Report of a Committee, consisting of Messrs. J. LARMOR and G. H. BRYAN, on the present state of our knowledge of Thermodynamics, specially with regard to the Second Law. [Ordered by the General Committee to be printed among the Reports.] PART I.-RESEARCHES RELATING TO THE CONNECTION OF THE SECOND LAW WITH DYNAMICAL PRINCIPLES. DRAWN UP BY G. H. BRYAN. Introduction. 1. The present report treats exclusively of the attempts that have been made to deduce the Second Law of Thermodynamics from purely mechanical principles. Before considering the several methods in detail it may be well to summarise the meaning of the various terms which enter into the mathematical expressions of the laws of thermodynamics, with a view of showing more fully what conditions must be kept in view in establishing the dynamical analogues. This has been done more or less fully by several authors of papers on the subject, but more especially by von Helmholtz in his paper on the 'Statics of Monocyclic Systems.' The substance of this paper will be dealt with more fully later on in the present Report, but we will now mention the principal points touched on in the introduction. 2. Meaning of the Second Law.-Let a quantity dQ of work in the form of heat be communicated to a body whose absolute temperature is 0. Let E be the internal energy of the body, dW the work done against external forces by the change in the configuration of the body which takes place during the addition of dQ. It is not assumed that the external forces are conservative. Then the First and Second Laws are expressed by the equations where dS is a perfect differential of a quantity S, called the entropy, whose value depends only on the state of the body at the instant considered. The essential principle involved in the Second Law does not lie solely in the fact that dQ has an integrating divisor 0. In fact, if we assume that the state of a body is completely defined by two variables ≈ and y, it must always be possible to put dQ in the form dQ=Mdx+Ndy, where M, N are functions of x and y only. And it is always possible to find an integrating factor for an expression of this form. Moreover, if one integrating factor can be found for dQ, an infinite number of such factors can be found. For in equation (2) let s be any arbitrary function of S; then we may write the equation in the form ds dQ=0 ds. ds 1 Crelle, Journal, vol. xcviii. so that ʼn as well as 0 is the reciprocal of an integrating factor of dQ, or, as we may call it, an integrating divisor' of dQ. Since dS/ds may be regarded as a function of S, we see that the product of the temperature into any arbitrary function of the entropy of a body is an integrating divisor of dQ, and therefore possesses properties analogous to in equation (2). Hence the absolute temperature 0 is not fully defined by equation (2), and the Second Law of Thermodynamics is not, therefore, completely proved by the establishment of an equation of this form. 3. It is, therefore, necessary to take into account the other property by which temperature is characterised, namely, that heat always tends to pass from a body of higher to one of lower temperature, and in particular that if two bodies in contact have the same temperature there will be no transference of heat between them. The Second Law of Thermodynamics consists in the fact that among the integrating factors of dQ there is one whose reciprocal, e, possesses the properties of temperature just mentioned. 4. But, nevertheless, without considering the properties of thermal equilibrium between different bodies we derive one very important inference from equation (2)—namely, that the thermal condition of a system whose parts are in thermal equilibrium can be completely defined by a single coordinate, or, in other words, that the consideration of thermal phenomena only adds one to the total number of coordinates otherwise required to fix the state of a dynamical system. 5. Impossibility of a Perfectly General Mechanical Proof. To reduce the First Law of Thermodynamics to the principle of Conservation of Energy it is only necessary to assume that heat is some form of energy; no hypothesis is required as to what particular form this energy takes. It was natural, therefore, that physicists should at a very early date endeavour to reduce the Second Law in like manner to a purely dynamical principle, and the principle of Least Action naturally suggested itself as the probable analogue of Carnot's principle. But here a limitation at once arises f.om the necessity of giving a dynamical meaning to dQ, the energy communicated to the system in the form of heat, and of separating dQ from -dW, the energy communicated in the form of mechanical work. 6. This