APPLICATIONS OF DYNAMICS TO PHYSICS AND CHEMISTRY BY J. J. THOMSON, M.A., F.R.S. FELLOW OF TRINITY COLLEGE AND CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS, CAMBRIDGE. (j UNJVEKSITY MACMILLAN AND CO. AND NEW YORK. 1888 [All Rights reserved.] PRINTED BY C. J. CLAY, M.A. & SONS, AT THE UNIVERSITY PRESS. 3 J PREFACE. THE following pages contain the substance of a course of lectures delivered at the Cavendish Labora- tory in the Michaelmas Term of 1886. Some of the results have already been published in the Philosophical Transactions of the Royal Society for 1886 and 1887, but as they relate to phenomena which belong to the borderland between two departments of Physics, and which are generally either entirely neglected or but briefly noticed in treatises upon either, I have thought that it might perhaps be of service to students of Physics to publish them in a more complete form. I have included in the book an account of some investiga- tions published after the delivery of the lectures which illustrate the methods described therein. There are two modes of establishing the connexion between two physical phenomena ; the most obvious as well as the most interesting of these is to start with trustworthy theories of the phenomena in ques- tion and to trace every step of the connexion between them. This however is only possible in an exceed- ingly limited number of cases, and we are in general compelled to have recourse to the other mode in VI PREFACE. which by methods which do not require a detailed knowledge of the mechanism required to produce the phenomena, we show that whatever their explanation may be, they must be related to each other in such a way that the existence of the one involves that of the other. It is the object of this book to develop methods of applying general dynamical principles for this purpose. The methods I have adopted (of which that used in the first part of the book was suggested by Maxwell's paper on the Electromagnetic Field) make everything depend upon the properties of a single function of quantities fixing the state of the system, a result analogous to that enunciated by M. Massieu and Prof. Willard Gibbs for thermodynamic pheno- mena and applied by the latter in his celebrated paper on the "Equilibrium of Heterogeneous Substances" to the solution of a large number of problems in thermodynamics. I wish in conclusion to thank my friend Mr L. R. Wilberforce, M.A., of Trinity College, for his kindness in correcting the proofs and for the many valuable suggestions he has made while the book was passing through the press. J. J. THOMSON. TRINITY COLLEGE, CAMBRIDGE, May 2nd, 1888. CONTENTS. CHAPTER I. PAGE Preliminary Considerations ........ i CHAPTER II. The Dynamical Methods to be employed ..... 8 CHAPTER III. Application of these Principles to Physics 16 CHAPTER IV. Discussion of the terms in the Lagrangian Function . . -31 CHAPTER V. Reciprocal Relations between Physical Forces when the Systems exerting them are in a Steady State 80 CHAPTER VI. Effect of Temperature upon the Properties of Bodies ... 89 CHAPTER VII. Electromotive Forces due to Differences of Temperature . . 106 CHAPTER VIII. On " Residual" Effects .... .128 Vlll CONTENTS. CHAPTER IX. PAGE Introductory to the Study of Reversible Scalar Phenomena . .140 CHAPTER X. Calculation of the mean Lagrangian Function . . . -151 CHAPTER XL Evaporation 158 CHAPTER XII. Properties of Dilute Solutions 1 79 CHAPTER XIII. Dissociation 193 CHAPTER XIV. General Case of Chemical Equilibrium . . . . .215 CHAPTER XV. Effects produced by Alterations in the Physical Conditions on the Coefficient of Chemical Combination 233 CHAPTER XVI. Change of State from Solid to Liquid 243 CHAPTER XVII. The Connexion between Electromotive P'orce and Chemical Change 265 CHAPTER XVIII. Irreversible Effects 281 APPLICATIONS OF DYNAMICS TO PHYSICS AND CHEMISTRY. CHAPTER I. PRELIMINARY CONSIDERATIONS. i. IF we consider the principal advances made in the Physical Sciences during the last fifty years, such as the extension of the principle of the Conservation of Energy from Mechanics to Physics, the development of the Kinetic Theory of Gases, the discovery of the Induction of Electric Currents, we shall find that one of their most conspicuous effects has been to intensify the belief that all physical phenomena can be explained by dynamical principles and to stimulate the search for such explanations. This belief which is the axiom on which all Modern Physics is founded has been held ever since men first began to reason and speculate about natural phenomena, but, with the remarkable exceptions of its successful application in the Corpuscular and Undulatory Theories of Light, it remained unfruitful until the researches of Davy, Rumford, Joule, Mayer and others showed that the kinetic energy possessed by bodies in visible motion can be very readily converted into heat. Joule moreover proved that whenever this is done the relation between the quantity of kinetic T. D. I 2 DYNAMICS. energy which disappears and the quantity of heat which appears in consequence is invariable. The ready conversion of kinetic energy into heat con- vinced these philosophers that heat itself is kinetic energy; and the invariable relation between the quantities of heat produced and of kinetic energy lost, showed that the principle of the Conservation of Energy, or of Vis- Viva as it was then called, holds in the transformation of heat into kinetic energy and vice versa. This discovery soon called attention to the fact that other kinds of energy besides heat and kinetic energy can be very readily converted from one form into another, and this irresistibly suggested the conclusion that the various kinds of energy with which we have to deal in Physics, such for example as heat and electric currents, are really forms of kinetic energy though the moving bodies which are the seat of this energy must be indefinitely small in comparison with the moving pieces of any machine with which we are acquainted. These conceptions were developed by several mathema- ticians but especially by v. Helmholtz, who, in his treatise Ueber die Erhaltung der Kraft, Berlin, 1847, applied the dynamical method of the Conservation of Energy to the various branches of physics and showed that by this prin- ciple many well-known phenomena are connected with each other in such a way that the existence of the one involves that of the other. 2. The case which from its practical importance at first attracted the most attention was that of the transformation of heat into other forms of energy and vice versa. In this case it was soon seen that the principle of the Conservation of Energy the First Law of Thermodynamics as it was called was not sufficient to obtain all the relations PRELIMINARY CONSIDERATIONS. 3 existing between the effects of heat on the various pro- perties of a body and the heat produced or absorbed when certain changes take place in the body, but that these relations could be deduced by the help of another principle, the Second Law of Thermodynamics which states that if to a system where all the actions are perfectly rever- sible a quantity of heat dQ be communicated at the absolute temperature 0, then r '!- the integration being extended over any complete cycle of operations. This statement is founded on various axioms by different physicists, thus for example Clausius bases it upon the " axiom" that heat cannot of itself pass from one body to another at a higher temperature, and Sir William Thomson on the " axiom " that it is impossible by means of inanimate material agency to derive mechanical effect from any por- tion of matter by cooling it below the temperature of the coldest of the surrounding objects. Thus the Second Law of Thermodynamics is derived from experience and is not a purely dynamical principle. We might have expected a priori from dynamical con- siderations that the principle of the Conservation of Energy would not be sufficient by itself to enable us to deduce all the relations which exist between the various properties of bodies. For this principle is rather a dynamical result than a dynamical method and in general is not sufficient by itself to solve completely any dynamical problem. Thus we could not expect that for the dynamical treat- ment of Physics the principle of the Conservation of Energy would be sufficient by itself, since it is not so in the much simpler cases which occur in ordinary Mechanics. 4 DYNAMICS. The resources of dynamics however are not exhausted even though the principle of the Conservation of Energy has been tried. Fortunately we possess other methods, such as Hamilton's principle of Varying Action and the method of Lagrange's Equations, which hardly require a more detailed knowledge of the structure of the system to which they are applied than the Conservation of Energy itself and yet are capable of completely determining the motion of the system. 3. The object of the following pages is to endeavour to see what results can be deduced by the aid of these purely dynamical principles without using the Second Law of Thermodynamics. The advantages of this method in comparison with that of the two laws of Thermodynamics are (1) that it is a dynamical method, and so of a much more fundamental character than that involving the use of the Second Law ; (2) that one principle is sufficient instead of two ; (3) that the method can be applied to questions in which there are no transformations of other forms of energy from or into heat (except the unavoidable ones due to friction), while for this case the other method degenerates into the principle of the Conservation of Energy, which is often not sufficient to solve the problem. The disadvantages of the method on the other hand are that, since the method is a dynamical one, the results are expressed in terms of dynamical quantities, such as energy, momentum, or velocity, and so require further knowledge before we can translate them in terms of the physical quantities we wish to measure, such as intensity of a current, temperature, and so on : a knowledge which in all cases we do not possess. PRELIMINARY CONSIDERATIONS. 5 The Second Law of Thermodynamics, on the other hand, being based on experience does not involve any quantity which cannot be measured in the Physical Laboratory. For this reason there are some cases where the Second Law of Thermodynamics leads to more definite results than the dynamical methods of Hamilton or Lagrange. Even here I venture to think the results of the application of the dynamical method will be found interesting, as they show what part of these problems can be solved by dynamics, and what has to be done by considerations which are the results of experience. 4. Many attempts have been made to show that the Second Law of Thermodynamics is a consequence of the principle of Least Action ; none of these proofs seem quite satisfactory ; but even if the connexion had been proved in an unexceptionable way it would still seem desirable to investigate the results of applying the principle of Least Action, or the equivalent one of Lagrange's Equations, directly to various physical problems. If these results agree with those obtained by the use of the Second Law of Thermodynamics, it will be a kind of practical proof of the connexion between this law and the principle of Least Action. 5. Considering our almost complete ignorance of the structure of the bodies which form most of the dynamical systems with which we have to deal in physics, it might seem a somewhat unpromising undertaking to attempt to apply dynamics to such systems. But we must remember that the object of this application is not to discover the properties of such systems in an altogether a priori fashion, but rather to predict their behaviour under certain circum- stances after having observed it under others. A dynamical example may illustrate what the application DYNAMICS. of dynamics to physical problems may be expected to do, and the way in which it is likely to do it. Let us suppose that we have a number of pointers on a dial, and that behind the dial the various pointers are connected by a quantity of mechanism of the nature of which we are entirely ignorant. Then if we move one of the pointers, A say, it may happen that we set another one, B, in motion. If now we observe how the velocity and position of B depend on the velocity and position of A, we can by the aid of dynamics foretell the motion of A when the velocity and position of B are assigned, and we can do this even though we are ignorant of the nature of the mechanism connecting the two pointers. Or again we may find that the motion of B when A is assigned depends to some extent upon the velocity and position of a third pointer C : if in this case we observe the effect of the motion of C upon that of A and B we may deduce by dynamics the way in which the motion of C will be affected by the velocities and positions of the pointers A and B. This illustrates the way in which dynamical considera- tions may enable us to connect phenomena in* different branches of physics. For the observation of the motion of B when that of A is assigned may be taken to represent the experimental investigation of some phenomenon in Physics, while the deduction by dynamics of the motion of A when that of B is assigned may represent the prediction by the use of Hamilton's or Lagrange's principle of a new phenome- non which is a consequence of the one investigated experi- mentally. Thus to take an illustration, suppose we investigate experimentally the effect of a current of electricity both steady and variable upon the torsion of a longitudinally magnetized iron wire along which the current flows, then we PRELIMINARY CONSIDERATIONS. 7 can deduce by dynamics the effects of torsion and variations of torsion in the wire upon a current flowing along it. The method is really equivalent to an extension and generalization of the principle of the equality of action and reaction, as when we have two bodies A and B acting upon each other if we observe the motion of B which results when A moves in a known way we can deduce by the aid of this principle the motion of A when that of B is known. The more general case which we have to consider in Physics is when instead of two bodies attracting each other we have two phenomena which mutually influence each other. CHAPTER II. THE DYNAMICAL METHODS TO BE EMPLOYED. 6. As we do not know the nature of the mechanism of the physical systems whose action we wish to investigate, all that we can expect to get by the application of dynamical principles will be relations between various properties of bodies. And to get these we can only use dynamical methods which do not require an intimate knowledge of the system to which they are applied. The methods introduced by Hamilton and Lagrange possess this advantage and, as they each make the behaviour of the system depend upon the properties of a single func- tion, they reduce the subject to the determination of this function. In general the way that we are able to connect various physical phenomena is by seeing from the behaviour of the system under certain circumstances that there must be a term of a definite kind in this function, the existence of this term will then often by the application of Lagrangian or Hamiltonian methods point to other phenomena besides the one that led to its detection. 7. We shall now for convenience of reference collect the dynamical equations which we shall most frequently have to use. The most generally useful method is Hamilton's principle DYNAMICAL METHODS. 9 of Varying Action according to which (see Routh's Advanced Rigid Dynamics, p. 245) where T and V are respectively the kinetic and potential energies of the system, / the time, and q a coordinate of any type. In this case / and /j are each supposed to be constant. In some cases it is convenient to use the equation in this form but in others it is more convenient to use Lagrange's Equations, which may be derived from equation (i) (Routh's Advanced Rigid Dynamics, p. 249) and which may be written in the form d dL dL - , x dtTq-^q-^ (2) " where L is written for T- V and is called the Lagrangian function and Q is the external force acting on the system tending to increase q. In the preceding equations the kinetic energy is sup- posed to be expressed in terms of the velocities of the coordinates. In many cases however instead of working with the velocities corresponding to all the coordinates it is more convenient to work with the velocities corresponding to some coordinates but with the momenta corresponding to the others. This is especially convenient when some of the coordinates only enter the Lagrangian function through their differential coefficients and do not themselves occur explicitly in this function. In a paper " On some Applica- tions of Dynamical Principles to Physical Phenomena" (Phil. Trans. 1885, Part n.) I have called these "kinos- thenic " coordinates. In the following pages the term "speed coordinates" will for the sake of brevity be used instead wherever it will not lead to ambiguity. 10 DYNAMICS. The most important property of such a coordinate is that whenever no external force of its type acts upon the system, the momentum corresponding to it is constant. For if x be a speed coordinate, since dL * = ' we have by Lagrange's equation since no external force acts on the system d dL as the momentum corresponding to x is dLldy^ this equa- tion shows that it is constant. 8. Routh {Stability of Motion, p. 61) has given a general method which enables us to use the velocities of some coordinates and the momenta corresponding to the remain- der, and which is applicable whether these latter coordinates are speed coordinates or not. The method is as follows : suppose that we wish to use the velocities of the coordinates q^ ^ 2 ...and the momenta corresponding to the coordinates n 2 ...then Routh has shown that if we use instead of L the new function L given by the equation. and eliminate 19 < 2 ...by means of the equations dT dT * i '';zr'*-;/r '-** #<, d

l5 < 2 ,...< w . Let us further suppose that there are no terms in the expression for the kinetic energy which involve the product of the DYNAMICAL METHODS. 13 velocity of a q and a coordinate and that the system has no potential energy. Then by Routh's method we can use Lagrange's equation for the q coordinates if instead of the ordinary Lagrangian function L which reduces in this case to the kinetic energy we use the modified function L' given by the equation L' = i-^~.. ....................... (6), d(f> or and where lt < 2 are to be eliminated by the aid of the equations Thus since the expression for L does not contain any terms involving the product of the velocity of a # and a tf> coordinate, L will be of the form T( 99 ) ~ ^W) where T {qq} is the kinetic energy arising from the motion of the q or positional coordinates, T^ that arising from the motion of the kinosthenic or speed coordinates. By Routh's modification of the Lagrangian equations we have but so that equation (8) reduces to d_ dT (qq} _ dT (99} = _ dTw / v dt dq, ' dq, ', dq, 14 DYNAMICS. If the system fixed by the positional coordinates q had possessed a quantity of potential energy equal to F, the equations of motion would have been of the type, * _ ._" systems, which we regard as the potential energy of the "q" systems, may be either that of parts of the system, or the surrounding ether, or both ; in many cases we should expect it to be mainly the ether. CHAPTER III. APPLICATION OF THESE PRINCIPLES TO PHYSICS. 13. IN our applications of Dynamics to Physics it will be well to begin with the cases which are the most nearly allied to those we consider in ordinary Rigid Dynamics. Now in this subject when there is no friction all the motions are reversible and are chiefly relations between vector quan- tities. We shall therefore begin by considering reversible vector effects and afterwards go on to reversible effects involving scalar as well as vector relations ; those for example in which a scalar quantity such as temperature is prominently involved : lastly we shall consider irreversible effects. Thus the order in which we shall consider the subject will be 1. Reversible vector phenomena. 2. Reversible scalar phenomena. 3. Irreversible phenomena. 14. We shall begin by considering the relations between the phenomena in elasticity, electricity, and magnetism and the way in which these depend upon the motion and configuration of the bodies which exhibit the phenomena. These phenomena differ from some we shall consider later on in that we have the quantities concerned in them entirely under our control and can by applying proper COORDINATES. I/ external forces make them take any value we please (sub- ject of course to such limitation as the strength of the material and the saturation of magnets may impose). The other phenomena on the other hand depend upon a multi- tude of coordinates over whose individual motion we have no control though we have some over their average motion. As the first kind of phenomena most closely resemble those we have to do with in ordinary dynamics we shall begin with them. 1 5. The first thing we have to do when we wish to apply dynamical methods to investigate the motion of a system is to choose coordinates which can fix its configuration. We shall find it necessary to give a more general meaning to the term "coordinate" than that which obtains in ordinary Rigid Dynamics. There a coordinate is a geometrical quantity helping to fix the geometrical con- figuration of the system. In the applications of Dynamics to Physics however, the configurations of the systems we consider have to be fixed, with respect to such things as distributions of electricity and magnetism, for example, as well as geometrically, and to do this we have in the present state of our knowledge to use quantities which are not geometrical. Again the coordinates which fix the configurations of the systems in ordinary dynamics are sufficient to fix them completely, while we may feel pretty sure that the coordi- nates which we use to fix the configuration of the system with respect to many of its physical properties, though they may fix it as far as we can observe it, are not sufficient to fix it in every detail ; that is they would not be sufficient to fix it if we had the power of observing differences whose fineness was comparable with that of molecular structure. T. D. 2 1 8 DYNAMICS. Hydrodynamics furnishes us with many very good illus- trations of this latter point. For example, when a sphere moves through an incompressible fluid, we can express the kinetic energy of the system comprising both sphere and fluid in terms of the differential coefficients of the three coordinates which fix the centre of the sphere, though it would require a practically infinite number of coordinates to fix the configuration of the fluid completely. Now Thomson and Tait (Natural Philosophy, vol. I. p. 320) have shown how we can "ignore" these coordinates when the kinetic energy can be expressed without them, and that we may treat the system as if it were fully determined by the coordinates in terms of whose differential coefficients the kinetic energy is expressed. And again Larmor (Proceedings of London Mathemati- cal Society, xv. p. 173) has proved that if Z' be Routh's modification of the Lagrangian function, that is q^ q 2 -,-- being the coordinates retained, < < 2 ,... those ignored, (? <2 2 .--, 3> 15 $2 the momenta corresponding to these coordinates respectively, if [ * L'dt=o .................. (IT). Jt then t If all the kinetic energy vanishes when the positional coordinates q^ ,,... are constant, as is the case when a number of solids move through a perfect fluid in which there is no circulation, Z' is the difference between the kinetic and potential energies of the system. If however the kinetic energy does not vanish when the velocities of the positional coordinates all vanish, as for example when a number of solids are moving through a fluid in which there is circulation, Z' no longer equals the difference between the kinetic and potential energies of the system. UJNIVERSITS COORDINATES. It follows from (i i), by the Calculus of Variations, that if L' be expressed in terms of a series of quantities q^ q z ,... and their first differential coefficients, then whatever these quantities may be, we must have a series of equations of the type d dL' dL' Thus we see that we may treat any variable quantities as coordinates if the modified Lagrangian function can be expressed in terms of them and their first differential coeffi- cients. We shall take this as our definition of a coordinate. 1 6. When we introduce a symbol to fix a physical quantity we may not at first sight be sure whether it is a coordinate or the differential coefficient of one with respect to the time. For example, we might feel uncertain whether the symbol representing the intensity of a current was a coordinate or the differential coefficient of one. The simplest dynamical considerations however will enable us to overcome this difficulty. Thus if when there is no dissipation of energy by irreversible processes, the quantity represented by the symbol remains constant under the action of a constant force tending to alter its value, the energy at the same time remaining constant, then the symbol is a coordinate. Again, if it remains constant and not zero when no force acts upon it, there being no dissipation and the energy remaining constant, the symbol represents a velocity, that is, the differential coefficient of a coordinate with respect to the time. Let us apply these considerations to the example men- tioned above ; as the intensity of a current flowing through a perfect conductor, the only circumstances under which there is no dissipation, does not satisfy the first of these 2 2 20 DYNAMICS. conditions, while it does satisfy the second, we conclude that the intensity of a current ought to be represented by the rate of change of a coordinate and not by the coordinate itself. Specification of Coordinates. 17. To fix the configuration of the system so far as the phenomena we are considering are concerned we shall use the following kinds of coordinates. (1) Coordinates to fix the geometrical configuration of the system, i.e. to fix the position in space of any bodies of finite size which may be in the system. For this purpose we shall use the coordinates ordinarily used in Rigid Dy- namics and denote them by the letters x lt x a , x 3 ...; and when we want to denote a geometrical coordinate generally without reference to any one in particular we shall use the letter x. (2) Coordinates to fix the configuration of the strains in the system. We shall use for this purpose, as is ordinarily done in treatises on elasticity, the components parallel to the axes of x, y, z of the displacements of any small portion of the body, and denote them by the letters a, (3, y respec- tively. For the strains *Y,df*\ fJ: + These terms may be divided into fifteen types. There are five sets which are quadratic functions of the velocity or momentum corresponding to one kind of coordi- nate. Each of these five sets must exist in actual physical systems if there is anything analogous to inertia in the phenomena which the corresponding coordinates typify. Again, there are ten sets of terms of the type (xy) xy or (x) x, involving the product of two velocities or a velocity and a momentum of two coordinates of different kinds. To determine whether any particular term of this type exists or not we must determine what the physical conse- quences of it would be ; if these are found to be contrary to experience we conclude that this term does not exist. 24 DYNAMICS. 20. We can determine the consequences of the exist- ence of a term of this kind in the expression for the kinetic energy in the following way. Let us suppose that we have a term in the modified Lagrangian function of the type where/ and q may be any of the five kinds of coordinates we are considering. Then we have by Routh's modification of Lagrange's equations idL' _dL' _ dt dp ~ ~ where P is the external force of type/ acting on the system. Thus the effect of the term is equivalent to the existence of a force of the type / equal to that is, -{(/?) ? + |(/?)? 2 + 2|.(/ ? )^ ...... (14); a force of type q equal to that is, - {(#)/ + | (/?)/+ S^(/ ? )r/} ...... (15); and a force of type r equal to LAGRANGIAN FUNCTION. 25 that is, pq-fc (pq) (16); where r is a coordinate of any type other than that of poiq. Each of the terms in these expressions would correspond to some physical phenomenon ; and as it is clearer to take a definite case to illustrate this, let us suppose that p is the geometrical coordinate symbolized by x, and q the electrical coordinate y. Then if the term (xy) xy occurred in the expression for the Lagrangian function, the mechanical force produced by a steady current would not be the same as that produced by a variable one momentarily of the same intensity. This is so because by the expression (14) there is the term (xy)y in the expression for the force of type x, that is the mechanical force, and as y is zero if the current is steady, there would be a mechanical force depending on the rate of variation of the current if this term existed. Again, we see from the term -=- (xy) y 2 in the expression (14), remembering that p stands for x and q for y, that if (xy) were a function of y the current would produce a mechanical force proportional to its square, so that the force would not be reversed if the direction of the current was reversed. Or again, if we consider the expression for the force of type y or q t that is the electromotive force, we see that the existence of this term implies the production of an electro- motive force by a body whose velocity is changing, depending upon the acceleration of the body ; this is shown by the existence of the term (xy) x in (15), the expression for the electromotive force. 