## Ion Exchange Kinetics

The rate of ion exchange is governed by the various diffusion processes in the system. In general,

Equivalent fraction of A in solution

Figure 3 Calculated ion exchange isotherms for a hypothetical di-univalent (A2 + /B+) exchange having a selectivity coefficient kA/B = 1. The isotherms have been generated using constant total ion concentrations: 1, 0.01 mol L"1; 2, 0.1 mol L"1; 3, 1 mol L"1; 4, 5molL"1. The isotherms show increasing preference of the ion exchanger for ion A with increasing dilution of the solution (electroselectivity effect.)

Equivalent fraction of A in solution

Figure 3 Calculated ion exchange isotherms for a hypothetical di-univalent (A2 + /B+) exchange having a selectivity coefficient kA/B = 1. The isotherms have been generated using constant total ion concentrations: 1, 0.01 mol L"1; 2, 0.1 mol L"1; 3, 1 mol L"1; 4, 5molL"1. The isotherms show increasing preference of the ion exchanger for ion A with increasing dilution of the solution (electroselectivity effect.)

diffusion can be described by Fick's first law. The flux of ion A(Ja) is given by:

where D is the diffusion coefficient. This equation describes purely statistical diffusion that is driven by the concentration gradient. In ion exchangers it is usually necessary to consider also the electric potential (q) and then the flux of ion A ( JA) is given by the Nernst-Planck equation:

where F is the Faraday constant. Most commonly, the kinetics of ion exchange reactions are interpreted in the terms of external or internal diffusion. As the external solution is usually agitated, there is essentially no concentration gradient in the bulk of the external solution. Gradients arise, however, within a thin layer of solution adhering to the surface of the exchanger particle. Diffusion across this layer is called film diffusion. This concept, developed first by

Nernst, often satisfactorily describes the diffusion processes at solid-solution interfaces. The internal diffusion of the ions in the exchanger phase is called particle diffusion. Most often, the particle is considered homogeneous, so that the different dif-fusional processes within the particle (pore diffusion, matrix diffusion) are represented by a single particle diffusion coefficient. Either particle or film diffusion may be the rate-determining step for the exchange process or both may contribute to the rate in intermediate cases. In general, film diffusion may dominate at early stages of exchange (A low in exchanger, B low in solution) when the concentration gradient in the particle is large (fast rate in particle). However, as the exchange proceeds further, the concentration gradients in the particle decrease and particle diffusion may become the rate-determining step.

In the exchange process, ions A and B move in opposite directions. Therefore, generally, the so-called interdiffusion coefficient (DA/B) must be used in eqns [13] and [14]. For particle diffusion:

Normally, the interdiffusion coefficient is not constant, but changes with the ionic composition (CA) of the exchanger. To calculate ion exchange rates in a given ion exchange system, the Nernst-Planck equations must be solved simultaneously for each diffusing species under boundary conditions specific to the system. In general, the resulting equations are nonlinear differential equations, which have analytical solutions only in some special cases. Such a case, for instance, is iso-topic ion exchange, for which a so-called self-diffusion constant can be used. Assuming also that the solution has indefinite volume - the concentration of ion in the solution remains essentially constant - it is obtained for the half-time (iy2) of the exchange reaction. In the case of particle diffusion:

where r0 is the radius of the exchanger particle and D is the particle diffusion coefficient. For film diffusion:

ion exchangers, cation diffusion coefficients are typically 2-5 orders of magnitude lower.

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