On the derivations of the Debye–Hückel equations

TLDR: In this paper , the authors present the derivations of basic thermodynamic properties and activity coefficients equations from the linearised Poisson-Boltzmann equation, and demonstrate the necessary steps to reach all original models derived from the Debye-Hückel model and further explore their capabilities and limitations concerning individual ion and mean ionic activity coefficients for different size dissimilarities scenarios between ions.

Abstract: This work presents the derivations of basic thermodynamic properties and activity coefficients equations from the linearised Poisson–Boltzmann equation. We consider two main approaches, the first one is based in classical thermodynamics, which has been used in the original work of Debye and Hückel, leading to the model which has been an important cornerstone of electrolyte thermodynamics since its original publication in 1923. The second approach relies on more modern derivations based on statistical mechanics, the so-called charging approaches. Both derivation routes have differences and shortcomings. We demonstrate the necessary steps to reach all original models derived from the Debye–Hückel model and further explore their capabilities and limitations concerning individual ion, and mean ionic activity coefficients for different size dissimilarities scenarios between ions. One immediate conclusion is that there is an unnecessary consideration in the Debye and Hückel derivation which is cancelled by another one they have made, leading to a correct expression for the activity coefficient. Also, the long-lasting consideration that both the Debye and Güntelberg charging processes lead to the same thermodynamic properties is demonstrated to be inaccurate, as it is rigorously true only when a common distance of closest approach is used. GRAPHICAL ABSTRACT