26 DYNAMICS. If (xy) were a function of x, the term (xy) x 2 in (15) shows that a moving body would produce an electromotive force proportional to the square of its velocity, and therefore one that would not be reversed when the direction of motion of the body was reversed. As none of these effects have been observed we conclude that this term does not exist in the expression for the Lagrangian function of a physical system (see Maxwell, Electricity and Magnetism, 574). 21. We shall now go through the various types of terms which involve the product of the velocities of two coordi- nates of different kinds, or a velocity of one kind and the momentum of another, in order to see whether they exist or not in the expression for the Lagrangian function. The reasoning to be used is of the same nature as that just given, and we may leave it to the reader to show by the consideration of the expressions (14) and (15) that the existence of the several terms carries with it the con- sequences we describe. Taking the terms in order we have 1. Terms of the form (xy) xy. We have just seen that terms of this kind cannot exist in the expression for the Lagrangian function. See also Max- well, Electricity and Magnetism, n. part iv. chap. 7. 2. Terms of the form (xw) xw. Terms of this form may exist in the case of a vibrating solid body which is also moving as a whole. For the velocity of any point in the solid equals the velocity of the centre of gravity plus the velocity of the point relatively to its centre of gravity. This latter velocity will involve w, so that the LAGRANGIAN FUNCTION. 2/ square of the velocity and therefore the kinetic energy may involve xw. 3. Terms of the form (xrj) xri cannot exist, for we can prove that they would involve the existence of a magnetizing force in a moving body depend- ing upon the acceleration of the body. It would also require that the mechanical force exerted by a magnet should depend upon the rate of change of the magnetization. None of these effects have been observed. 4. Terms of the form (*) ^ apparently do not exist, for they would require that the mechanical force exerted by a magnet should depend upon the rate of variation of the magnetic intensity, and this effect has not been observed. 5. Terms of the form (yw) yu>. If these terms existed it would be possible to develop electromotive forces by vibration, and these forces would depend upon the acceleration of the vibration and not merely upon the velocity ; as these have not been observed we conclude that this term does not exist in the Lagrangian function of physical systems. 6. Terms of the form (yn) yn- If these terms existed there would be electromotive forces depending upon the rate of acceleration of the changes in the magnetic field. They also indicate magnetic forces depending upon the rate of change of the current. As neither of these effects 28 DYNAMICS. have been observed we conclude that terms of this form do not exist in the expression for the Lagrangian function. 7. Terms of the form 00 # Terms of this kind only involve the production of an electromotive force in a varying magnetic field, the electro- motive force varying as the rate of change of the magnetic field. This is the well-known phenomenon of the produc- tion of an electromotive force round a circuit whenever the number of lines of magnetic force passing through it is changed. As the term we are considering is the only one in the Lagrangian function which could give rise to an effect of this kind without also giving rise to other effects which have not been verified by experience, we conclude that this term does exist. 8. Terms of the form (wrj) wfi. If we take any molecular theory of magnetism, such as Ampere's, where the magnetic field depends upon the arrangement of the molecules of the body, we should rather expect this term to exist. The consequences of its existence have however not been detected by experiments. If this term existed, then considering in the first place its effect upon the magnetic configuration we see that a vibrating body should produce magnetic effects depending upon the vibrations. Secondly, considering the effects of this term on the strain configuration we see that there should be a distorting force depending upon the rate of acceleration of the magnetic field. As neither of these effects have been observed there is no evidence of the existence of this term. LAGRANGIAN FUNCTION. 29 9. Terms of the form (wg) w. These would involve the existence of distorting forces depending upon the rate of change of the magnetic field, and we have no evidence of any such effect. 10. Terms of the form (of) tf . If we assume Ampere's hypothesis of molecule currents this term is of the same nature as the term (xg) xg which we discussed before, so that unless the properties of these molecular circuits differ essentially from those of finite size with which we are acquainted this term cannot exist. 21. Summing up the results of the foregoing considera- tions, we arrive at the conclusion that the terms in the Lagrangian function which represent the kinetic energy depending upon the five classes of coordinates we are considering must be of one or other of the following types : (xx) x 2 (17). (ww) w* (m)^ () f (xw)xw 22. We might make a model with five degrees of freedom which would illustrate the connection between these phenomena which are fixed by coordinates of five types. And if we arrange the model so that its configuration being defined by the five coordinates x, y, w^ 77, , only those terms which are in the expression (17) shall exist in 3O DYNAMICS. the expression for its kinetic energy, and make the potential energy of the model corresponding to each coordinate analogous to that possessed by the physical system, then the working of this model will illustrate the interaction of phenomena in electricity, magnetism, elasticity &c., and any phenomenon exhibited by the model will have its counter- part in the phenomena exhibited in these subjects. When however we know the expression for the energy of such a model, there is no necessity to construct it in order to see how it will work, as we can deduce all the rules of working by the application of Lagrange's Equations. And from one .point of view we may look upon the method we are using in this book as that of forming, not a model, but the expression for the Lagrangian function of a model every property of which must correspond to some actual physical phenomenon. CHAPTER IV. DISCUSSION OF THE TERMS IN THE LAGRANGIAN FUNCTION. 23. WE must now proceed to examine the terms in the expression (17) more in detail, and find what coordinates enter into the various coefficients (xx), (yy) When we have proved that these coefficients involve some particular coordinates we must go on to see what the physical consequences will be. In this way we shall be able to obtain many relations between the phenomena in electri- city, magnetism and elasticity. 24. The first term we have to consider is {xx} x 2 , which corresponds to the expression for the ordinary kinetic energy of a system of bodies. We know that {xx} may be a function of the geometrical coordinate typified by x, but we need not stop to consider the consequences of this as they are fully developed in treatises on the Dynamics of a System of Rigid Bodies. Next {xx} may involve the electrical coordinate y, for in a paper "On the Effects produced by the Motion of Electrified Bodies," Phil. Mag. Apr. 1881, I have shown that the kinetic energy of a small sphere of mass m charged 32 DYNAMICS. with a quantity of electricity e and moving with a velocity v 15 a where a is the radius of the sphere and /u, the magnetic permeability of the dielectric surrounding it. The existence in the kinetic energy of this term, which is due to the "displacement currents" started in the surrounding dielectric by the motion of the electrification on the sphere, shows that electricity behaves in some respects very much as if it had mass. For we see by the expression (18) that the kinetic energy of an electrified sphere is the same as if the mass of the body had been increased by 4fjLe 2 /i$a. Thus whenever a moving body receives a charge of electricity its velocity will be impulsively changed, for the momentum will remain constant, and as the apparent mass is suddenly increased the velocity must be impulsively diminished. The apparent increase in mass cannot exceed a very small quantity because air or any other dielectric breaks down when the electric force gets very intense. If we take 75 as the intensity in electrostatic measure in C.G.S. units of the greatest electric force which a fairly thick layer of air can stand, which is the value given by Dr Macfarlane (Phil. Mag., Dec. 1880), we have, since the electric force at the surface of the sphere must be less than 75, K being the specific inductive capacity of the medium. So that the ratio of the increase in mass to the original mass, which by (18) is equal to 15 a ELECTRIFIED SPHERES. 33 cannot exceed K* 1500 and since in air i / ^K= 9 x io 2 we see that the ratio cannot exceed i . 6 x icT 18 a 3 /m, or about 4 x io~ 19 /p, where p is the mean density of the substance enclosed by the electrified surface. Thus the alteration in mass, even if the mean density inside the surface is as small as that of air at the atmospheric pressure and oC., is only about 5 x io~ 16 of the original mass, and is much too small to be observed. Let us now consider the electrical effects of this term. Let Q be the electromotive force acting on the sphere. The energy of the system, using the same notation as before, is 1 2 uu?' 2 /A ee' \ dx' 36 DYNAMICS. is constant, along with symmetrical expressions for the y and z coordinates. 26. Other electrical theories besides Maxwell's lead to the conclusion that the coefficient [xx] is a function of the electrification of the system. Thus according to Clausius' theory (Crelle, 82, p. 85) the forces between two small electrified bodies in motion are the same as if, using the same notation as before, there was the term , cos e ee' in the expression for the Lagrangian function. The first of these is the same as the term we have just been considering. The forces which according to Weber's theory (Abhand- limgen der Koniglich Sdchsischen Gesellschaft der Wissen- schaften, 1846, p. 211. Maxwell's Electricity and Magnetism, 2nd Edit. vol. n. 853) exist between two electrified bodies in motion may easily be shown to be the same as those which would exist if in the Lagrangian function there was the term ee'(x-x' .. y-y , z z' .. Y ee (U - U') (v - ,<) + __._ (W _ ,) _ _ , -g __._ where x, y, z, x, /, z' are the coordinates of the centres of the electrified bodies and w, v, w, u', v', w' the components of their velocities parallel to the axes of coordinates. This term leads however to inadmissible results, as we can see by taking the simple case when the bodies are moving in the same straight line which we may take as the axis of x. In this case the term in the kinetic energy reduces to ee' or -- (u* -2?tu'+ u' 2 ) K ELECTRIFIED SPHERES. 37 so that the electrified bodies will behave as if their masses were in consequence of the electrification increased by zeelR since the coefficients of u* and it' 2 are each increased by half this amount. Hence if we take e and / of opposite signs and suppose the electrifications are great enough to make zee'lR greater than the masses of one or both of the bodies, then one of the bodies at least will behave as if its mass were negative. This is so contrary to experience that we conclude the theory cannot be right. This consequence of Weber's theory was first pointed out by v. Helmholtz ( Wissenschaftliche Abhandlnngen, i. p. 647). The forces which according to Riemann's theory, given in his posthumous work Schwere, Elektricitdt und Magnetis- mus, p. 326, exist between two moving electrified bodies may easily be shown to be the same as those which would exist if there were the term in the expression for the Lagrangian function. We can easily see that this theory is open to the same objection as Weber's, that is, it would make an electrified body behave in some cases as if its mass were negative. 27. If we regard the expression for the kinetic energy from the point of view of its bearing on electrical phenomena we shall see that it shows that if we connect the terminals of a battery to two spheres made of conducting material, the quantity of electricity on the spheres will depend upon their velocities. We see from the expression (22) for the kinetic energy of a moving conductor that if we have a number of conductors moving about in the electric field there will be a positive term in the Lagrangian function depending upon the square of the electrification. And the same is true to a smaller 38 DYNAMICS. extent if the moving bodies are not conductors but substances whose specific inductive capacity differs from that of the surrounding medium. This is equivalent to a decrease in the potential energy produced by a given electrification, since an increase in the potential energy corresponds to a decrease in the Lagrangian function. Thus the presence of the moving conductors is equivalent to a diminution in the stiffness of the dielectric with respect to alterations in its state of electrification. And therefore the speed with which electrical oscillations are propagated across any medium will be diminished by the presence of molecules moving about in it ; the diminution being pro- portional to the square of the ratio of the velocity of the molecules to the velocity with which light is propagated across the medium. Thus if the electromagnetic theory of light is true the result we have been discussing has an important bearing on the effect of the molecules of matter on the rate of propagation of light. 28. We can see that {xx} may be a function of the strain coordinates, for let us take the case when {xx} is the moment of inertia of a bar about an axis through its centre : then it is evident if the bar be compressed in the middle and pulled out at the ends that the moment of inertia will be less than if the bar were unstrained, for the effect of the strain has practically been to bring the matter forming the bar nearer to the axis. Thus the moment of inertia and therefore {xx} may depend upon the strain coordinates. These coordinates will in general only enter {xx} through the expression for the alteration in the density of the strained body, i.e. through da dB dy +-+-/- (23), dx dy dz and this will only enter {xx} linearly. ELECTRIC CURRENTS. 39 If we form the equations of elasticity by using Hamilton's principle 3 (T- V) dt = o Jta \ve shall easily find that the presence of (23) in {xx} leads to the introduction of the so-called "centrifugal forces" into the equation of elasticity for a rotating elastic solid. This however we shall leave as an exercise for the reader. 29. Let us now consider that part of the Lagrangian function which depends upon the velocities of the electrical coordinates, i.e. the part denoted by Let us take the case of two conducting circuits whose electrical configuration is fixed by the coordinates y } , y 2 , where j,, j> 2 are the currents flowing through the circuits respectively. This part of the Lagrangian function may in this case be conveniently written Now we can fix the geometrical configuration of the two circuits if we have coordinates which can fix the position of the centre of gravity and the shape and situation of the first circuit, the shape of the second circuit and its position relatively to the first. Let us denote by x 2 - x l any coordinate which helps to fix the position of one circuit relatively to the other, and by ,, 2 coordinates helping to fix the shape of the first and second circuits respectively. It is evident that the kinetic energy must be expressible in terms of these coordinates, for the only coordinates neces- sary to fix the system which we have omitted are those fixing the centre of gravity and situation of the first circuit, and 40 DYNAMICS. since a motion of the whole system as a rigid body through space cannot alter this part of the kinetic energy of the system, the expression for the kinetic energy cannot involve these coordinates. If we write for a moment x instead of x z - x l (a coordi- nate helping to fix the position of one circuit relatively to the other) then by Lagrange's Equations we see that these terms in the kinetic energy correspond to the existence of a force tending to increase x equal to i dL . dM . i dN . --X (24) - We see from this expression that dLldx, and dNIdx must vanish, otherwise there would be a force between the two circuits even though the current in one of them vanished. The quantities L and N are by definition the coefficients of self-induction of the two circuits, and hence we see that the coefficient of the self-induction of a circuit is independent of the position of other circuits in its neighbour- hood and is therefore the same as if these circuits were re- moved 1 . By (16) the force tending to increase x is dM . . that is there is a force between the two circuits proportional to the product of the currents flowing through them, and also to the differential coefficient with respect to the coordinate along which the force is reckoned of a function which does not involve the electrical coordinates. This corresponds exactly to the mechanical forces which are actually observed 1 This is quite consistent with the apparent diminution in the self- induction caused by a neighbouring circuit when an alternating current is used. ELECTRIC CURRENTS. 4! between the circuits, and a little consideration will show that these forces could not arise from any other terms in the Lagrangian function. Thus the consideration of the mechan- ical forces which two circuits carrying currents are known to exert upon each other proves that the term My\y 2 exists in the expression for the Lagrangian function. Let us now go on to consider the effect of these terms on the electrical configuration of the two circuits. By Lagrange's Equation for the coordinate^, we have d dL 1 dL' where Y } is the external electromotive force tending to increase y^. Now as we shall prove directly dL'\dy\ = o, so that the effects on the electrical configuration of the first circuit, arising from the term are the same as would be produced by an external electro- motive force tending to increase y } equal to -~(l^ + My,) ..................... (26). Thus if any of the four quantities Z, M, j\, y. 2 vary in value there is an electromotive force acting round the circuit through which the current jy, flows. And the expression (26) gives the E. M. F. produced either by the motion of neighbouring circuits conveying currents or by alterations in the magnitudes of the currents flowing through the circuits. This example is given in Maxwell's Electricity and Magnetism, vol. n. part iv. chapter vi., and it is one -which illustrates the power of the dynamical method very well. The existence of the mechanical force shows that there is 42 DYNAMICS, the term My* y in the expression for the Lagrangian function and then the law of the induction of currents follows at once by the application of Lagrange's Equations. The problem we have just been considering is dynamic- ally equivalent to finding the equations of motion of a particle with two degrees of freedom when under the action of any forces. We know that these cannot be deduced by the aid of the principle of the Conservation of Energy alone, for to take the simplest case of all, that in which no forces act upon the particle, the principle of the Conservation of Energy is satisfied if the velocity is constant whether the particle moves in a straight line or not. From this analogy we see that when we have two circuits the principle of the Conservation of Energy is not sufficient to deduce the equations of motion, and that some other principle must be assumed implicitly in those proofs which profess to deduce these equations by means of the Conservation of Energy alone. 30. There is no experimental evidence to show that {yy} is a function of the electrical coodinates y, and it certainly is not when the electrical systems consist of a series of conducting circuits, for if it were the coefficients of self and mutual induction would depend upon the length of time the currents had been flowing through the circuits. And in any case it would require the existence of electro- motive forces which would not be reversed if the direction of all the electric displacements in the field were re- versed. 31. Similar reasoning will show that {yy} cannot be a function of the magnetic coordinates, for if it were there would be magnetic forces produced by electric currents EFFECT OF STRAIN. 43 which would not be reversed if the directions of all the currents in the field were reversed. 32. We- must now consider whether {yy} is a function of the strain coordinates or not. If it is then the coefficients of self and mutual induction of a number of circuits must depend upon the state of strain of the wires forming the circuits. This result though not impossible has never been detected, and it is contrary to Ampere's hypothesis that the force exerted by a current depends only upon its strength and position and not upon the nature or state of the material through which it flows. Then again, if we consider what the effect on the elastic properties of the substance would be if {yy} were a function of the strain coordinates, we see at once that it would indicate that the elastic properties of a wire would be altered while an electric current was passing through it. The evidence of various experimenters on this point is somewhat conflicting. Both Wertheim (Ann. de Chim. et de Phys. [3] 12, p. 610, Wiedemann's Elektricitcit, n. p. 403) and Tomlinson have observed that the elasticity of a wire is diminished when a current passes through it and that this diminution is not due to the heat generated by the current. Streintz ( Wien. Ber. [2] 67, p. 323, Wiedemann's Elektrititat, n. p. 404) on the other hand was unable to detect any such effect. But even if this effect were indisputably established it would not prove rigorously that {yy} is a function of the strain coordinates, for as we shall endeavour to show when we consider electrical resistance this effect might have been due to another cause. To sum up we see that {yy} is a function of the geometrical coordinates but not of the electric or magnetic ones and probably not of the strain ones. 44 DYNAMICS. 33. We shall now consider the part of the Lagrangian function which depends upon the magnetic coordinates and which does not involve the velocities of the geometrical, electrical or strain coordinates. Thus the terms we are about to consider in the Lagrangian function of unit volume of a substance are those we have denoted by we may have in addition to these terms arising from the potential energy. In order to begin with as simple a case as possible let us suppose that all the magnetic changes take place indefi- nitely slowly ; in this case we may neglect the term and confine our attention to the terms ijgtf'+jtfte or as it is more convenient to write them (27). Let us take first the case when the magnetization is parallel to one of the axes, x for example, and let us denote the magnetic force parallel to this direction by H and the intensity of magnetization by /, where by definition /=irf ........................ (28). The investigation in 389 of Maxwell's Electricity and Magnetism shows that if we suppose that all the energy in the magnetic field is resident in the magnets, there is in the Lagrangian function for unit volume of a magnet the term HI. The result of this investigation is stated in the Electricity MAGNETIZATION. 45 and Magnetism to be that the potential energy of unit volume of the magnet is -HI, but we have seen in 9 that the question whether energy determined in this manner is kinetic or potential is really left unsettled : what is actually proved is that a certain term exists in the Lagrangian function. If we suppose that the energy is distributed throughout the whole of the magnetic field, including unmagnetized substances as well as magnets, then the investigation in 635 of the Electricity and Magnetism shows that the Lagrangian function of unit volume anywhere in the magnetic field con- tains the term _ where B is the magnetic induction. These two ways of regarding the energy in the magnetic field lead to identical results ; and as we shall for the present confine our attention to the magnetized substances we shall find it more convenient to adopt the first method of looking at the question. We have seen that the Lagrangian function for unit volume of a magnet contains the term HI, or in our notation and this is the term we previously denoted by Mflfc Since the magnetic changes are supposed to take place indefinitely slowly, Lagrange's equation for the >? coordinate reduces to 46 DYNAMICS. Applying this to the expression (27) and substituting for i {} we get ^=o ............... (30), and since is supposed to remain constant and therefore &fy=*r/, this may be written + H = o ............... (31). So that if k be the coefficient of magnetic induction and defined by the equation I we have by (31) -/; and therefore (33). If we know the way in which 7 varies with H we could by this equation express A as a function of /. The relation between / and H is however in general so complicated that there seems but little advantage to be gained by taking some empirical formula which connects the two and determining A by its help. For small values of H, Lord Rayleigh (Phil. Mag. 23, p. 225, 1887) has shown that II H is constant, so by equation (33) A in this case is also constant. 34. The mechanical force parallel to the axis of x acting on unit volume of the magnet is dIJ MAGNETIZATION. 47 The only quantity in the terms we are considering which involves x explicitly is J7, so that dL'Idx reduces to ,dH *-fc T dH or I -^ (34), and this is the mechanical force parallel to the axis of x acting on unit volume of the magnet. This expression may also be written 1 , dH* - k j 2 ax with similar expressions for the components parallel to the axes ofy and z. These are the same expressions for this force as those given in Maxwell's Electricity and Magnetism, vol. IT. p. 70, the consequences of which are as is well known in harmony with Faraday's investigations on the way in which para- magnetic and diamagnetic bodies move when placed in a variable magnetic field. 35. We have just investigated the mechanical forces produced by a magnetic field ; we shall now proceed to investigate some of the stresses produced by it. Let us take the case of a cylindrical bar of soft iron whose axis coincides with the axis of x, and suppose that it is magnetized along its axis. Let e, /, g be the dilatations of the bar parallel to the axes of x, y, z respectively. We shall at present assume that there is no torsion in the bar. We shall suppose that the changes in the strains take place so slowly that we may neglect the kinetic energy arising from them. The- potential energy due to these strains is }* + \n {e 2 +/ 2 +/ - 2ef- 2eg - 2/g\, 48 DYNAMICS. where n is the coefficient of rigidity and m - ;//3 the modulus of compression. Thus the terms in the Lagrangian function involving the magnetic and strain coordinates are - \m (e +/+ g} 2 - \n (e 2 +f 2 + g 2 - 2ef- 2eg neglecting those depending on the rate of variation of these quantities which rate we shall assume to be indefinitely small. The experiments of Villari and Sir William Thomson (Wiedemann's Elektricitdt, in. p. 701) have shown that k depends upon the strain in the magnet, hence by equation (32) A will be a function of the strains. We shall pro- ceed to investigate the stresses which arise in consequence of this. Using the Hamiltonian principle and substituting da-ldx, dfi/dy, dyfdz for e, /, g respectively, we get the following equations by equating to zero the variation caused by changing a into a + Sa dL d dL -dx-dx-de =Q > ................... (35) inside the bar, dL at the boundary. By equating to zero the variation caused by changing ft into ft + 8/3 we get dL d dL inside the bar, dL at the boundary. STRAIN AND MAGNETIZATION. 49 And by equating to zero the variation caused by changing y into y + By we get dL d dL - = (39) inside the bar ; at the boundary. The first and second terms in the equations (35), (37) and (39) may conveniently be considered separately. Since H is the only quantity in the expression for L which can involve the coordinates x, y, or z explicitly the terms dL dL dL dx ' dy ' dz reduce to ^dH .dH .dH ^ ^ or kH ,-, kH -y- , dx dy dz respectively. These are the expressions for the components of the mechanical force acting on the body, and it is shown in Maxwell's Electricity and Magnetism, 642, that this dis- tribution of force would strain the body in the same way as "a hydrostatic pressure H 2 j%ir combined with a tension BHI^T? along the lines of force," B being the magnetic induction. Thus we may suppose that the strains arising from these terms are known. If e, f, g are the strains due to the second term in equations (35) (37) and (39), we have T. D. 4 DYNAMICS. if? -y-m(' +f+g) - n (f-e-g) = o dA if? -- - m (e+f+g] -n(g-e-f) = o -.(41). Solving these equations and putting r; = / we get 2** = -^-4 (AF) -^T n {^( A ^ 2 ) + ^ (4J 2 )} _m^n_(d a r J $mn {de x ' dg ^ ' ) 2m 2Jlg = 2m d m-n (d dg d \ / x (42). and If the magnet is symmetrical about its axis we have */ / , ^ So that equations (42) reduce to m i T ne = 3;^ n de -n df \ .-.(43). - n df The dilatation e + zf is equal to 3;^ - 11 (de ' df Differentiating equations (43) with regard to / 2 , we get approximately since ^ (AI 2 ) and -^.