Marcus Rate theory

Electron-transfer pathways In spite of the success of the Marcus theory, rates of electron-transfer from the iron of cytochrome c have been found to vary for different pathways.150 153 155 For example, transfer of an electron from Fe(II) in reduced cytochrome c to an Ru(III) complex on His 33 was fast ( 440 s-1)157 but... 

Eig. 2. Electron-transfer reaction rate, vs exoergicity of reaction the dashed line is according to simple Marcus theory the soUd line and data poiats are... 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

Other measures of nucleophilicity have been proposed. Brauman et al. studied Sn2 reactions in the gas phase and applied Marcus theory to obtain the intrinsic barriers of identity reactions. These quantities were interpreted as intrinsic nucleo-philicities. Streitwieser has shown that the reactivity of anionic nucleophiles toward methyl iodide in dimethylformamide (DMF) is correlated with the overall heat of reaction in the gas phase he concludes that bond strength and electron affinity are the important factors controlling nucleophilicity. The dominant role of the solvent in controlling nucleophilicity was shown by Parker, who found solvent effects on nucleophilic reactivity of many orders of magnitude. For example, most anions are more nucleophilic in DMF than in methanol by factors as large as 10, because they are less effectively shielded by solvation in the aprotic solvent. Liotta et al. have measured rates of substitution by anionic nucleophiles in acetonitrile solution containing a crown ether, which forms an inclusion complex with the cation (K ) of the nucleophile. These rates correlate with gas phase rates of the same nucleophiles, which, in this crown ether-acetonitrile system, are considered to be naked anions. The solvation of anionic nucleophiles is treated in Section 8.3. 

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). 

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

More recently, a value of 3x 10 l.mole . sec has been calculated for the exchange rate coefficient at 10 °C and zero ionic strength by Campion et al. using the Marcus theory and rate coefficients for the reactions... 

The isotopic method has been used in conjunction with a flow apparatus by Stranks, to measure the exchange between the cyclopentadienyl complexes of iron (III) and iron (II) in methanol. Separation was based on the insolubility of Fe(C5H5) in petroleum ether at —80 °C. Using Fe(II) and Fe(III) 10 M and short reaction times ( msec), a rate coefficient 8.7 x 10 l.mole .sec at — 75 °C was obtained. The rate of exchange in the presence of chloride ions and inert electrolytes was found to be more rapid. Calculations using Marcus Theory showed reasonable agreement with the experimental observations. In deuterated acetone, line broadening measurements have led to an estimate of this rate coefficient of > 10 l.mole . sec at 26 °C. 

Wahl et al ° have suggested that the rate coefficient obtained by the nmr method" at cation concentrations 5x 10 M is that for the step k2 in the above mechanism. There is reasonable agreement between the data obtained by these two different procedures. A comparison has been made of the experimental results with those obtained from the Marcus theory °. 

Recently, Hurwitz and Kustin have reinvestigated this exchange reaction using the same isotopic procedure and the 2-butanone separation method, in conjunction with a stopped flow apparatus. A rate coefficient of 2.3 x 10 l.mole . sec was obtained for the conditions, temperature 25 °C and ionic strength 0.1 M. Application of the Marcus theory to results obtained for the reaction... 

Comparison of equations (2.11) and (2.15) reveals q and r to be kikilk i and A 2//r i, respectively. This enables k to be calculated from qjr. In its simplest forms the structure of the reactive intermediate can be viewed as V(OH)Cr " (when n is 1) or as VOCr (when n is 2). Similar species which have been characterized or implied kinetically are CrOCr (ref. 33), Np02Cr (ref. 37), U02Cr (ref. 31), VOV " (ref. 34), U0Pu02 + (ref. 41), Pu02pe + (ref. 42) and FeOFe + (ref. 38). Predictions on the rate of the V(III)- -Cr(lI) system, based upon Marcus theory", have been made by Dulz and Sutin on the assumption that an outer-sphere process applies. The value arrived at by these authors is 60 times lower than the experimental value. 

On this basis Cr(V), not Cr(IV), is the kinetically important intermediate such that k = 3 k4 and k = k Jk. The hydrogen-ion dependence of the reaction rate has been discussed. Furthermore, comparisons are drawn with the rate of the Cr(VI)+Fe(phen)3 reaction, and Sullivan has speculated on the intimate nature of both mechanisms in the light of Marcus theory... 

Campion et al have compared the rates of oxidation of W(CN)g ", Mo(CN)8 , and Fe(CN)6 by Ce(IV) with those calculated from the Marcus theory. The results on these systems are given in Table 35 together with data on a number of related reactions. Agreement between observed and calculated rate coefficients is good. 

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.)... 

Rates of addition to carbonyls (or expulsion to regenerate a carbonyl) can be estimated by appropriate forms of Marcus Theory. " These reactions are often subject to general acid/base catalysis, so that it is commonly necessary to use Multidimensional Marcus Theory (MMT) - to allow for the variable importance of different proton transfer modes. This approach treats a concerted reaction as the result of several orthogonal processes, each of which has its own reaction coordinate and its own intrinsic barrier independent of the other coordinates. If an intrinsic barrier for the simple addition process is available then this is a satisfactory procedure. Intrinsic barriers are generally insensitive to the reactivity of the species, although for very reactive carbonyl compounds one finds that the intrinsic barrier becomes variable. ... 

According to the Marcus theory [9], the electron transfer rate depends upon the reaction enthalpy (AG), the electronic coupling (V) and the reorganization energy (A). By changing the electron donor and the bridge we measured the influence of these parameters on the charge transfer rate. The re-... 

Table 6.6 lists some reactions of the electron in water, ammonia, and alcohols. These are not exhaustive, but have been chosen for the sake of analyzing reaction mechanisms. Only three alcohols—methanol, ethanol, and 2-propanol—are included where intercomparison can be effected. On the theoretical side, Marcus (1965a, b) applied his electron transfer concept (Marcus, 1964) to reactions of es. The Russian school simultaneously pursued the topic vigorously (Levich, 1966 Dogonadze et al, 1969 Dogonadze, 1971 Vorotyntsev et al, 1970 see also Schmidt, 1973). Kestner and Logan (1972) pointed out the similarity between the Marcus theory and the theories of the Russian school. The experimental features of eh reactions have been detailed by Hart and Anbar (1970), and a review of various es reactions has been presented by Matheson (1975). Bolton and Freeman (1976) have discussed solvent effects on es reaction rates in water and in alcohols. 

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. 

Electron Transfer Far From Equilibrium. We have shown how the Marcus Theory of electron transfer provides a quantitative means of analysis of outer-sphere mechanisms in both homogeneous and heterogeneous systems. It is particularly useful for predicting electron transfer rates near the equilibrium potential,... 

The extent to which steric effects adversely affect the attainment of such intimate ion-pair structures would be reflected in an increase in the work term and concomitant diminution of the inner-sphere rate. This qualitative conclusion accords with the reactivity trend in Figure 16. However, Marcus theory does not provide a quantitative basis for evaluating the variation in the work term of such ion pairs. To obtain the latter we now turn to the Mulliken theory of charge transfer in which the energetics of ion-pair formation evolve directly, and provide quantitative informa-... 

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