Kinetic theory initial density dependence

The bimolecular reaction rate for particles constrained on a planar surface has been studied using continuum diffusion theory " and lattice models. In this section it will be shown how two features which are not taken account of in those studies are incorporated in the encounter theory of this chapter. These are the influence of the potential K(R) and the inclusion of the dependence on mean free path. In most instances it is expected that surface corrugation and strong coupling of the reactants to the surface will give the diffusive limit for the steady-state rate. Nevertheless, as stressed above, the initial rate is the kinetic theory, or low-friction limit, and transient exp)eriments may probe this rate. It is noted that an adaptation of low-density gas-phase chemical kinetic theory for reactions on surfaces has been made. The theory of this section shows how this rate is related to the rate of diffusion theory. 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

Density fimctional theory (DFT) calculations show that this mechanistic hypothesis is energetically reasonable. The generation of the HO-Fe =0 oxidant from the [(TPA)Fe -OOH(OH2)] intermediate was found to have a thermodynamic cost of only 5 kcal/mol and a kinetic barrier of 20 kcal / mol (Fig. 18.5) [52]. Furthermore, the HO-Fe =0 oxidant could carry out either epoxidation or cis-dihydroxylation of the olefin, depending on which oxygen atom of the oxidant initiated attack of the substrate [53]. Thus, epoxidation occurs by 0x0 attack on the olefin, forming the first C-O bond and an intermediate carbon-based radical, which then reboimds to form the second C-O bond. On the other hand, czs-dihydroxylation is initiated by hydroxo attack to form the first C-O bond and an intermediate carbon-based radical, followed by reboimd with the 0x0 group to form the second C-O bond. 

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