The value 68 for H, is the experimental one, which is very uncertain; the value 46 would bring H2 into accord with all other bodies, as may be seen in the "Viscosity of Gases and Molecular Force." 2 The most natural relation to expect between‚Âm12, ¿a1⁄2m1⁄23, andАmm is ¡Аmm—(¡Аm2 2Аm2)3, and accordingly we now give values of 10-1(M1ẞ1 1C1M2ß2 2C2)3, obtained from the numbers just given, and the ratio of 10−11C2{(M1ẞ1)3/2 + (M2ß2)3/2}3 given above to The mean value of the ratio is 1.2, and the departures of some of the individual numbers from the mean, though large, are not larger than could be caused by only slight error in the experiments or the theory, for it must be remembered that the constants C2 occur in equation (8) in such a manner as to make their values when calculated from that equation very sensitive to small errors in the ratio of the diffusion-coefficients at two temperatures. Thus, notwithstanding the high degree of accuracy attained by v. Obermayer in his elaborate experiments, it must be allowed that the last series of numbers is as nearly constant as can be expected. To show this clearly it will be best to assume that the ratio is 1, and calculate Č2 from the equation 1C2{(M1ß1)3/2 | (M2ß2)3/2}3=(M1ß1 1C1 M2ß2 2C2)3, . (9) and then by the equation (8) calculate values of the ratio of the diffusion-coefficients at v. Obermayer's two temperatures for comparison with his experimental values:— Exper.... Theory... CO, & N, CO, & H. CO,&N,O. O, & H. O, & Ng. 0, & CO, The largest discrepancy between theory and experiment amounts to 2 per cent., and it cannot be claimed that the ratio of the diffusion-coefficients at two temperatures, as measured experimentally, can be guaranteed correct to within 2 per cent., especially as the experimental measurements only yield values of the diffusion-coefficients by the intervention of quite an elaborate theory of the experiment. The outcome of the investigation so far, then, is that v. Obermayer's experiments (in continuation of Loschmidt's) on the temperature variation of diffusion establish at least the approximate truth of the law that the parameter A1⁄2m ̧m, in the attraction of two unlike molecules of masses mi and m2 is equal to the square root of the product of the parameters Am and Am2 for the like molecules. To test the truth of the law in an independent manner, some experiments have been carried out on the surface-tension of mixed liquids and will be described in another paper. 2 Meanwhile there is interesting matter to discuss in connexion with diffusion. It can be seen how desirable are experiments on the temperature variation of the diffusioncoefficients of many more pairs of gases. This variation could be prophesied for a large number of pairs of gases by calculating C2 according to equation (9), using therein the values of B and C1 given in the paper on the Viscosity of Gases and Molecular Force, but the calculations would possess more interest if carried out in connexion with the experiments than at present. However, as the diffusioncoefficients of many more pairs of gases have already been determined experimentally at one temperature, it seems at first sight to be possible to determine the corresponding values of C2 from them in the following manner. write our relation (7) in the form D=BT (1/M1+1/M2) * Let us (10) where B is a constant the same for all pairs, then for the six pairs of gases already studied, as we know all the variables, we can obtain values of B which ought to be all nearly the same. But when the calculation is made, using the lower value of T in each of v. Obermayer's experiments, which is about 284, it is found that B, instead of being constant, is closely proportional to 1+1C2/T, as the following values show: 1.53 2' CO2 & N2. CO2 & H2. CO2 & N2O. O2&H2. O2 & N2. O2 & CO. 103B/(1+,C2/T).. 1·11 1.12 This curious result has some interesting bearings. In the first place, it means that the diffusion-coefficients of actual gases at temperatures about 284 absolute are related to one another almost as they would be if the molecules were forceless spheres, and this explains why the investigators of diffusion have hitherto found fair agreement between the results of experiment at ordinary temperatures and the kinetic theory of forceless perfectly restitutional spherical molecules. The immediate effect of the result on our present inquiry is to render illusory the hope of obtaining values of 1C2 for the various pairs of gases for which Loschmidt and v. Obermayer have found values of the diffusion-coefficient at only one temperature near 284, for as regards these values we have just seen that the molecules behave almost as if forceless. It may be suggested that the failure of B to prove constant is due to inadequacy of Stefan's theory of diffusion, but the expressions for the diffusion-coefficient given by Meyer and Tait gave on trial about the same results as Stefan's; so that the failure of B to prove constant is not due to any peculiarity of Stefan's theory. We have to go deeper for the reason, and in doing so have to open up a very important department of molecular dynamics of which at present we know but little, namely, the nature of collisions between molecules. Hitherto in the kinetic theory it has been assumed that the forces called into play during the collision of two molecules are such as they would be if the molecules were perfectly restitutional spheres, and the assumption seems to have worked well as regards the general phenomena of gases; but in