Double-sphere model

A very simple model for reaction between two ions in solution is shown in Fig. 8.1. This model is referred as double sphere model. ZA and ZB indicate the number of positive or negative charges on the ions. Initially the ions are at infinite distance from each other and in activated state they are considered to be intact and they are dAB distance apart. 

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] ... 

The interfacial electric double layer is represented by a combination of the improved jeUium model on the metal side and the hard sphere model of ions and dipoles on the aqueous solution side. Then, the electric capacity, Ch, of the compact layer is given by Eqn. 5-34 ... 

The simplest model of the structure of the metal-solution interphase is the Helmholtz compact double-layer model (1879). According to this model, all the excess charge on the solution side of the interphase, qs. is lined up in the same plane at a fixed distance away from the electrode, the Helmholtz plane (Fig. 4.4). This fixed distance xH is determined by the hydration sphere of the ions. It is defined as the plane of the centers of the hydrated ions. All excess charge on the metal, qM, is located at the metal surface. 

Hard-sphere electrolyte model - double-layer models... 

For one-electron transfer reactions occurring via outer-sphere mechanisms, wp and ws can be estimated on the basis of electrostatic double-layer models. Thus, if the reaction site lies at the outer Helmholtz plane (o.H.p.), wp = ZFd and ws = (Z - 1 )Fcharge number of the oxidized species and (j>d is the potential across the diffuse layer. Rewriting eqn. (7) in terms of rate constants rather than free energies yields the familiar Frumkin equation [8]... 

Few outer-sphere electrode reactions have precursor-state concentrations that are measurable [21] so that it is usual to estimate wp and ws from double-layer models. The simplest, and by far the most commonly used, treatment is the Frumkin model embodied in eqns. (8) and (8a) whereby, as noted in Sect. 2.2, the sole contributor to wp and ws is presumed to be electrostatic work associated with transporting the reactant from the bulk solution to the o.H.p. at an average potential Gouy-Chapman (GC) theory [58],... 

Thirdly, it will be important to gain more direct information on the stability of outer-sphere precursor states, especially with regard to the limitations of simple electrostatic models (Sect. 4.2). One possible approach is to evaluate Kp for stable reactants by means of differential capacitance and/or surface tension measurements. Little double-layer compositional data have been obtained so far for species, such as multicharged transition-metal complexes, organometallics, and simple aromatic molecules that act as outer-sphere reactants. The development of theoretical double-layer models that account for solvation differences in the bulk and interfacial environments would also be of importance in this regard. 

The results are given in Figures 14.lid and 14.lie. The semiquantitative agreement between experimental data and calculated data is obvious. The surface charge estimated can be converted into a surface potential on the basis of the diffuse double-layer model from which a stability could be calculated. Alternatively, a Stem model approach may be used, incorporating a distance of closest approach of outer-sphere ions (Section 9.5). 

Fig. 4 a) STM image of the NiAl(100)-c(V2 x 3V2)R45° surface. Two domains A and B could be seen on the upper terrace, which is separated by a double step from the lower terrace on the right side (from ref [47]). b) Hard sphere model of the corresponding surface, (from ref [37]). 

The attractive pressure of the chain-of-spheres model is the sum of the double-series attractive pressures of the c external degrees of freedom,... 

It is well known that lymphocytes are spherical, and have a thin cell membrane and a spherical nucleus (surrounded by a thin nuclear envelope) that occupies about 60% of the cell volume (188). Therefore, the dielectric properties of lymphocytes can be described by the double-shell model (188, 190, 191)(see Fig. 49). In this model the cell is considered to be a conducting sphere covered with a thin shell, much less conductive than the sphere itself, in which a smaller sphere with a shell (i.e., the nucleus) is incorporated. In addition, one assumes that every phase has no dielectric losses and the complex dielectric permittivity can thus be written as ... 

The middle of the twentieth century marked the end of a long period of determining the building blocks of chemistry chemical elements, chemical bonds, and bond angles. The lists of these are not definitely closed, but future changes will be more cosmetic than fundamental. This made it possible to go one step further and begin to rationalize the strucmre of molecular systems, as well as to foresee the structural features of the compounds to be synthesized. The crucial concept is based on the Bom-Oppenheimer approximation and on the theory of chemical bonds and resulted in the spatial structure of molecules. The great power of such an approach was first proved by the construction of the DNA double helix model by Watson and Crick. The first DNA model was built from iron spheres, wires, and tubes. 

Ostwald first modelled catastrophic inversions as being caused by the complete coalescence of the dispersed phase at the close packed condition (corresponding to a dispersed phase fraction of 0.74 in Ostwald s uniform hard sphere model). Other studies, e.g. Marzall, have shown that catastrophic inversions (though these inversions were not called catastrophic inversions by that author) can occur over a wide range of WOR. It has been suggested that this may be due to the formation of double emulsion drops (0/W /0), boosting the actual volume of the dispersed phase. 

The bi-spherical model in which the activated complex is a double-sphere, eaeh keeping the same charge as its origin molecule, the total eharge of the activated complex remaining the same as those of the ions reactants. 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

The Ball and Wire model is identical to the Wire model, exeept that atom positions are represented by small spheres. This makes it possible to identify all atom locations in all molecules. The Tube model is identical to the Wire model, except that bonds, whether single, double or triple, are represented by single colored tubes. The tubes are useful because they better eonvey the three-dimensional shape of a molecule. The Ball and Spoke model is a variation on the Ibbe model atom positions are represented by colored spheres, making it possible to see all atom locations in all molecules. 

Figure 6 Model structure of CHMO. The FAD cofactor is shown as sticks. Mutations observed in a directed evolution study resulting in altered enantioselectivities are highlighted (spheres). " Note that a number of mutations were found in double/triple mutants and cannot be linked with certainty to the altered catalytic properties. Mutations in double/triple mutants that are on the...

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