Distance of the closest approach

In Equation 9.14 is the valance of thcth ion, e is the electron charge, D is the dielectric constant of the solvent mediuraj, is the distance of the closest approach toitheion, r is the distance from the ion at where the potentiatM and is the Debye kappa. The Debyeis... 

Unfortunately, the experimental data concerning the distances at which electron exchange reactions in the membranes take place are very scarce. Tsuchida et al. have shown [147], that even when the photoexcited Zn porphyrin embedded in the membrane cannot approach the membrane // water interface closer than 12 A, the electron transfer is still possible to MV2+ located in the water phase outside the membrane. However, when the distance of the closest approach of these reactants is increased up to 17 A, the electron transfer is totally stopped. Examples of electron transfer proceeding presumably via electron tunneling across molecular layers about 20 A thick, which separate electron donor and acceptor molecules, can be found in papers by Mobius [230, 231] and Kuhn [232, 233]. Note, that in... 

The maximum value of the current density lies at the opposite side of the electrocatalyst, considering it is positioned as in Figure 13.1, that is, at the nearest distance of the closest approach. 

The simplest interpretation of the compact-layer capacitance is represented by the Helmholtz model of the slab filled with a dielectric continuum and located between a perfect conductor (metal surface) and the outer Helmholtz plane considered as the distance of the closest approach of surface-inactive ions. Experimental determination of its thickness, zh, may be based on Eq. (12). Moreover, its dielectric permittivity, h, is often considered as a constant across the whole compact layer. Then its value can be estimated from the values of the compact-layer capacitance, for example, it gives about 6 or 10 (depending on the choice of zh) for mercury-water interface, that is, a value that is much lower than the one in the bulk water, 80. This diminution was interpreted as a consequence of the dielectric saturation of the solvent in contact with the metal surface, its modified molecular structure or the effects of spatial inhomogeneity. The effective dielectric permittivity of the compact layer shows a complicated dependence on the electrode charge, which cannot be explained by the simple hypothesis of the saturation effects on one hand or by the unperturbed bulk-solvent nonlocal polarizability on the other hand. 

This distinction can be checked experimentally if the measurements are performed for different surface-inactive electrolytes but for the same electrode and solvent. According to the general theory [42-44] discussed in Sect. 2.1.7, the distance of the closest approach, which should be noticeably different for these electrolytes, has got almost no effect in the compact-layer capacitance if the ions do not penetrate into the region of the reduced dielectric response near the surface. This theoretical prediction turns out to be in conformity with experimental data [35, 37, 45, 46] for three mercury-aqueous solution interfaces for which the PZ plot at the p.z.c. gives practically identical values for the compact-layer capacitance, Ch(0) = 29gFcm-2 (Fig. 6). 

The profile of an ideally smooth interface is sketched in Fig. 13.Thehalf-spacez < 0 is occupied by the ionic skeleton of the metal. This can be described, roughly, in a jellium model, as a continuum of positive charge n+ and the effective dielectric constant due to the polarizability of the bound electrons (this quantity is, with rare exceptions (Hg Sh = 2, Ag 5 = 3.5), typically close to 1 [125]). The gap 0 < z < a accounts for a nonzero distance of the closest approach of solvent molecules to the skeleton. The region of a < z < a + d stands for the first layer of solvent molecules, while z > a + d is the diffuse-layer region. n(z) denotes the profile of the density of free electrons. This is, of course, an extremely crude picture, but it eventually helps to rationalize the results of the various theoretical models and simulations. 

The above result may raise questions, because Ihe properties of adsorbed monolayers at charged interfaces should be governed by the long-range conlombic interactions. However, it can be easily explained if we take into account that, when we adopt an adsorption mechanism like that represented by Eq. (2), in fact, we model the adsorbed layer as a mixture of adsorbate A molecules and solvent clusters Sy with dimensions equivalent to A. That is, the adsorbed layer consists of species with dimensions greater than 0.25 nm and therefore the distance of the closest approach between two adsorbed dipoles cannot fall below 0.5 - 0.6 nm. Thus when we model the adsorbed layer as a mixture of adsorbate A molecules and solvent clusters Sa, the coulom-bic interactions stop to play the dominant role regarding the properties of this layer. This result is independent of whether we have polarizable or non-polarizable adsorbed molecules and, in fact, verifies the use of... 

This relationship is derived from the fact that, at the distance of the closest approach, all energy of the two-particle system is potential energy. 

It is possible to reduce the number of free parameters by replacing a by the Bjerrum distance of the closest approach of free ions given by... 

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