Application of the Stefan-Maxwell equations to diffusion in ion exchangers. 1. Theory

Abstract

Starting with the Stefan-Maxwell equations, general expressions for the ionic flux rates for binary exchange in ionsxchange resins have been developed. These equations have been shown to reduce to the Nernst-Planck equations exactly only as the concentration of either exchanging ion approaches unity. Furthermore, the single-ion diffusion coefficients used in the Nernst-Planck equations are shown to be certain combinations of the Stefan-Maxwell interaction coefficients. Most importantly, these combinations of the Stefan-Maxwell interaction coefficients are shown to reduce to the tracer diffusion coefficient of each exchanging ion, measured in ion-exchange resin completely in the competing ion form. As these limiting tracer ion diffusion coefficients may be very different from the usual pure selfdiffusion coefficients, this result may explain existing anomalies resulting from the use of the Nernst-Planck equations to describe diffusion in ion-exchange resins and related ion-exchange systems.The Stefan-Maxwell relations were used to analyze the ion exchange of Na' and Cs' in Dowex 50-X8. Tracer diffusion coefficients for both ions were measured as a function of composition. This allowed certain combinations of the Stefan-Maxwell interaction coefficients to be determined which in turn allowed limiting single ion tracer coefficients (Na' diffusing in Cs' form resin only and Cs' diffusing in Na' form resin only) to be estimated. Resultsshow that use of these limiting tracer coefficients in the Nernst-Planck equations gives much improved prediction of ion exchange than does the use of usual selfdiffusion coefficients. The improved results are attributed to the fact that the Stefan-Maxwell relations take into account important ion-ion interactions.

Entropy generation analysis of electrodialysis

“…The phenomenological coefficients in irreversible thermodynamics and the MS diffusion coefficients can in fact be theoretically related to one another [17][18][19]. It has been shown, however, that the MS coefficients are less dependent on composition (total dissolved solids, as well as the different ions present) than are the phenomenological coefficients [20].…”

Multicomponent ion transport in a mono- and bilayer cation-exchange membrane at high current density

“…The fixed ionic group concentration is defined with the logistic function as presented in Eq. (13) in which x 0 is zero and the slope of the curve is chosen 120. The slope of the curve defined the thickness of the transition state in the logistic function.…”

“…Also, virtually, no data exist in the literature that compares the performance of mono-and bilayer cation-selective membranes especially at high current densities. There are various methods to model ion transport in ion-exchange membranes, and these have been reviewed and modeled by several authors [11][12][13]. In our earlier paper [14], we developed a NernstPlanck model of multicomponent ion transport through a cation-exchange membrane for a monolayer membrane.…”

Final report on LDRD project: A phenomenological model for multicomponent transport with simultaneous electrochemical reactions in concentrated solutions

“…The theory has been applied to determine activity coefficients in dilute electrolyte solutions and the results are valid only in the limit of infinite dilution due to the neglect of shortrange interaction forces between ions in concentrated electrolyte solutions (Newman, 1991;Phto and Graham, 1986). In comparing the commonly used form of the electrolyte diffusion equations derived from the theory of irreversible thermodynamics with the Stefan-Maxwell Equations (4.8), Graham and Dranoff (1982) found that the Stefan-Maxwell diffusion coefficients were (a) less concentration dependent than the phenomenological coefficients of irreversible thermodynamics and (b) less dependent on the presence of other ions in the system. Pinto and Graham (1986) relate the StefanMaxwell coefficients to the viscosity of the solution and limiting ionic nobilities; they state that "the coefficients of the Stefan-Maxwell equations, unlike the coefficients of other flux equations, can be individually identified with physical phenomena occurring during the diffusion process.…”

“…Some advantages of using the Stefan-Maxwell formulation (Lightfoot et al, 1962;Riede and Schliinder, 1991, p. 611;Krishna and Wesselingh, 1997, p. 869;Pinto and Graham, 1986;Graham and Dranoff, 1982) where f12 (defined as the force required to give species 1 a speed of unity, i.e., force per unit speed) is the friction coefficient of a solute (designated species 1) in a solvent (designated species 2), k is the Boltzmann constant, and T is the temperature. where Rz is the radius of the solvent species; similarly when species 1 is the solvent and species 2 is the solute then the analogous formula (Wesselingh and Bollen, 1997) is:…”