

572 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



senting neutral complexes are unable to respond to the applied field and 

 so do not contribute to the overall mobility. The mobility of unpaired 

 ions is assumed to be no , the mobility observable at infinite dilution. The 

 apparent mobility n at any finite concentration is then no reduced by 

 the fraction P/N of ions paired. Thus 



M = [1 - iP/N)U (7.15) 



The Bjerrum-Fuoss theory when applied to real systems reproduces 

 the experimental data very well, although the parameter a, the distance 

 of closest approach, needs to be determined from the data itself. : 



The concept of a pair defined in terms of the minimum occurring at b, 

 becomes rather vague when that minimum vanishes in favor of a point 

 of inflection. At this stage triplets and other higher order clusters form; 

 and the situation becomes very complicated. 



In Reference 44, Reiss has developed a more refined theory of pairing. 

 Instead of avoiding the use of an inconsistent g(r) by introduction of the 

 mass action principle, an attempt is made to provide a rigorous form for 

 g(r), which proves to be the following 



g(r) = exp [-47rr^A^/3] 47rr\ exp [q^/KkTr] (7.16) 



in which 



! 



h = I / j exp [- 47rr'i\r/3] 4xr' exp [q'/KkTr] dr (7.17) i 



It is also shown that the activity of an ionic species, measured by A^ — P 

 in the Bjerrum-Fuoss theory, is measured by y/hN in the more rigorous , 

 theory. The distribution (7.16) suffers neither from an inability to con-d 

 serve charge in the volume 1/N (as does (7.4)) nor from any inconsistency 

 involving the interaction of a nearest neighbor with other ions than the 

 one to which it is nearest neighbor [as does (7.4)]. 



When -s/hN computed by (7.17) is compared with {N — P) computed 

 according to (7.10) and (7.14), for arbitrary values of /c, a, T, and N, 

 the results are almost identical. This shows the virtue of the Bjerrum- 

 Fuoss theory, and in fact, suggests that in most cases it should be used 

 for calculation rather than the more refined theory, for the latter involves 

 rather complicated numerical procedures. 



The refined theory can also be adapted to the treatment of transport 

 phenomena. Thus in place of g{r) it is possible to write a distribution 

 function r(r), specifying the fraction of nearest neighbors lying in the \o\- 

 ume element dr, in a system in the steady state rather than at equilib- 

 rium. In the presence of an applied field the distribution loses its spheri- 



