising or lowering A. At the bottom of B there is a small nozzle with a capillary opening through it (shown on an enlarged scale in Fig. 36). The liquid in B runs down * See his Popular Lectures and Addresses, Vol. I, p. 45. UNIVERSITY 94 THE MOLECULAR THEORY OF MATTER. through the nozzle, and if the height h is not too great, a drop will form at the bottom of the nozzle, whose radius of curva- ture will be just sufficient to enable the surface tension at that point to sustain a column of liquid of height h. This radius of curvature is next determined by any of the methods given in the books on physics that are applicable to the case some optical method being preferable and the surface tension is then given by the equation pr whr ~ (36) where r is the radius of curvature of the drop, and w is the weight of a unit volume of the liquid. In his article on Capillary Action in the Encyclopedia Britannica, Maxwell gives the following values of the surface tensions of various liquids at 20 C. (68 Fahr.), as determined by Quincke, the TABLE OF SURFACE TENSIONS AT 20 C. SURFACE TENSION. LIQUID. Specific Dynes Grammes Grains Gravity. per Centimeter. per Centimeter. per Inch. Water . . . 1 0000 81 083 324 Mercury 13 5432 540 551 O1 K.Q Bisulphide of carbon . 1.2687 32.1 .033 1.28 Chloroform .... 1.4878 30.6 .031 1.22 Alcohol 7906 25 5 026 1 02 Olive oil 0.9136 36.9 .038 1.47 Turpentine . . 0.8867 29.7 .030 1.19 Petroleum .... 0.7977 31.7 .032 1.27 Hydrochloric acid . 1.1 70.1 .071 2.80 Solution of hyposul- ) phite of sodium ) 1.1248 77.5 .079 3.10 LATENT HEAT OF VAPORIZATION. 95 free surface of the liquid being in contact with air in every case. He gives the tensions as expressed in dynes per linear centimeter, and I have reduced them also to grammes per linear centimeter and to grains per linear inch ; so that the last column gives the normal pull, in grains, exerted by a liquid on a straight line, one inch long, lying in its surface. (Of course there is an equal and opposite pull on the other side of the line, unless it happens to form one of the boundaries of the surface, as shown at W in Fig. 33.) The value of the surface tension of water given in this table is certainly too great. Brunner found it to be 75.2 dynes, and Wolf found 76.5 and 77.3. Rayleigh's deter- mination, based on a study of the wave-length of ripples, gave 73.9 dynes at 18 C. The latest trustworthy deter- mination that I have seen is that given by Mr. T. Proctor Hall in the Philosophical Magazine for November, 1893. He finds that at t centigrade the surface tension of water, in dynes per linear centimeter, is 75.48 .140 t, which is equivalent to .07694 .000143 J when expressed in grammes per linear centimeter. Latent Heat of Vaporization. To vaporize a given mass of liquid we have to tear all its molecules apart, in opposition to the attractive forces that exist among them, and we have also to increase the average speed of the molecules. Each of these operations necessitates the expenditure of energy, and the total amount of energy thus required is surprisingly large. For example, to vaporize one pound of water (about a pint) at 212 Fahr., we have to expend no less than 966 British units of heat, or about 753,000 foot-pounds of energy. This shows that although the range of sensible molecular attrac- tion, in water and other liquids, is quite small, the attrac- tive forces, where they exist, must be very great. Following are the latent heats of vaporization of a few familiar liquids. 96 THE MOLECULAR THEOEY OF MATTER. LATENT HEATS or VAPORIZATION.* LIQUID LAI ENT HEAT. Heat Units. Centimeter-Kilograms. Water 535.9 22,883 Alcohol 202 4 8 642 Bisulphide of Carbon . . . Mercury 86.7 62 3,702 2 647 Investigation of the Work done in bringing a Molecule to the Surface. A molecule in the interior of a liquid is always under the influence of the attractive forces of its neighbors, and the resultant force exerted upon it at any instant must be found by compounding the attractions of all the molecules that are near enough to have any sensible effect upon it. The ceaseless change of position that goes on among the molecules of a liquid causes this resultant force to assume all imaginable directions and magnitudes in rapid succession ; and it follows that the molecule under consideration is in a sort of statistical equilibrium, if I may use the phrase. This is the kind of equilibrium that is contemplated when we say that molecules in the interior of a liquid are attracted equally in every direc- tion. With this much premised I am going to tell you of a property of liquids that we shall use, presently, in estimating the sizes of molecules. The property that I refer to is this : Whatever the law of molecular attraction may be, the work that has to be done against molecular attractive forces in bring- ing a particle of a liquid from the interior to the surface is precisely one half of the work that would have to be done to transport that particle from the interior of the liquid to an infinite distance. In proving this proposition we shall con- *At the boiling points of the respective liquids. A gramme of the liquid is considered in each case, and the unit of heat is the heat required to raise the temperature of one gramme of water from 3 C. to 4 C. BRINGING A MOLECULE TO THE SURFACE. 97 ceive a sphere to be described about the particle as a center, the radius of the sphere (which we will denote by R) being such that the attractions of all the particles that lie outside of it can be neglected. So long as the sphere is wholly sub- merged, the particle is in equilibrium ; but when the distance of the particle from the surface of the liquid becomes less than Rj a portion of the sphere will project above the liquid, and the particle will no longer be in equilibrium, but will be attracted downward, towards the interior. For example, con- sider this diagram (Fig. 37), the upper part of which illus- FIG. 37. DIAGRAM TO ILLUSTRATE THE WORK DONE IN BRINGING A PARTICLE OF LIQUID TO THE SURFACE. trates six successive stages of the approach of the particle to the surface. In the first stage, shown on the left, the sphere just touches the surface. In the next stage a part of the sphere projects above the liquid, and therefore there is a resultant downward attraction on the particle. To find the magnitude of this resultant, conceive a horizontal plane to be passed through the lower part of the sphere so as to cut off a segment equal to that which projects above the surface of the liquid. (The plane in question is indicated in the diagram by a short, straight line.) It follows from the symmetry of the figure that the component attractions of all the molecules that lie above this plane are perfectly balanced ; and hence the resultant attraction of the liquid on the particle we are considering is equal to the attraction exerted upon it by the 98 THE MOLECULAR THEORY OF MATTER. segment that the imaginary plane cuts off from the lower part of the sphere. You will see that the same reasoning applies at every stage, until the particle finally lies in the surface, as shown on the right. Now, if you please, turn your attention to the lower half of the diagram, which represents six corre- sponding stages in the removal of the particle from ths sur- face of the liquid to an infinite distance. As we pass from right to left, the particle, in each position, is attracted down- ward by the segment of the sphere which lies within the liquid. By comparing the upper and lower parts of the diagram, you will see (1) that the particle is attracted down- ward, whether it is above the surface or below it ; and (2) that the magnitude of the attraction is the same, when the particle is at a given distance from the surface, whether the particle is inside the liquid or outside. It follows from this symmetry of the attractions, that the work required to bring the particle to the surface of the liquid is precisely equal to the work required to carry it from the surface to a point out- side where the attraction of the liquid is no longer sensible ; and that is the same thing as saying that the work required to bring the particle to the surface is one half of the work required to transport it from the interior of the liquid to an infinite distance. Numerical Estimation of the Work done in bringing a Molecule to the Surface. We are now in position to com- pute the work that would have to be expended, against inter- molecular forces, in bringing to the surface of any given liquid a small portion of it, of given weight, originally situated in the interior. For, let us consider what happens when a unit weight of the liquid, originally at (say) 20 C., is heated to its boiling point and then entirely evaporated. In the first place, the mass of liquid contains a certain quan- tity of kinetic energy, because its molecules are certainly in motion. We cannot calculate this energy, because we do not know enough about the velocities of molecules in liquids. BRINGING A MOLECULE TO THE SURFACE. 99 For the present, therefore, we shall merely denote it by ki. We next proceed to raise the temperature of the liquid from 20 C. to the boiling point. In doing so we accelerate the molecules of the liquid, and (since the liquid expands upon being heated) we also do a certain amount of work in pulling the molecules apart, against the attractive forces they exert upon one another. The kinetic energy that has been added, in increasing the speed of the molecules, will be denoted by & 2 > and the total amount of heat-energy that we had to add in order to raise the temperature from 20 C. to the boiling point will be denoted by h. The next step is to evaporate the liquid ; and in order to effect the evaporation we have to supply the quantity of heat known as the "latent heat of vaporization," which we shall represent by H. The unit weight of liquid has now been transformed into a unit weight of vapor, and obviously we have the relation h + H= k z + k z + W+ w, (37) where & 2 is the kinetic energy added in heating the liquid from 20 C. to the boiling point, k s is the kinetic energy added in transforming the boiling-hot liquid into vapor, W is the total energy expended in overcoming molecular at- tractions, and w is the so-called " external latent heat " that is, the work done by the vapor in expanding from the liquid-volume to the vapor-volume against atmospheric pressure. Of these quantities we know H, h, and w, and although we do not know k 2 and k s separately, we can com- pute their sum with a sufficient degree of approximation. In fact, where K is the total kinetic energy of the vapor, which can be computed when we know the number of degrees of freedom of the molecules, and their average velocity of translation. We may observe that k l is the kinetic energy of the mole- cules in the liquid state ; and since the kinetic energy of a 100 THE MOLECULAR THEORY OF MATTER. body varies as the square of the body's velocity, and we have good reason to believe that the molecular velocity in a liquid is far less than it is in the vapor of that liquid, it follows that &! is extremely small in comparison with K. We shall therefore neglect it, and consider k z -\- k 3 to be equal to K. With this modification (37) becomes equivalent to W=H+h K w. (38) The quantities H and h are given in the books on heat, from which source data can also be had for computing w. To determine K we shall treat the vapor as a perfect gas ; for although this mode of procedure is not strictly correct, it will give results accurate enough for our purpose. We have, as the kinetic energy of translation in a given mass of vapor, (39) where M is the mass of the vapor and u* is the mean-square velocity of its molecules, as defined in equation (5). But we also have w*A * ft p = -g-> (40) where A, being the absolute density of the vapor, is equal to its mass divided by its volume that is, M A =y Substituting this value of A in (40), we find that which, in (39), gives Now k, being the total kinetic energy of translation of the vapor, involves three degrees of freedom ; and hence K, the 'TL total kinetic energy, is found by multiplying k by > where n o * See equation (13). BRINGING A MOLECULE TO THE SURFACE. 101 is the number of degrees of freedom in the vapor under con- sideration. When n is not certainly known (as in most cases it is not), we shall be obliged to substitute for it its value in terms of y, as given by equation (22). I am aware that grave objections could be urged against such a substitution in the case of vapors, but it is the best we can do. It gives us n]c_ ~ 3 - 2k _ pV (41) By computing K in accordance with this equation for the four liquids whose latent heats of vaporization are given on page 96, we obtain the values given in the fourth column of the accompanying table. COMPUTATION OF THE WORK DONE AGAINST MOLECULAR ATTRACTION IN BRINGING A GRAMME OF LlQUID TO THE SURFACE, AT 20 C. LIQUID H h K w W \W Water 22,880 3,420 5910 1,750 18640 9320 Alcohol 8,640 1,690 6,180 630 3,520 1,760 Bisulphide of Carbon . 3,700 290 1,790 350 1,850 925 Mercury 2,650 470 390 260 2,470 1,235 All of the quantities in the table are expressed in centi- meter-kilograms, and, as the heading indicates, the computa- tion is performed for one gramme of liquid in each case. The various quantities are given only approximately, and the results have the same order of certainty (or uncertainty) as the values of y that I have used in equation (41). The quan- tity W is obtained by substituting H, h, K, and w in equation (38) ; and as Wis the work that would have to be expended, against molecular attraction, in order to transport a gramme of the liquid from the interior of the liquid to an infinite distance (taking it away a molecule at a time), it follows, from the preceding section, that ^ W is the work that we should have to do in order to bring to the surface a gramme 102 THE MOLECULAR THEORY OF MATTER. of liquid that was previously in the interior. I have dwelt upon this computation at greater length than its intrinsic importance would warrant, because, as I have already told you, we shall presently make use of it in determining the sizes of molecules. IV. THE MOLECULAR THEORY OF SOLIDS. Condition of the Theory. The molecular theory of solids is still in its veriest infancy ; in fact it might almost be said that there is no such theory. The phenomena exhibited by solids are extremely complicated, and this fact implies, with- out doubt, a corresponding complexity in molecular structure. In order to give you a clear idea of the formidable task that lies before the philosopher who would penetrate the inner secrets of the solid condition of matter, let me quote a few words from Maxwell. " The stress [in a solid body] at any given instant," he says, "depends not only on the strain at that instant, but on the previous history of the body. Thus the stress is somewhat greater when the strain is increasing than when it is diminishing, and if the strain is continued for a long time, the body, when left to itself, does not at once return to its original shape, but appears to have taken a set, which, however, is not a permanent set, for the body slowly creeps back towards its original shape with a motion which may be observed to go on for hours and even weeks after the body is left to itself. . . . The phenomena are most easily observed by twisting a fine wire suspended from a fixed sup- port, and having a small mirror suspended from the lower end, the position of which can be observed in the usual way by means of a telescope and scale. If the lower end of the wire is turned round through an angle not too great, and then left to itself, the mirror makes oscillations, the extent of which may be read off on the scale. These oscillations decay much more rapidly than if the only retarding force were the resistance of the air, showing that the force of torsion in the ARRANGEMENT OF MOLECULES IN SOLIDS. 103 wire must be greater when the twist is increasing than when it is diminishing. This is the phenomenon described by Sir W. Thomson under the name of the viscosity of elastic solids. But we may also determine the middle point of these oscilla- tions, or the point of temporary equilibrium when the oscilla- tions have subsided, and trace the variations of its position. If we begin by keeping the wire twisted, say for a minute or an hour, and then leave it to itself, we find that the point of temporary equilibrium is displaced in the direction of twist- ing, and that this displacement is greater the longer the wire has been kept twisted. But this displacement of the point of equilibrium is not of the nature of a permanent set, for the wire, if left to itself, creeps back towards its original position, but always slower and slower. This slow motion has been observed by the writer going on for more than a week, and he has also found that if the wire were set in vibration the motion of the point of equilibrium was more rapid than when the wire was not in vibration. We may produce a very com- plicated series of motions of the lower end of the wire by previously subjecting the wire to a series of twists. For instance, we may first twist it in the positive direction, and keep it twisted for a day, then in the negative direction for an hour, and then in the positive direction for a minute. When the wire is left to itself the displacement, at first posi- tive, becomes negative in a few seconds, and this negative displacement increases for some time. It then diminishes, and the displacement becomes positive, and lasts a longer time, till it, too, finally dies away."* I think I need not add anything to the passage I have just quoted. The facts to be explained by the molecular theory of solids are certainly for- bidding enough, and it is not strange that so little progress has been made. Arrangement of the Molecules in Solids. The most obvious property of a solid is, that it preserves its shape so * Encyclopaedia Britannica, article Constitution of Bodies. 