At the time when Clausius laid the foundation of the kinetic theory the low conductivity of gases and vapours was adduced, as has been already mentioned in § 61 at the beginning of Part II., as the most weighty objection to the correctness of the new hypothesis. It was asserted that in a medium the molecules of which are moving with great speed the heat, which consists in the energy of motion of the molecules, must also be transferred with similar speed from one spot to another. This objection we may still attempt to maintain, even if the similar objection which has been raised in respect to the diffusion of gases is already removed. For although it has been shown that on this theory the mixing of two gases does not proceed with the speed at which the molecules move, but very slowly, because the molecules frequently collide and therefore traverse only very short distances in spite of their large speed, yet this answer to the doubts derived from the phenomenon of diffusion cannot without more consideration be applied to the phenomenon of conduction, since in diffusion there is a question of a transference of mass, and in conduction, on the contrary, of a transference of energy. This difference is best illustrated by an example which seems to lead to a conclusion by analogy. A very common piece of physical apparatus consists of a row of elastic balls which hang in contact and are used for experiments on the collision of balls. If, for instance, we allow one of these balls to fall against the remainder of the row, it comes to rest itself on the collision if all the balls are of the same size, and all the others remain at rest except the last, which flies off from the rest with the same speed with which the first struck the row. In this experiment the energy alone of the motion has been propagated, and, indeed, with very great speed, without the mass through which the transmission was effected-that is, the row of intermediate balls -being carried on with the energy. We might likewise conclude that the energy of the heat-motion may be very rapidly transmitted by means of a group of molecules, which now and then collide and mutually transmit energy to each other in the collisions, without the molecules which effect the transmission having themselves to move in the direction of the motion of the heat. But this instance in illustration of the objections contains in itself its own refutation, for it does not correspond in all points to actuality. In the collision-apparatus the transmission of energy takes place at all the collisions in the same direction, and travels therefore over a wide stretch in a short time. But in the gaseous medium in which the molecules collide now in this direction and now in that, the energy is carried over now here and now there, and is jerked about in the same zigzags as the molecules. The transmission of heat therefore goes on in a fixed direction with slowness similar to that of the forward motion of the molecules. 104. Kinetic Theory of Conduction Starting from this conception Clausius,' a short time after Maxwell, who first treated the problem, gave a detailed analysis of the process of the conduction of heat, by which he has removed the last doubts before mentioned regarding his hypothesis. Stefan and von Lang1 have later given elementary demonstrations of this theory. The same question has been mathematically treated by Boltzmann on the basis of a later hypothesis of Maxwell's, according to which the molecules of gases repel each other with forces that are inversely as the fifth power of the distance. The theory given in the Mathematical Appendices of this book starts from Maxwell's older view and rests on Maxwell's law of distribution of speeds. 5 Pogg. Ann. 1862, cxv. p. 1; Abhandl. ü. Wärmetheorie, 1867, Abth. 2, p. 277; Mechanische Wärmetheorie, iii. p. 105. 2 Phil. Mag. 1860 [4] xx. p. 31; 1868, xxxv. p. 214. Scient. Papers, i. p. 403; ii. p. 74. Wien. Sitzungsber. 1863, xlvii. Abth. 2, p. 81. Ibid. Abth. 2, 1871, lxiv. p. 485; 1872, lxv. p. 415; Pogg. Ann. 1871, cxlv. p. 290; 1872, cxlviii. p. 157; Einleitung in die theor. Physik, 1867, p. 529. 5 Wien. Sitzungsber. 1872, lxvi. Abth. 2, p. 330; 1875, lxxii. Abth. 2, p. 458. It is easy to give an idea of the matter contained in these mathematical theories without repetition of the calculation, because the analogy of the problem before us with the theories earlier discussed is obvious. Diffusion consists in a transference of mass effected by means of the molecular motions and viscosity in a transfer of forward momentum caused by the same means; conduction of heat is likewise a transfer of energy, which is effected as before by the motion of the molecules. This similarity goes so far that the propagation of heat may be directly looked on as a diffusion-phenomenon in which the warmer and colder particles diffuse among each other. For, since the molecules as they pass from an upper and warmer layer to a lower and colder one retain their energy till a collision, the process of conduction of heat is completely identical with that of diffusion; and we have no further difference to take into account than this, that we have now to find, not the number of the diffusing particles, but the sum of their energy. In order to form a distinct idea of the arrangement of the experiment that shall correspond as nearly as possible to that chosen before, let us consider a gas enclosed between two unlimited, or, at least, very widely extending, parallel plane walls which lie horizontally with the distance between them equal to the unit of length; and consider the lower to be kept at the constant temperature 0° C., and the upper at the temperature 1° C. Under these circumstances a distribution of temperature is produced of itself between the walls which is independent of the time, and such that at the height x above the lower limiting plane the temperature is A constant flow of heat in the direction from above to below takes place in the gas, and this is such that through each imaginary horizontal plane in the space occupied by the gas there flows an equal amount of heat in unit time. In accordance with the usual definition we denote as the coefficient of conductivity of the gas [or simply its conductivity] that amount of heat which in unit time passes through unit area of such a horizontal plane under the given circumstances. If we now go back to the consideration respecting the current of diffusion which was worked out in § 95, we have merely to change the meaning of the symbol n which occurs in it in order to apply it to the flow of heat. There n denoted the difference of the values which the number of molecules per unit volume of one of the two diffusing gases has in two different layers separated by unit length. We may take over this signification to the present problem in so far as we can refer it to the number of warmer or colder particles which meet each other; we understand therefore by n the difference of the values of the number of the, for instance, warmer particles in two different layers which are separated by unit of length. Then it follows that the number of particles which in unit time carry heat over unit area may be expressed by the product nD, where D denotes the coefficient of diffusion. We have, however, yet another alteration to consider; for we have no longer to do with the number of particles that pass across, but, as we have already said, with the energy carried over by them. Instead, therefore, of the number n, we must introduce the difference of the heatenergy per unit volume at two layers which are distant from each other by unit length. We have taken the difference of temperature corresponding to this distance to be 1 degree; hence the difference in the thermal energy of a molecule in two layers separated by unit length is mc calories, if c denotes the specific heat at constant volume and m the mass of a molecule. It thus follows that the difference of the energies per unit volume for which we are looking is Nmc in thermal units, if N denotes, as before, the number of molecules in unit volume, and the expression t = NmcD results on our theory for the quantity of heat which is termed the coefficient of conductivity. The coefficient of diffusion D, which was determined in § 95 by the formula in our special case in which the two kinds of gas are the same, for we may neglect the small difference in the values of and that arise from the inequality of temperature, and therefore by §§ 96 and 97 put Ω, Ω, 2 and by § 95 we have also N1 + N2 = N. The conductivity of a homogeneous gas for heat is therefore t = πNmQLc. 105. Relation of the Conduction of Heat to the Viscosity The formula shows a simple relation of the conductivity to the coefficient of viscosity which by § 78 is given by the formula n = 0·30967 NmQL. From this we find that the conductivity can be expressed in terms of the viscosity and the specific heat of the gas as in the equation = = (/030967) nc The factor which occurs in this formula is not much greater than 1, and we should not be unjustified if in this theory, which depends only on approximation, we were to put it equal to 1, and thus obtain This value would, indeed, have directly resulted if we had |