Intrinsic barrier rate constant

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... 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

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 these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

For species 11 we will use the intrinsic barrier for hydroxide addition to trimethyl phosphate, G = 19 (calculated using rate and equilibrium data from reference 100) and assume the same value for the attack of hydroxide at sulfur on dimethyl sulfate. This (nonobservable) rate will be estimated using a Brpnsted type plot from the rate constants for diaryl sulfates (diphenyl sulfate,and bis p-nitrophenyl sulfate), estimated from the rate for phenyl dinitrophenyl sulfate,assuming equal contributions for the two nitro groups. This gives ftg = 0.95, and thus for dimethyl sulfate log k = 11.3... 

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)°... 

The intrinsic barriers for the reaction of [12+] correspond to intrinsic rate constants of ( mcoh)o = 1 x 108m-1 s-1 and (kp)0 = 450 s-1 (equation 4). This analysis shows that the thermoneutral addition of methanol to [12+] is an intrinsically fast reaction, with a rate constant that is only 50-fold smaller than that for a diffusion-limited reaction.16... 

The partitioning of ferrocenyl-stabilized carbocations [30] between nucleophile addition and deprotonation (Scheme 18) has been studied by Bunton and coworkers. In some cases the rate constants for deprotonation and nucleophile addition are comparable, but in others they favor formation of the nucleophile adduct. However, the alkene product of deprotonation of [30] is always the thermodynamically favored product.120. In other words, the addition of water to [30] gives an alcohol that is thermodynamically less stable than the alkene that forms by deprotonation of [30], but the reaction passes over an activation barrier whose height is equal to, or smaller than, the barrier for deprotonation of [30], These data require that the intrinsic barrier for thermoneutral addition of water to [30] (As) be smaller than the intrinsic barrier for deprotonation of [30] (Ap). It is not known whether the magnitude of (Ap — As) for the reactions of [30] is similar to the values of (Ap - As) = 4-6 kcal mol 1 reported here for the partitioning of a-methyl benzyl carbocations. 

This system illustrates the importance of both the thermodynamic and intrinsic barriers in determining the direction of electron transfer within a given reactant pair. In addition, systems such as the one considered here in which the oxidative and reductive pathways possess comparable rate constants afford an opportunity of controlling or switching the direction of electron transfer by modifying one of the barriers. 

AGq is the standard activation free energy, also termed the intrinsic barrier, which may be defined as the common value of the forward and backward activation free energies when the driving force is zero (i.e., when the electrode potential equals the standard potential of the A/B couple). Expression of the forward and backward rate constants ensues ... 

Figure 2.29. If the intrinsic barrier for electron transfer is small, the potential range within which the activation control prevails is accordingly narrow and the corresponding asymptote is approximately linear, as represented in the figure, where ks is the standard rate constant (i.e., the rate constant at zero driving force). Under these conditions, redox catalysts that offer a small driving force resulting in counter-diffusion control can be found. This behavior is identified by the value of the slope (F/TIT In 10). The intersection of the counter-diffusion and the diffusion asymptotes provides the value of the standard potential sought, , B.

The rate constants, efficiencies, and thermodynamic data used to extract intrinsic barriers via the analysis outlined above appear in Table I. Sample input parameters for KRKM calculations have been published elsewhere. Table II and Figure 4 contain the intrinsic barriers for the systems we have examined. 

Relationships having the same form as eq 14 can also be written for the enthalpic and entropic contributions to the intrinsic free energy barriers (10). Provided that the reactions are adiabatic and the conventional collision model applies, eq 14 can be written in the familiar form relating the rate constants of electrochemical exchange and homogeneous self-exchange reactions (13) ... 

The same relationships apply, in principle, to the concerted pathway (60). However, the intrinsic barrier is now so high, because of the contribution of bond breaking, that the possibility of observing a region that is counterdiffusion controlled is quite unlikely since the rate constant of the forward reaction would then be immeasurably small in most cases. At any rate, even if one conceives that such a situation might occur, and AGq then determined would be so different from that of the outer sphere electron transfer (59) in the stepwise pathway that the confusion would hardly be possible. 

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. ... 

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