104 THE MOLECULAR THEORY OF MATTER. long as it is not acted upon by external forces. Moreover, when such forces are applied, the solid indeed becomes deformed, but it eventually regains its original shape after the forces have been removed, provided they did not exceed a certain magnitude called the " elastic limit," which is peculiar to the solid under examination and to the way in which the forces were applied. We are obliged to conclude, from these facts, that the molecules of a solid are not free to roam about, but that some or all of them have determinate mean positions about which they may oscillate and rotate, but from which they never permanently depart except when constrained to do so by an external force greater than whatever the forces may be that determine the mean positions. In order to gain a clearer insight into the nature of solids, let us now regard them from a somewhat different point of view let us con- sider the transition of a liquid into a solid by congelation. It is well known that when a liquid is cooled to a certain point peculiar to itself, it solidifies, changing its volume at the same time and giving out a certain amount of heat, called the "latent heat of fusion." We are therefore impelled to believe that when the average kinetic energy of translation of the liquid molecules is reduced to a certain value by the abstraction of heat, the inter-molecular attractive forces become able to restrict the migrations of the molecules, and to confine them, as I have already said, to certain mean positions. Furthermore, since energy cannot be created or destroyed, the fact that the body gives up a certain amount of " latent heat " when it solidifies must necessarily imply that there has been a diminution in the kinetic energy of the molecules, or in their potential energy, or in both. Now there undoubtedly is some loss of kinetic energy when a liquid solidifies, but I think we can assert that there is a loss of potential energy also, just as there is when a gas condenses into a liquid. If all liquids contracted upon solidifying it would be plain that potential energy disappears during the process ; for contraction would imply that the molecules ARRANGEMENT OF MOLECULES IN SOLIDS. 105 approach one another. The fact is, however, that many liquids expand upon solidification, water being a familiar example of this. Hence we are called upon to explain how it can be that the potential energy in a body can grow less when the molecules of the body go farther apart. I believe that this question has a close bearing on the constitution of molecules a subject of which we know practically nothing, at present. If you will glance at these diagrams, however, (Figs. 38 and 39,) I think you will see that the condition of FIGS. 38 and 39. ILLUSTRATING MINIMUM POTENTIAL ENERGY AND MINIMUM VOLUME. least potential energy in a system of bodies does not neces- sarily coincide with that of closest approach of the centers of those bodies. The diagrams are not intended to represent molecules in any sense, but they may serve to illustrate the point under consideration. I have assumed that each of these ideal bodies has four attractive poles, which are represented by the black spots. The centers of attraction, or poles, of such bodies would tend to approach one another as closely as possible, and the potential energy of the system would be least when the poles were nearest together that is, when the arrangement was like that shown in Fig. 38. If the bodies were placed as shown in Fig. 39 the total volume would be least, but the potential energy in that case would be greater than in the arrangement shown in Fig. 38, because 106 THE MOLECULAR THEORY OF MATTER. the attractive poles are further apart. It is plain, therefore, that the potential energy of a system of bodies is not neces- sarily least when the space occupied by the system is least. In other words, the equipotential surfaces of molecules are not necessarily spherical. This conclusion, derived from a consideration of the phenomena of congelation, is confirmed by a study of the physical properties of crystals with unequal axes. Maxwell's Views Concerning the Molecular Constitution of Solids. In speaking of the classification of bodies under the three heads of solids, liquids, and gases, I said, in the early part of the evening, that such a division is not entirely satis- factory ; and we have now come to a point where we must examine this classification more carefully. Many bodies are undoubtedly solids, and many others are undoubtedly liquids ; but there are bodies, such as wax, pitch, and tar, which possess some of the properties of each, and which are quite difficult to classify satisfactorily. A mass of pitch, for example, may be brittle, so that it is easily shattered by a blow, and yet this same mass, when placed on an inclined plane, loses its shape and flows slowly down the plane. It may not reach the bottom for months, or perhaps years ; but a true solid would remain where it was placed, for all eternity.* To account for the properties of pitch-like bodies, and for the peculiar behavior of wires and other solids when submitted to strains, Maxwell has proposed the following hypothesis: "We know that the molecules of all bodies are in motion. In gases and liquids the motion is such that there is nothing to prevent any molecule from passing from any part of the mass to any other part ; but in solids we must suppose that some, at least, of the molecules merely oscillate about a certain mean position, so that, if we consider a certain group of molecules, its configuration is never very different * It is assumed, of course, that the inclination of the plane is not great enough for either body to slide upon it. THE MOLECULAR CONSTITUTION OF SOLIDS. 107 from a certain stable configuration, about which it oscillates. This will be the case even when the solid is in a state of strain, provided the amplitude of the oscillations does not exceed a certain limit, but if it exceeds this limit the group does not tend to return to its former configuration, but begins to oscillate about a new configuration of stability, the strain in which is either zero, or at least less than in the original configuration. The condition of this breaking up of a con- figuration must depend partly on the amplitude of the oscil- lations, and partly on the amount of strain in the original configuration ; and we may suppose that different groups of molecules, even in a homogeneous solid, are not in similar circumstances in this respect. Thus we may suppose that in a certain number of groups the ordinary agitation of the molecules is liable to accumulate so much that every now and then the configuration of one of the groups breaks up, and this whether it is in a state of strain or not. We may in this case assume that in every second a certain proportion of the groups break up, and assume configurations corresponding to a strain uniform in all directions. If all the groups were of this kind, the medium would be a viscous fluid. But we may suppose that there are other groups, the configuration of which is so stable that they will not break up under the ordinary agitation of the molecules unless the average strain exceeds a certain limit, and this limit may be different for different systems of these groups. Now if such groups of greater stability are disseminated through the substance in such abundance as to build up a solid framework, the substance will be a solid, which will not be permanently deformed except by a stress greater than a certain given stress. But if the solid also contains groups of smaller stability and also groups of the first kind which break up of themselves, then when a strain is applied the resistance to it will gradually diminish as the groups of the first kind break up, and this will go on till the stress is reduced to that due to the more permanent groups. If the body is now left to itself, it will 108 THE MOLECULAR THEORY OF MATTER. not at once return to its original form, but will only do so when the groups of the first kind have broken up so often as to get back to their original state of strain. This view of the constitution of a solid, as consisting of groups of molecules some of which are in different circumstances from others, also helps to explain the state of the solid after a permanent deformation has been given to it. In this case some of the less stable groups have broken up and assumed new configura- tions, but it is quite possible that others, more stable, may still retain their original configurations, so that the form of the body is determined by the equilibrium between these two sets of groups ; but if, on account of rise of temperature, increase of moisture, violent vibration, or any other cause, the breaking up of the less stable groups is facilitated, the more stable groups may again assert their sway, and tend to restore the body to the shape it had before its deformation.' 77 * I have quoted Maxwell's words at length, both because he has stated his views with remarkable lucidity, and because his hypothesis, so far as I know, is the only one yet proposed that is at all adequate to explain the complicated phenomena exhibited by solid bodies. Even here we should proceed with caution, however, for the hypothesis that he offers us, pro- found and beautiful as it is, cannot logically be regarded as anything more than a hypothesis until its consequences have been worked out mathematically, and compared with the facts. % Sublimation. Some solids exhibit a phenomenon quite analogous to evaporation, to which phenomenon the name of sublimation has been given. " There are doubtless frequent collisions among the molecules of solids, preciselyas there are among those of liquids and gases. "Such being the case, we must conclude that the velocities of oscillation and rotation of the molecules of solids are not all equal ; and doubtless there is also some undiscovered law of distribution of these * Encyclopaedia Britannica, article Constitution of Bodies. SUBLIMATION. 109 velocities, analogous to Maxwell's law for gases. If this be admitted, we may infer that every now and then a molecule on the exterior of a solid will have an impetus sufficient to overcome the restraining forces, and to tear itself away and escape from the solid, just as molecules escape from a liquid in evaporation. "With some solids, such as camphor, this action is very noticeable. Ice also evaporates (or sublimes), in dry air, at temperatures below the melting point. These two substances, as well as others, even have a definite vapor pressure corresponding to each temperature, just as liquids do when placed in closed vessels. The reason for this definite relation between temperature and vapor pressure need not be repeated, for it is precisely the same as I have already given you in speaking of evaporation from liquids. ' Many solids, such as iron and stone, do not sensibly sublime at any tem- perature, though metals and argillaceous earths and other non-volatile solids often have characteristic odors which may possibly be due to a slight loss of substance by sublimation. Other solids, such as arsenic, sublime freely at elevated tem- peratures, mercury bichloride, or " corrosive sublimate/ 7 taking its popular name from this property. There are reasons for believing that even carbon is slightly volatile at high tem- peratures at least under the conditions to which it is exposed in the bulbs of incandescent electric lamps. "* It must be admitted that solids, in general, do not sublime as freely as one might naturally expect them to ; and the reason for this probably is, that the intermolecular forces in these bodies are so enormous that it is seldom that a molecule happens to have a velocity great enough to enable it to fly away. (That the attractive forces between neighboring molecules are great, in solids, is shown by the tensile strength of these bodies.) When the intermolecular forces in solids are counterbalanced in some degree by other forces, acting in opposition to them, there is often a loss of molecules from the solid, and the resulting phenomena correspond, to some extent, with those of sublimation. Examples of this are afforded by the phenom- 110 THE MOLECULAR THEORY OF MATTER. enon known as solution, and also by that known as dissociation. We shall consider dissociation first. Dissociation. If steam be passed through a tube contain- ing red-hot iron-filings, the steam is decomposed, black oxide of iron is formed, and hydrogen escapes from the free end of the tube. On the other hand, if hydrogen be passed through a red-hot tube containing black oxide of iron, the oxide is reduced to metallic iron, the liberated oxygen combines with the hydrogen, and steam escapes from the free end of the tube. These apparently contradictory phenomena were first adequately explained by Deville ; and in order to understand his explanation properly, let us first consider what forces are involved in the problem. (1) The hydrogen molecule being double, we have to consider the attraction of its two compo- nent atoms for each other. (2) We have likewise to consider the attraction that the atoms composing an iron molecule exert upon one another, and also (3) the attractions existing among the atoms of a steam molecule, and (4) those existing among the atoms of a molecule of the oxide of iron. Let us now return to the first experiment, in which steam is passed over red-hot iron. In this case the motions of the steam molecules are accelerated by the heat to s-uch a degree that some of them are torn asunder, and oxygen and hydrogen atoms are liberated, and mingle with the molecules of steam. If no iron were present, these dissociated atoms of oxygen and hydrogen would occasionally chance to collide with one another again, and in some of these cases the speed of the colliding atoms would not be great enough to prevent them from recombining and forming new molecules of steam. The steam-molecules are therefore always dissociating and re-form- ing, and it is plain that at any given instant the proportion of such molecules that are in a state of dissociation will be greater, the higher the average speed of the steam-molecules that is, the higher the temperature. As a matter of fact, however, some of the liberated oxygen atoms come in contact DISSOCIATION. Ill with the iron that is present, and combine with it to form the black oxide ; and the hydrogen atoms that were formerly their partners pass on with the steam. In the second experi- ment the phenomena are very similar. The heat partially dissociates the iron oxide, the oxygen atoms thus- liberated find partners among the hydrogen atoms flowing overhead, and the steam thus formed is swept away in the stream of hydrogen, and is prevented from again coming in contact with the reduced iron that has been left behind. Now let us con- sider the experimental tube to be stopped up at both ends, so that nothing can enter it or leave it. All the various phenom- ena I have described will then take place simultaneously, and a permanent distribution will presently be reached, in which molecules of iron oxide are dissociating in some places just as fast as molecules of oxygen are combining with metallic iron in other places, and molecules of steam are dissociating just as fast as other molecules of steam are forming. When this condition of equilibrium is established, the composition of the contents of the tube will appear to be constant ; but the constancy is only apparent it is of a statistical nature, just as the constancy of the vapor-density over a liquid is statistical. If the distribution of the various substances in the experimental tube is altered, the balance will be destroyed. For example, if some of the hydrogen is removed, more steam will be dissociated and more iron oxide formed, until the proportion of hydrogen to steam becomes the same as before. Conversely, if some of the steam is removed the iron oxide will be dissociated faster than it is formed, the oxygen thus liberated will combine with some of the hydrogen, and more steam will be formed ; and this readjustment will continue until the original proportion of hydrogen to steam is again restored. Hence it is plain that if the steam that is formed is continually swept away by a current of hydrogen, the result will be that the oxide of iron will be all ultimately reduced ; while if the hydrogen that is liberated is swept away by a current of steam, the iron will be continuously oxidized, until 112 THE MOLECULAR THEORY OF MATTER. it is all converted into oxide. This beautiful explanation of the phenomena is due to Deville, and the principles that underlie his theory have been applied to a wide range of other phenomena, and upon them a complete theory of chemical equilibrium has been erected. I regret that time will not allow me to discuss this most interesting subject further, nor even to give more examples of dissociation ; but you will find Guldberg and Waage's generalized theory of chemical equi- librium given in Ostwald's Outlines of General Chemistry. Solutions.* Many solids, when brought into contact with certain liquids under suitable conditions, cease to exist as solids, and diffuse throughout the liquid in which they are submerged. This process, by which the solid is caused to disappear, is called solution; and the same word is also used, as a noun, to signify the liquid that results from the process. In some cases the liquid acts chemically upon the solid, form- ing a new compound, which remains behind when the solvent is evaporated an instance of such action being the formation of chloride of sodium when caustic soda is placed in hydro- chloric acid. We shall not discuss these cases, but shall con- fine our attention exclusively to those in which the solvent is not positively known to act chemically upon the dissolved substance as when sugar is dissolved in water, or silver chloride in dilute ammonia. In such instances we have to think of the molecules of the liquid as exerting a certain attraction on the external molecules of the solid ; though we * In accordance with the general plan of this book, I have included, in the present section, only such points as seem to be well established, and to have an immediate bearing on the molecular structure or deport- ment of bodies. Our knowledge of solutions is still far from perfect, and the reader will find that chemists are divided, upon this subject, into two great schools or factions. I will not pretend to decide between these factions. The reader may profitably consult the article on Solutions in Watts's Dictionary of Chemistry (last edition, 1894), where he will find the rival doctrines ably expounded. Ostwald's Solutions (London, Long- mans, Green & Co., 1891) can also be recommended. SOLUTIONS. 