Exchanger activity coefficient models

The clay ion-exchange model assumes that the interactions of the various cations in any one clay type can be generalized and that the amount of exchange will be determined by the empirically determined cation-exchange capacity (CEC) of the clays in the injection zone. The aqueous-phase activity coefficients of the cations can be determined from a distribution-of-species code. The clay-phase activity coefficients are derived by assuming that the clay phase behaves as a regular solution and by applying conventional solution theory to the experimental equilibrium data in the literature.1 2 3... 

This influence of the valence and activity coefficients of the displacer salt on the retention behavior of polypeptides and proteins can be anticipated from theoretical treatments of the ion-exchange chromatographic separation of proteins. According to the nonmechanistic stoichiometric model of protein retention behavior in HP-IEX80,82-85 the influence of a divalent cation salt such as CaCl2 on the retention behavior of a protein in HP-IEC can be evaluated in terms of the following relationships ... 

The other interpretation of surface activity coefficients supposes that the interactions between the pairs of the exchanging and exchanged ions (Me, - Mex, Me2-Me2, and Mex - Me2, respectively) are not equivalent, so the surface activity is necessarily the function of surface composition (Hogfeldt 1984, 1988 Konya et al. 1989). This is the Hogfeldt three-parameter model, in which the selectivity coefficient depends on the surface molar factions of the ions and their interactions, as follows ... 

The rational activity coefficients cannot be evaluated in any simple manner. Following the model of Truesdell and Christ (16), a regular solution approach to the problem can lead to expressions for the rational activity coefficients. If the exchange sites have the same charge and approximately the same size, then a symmetrical solid solution will be formed where the rational activity coefficients for the two components are given by ... 

In the above expressions, F2 (/) is the preferential binding parameter for site /, yj and 73 are the activity coefficients of water and cosolvent (in a molar fraction scale) in a protein-free mixed solvent, and is the equilibrium constant for the exchange equilibrium (eq 26) on site i. For an ideal mixed solvent, Ki = K-. In the Schellman model, it is necessary to know how the protein surface is subdivided into various kinds of sites. Such information is not available for real protein solutions. However, when some simplifications are made, the Schellman model can provide some information regarding the effect of various cosolvents on the protein stability. 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

The Nernst-Planck model is based on limiting laws for ideal systems. It accounts only for diffusion and electric transference of ions, not for electroosmotic solvent transfer in the ion-exchanger phase, swelling or shrinking of the ion-exchange material, variations of activity coefficients and diffu-sivities, and possible slow structural relaxation of the exchanger matrix. It also postulates the existence of individual diffusion coefficients for ions. 

The degree of success of the ideal ion mixture model of ion exchange, which allows for nonideal behavior by introducing activity coefficients for individual ions in much the same way that the activities of ions in aqueous solutions are adjusted for nonideality. 

Currently available thermodynamic and kinetic data bases are incomplete to support quantitative modeling of many corrosion systems, particularly those where predictions of behavior under extreme conditions or over extended periods of time are desired. Because the unavailability of data limits the use of models, a critical need exists to upgrade and expand the sources of information on the thermodynamic properties of chemical species, exchange current densities, activity coefficients, rate constants, diffusion coefficients, and transport numbers, particularly where concentrated electrolytes under extreme conditions are involved. Many of these data are obtained in disciplines that traditionally have been on the periphery of corrosion science, so it will be necessary to encourage interdisciplinary collaboration to meet the need. 

It has been well documented by the research performed in this laboratory that use of the Gibbs-Donnan model for the interpretation of ion-exchange equilibria provides a most useful avenue for the accurate anticipation of counter ion distribution in charged polymeric systems. The activity coefficient ratio of competing ions in the polymeric phase which is the essential parameter to be assessed (see Equation 1) for successful use of this model has, as we have pointed out, so far been unaccessible by a straight-forward computation. This deficiency of our... 

Solvents. Solvent transfer parameters have been used from time to time in diagnosis of mechanisms of organic reactions. Now their first use in an inorganic system has been discussed. Comparisons between kinetic results and solvent transfer activity coefficients for solvolysis of [Cr(NCS)e] in DMSO, DMF, and dimethylacetamide have been considered in terms of a dissociative mechanism. The DMSO results can be accommodated by a dissociative model for the transition state, but those in DMF and dimethylacetamide fit less well. It is interesting to compare these conclusions with, for instance, the dissociative mechanism proposed by the same authors for anation of the [Cr(DMSO)e] + and [Cr(DMF)6] + cations, and the associative mechanism suggested on the basis of the determined activation volume for exchange of DMSO with [Cr(DMSO)e +. ... 

Thus, the mathematic models for reactor design are also classified into continuous heat exchange bed and adiabatic one. Usually, the design of reactor adopts one-dimension quasi-homogeneous model which considers that when reactive gas passes the catalyst bed like a plug-flow, there exist no radial and axial return mixture, and microkinetics can be treated in intrinsic kinetics multiplied by an effective factor that involves the effects of transfer processes, and by an active coefficient that involves the effects of reduction, poisoning and declining etc. Macrokinetics can be... 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

Figure 28. Svensson s macrohomogeneous model for the i— 1/characteristics of a porous mixed-conducting electrode, (a) The reduction mechanism assuming that both surface and bulk diffusion are active and that direct exchange of oxygen vacancies between the mixed conductor and the electrolyte may occur, (b) Tafel plot of the predicted steady-state i— V characteristics as a function of the bulk oxygen vacancy diffusion coefficient. (Reprinted with permission from ref 186. Copyright 1998 Electrochemical Society, Inc.)...

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