Transition probability chemical reaction rates

How does the solvent influence a chemical reaction rate There are three ways [1,2]. The first is by affecting the attainment of equilibrium in the phase space (space of coordinates and momenta of all the atoms) or quantum state space of reactants. The second is by affecting the probability that reactants with a given distribution in phase space or quantum state space will reach the dynamical bottleneck of a chemical reaction, which is the variational transition state. The third is by affecting the probability that a system, having reached the dynamical bottleneck, will proceed to products. We will consider these three factors next. 

A central concept for the derivation is again the transition probability [47,57] introduced in Eq. (29). Within the framework of the deterministic description, the equalities of Eq. (29) and Eq. (30) is always valid for all chemical reactions, so that all one needs is to evaluate individual chemical reaction rates, say v, and vR. Let ps be the fraction of sites to be occupied by monomer units. The possible bond number is then /M0 ps2. Considering that the typical percolation model is the R-Ay type, and familiar branching reactions are irreversible, it suffices to solve the following equality ... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

If an electron rather than an atom tunnels, the transition probability for this particle is much larger in view of its smaller mass therefore the tunnel effect may account for the observed rates of such chemical reactions (22) in which an electron transition can be considered an essential step. Detailed calculations based on the tunneling theory have been... 

These equations are similar to those of first- and second-order chemical reactions, I being a photon concentration. This applies only to isotropic radiation. The coefficients A and B are known as the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. These coefficients play the roles of rate constants in the similar equations of chemical kinetics and they give the transition probabilities. 

If we compare the rate equation above with the phenomenological one for desorption, we see that the transition probability is nothing but the reaction rate constant. This is not always the case. It is quite common that the transition probability and the reaction rate constant differ by a factor that depends on the number of neighbors of a site. It s very important to know that factor when one tries to calculate rate constants quantum chemically, because one can only calculate transition probabilities using quantum chemical methods, and one needs the derivations we present here to obtain the rate constants. 

A further point worth noting is the reaction rate constant in the macroscopic rate equation. We see that it differs from the transition probability by a factor Z. This means, all other things being equal, that surfaces with high Z are more reactive than those with low Z. It is also important not to forget this factor when one want to derive the reaction rate constant using quantum chemical methods. 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

We consider an ensemble of reactant molecules with quantized energy levels to be immersed in a large excess of (chemically) inert gas which acts as a constant temperature heat bath throughout the reaction. The requirement of a constant temperature T of the heat bath implies that the concentration of reactant molecules is very small compared to the concentration of the heat bath molecules. The reactant molecules are initially in a MaxweD-Boltzmann distribution appropriate to a temperature T0 such that T0 < T. By collision with the heat bath molecules the reactants are excited in a stepwise processs into their higher-energy levels until they reach "level (2V+1) where they are removed irreversibly from the reaction system. The collisional transition probabilities per unit time Wmn which govern the rate of transfer of the reactant molecules between levels with energies En and Em are functions of the quantum numbers n and m and can, in principle, be calculated in terms of the interaction of the reactant molecules with the heat bath. 

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