Outer-sphere PRE

B. Outer-sphere PRE and electron relaxation recent development... 

Another limiting situation arises when the paramagnetic species interact only weakly with the molecules carrying the nuclear spins. In such a case, it is not meaningful to speak about exchange between discrete sites, but rather about free diffusion or diffusion in a potential. One then speaks about outer-sphere PRE, still referring to the enhancement of the spin-lattice relaxation rate. The outer-sphere PRE is also proportional to the concentration of... 

B. Outer-Sphere PRE and Electron Relaxation Recent Development... 

An analytical theory of the outer-sphere PRE for slowly rotating systems with an arbitrary electron spin quantum number S, appropriate at the limit of low field, has been proposed by Kruk et al. (144). The theory deals with the case of axial as well as rhombic static ZFS. In analogy to the inner sphere case (95), the PRE for the low field limit could be expressed in terms of the electron spin spectral densities s ... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

The exact form of the pre-exponential factor A (see Chapter 5) is still being debated from the preceding considerations it is apparent that we must distinguish two cases If the reaction is adiabatic, the pre-exponential factor will be determined solely by the dynamics of the inner and outer sphere if it is nonadiabatic, it will depend on the electronic overlap between the initial and final state, which determines the probability with which the reaction proceeds once the system is on the reaction hypersurface. 

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

Cerium(IV) oxidations of organic substrates are often catalysed by transition metal ions. The oxidation of formaldehyde to formic acid by cerium(IV) has been shown to be catalysed by iridium(III). The observed kinetics can be explained in terms of an outer-sphere association of the oxidant, substrate, and catalyst in a pre-equilibrium, followed by electron transfer, to generate Ce "(S)Ir", where S is the hydrated form of formaldehyde H2C(OH)2- This is followed by electron transfer from S to Ir(IV) and loss of H+ to generate the H2C(0H)0 radical, which is then oxidized by Ce(IV) in a fast step to the products. Ir(III) catalyses the A -bromobenzamide oxidation of mandelic acid and A -bromosuccinimide oxidation of cycloheptanol in acidic solutions. ... 

Bayburt and Sharp 143) formulated a low-field theory (i.e. a theory for the case of ZFS dominating over the electron spin interaction) for the outer-sphere relaxation, treating also the electron spin relaxation in the simplified manner expressed by Eq. (52). That model predicted only a weak dependence of the PRE on the magnitude of the static ZFS and its application to the cases of high static ZFS is problematic. 

Theoretical models for outer-sphere nuclear spin relaxation in paramagnetic systems, including an improved description of the electron spin relaxation, have been developed intensively for the last couple of years. They can be treated as counterparts of the models of inner-sphere PRE, described in the Section V.B and V.C. 

Ru Ru step and a self-exchange rate of 2xlO" M s for the c -[Ru 0)2(L)] " /cw [Ru (0)2(L)]+ couple has been estimated a mechanism involving a pre-equilibrium protonation of ci5-[Ru (0)2(L)]+ followed by outer-sphere electron transfer is proposed for the Ru Ru step. For reduction by [Fe(H20)6] +, an outer-sphere mechanism is proposed for the first step and an inner-sphere mechanism is proposed for the second step. ... 

The E° difference is a necessary but not a sufficient condition. The rate constant for either ET (in general, / et) may be described in a simple way by equation (4). The activation free energy AG is usually expressed as a quadratic function of AG°, no matter whether we deal with an outer-sphere ET or a dissociative ET. However, even if the condition (AG")c < (AG°)sj holds (hereafter, subscripts C and ST will be used to denote the parameters for the concerted and stepwise ETs, respectively), the kinetic requirements (intrinsic barriers and pre-exponential factors) of the two ETs have to be taken into account. While AGq depends only slightly on the ET mechanism, is dependent on it to a large extent. For a concerted dissociative ET, the Saveant model leads to AG j % BDE/4. Thus, (AGy )c is significantly larger than (AG )sj no matter how significant AGy, is in (AG( )gj (see, in particular. Section 4). In fact, within typical dissociative-type systems such as... 

