The Cross Relation

The Cross Relation.—The equations (16) and (17) due to Marcus relate the specific 

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A powerful application of outer-sphere electron transfer theory relates the ET rate between D and A to the rates of self exchange for the individual species. Self-exchange rates correspond to electron transfer in D/D (/cjj) and A/A (/c22)- These rates are related through the cross-relation to the D/A electron transfer reaction by the expression... 

The cross relation has proven valuable to estimate ET rates of interest from data tliat might be more readily available for individual reaction partners. Simple application of tire cross-relation is, of course, limited if tire electronic coupling interactions associated with tire self exchange processes are drastically different from tliose for tire cross reaction. This is a particular concern in protein/protein ET reactions where tire coupling may vary drastically as a function of docking geometry. 

Once we have obtained these barriers, we are now able to go back, insert these values into the cross-relation, and test the... 

A model has been considered for Sn2 reactions, based on two interacting states. Relevant bond energies, standard electrode potentials, solvent contribntions (nonequi-librinm polarization), and steric effects are included. Applications of the theory are made to the cross-relation between rate constants of cross- and identity reactions, experimental entropies and energies of activation, the relative rates of Sn2 and ET reactions, and the possible expediting of an outer sphere ET reaction by an incipient SN2-type interaction (Marcus, 1997). 

From Eqn. (14.3), one can easily derive the cross relations between extensive and intensive state variables as, for example,... 

For convenience we will make a simple demonstration of how to transform a 2x2 matrix problem to complex symmetric form. In so doing we will also recognise the appearence of a Jordan block off the real axis as an immediate consequence of the generalisation. The example referred to is treated in some detail in Ref. [15], where in addition to the presence of complex eigenvalues one also demonstrates the crossing relations on and off the real axis. The Hamiltonian... 

There were many predictions arising from the theory and its extension to electrochemical and other systems [11, 23, 24]. One such prediction, the cross-relation , was based on the relation between the A for reactions between two different redox systems, A 2, to the A s of the self-exchange reactions, An and A12, for each of the two systems (A,2 = l/2(An + A22)). The result for k12, the rate constant for the cross-reaction, is... 

The cross-relation (equation (1.6)), has also been applied successfully to transfers of CH3 [31] and to transfers of H" [32-33], while equation (1.4) has been used to treat proton transfers [34] and proton bound dimers [34d]. As already noted, the intersecting parabolas of Fig. 1.3 would not be applicable and so some other treatment was needed to understand the... 

As for estimates of individual rate constants via the cross relations, this procedure seems to work well for organic electron-transfer processes, and the few existing limitations are of the same kind as those encountered for inorganic redox processes. 

Maxwell first noted the cross relations based on a property of the total differentials of the state functions. The cross differentiations of a total differential of the state function are equal to each other. Table 1.14 summarizes the total differentials and the corresponding Maxwell relations. The Maxwell relations may be used to construct important thermodynamic equations of states. 

The cross relations can be seen in a reversible change of a rectangular rubber sheet subjected to two perpendicular forces Fx and Fy under isothermal conditions. If the extent of stretching in both directions of x andy are Ax and Ay, we have... 

Some of the new theoretical relations, the cross-relation between the rates of a cross-reaction of two difierent redox species with those of the two relevant selfexchange reactions, were later adapted to non-electron transfer reactions involving simultaneous bond rupture and formation of a new bond (atom, ion, or group transfer reactions). The theory had to be modified, but relations such as the crossrelation or the effect of driving force (—AG°) on the reaction rate constant were again obtained in the theory, in a somewhat modified form. For example, apart from some proton or hydride transfers under special circumstances, there is no predicted inverted effect. Experimental confirmation of the cross-relation followed, and an inverted effect has only been reported for an H+ transfer in some nonpolar solvents. The various results provide an interesting example of how ideas obtained for a simple, but analyzable, process can prompt related, yet different, ideas for a formalism for more complicated processes. 

