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Generalized Debye--Hückel Equation From Poisson--Bikerman Theory

Research paper by Chin-Lung Li, Jinn-Liang Liu

Indexed on: 25 Dec '20Published on: 02 Sep '20Published in: SIAM journal on applied mathematics



Abstract

SIAM Journal on Applied Mathematics, Volume 80, Issue 5, Page 2003-2023, January 2020. The Debye--Hückel (DH) equation is a fundamental physical model in chemical thermodynamics that describes the free energy (chemical potential, activity) of an ion in electrolyte solutions. We derive, analyze, and verify a generalized DH equation from a fourth-order Poisson--Bikerman theory that accounts for steric and correlation effects of ions and water treated as nonuniform spheres with voids. The derivation yields Debye and correlation lengths that include the steric effect. We perform asymptotic analyses in detail and show that generalized DH and Debye models reduce to their classical counterparts when these effects vanish in limiting cases. Moreover, the generalized DH model is shown to differ much from Hückel's model as their approximations of Born solvation energies are inverse of each other in terms of fitting parameters and ionic strength. Numerical evidence also verifies this finding which may explain why extended DH models need more parameters generally without physical hints to fit experimental data over wide ranges of composition, temperature, and pressure. The generalized DH model needs only the same three parameters for all different ions in a binary or ternary electrolyte solution to fit each experimental data curve of mean activities at various concentrations (up to 6 mol/kg), temperatures (25 to 300 $\operatorname{^{\circ}{C}}$), and pressures (1.01 to 85.5 bars). These parameters model the unknown Born energy of an ion in electrolyte solutions under variable conditions and show orderly values in a total of 17 data curves.