Predicting activity coefficients with the Debye–Hückel theory using concentration dependent static permittivity

Abstract

In order to investigate the impacts of a concentration dependent static permittivity in the Debye-Hückel theory, two electrostatic Helmholtz free energy models and four activity coefficient models, with the ion-solvent interactions naturally included, are derived under different assumptions. The effects of static permittivity and model parameters are analyzed by predicting the mean ionic activity coefficients. It is found out that it is reasonable to assume a constant static permittivity in deriving the electrostatic Helmholtz free energy model but it is highly recommended to take the concentration dependence of static permittivity into account in subsequent calculations of thermodynamic properties. The activity coefficient model derived in this way accurately predicts the mean ionic activity coefficients of the investigated systems up to 0.1 mol/kg H 2 O, which indicates that the size parameters in the Debye-Hückel theory might be determined in advance before it is integrated into a more complete thermodynamic model.

Comment on “Predicting activity coefficients with the Debye–Hückel theory using concentration dependent static permittivity”

“…The purpose of this comment is to unveil the curious finding by Lei et al, 1 providing an exact mathematical proof that the two models are equivalent in a certain respect and thus clarifying the classification of extended DH models.…”

“…In a recent article, 1 Lei et al compared four extended Debye-Hückel (DH) models for the mean ionic activity coefficients (MIACs) in a binary electrolyte solution. All models use concentration-dependent static permittivity of the solution, and the authors abbreviated them as DHFULL, EDH, EDH-M, and EDH2015.…”

“…Besides, they can also differ in ion-ion interaction and solvation terms. Numerical calculations performed by Lei et al 1 have led to a surprising finding that "EDH2015 and EDH always give the same mean ionic activity coefficients … as long as the same input information is used" despite apparently different assumptions made at the derivations of the corresponding equations (see fig. 1 in Reference 1).…”

“…This corresponds to the EDH model in the work by Lei et al 1 Equations ( 13) and ( 14) can be derived from the expression for the excess Gibbs energy of solution:…”

Investigation of the Limits of the Linearized Poisson–Boltzmann Equation

“…Thus, the final equation is different from DHFULL. Liang and co-workers 34,35 have demonstrated numerically that the original EDH and the DH version of Shilov and Lyashchenko (with variable relative permittivity before the charging process) give identical results for mean ionic activity coefficients, no matter how the ion-size parameters are evaluated. In a follow-up manuscript, Shilov and Lyanschenko 36 show that this equality can be attributed due to an "extended" Debye charging process where the relative permittivity dependence of the charging parameter is considered.…”