Multicomponent Ion Exchange Equilibria

Most of the ion exchange reactions that occur in soils involve three or more principal ions. For example, in soils at pH 4, the trio H+, Ca2+, Al3+ might be considered, and, in soils at pH 7, the set Na+, Mg2+, Ca2 is important.1 Multicomponent cation exchange is the norm for reactions on humic substances13 and the same is true for anion exchange on metal oxides (e.g., CE, NOJ, and SO4 on iron or aluminum oxyhydroxides).14 

In a multicomponent ion exchange system, the reactions between exchanging ions still can be expressed as in Eqs. 5.1 and 5.2, which are binary processes, but the description of exchanger composition in terms of mole or charge fractions requires a generalization of expressions like Eq. 5.12  

As in Section 4.1, it is convenient to label exchanging ions with numerical indices in describing multicomponent ion exchange equilibria. The thermodynamic approach will be illustrated for a ternary cation exchange system (e.g., Na+, Mg2+, Ca2+), but the extension to an N-ary system is direct. This is evident, for example, in the Gibbs-Duhem equation for a ternary system  

For each additional exchanging ion, one simply adds a term, X d In fi with the understanding that the sum of the mole fractions Xj (i = 1, 2, 3,. . . ) must always equal 1.0. Similarly, for each possible cation exchange reaction (Eq. 5.1), one defines a conditional exchange equilibrium constant 

Strictly speaking, both Kiic and the adsorbate species activity coefficients should carry a designation of the number of exchanging cations (e.g., T for ternary) to emphasize their composition dependence. (This designation is not necessary for the exchange equilibrium constant Kl( because it is independent of composition.) 

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The theoretical treatment of multicomponent ion exchange is quite complex, especially if it involves species of different valency. The treatment of ion-exchange equilibria involving only ions of the same valence is quite straightforward since it can be assumed that the selectivity coefficient or separation factor is constant over the complete range of ionic concentration. 

Mazzotti et al. (1996) studied the esterification of acetic acid with ethanol on a highly cross-linked sulphonic ion exchange resin (Amberlyst-15) in a continuous simulated moving bed reactor. The resin acted as catalyst as well as a selective sorbent simultaneously. They have studied the multicomponent sorption equilibria and swelling of the resin, as well as esterification kinetics with appropriate models. The thermodynamic and kinetic descriptions of the system have been combined to develop a fully predictive mathematical model of the chromatographic reactor. 

Chapter 11 of this undergraduate textbook devotes 77 pages to adsorption and ion exchange. Single and multicomponent equilibria, for both gaseous and liquid systems are described qualitatively. Stagewise operation, which is... 

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