113 shall be careful not to say much about the ultimate nature of this attraction, lest we might give offense to the advocates of one or other of the two schools into which, on this point, chemists are divided. If the attraction of the liquid for the molecules of the solid is slight, the attraction of the molecules of the solid for one another may still greatly preponderate, and in that case there will be no solution, and we shall have merely a solid submerged in a liquid. On the other hand, if the external molecules of the solid are attracted outward by the liquid more powerfully than they are attracted inward by the solid itself, they will be detached, and the solid will rapidly dissolve. In case the attraction towards the liquid is less than that towards the solid, but yet comparable with it, particles will tear themselves loose from the solid and pass into solution, whenever their velocities chance to be great enough to enable them to overcome the excess of attraction towards the solid. In some cases heat will be evolved during the process of solution, and in other cases it will disappear, or become latent ; and we cannot say, in advance, which of these phenomena will take place in any given instance, until we fully understand the nature and magnitude of the various forces that are called into operation. I believe that many of the phenomena of solutions, as well as those of crystallization and other changes of state in bodies, can be explained by means of the single assumption that in every case the mole- cules of the system tend towards that configuration in which their united potential energy is as small as possible, just as a marble when thrown into a wash-bowl tends continually towards the bottom of the bowl, where its potential energy is least. Thus, when we find that a given solid dissolves in a given liquid, we are to infer that the potential energy of the system is less when the solid is in solution than it is when the solid is intact, and merely submerged in the pure liquid ; and when we find that a given solid does not dissolve in a given liquid, the converse is to be inferred. I can imagine that you will be ready with the objection that if the potential 114 THE MOLECULAR THEORY OF MATTER. energy diminishes, the kinetic energy must increase ; and that this would imply that solution is always accompanied by y a rise in temperature, which is by no means the fact. You must bear in mind, however, that the properties of solids show that the motions of the molecules of these bodies are restricted in some way, and that the process of solution undoubtedly removes these restrictions, so ' that when the molecules of the solid mingle with those of the liquid, they acquire a certain additional number of " degrees of freedom." Hence, although the diminution of potential energy certainly does imply an increase in the total kinetic energy, it does not by any means imply an increase in the kinetic energy of translation. That is, it does not of necessity imply an increase in temperature. Diffusion. We may apply this same principle of minimum potential energy in discussing the diffusion of a dissolved substance through the solvent. Suppose, for example, that we have a solution in which the concentration is different in different regions ; then the potential energy of the system, per unit volume, must also be different in different regions ; and hence the molecules of the dissolved solid will tend towards the more dilute parts or towards the more concen- trated parts, according as the potential energy of the system would be lessened by the one process or by the other. In fact it can be easily shown, by mathematical reasoning, that if the temperature of the solution be maintained constant at all points (so as to avoid thermodynamical considerations), the potential energy of the system will necessarily be least when the concentration is everywhere the same that is, when the solution is homogeneous. To prove this, let us assume that at the outset the solution proposed for considera- tion has different degrees of concentration in different parts, and let us conceive it to be divided by imaginary surfaces into a great number of elementary portions, whose volumes are v l} v zj v s , . . . Furthermore, let us represent the quantity of dis- DIFFUSION. 115 solved substance, per unit volume, in these respective elemen- tary portions, by c b c 2 , c s , . . . , and the corresponding potential energies by p lt p 2) p s , ... Then, representing the total quantity of solid in solution by Q, and the total potential energy of the system by P, we have Q = Wi + c 2 v 2 + c s v s +... (42) and P =p 1 v 1 + p 2 v 2 + p s v s + . . . (43) We are required to discover the relations that must exist among the c's, in order that P shall be as small as it can be, consistently with the condition (42). It is shown, in works on the differential calculus, that such problems can be solved by differentiating the expression P-kQ (44) with respect to each of the variables .0000054 cm., I being the thick- ness in question; with a wedge of sulphide of silver, and mercury as the liquid, he found 1= . 0000048 cm.; with mercury and a wedge of iodide of silver, 1 = . 0000059 cm.; and with mercury and a wedge of collodion, I < .0000080 cm. M. Quincke concludes from his experiments that one may adopt, as the mean value of the radius of sensible molecular attraction in these cases, l = . 0000050 cm." Other Methods of Investigation. Many other ingenious methods have been devised for determining the sizes of mole- cules and the radius of sensible molecular attraction, but I can consider only a few of them. First of all, we must not omit to mention the result obtained by Plateau himself. He OTHEK METHODS OF INVESTIGATION. 147 found, from a study of soap-bubbles, that the limiting thick- ness of a film of his glycerine liquid that is, the thickness .at which the film becomes unstable is about .0000114 cm.; and as he believed that the film would be unstable when its thickness was reduced to twice the radius of sensible molecular attraction, he concluded that the value of this radius, in the liquid composing his bubbles, was .0000057 cm. Maxwell, however, has endeavored to show that a liquid film is stable until its thickness is reduced, not to double the radius of attraction, but to that radius itself. If his theory is correct, we must conclude from Plateau's data that the radius of sensible attraction, in the glycerine liquid, is .0000114 cm. More recently the limiting thickness of soap-films has been accurately measured by Eeinold and Eticker, who find it to be about .0000012 cm. Oberbeck found that a coating of metal .0000003 cm. thick is sufficient to polarize platinum that is, this thickness is sufficient to alter the character of the surface of platinum so that a distinct difference in electrical potential is observed between a plate so prepared and another plate of the pure metal. Lord Kelvin found that " a quite infinitesimal whiff of iodine vapor " is sufficient to alter the surface of a silver plate so that a difference in potential is observable ; but I do not know that he has given any numerical estimate of the thickness of the resulting film of iodide of silver. Wiener found that a film of silver .000.000,02 cm. thick produces a sensible effect on the phase of reflected light. He also found that when a silver plate is reduced in thickness to .0000012 cm., it no longer produces the same effect on the phase of reflected light that a thick plate does. There is no very close agreement among these various results, nor could we reasonably expect that there would be, when the quantities measured are so imperfectly defined, and the methods and substances used are so different. Nevertheless, the results are all of the same general order of magnitude, and when taken in connection with those obtained by the methods that I have already described, they suffice to 148 THE MOLECULAR THEOHY OF MATTER. give us a very good general idea of the sizes of molecules, and of the distance at which molecular forces cease to be sensible. Number of Molecules in a Unit Volume of Gas. Accord- ing to Avogadro's law, all gases contain the same number of molecules per unit volume, when they are subject to the same conditions ; and it will be interesting to calculate this number for some given temperature and pressure. If we continue to regard the molecules of a gas as spherical, the volume of a single molecule is 7r8 8 ; and if -ZV is the number of them in a cubic centimeter of the gas, the combined volume of all the molecules in this cubic centimeter will be NvS 8 . But this is what we have called the " aggregate molecular volume," and have denoted by e. Hence or = (55) By substituting in this equation the values of e and 8 that we have found for various gases, we obtain the values of N given in the table. NUMBER OF MOLECULES IN A UNIT VOLUME OF GAS, AT 0C., AND ATMOSPHERIC PRESSURE. GAS. NUMBER OF MOLECULES. In a Cubic Centimeter. In a Cubic Inch. Hydrogen Nitrogen 6.1X10 18 5.5X10 18 6.2X10" 23.2X10 18 95X10 18 86X10 18 97 X 10 18 363 x 10i 8 Oxygen Carbon dioxide . . These numbers are as nearly equal as could well be expected. It must be remembered that in the determination of N we have the accumulated effects of all the errors that have been ILLUSTRATIONS OF MOLECULAR MAGNITUDES. 149 committed in obtaining the coefficients of viscosity, the free paths, the " aggregate molecular volumes/' and the molecular diameters. The values of ^V for hydrogen, nitrogen, and oxygen agree among themselves remarkably, but the value for carbon dioxide is apparently too large. You will observe that in determining e I have used Mr. Sutherland's data for the first three gases, and Clausius's data for the last one. If we take Mr. Sutherland's results for all the gases, we should have, for carbon dioxide, = .00117,* which is about sixty per cent greater than the value given by Clausius's equation (30). By using Mr. Sutherland's value of e in equation (53) we find 8 = 63 X 10~ 9 cm. ; and a further substitution, in (55), gives N= 8.9 X 10 18 * as the number of molecules in a cubic centimeter of the gas, under the stated conditions. This agrees much better with the results given in the table for hydrogen, nitrogen, and oxygen ; and, as these four values of JVhave been deduced from independent observations, I think we may reasonably conclude that Clausius's theorem will give us a fairly accurate knowledge of molecular diameters, and hence also of N, when the mathematical investigation of the kinetic theory of gases has been carried far enough to enable us to construct a theoretically perfect gas equation, from which to deduce accurate values of e. At present I think we can only say that at C. and under one atmosphere pressure, one cubic centimeter of gas contains something like 7,000,- 000,000,000,000,000 molecules. This corresponds to about 110,000,000,000,000,000,000 per cubic inch. Illustrations of Molecular Magnitudes. I have given you a lot of numbers with strings of ciphers attached to them, sometimes running away off to the right, and at other times off to the left. I should like, now, to give you a few homely * See the Philosophical Magazine for March, 1893, page 232. The quantity that I have denoted by e is found by dividing Mr. Sutherland's ft by the volume of a gramme of the corresponding gas, measured at C. and one atmosphere, and taking f of the quotient. In the case of CO 2 we have c =f X (0.692 -j- 505) =.00117. 150 THE MOLECULAR THEORY OF MATTER. illustrations of what these numbers mean. We have found that in gaseous bodies the molecules are moving with speeds comparable with that of a cannon ball indeed, we found that the average velocity of a hydrogen molecule, at C., materially exceeds a mile a second, which speed has never been attained with the best modern artillery. We have found that the average distance that a gas-molecule travels, under ordinary circumstances, between successive collisions with its fellows, is only a few millionths of an inch, a distance far too small to be seen with certainty even by the most powerful microscopes. The number of collisions experienced by a single molecule of gas in one second was also found to be very great. Every nitrogen molecule, for example, is struck by its fellows, under the standard conditions, something over 5,000,- 000,000 times per second. It would take you fifty-three years to count that many, if you should count three every second, and twenty-four hours every day. We have found that molecules are amazingly small so small,. for example, that if every man, woman, and child in this world were to lay down a molecule of carbon dioxide so that all these molecules should lie in a straight line, and each should touch its neigh- bors, the row thus formed would hardly be more than a yard long.* A cube formed of a hundred thousand million mole- cules of hydrogen in contact with one another would hardly be visible in the finest microscopes that we have. The number of molecules in a cubic inch of gas at the freezing point and under atmospheric pressure is so great, that not- withstanding the fact that it would take about 55,000,000 hydrogen molecules to make a row one inch long, and that only one three-thousandth of the volume of the gas is really filled by matter (the rest being vacuous space around the molecules), if all the molecules that exist in a cubic inch of this gas were placed in a row, touching one another, they would form a line about 32,000,000 miles long, or long enough * According to the Statesman's Year-Book^ the population of the world, in 1890, was about 1,468,000,000. THE CONSTITUTION OF MOLECULES. 151 to wind around the earth more than a thousand times. If the molecules contained in a cube whose edge is one inch and a half were similarly arranged, they would reach from the earth to the sun. To state the same thing in still another way, let me say that if the molecules in a cubic inch of gas (under the assumed conditions of temperature and pressure) were spread out uniformly so as to form a single sheet or layer, and if they were distributed so that their average distance from center to center was about the same as the corresponding distance between the letters on this page, then the layer in question would cover all the six continents of the earth, six times over. It would be easy to investigate the speed of rotation of molecules in gases, by the aid of Boltzmann's theorem, and to show, by making certain assumptions about their shapes, that hydrogen molecules (for example) rotate so swiftly that they perform one entire revolution while light, with its prodigious velocity of 186,000 miles per second, is traveling a few millionths of an inch. It might also be interesting to speculate on the tensile strength of a substance that could hold together while rotating with so fearful a speed. I have not discussed these points, however, for I was afraid that if I should do so, I could not help conveying the impression that we know far more about them than we do. Some day it may be profitable to theorize about these things, but certainly the time is not yet come. VI. THE CONSTITUTION OF MOLECULES. Preliminary Remarks. We are now about to leave the field of knowledge altogether, and to enter the vast domain of speculation ; for I think I can safely say that at present we do not possess the least particle of positive information con- cerning the constitution of molecules. I have even had doubts about the propriety of discussing the subject at all, so profound is our ignorance of it ; but upon more mature reflection it seemed hardly proper to omit all mention of the 152 THE MOLECULAR THEORY OF MATTER. views that have been advanced by physicists from time to time, especially as we have made certain tacit assumptions about the constitution of molecules in the preceding sections assumptions whose uncertainty should at least be pointed out. The chief difficulty before the philosopher who would investigate the structure of molecules lies in the fact that it is impossible to see or measure a molecule directly. We know them, not individually, but only in the aggregate ; and we can infer their structure only by observing the gross results of their interaction in large numbers. Our situation may be best illustrated, perhaps, by imagining a huge being who could observe only the general trend of events upon the earth the march of civilization and the rise and fall of empires and who desired to infer from these data the anatomy and mental characteristics of the invisible creatures whose indi- vidual acts, when summed up, had given rise to these phenomena. It is obvious that in either case the problem is of enormous difficulty, and that, to a certain extent, it may prove to be indeterminate. "VVe must proceed by making various assumptions about the constitution of molecules, and we must then rigorously follow out the results of these assumptions and compare them with the facts. Those that are found to disagree with the facts will then be rejected, and the others will be tentatively retained until new and crucial facts may be discovered, which shall make a further rejection possible. By such a process of exclusion we may hope to arrive, some day, at a fair knowledge of the structure of molecules ; but the progress of the molecular theory in this direction is likely to be extremely slow, on account of the mathematical difficulties that continually present themselves, and which can be overcome only by a great amount -of patient labor. General Facts to be Explained. The successful molecular theory or " final theory," as we may call it, must explain a vast range of phenomena, extending all the way from those GENERAL FACTS TO BE EXPLAINED. 