Surface complexation models equation

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

The monoprotic model is appealing since it is very simple, realistic, and based on one less adjustable parameter than the diprotic model. The value of the parameter Ka can be found directly from the H+ concentration in the bulk of solution at Oq =0, since Ka = a + at this condition, according to Equation 6. Since the surface complexation models are already recognized as being underdetermined, any physically realistic model with fewer adjustable parameters is welcomed. 

The adsorption data is often fitted to an adsorption isotherm equation. Two of the most widely used are the Langmuir and the Freundlich equations. These are useful for summarizing adsorption data and for comparison purposes. They may enable limited predictions of adsorption behaviour under conditions other than those of the actual experiment to be made, but they provide no information about the mechanism of adsorption nor the speciation of the surface complexes. More information is available from the various surface complexation models that have been developed in recent years. These models represent adsorption in terms of interaction of the adsorbate with the surface OH groups of the adsorbent oxide (see Chap. 10) and can describe the location of the adsorbed species in the electrical double layer. 

The surface complexation models differ from the above equations in that they explicitly define the chemical reaction involved in the adsorption process. A crucial feature of these models is the treatment of adsorption as an interaction of adsorbing species with well defined coordination sites (the surface OH groups) in a manner analogous to complexation reactions in solution. A further feature of these models is that the chemical free energy of adsorption predominates with electrostatic effects having but a secondary role. 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

The above surface complexation models enable adsorption to be related to such parameters as the number of reactive sites available on the oxide surface, the intrinsic, ionization constants for each type of surface site (see Chap. 10), the capacitance and the binding constants for the adsorbed species. They, therefore, produce adsorption isotherms with a sounder physical basis than do empirical equations such as the Freundlich equation. However, owing to differences in the choice of adjustable... 

As seen from Equations 1.54-1.56, the intrinsic stability constants of surface reactions are dependent on two factors a chemical and an electric contribution. The chemical contribution is taken into consideration by the mass balance the electric contribution is treated by the charge balance. There are several surface complexation models that mainly differ in the description of the electric double layer that is used to calculate the surface potential, which is done by different double-layer models. These models have been mentioned previously in this chapter. Since, however, the terminology usually used in electrochemistry, colloid chemistry and, especially, in the discussions of surface complexation models is different, they are repeated again ... 

The thermodynamic equilibrium models, including surface complexation models, require the solution of a complex mathematical equation system. For this reason, many computer programs (e.g., CHEAQC, CHEMEQL, CHESS, EQ3/6, F1TEQL, Geochemist s Workbench, H ARPHRQ, JESS, MINTEQ and its versions, NETPATH, PHREEQC, PHRQPITZ, WHAM, etc.) have been developed to calculate the concentration and activity of chemical species, estimate the type and amount of minerals formed or dissolved, and the type and amount of sorbed complexes. 

Using surface complexation models is another way to quantitatively describe the ion-exchanges processes. Surface complexation models also apply the law of mass action, combined with surface electric work (Table 1.7). However, there are some theoretical problems with the calculations, namely, that the equations of intrinsic stability use concentrations instead of activities, without discussing... 

A significant problem in surface complexation models is the definition of adsorption sites, The total number of proton-exchangeable sites can be determined by rapid tritium exchange with the oxide surface (25). Although surface equilibria are usually written in terms of one surface site, e.g. Equations 5, 6, 8, 9, adsorption isotherms for many ions show that the number of molecules adsorbed at maximum surface coverage (fmax) is less than the total number of surface sites. For example, uptake of Se(VI) and Cr(VI) ions on Fe(0H)3(am) at T ax 1/3 and 1/4 the total... 

A)jS, whether sampled from probability distribution functions or calculated by regression equations or surface-complexation models, can be used in many contaminant transport models. Alternate forms of the retardation factor equation that use a (Equation (3)) and are appropriate for porous media, fractured porous media, or discrete fractures have been used to calculate contaminant velocity and discharge (e.g., Erickson, 1983 Neretnieks and Rasmuson, 1984). An alternative approach couples chemical speciation calculations... 

Figure 9.19. The diffuse double layer, (a) Diffuseness results from thermal motion in solution, (b) Schematic representation of ion binding on an oxide surface on the basis of the surface complexation model, s is the specific surface area (m kg ). Braces refer to concentrations in mol kg . (c) The electric surface potential, falls off (simplified model) with distance from the surface. The decrease with distance is exponential when l/ < 25 mV. At a distance k the potential has dropped by a factor of 1/c. This distance can be used as a measure of the extension (thickness) of l e double layer (see equation 40c). At the plane of shear (moving particle) a zeta potential can be established with the help of electrophoretic mobility measurements, (d) Variation of charge distribution (concentration of positive and negative ions) with distance from the surface (Z is the charge of the ion), (e) The net excess charge.