(AI 2 ) must be small compared with m or n, or the changes in the elasticity caused by magnetization would not be so small as to have escaped detection, STRAIN AND MAGNETIZATION. 51 de m d d m-n d d ^ ' ' ( ' _ m-n ^ ' ^ ' ' \ ( ' df m-n d d tjr^.n + nd d (AP\\ ~dl^~Z^-n de dl* ( ' + -^n JfTP^ ' } J Now by equation (32) So that these equations become de m i dk m-n i dk dl* ~ yn-nk* de yn-n k 2 df , . . df _ m-n i dk m + n i dk \ Now if the coefficient of magnetization depends upon the strains, the intensity of magnetization of the bar when under the action of a constant magnetizing force will be altered by strain, and in order to compare the formulae with the results of experiments we shall find it more convenient to express defd! 2 , dfldl* in terms of the changes which take place in the intensity of magnetization when the bar is stretched rather than in terms of dk'de and dkjdf. We have /=//, so that when H is supposed to be constant dl r^dk r - r dk dl and equations (46) may be written de _/ dk\f m i dl m-n i dl\ 1 * dl* \ di)\yn-nklde yn-nkl d/)\ , ox , , . ,, , T i-r r (4w df (IT "k\ ( tn n \ dl m + n i dl\ dl 2 \ dl )\ yn n kl de yn n kl df) j These expressions give the strains which result from the dependence of the intensity of magnetization on the state 42 52 DYNAMICS. of strain of the magnetized body. In addition there are the strains arising from Maxwell's distribution of stress. Kirchhoff (Wied. Ann. xxiv. p. 52, xxv. p. 601) has investigated the effect of this on a small soft iron sphere placed in a uniform magnetic field and has shown that it would produce an elongation of the sphere along the lines of force and a contraction at right angles to them. We may therefore assume that in general this distribution of stress causes an expansion of the magnet in the direction of the lines of force and a contraction in all directions perpen- dicular to this. The expressions for the strains in a magnetizable substance placed in the magnetic field have also been in- vestigated by v. Helmholtz (Wied. Ann. xin. p. 385). The object of the investigations of v. Helmholtz and Kirchhoff was rather different from that of Maxwell. Maxwell's object was to show that his distribution of stress would produce the same forces between magnetized bodies as those which are observed in the magnetic field, while v. Helmholtz and Kirchhoff's object was to show that it follows from the prin- ciple of the Conservation of Energy that, whatever theory of electricity and magnetism we assume, the bodies in the electric or magnetic field must be strained as if they were acted upon by a certain distribution of stress which in the simplest case is the same as that given by Maxwell. We have in addition to the strain produced by these stresses, the strains depending upon the alteration of the intensity of magnetization with stress along and perpen- dicular to the lines of force. The effect of stress along the lines of force on the magnetization of iron has been investigated by Villari (Pogg. Ann. 126, p. 87, 1868) and Sir William Thomson (Proc. Roy. Soc. 27, p. 439, 1878) ; both these physicists found that STRAIN AND MAGNETIZATION. 53 the intensity of magnetization was increased by stretching when the magnetizing force was small, but that when the magnetization exceeds about 10 when measured in C.G.S. units the intensity of magnetization is diminished by stretching. Sir William Thomson also investigated the effect of stress at right angles to the lines of magnetic force on the intensity of magnetization and found that this was in general opposite to that of tension along the lines of force, so that for small values of the magnetizing force extension at right angles to the lines of force diminishes the mag- netization, while for larger values of this force it increases it. The critical value of the force in this case however is higher than that for tension along the lines of force. Thus, except when the magnetizing force is between the critical values, dljde and dlldf have opposite signs, hence we see by equation (48) that except in this case, since Prof. Ewing's measurements show that Hdkldl is always less than unity, de , dl dP and de have the same sign, and df have opposite signs. Now dllde is positive or negative according as the magnetizing force is less or greater than the critical value, so that when the magnetizing force is less than the critical value the extension we are investigating will increase with the magnetic force, but when the magnetizing force is greater than this value the extension will diminish as the force increases. As we mentioned before the strain produced by Maxwell's distribution of stress, which is the other cause tending to 54 DYNAMICS. strain the body, has been shown by Kirchhoff to produce an expansion along the lines of force and a contraction at right angles to them. Thus when the magnetizing force is less than the critical value this strain and the strain we have just investigated act in the same way, but when the force is greater they act in opposite directions. Joule's investigations (Phil. Mag. 30, pp. 76, 225, 1847) prove that the length of an iron bar increases when it is magnetized and as far as the experiments went the increase in the length was proportional to the square of the magnetizing force. Mr Shelford Bidwell (Proc. Roy. Soc. XL. p. 109) however has lately shown that when the mag- netizing force is very large the magnet shortens as the magnetizing force increases. Comparing these experimental results with our theoretical conclusions we see that they are in accordance when the magnetizing force is small, and that when the magnetizing force is large they indicate that the strains due to the same cause as that which causes the intensity of magnetization to alter with strains are more powerful than those arising from Maxwell's distribution of stress. Prof. Ewing's experiments on the effect of strain on magnetization (" Experimental Researches in Magnetism," Phil. Trans. 1885, part n. p. 585) would seem to show that this must be the case. For Kirchhoff (Wiedemann's Annalen, xxv. p. 60 1) has shown that the greatest increase in length which Maxwell's stresses can produce in a soft iron sphere whose radius is R, placed in a uniform magnetic field where the force at an infinite distance from the sphere is H, is '53 H* R I 7 67T E where E is one of the constants of elasticity for soft iron and is equal in the c. G. s. system of units to r8x io 12 . STRAIN AND MAGNETIZATION. 55 Thus in this case supposing k to be constant we have de i53 i 1- Now according to Prof. E wing's experiments the intensity of magnetization of a soft iron wire which was represented by 181 when there was no load was increased to 237 when the wire was loaded with a kilogramme, so that in this case 87 i 7 = -.--nearly ............ (50). The diameter of the wire was such that the load of a kilogramme corresponded to a stress of about 2 x io 8 per square centimetre in C.G.S. units, so that if q be Young's modulus for the wire and Se the extension produced by the load q^e = 2 x i o 8 : for wrought iron q\n is about 2-5, so that nBe 8 x io 7 and therefore by (49) \_dl__ i nl de 2 '4 x io 8 ' so that by equation (48) if e be the elongation due to the magnetization de i i , , 7xio 8 l " Comparing this with (49) we see that the part of de\dl* due to the cause we are now considering is very much greater than that due to Maxwell's distribution of stress. The value of dl\de is probably exceptionally large in this case, and near the critical value it is doubtless very much less, so that in this case it is conceivable that the effect of the Maxwell 5 6 DYNAMICS. stress may be comparable with that due to the alteration of intensity of magnetization with strain. Since the Maxwell effect is in general so small compared with the other we should expect the critical value of the magnetizing force to be approximately the same as the value of the force when the extension is a minimum ; it is however much less. There seems however to be reason to think that the critical value when the magnet is free from strain has been very much underestimated. Indeed Prof. Ewing (loc. tit.) expresses his opinion that "if we deal only with very small stresses it is doubtful whether any reversal of the positive effect of stress would be reached even at the highest obtainable value of the magneti- zation." By the positive effect of stress Prof. Ewing means an increase of magnetization with an increase of stress, the magnetizing force remaining constant. Bid well's discovery that dejdl* is negative when the magnetization exceeds a certain value, in conjunction with the theoretical results we have been investigating in this paragraph, shows that when the magnetization reaches this value the positive effects of stress must be reversed. The magnet in this case however is not free from stresses as it is acted on by those called into play by the magnetization. 36. If the dilatation in volume e + 2/ be denoted by 8, then the part of 8 due to the same cause as that which makes the intensity of magnetization depend upon strain is by (48) given by the equation must be negative when / is large and hence that the co- efficient of in A must be negative. Let us call this coefficient /, the coefficient of c 2 in the Lagrangian function is . but the apparent coefficient of rigidity is twice the coefficient of - f in the Lagrangian function so that in this case the apparent coefficient of rigidity is Thus in this case the effect of strong magnetization is to increase the rigidity, so that the same couple will not twist the wire as much when it is strongly magnetized as when it is unmagnetized. When the intensity of magnetization is small the opposite will be the case, as in this case the twist in a wire increases when it is longitudinally magnetized. Since =- TORSION AND MAGNETIZATION. 6l and by (47) and also we have dc -JTT - - -j* < = r- r -J- \--T dl \ 11 dc v ' } nkl dc \ dc or approximately since d*(AI z )lndc 2 is very small dc i dl / __ ^\ ............ (6o) " We see by this equation that when the magnetization is so strong that magnetizing the wire diminishes the twist in it, then twisting the wire will diminish the intensity of magnetization. On the other hand when the intensity of magnetization is so small that magnetizing the wire increases the twist in it then twisting the wire will increase the intensity of magnetization. The reciprocal relations between torsion and magnetiza- tion have been experimentally investigated by Wiedemann (Lehre von der Elektriritat, in. p. 692) and he arranges the corresponding results in parallel columns. These are also quoted in Prof. Chrystal's article on Magnetism in the Encyclopedia Britannica. The following is one set of the corresponding statements. " 5. If a wire under the influence of a twisting strain is magnetized, the twist increases with weak but diminishes with strong magnetization." "V. If a bar under the influence of a longitudinal magnetizing force is twisted the magnetization increases with small twists but decreases with large ones." Comparing these statements with the results we have previously obtained we see that whether the first part of V is 62 DYNAMICS. true or not depends upon the intensity of magnetization. If the twist be of such a magnitude that 5 is true, then the first part of V is true if the magnetization is weak, but the opposite is true if the magnetization is strong. Further since by V the influence of twist on magnetization depends upon the size of the twist, it follows by equation (60) that the influence of magnetization upon twist must depend upon the size of the twist so that 5 is only true when the twist is on one side of a critical value, when it is on the other side the contrary is true. The existence of a critical twist as well as a critical magnetization makes the verbal enunciation of the relations between torsion and magnetization cumbrous; they are all however expressed by equation (60). 39. Strains in a dielectric produced by the electric field. The strains produced in a dielectric by the electric field can be found by a method so similar to that used in the last two paragraphs that we shall consider them here though they have no connexion with the terms in the Lagrangian function which we have been considering. Let/, , r be the electric displacements parallel to the axes of x t y, z respectively, then if the body is isotropic, the terms in the Lagrangian function of unit volume of the dielectric which depend upon the coordinates fixing the strains and electric configuration if the dielectric is free from torsion are, where e, /, g are the dilatations parallel to the axes of x, y, z respectively, ^Tthe specific inductive capacity of the dielectric and X, Y, Z the electromotive forces parallel to the axes STRAIN AND MAGNETIZATION. 63 of x t y, z. Then we see as in 35 that e,f, g the strains due to the dependence of K upon the strains in the dielectric are given by the equations = - Z = o ~ = o where L stands for the expression (61). Substituting for L its value these equations become re- spectively 27r{/ 2 + q* + r 2 } ^ ~ 4- ;// (e +/+ g) + n (e -f-g) - o n (f-. e -X\ = Q v. (62). \*/ 3 / / \ / Now / = KX, q=~ KY, r = -~ KZ. 47r 4?r 4?r So that if R 2 =X 2 + Y 2 + Z 2 , we get from equations (62) dK , x (dK . dK\\ de , . ,, . -(m-n) (-^. + -- )\ -..(63) \df dgj) with symmetrical expressions for g and //. The expansion in volume is given by the equation dK dK Just as in the analogous case of magnetism these are not the only strains produced in the dielectric by the electric field. The term (Xp + Yq + Zr) which occurs in the Lagrangian function can be shown to involve the same 64 DYNAMICS. distribution of strain in the dielectric as would be produced by the distribution of stress which Maxwell supposes to exist in the electric field, viz. a tension KR 2 !^ along the lines of force and a pressure of the same intensity at right angles to them. The effect of this distribution of stress will be of the same character for all dielectrics, and its nature depends more upon the distribution of force throughout the electric field than upon the nature of the dielectric. The experiments of Quincke (Phil. Mag. x. p. 30, 1880) and others show that the behaviour of different dielectrics when placed in the same electric field is very different. Thus, for example, though most dielectrics expand when placed in an electric field, the fatty oils on the contrary contract. This difference of behaviour shows that in many cases at any rate, the strains due to the same cause as that which makes the specific inductive capacity depend upon the strain are greater than those produced by Maxwell's distribution of stress. Quincke has shown that the coefficients of elasticity of a dielectric are altered when an electric displacement is pro- duced in it, this shows that \IK when expanded in powers of e must contain a term in e z and is another proof that the specific inductive capacity depends upon the strain in the dielectric. Since part of the strain of a dielectric in an electric field is due to the same cause as that which makes the specific inductive capacity depend upon strain, the expression for i/^when expanded in powers of e must con- tain the first power of the strains as well as the second, as if it only contained the second powers placing the dielectric in an electric field would merely be equivalent to changing the coefficients of elasticity of the body and so could not strain the body if it were previously free from strain. No experiments seem to have been made to determine MAGNETIC INERTIA. 65 directly the values of dKlde, dK/df&c., and the experimen- tal difficulties which would have to be overcome in order to do this are much greater than those in the corresponding case in magnetism. The dependence of K upon strain is probably much less than that of /, the coefficient of magnetic induction. For the specific inductive capacity seems to be much more independent of the molecular state of the dielectric than the coefficient of magnetic induction is of the molecular state of soft iron. Thus there is a com- paratively small difference between the specific inductive capacities of various substances, while the coefficient of magnetic induction of iron is enormously greater than that of any other substance. Again, the coefficient of magnetic induction is known to be much affected by changes in temperature; while some recent experiments made by Mr Cassie in the Cavendish Laboratory have shown that the effect of changes of temperature on the specific inductive capacities of ebonite, mica and glass is small, amount- ing in the case of glass, for which it is largest, to i part in 400 for each degree centigrade of temperature. No experi- ments seem to have been made on the effect of torsion on electrification or of electrification upon torsion. 40. Influence of inertia on magnetic pheno- mena. In the preceding investigations we have supposed the magnetic changes to take place so slowly that the effects of inertia may be neglected. If however a change in the magnetization involves, as it does according to all molecular theories of magnetism, motion of the molecules of the magnet, then magnetism must behave as if it possessed inertia. In soft iron and steel the conditions are made so com- plex by the effects of magnetic friction, magnetic retentive- ness and permanent magnetism, that it would be difficult to T.D. 5 66 DYNAMICS. disentangle the effects of inertia proper from other compli- cations. The effect, if it exists, would probably be detected most easily in the case of crystals, as only one of these, quartz, has ever been suspected of showing residual mag- netism (see Tumlirz, Wied. Ann. xxvn. p. 133, 1886). The effect of inertia would be to introduce into the equations of magnetization a term as** 1 M w* where / is the intensity of magnetization. The equations of magnetization would therefore be of the form where 77 is the external magnetic force. If ffis periodic and varies as e ipt then by (65) (60, so that if/ be so large that kMp 2 >i, the crystal if para- magnetic for a steady magnetic force will be diamagnetic for a variable one and vice versa. Changes of this kind could be detected very readily if the crystal were freely suspended in the magnetic field, for when / 2 passed through the value \lkM the crystal would swing through a right angle. 41. The term (y) y in the Lagrangian func- tion. We have considered the terms depending upon the squares of the velocities of the electrical coordinates, and those depending solely on the magnetic coordinates, let us now consider those terms in the expression for the kinetic energy which involve the product of the velocities of a mag- netic and an electrical coordinate. It is proved in Maxwell's Electricity and Magnetism ELECTROMOTIVE FORCE. 6/ { 634) that when a current whose components are #, v, w flows through the element of volume dxdydz and the volume dxdy'dz' is magnetized to the intensities A, B, C parallel to the axes of x, y, z respectively, then the kinetic energy L possessed by the system is where / is the reciprocal of the distance between the elements dxdydz and dx'dy'dz. Now we represent the intensity of magnetization by TJ where is the momentum corresponding to a kinosthenic or speed coordinate and rj is a vector quantity. Since rj is a vector quantity it may be resolved into com- ponents parallel to the axes of x, y, z. Let us denote these components by X, //,, v respectively, then we may put Making this substitution we have . v f\ +v ( v ^. x ^ dz' dy } \ dx' dz So that these terms are of the form Considering the Lagrangian Equation for the electrical coordinate, we see that there is an electromotive force parallel to the axis of x on the element dxdydz equal to dtdu 9 52 68 DYNAMICS. so that, per unit volume, this force equals -'*{*%- c with corresponding expressions for the electromotive forces parallel to the axes of y and z. These are the usual expressions for the electromotive forces due to the variations of the magnetic field. The magnetic force parallel to x acting on the element dx'dy'dz' is by 33 equal to idL d\ so that the magnetic force parallel to x per unit volume is equal to * -*** ............... (7o)> with similar expressions for the magnetic forces parallel to the axes of y and z. These expressions agree with those given by Ampere for the magnetic force produced by a system of currents. Again there is a mechanical force acting on the element dxdydz whose component parallel to the axis of x is d_L Jx' If we call . G, H respectively, then F, G, H are the same as the MECHANICAL FORCE DUE TO A CURRENT. 69 quantities denoted by the same symbols in Maxwell's Electricity and Magnetism. Since the force on the element dxdydz is dL dx' we see that the force on unit volume may be written dF dG dH U -- + v -j + W r- , dx dx dx or (dG dF\ (dF dH\ dF dF dF V\ . -j- \-W \-j } + U -j- + V-j- + W- r ...( f J2\ {dx dy } {dz dx } dx dy dz w This differs from Maxwell's expression for the same force by the term dF dF dF U +V -j- +W -T-. dx dy dz du dv dw Since -j- + -j- + --j- = o ax ay dz it follows that MdF dF dF\ , u +v -j- +w ^ ) dxdydz = o dx dy dz) if all the circuits are closed. So that as long as the circuits are closed the effect of the translator}' forces is the same as if they were given by Maxwell's expressions. In the above investigation we have assumed that we could move the element without altering the current ; if we suppose the current to move with the elements we shall get Maxwell's expression exactly. The components parallel to y and z of the force on the ele- ment dxdydz are given by expressions corresponding to (72). The force parallel to x on the magnetized volume Jx'dy'dz', is *L dx' 7O DYNAMICS. so that the force parallel to oc per unit volume is with corresponding expressions for the forces parallel to y and z. Thus the force on the magnet is equal and opposite to that on the current. We see by this example how from the existence of a single term in the expression for L we can deduce the laws of the induction of currents, the production of a magnetic field by a current, the mechanical force on a current in a magnetic field and the mechanical force on a magnet placed near a current. 42. Twist in a magnetized iron wire produced by a current. Prof. G. Wiedemann (Elektricitat, in. p. 689) has shown that when a current flows along a longi- tudinally magnetized wire, it produces a couple tending to twist the wire. This shows that there must be a term in the Lagrangian function for the wire of the form /*rf ........................ (74), where y is the current flowing along the wire, >; the intensity of magnetization /, and c the twist about the axis of the wire,/(<:) being some function of c. Applying Hamilton's principle to this term we see that it indicates the existence of a couple tending to twist the wire equal to >l ........................ (75). Applying Lagrange's equation for the ^coordinate to this term we see that since the electromotive force tending to TORSION AND MAGNETIZATION. 7 1 increase y is =*K) 'dp\ f v\ dk when y is constant, k is also constant so that koV o and therefore from (93) >= v Ji-l\ d d A vj dy k I dk so that if the pressure is increased from P l to P 2 the increase E in the electromotive force required to keep the current constant is given by T fib *E=\v a \P*-P?\ ............ (94). To get an idea of the magnitude of this effect let us take the case of a solution of chloride of lithium, the volume of the solution being i cubic centimetre. The data for calculating dkjdy in this case are the fol- lowing : The passage of unit quantity of electricity corresponds to the decomposition of about 4'3xio~ s grammes of lithium chloride, we shall suppose that none of this is redis- solved, then the passage of a unit quantity of electricity will withdraw this quantity of salt from the solution. Rontgen's and Schneider's experiments (Wiedemann's Annalen, xxix. p. 186, 1886) show that the addition of 6 grammes of lithium chloride to 100 cubic centimetres of 62 84 DYNAMICS. water increases the modulus of compression by about 15 parts in 100, so that if the increase in the modulus is proportional to the quantity of salt, then the subtraction of 4-3 x 10 ~ 3 grammes from i cubic centimetre will diminish the modulus by about i part in 100, hence i dk T -r =- 10 , approximately. Now k for water is about 2-2 x io 10 , so that if B is the change produced by a pressure of 1000 atmospheres, which in absolute measure is about io 9 , we have TO 18 T & = -J- ^ TO 2 = -Jxi0 6 , 2 2'2 X IO 10 IO 2 that is the pressure of 1000 atmospheres would diminish the counter electromotive force by about 1/400 of a volt. The numbers given by Rontgen and Schneider for the effect of carbonate of soda on the coefficient of compres- sibility, show that the effect of pressure on a solution of this salt would be much greater than that on the lithium chloride solution. Let us now suppose that the volume of the solution is altered by the passage of an electric current, but that the coefficient of compressibility is unaltered. Then since if the passage of the unit of electricity increases the volume by dvjdy we must apply an additional pressure kdvjv Q dy to- keep the volume constant, so that /dp\ \, we see from this equation that the change BE in the counter electromotive force is given by the equation J B = S/g (95)- When the electric current goes through a salt solution the changes which take place and which alter the volume are so numerous that it is not possible to calculate from existing data the change which takes place in the volume when unit quantity of electricity passes through the solution. In order to see of what order this effect is likely to be, let us suppose that the change in the volume is comparable with the volume of the salt electrolysed. When unit quantity of electricity goes through a solution of sulphate of potassium it electrolyses about 9 x io~ 3 grammes of salt, and since the specific gravity of the salt is 2*6, the volume of this is about 3 -5 x zo" 3 , hence in this case we may suppose that dvldy is comparable with 3*5 x io~ 3 and that the change in the counter electromotive force produced by 1000 atmospheres is of the order 3'5 x i 6 , or about 1/28 of a volt. We will now consider the case when gas is given off. Let us suppose we are electrolysing water, above which we have air, enclosed by a cylinder with a moveable piston. If unit quantity of electricity goes through the water, 9 x io~ 4 grammes of water are electrolysed, the volume of the water therefore diminishes by 9x10** cubic centi- metres. At one terminal io~ 4 grammes of hydrogen will be liberated, and 8 x io~ 4 grammes of oxygen at the other. 86 DYNAMICS. Let us proceed to find the change in the pressure, the volume remaining constant when unit of electricity passes. The diminution in pressure due to the disappearance of the water is, if v be the volume of the gas above the water the increase in pressure due to the io~ 4 grammes of hydro- gen is if the temperature is o C. and the increase due to the oxygen is one half of this, hence so that by (95) dE But so that E i,., K _ 41 =- = -- {1-65 x io - 9 x/ x 10 }. dv v l dE i'6q x io 6 - 9 x io . dp p If the pressure is increased from P l to P a the change & in E is given by the equation &E= 1-65 x io 6 x log 5* - 9 x io- 4 x (P 2 -P,). *1 For a thousand atmospheres the counter electromotive force is increased by i '65 x i o 6 x 6*9 - 9 x io~ 4 x io 9 approximately, = i'2 x io 7 9 x io 5 , EFFECT OF SURFACE TENSION ON E. M. F. 8/ so that the counter electromotive force is increased by about one-eighth of a volt. The effect of surface tension is given by I 7 Ql / I 7 I \o / y constant \ #y /^S 1 constant This effect will in general be very small, for example in the case of chloride of lithium, the experiments of Rontgen and Schneider (Wiedemann's Annalen, xxix. p. 209, 1886), show that the addition of 6 parts by weight of lithium chloride to 100 of water increases the surface tension by about 3 parts in 100. The passage of i unit of electricity decomposes about 4 '3 x io~ 3 grammes of lithium chloride, so that if v be the volume of the solution i dT 13 io 2 r- = - X -^-_ X X 4'7 X IO T dy v io 2 6 = - 2 x io~ 3 - approximately, and for water T= 81, so that dT 16-2 x io~ 2 and therefore by (96) dE 16-2 x io or if the volume remains constant the effect of increasing the surface by S is to diminish the counter electromotive force by l6'2 X I0~ 2 6* v Suppose that the liquid is squeezed out into a thin film 88 DYNAMICS. whose thickness is / then v = St and ~ l6'2 X I0~ 2 S= - - If t were of the order of molecular distances say io~ 7 then &" = - 1 6*2 x io 5 , or the counter electromotive force is diminished by about '016 volts. The preceding investigation is on the supposition that the electrolyte is in contact with the air; if it were in contact with a solid such as glass the withdrawal of the electrolyte from the solution on the passage of the current would increase the surface tension between the liquid and the solid, so that the electromotive force required to decompose an electrolyte in a porous plate would be larger than that required to decompose it when it is in bulk. Again, the surface tension of liquids is altered when they absorb gases, so that the electromotive force required to decompose an electrolyte which absorbs a gas produced by the passage of the current will be different when the electrolyte fills the interstices of a porous plate from that required when it is in an ordinary electrolytic cell. CHAPTER VI. EFFECT OF TEMPERATURE UPON THE PROPERTIES OF BODIES. 