153 that thrust themselves upon us in our everyday life to those more recondite ones that can be observed only in the labora- tory, under specially favorable conditions. Some physicists demand that the final theory shall explain even the inertia of matter ; but it seems to me that this is asking too much, and that such a demand implies an imperfect understanding of what " explanation " means. To " explain " a fact is to show that it is a necessary consequence of other facts that are more readily grasped by the human understanding ; and as inertia is perhaps the most fundamental of the known properties of matter, it seems as though any attempted explanation of it would only involve us in a sea of words and ink, without enlightening us in the least. Elasticity, however, is certainly capable of explanation. When it is manifested by bodies of sensible size, we refer it to the interaction of the molecules of which the bodies are composed ; but when it is manifested by the molecules themselves, some further explanation is required. I have told you that molecules are perfectly elastic ; but I have merely postulated the elasticity, and have not attempted to explain it. We shall see, presently, that Lord Kelvin's vortex theory affords a mechanical explanation of the elasticity of molecules, and although I should not like to say that his explanation is the correct one, it proves to us, at least, that .such an explanation is possible. The final molecular theory must unravel the mysteries of chemical and physical attraction, and must also show why an atom of A has a powerful affinity for an atom of B, but is comparatively indifferent towards an atom of C. It must unify the conceptions of chemistry and physics, and consolidate these sciences into one grand Science of Matter. It will also have to explain the possibility of such enormous and comparatively stable aggregates of atoms as occur in proteid bodies and gums, whose molecular weights, in some instances, are believed to be as high as 13,000 or 14,000. These are formidable things to require of an infant theory, and yet no theory which does not cover them all can possibly hope for final acceptance. It may be well to indicate 154 THE MOLECULAR THEORY OF MATTER. some of the general facts of chemistry and physics that are likely to serve as stepping-stones towards the great theory of matter that future generations will undoubtedly evolve, and I shall therefore refer briefly to Dulong and Petit's law, to Prout's hypothesis, to the periodic law of Mendeleieff and Meyer, and to the theory of radiation. Dulong and Petit's Law. In the early part of the present century two distinguished French physicists, MM. Dulong and Petit, announced that the specific heats of certain elements upon which they had experimented are inversely proportional to the respective atomic weights of those elements, and from this they concluded that -the amount of heat required to raise the temperature of N atoms by 1 is the same for all the elementary bodies. This remarkable generalization did not meet with universal and immediate acceptance, because it failed in numerous cases* unless the atomic weights of the corresponding elements were changed somewhat from the values that had been assigned to them from purely chemical considerations. Moreover, it could not possibly be an exact law, because the specific heats of bodies are not constant, but vary with the temperature, and some- times to a considerable extent. Subsequent experimenters have paid great attention to Dulong and Petit's law, however, and now that the atomic weights of the more familiar elements have been pretty well determined, the law is found to be surprisingly near the truth. Dr. L. Meyer gives a list of fifty elements which accord with it very well, and I have selected ten of them, almost at random, to show you its wide applicability, and the order of its accuracy.* The atomic weights in these elements range from 7 to 240, and yet when we multiply each of them by the corresponding specific heat we find that the product remains constant, or nearly so. Furthermore, the small variations that do occur do not appear * Dr. Lothar Meyer, Modern Theories of Chemistry (London, Long- mans, Green & Company, 1888), page 73. DULONG AND PETIT S LAW. 155 to follow any particular law, but have rather the character of "errors of observation"; and although they are quite too large to be attributed to this cause, their irregularity forces ILLUSTRATIONS OF DULONG AND PETIT' s LAW. S S p 1 w H e ELEMENT. Is 1 1 ELEMENT. H S | | Q ^ & " PH < CO PH Lithium . 7.0 .941 6.6 Antimony . . 120 .0508 6.1 Aluminium . . 27.0 .214 5.8 Tungsten . . 184 .0334 6.1 Potassium . . 39.0 .166 6.5 Gold . . . 196 .0324 6.4 Copper . . . 63.2 .0952 6.0 Bismuth . . 207 .0308 6.4 Silver . . . 107.7 .0570 6.1 Uranium . . 240 .0277 6.6 upon us the conviction that Dulong and Petit's law is not a- mere " first approximation " to the relation between specific heat and atomic weight, but that it expresses that relation accurately, and that the outstanding differences are due to causes which are not functions of the atomic weights alone. Dulong and Petit's law has been thought, by some, to indicate that matter, in the last analysis, is of only one kind. To my own mind the law seems to controvert this hypothesis, rather than to sustain it ; for if there is really but one ultimate kind of matter, we might indeed expect to find the specific heats of bodies equal, but I think we should hardly look for them to be inversely proportional to the molecular weight. Dulong and Petit's law seems rather to be an experimental indication that some modification of Boltzmann's law of the partition of kinetic energy in the molecules of gases will also be found to be true of the molecules of liquids and solids. To illustrate, let us consider a gas in which the effects of intermolecular attraction are insensible. Maxwell has shown that if two* such gases have the same temperature, their kinetic energies.; of translation (per molecule) will also be equal [page 35]i. Since this is true for any temperature, it follows that if the two gases be raised through the same range of temperature 156 THE MOLECULAR THEORY OF MATTER. (say from ti to 2 ), the increase in the average kinetic energy of translation will be the same in each case. For the sake of clearness let us name the two gases "A" and "JB," respect- ively ; and let us denote by k the average increase in kinetic energy of translation experienced by a molecule of either gas when the temperature is raised from t to (-|-l) . Kow if n-L and n 2 are the number of degrees of freedom of the mole- cules of A and B, respectively, then the total increase in kinetic energy of N molecules, when the temperature is raised from t to (< + l), will be ^ Nfc and ^ Nk, o o respectively. But since we assume that the change in potential energy is insensible, it follows that these quantities are equal to s^W^ and s 2 W 2 , respectively, where s is the specific heat of the gas, and W is the weight of N molecules. Hence, ^n l Nk = s 1 W l) and %n 2 Nk = s 2 W 2 . (56) Now although we do not know N with any considerable approach to accuracy, we do know that and W where iv and w 2 are the weights of a molecule of A and B, respectively. Substituting these values of TFi and W 2 in (56) and dividing by N, we have ^n i k = s l Wi and ^ n 2 k = s 2 w 2 . . Hence, finally, S 1 W l _S 2 W 2 - \ (ot ) W! n 2 and this, I imagine, is the true form of Dulong "and Petit's law. As the absolute values of w are not known with pre- cision, we may advantageously use, instead of them, those relative values which chemists call the "molecular weights." ^Equation (57) may be written sw = Cn } (58) DULONG AND PETIT S LAW. 157 where s represents specific heat, w represents molecular weight, n is the number of degrees of freedom of a molecule, and C is a constant. You will notice that this differs from Dulong and Petit's law, as ordinarily stated, in the intro- duction of n in the second member. It may be interesting to determine C by examining some of those gases and vapors for which n has been determined by means of equation (22). DETERMINATION OF C. SUBSTANCE. SPECIFIC HEAT. MOLECULAR WEIGHT. PRODUCT. n. C. Mercury vapor . .".*.'..'*..' .015 200 3.00 3 1.00 Hydrogen 2.43 2 4.86 5 .97 Oxvsren .154 32 4.93 g .99 Nitrogen .173 28 4.84 5 .97 Carbon monoxide .... .173 28 4.84 5 .97 Hydrochloric acid vapor . .132 36 4.75 5 .95 Water vapor .371 18 6.68 6 1.11 Carbon dioxide .... .15 44 6.60 7 .94 The average of these values of C is 0.98 ; but 0.98 is so near to 1.00 that the data that we have are not sufficient to distinguish between the two with certainty. We may there- fore consider C to be unity, and may write equation (58) thus : sw = n. (59) Now if some law closely analogous to Boltzmann's holds true for solids, then Dulong and Petit's law might follow as a natural consequence. If we strain the point a little and apply Boltzmann's law itself to solids, we see that the constancy of the product of the specific heat and molecular weight would merely imply that the . molecules of the substances under discussion have the same number of degrees of freedom. We could even deduce the number of degrees of 158 THE MOLECULAR THEORY OF MATTER. freedom in any given case, by substituting in (59) the proper values of s and w, remembering (in case the body is an element) that w is the molecular weight, and not the atomic weight. The elements in Meyer's list would then appear to consist, for the most part, of molecules having 11, 12, or 13 degrees of freedom. Of course the explanation of Dulong and Petit's law that I am offering you is merely speculative at present ; but if it be accepted for the moment, then certain of the difficulties that beset other explanations of the law disappear. For example, it is probable, beforehand, that the molecules of all bodies do not have the same number of degrees of freedom ; and hence, in accordance with (59), we could not reasonably expect the product of the specific heat and the molecular weight to be always the same, but should rather expect that bodies would be divisible into classes in this respect, the product being the same throughout any one class, but different in different classes ; and this is the fact, provided the law be extended (as I think it should be) so as to include compounds.* Again, we can understand why the law is not exact ; for Boltzmann's theorem (and presumably its analogue for solids also) relates only to the partition of kinetic energy among the various degrees of freedom of the molecules. The law of distribution of potential energy is doubtless quite different ; and as the specific heat of a body corresponds to the increase in its total energy, it follows that (59) will not be strictly true. The wide divergence from Dulong and Petit's law exhibited by certain bodies (such as boron) at some temperatures, and the comparatively good agreement of these bodies with it at other temperatures, may be due either to a change in the number of degrees of freedom of the molecule, or to a local variation in potential energy great enough to mask the effects of the equable distribution of kinetic energy. * Of course numerous attempts to so extend it have been made already ; but so far as I know, they have not been based on the considerations here presented. PROUT'S HYPOTHESIS. 159 Prout's Hypothesis. The idea that matter is not really of seventy kinds or so, but that it consists of only one funda- mental kind, is quite ancient ; but " in 1815, soon after Dalton's atomic theory had met with general recognition, Prout brought forward the view that the primordial matter of which all elements are composed is hydrogen, and that consequently the atomic weights of all the other elements are simple multiples of the atomic weight" of that substance.* This hypothesis has provoked much discussion, and since it was first proposed it has been attacked and defended by many distinguished chemists ; and although it is rather in disfavor at present, I think we cannot yet say that it has been finally laid to rest. One can hardly glance at a table of atomic weights without being impressed by the close approach of these quantities to integral values. Of course there are conspicuous exceptions chlorine, for example, to Front's hypothesis in its original form, and to reconcile these it has been assumed that the various elements are composed, not of hydrogen, but of some unknown and still simpler substance whose atomic weight is ^ or 1 that of hydrogen ; but this seems like a very artificial extension of the hypothesis, because by a further extension of the same kind we could easily account for any exceptions whatever. The fact that many of the atomic weights are nearly integral demands some sort of an explanation, however, for it can hardly be acci- dental. When chemical science was in a less developed condition it was easy to believe that the atomic weight of nitrogen (for example) is 14.00, instead of 14.02 as indicated by experiment, and that the atomic weight of carbon is 12.00 instead of 11.97 ; but we can 110 longer entertain any such hypothesis. This point was strongly emphasized by Stas's magnificent researches, for his results are apparently of such extraordinary accuracy that an error of one-tenth of one per cent is quite out of the question in them. " It is possible," says Dr. Lothar Meyer, " that the atoms of all or many of the * Meyer, Modern Theories of Chemistry, page 113. 160 THE MOLECULAR THEORY OF MATTER. elements chiefly consist of smaller particles of matter of one distinct primordial form, perhaps hydrogen, and that the weights of the atoms do not bear a simple relation to one another because the atoms contain, in addition to the particles of this primordial matter, varying quantities of the matter which fills space and is known as the luminif erous ether, which is perhaps not quite devoid of weight. This appears to be the only permissible hypothesis." Dr. Meyer's surmise may possibly be correct, although certain grave difficulties would have to be overcome before we could accept it. If you will bear in mind what I said a few moments ago about all these points being purely speculative, I will offer another hypothesis which may not be better than Dr. Meyer's, but which appears to be at least as good, and quite as defensible. There is no harm in letting one's fancy loose in this way, any more than there is in reading a fairy tale ; but it is of the first impor- tance, in either case, that we should carefully remember what we are doing, so that possibility may not be confused with probability. There is one point which is everywhere taken to be self-evident by writers on chemistry, but which is not so, to me, by any means. I cannot see what warrant there is for assuming that when an atom whose weight is A combines with another atom whose weight is B, the weight of the resulting molecule is universally and necessarily A + B. This principle, instead of being a truism, must receive a most exact explanation by the final molecular theory. It appears to be true in such reactions as we can observe, but as we have never split an element up into its constituent hydrogen-atoms (if indeed it contains such atoms !) there is no evidence that in such a case the " law of conservation of weight " would still hold true. When we know more about the nature of gravita- tion we shall be in a better position to discuss this point ; but at present I think we may say that it is just possible that there may be cases in which an atom of weight A, when com- bining with another of weight B, does not produce a molecule of weight A-\-B. I am well aware that this would make PERIODIC LAW OF MEYER AND MENDELEIEFF. 161 perpetual motion possible, for if the weight of the given substances happened to be greater in the combined state than in the uncombined one, we should only have to let them fall some convenient distance while they are combined, and raise them again while they are uncombined, and we should gain a little energy every time the cycle was repeated; while if combination should cause a loss of weight instead of a gain, we could attain the same end by performing the cycle in the opposite direction. Now I am sure that nobody has greater faith in the conservation of energy than I have, and yet we should remember that that grand principle, the discovery of which will cause the nineteenth century to be remembered forever, is nevertheless merely an abstraction from our experience; and that it teaches us nothing except that we have never known energy to be created or destroyed, and that with the means at our command we cannot create it nor destroy it. If it be true, therefore, that matter is composed of some fundamental substance combined with itself in varying degrees of complexity, then whenever the law of conservation of weight would be violated upon splitting a body up into its constituents, or in forming it from them, the means at our disposal can never enable us to effect either the separation or the combination ; and so far as we are concerned, such a body would forever remain an element. On the other hand, when- ever the law of conservation of weight would not be violated upon splitting a body up, the body in question is not an element, but a compound; and we can reasonably hope to effect its separation into two or more simpler bodies. This hypothesis explains both the existence of "elements," and the slight deviations from integral values that we find in their atomic weights. I offer it for what it is worth, and have nothing further to say in defense of it. Periodic Law of Meyer and Mendeleieff . It has long been known that certain approximate numerical relations exist among the atomic weights of elements that have similar 162 THE MOLECULAR THEORY OF MATTER. properties. For example, if we consider the three closely related elements lithium, sodium, and potassium, we find that the atomic weight of sodium is almost exactly the arithmetic mean between the other two. Thus we have Li = 7.01 and K = 39.02, the mean of which is 23.02 ; and the atomic weight of sodium is 23.00. Again, in the triad calcium, strontium, and barium, we have Ca = 39.99 and Ba = 136.76, the mean of which is 88.38 ; and the atomic weight of strontium is 87.37. The triad chlorine, bromine, and iodine, affords another familiar example, for we have Cl = 35.37 and I = 126.56, giving a mean of 80.96 ; and the atomic weight of bromine is 79.77. As I have said, these peculiarities, and numerous other similar ones, were known many years ago ; but we are still ignorant of their true meaning. In recent years a great multitude of relations of this sort among the elements have been brought to light, and two distinguished chemists, one a Russian and the other a German, have shown, by independent investigations, that if the elements are arranged in the order of their atomic weights, many of their properties recur, at intervals, in a sort of "periodic" manner, as we pass from one end of the array to the other. Tables have also been prepared, in which the elements are arranged in rows and columns in such a manner that their relations to one another can be plainly seen. I shall not dwell at length upon this " periodic law," because an adequate discussion of it would require a great deal of time. It has been set forth very simply and clearly, however, and also in considerable detail, in Meyer's Modern Theories of Chemistry, and in Mendeleieff's Principles of Chemistry. When the attempt was made to arrange the elements in tabular form, it was found necessary to leave certain spaces in the table vacant. It was easy to imagine that elements would some day be found, which would fit into these spaces ; but it was much more difficult to predict the exact properties that these hypothetical elements would have. Nevertheless, Mendeleieff undertook the task, and in several instances his predictions PERIODIC LAW OF MEYER AND MENDELEIEFF. 