Self-exchange reactions rate constants calculation

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

Ky and ky are the equilibrium constant and cross reaction rate constant for Eq. 2, kii and kjj are the self-exchange ET rate constants, and Z is a preexponential factor usually set at 10 (results are quite insensitive to its value). Because Ky can be calculated from the difference in formal oxidation potentials for the components, Eq. 2 states that ky only depends upon the formal oxidation potential and intrinsic (AG° = 0, or self-ET) rate constant for each couple involved. 

The Marcus therory provides an appropriate formalism for calculating the rate constant of an outer-sphere redox reaction from a set of nonkinetic parameters1139"1425. The simplest possible process is a self-exchange reaction, where AG = 0. In an outer-sphere electron self-exchange reaction the electron is transferred within the precursor complex (Eq. 10.4). 

Calculation of rate constants for cross reactions from known data on self-exchange reactions. 

Such self-exchange reactions cannot be followed by optical spectroscopy since no chemical change is involved. However, ESR spectroscopy provides a unique capability to do so because the spin states of the protons in the radical effectively label a particular radical. The reaction leads to broadening of the ESR lines, which is detectable at rates of transfer >10 s. From the dependence of line width on phenolate concentration it was calculated that the rate constant for this self-exchange reaction is 1.9 x 10 ... 

Because AG is not directly measurable, 2, can be calculated from the observed rate constant k for the self-exchange reaction by using Equation 1.18. This is obtained by substituting Equation 1.17 into Equation 1.11 and assuming that k = 1. 

Rate constants for outer-sphere electron transfer reactions that involve net changes in Gibbs free energy can be calculated using the Marcus cross-relation (Equations 1.24—1.26). It is referred to as a cross-relation because it is derived from expressions for two different self-exchange reactions. 

Although we shall not delve into the theory, it is possible to calculate the terms on the right-hand side of equation 25.56, and thus to estimate values of AG for self-exchange reactions. The rate constant, k, for the self-exchange can then be calculated using equation 25.57 the results of such calculations have been checked against much experimental data, and the validity of the theory is upheld. 

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The equation is best understood by using a specific example. Suppose we wish to calculate the rate constant at 0°C for the reduction of [Co(bipy)j] " by [Co(terpy)3]. The required self-exchange reactions are... 

A more rigorous way of testing theoretical calculations of the rate constant is to consider a series of related reactions and compare variations in the observed values with those predicted by variations in the molecular properties of the reactants, especially where these are known experimentally. Self-exchange reactions offer the best prospects for such a test. For these reactions, the free energy of activation may be expressed (by applying Equations (9.18) and (9.12)) by ... 

Using Sutin s semi-classical form of the Marcus model [34], Kozik and Baker predicted (calculated) a rate constant for the self-exchange reaction between Iqx and lie of (3.6 0.1) X 10 M s (at zero ionic strength), which was in close agreement with their experimental (measured) value of (1.1 0.2)x 10 M s after extrapolation to zero ionic strength. 

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