Vanselow model

This factoring of In f produces a scale-invariant part that serves to interpolate smoothly between the value of f" and f = 1.0. Two well-known special cases of the van Laar model are the regular solution model, which results by setting Iab = Ca > aHd the Vanselow model, which results by setting fAB = fBA = l.O.1 The Vanselow model thus corresponds to an ideal solid solution (f = fd = 1.0 cf. Section 3.3). 

When heterovalent exchange (e.g., exchange involving both mono- and bivalent ions) takes place, which is frequently seen in soil-solution systems, the quantitative treatment runs into further difficulties. In this case, the determination of mole fractions also becomes problematic. The Vanselow equation assumes that the monovalent and bivalent ions are equivalent when calculating mole fractions. Other empirical equations, however, simply introduce some factors for the ions with different valencies. For example, according to Krishnamoorthy and Overstreet (1949), this factor is 1 for monovalent ions, 1.5 for divalent ions, and 2 for trivalent ions, which is not in agreement in stoichiometry. Another model... 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

With an arbitrary definition of KNaX as equal to unity, thus establishing a reference half reaction, the equilibrium constant for any other half reaction can be determined from measured selectivity coefficients. The Gapon equation can be readily implemented in this manner. Implementation of the Vanselow equation, however, requires modification of the general equilibrium models to account for the more complex dependence of mole fractions on the molar concentrations. An example ion-exchange calculation using the half reaction approach to represent the Gapon equation is presented in Appendix 2. 

Adsorption of a dissolved ionic species is always part of an exchange reaction that involves a competing ionic species. The desorbing species creates the vacant site to be occupied by the adsorbing one. Soil scientists use a variety of ion exchange models (e.g., the Gaines-Thomas, Gapon, Vanselow, and Rothmund-Kornfeld models) in which different conventions are used to write the concentrations of dissolved and adsorbed species. These models are adequately described and compared elsewhere (cf. Bolt 1979 Sposito 1981, 1989 Appelo and Postma 1993). 

Davis developed an equation similar to the Vanselow equation from statistical thermodynamics. Electrostatic forces between colloid surfaces and adsorbed cations were calculated for various surface configurations of charge sites. These sites were assumed to be neutralized by individual adsorbed ions. Hence, the model resembles most closely the Helmholtz model of the double layer with the charge of cations on the surface assumed to be just equal to the number of colloid charges. The resultant equation is... 

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