described using surface complexation models. These models have been widely used to describe an array of chemical reactions including proton dissociation, metal cation and anion adsorption reactions on oxides, clays, and soils, organic ligand adsorption on oxides, and competitive adsorption reactions on oxides. Applications and theoretical aspects of surface complexation models are extensively reviewed in Goldberg (1992). Surface complexation models often are chemical models based on molecular descriptions of the electric double layer using equilibrium-derived adsorption data (Goldberg, 1992). They include the constant capacitance (CCM), triple-layer (TLM), and modified triple-layer, Stern variable surface charge-variable surface potential (VSC–VSP), generalized two-layer, and the one-pK models. Differences in the surface complexation models lie in the descriptions of the electrical double layer, i.e., in the definition and assignment of ions to the planes or layers of adsorption and in differences in the electrostatic equations and the relations between the surface potential and surface charge. These models provide some information on the physical description of the electric double layer, including the capacitance and location of adsorbed ions, and they can describe data over a broad range of experimental conditions such as varying pH and I. Two kinds of data can be derived from these models: material balance data, i.e., the quantity of a material adsorbed, and information that can be used to describe electrokinetic phenomena (Westall and Hohl, 1980). Common characteristics of surface complexation models are consideration of surface charge balance, electrostatic potential terms, equilibrium constants, capacitances, and surface charge density. A summary of characteristics of the different surface complexation models is given in Table 5.4.

The general balance of surface charge model is

(5.16)σo+σH+σis+σos+σd=0,

where σo is the constant charge in minerals due to ionic or isomorphic substitution, σH is the net proton charge, which is equal to ΓH−ΓOH, where Γ is the surface excess concentration (σH is equivalent to the dissociation of H+ in the diffuse layer), σis is the inner-sphere complex charge as a result of inner-sphere complex formation, σos is the outer-sphere complex charge as a result of outer-sphere complexes, and σd is the dissociated charge or the charge in the bulk solution that balances the surface charge (these ions do not form any complex with the surface). The σo is negative while σH, σis, σos can be positive, negative, or neutral. Another way to describe the net total particle charge (σp) on a colloid is

(5.17)σP=σo+σH+σis+σos.

The σp can be positive or negative but must be balanced by σd or ions in the soil solution or in the dissociated form. As pointed out earlier, all surface complexation models contain a balance of surface charge equation and general surface complexation reactions (Hohl et al.,1980; Goldberg, 1992),

(5.18)SOH+H+≡SOH2+

(5.19)SOH≡SO-+H+

(5.20)SOH+Mn+≡SOM(n-1)+H+

(5.21)2SOH+Mn+≡(SO)2M(n-2)+2H+

(5.22)SOH+Ll−≡SL(l-1)-+OH-

(5.23)2SOH+Ll−≡S2L(l-2)-+2OH-,

where SOH is the surface functional group and S represents the metal bound to the surface functional group, e.g., OH of an oxide surface or of the aluminol or silanol group of a clay mineral, M is the metal ion, n+ is the charge on the metal ion, L is a ligand, and l− is the ligand charge. The intrinsic equilibrium constants (see Box 5.4 for a discussion of intrinsic and conditional equilibrium constants) for reactions in Eqs. (5.18)–(5.23) are (Hohl et al., 1980; Goldberg, 1992)

(5.24)K+int=[SOH2+][SOH][H+]exp[Fψi/RT]

(5.25)K+int=[SO-][H+][SOH]exp[−Fψi/RT]

(5.26)KM1int=[SOM(n-1)][H+][SOH][Mn+]exp[(n-1)Fψi/RT]

(5.27)KM2int=[(SO)2M(n-2)][H+][SOH]2[Mn+]exp[(n-2)Fψi/RT]

(5.28)KL1int=[SL(l-1)-][OH-][SOH]2[Ll-]exp[-(l-1)Fψi/RT]

(5.29)KL2int=[S2L(l-2)-][OH-]2[SOH]2[Lℓ-]exp[(-l-2)Fψi/RT],

where brackets indicate concentrations in mol liter−1, ψi is the surface potential in V in the ith surface plane, F is the Faraday constant in C mol−1, R is the gas constant in J mol−1 K−1, and T is absolute temperature in degrees Kelvin. The log of the intrinsic equilibrium constants can be obtained by plotting the log of the conditional equilibrium constants versus surface charge (σ) and extrapolating to zero surface charge (Stumm et al., 1980). The term ei-Fψi/RT considers surface charge effects on surface complexation. Surface complexation models also contain several adjustable parameters including Ki, equilibrium constants; Ci, capacitance density for the ith surface plane; and [SOH]T, the total number of reactive surface hydroxyl groups. Details on the determination of these parameters can be found in Goldberg (1992).

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