A Multiresponse method for fitting solid-phase activity coefficient models to ternary ion-exchange data

A popular method for fitting solid-phase activity coefficient models in the thermodynamics of ternary ion-exchange has been the Rational method. The standard statistical tests used to compare model functional forms are undefined in the Rational method because all of the dependent variables in the regression (Gibbs Energies) take the value of zero. The present study develops the mathematically equivalent but statistically improved approach of directly fitting Vanselow selectivity coefficients with multiresponse regression. In this multiresponse approach, the Gibbs Energy at equilibrium is set to zero, a selected model is substituted into the equilibrium expression in place of the solid-phase activity coefficients, and the activity coefficient model parameters are optimized numerically until the predicted Vanselow selectivity coefficients are as close as possible to the measured values. This Multiresponse method requires the use of restricted regression to ensure that the equilibrium constants obey the Triangle Rule. An advantage of the Multiresponse method is that it allows for calculation of the coefficient of determination (R2) and R2adj, which are statistics that can be used to compare solid-phase activity coefficient models. The example solid-phase activity coefficient models compared in this study are the Regular Solution model and a Cox mixture model. When applied to NH4+–Ca2+–K+ exchange on vermiculite, the R2adj statistics were found to be 0.944 for the Cox mixture model and 0.826 for the Regular Solution model. These results from the Multiresponse method indicate that the Cox mixture model is superior for this dataset.