Gaines-Thomas equation

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-. 

In the Gaines-Thomas equation (22), equivalent fractions are employed to represent the activities of ions in the exchanger phase, that is,... 

Substituting, the mass action equation (Eqn. 9.34) becomes, for the Gaines-Thomas convention,... 

Within the framework of the method of Gaines and Thomas, Equation 7.10 is differentiated, and then the Gibbs-Duhem equation is applied to obtain the equilibrium constant ... 

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... 

Apart from this definition, the form of the Krishnamoorthy-Overstreet conditional constant would be similar to those of Vanselow and Gaines-Thomas (see Table 5.1) it should be noted, by inspection of Equations 5.43 and 5.44, that the... 

Becanse of the complexity of soils, there are no general relations between the proportions of two cations on the exchange complex and their reduced activity ratio in solntion. Bnt two equations are commonly used the Gaines and Thomas... 

We have used the phase distribution data shown in Figures 1-4 to calculate the standard free energies of the ion exchange reaction using a simplified version of the equation of Gaines and Thomas (12). 

Nevertheless, the thermodynamic formulation normally used for zeolites is the one developed by Gaines and Thomas [34], who expressed the equilibrium constant (Equation 7.5) as follows. 

There are many other types of exchange equations, such as the Gaines and Thomas, the Kerr, and the Krishnamoarthy-Overstreet (Table 4.3). Generally, however, the Capon and Vanselow equations are the most widely used. 

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) ... 

Kc is known as the Kielland coefficient and is related to Ka by the simplified Gaines and Thomas [11] equation ... 

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