Self exchange

A powerful application of outer-sphere electron transfer theory relates the ET rate between D and A to the rates of self exchange for the individual species. Self-exchange rates correspond to electron transfer in D/D (/cjj) and A/A (/c22)- These rates are related through the cross-relation to the D/A electron transfer reaction by the expression... 

The cross relation has proven valuable to estimate ET rates of interest from data tliat might be more readily available for individual reaction partners. Simple application of tire cross-relation is, of course, limited if tire electronic coupling interactions associated with tire self exchange processes are drastically different from tliose for tire cross reaction. This is a particular concern in protein/protein ET reactions where tire coupling may vary drastically as a function of docking geometry. 

Figure C3.2.11. Log of the ET rate (A) against (l/s p-l/E ) for tire bis(biphenyl) cliromium self-exchange reaction. From 1341.

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

Eree energy curves for the self-exchange reaction between two rubredoxins (Rdi and Rd2) were generated from MD simulations [86,87]. 

If the film is nonconductive, the ion must diffuse to the electrode surface before it can be oxidized or reduced, or electrons must diffuse (hop) through the film by self-exchange, as in regular ionomer-modified electrodes.9 Cyclic voltammograms have the characteristic shape for diffusion control, and peak currents are proportional to the square root of the scan speed, as seen for species in solution. This is illustrated in Fig. 21 (A) for [Fe(CN)6]3 /4 in polypyrrole with a pyridinium substituent at the 1-position.243 This N-substituted polypyrrole does not become conductive until potentials significantly above the formal potential of the [Fe(CN)6]3"/4 couple. In contrast, a similar polymer with a pyridinium substituent at the 3-position is conductive at this potential. The polymer can therefore mediate electron transport to and from the immobilized ions, and their voltammetry becomes characteristic of thin-layer electrochemistry [Fig. 21(B)], with sharp symmetrical peaks that increase linearly with increasing scan speed. 

Marcus and Hush have developed a theory, which bears their names, that relates the value of kj2 to the rates (ku and 22) of the self-exchange reactions of the two... 

There is a very special case for self-exchange reactions in which the left side of the equation is identical to the right side. Accordingly, there is no free energy change in the reaction, and the equilibrium constant ( fn) must be unity (Eq. 9.29). 

Figure 9-6. The consequences of a self-exchange electron transfer between a ground state cobalt(ii) and a ground state cobalt(iii) complex.

Figure 9-7. The self-exchange electron transfer reaction between vibrationally excited cobalt(ii) and cobalt(iii) complexes.

This is the origin of the various values for self-exchange rate constants. We may now attempt to rationalize some of these in terms of the /-electron configurations of the various oxidation states. Consider the self-exchange rate constants for some iron complexes. 

We conclude with a consideration of a few other cobalt self-exchange reactions. The reaction in Eq. (9.33) is faster than that involving the ammine complexes (Eq. 9.30) because the water is a weaker-field ligand than ammonia. Thus, the activation energy for the formation of the electronically excited states is lower, as is the change in Co-ligand distances in the two oxidation states. 

The self-exchange velocity (SEV), which can be calculated from molecular dynamics simulation, has reproduced the Chemla effect. Fur-... 

Okada et al. have presented a dynamic dissociation model, which is schematically shown in one dimension in Fig. 4. They assumed that the separating motion of a cation (or anion) of interest from the reference anion (cation), which is called the self-exchange velocity,is the electrically conducting process, which will be considered in Section III.7( ) in more detail. The Chemla effect can also be reproduced by the SEV. 

Okada et al. have found that internal mohilities are strongly related to the separating motion of unlike ion pairs defined by the self-exchange velocity, which can be easily calculated from MD simulation ... 

Figure 17. Self-exchange velocities vs. internal mobilities calculated from the same MD runs for pure LiCl and (Li, Cs)Cl mixture(xcs, = 0.90). experimental for u. (Reprinted from Ref 47 with permission of Trans Tech Publications.)...

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

The reduced poly-[Fe(II)TPP] porphyrin site now finds itself next to a fresh poly-[Fe(III)TPP(X)] site one polymer lattice unit further into the polymer. An electron hopping - or self exchange - reaction can then ensue, repeatedly, in successive layers and sites ... 

The reaction amounts to a vectorically directed current in the sense of occurring down a concentration gradient of reduced poly-[Fe(II)TPP] sites emanating from the reducing electrode/polymer interface. The magnitude of the current clearly conveys information about the rate of the poly-[Fe(III)TPP(X)] - poly-[Fe(II)TPP] self exchange reaction. 

In the present case, the electron hopping chemistry in the polymeric porphyrins is an especially rich topic because we can manipulate the axial coordination of the porphyrin, to learn how electron self exchange rates respond to axial coordination, and because we can compare the self exchange rates of the different redox couples of a given metallotetraphenylporphyrin polymer. To measure these chemical effects, and avoid potentially competing kinetic phenomena associated with mobilities of the electroneutrality-required counterions in the polymers, we chose a steady state measurement technique based on the sandwich electrode microstructure (19). 

Figure 4 shows the application (6) of potentials to the Pt and Au electrodes of the sandwich (vs. a reference electrode elsewhere in the contacting electrolyte solution) so that they span the E° of the poly-[Co(II/I)TPP] couple (Fig. 4B). There is a consequent redistribution of the concentrations of the sites in the two oxidation states to achieve the steady state linear gradients shown in the inset. Figure 4C represents surface profilometry of a different film sample in order to determine the film thickness from that the actual porphyrin site concentration (0.85M). The flow of self exchange-supported current is experimentally parameterized by applying Fick s first law to the concentration-distance diagram in Fig. 4B ...

Fig. 5. Plot of apparent electron self exchange rate constants kf P, derived from polymer De values for films containing the indicated metals, mixed valent states, and ligands, all in acetonitrile, using Equation 2, vs. literature heterogeneous electron transfer rate constants k° for the corresponding monomers in nitrile solvents. See Ref. 6 for details. (Reproduced from Ref. 6. Copyright 1987 American Chemical Society.)...

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

Outer-sphere electron transfer reactions involving the [Co(NH3)6]3+/2+ couple have been thoroughly studied. A corrected [Co(NH3)6]3+/2+ self-exchange electron transfer rate (8 x 10-6M-1s-1 for the triflate salt) has also been reported,588 which is considerably faster than an earlier report. A variety of [Co(NH3)g]3+/2+ electron transfer cross reactions with simple coordination compounds,589 organic radicals,590,591 metalloproteins,592 and positronium particles (electron/ positron pairs)593 as redox partners have been reported. 

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