MOLECULE 619 inv As Hence, if we attempt to cause one stratum of gas to pass over another in parallel planes, we experience a resistance due to the interchange of molecules between the portions of gas separated by the plane. This is in some respects analogous to sliding friction between solid bodies, and is called by German writers the " friction " (Reibun<f), by Max well and others the " viscosity," of the gas. Meyer : investi gates this effect of friction in a manner somewhat similar to that employed in case of diffusion, and obtains for the coefficient of viscosity J mNul. Relation of the Coefficient of Viscosity to Density and Tempera ture. The viscosity of a gas is independent of the density t being, m v according to 0. E. Meyer, f- ul. Now, for any one gas, l w is, as o we have seen, inversely proportional to the density, and therefore <al is inversely proportional to the density. On the other hand, N is directly proportional to the density. Hence the viscosity is in dependent of the density. This agrees with the result obtained by Maxwell from the kinetic theory in 1860, and with the results of experiments by Maxwell 2 and 0. E. Meyer. 3 Also, experiments by 0. E. Meyer and Springmiihl 4 on the transpiration of gases show that the times in which two different gases under similar circum stances flow through a tube maintain the same constant ratio to one another. As in the case of the coefficient of diffusion, ul is tersely proportional to the square root of the absolute temperature, both the coefficient of diffusion and that of viscosity depend on the same function ul, it should be possible from experiments on viscosity to determine the rate of diffusion. Experiments with this object have been conducted by Stefan 5 with very satisfactory results, his calculated values for the coefficient of diffusion agreeing very closely with those determined by Loschmidt s direct experiment. 6 We have given the above results for the coefficients of diffusion and viscosity from 0. E. Meyer s work, because his method has met with very general acceptance. It has been shown, however, by Boltzmann, 7 that the method is incomplete. Meyer s results can only be obtained on the assumption that the molecules of a gas undergoing diffusion or internal friction, which have any given velo city, a.sio, are moving with that velocity in all directions indifferently. We may calculate the number of molecules having velocity w that pass through a given plane during a short time dt, starting from encounters at any given distance from the plane. If we assume that the molecules, issuing from such encounters with velocity w, move indifferently in all directions, we obtain Meyer s result. This assumption is true only of a gas at rest that is, having no velocity of translation so that our result so obtained would express, in case of diffusion, the rate at which two gases begin to diffuse, if given at any instant both at rest that is, with no stream velocity but mixed in unequal proportions in different parts of space. In any actual case of diffusion, either of the two diffusing gases acquires a small velocity of translation. If we take this velocity into account in calculating the number of molecules of the gas passing through a plane, according to Meyer s method, we shall find that it introduces two new terms, one of which, when the motion becomes steady, is equal and opposite to the result obtained by Meyer. This is proved by Boltzmann in the case of viscosity in the treatise- above referred to. The same proof is easily applied in the case of diffusion. Stefan s Method. Stefan 8 regards the two diffusing gases as having small velocities of translation, or stream velocities, u^ and u y in opposite directions, so that the molecules of one gas, of mass ij, have an average momentum m^ in direction from left to right, and those of the other gas, of mass m.-,, an average mo mentum ioM 2 from right to left. By virtue of encounters between the two sets of molecules, each gas is always imparting to the other a portion of its own average momentum, and receiving from the other a corresponding momentum in the opposite direc tion. The momentum so transferred or interchanged is what Stefan calls the resistance which one gas offers to the other s diffusion. In this investigation Stefan assumes that all classes of molecules of one gas, whatever their molecular velocity in space, have the same average velocity in the direction of diffusion that is, the same stream velocity so that the motion of the molecules of a diffusing gas would be exactly represented by considering the molecules of a gas at rest that is, with only its molecular velocity at the same 1 See pp. 311-325 of the work above referred to. 2 Proceedings of the Royal Society, 8th February 1866. 3 Poggendorfs Annalen, 1871, cxliii. 14.
- Fogg. Ann., 1873, cxlviii. 1 and 526.