163 have been fully verified by subsequent discovery. One of the most striking instances of such verification is afforded by the metal known as scandium. Mendeleieff's predictions, and the actual facts as they were afterwards discovered, are here presented in parallel columns, and you will see that the corre- spondence is extremely close. (The unknown element was provisionally called "eka-boron," from its position in the table of elements.*) EKA-BORON (hypothetical). 1. Atomic weight about 44. 2. Oxide will have formula Eb 2 - O 3 ; will be soluble in acids, but in- soluble in alkalies ; specific gravity about 3.5 ; analogous to A1 2 3 , but more basic ; less basic than MgO. 3. Salts of Eb will be colorless, and will yield gelatinous precipi- tates with KOH, K 2 C0 3 , Na 2 HPO 4 , etc. 4. Sulphate will have the formu- la Eb 2 -3SO 4 , and will form, with K 2 S0 4 , a double salt which will probably not be isomorphous with the alums. SCANDIUM (actual). 1. Atomic weight = 44. 2. Oxide has formula Sc 2 3 ; is soluble in strong acids, but in- soluble in alkalies; specific gravi- ty = 3.8 ; analogous to A1 2 3 , but more decidedly basic. 3. Solutions of Sc salts are color- less, and yield gelatinous precipi- tates with KOH, K 2 C0 3 , and Na 2 - HP0 4 . 4. Sulphate has the formula Sc 2 3 S0 4 , and forms, with K 2 - S0 4 , the double salt Sc 2 -3S0 4 -3 K 2 - SO 4 , which is not an alum. There is a certain analogy, historically at least, between this " periodic law " and the rough arithmetical progression known to astronomers as " Bode's law." Bode's law success- fully predicted the asteroids, and assigned them their place in the solar system. Afterwards, when Adams and Leverrier undertook their famous labors which ended in the discovery of Neptune, they assumed, naturally enough, that the unknown planet would also conform to the law of Bode, and they arranged their computations accordingly. When the planet had been discovered, and its orbit investigated, it was found to be much nearer the sun than had been anticipated. In 164 . THE MOLECULAR THEORY OF MATTER. fact, it does not agree with Bode's law at all, and the " law " has therefore been rejected, and is now regarded only as a curiosity. It may be that the " periodic law " of chemistry is destined to a like fate, through the discovery of new elements that cannot be placed in the scheme of classification as it now stands, nor in any modified form of it ; yet this is quite improbable, because the periodic law is based on a vast assemblage of facts of different kinds instead of upon a mere observed arithmetical progression, and also because the law is more or less elastic, as will be evident to any one who takes the trouble to look up its history and note the various modifi- cations that it has undergone since it was first proposed. There was nothing elastic about Bode's law, and when the adverse fact came, the " law " gave way before it. Elastic-Solid Theory of Light. It has long been known that light is not propagated with infinite rapidity, but that it travels with a finite (though prodigious) speed. It has also been long known that light is not a substance, but that it is a mere periodic disturbance of some kind or other. Now if it be admitted that light is a disturbance of some kind, and also that it can travel through all the interstellar spaces of the sidereal universe, it follows that throughout these spaces there must be some body that is disturbed; for it would be highly absurd to suppose that a disturbance of any kind could take place in an absolute vacuum. The hypothetical body that is thus assumed to fill all space is called the luminiferous (or " light-bearing ") ether. Of course it is not related in any way to the volatile liquid that chemists know as "ether," and it is unfortunate that the same word should be applied to such different things. The physicists have the right of way here, however, for their ether was discovered and named long before chemists produced the other kind of ether. Although the ether of the physicist does not directly affect our senses in any way, and is quite imperceptible to any chemical or physical tests, its general properties have nevertheless been ELASTIC-SOLID THEORY OF LIGHT. 165 deduced (by making certain special assumptions about the nature of light), and, as might be expected, they have been found to be very remarkable. In telling you of them I will outline, in a general way, what is known as the " elastic-solid " theory of light ; but I must state, before doing so, that this theory has been recently abandoned and replaced by another one that we shall consider presently. The ether being assumed to fill all space (or to extend, at least, to the most remote visible star), it was conceived, by the advocates of this theory, to be thrown into a state of vibration by the violent move- ments of the molecules of bodies. If the ethereal vibrations thus set up were comparatively slow, they were believed to produce the phenomena of radiant heat / while if they suc- ceeded one another rapidly enough to affect the retina of the eye, they were believed to constitute light. The first con- clusion with regard to the ether was, therefore, that it is elastic in some sense or other ; because an inelastic body can- not transmit vibrations. Moreover, since many of the stars that are visible to us are certainly more than a thousand mil- lion million miles away, there was good reason for believing not only that the ether is elastic, but that it is perfectly so. Otherwise the light of the fainter stars would be entirely extinguished by absorption in those tremendous wastes of space. Now an elastic body is something that we know a good deal about, from experience and experiment ; and it was therefore easy to investigate the kinds of waves that such a body as the ether can transmit. It was found that such waves would fall into two general classes. In the first place, if the ether were compressible there could be waves of alternate compression and rarefaction ; very similar to the sound-waves that we know so well in our own air. Again, if the ether could sustain a shearing stress, as solid bodies can, there could be what I may call a wave of propagation of shearing strain. In the first of these cases the to-and-fro displacements of the ether-particles would take place in a direction parallel to the direction of propagation of the wave ; 166 THE MOLECULAR THEORY OF MATTER. and in the second case the displacements would be perpen- dicular to the direction of propagation, just as they are when an undulation travels along a stretched rope or wire. Upon comparing these mathematical results with the facts of nature, it was easily seen, from the phenomena of polarized light, that waves of this second kind do actually occur in the ether ; but no phenomena could be discovered which could be attributed to waves of the first kind,* and hence it was con- cluded that waves of the first kind do not occur. You will see that the reasoning I have indicated leads to the strange results (1) that the ether cannot transmit waves of com- pression, and that it is therefore probably absolutely incom- pressible (I say probably, because the non-existence of waves of compression and rarefaction could also be explained by merely supposing the ether to be devoid of elasticity of volume), and (2) that it can sustain a shearing stress, and is therefore of the nature of an elastic solid. Planets, comets, and even such tiny things as atoms, can plunge onward through the ether without experiencing the least retarding effect ;f and yet certain kinds of molecular vibration are picked up by it and borne away with prodigious speed into the endless depths of space. The non-resistance offered by the ether is probably due to its incompressibility, which property prevents the establishment of a wave of condensation in front of the body, or of rarefaction behind it. There is no friction, and no eddies are produced. The great velocity of ether- waves (186,000 miles a second) shows either that the density of this body is very small, or that its rigidity is very great. Attempts have been made to determine both its rigidity and * Unless gravitation is such a phenomenon. This point is considered in a subsequent section. t The retardation experienced by Encke's comet has often been attrib- uted to the resistance of the ether ; but as this retardation has been only two-thirds as great, since 1871, as it was in former years, it must be referred to other causes probably to perturbations from some unknown meteoric stream. ELASTIC-SOLID THEORY OF LIGHT. 167 its density, and although no accurate results have been obtained, certain limits have been assigned, within which they are likely to lie. You will find the method of calcula- tion given in Maxwell's article on Ether in the Encyclopaedia Britannica. The minimum limit to the density, as obtained by this method, is 5.36 X 10~ 19 , the density of water being unity.* Although the ether does not directly retard the motion of material particles, it is undoubtedly influenced by them in some manner. Thus we believe that it penetrates all bodies, and fills up the spaces between their molecules ; and as the phenomena of refraction show that the velocity of light is less in a transparent body (say in glass) than it is in a vacuum, it follows that the ether in the glass has either a greater density or a less rigidity than it has in free space. Either of these suppositions will fit the case under considera- tion very well ; but there are other phenomena that will not be satisfied so easily, and it has been found to be impossible to make any single set of consistent assumptions, which shall reconcile the " elastic-solid " theory with the facts. For example, when we come to investigate certain problems in partial reflection from transparent media, and others relating to diffraction from small particles, we are obliged to conclude that it is the density of the ether that varies, the rigidity remaining practically constant. On the other hand, the phenomena of double refraction require us to admit that the rigidity of the ether in a doubly-refracting body is different in different directions ; and hence we conclude that the rigidity of the ether is altered by the presence of molecules of matter a conclusion at variance with that previously reached by considering the phenomena of diffraction and partial reflection. This is one of the rocks upon which the * In the article referred to, the density is stated to be 9.36 X 10- 19 . This result is erroneous on account of an arithmetical blunder, as any one can easily see by trying to verify Maxwell's computation. Of course the error is of no importance, and I should not have referred to it had I not noticed that whenever the result is quoted, the erroneous value is given. 168 THE MOLECULAR THEORY OF MATTER. " elastic-solid" theory went to pieces. There are numerous other objections to it, which are fully as serious as this one'. For example, the full mathematical theory of a non-isotropic elastic body involves the consideration of no less than twenty- one coefficients ; and if the ether within a doubly-refracting crystal really has different rigidities in different directions, we should expect the phenomena of double-refraction to be much more complex than they really are. In other words, the " elastic-solid " theory is too ponderous. It tends to pre- dict things that do not exist, and in order to prevent it from doing so we have to make certain arbitrary assumptions about the coefficients that occur in the " equations of motion " a proceeding which is repugnant, I think, to every philosophical mathematician. Taking into account the various difficulties that have arisen in the course of its development, we must admit, with Lord Eayleigh, that "the elastic-solid theory, although valuable as a piece of purely dynamical reasoning, and probably not without mathematical analogy to the truth," is no longer tenable. If we dismiss it from further con- sideration as being incompetent to explain the entire range of optical facts, of course we are at liberty to form a new conception of the ether also ; for the properties that I have already assigned to this body were merely those that the " elastic-solid " theory demanded ; and in abandoning that theory we abandon all its consequences at the same time, and prepare ourselves to take a fresh start in a new direc- tion which direction, fortunately, has already been pointed out. Electro -Magnetic Theory of Light. Many years ago Faraday, discussing the supposed phenomena of " action at a distance " as manifested by a magnet, said that he believed that there is some mechanism by which the magnetic influence is enabled to extend itself through a space apparently vacuous. " Such an action," he said, " may be a function of the ether ; for it is not unlikely that, if there be an ether, it should have ELECTRO-MAGNETIC THEORY OF LIGHT. 169 other uses than simply the conveyance of radiation." * These are profound words how profound they are, it was reserved for Maxwell to show. The old corpuscular theory of light, defended with such ingenuity by Sir Isaac Newton, was thrown overboard long ago because it could not explain the phenomena of interference, and for other reasons that I need not here repeat. Following it came the elastic-solid theory, which we have just considered, and which we have also been obliged to abandon, though perhaps with some reluctance. One would almost be ready to say that the truth must lie with one of these two theories, for it would appear that light must be either some kind of a substance, or some kind of a motion in a substance. Undoubtedly this is true, but it now appears that we were thinking of the wrong kind of a motion, altogether. Maxwell, after reading Faraday's experimental researches, was so impressed by them and by that marvelous insight into things which seemed, in Faraday, almost like intuition, applied his own ingenious and powerful mind to the problems whose solution Faraday had dimly glimpsed, and succeeded in completely re-volution izing our notions of light, and showing us the whole subject from an entirely new point of view. I shall not attempt to discuss the general theory of electricity, because it would lead us too far away from our subject ; but I must indicate, as briefly as I can, the nature of Maxwell's theory of light. He agrees with previous writers that light is some sort of a periodic disturbance in some sort of an ether, and that the displacements that occur as the wave progresses are indeed perpendicular to the direction in which the wave travels ; but he teaches us that these displacements are not analogous to those that are pro- duced in an elastic-solid when that solid is deformed. He considers that they are of an electrical nature, and that we must learn about them, not by observing the behavior of elastic bodies under stress, but by observing the phenomena exhibited by electrified bodies ; and this, you will see, is an * Experimental Researches in Electricity, Vol. Ill, p. 331. 170 THE MOLECULAR THEORY OF MATTER. entire change of base. Maxwell has given us the funda- mental equations that must be satisfied when an electrical wave is propagated through the ether equations analogous to the " equations of motion " of the old elastic-solid theory and by means of these equations the entire theory of light can be constructed on the new basis. The theory thus con- structed agrees well with the facts of observation, and it is free from the numerous objections that beset the old elastic- solid theory. Moreover, it successfully withstood the search- ing experimental tests devised by the late Professor Hertz, whose labors have shown us in a very direct manner that electrical radiations are propagated with the same speed as light, and that they can be reflected, refracted, diffracted, polarized, and made to interfere ; so that we are now quite ready to admit that light consists in a rapid succession of such radiations. It is not at all essential to Maxwell's theory of light that we should know what an " electrical displace- ment" really is. We derive his fundamental equations from a study of electrical phenomena as observed in gross matter, and we then apply these equations to the ether, and deduce, by means of them, the laws that govern the propagation of electrical disturbances in that body. We then obsej've that the laws so deduced are precisely the same as those that have long been known to hold true for light ; and hence we con- clude that light is an electrical phenomenon. This is the whole story, so far as we have any positive knowledge of it at present. We have some dim ideas about the nature of electric and magnetic displacements, but I think it is safe to say that we know little or nothing about them that is not liable to be profoundly modified by subsequent research. It is quite probable that there is some kind of an ethereal rotation going on in a magnetic field, because it is hard to account for magnetic rotation of the plane of polarization on any other hypothesis. However, it should be remembered that even this assumption is by no means beyond controversy, for the plane of polarization of light is affected by magnetism only PROVISIONAL ASSUMPTIONS. 