Proton transfers Quadratic equation

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. 

This is precisely the behaviour predicted by the Kira mechanism, provided that the formation of the silene-ROH complex is reversible and the proton transfer steps are rate-limiting. The complete mechanism is shown in Scheme 4, while equation 27 gives the predicted expression for the pseudo-first-order rate constant for decay of the silene, derived assuming the steady-state approximation for the silene-alcohol complex. Equation 27 reduces to the quadratic expression in [ROH] of equation 28 when k c (A h + A h [ROII ), i.e. under the conditions of the equilibrium assumption for the complex. In practice, it is difficult to distinguish between the two situations given by equations 27 and 28. The experimentally determined second- and third-order rate constants roh and k2ROH are defined in equations 29 and 30, respectively, in terms of the mechanism of Scheme 4 and using the... 

Values of a or may be determined in different ways. Since proton transfer in the transition state is expected to vary with the strength of the Bronsted acid or base, a and p should in general represent tangents to a curved log /c-log K plot. For small variations in pK systematic deviations from linearity are usually too small to be observed and only an average value of a or can be obtained, but for larger changes the curvature may be sufficient to evaluate a quadratic term in the dependence (equation 33) ... 

It is important that in Marcus s expression, X and AGr in fact refer not to the overall reaction but to the proton-transfer step occurring within the encounter complex formed between the acid and substrate w is the energy of formation of this complex [69, 70]. The terms of the expression may be evaluated from the parameters of a quadratic fit to the variation of AG with AGr, the directly-measurable overall free energy of reaction (i.e. from a free-energy analogue of equation 33), by using a structurally-related family of acids or bases and... 

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