Soil Science Society of America Journal 65:1426-1429 (2001)
© 2001 Soil Science Society of America
DIVISION S-2 - NOTES
An algorithm for generating cation exchange isotherms from binary selectivity coefficients
S. A. Bloom* and
R. S. Mansell
Soil and Water Science Dep., Univ. of Florida, P.O. Box 110290, Gainesville, FL 32611-0290
* Corresponding author (sab{at}mail.ifas.ufl.edu)
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ABSTRACT
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An algorithm for simulating ion exchange batch mode experiments utilizing a known binary or ternary linear or nonlinear selectivity matrix and disequilibrium concentrations in exchange and solution phases was developed, and can be used to calculate initial equilibrium conditions for the initiation of numerical transport simulations, to generate binary isotherms (by repeated applications), and to separate ion-exchange from other solute reactions in soils. The algorithm is an iterative process utilizing a binary search to determine the sorbed phase concentrations defined by the selectivity values and dissolved phase concentrations while maintaining the correct total mass of all ions. It was developed to aid in the separation of ion-exchange from other retention processes when empirical determination of exchange isotherms are difficult or impossible.
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INTRODUCTION
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INVESTIGATIONS OF CATION TRANSPORT in soil of species undergoing simple ion exchange typically follows two approximately independent approaches, repetitive batch mode experiments to generate exchange isotherms, and transport experiments with columns to explore the interactions of the ions with the soil matrix during water flow (Bond and Phillips, 1990a,b; Mansell et al., 1993b). These pathways converge when selectivity coefficients derived from the isotherms and the physical constraints of the column experiments are used to model observed breakthrough curves and/or distribution profiles. In some cases, exchange isotherm data are not available or are confounded by other processes, thus requiring that exchange selectivity coefficients be taken from similar systems or be estimated by sensitivity analyses from all or part of the observed results (Downs, 1993; Mansell et al., 2001). In the presence of confounding processes (such as chemisorption, microbial degradation, precipitation, and steric effects), selectivity coefficients can be used to generate exchange isotherms utilizing a reverse order to the usual procedure. Such calculated isotherms can then be compared with observed isotherms to help elucidate the influence of the confounding processes. The difficulty with reversing the procedure is that an equilibrium solution for N number of ions must be simulated from a disequilibrium starting condition. Algorithms for accomplishing this task exist, and are incorporated into models such as LEACHC (Robbins et al., 1980), SOWACH (Dudley and Hanks, 1991), and UNSATCHEM (Simunek et al., 1996), though these models do not appear to accommodate nonlinear selectivities. We present here a simple algorithm for binary and ternary cases which provides a solution to the problem, can be readily implemented in any desired computer language, and is a straightforward application of the definitions of the controlling parameters. We then extend its use to isotherm generation and suggest its use as a mechanism for separating ion-exchange from other sorption processes.
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Algorithm Description
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The purpose of the numerical algorithm is to determine the equilibrium proportions of two or three ions in both exchange and sorbed phases, based on three sets of parameters: physical and chemical constants, a pre-defined selectivity matrix, and the mol charge for all relevant ions. The physical and chemical parameters are: CT, the normality of the soil solution (molc m-3); ST, the cation exchange capacity (molc Mg-1); 
, the water content (m3 m-3); and 
, the bulk density (Mg m-3).
A selectivity matrix is the set of all binary selectivity coefficients between all ion pairs; for example, a binary system requires a single coefficient, a ternary system requires three coefficients, and a system with N ions requires (N2 - N)/2 coefficients. The form of the coefficient is immaterial but can be either the physically-based Gaines-Thomas model or the empirical Rothmund-Kornfeld highly nonlinear model. The Gaines-Thomas coefficient (Gaines and Thomas, 1953) is defined as:
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where ri and rj are the valences of ions i and j, Ci and Cj are the equilibrium dissolved phase concentrations (molc m-3), Si and Sj are the equilibrium sorbed phase concentrations (molc Mg-1) for ions i and j, where the activities of ions in the solution phase are approximated by the concentrations. An empirical alterative is the Rothmund-Kornfeld exchange coefficient (Bond and Phillips, 1990a, b; Mansell et al., 1993a), and is defined as
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where, nij and kij are empirical constants generated by a nonlinear regression of the isotherm data. If nij is equal to unity, Kij is identical to the Gaines-Thomas coefficient.
