Transition state theory expression

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

Pi)Ay /Rr. Thus, In 2 = (3000 - l)Ay /(82.05 X 298) = 5.7 cm. This exercise indicates that reaction rate is relatively insensitive to pressure changes if Ay is small. See Transition-State Theory Expressed in Thermodynamic Terms Gibbs Free Energy of Activation Enthalpy of Activation Entropy of Activation lUPAC (1979) Pure Appl. Chem. 51, 1725. 

TRANSITION STATE THEORY EXPRESSED IN THERMODYNAMIC TERMS... 

Our calculations of the activation free energy barrier for the cuprous-cupric electron transfer were not precise enough to permit a very accurate estimate of the absolute value of the exchange current for comparison with experiment. In principle, a determination of the absolute rate from the activation energy requires a calculation of the relevant correlation function [82] when the ion is in the transition region within the molecular dynamics model. We have not carried out such a calculation, but can obtain some information about the amplitude by comparing experiments with the transition state theory expression [84]... 

DERIVATION OF THE TRANSITION STATE THEORY EXPRESSION FOR A RATE CONSTANT ... 

Derivation of Transition State Theory Expression for a Rate Constant 115... 

In this section, we present a derivation of the conventional transition-state theory expression for the rate constant, Eq. (6.8), that avoids the artificial constructs of the... 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

In order to compare the experimental results at 416 K with theoretical predictions, / 4aatlhe higher temperature was calculated from the transition state theory expression. For model A and A this indicated that k should increase by a factor of 1.6 and for model B by 1.2 times. The positive temperature coefficient is associated with the low bending frequencies in the complex and decreases if these are replaced by rotations. Where the radicals are completely free to rotate in the complex then the high pressure rate coefficient becomes proportional to T. A comparison of the curves in fig. 5 and 6 indicates that the temperature dependence of the rate in the transition range is well represented only when it is assumed that k4. is approximately the same at 416 K as at 300 K. 

The electron transfer is assumed to be an adiabatic process in the Ehrenfest sense so that the transmission coefficient, k, in the transition-state theory expression (Section 1.6.2), Eq. (6.7), is equal to one. 

A simple formula for the canonical flexible transition state theory expression for the thermal reaction rate constant is derived that is exact in the limit of the reaction path being well approximated by the distance between the centers of mass of the reactants. This formula evaluates classically the contribution to the rate constant from transitional degrees of freedom (those that evolve from free rotations in the limit of infinite separation of the reactants). Three applications of this theory are carried out D + CH3, H + CH2, and F + CH3. The last reaction involves the influence of surface crossings on the reaction kinetics. 

In this section, the basic components of a canonical FTST theory are reviewed. A more detailed derivation can be found in paper I. The canonical variational transition state theory expression for the high pressure limiting rate constant is (with p =... 

We noted in Section 9.1.2.5 that it would not be correct to use the transition-state theory expression k = (IqsT/h) exp(—AG /k T), since in the Marcus model the transition-state assumptions do not apply to the conditions in the transition region. 

For dissociation of a molecule larger than a diatomic molecule the transition-state theory expression of dissociation becomes... 

The rate constant becomes independent of pressure and can be computed using the transition-state theory expression. The reaction is first order in concentration [A], as expected for a monomolecular reaction. 

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