Equilibrium Rate Constants. Transition-State Method

11 Equilibrium Rate Constants. Transition-State Method 

The rate constant for an elementary reaction is readily obtained if the reaction cross section and the distribution functions over the reactant states are known. In this connection, two problems have to be solved. One is dynamical (calculation of cross section) and the other is statistical (calculation of the distribution func- 

For the calculation of the rate constants of such processes, the transition-state method has been proposed [129, 131, 365, 518]. Its relative simplicity (permitting calculation of the rate constants for many reactions) lies in that it does not attempt taking into account all the dynamical features of the elementary processes. It introduces instead the activated complex concept. However, it does not give unambiguous indication as to how the activated complex properties are connected with those of the reactants, thus leaving aside the dynamical problem. For this reason, the transition-state approach is sometimes opposed to the collision theory, though very often they are correlated. 

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Transition state theory is a method for predicting the rate of chemical reactions. Technically, what the theory provides is the rate of crossing of a barrier. If there is only one barrier between reactants and products, then transition state theory specifies how to compute the reaction rate constant. Transition state theory assumes the vahdity of only one condition, but a cardinal one, namely that on one side of the barrier, the states of the system are in equilibrium. If there is only one barrier between reactants and products, then it is the reactants that should be kept at equihbrium. The simplicity of transition state theory is lost if the reactants are state-selected. ... 

So we see that the rate constant depends on the pressure of the bath gas. At the high-pressure limit, the rate constant (denoted as kco) naturally coincides with the result given by the transition-state method because in this case the distribution function is almost equilibrium... 

Both these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

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 (TST) (4) is a well-known method used to calculate the kinetics of infrequent events. The rate constant of the process of interest may be factored into two terms, a TST rate constant based on a knowledge of an equilibrium phase space distribution of the system, and a dynamical correction factor (close to unity) used to correct for errors in the TST rate constant. The correction factor can be evaluated from dynamical information obtained over a short time scale. 

In Chapter 5, attention is directed toward the direct calculation of k(T), i.e., a method that bypasses the detailed state-to-state reaction cross-sections. In this approach the rate constant is calculated from the reactive flux of population across a dividing surface on the potential energy surface, an approach that also prepares for subsequent applications to condensed-phase reaction dynamics. In Chapter 6, we continue with the direct calculation of k(T) and the whole chapter is devoted to the approximate but very important approach of transition-state theory. The underlying assumptions of this theory imply that rate constants can be obtained from a stationary equilibrium flux without any explicit consideration of the reaction dynamics. 

But if we examine the localized near the donor or the acceptor crystal vibrations or intra-molecular vibrations, the electron transition may induce much larger changes in such modes. It may be the substantial shifts of the equilibrium positions, the frequencies, or at last, the change of the set of normal modes due to violation of the space structure of the centers. The local vibrations at electron transitions between the atomic centers in the polar medium are the oscillations of the rigid solvation spheres near the centers. Such vibrations are denoted by the inner-sphere vibrations in contrast to the outer-sphere vibrations of the medium. The expressions for the rate constant cited above are based on the smallness of the shift of the equilibrium position or the frequency in each mode (see Eqs. (11) and (13)). They may be useless for the case of local vibrations that are, as a rule, high-frequency ones. The general formal approach to the description of the electron transitions in such systems based on the method of density function was developed by Kubo and Toyozawa [7] within the bounds of the conception of the harmonic vibrations in the initial and final states. 

Less commonly used measurement techniques include the pH dependence of partition coefficients [74], fluorescence spectra [75], nuclear magnetic resonance chemical shifts or coupling constants, HPLC or CE retention volumes [76,77], and the dependence of reaction rates for ionizable substrates on pH (also called kinetic methods). Kinetic methods were amongst the earliest methods to be used for pKg determination. In some cases, they may be the only feasible method, for example, extremely weak acids (pKa > 12) without suitable absorption spectra. The difficulty with kinetic methods is that they may not actually measure the pKg value for the substrate, but that of the reaction transition state. If the electronic configuration of the transition state is similar to that of the reactant (early transition state), then the kinetic may be quite close to the equilibrium value. However, if the transition state more nearly approximates the reaction products (late transition state), then the kinetic value may bear little resemblance to that for the reactant. This explanation might account for the lack of agreement between the first apparent kinetic pK = 4.0) and equilibrium (pK = 8.6) pKg values for hydrochlorothiazide at 60 °C [78]. Similar restrictions may be placed on the use of pKa values from the pH dependence of fluorescence spectra, as these reflect the properties of the first excited state of the molecule rather than its ground state [75]. 

In several cases the reactant to product transition involved a complicated pathway that included a number of transition states and intermediates on the potential energy surface. Examples are given for a number of such reaction systems. In order to evaluate the final rate parameters for the reactant to product transition in such cases, the overall pathway on the potential energy surface had to be treated as an independent kinetics scheme to be computer modeled by normal numerical integrations. This has to take into account all the forward and backward rate constants. The latter are determined by equilibrium constants between the various steps on the surface. All of these methods of calculations and the experimental methods used are described in Section 6.1. 

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