Gaines—Thomas selectivity coefficient

Both approaches lead to identical standard thermodynamic values of exchange (9-10). Such a difference in the choice of the surface concentration scale is of course only important for heterovalent exchange equilibria. For the heterovalent case the numerical value for both selectivity coefficients, Kg (Gaines Thomas) and (Vanselov) differ and, consequently, their variation with surface composition also differs. 

The equilibrium constant (Equation 1.81) and the selectivity coefficients (Equations 1.82 and 1.83) have been calculated by the selectivity function, by the integration of the selectivity coefficient versus the surface molar or equivalent fraction (Gaines and Thomas 1953 Howery and Thomas 1965 Argersinger et al. 1950 Ekedahl et al. 1950 Hogfeldt 1950) ... 

Since the cation-exchange processes of the main cations (sodium, potassium, magnesium, calcium) have a significant role in the nutrient cycle of soils, the classical literature has discussed their cation-exchange processes in detail (e.g., Boyd et al. 1947 Gaines and Thomas 1953 Howery and Thomas 1965 Sposito 1981 Filep 1999). As described earlier, cation exchange is characterized by the selectivity coefficients and... 

The thermodynamic equilibrium constant can be evaluated by using the simplified form of the Gaines-Thomas equation [201], assuming that the change in water content in the exchanger is negligible. The equation, with defined as the corrected selectivity coefficient by Eq. (4), follows-. 

If the equivalent fraction convention for exchanger phase activities is adopted (i.e., the Gaines-Thomas approach), the molar concentrations are proportional to the equivalent fractions, and it can easily be shown that the half reaction constants and the binary-exchange constants (selectivity coefficients) are related by... 

Such a normalization assumes that the residual Na+ ions are inaccessible for exchange and have no influence on the selectivity coefficient. That this assumption is incorrect results from the fact that the maximum limit to exchange is influenced by the nature of the outgoing and ingoing ions, and also by the Al/Si ratios of the zeolites. In order to obtain comparable values of the equilibrium constant, we modified the Gaines and Thomas equation. The corrected and normalized selectivity coefficient is defined as... 

Note that no individual sites are considered, just the surface taken as an exchanger. The activity convention is to use equivalent fractions, as in the Gaines-Thomas case thus, the corresponding selectivity coefficient is given by... 

The results (Bond 1995) are shown in Figure 5.8a. Here the Na+ isotherm only is plotted, as an equivalent fraction in the solid and as a function of both mole fraction and equivalent fraction in solution phase the second way is often found in the literature. Obtaining the nonpreference isotherm is a bit more complex here, starting from the equilibrium expression for reaction 5.56 in terms of the Gaines-Thomas selectivity coefficient, because the exchanger phase activity coefficients are not known. 

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