﻿474 Messrs. C. G. Darwin and R. H. Fowler 



on 



It does not appear profitable to examine these expressions 

 further here, since the agreement with experiment is not 

 very good at all temperatures. It is to be presumed that 

 the assumption of constant moments of inertia is at fault, 

 and this is supported by some of the evidence from band 

 spectra ; further, it is probable that the case of no rotation 

 must be excluded, involving the omission of the first term 

 in the partition functions. The discussion of the practical 

 applications of these formulae cannot be entered into here. 



§ 12. Assemblies containing free molecules. 



The problems we have so far discussed have all possessed 

 the distinguishing characteristic that the temperature is the 

 only independent variable. As soon as we treat of free 

 molecules this is no longer the case, for now the volume 

 must be another independent variable. Nevertheless, as we 

 shall see, the same methods of calculation are available. 

 The partition is no longer represented exactly by integrals, 

 as it was for the quantized motions, but from the nature of 

 the case some form of limiting process is required. The 

 free molecules cannot of course be regarded as the limit of 

 three-dimensional vibrators of low frequency, for they have 

 no potential energy to share in the partition. We must 

 proceed by the method common to most discussions of the 

 distribution laws of classical assemblies — divide up into cells 

 the six-dimensional space in which the state of any molecule 

 is represented, associate with each cell a certain constant 

 value of the energy, and in the limit make all the dimensions 

 of all the cells tend to zero *. 



We take an assembly composed of M systems of the type 

 A of § 9 and P free-moving monatomic molecules of mass m 

 and of small size, the whole enclosed in a vessel of volume V. 

 The energy of the molecules is solely their energy of trans- 

 lation ; they are supposed to obey the laws of classical 

 mechanics (except during their collisions with the A's). In 

 order to specify the state of the assembly, we*fcake a six- 

 dimensional space of co-ordinates q u q 2 ,...p 3 , the three 

 rectangular co-ordinates and momenta of a molecule in the 

 vessel. We divide up this space into small cells, 1, 2, 3, ..., 

 t ..., of extensions (dq l ...dp z ) u which may or may not be 



* That the limit of the distribution laws worked out for the cells is 

 the true distribution law for the actual assembly is an assumption 

 implicit in all such discussions. 



