Marcus Curvature

Marcus was the first to investigate how the energy profile might vary with AG (equivalent to ApAT) the assumption was that the intersecting curves are two parabolae and the effect of changing AG is to shift the vertical relationship [23,24]. Using this model the relationship (Eqn. 33) was obtained. 

The term AG is the intrinsic barrier namely the activation energy for AG = 0. 

The problem lies in the model representing the forces on the proton in its transfer and two other models give results qualitatively similar to the Marcus theory. One of these involves intersecting Morse curves and the other the proton moving in an electrostatic force field between two negative charges a fixed distance apart [25]. 

In principle it should be possible to investigate the role of the solvent in proton-transfer reactions by the use of the Marcus curvature . Let us treat the encounter and separation of products as discrete steps (Eqn. 34) with the associated energies and respectively. 

We may rewrite Eqn. 34 in terms of these quantities (Eqn. 35) assuming w wP and AGl are constant in a series they may be determined from the experimental data from the coefficients of the quadratic equation in AG.  

An excellent and up-to-date treatment of Marcus theory is given in Chapters 8 and 9 of E.F. Caldin, The Mechanisms of Fast Reactions in Solution, IOS Press, Amsterdam, 2001. 

Curvature can result from non-linearity of the change in the shape of potential energy surfaces in the region where they intersect. A reasonable assumption (over a small range of pK variation) is that the two intersecting curves are parabolic and that the change in entropy within the series is constant. The equations for free energy may therefore be written for the reactant parabola (Equation 4) and product parabola (Equation 5).  

Substituting in Equation (4) gives y at the saddle-point (AG Equation 8)  

This approach is a mathematical alternative to the qualitative arguments given in Chapter 1 and in Appendix 1 (Sections Al.1.2 and Al.1.3) for the deduction of Class I free energy relationships and the variation of the selectivity coefficient, a, with ACq. 

When the reaction is thermoneutral c = 0 and AG = the corresponding transition state energy is given by Equation (9). 

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Mayr has also commented on the need for compensation for Marcus curvature in an extended free energy relationship. In the context of a discussion of the reactions of carbocations with alkenes, he suggests the alternative possibility that this compensation might arise from a log A-dependent change in the relative energies of frontier orbitals on the carbocation and the nucleophile.30... 

Amyes and Richard49 deduced the presence of a transition state imbalance in the deprotonation of methyl and benzylic mono carbonyl compounds by HO- from the linearity of the BrlTnsted plot of the rate constants versus the pK of these carbonyl compounds. They argued that because of the large reactivity range the Br Ansted plot should have shown Marcus curvature 5 8 if the intrinsic barriers for these reactions were all the same and hence the absence of such curvature indicates changes in the intrinsic barriers. They... 

Figure 20 Marcus curvature for proton transfer between carbon acid and heteroatom bases...

Extremely strong solvational imbalances in the transition state for deprotonation of a trinitrobenzylic carbon acid (96) by oximate bases in 1 1 (v/v) H20-Me2S0 accounts for the occurrence of rapid Marcus curvature in the corresponding Brpnsted plots and the consequent levelling of oximate reactivity in the proton transfer process. " The desolvation of the oximate ion pair prior to actual proton transfer ensures that the intrinsic barrier becomes dominated by the work term for formation of the encounter complex. 

Here X is tire reorganization energy associated witli the curvature of tire reactant and product free energy wells and tlieir displacement witli respect to one another. Assuming a stmctureless polarizable medium, Marcus computed the solvent or outer-sphere component of tire reorganization energy to be... 

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... 

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. ... 

Hammond behaviour may be observed if the intersecting parabolae are of different curvature. The physical interpretation of this difference in curvature is that bond-making and bond-breaking processes are not necessarily synchronous as is assumed in conventional Marcus theory. While such a modification in the theory may overcome the inherent problem of treating anti-Hammond effects it does make application of the Marcus theory more difficult by the introduction of additional unknowns into the free energy relationship of (112). 

In conclusion, it can also be pointed out that in principle a large value of A is itself sufficient to account for an extended linear free energy relationship. However, as Mayr has noted this is only true if the slope of the plot is O.5.238 Moreover, if the Marcus expression offers a quantitative guide to the degree of curvature of a free energy relationship (and it is by no means clear that it does),228 it is evident that the intrinsic barriers to reactions of carbocations with familiar nucleophiles are insufficiently large to account for the lack of curvature. 

Notwithstanding the possibility of variation of an intrinsic barrier within a reaction series, for comparisons between different reactions it is often convenient to assume that an unmodified Marcus expression applies. This approximation is justified partly by the high intrinsic barriers and small amounts of curvature characteristic of most reactions at carbon, including reactions of carbocations. The Marcus relationship then provides a common framework for comparisons between reactions based on the measurement of even a single combination of rate and equilibrium constants. Thus, calculation... 

This has important implications with respect to the shape of the voltammetric response predicted by the different models. Thus, the MHC model has been proven, theoretically and experimentally, to be unable to fit the voltammetric response of redox systems that show BV transfer coefficients significantly different from 0.5 [30]. In such cases, as well as in the analysis of surface-confined redox systems, the use of the asymmetric Marcus-Hush theory has been recommended [35] which considers that the force constants for the redox species can be different leading to Gibbs energy curves of different curvatures. 

Eigen (1964) found that a plot of ApR against the rate constant for proton transfer between acetylacetone and a series of bases gave a curved plot. It should be noted, however, that Eigen s explanation for curvature is quite different from the one based on Marcus theory and the reactivity-selectivity principle. The curvature discussed by Eigen is attributed to a change from a rate-determining proton transfer to a diffusion controlled reaction which is independent of the catalyst p. 

The Br0nsted plots (Fig. 3) give information on this point. The higher curvature of the plot for DMSO compared to methanol is indicative of a lower intrinsic barrier to proton transfer for the dipolar aprotic solvent. Since in the extended Marcus theory the solvent effect has already been taken into account, one would expect the intrinsic barrier for proton transfer to be identical in the two systems. This is not the case. Therefore it appears that separation of the mechanism into reagent positioning with concomitant solvent reorganization is not warranted. 

Another approach used to interpret curvature of Br0nsted plots has been given by Murdoch (1972). This model, which incorporates Marcus theory, shows that the diffusive steps (10a, c) of the three-stage Eigen mechanism can also influence curvature. It is shown mathematically that increased difficulty of diffusion has the same... 

Marcus, 1969) by assuming that the point which represents the transition state on the potential energy profile is the intersection of two parabolas of equal curvature. It was also considered that the proton abstraction reaction can be divided up [eqn (7)] into three discrete steps (0 encounter of the reactants, (ii)... 

Attempts were made to observe a curvature of the Bronsted plots for ketone ionisation. Cohen and Marcus (1968) and Bell (1973) (see also J. R. Jones, 1973 Kresge, 1975b) collected data for the reaction of carbonyl compounds (including ketones, esters and keto-esters) with bases and have observed a slight curvature. The data fit the Marcus equation with AG% = 10 kcal mol-1 and Wr = 4 kcal mol-1 (Hupe and Wu, 1977). 

Figure 2 Two parameters defining the Marcus-Hush model of two intersecting parabolas the equilibrium free energy gap AFo and the classical reorganization energy Xci- The parabolas curvature is l/(2Xd).

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