It is called the cross-relation because it is algebraically derived from expressions for the two related electron self-exchange reactions shown inEquations 1.21 and 1.22.. Associated with these reactions are two self-exchange rate constants k and k22 and reorganization energies Xu and 22-... 

When nuclear tunneling occurs, the system passes from the R surface to the P surface by crossing horizontally from the first to the second of these surfaces. This is depicted schematically by the horizontal line that extends from a to b in Figure 1.4. In practice, at room temperature, and for reactions in the normal region, nuclear tunneling usually accounts for only minor contributions to rate constants. The cross-relation in the normal region is even less affected by nuclear tunneling, due to partial cancellation of the quantum correction in the ratio 12/( 11 22 12)... 

Both involve high-pressure electrochemistry. One is the measurement of the pressure dependence of the rate constant for electron transfer in a given couple at an electrode, but it is not immediately clear how feg] and the corresponding volume of activation relate to feex and AV, respectively, for the self-exchange reaction of the same couple. This is a major theme of this chapter, and is pursued in detail below. The other method involves invocation of the cross relation of Marcus [5], which expresses the rate constant ku for the oxidation of, say, A by B+ in terms of its equilibrium constant and the rate constants kn and fe22 for the respective A+/A and B+/B self-exchange reactions ... 

When applied to a side coefficient, the end result of this operation is a new differential quotient and therefore one of the cross relations of interest to us. Let us look at the approach using our concrete example. Again, we will start with the differential quotient dSld —p))j. ... 

The cross relation in question, which is raie of Maxwell s relations, is here obtained directly through flipping. If a main coefficient is flipped, it is generally reproduced there is no new informatiOTi as we can easily see ... 

For the mathematically inteiested In order to derive Eq. (12.2), we will refer back to the cross relation discussed in Sect. 9.3 known as n n coupling. When one substance tries to displace (or favor) another one, this happens reciprocally and with equal strength. The corresponding displacement coefficients are equal as can easily be shown by applying the flip rule (main equation dW = —pdV + TdS + ji drif + p dn ) ... 

The "cross-relation which relates the rate constant fcj, for an electron transfer reaction between two different redox species -I- fi to the rate constant k. for the self-exchange reaction (isotropic excKange) A -t- A, and to that of B, 22. the equilibrium constant... 

It has not been possible, of course, to determine any self-exchange ft s in the reaction center and so pursue tests such as that of the cross-relation, eq. (2.3). Effects of temperature on some of the ft s and on AJBg have been determined and provide useful information. Reactions such as (3.1)-(3.2), (3.5), (3.7) and (3.8) are activationless , and so their rates are, interestingly enough, "maximized in nature, for the given separation distances. 

The adaptation of the familiar cross relation of Marcus to high-pressure kinetics provides an example of the utility of AFceu data. The cross-relation connects the rate constant A ab of redox reaction (A + B products) in homogeneous solution to those (kAA, bb) for the related selfexchange reactions (A+ + A A -b A, etc.) and the equilibrium constant Kab-... 

II is of interest that the slopes of the Br0nsted plots of these reactions where X and Y are NO2, Cl , Br and I are close to 0.5 (the expected value if the transition state is midway between reactants and products), but smaller for OH and CN which kinetically are poor nucleophiles, i.e., have relatively high energy-barriers and hence product-like transition states (in accordance with the Hammond postulate). A similar pattern is observed for the a-deuterium isotope effects for methyl transfers, for which the cross-relation predicts that the fractionation factors will be related by the equation the view that the theoretical... 

The cross-relation may be applied to systematised calculations of electron-transfer rate constants. Assuming that it is a good approximation, the rate constant of an electron-transfer reaction (9.31) can be estimated from a knowledge of the equilibrium constant and the rate constants of the two related exchange reactions ((9.32) and (9.33)). Likewise, if the rate constants of the electron-transfer reaction and of one of the exchange reactions... 

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