The electron transfer rate

At this stage, the theoiy did not yield the mass (a parameter of dimensionality [mass] X [length] ) associated with the dielectric fluctuations, which is needed to determine )s that appears in the adiabatic rate expression. On the other hand, short of the nonadiabatic coupling itself, all parameters needed to calculate the nonadiabatic rate from Eq. (16.51) have been identified within this dielectric theory. 

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At low currents, the rate of change of die electrode potential with current is associated with the limiting rate of electron transfer across the phase boundary between the electronically conducting electrode and the ionically conducting solution, and is temied the electron transfer overpotential. The electron transfer rate at a given overpotential has been found to depend on the nature of the species participating in the reaction, and the properties of the electrolyte and the electrode itself (such as, for example, the chemical nature of the metal). 

The electron transfer rates in biological systems differ from those between small transition metal complexes in solution because the electron transfer is generally long-range, often greater than 10 A [1]. For long-range transfer (the nonadiabatic limit), the rate constant is... 

M Tachiya. Relation between the electron-transfer rate and the free energy change of reaction. J Phys Chem 93 7050-7052, 1989. 

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

As an illustration of these considerations, the Arrhenius plot of the electron-transfer rate constant, observed by DeVault and Chance [1966] (see also DeVault [1984]), is shown in fig. 13. 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

Further improvements can be achieved by replacing the oxygen with a non-physiological (synthetic) electron acceptor, which is able to shuttle electrons from the flavin redox center of the enzyme to the surface of the working electrode. Glucose oxidase (and other oxidoreductase enzymes) do not directly transfer electrons to conventional electrodes because their redox center is surroimded by a thick protein layer. This insulating shell introduces a spatial separation of the electron donor-acceptor pair, and hence an intrinsic barrier to direct electron transfer, in accordance with the distance dependence of the electron transfer rate (11) ... 

The reason for the exponential increase in the electron transfer rate with increasing electrode potential at the ZnO/electrolyte interface must be further explored. A possible explanation is provided in a recent study on water photoelectrolysis which describes the mechanism of water oxidation to molecular oxygen as one of strong molecular interaction with nonisoenergetic electron transfer subject to irreversible thermodynamics.48 Under such conditions, the rate of electron transfer will depend on the thermodynamic force in the semiconductor/electrolyte interface to... 

EPR studies on electron transfer systems where neighboring centers are coupled by spin-spin interactions can yield useful data for analyzing the electron transfer kinetics. In the framework of the Condon approximation, the electron transfer rate constant predicted by electron transfer theories can be expressed as the product of an electronic factor Tab by a nuclear factor that depends explicitly on temperature (258). On the one hand, since iron-sulfur clusters are spatially extended redox centers, the electronic factor strongly depends on how the various sites of the cluster are affected by the variation in the electronic structure between the oxidized and reduced forms. Theoret-... 

Figure 23. Arrhenius plot of the electron transfer rate. The electronic coupling strength is TIad = 0.0001 a.u. Solid line-Bixon-Jortner perturbation theory Ref. [109]. FuU-circle present results of Eq. (26 ). Dashed line-results of Marcus s high temperature theory [Eq.(129)]. Taken from Ref. [28].

Recently five monometallic (Au, Pd, Pt, Ru, Rh) nanoparticles were investigated as electron mediators together with four core/shell bimetallic (Au/Pd, Au/Pt, Au/Rh, Pt/ Ru) nanoparticles [53,194-196]. The linear relationship was observed between the electron transfer rate coefficients and the hydrogen generation rate coefficient as shown in Figure 15. 

FIG. 10 Potential dependence of the electron-transfer rate constant k i) normalized to the value at the potential of zero charge TCNQ reduction by hexacyanoferrate at the water-DCE... 

According to the Marcus theory [9], the electron transfer rate depends upon the reaction enthalpy (AG), the electronic coupling (V) and the reorganization energy (A). By changing the electron donor and the bridge we measured the influence of these parameters on the charge transfer rate. The re-... 

