Marcus equation quantities

Figure 3. Relationship between the Marcus equation quantities AE, AE and...

Here, the first two terms jointly define an intrinsic barrier quantity that is determined by the averaged/and G quantities and by the resonance energy of the transition state, due to the avoided crossing, while the third term gives the effect of the reaction thermodynamics (taken only to first order). Equations 6.10 and 6.11 also form a bridge to the popular Marcus equation that is used in physical organic chemistry (13) to analyze the barrier in terms of an intrinsic barrier, Aand a thermodynamic attenuation factor ... 

Complex character of the energy profile of the gas-phase reactions 8 2 (see Fig. 5.2) explains the lack of direct correlations between the experimentally assessed activation barriers AE (when they are positive) and the exothermicity of the reaction AH. A better-grounded correlation should be sought between the values of the intrinsic barriers AE and the quantity AE = AH -I- (AEx — AEyx). Wolfe et al. [26] treated, using the 4-3IG basis set, the results of their calculations on over twenty 8 2 reactions and found that the data obey quite satisfactorily the well-known Marcus equation [48] which describes reactions of electron and proton transfer as well as those of the alkyl groups transfer [49] in solution. 

This quantity A./4, known as the intrinsic barrier, for the series of reactions, is the height of the activation barrier in the case where AGp,Q is not affected either by exergonic pull or endergonic drag , and may be regarded as a purely kinetic quantity, not directly related to thermodynamic quantities (though of course determined by the same forces) [17,a]. The Marcus equation can be expressed in terms of this intrinsic barrier by combining Equations (8.29) and (8.27). We obtain ... 

The extent to which the effect of changing substituents on the values of ks and kp is the result of a change in the thermodynamic driving force for the reaction (AG°), a change in the relative intrinsic activation barriers A for ks and kp, or whether changes in both of these quantities contribute to the overall substituent effect. This requires at least a crude Marcus analysis of the substituent effect on the rate and equilibrium constants for the nucleophile addition and proton transfer reactions (equation 2).71-72... 

Most systematic studies on gas-phase SN2 reactions have been carried out with methyl halides, substrates which are free of complications due to competing elimination. Application of the Marcus rate-equilibrium formalism to the double-minimum potential energy surface led to the development of a model for intrinsic nucleophilicity in S 2 reactions233. The key quantities in this model are the central energy barriers, Eq, to degenerate reactions, like the one of equation 22, which are free of a thermodynamic driving force. 

The free energy of reaction, AG°, is an important parameter in Marcus theory (as in, for example, eq. 4.4). Fortunately, this is an easy quantity to calculate from the redox potentials of the couples involved. For forward ET between ground-state D and A (eq. 4.1), the Nemst equation takes the form... 

Here, most quantities are defined above and k(e + Ei) = k(E ) is the unimolecular dissociation rate constant, evaluated using modern statistical theories, such as Rice-Ramsperger-Kassel-Marcus (RRKM) theory. Note that Equation (8) combines the distribution of deposited energies (Equation (5)) with the probability that the complex dissociates in time r (term in square brackets), and a summation over the internal energy available to the reactants. Importantly, the integration recovers Equation (2) when the dissociation rate, A ( ), is faster than the experimental time scale, such that the term in brackets is unity. 

The molar volumes of the reactants, V, V, are presumably known. The ds are known for many substances or can be calculated from known quantities (Hildebrand and Scott, 1962 Marcus, 1985). The molar volume of the activated complex is not usually known may be obtained through the effect of pressure on the reaction rate (see earlier. Section 1.10). Finally, may be obtained from the best fit of data for the rate constant of the reaction in a variety of solvents to Equation 2.14. 

The effects of the ions on the structure of the water were then described by Marcus [51, 53] as the ratios AG bj, = A i" °according to Equation 5.13. The water stracture effects of ions according to this approach are shown in Table 5.2 — structure makers having positive values and structure-breakers negative values. These results are unsatisfactory, due to the inaccuracy of the AjU,°° ° data, making the divalent cations Ba and Cd appear as strong water-structure breakers and LP as a mild structure breaker, contrary to aU other information concerning these ions. The available data for the nine alkah metal and hahde ions appear to be the most accurate, and their correlations with other quantities that describe the water structural effects of ions are ... 

Another quantity of interest is the ionic viscosity -coefficient. Equation 2.29, that in aqueous solutions describes the effect of the ion on the structure of the solvent water. Some values of in nonaqueous solvents have been compiled by Jenkins and Marcus [9] and are reproduced in Table 5.7. It should be noted that practically in all the solvents (except light and heavy water), all the Brj values are positive and the ions appear to enhance the structure of the solvent. However, the splitting of the... 

---