171 when molecules of gross matter are present. There is a great deal of work to be done before we can form a clear and true conception of the ether, and of what goes on in it in the vicinity of an electrified body, or a magnet, or a ray of light ; * but when such a conception has been attained, we shall inci- dentally learn a great deal about the relation of ether to matter, and about the constitution of the molecules of which matter is composed. Provisional Assumptions about the Constitution of Mole- cules. In discussing the molecular theory of matter I have made certain assumptions about the constitution of molecules, which are perhaps the most natural ones to make, and which ought therefore to serve as a basis for our investigations, at least until it appears that they are inadequate, or that some other assumptions would be better. Thus I have assumed that molecules are composed of smaller bodies called atoms, which are held together (but not necessarily in contact) by certain attractive forces, whose precise nature we have not determined. The atoms have^ been assumed, furthermore, to have definite forms and sizes, to be perfectly elastic, and to be subject to the same laws of mechanics that govern the, larger masses of matter that we can observe directly. These assumptions are simple enough, but it is far from certain that they correspond with the facts ; and at all events they are open to certain grave philosophical objections which we shall presently consider, in connection with Lord Kelvin's vortex- theory. The attractive forces that undoubtedly exist between molecules must receive some kind of a mechanical explanation, but although some attempts have been made to provide such * Dr. Oliver J. Lodge's Modern Views of Electricity (Macmillan & Co., 1889) is an excellent book on this subject, though I fear that his ingen- ious models tend to give students a too mechanical conception of the ether and of electric action, and to deceive them into the belief that we know much more about these things than we really do. It is possible, however, that I am too conservative on this point. 172 THE MOLECULAR THEORY OF MATTER. an explanation, the subject is still quite obscure.* These forces are sometimes thought to be of an electrical nature ; but until we know more about the ether and the mechanism of electric attraction such an assumption can hardly be con- sidered satisfactory. The elasticity of molecules certainly admits of explanation, and something has already been done in this direction, as we shall presently see. It is far from certain that molecules even have definite dimensions ; for they may be mere centers of condensation of the ether, or of some non-ethereal substance distributed through it, and they may be as indefinite in their boundaries as the nebulae that we see in the heavens. We do not even know that those general principles of mechanics that we call "axioms," and which are derived from our observation of vast aggregates of mole- cules, are applicable, without modification, to the molecules themselves ; and yet I think it is logical for us to adopt them until it can be shown that they lead to false results. So far as the general assumptions that I have made this evening are concerned, I think it can be said that there is no gross failure in the molecular theory that can be attributed to them. The present fragmentary state of the theory appears to be attrib- utable to the enormous mathematical difficulties that are involved, rather than to error in the premises. As I have told you, the assumptions that we have made are liable to certain philosophical objections, and for this reason we must consider them to be merely provisional, accepted for the present as convenient stepping stones, but subject to revision as the growth of the molecular theory proceeds. We are not yet prepared to develop the molecular theory from any point of view that differs materially from that which I have given you, but it will be interesting to note the direction in which future research is likely to lead us, and for this reason I must notice two or three of the more important hypotheses that have been advanced, concerning the constitution of molecules. * Some of these attempts are considered in a subsequent section on "Gravitation." RANKINE'S HYPOTHESIS. 173 Rankine's Hypothesis. I do not quite know what I ought to say about Eankine's views concerning the constitution of molecules. He certainly did deduce many of the known properties of bodies from his "hypothesis of molecular vortices," but I am not aware that any other mathematician has found that hypothesis promising enough to call for further investigation. He attributes the hypothesis to Sir Humphrey Davy, but it has long been known by Eankine's name because he was the first to develop it by mathematical methods. The hypothesis of molecular vortices assumes "that each atom of matter consists of a nucleus or central point enveloped by an elastic atmosphere, which is retained in its position by attractive forces, and that the elasticity due to heat arises from the centrifugal force of those atmospheres, revolving or oscillating about their nuclei or central points."* He does" not attempt to decide " whether the elastic atmos- pheres are continuous, or consist of discrete particles"; nor did he find it necessary to determine whether the nucleus of a molecule " is a real nucleus having a nature distinct from that of the atmosphere, or a portion of the atmosphere in a highly condensed state, or merely a center of condensation of the atmosphere, and of resultant attractive and repulsive forces." He believed "that the vibration which, according to the undulatory hypothesis, constitutes radiant light and heat, is a motion of the atomic nuclei or centers, and is propagated by means of their mutual attractions and repul- sions." This form of the theory of light receives some considerable support from the phenomena of double refraction and polarization, but of course it has no bearing on the mode of propagation of light through free space, unless the ether itself is conceived to have a similar constitution. Eankine perceived this fact very clearly, and in an interesting paper on light, read before the British Association in 1853,t he * Rankine, Miscellaneous Scientific Papers (London, Charles Griffin. &Co., 1881), page 17. t Miscellaneous Scientific Papers, page 156. 174 THE MOLECULAR THEORY OF MATTER. assumes "that the luminiferous medium is composed of detached atoms or nuclei distributed throughout all space, and endowed with a peculiar species of polarity, in virtue of which three orthogonal axes in each atom tend to place them- selves parallel respectively to the corresponding axes in every other atom ; and that plane-polarized light consists 'in a small oscillatory movement of each atom round an axis transverse to the direction of propagation." A serious objection to Eankine's hypothesis of molecular vortices is, that it seems to hold out little promise of eventually offering a mechanical explanation of attraction. He merely postulates the attrac- tion, and when we look for some sufficient cause for it, the hypothesis is barren, and its prophet dumb. The facts of chemistry are also hard to explain from Rankine's point of view ; and if we judge the hypothesis of molecular vortices according to its fruits, we must pronounce it a step in the wrong direction ; for it still remains where it was some forty years ago. Lord Kelvin's Vortex Theory. We have now to consider a very curious and interesting theory of the constitution of molecules, which was originally proposed by Lord Kelvin. Most of the theories that have been advanced have assumed that there are two kinds of matter, one being that which we ordinarily call " matter," and the other being the imponderable " ether," whose existence we have been obliged to admit in order to account for the phenomena of electricity, light, and radiant heat. Lord Kelvin has dispensed with one of these substances altogether, by assuming that molecules (or the atoms composing them) are merely definite portions of the ether itself, which are distinguished from the remainder of that vast body by being endowed with a peculiar kind of motion, called " vortex motion." I should like to give you a clear idea of vortex motion, and perhaps it will be best for me to begin by calling attention to certain cases of it that you are likely to have seen. The grandest example occurring in LORD KELVIN'S VORTEX THEORY. 175 nature is the cyclone, which consists, as you know, in a violent rotation of the air about a central vertical axis, accompanied by a translation of the axis in a direction perpendicular to its length. You have probably often seen smoke-rings blown from the stack of a locomotive, and if you have observed them closely you have noticed that they are in rapid rotation, the smoke-particles passing up through the ring on the inside and down again on the outside. These rings are good examples of vortex motion in which the axis of rotation returns into itself so as to form a closed curve. Experienced smokers can often produce similar rings, on a small scale, with their lips ; and very good ones can be made by the simple apparatus devised by Professor Tait and shown in Fig. 50. This apparatus consists of a box with a round hole several inches in diameter in one end of it, the other end being removed and replaced by a tense sheet of india rubber. B' A FIG. 50. TAIT'S APPARATUS FOR PRODUCING SMOKE-RINGS. In order to make the rings visible, the box may be filled with smoke or some similar substance. Professor Tait, for this purpose, makes use of the dense cloud of sal ammoniac particles that is produced when vapors of ammonia and hydrochloric acid are allowed to mingle with each other. The ammonia is sprinkled over the inside of the box, and the hydrochloric acid is generated by pouring strong sulphuric acid over some salt contained in a saucer which rests upon the bottom of the box. If the stretched sheet of rubber be now gently struck, a beautiful smoke-ring issues from the front of the box. The constitution of the smoke-rings pro- duced in this manner is indicated in Fig. 51, where the small, 176 THE MOLECULAR THEORY OF MATTER. curved arrows indicate the direction of the rotation, and the large, straight one shows the direction in which the ring travels.* A great variety of beautiful experiments may be tried by means of this simple apparatus. For example, we may study the action between a pair of rings by producing two of them in quick succession, as shown at A and B, in Fig. 50. When the experiment is suc- FiG.51. DIAGRAM OF cessful we shall see the first ring enlarge A SMOKE-RING. and s i ac ^ en j ts S p ee d of translation, while the second one grows smaller and moves faster, so that it presently passes through the first one, and the rings take the relative positions indicated at B ' and A'. The ring A ' being now the foremost one, tends to slow down and enlarge, and the ring B ' tends to grow smaller and move faster so as to pass through A', and so on perpetually ; but it is difficult to realize more than one such passage, in the actual experiment, because the viscosity of the air soon stops the rotation of the ring, and when the rotation has ceased the ring is no longer a vortex, but merely a wreath of smoke. Helmholtz was the first man to investigate vortex motion by rigid mathematical methods, and some of his results are very interesting. In his researches the fluid in which the vortices exist was assumed to be frictionless, homogeneous, and incompressible. These properties being admitted, he showed (1) that a vortex can never be produced nor destroyed in a medium of this character, so that if such vortices exist, they will continue to exist for- ever ; (2) that a vortex cannot have a free end within the fluid, and hence every vortex must either return into itself so as to form a closed curve (like a smoke-ring), or be infinite in * It seems hardly necessary to say that the smoke and the sal ammoniac fumes play no part whatever in these experiments, except to make the rings visible. The existence of the rings can be demonstrated when the smoke is entirely absent, by the effects produced on a distant sheet of tissue paper, or a candle-flame. LORD KELVIN'S VORTEX THEORY. 177 length, or have its ends upon a bounding surface of the fluid ; (3) that a vortex always consists of the same portion of fluid, so that when it travels through the surrounding fluid it is not alone the motion which progresses (as would be the case in a wave) ; the vortex does not lose its identity, but " moves " in the same sense that a projectile moves when propelled through the air; (4) no two vortices can ever intersect each other, and no vortex can ever intersect itself. It is also known that a vortex in a frictionless, incompressible fluid would behave like a perfectly elastic body. Many other properties of vortices have been deduced, but those that I have mentioned will be sufficient for our present purposes. Some eight years after the publication of Helmholtz's paper on vortex motion, and while watching Professor Tait's beautiful experiments on smoke-rings, Lord Kelvin conceived the idea that atoms may possibly be vortices in the luminiferous ether. There is much to be said in favor of this hypothesis, from a philosophic point of view. As we have seen, it enables us to dispense with one of the two " kinds of matter " entirely, for it teaches that all things are composed primarily of ether, and that " gross matter " is distinguished from the surrounding medium solely by its being endowed with the peculiar kind of motion that we have just been considering. It explains the perma- nence of "gross matter," because Helmholtz's investigations prove that a vortex-atom in a frictionless fluid can never be created nor destroyed. It explains the elasticity of molecules, because it shows that an ether-vortex would behave like a perfectly elastic body, even though the ether itself were entirely devoid of elasticity. Most of the theories of matter leave the existence of some 70 or so elements as much of a mystery as the existence of " species " was, in the biological world, before the time of Spencer and Darwin ; but the vortex-theory gives us at least a suggestion on this point. I have already told you that a finite vortex, in an infinite, frictionless fluid, must return into itself; but there is no reason for assuming that it must form a simple ring, like a 178 THE MOLECULAR THEORY OF MATTER. smoke-ring. There is no assignable reason why it could not have the form suggested in Fig. 52, or any other more compli- cated form ; and since a vortex can never intersect itself, the degree of knottedness of a vortex-atom can never change. It may be, there- fore, that the elements differ in their properties, on account of their atoms differing in knottedness. The vortex- theory of Lord Kelvin also holds out FiG~"52. A KNOTTED some faint promise of explaining VORTEX. other facts of chemistry ; and in this respect, at least, it is decidedly superior to Eankine's hypothe- sis. I will not attempt to say just what explanation of chemical combination might prove to be best, but there is a certain suggestiveness in the behavior of a pair of similar and nearly parallel smoke-rings, which tend to thread through and through each other perpetually, as illustrated in Fig. 50. A host of other possibilities lie before the vortex-theory, but it is doubtful if further speculation would be profitable for us. The consequences of the vortex-theory can be deduced by rigid mathematical methods, and it is idle to try and guess them in advance. In fact, one of the greatest philosophical advantages of the vortex-theory is, that it admits of so few assumptions. Other theories are more or less elastic, and can be modified so as to bring them into harmony with each new phenomenon ; but when the fundamental assumptions of the vortex-theory have once been made, we are bound to adhere to them, and to deduce from them, by exact analysis, all the known properties of matter. As Maxwell says, " When the vortex-atom is once set in motion, all its properties are abso- lutely fixed and determined by the laws of motion of the primitive fluid, which are fully expressed in the fundamental equations. The disciple of Lucretius may cut and carve his solid atoms in the hope of getting them to combine into worlds ; the follower of Boscovich may imagine new laws of force to meet the requirements of each new phenomenon ; but 179 he who dares to plant his feet in the path opened up by Helmholtz and Thomson [Lord Kelvin] has no such resources. His primitive fluid has no other properties than inertia, invariable density, and perfect mobility, and the method by which the motion of this fluid is to be traced is pure mathe- matical analysis. The difficulties of this method are enor- mous, but the glory of surmounting them would be unique."* The vortex-theory is inseparably united to the theory of electricity and light, since both of these theories involve a discussion of the ether ; and it remains to be seen whether a constitution can be imagined for that body which shall explain the propagation of radiant energy, without excluding the possibility of vortex-motion. Before the old elastic-solid theory of light was abandoned, the vortex-atom could hardly be seriously considered ; for a vortex in an elastic-solid is a manifest absurdity. The electro-magnetic theory cleared the way for the vortex-atom, however, by teaching us that our elastic-solid analogy was erroneous. We are now free to form a new conception of the ether, which may possibly reconcile the vortex-atom with the theory of light ; but our past experience in this direction has shown us that we should proceed with extreme caution. The advocates of the vortex- theory are extending their theory to the ether itself, in an attempt to explain how that body may be a frictionless, incompressible fluid, and yet have elasticity. For this purpose the ether is regarded as a perfect snarl of minute, interlacing vortices, which are normally in equilibrium, but which serve as an elastic framework for the transmission of radiant energy. This branch of the vortex-theory is too abstruse and too imperfectly developed to be considered further in this place ; and it may be said of the vortex-theory in general, that it is full of enormous mathematical difficulties, and that for this reason we can regard it, at present, only as a highly interesting possibility, whose consequences must be traced out by future generations. * Encyclopaedia Britannica, article Atom. 180 THE MOLECULAR THEORY OF MATTER. Dr. Burton's Strain-Figure Theory. Numerous other theories of the constitution of molecules have been advanced, but most of them are open to so many objections that they cannot be considered to be tenable at present, and need not be discussed in this place. As an example I may mention the complicated theory of Lindemann, which considers a molecule to consist of a series of concentric spherical shells, each of which is elastically connected with its neighbors. Dr. C. V. Burton's theory must be briefly mentioned, however, for although it is in a very imperfectly developed condition, it presents points of novelty that cannot fail to broaden our conception of what a molecule may be.