Gauss's divergence theorem in the theory of Wien dissociation of weak electrolytes

Abstract

The central role played by Gauss's divergence theorem in Onsager's theory of Wien dissociation of a weak electrolyte in an applied electric field, X, is indicated. The screening effects of the “ionic atmospheres” are assumed negligible in the region of interest. It is first shown that the steady state continuity equation for ionic motion splits into two separate steady state continuity equations, one describing the state of complete dissociation and the other describing undissociated ion-pairs. The association rate constant A and the dissociation constant K(X) can both be expressed in terms of flux integrals over a closed surface S; it is proved using Gauss's theorem and the two separate steady state continuity equations, that A and K(X) are both independent of the choice of S, provided S is closed and surrounds the origin r= 0 or the sphere r⩽a if a, the distance of closest approach of two ions of the electrolyte of opposite sign, is taken to be non-zero. In particular, S need not be spherical and the theory is still valid if S is only piecewise smooth. These results are illustrated by considering briefly the derivation of Onsager's expressions for A and K(X)/K(0).