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CHAPTER 2
Foundations of the Theory, Its First Formulation
In this chapter I have summarized the two Marcus’ landmark papers of 1956 – Refs. [1] (Part I of the theory) and [2], which laid the foundation of the whole theory. References have been given in keeping with the educational character of this work. The most important results have been emphasized and mathematical proofs have been given in separate starred sections following the results, so they can be omitted by the noninterested reader. In Chapters 5 and 6, the more general and more abstract formulations of the theory in terms of potential energy surfaces and statistical mechanics is dealt with, but the present earlier treatment in terms of the dielectric continuum theory represents an easier introduction to Marcus theory, with its detailed explanation of all the different steps involved in the ET process, and by itself allows an understanding of the applications given by M. in Part II and in Part III. Similarly, one deals with ideal gases before studying the real ones.
2.1.The Reaction Scheme for Bimolecular ET Reactions
The usual way of calculating a reaction rate is that of first determining the free energy of activation and of then introducing it in the absolute reaction rate theory formula [3–7] for the rate constant. But the ET overall reaction scheme corresponds in general to a sequence of steps, some of which may be slow and so must be considered in calculating the overall rate constant to be compared with the observed rate constant of the reaction sequence. The reaction scheme considered by M. for the bimolecular ET reactions is the following:
The reverse step of Eq. (2.3) is not considered because we are interested in calculating the rate constant of the overall forward reaction and, moreover, the concentration of the products, starting from a very dilute solution of reactants, would be very low.
The process relative to the constant k−1 corresponds to the probability of “a disorganizing motion of the solvent, destroying the polarization appropriate to the intermediate state” so leading to deactivation. Likewise, X can go back to X∗ and so a rate constant k−2 must be considered. Because of the small electronic interaction that was supposed to exist between reactants, the X∗ → X process can be slow and is the reason why two electronic structures of the TS were considered.
Note that A and B are not necessarily the actual compounds introduced in the reaction system, they may rather be “active entities formed from them” (M.).
The overall rate constant of the reaction sequence is kbicacb where the c’s denote concentrations and kbi is the observed bimolecular rate constant. According to Eq. (2.3), the rate is also given by k3cx, so that:
The steady-state equations [3–7] for the concentrations of X∗ and X are given by Eqs. (2.5) and (2.6):
The meaning of Eq. (2.5) (standard chemical kinetics. . .) is the following: if during the reaction the concentration of X∗ remains constant, this means that the effect of the two reactions leading to formation of X∗, with velocities k1cacb and k−2cx is equal to the effect of the reactions leading to depletion of it with velocities
Introducing the value obtained for cx in Eq. (2.4), Eq. (2.7) is obtained [1]:
M. has shown that when the probability of forming X from X∗ is ≥ than the probability of X∗ reforming A and B, then
If the above condition is not valid, the more complex Eq. (2.7) is to be used. A more detailed analysis, given in Marcus’ later papers [8–10] uses a nonadiabatic/adiabatic transition probability, replacing the factor 1/[1 + (1 + k−2/k3)k−1/k2].(1)
In an impressive tour de force, M. estimated in his landmark paper [1] all of the rate constants appearing in Eq. (2.7). The description of their calculation is the main purpose of this chapter.
2.2.A Model for