Marcus’ rate constant

The TST rate constant for electronically adiabatic ET reactions is the well-known Marcus rate constant kjjj [27-29], In the language of this chapter, solvent dynamical effects can alter the actual rate from this limit due to the friction influence. The corresponding GH equations for kct = / kfj are strictly analogous... 

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

The symmetry coefficient = —P d nk/dAE is usually close to j, in agreement with the Marcus formula. Turning to the quantum limit, one observes that the barrier transparency increases with increasing AE as a result of barrier lowering, as well as of a decrease of its width. Therefore, k grows faster than the Arrhenius rate constant. At 7 = 0... 

If the intrinsic barrier AGq could be independently estimated, the Marcus equation (5-69) provides a route to the calculation of rate constants. An additivity property has frequently been invoked for this purpose.For the cross-reaction... 

According to the Marcus theory [64] for outer-sphere reactions, there is good correlation between the heterogeneous (electrode) and homogeneous (solution) rate constants. This is the theoretical basis for the proposed use of hydrated-electron rate constants (ke) as a criterion for the reactivity of an electrolyte component towards lithium or any electrode at lithium potential. Table 1 shows rate-constant values for selected materials that are relevant to SE1 formation and to lithium batteries. Although many important materials are missing (such as PC, EC, diethyl carbonate (DEC), LiPF6, etc.), much can be learned from a careful study of this table (and its sources). 

Both Marcus27 and Hush28 have addressed electron transfer rates, and have given detailed mathematical developments. Marcus s approach has resulted in an important equation that bears his name. It is an expression for the rate constant of a net electron transfer (ET) expressed in terms of the electron exchange (EE) rate constants of the two partners. The k for ET is designated kAS, and the two k s for EE are kAA and bb- We write the three reactions as follows ... 

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,... 

The Marcus treatment applies to both inorganic and organic reactions, and has been particularly useful for ET reactions between metal complexes that adopt the outer-sphere mechanism. Because the coordination spheres of both participants remain intact in the transition state and products, the assumptions of the model are most often satisfied. To illustrate the treatment we shall consider a family of reactions involving partners with known EE rate constants. 

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. Show that a key result is that the reaction rate constant is... 

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

In subsequent works, Marcus developed his theory further in a series of papers providing expressions for the work terms, the reorganization energy and the macroscopic ET rate constants [3 6]. Assuming a sharp liquid-liquid boundary, the solution of the mean molar volume of reactants yields an expression for of the form... 

FIG. 21 Plot of log ki2 vs. AEi/2 showing the dependence of ET rate on the driving force for the reaction between ZnPor and four aqueous reductants. The difference between the half-wave potentials for an aqueous redox species and ZnPor, AE-i/2 = AE° + A°0, where AE° is the difference in the formal potentials of the aqueous redox species and ZnPor and A° is the potential drop across the ITIES. The solid line is the expected behavior based on Marcus theory for X = 0.55 eV and a maximum rate constant of 50 cm s M . (Reprinted from Ref. 49. Copyright 1999 American Chemical Society.)... 

Fig. 4 Free energy dependence of the rate constants for charge separation and charge recombination for hairpins in which two A T base pairs separate the linker acceptor from the nucleobase donor. The dashed line is a fit of the charge separation data to the Marcus-Levitch-Jortner equation...

Marcus RA (1963) On the theory of oxidation-reduction reactions involving electron transfer. V. Comparison and properties of electrochemical and chemical rate constants. J Phys Chem 67 853-857... 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

In the previous section we have shown that the Marcus equation can be derived from Eq. (3.40). In this section, other forms of rate constants used in literatures will be derived. Notice that at T = 0, Eq. (3.40) reduces to... 

To what extent are the variations in the rate constant ratio /cs//cpobserved for changing structure of aliphatic and benzylic carbocations the result of changes in the Marcus intrinsic barriers Ap and As for the deprotonation and solvent addition reactions It is not generally known whether there are significant differences in the intrinsic barriers for the nucleophile addition and proton transfer reactions of carbocations. 

Table 2 Rate constants, equilibrium constants, and estimated Marcus intrinsic barriers for the formation and reaction of ring-substituted l-phenylethyl carbocations X-[6+] (Scheme 8)°... 

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