﻿Virial of a Mixture of Ions. 553 



stand for the probabilities o£ the occurrence of configura- 

 tions of which the potential energies are E, E' respectively 

 in a system of molecules where intermolecular forces exist, 

 and P, F' stand for the probabilities of the occurrence of the 

 same configurations in the same system of molecules if there 

 were no intermolecular forces, then 



p ( E ) _P .-(E-E')/w m 



P'(E')~P' ' " " * ■'" 



where w stands for the most probable value of the kinetic 

 energy of, a molecule. If R is the gas constant, T the 

 absolute temperature, and v the number of molecules in a 

 gram molecule, then 



RT 



w =~v (-> 



In applying Boltzmann's theorem to our system of ions 

 we assume that if there were no inter-ionic forces the distribu- 

 tion of the ions would be a random one ; P and P(E) will 

 stand respectively for the probabilities of the occurrence of 

 a given configuration in a random distribution, and in the 

 distribution as modified by the electrical forces. Equation (1) 



may be written in the form — W <?~ ' w = constant for all 

 configurations =k, say, hence 



P(E)-/,-P^- E/w , (3) 



where k is a quantity which is independent of E, and is 

 theoretically determined from the consideration that 2P(E) 

 for every possible configuration of the system must be equal 

 to unity. 



We next proceed to determine P, and we shall calculate it 

 for a configuration which is defined as follows : — A is a 

 given ion in the mixture ; the nearest ion to it, A l3 lies in a 

 given elementary volume dv 1 at a distance i\ away from A ; 

 the second nearest ion to it, A 2 , lies in a volume dv 2 at a 

 distance r 2 away from A ; ... the with nearest ion A,„ lies in 

 a volume dv m at a distance r m away from A . 



In a random distribution the chance that any individual 

 ion is not at a given instant included in any volume o 

 imagined in the whole volume is 1 — v/V. The chance that 

 none of the 2N ions are in v is therefore 



(.-vr. 



or. on expansion, 



- 2N , 2N(2N-1) . 