Under the assumption that the initial dissolved phase concentration (Ci) and sorbed phase concentration (Si) are known for all three ions (though the phases may not be in equilibria), the total mass of any ion i, Mi, is defined as:
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We also assume that the system is closed, that is, mass neither enters or leaves the system, and ST is constant. These assumptions imply that CT is also constant. We therefore know that the limits of Ci are 0 to CT, and the limits of Si are 0 to ST for any ion i. It follows then that there is at least one set of values for Ci and Si for all three ions that simultaneous comply with the requirements that:
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 | (5) |
where C'i, S'i, and M'i represent the equilibrium dissolved phase concentration, sorbed phase concentration, and total mass for ion i. These constraints are equivalent to stating that, regardless of the distribution of the three ions between dissolved and sorbed phases, the mass of each ion, and the sum of the concentrations in dissolved and sorbed phases, are constant.
The set of C'i and S'i values can be found by a nested binary search procedure. C'1 and S'1 are estimated in the outer loop, C'2 and S'2 are estimated in the inner loop, and C'3 and S'3 are determined by difference (using Eq. [4]). Since the limits of C1 are 0 to CT, we initially set C'1 to (CT + 0)/2. This implies that the limits for C2 are 0 to CT/2, and we initially set C'2 to (CT/2 + 0)/2. We can then set C'3 to CT - C'1 - C'2. Using the standard procedure for projecting dissolved phase concentrations into sorb phase utilizing the defined selectivity matrix (Valocchi et al., 1981; Mansell et al., 1993a,b), we can generate estimates of S'1, S'2, and S'3 based on the given physical constants and selectivity matrix. Using Eq. [5], we can then calculate M'2 and compare that value to the known mass, M2. If the M'2 is larger than M2, then the value of C'2 was too large and the correct answer lies between the current lower limit (0) and the current guess (CT/2 + 0)/2, so the upper limit is set to the current guess. If the M'2 is smaller than M2, then the value of C'2 was too small and the correct answer lies between the current guess (CT/2 + 0)/2 and the current upper limit (CT/2), so the lower limit is set to the current guess. The process continues until the difference between upper and lower limits for ion 2 reaches a predefined termination criterion (here equal to 10-15). When the inner loop thus terminates, an identical procedure is then applied in the outer loop to refine the estimate of C'1. By the end of the total procedure, the masses of N-1 of the ions will have converged to the actual masses (with the Nth ion effectively converged, since CT and ST are constants) and the dissolved and sorbed phase will be in equilibrium, as defined by the selectivity matrix.
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Sample Simulations
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The algorithm can be demonstrated using a binary isotherm from a ternary ion exchange system (Mansell et al., 1993b), consisting of a K+, Na+, and Ca++ system in a Brucedale subsoil from Australia. The Rothmund-Kornfeld form of Kij was used for this soil because of steric effects observed for ternary ion exchange. The normality was 200 molc m-3, the cation exchange capacity was 228 molc Mg-1, the water content was 0.53 m3 m-3, and the bulk density was 0.97 Mg m-3; thus, the total mol charge in the system was 327.16 molc. The soil-to-solution ratio (/) here is 1.8 m3 Mg-1. The coefficients for the Rothmund-Kornfeld model are given in Table 1 and were derived from the observed isotherm using a nonlinear fitting procedure. The observed isotherm for the K+ to Ca++ case is shown in Fig. 1
. One hundred equilibria were calculated using the described algorithm, utilizing systematically increasing proportions of K+ from 1 to 99% of total mass, and are presented as the solid line in Fig. 1. The average error between the equilibria predicted by the algorithm and the calculated values of the isotherm based on the given Rothmund-Kornfeld coefficients was 0.04% (standard deviation of 0.188%).
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Fig. 1. K+ to Ca++ isotherm with observed (symbols) and predicted equilibria (line) generated by the described algorithm.
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An example of a ternary case is given in Table 2. The initial conditions posit a Na+ saturated sorbed phase with the dissolved phase equally divided between the other two cations. The equilibrium solution shown was generated in 0.82 s, taking 56 iterations in the outer loop and 3012 iterations in the inner loop with a 0.12% error (in total mol charge) (using a 600 MHz Pentium III with an implementation coded in Pascal).
