Model, multistep activation energy

In principle, a multistep reaction with any network is fully described by the complete set of rate equations of all participants, compiled as shown in Section 2.4. However, if the reaction is complex, the experimental work required to verify the assumed mechanism and to determine all its coefficients and their activation energies can get out of hand. A reduction of complexity then is imperative, and is also desirable for both a better understanding of reaction behavior and more efficient numerical modeling. The present chapter describes ways of achieving this. 

For multistep intrachannel ion-binding models, the formulation of the rate constants for individual transitions is complicated by the existence of two parameters in the flux expression. Parlin and Eyring (5) and Starzak (6) developed a flux expression for ion flow through membranes or channels that used transition-state theory where the rate constant was characterized by an activation energy barrier. For dimensional consistency, this rate constant, kif appeared in conjunction with a length, Xs, the distance in the channel that the ion must move to cross the energy barrier. 

The overview provided above covers the classical perspective on crystallization and particularly nucleation. A rapidly growing body of work shows that this view is not complete. The alternative model for nucleation is often termed the two-step nucleation theory. In brief, this model sng-gests that nucleation is a multistep process where the first step involves phase separation via the formation of liquid or amorphous nanoparticles. This is then followed by crystallization within this particle. The activation energy for each of these steps is relatively small, and it is expected that the overall process would be faster when compared with a single-step (classical) process with the same overall activation energy. Experimental and theoretical evidence supports this mechanism, and it has been suggested that the nucleation process is likely to proceed in this way in most, if not all, cases. ... 

Given that electrochemical rate constants are usually extremely sensitive to the electrode potential, there has been longstanding interest in examining the nature of the rate-potential dependence. Broadly speaking, these examinations are of two types. Firstly, for multistep (especially multielectron) processes, the slope of the log kob-E plots (so-called "Tafel slopes ) can yield information on the reaction mechanism. Such treatments, although beyond the scope of the present discussion, are detailed elsewhere [13, 72]. Secondly, for single-electron processes, the functional form of log k-E plots has come under detailed scrutiny in connection with the prediction of electron-transfer models that the activation free energy should depend non-linearly upon the overpotential (Sect. 3.2). 

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