Models constant capacitance

Goldberg, S. Reanalysis of boron adsorption on soils and soil minerals using the constant capacitance model. Soil Sci Soc Am J 1999 63 823-829. 

Alternatively, in the literature, the constant capacitance model and the Stern model were used to describe the dependence of the surface charge density on the surface potential. In the constant capacitance model, the surface charge is defined as ... 

Vaughan PJ, Suarez DL. Constant capacitance model computation of boron speciation for varying soil water content. Vadose Zone J. 2003 2 253-258. 

The diffuse double layer model is used to correct for Coulombic effects. The constant capacitance model depends on the input of a capacitance but the result obtained is not very different. 

In addition to the diffuse double layer and the constant capacitance model dis-... 

In Fig. C microscopic acidity constants of the reaction AlOHg =AIOH + H+ for y-AI203 are plotted as a function of AIOH. The data are for 0.1 M NaCICV This figure illustrates (within experimental precision) the conformity of the proton titration data to the constant capacitance model. Calculate the capacitance. 

The term F2/CsRT is obtained from the constant capacitance model (Chapter 3.7). Fig. 4.6 gives a plot of the linear free energy relation between the rate constants for water exchange and the intrinsic adsorption rate constant, kads. 

Diprotic Surface Groups. Most of the recent research on surface hydrolysis reactions has been interpreted in terms of the diprotic surface hydrolysis model with either the triple layer model or the constant capacitance model of the electric double layer. The example presented here is cast in terms of the constant capacitance model, but the conclusions which are drawn apply for the triple layer model as well. 

An example of the use of this method with the constant capacitance model on the data for TiC>2 in 0.1 M KNO is illustrated in Figure 6. It appears from the figure that the problem is perfectly well determined, and that unique values of Ka and Ka2 can be determined. However, as is shown below, the values of Ka and Ka2 determined by this method are biased to fulfill the approximations made in processing the data (i) on the acidic branch, nx+, nx nx-, which yields a small value for Ka2, and (ii) on the basic branch, nx-, nx nx+, which yields a large value of Ka. Thus the approximation used to find values for Qa and Qa2 leads to values of Ka and Ka2 consistent with the approximation of a large domain of predominance of the XOH group. This constraint arose out of the need for mathematical simplicity, not out of any physical considerations. 

Figure 7. Covariability between values of C and Kd yielding best fit of diprotic surface hydrolysis model with constant capacitance model to titration data for TiC>2 in 0.1 M KNOj (Figure 5). The line is consistent with Equation 29. The crosses represent values of C and log found from a nonlinear least squares (NLLS) fit of the model to the data, with the value of capacitance imposed in all cases the fit was quite acceptable. The values of and C found by Method I (Figure 6) also fall near the line consistent with Equation 29. The agreement between these results supports the use of the linearized model (Equation 29) for developing an intuitive feel for surface reactions.

Equilibrium Calculations. The computer program SURFEQL (29) was used to calculate the equilibrium distrubution of chemical-species. The constant capacitance model (30, 1) was used for the surface equilibria calculations. The equilibrium constants used in these calculations are given in Davies (26). 

As shown in Figure 1, the adsorption of Mn(II) on y-FeOOH can be successfully described using a constant capacitance model. In these calculations the hydrolysed surface complex =FeO-Mn-OH was not considered. The reason for not considering both the bidentate (sS0)2Mn and hydrolysed surface species is that both have virtually the same pH dependence, so it is impossible using the available data to make anything other than an arbitrary choice about the relative proportions of these two species. Based on the model calculations, in the pH range 8-9, the predominant Mn(II) species on the y-FeOOH surface is the bidentate surface complex or the hydrolysed surface complex. 

Figure 1. Mn(II) adsorption as a function of pH. The solid lines are calculated using the constant capacitance model.

They used the constant capacitance model (surface capacitance of 18F/m2) to fit the following three sorption reactions to observed absorption edge data ... 

CASH CBM CBO CBPC CC CCB CCM CCP CDB CEC CFBC CFC CFR CMM COP CSH CT Calcium aluminosilicate hydrate Coal bed methane Carbon burn-out Chemically-bonded phosphate ceramics Carbonate carbon Coal combustion byproducts Constant capacitance model Coal combustion product Citrate-dithionate-bicarbonate Cation exchange capacity Circulating fluidized bed combustion Chlorofluorocarbon Cumulative fraction Coal mine methane Coefficient of performance Calcium silicate hydrate Collision theory... 

The elegance of the surface complexation approch lies in the fact that it can be incorporated into the thermodynamic speciation models used for soluble complexes. Consequently many of the computer models, e.g. SOILCHEM, HYDRAQL, MINTEQA2 and ECOSAT, include several different SCMs. Some commonly used SCMs are the diffuse-double-layer model, DDLM (Huang and Stumm, 1973 Dzombak and Morel, 1990), the constant capacitance model, CCM (Stumm et al., 1970 1976 1980 Schindler et al., 1976), the triple-layer model, TLM (Davis etal., 1978 Davis and Leckie, 1978,1980 Hayes and Leckie, 1987 Hayes et al., 1988) and the 1 pK basic Stern model (Bolt and Van Riemsdijk, 1982 Van Riemsdijk et al., 1986 1987). 

The constant capacitance model. Problems involving surface species can be described using the format used for the soluble species in Section 5.2.5. 

Goldberg, S. and Traina, S.J. (1987) Chemical modelling of anion competition on oxides using the constant capacitance model-mixed-ligand approach. Soil Sci. Soc. Am.J., 51,... 

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 characteristic features of parameter estimation in a molecular model of adsorption are illustrated in Table 9.9, taking the simple example of the constant-capacitance model as applied to the acid-base reactions on a hydroxylated mineral surface. (It is instructive to work out the correspondence between equation (9.2) and the two reactions in Table 9.9.) Given the assumption of an average surface hydroxyl, there are just two chemical reactions involved (the background electrolyte is not considered). The constraint equations prescribe mass and charge balance (in terms of mole fractions, x) and two complex stability constants. Parameter estimation then requires the determination of the two equilibrium constants and the capacitance density simultaneously from experimental data on the species mole fractions as functions of pH. 

Table 9.9 Surface acid-base reactions in the constant-capacitance model...

The site binding model based on reactions (1), (2), (14) and (15), often called surface complexation model (SCM), was, beside the simple site binding models (for example two layer model or constant capacitance model) readily applied to a description of the edl on the metal oxide-electrolyte solution interface. Reactions (14) and (15) describe the adsorption of so-called back-... 

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