Vanselow equation

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. 

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... 

The equilibrium constant can be written similarly to the Vanselow equation (Equation 1.79), expressed by t, for the monovalent cation ... 

The mole fraction convention is employed in the Vanselow equation (20)... 

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. 

This form of exchange equation, which equates the active mass of adsorbed cations to the mole fraction of these cations on the exchange sites, is often referred to in the soil chemistry literature as the Vanselow equation. Ion exchange equations of this form are the mathematical expression of the important hypothesis of ion exchange, that ... 

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... 

Here Y is the number of nearest-neighbor (closest) charge sites, and Z the cation valence. The main difference between the Davis equation (8.18) and the Vanselow equation (8.16) is the specific ion factor q.i for the divalent cation. For monovalent ions, q is unity. 

Under the Vanselow convention, as previously mentioned, extra work is needed to figure this set of derivatives. Differentiating Equation 9.38 gives,... 

In the quantitative treatment of ion-exchange processes, several authors used the law of mass action. The main difference among these approaches is how the activities and surface concentration of the ions are treated. The first such approach was the Kerr equation, which uses the concentration of the ions on the solid and liquid as well but totally neglected the activity coefficients (Kerr 1928). The Vanselow (1932) equation applied activities in the solution and expressed the concentration of the ions on the solid phase in mole fraction, and in this way, it defined the selectivity coefficient (Equation 1.79). 

This equation is reciprocal to Equation 1.79, except that concentrations are used here instead of activities (y 1) (Vanselow [1932] equation for homovalent exchange), that is, cM<, = aMei and cMei = aMei are the concentration or the activities of the ions in the solution. 

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. 

As indicated in Section 8.4, the primary difference between various cation exchange equations is their differing treatment of the activities of exchangeable cations. Vanselow, for example, assumed that the activities of exchangeable cations were proportional to their mole fractions. This is equivalent to saying that ions on soil colloid surfaces behave as if in ideal solution (Appendix 3.2). The mole fraction of an ion in a binary system is... 

In the Vanselow approach, the selectivity coefficient (conditional constant) given by Equation 5.18 is used, which effectively assumes ideality in the exchanger phase, in an attempt to overcome the lack of ability to obtain the corresponding activity coefficients. However, Vanselow selectivity coefficients show a dependence on the composition, especially for heterovalent exchange (i.e., exchange between ions of different charge, such as Na+-Ca +). Other forms of selectivity coefficients have been proposed, as detailed in the following discussion. Table 5.1 summarizes these coefficients. 

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... 

Equation 5.59 is plotted in Figure 5.8a as a dashed line. The Vanselow coefficients can be calculated and are plotted in Figure 5.8b here only one extrapolated point is required for y a = 1- The integration yields = 0.175, comparing reasonably well with values of 0.182 and 0.197 reported by Bond (1995). 

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