The Helmholtz capacity

At low electrolyte concentrations, up to about a 10 3 M solution, the Gouy-Chapman theory agrees quite well with experimental values of 

Several theories have been proposed to explain the origin and the order of magnitude of the Helmholtz capacity. Though differing in details, recent theories agree that the Helmholtz capacity contains contributions both from the metal and from the solution at the interface  

Due to the finite size of the ions and the solvent molecules, the solution shows considerable structure at the interface, which is not accounted for in the simple Gouy-Chapman theory. The occurrence of a decrease of C from the maximum near the pzc is caused by dielectric saturation, which lowers the dielectric constant and hence the capacity for high surface-charge densities. 

The latter effect can be understood within a simple model for metals the jellium model, which is based on the following ideas As is 

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Compare this result with Eq. (3.13). For most metals Ltf 0.5 A. By examining the experimental data in Fig. 3.5, show that this model cannot explain the origin of the Helmholtz capacity. 

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. 

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]. 

As with the jellium model, the main significance of these calculations lies in the physical insight that they give into the structure of the solution at the interface, and the origin of the Helmholtz capacity. 

The Helmholtz capacity of a metal electrode is typically on the order of 10 laF/cm, " " which is much greater than the value for a silicon electrode. " It has been reported that the Hehnholtz layer capacitance of silicon in 0.1 M JMFe(CN)6 + 0.5 M KCl aqueous solution is about 3pF/cm. ln 0.1 M tetrabutylammonium perchlorate (TBAP) in acetonitrile it is found to be about 1.5 aF/cm. " A larger Ch value, 22 p,F/cm, has been found for deep accumulation of a silicon in acetonitrile and the value varies with the accumulated charge. " ... 

It should be mentioned here, that the capacity of the space charge layer in an intrinsic semiconductor looks very similar to that of the diffuse Gouy layer in the electrolyte (compare with Eq. 5.8). This is very reasonable because the Gouy layer is also a kind of space charge layer with ions instead of electrons as mobile carriers. Q was actually derived by the same procedure as given here for Csc- Similarly as in the case of Cn and C(j, the space charge capacity Cjc and the Helmholtz capacity Ch can be treated as capacitors circuited in scries. We have then... 

The Helmholtz capacity always depends strongly on the charge density au, different metals may have quite different capacities, but typically there is a maximum, also known as the hump, near the pzc (see Fig. 2). Since the strength of the electric field increases on both sides of the pzc, the occurrence of the hump can he explained by dielectric saturation. Watts-Tobin [6] elaborated this idea into... 

Fig. 2 The Helmholtz capacity for Ag(ni) and Hg in contact with an aqueous solution after Ref. [77] with permission from the author.

The first term in Eq. (10) is just the GC capacity at the pzc the second term is independent of the ionic concentration, and can be identified with the Helmholtz capacity. However, in this model the Helmholtz capacity is not caused by a single monolayer of solvent with special properties, like in the Stern model, but results from an extended boundary layer. It depends on the dielectric properties of the solvent and on the diameters of the particles. Since A. C e, the influence of the ions on the capacity is predicted to be small. This is in line with the experimental... 

For aqueous solutions, with SA and e 80, Eq. (10) predicts a Helmholtz capacity of the order of 16 xFcm at the pzc this is much smaller than any experimental values. The contribution of the metal is missing in this model. We will see later that the electronic polarizability of the metal surface increases the Helmholtz capacity. 

Trasatti [28] has noticed that in aqueous solutions the Helmholtz capacity Ch of the simple sp metals, taken at the pzc, correlates with their electronic densities (see Fig. 5). This indicates that the surface polarizability of simple metals increases with their electronic densities, a trend that can be explained by the jellium model presented below. 

As one might expect, the polarizability increases with the electronic density (see Fig. 6), which qualitatively explains the corresponding increase of the Helmholtz capacity, though for jellium in vacuum... 

To arrive at a complete double-layer model, Amokrane and Badiali proceeded in a semiempirical manner. Assuming that the contributions of the metal and the solvent to the inverse capacity are additive (see Eq. 14), they obtained the solvent capacity from experimental data for the Helmholtz capacity of Ag(lll). In this way they obtained a solvent capacity for water, which shows a pronounced maximum near the pzc (see Fig. 9) this curve can be fitted to a model in which the solvent is represented by a layer of dipoles. 

This procedure can be carried further by taking experimental data for the Helmholtz capacities of other metals and extracting the metal contribution by assuming that the solvent part is not affected by the metal. In this way, Amokrane and Badiali arrive at a consistent interpretation of all experimental data. However, their theory hinges on the calculations for Ag(lll). In fact, the variation of the calculated metal capacity is so strong that it practically determines the solvent capacity... 

FIGURE 2.41. Electrode impedance for charge transfer plus diffusion of the redox components wMiout the conhibution of the Helmholtz capacity as a function of die angular frequency. 

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