Determination of Rate Coefficients for Potassium-Calcium Exchange on Vermiculite using a Stirred-Flow Chamber

A stirred-flow technique has been widely used to study the rate of ion adsorption on clays and soils. However, no mathematical analysis of chemical kinetics combined with mass transport has been used to derive rate coefficients for ion exchange phenomena. Accordingly, the objectives of this study were to: (i) develop a mathematical model of the kinetics of an elementary cation-exchange reaction combined with transport processes in a stirred-flow chamber and (ii) determine rate coefficients based on this model for K-Ca exchange on a vermiculite. The kinetics of exchange were described by a second-order mechanistic rate law from which the equilibrium exchange equation was derived. The second-order mechanistic rate law combined with an equation of mass balance describing transport were solved numerically. The equilibrium K-Ca exchange data were described by the Gaines and Thomas approach with a therrnodynamic exchange equilibrium constant Ke, = 754.6 L mol". The values of the apparent adsorption rate coefficient (fc,) ranged from 84 to 185 L mol~ min~'? as influent rate increased from 0.3 to 2.0 mL min". The calculated K-Ca exchange half-life (tm) ranged from 0.5 to 1.5 min, in agreement with published data based on batch methods. A ARRAY of techniques can be employed to measure the rates of soil chemical reactions (Sparks, 1989). These techniques can be categorized into chemical relaxation techniques that monitor reactions rates on millisecond time scales and batch and flow methods that can be used to measure slower reactions (>15 s). The stirredflow technique that was developed by Carski and Sparks (1985) is a combination batch and flow method, and it has been extensively used to measure adsorptiondesorption reactions on soils and soil components. Seyfried et al. (1989) showed that perfect mixing, i.e., the concentration of the adsorptive hi the chamber equals the effluent concentration occurring in the stirred-flow chamber. Bar-Tal et al. (1990) used a stopped-flow test to determine if tune-dependent reactions could be measured with the stirred-flow method and employed varying influent concentrations and flow rates to elucidate kinetic models. Eick et al. (1990), using the tests of Bar-Tal et al. (1990), showed that K-Ca exchange on montmorillonite was too rapid to be measured with the stirred-flow A. Bar-Tal and S. Feigenbaum, Inst. of Soils and Water, and S. Fishman, Dep. of Statistics, Agricultural Research Organization, The Volcani Center, Bet Dagan, 50250, Israel; and M.J. Eick and D.L. Sparks, Dep. of Plant and Soil Sciences, Univ. of Delaware, Newark, DE 19717-1303. Contribution from the Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel, no. 1260-E, 1993 series, and the Univ. of Delaware. Received 29 Jan. 1994. "Corresponding author (vwarie® volcani). Published in Soil Sci. Soc. Am. J. 59:760-765 (1995). technique, while exchange on vermiculite was kinetically controlled and could be measured. A number of kinetic models have been used to describe tune-dependent soil chemical reactions; a simple firstorder equation has been the most widely used model. However, this model is most appropriate for reactions from equilibrium where reverse reactions are not occurring. Sivasubramaniam and Talibudeen (1972) noted that the assumption of a first-order reaction for K desorption is valid as long as the concentration of the displaced K is negligible relative to that of the extracting ion. They also showed that K absorption on soil deviated from a linear first-order reaction when the ratio of K to Al exceeded a certain value. Moreover, a simple first-order model is not appropriate to describe reactions where both chemical kinetics and mass transport processes occur simultaneously and cannot be easily separated. Such conditions are usually the case when one is measuring reaction rates in soils and soil components. In such systems, the best model would be one that combines both chemical kinetics and mass transport. Bar-Tal et al. (1990), using a stirred-flow technique, showed that an instantaneous equilibrium ion exchange reaction combined with mass transport could be integrated analytically such that a pseudo-first-order model was obtained. Higher ordered kinetic equations are often required when two or more reactants affect the reaction rate. Consider the reversible binary K-Ca exchange reaction: Ca-X2 k. 2K = 2(K-X) Ca [1] where X is the charged surface, and ka and fcd are adsorption and desorption rate coefficients, respectively. If one assumes that the cation-exchange reaction in Eq. [1] is an elementary reaction and diffusion is not the limiting factor, then the reaction rate, r, can be defined as follows (Tang and Sparks, 1993): r = dCK/2d/ = -fa [K-X] CCa + fa [Ca-X]CK [2] where C and [ ] denote concentration in the solution and exchanger phases, respectively, t is the time, and Ck and Cca are the K and Ca concentrations in the chamber solution, respectively. Recently Eq. [2] has been employed hi a study of cation-exchange kinetics on montmorillonite using pressure-jump relaxation (Tang and Sparks , 1 993) . This equation has not been used in analysis of data obtained with the stirred-flow technique that requires a mathematical expression for the mass transport (Bar-Tal et al., 1990). The objectives of this study were to: (i) develop a