Dielectric continuum, spheres

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

In the so-called primitive double-layer model the solvent is represented as a dielectric continuum with dielectric constant e, the ions as hard spheres with diameter a, and the metal electrode as a perfect conductor. For small charge densities on the electrode the capacity of the interface is given by [15] ... 

Both the electronic couphng matrix element and the outer-sphere component of the nuclear reorientation parameter are thought to vary with donor-acceptor separation and orientation [29, 49]. It has been shown in studies of Os and Ru-ammines bridged by polyproline spacers that the distance dependence of X can be greater than that of [50]. Dielectric continuum models of solvent reorganization predict that Xg will increase with... 

Marcus calculates the outer sphere contribution to the reorganization energy from non-equilibrium dielectric continuum polarization theory for the solvent environment (— 5 nm)... 

An expression has been derived by Marcus34 and Hush35 for A0 assuming the solvent to be a structureless dielectric continuum characterized by the macroscopic dielectric constants Dop and Ds. D0p and Ds are the optical and static dielectric constants, respectively, and Dop = n2 where n is the index of refraction in the visible spectral region. In the limit that the reactants can be treated as two non-interpenetrating spheres, AQ is given by equation (23). 

The most striking application of electron transfer theory has been to the direct calculation of electron transfer rate constants for a series of metal complex couples.36 37 46 The results of several such calculations taken from ref. 37b are summarized in Table 2. The calculations were made based on intemuclear separations appropriate to the reactants in close contact except for the second entry for Fe(H20)j3+/2+, where at r = 5.25 A there is significant interpenetratidn of the inner coordination spheres. The Ke values are based on ab initio calculations of the extent of electronic coupling. k includes the total contributions to electron transfer from solvent and the trapping vibrations using the dielectric continuum result for A0. the quantum mechanical result for intramolecular vibrations, and known bond distance changes from measurements in the solid state or in solution. 

Here, r denotes the position vector of the charges qt with respect to the center of the sphere, and r, the distance from the center. By applying the dielectric scaling function for dipoles (Eq. (2.3)), which—as we have seen in Fig. 2.1—is also a good approximation for most other multipole orders, it was immediately clear that the idea of using a scaled conductor instead of the EDBC leads to a considerable simplification of the mathematics of dielectric continuum solvation models, with very small loss of accuracy. It may also help the finding of closed analytic solutions where at present only multipole expansions are available, as in the case of the spherical cavity. Thus the Conductor-like Screening Model (COSMO) was bom. 

With respect to the solvation energy, this is usually approximated by modeling the reactants and products as spheres and the solvent as a dielectric continuum (Bom theory), which in the case of an interface electron transfer gives rise to the following expression [30, 36] ... 

Rather full calculations of /. (r) vs. r for various p values must be compared to the experimental results to determine p. Equation (6) gives a widely used expression for solvent reorganization energy that can be substituted into k expressions. It was derived by Marcus over 40 years ago and is both simple and useful [61]. It models the donor and acceptor as two conducting spheres imbedded in a dielectric continuum. 

In summary, it appears from this discussion that Franck-Condon energies can now be calculated for a diverse group of inorganic charge-transfer systems and that, although the accuracy of individual values is uncertain, it is possible qualitatively to rationalize the differences between analogous systems. Absolute predictions are much less satisfactory at the present time, and the electrostatic theory based on a dielectric continuum has only very limited applicability to the systems that have so far been studied. When inner-sphere reorganization... 

In 1954 Weiss32 used Bernal and Fowler s simplified solvation model,16 with an Inner Sphere of ionic coordination, i.e., a small spherical double layer around the ion of charge ze, followed by a sharp discontinuity at radius q, the edge of the Outer Sphere or Dielectric Continuum. He used a simple electrostatic argument to determine the energy to remove an electron at optical frequency from the Inner Sphere ... 

Since Bernal and Fowler,16 the charging radius r0 in the Born equation has been put equal to the Inner Sphere radius, or approximately the ion to water molecule center distance plus 1.4 A. At least for 1+ ions, this gives a fairly good approximation to the Gibbs energy of interaction of the ion with the outer Dielectric Continuum if aT and s are constant throughout the medium. High-valency ions are discussed in Section IV. 

Such considerations also provide a rationale of the inability of the dielectric continuum model, as conventionally expressed with a fixed frequency factor, to describe the solvent-dependent kinetics of some other outer-sphere reactions [45b, 95b]. As noted in Sect. 3.3.1, the influence of solvent dynamics upon vn should disappear for reactions having moderate or large inner-shell barriers, the frequency factor being determined by vis instead [eqns. (22) and (25)] this can account for the success of the conventional (fixed-frequency) dielectric-continuum treatment in describing solvent-dependent kinetics for some reactant systems [45],... 

In case of ionophores the formation of ion pairs is dependent on the electrical charge e, the dielectric constant e and the center-to-center distance a. The association constant Ka was calculated for rigid charged spheres with diameter a in a dielectric continuum ... 

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