Diffusion diffuse double-layer model

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

There is no experimental way to measure y. (As we mentioned before, the zeta potential - as obtained, for example, from electrophoretic measurments - is in a not readily definable way - smaller than y.) But as discussed in section 3.3 we can obtain the surface charge (Eq. 3.2) and then compute the surface potential y on the basis of the diffuse double layer model with Eq. (3.8a) Eq. (3.8a) in simplified form for 25° C is... 

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. 

Metal binding by a hydrous oxide from a 10 7 M solution (SOH + Me2+ OMe+ + H+) for a set of equilibrium constants (see Eqs. (i) - (iii) from Example 2.3) and concentration conditions (see text). Corrected for electrostatic interactions by the diffuse double layer model (Gouy Chapman) for 1 = 01 The hydrolysis of Me2+ is neglected. 

For charge correction we use the diffuse double layer model and assume I = 0.01. 

Specific surface area 40 m2 g 1, acidity constants of FeOHg pK., (int) = 7.25, K 2 = 9.75, site density = 4.8 nrrr2, hematite cone = 10 mgle. Ionic strength 0.005. For the calculation the diffuse double layer model shall be used. 

The results are given in Figs. 7.5 d) and e). The semiquantitative agreement between experimental data and calculated data is obvious. The surface charge estimated can be converted into a surface potential on the basis of the diffuse double layer model from which a stability could be calculated. 

HFOa Adsorption isotherms for arsenite and arsenate over free concentration range from 10-7 to 10-3 M (pH 4-10). Fitted to Langmuir isotherm at low concentrations and linear isotherm at higher concentrations. Dzombakand Morel (1990) fitted this data to their diffuse double layer model Pierce and Moore (1982)... 

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). 

Diffuse-double-layer model. The DDLM is similar to the CCM but the charge potential is formulated from the Gouy-Chapman theory ... 

At low ionic strength, a diffuse double-layer model (Gouy-Chapman model) is used. 

Tonkin, J. W., L. S. Balistrieri, and J. W. Murray. 2004. Modeling sorption of divalent metal cations on hydrous manganese oxide using the diffuse double layer model. Appl. Geochem. 19 29-53. 

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... 

The strong affinity of iron oxides for Se(IV) has been well documented (Dzombak and Morel, 1990) and calculations based on the Dzombak and Morel (1990) diffuse double-layer model and default HFO database show the principal response to pH and redox speciation changes (Figure 11). The selenate species is less strongly adsorbed by iron oxides at near-neutral pH than the selenite species (Figure 11). Clay minerals (Bar-Yosef and Meek, 1987) also adsorb Se(IV). 

There has been no lack of attempts to correct existing 2D equations of state for the effect of the double layer. Most of these are based on the diffuse double layer model, and therefore remain limited to the low pressure range, i.e. to the least interesting parts of x(A) curves. Henderson-Hasselbalch interpretations, using [II.3.6.53 or 54] cannot be carried out for lack of information on the degree of dissociation a. Basically, the route for incorporation of double layer formation into 2D-equations of state is embodied in (3.4.53] in combination with [3.4.48] and [3.4.16]. Ekjuation [3.4.53] has four parameters, viz. a°,, (or y/, the two are... 

Compute the equilibrium composition, and the extent of Pb(II) sorption to a-Fe203, as a function of pH, using the Gouy-Chapman diffuse double-layer model. 

A m, the apparent constant, is the product of an intrinsic constant (a constant valid for a hypothetical uncharged surface) and a Boltzmann factor. / is the surface potential, F the Faraday constant, and AZ the change in the charge number of the surface species of the reaction for which the equilibrium constant is defined (in this case AZ = -bl). The intrinsic constant is experimentally accessible by extrapolating experimental data to the surface charge where op = 0 and where l/ = 0. The correction, as given above, assumes the classical diffuse double-layer model (a planar surface and a diffuse layer of counterions). 

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. 

Figure 7. The effect of ligands and metal ions on surface protonation of a hydrous oxide is illustrated by two examples (1). Part a Binding of a ligand (pH 7) to hematite, which increases surface protonation. Part h Adsorption of Pb2+ to hematite (pH 4.4), which reduces surface protonation. Part c Surface protonation of hematite alone as a function of pH (for comparison). All data were calculated with the following surface complex formation equilibria (1 = 5 X 10"3 M >. Electrostatic correction was made by diffuse double layer model.

The expressions for low potentials are the so called Debye-Huckel (DH) limits of the diffuse double layer model. 

McBride, M. B. (1997). A critique of diffuse double layer models applied to colloid and surface chemistry. Clays Clay Miner. 45, 598-608. 

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