45. WE have only considered so far the relations between the phenomena in electricity, magnetism and elas- ticity and have not discussed any phenomenon in which temperature effects occur. We shall now go on however to endeavour to extend the methods we have hitherto used to those cases in which we have to consider the effects of temperature upon the properties of bodies. Before doing this however we must endeavour to arrive at some dynamical interpretation of temperature. The only case in which a dynamical conception of temperature has been attained is in the Kinetic Theory of Gases, and there the temperature is the mean energy due to the translatory motion of the molecules of the gas. So that if N be the number of molecules of the gas in unit volume NB is the energy of translatory motion of the molecules at the tempe- rature 0. There seems good reason for believing that NO is a part of the kinetic energy of the molecules when these are aggregated so as to form a solid or liquid as well as when .they form a gas. 9O DYNAMICS. The experiments and ideas which led to the establish- ment of the principle of the Conservation of Energy at the same time led to the conclusion that the energy of sensible heat is energy due to the motion of the molecules and is therefore part of the kinetic energy of the system. The reader should refer on this point to Maxwell's Theory of Heat, p. 301. Another reason for supposing that the temperature in the liquid as well as in the gaseous condition is measured by the mean energy of translation of the molecules is, that Van der Waals (Die Continidtdt des gasformigen und fliissigen Zustandes] has given a theory of the molecular constitution of bodies in those states which are intermediate between the liquid and gaseous, in which this supposition is made, and that this theory agrees well with the facts in many important respects. And again since most solids and liquids are capable of getting into a state where their specific heat is constant, that is, where the rise in temperature is proportional to the energy communi- cated to the system, we are led to suppose that the kinetic energy of some particular kind is a linear function of the temperature. This following illustration will show that it is probable that when we have two bodies in contact the collisions- between the molecules will tend to equalize the mean energy of this translatory motion when these bodies are solids and liquids as well as when they are gases. The mean translatory energies of two substances in contact thus tend to become equal, so that in this important respect the mean translatory energy has the same property as tempe- rature. Let us suppose that we have two different substances, composed of molecules A and B respectively, and that the molecules of the two substances are separated by a material TEMPERATURE. 91 plane surface. Let us also suppose that the mass of this, plane is large compared with that of a molecule of either substance and that it is prevented by perfectly elastic stops from moving through more than a distance comparable with molecular distances. Since the mass of the plane is very much greater than that of a molecule and since it can only move through a small distance in one direction the velocity of the plane will be very small compared with that of the molecules we shall suppose that it is so small that the number of molecules which are moving more slowly than the plane may be neglected, or what amounts to the same thing that all the molecules on the surface of the substances which are moving towards the plane strike it, and that none of those which are moving away from the plane do so. Let us suppose that the action between the molecule and the plane is the same as that between a perfectly elastic sphere and plane. Let m be the mass of an A molecule, v the velocity of the molecule, and a the angle its direction of motion makes with the normal to the plane before impact, V the velocity after impact, M the mass of the plane, w and W its velocity before and after it is struck by the molecule. Then we may easily show that - m{ V 2 -v*}= -r-rj -AMw 2 - mv 2 cos 2 a-(M- m) vw cos a}. 2 (M+m)* { Let us take the sum of the equations representing the effects of all the collisions which take place in unit time, we have y$m { V 2 - v*\ = (M+ m} 2 ^ ^ Mw * ~ v 2cosSa -( M - m ) zwcos a}... (9 7). If N be the number of A molecules which come in con- 92 DYNAMICS. tact with the plane in unit time and O l the mean translatory kinetic energy of such molecules, then if B0 l denotes the change in 1 in unit time If N' be the number of collisions and the mean kinetic energy of the plane, then Since the directions of motion of the A molecules are equally distributed Since the plane is supposed to move so slowly that all the molecules moving towards it strike it, and since its average velocity is zero, we have 5 (M m) vw cos a = o, so that equation (97) becomes {2N'0 - J W0\ ..... (98). m) 2 * If 2 be the average translatory kinetic energy of the molecules which strike the plane in unit time, ^ the num- ber of such molecules and TV/ the number of collisions, m' the mass of a molecule, we have similarly ' e - and we have also 2 Mm TEMPERATURE. 93 Now we can make the average kinetic energy of the plane what we please by giving it the proper initial velocity. For our purpose we wish the plane to act as a transmitter and not as a storer of energy, and it will do so if we give it such an initial velocity that the mean kinetic energy of the plane does not alter in unit time. If this is the case 8# vanishes and we have by (100) so that m'N ~ mN' b (M+m) 2 ' Substituting these values for 20-^/3, and 20-0 2 /3 in equations (98) and (99), we have Thus if 2 is greater than W O l will increase and 2 will diminish, and vice versa, and if O l is equal to 2 they will remain equal ; thus the mean translatory energy behaves in these respects exactly like temperature. There seems nothing in the above illustration to restrict it to the case of gases, and we should expect it would hold equally well for solids or liquids. 94 DYNAMICS. 46. We are thus led to assume that part of the kinetic energy of a system, whether that system be a portion of a solid, liquid or gas, is proportional to the temperature. Let us denote this part of the kinetic energy by where u is a coordinate helping to fix the position or con- figuration of a molecule. We see at once that there is an essential difference between these coordinates and those we have hitherto been considering and which fix the geometrical, strain, electric and magnetic configuration of the system. We have these latter coordinates entirely under our control and subject to certain limitations imposed by the finite strength of materials, the strength of dielectrics, and magnetic saturation ; we may make them take any value we please. We may therefore fitly call these coordinates con- trollable coordinates. It is quite different, on the other hand, with the coordinates fixing the separate moving parts of the systems whose kinetic energy constitutes the tempera- ture of the body. We can it is true affect the average value of certain functions of a large number of these coor- dinates, but we have no control over the coordinates indivi- dually. We may therefore call these coordinates "uncon- strainable" coordinates. Their fundamental property is that we can not oblige any individual coordinate to take any value which may be assigned. Since we have no power of dealing with individual molecules, the "controllable" coordinates must merely fix the position of a large number of molecules as a whole. If the term involves any "controllable" coordinate <, then it is evident TEMPERATURE. 95 that this coordinate must enter as a factor into all the terms in the form expressed by the equation J {(uu)u* + ...} = J/(0) \(uu)'u* + ...} (101), where the coefficients (uu)' do not involve < : otherwise the phenomenon would be influenced more by the motion of some particular molecule than by that of others. We shall assume that 0, the temperature, is proportional to that is that = JC" {()** + } , (102), where C does not involve any of the "controllable" coordi- nates which fix the configuration of the system. 47. We may conveniently divide the kinetic energy of a system into two parts, one depending on the motion of "unconstrainable" coordinates, which we shall denote by T u , and we shall assume that this is proportional to the absolute temperature 6, the other depending on the motion of the "controllable" coordinates, we shall denote by T c , T c corresponds to what v. Helmholtz in his paper on 11 Die Thermo dynamik chemischer Vorgange" ( Wissenschaftliche Abhandlungen, n. p. 958) calls "die freie Energie." There will not be any terms in the kinetic energy involving the product of the velocities of an " unconstrainable" and a "controllable" coordinate, otherwise the energy of the system would be altered by reversing the motion of all the "uncon- strainable" coordinates. Let us suppose that is a controllable coordinate which enters into the expression for that part of the kinetic energy which expresses the temperature, then if & be the. external force of this type acting on the system we have by Lagrange's equations, V being the potential energy, 96 DYNAMICS. a_d_dT_dT dV ~ dt~d$ d<$>* d<$>' Now T=T C +T H dT u and p- = o, #(/> so that to_d_dT c _dT c __dT d_V ~ dt d$ d$ d + d" Now by equation (103) T u is of the form where (ww)' does not involve <, so that we have dT u and therefore dV . '" differentiating this equation on the supposition that all the controllable coordinates are constant and that the only variable is the energy depending on the motion of " uncon- trollable " coordinates, we have and therefore by (104) 48. Now let us suppose that energy is communicated to the system, partly by the action of the external forces on the "controllable" coordinates, and partly through the "uncon- strainable" coordinates: let the quantity of work commu- nicated in the latter way be S<2- If the motion of the DYNAMICAL EQUATIONS. 97 "unconstrainable" coordinates is that which gives rise to the energy corresponding to temperature, SQ may be regarded as a quantity of heat communicated to the system. We have by the Conservation of Energy, if denotes a " controllable " coordinate, S7; + S7;+SF ......... (107). Now *T e =$8 + :S .......... (108), I 9 a ) and since T c is a quadratic function of the velocities of the " controllable " coordinates, we have and therefore ........... (109); a ) so that by subtracting (108) from (109) we get Since the change in the configuration is that which actually takes place in the time S/> we have <8/=S0, so that ddT e dT\ - and therefore equation (107) becomes Now, if V be completely fixed by the controllable coordinates, we have T. D. 98 DYNAMICS. So that Substituting for the value given by (103) we have S<2 = 2S constant 87; ......... (114). Let us suppose that the quantity of work communicated to the system is just sufficient to prevent T u from changing, then or V d } T U constant \dT u ) $ constant Remembering that T u is proportional to the absolute temperature 0, we see that equation (115) becomes /dQ\ _ \d/6 constant dO $. constant where in finding d^fdO we must take care that is the only quantity which varies. In this form equation (116) is identical with the third thermodynamical relation given in Maxwell's Theory of Heat, p. 169, and v. Helmholtz in his paper "Die Thermo- DYNAMICAL EQUATIONS. 99 dynamik Chemischer Vorgange" ( Wissenschaftliche Abhand- hmgen, 2, p. 962) deduces this equation from the Second Law of Thermodynamics and applies it to the case of the variation of the electromotive force of galvanic cells with temperature. The conclusions at which he arrives have been verified by the experiments of Czapski (Wied. Ann. 21, p. 209) and Jahn (Wied. Ann. 28, pp. 21, 491). If 8(2 = o, that is if all the work done on the system is done by means of forces of the types of the various controllable coordinates, then we have by equation (114) dT H ) $ constant \ d<$> /> constant 49. Since we see by (113) that or ^=28 log/ W + Slog T; ......... (118), * u so that BQ T u is a perfect differential. This is analogous to the Second Law of Thermodynamics, and we see by the analogy that it shews that energy arising from the motion of quantities fixed by "unconstrainable" coordinates can only be partly converted into work spent in moving the quantities fixed by the "controllable" coordinates. The amount which can be converted follows laws analogous to those which regulate the conversion of heat into mechanical work. 72 100 DYNAMICS. In the preceding work we have assumed that the potential energy of the system was not changed if the "controllable" coordinates remained unchanged. When however the system is a portion of a solid or liquid the potential energy may by some alteration in the state of aggregation be changed without there being any corre- sponding change in the controllable coordinates. To include this case we must suppose that V is a function of the temperature as well as of the <'s, and that its value in the neighbourhood of the temperature corresponding to a change of state in the substance varies very rapidly. In this case we have 6 being the temperature, v dV dV 8F =^ + ^ 8 ' and instead of (114) Since 80 and &T H vanish together we see that equation (116) still holds. Equations (117) and (118) however require modification. We have now (&Q-S V($> constant))/ 7L a perfect differential instead of $Q/T, e . 50. Relations between heat and strain. We shall now apply equation (116) to determine the effects due to the variation of various physical quantities with tempera- ture, and shall begin by considering the effects produced by the variation of the coefficients of elasticity m and n with temperature. In equation (116) let us suppose that is a stress of type e, then using the same notation as in 35, we have $ = m (e +f + g) + n(e -f-g\ dn . f . THERMAL EFFECTS DUE TO STRAIN. IOI So that by equation (116), 8Q, the heat which must be supplied to unit volume of the bar to keep its temperature from changing when e is increased by Be is given by the equation and thus if the coefficients of elasticity diminish as the temperature increases, heat must be supplied to keep the temperature of a bar constant when it is lengthened, and hence if the bar is left to itself and not supplied with heat it will cool when it is extended. If $ is a couple tending to twist the bar about the axis of x, we have, if a is the twist about that axis, dn and therefore by (116) 8(), the heat required by unit volume of the bar to keep the temperature from changing when a is increased by &a is given by the equation so that if a rod which is already twisted is twisted still further it will cool if left to itself, provided, as is usually the case, the coefficient of rigidity diminishes as the temperature increases. The preceding results were first obtained by means of the Second Law of Thermodynamics by Sir William Thomson in his paper on the Dynamical Theory of Heat (Collected Papers, Vol. i. p. 309). 51. Thermal Effects produced by Electrifica- tion. Let us now consider the case when < is an electric IO2 DYNAMICS. force parallel to the axis of x, producing an electric displace- ment / in that direction. In this case if K be the specific inductive capacity of the dielectric, we have _ )f constant ~ ~ K* dQ J ' so that 8(2, the heat which must be supplied to unit volume of the dielectric in order to prevent its temperature changing when the electric displacement is increased by S/j is by (116) given by the equation Some recent experiments made by Mr Cassie in the Caven- dish Laboratory on the effect of temperature on the specific inductive capacities of glass, mica and ebonite, have shewn that the specific inductive capacity of these dielectrics increases as the temperature increases, and that at about 30 C. i dK , . for glass, i dK T r --, = '0004 for mica, li. uv i dK -^j:- = '0007 for ebonite. A av Thus the heat which must be supplied to unit volume of a piece of glass to enable its temperature to remain constant when it is electrified is by (123) 002 . and this at 30 C. THERMAL EFFECTS DUE TO MAGNETIZATION. IO3 /' is the work supplied from electrical sources, hence in charg- ing a Leyden jar, we see that the mechanical equivalent of the heat absorbed by it during charging, if its temperature remains constant, is about two-thirds of the work supplied to it from electrical sources. We see also by equation (123) that a piece of glass will be cooled when it moves from a place where the electric force is weak to one where it is strong. 52. Thermal effects of Magnetization. Let us now suppose that $ is a magnetic force magnetizing a piece of soft iron or other magnetizable substance to the intensity /. Then if k be the coefficient of magnetic induction so that \d(t. /constant %) + (j, v z ... corresponding to the various coordinates u^ u z , we see, since dT that it will be of the form where f(v lt v 2 --) denotes a quadratic function of z/ x , v 2 &c., which does not involve p. As this expression is pro- portional to the temperature 0, we see that if p be suddenly increased by Sp, the increase BO in the temperature is given 112 DYNAMICS. by the equation 80 so that if bo- is positive the temperature of the body is increased by communicating a charge of electricity to it, that is the electricity behaves like a body whose specific heat was negative. But we saw that bo- was of the opposite sign to what Sir William Thomson has defined as the specific heat of electricity in the substance. Hence we see that the analogy between the behaviour of electricity and that of a fluid possessing either positive or negative specific heat can be extended to cover the case when a bodily charge of electricity is communicated to the body. We can shew however that if the charge of electricity be of the same order of magnitude as those which occur in electrostatic phenomena this heating effect must be ex- tremely small. For multiplying both sides of equation (132) by /?, we have ($W fibo- %- = - op, approximately. u a Now fSba-ja is by equation (130) the "specific heat" of electricity. The value of this for antimony at the tempera- ture 27C. is (see Tait's Heat, p. 180) about io~ 2 x 300 when the unit is io~ 5 of the E. M. F. of a Grove's cell. As the E. M. F. of a Grove's cell is about 2 x io 8 in absolute measure the "specific heat" of electricity in antimony in absolute measure will be about 6000. We must now find a limiting value for 8p. Let us sup- pose that electricity is uniformly distributed through a sphere of radius r, then if p be the density of the electrical distribu- THERMOELECTRICITY. 1 1 3 tion, K the specific inductive capacity, the force just outside the sphere is Now the greatest value this can have in air is (see Everett's Units and Physical Constants, p. 142) about 4 x io 12 , so that a limiting value of p will be given by Now so that P = - o- approximately. 9 x zoV J Hence substituting this value of p for Sp, we get at the temperature 27C. 9 x io x r 2 Now (3S6 is the mechanical equivalent of the heat available for changing the temperature, so that the change in temperature will be of the order since 4-2 x io 7 is the mechanical equivalent of heat. Thus the change of temperature which can be produced in this way by any statical charge of electricity is infinitesimal. 55. Thermoelectric effects of strain. If the quantity b in the expression for / (e) is a function of the strain in the wire along which the current is passing, then putting b=f (e\ where e denotes a strain in the wire, we T. D. 8 1 14 DYNAMICS. see by equation (128) that at each point of the wire there is the electromotive force acting along the wire, ds being an element of its length. Now if we have a closed circuit made of one metal, in which 2 ' the amplitude of the vibrations of x are thus given by the expression where exp (x) = e x . So that the ratio of the amplitudes of two successive swings is where T is the time of a complete oscillation, and is given by the equation, T 27r{A/fj} 2 approximately. Substituting this value of T in (151) we get for the ratio of the two amplitudes the expression exp r / ( \ ~j~ . Aa 2 + fib 2 ) Now if the motion of x were resisted by a frictional force proportional to the velocity, the equation for x would be d 2 x dx the solution of which is _\t , , A 2 \2 >S \V4~4^Tv where C and c are constants. The ratio of the amplitudes of two successive swings in this case is ' \T\ 2AJ ' RESIDUAL EFFECTS. 139 or approximately When the decrease in the amplitude is due to the connexion with the secondary system, the ratio of two successive amplitudes is :so that the logarithmic decrement when the resistance is frictional varies as when it is due to the secondary system it varies as Hence we see that if the mass of the vibrating body is altered, the variation of the logarithmic decrement will be less in this case than it would if the decay in the oscillations were due to friction. This agrees with the results of Sir William Thomson's experiments on the decay of the tor- sional vibrations of wire, as he found that the loss was greater with the longer periods than that calculated ac- cording to the law of square roots from its amount in the experiment with shorter periods. In fact if A were much smaller than /^ 2 /# 2 the rate of decay would be increased instead of diminished by increasing the vibrating mass, as ihe rate of decay has its maximum value when A CHAPTER IX. INTRODUCTORY TO THE STUDY OF REVERSIBLE SCALAR PHENOMENA. 68. So far we have been dealing with phenomena in which as in ordinary dynamics the quantities concerned were mainly of a vector character. We shall now how- ever go on to consider phenomena when the quantities- we have to deal with are chiefly scalar, such as the phenomena of evaporation, dissociation, chemical combi- nation, etc. where the quantities which have to be considered are such things as temperature, vapour density, or the num- ber of molecules in a particular state. The chief difference between these cases and those we have been considering is that in these we have as in the kinetic theory of gases to- deal chiefly with the average values of certain quantities and cannot attempt to follow the variations of the individual members which make up the average, while in the previous cases we have been able to follow in all detail the changes in most of the quantities introduced. In these new cases all that we can get by the application of the Hamil- tonian principle are relations between the averages of a series of quantities ; as however these averages are all that we can SCALAR PHENOMENA. 141 observe in these cases, this limitation is not serious from a practical point of view. The relations we shall deduce are those which exist when the body is in a steady state. 69. The systems we shall have to consider are portions of matter in the solid, liquid or gaseous state, and consist, according to the molecular theory of bodies, of a very large number of secondary systems or molecules. Now we can control a primary system in many ways, we can fix its geo- metrical position, we can within certain limits strain it in any way we please, we may establish electric currents or electric displacements through it, and if the body is magnetic we can magnetize it within the limits of saturation : so that the coordinates fixing the geometrical, the strain, the electric and the magnetic configurations are under our control and have therefore been called ( 46) controllable coordinates. The coordinates fixing the positions of the several secondary systems are not however within our control and we have not the power of altering any one of them ; v/e have called these unconstrainable coordinates. 70. When we say that a system consisting of a great number of molecules is in a steady state we mean that the state is steady with respect to the controllable coordinates and make no supposition as to whether it is so or not with respect to the unconstrainable ones, all that we shall assume is that the mean values which we can observe and which depend upon the unconstrainable coordinates are steady. Thus when the system is in a steady state the velocity of each controllable coordinate must be constant, and if the coordinate enters explicitly into the expression for either the kinetic or potential energy, that is if the coordinate is not a " kinosthenic " or speed one, the velocity must vanish. 142 DYNAMICS. 71. We shall now proceed to prove that when a system consisting of a great number of molecules is in a steady state the mean value of the Lagrangian function has a stationary value so long as the velocities of the controllable coordinates are not altered. Let us denote the controllable speed coordinates by the symbol q^ the controllable positional coordinates by q a and the unconstrainable coordinates by ^ 3 , then we have by the Calculus of Variation f I Jt / 2 . When the system is acted on by gravity the potential energy is mgl cos S so that the Lagrangian function is \ml 2 sin 2 3< 2 + mgl cos 3 which may be written - $dO + mgl cos S and this possesses the property of being stationary. If however the Lagrangian function is expressed in terms of 3> the momentum corresponding to < and given by the equation $ = mP sin 2 $ the Lagrangian function becomes ^ 2 < 2 cosec 2 3- + mgl cos &, and this expression is not stationary. 10 2 148 DYNAMICS. The function which possesses this property is the "modified" Lagrangian function ' - <1> 2 cosec 2 S + mgl cos S. 2ml 2 Since however when expressed in terms of 6 and < equals I 2 COS S ~^? sin 2 3' we see that the " modified " Lagrangian function again equals - \dO + mgl cos S. Thus the expression is stationary however may be expressed, whether in terms of or <. This example illustrates the principle that if we calculate the Lagrangian function from the forces necessary to pre- serve equilibrium we need not consider whether it is ex- pressed in terms of velocities or momenta. 78. If we consider the proof by which the equation was established we shall notice one point which we must continually bear in mind when we are calculating the value of the potential energy. By the Calculus of Variations dL PI T ,, f'i (dL d dL} (dL Ldt=\ <-j -T.-JT-I fydt + 2 ( -=- . It, !t, \dq dt dq } y \dq LAGRANGIAN FUNCTION. 149 and thus if equation (156) holds we must have dl^_d_ dL_ dq dt dq Now in ordinary Rigid Dynamics perhaps the most usual form of Lagrange's equation is d_ dJL_dJ^_ dt dq dq ~ V> where Q is the external force of type q tending to increase this coordinate. In this case L = T - V where V is the potential energy when the coordinates have their assigned value and the system is free from the action of external forces. If however we are to use equation (153) we must put L = T V where that is we must add to the potential energy we are con- sidering the potential energy of the system which produces the external forces. Lagrange's equation may now be written and equation (153) is true. Thus to take an example, in Electrostatics we often assume that the potential energy V of unit volume of a dielectric whose specific inductive capacity is K and through which the electric displacements parallel to the axes of x, y. z are respectively /j , h is f {/+*+*} and that the equations of equilibrium are _ __ df ' dg ' dh~ ' 150 DYNAMICS. where X, Y, Zare the components of the electromotive force parallel to the axes of x, y, z respectively. If however we wish to apply the theorem we are now considering we must put for then the equations of equilibrium are dV _dV___dV_ ~df~~4i~J/i~ ' The necessity of choosing V so that the equations of motion are of the form d dT d IfTj-T^-^^' is one to which we must always be alive in dealing with this subject. CHAPTER X. THE CALCULATION OF THE MEAN LAGRANGIAN FUNCTION. 