﻿Chemical Constants of some Diatomic Gases. 991 



5. In the first case we suppose the gas molecule to consist 

 of two identical atoms rigidly attached to each other at a 

 fixed distance. In addition to the coordinates of the centre 

 of gravity we require two angles, 6 and <£, defining the 

 direction of the molecular axis. Rotation about this axis is, 

 as usual, ignored. We then have 



qi = x ; q 2 =y ; q~ = z ; q± = ; £ 5 = </> : 



p l = m l r; p 2 = my ; p 3 = mz ; p± = mK 2 6 ; p 5 = mK 2 sin 2 0$, 



where K is the radius of gyration. 



The energy of this molecule is given by 



e=f (&+f+h*)+~(e s +sm*0<j>*)+e o , . (10) 



where e is the energy of the molecule at rest in the 

 generalized space. Thus 



%e ~ ** = Y ~ \\ . . . sin 2 6 cLv dy dz dx dy dz d$ d<f> d0 d$e ~' m . 



.... (11) 



The limits of the multiple integral are the boundaries of 

 the element of volume for x, y, and z; the angles and 2tt 

 for 0; the angles and ir/2 for 0, and all the velocities 

 from — oo to + go . Hence, if V is the total volume, 



H 



The free energy, sfr, of the system is then given by * 



^=-kTxhog e le~^-log e ~\, . . (12) 



where N is the total number of molecules in the volume V, 

 say N = N , the number of molecules in one gram-molecule. 

 In the above case 



t=-«N log.^^?(2^mCTf 3 + N 6 . . (13) 



For equilibrium between the vapour and the condensed 

 phase 



^'-^=p(V-V'), (14) 



in which dashed symbols refer to the condensate. Substi- 

 tuting in (14) from (13), and neglecting small terms, we find 



zl 2ttK 2 V., 9 « € -^7No h 

 k log e -jj-g— (2iTkmT) * =A\ 



* Planck, loc. cit. p. 210. 



