DIFFUSION these two planes, i.e., G(z0 + cos 9). Hence the total quantity of G carried across the plane z—z0 in the negative direction is T, where rcc rir r& p= dw I d6 d wf(w) ^ sin0 cos^ —-— G(z0 + X cos6), J0 J0 J0 l the integration between 9 = 0 and 9 = ^tt representing the part due to molecules coming from the positive side and between 6 = ^tt and d = Tr, that due to molecules coming from the negative side. Making the assumption that G varies uniformly, or neglecting differential coefficients above the first, we obtain G(z0 + cos0) = G(zo) + A cos0 therefore
r=i2> ^ dzjfo f{w)wl dw— 6 dz For the diffusion of two gases, A and B, in a mixture containing and N6 molecules per unit volume of the two components, we put G = Na and G = INr6 in succession, and obtain d1Sh r„= — V'l a dz dz ’
If wla and wlb are unequal, Fa and rs will be unequal. The formula gives more molecules flowing in one direction than in the other, and the pressure (which is proportional to + would on this hypothesis soon cease to be uniform. Meyer assumes, therefore, that there is a counter current F6 - Fa, and, as in the article Molecule, the coefficient of diffusion becomes 3 Na+Nft • If, however, we assume that the diffusion of the gas A is unaffected by its collisions with molecules of the same gas, and only depends on collisions with the B molecules, and similarly for the gas B, we get for the coefficient of diffusion 1 wla + w/,b 3 2 The last formula has been used by Stefan, but is objected to by Meyer. Both formulae have been tested experimentally, but practical difficulties give rise to discrepancies in the observed results quite as great as the differences given by the formulae. Where the gases are mixed in equal proportions the two formulae become identical; and to decide between them it is necessary to examine diffusion in mixtures where one gas preponderates largely over the other. Such cases are, however, difficult of observation. • When the molecules are of the same size, shape, and mass wla~wlb, and the coefficient of diffusion is wl on either hypothesis. It is therefore everywhere constant. Taking unequal spherical molecules, Tait, by evaluating the integrals depending on his form of l, arrives at the following conclusions:—(1) For molecules of elusions11’ eclual mass, a difference of size, the mean of the diameters being unchanged, favours diffusion. (2) Diffusion is, however, but little affected by even a considerable disparity in size of the molecules, but depends mainly on the mean of the diameters. (3) Taking molecules of masses in the ratio 16:1, and of diameters in the ratios 3:1, 1:1, and 1 : 3, it is found that if the sum of the diameters is kept constant, diffusion is most rapid when the molecules of greater mass have the greater diameter. (4) A gas diffuses more quickly into one of different than into one of the same specific gravity. (5) If the diameters of the more massive molecules are decreased and of the lighter ones increased, keeping their sum constant, the rate of diffusion decreases to a minimum at first and then increases before the more massive molecules become infinitesimal compared with the others. (6) Owing to the smallness of the variations of the diffusion-coefficient, experiments on diffusion are not well suited for determining the relative size of the molecules of different gases. Next taking G to represent translational velocity in a plane perpendicular to the axis of z and assuming it to be proportional 2 Viscosity ’^ density be p, then pF will represent the and thermal1110111611 turn carried across the plane z = zb per unit conduce time, i.e., the shearing force, and the coefficient of tivity. dGjdz in pF will be the coefficient of viscosity, which therefore = h,pwl. . I11 conduction of heat the mean translational energy T or mw1 is a function of z. If f(w)dw denote the proportion of molecules
OF
GASES
461
with speeds between w and w + dw at a point where this mean energy is T0, f[w)dw will at other points denote the proportion of molecules with speeds between w ^/(T/To) and (w + dw) */(T/T0). We assume these molecules to carry with1 them translational energy 4mw2T/T0, and rotational energy ffmw T/T0 where /3— 2/(3& - 3) - 1, and k is the specific heat ratio. In calculating the true thermal conductivity we have to separate the transference of heat-energy due to conduction proper from that due to motion of the medium. By a method resembling Boltzmann’s we find Putting rj — mivl we see that the thermal conductivity of a gas is proportional to its coefficient of viscosity rj. The above, and other formulae based on alternative assumptions, give in general for the conductivity an expression of the form K = (Ak + B)r]cv, cv being the specific heat at constant volume and A, B, con slants. Meyer finds A = 0,795, B = 0’205, giving for air at 0° C., K^IOG.IO-7 centigrade C.G.S. units. Boltzmann finds A = !"!-, B= — §£, giving K = 536.10-7. The values observed for air by Stefan and Winkelmann are 558.10-7 and 525.10-7 respectively. The thermal conductivity of gases is treated at considerable length by Y erdet-Buhl mann. In the equation F = ±wl dGjdz the coefficient of dGjdz, viz. ^wl, varies as the mean velocity iv and as the mean free path l. Now from the expression nirs2 in the denominator of Tait’s formula, or otherwise, we infer that for the same gas or mixture of gases l varies inversely as n, and therefore directly as the volume. Hence on the hypothesis that the molecules are elastic bodies the coefficients of diffusion, viscosity, and conductivity vary as the volume and the square root of the absolute temperature. If the pressure p and temperature T be taken as variables they will vary inversely as p and directly as Tl Now it appears from Maxwell’s experiments that the coefficient of viscosity at constant density is proportional to T instead of T^, and from Loschmidt’s experiments it is not improbable that the coefficient of diffusion at constant pressure is proportional to T2 instead of Tl These considerations led Maxwell to consider a kinetic theory based on the hypothesis that the molecules of a gas repel one another with finite forces which are functions of the distance between them, and in particular to consider the case when the force varies inversely as the fifth power of the distance, in whiqh case the viscosity varies as T. The phenomena of diffusion, viscosity, and conductivity and fluid motions, have been worked out on this hypothesis very fully by Boltzmann and others. The relation of the coefficient of diffusion to the temperature appears, however, difficult to determine experimentally. A “pressure balance ” for this purpose has been described by M. Toepler. Authorities.1—L. Boltzmann. Vorlesungen iiber Gastheorie, Leipzig, Barth, vol. i. 1896, vol. ii. 1898 ; “ Bemerkungen iiber Warmeleitung der Gase,” W.S. Ixxii. 1875, and Fogg. Ann. clvii. 1876 ; “ Zur Theorie der Gasreibung,” W.S. Ixxxi. 1880, Ixxxiv. 1881; “Zur Theorie der Gasdiffusion,” W.S. Ixxxvi. 1882, Ixxxviii. 1883; “ Ueber einige Fragen der Gastheorie,” W.S. xevi. 1887 ; “Zur Integration der Diffusionsgleichung,” Sitzung. der Jc. layer, math.-phys. Classe, May 1894.—L. Boltzmann and G. H. Bryan. “ Warmegleichgewicht zweier sich beriihrender Kbrper,” W.S. cii. 2 a. Dec. 1894.—G. H. Bryan. . “Reports on Thermodynamics,” Reports, Brit. Assoc., 1891, 1894 ; “On certain Applications of the Theory of Probability to Physical Phenomena, ” Am. Jour. Math. xix. 3.—S. H. Burbury. A Treatise on the Kinetic Theory of Gases, Camb. Univ. Press, 1899 ; “On some Problems in the Kinetic Theory of Gases,” Fhil. Mag., Oct. 1890. —Des Coudres. “ Diffusionsvorgange in einem Cylinder,” Wled. Ann. Iv. 1895, p. 213.—Kundt and Warburg. “ Ueber die 1
W.S. = Sitzungsherichte der k. k. Wiener Akademie.
