Double-layer model

The necessity to calculate the electrostatic contribution to both the ion-electrode attraction and the ion-ion repulsion energies, bearing in mind that there are at least two dielectric ftmction discontinuities hr the simple double-layer model above. 

Various proposed values for the constants can be found in the literature [8]. Despite double-layer model predictions [148,149] that exponents Jt and y are both unity, and a dimensional analysis model [204] giving x as 1.88 andy as 0.88, test work on a practical scale [202,203] has indicated that both exponents are approximately equal to 2. This implies that a is roughly independent of pipe diameter and that the ratio //3 s 4/jt s 1. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

Various pc electrode models have been tested.827 Using the independent diffuse layer electrode model74,262 the value of E n = -0.88 V (SCE) can be simulated for Cd + Pb alloys with 63% Pb if bulk and surface compositions coincide. However, large deviations of calculated and experimental C,E curves are observed at a 0. Better correspondence between experimental and calculated C,E curves was obtained with the common diffuse-layer electrode model,262 if the Pb percentage in the solid phase is taken as 20%. However, the calculated C, at a Ois noticeably lower than the experimental one. It has been concluded that Pb is the surface-active component in Cd + Pb alloys, but there are noticeable deviations from electrical double-layer models for composite electrodes.827... 

In general Figures 6.18 to 6.25, and in particular figures 6.18, 6.19, 6.20, 6.24 and 6.25 show, beyond any reasonable doubt, that the effective double layer model of promotion, expressed mathematically by Equations 6.65 and 6.66, grasps the essence of promotional kinetics. 

As already discussed in Chapter 6 (Figure 6.25) the observed complex rate dependence of CO oxidation on pco, P02 and UWR (O) (Figs. 4.16, 4.31, 9.6 and 9.7) can be described in a semiquantitalive fashion by the effective double layer model presented in Chapter 6. The system provides an excellent paradigm of the promotional rules Gl, G2 and G3 which are summarized by the general inequalities (6.11) and (6.12) written specifically here for the CO oxidation system ... 

INTERRELATION OF PROMOTION, ELECTROCHEMICAL PROMOTION AND METAL-SUPPORT INTERACTIONS THE DOUBLE-LAYER MODEL OF CATALYSIS... 

The basic double-layer model considers the solid as a perfect conductor, so that gM(dip) is charge independent and the potential... 

To be useful in modeling electrolyte sorption, a theory needs to describe hydrolysis and the mineral surface, account for electrical charge there, and provide for mass balance on the sorbing sites. In addition, an internally consistent and sufficiently broad database of sorption reactions should accompany the theory. Of the approaches available, a class known as surface complexation models (e.g., Adamson, 1976 Stumm, 1992) reflect such an ideal most closely. This class includes the double layer model (also known as the diffuse layer model) and the triple layer model (e.g., Westall and Hohl, 1980 Sverjensky, 1993). 

The double layer model is of the two the more fully developed in the literature (e.g., Dzombak and Morel, 1987) and hence currently the most useful in geochem-... 

Despite the seeming exactitude of the mathematical development, the modeler should bear in mind that the double layer model involves uncertainties and data limitations in addition to those already described (Chapter 2). Perhaps foremost is the nature of the sorbing material itself. The complexation reactions are studied in laboratory experiments performed using synthetically precipitated ferric oxide. This material ripens with time, changing in water content and extent of polymerization. It eventually begins to crystallize to form goethite (FeOOH). 

Because it is based on chemical reactions, the double layer model can be integrated into the equations describing the equilibrium state of a multicomponent system, as developed in Chapter 3. The basis appears as before (Table 3.1),... 

In the so-called primitive double-layer model the solvent is represented as a dielectric continuum with dielectric constant e, the ions as hard spheres with diameter a, and the metal electrode as a perfect conductor. For small charge densities on the electrode the capacity of the interface is given by [15] ... 

Let us now extend the long-period hydronium ice-like model for the IHP on Pt(lll) to explain the observations in electrolytes other than sulphate. In acid chloride, both the observations and the model carry-over directly from the case of sulphate. In fluoride, perchlorate, bicarbonate and hydroxide, in Which the anomalous features shift considerably in both potential and appearance (especially in the basic media) from sulphate, another model is needed. Both (bi)sulphate and chloride are large weakly hydrated anions, and in the double-layer model of Figures 4-5, they interact strongly with both the hydronium ions and the Pt surface. The contact adsorption... 

Fig. 1 Double layer model for a cathode, (a) Helmholtz model (b) Gouy-Chapman model (c) Stern model. 

The description of the sorption of charged molecules at a charged interface includes an electrostatic term, which is dependent upon the interfacial potential difference, Ai//(V). This term is in turn related to the surface charge density, electric double layer model. The surface charge density is calculated from the concentrations of charged molecules at the interface under the assumption that the membrane itself has a net zero charge, as is the case, for example, for membranes constructed from the zwitterionic lecithin. Moreover,... 

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. 

A Simplified Double Layer Model (Constant Capacitance)... 

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. 

Equations 9-14 provide the framework for combining either of the two surface hydrolysis models that were presented with any of the four electric double layer models to define the interface model completely and to solve for all unknown potentials, charges, and surface concentrations. In the following section some specific limiting cases are considered. 

Two models of surface hydrolysis reactions and four models of the electrical double layer have been discussed. In this section two examples will be discussed the diprotic surface group model with constant capacitance electric double layer model and the monoprotic surface group model with a Stern double layer model. More details on the derivation of equations used in this section are found elsewhere (3JL). ... 

The total potential in the double layer also follows from the double layer model of Figure 2, and can be calculated to be ... 

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