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1984_NkediKizza_Two site equivalence

1984_NkediKizza_Two site equivalence - WATER RESOURCES...

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WATER RESOURCES RESEARCH, VOL. 20, NO. 8, PAGES 1123-1130, AUGUST 1984 On the Equivalence of Two Conceptual Models for Describing Ion Exchange During Transport Through an Aggregated Oxisol P. NKEDI-KIZZA, 1'2 J. W. BIGGAR, ! H. M. SELIM, 3 M. TH. VAN GENUCHTEN, 4 P. J. WIERENGA, 5 J. M. DAVIDSON, 6 AND D. R. NIELSEN ! Breakthrough curves (BTCs) of thecation '•5Ca2+, an anion 36C1-, and3H20 were measured during miscible displacement through water-saturated soil columns packed with aggregates of an Oxisol. Two conceptual models were used to simulate the observed asymmetry and tailingin the BTCs caused by an apparent nonequilibrium situation in the porous medium. In both modelsthe exchange process on one type of site was assumed to be instantaneous whilethe rate of isotopic exchange on another type of site was assumed to be either a diffusion-controlled process (model 1) or a first-order reversible kinetic process (model 2). Isotopic exchange in both models was described with a linear isotherm. It is shown that the two models are mathematically equivalent with respect to the derived BTCs. INTRODUCTION Kinetic studies with the batch equilibration method have suggested that ion exchange is an instantaneous process [Hiss- ink, 1924; Borland and Reitemeier, 1950]. This should be ex- pected since ion exchange is a physical stochiometric process with AH values of about 2 kcal mol-•, resembling dipole- dipole interactions [Helfferich, 1962]. The interdiffusion of counter ions from the solution to the exchanger and vice versa, called ion exchange, is probably the main step that limits instantaneous equilibrium. This suggests that diffusion to and from the exchangesites, rather than the actual ex- change reaction itself, determines the apparent exchange rate [Helfferich, 1962]. Because of relatively small particles nor- mally used in batch ion-exchange kinetic studies, and because of vigorous shaking, diffusion often can be eliminated as a rate-limiting step [Boyd et al., 1947; Helfferich, 1962]. How- ever, in a system with flowing water, and especially in aggre- gated soils, the ion-exchange rate can be limited by the rate at which the ions are transported by diffusion to the exchange sites. Several researchers [Smith, 1968; Nkedi-Kizza et al., 1982] have shown that even if the point reaction is instanta- neous, the global or space kinetics can be influenced by such variables as fluid velocity (which is related to residence time) or aggregate size (which determines the diffusion path length). Boyd et al. [1947] recognized two potential rate-limiting stepsfor ion-exchange kinetics: (1) intradiffusion of counter ions within the ion exchanger (particlediffusion) and (2) in- terdiffusion of counter ions in the adherent films (film diffu- sion). In practice, either step can be rate limiting, and for some cases the rate may be affected by both steps [Boyd et al., 1947; Hiester and Vermeulen, 1948 ; Lapidus andAroundson, 1952].
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