Bigeleisen theory

Since the discovery of the deuterium isotope in 1931 [44], chemists have long recognized that kinetic deuterium isotope effects could be employed as an indicator for reaction mechanism. However, the development of a mechanism is predicated upon analysis of the kinetic isotope effect within the context of a theoretical model. Thus, it was in 1946 that Bigeleisen advanced a theory for the relative reaction velocities of isotopic molecules that was based on the theory of absolute rate —that is, transition state theory as formulated by Eyring as well as Evans and Polanyi in 1935 [44,45]. The rate expression for reaction is given by... 

Rees and Thode (1966) reduced Se(VI) to Se(IV) with room temperature 8M HCl, and obtained a fractionation factor (ese(vi)-se(iv)) of 12%o ( 1). Interpretation of the results was complicated somewhat by the fact that the Se(IV) product was recovered by reduction to Se(0). The authors reported that some back-reaction of this precipitate may have occurred. They also presented some theoretical estimates of a, based on the theory of Bigeleisen (1949) and some assumptions about the nature of the reaction, that were consistent with the experimental results. In more recent experiments (Johnson et al. 1999), Se(VI) reduction by 4N HCl at 70°C yielded an ese(vi)-se(iv) value 5.5%o ( 0.3). The difference between this result... 

The theory of isotope effects and a related isotope fractionation mechanism will be discussed very briefly. For a more detailed introduction to the theoretical background, see Bigeleisen and Mayer (1947), Urey (1947), Melander (1960), Bigeleisen (1965), Richet et al. (1977), O Neil (1986), Criss (1999), Chacko et al. (2001), Schauble (2004), and others. 

The theory of kinetic isotope fractionations has been discussed by Bigeleisen and Wolfsberg (1958), Melander(1960), and Melander and Saunders (1980). Knowledge of kinetic isotope effects is very important, because it can provide information about details of reaction pathways. 

Most interestingly, the lifetimes for 3,3-isotope effect on the lifetime of the vibrational energy redistribution requirements, not the sorts of factors associated with standard Bigeleisen and Mayer theory. °... 

The theory of IEs was formulated by Bigeleisen and Mayer.9 The IE on the acid-base reaction of Equation (1) is defined as the ratio of its acidity constant KA to the acidity constant of the isotopic reaction, Equation (2). The ratio KJ KA is then the equilibrium constant XEIE for the exchange reaction of Equation (3). That equilibrium constant may be expressed in terms of the partition function Q of each of the species, as given in Equation (4), which ignores symmetry numbers. 

H)+ and (D)+ represent the transition states for proton and deuteron transfer, respectively. To an excellent approximation (Bigeleisen, 1955) /cH//cD should be uninfluenced by secondary effects. It has a value very close to that suggested as a maximum by simple theory (Bell, 1959c). 

The application of isotope effects studies of reaction mechanism includes comparison of experimental values of isotope effects and predicted isotope effects computed for alternative reaction pathways. On the basis of such analysis some of the pathways may be excluded. Theoretical KIEs are calculated using the method of Bigeleisen and Mayer.1 55 KIEs are a function of transition state and substrate vibrational frequencies. Equilibrium isotope effects are calculated from substrate and product data. Different functionals and data sets are used in these calculations. Implementation of a one-dimensional tunnelling correction into conventional transition-state theory significantly improved the prediction of heavy-atom isotope effects.56 Uncertainty of predicted isotope effect can be assessed from the relationship between KIEs and the distances of formed or broken bonds in the transition states, calculated for different optimized structures.57 Calculations of isotope effects from sets of frequencies for optimized structures of reactants and transition states are facilitated by adequate software QUIVER58 and ISOEFF.59... 

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... 

Condensed Phase Isotope Effects. The chromatographic results discussed later in the paper will be interpreted with the use of the statistical theory of isotope effects in condensed systems attributed to Bigeleisen (6). With the application of a cell model to the condensed phase and the assumption of harmonic frequencies for all 3N modes the theory leads to ... 

Kinetic isotopic effect studies of this kind permit a test of the mass dependence predicted by transition state theory or other theories of bimolecular rate coefficients. A qualitative test of various potential energy surfaces can also be made using such data. Using the theory of Bigeleisen and Wolfsberg of kinetic isotopic effects and several methods of constructing potential energy surfaces (Wheeler et al, Sato, Johnston and Parr ) Timmons and Weston ° found no surface which predicted rates to within better than 40 % of the measured values taken as a whole. 

The theory of calculation of ratios of isotopic partition functions has been developed in detail for exchange equilibria in gases. The formulation of Bigeleisen and Goeppert-Mayer,10 which writes everything in terms of the vibrational frequencies of the molecules, will be reviewed. Classically QAJQAx is where the tn/s are... 

In the theory of Bigeleisen [6], a combination of the theory of equilibrium isotope effects with Eyrings transition state theory [5], kinetic H/D isotope effects can be expressed by... 

The theory of stable isotope exchange was described by Urey (1947) and Bigeleisen and Mayer (1947), and has been reviewed and elaborated many times (Javoy 1977, Clayton 1981, O Neil 1986, Hoefs 1997, Criss 1999 Chacko et ak, this volume Cole and Chakraborty, this volume). Only the basic equations need be given here. 

The earlier contributors to this symposium have outlined the basic theory of the interpretation of the effects of isotopic substitution on reaction rates originally developed independently by Bigeleisen and Mayer (2) and Melander. ( ) They showed that it is the geometrical structure, nuclear masses and, most importantly, the vibrational force fields of initial and transition states that determine the magnitude of isotope effects on reaction rates. 

Isotope effects on reaction rates have been studied intensively, and will receive the major share of the space in this chapter. This field has been reviewed on a number of previous occasions (Me-lander, 1960 Saunders, 1961 Bigeleisen and Wolfsberg, 1959). The present chapter is not intended to be comprehensive or rigorous its aim is to present qualitatively the theory of isotope effects, and to demonstrate with appropriate examples the use of isotope effects in reaction mechanisms studies. 

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