An analytic solution of the non-linear equation ∇2λ(r)=f(λ) and its application to the ion-atmosphere theory of strong electrolytes

Abstract

For a long time the formulation of a mathematically consistent statistical mechanical theory for a system of charged particles had remained a formidable unsolved problem. Recently, the problem had been satisfactorily solved, (see Bagchi [1] [2]) ,by utilizing the concept of ion-atmosphere and generalized Poisson-Boltzmann (PB) equation. Although the original Debye-Hueckel (DH) theory of strong electrolytes [3] cannot be accepted as a consistent theory, neither mathematically nor physically, modified DH theory, in which the exclusion volumes of the ions enter directly into the distribution functions, had been proved to be mathematically consistent. It also yielded reliable physical results for both thermodynamic and transport properties of electrolytic solutions. Further, it has already been proved by the author from theoretical considerations (cf. Bagchi [4])as well as from a posteriori verification (see refs. [1] [2]) that the concept of ion-atmosphere and the use of PB equation retain their validities generally. Now during the past 30 years, for convenice of calculations, various simplified versions of the original Dutta-Bagchi distribution function (Dutta & Bagchi [5])had been used successfully in modified DH theory of solutions of strong electrolytes. The primary object of this extensive study, (carried out by the author during 1968-73), was to decide a posteriori by using the exact analytic solution of the relevant PB equation about the most suitable, yet theoretically consistent, form of the distribution function. A critical analysis of these results eventually led to the formulation of a new approach to the statistical mechanics of classical systems, (see Bagchi [2]), In view of the uncertainties inherent in the nature of the system to be discussed below, it is believed that this voluminous work, (containing 35 tables and 120 graphs), in spite of its legitimate simplifying assumptions, would be of great assistance to those who are interested in studying the properties of ionic solutions from the standpoint of a physically and mathematically consistent theory.