Transmission coefficient isotope effect

Specifically, following the rate expression of QTST in Eq. (4-1) and assuming the quantum transmission coefficients the dynamic frequency factors are the same, the kinetic isotope effect between two isopotic reactions L and H is rewritten in terms of the ratio of the partial partition functions at the centroid reactant and transition state... 

According to transition-state theory it is possible to consider reaction velocities in terms of a hypothetical equilibrium between reactants and transition state. It follows that the influence of the isotopic composition of the medium on reaction velocity can be considered to be the same as its influence on the concentration of transition states. The kinetic formulation of the problem can thus be replaced by one couched in equilibrium terms, and the equilibrium theory of the preceding section can be applied with a minimum of modification (Kresge, 1964). The rate constant, or catalytic coefficient, (k) for a catalysed reaction can be written as the product of three factors, viz. the equilibrium constant (K ) for the process forming the transition state from the reactants, the transmission coefficient, and the specific rate of transition state decomposition (kT/h). We recognize that the third factor is independent of the isotopic nature of the reaction and assume that there is no isotope effect on the transmission coefficient. It follows that... 

Equation VI. 10 leads to transmission coefficients of one-half for symmetrical reactions but other factors for unsymmetrical reactions. These transmission coefficients must be taken into account in calculating isotope effects. The agreement between the values calculated by Hirschfelder, Eyring, and Topley35 and the recently deduced experimental values of Boato, et al.1 is fair. 

The K is a transmission coefficient which expresses the fraction of transition-state species going on to products relative to those returning to reactant and is usually considered, for no very good reason, to be insensitive to isotopic substitution. Boltzmann s constant, absolute temperature and Plancks constant are k, T and h respectively. The isotope effect is simply the ratio of the two rate constants (Eqn. 3). 

According to absolute rate theory , the kinetic isotope effect is equal to the quotient of the equilibrium constants for the formation of the transition states in the labelled and unlabelled substrates, respectively, if symmetry factors are neglected and the transmission coefficients are independent of isotopic substitution (60). Substitution of equilibrium constants reduces the calculation to solving the complete partition functions (61). 

While the quantum instanton approximation cannot describe the recrossing effect, this effect, which is predominantly classical, could be included empirically as a multiplicative factor (a transmission coefficient ) obtained in a classical molecular dynamics simulation. The main challenge for future research is to develop accurate and efficient approximations for the rate constant, its temperature dependence, and the kinetic isotope effect that include the quantum and recrossing effects simultaneously and rigorously. 

Where k is the rate constant, k is the transmission coefficient from the collision theory, Piunn is the tunneling correction, and the Qs are the standard partition functions (see Felipe et al. in this volume). By substituting appropriate carbon atoms with different isotopes on the molecule, one can calculate the ratio of the rate constants (kinetics isotopic effect). 

In this equation is the reduced mass for motion along the reaction co-ordinate (in this case close to the mass of the hydrogen isotope in question), k is a transmission coefficient which will be referred to later, and K is the complete equilibrium constant for the formation of the transition state from AH-hB. As before, the equilibrium constant is expressed in terms of partition functions and the expression for the effect of isotopic substitution simplified by using a product rule, but the special nature of the transition state introduces some modifications. In place of... 

The invocation of non-RRKM behavior in thermal reactions is ordinarily frowned upon by the chemical kinetic community. Here, however, we are dealing with rapid non-adiabatic isomerization of two constitutionally and structurally identical, but energetically different, triplets. It may well be a case in which strong and sometimes peculiar effects (e.g. isotope effects) can be brought about by. .. the non-constancy of the transmission coefficient as a function of kinetic energy [47, 48]. 

Let us dwell now on the dependence of the rate constant on the isotopic composition of the reactant molecules which is usually called kinetic isotope effect. Various types of isotope effects are illustrated in Table 3. Assuming that the transmission coefficient is independent of the isotopic composition, Eq. (11.1) would yield the ratio of the rate constants ki/kg for reactions of molecules with a different isotopic composition. This ratio depends on the symmetry of reactants, their zero-point vibrational energies and effective masses corresponding to motions along the reaction coordinate (for detail see [222, 304]). In the classical limit (E2, EJ < kT), the ratio kj/kg depends on the ratio of effective masses rather than on temperature. In the essentially quantum case (E, EJ kT), the value of kj/kg is influenced mainly by the change in zero-point energies however, the ratio X1/X2 Iso substantially differ from unity [308, 309] as demonstrated for different isotopic variants of reaction H + Hg H2 + H. It is just the difficult calculation of the transmission coefficient that limits the applicability of the transition-state method to the calculation of the isotope effect. 

Table 3 Reaction rate coefficients, kinetic isotope effects, and reiative transmission coefficients for the reactions of H and Mu with the haiogen gases.

B. C. Garrett, D. G. Truhlar, and A. W. Magnuson, Variational transition state theory and vibrationally adiabatic transmission coefficients for the kinetic isotope effects in the Cl-H-H reaction system, J. Chem. Phys. 74 1029 (1981). 

---