Plane of closest approach

The application of the overpotential t] can be considered to be equivalent to the displacement of the potential energy curves by the amount 7]F with respect to each other. The high field is now applied across the double layer between the electrode and the ions at the plane of closest approach. It is apparent from Fig. 12 that the energy of activation in the favoured direction will be diminished by etrjF while that in the reverse direction will be increased by (1 — ac)r]F where the simplest interpretation of a is in terms of the slopes of the potential energy curves (a = mi/ mi+m )) at the points of intersection electrode processes indeed are the classical example of linear free energy relations. 

FIG. 10 Schematic representation of the proposed surface model (a) the concentration and (b) the electrical potential profiles at the interface of the membrane and aqueous sample solution, x = 0 and 0 are the positions of ions in the planes of closest approach (outer Helmholtz planes) from the aqueous and membrane sides, respectively. (From Ref. 17.)... 

This plane of the center of mass of the excess ionic charge o,(x) is the effective excess charge plane on the solution side, which may be compared with the effective image plane on the metal side. In simple cases, the effective excess charge plane coincides with the outer Helmholtz plane (the plane of closest approach of hydrated ions) as shown in Fig. 5-21. 

Next, we discuss the plane of the closest approach (x = of water molecules to the jellium metal edge (x = 0). At the zero charge interface, this plane of closest approach of water molecules is separated by a distance equal to the radius of water molecules from the metal siuface. As the interfadal excess charge increases, the electrostatic pressure (electrostriction pressiue) reduces the distance of Xdip in prop>ortion to the square of the interfadal charge, a (= om = - os) the electrostatic force in the compact layer is proportional to om x as. The change in Xitp due to the interfadal charge is then given by Eqn. 5-32 ... 

Fig. 6-22. Effect of interfadal excess charge, om, on the effective image plane, Xi , the shift of the plane of closest approach of water molecules, the inverse interfacial capadty on the metal side, 1 /Cm and the interfadal capacity on the solution side, Cs. M/vac = metaWacuum interface M/sol = metal/solution interface.

The reaction of electron transfer at electrodes in aqueous electrolytes proceeds either with hydrated redox particles at the plane of closest approach of hydrated ions to the electrode interface (OHP, the outer Helmholtz plane) or with dehydrated and adsorbed redox particles at the plane of contact adsorption on the electrode interface (IHP, the inner Helmholtz plane) as shown in Fig. 7-2. 

The plane of closest approach of hydrated ions, the outer Helmholtz plane (OHP), is located 0.3 to 0.5 run away from the electrode interface hence, the thickness of the interfacial compact layer across which electrons transfer is in the range of 0.3 to 0.5 nm. Electron transfer across the interfacial energy barrier occurs through a quantum tunneling mechanism at the identical electron energy level between the metal electrode and the hydrated redox particles as shown in Fig. 8-1. 

Thus, Grahame modified Stern s model by introducing the inner plane of closest approach [inner Helmholtz plane (IHP)], which is located at a distance X from the electrode (Fig. 4.11). The IHP is the plane of centers of partially or fully dehydrated... 

The electric field which actually affects the charge transfer kinetics is that between the electrode and the plane of closest approach of the solvated electroactive species ( outer Helmholtz plane ), as shown in Fig. 2.2. While the potential drop across this region generally corresponds to the major component of the polarization voltage, a further potential fall occurs in the diffuse double layer which extends from the outer Hemlholtz plane into the bulk of the solution. In addition, when ions are specifically absorbed at the electrode surface (Fig. 2.2c), the potential distribution in the inner part of the double layer is no longer a simple function of the polarization voltage. Under these circumstances, serious deviations from Tafel-like behaviour are common. 

The plane of closest approach at x = x2 or the outer Helmholtz plane OHP is depicted in Fig. 2 together with its potential with respect to the solution A02-... 

In the absence of specific adsorption and dipolar contributions, there is no excess charge in the whole double layer when positive and negative ions are equally distributed at the plane of closest approach qM and A02 will be both zero. The corresponding electrode potential is the potential of zero charge (pzc) which can be evaluated from the minimum in the differential capacity—potential curve for a metal electrode in contact with a dilute electrolyte [6]... 

