Helmholtz capacity

Figure 3.3 Helmholtz capacity for Ag(lll) and mercury in aqueous solutions the notation for single crystal surfaces will be explained in the next chapter.

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 ... 

Figure 3.5 The inverse Helmholtz capacity at the pzc as a function of the electronic density the latter is plotted in atomic units (a.u.), where 1 a.u. of density = 6.76 x 1024 cm-3. The dashed line is based on a model calculation of Schmickler and Henderson [3].

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. 

Figure 17.4 Calculated values of the inverse metal capacity versus experimental values of the inverse Helmholtz capacity at the pzc.

Numerical values for the effective position of the image charge at the pzc (cr = 0) are given in Table 17.1. Figure 17.4 shows a plot of 1 /Cm, calculated from Eq. (17.18), versus experimental values for the inverse Helmholtz capacity of a few sp metals in contact with an aqueous solution. Since the electrons penetrate only a short distance into the solution, we expect the interaction between the metal electrons and the solvent to be comparatively weak. If this is true, we expect from Eq. (17.5) ... 

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]. 

If we combine these ideas with those developed in the previous section, we conclude that there are two contributions to the inverse Helmholtz capacity one from the structure of the solution, and one from the response of the surface electrons. It is natural to combine the... 

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. 

Fig. 6.70. (a) Parsons-Zobel plots of bare Au(111), covered with decanethiol (DT), co-hydroxydecanethiol (HDT), and 4 -hydroxy-4-mercaptobiphenyl (HBT). (Reprinted with permission from R. P. Janek, W. R. Fawcett, and A. Ulman, J. Rhys. Chem. B 101 8550, Fig. 12, copyright 1997, American Chemical Society.) (b) Helmholtz capacity (CH), as a function of the electrode charge for Ag(111) in contact with an aqueous solution of ions that are not specifically adsorbed. (Reprinted with permission from W. Schmickler, Chem. Rev. 96 3177, Fig. 3, copyright 1996, American Chemical Society.)... 

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. " ... 

This is illustrated in Fig. 4.7. In the simplest case the solid-liquid interface can be described by a charge transfer resistance R and a capacity in parallel (Helmholtz capacity for metal electrodes, Ch, and a space charge capacity for semiconductor elec-... 

Using e = 20 and xj = 5 x 0 cm, one obtains Ch 3 x 10 F cm 2. This capacity value agrees with experimental data within an order of magnitude. This Helmholtz capacity is independent of the electrode potential, i.e. in the case of a metal electrode any external variation of the electrode potential leads only to a corresponding change of the charges on both sides of the interface. 

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... 

If all surface states are occupied by electrons their density is given by A, = AQs/e (Eq. 5.35). Assuming a Helmholtz capacity of Ch 10F cm" one obtains /V, in the order of about lO cm 2 for a shift of ACfb = 0.2 V. It is interesting to note that the shift of the flatband potential occurs mostly at very low light intensities, and it saturates at higher light intensities because then all surface states are filled [45]. In Fig. 5.20 the shift of energy bands is indicated for some semiconductors. 

Electrode area Richardson constant Activity coefficient Differential Helmholtz capacity Differential space charge capacity Concentration of species j in solution... 

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... 

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