Gibbs-Duhem Equation for Ion-Exchanging Mixtures in Donnan Equilibrium with Electrolyte Solutions

Abstract

An important chemical reaction in soil–water systems is ion exchange between electrically charged surfaces and bathing electrolyte solutions. Characterization of ion exchange by classical thermodynamic methods has been difficult, however, due to limitations in measuring activities of exchanger species. In the typical case of insoluble exchanger species, formulations based on Donnan membrane equilibria are limited by lack of infinitely dilute reference states for defining activity coefficients. Gaines and Thomas determined activities of insoluble exchanger species by integrating a Gibbs–Duhem equation for a solid exchanger phase. Integration is difficult, however, for multielectrolyte systems with variable solvent activity. Snyder and Cavallaro merged the Donnan and Gaines–Thomas theories by integrating a Gibbs–Duhem equation for a macroscopically homogeneous Donnan-like mixture of solvent, electrolytes, and soluble or insoluble exchanger species equilibrated with an electrolyte solution across a semipermeable membrane. This formulation resolved many previous difficulties but still did not adequately account for osmotic pressure and electrolyte concentration differences occurring across the semipermeable membrane. Furthermore, the method of determining independent component masses, necessary for integrating the Gibbs–Duhem equation, became complicated in systems with many chemical reactions. These problems were addressed in the current study by reformulating the theory using an extension of the De Donder chemical affinity theory. The formulation reduces to that of Gaines and Thomas in the limiting case of vanishing osmotic pressure and electrolyte concentration differences. Colloidal instability phenomena are treated as phase transitions, manifested by discontinuities in osmotic pressure–volume relations in the Gibbs–Duhem equation.