Thermal reactions transition state theory

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

As discussed by Miller and co-workers [52,53], it is worthwhile to develop theories that enable us to evaluate thermal reaction rate constants directly and not to rely on the calculations of the most detailed scattering matrix or the state-to-state reaction probabihty. Here, our formulation of the nonadiabatic transition state theory is briefly described for the simplest case in which the transition state is created by potential surface crossing [27]. 

Figure 22 shows an application of the present method to the H3 reaction system and the thermal rate constant is calculated. The final result with tunneling effects included agree well with the quantum mechanical transition state theory calculations, although the latter is not shown here. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The kinetic model for proton transfer based upon transition state theory that incorporates a tunneling contribution to the overall reaction rate assumes that tunneling occurs near the region of the transition state (pathway a in Scheme 2.5). There is, however, another possibility for the reaction path for proton transfer. In lieu of thermally activating the vibration associated with the proton-transfer coordinate to bring it into the region of the transition state, the proton may instead... 

As is implied by the name, a unimolecular reaction is one in which a single molecule of reactant decomposes or rearranges to give rise to product molecules. Ordinary thermal reactions can be modeled by a process which considers the reactant to be in thermal equilibrium with a transition state which then decomposes (rearranges) to give products. One can theoretically describe the process and its isotope effects using transition state theory. For unimolecular reactions, on the other hand, while there is still a transition state, it is not in thermal equilibrium with the reactant except for systems at high pressure. Consequently, a more elaborate theoretical framework is required to understand unimolecular reactions and their isotope effects. 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

Corrections to transition-state theory due to quantum tunneling along the reaction coordinate give a thermal rate constant that is larger than the prediction obtained from classical transition-state theory. 

As shown in the previous example, for the reaction F + H2 —> HF H, transition-state theory is sufficiently accurate to reproduce the experimental result for the thermal rate constant at T = 300 K. 

Table 6.3 A comparison of different theoretical approaches to the evaluation of the thermal rate constant for the F + H2 —> HF + H reaction at T = 300 K. TST is transition-state theory (Example 6.2), QCT is the quasi-classical trajectory method [Chem. Phys. Lett. 254, 341 (1996)], and QM is (exact) quantum mechanics [J. Phys. Chem. 102, 341 (1998)].

In this chapter, we discuss TPR and reduction theory in some detail, and show how TPR provides insight into the mechanism of reduction processes. Next, we present examples of TPO, TP sulfidation (TPS) and TPRS applied on supported catalysts. In the final section we describe how thermal desorption spectroscopy reveals adsorption energies of adsorbates from well-defined surfaces in vacuum. A short treatment of the transition state theory of reaction rates is included to provide the reader with a feeling for what a pre-exponential factor of desorption tells about a desorption mechanism. The chapter is completed with an example of TPRS applied in ultra-high vacuum (UHV), in order to illustrate how this method assists in unraveling complex reaction mechanisms. 

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