Partition function reaction rate theory

Shannon calculated the rate constant for thermal decomposition of a solid from absolute reaction rate theory. The resulting equation is of the same form as Equation 1.27, but v is replaced by a partition function ratio ... 

Absolute reaction rate theory is a theory that aims to provide explanations for both the activation energy and the pre-exponential factor A (the frequency factor ) in the rate equation from first principles. The underlying theories that it uses are quantum mechanics and statistical mechanics. The rate formula of the absolute theory of reaction rates is given in terms of the partition functions Z of the reactants and the transition state by... 

The basic assumption of absolute reaction rate theory is that a transition state can be treated in almost all respects as if it were a molecule in equilibrium with its surroundings. Kassel s original criticism, recently restated by Johnston and Rapp (57), that the use of quantum mechanical state partition functions for a species as transient as an activated complex is highly questionable still remains unanswered. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

The rate of reaction from transition state theory is given by equation (4.31) as Rate of reaction = A v Therefore, in terms of partition function... 

In earlier sections of this chapter we learned that the calculation of isotope effects on equilibrium constants of isotope exchange reactions as well as isotope effects on rate constants using transition state theory, TST, requires the evaluation of reduced isotopic partition function ratios, RPFR s, for ordinary molecular species, and for transition states. Since the procedure for transition states is basically the same as that for normal molecular species, it is the former which will be discussed first. 

The rate constants, k+ and k of the forward and backward reactions are finally derived from (12) and (13) according to the transition-state theory, i.e. assuming that the transition and the initial states, on the one hand, and the transition and final states, on the other, are in equilibrium (Glasstone et al., 1941). Thus, estimating the partition function of these three states in the classical way gives (18) and (19), where p is the reduced mass of the two reactants in the homogeneous case and m the mass of the reactant in the electrochemical case. 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

In this approach properties of potential energy surfaces are investigated from the point of view of all possible monomolecular transformations of the given reactants. A plausible suggestion concerning the mechanism of the reaction under study is usually made on the basis of reaction barriers or activation energies. Moreover, in some studies, partition functions are evaluated and rate constants are obtained within the framework of the absolute rate theory. 

All of the ab initio calcnlations that include electron correlation to some extent clearly favor the concerted pathway for Reaction 4.1. All of these computations also identified a transition state with Q symmetry, indicating perfectly synchronons bond formation. One method for distinguishing a synchronous from an asynchronous transition state is by secondary kinetic isotope effects (KIEs). Isotopic snbstitution alters the frequencies for all vibrations in which that isotope is involved. This leads to a different vibrational partition function for each isotopicaUy labeled species. Bigeleisen and Mayer determined the ratio of partition functions for isotopicaUy labeled species. Incorporating this into the Eyring transition state theory results in the ratio of rates for the isotopicaUy labeled species (Eq. (d. ))." Computation of the vibrational frequencies is thus... 

There are two classes of reactions for which Eq. (10) is not suitable. Recombination reactions and low activation energy free-radical reactions in which the temperature dependence in the pre-exponential term assumes more importance. In this low-activation, free-radical case the approach known as absolute or transition state theory of reaction rates gives a more appropriate correlation of reaction rate data with temperature. In this theory the reactants are assumed to be in equilibrium with an activated complex. One of the vibrational modes in the complex is considered loose and permits the complex to dissociate to products. Figure 1 is again an appropriate representation, where the reactants are in equilibrium with an activated complex, which is shown by the curve peak along the extent of the reaction coordinate. When the equilibrium constant for this situation is written in terms of partition functions and if the frequency of the loose vibration is allowed to approach zero, a rate constant can be derived in the following fashion. 

The first stage in the calculations is to select a transition complex so that the rate coej0 cient in the high pressure limit is equal to that predicted by transition state theory. This requires that g+, the total partition function per cm for the complex with the contribution from motion along the reaction coordinate removed, is given by... 

The rate constants kdiss and kadd were estimated using transition state theory. The crucial difference between the two processes is that while the partition functions of the reactant and transition states are quite similar for the addition reaction, the same is not true for N2 dissociation. Writing each rate as k = Aexp(—Ea/kT), where Ea is the activation energy, Logadottir and Norskov... 

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