Marcus equation discussion

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... 

From the above discussion, one can see that whether or not the Marcus equation can be used to analyze experimental data, it is important to test its temperature dependence to see if it is of the Arrhenius type. 

Discuss the so-called turn-over feature in Marcus equation. Draw plots for the temperature and the AGfi dependence of Eq. (99), respectively. 

The applicability of the Marcus equation (Eq. 14) to MCET reactions was examined using a ferrocene-naphthoquinone dyad (Fc—NQ) as discussed below (Scheme 19)(121,122). No ET from the Fc to NQ moiety occurs in Fc—NQ with a... 

The kinetics are much more complex and depend on the reorganization of the molecular framework [125], the solvation shell, and the electrostatic interaction. A semi-quantitative estimation of rate constants may be obtained with the well-known Marcus equation [126]. The calculated data compare quite well with experimental values. Most of the experimental hydrocarbon data have been provided by Szwarc and his school [5]. The state of the art has been discussed in an excellent review [121]. 

Here, AG is defined by Equation 1.4, k is the transmission coefficient, and Z is the collision frequency in units of M 1 s 1. The transmission coefficient is discussed above. In practice, k is often set equal to 1. Although this gives reasonable results in numerous cases, this is one of the many assumptions embedded within the familiar, classical form of the Marcus equation (Equation 1.4). Expansion of Equation 1.4 gives Equation 1.12, in which the Coulombic work term (i.e., the first term on the right-hand side of Equation 1.4) is abbreviated as W(r). 

The Marcus equation is also examined in Chapter 9. As discussed previously regarding the Lewis chapter, the quadratic term of the Marcus equation leads to a dependency of rate on the square of pKa, so that Brpnsted plots would be expected to be curved. Bordwell and co-workers observe curvature in some of their Br0nsted plots but conclude that the curvature is too large to be a Marcus effect and actually results from a solvation effect for some heteroatom substituents. These workers suggest that the curvature observed for Brpnsted plots in water results from differential solvation. 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

The rate constants of the reaction CH3Y + X- CH X + Y in sulfolane solution are described by the Marcus equation the quadratic term contributes very little. The Marcus equation then reduces to the expression log kyx = My + Nx, where My is a property of CH3Y only and Nx is a property of CG X only. Each term includes only the identity rates and the equilibria for methylation of a reference nucleophile. The two terms are determined independently of unsymmetric rate measurements, in contrast to the Swain-Scott equation. Short tables of both terms are presented. Extension to other solvents and to other reactions including group transfers is discussed. With other alkyl groups, the simple expression may cover the continuum from elimination-addition to addition-elimination and may also cover other group transfers. 

Mention has been made previously of the possibility that the maximum electron-transfer rate observed in a series of reactions, as the driving force is increased, may be due to the onset of diffusion control, unrelated to the redox properties of the reactants. This has usually been discussed by amending the value of Z in the Marcus equation (1). Thus for uncharged reactants in water at room temperature Za 10 M s, but for like-charged reactants it may be very much less. Brunschwig and Sutin have now proposed a different formulation in which Z is a parameter independent of the encounter rate. Writing the general bimolecular electron-transfer mechanism as... 

So far our discussion of slow proton-transfers and Bronsted exponents has been a qualitative one, apart from the crude electrostatic model represented by Figures 15 and 16. Recently considerable use has been made of a general equation relating the rate of a reaction to its standard free energy change, which was first derived by Marcus for electron-transfer reactions, and later applied by him and by others to reactions involving the transfer of atoms or protons. For present purposes the Marcus equation can be written as ... 

In the calculation of rate constants for electron transfers, the Marcus equation takes a slightly different form than given in Eq. 7.63. As shown in Eq. 7.65, there is still a quadratic dependence upon the free energy change for the reaction, but now there is a new term, 2, which is a value that reflects the required reorganization energy. This term takes into account the rearrangement of the system of reactants and solvents discussed above that are neces-... 

The effect of solvation is not spiecifically included in the Marcus equation (Eq. 7.63). However, one expects a solvent effect on the reaction of hydroxide with methyl bromide (the example used in this chapter when Marcus theory was discussed). In what manner does solvation come into this equation, such that it works in a variety of solvents ... 

The requirements for application of the Marcus equation to reactions other than electron transfer are expressed in precise and succinct form hy J. Jortner, Faraday Discuss. Chem. Soc. 74 (1982) 306, 307. See also the contributions to that Discussion by R.A. Marcus, p. 306 and J.R. Murdoch, pp. 297 seq. 

The Marcus equation for nonadiabatic electron-transfer reactions (Eq. B5.3.4), and the Forster theory that we discussed in Chap. 7 apply only to systems with weak intermolecular interactions, which we now can define more precisely as meaning that H21 lh steady-state approximation to the stochastic Liouville equation for a two-state reaction in this limit From Eqs. (BIO.1.15), (10.29a), (10.29b), and (10.30), we have... 

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. 

The theoretical results obtained for outer-sphere electron transfer based on self-exchange reactions provide the essential background for discussing the interplay between theory and experiment in a variety of electron transfer processes. The next topic considered is outer-sphere electron transfer for net reactions where AG O and application of the Marcus cross reaction equation for correlating experimental data. A consideration of reactions for which AG is highly favorable leads to some peculiar features and the concept of electron transfer in the inverted region and, also, excited state decay. 

These comments complete the discussion of the basic equations of the Marcus- Rice theory and also link them to other familiar concepts. 

In his 1961 paper,67 Hush modified his outer sphere energy equation by changing the (1 - l/s0) term to the Marcus expression (1/n2 - 1/e0) because the electron transition time, though longer than that under FC conditions, would still be short compared to solvent molecule motion. This proposition will be discussed later. The q value... 

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