Adiabatic redox reactions

The adiabatic redox reactions at electrodes were first considered by MARCUS /40a,145/ in a classical (semiclassical) framework. lEVICH, DOGONADZE and KUSNETSOV /146,147/, SGHMICKLER and VIELSTICH /169/ a.o. have developed a quantum theory for non-adiabatic electron transfer electrode reactions based on the oscillator-model. The complete quantum-mechanical treatment of the same model by CHRISTOV /37d,e/ comprises adiabatic and non-adiabatic redox reactions at electrodes. 

FIGURE 35.2 Scheme of diabatic (solid line) and adiabatic (dashed line) free-energy curves for a simple electrochemical redox reaction Ox —> Red. 

Pig. 2-37. Redox reaction cycle FeJ5 - Fejj + ei iD, - FeJ in aqueous solution solid arrow=adiabatic electron transfer, dotted arrow = hydrate structure reorganization X = reorganization energy ered.d = most probable donor level eox.a = most probable acceptor level. 

In addition to redox reactions due to the direct transfer of electrons and holes via the conduction and valence bands, the transfer of redox electrons and holes via the surface states may also proceed at semiconductor electrodes on which surface states exist as shown in Fig. 8-31. Such transfer of redox electrons or holes involves the transition of electrons or holes between the conduction or valence band and the surface states, which can be either an exothermic or endothermic process occurring between two different energy levels. This transition of electrons or holes is followed by the transfer of electrons or holes across the interface of electrodes, which is an adiabatic process taking place at the same electron level between the surface states and the redox particles. 

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... 

As a generalization, nonadiabaticity tends to be greater in redox reactions than for the ion-transfer reactions (e.g., Oz + 4H+ + 4e - 2H20) where the value of VI2 can be as much as 0.5 eV (Newton, 1986 Bockris and Sidik, 1998). Such reactions will be largely adiabatic. 

The essentials of quantum kinetics were in place by 1954, Weiss having added to the Gurney theory a comprehensive theory of redox reactions. By this date, tunneling, adiabatic and non-adiabatic electron transfer, the simplicity introduced by considering redox reactions between isotopes, the separate contribution from outer sphere and inner sphere, and in particular the equation for the reorganization energy involving and stat had all been published. 

The transmission coefficient k — 1 for weak overlap of electronic states of reactants and products in the transition state. It is strong enough to be adiabatic but yet weak enough for the free energy of activation not to have an appreciable contribution from the resonance energy. This condition is almost fulfilled by outer sphere redox reactions at electrodes. 

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. 

In a coupled electron transfer reaction, the preceding adiabatic reaction step influences the experimentally-determined rate constant even though the electron transfer step is the slowest for the overall redox reaction. This occurs when the relatively fast reaction step which precedes electron transfer is very unfavorable (i.e. Kx (kx/kfix) l)- In Hii case, ks will be influenced by the equilibrium constant for that non-electron transfer process such that ks = kgT Kx (Harris et ah, 1994 Davidson, 1996). It follows that the experimentally-derived X ( lobs) may contain contributions from both the electron transfer event and the preceding reaction step (i.e. obs [ ET. x])- For example, lo sfor interprotein electron transfer reactions may reflect contributions from an intracomplex rearrangement of proteins after binding to achieve an optimum orientation for electron transfer. As with a true electron transfer reaction, k3 will vary with AG° since ks is proportional to ksT, although H b may also be affected by the coupling. 

Fn in Eq. (33) takes into account the fact that the electron transfer can also occur for reactant molecules without fully overcoming the energy barrier. At normal temperatures for electrode reactions F approaches 1 [131]. Therefore Eq. (33) is often used without the F term, in this equation was estimated to have a value of the order of 60 pm [138]. However, the quantitative estimation of depends on the adiabaticity factor k (see Eq. (24)), which varies with the reactant-electrode separation distance. It was suggested [139] that the electrode reactions should be more adiabatic than the corresponding homogeneous redox reactions. But as the reaction site, for outer-sphere systems, is probably separated from the electrode surface by a layer of solvent molecules [129], k may drop below 1 since the reaction may become less adiabatic. These problems were more extensively discussed in [131]. 

The above results for outer-sphere redox reactions were obtained by CHRISTOV /37d/ using an essentially different approach based on the general formulation of reaction rate theory. Equation (63.IV), however, differs from a similar result of DOGONADZE /147,151/ for adiabatic processes, which incorrectly gives the same activation energy as for non-adiabatic processes i.e., it neglects the large resonance interaction. 

We now turn to the inner-sphere redox reactions in polar solvents in which the coupling of the electron with both the inner and outher solvation shells is to be taken into account. For this purpose a two-frequency oscillator model may the simplest to use, provided the frequency shift resulting from the change of the ion charges is neglected. The "adiabatic electronic surfaces of the solvent before and after the electron transfer are then represented by two similar elliptic paraboloids described by equations (199.11), where x and y denote the coordinates of the solvent vibrations in the outer and inner spheres, respectively. The corresponding vibration frequencies and... 

In this situation the rate constant of inner-sphere redox reactions can be evaluated by applying the adiabatic formulation of reaction rate theory and by using, for instance, the basic equation (103.III) which yields (since g = 1)... 

The adiabatic inner-sphere redox reactions were first treated by MARCUS /145/, who made use of the classical and semiclassical statistical theory A quantum-mechanical treatment of the two-frequency oscillator model by DOGONADZE and KUSNETSOV /147/ provides tractable rate expressions for non-adiabatic processes in both high and low temperature ranges. Similar results were obtained by KESTNER, LOGAN and JORTNER /148/. 

For outer-sphere electron transfer, the rate equations (62.IV) and (63.IV) for non-adiabatic and adiabatic reactions may be used by introducing the electrode potential cp through the relation (107.IV) or the overvoltage through equation (114.IV), instead of the reaction heat Q. For inner-sphere redox reactions the expressions (83 IV) and (85.IV) can be used in a similar way for electronically non-adiabatic and adiabatic reactions, respectively. The conditions of validity of the Tafel equation are then given by ( 12.IV) or (116.IV). 

Non-adiabatic multiphonon electron tunnelling theory has been used to interpret kinetic data on a number of redox reactions of cytochrome c and yields a self-exchange rate constant for the protein of 1.7 x 10 s (ATT =4.9 kcal... 

The question arises above which interaction energy must a reaction be considered to be adiabatic This is difficult to answer, especially for electrode reactions, because it depends on the distance of the reacting species during the electron transfer. In the case of reactions in homogeneous solutions, Newton and Sutin [10] have estimated for typical transition-metal redox reactions that V p 0.025 eV is a reasonable limit above which a reaction must be considered to be adiabatic. This problem will be discussed again later in connection with some quantum mechanical models for electron transfer. 

For example, assuming anhydrite-magnetite-calcite-pyrite-pyrrhotite buffers redox in sub-seafloor reaction zones and a pressure of 500 bars, dissolved H2Saq concentrations of 21 °N EPR fluid indicate a temperature of 370-385°C. However, the estimated temperatures are higher than those of the measurement. This difference could be explained by adiabatic ascension and probably conductive heat loss during ascension of hydrothermal solution from deeper parts where chemical compositions of hydrothermal solutions are buffered by these assemblages. 

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