Spheres in a dielectric continuum

Association Phenomena According to the theoretical model of spheres in a dielectric continuum the ions are represented as rigid, charged spheres that do not interact with solvent, which is considered to be a medium without any kind of structure. The only interaction is that which occurs between the ions, and the formation of ion pairs is controlled only by electrostatic forces. On these bases, the association constant may be expressed by the Fuoss equation (29) ... 

The generic model for a ionic fluid is the primitive model ( or RPM ) of charged hard spheres in a dielectric continuum with the dielectric constant e. Thus, the potential is just the Coulomb potential between the ions labelled i and k, which is cut-off at the collision diameter [Pg.150]

The potential profile is the least reliable feature of the GC model. Certainly, Monte Carlo calculations in which the ions are represented as charged hard spheres in a dielectric continuum show that the GC potential profile is seriously in error at high electrolyte concentrations. However, it is sometimes used at very low concentrations to obtain an approximate idea of potential variation in the diflhse layer. 

One of the simplest realistic molecular models of an ionic solution is the restricted primitive model. This consists of an equimolar mixture of oppositely charged but equal-sized hard spheres in a dielectric continuum. 

The classical theory of electron transfer developed by Marcus starts with the same kind of hard spheres in a dielectric continuum model that is used to derive the free energy of solvation of an ion. A central role in the theory is played by the reorganisation energy X, which in its simplest definition is given by... 

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 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 ... 

The latest models propose to represent electrolyte solutions as a collections of hard spheres of equal size, ions, immersed in a dielectric continuum, the solvent. For such a system, what is called the Mean Spherical Approximation, MSA, has been successful in estimating osmotic and mean activity coefficients for aqueous 1 1 electrolyte solutions, and has provided a reasonable fit to experimental data for dilute solutions of concentrations up to -0.3 mol dm". The advantage in this approach is that only one... 

In the absence of a reliable theory, computer simulations have become the most important means of tackling the question of ionic criticality. In the last ten years there have been numerous attempts to identify the universality class of what is perhaps the most basic model of ionic fluids, the restricted primitive model (RPM). The RPM consists of an equimolar mixture of positively and negatively charged hard spheres with diameter a, immersed in a dielectric continuum with dielectric constant D. The pair potential is,... 

Fuoss developed a new theory of ion association in 1958 [27] which overcame some of the difficulties associated with the Bjerrum approach. The cations in the solution were assumed to be conducting spheres of radius a and the anions to be point charges. The ions are assumed to be immersed in a dielectric continuum of permittivity Sj. Only oppositely charged ions separated by the distance a are assumed to form ion pairs. The resulting expression for the association constant is... 

The GC results are compared in fig. 10.18 with Monte Carlo calculations of Boda et al. [32]. These were carried out assuming that the electrolyte ions are hard spheres with a diameter of 300 pm in a dielectric continuum. The estimates of < ) using the Monte Carlo technique fall below the GC estimates. They demonstrate the importance of including finite ion size in a model of the diffuse layer. 

The lowest level of the integral equation approach treats the ions as charged hard spheres embedded in a dielectric continuum. It fulfills the conditions... 

When applied to the primitive model of electrolyte solutions (i.e., charged hard spheres of arbitrary diameters in a dielectric continuum), the HNC equation is superior to the PY equation because it preserves the correct long-range behavior. On the other hand, in fluids with only short-range forces the PY equation can be successfully applied because some of the omitted terms cancel one another. 

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] ... 

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] ... 

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... 

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