Marcus’ equation

Marcus equation Marezine Marfey s reagent Margaric acid [506-12-7]... 

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... 

There are several equations other than the Marcus equation that describe rate-equilibrium relationships. Murdoch writes all of these equations in the general form... 

For a system describable with a single progress variable, we derived the Marcus equation, Eq. (5-76). 

Let X be the normalized progress variable in a system subject to the Marcus equation (Eq. 5-69), so jc = -t- AG°/8AGo, where has the significance of a in Eq. (5-67). Then deduce this equation, which describes the energy change, relative to the reactant, over the reaction coordinate ... 

In Eq. (7-21), AGo is the intrinsic barrier, the free energy of activation of the (hypothetical) member of the reaction series having AG" = 0. It is evident that the Marcus equation predicts a nonlinear free energy relationship, although if a limited... 

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... 

The latter is, except for a couple of terms related to solvent reorganization, the Marcus equation. The central idea is that the activation energy can be decomposed into a component characteristic of the reaction type, the intrinsic activation energy, and a correction due to the reaction energy being different from zero. Similar reactions should have similar intrinsic activation energies, and the Marcus equation obeys both the BEP... 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

Table 15.2 Comparing experimental activation barriers with those calculated by the Marcus equation...

Again this averaging procedure can only be expected to work when the reactions are sufficiently similar . This is difficult to quantify a priori. The Marcus equation is therefore more a conceptual tool for explaining trends, than for deriving quantitative results. 

Equation (3.34) without the H X ) and the H 2la terms is identical to the Marcus equation for methyl transfer reactions (Ref. 13). This equation predicts, at the range ( AG0 < a), a linear relationship between AAG0 and AAg by... 

The Marcus equation says that the overall AG for a one-step reaction is ... 

One type of process that can successfully be treated by the Marcus equation is the Sn2 mechanism (p. 390)... 

When R is CH3 the process is called methyl transfer. For such reactions, the work terms and are assumed to be very small compared to AG° and can be neglected, so that the Marcus equation simplifies to... 

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... 

The Brpnsted law is therefore a special case of the Marcus equation. 

Excited state electron transfer also needs electronic interaction between the two partners and obeys the same rules as electron transfer between ground state molecules (Marcus equation and related quantum mechanical elaborations [ 14]), taking into account that the excited state energy can be used, to a first approximation, as an extra free energy contribution for the occurrence of both oxidation and reduction processes [8]. 

Equation (3.44) (in the Arrhenius form) is usually called the Marcus equation [74,75]. A special feature of the Marcus equation is that it predicts the parabolic dependence of the activation energy AEa on the free energy change AG/, that is, AEa is related to the free energy change AG/ in a parabolic form. 

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... 

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