The constant capacitance model

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

Therefore, from a knowledge of the intrinsic acidity constants, Ns and the value of kj, the distribution of surface species, xs, and can be calculated as a function of pH. The results of such a calculation for a goethite suspension are shown in Fig. 5.5 using data from Lumsdon and Evans (1994). 

Complexation of cations and anions is achieved by introducing mass action expressions for the formation of the cation or anion surface complex into the mass balance for the total number of surface complexation sites and, if the new surface species is charged, into the charge balance equation, e.g. metal surface complexes, Cd2+. 

Using the equations and input parameters in Table 5.8 the results of a calculation involving phosphate adsorption on a goethite are shown in Fig. 5.6. 

The constant capacitance modeP is a molecular description of surface complexation reactions involving the inorganic hydroxyl group. The chemical basis of the model can be developed from its three principal assumptions concerning the interfacial region  

Inorganic hydroxyl groups form only iniier-sphere surface complexes with adsorbed species. 

The chemical reactions that describe surface complexation are 

When the total particle charge is small in absolute magnitude, it is proportional to the inner potential at the particle surface  

THE NET PROTON CHARGE. In the Constant capacitance model, the surface complexation reactions that involve or OH alone are special cases of Eq. 5.37a  

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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 ... 

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. 

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). 

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 ... 

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). 

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...

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

The constant-capacitance model assumes that the double layer on the solid-liquid phase boundary can be regarded as a parallel-plate capacitor (Fig. 14b). 

Fig. 14 Idealized distribution of the electrical potential in the vicinity of hydrated oxide surfaces after the (a) diffuse-layer model (b) the constant-capacitance model (c) triple-layer model (after Drever 1997).

Since charged particles involve all these processes, including the formation of edge charges (Equations 2.3-2.5), first, the electric properties of interfaces have to be determined. A simple way to do so is the application of a support electrolyte in high concentration. The electric double layer, in this case, behaves as a plane and, as a first approach, the Helmholtz model, that is, the constant capacitance model, can be used (Chapter 1, Section 1.3.2.1.1, Table 1.7). It is important to note that the support electrolyte has to be inert. A suitable support electrolyte (such as sodium perchlorate) does not form complexes (e.g., with chloride ions, Section 2.3) and does not cause the degradation of montmorillonite (e.g., potassium fixation in the crystal cavities). In this case, however, cations of the support electrolyte, usually sodium ions, can also neutralize the layer charges ... 

To study nonexchangeable acidity, active and exchangeable acidity has to be blocked (Hargrove and Thomas 1982 Thomas and Hargrove 1984). In potentio-metric titration, the main source of exchangeable acidity is the permanent charge of aluminosilicates, which can be neutralized by a support electrolyte in high concentration (e.g., 0.1 mol/dm3 sodium salt) (Chapter 2, Section 2.4.3). The application of the support electrolyte makes it possible to use the constant capacitance model, too. 

Goldberg, S., S. M. Lesch, D. L. Suarez, and N. T. Basta. 2005. Predicting arsenate adsorption by soils using soil chemical parameters in the constant capacitance model. Soil Sci. Soc. Am. J. 69 1389-1398. 

A more mechanistic and robust depiction of reversible metal adsorption is provided by SCMs that account explicitly for competitive speciation reactions using an equilibrium thermodynamic framework. Examples of SCMs in current use include the constant capacitance model (CCM), the diffuse double-layer model (DDLM), and the triple-layer model (TLM) (Stumm Morgan, 1996 Koretsky, 2000). Each of these models envisages... 

Several SCM s have been described in the literature. The more commonly used models include the Constant Capacitance Model (Schindler and Stumm, 1987), the Diffuse Double Layer Model (Stumm et al., 1970) and the Triple Layer Model (Davis et al., 1978 Yates et al, 1974). All are based on electric double layer theory but differ in their geometric description of the oxide-water interface and the treatment of the electrostatic interactions. 

Because the various SCM s have different formulations for treating adsorption reactions and the electrostatic terms, parameters fit to one model may not he applicable to other models (Morel et al, 1981). For example, Gao and Mucci (2001) determined different Log K s for As(V) adsorption by goethite when the data were fit to the Constant Capacitance Model, the Basic Stem Model, and the Triple Layer Model. 

Although the first three articles in the Croatica Chemica Acta series provided a conceptual framework for the constant capacitance model, a number of theoretical loose ends remained. One was the vexing problem of how best to estimate model parameters from experimental data. This surprisingly complicated issue was reviewed critically by Westall and Hohl (13) and, more recently, by Hayes et al. (18) and Goldberg (19), so it will not be discussed in this review. 

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