79. SINCE we can observe and regulate the forces of the types of the "controllable" coordinates we can determine how they depend upon the values of these coordinates and then by means of the expression (155) calculate all those terms in the mean Lagrangian function which involve such coordinates. There may however be some terms in the Lagrangian function which do not involve these quantities and if we require these we must determine them by other considerations ; a large number of problems can however be solved even though we do not know the values of these terms. To get some idea of the different kinds of terms which may exist in the Lagrangian function let us consider the energy of a system consisting of a large number of molecules. In the expression for the energy we can calculate all the terms involving the coordinates which fix the electric, magnetic or elastic configuration of the system, and in the terms depending upon the strain coordinates we may include those terms which involve the average distance between the molecules. There may however be some terms left which 152 DYNAMICS. each molecule contributes independently of its neighbours and which do not involve any of the controllable coordinates. The sum of these contributions will be proportional to the number of the molecules and must also be a function of the temperature, because the mean state of the system is fixed by the controllable coordinates and the temperature, and the mean kinetic energy must therefore be a function of these quantities. By hypothesis the terms we are considering do not involve the controllable coordinates, so that the only quantity they can depend upon is the temperature. The potential energy of the molecules may also contribute terms to the Lagrangian function which do not involve the con- trollable coordinates and which therefore we cannot calculate by equation (155)- For the purposes for which we use the Lagrangian function all that we require to know about it is the change in its value when the system is changed in some definite way. Now if we measure the amount of heat absorbed or evolved when the change takes place and know the change, if any, which takes place in the kinetic energy, we can calculate the alteration in the part of the potential energy which is independent of the controllable coordinates. The methods of calculating the mean value of the Lagrangian function will be best illustrated by working out some particular cases. Let us begin with that of a perfect gas. Mean value of the Lagrangian function for a perfect gas. 80. Let us suppose that unit mass of the gas is enclosed in a cylinder furnished with a piston, whose distance from the base of the cylinder is represented by the coordinate x, then since the pressure of the gas is a force tending to alter LAGRANGIAN FUNCTION FOR A GAS. 153 the value of x, the mean Lagrangian function for the system of molecules forming the gas must involve the coordinate x. If H denotes the mean value of the Lagrangian function of the system, the mean value of the force of type x pro- duced by the system when in a steady state is by Lagrange's equations dH dx' Since there is equilibrium between the pressure due to the gas and the external pressure dH ^c= A *> where / is the pressure of the gas and A the area of the piston. But if the gas obeys Boyle's law Re * m T' where v is the volume of unit mass of the gas, the absolute temperature and J? a constant such that RQ equals the square of the velocity of sound in the gas. Now so that dff_RB dv_ dx v dx' Integrating this equation we have in so far as H depends upon v and 0, (157), where v is an arbitrary constant and /(0) an arbitrary func- tion of 0, which does not involve x. It corresponds to the 154 DYNAMICS. part of the kinetic energy which depends entirely upon un- constrainable coordinates. We shall find in the course of this work that a great many problems can be solved without a knowledge of the value of / (0). As far as f (0} is linear it may be included in the first term, as we may regard v as quite arbitrary. The expression (157) will give the value of the mean Lagrangian function so far as it involves x, it also includes that part of the kinetic energy which is expressed entirely in terms of unconstrainable coordinates, for this can be included in the term f(6) ; to complete its value we must subtract from it w the potential energy of unit mass of the gas when its particles are infinitely distant from each other, as this is the part of the potential energy which depends upon uncon- trollable coordinates. Thus for unit mass of the gas or if p be the density of the gas H=RQ log Po p We shall see later on, when we consider the phenomenon of evaporation, that f(0) is of the form AO + 0\ogO .................. (158). The value of H for a mass m of gas whose density is p is given by the equation ff=m0log^ + mf(6)-mw ......... (159). P This is the Lagrangian function for the gas itself; when an external pressure acts upon it we must add to this value the mean Lagrangian function of the system producing the pressure. We may suppose that this system is a weight LAGRANGIAN FUNCTION FOR A SOLID. 155 placed upon the piston, the variable part of the potential energy of this is, if V be the volume occupied by the gas pv. So that its mean Lagrangian function is -pv and the Lagrangian function of the two systems is therefore mRQ log'' + mf(Q)-mw-pV ......... (160). P Mean value of the Lagrangian function for a liquid or solid. 81. We must now proceed to find the mean value If of the Lagrangian function for a liquid or solid. Let us suppose that we have a piston whose distance from a fixed plane is x pressing upon a bar of the substance. Then we have by Lagrange's equations when the motion is steady dH = mean force tending to increase x. produced by the substance, so that dp where p is the pressure required to balance this force and a the area of the cross section of the bar. The differential coefficient dpIdQ is obtained on the supposition that the volume is constant. Since adx = dv, fdp\ we have ^ ,- = ( -^ ) d6dv \dO) v constant 156 DYNAMICS. Thus dv v constant where ft is the mean value of (dpIdO) between zero and 0. Thus H= = Oy + S l (0) say. Where / (0) is an arbitrary function of the temperature, it .is unnecessary to add an arbitrary function of v on integration as this will be included in the potential energy due to strain. If the mass of the substance is unity a where o- is the density, so that in the expression for H for unit mass of the substance there are the terms From this we must subtract w' the potential energy of unit mass of the substance. Thus in the Lagrangian function for a mass m of the substance there are the terms f " B - mO I da- + ;/// (0) - mw . " If there is any external pressure we must add to this the expression for the mean value of the Lagrangian function of the system producing this pressure. This, as in the case of the gas, will be -pV, where p is the external pressure and V the volume of the solid or liquid. Adding this term we get -pr' (161). LAGRANGIAN FUNCTION FOR A SOLID. 157 It must be remembered that we have only calculated the value of the Lagrangian function in the simplest case when the body is in a steady state, when it is free from all strain except that inseparable from the body at the tempe- rature we are considering, and when it is neither electrified nor magnetized. The change in the Lagrangian function due to any additional strain or to electrification or mag- netization can be at once determined by finding the energy required to establish this particular condition. For example, the change in H produced by statical electrification equals minus the potential energy of the electrical distribution, the change due to any system of electric currents flowing through solids or liquids is the kinetic energy due to this distribution of currents, and can be calculated by the ordinary formulae of electrokinetics. 82. The problems which we shall now proceed to solve, making use of the principle that the mean value of the Lagrangian function is stationary, are those which can often be solved on thermodynamical principles by using the condition that the value of the entropy of the system is stationary. The value of H must therefore be closely connected with that of the entropy, and in fact we see from its value for a perfect gas in equilibrium under external pressure that, with the exception of the term p V, those terms in H which depend upon the controllable coordinates occur also in the expression for the entropy. It seems however preferable to use the function H which has a direct dynamical significance, rather than the entropy which depends upon other than purely dynamical considerations. CHAPTER XI. EVAPORATION. 83. WE shall now go on to apply the principle that the value of H is stationary to solve some special problems in Physics. The first problem we shall consider is that _of finding the state of equilibrium when a given mass of some liquid is placed in a closed vessel from which the air has been exhausted; some of the liquid will be vaporized and we wish to find how far the vaporization will proceed before equilibrium is obtained. This of course is equivalent to finding the density of a vapour when in equilibrium in presence of the liquid. Let v, v' be the volumes occupied by the vapour and liquid respectively, the mass of the vapour, 17 that of the liquid, the rest of the notation being the same as that used in 80 and 81. Then assuming that the vapour obeys Boyle's law we see from equation (158) that the vapour contributes to the expression for H for the whole system the terms io g + &(0) - &> (162), since p the density of the vapour equals /z/. EVAPORATION. 159 From equation (161) we see that the liquid when it is free from surface tension, electrification and the like, furnishes to the same expression the terms -r)0 I -. 2 da- + r;/, (Q) - rjw' ( J 63). Thus H the mean value of the Lagrangian function for both the liquid and vapour is the sum of (162) and (163) so that we have H = IRQ log ^ + / (0) + ^ (6i) - ^0 I *-td = p we may write (165) as ^-JWtog& + *-( -0+W Since dHld vanishes in the state of equilibrium we have then J?0 log & = - ^ + (w-w] - if/ (0) ...(166), P (T or p ~ p f. a e ^ since p/cr is very small we may write this as p=*0)- * (l6 7 ), where <^> (0) is some function of 9. EVAPORATION. l6l Bertrand (Thermodynamique, p. 93) has shown that the results of Regnault's experiments on the vapour pressures of different liquids can be represented by the following expressions, / being the pressure in millimetres of mercury : water; log/ = 17-44324 - 2795/0 - 3' 8682 log 0> ether; log/ = 13 -42311 - 1729/0 - 1-9787 log 0, alcohol; log/ = 21-44686 - 2743/0 - 4-2248 log 0, chloroform; log/ = 19-29792 - 2179/0 - 3*91583 log 0, bisulphide of carbon Iog/ = i2'58852 -1684/0- 1*7689 log 0. This form of expression was originally used by Dupre. (Theorie Mecanique de la Chaleur, p. 97.) The coefficient of i/0 in each of these expressions is nearly \\R, where X is the latent heat of the substance at the absolute zero of temperature. This is the term (w - w')j6 in our expression (166) and w-w' is the latent heat at absolute zero, hence by comparing the other terms in these expressions we see that/(0) must be of the form A6 + BQ log 0. 84. We can by the aid of the preceding formulae very easily determine the effect upon the vapour pressure of any slight change in the physical condition of the liquid or vapour. Let us suppose that the physical conditions are so changed that the mean Lagrangian function exceeds the value we have hitherto assumed for it by \. Then instead of equation (167) we have evidently RO log + R6 + \l/ (0) (w w") + - = o, p cr dc, so that Sp the change in the vapour density due to the cause T. D. II 162 DYNAMICS. which produced the change x i n tne mean Lagrangian function is given by the equation or so that if x increases with the vapour pressure in the state of equilibrium is increased, while if x diminishes as increases the equilibrium vapour pressure is diminished. This very important principle is a particular case of the more general one that ; when the physical environment of a system is slightly changed and the consequent change in the mean Lagrangian function increases as any physical process goes on, then this process will have to go on further in the changed system before equilibrium is reached than in the unchanged one, while if the change in the mean Lagrangian function diminishes as the process goes on it will not have to proceed so far. We shall have numerous examples of this principle in the course of the following pages. 85. Let us now consider the effect of surface tension upon the vapour pressure. In order to take a definite case let us suppose that the liquid is a spherical drop. It will possess in consequence of surface tension potential energy proportional to its area, and as the area of the drop diminishes as the water evaporates the energy due to the surface tension changes, and since anything which causes the energy to change as evaporation goes on alters the state of equilibrium, the vapour pressures when there is equi- librium in this case cannot be the same as when evaporation produces no change in the area and therefore no change in the energy due to surface tension. EVAPORATION. 163 If a be the radius of the drop and T the energy per unit area due to surface tension then in the expression for the potential energy of the liquid there will be in addition to the terms we have already considered the term and therefore in the mean Lagrangian function for the liquid and vapour the additional term So that with our previous notation x = -47ra 2 T. i da ii dv' NOW - -jr = -- , TT , ad 3 z'' d% and therefore = --- -,- , da i hence ^> = ~ 2" df 4iKr = 200 ^ T p ~ 1-3 x io 9 since T '- 81 we have -- = 1*2 X I0~ 5 . P We see that the energy due to surface tension makes the Lagrangian function increase as evaporation goes on, so that by the principle given at the end of 84, the effect of it will be to make evaporation go on further than it otherwise would. If we have the water in narrow capillary tubes then when it evaporates the area of the surface of contact of the tube with water is diminished but that of the surface of contact of the tube with air is increased. Since the surface tension of the surface of separation of the tube and air is greater than that of the tube and water, the potential energy due to surface tension increases as evaporation goes on, thus the mean Lagrangian function diminishes as the liquid evaporates, so that by the principle of 84 the effect of surface tension in this case will be to stop evaporation and promote condensation. We can easily shew that if a is the radius of the tube then in this case substituting the value of dald given by (169) we have dx i e 2 d 87rA" a* a- ' so that if Sp be the change in the vapour density produced by the electrification we have by (168) 1 66 DYNAMICS. To calculate the magnitude of this effect let us suppose K= i ; then e/a 2 is the electric force just outside the sphere, and this cannot exceed a certain value, otherwise the insu- lating power of the air would break down and the electricity escape. The maximum value of e/a 2 when the sphere is surrounded by air at the atmospheric pressure is about 120 in electrostatic measure : and as o- is unity, p a small fraction, the maximum alteration in the vapour density will be given by the equation So i ' now RB for water vapour at ioC. is about 1*3 x io 9 , so that this value will be independent of the size of the drop. Comparing equations (170) and (172) we see that the maximum effect due to electrification is about equal in magnitude though opposite in sign to that due to a curvature of 1/4 of a centimetre. The effect of electrification is to diminish the vapour density when there is equilibrium between the liquid and the vapour, it therefore increases the tendency of the vapour to deposit on the liquid. We should therefore expect an electrified drop of rain to be larger than an unelectrified one, so that this effect may help to produce the large drops of rain which fall in thunderstorms. 87. Effect of an electric field upon the vapour pressure. Electricity also produces an effect upon the vapour pressure when the drop is not charged but merely placed in an electric field. Let us suppose that the field is due to a charge of electricity e collected at a point EVAPORATION. l6/ P, let the radius of the drop of water which we shall suppose spherical be a and let/ be the distance of the centre from P. Then (Maxwell's Electricity and Magnetism, Vol. I. p. 232), the potential energy due to the mutual action of the electrified point and the drop of water is so that the increase in the mean Lagrangian function is and therefore by (168) the change Sp in the density of the vapour when in a state of equilibrium is given by the equation Sp i (r i j 3 ^a 2 e 2 ^ "~> \da 7 = RV o--p ~K \ 2f 2 (f 2 ~a 2 ) + f*(J*-a*f I d ' Substituting for dajdt; from (169) we have 3p_ i / (3 i a 2 I i if # be small compared with / then approximately p Now ^/Ay 2 is the force at the centre of the drop due to the electrified point, calling this F and remembering that " a- and thus the alteration in the vapour pressure produced by an external pressure of an atmosphere is given by the equation & _ density of steam at atmospheric pressure p density of water = atoC. 1200 So that for each atmosphere of pressure the vapour pressure of water is increased by about one part in twelve hundred. For ether the increase would be about one part in 220. 90. The other case we shall consider is when the pressure acting on the system remains constant. We shall use the same notation as before. The only change we shall have to make in the mean Lagrangian function is to add to it that of the system producing the steady external pressure. We may suppose in order to fix our ideas that this system is a quantity of mercury placed on the piston, which may be supposed to move vertically up and down, then if P be the steady pressure per unit area the potential energy is equal to so that the mean Lagrangian function of this system is -P(v+v'\ where if is the volume of the liquid. I7 2 DYNAMICS. ^ dH \ he condition - = o gives M log - ^ + Now and i=/' where/ and/' are the pressures due to the water vapour and air respectively. Since />=/ +p' and 'J' = - 1 tff 07- the above equation reduces to RQ log ^-RQ +/(#) _/ x (0) -yO-(w- Wj + ^=0, or if S/> be the change produced by the external pressure, = P o- a similar result to the one we obtained before. We see from this result that (apart from any other cause) rain drops will form more easily when the barometer is low than when it is high. Regnault's experiments seem to show that the vapour pressure in a vacuum is greater by nearly 5 per cent, than when there is air at atmospheric pressure above the liquid (Wullner's Lehrbuch der Physik, in. p. 703), but he attributed this difference to the condensation of the liquid on the sides EVAPORATION. 1/3 of the vessel ; the absorption of the air by the liquid might also- tend to produce an effect in this direction, though, as the following investigation will show, to nothing like the extent of 5 per cent. 91. Effect of absorbed air on the vapour pressure. When the liquid contains some gas diffused through its volume which remains behind when it evaporates, the evaporation of the liquid will cause the volume occupied by the gas to diminish and its density to increase. Thus by (157) the mean value of its Lagrangian function will diminish as evaporation goes on, so that by 84 the presence of the gas dissolved throughout the volume will diminish the equilibrium vapour pressure. Let e be the mass of the dissolved gas, v' the volume of the liquid in which it is dissolved, then the Lagrangian function of the gas is e^log^-ew'-K/W (178), where w' is the intrinsic potential energy of unit mass of the dissolved gas. The expression (178) is the quantity we denoted by x i n 84. The only variable in x which involves is v and dv'jd = - i/cr, so that we have d x _ R'Oe d v'cr and therefore by (168) ,-Sp i It'Qe , . 6-?- = - r (i79)- p a- p V If 8/ be the increase in the vapour pressure caused by the dissolved gas, P the pressure this gas would produce if it were free from the liquid and filled the volume z/, then since 1/4 DYNAMICS. R'Qt and - = P. v equation (179) may be written tr-p So that since p is very small compared with a- we have approximately or if/ be the vapour pressure, and p the density of steam at the atmospheric pressure w, equation (180) may be written ty = _pP p a- TT And since p' I ) is the pressure P expressed in atmospheres. The volumes of the various gases which one volume of water will absorb at oC. under the pressure of 760 milli- metres of mercury were determined by Bunsen and are given in the following table : Hydrogen '019 Nitrogen -0203 Air '0247 Carbonic Acid 179 Chlorine 3*0361 so that according to equation (180) the vapour pressure of water saturated with air will be lowered by about one part in 50,000, when saturated with carbonic acid by about i part in 660 and when with chlorine by about i part in 400. In this investigation we have assumed that the properties of the liquid are not altered by the presence of the gas ; if they are, then we must regard y and w' as functions of e, and this would lead to the introduction of several additional EVAPORATION. 1/5 terms into- the equation for S/. We have assumed too that the gas remains behind as the liquid evaporates, as in this case the diminution of the vapour pressure is greater than if some of the gas were to be set free when the liquid evapo- rates. 92. The effect of dissolved salt on the vapour pressure. Van J t Hoff (" L'equilibre chimique dans les systemes gazeux ou dissous a Petat dilue," Archives Neer- landais, xx. p. 239, 1886) has pointed out that Pfeffer's experiments on the osmotic pressure produced by salts dissolved in water (PfefTer, Osmotische Untersuchungen, Leip/ig, 1877) and Raoult's experiments on the effect of dissolved salts on the freezing point of solutions (Annales de Chimie, 6 me serie, iv. p. 401). show that the molecules of a salt in a dilute solution exert the same pressure as they would exert if they were in the gaseous state at the same temperature and occupying a volume equal to that of the liquid in which the salt is dissolved, and that the pressure exerted by these molecules obeys Boyle's and Gay Lussac's law. This being so, the mean Lagrangian function for the salt dissolved in the liquid is the same as that of an equal mass of the salt in the gaseous state filling the volume occupied by the liquid. Thus if the properties of the liquid are not altered by the presence of the salt the results of the preceding section will apply, and we shall have, supposing that the salt remains behind when the liquid evaporates, where cr is the density of the liquid, p the density of its vapour at the atmospheric pressure, (P) the pressure in atmospheres which would be exerted by the dissolved salt if it were in the gaseous state. Thus, for example, suppose that we have n grammes of i;6 DYNAMICS. salt in a litre of the solvent, where n is the molecular weight of the salt. This strength of solution is often called for brevity a strength of one equivalent per litre. This quantity of salt will by Avogadro's law produce the same pressure as 2 grammes of hydrogen per litre, that is about 22 atmospheres ; if this quantity of salt were dissolved in water it would by equation (181) since p/o- is about 1/1200 diminish the vapour pressure by about i part in 55, if it were dissolved in ether, C 4 H 10 O, where p/o- is about 1/2 20,. then the vapour pressure would be diminished by about i part in 10, if it were dissolved in alcohol, C 2 H 6 O, where p/o- is about 1/380, the vapour pressure would be reduced by about one part in 17. We see from equation (181) that the diminution in the vapour pressure is proportional to the quantity of salt dissolved. We can also express the result of this equation as follows. If P is the pressure due to one equivalent in grammes of the salt dissolved in a kilogramme of the solvent, then P/o; where o- is the density of the solvent, is the pressure in atmospheres due to one equi- valent of the salt dissolved in a litre of the solvent. Hence P 2 - = J(density of hydrogen at atmospheric pressure.) So- o- 10 that we may write equation (181) as = (molecular weight of solvent) x i x io~ 3 , where p is the diminution in vapour pressure when one equivalent of the salt is dissolved in a kilogramme of the solvent. Another way of expressing the same thing is that when one equivalent of the salt is dissolved in 1000 equivalents of the solvent, i.e. in 1000 m grammes where m is the molecular weight of the solvent, the diminution in the vapour pressure amounts to i part in 1000, whatever be the nature of the salt or solvent. VAPOUR PRESSURE. *P P P Since - , or TT and since P is directly, while p' is inversely, proportional to the absolute temperature, we see that the ratio Bp/p ought to be nearly independent of the temperature since o- only varies very slowly with it. 93. In the preceding investigation we have assumed that the properties of the solvent were unaltered by the presence of the salt, and that all the solvent did was to enable the salt to exist in a condition in which the molecules were very far apart. If however the properties of the solvent are altered by the presence of the salt, then we must regard w as a function of the quantity of salt dissolved. In this case instead of equation (179) we shall have K" " 8 "' where W is the change in it/ produced by the presence of the salt. Now if s be the strength of the solution, i.e. the quantity of salt in unit volume of the solvent, dw' dw' 77 ~T S j , arj aS so that equation (182) becomes dw' P - s =- + ~ = o. as o- If the change in w is proportional to the strength of the solution then 8w' s : o. as T. D. 12 178 DYNAMICS. In the general case the lowest power of s which occurs in the expression s , du* OW S 7 ds is the second, so that the effect produced by the alteration of the properties of the solvent depends upon the squares and higher powers of the concentration, while the effect we investigated in the preceding section was proportional to the first power, and therefore when the solution is dilute is relatively the more important. Raoult, Comptes Rendus 104, p. 1433, has recently found that when one equivalent of a substance is dissolved in 100 equivalents of a solvent the vapour pressure is reduced by 1-05 parts in 100, which agrees very well with the results we have just obtained. CHAPTER XII. PROPERTIES OF DILUTE SOLUTIONS. 94. THE effect produced on the vapour pressure of any solvent by dissolving other substances in it has been discussed in the last chapter ; in this chapter we shall consider some other properties possessed by dilute solutions. Absorption of gases by liquids. Let us suppose that we have a closed cylinder containing a gas and a liquid and that we wish to find how much of the gas will be ab- sorbed by the liquid. In this case we have four substances to consider, 1. The liquid. 2. The vapour of the liquid. 3. The free gas. 4. The gas dissolved in the liquid. The variation which we shall suppose to take place, and which will not by the Hamiltonian principle alter the value of H when the system is in equilibrium, is that corre- sponding to the escape of a small quantity of gas from the liquid. This will not affect the value of the mean Lagrangian function of the vapour of the liquid, so that we may leave this 12 2 l8o DYNAMICS. out of account in solving this problem. Let the mass of the liquid be 17, that of the free gas , and that of the absorbed gas . Then using the same notation as we have hitherto employed, the mean Lagrangian function for the liquid is ............... ( l8 3), for the free gas &0 log?* + tf (&)-&, ............ (184), for the dissolved gas W ............ (185), where v is the volume occupied by the free gas, and v' the volume of the liquid or that occupied by the dissolved gas, and where w and f'(6) are the quantities for the dissolved gas which correspond to w and/(#) for the free gas. If we denote the sum of the expressions (183), (184) and (185) by If, then by the Hamiltonian principle H is stationary when the system is in equilibrium, so that if we suppose a small variation to be caused by a quantity of gas S escaping from the liquid then we must have for equilibrium dH this is equivalent to RO log -?- - RO +f(0) -w-RO log ->- + .0 -/' (0) + a-' o (186). The last term when the amount of gas absorbed is not large will be very nearly independent of the quantity of gas dissolved. Equation (186) may be written DIFFUSION OF SALTS. l8l RB log c^ = w-w' +f (0) -f(6) where ^ is a constant and p arid p are the densities of the free and dissolved gas respectively. Since the temperature is constant, we see from this equation that p'/p is constant, that is, the quantity of gas in unit volume of the liquid is proportional to the density of the free gas. This is Henry's law of the absorption of gases by liquids and it has been verified by the researches of Bunsen and others. Bunsen's experiments showed that the value of the ratio p'/p depends upon the temperature, hence we see from equation (187) that w-0-w'-'0 + r& + n0-r7ii cannot be zero, otherwise p'/p would be the same at all tempera- tures. Thus either the properties of the free gas can not be quite the same as those of the dissolved gas, or else the properties of the water are altered by the gas dissolved in it. 95. A similar investigation will apply to the case of a solid or gas which can dissolve in two fluids which do not mix. We can prove in this way that when there is equilibrium when the fluids are shaken up together then, provided the solutions are dilute, the amount dissolved in unit volume of one fluid will bear a constant ratio to that dissolved in the same volume of the other (see Ostwald's Lehrbueh der allgemeinen Chemie, I. p. 401). 96. The diffusion of salts through the solvent, a process which goes on until the solution acquires a definite state, can be explained by the same principles. In the following investigation of this problem we include the consideration of the effect of gravity upon diffusion. Let us suppose that 182 DYNAMICS. we have a shallow vessel whose volume is v and that this is connected by a capillary tube of fine bore with another shallow vessel whose volume is v' situated at a height h above the lower vessel. Let the two vessels be filled with water containing a certain quantity of salt dissolved in it, then we wish to find how the salt is divided between the vessels when equilibrium is established. Let and f\ be the quantities of salt in the lower and upper vessels respectively, then if h' be the height of the lower vessel above some fixed plane, the potential energy of the salt in the lower vessel may be taken to be g/i', so that there is the term - gh f in the expression for the mean Lagrangian function of this salt, similarly there is the term rjg(/i + h') in the expression for the mean Lagrangian function of the salt dissolved in the upper vessel. Thus using the same notation as before the expression for the mean Lagrangian function of the salt dissolved in the lower vessel is (188), the mean Lagrangian function of the salt dissolved in the upper vessel is + if (6) - ifw - ng (h' + h) . . . ( 1 89). " Let us' suppose that a quantity 8rj of salt goes from the lower to the upper vessel, then if there is equilibrium this change must not alter the value of H^ the mean Lagrangian function of the salt and solvent in the two vessels. If the solutions are dilute the only part of H which varies is the sum of the expressions (188) and (189), and the condition dH COMPRESSIBILITY OF SALT SOLUTIONS. 183 leads to the equation or if ', rj f are the masses of salt in unit volume in the lower and upper vessels respectively -,-#4 = 0, ' _** or | = * ..................... (190). So that the concentration of the solution when there is equilibrium varies in the same way with the height as the density of a gas under the action of gravity. 