Marcus quadratic relationship

Figure 7.2 Rate constants of proton transfers from enolate ions and water. The curve represents an application of the Marcus quadratic relationship, with AG (0)=57 kJ mol" and k f= 10 mol" dm sec". ...

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

Eq. (7.6), known as the equation of Marcus, leads to a quadratic relationship between the barrier of reaction and the reaction energy, which shows up also in the modified form... 

In Marcus original formulation of ET theory, the free energy curves Gj and Gj are assumed to be quadratic in x (linear response approximation). Using this assumption, Marcus derives the relationship between the activation free energy and the reaction free energy... 

Another important aspect of the Marcus theory has also been systematically investigated with organic molecules, namely the quadratic, or at least the non-linear, character of the activation-driving force relationship for outer sphere electron transfer. In other words, does the transfer coefficient (symmetry factor) vary with the driving force, i.e. with the electrode... 

There are two major concepts involved in the physico-chemical description of a chemical reaction the energetics, which determines the feasibility of the reaction, and the kinetics which determines its rate. In general these two concepts are independent and the rate of a chemical reaction can be varied according to the mechanism (e.g. catalysis) but within certain assumptions there is a mathematical relationship between the rate constant and the reaction free energy difference. These relationships are either linear (linear free energy relationship, LFE) or quadratic (QFE), the latter being often referred to as the Marcus model — a description which should not hide the important contributions of other workers in this field [1],... 

The Marcus relation, Eq. (19.6), is clearly not a linear relationship between the activation energy and the reaction asymmetry but a quadratic one. The first derivative of AEt with respect to is equivalent to the Bronsted coefficient a in Eq. 

At about the same time, we (8) made some equilibrium measurements in sulfolane, augmenting the earlier data of Jackman and co-workers (9), and new rate measurements allowed us to estimate one identity rate (that for PhSMe2+) by a linear free energy relationship (LFER) extrapolation. Assuming that the quadratic term in the Marcus equation was negligible, we estimated identity barriers for a number of other leaving groups. 

One of the most fundamental achievements of the seminal Marcus theory on electron-transfer reactions is the nowadays widely used quadratic driving force-activation free energy Marcus relationship... 

Beginning in the 1950s, Marcus developed and refined a classical theoretical description of redox reactions occurring in homogeneous solution [79-84]. He also extended the treatment to include electrode processes [81-86]. Related treatments are the quantum models of Levich and Doganadze and the works of Gerisher. More recently, Chidsey [29] presented a derivation of Marcus nonadiabatic (MNA) behavior, characterized by the quadratic free energy relationship for electrode processes ... 

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