The influence of the electronic coupling on the electron transfer rate was determined by changing the length of the (A T)n bridge. As expected, the rate decreased as the number n of the A T base pairs between electron donor and electron acceptor increased [4, 7]. But, surprisingly, the exponential correlation of Eq. (1) between the rate kEr and the distance is not valid for short distances. The plots in Fig. 3 and Fig. 4 show that at 6 A the electron transfer rate /cEt is much faster than expected [4, 7]. 

Fig. 4 Distance dependence of the electron transfer rate from 7 -deazaguanine Gz (see Fig. 2) to the enol ether radical cation...

When the electron coupling between locally excited-state (LE) and charge transfer state (CT) is weak, the electron transfer rate kcl can be expressed as (7)... 

Moreover, it has been demonstrated that CNTs promote the direct electrochemistry of enzymes. Dong and coworkers have reported the direct electrochemistry of microperoxidase 11 (MP-11) using CNT-modified GC electrodes [101] and layer-by-layer self-assembled films of chitosan and CNTs [102], The immobilized MP-11 has retained its bioelectrocatalytic activity for the reduction of H202 and 02, which can be used in biosensors or biofuel cells. The direct electrochemistry of catalase at the CNT-modified gold and GC electrodes has also been reported [103-104], The electron transfer rate involving the heme Fe(III)/Fe(II) redox couple for catalase on the CNT-modified electrode is much faster than that on an unmodified electrode or other... 

The electron-transfer rate between large redox protein and electrode surface is usually prohibitively slow, which is the major barricade of the electrochemical system. The way to achieve efficient electrical communication between redox protein and electrode has been among the most challenging objects in the field of bioelectrochemistry. In summary, two ways have been proposed. One is based on the so-called electrochemical mediators, both natural enzyme substrates and products, and artificial redox mediators, mostly dye molecules and conducted polymers. The other approach is based on the direct electron transfer of protein. With its inherited simplicity in either theoretical calculations or practical applications, the latter has received far greater interest despite its limited applications at the present stage. 

Nanotechnology has provided a novel way to enhance the electron-transfer rates between Hb and the electrode. As in the case of cyt c and Mb, nanocrystalline Ti02 film has been proposed to be a promising interface for the immobilization of Hb. GNPs are renowned for their good biocompatibility. With the help of these GNPs, Hb can exhibit a direct electron-transfer reaction without being denatured. To improve the... 

Z.Q. Feng, S. Imabayashi, T. Kakiuchi, and K. Niki, Electroreflectance spectroscopic study of the electron transfer rate of cytochrome c electrostatically immobilized on the w-carboxyl alkanethiol monolayer modified gold electrode. J. Electroanal. Chem. 394, 149-154 (1995). 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

The distance dependence of the electron-transfer rate can also be demonstrated for the non-benzenoid systems [14] and [15], which... 

It has been shown so far that internal and external factors can be combined in the control of the electron-transfer rate. Although in most cases a simple theoretical treatment, e.g. by the Marcus approach, is prevented by the coincidence of these factors, it is clear that the observed features for the isoenergetic self-exchange differ by the electronic coupling and the free energy of activation. Then it is also difficult to separate the inner- and outer-sphere reorganization energies. 

Figure 4. Correlation of the ionization potentials of alkylmetal donors with the electron-transfer rate constant (log kFe) for Fe(phen)s3+ (%), Fe(bpy)s3+ (O), and Fe(Cl-phen)s3+ ((D), (left). The figure on the right is the same as the left figure for Fe(phen)s3+ except for the inclusion of electron-transfer rates for some alkyl radicals as identified, (Note the expanded scale,)...

One striking prediction of the energy gap law and eq. 11 and 14 is that in the inverted region, the electron transfer rate constant (kjjj. = ket) should decrease as the reaction becomes more favorable (lnknr -AE). Some evidence has been obtained for a fall-off in rate constants with increasing -AE (or -AG) for intermolecular reactions (21). Perhaps most notable is the pulse radiolysis data of Beitz and Miller (22). Nonetheless, the applicability of the energy gap law to intermolecular electron transfer in a detailed way has yet to be proven. 

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