97. A large number of experiments have been made on the effect of dissolved salts on the coefficients of com- pressibility of various solutions (see Schumann " Compressi- bilitat von Chlorid Losungen," Wied. Ann. xxxi. p. 14, 1887 and Rontgen and Schneider, Wied. Ann. xxix. p. 165, 1886), we shall therefore investigate an expression for this effect and see what information can be gained by comparing it with the results of the above-mentioned experiments. Let us suppose that the solution whose original volume is v is subjected to a hydrostatic pressure p which reduces its volume to v, and that k' is its coefficient of compressi- bility. Then the mean Lagrangian function of the solution and the system producing the pressure is the mean Lagrangian function of the dissolved salt is, using the same notation as hitherto, 184 DYNAMICS. where is the mass of the salt. If H is the sum of these expressions then by the Hamiltonian principle H must be stationary when there is equilibrium. Let us suppose that the volume is increased by dv, then since H is stationary we must have dH now %RQ\v is the pressure due to the molecules of the salt, let us call this P. If p be increased by 8/, the correspond- ing diminution in volume v is by (191) given by the equation i Sv - or since v is very nearly equal to v we may write this equation in the form So that the apparent coefficient of compressibility is K i^Pk'' thus the pressure due to the molecules of the dissolved salt produces a decrease in the coefficient of compressibility. Let us see what the magnitude of this effect would be if the pressure of the molecules were the only way in which the dissolved salt affected the resistance to compression. If we make this assumption k' 1/2*2 x io 10 , this being the COMPRESSIBILITY OF SALT SOLUTIONS. 185 value when measured in c. G. s. units of this constant for pure water at i5C. If there is one equivalent of salt in a litre of water, P is 22 atmospheres or in absolute measure 2-2 x io 7 . Since the reduction in the coefficient of com- pressibility is very nearly equal to or to one part in i /-/%', we see that when the strength of solution is one equivalent per litre the reduction in the coefficient of compressibility ought to amount to one part in 2'2 X IO 7 that is to one part in 1000. The following table taken from Rontgen's and Schneider's paper will show that the effect of dissolved salts is some- times more than a hundredfold that calculated on the above assumptions, and hence we conclude that in addition to producing a pressure in the solvent the dissolved salt must directly alter its elastic properties. Reduction in com- Names of salt Strength of solution in pressibility found by Rontgen and or acid. equivalents per Schneider reckoned litre. in parts per thousand. HNO 3 49 42 HBr 49 40 HC1 52 52 H 2 S0 4 48 79 NH 3 I 49 90 NH 3 NO 3 5 94 NH 3 Br 5 90 NH 8 C1 Na 2 C0 3 t '97 1 86 DYNAMICS. 98. The pressure due to the molecules of the dissolved salt will explain many of the phenomena exhibited by solutions. The molecules of the salt may be regarded as confined within a limited volume by the solvent, and they will take any opportunity of expanding even though they may have to do work to enable them to do so. Thus if the solution was contained in a vessel provided with a bottom pervious to water but impervious to the substance dissolved in it, then if the vessel is placed in water with its top above the surface water will flow up into the vessel through the bottom, the work required to lift the water being supplied by the expansion of the molecules of the dissolved salt. This constitutes the well-known phenomenon of osmosis. A diaphragm which is said to be impervious to all salts though it allows water to pass through it can be made by allowing weak solutions of sulphate of copper and ferrocyanide of potassium to diffuse into a porous plate from opposite sides, these solutions when they meet form a membrane of the kind desired. . Detailed instructions for making these membranes are given in Pfeffer's Osmotische Untersuchungen, Leipzig, 1877. By following his directions I have succeeded in making such membranes though the number of failures was very large compared with the number of successes. Mr Adie, who is making some investigations on this subject at the Cavendish Laboratory, finds that the membranes are formed more readily if ferric chloride is used instead of copper sulphate. We shall now attempt to find by means of Hamilton's principle the height to which the fluid will rise in the osmometer. Let us suppose that the osmometer is a long tube with a diaphragm of the kind we have been describing at the bottom, and that it contains water and salt. Let OSMOSIS. IS/ be the mass of the salt, rj that of the water inside the tube, that of the water outside, and let v be the volume of the tube occupied by the solution. Then, using the same notation as hitherto, the mean Lagrangian function for the salt is *# log ^ + /(*) -*(, + *) (192), where z is the height of the centre of gravity of the salt molecules above some fixed plane. The mean Lagrangian function for the liquid in the tube is rjy'O + -nfl (0) - rj (wj +&) (193), and for the liquid outside the tube y<9 + t/;W-K + ^) (194), where y is the height of the centre of gravity of the water outside the osmometer : the quantities for the liquid inside the tube are denoted by affixing dashes to the symbols denoting the corresponding quantities for the water outside the tube. By the Hamiltonian principle the value of H, the sum of (192), (193) and (194), is stationary when there is equili- brium. Let us suppose that a quantity of water Srj flows into the osmometer. Then since if there is no contraction dv dz i di\ d-rj 20. ' where a is the area of the cross section of the osmometer, and 1 88 DYNAMICS. where h is the height of the top of the fluid in the osmometer above the level of that outside : the condition dH ^ = leads to the equation if the properties of the solution are not altered by the pre- sence of the salt then and equation (195) becomes where/ is the pressure due to the molecules of the dissolved salt, and h' the height of a column of water whose mass is the same as that of the salt dissolved in the osmometer. If the strength of the solution in the osmometer is one equiva- lent per litre, / is about 22 atmospheres, so that in this case h + \ti is about 660 feet ; that is the water would flow into the osmometer until the height of the liquid in the tube is nearly an eighth of a mile above the level of the water outside. If the liquid is not allowed to expand but confined in a constant volume we can easily prove in a similar way that if the properties of the solvent are not changed by the addition of the salt then when there is equilibrium the pressure exerted by the fluid in the osmometer must be the same as that due to the molecules of the salt. This result is given by Van t' Hoff {L'equilibre chimique, Archiv Neerlandais t. 20, p. 239). Pfeffer (Osmotische Untersuchungen, p. 12) gives as the OSMOSIS. 189 pressure for a i % solution of potassium sulphate that due to 192-6 centimetres of mercury and for a i / solution of potassium nitrate that due to 178-4 centimetres. The pressure calculated on the above principles for potassium sulphate is 97 centimetres if we assume that the molecule is K 2 SO 4 and 194 if the molecule is J (K 2 SO 4 ), for potassium nitrate it is 167 if the molecule is KNO 3 . We see as in 90 that the terms in (195) depending upon the alteration of the properties of the solvent by the addition of the salt do not contain any powers of the strength of the solution below the second. A measurement of the osmotic pressure produced by any salt solution will on the above assumptions give the same information about the structure of the molecule of the salt in the solution as a vapour density determination does about the structure of the gas whose vapour density is determined, for it enables us to find the number of molecules in a given mass of the gas. Thus Pfeffer's measurement of the os- motic pressure due to potassium sulphate suggests that the relation between the composition of the molecule of this salt and that of potassium nitrate is represented by ^K 2 SO 4 and KNO 8 , and not by K 2 SO 4 and KNO 3 . Even if we do not assume that the molecules of a salt produce a pressure analogous to that of a gas, it would still follow from the Hamiltonian principle that there would be a rise in the osmometer if the increase in the mean Lagran- gian function of the liquid inside the osmometer caused by the addition of unit mass of water is greater than the diminution in the mean Lagrangian function in the water outside the osmometer caused by the abstraction of unit mass of water. Anything that causes a change of this kind will increase the height to which the fluid will rise in the osmometer ; 1 90 DYNAMICS. thus, if the addition of water to the solution inside the os- mometer is attended by an evolution of heat, the solution will rise higher in the osmometer than one of similar strength in which no heat was evolved in dilution. On this account the indications of the osmometer are somewhat ambiguous, and before coming to any definite conclusion as to the structure of the molecule of the salt it would be necessary to use several solvents and to show that the osmotic height varied as the absolute temperature. 99. Surface Tension of Solutions. The experi- ments of Rontgen and Schneider already alluded to have proved that for most solutions the product of the height to which the solution rises in a capillary tube into the density of the solution is greater for a solution of a salt than for pure water, and that for dilute solutions of most (though not all) substances this product increases with the strength of the solution. It follows from this that the tension of the surface of contact of the solution with air increases with the strength of the solution, while the tension of the surface of contact of the solution with glass or any other solid body diminishes as the solution gets stronger. The variation of the surface tension with the strength of solution may cause the strength of the solution to vary near the surface. To investigate the magnitude of this effect let us suppose that we have a thin film whose area is S and surface tension T, connected with the bulk of the liquid by a capillary thread. Let be the mass of salt in the thin film, f] that in the rest of the liquid ; then if is the mass of water in the film, that of the rest of the water, the mean Lagrangian function of the liquid and salt in the film is, using the same notation as before, SURFACE TENSION, RO log - + tf(0) - fa + 7 + /; (0) - where v is the volume of the film. The mean Lagrangian function for the rest of the liquid is t]RB log VP + ij/(0) - 77?^ + eyO + e/;(0) - ?/,. Let us suppose that a mass of salt S goes into the film, the change in the mean Lagrangian function is and this by the Hamiltonian principle must vanish; thus if p, p are the masses of salt in unit volume of the film and liquid respectively, we get P or if T is the increase in the surface tension when the mass of salt in unit volume is increased by unity , zT' p r e > where t is the thickness of the film. Thus if the surface tension is increased by the addition of the salt there will be less salt per unit volume in the film than in the liquid in bulk, while if the surface tension is diminished by the addi- tion of salt there will be more salt in unit volume of the film than in unit volume of the rest of the liquid. We saw that the surface tension of a solution in contact with a solid di- minished as the strength of the solution increased, thus if we had a film in contact with a solid there would be more salt in unit volume of the film than in unit volume of the bulk of I Q2 DYNAMICS. the liquid ; if we dipped for example a piece of filter paper in such a solution, the solution in the filter paper would be stronger than the rest. Or, again, if such a solution were to flow through a capillary tube the salt would have a tendency to flow to the sides, so that the more quickly moving fluid at the centre would get weaker and weaker. Many experi- mental illustrations of this could be given ; one of these is an experiment tried by Dr Monckman and myself at the Cavendish Laboratory, in which a deep coloured solution of potassium permanganate emerged almost colourless after trickling through finely divided silica. Again, if a piece of filter paper be dipped into a coloured solution of a salt such as potassium permanganate, unless the salt has a very strong affinity for the water the solution after rising some height in the filter paper becomes colourless. If a small quantity of paraffin oil be mixed with water the surface tension of the solution against a solid is greater than that of water, and such a solution will increase in strength when it flows through finely divided silica. CHAPTER XIII. DISSOCIATION. 100. THE Hamiltonian method can t>e used for the pur- pose of obtaining the laws which govern the phenomena of dissociation, i.e. the splitting up of a molecule into its atoms, such as the iodine molecule I 2 into the atoms I and I ; or of a complex molecule into simpler ones, as in the case of nitrogen tetroxide, where the molecule N 2 O 4 splits up into two molecules of NO 2 , or when the molecule of chloride of ammonium splits up into ammonia and hydrochloric acid. This phenomenon has some analogy with that of evaporation ; as in the latter case we have equilibrium between portions of matter in two different states, the gaseous and the liquid, matter being able to pass from the one state to the other by evaporation and condensation, so in dissociation we have also equilibrium between portions of the same substance in two different conditions, both in the gaseous state, the molecules in the one condition being more complex than those in the other, and matter being able to pass from one condition into the other by the more complex molecules splitting up, " dissociating" as it is called into the simpler ones, while on the other hand some of the simpler ones combine and form the more complex molecules. Equilibrium is attained when the number of the complex T. D. 13 IQ4 DYNAMICS. molecules which split up in any time is the same as the number formed in the same time. Let . us first investigate the case when the complex molecules contain two of the simpler ones ; this is the case when, as in iodine, the more complex systems are di-atomic molecules and the simpler ones atoms, as well as in such cases as the dissociation of N 2 O 4 . Let us suppose that the system is contained in a closed vessel and that is the mass of the complex molecules, 77 that of the simpler ones. We shall for the present assume that both gases obey Boyle's law and that the fundamental equation for the complex gas is f = J? lP 0, and for the simpler gas / = *,P0, where / is the pressure, p the density and the absolute temperature. Since the molecules of the complex gas consist of two of those of the simpler gas, the density of the simpler gas will at the same pressure and temperature be half that of the complex gas and therefore ^ 2 =2^. The mean Lagrangian function of the complex gas is fl^log^a+S/iW-frr, (196), where v is the volume of the vessel in which the gas is contained and w l the potential energy of unit mass of the complex gas. The mean Lagrangian function for the simpler gas is ^ 2 01og^W 2 W-W 2 (197), where w 2 is the potential energy of unit mass of this gas. DISSOCIATION. 195 The mean Lagrangian function H for the two gases, assuming that the properties of each are not modified by the presence of the other, is the sum of the expressions (196) and (197). By the Hamiltonian principle the value of H is stationary when the system is in equilibrium. Let us suppose that the state of equilibrium is disturbed by a mass S of the simpler molecules combining to form complex ones. Then since the value of H is stationary we must have dH_ Since the mass of the gas is constant and the condition ~=o is equivalent to log - xf +/ (0) -w,- R 2 e log p or since we have vp 2 = o ...... (198), $ log This can be written 2 Wj-TOg ^=*(). *. .................. (199), where (0) is a function of 6 but not of , rj or v. In experiments on dissociation the quantity usually measured is the vapour density of the mixture at some standard pressure TT. 132 196 DYNAMICS. Let A be the density of the mixture of the two gases at this pressure, D that of the complex or undissociated gas at the same Dressure. the same pressure. The pressure in the vessel is or since R z - 2^, the pressure equals Rfl. The density of the gas at this pressure is v ' so that A the density at the pressure TT is given by the equation A= ll _^_ . the density D of the complex gas at this pressure is given by the equation so that u r and therefore +2rj~ D and fi^T 4H since PV ( + 2rf) Rfl, pv Z>-A we have *) -^-^ j^~ pv 2^-D DISSOCIATION. 197 so that equation (199) becomes where 10 1. Before discussing this equation we shall investigate the way in which it must be modified if the gas does not obey Boyle's law. Formulae connecting the pressure and volume in such gases have been given by Van der Waals (Die Continuitat des gasformigen una flilssigen Zustandes] and Clausius (Wied. Ann. ix. p. 337). Van der Waals' formula, which is rather the simpler of the two, is RQ a where R is the value of pfpO for a perfect gas of the same specific gravity, and b and a constants depending upon the nature of the gas. Clausius' formula is RB__ *_ P ~V-a 6(v + p)*' where R is the same as in Van der Waals' formula for the same gas, and a, /?, K are small constants depending upon the nature of the gas. We shall now investigate the differ- ence produced in the state of equilibrium of a dissociable gas if it and the components into which it is decomposed obey Van der Waals' law instead of Boyle's. 198 DYNAMICS. Let the fundamental equations of the complex and simple gases be respectively and where as before. Then we can easily prove that instead of the term P in the mean Lagrangian function, we have the term where p is the density of the complex gas; with a corre- sponding term in the expression for the Lagrangian function of the simpler gas. The condition dH =0 will now lead to the equation log (I - b lP ) + log P - - P I - - R 2 | log ( i - b 2 p') + log ^ - ^-^1 -f +f l (0)~f 2 (6)-(w l -w a ) = o .......... (201) and not to the equation (198). DISSOCIATION. 199 Equation (201) may be written, since J? a =2jRj, = / - w Now if we suppose that the deviations from Boyle's Law are slight, so that b l and b z are so small that their squares may be neglected, we may write this equation as = dt, are the alterations in the masses of PC1 8 , PC1 3 , and C1 2 respectively ; hence, remembering that we see that the condition dH leads to the equation it (208), and thus ifQ^v is constant as long as the temperature is constant. Let us suppose that the values of , -rj, before dissoci- ation commenced were , iy , and that the mass c^p of PC1 5 gets decomposed, then we have and the equation to find / is (% + ^ 2 ^) (^o + 'a/) ^ ^ tfo - 'i/) ...... ( 2 9), where ^ is a function of the temperature. We shall now discuss the effect upon / of alterations in the values of v, r] and . Differentiating (209) we get, writing y for T. D. 14 210 DYNAMICS. dp I ........................ (210), 1 dv v dp_ _ i , = * We see from (210) that dpjdv is positive, so that dissociation will be promoted by increasing the volume in which a given quantity of gas is confined. From equations (211) and (212) we see that both dpjd-q^ and dpjd^ are negative, so that the presence of free PC1 3 and C1 2 tends to stop the dissociation. Wiirtz proved experimentally that there was very little dissociation of PC1 5 when it was placed in an atmosphere of PC1 3 . We can also see from general principles that this must be so, for as soon as the molecule PC1 5 breaks up the free chlorine will be surrounded by such a multitude of mole- cules of PC1 3 that most of it will recombine and form PC1 5 , and in this way stop the dissociation. In this case, as in the former, theory indicates that if there is no catalytic action the presence of a neutral gas would not produce any effect. In some cases, though the results of the dissociation are in the gaseous state, the body which dissociates is in the solid or liquid state instead of, as in the previous instances, the gaseous. The dissociation of NH 5 S into H 2 S and NH 3 is an example of this kind. We have only to slightly modify the preceding work to make it applicable to this case. Let as before be the mass of the dissociating body, rj and those of the components into which it is dissociated. Then the mean Lagrangian function for the solid or liquid dissociating body is by (81) DISSOCIATION. 211 The mean Lagrangian functions of the gases into which it dissociates are respectively +. -*., and Then from the condition dH <*r> we get, since d% = - (drj + d), dv i # = ~a' and *-*, ', ',' where ^ 2 and , while C by its action on D produces A A well-known example of this kind of action is the case in which the four substances A, B, C, D are respectively nitric acid, sodium sulphate, sulphuric acid and sodium nitrate : the nitric acid acts on the sodium sulphate and forms sodium nitrate and sulphuric acid, while the sulphuric acid acts on the sodium nitrate and forms sodium sulphate and nitric acid. The problem we have to discuss is to find, when any quantities of four such substances are mixed together, the quantity of each when there is equilibrium. Let , -T), , be the masses of the substances A, B, C, D respectively, w lt w 2 , w 3 , w 4 the mean potential energy of unit mass of each of these substances, w the mean potential energy of the mixture. Let us suppose that each of these substances obeys Boyle's law; and p denoting the density 2l6 DYNAMICS. . of any one at the temperature 6 and pressure /, let the fundamental equation of A be that of B p = that of C p = and that of D p = R^B. Then the mean Lagrangian functions of A, B, C and D are respectively where z; is the volume in which the substances are confined, The above expressions represent the mean Lagrangian functions equally well whether the substances A, B, C, D are gases or dilute solutions, provided the solutions are so dilute that the molecules of the substances dissolved in them exercise the same pressure as they would if placed at the same temperature in the same volume when empty. If we are considering solutions we shall require the mean Lagrangian function of the solvent, for the properties of this may alter as chemical combination goes on. If TT is the mass of the solvent, w 5 the potential energy of unit mass, then its mean Lagrangian function will be of the form Try 6 + 7r/ 5 (0) TTW & . We must now investigate the relations between the changes in , 17, , c as chemical action goes on. CHEMICAL EQUILIBRIUM. 217 Let us denote by (A) the molecule of the substance A, with a similar notation for the other molecules, and let the chemical action which goes on between the four substances be represented by a(A) + b(B} = c(C) + d(D] ............ (213). Thus, in the case of the mixture of sulphuric and nitric acids, sodium nitrate and sodium sulphate, since the equation which expresses the reaction is 2 HNO 3 + Na 2 SO 4 - H 2 SO 4 + 2 NaNO 3 , if the molecules of nitric acid, sodium sulphate, sulphuric acid and sodium nitrate in the solution are represented respectively by HNO 3 , Na 2 SO 4 , H 2 SO 4 and NaNO 3 , then a = 2, b = i, c= i, d 2. If however the molecules of these substances are represented by H N 2 O 6 , Na 2 SO 4 , H 2 SO 4 , Na 2 N 2 O 6 , then a = b = c = d=\. Thus we see that it is necessary to know the structure of the molecule as well as its relative composition. From equation (214) we see that if a molecules of A disappear it must be because they have combined with b of B to produce c of C and d of Z>, so that b molecules of B have also disappeared, while c of C and d of D have appeared. Let 8 1} 8 2 , 8 3 , 8 4 represent the relative densities of A, B, C, D at the same temperatures and pressures, then If the masses of A, B, C, D are altered by d^ dv\, d^ de respectively, then the alterations in the number of molecules of A, B, C, D are respectively proportional to dj_ drj d di V V V V 218 DYNAMICS. Thus, since the alterations in the number of molecules are proportional to a, b,-c,-d respectively, we have So that d* Now when the system is in equilibrium the value of the Hamiltonian function must be stationary, so that if we suppose the equilibrium displaced by the quantity d of A combining with the proper quantity of B the change in the Hamiltonian function must be zero, hence we must have dH . ,. - . =o (216). Let us take first the case when A, J3, C, D are gases, then since H is the sum of the mean Lagrangian function for these substances the condition (216) with the help of equations (215) gives the equation R$ log - a\R$ + b^R 2 log - - c^Rf log p - d^R log V ^ where w = w l + rpv a + w a + cw 4 . Then by (214) we may write this equation in the form CHEMICAL EQUILIBRIUM. 2IQ a dw # # ......... ( 2I y) when <(0) is a function of & but does not involve , ;, r - 112. In the case of dilute solutions the equation corre- sponding to (217) is easily seen to be Y c d a ^ w a e Q J_l = ^) 7J c+d-a-6 JWl*t R# d ..... ( 2I g) where Q is the mean Lagrangian function of the solvent and equals The value of dQjd will be zero if the properties of the solvent do not change as chemical action goes on; in any case since the solutions are very dilute the properties of the solvent may be assumed to be changed by an amount pro- portional to the quantity of salt dissolved, Q will therefore be a linear function of , rj, , e and dQfd^ will not involve any of these quantities, and in this case as in the former one we have -* K* & ......... (219) so that the equations of equilibrium for gases and dilute solutions are of exactly the same form. 113. The value of dw\d% measures the increase in the potential energy of the system when the mass of is increased by unity. Now if heat is produced when C and D combine to form A and B, the potential energy diminishes as in- creases, and when the quantity of heat is large its mechanical equivalent may be taken as a measure of the decrease in the potential energy. If the combination of C and D is accompanied by the production of heat, dwjd^ is negative, and we see therefore 220 DYNAMICS. that if 6 = o or either or e must vanish, that is, the combination of C and D will go on until one of these substances gives out, in other words the reaction attended by the production of heat will at the zero of absolute temperature go on as far as possible. According to Berthelot's law of " Maximum Work " the reaction accompanied by the formation of heat goes on as far as possible at all temperatures, the equation (218) however shows that this is strictly true only at the zero of temperature. For substances which give out large quantities of heat when they combine equation (218) shows that the com- bination increases so rapidly as the temperature diminishes, that if there is any combination at all at temperatures as high as 1000 C., Berthelot's law will be practically true at all ordinary temperatures. To illustrate this let us take the case of hydrogen and oxygen, where the combination is represented by the equation 2 H 2 + 2 =2H 2 0. Let , y, be the quantities of hydrogen, oxygen and water respectively, then a = 2, =i, c=2, d=o, and equa- tion (218) becomes c>2 2 (JV! <--I*(0).**. If we substitute for/(0) its value given on page 270 we shall find that (0) in this case = C/0 3 ' 5 . For hydrogen at oC. JK^O T'i x io 10 , and since in the combination of one gramme of hydrogen with oxygen 34000 calories are given out, dw 19 ^-=1.43x10". Let us suppose that equivalent quantities of hydrogen and oxygen are mixed together, and that the number of equivalents which combine to form water is to the whole CHEMICAL EQUILIBRIUM. 221 number of equivalents of either oxygen or hydrogen present initially as x to i, then x is given by the equation 2 fl 2 dlV Suppose that at 1092 C. one half of the equivalents combine, then the value of x at 546 C. is given by the equation 3-5 /l, 30 _ 130\ = 2.1 1 thus approximately i = - x 10 5 So that at this temperature only about one in five hun- dred thousand of the molecules will be left uncombined. Thus in a case like this very considerable dissociation at one temperature is compatible with almost complete combina- tion at a temperature not very much lower. 114. The effect of pressure on chemical equi- librium. We have by equation (219) thus if a + b the ratio gJjg'irf is independent of the volume, so that if we mix given quantities of the four substances the amount of chemical action which will go on will be independent of the volume into which the substances are put. Since the chemical reaction is such that when A acts on B, a molecules of A and b of B disappear while c of C and d of JD are produced, we see that if a + b = c+d the number of mole- cules in the vessel does not change as the reaction goes on. This is sometimes expressed by saying that the combination 222 DYNAMICS. takes place without change of volume, and in this case, as we have just seen, the amount of chemical combination is not affected by the volume in which the combining sub- stances are placed. If a + b is greater than c+d then the larger the volume z/, the smaller will be the ratio of g^ to t^rf. Now the action of C on D tends to diminish this ratio, while that of A on B tends to increase it, and if a + b is greater than c + d the number of molecules is increased when C acts on D and diminished when A acts on B. Thus we see from equation (219) that when chemical com- bination alters the number of molecules the state of equi- librium depends upon the volume within which the substances are confined, and that the effect of increasing the volume is to favour that reaction which is accompanied by an increase in the number of molecules. In other words, the chemical action which produces an increase in volume is hindered by pressure, while that which produces a diminution is helped by it. This is another example of the law stated in (84). 