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Discussion
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The algorithm presented here has two possible applications. It can be used to set initial equilibrium (boundary) conditions in numerical simulations of transport when such conditions are not available as observed parameters. The algorithm can also be used repeatedly to generate isotherms predicated on a given selectivity matrix which can, in turn, be used to help separate ion-exchange from other sorption reactions in soils since empirical determination of exchange isotherms are extremely difficult when retention processes other than cation exchange are active.
Two obvious cases in which an investigator might wish to generate isotherms from a set of selectivity coefficients rather than the reverse are when no isotherm exists or when an existing isotherm is actually a summation of ion exchange and some other process. If there is a confounding process that removes mass (or mol charge) from the dissolved phase without that mass appearing in the exchange phase, for example, decay, biological degradation, or chemisorption, then the values of Si and the magnitude of Kij will be overestimated. A more subtle case with similar implications is exchange-mediated chemisorption of hydrazinium cations, in which exchange sites in close proximity to chemisorption sites act in a quasi-catalytic fashion to facilitate the chemisorption of ions on the binding sites while being deactivated in the process, thus reducing the effective cation exchange capacity (Mansell et al., 2001). In such cases, simulation of the exchange using the erroneous numerically-magnified selectivity coefficients will predict breakthrough curves at odds with reality.
Conversely, if a reasonable hypothesis as to the nature of the confounding process is available, then an ion-exchange model can be modified to explicitly include the effects of that process and simulations can be run to allow the estimation of selectivity and the critical parameters controlling the confounding process. If this is done, then the algorithm presented here can be used to project an isotherm influenced by ion exchange only in contrast to an isotherm influenced by exchange and the confounding process.
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Appendix
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Pseudocode for the equilibrium solution algorithm. The following code assumes that certain variables such as (thetaS), (BulkDensity), the selectivity matrix (Kij), normality (CT), and the original masses of each ion, Mi (total mass, i) are global variables:
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NOTES
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Florida Agric. Expt. Stn. Journal Series No.R-07882.
Received for publication October 27, 2000.
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REFERENCES
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- Bond, W.J., and I.R. Phillips. 1990a. Cation exchange isotherms obtained with batch and miscible displacement techniques. Soil Sci. Soc. Am. J. 54:722–728.[Abstract/Free Full Text]
- Bond, W.J., and I.R. Phillips. 1990b. Approximate solutions for cation transport during unsteady, unsaturated water flow. Water Resour. Res. 26:2195–2205.
- Downs, W.C. 1993. Fate and transport of hydrazine through columns of saturated sandy soil. Ph.D. Dissertation, University of Florida, Gainesville, FL.
- Dudley, L.M., and R.J. Hanks. 1991. Model SOWACH: Soil-Plant-Atmosphere-Salinity Model (User's Manual). Rep. 140. Utah Ag. Exp. Sta., Logan, UT.
- Gaines, G.L., Jr., and H.C. Thomas. 1953. Adsorption studies on clay minerals, II. A formulation of the thermodynamics of exchange adsorption. J. Chem. Phys. 21:714–718.
- Mansell, R.S., S.A. Bloom, and W.J. Bond. 1993a. A tool for evaluating a need for variable selectivities in cation transport in soil. Water Resour. Res. 29:1855–1858.
- Mansell, R.S., S.A. Bloom, and W.C. Downs. 2001. A transport model with coupled ternary exchange and chemisorption retention for hydrazinium cations. J. Environ. Qual. 30:1540–1548.[Abstract/Free Full Text]
- Mansell, R.S., W.J. Bond, and S.A. Bloom. 1993b. Simulating cation transport during water flow in soil: Two approaches. Soil Sci. Soc. Am. J. 57:3–9.
- Robbins. C.W., J.J. Jurinak, and R.J. Wagenet. 1980. Calculating cation exchange in a salt transport model. Soil Sci. Soc. Am. J. 44: 1195–1199.[Abstract/Free Full Text]
- Simunek, J., D.L. Suarez, and M. Sejna. 1996. The UNSATCHEM Software Package for Simulating One-Dimensional Variably Saturated Water Flow, Heat Transport, Carbon Dioxide Production and Transport, and Solute Transport with Major Ion Equilibrium and Kinetic Chemistry, Version 2.0. Research Report No. 141, U.S. Salinity Laboratory, USDA-ARS, Riverside, CA.
- Valocchi, A.J., R.L. Street, and P.V. Roberts. 1981. Transport of ion-exchanging solutes in groundwater: chromatographic theory and field simulation. Water Resour. Res. 17:1518–1527.
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ABSTRACT
INTRODUCTION
