Theory of Potentials of Average Force and Radial Distribution Functions in Ionic Solutions

The procedure employed by Mayer for the calculation of osmotic pressure and activity coefficients of ions in solution from the Fuchs expansion of the grand potential in multicomponent systems is applied to the recently obtained expansion of the potential of average force in these systems. The resulting multiply infinite series are summed to give a series analogous to the original expansion, involving the Debye-Hückel potential of average force in a remarkably simple way. A similar expansion for the radial distribution function is presented. Explicit expressions for both quantities, exact through terms of first order in the ionic concentration, are given. An expansion for the osmotic pressure, analogous to the virial expansion, is presented. The presence of the Debye-Hückel potential of average force in all these expansions is discussed, and the possibility of obtaining the potential of average force exact through terms of second order in the ionic charging parameters is indicated. Possible applications to ionic conductivity, diffusion, and viscosity are discussed. Additional applications to the theory of defects in ionic crystals, impurity conduction in semiconductors, etc., are suggested. The significance of the individual terms in the new expansions is discussed. Proof of an n-fold convolution theorem for Fourier transforms is presented.