reality it was not required there, and could be replaced by the assumption that the translatory kinetic energy of a number of molecules is a constant fraction of their total kinetic energy The usual assumption of perfect restitutionality causes no difficulty in connexion with the theory of the viscosity of a single gas, because the nature of the collisional forces between molecules is not directly involved in that theory; but in the theory of diffusion, as well as in that of the characteristic equation of the element gases, the forces involved in collision enter as an essential element of the calculation. Now in the paper on the Viscosity of Gases and Molecular Force, in connexion with the theory of the characteristic equation of the element gases, just such a discrepancy as we have encountered in diffusion cropped up between the behaviour of actual gases and the theory of a medium composed of attracting perfectly restitutional spheres; and it was pointed out that in some way, which at present must be called accidental, the departure from perfect restitutionality in the collisions compensated for a certain effect of molecular attraction in such a way as to make the molecules behave in one respect as if they were forceless. It seems desirable, therefore, to bring out clearly the parallelism of the two cases. In the theoretical characteristic equation of a medium made of attracting smooth perfectly restitutional spheres, one term is the virial of the collisional forces of all the spheres in unit mass which takes the form Zauv/2 (see Viscosity of Gases and Molecular Force), where a is the radius of a sphere, μ the average momentum imparted to a sphere in a collision, and the average number of collisions per sphere per second, the summation to extend to all spheres in unit mass. This is closely similar to the expression which comes in in diffusion for the resistance experienced by one medium in passing through another, and which was written μv. In the virial expression μ is momentum due to velocity of agitation, while in the diffusion resistance μ is the momentum due to relative motion of the two media, which is very slow compared to the velocities of agitation. In the diffusion resistance v denotes the number of collisions per second of a sphere of one set with the spheres of the other in unit volume. It was shown that Σapv/2 when evaluated takes the form whereas Amagat's experiments on H2, O2, N2, and CH4 above the critical volume can be represented by the form so that the factor (1+1C1/T) due to molecular force seems to fall out. Now in the diffusion expression it is a factor approximately equal to 1+1C2/T that appears to drop out; and the main difference between the two cases is that in diffusion the velocity of diffusion involved in the momentum is small compared to the average velocity of agitation involved in the μ of the collisional virial. Thus it appears that the momentum communicated from molecule to molecule in a collision is not transmitted in the same manner as with smooth perfectly restitutional attracting spheres, but that there is some mechanism by which the transmission is made to depend on the ratio of the potential energy at contact to the mean kinetic energy in such a manner as to make the final effect of the forces acting during the collision of molecules the same as if the molecules were both forceless and perfectly restitutional smooth spheres. The mechanism is probably that which preserves proportionality between the mean translatory kinetic energy and the mean vibratory energy of a molecule; and the difference between the diffusion case and that of the collisional virial may perhaps lie in the fact that the mechanism does not operate in the same manner as regards the mass motion of diffusion and the molecular motion of heat. But the whole question of molecular collision is so large a one that it will require considerable research to itself; from the glimpse we have got into it, it appears that the momentum imparted to a molecule during a collision, instead of being u as calculated on the assumption that the molecules are smooth perfectly restitutional spheres, is hp, where h is a parameter characteristic of the pair of molecules colliding, and which we have found empirically in the case of diffusion to be approximately proportional to 1+1C2/T. It may be worth while noting a certain regularity in the departure from strict proportionality, or in the departure of 103B/(1+1C2/T) from constancy for the two triatomic molecules CO2 and N2O its value is least, namely, 94; for the triatomic CO2 with diatomic N, and H, it is 111 and 1.12; while for the three diatomic pairs O2 with H2, N2, and CO, it is 1.17, 1.17, and 1.16. With Loschmidt's and v. Obermayer's diffusion-coefficients for a number of pairs of gases at about 15° C., we can test more extensively our empirical relation that at about that temperature the diffusion-coefficient is proportional to (1/M2+1/M2)3/{(M1ß1)3/2 + (M2ß2)*/2}2. The data are available for sixteen pairs including the six already considered, and as the experimenters have reduced their results to values at 0° C. by the approximate formula D273/D=(273/T)2, which is near enough to the truth for small differences between T and 273, we will take the values D273 as suitable for our present purpose. The additional values required for (MB)/2 are: With these and the values already given, and the values of D reproduced from v. Obermayer and Loschmidt, the values of (1/M1+1/M2)3/{(M1ß1)*/2 + (M2ß2)*/2}2D |