Adsorbed-Ion Activity Coefficients

An experimentally variable Kv with respect to exchangeable Na+ load (Eq. 4.53) can be transformed to the thermodynamic exchange constant (Keq) as follows  

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For the direction and the stoichiometry of the exchange reaction used in this text (Reaction 4.38), the equations for estimating adsorbed-ion activity coefficients according to Argersinger et al. (1950) are... 

The formal similarity between adsorption and complexation reactions can be exploited to incorporate adsorbed species into the equilibrium speciation calculations described in Sections 2.4 and 3.1. To do this, a choice of adsorbent species components (SR r in Eq. 4.3) must be made and equilibrium constants for reactions with aqueous ions must be available. A model for computing adsorbed species activity coefficients must also be selected.8 Once these choices are made and the thermodynamic data are compiled, a speciation calculation proceeds by adding adsorbent species and adsorbed species (SR Mp(OH)yHxLq in Eq. 4.3) to the mole-balance equations for metals and ligands, and then following the steps described in Section 2.4 for aqueous species. For compatibility of the units of concentration, njw) in Eq. 4.2 is converted to an aqueous-phase concentration through division by the volume of aqueous solution. 

One conceptually simple approach is to express all binary ion exchange reactions as combinations of the special case of Eq. 4.3, in which the species Q is deleted and SR = SR, with the possibility that more than 1 mol of SR may combine with adsorptive ions to form the adsorbate species. This approach portrays ion adsorption formally as a complexation reaction and builds ion exchange reactions as combinations of these reactions.7 Since the adsorbate species may be formed from both cations and anions (cf. Eq. 4.3), ion exchange reactions involving charged complexes [e.g., CaCT(aq) as well as monatomic ions [e.g., Na (aq)J can be described. If the further simplification is made that adsorbate species activity coefficients do not depend on exchanger composition, then equilibrium spccialion calculations can be performed exactly as described... 

It is important to emphasize that N-ary ion exchange relationships which do not enjoy thermodynamic status must be examined case by case to determine whether they can be built up from binary exchange data. For example, there is no reason to expect any conditional equilibrium constant Kijc to remain invariant under changes in the composition of an ion exchange system from binary to ternary. As another example, one can pose (he question as to whether binary exchange data for adsorbed species activity coefficients can be used, in... 

The constant-capacitance model (Goldberg, 1992) assigns all adsorbed ions to inner-sphere surface complexes. Since this model also employs the constant ionic medium reference state for activity coefficients, the background electrolyte is not considered and, therefore, no diffuse-ion swarm appears in the model structure. Activity coefficients of surface species are assumed to sub-divide, as in the triplelayer model, but the charge-dependent part is a function of the overall valence of the surface complex (Zk in Table 9.8) and an inner potential at the colloid surface exp(Z F l,s// 7). Physical closure in the model is achieved with the surface charge-potential relation ... 

The total net surface charge density is proportional to net valence times concentration for each reactant adsorbent species and product adsorbent-adsorbate species in Eq. 4.15. Its presence in the equation for the activity coefficient reflects a model concept, that these charged surface species create an average electric field that influences ion adsorption. See, for example, G. Sposito, op. cit.7... 

A large body of experimental research exists concerning two-component ion exchangers, whose behavior is described by Eq. 5.1 or 5.2.5 These systems thus exhibit binary ion exchange equilibria. The central problem in applying chemical thermodynamics to them is to derive equations that permit the calculation of and the activity coefficients of the two adsorbate species.6 Several approaches have been taken to solve this problem, each of which reflects a particular notion of how exchanger composition data can be utilized most effectively to calculate thermodynamic quantities. 

The activity coefficients are affected from differences in size, the short-range lateral interactions among the adsorbed particles and in the case of adsorbed ions from the repulsive Coulombic interactions among these ions. Analytical expressions for the activity coefficients may be obtained either from statistical mechanical models or experiment. Thus, an approximate expression, arising from monolayer models under mean field approximation, is the following [12,13,21] ... 

Various chemical surface complexation models have been developed to describe potentiometric titration and metal adsorption data at the oxide—mineral solution interface. Surface complexation models provide molecular descriptions of metal adsorption using an equilibrium approach that defines surface species, chemical reactions, mass balances, and charge balances. Thermodynamic properties such as solid-phase activity coefficients and equilibrium constants are calculated mathematically. The major advancement of the chemical surface complexation models is consideration of charge on both the adsorbate metal ion and the adsorbent surface. In addition, these models can provide insight into the stoichiometry and reactivity of adsorbed species. Application of these models to reference oxide minerals has been extensive, but their use in describing ion adsorption by clay minerals, organic materials, and soils has been more limited. 

X is the concentration of the adsorbate A which is varied in order to study the effect of bulk concentration on the interfacial surface excess y is the concentration of the electrolyte whose activity is kept constant. The corresponding electrolyte concentration is kept reasonably high to provide electrical conductivity to the solution. For low values of x, the electrolyte concentration is constant. However, at higher concentrations of organic solute, the activity coefficient of the electrolyte varies with organic solute concentration. Thus, in general, the concentration of the electrolyte must also be varied in order to keep its activity constant. It is also important that the ions of the electrolyte not adsorb on the electrode to a significant extent. 

Van Cutsem P and Gillet C (1982) Activity coefficient and selectivity values of Cu, Zn and Ca ions adsorbed in the Nitella flexilis L. cell wall during triangular ion exchanges J Exp Bot 33 847-853. 

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