5 Sitzungtber. d. k.-k. Akcul., 1872, Ixv. 323. 6 For a full account of these and other experiments on diffusion and vis cosity, see O. E. Meyer, Kinetische Theorie d. Case, under the heads "Reibung" and "Diffusion." 7 " Zur Gas-Reibung," in the Sitzungsber. (J. k.-k. Akad., 1881. 8 Memoir "On the Dynamical Theory of Diffusion " (Sitzungsber. d.k.-k.Akad., Ixv.) temperature and pressure, and then giving to each molecule tho additional common velocity u in the direction of diffusion. Boltz mann, however, shows that, in order correctly to represent the motion of the diffusing gas, we must impart to molecules having different molecular velocities independent of direction different common velocities in the direction of diffusion. And it will be found that the resistance of the gases is sensibly modified by this property. 9 The complete solution of the problem, that is, the determina tion of u as a function of w, on the hypothesis that the molecules are elastic spheres, is difficult. If we assume molecules to be centres of force varying inversely as the wth power of the distance, so that the force at distance r is - r n where /j. is constant, we obtain the following result. We assume the molecules of gas A whose absolute velocities are between w and w + dw to have an average stream velocity u in direction of the tube, where u is a function of w. Then, if the terminal con dition at the ends of the tube be maintained constant, we obtain an equation of the form of - _, 2 3;i-3 m-! + m 8 __ N dx 3;i-3 m-! + m 8 unit volume multiplied by the average value for all molecules of gas A 7t-5 u V n ~ l , where V is the relative velocity of two molecules, one taken from each gas, and C is a constant, and m l} m 2 the masses of the molecules of gas A and gas B respectively. By making n infinite we obtain the result for elastic spheres : in 7J.-5 that case V n ~ l =V, and the problem is to find the average value of uV. Since p varies as the absolute temperature, and the average value of V varies as the square root of the absolute temperature, we may infer that the average value of u that is, the stream velocity will vary approximately as the square root of the temperature, as it appears to do from experimental evidence. If, on the other hand, 4n-8 n=5, V disappears, and 3 - 3 " In this case the analytical de termination of u presents no difficulty ; but in the result the stream velocity varies as the absolute temperature, which accords less satis factorily with experiments. ON MOLECULAR DIMENSIONS. Many attempts have been made in recent years to form an estimate or conjecture, more or less accurate, of the numerical value of the dimensions of a molecule and the absolute force between molecules. 10 In accordance with the view of the subject considered in this article, we are here concerned with such specula tions only in so far as they are founded upon the kinetic theory of gases, or supported by it. The phenomena of diffusion and viscosity especially have afforded grounds for estimates of molecular dimensions. It is first necessary to define what is meant by the dimensions of a molecule. Regarded as an elastic sphere, it has dimensions with the conception of which we are familiar. It is not, of course, seriously contended by any physicists that the molecules of a gas are actually hard elastic spheres, exerting no force on each other at any dis tance greater than that of actual contact, and then an infinite force. It is necessary to conceive the forces as finite, although they may diminish so rapidly with the dis tance as that the motions of molecules in the aggregate differ little from what they would be if the molecules were ideal elastic spheres. Nevertheless, they must be finite forces ; and, that being the case, it is difficult, if not im possible, to frame a definition of the boundary of a mole cule, except as a certain surface at which the forces acting between the molecule in question and other molecules attain a certain value. If, for instance, we were to regard a molecule as a centre of force, 9 For Boltzmann s own treatment of the subject we cannot, within the limits of this article, do more than refer the reader to the memoir above mentioned, "Zur Gas-Reibung," and another as yet unfinished memoir "On Diffusion," in the Sitzungsber. d. k.-k. Akad., 1882. 10 An account of these will be found in O. E. Meyer s Kin. Theorie d. Gasf, in Professor Tait s Recent A h-ances in 1 hysiml Science, lect. xii., and in the following memoirs :Phil. Mag., July 1879, " On the Size of Molecules," by N. D. C. Hodges; Phil. Mag., March 1880, "On the Mean Free Path of Molecules," by the same author. See also, lecture delivered by Sir W. Thomson at the
Royal Institution, 2d Feb. 1883.