Figure 1. Schematic picture of the metal/solution interphase in the case of nonspecific (a) and specific (b) anionic adsorption, x = 0, x = P and x = d are die electrode surface plane, the plane of closest approach for the specifically adsorbed anions, and that for the nonspecifically adsorbed ions. Curve 1 represents the potential-distance profile. In (b), curve 1 results from the combination of curve 2, expressing die contribution from the charge density as of the specifically adsorbed anions, and curve 3, expressing die contribution from die charge density Om on the metal. The potential difference, ft1 — d> across die inner layer is the same in (a) and (b). (Reprinted from Ref.7 with permission from the Am. Chem. Soc.)...

In terms of k (r), the effective value of Sr according to eqn. (34) equals /J thus if (S = 1.5 A, then <5r % 0.7 A. If the reaction is adiabatic at the plane of closest approach, larger values of Sr will arise since k (r) will then only decrease exponentially with increasing r for r > a, i.e. for sufficiently large electrode-reactant distances so that non-adiabaticity is encountered. However, in terms of eqn. (33), the effective Sr values can be affected substantially by the dependence of g (r) upon r. For attractive electrode-reactant interactions, Sr will be decreased since the reactant concentration will decrease sharply as r increases, whereas for repulsive interactions, Sr will be increased. 

However, since insufficient information is available regarding the dependence of gx, and especially k, upon r, eqn. (33) is of only formal rather than practical value at the present time. Since the reaction zone thickness is liable to be of molecular dimensions, the conventional procedure of estimating the effective work terms for a single reaction site at the expected plane of closest approach is probably acceptable, although it is nonetheless important to recognize its limitations. 

As noted in Sect. 3.1, the pre-exponential factor for outer-sphere reactions is sensitive to the "effective electron-tunneling distance , SrK%. If the reaction is non-adiabatic at the plane of closest approach (i.e. if 1), given the functional dependence of fcel upon r one deduces that 5r 0.5 A if P 1.5 A 1 (vide supra), and hence drfCe, < 0.5 A. If, however, adiabaticity is maintained for sites beyond the plane of closest approach (i.e. for r > [Pg.43]

The resulting estimates of <5r/c i for Cr(III)/Cr(II) aquo couples, ca. 0.1-0.3 A, are close to that extracted from temperature-dependence measurements and indicate that these strongly hydrated species follow marginally non-adiab-atic pathways even at the plane of closest approach [30]. Markedly larger values of 5rK°eU ca. 5 A, were obtained for similar ammine complexes, as from a related comparison involving unimolecular outer-sphere reactivities [21], This is consistent with the smaller hydrated radii of the ammines allowing a closer approach to the metal surface, whereupon a 1 [15a, 30]. 

Conversely, according to the description of the electrical double layer based on the Stern-Gouy-Chapman (S-G-C) version of the theory [24], counter ions cannot get closer to the surface than a certain distance (plane of closest approach of counter ions). Chemically adsorbed ions are located at the inner Helmholtz plane (IHP), while non-chemically adsorbed ions are located in the outer Helmholtz plane (OHP) at a distance x from the surface. The potential difference between this plane and the bulk solution is 1 ohp- In this version of the theory, Pqhp replaces P in all equations. Two regions are discernible in the double layer the compact area between the charged surface and the OHP in which the potential decays linearly and the diffuse layer in which the potential decay is almost exponential due to screening effects. 

For nonadiabatic reactions, when one moves from the plane of closest approach away from the electrode, it is assumed [204] that the transmission coefficient decreases according to the equation... 

The Triple-Layer Model (TLM) This model, first developed by Yates et al. (1974) and Davis (1978), is essentially an expanded Stem-Grahame model the specifically adsorbed ions are placed as partially solvated ions at a plane of closest approach. Additional capacitances are introduced. Subsequently, Hayes (1987) had to modify the earlier TLM by moving specifically adsorbed ions to the mean plane of the surface. 

A plane (m) is associated with an excess concentration of elections near the physical surface of the electrode, represented by a solid line. The inner Helmholtz plane (ihp) is associated with ions that are specifically adsorbed onto the metal surface. The outer Helmholtz plane (ohp) is the plane of closest approach for solvated ions that are free to move within the electrolyte. The ions within the electrolyte near the electrode surface contribute to a diffuse region of charge. The diffuse region of charge has a characteristic Debye length. 

The equations do not take into account the finite size of the ions the potential to be used is ipi, the potential at the Stern plane (the plane of closest approach of ions to the surface), which is difficult to measure. The nearest experimental approximation to is often the zeta potential (0 measured by electrophoresis. 

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