limitation requires that some special assumption shall be made regarding the nature of heat, and the natural and almost inevitable assumption is that every finite portion of matter is built up of a very large number of elementary portions, called molecules, and that the form of energy known as Heat is due to the relative motion of the molecules among themselves. But, further, these molecules must be characterised by some peculiar property, such as their (practically) infinitely large number whereby their dynamical properties differ in some manner from those of a finite number of particles or rigid bodies. For without such a distinction it would be impossible to deduce any dynamical equations involving dQ, the work performed on the system through the coordinates defining the positions of the molecules and not involving -dW, the work performed through the coordinates determining the external configuration of the system. The two portions of the work could only enter together into the equations in the form dE. In other words, it is impossible to deduce the Second Law of Thermodynamics from purely mechanical principles without making some axiomatic assumption regarding the nature of the molecules whose motion produces the phenomenon of heat. 7. The question now arises as to what dynamical quantity represents temperature. We have good reasons for believing that, in gases at least, the absolute temperature is proportional, either to the total mean kinetic energy, or to the mean kinetic energy of translation of the molecules. But if this or indeed any other hypothesis be adopted it will be necessary, before the mechanical theory of heat is complete, to prove that (1) the molecular kinetic energy is an integrating divisor of dQ; (2) it determines the thermal state of a body in relation to other bodies. Most of the earlier writings are concerned only with the first property. But a complete mechanical proof of the Second Law would involve a mechanical definition of temperature applicable to all kinds and states of matter, together with an explanation on dynamical or statistical laws of the principle of degradation of energy in non-reversible processes; and we are still far from arriving at a satisfactory solution of either of these problems. 8. It will be convenient to classify the methods by which the problem has been attacked as follows, under three headings corresponding to the three different fundamental hypotheses which underlie them : : I. The Hypothesis of Stationary' or 'Quasi-Periodic' Motions as adopted by Clausius and Szily. II. The Hypothesis of Monocyclic Systems' of von Helmholtz, and similar hypotheses. III. The Statistical Hypothesis of Boltzmann, Clerk Maxwell, and other writers on the Kinetic Theory of Gases. 1 9. Rankine seems to have been the first who attempted to deduce the Second Law from dynamical principles. As early as 1855 he published a paper On the Hypothesis of Molecular Vortices,' in which he obtained. equations analogous to those of thermodynamics; and in a paper read at the British Association in 18652 he explained the Second Law on the hypothesis that heat consists in any kind of steady molecular motion within limited space,' such as that due to circulating streams. Both of Rankine's hypotheses are special cases of Helmholtz's Monocyclic Systems.' Boltzmann seems to have been the next to take up the subject, but his claim to priority has been disputed by Clausius, whose investigations appeared about five years later. Boltzmann was undoubtedly the first to regard the subject from a statistical point of view. Szily laid claim to the discovery of the connection of the Second Law with Hamilton's Principle of Least Action, and he may fairly be entitled to the credit of having propounded this connection. But most of his early investigations are not only wanting in rigour, but in many cases so inaccurate that they do not prove the connection at all. 1 Phil. Mag. 1855, pp. 354, 411. 2 Ibid. 1865, p. 241. Clerk Maxwell's theorem, named after its discoverer, was the first attempt at a kinetic analogue of thermic equilibrium. It was generalised by Boltzmann, and afterwards further generalised by Maxwell himself; but the latter extensions are probably incorrect, as we shall see hereafter. Having thus briefly mentioned the earliest researches on the present subject, let us turn to a consideration of the papers themselves, beginning with the writings of Clausius and Szily. SECTION I. The Hypothesis of Stationary or Quasi-Periodic Motions. 10. Clausius and Szily.