* His theory bears a superficial resemblance to Lord Kelvin's, inasmuch as it con- siders an atom to consist of a modified portion of the ether ; but further than this the two theories are radically different. Dr. Burton conceives that the ether, although possibly of a fluid nature, is nevertheless endowed with some sort of elasticity (which is no doubt the case, since otherwise it could not transmit radiant energy). He further assumes that small strains in the ether are always proportional to the stresses that accompany them ; but that when the strains exceed a certain limit the ether takes a sort of " permanent set," after which it never returns to its primitive condition. Dr. Burton assumes that atoms are merely deformations in the ether that have been produced by such a process of over- straining. Lord Kelvin's vortex-theory has been facetiously called the " doughnut-theory," and perhaps we may designate Dr. Burton's theory, in the same spirit, as the "ether-dent theory " ; though neither term is very apt, for a vortex is not necessarily a simple ring, and a " strain-figure " in the ether is far from being a mere dent. To give you a clearer idea of Dr. Burton's fundamental assumption I will quote his own description of it. (i Consider a region," he says, "either * For a full account of Dr. Burton's theory see his paper entitled A Theory Concerning the Constitution of Matter, in the Philosophical Magazine for February, 1892. DR. BURTON'S STRAIN-FIGURE THEORY. 181 infinite or having very distant boundaries, and filled with a homogeneous isotropic elastic medium, whose condition throughout is one of stable equilibrium for small strains of any type. Let the medium now be strained, and held in its strained condition by some compelling agency : there will be a corresponding distribution of stress in the medium, and, provided the strain has at no point too great a value, the original condition will be completely regained after the com- pelling agency has been removed. But suppose that, instead, the medium is strained further and further from its initial state, and suppose that the restoring stresses do not always increase with the strain, but that beyond a certain point in the process they begin to fall off in value, until at last a point is reached at which the general tendency of the stress is to further increase the strain. If the compelling agency is now withdrawn, the medium will subside into a new condition of stable equilibrium, involving stress and strain at every point. The state of things thus impressed on the medium is, accord- ing to my view, an atom or a constituent of an atom ; it will hereafter be referred to as a strain-figure" This passage does not purport to explain the origin of matter ; it is intended merely to convey to the reader the meaning of the term " strain-figure " as used in Dr. Burton's paper. It is sug- gested, however, that if the ether " had long ago possessed motion of the most general kind, we might imagine its present condition to be due to the degeneration of that motion into a fine-grained turbulence ; and if, in the quasi-solid so consti- tuted, the existence of strain-figures were possible, it seems not unlikely that such would incidentally have been formed, unless the motion fulfilled special conditions." If such special conditions were absent, it is therefore possible, on the strain-figure hypothesis, that atoms would have resulted, from time to time, whenever and wherever the motion of the ether should chance to be such as to produce a strain sufficient to give rise to a "permanent set." Dr. Burton's theory, there- fore,, holds out some hope of even explaining the origin of 182 THE MOLECULAR THEORY OF MATTER. matter ; and in this respect it differs from every other theory with which I am familiar. A mathematical analysis of strain-figures shows that they would possess many of the characteristics that atoms are supposed to have. They could be moved about in the ether without encountering resistance ; but we are to consider that when such motion occurs it is not the ether itself that moves, but that the modification of structure that constitutes a strain-figure is merely transferred from one part of that body to another part. (The vortex- theory, you will remember, requires us to suppose that an atom always consists of the same portion of ether ; and in this respect it is diametrically opposed to Dr. Burton's theory.) Gravitation and other inter-moleQular and inter- atomic forces are assumed to arise from the distribution of stress that accompanies the strains in the strain-figures. Other consequences of the strain-figure hypothesis have been examined, but the hypothesis is so new and so imperfectly developed that it will hardly be profitable to discuss it further in this place. It is extremely ingenious and interesting, but we must wait for further researches before we can pronounce upon its adequacy or inadequacy. Internal Vibration of Molecules. The various phenomena of physics and astronomy compel us to admit that matter can move through the ether freely, without experiencing the least resistance. But since we know that matter can emit radiant energy, it follows that there are modes of molecular motion that can be communicated to the ether ; and we are impelled to the belief that it is the internal vibratory energy that is transmitted in this way. That such energy exists, is quite evident ; for if the molecules of bodies are elastic, we must suppose them to be capable of some sort of internal vibration. Let us consider a gas, assuming that its molecules have definite masses and definite sizes, and that for each of them there is a certain shape in which the internal stresses are either zero, or at least a minimum. When two such mole- INTERNAL VIBRATION OF MOLECULES. 183 cules collide, we must suppose that the collision throws each of them into violent vibration, just as a stretched string is thrown into vibration upon being struck with a hammer. We know that the string can vibrate in different ways : it may vibrate as a whole, or in two equal segments separated by a node, or in three such segments, or four, or, in general, in any number of them. When such a string is struck we usually find that all of these possible modes of vibration occur simul- taneously, so that the actual motion is very complex. In the stretched string the periodic times of the various possible vibrations are proportional to the roots of the equation sin |- 1 = 0; that is, they are inversely proportional to the simple numbers 1, 2, 3, 4, ... Doubtless there is some similar law connecting the periodic times of the possible vibrations that may occur in elastic molecules ; though we cannot suppose that law to be as simple as the one connecting the various periodic times of the string. In discussing this question of the vibration- periods of elastic systems, Professor J. J. Thomson tells us that "if the vibrating system . . . were like a bar, the periods would be proportional to the natural numbers for the longitudinal and torsional vibrations," and to the reciprocals of the roots of the equation for the transverse vibrations. " If the system were a circular membrane/ 7 he continues, "the frequencies would be pro- portional to the roots of an equation formed by equating a Bessel's function to zero. If the system were a uniform elastic sphere, the frequencies would be the roots of a compli- cated equation given by Chree in the Transactions of the Cambridge Philosophical Society. Other periods which have been worked out are those of circular vortex rings. The 184 THE MOLECULAK THEORY OF MATTER. frequencies of the higher vibrations [of such rings] about the circular form are proportional to where n is a large natural number [i.e. integer], and the vibrations about the circular cross-section are proportional to the natural numbers."* We have to think of a gas-molecule as colliding with another one, and then flying off through the ether in a sensibly straight line until it again experiences a collision. At each collision vibrations are set up within the molecule, and in the interval between successive collisions that is, while the molecule is describing its " free path " it vibrates according to its own proper periods, and the ether in which it is submerged picks up these vibrations and carries them away as radiant heat, or as light. You must not suppose, however, that the process is exactly analogous to what occurs when a particle immersed in a jelly is caused to vibrate. This was indeed believed to be the case when the elastic-solid theory of light was held to be true, but when that theory was discarded it became evident that the real phenomena are essentially different from those suggested by the jelly-analogue ; and I think we are still a long way from knowing precisely what does take place when a vibrating molecule gives up its energy to the ether. You will observe that writers on the theory of light merely consider what occurs in the ether as it transmits the light, and do not attempt to trace the exact processes by which the ethereal motions originate. There can be no doubt that here is a fruitful field for investigation, but at present we are hardly prepared to enter upon it. Thus far I have referred only to the vibrations of #as-molecules, and I have said that while a molecule is traversing its free path, the vibrations that occur are performed in accordance with its natural vibration-periods. At the instant of collision, and for an extremely short time * Watts's Dictionary of Chemistry (new edition), article Molecular Constitution of Bodies. INTERNAL VIBRATION OF MOLECULES. 185 afterwards, the vibrations probably differ more or less from these periods, on account of the extreme violence of the intra- molecular shocks. Careful attention should be paid to this point, since it shows that in solids or liquids, where there is practically no free path, and in highly compressed gases where the free path is very short, we cannot expect to find the com- parative simplicity of vibration that we do find in gases under ordinary conditions of density. The data for investigating the vibration-periods of molecules are furnished by that simple yet ingenious and powerful instrument known as the spectroscope, which enables us to analyze the complicated vibratory motion that they have impressed upon the ether, and to examine separately the constituent simple vibrations of which it is composed. Professor E. C. Kedzie says of this wonderful instrument, " If there was ever a flank movement on nature by which she has been compelled to surrender a part of her secrets it was the discovery of the spectroscope, ' which enables us to peer into the very heart of nature ' " ; * and Maxwell says, though I think his statement is far too strong, " An intelligent student, armed with the calculus and the spectroscope, can hardly fail to discover some important fact about the internal constitution of a molecule." f By the aid of this instrument the vibration-periods of molecules have been patiently studied and tabulated, $ and many attempts have been made to find some relation among them, analogous to the integer-law that holds for stretched strings, and to the other laws that I have told you about in connection with bars and membranes and spheres. No very great success has rewarded these efforts, yet something has been done, and more is sure to follow. Hydrogen, on account of its chemical and physical properties, has long been regarded as a compara- tively simple substance ; and especial attention has been paid * Proceedings of the American Association for the Advancement of Science, August meeting, 1891, page 162. t Nature, March 11, 1875. I See, for example, Watts's Index of Spectra (Manchester, England, Abel Hey wood & Son, 1889). 186 THE MOLECULAR THEORY OF MATTER. to it on that account, in the hope that the relation among its various vibration-periods might prove to be comparatively simple, and might therefore be the more readily found. Experience has shown that this hope was justifiable ; for a remarkable relation among the vibration-periods of hydrogen has been discovered by Balmer. The relation in question is this : If the different lines in the spark-spectrum of hydrogen be numbered consecutively, calling the Ha. line 3, the next one 4, and so on, then the wave-length of the line whose, number is m is _ 2 X = 3645.42 4* (60) In the following table the results of this formula are com- pared with the observed facts. SPARK-SPECTRUM OF HYDROGEN. BALMER' s LAW. LUTE. m. WAVE-LENGTH. DIFFERENCE. CALCULATED. OBSERVED. Ha 3 6562.8 6563.1 + .3 H{3 4 4860.6 4860.7 + .1 Hy 5 4339.8 4339.5 .3 H3 6 4101.1 4101.2 +.1 He 7 3969.5 3969.2 -.3 H{ 8 3888.4 3888.1 -.3 H-n 9 3834.8 3834.9 +.1 He 10 3797.3 3797.3 .0 Hi 11 3770.0 3769.9 -.1 HK 12 3749.6 3750.2 + .6 H\ 13 3733.8 3734.1 + .3 Hn 14 3721.4 3721.1 -.3 Hv 15 3711.4 3711.2 -.2 * The vibration-period is proportional to the wave-length being equal, in fact, to the wave-length divided by the velocity of light. Balmer 's law can easily be written so as to give the vibration-periods directly ; but most writers state it in connection with the wave-lengths, and I have thought best to follow the custom thus established. INTERNAL VIBRATION OF MOLECULES. 187 The agreement between the calculated and observed wave- lengths is very good ; and we may conclude that the hydrogen molecule is so constituted that equation (60) represents all the various kinds of elastic vibration that are possible within it, under the conditions that prevail when the spark-spectrum of the gas is being examined. Further than this we cannot go, at present, because no one has shown what sort of a body would have the series of vibration-periods that is represented by (60). I have compared the light-producing vibrations of a molecule to the sound-producing vibrations of a sonorous body ; but I must caution you against supposing that there is any very close analogy between the two. I have referred to the familiar phenomena of sound, in order to assist your imaginations a little ; but you should understand clearly that the ultimate phenomena of light are probably quite different. Since the electrical nature of light has been recognized, the suggestion has been made that molecules behave like con- ductors in which oscillatory electrical discharges take place, the form, capacity, and resistance of the molecules determin- ing the rapidity of the discharges, and hence also the positions of the spectral lines. I do not think this conception adds much to our knowledge of the molecule, since I can think of an " electrical discharge " only as a motion of some kind, in which the molecule and the ether probably both participate; nor do I think that any other hypothesis is likely to help us much until we have a more exact knowledge of the kind of motion that occurs in the free ether when a ray of light is passing through it. Many physicists appear to regard molecules as aggregates of smaller particles, which are held together by a system of attractive forces, and which execute to-and-fro oscillations of definite periods when the equilibrium of the system is disturbed by a collision, or in any other manner ; the light-waves being supposed to arise from these oscillations. Thus Professor J. J. Thomson says, that Balmer's law seems to show " that the hydrogen molecule; is a system possessing an infinite number of degrees of free- 188 THE MOLECULAR THEORY OF MATTER. dom, and not a finite number of rigid particles mutually attracting each other." So far as the to-and-fro theory of the origin of light is concerned, I think we may dismiss it altogether ; for we have already assumed that the ether does not absorb energy from a particle moving through it, and it is therefore difficult to conceive how light can be produced by any combination of such to-and-fro motions. Professor Thom- son would probably admit this point readily enough, and I am inclined to think that his remark has a deeper significance that it is aimed, in fact, at Boltzmann's law of the partition of kinetic energy among the different degrees of freedom of gas-molecules ; for if that law be true, it follows that the number of degrees of freedom of a molecule is finite.* For some reason or other Boltzmann's theorem has given rise to a good deal of controversy, and English mathematicians, as a rule, appear to be distinctly hostile to it, although I cannot quite see why. As I understand the theorem, it relates only to the kinetic energy of translation and rotation of the parts of a molecule, and not to those modes of vibration which are analogous to the motions of a sounding wire or bell or other elastic body. The general facts that have been ascertained about the molecules of gases appear to be these : The kinetic energy of the molecules is divisible into two parts, one of which is enormously greater than the other. The greater part consists in various motions of translation and rotation, either of the molecule as a whole or of its parts among them- selves ; and it is this portion of the kinetic energy which is divided up equally among the different " degrees of freedom " of the molecules (using the phrase in its restricted sense). The remaining small part of the total kinetic energy consists in elastic vibratory motion, within the very substance of the molecule or its parts ; and to this almost infinitesimal portion of the kinetic energy Boltzmann's law does not apply. The motions of translation and rotation are not impeded by the * See the section on "Generalized Theorems" (page 34), and that on .the "Ratio of the Specific Heats of Gases" (page 47). INTEKNAL VIBRATION OF MOLECULES. 189 ether ; but the internal vibratory motion is of such a character that the ether absorbs it, and bears it away as radiant energy. If a gas receives no heat from without, it will be cooled by the continual abstraction of vibratory energy until its tempera- ture falls to the absolute zero. It is true that only a small part of the total kinetic energy exists at any one time in the form of energy of vibration, but it is also true that the vibra- tory energy can never entirely disappear while the gas con- tains any kinetic energy whatever ; for some vibratory energy must exist, so long as there are molecular collisions. As the vibratory energy is removed by the ether, more will be pro- duced, at the expense of energy of other kinds ; and the end of the process will come only when the molecules have all come to rest. If the gas is exposed to radiations emanating from an external heat-source, the phenomena are reversed the vibratory energy of the molecules increases by absorption of the ethereal vibrations, and the energy so gained becomes presently transformed into energy of other kinds, by means of the collisions that are constantly occurring among the molecules ; and when the kinetic energy of translation of the molecules has become sensibly increased by this process, we say that the gas has grown " warmer. 