The dissolution rate equation for silica in nonflnoride solntions may also be expressed according to the surface complexation model described by Eq. (4.3) and Fig. 4.32 considering the contributions of different surface complexes sSiOHJ, sSiOH, =SiO-Na% and =SiO. The rate equation for the dissolution of silica in NaCl solutions at pH 2-13 at 25 °C can be described as ... 

Surface complexation models (SCM s) provide a rational interpretation of the physical and chemical processes of adsorption and are able to simulate adsorption in complex geochemical systems. Chemical reactions at the solid-solution interface are treated as surface complexation reactions analogous to the formation of complexes in solution. Each reaction is defined in terms of a mass action equation and an equilibrium constant. The activities of adsorbing ions are modified by a coulombic term to account for the energy required to penetrate the electrostatic-potential field extending away from the surface. Detailed information on surface complexation theory and the models that have been developed, can be found in (Stumm et al., 1976 ... 

Methods for measurement of parameters used in SCM s have been described in the literature. Only a brief summary is presented here. Surface complexation model parameters that can be measured directly include, (1) the solid concentration, (2) surface site density, (3) surface area, and (4) equilibrium constants for the mass action equations describing all relevant adsorption reactions. The relation between surface charge and potential is calculated in geochemical equilibrium models. 

The evaluation of qAB is a formidable task if done exactly, but the constant capacitance model (as well as other surface complexation models) is the special case that results from equation 12 by a simple approximation (22) ... 

The models describing hydrolysis and adsorption on oxide surfaces are called surface complexation models in literature. They differ in the assumptions concerning the structure of the double electrical layer, i.e. in the definition of planes situation, where adsorbed ions are located and equations asociating the surface potential with surface charge (t/> = f(5)). The most important models are presented in the papers by Westall and Hohl [102]. Tbe most commonly used is the triple layer model proposed by Davis et al. [103-105] from conceptualization of the electrical double layer discussed by Yates et al. [106] and by Chan et al. [107]. Reviews and representative applications of this model have been given by Davis and Leckie [108] and by Morel et al. [109]. We will base our consideration on this model. 

Surface complexation models of the solid-solution interface share at least six common assumptions (1) surfaces can be described as planes of constant electrical potential with a specific surface site density (2) equations can be written to describe reactions between solution species and the surface sites (3) the reactants and products in these equations are at local equilibrium and their relative concentrations can be described using mass law equations (4) variable charge at the mineral surface is a direct result of chemical reactions at the surface (5) the effect of surface charge on measured equilibrium constants can be calculated and (6) the intrinsic (i.e., charge and potential independent) equilibrium constants can then be extracted from experimental measurements (Dzombak and Morel, 1990 Koretsky, 2000). 

The basis for the discussion of adsorption on charged surfaces is the surface complexation model. The precept for this model is the use of the standard mass-action and mass-balance equations from solution chemistry to describe the formation of surface complexes. Use of these equations results in a Langmuir isotherm for the saturation of the surface with adsorbed species. There are of course other models that satisfy these precepts, but which are not generally referred to as surface complexation models, for example, the Stern model (J). 

While there are similar mass-balance and mass-action equations in all surface complexation models, there are a great number of ways to formulate the electrostatic energy associated with adsorption on charged surfaces. Customarily the electrostatic energy of an adsorbed ion of formal charge 2 at a plane of potential is taken by Coulomb s law to be zFt/r, but the relationships used to define surface potential t/r as a function of surface charge a, or any other experimentally observable variable, are different. In addition, different descriptions of the surface/solution interface have been used, that is, division of the interface into different layers, or planes, to which different ions are assigned formally. 

This equation and Eq. 5.82 provide the mass and charge balance constraints on any surface complexation model. 

Most of the research on metal sorption at the mineral/water interface has dealt with equilibrium aspects. Numerous studies have used macroscopic approaches such as adsorption isotherms, empirical and semi-empirical equations (e.g., Freundlich, Langmuir), and surface complexation models (e.g., constant capacitance, triple layer) to describe adsorption, usually based on a 24 hour reaction time. 

Table II. Mass action equations for the surface complexation model...

In the following sections, different surface complexation models will be introduced. General aspects and specific models will be discussed. The components of surface complexation theory will be presented, as well as some recent developments covering, for example, the use of equations for the diffuse part of the electrical double layer for electrolyte concentrations, for which the traditional Gouy-Chapman equation is not recommended or a generalization of Smit s compartment model [6] for situations in which the traditional models are at a loss. 

Two potential improvements compared to the common practice are introduced. Both refer to the description of the diffuse layer. The commonly applied surface complexation models involve the Poisson-Boltzmann approximation for diffuse-layer potential of the electric double layer (resulting in the Gouy-Chapman equation for flat plates in most apphcations). 

For comparison reasons, the results derived from the simulation were additionally calculated by means of the Thiele modulus (Equation 12.12), i.e., for a simple first-order reaction. The reaction rate used in the model is more complex (see Equation 12.14) thus, the surface-related rate constant kA in Equation 12.12 is replaced by... 

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