Marcus theory equation

Note that calculated energies have been substituted for the more normal free energies of Marcus theory. Equation (2) predicts a low 7.2 kcalmoPM activation energy for the 1 -+ 2. The experimental work in Freon glasses did not reveal the existence of 1, although recent work has shown that it does have a finite lifetime in solution and that the estimated rearrangement barrier is 4.8 kcalmo] [12],... 

V. Like the classical Marcus theory, equation 1 predicts an inverted region, although the decrease of rates for highly exoergic reactions may be less pronounced than in the Marcus theory. The quantum mechanical theory also predicts modifications of the effects of temperature and polarity. Some principal features of these predictions have been verified by experiments using both pulse radiolysis and laser photoexcitation. 

The subscrips in and out introduced here refer to inner- and outer-sphere by analogy with the two reorganization energies of the Marcus theory [equation (4)J. Actually the theory does not assume that high frequencies are associated exclusively with inner-sphere bond vibrations, though this is probably a good approximation. In Jortner s notation vjn and vout are written cos and [Pg.8]

This model is formulated by assuming that the two fundamental steps in the charge percolation process occur in a sequential manner. However this mechanism implies that the system passes through two rather energetically unfavorable states bottom left and top right in Fig. 1.16). Hence it appears that an energetically more favorable situation involves a concerted pathway (the dotted path in Fig. 1.16). Saveant has modeled this situation in terms of Marcus theory. Equations 78 and 79 are still valid. However in this case the equation for the diffusion coefficient must be replaced by defined in Eqn. 91. 

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... 

Marcus theory. Prove the point that A = 4AGJ by making use of the analytic expressions for the equation of a parabola. The two equations should be those that describe the curves on the left side of Fig. 10-11. 

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. 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

From the discussion of the Marcus theory above and equations (20) and (21), we see that the experimental data needed to judge the feasibility of ET steps involving spin traps and spin adducts are the redox potentials and A values of the ST +/ST and ST/ST - couples, as well as those for hydroxylamine derivatives related to the operation of reactions (4) or (5). The electroactivity of the spin adducts themselves is also of interest since it must somehow be related to their lifetimes in a redox-active environment. Moreover, the excited-state redox potentials (of ST /ST and ST,+/ST ) are also necessary for the understanding of photo-ET processes of spin traps. 

A different electrochemical approach was applied to the cathodic reduction of sulfones in W,JV-dimethylformamide (Djeghidjegh et al., 1988), for example t-butyl phenyl sulfone, which is reduced at a more negative potential ( pc = -2.5 V) than is PBN (-2.4 V). Thus, the electrolysis of a mixture of PBN and the sulfone would possibly proceed via both true and inverted spin trapping. If a mediator of lower redox potential, such as anthracene (-2.0 V), was added and the electrolysis carried out at this potential, it was claimed that only the sulfone was reduced by anthracene - with formation of t-butyl radical and thus true spin trapping was observed. It is difficult to see how this can be reconciled with the Marcus theory, which predicts that anthracene - should react preferentially with PBN. The ratio of ET to PBN over sulfone is calculated to be 20 from equations (20) and (21), if both reactions are assumed to have the same A of 20 kcal mol-1. 

In a series of important papers Marcus and coworkers applied the RRKM (Rice-Ramsberger-Kassel-Marcus) theory of unimolecular reactions to the ozone problem in a successful effort to rationalize the MIF s described above (see Historical Vignette 14.1). The 2002 paper of Gao and Marcus (reading list) considered a kinetic scheme which mildly elaborates that of Equation 14.1... 

Within exact C2 symmetry, the two states may cross. The system can, however, deviate from this ideal symmetry to allow coupling between the two states so that the crossing is avoided. If we equate the 81 -> 82 excitation energy in 1 to the reorganization energy. A, of Marcus theory [11], we can use the Marcus-Hush expression [Eq. (2)] to estimate the activation energy, AE, based on the calculated values for A and the heat of reaction, AE —12kcalmol" ) ... 

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. ... 

Equation (82) predicts that for reactions with zero free energy change a = 0.5, while for exothermic reactions, a < 0.5 and for endothermic reactions, a > 0.5. Since according to both Marcus theory and the Bell— Evans-Polanyi model early transition states are related to exothermic reactions and late transition states to endothermic reactions, a may be interpreted as a relative measure of transition state geometry. However, in our view even this interpretation should be treated with a measure of healthy scepticism. Even if one accepts Marcus theory without reservation, the a... 

The Marcus equation allows AG for RX + Y —> RY + X to be calculated from the barriers of the two symmetrical reactions RX + X - RX + X and RY + Y — RY + Y. The results of such calculations are generally in agreement with the Hammond postulate. Marcus theory can be applied to any single-step process where something is transferred... 

We now explore whether the pattern of reactivity predicted by the Marcus theory is found for methyl transfer reactions in water. We use equation (29) to calculate values of G from the experimental data where, from (27), G = j(JGlx + AG Y). The values of G should then be made up of a contribution from the symmetrical reaction for the nucleophile X and for the leaving group Y. We then examine whether the values of G 29) calculated for the cross reactions from (29) agree with the values of G(27) calculated from (27) using a set of values for the symmetrical reactions. The problem is similar to the proof of Kohlrausch s law of limiting ionic conductances. 

In order to calculate values of G from equation (29) we have to know not only the kinetic parameters but also the thermodynamic driving force for the SN2 reaction. We are grateful for Dr Abraham s advice and help in calculating these values. His values for the reactants and products are collected together in Table 1 (Abraham and McLennan, 1977). The results for the calculation of G are displayed in Table 2 which has been arranged like a football league table. Only half the table needs to be filled in, since, as shown in (31), the Marcus theory does obey the proper thermodynamic constraint that the ratio of the rates of the forward and backward reactions is given by the equilibrium... 

Fig. 12 Test of the Marcus theory for methyl transfers in H,0. The graph compares the values of G(29) calculated by equation (29) from experimental free energies with C,7) calculated by equation (27) from the Cx x for the symmetrical reactions...

---