Steric Control of the Inner/Outer-Sphere Electron Transfer 461 Thermal and Photochemical ET in Strongly Coupled CT Complexes 463 Electron-Transfer Paradigm for Arene Transformation via CT Complexes 465 Electron-Transfer Activation of Electrophilic Aromatic Substitution 469 Structural Pre-organization of the Reactants in CT Complexes 470 CT Complexes in Aromatic Nitration and Nitrosation 472 Concluding Summary 475 References 475... 

Relatively little attention has been given in the literature to the electronic transmission coefficient for electrochemical reactions. On the basis of the conventional collisional treatment of the pre-exponential factor for outer-sphere reactions, Kel has commonly been assumed to equal unity, i.e. adiabatic reaction pathways are followed. Nevertheless, as noted above, the dependence of xei upon the spatial position of the transition state is of key significance in the "encounter pre-equilibrium treatment embodied in eqns. (13) and (14). Thus, the manner in which Kel varies with the reactant-electrode separation for outer-sphere reactions will influence the integral of reaction sites that effectively contribute to the overall measured rate constant and hence the effective electron-tunneling distance, Srx, in eqn. (14). 

It is of interest to estimate likely values of the combined pre-exponential factor in eqn. (14) for outer-sphere reactions... 

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]

Such an energy transfer is taken into account in the encounter pre-equilibrium model [131, 133], which considers the outer-sphere electrode reaction to be a two-step process. In the first step the reactant diffuses to the reaction zone with a thickness dr at the electrode surface, where the probability of the charge transfer process between reactant and electrode is significant [133]. Here the electrode and reactant in the reaction zone are similar to a pair of reactants which exchange the electrons in a homogeneous reaction. 

Khan [174] studied the electrooxidation of ferrocene at a Pt electrode in polar solvents ranging from methanol to heptan-l-ol. Experimental data concorded well with the calculated results when solvent influence on the pre-exponential coefficient was considered. In calculations v = rb was used. Khan [174] points out that expressed by Eq. (36) exhibits a temperature dependence different from that predicted by the classical expression = k T/h. Another conclusion which may follow from the same paper is that the transmission coefficient for the electrochemical outer-sphere electron-transfer reactions in polar alcoholic solvents may not be equal to unity. 

The distance decay constant / (see below) in Miller et al. s original study was 0.9 per CH2, using ferricyanide and iron(IH) hexahydrate [44]. In a later study which accounted more thoroughly for double layer effects, 2 was determined to be 1 eV for kinetically facile redox probes such as ferricyanide, 1.3 eV for Ru-hexamine and 2.1 eV for iron(III) hexahydrate. With a better understanding of the redox probe behavior, f was found to be 1.08 + 0.20 per CH2 and independent of the redox couple and electrode potential [96]. Pre-exponential factors were also extracted from the Tafel plots. The edge-to-edge rate constants (extrapolated) are approximately 10 -10 s for all redox probes, which is reasonable for outer-sphere electron transfer. The pre-exponential factors are 5 x lO s [96]. 

An Id mechanism in which an ion-pair precursor complex is formed in a rapid pre-equilibrium step, equation (4.12), followed by a rate determining loss of H20 and entry of Y- from the outer-sphere of the complex. 

In general, many kinetics data are accumulated prior to proposing a reaction mechanism. In our case, we will simply use the stoichiometry information obtained in Experiment 5.4 along with intuition based on past work in the field. The following is an interactive pre-lab exercise for proposing the rate law for the electron transfer between [Co(en)3)]2+ and [Co(ox)2(en)]. The kinetics will then be investigated using conventional visible spectroscopy. Experimental data, in combination with the rate law, will be used to determine the outer-sphere electron rate constant. 

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