Rate constant Marcus equation

The composite rate constant shows a large kinetic isotope effect for ArCDj (kii/ko = 4.5-7.3), strong salt effects, specific cation accelerations (Na" ) and inhibitions (BU4N ), and a marked dependence on solvent ratio and dielectric constant. Marcus-equation estimates of k, and fc i, the latter diffusion controlled, yield 2 = 3.6 x 10 (OAc ) and 80 (H2O) M s". ... 

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... 

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 Brpnsted equations relate a rate eonstant k to an equilibrium constant K. In Chapter 6, we saw that the Marcus equation also relates a rate term (in that case AG ) to an equilibrium term AG°. When the Marcus treatment is applied to proton... 

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...

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... 

Figure 13. Test for the consistency of AG evaluated from the intrinsic rate constant (log kj using Marcus Equation 4 (a), Rehm-Weller Equation 17 (b), and Marcus-Levine-A gmon Equation 18 (c) at various potentials.

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... 

The kinetics are much more complex and depend on the reorganization of the molecular framework [141], the solvation shell, and the electrostatic interaction. A semi-quantitative estimation of rate constants may be obtained with the well-known Marcus equation [142]. The calculated data compare quite well with experimental values. Most of the experimental hydrocarbon... 

On the basis of the very negative activation entropies, the transition states for the addition are highly ionic, i.e. there is a large degree of electron transfer in the transition state as with the hydroxyalkyl radicals (Sect. 2.1.1). In support of this is the fact that the rate constants for addition depend on the reduction potentials of the nitrobenzenes, varied by the substituent R3 in a way describ-able by the Marcus equation for outer-sphere electron transfer [19]. 

A simple diagram depicting the differences between these two complementary theories is shown in Fig. 1, which represents reactions at zero driving force. Thus, the activation energy corresponds to the intrinsic barrier. Marcus theory assumes a harmonic potential for reactants and products and, in its simplest form, assumes that the reactant and product surfaces have the same curvature (Fig. la). In his derivation of the dissociative ET theory, Saveant assumed that the reactants should be described by a Morse potential and that the products should simply be the dissociative part of this potential (Fig. Ib). Some concerns about the latter condition have been raised. " On the other hand, comparison of experimental data pertaining to alkyl halides and peroxides (Section 3) with equations (7) and (8) seems to indicate that the simple model proposed by Saveant for the nuclear factor of the ET rate constant expression satisfactorily describes concerted dissociative reductions in the condensed phase. A similar treatment was used by Wentworth and coworkers to describe dissociative electron attachment to aromatic and alkyl halides in the gas phase. ... 

The reversibility of the [Os(bpy)3]3+/2+ couple makes it useful for the determination of the electron self-exchange rates of other couples by application of the Marcus cross-reaction equation. Recently, this has been applied to the oxidation of S032- to S042- (622). The new rate constant for this reaction of 1.63 x 107 M-1 sec-1 is consistent with the... 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

The rate constant expression in equation (18) derived by Marcus is complete in the sense that it includes a pre-association between reactants, a time dependence arising from the frequency with which the reactants collide, and the thermal activation required for electron transfer to occur. On the other hand, the quantum mechanically derived expressions for fcet are dependent upon the interreactant separation, and the dependence on V must be included explicitly. 

The second and far more common approach to testing the predicted dependence of kob on AG has been based on the so-called Marcus cross-reaction equation. The cross-reaction equation interrelates the rate constant for a net reaction, D+A- D++A ( el2), with the equilibrium constant (Kl2) and self-exchange rate constants for the two-component self-exchange reactions D+ 0 (Zen) and A0/- (k22). Its derivation is based on the assumption that the contributions to vibrational and solvent trapping for the net reaction from the individual reactants are simply additive (equation 63). The factors of one-half appear because only one of the two components of the self-exchange reactions is involved in the net reaction. The expression for A0 in equation (63) is an approximation. Note from equation (23) that k is a collective property of both reactants and the approximation in equation (63) is valid only if the reactants have similar radii. 

The interpretation of reactivities here provides a particular challenge, because differences in solvation and bond energies contribute differently to reaction rates and equilibria. Analysis in terms of the Marcus equation, in which effects on reactivity arising from changes in intrinsic barrier and equilibrium constant can be separated, is an undoubted advantage. Only rather recently, however, have equilibrium constants, essential to a Marcus analysis, become available for reactions of halide ions with relatively stable carbocations, such as the trityl cation, the bis-trifluoromethyl quinone methide (49), and the rather less stable benzhydryl cations.19,219... 

Marcus5 8 taught us that the most appropriate and useful kinetic measure of chemical reactivity is the intrinsic barrier (AG ) rather than the actual barrier (AG ), or the intrinsic rate constant (kQ) rather than the actual rate constant (k) of a reaction. These terms refer to the barrier (rate constant) in the absence of a thermodynamic driving force (AG° = 0) and can either be determined by interpolation or extrapolation of kinetic data or by applying the Marcus equation.5 8 For example, for solution phase proton transfers from a carbon acid activated by a ji-acceptor (Y) to a buffer base, Equation (1), k0 may be determined from Br A ns ted-type plots of logki or... 

In most cases, the rate constants kucK were converted to k [Equation (18)] assuming that mechanism (b) of Scheme 6 accounts for the uncatalyzed reaction. Clearly, the rate constant kfc for phorone should not be converted to k e, because the uncatalyzed reaction is due to an intramolecular 1,5-H shift rather than to pre-equilibrium ionization of the enol. Conversion of kjf = 2.6 s-1 would give k = 1.8 x 10um-1s-1, which is higher than any of the values observed for simple enols and more than two orders of magnitude higher than that predicted by the Marcus equation for k . 

Similar arguments apply to the six a-carboxy-substituted ketones that have been studied by Kresge and coworkers (entries acetoacetate to oxocyclobutane-2-carboxylate in Table 1). Kresge already noted that the rate constants kucK observed for the uncatalyzed ketonization of some of these compounds would give unrealistically high calculated values for k e near or above 1011 m-1 s-1 using Equation (18). Indeed, these calculated values of k are about two orders of magnitude above those expected from the Marcus relation except that for 4,4,4-trifluoroacetate. The rate constants k c observed for the formation of these a-carboxy-substituted ketones are, however, close to those expected for the protonation of the neutral enols by water, k = kf. 

The acidity constants of protonated ketones, pA %, are needed to determine the free energy of reaction associated with the rate constants ArG° = 2.3RT(pKe + pK ). Most ketones are very weak bases, pAT < 0, so that the acidity constant K b cannot be determined from the pi I rate profile in the range 1 < PH <13 (see Equation (11) and Fig. 3). The acidity constants of a few simple ketones were determined in highly concentrated acid solutions.19 Also, carbon protonation of the enols of carboxylates listed in Table 1 (entries cyclopentadienyl 1-carboxylate to phenylcyanoacetate) give the neutral carboxylic acids, the carbon acidities of which are known and are listed in the column headed pA . As can be seen from Fig. 10, the observed rate constants k, k for carbon protonation of these enols (8 data points marked by the symbol in Fig. 10) accurately follow the overall relationship that is defined mostly by the data points for k, and k f. We can thus reverse the process by assuming that the Marcus relationship determined above holds for the protonation of enols and use the experimental rate constants to estimate the acidity constants A e of ketones via the fitted Marcus relation, Equation (19). This procedure indicates, for example, that protonated 2,4-cyclohexadienone is less acidic than simple oxygen-protonated ketones, pA = —1.3. 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

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