Outer-sphere model

Fig. 14. Theoretical dependence of relaxivity on calculated on the basis of the classical inner-/outer-sphere model (20 MHz, o = 1, = 5 x 10 s, Ty = 20 ps,...

C.l. Limitations of the Current Theoretical Models Outer-sphere model... 

The electron transfer (ET) may take place via n orbitals of the ligands at the silicon atom (outer-sphere model). Inner-sphere ET could involve a chloro bridge between silicon and tin. It leads to a four-centered transition state in a special case. 

Many electron transfer reactions of inorganic radicals conform to the outer-sphere model and hence can be modeled with the Marcus theory of electron transfer.71 This model relies, in part, on the concept of self-exchange reactions, and the inference that self-exchange reactions can be defined for radicals. For many years, it was simply... 

The interaction between and NOj" ions in aqueous solutions has been studied by several techniques during the past 20 years. A literature survey up to the early 1970s is presented in Gmdin (1974). The main question that has been debated concerns the nature of the complexation is it inner or outer sphere From thermodynamic considerations, Choppin (1971) arrived at a predominantly outer-sphere model see also Choppin and Bertha, 1973. However, in a recent study of fluorescence spectra and lifetimes, Biinzli and Yersin (1979) suggested inner-sphere complexation for europium nitrate, in contrast to the corresponding perchlorate system, see fig. 70. After studying the effect of various ligands on the " Fq transition. 

The Marcus treatment applies to both inorganic and organic reactions, and has been particularly useful for ET reactions between metal complexes that adopt the outer-sphere mechanism. Because the coordination spheres of both participants remain intact in the transition state and products, the assumptions of the model are most often satisfied. To illustrate the treatment we shall consider a family of reactions involving partners with known EE rate constants. 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

Fig. 5. A model for S042- substitution on [Be(H20)4]2+ proceeding from the outer-sphere complex on the left through the transition state at center to the inner-sphere complex at the right of the figure (16, 66).

As with the Marcus-Hush model of outer-sphere electron transfers, the activation free energy, AG, is a quadratic function of the free energy of the reaction, AG°, as depicted by equation (7), where the intrinsic barrier free energy (equation 8) is the sum of two contributions. One involves the solvent reorganization free energy, 2q, as in the Marcus-Hush model of outer-sphere electron transfer. The other, which represents the contribution of bond breaking, is one-fourth of the bond dissociation energy (BDE). This approach is... 

This problem has been addressed recently197 with model reactions in which the nucleophile (viz. electron donor) is NO-. It has been found that outer-sphere transition states are easier to compute with NO- than with other electron donors, thus allowing a complete characterization of the competition between the various pathways. 

N. Sutin, Brookhaven National Laboratory Strictly speaking, the outer-sphere and inner-sphere designations refer to limiting cases. In practice, reactions can have intermediate outer-sphere or inner-sphere character this occurs, for example, when there is extensive interpenetration of the inner-coordination shells of the two reactants. Treating this intermediate situation requires modification of the usual expressions for outer-sphere reactions — particularly those expressions that are based upon a hard-sphere model for the reactants. 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

To develop these ideas into a quantitative theory, we require models for the inner and outer sphere and their reorganization. The problem is similar to that encountered in infrared and Raman spectroscopy, where... 

To obtain an estimate for the energy of reorganization of the outer sphere, we start from the Born model, in which the solvation of an ion is viewed as resulting from the Coulomb interaction of the ionic charge with the polarization of the solvent. This polarization contains two contributions one is from the electronic polarizability of the solvent molecules the other is caused by the orientation and distortion of the... 

However, a closer inspection of the experimental data reveals several differences. For ion-transfer reactions the transfer coefficient a can take on any value between zero and one, and varies with temperature in many cases. For outer-sphere electron-transfer reactions the transfer coefficient is always close to 1/2, and is independent of temperature. The behavior of electron-transfer reactions could be explained by the theory presented in Chapter 6, but this theory - at least in the form we have presented it - does not apply to ion transfer. It can, in fact, be extended into a model that encompasses both types of reactions [7], though not proton-transfer reactions, which are special because of the strong interaction of the proton with water and because of its small mass. 

There are three points of significance of this result. One is that it provides strong support for the 10-step mechanism originally proposed for reaction 1. Another is that it facilitates a more robust fitting of the mechanism to the kinetic data obtained for that reaction. Thirdly, it confirms that reaction 2 has a rate constant that is four orders of magnitude greater than predicted by Marcus theory. It is concluded that reaction 2 is poorly modeled as an outer-sphere process and is better described as... 

Iron(III)-catalyzed autoxidation of ascorbic acid has received considerably less attention than the comparable reactions with copper species. Anaerobic studies confirmed that Fe(III) can easily oxidize ascorbic acid to dehydroascorbic acid. Xu and Jordan reported two-stage kinetics for this system in the presence of an excess of the metal ion, and suggested the fast formation of iron(III) ascorbate complexes which undergo reversible electron transfer steps (21). However, Bansch and coworkers did not find spectral evidence for the formation of ascorbate complexes in excess ascorbic acid (22). On the basis of a combined pH, temperature and pressure dependence study these authors confirmed that the oxidation by Fe(H20)g+ proceeds via an outer-sphere mechanism, while the reaction with Fe(H20)50H2+ is substitution-controlled and follows an inner-sphere electron transfer path. To some extent, these results may contradict with the model proposed by Taqui Khan and Martell (6), because the oxidation by the metal ion may take place before the ternary oxygen complex is actually formed in Eq. (17). 

Most of the kinetic models predict that the sulfite ion radical is easily oxidized by 02 and/or the oxidized form of the catalyst, but this species was rarely considered as a potential oxidant. In a recent pulse radiolysis study, the oxidation of Ni(II and I) and Cu(II and I) macrocyclic complexes by SO was studied under anaerobic conditions (117). In the reactions with Ni(I) and Cu(I) complexes intermediates could not be detected, and the electron transfer was interpreted in terms of a simple outer-sphere mechanism. In contrast, time resolved spectra confirmed the formation of intermediates with a ligand-radical nature in the reactions of the M(II) ions. The formation of a product with a sulfonated macrocycle and another with an additional double bond in the macrocycle were isolated in the reaction with [NiCR]2+. These results may require the refinement of the kinetic model proposed by Lepentsiotis for the [NiCR]2+ SO/ 02 system (116). 

FIGURE 1.13. Free-energy profiles in outer-sphere electron transfer according to the Butler-Volmer approximation (a) and to the Marcus-Hush model (b). 

---