174 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
In sharp distinction from Meyer’s theory, Stefan* * * § and Maxwell! put forward a 
theory of diffusion which led to the result that D 12 is entirely independent of the 
ratio vi‘.v 2 , the formula involving only the sum vi + v 2 or v 0 . Stefan’s theory was 
based on the hypothesis of rigid elastic spherical molecules, and agrees with the 
corresponding form of our first approximation to D 1S (13'06) ; the same result was 
subsequently obtained by Langeyin| and the present author § on the most general 
molecular hypothesis, in the form (13'06). Maxwell’s first formula related to the 
elastic sphere theory, and was greater than that of Stefan by one-third, Maxwell 
being considerably in error : his second formula related to Maxwellian molecules, and 
was identical with the exact formula for such molecules given in § 13 ( b ). All these 
results share the property that they are independent of p 1 : v 2 , and as the present theory 
shows, all of them are in error in this respect, except Maxwell’s second formula. 
They require to be multiplied by the correcting factor V'/V of (13’03) : this factor is 
unity in the case of Maxwellian molecules, so that for a mixture of two sets of such 
molecules D 12 is independent of their numerical proportions. This is one of the few 
properties of a gas which depends in its very nature (i.e., not merely in absolute 
magnitude) on a particular molecular model : another such property will be noticed 
in § 14. The fact that the absence of variation is in disagreement with experimental 
results confirms the conclusion drawn from other sources of evidence that Maxwellian 
molecules are unsatisfactory representations of actual molecules. 
The only other general case of independence of D 12 or v l : v x is that in which the 
dynamical properties of the two sets of molecules are alike, as in the case of self¬ 
diffusion (§13 (/’)). Hence we may speak of the coefficient of self-diffusion of a gas 
without specifying the ratio v x : v 2 in contemplation ; in general, on the contrary, D 12 
is defined only when the value of v x : v 2 is assigned. 
In the case when m 2 /m x is negligible (§13 (e)) the mutual encounters of the light 
molecules are neglected, with the result that in this case also D 12 depends only on r 0 . 
This, however, is perhaps hardly to be regarded as an additional exception to the 
general rule. 
The error in the formulae of Stefan and the other authors mentioned arose from 
the neglect of the difference between the actual law of distribution of the peculiar 
velocities of the molecules and the assumed Maxwellian law. This is taken account 
of in the present paper, and leads to the correction factor V'/V in ( 13‘03 ); we will 
now proceed to consider how this factor varies with the ratio v x : v 2 by studying the 
variation of the second approximation to it, i.e., (l—e 0 ) _1 . From this we may readily 
* Stefan, ‘ Wien. Sitzb.,’ 63 (2), p. 63, 1871 ; 65, p. 323, 1872. 
f Maxwell, ‘Scientific Papers,’ i., p. 392 ; ii., p. 57 and p. 345. Of. also Boltzmann, ‘Wien. Sitzb.,’ 
66 (2), p. 324, 1872; 78, p. 733, 1878 ; 86, p. 63, 1882; 88, p. 835, 1883. Also ‘ Vorlesungen fiber 
Gastbeorie,’ i., p. 96. 
I Langevin, loc. cit. ante. 
§ ‘ Phil. Trans.,’ A, vol. 211, p. 449. 