-In 1870 Clausius showed that when a system of particles is in stationary motion, the mean vis viva of the system is equal to its virial. About a year later he gave a proof of the Second Law, based on the laws of motion, in a paper entitled "On the Second Axiom in the Mechanical Theory of Heat.' ? The methods of proof employed by Clausius in this paper are very laborious and complicated, while his arguments are artificial and, in places, not very intelligible. Soon after Clausius' paper had appeared, Szily endeavoured to show that what in the mechanical theory of heat is called the Second Law is nothing other than Hamilton's Principle of Least Action.' The proofs which Szily gave are, in many places, quite at variance, not only with the principles of dynamics, but also even with the laws of Thermodynamics themselves. Thus he repeatedly mistook dE for dQ, and tried to show that dE/T is a complete differential (a result not in general true); moreover, in endeavouring to account for the principle of degradation of energy in a non-reversible cycle, he altogether ignored the First Law, and supposed some of the molecular energy of the system to be actually lost or annihilated by friction, viscosity, or imperfect elasticity of the molecules, or by other similar resistances. In consequence he had to employ methods of proof that were far from rigorous, and even, in many instances, illogical. Szily's papers seem, however, to have had one good effect-namely, that of stimulating Clausius to remodel his investigations in a simpler and more intelligible form. Those who care to examine the original papers of these writers will find them translated in the volumes of the Philosophical Magazine' from 1871 to about 1876. Among them is a paper by Szily, in which he claimed to have deduced the Second Law from the First 'without any further hypothesis whatever.' Yet Szily based this investigation on two hypotheses which are hardly more axiomatic than Carnot's principle. 11. Clausius' Methods.-It would be useless to enter into further criticism. We now proceed to give a proof of the Second Law based on the methods of Clausius, with the object of bringing into prominence the more salient features of his investigations, and of presenting them in a concise form. The assumptions which form the basis of Clausius' proof may be stated as follows: (i.) In the steady or undisturbed state of the system the motion of the molecules shall be stationary or quasi-periodic; in other words, the potential and kinetic energies of the molecules shall fluctuate rapidly 2 Ibid. vol. xlii. (1871) (September). Ibid. V. series, vol. i. (1876), p. 22. 1 Phil. Mag. vol. xl. (1870), p. 122. Ibid. vol. xliii. (1872), p. 339. about their mean values, and there shall be one or more 'quasi-periods,' i, satisfying the definition which will be given in the course of the proof (equation 13, infra). (ii.) When the state of the system is changed (as by the communication of heat or by changes in the volume or external configuration of a body), such changes shall be capable of being treated as small variations of the motion from the state of steady motion. Helmholtz, in his paper on Monocyclic Systems, makes a similar assumption—namely, that the changes in the state of the system shall take place so very slowly that the motion of the system at any instant differs infinitesimally little from a possible state of steady motion. This is the exact equivalent of the assumption always made in treating the Second Law from a physical point of view-namely, that heat is communicated to or taken from the working substance so slowly that at every instant of the process the temperature of the body is sensibly uniform through out. 12. With these assumptions, let the positions of the molecules be determined in the first instance by the Cartesian coordinates (x, y, z) of the particles (m) forming them. Suppose that the state of the system also depends on the values of certain other coordinates, P1, P2, &c., which, as suggested by J. J. Thomson, we shall call the controllable coordinates' of the system; to this class belong the volume of the body, the charge of electricity present on it, or any coordinates which can be acted on directly from without. The values of these latter coordinates will enter into the expression for the potential energy of the system. Let T=kinetic energy of system-m(¿2 + y2+ż2). .E=total energy=T+V. In Thomson and Tait's Natural Philosophy,' part i. § 327, it is shown that ST - ż82) ôf "2Tdt=[Σm(rôx + ijôy + žôz)] + ["{ôTM – Σm(ëâx+yôy+žô=)} dt (5) But by D'Alembert's Principle we always have for the motion of the Now, V is a function not only of the molecular coordinates (x, y, z) but also of the controllable coordinates P1, P2, . . . and these latter are also liable to variation. Hence for the complete variation of V we have 1 Applications of Dynamics to Physics and Chemistry, p. 94. (7) |