Integrable Gibbs Duhem Equation for Ion Exchange Equilibria at Variable Water Potential.

Thermodynamic characterization of ion exchange equilibria in soils has been limited by lack of a general method for determining activities of solid exchanger species.  Existing methodology, due to Gaines and Thomas, requires integrating the Gibbs-Duhem equation for a solid exchanger phase at fixed water potential, a severely restrictive condition.  Furthermore, the method utilizes an ideal solid solution as reference state for the exchanger phase, assuming that the idealized solution consists only of exchanger species with no aqueous component. This formalism does not contemplate macroscopic pressure-volume relationships, punctuated by sharp discontinuities due to flocculation or dispersion, which are typcally exhibited by ion exchangng colloidal systems. Langmuir suggested, many years ago, that a more realistic reference state for such systems would be an ideal aqueous solution, where charged solid particles and counter-ions are treated on the same footing as electrolyte species. Here this idea is developed more fully, for ion exchange systems ranging from dilute colloidal suspensions to flocculated systems and unsaturated soils. An integrable Gibbs-Duhem equation, derived from De Donder’s theory of chemical affinities, is introduced which is easily solved for the chemical potential of any arbitrary exchanger species, with no restriction on variations of solvent and electrolyte chemical potentials. Possible standard states are discussed.  The formulation is in principle applicable to both constant- and variable- charge systems. Applications to published data are briefly discussed.