Electron-exchange reactions transmission coefficient

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

Table 11.2 Transmission coefficients for electron-exchange reactions." ...

The transmission coefficient k is approximately 1 for reactions in which there is substantial (>4kJ) electronic coupling between the reactants (adiabatic reactions). Ar is calculable if necessary but is usually approximated by Z, the effective collision frequency in solution, and assumed to be 10" M s. Thus it is possible in principle to calculate the rate constant of an outer-sphere redox reaction from a set of nonkinetic parameters, including molecular size, bond length, vibration frequency and solvent parameters (see inset). This represents a remarkable step. Not surprisingly, exchange reactions of the type... 

The electron exchange rate k (Eq. 10.5) is a function of the transmission coefficient k (approximately 1 for reactions with substantial electronic coupling (>4 kJ), i.e., for adiabatic reactions), the effective collision frequency in solution (Z 1011 M 1 s 1 Ar2) and the free energy term AG. ... 

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2> where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s"1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

The electron exchange rate k [Eq. (H.6)] is a function of the transmission coefficient k [approximately 1 for reactions with substantial electronic coupling... 

The calculation of the Grote-Hynes transmission coefficient has been performed for a wide variety of reaction systems in both weakly interacting and strongly interacting solvents atom exchange in rare gas solvents, and in water, electron transfer in dipolar solvents, ion pair... 

We have assumed so far that all the mass points reaching the top of the pass cross it. It is conceivable that some of them may fall back on the same side, hence the introduction of a transmission coefficient, a. It is also conceivable that the value of a may depend on the shape of the summit, in particular the reaction path (Figure 10.8). In the absence of a more accurate initial value, the approximation of a = 1/2 may be selected for an almost symmetrical barrier. It appears that this value can be much lower for reactions involving the exchange of electrons. 

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