Electron transfer rate model rates

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

Fig. 16.3 Quantum yield (QY) for electron and hole transfer to solution redox acceptors/donors as a function of the reduced variables y (related to the surface properties of the catalyst, i.e., ratio between interfacial electron transfer rate and surface recombination rate) and w (related to the ratio between surface migration currents of hole and electrons to the rate of bulk recombination), according to the proposed kinetic model [23],...

The value of log rn for the Fe(H20) 2+ - Fe(H20)6 + exchange (which features a relatively large inner-sphere barrier) is plotted as a function of 1/T in Figure 5. The nuclear tunneling factors are close to unity at room temperature but become very large at low temperatures. As a consequence of nuclear tunneling, the electron transfer rates at low temperatures will be much faster than those calculated from the classical model. 

We have investigated the ferrocene/ferrocenium ion exchange to determine the effects of different solvents on electron-transfer rates. There is probably only a very small work term and very little internal rearrangement in this system. Thus the rates should reflect mostly the solvent reorganization about the reactants, the outer-sphere effect. We measured the exchange rates in a number of different solvents and did not find the dependence on the macroscopic dielectric constants predicted by the simple model [Yang, E. S. Chan, M.-S. Wahl, A. C. J. Phys. Chem. 1980, 84, 3094]. Very little difference was found for different solvents, indicating either that the formalism is incorrect or that the microscopic values of the dielectric constants are not the same as the macroscopic ones. 

A vibronic coupling model for mixed-valence systems has been developed over the last few years (1-5). The model, which is exactly soluble, has been used to calculate intervalence band contours (1, 3, 4, 5), electron transfer rates (4, 5, 6) and Raman spectra (5, 7, 8), and the relation of the model to earlier theoretical work has been discussed in detail (3-5). As formulated to date, the model is "one dimensional (or one-mode). That is, effectively only a single vibrational coordinate is used in discussing the complete ground vibronic manifold of the system. This is a severe limitation which, among other things, prevents an explicit treatment of solvent effects which are... 

From the expressions given for example in Refs. [4,9,29], it can be seen that the nuclear factor, and consequently the electron transfer rate, becomes temperature independent when the temperature is low enough for only the ground level of each oscillator to be populated (nuclear tunneling effect). In the opposite limit where IcgT is greater than all the vibrational quanta hco , the nuclear factor takes an activated form similar to that of Eq. 1 with AG replaced by AU [4,9,29]. The model has been refined to take into account the frequency shifts that may accompany the change of redox state [22]. 

Because electron transfer is a rate phenomenon, considerations of timescale are unavoidable in the modeling and understanding of ET reactions. Indeed, even in simple polaron theory (vibronic theory for electron transfer between two sites), there are several timescales including... 

The key finding of the preliminary investigations of such molecules is that the non-protein diporphyrin models can react many times faster (ca. 104) than similar protein-protein systems at similar distances between the reaction sites and similar AG°. This result minimally suggests that the protein matrix does not accelerate the electron transfer rate [104], although a final conclusion cannot be drawn on the basis of these very limited data. Further studies in this area are necessary. 

A number of covalently bound porphyrin-quinone systems have recently been synthesized as models for carefully spaced donor-acceptor systems. These are designed to test the possibility of controlling electron transfer rates by spatial separation of donor-acceptor pairs. Among these exceedingly clever studies (30) are molecules designed to separate the donor and acceptor by rigid insulating molecules, for example, 2 (31),... 

To compare these two mechanisms, an NADH model without the recognition site was synthesised. The contribution of the flavin binding to the rate constant was thus evaluated and it was shown that the proximity of flavin and NADH model influenced the electron transfer rate. Mechanistic computations helped to show that with the appropriate NADH model system, both components were optimally arranged for the electron transfer. Although the exact mechanism of the reaction is still under debate, the kinetic isotope effect experiment indicated that in this case, the hydrogen at 4-position was transferred in the rate determining step which supported the hydride mechanism. 

Figure 3.6. The bell-shaped and rectangular models of the electron transfer rate W r).

Figure 2 Plots of the logarithm of electron transfer rate vs. the negative of the free energy of the reaction for three ET models and six rate measurements. The data are from Refs. 54, 55, 57, 59, 60 for a Zn-substituted Candida krusei cytochrome c that was also successively substituted at histidine 33 by three Ru(NH3)4L(His 33)3+ derivatives with L = NH3, pyridine, or isonicotinamide. The shortest direct distance between the porphyrin and imidazole carbon atoms was 13 A corresponding to the 10-A edge-to-edge D/A distance. Table 1 presents a summary of the parameters used in the three calculations plotted in this figure. For a (3 of 1.2 A-1, Eq. (5) yields HAB values ( 10 cm-1) of 80 cm-1,50 cm-1, and 75 cm-1, respectively, for Eq. (1), the semiclassical model [Eq. (4)], and the Miller-Closs model at the above D/A separation distance. The s values were calculated using Eq. (6) with the following parameters aD = 10 A, aA = 6 A, and r = 13 A. The kj and H°B parameters were varied independently to produce the plotted curves.

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