The Gouy-Chapman capacity

According to Gauss s theorem, E( 0) = a jet o using the identity cosh x— 1 = 2 sinh2 x/2 gives  

Differentiation then gives the Gouy-Chapman expression Eq. (2.10). 

Sometimes one requires not only the capacity but the potential 4 x) we sketch the derivation. If we integrate Eq. (A.l) from x to oo, we obtain by the same arguments for the derivative f (x)  

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This differential capacity is known as the Gouy-Chapman capacity. It has a pronounced minimum at the pzc, and it increases with the square root of the electrolyte concentration. Figure 3.1 shows the Gouy-Chapman capacity calculated for several electrolyte concentrations. 

The potential of zero charge (pzc) is a characteristic potential for a given interface, and hence is of obvious interest. In the absence of specific adsorption, it can be measured as the potential at which the Gouy-Chapman capacity obtains its minimum this value must be independent of the electrolyte concentration, otherwise there is specific adsorption. For liquid metals the pzc coincides with the maximum of the surface tension (see Section 3.5). 

Here Cac denotes the Gouy-Chapman capacity, Agm gives the first order deviation from the Gouy-Chapman theory in the absence of... 

Fig.l The Gouy-Chapman capacity for several ionic concentrations of a 1 1 electrolyte. 

Fig. 15 Interfacial capacity in the lattice gas model for various ion-solvent interactions. The data were obtained from an MC simulation [85]. The crosses denote the Gouy-Chapman capacity.

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

Figure 2.11 According to the Gouy-Chapman theory, the capacity of the electrode/electrolyte interface should be a cosh function of the potential difference across it (see text). Concentration of electrolyte in (b) > than that in (a).

Figure 12.3 Capacity of the interface between a solution of NaBr in water and TBAs/TPB in nitrobenzene. The upper points are for 0.1 M solutions, the lower for 10 2 M in both phases. The two curves have been calculated from the Gouy-Chapman theory. The sign convention for the potential is A = (py, — (po + const., where the index w stands for the aqueous and o for the organic phase. Data taken from Ref. 1.

On the whole, the Gouy-Chapman theory seems to work well for ITIES, indicating that any contribution from the dipole potential is small. In particular the interfacial capacity exhibits a minimum at the potential of zero charge for low electrolyte concentrations (see Fig. 12.3). 

The Gouy-Chapman theory treats the electrolyte as consisting of point ions in a dielectric continuum. This is reasonable when the concentration of the ions is low, and the space charge is so far from the metal surface that the discrete molecular nature of the solution is not important. This is not true at higher electrolyte concentrations, and better models must be used in this case. Improvements on the Gouy-Chapman theory should explain the origin of the Helmholtz capacity. In the last section we have seen that the metal makes a contribution to the Helmholtz capacity other contributions are expected to arise from the molecular structure of the solution. 

The most important result is the existence of an extended boundary region, where the structure of solution differs significantly from the bulk, and where the potential deviates from the predictions of the Gouy-Chapman theory. In this model the interfacial capacity can be... 

For water at room temperature, A 2.65. The natural interpretation of Eq. (17.20) is this The structure of the solution at the interface causes deviations from the Gouy-Chapman theory. The leading correction term is independent of the electrolyte concentration and therefore contributes to the Helmholtz capacity for water (s 3 A) one obtains a contribution of about 7.1 A (0.64 cm2//F-1). At very high concentrations terms of order k and higher become significant. These should cause deviations from a straight line in a Parsons and Zobel plot, which have indeed been observed [10]. 

The Gouy-Chapman theory was tested experimentally on the basis of the doublelayer capacity measurements. This theory predicts a parabolic capacitance-potential relationship and a square-root dependence on concentration at constant e and T ... 

Variation of the double-layer capacity with applied potential according to the Gouy-Chapman theory is shown in Figure 4.8. Equation (4.10) includes the approximation ifjyi = iIj(x = 0), which is in harmony with the basic assumption of this... 

Figure 4.8. Variation of the doublelayer capacity with applied potential according to the Gouy-Chapman theory.

The potential dependence of C is the basic improvement of the model, in comparison with the Hehnholtz model, which predicts potential-independent capacity. However, a comparison of experimental data and values calculated on the basis of Eq. (4.10) shows that the function C = behaves according to the Gouy-Chapman model in very dilute solutions and at potentials near the minimum (Fig. 4.8). In concentrated solutions, on the other hand, and at p>otentials farther away from the minimum, the theory is in disagreement with experimental results. Once again, a new theory is called for. 

Fig. 6.65. (a) Representation of the Gouy-Chapman theory according to Eq. (6.130). (b) Plots of the differential capacity of the mercury/solution interface vs. electrode potential for various concentrations of A/,A/dimethylacetamide in 0.15 M Na2S04-water.(Reprinted with permission from W. R. Fawcett, G. Y. Champagne, S. Komo, and A. J. Motheo, J. Phys. Chem. 92 6368, Fig. 6, copyright 1988, American Chemical Society.)... 

What happens when the concentration c0 of ions in solution is very large Equations (6.124) and (6.130) indicate that while CG increases with increasing c0, CH remains constant. Thus, as c0 increases, (1/CG) (1/CH), and for all practical purposes, C CH. That is, in sufficiently concentrated solutions, the capacity of the interface is effectively equal to the capacity of the Helmholtz region, Le., of the parallel-plate model. What this means is that most of the solution charge is squeezed onto the Helmholtz plane, or confined in a region vety near this plane. In other words, little charge is scattered diffusely into the solution in the Gouy-Chapman disarray. 

The capacity of the inner layer can be obtained from eqn. (38) and the experimental total double layer capacity if Cd is calculated from the Gouy—Chapman theory [24] using eqn. (45). 

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