The Marcus cross-relation

Because electron transfer reactions are of such importance for metabolism and other biological processes, to discuss them quantitatively we need to be able to predict their rate constants Marcus theory provides a way. 

It follows from eqns 8.26 and 8.27 that the rate constant may be written as 

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

For the net reaction and the self-exchange reactions, the Gibbs energy of activation may be written from eqn 8.29 as 

For the self-exchange reactions A Ggo = AjGJa = 0 and hence A Gdd = Add and A Gaa = Haa. It follows that 

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These equations do not necessarily show the actual charges the important point is that all three are single-electron events. The asterisks can be thought of as an isotopic label, but need not be anything that concrete, since certain line-broadening techniques (Section 11.5) provide EE rate constants without them. The Marcus cross relation is an expression for kA% as a function of kAA, bb> and A, the equilibrium constant for Eq. (10-67). It reads,... 

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. 

Marcus theory. Consider that the reorganization energy for the ET reaction, AAb, can be approximated as the mean of the reorganization energies for the EE reactions Aab = (Aaa + ABb)/2. Show that substitution of this expression into Eq. (10-63) gives the usual form of the Marcus cross relation. 

X 10 M at 1.0 M ionic strength. A simple outer-sphere mechanism can be ruled out since k is more than two orders of magnitude greater than predicted by the Marcus cross-relation, and an oxygen atom transfer mechanism has been proposed. ... 

Hydrogen atom abstraction by non-radical, metal- species, once considered unlikely, has been shown in recent years to operate in a large number of reactions (113-115) with the kinetics responding to the thermodynamic driving force and intrinsic barriers as predicted by the Marcus cross relation (116). 

It was recently shown (Ratner and Levine, 1980) that the Marcus cross-relation (62) can be derived rigorously for the case that / = 1 by a thermodynamic treatment without postulating any microscopic model of the activation process. The only assumptions made were (1) the activation process for each species is independent of its reaction partner, and (2) the activated states of the participating species (A, [A-], B and [B ]+) are the same for the self-exchange reactions and for the cross reaction. Note that the following assumptions need not be made (3) applicability of the Franck-Condon principle, (4) validity of the transition-state theory, (5) parabolic potential energy curves, (6) solvent as a dielectric continuum and (7) electron transfer is... 

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. 

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

Similarly, as for A-D systems, values of Ao and Aj have been estimated on the basis of the Marcus cross-relation, leading to the conclusion that Ai is very small, probably not exceeding 0.05-0.10eV. The Ao values for arromatic hydrocarbons lie in the range 0.32-0.43 eV [94] (determined for the A/A couple, but similar values... 

The Marcus cross-relation can be expressed as a LFER since the logarithms of the rate constants are proportional to the free energies of activation. Thus... 

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 =... 

The calculations presented here show that many different factors must be considered in estimating the rate constant. Nevertheless, electron transfer theory is remarkably successful in describing this elementary solution reaction. Theory has gone much further than described here, especially in developing the quantum-mechanical description of electron transfer. More details can be found in recent reviews [29, 30]. There are other related topics which have not been discussed in this section. They include, for example, photo-induced electron transfer [30], and the Marcus cross-relation [5]. 

The Marcus Cross-Relation The rate constant, /c12, for electron transfer between two species, Am and B" (Equation 1.20) that are not related to one another by oxidation or reduction, is referred to as the Marcus cross-relation (MCR). 

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. 

When the self-exchange rates ki i are corrected for work terms or when the latter nearly cancel, the cross-reaction rate ki2 is given by the Marcus cross relation,... 

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. 

The success of the Marcus cross relation with stellacyanin indicates that the copper site in the protein is accessible to inorganic reagents. The rate constants... 

The Marcus cross relation can in general be applied to correlate the rate constants of self-exchange reactions, Ku and K12, with the rate constant, K12, and equilibrium constant, JC12, of a cross reaction between two redox systems. For the self-exchange reactions... 

If the reaction would proceeded via the outer-sphere mechanism, reaction (45), the calculated electron transfer rate constant applying the Marcus cross-relation 85-87) would be 1.6xlO M s (35). However, the measured rate constant is 3.5xlO M s (31,88), which indicates that the reaction proceeds via the inner-sphere mechanism. 

Mayer has observed transfer of H to TEMPO from an N-H bond in the tris iron(II) complex of 2,2 -bi(tetrahydropyrimidine) (1.11), and has shown that the Marcus cross relation accurately models its negative enthalpy of activation [38]. As previously suggested in another context [39], the high point on the enthalpy surface appears to occur before the transition state. 

Actinides. The reactions [Ru(NH3)6] +U +, [R.u(NH3)5H20] +-l-U +, and [Ru(en)3] ++U + arc second-order, are independent of H+ concentration, and ha> e almost zero enthalpy of activation.The latter feature is attributed to the concentration of charge in the transition state, as with other reactions between highly charged cations which have been noted to have low, even negative A/f, and highly negative By means of the Marcus cross-relation, the self-... 

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