7 ' The well-known difficulty of heating gases by simple radiation shows, how- ever, that the molecules of these bodies do not readily absorb vibratory energy from the ether. I do not know why this is so : solid bodies take up such energy readily enough, and it is not easy to account for the diathermacy of gases. Before leaving this subject I must say that equation (60) is by no means general. It applies only to hydrogen, and we cannot, by changing the constants, make it represent the vibration- periods of other bodies. Numerous investigators have sought for similar relations, however, that should hold true for other elements, and with some slight degree of success. Thus Kayser and Kunge found that the various wave-lengths of lithium, sodium, potassium, rubidium, and caesium are expres- sible by equations of the form 190 THE MOLECULAR THEORY OF MATTER. . m m Numerous formulae of this sort have in fact been proposed, but, so far as I am aware, none of them is as satisfactory as Balmer's. The spectroscope offers us a vast fund of informa- tion, and yet for theoretical purposes it is nearly useless, because the key by which the riddle is to be read still remains unfound. Gravitation. It is strange that no satisfactory theory of the nature of gravitation has yet been proposed; for the phenomena to be explained are simple, and they are familiar to everybody. The general fact of gravitation is, that between every pair of material particles there exists an attractive force which is proportional to the product of the masses of the two particles, and to the reciprocal of the square of the distance between them. Various attempts to account for this attraction have been made, and although all the theories yet proposed have been eminently unsatisfactory, it may be of interest to review a few of them briefly. Newton naturally gave some attention to the problem, and suggested that the ether is everywhere in a state of pressure, but that for some reason or other this pressure is less in the neighborhood of dense bodies than it is elsewhere. He showed that such a state of things would cause two bodies immersed in the ether to be urged towards each other, and he also showed that if the diminution of pressure at any point, due to the presence of the dense body, were inversely proportional to the distance of that point from the body, the apparent attraction would obey the law of inverse squares. He was unable, however, to imagine any physical cause for such a distribution of ether- pressure. The most famous theory of gravitation is undoubt- edly that of the Swiss philosopher Le Sage. According to him there is an enormous number of extremely small, " ultra- mundane " corpuscles of some sort or other, flying about through space with tremendous speed, and in every conceiv- GRAVITATION. 191 able direction. If there were only one body in space it would be bombarded equally on all sides by the corpuscles, and the impact-forces exerted upon it would be in equilibrium. But Le Sage conceived that if there were two bodies in space, each of them would shield the other one to a certain extent, so that the bombardment that either body received would be most severe on the side remote from the other one, and hence the two bodies would be urged together, and there would be an apparent attraction between them. There are a host of objections to this theory, and we cannot even momentarily consider it to be true. For example, in order to explain why a great amount of heat is not produced by the collisions, we have to assume that the corpuscles rebound with undiminished energy. But in that case, although the bodies would still shield each other, as before, from the impact of corpuscles coming directly from the depths of space, we must note that each body would also reflect corpuscles in all directions, so that its neighbor would be struck by a certain number of corpuscles that would otherwise have missed it ; and it has been shown that the impact against one body of the corpus- cles reflected from the other one would just suffice to annul the effect of the direct shielding action, and to prevent the realization of any gravitative tendency whatever. Le Sage's original theory can be modified so as to avoid this difficulty, but it will hardly be profitable for us to discuss such modifi- cations, because there are so many other objections to the theory that it now has only a historic interest. For example, in order to account for the fact that gravitative action varies as the mass of a body, and not as its surface, we have to sup- pose that the " ultramundane corpuscles " can pass through ordinary matter quite freely, so as to strike all its molecules with substantially the same frequency ; and this implies a more open structure of matter than we can readily reconcile with what we know of the sizes of molecules, and of inter- molecular distances. The "ether-squirt" theory of gravita- tion assumes that each particle of matter is a sort of center at 192 THE MOLECULAR THEORY OF MATTER. which ether is continuously created, so that from every such particle a ceaseless stream of ether flows out in all directions. Such a state of things would give rise to an apparent attrac- tion, similar to gravitation ; but as the theory requires us to admit a wholesale and perpetual production of ether from nothing, we must conclude that the probability of its truth is no greater than the euphony of its name. Similar remarks apply to the " ether-sink " theory, which differs from the one we have just considered only in assuming that at each mole- cule there is a destruction of ether, instead of a creation of it. The so-called " vibratory theory" of gravitation is more interesting than any that we have yet considered. You will remember that I said that the ether is believed to be incapable of transmitting waves of compression and rarefaction (analo- gous to sound-waves), because no phenomena could be dis- covered which could be attributed to such waves. It has been suggested that gravitation may transpire to be the missing phenomenon, and that the attraction between two bodies may be due to the mutual action of the compression- waves and rarefaction-waves that emanate from them. Con- siderable attention has been paid to this hypothesis, both mathematically and experimentally. It has been shown that a tuning-fork vibrating in the air can attract a light pith ball, and other phenomena of a like nature have been observed. The mathematical investigations that the theory calls for are exceedingly difficult, and I am not aware that they have yielded any conclusive results. It appears to be true, how- ever, that a particle would be attracted toward the center of disturbance if its density were greater than that of the sur- rounding medium, and its dimensions small in comparison with the length of the waves. This result, so far as it goes, is favorable to the wave-theory ; for matter is universally admitted to be denser than ether, and there is good reason for believing that if gravitation-waves exist at all, they are very long. In connection with this theory Lord Kelvin has called attention to " the general principle that in fluid motion the GRAVITATION. 193 average pressure is least where the average energy of motion is greatest." Now if the wave-theory of gravitation is true, the vibratory energy of the ether would be greatest in the immediate neighborhood of molecules of matter ; and hence the ether-pressure would be least at such places, and we should have a distribution of pressure something like that demanded by Newton's theory of gravitation. The wave- theory is quite interesting, but unfortunately there are certain grave objections to it, some of which appear to be absolutely fatal. In the first place the theory requires us to admit that a body exerts a greater attractive power when hot than it does when cold ; because when the body is hot its molecules are vibrating more energetically, and hence the waves that they emit have a greater amplitude. No such phenomenon has ever been detected. Again, if gravitative attraction is due to a wave-motion in the ether, we should have to admit that it has a finite speed of propagation. Nothing of the kind has been observed ; but we know from astronomical con- siderations that if there is any such a finite speed of propa- gation, it is certainly greater than a million times the speed of light. This naturally suggests that gravitation is not due to any sort of wave-motion in the medium that transmits light. In Maxwell's theory of gravitation it is assumed that bodies produce a stress in the ether about them, of such a nature that there is a pressure along the lines of gravitative force, combined with an equal tension in all directions at right angles to those lines. (In the case of a single body in space, the pressures would be radial, and the surfaces of equal tension would be concentric spheres described about the body as a center ; the stresses in the ether about the body being somewhat similar to those that exist in a cannon at the moment of discharge.) " Such a state of stress,' 7 says Max- well, "would no doubt account for the observed effects of gravitation. We have not, however, been able hitherto to imagine any physical cause for such a state of stress. It is easy to calculate the amount of this stress which would be 194 THE MOLECULAR THEORY OF MATTER. required to account for the actual effects of gravity at the surface of the earth. It would require a pressure of 37,000 tons' weight on the square inch in a vertical direction, com- bined with a tension of the same numerical value in all hori- zontal directions. The state of stress, therefore, which we must suppose to exist in the invisible medium, is 3,000 [1,000] times greater than that which the strongest steel could support." * Maxwell's theory is somewhat promising, but I think we cannot say more than this of it until a sufficient cause for the ether-stresses can be found. We may now pass to the consideration of Lord Kelvin's vortex-theory of the constitution of molecules ; and here we find that there is some small hope of explaining gravitation, though it is of a purely negative character. It is known that vortices exert a sensible influence upon one another, even when they are a considerable distance apart, and for certain special cases this influence has been investigated with a considerable approach to precision ; but the vortex-theory has not yet been developed sufficiently to enable us to investigate the interaction of vortices with absolute precision, and there are, undoubtedly, certain residual effects which are not included in the approximate equations that form the basis of what we now know about vortices. t It is possible that these neglected residual effects will prove to be sufficient to account for gravitative attraction ; for gravi- tation, as is well known, is an extremely weak force, becoming sensible only when bodies of enormous size are involved. The difficulty of deciding this point is tremendous, and at present we can only say that the vortex-theory may possibly be competent to explain gravitation. Professor Thomas Preston makes an interesting suggestion about gravitation, which I will quote to you, although I do not think it is put forward as being in any degree probable. I have told you * Encyclopaedia Britannica, article Attraction. t For a mathematical discussion of the interaction of vortices, see Professor J. J. Thomson's Treatise on the Motion of Vortex Eings (London, Macmillan & Co., 1883). GRAVITATION. 195 that a finite ether-vortex must either return into itself so as to form a closed curve, or must have its ends against a bound- ing surface of the ether. Thus it would be possible to have vortices that do not return into themselves, but which have ends that abut against molecules of dense matter. With this possibility in mind Professor Preston says, "We might sup- pose a body connected to the earth by vortex filaments in the ether, which would replace the lines of force. The ether is spinning round these lines, and when the body is lifted from the earth the work done is expended in increasing the length of the vortex filaments. The work is thus being stored up as energy of motion of the ether, and when the body falls to earth the vortex lines diminish in length, and their energy of motion passes into the body and is represented by the kinetic energy of the mass.' 7 * Professor Preston probably intended this suggestion as a mere illustration of the possibility of explaining gravitation ; for it would be quite extravagant to imagine every molecule in the universe to be united to every other one by a vortex filament space could hardly contain such a tangle. Moreover, since vortices cannot intersect, a few seconds of intermolecular motion would suffice to tie up the vortex-system into a mass of knots that would drive Gordius mad with envy, and render the " first law of motion " impossible. Dr. Burton, in connection with his strain-figure theory of the constitution of molecules, has suggested that molecules do not have definite sizes, but that in the ether surrounding each molecular center there are stresses and strains which grow continually less as we pass away from that center, but which never absolutely vanish. In this case every molecule in the universe would exert some influence on every other molecule. Dr. Burton shows that in its most general form his theory could account for either attraction or repulsion, but that by making a certain very simple assump- tion about the nature of the strain-figures, the theory could be * Thomas Preston, Theory of Heat (London and New York, Mac- millan & Co., 1894), page 90. 196 THE MOLECULAR THEORY OF MATTER. modified so that the forces acting between distant molecules should always be attractive ; yet even if the strain-figure theory should afford a perfectly satisfactory explanation of gravitation, we could not logically accept that explanation until the fundamental conceptions of the original hypothesis were shown to be in harmony with all the other phenomena of matter. I have now reviewed, briefly, some of the more famous and interesting theories that have been proposed to account for gravitative attraction, and it is easy to see that they all fail in some important particular. It may be that the future will bridge over some of these failures. It may be, on the other hand, that we have been looking at the problem from the wrong point of view, altogether. Conclusion. We have examined the molecular theory of matter as it stands to-day, and we have found that something is known of the constitution of gases, a little about the constitution of liquids, and almost nothing, for certain, of solids. So far as the sizes of molecules are concerned, we have seen that it is possible to discover the general order of their magnitude ; and we have also seen that nothing what- ever is known about their constitution, or about the mechani- cal nature of intermolecular forces. We are still very far from having a complete theory of matter our knowledge of it is in fact very fragmentary and yet there is strong reason to believe that we are working in the right direction. A great deal has been done in the last forty years, and it is just possible that within the next forty the world will be fortu- nate enough to produce a great genius who shall coordinate the isolated facts that we now have, fill up the vast gaps in our present knowledge, and provide us with a classical treatise on The Constitution of Matter, which shall be worthy to stand on our shelves beside the immortal Principia of Sir Isaac Newton. APPENDIX. On the Integration of Certain Equations in the Text. In deriving equations (2) and (4), and in constructing the table on page 26, it is necessary to perform certain integrations that will doubtless prove troublesome to the student unless he has devoted more time to the integral calculus than is usual in our colleges and scientific institutions. The processes by which these integrations may be effected will therefore be briefly indicated. It is easily shown, by direct differentiation, that (L (x ' c ) ( fi J. ) X dx Zi e X dx. If we integrate this equation, term by term, and then trans- pose and divide by 2, we have the formula /V* 2 x n dx = 4-cc"- 1 - c-* 2 + ?t i fx n ~ 2 . (63) The volume included between this surface and the plane = 0, and between the x-limits and oo , and the y-limits and oo , is V =CC dx dy. (64) If we represent the integral (65) by u, then, integrating (64) with respect to #, we have *dy. (66) Now the integral expressed in (66) is evidently the same as (65), since its value cannot depend upon the particular symbol that we use to represent the variable quantity. But we have represented the value of (65) by u ; and hence (66) becomes V = u\ (67) If we now return to (63), and express the value of z in polar coordinates, we have (since # 2 + y 2 r 2 ). The elementary area in the plane 2 = becomes rdO . dr instead of dxdy, and the limits of the inte- gration (in order to include the same part of the solid as APPENDIX. 199 before) are and oo for r, and and ^-TT for 0. Hence the volume of the solid, expressed in polar coordinates, is y= T r\-r* rdOdr. (68) Integrating with respect to 6 we have 00 V = ^ r or M = jVi = 0.8862269..., which is therefore the value of (65). Returning now to the integral / (71) let us make successive applications of equation (61), until the exponent of x has been reduced to either unity or zero (according as n is odd or even). We shall then have (71) expressed as a series, whose terms, with the exception of the last one, will all be of the form Ax m e~< 200 APPENDIX. Evidently this expression is zero when x = 0, and it can be shown, by the usual methods employed in the differential calculus for evaluating indeterminate quantities, that it is also zero when x = oo . Hence when (71) is integrated between the limits and oo all the terms of the series obtained by successive applications of (61) disappear, with the exception of the last one, which is ~ f W0d L*% >M WAY 11 1940 e.^ o LD JUL 23 1941 MAY 13 '65-8 PM ' JUL 2% 1942 OCT 28 t943 2? W ' ,I6Mar'62RH LD 21-100m-7,'33 75/fl