115. Let us now consider a little more closely some of the results of equation (219), taking for the sake of simplicity the case when a = b = cd i. Let us suppose that the masses of the four substances A, -B, C, D before combination begins are , iy , , e , and that when they have reached the state of equilibrium a quantity 8,/ of A has disappeared, then by equations (215) we have and equation (219) becomes MASS ACTIONS. 223 Let us suppose that the quantities of the substances mixed together initially were proportional to their combining weights, i.e. that initially equivalent quantities of the four substances were taken, then we may put And equation (220) becomes if we put t+p then * = and is called the affinity coefficient of the reaction (Muir's Principles .of Chemistry, p. 417). Thus we may write equa- tion (220) in the form where / is constant as long as the temperature remains unchanged. The effects due to what are called " mass actions," that is the effects produced by varying the quantities of the four substances initially present may be deduced at once from this equation. Let S/j be the increase in / when is increased by S , the quantities r/ , > e o remaining constant; and let S/ 2 , 8/ 3 , 8/ 4 be the respective increases when 77 , ^ , e are increased by 877,,, 8^ , 8c respectively. Then we get at once from equation (221) 224 DYNAMICS. where y is the positive quantity or We see from these equations that tyJ8 and fy>J$r) are positive while S/ 3 /S , S/ 4 /^ are negative, so that any increase in the quantity of A and B initially present increases the amount of combination that goes on between these sub- stances, while any increase in the quantities of C and D initially present decreases the amount of combination, and further that the effects of equal small changes in the masses of A, B, C, D before combination takes place are inversely proportional to the amount of these substances present in the state of equilibrium. In the more general case, where a, b, c, d are not each put equal to unity, we may easily prove that A- f , CHEMICAL COMBINATION. 22$ and that if 8/ be the change in p due to an increase 87' in volume, everything else being constant, , 2 where y = y + __*+ _a + _, . here a&^p is the mass of A which has disappeared. 1 1 6. The expression (221) agrees with the formula ob- tained by Guldberg and Waage from quite different principles (see Muir's Principles of Chemistry, p. 407, and Lothar Meyer, Modernen Theorien der Chemie, chap. xm). The case when a = b = c = d\s the only one however in which the expression deduced from Hamilton's principle agrees with that given by Guldberg and Waage. According to their theory, as given in the works we have just cited, the equation (221) is always true, while according to the theory we have been explaining it is only true when a-b = c = d. It would seem however that the principles from which Guldberg and Waage deduced their equations would when a, b, c and d are not all equal lead to equation (219) rather than (221), for their point of view seems to be as follows. Consider first the case when a = b = c = d =- i, then in a certain proportion of the collisions which occur between the molecules of A and B, chemical combination between A and B will take place. The number of collisions in unit time is propor- tional to the product of the numbers of molecules of A and B, and so is proportional to jjt\. The number of cases in which combination takes place may be taken therefore to be k&\ when k is a quantity which is independent of the quantities of A, B, (7, D present. In other words, the number of molecules which leave the A, B states and enter those of C and D is k&\ ; in a similar way we can see that the number of molecules of C and D which become A and T. D. 15 226 DYNAMICS. B is. 'e. Now when the system is in a steady state the number of molecules of A and B formed must be the same as the number which disappear, and therefore V = which is Guldberg's and Waage's equation. We can easily see however that the above reasoning is only applicable when chemical combination takes place between one molecule of A and one of B, and again between one of C and one of D, or in other words when a = b = c = d=i. If on the other hand the equation which represents the chemi- cal reaction is 2 (A) + (B) = (C) + 2 (D\ then chemical combination will take place when one mole- cule of B is in collision with two of A simultaneously ; the number of such combinations will be proportional to i/* and not to rjg, and thus the number of molecules of A which disappear owing to their combination with B molecules may be represented by ktff \ similarly the number of molecules of D which disappear and of A which appear by the combination of C and D may be represented by k'^ 2 ; and since in the state of equilibrium the number of molecules of A which disappear must be the same as the number which appear we must have hff = Ktf, which agrees with equation (219) but not with Guldberg and Waage's equation. 117. As we noticed before in (107), there is some ambiguity as to what the molecule of the dissolved salt or acid really is. For example, take the case already mentioned where the reaction is represented by the equation H 2 SO 4 + 2NaNO :j = 2 HNO 3 + Na.SO,, we do not know whether the molecule of sodium nitrate is CHEMICAL COMBINATION. 227 represented by NaNO 3 or by Na 2 N 2 O 6 , or whether the molecule of nitric acid is represented by HNO 3 or H 2 N 2 O 6 . This point could probably be settled by experiments on osmotic pressure, the lowering of the vapour pressure of the solution and the effect of the salt or acid upon the freezing point. If the molecules are represented by Na 2 N 2 O 6 , H 2 N 2 O 6 and not by NaNO 3 , HNO 3 , it would be necessary to dissolve 170 and 126 grammes of these substances in a litre of water, instead of 85 and 63 to produce the effects observed in solutions of one gramme equivalent per litre. We can however use the formula (219) giving the amount of chemical action between these substances to decide this point. If the molecules are represented by HNO 3 , Na 2 SO 4 , H 2 SO 4 and NaNo 3 then by equation (219) e 2 /?7 2 is constant provided the temperature remains unaltered, if however the molecules are represented by H 2 N 2 O 6 , Na 2 SO 4 , H 2 SO 4 , and Na 2 N 2 O 6 (or by HNO 3 , JNa a SO 4 , JH 2 SO 4 , NaNOJ then is constant as long as the temperature is unaltered, where , 17, , e are the masses of the sulphuric acid, sodium nitrate, nitric acid and sodium sulphate respectively. This reaction has been investigated by Thomsen (Thermo- chemische Untersuchungen i. p. 121) and in the following table n ef/& *W 8 4 2.07 2.6 40.5 33 2 I 2-5 3-3 13-05 8 J 4.1 3-2 * 4.1 I.O the values of eg/grf, /??, calculated from his experiments 2 228 DYNAMICS. for different proportions of the substances, are given, n is the ratio of the number of equivalents of sodium sulphate to the number of equivalents of nitric acid before chemical combination commences. It will be seen from this table that when there is only a very small quantity of nitric acid present initially, formula (219) seems to agree with the observations as well as (221), but that it ceases to be any approximation when the solution gets stronger, and that now equation (221) agrees better with the experiments. From this we should conclude that in very dilute solutions the molecules of nitric acid and sodium nitrate rnay possibly be represented by HNO 3 , NaNO 3 , but that in stronger solutions either they are re- presented by H 2 N 2 O 6 , Na 2 N 2 O ti , or else that the molecules of sulphuric acid and sodium sulphate are represented by 1H 2 SO 4 , JNa 2 SO 4 . Pfeffer's determination (98) of the osmotic pressure produced by a potassium sulphate solution suggests that the molecule is represented by J (K 2 SO 4 ). We ought not however to attach as much weight to the experi- ments with dilute solutions as to those with strong, because in the weak solutions a very small error in the determina- tions will produce a considerable error in the value of 2 /W or eC/ft. If there was any change of this kind in the constitution of the molecules as the strength of the solution increased it would probably show itself in the effect of the substance on the osmotic pressure, on the vapour pressure, and on the lowering of the freezing point, even though these effects were complicated by the alteration in the properties of the solvent produced by the addition of the salt. 118. In the case we have just been considering the four substances A, B, C, D were supposed to be either CHEMICAL COMBINATION. 22Q gaseous or soluble. We must now see how the equations have to be modified when one or more of the substances is a solid, and if we are considering the case of solutions an insoluble one. Let us take first the case when only one of the substances D is an insoluble solid, for example when the four bodies are oxalic acid, calcium chloride, hydrochloric acid, and calcium oxalate. The mean Lagrangian function for D will now be of the form and the condition dH -rr will lead to the equation ** ............ (222 ). If two of the substances are insoluble solids, as for example when A is potassium carbonate, B barium sulphate, C potassium sulphate, D barium carbonate, then we can easily prove that | = ^-^(0)/i<> .............. (223). We see from these equations that the amount of com- bination which goes on does not depend on the masses of the insoluble substances. 119. As an example of a case where the conditions are rather more complicated than in those discussed in the last paragraph, we shall consider a case investigated by Horstmann ( Watts' Dictionary of Chemistry, 3rd Supple- ment, p. 433) where hydrogen, carbonic oxide and water were exploded, and water and carbonic acid produced. 230 DYNAMICS. Here we have to consider five substances, hydrogen, carbonic oxide, oxygen, water, and carbonic acid ; let , ?, , c, TT, be the masses of these substances respectively, and let c lt r 2 ,...<: 5 , be their molecular weights. Let the relation between the pressure /, the density p, and the absolute temperature be for hydrogen p-Rfa for carbonic acid with a corresponding notation for the others. Let the mean Lagrangian function for the hydrogen be where v is the volume in which the gases are confined, and o/j the mean potential energy of unit mass of hydrogen. The mean Lagrangian function of the other gases will be given by analogous expressions. Now whatever changes go on among the various gases we have since the quantity of hydrogen is constant - + = a constant ; fi ^ since the carbon is constant -n TT + - = a constant : ', '5 since the oxygen is constant 1 f] i 7T i-+ ~ + i-+-=a constant ; 2 ' 2 ^3 ^ ^ these are three equations between five unknown quantities, so that if we give arbitrary variations to two of them the variations of the others will be determinate. CHEMICAL COMBINATION. 231 Let us choose and rj as the independent variables, then when 1} is constant we have and when is constant When the system is in equilibrium the mean Lagrangian function is stationary for all possible variations, so that we must have fdff\ = Q idff\ \ dc, / 17 constant \ drf /^constant Remembering that the first equation gives e ,k - 1 ( d ^L\ -! = d> (6) g^Ki 6 ^ />? constant .......... (224), 8* and the second i i fdw\ ' lA^V^ /Constant where w is the mean potential energy of the mixture of gases. These are of the same form as the equations I obtained from kinematical considerations alone in my paper on the Chemical Combination of Gases already referred to. If we divide (224) by (225) we get > MKf )-(!")! 232 DYNAMICS. so that, as long as the temperature is constant, the ratio of the quantity of water formed to the quantity of carbonic acid always bears a constant ratio to the ratio of the quantity of free hydrogen to that of free carbonic oxide. This was the result obtained by Horstmann in the experi- ments before mentioned. CHAPTER XV. EFFECTS PRODUCED BY ALTERATIONS IN THE PHYSICAL CONDITIONS ON THE COEFFICIENT OF CHEMICAL COMBINATION. 120. SINCE the value of *c IV is independent of the values of , rj, , e, and since when it is known the amount of chemical combination can be determined, it is convenient to have a name for it, we shall therefore call it the coefficient of chemical combination for A and B and denote it by k. The more intense the chemical action between A and B the smaller the values of , 77 in the state of equilibrium and therefore the larger the value of k. We have by equation (220) a dQ a dw k = <^> l (ff)e **# e*i* ^ (226). The alterations which we shall suppose to take place in the physical conditions can be represented by changes 8(2 and 8w in the values of Q and w, and we see from 234 DYNAMICS. equation (226) that if k be the corresponding change in k a d&Q a *" " If the substances with which we have to deal are gases we must put Q and &Q equal to zero. We considered when we were discussing dissociation in chapter xiv. most of the changes in the physical conditions which could influence the state of chemical equilibrium in this case, and the results obtained then will apply to the more general problem we are discussing now. We see from (227) that any cause producing a change in the potential energy which increases as any chemical action goes on will tend to stop this action which will not have to go on so far before attaining equili- brium as it would if the disturbing cause had been absent and vice versa. We shall now go on to consider more particularly the cases of dilute solutions and the effects produced upon chemical equilibrium by changes in the properties of the solvent arising from the progress of chemical change. 121. Effect of Surface Tension. The first effect we shall consider is that due to the surface tension of the solution. We know that the surface tension depends upon the strength and the nature of the solution, so that since the composition changes as chemical action goes on the surface tension of the solvent and therefore its mean Lagrangian function will change ; and therefore by the principle we have just stated the conditions for equilibrium will be altered by the surface tension. Let A be the area of the surface of the solution, T the surface tension, then the potential energy due to the surface CHEMICAL COMBINATION. 235 tension is TA and there is therefore in the expression for the mean Lagrangian function the term TA, so that by equation (227) the effect of the surface tension on the coefficient of chemical combination is given by the equation - -- d (AT*} k ~ Rfd^ Let us endeavour to get some idea of the magnitude of this effect. If c is the molecular weight of the substance whose mass is , then since at o C. iT x io, we have, if for simplicity a be put equal to unity, No\v cd (A T}jd is the increase in A T when the quantity in the solution is increased by one gramme-equivalent. If v be the volume of the vessel whose surface we shall suppose to remain constant as combination goes on, then where T' is the increase in T when the quantity is increased by one gramme-equivalent per litre. Now the experiments of Rontgen and Schneider (" Oberflachen Spannung von Fliissigkeiten," Wied. Ann. xxix. 165) show that T' even in the case of simple salts may be as much as 5 or 6 so that 8& . ? A T is of the order -=- . k 10' v and if the solution be spread out in a film of thickness /, 236 DYNAMICS. A/v = 2// so that 6 6/e . 01 T is of the order,-; thus if the thickness of the film is i/ioooo of a centimetre the value of k is altered by about '6 per cent. If the thickness of the layer is comparable with molecular distance, say about io~ 7 , then kjk might be as large as 6. This of course implies that the conditions of equilibrium would be completely altered. Thus in very thin films the in- fluence of capillarity might be sufficient to modify com- pletely the nature of chemical equilibrium, though we should not expect it to do much in the body of a fluid. If the surface tension increases as the chemical action goes on the capillarity will tend to stop the action, while if the surface tension diminishes as the action goes on, the capillarity will tend to increase the action. Thus the chemical action in a space such as a thin film throughout which the forces producing capillary phenomena are active might be very different from the chemical action in the same substance in bulk when most of it would be free from the action of such forces. This point does not seem to have received as much attention as it deserves, but there are some phenomena which seem to point to the existence of such an effect. One of these is that called by its discoverer Liebreich " the dead space in chemical reactions," which is well illustrated by the behaviour of an alkaline solution of chloral hydrate. If the proportion of alkali to chloral is properly adjusted, chloroform is slowly deposited as a white precipitate, and if this solution is placed in a test-tube, then at the top of the liquid there is a thin film which remains quite clear and free from chloroform, showing that, unless this effect is due to , T He fNIVERSITl CHEMICAL COMBINATION. iF^MN^V some chemical action of the air, the alkali and chloral do not combine, or if they do chloroform is not precipitated. In fine capillary tubes too, no deposit seems to be formed. This phenomenon could be explained on the above principles if the surface tension of the alkaline solution increases when the alkali combines with the chloral and chloroform is deposited, for in this case the surface tension would increase as chemical action went on, and would therefore tend to stop this action. Dr Monckman made some experiments in the Cavendish Laboratory on the changes in the surface tension of the solution as the reaction went on, and he found that it increased to a very considerable extent, so that this case is in accordance with our theory. The thickness of the dead space (from i to 2 mm. in Liebreich's experiments) is somewhat greater than we should have expected, but any want of uniformity in the liquid such as that produced by the deposition of chloroform itself would increase the thickness of the dead space. Some other effects produced by surface tension are discussed by Prof. Liveing in his paper " On the Influence of Capillary Action in some Chemical Decompositions " (Proceedings Camb. Phil. Soc. vi. p. 66). 122. Effects due to pressure. Pressure can pro- duce effects of two kinds upon chemical action. The first is when the volume of the liquid under pressure alters as chemical action goes on, the effect of pressure in this case is proportional to the amount of the pressure: the second effect is when the coefficient of compressibility of the liquid changes as the chemical action goes on, the effect of pressure due to this cause is proportional to the square of the pressure. Let us suppose that P is the external pressure, v 238 DYNAMICS. the volume, we may regard the external pressure as produced by an external system whose mean Lagrangian function is and we have by equation (227) ?*- a - k ~' dv Thus if v increases with , k is positive, in other words the value of Tg^l^if is increased and therefore and TJ are less than they would be if there were no external pressure. Thus the external pressure tends to stop that action which is accompanied by an increase in volume, and vice versa. Let us now endeavour to form some estimate of the probable size of this effect. If the molecules of the substance produce the same pressure as if they were in the gaseous state, then at o C. 2-2 where c is the combining weight of the substance. Thus if the volume increases by y cubic centimetres per gramme of A formed we have by (228) if the pressure is x atmospheres, cxya k 2'2 X I0 4 ' The cases in which in general y will have the greatest value are those in which we have some of the bodies in solution while others are precipitated, if we suppose that when a salt is dissolved the volume of the solvent is not altered then y will in general be not greatly different from unity, and in this case we have CHEMICAL COMBINATION. 239 S/ cax so that it would require a pressure of 22ofac atmospheres to change the coefficient of combination by one per cent, thus if the substances taking part in the reaction have large combining weights, the reaction will be sensitive to the influence of pressure. Let us now consider the effect on the chemical equi- librium when the coefficient of compressibility changes as the chemical action goes on. Let o- be the expansion or contraction of the solution, K its bulk-modulus, v its volume, then in the expression for the potential energy of the solvent there is the term ir/o-V, and therefore in the expression for the mean Lagrangian function the term If S/ be the change in the coefficient of combination due to the change in K as the chemical action progresses we have by equation (227) Now if P be the external pressure K'. Let a/,, w 2 be the potential energies of unit masses of the salt and solution respectively. Then the mean Lagrangian function for the salt is -to, 16 2 244 DYNAMICS. where / (0) is the part of the mean kinetic energy of unit mass which does not depend upon the controllable coordi- nates. If v be the volume of the salt and we put then the Lagrangian function for the salt may be written The mean Lagrangian function for the solution is with a similar notation where f a (0) is the part of the kinetic energy of unit mass of the liquid which does not depend upon the controllable co- ordinates, and v is the volume of the solution. We must remember that though O and w l do not depend upon the values of and ij yet the values of O', w z and f a (9) may do so as the properties of the solution may and generally do alter when the amount of salt the solution contains is altered. By the Hamiltonian principle the value of the mean Lagrangian function of the salt and solution when in equilibrium is stationary. Let us suppose that when the system is in equilibrium the conditions are disturbed by a mass 8$ of the salt melting, then the change in the value of H is, if (0), which when equated to zero gives the temperature at which melting takes place. Let us now consider the effect of a slight change in the physical conditions. If this change increases the Lagrangian function by x and does not affect appreciably the values of O/cr, O'/p, we have if the melting point is now 6 + 30, or since (0) = o Let us consider the effect of pressure upon the freezing point. If the external pressure is / then x = -p(v + i/) 9 and since T. D. 258 DYNAMICS. so that But from equation (116) we have S C = ' the volumes, tn e uniform twist produced by a force P acting at an arm , then in the mean Lagrangian function of the tube there are the terms - - Itfn (a 4 - 0*) 4 where v is the volume of the ice. So that if 80 be the rise in the freezing point produced by the torsion we have = i [Pbl- }*** (a' + If)] - J** If the sides melt equally we have since a and b are approximately equal so that 20-X since dv _i d^ IT' To get some estimate of the magnitude of this effect let us suppose that the cylinder is i centimetre in radius, and that < is 1/40. Since Young's modulus for ice is 6 x io 10 , n is probably about 2 -4 x io 10 , substituting these values we 262 DYNAMICS. find 80 i -5- = approximately, so that 80 = - -68 C. So that in this case the ice on the surface would melt unless the temperature was lower than - -68 C. 130. Effect of surface tension on the freezing point. If a portion of a drop of water freezes, the form- ation of the ice will cause a diminution in the surface of separation of the water and air if the ice rises to the surface of the drop, to balance this however we have two fresh surfaces formed where the ice meets the water and air; the diminution in the first surface would tend to promote freezing, the formation of the other two would tend to pre- vent it, but as we do not know the surface tension between ice and water and between ice and air we cannot calculate which of these tendencies would have the upper hand. 131. The effect of dissolved salt on the freezing point. When a salt solution freezes the salt appears to remain behind, and the ice from such a solution is identical with that from pure water. Thus when a portion of a salt solution freezes, the particles of salt are brought closer to- gether, and work has therefore to be done upon them, the Lagrangian function therefore diminishes, and we see by equation (239) that the presence of the salt will tend to> prevent the water from freezing. To calculate the magni- tude of this effect, let be the mass of the salt, then using the same notation as before, the mean Lagrangian function for the salt if the solution is dilute is LIQUEFACTION. 263 where w 3 is the mean potential energy of unit mass of the salt. When the mass of ice is increased by 8 the only quantity which changes in the expression is if which dimin- ishes by S//o. Thus equation (239) becomes W i (Oft /Q' - - H where 8 (Q'/p) and B/ a (6) are the changes in O'/p and f a (6) due to the salt. If zzr be the pressure due to the molecules of salt in the solution, so that If we suppose that the salt does not alter the properties of the solvent we have Let us first suppose that the solvent is water; if we consider solutions whose strength is such that a number of grammes equal to the formula weight is dissolved in one litre of water, then w is about 22 atmospheres, or in absolute measure about 2 2 x io 7 , X = 80 x 4 2 x io 7 , = 273, and p is unity ; substituting these values we get 86 = -i-8C. Raoult, Annales de Chimie et de Physique, v. in. p. 324, 1886, found that solutions of this strength of many substances, chiefly organic salts, froze at --1*9, but that the freezing points of solutions of salts and acids were generally lower than this ; he attributed the increased effect to the dis- 264 DYNAMICS. sociation of the molecules; it might however, as in the analogous cases we considered before, be due to the altera- tion of the properties of the solvent by the addition of the salt. It would also take place if there were any chemical action between the salt and solvent of such a nature that heat is evolved when the solution is diluted. When the solvent is acetic acid, X 44*34 x 4-2 x io 7 (Landholt and Bornstein Tabellen) p = i-o5 and $ = 290; substituting these values we get for the lowering of the freezing point of any solution of the same strength as before In this case Raoult found 80 = 3*9. When the solvent is benzine, A=29x4'2xio 7 , p = '9 and 0= 275, so that the lowering of the freezing point of a solution of the same strength as before is 80= - 5-4 C. Raoult found in this case that 80 was - 4*9 C. Raoult found that the effect of dissolved salts on the freezing points of acetic acid and benzine was much more regular than their effect on the freezing point of water. CHAPTER XVII. THE CONNEXION BETWEEN ELECTROMOTIVE FORCE AND CHEMICAL CHANGE. 132. THE principle that when a system is in equilibrium the Hamiltonian function is stationary can be applied to de- termine the connexion between the electromotive force of a battery and the nature of the chemical combination which takes place when an electric current flows through it. We shall begin by considering Grove's gas battery, as this is the case where the chemical changes seem on the whole to be the least complex. In this battery the two electrodes are covered with finely divided platinum, the upper half of one is surrounded by some gas, say hydrogen, while the lower half dips into acidulated water ; the upper half of the other electrode is surrounded by some other gas, say oxygen, the lower half again dipping into acidu- lated water. The two electrodes are well coated with hydrogen and oxygen respectively. If the electrodes are connected a current will flow through the battery and the hydrogen and oxygen above the electrodes will gradually disappear, while the water will increase during the passage of the current. To investigate the electromotive force of a battery of this kind let us suppose that the electrodes have got into a 266 DYNAMICS. permanent condition, so that the gases attached to them are not altered during the passage of the current, let us also suppose that the electrodes are connected with the plates of a condenser whose capacity is C, these plates being made of the same material. Then if unit quantity of positive electricity flows from the plate of the condenser which is connected with the hydrogen electrode through the cell to the other plate, by Faraday's Law an electrochemical equiva- lent of hydrogen will appear at the electrode covered with oxygen and one of oxygen at the electrode covered with hydrogen; the hydrogen and the oxygen will combine and the result of the passage of the unit of electricity will be that an electrochemical equivalent of hydrogen and one of oxygen will disappear and an electrochemical equivalent of water will appear. The systems whose mean Lagrangian functions change during this process are (i) the condenser, (2) the hydrogen above one electrode, (3) the oxygen above the other, and (4) the water. Let Q be the quantity of positive electricity on the plate of the condenser connected with the oxygen electrode, and let , rj, be the masses of the hydrogen and oxygen above the electrodes and of the acidulated water respectively. The mean Lagrangian function for the condenser is The mean Lagrangian function for the hydrogen is where using the same notation as hitherto, The mean Lagrangian function for the oxygen is where Za = *,0 log ^ ELECTROMOTIVE FORCE. 