Using the Marcus cross-relation

The following data were obtained for cytochrome c and cytochrome C55, two proteins in which heme-bound iron ions shuttle between the oxidation states Fe(II) and Fe(III)  

Then compare the estimated value with the observed value of 6.7 x 10 dm mol s.  

Strategy We use the standard potentials md eqns 5.16 (In K = vf (ju/ilT) and 5.17a ( ceu = hi Ef) to calculate the equihbrium constant K. Then we use eqn 8.34, the calculated value of K, and the self-exchange rate constants fcji to calculate the rate constant fcots- 

Solution The two reduction half-reactions ue Right cytochrome c(ox) -1- e — cytochrome c(red) Eg = -1-0.260 V 

Therefore, K = 0.36. From eqn 8.34 and the self-exchange rate constants, we calculate 

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Calculation of rate constants (k]2) for organic electron transfer processes, using the Marcus cross relations (62) and (63)... 

E21.20 Using the Marcus cross-relation (Equation 21.16), we can calculate the rate constants. In this equation [k i = [kn k7i K. 2 f 2], the values of k and A22 can be obtained from Table 21.12. We can assume/i2 to be unity. The redox potential data allows us to calculate because = RTNF] nK. The value of can be calculated by subtracting the anodic reduction potential (the couple serves as the anode) from the cathodic one. 

E2I.21 Using the Marcus cross-relation (Equation 21.16), we can calculate the rate constants. In this equation [1ti2 =... 

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. 

Estimated using the Marcus cross relation see references 51 and 91. 

Data are given in Table 10-7 to illustrate certain facets of the Marcus cross relation. They refer to six reactions in which the cage complex Mn(sar)3+ is reduced or Mn(sar)2+ oxidized.34 These data were used to calculate the EE rate constant for this pair. The results of the calculation, also tabulated, show that there is a reasonably self-consistent value of fcEE for Mn(sar)3+/Mn(sar)2+ lying in the range 3-51 L mol-1 s-1. When values34 for an additional 13 reactions were included the authors found an average value of kEE = 17 L mol 1 s l. 

Even in the domain of inorganic redox chemistry relatively little use has been made of the full potential of the Marcus theory, i.e. calculation of A, and A0 according to (48) and (52) and subsequent use of (54) and (13) to obtain the rate constant (for examples, see Table 5). Instead the majority of published studies are confined to tests of the Marcus cross-relations, as given in (62)-(65) (see e.g. Pennington, 1978), or what amounts to the same type of test, analysis of log k vs. AG° relationships. The hesitation to try calculations of A is no doubt due to the inadequacy of the simple collision model of Fig. 4, which is difficult to apply even to species of approximately spherical shape. 

X 10 s ), the Marcus cross relation (Equation 6.26a) can be used to calculate the reaction rates for the reduction of Cu -stellacyanin by Fe(EDTA) and the oxidation of Cu -stellacyanin by Co(phen)3 +. E°(Cu ) for stellacyanin is 0.18 V vs. NHE, and the reduction potentials and self-exchange rate constants for the inorganic reagents are given in Table 6.3. For relatively small AE° values,/12 is 1 here a convenient form of the Marcus cross relation is log k,2 = 0.5[log kn + log 22 + 16.9AE°2]. Calculations with kn, 22, and AE°2 from experiments give k,2 values that accord quite closely with the measured rate constants. 

Figure 8.9 Test of Marcus cross-relation for methyl-transfer reactions in sulpholane. Comparison of measured rates of nine methyl-transfer reactions in sulpholane compared to those calculated from the Marcus cross-relation (closed circles). (The open circles refer to some experimental identity rates, used in the Marcus calculation, that necessarily fit the line of unit slope.) From E.S. Lewis, Ref. [27,a,c].

There is growing support for the approximations that, in bimolecular electron transfer at least, A can be divided into independent contributions A (A" ), A (B) from the two reactants (c./. an earlier derivation of the Marcus cross relation on this basis), and moreover that A(A ) = A(A) (c/. Ref. 16). Using these assumptions, Frese has calculated reorganization energies for a large number of self-exchange and cross-reactions. In many cases values of A for individual redox couples are consistent from one reaction to another. Of interest are the different values of A for... 

To derive the Marcus cross-relation, we use eqn 8.33 to write the rate constants for the self-exchange reactions as... 

Electron self-exchange reaction between O2 and 02 was then discussed, and developments before and after an experimentally determined rate constant for this reaction was published, were also summarized. Related to this, the problem of size differences between O2 or 02 and their typical metal-complex electron donors or acceptors was recently solved quantitatively by addition of a single experimentally accessible parameter, A, which corrected the outer-sphere reorganization energy used in the Marcus cross relation. When this was done, it was found that rate constants for one electron oxidations of the superoxide radical anion, 02 , by typical outer-sphere oxidants are successfiiUy described by the Marcus model for adiabatic outer-sphere electron transfer. 

The fact that methyl transfers in water are elementary reactions suggests that they should be a better ground to test the applicability of free-energy relationships in 8 2 reactions. Albery and Kreevoy made an extensive smdy of such reactions and attempted to interpret their free-energy dependence using the Marcus cross-reaction scheme. According to the Marcus cross-relation, the free energy of activation of the cross-reaction... 

Using the Marcus equations, it is possible in some cases to obtain good correlation between the rate constants of isotopic exchange, i.e. the rate constants for redox reactions for two pairs of ions that are identical in composition (e.g. Fe(CN) "/Fe(CN)g, and Mo(CN)g /Mo(CN) "), and the rate constant for a reaction between different ions (Fe(CN) "" +Mo(CN) ), called cross-relations (see, for example [206,233,2j4]. 

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