267 and for the acidulated water Z W where Now .when unit of electricity passes from the one plate to the other of the condenser, the electrochemical equivalent of hydrogen is carried to the oxygen and there combines with it at one electrode, while the electrochemical equivalent of oxygen is carried to and combines with the hydrogen at the other electrode. Thus if c l and <= 2 are the electrochemical equivalents of hydrogen and oxygen, the net result of the process is that Q has increased by unity, and t] diminished by e 1 and e 2 respectively, while has increased by (ej + 2 ). Hence by the principle that the Hamiltonian function is stationary when there is equilibrium we must have but Q/C is the amount by which the potential of the plate connected to the oxygen electrode exceeds that of the one connected to the hydrogen electrode, in other words it is the electromotive force of the battery, which we shall call /, hence If LW> be the mean Lagrangian function of unit mass of aqueous vapour above the acidulated water and in equilibrium with it, we have by 83 where is the mass of the aqueous vapour, and Z W ^ 8 01og^Va'W-<. Substituting these values in (240) we get 268 DYNAMICS. = t&O log ^ + t 2 X 2 log^ - (c, + c,) ^ log { l ^ + V? 2 - (c, + c But and by (83) - is of the form Lastly e^/j + e 2 o/ 2 - (, + 2 ) w^ is the loss of potential energy which occurs when an electro- chemical equivalent of hydrogen combines with one of oxygen and may be measured by the quantity of heat developed by the combination of an electrochemical equiva- lent of hydrogen at the zero of absolute temperature; we shall denote it by e^, making these substitutions we see /= ^0 lOg ^ + ^0 + ^0 lOg +,?... (241)^ /> where A - A' + e l J? 1 log -^, PoPo" hence we have 0* - - 6 -^ +p = t.q ............. (242). u-u au Thus if we know the way in which/ depends upon 6 we can determine g, so that by measurements of the electromotive force of a cell and the variations of this force with the tem- perature we can calculate the mechanical equivalent of the heat developed in the combination which takes place in the cell. 133. Equations (241) and (242) are not confined to the case of the Gas Battery. We can prove in a similar way that if p is the electromotive force of any battery where the solutions used are dilute, then ELECTROMOTIVE FORCE. 269 where e l is the electrochemical equivalent of hydrogen, } the value of jR for this gas, /o 1? /o 2 ... the masses in unit volume of those substances which disappear as the chemical action which produces the current goes on, while o^, o- 2 ... are the masses in unit volume of those which appear, a, b, ...c, d, ... are the ratios of the electrochemical equivalents of the sub- stances to that of hydrogen, divided by the molecular weight of the substance, f.q is the mechanical equivalent of the heat which would be evolved at the absolute zero of tempera- ture by the chemical action which takes place when unit of electricity passes through the cell. From this equation we get as before *-+'-* ............... (*> By v. Helmholtz's principle (48) OdpjdB is the heat which must be supplied to the cell in order to keep the temperature constant when the unit of electricity passes through the cell, or in other words BdpjdO is the mechanical equivalent of the heat which is "reversibly generated when unit of electricity passes through the cell. Now/ the work done in driving this quantity of electricity through the cell plus OdpfdO the heat reversibly generated must be equal to fw the heat equivalent of the chemical action which takes place in the cell, hence by (243) we have 2 Now W and tq are the mechanical equivalents of the heat developed by the same combination when it takes place at the temperatures and absolute zero respectively, and the difference between these quantities must be the differ- ence between the mechanical equivalents of the quantities of heat required to raise them from zero to 6 degrees in their 2/0 DYNAMICS. combined and uncombined states. If we consider the case when two gases A and B combine to form two others C and D y then if c lt c z , c# , C, D, we see that whenever the combination leaves the number of molecules unaltered B will vanish and the equation will be rigorously true. We see by this equation that when the electromotive force increases as the temperature in- creases the electromotive force is greater, while when the electromotive force diminishes as the temperature increases it is less than that calculated from the formula p = c^, which is often employed. If k be the coefficient of the chemical combination (115) which goes on in the cell, i.e. the value of when the densities of the gases or solutions have the values they possess when in chemical equilibrium with each other, then since any small change cannot alter the value of the mean Lagrangian function of the gases or dilute solutions when in equilibrium, we get if we suppose the change is that which would take place if unit of electricity were to pass through the solutions o = e l R,P\Qgk + AO + 0\o%0 + cq ...... (247); combining this with (245) we get or l og = l og ............ (248) . 2/2 DYNAMICS. This equation affords a very easy method of finding the coefficient of any chemical combination if we can make a cell in which this combination takes place, for then if we measure the electromotive force and the densities of the solutions, equation (247) will at once give k. Thus the Daniell's cell enables us to calculate the coefficient of the combination Zn + H 2 SO 4 + CuSO 4 = ZnSO 4 + H 2 SO 4 + Cu. Here if p and o- are the masses per unit volume of the CuSO 4 and ZnSO 4 respectively when there is chemical equilibrium so that if p' and o-' are the densities .of the CuSO 4 and the ZnSO 4 when the electromotive force is p we have Now at o C. t^Rfi is nearly io 6 and/ is about io 8 so that log k = ^ log - 100, or approximately since for ordinary strengths of solution log/// "' is small compared with 100 log e - = -200, hence we see that in this case when there is equilibrium practically all the sulphuric acid goes to the zinc. If we determined the electromotive force of a battery when lead wire dipped respectively into acid solutions of lead nitrate and lead chloride, we should be able by equation (247) to determine the coefficient of the action 2 HC1 + Pb(NO d ) 2 = 2HNO 3 + Pb C1 2 , ELECTROMOTIVE FORCE. 273 and so determine the way in which lead divides itself between hydrochloric and nitric acids. If we return now to the hydrogen and oxygen gas battery, equation (247) is for this case (249). We can easily deduce from this equation the way in which the electromotive force of a gas battery depends upon the pressure of the gases in the vessels above the electrodes. If p l is the electromotive force of the battery when the densities of the hydrogen and oxygen are p, p respectively, / 2 the electromotive force when the densities are a- and V V 3rd case. When a + b > c + d, i.e. when there is a diminu- tion in volume after combination. In this case as 6 increases from zero to infinity p l a p a */cr ] c cr a rf also increases from zero to infinity, so that in this case it is always possible to find a temperature which should be one of equilibrium for any arbitrarily chosen set of values of p lt p 2 , cr^ Now unless the structure of the system is steadily chang- ing \dL'ldx\* will either vanish or be exceedingly small, so that in general we may neglect it and write dL dL\ ( T dL' . Tr~ -j- ) dt- I -j-dt dt dx dx) J dx 286 DYNAMICS. Let us choose T so that though a great many collisions occur in this time, yet the values of x, x, x are not changed in it by a finite amount. Now if T be the time a collision lasts and if there are n of them per second if as in a numerous class of cases x may be supposed to remain constant during the collision, we may write (252) as , dx dx where X Since Jo \dtdx dx) dt T \dtdx~ dx}* equation (251) becomes dtdx dx dx' Thus the effect produced by these intermittent forces is the same as that which would be produced by a steady force X of type x and given by the equation X=n dx' Similarly they would produce the same effect as a force Y of type y where Y=n ~dy' If dxldx, dxldy do not involve the velocities x, y and if n the number of collisions per second is a linear function IRREVERSIBLE EFFECTS. 287 of these velocities, these forces will be of the character of frictional forces. If n does not involve the coordinates x, y explicitly then we have - dY dX the consequences of this equation will be similar to those developed in 44. Thus for example suppose we were to find that the logarithmic decrement of the torsional vibra- tions of a wire depended on the extension of the wire, then it would follow from (252) that when the wire was vibrating there would be a force tending to alter its length. If the frictional resistance to the torsional vibrations were /x0, where is the angular velocity of a pointer attached to the wire, then if the above equation is true, there would be a force X tending to lengthen the wire and given by the equation - = - 6 -_- , where x is the length of the wire. au ax Thus if -~ is constant we have dx x=-ei>f. dx Whence it follows that if the torsional vibrations were periodic there would be a force tending to produce longi- tudinal vibrations of half their period ; or again, if the viscosity of an iron wire were altered by magnetization there would be a periodic magnetizing force acting on a vibrating wire whose period would be half that of the torsional vibrations. The relation (253) is only satisfied when n is in- dependent of x and y, if n is a function of these quantities we shall have the relation 288 DYNAMICS. dY dX v d\o>gn --- -j- = A j -- - ax ay ay ax instead, and if we consider forces of a third type z, the two additional relations d\ogn _~ dz dx dx dz and dY_ dtegn d\ogn .dllvl t " ' -* T ~ ' -* dy dz dz dy so that 7 (dY dX\ ^(dX dZ\ v (dZ dY\ z(-_ j- }+y\~*-- j- ) + x (-j -j- )-o. ..(254). \dx dy J \dz dx J \dy dz J In these relations X, Y, Z are only those parts of the forces of types x, y, z which are intermittent in their action. If from the nature of the case we can see that the number of collisions is independent of some one coordinate x, then it follows from the above equations that (i/dY dX\, [i/JZ dX\j log n = l-=( j -- -=- ) dy + I -=, ( - ---- }dz. ) X\dx dy J ' ] X\dx dz J If the viscous forces arise from collisions with several distinct systems, instead of with one as we have hitherto assumed, we shall have = 1 ,-... dy * ay where n lt n a are the numbers of collisions per second with the systems (i), (2)... respectively, and where Z/ is the Lagrangian function of the rth system. VISCOUS FORCES. 289 If ... are independent of x, y then as before dX_dY dy ~ doc ' but if j, , involve the coordinates x, y, then the relation (253) must be replaced by one involving higher differential coefficients. The preceding considerations show that in those cases where the viscous forces are due to "collisions" we have several criteria the fulfilment or non-fulfilment of which will afford us information about the constitution of the system. Thus if (252) is not fulfilled we conclude that the number of collisions depends upon the value of the coordinates, if (253) is not fulfilled we conclude that the viscous forces are due to collisions with more systems than one and so on. There is a great dearth of experiments on the influence of various physical conditions on viscous forces except when these forces are those which resist the passage of electricity through conductors. It does not seem probable however that in this case the resistance can be due to a suc- cession of impulses whose number is proportional to the strength of the current ; for the case is not analogous to that of a viscous force depending on the change of shape or configuration of a system, where we might reasonably expect the number of effective collisions to be propor- tional to the velocity of the change. In order to get some idea as to how discontinuous forces can produce the effect of electric resistance, let us consider some cases in which effects analogous to resistance are produced by a succession of changes following one another in quick succession. A very good example of a case of this kind is the arrangement given by Maxwell (Electricity and Magnetism, n. p. 385) for measuring in T. D. 19 DYNAMICS. electromagnetic measure the capacity of a condenser, in which by means of a tuning fork interrupter the plates of a condenser are alternately connected with the poles of a battery and with each other. If the rate of discharge is very rapid, this arrangement of condenser and tuning fork produces the same effect as a resistance ijnC where C is the capacity of the condenser and n the number of times it is discharged per second. Thus in this case a combination of induction and discharge produces the same effect as a resistance. Another case in which the conditions are plainly discontinuous but which produces the same effect as a continuous current, if the rate of alternation is sufficiently rapid, is when electricity passes through a closed glass tube filled with air. If electrodes are fused into the tube and connected to an electrical machine in action there will be no discharge of electricity across the tube until the electro- motive force gets large enough to break down the electric strength of the air, when a spark will pass, an interval will elapse before the second spark passes, during which the electromotive force inside the tube will be increasing to the value necessary to overcome the electric strength of the air. If this interval is very short then the successive discharges will produce the same effect as a continuous current through the tube. The consideration of this case may also throw some light on the mechanism by which the discharge is effected, for there are many reasons for believing that in this case the discharge is accomplished by the decomposi- tion of the molecules of the gas, the energy required for this decomposition coming from the electric field, and the consequent exhaustion of the electric energy producing the electric discharge. The reasons which lead us to this con- clusion are as follows : (i) Different gases differ much more in their electric ELECTRIC RESISTANCE. 2QI strengths than they do in other physical qualities, the difference is much more comparable with the differences between their chemical properties than their physical ones, and the difference between a chemical and a physical pro- cess seems to be that in the chemical process the mole- cules are split up while in the physical one they are not. (2) In many cases there is direct evidence from both spectroscopic and chemical analysis that this decomposition takes place, and again gases of complex composition whose molecules are easily split up are also electrically very weak. (3) We can explain by this hypothesis in a general way (Proc. Camb. Phil. Soc. v. 400) why the electric strength should gradually diminish as the gas gets rarer and rarer, until when the pressure is about that due to a millimetre of mercury the electric strength is a minimum, when the pressure falls below this value the electric strength increases again until at the highest exhaustion which can be got by the best modern air pumps the strength is so great that it is almost impossible to get a spark through the gas. (4) Dr Schuster has shown (Proc. Royal Society, xxxvu. p. 318) that the electrical discharge through mercury vapour which is supposed to be a monatomic gas presents a peculiar appearance and passes with great difficulty, and quite recently Hertz (Wied. Ann. xxxi. p. 983, 1887) has shown that the electric discharge passes more easily through a gas when it is exposed to the action of violet or ultra-violet light than when it is in the dark ; since ultra-violet light has a strong tendency to decompose the molecules of a gas through which it is passing, this is very strong evidence in favour of the view that the discharge is caused by the splitting up of the molecules of the gas. In the case of the electric discharge through gases the 192 2Q2 DYNAMICS. insulation seems to be perfect until the electromotive force reaches a definite value, when a spark passes. Thus the field can apparently not be discharged by a rearrangement of the molecules unaccompanied by decomposition. There is evi- dence however that when the molecules are split up into constituents a state of molecular structure is produced in which the discharge may be produced by rearrangement without further decomposition. Thus Dr Schuster has shown (Proc. Roy. Soc. XLII. p. 371) that when a strong electric dis- charge passes through a gas, a very small electromotive force is sufficient to produce a current in a region of the gas screened off from the electrical influence of the primary dis- charge. Again Hittorf found that a gas was weakened for discharges in the horizontal direction by passing a vertical discharge through it. The diminution in the electric strength of a gas after the passage of a spark can be accounted for in the same way. Again in Mr Varley's experiments on the electric discharge through gases (Proc. Roy. Soc. xix. 236) the quantity of electricity which passed through a tube filled with gas was proportional to E - Q where E is the difference between the potentials of the electrodes and a constant electromotive force, in other words the quantity of electricity which flowed through the tube was proportional to the excess of the electromotive force above that which broke the dielec- tric down ; this seems to indicate that the electromotive force JS produces a supply of atoms in the nascent condi- tion and that the rearrangement of these atoms discharges the field. In the case of fluid insulators the insulation for low electromotive forces is not as in the case of gases perfect. A condenser the plates of which are separated by a liquid dielectric always leaks however small the difference between the potentials of the plates may be. Some experiments ELECTRIC RESISTANCE. 2Q3 recently made by Mr Newall and myself (Proc. Roy. Soc. XLII. p. 410) showed that for small electromotive forces the leakage obeyed Ohm's law, that is, was proportional to the difference of potential between the plates. This indicates that the leakage is produced by the rearrangement under the electromotive force of some molecular condition, and that this condition is not produced by the electric field, for if it were the leakage would vary as a higher power than the first of the electromotive force. Quincke, who investigated the passage of electricity through the same liquids, using however electromotive forces comparable with those which would produce sparks through the dielectric, found that under these circumstances the quantity of electricity passing through the dielectric varied as a higher power than the first of the electromotive forces, which is just what we should have ex- pected if the electric field split up the molecules of the fluid. There are many liquids which, though they only conduct electricity with great difficulty when pure, yet when salts or other substances (which may themselves be non-conductors) are dissolved in them, conduct readily. This kind of con- duction is called electrolytic and is accompanied by effects which are not observed in other cases. Since the solvent is not a conductor, the discharge of the electric field which constitutes conduction must in some way or other be due to the action of the substance dissolved in it. The consideration of the discharge through gases as well as the chemical decomposition which always accom- panies this kind of conduction suggests that in this case the discharge is caused either by the splitting up of the mole- cules of the salt by the electric field, or else by the rearrangement when in a nascent condition of the atoms of a molecule of the salt or the constituents of a more complex molecule containing both salt and solvent, the splitting up of 294 DYNAMICS. the molecule being done independently of the electric field. The first of these methods is unlikely for the following reasons. (1) If it were true it would require a finite electro- motive force to start a current through an electrolyte, just as to send a spark through a gas, whilst from the evidence of many experiments it seems clear that the smallest electro- motive force is sufficient to start a current through an electrolyte. (2) The experiments of Prof. Fitzgerald and Mr Trouton (Report of the British Association Committee on Electrolysis, 1886, p. 312) have shown that Ohm's Law is obeyed with great exactness by a current flowing through an electrolyte, whereas if the electromotive force had to break up the molecules the current would be proportional to a higher power than the first of the electromotive force. (3) If the molecules were split up by the current then the salt will form a greater number of individual systems when the current is flowing than when it is not. Now the rise of the solution in an osmometer and the lowering of its vapour pressure depend upon the number of molecules in unit volume of the liquid and not upon their kind, so that if the number of separate systems is increased by the passage of the current these effects ought to be increased by the passage of a current through the solution. I have lately made some experiments on both these effects and have not been able to detect that the slightest change was made by the current. For these reasons we conclude that the splitting up of the molecules which allows the current to pass is not caused by the electromotive force but takes place quite indepen- dently of the electric field. The forces between the atoms in a molecule are usually too strong to allow of any arrangement under the electric field, but when the molecule breaks up and these interatomic ELECTRIC RESISTANCE. 295 forces either vanish or become very small the constituents of the molecule are free to move under the electro- motive force, and they will move so as to diminish the strength of the electric field. In order to form a definite idea of the way in which the field gets discharged we may take the usual view that the constituents into which the molecule splits up are charged with opposite kinds of electricity, and that when the molecule splits up the positively charged constituent travels in one direction, the negatively charged one in the other; in this way we get two layers of positive and negative electricity formed, the electric force due to which neutralizes in the region between the layers the external electric force. The positively charged molecules soon come into the neighbourhood of some negatively charged ones travelling in the opposite direction and they recombine, while the negatively charged ones do the same with some positive molecules, thus the force due to the layers vanishes and the external electric field is re-established to be soon demolished again by the decomposition and rearrangement of other molecules. Although we suppose that the current is transmitted by the molecules of the electrolyte breaking up, this does not necessarily imply that the electrolyte should when free from electromotive force be largely dissociated, for all that is necessary on this view for the passage of a current is that the molecules of the electrolyte should split up, and there is nothing to prevent our supposing, if other reasons render it probable, that they would instantly re-unite if no electromo- tive force acted upon them. And since the state of dissoci- ation depends upon the ratio of the time the atoms remain dissociated to the time during which they are combined, we may make this as small as we please and yet have continual splitting up of the molecules. 296 DYNAMICS. There does not seem any necessity for supposing that the passage of electricity through metals and alloys is accomplished in a fundamentally different way from that through gases and electrolytes. For the chief differences between conduction through metals and through electrolytes are (i) that in electrolytic conduction the components of the electrolyte appear at the electrodes, and we have polar- ization, and (2) that the conductivities of electrolytes in- crease while those of metals diminish as the temperature increases. Let us begin by considering the first of these differences, that of polarization. A little consideration will show that we could hardly expect to detect it in the case of metals or alloys, for here instead of, as in electrolytes, the property of splitting up being confined to a few molecules sparsely scat- tered through a non-conducting solvent, the whole of the molecules can split up, thus the rate of disappearance of any abnormal condition would be almost infinitely greater than in the case of electrolytes, so that if any polarization were produced it would probably die away before it could be detected. Let us next consider the appearance of the constituents of the conductor at the electrodes. The only case in which we could expect to detect this is that of the alloys, but even in this case Prof. Roberts-Austen was unable to detect any change of composition in the alloy round the electrodes ; we must remember however that an alloy differs very materially from an electrolyte because while in the latter we have a few "active" molecules embedded in a non-conductor, in the former it is as if the solvent as well as the salt conducted, so that the discharge is not concentrated on a few molecules of definite com- position but can travel by an almost infinite variety of paths. ELECTRIC RESISTANCE. 2Q/ Then again the statements about the effect of heat on the conductivity of elements and electrolytes though true in general are subject to exceptions, thus the conductivities of selenium, phosphorus and carbon increase as the tempera- ture increases ; that of bismuth is said to increase at certain temperatures, and I have lately found that the conductivity of an amalgam containing about 30 per cent, of zinc and 70 of mercury is greater at 80 C. than at 15 C. We must remember too that the rate of increase of conductivity with temperature for electrolytes diminishes as the concentration increases. No sharp line of demarcation can therefore be drawn between the two classes of conductors on this account. There does not seem any difference between metallic and electrolytic conduction which could not be attributed to the vastly greater number of molecules taking part in metallic conduction, whilst assuming that in all cases the current consists of a series of intermittent discharges caused by the rearrangement of the constituents of molecular systems. We shall therefore proceed to examine the dynamical results to which such a conception of the electric current leads. Let us consider the case of an electric field where the electromotive force is everywhere parallel to the axis of x. Let the electric displacement in this direction be /, then in the Lagrangian function of unit volume of the medium there is the term where K is the specific inductive capacity of the medium. 298 DYNAMICS. This term gives rise to the force _ K parallel to the axis of x. In consequence of the continual rearrangement of the molecular systems//^ is not uniform but keeps alternately vanishing and rising to a maximum value. If these alterations are sufficiently rapid the effect represented by this term will be the same as that of a steady force equal to its mean value, that is to K Let us suppose that in consequence of the rearrange- ment of molecular systems /vanishes n times a second, and that T is the period which elapses between the end of one period of extinction and the end of the next, then where i is the maximum value of/, and ft a quantity which depends upon the ratio of the time the field is destroyed to that during which it exists. When the molecular systems rearrange themselves so as to discharge the electric field molecules charged with I units of electricity pass through unit area in one direction, while I units of negative electricity are carried by molecules moving in the opposite direction. Thus 2f is the sum of the positive electricity moving in one direction and of the negative in the opposite passing through unit area in unit time, it is therefore equal to u where u is the intensity of the current, and since nr is equal to unity, the force we are considering equals nK ELECTRIC RESISTANCE. 299 so that this continual breaking down of the field produces the same effect as if the substance possessed the specific resistance zirfilnK. Thus the greater the number of times per second the displacement breaks down &c., the better the conductivity. Now the breaking down of the displacement is caused by the rearrangement of the molecules, and the rearrange- ment of the molecules in a solid will produce much the same effects as the collisions between the molecules of a gas, and will tend to equalize the condition of the solid, thus we might expect the rate of equalization of temperature to increase with the number of molecular rearrangements. The electrical conductivity would also increase in the same way, so that this view fits in with the correspondence which exists between the orders of the metals when arranged ac- cording to thermal and to electrical conductivities. The preceding investigation of the resistance of such a medium is only valid when the electromotive force is ap- proximately constant over a time which includes a great many discharges. If the displacement were to be reversed during the interval between two successive rearrangements of the molecules the substance would behave like an insula- tor and not like a conductor. If