The Transition State Theory

While this collision theory of reaction provides a good physical explanation of the observed variations in the rates of organic reactions with temperature and. the concentrations of the reactants, it is difficult on this basis to estimate the vital parameters A and E in the Arrhenius equation. We will now consider an alternative picture of the processes leading to chemical reaction which will serve us better in this respect. 

A chemical reaction involves a reorganization of a given collection of n atoms from the grouping corresponding to the molecule or molecules of reactant to that corresponding to the product. We can in principle calculate 

The minima in our 3n-dimensional contour map (or potential surface) will correspond to possible stable arrangements of the atoms, i.e., to a stable molecule or collection of molecules that can be formed from them. In particular, one such minimum (R) will correspond to the reactants in our reaction and another (P) to the products. The situation is indicated schematically in Fig. 5.1 for a simplified case with just two geometric variables. 

If then we can calculate the potential surface for the reaction and so identify the transition state, we can estimate the parameter E (i.e., the activation energy) in the Arrhenius equation (5.13) what about the other parameter, A1 

It is easily seen that the transition state for a reaction also represents an equilibrium configuration of the reacting system. The energy is a minimum for any geometric displacement other than motion to and fro along the reaction path, while for the latter it is a maximum. The equilibrium is stable except for displacements along this one reaction coordinate. We can therefore treat the formation of the transition state from the reactants as an 

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The quasi-equilibrium assumption in the above canonical fonn of the transition state theory usually gives an upper bound to the real rate constant. This is sometimes corrected for by multiplying (A3.4.98) and (A3.4.99) with a transmission coefifiwient 0 < k < 1. 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

There is still some debate regarding the form of a dynamical equation for the time evolution of the density distribution in the 9 / 1 regime. Fortunately, to evaluate the rate constant in the transition state theory approximation, we need only know the form of the equilibrium distribution. It is only when we wish to obtain a more accurate estimate of the rate constant, including an estimate of the transmission coefficient, that we need to define the system s dynamics. 

For 9 < 1 there can be difficulties which arise from distributions which have zero probability in the barrier region and zero rate constant. In our analysis we assume that for any q the zero of energy is chosen such that the probability Pq r) is positive and real for all F. The transition state theory rate constant as a function of the temperature and q is... 

Returning to the more general expression, in the low temperature limit we find that the transition state theory estimate of the rate is... 

The elements of the transition matrix from state j to state i can be estimated in the transition state theory approximation... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

The transition state theory of Eyring or its extensions due to Truhlar and coworkers (see, for example, D. G. Truhlar and B. C. Garrett, Ann. Rev. Phys. Chem. 

For tme first-order bond mpture reactions, the activation energy, E, is equal to the energy of the mptured bond, and following the transition-state theory... 

Electrode kinetics lend themselves to treatment usiag the absolute reaction rate theory or the transition state theory (36,37). In these treatments, the path followed by the reaction proceeds by a route involving an activated complex where the element determining the reaction rate, ie, the rate limiting step, is the dissociation of the activated complex. The general electrode reaction may be described as ... 

A more general, and for the moment, less detailed description of the progress of chemical reactions, was developed in the transition state theory of kinetics. This approach considers tire reacting molecules at the point of collision to form a complex intermediate molecule before the final products are formed. This molecular species is assumed to be in thermodynamic equilibrium with the reactant species. An equilibrium constant can therefore be described for the activation process, and this, in turn, can be related to a Gibbs energy of activation ... 

The original microscopic rate theory is the transition state theory (TST) [10-12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to fonn product. 

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momenrnm along the reaction coordinate directed from reactant to product. Stated another way, is the equilibrium flux of reactant states across... 

The transition state theory rate constant can be constructed as follows. The total flux of trajectories across the transition state dividing surface will be equal to the rate of transition times the population of reactants at equilibrium N, or... 

It is a remarkable fact that the microscopic rate constant of transition state theory depends only on the equilibrium properties of the system. No knowledge of the system dynamics is required to compute the transition state theory estimate of the reaction rate constant... 

We must be able to define the reaction coordinate along which the transition state theory dividing surface is defined. 

Figure 3 Dynamic recrossmgs m the low and high friction regimes. Recrossmgs back to the reactive state lead to a lowering of the rate constant below the transition state theory value.

The transition state theory deseribes reaetants eombining to form unstable intermediates ealled aetivated eomplexes, whieh rapidly... 

The collision theory considers the rate to be governed by the number of energetic collisions between the reactants. The transition state theory considers the reaction rate to be governed by the rate of the decomposition of intermediate. Tlie formation rate of tlie intermediate is assumed to be rapid because it is present in equilibrium concentrations. 

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. 

If the transition state theory is applied to the reaction of two hard spheres, the result is identical with that of simple collision theory. - pp Because transition state theory is an equilibrium theory, it can be inferred that collision theory is also an equilibrium theory. 

Equation (5-69) is an important result. It was first obtained by Marcus " in the context of electron-transfer reactions. Marcus derivation is completely different from the one given here. In electron transfer from one molecule (or ion) to another, no bonds are broken or formed, so the transition state theory does not seem to be applicable. Marcus assumed negligible orbital overlap in the electron-transfer transition state, but he later obtained the same equation for group transfer reactions requiring significant overlap. Many applications have been made to proton transfers and nucleophilic displacements. ... 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

The transition state theory allows us to apply results of equilibrium thermodynamics, and we, therefore, write, for a reactant or transition state species i. 

After an introductory chapter, phenomenological kinetics is treated in Chapters 2, 3, and 4. The theory of chemical kinetics, in the form most applicable to solution studies, is described in Chapter 5 and is used in subsequent chapters. The treatments of mechanistic interpretations of the transition state theory, structure-reactivity relationships, and solvent effects are more extensive than is usual in an introductory textbook. The book could serve as the basis of a one-semester course, and I hope that it also may be found useful for self-instruction. 

As such, it could be treated with the Eyring s transition state theory. When stated in general terms, the transition state theory is applicable to any physico-chemical process which is activated by thermal energy [94] ... 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

Several attempts to relate the rate for bond scission (kc) with the molecular stress ( jr) have been reported over the years, most of them could be formally traced back to de Boer s model of a stressed bond [92] and Eyring s formulation of the transition state theory [94]. Yew and Davidson [99], in their shearing experiment with DNA, considered the hydrodynamic drag contribution to the tensile force exerted on the bond when the reactant molecule enters the activated state. If this force is exerted along the reaction coordinate over a distance 81, the activation energy for bond dissociation would be reduced by the amount ... 

The yielding of polymers can be understood analogue to the transition-state theory [15, 45, 60, 66-71]. In viscous liquids the molecules pass each other with some... 

Reaction temperature is one of the parameters affecting the enantioselectivity of a reaction [16]. For the oxidation of an alcohol, the values of kcat/fQn were determined for the (R)- and (S)-stereodefining enantiomers E is the ratio between them. From the transition state theory, the free energy difference at the transition state between (R) and (S) enantiomers can be calculated from E (Equation 2), and AAG is in turn the function of temperature (Equation 3). The racemic temperature (% ) can be calculated as shown in (Equation 4). Using these equations, % for 2-butanol and 2-pentanol of the Thermoanaerobacter ethanolicus alcohol dehydrogenase were determined to be 26 and 77 °C, respectively. 

The transition state theory is likely an oversimplification when applied to enzyme catalysis - it was originally developed to account for gas phase... 

Building on the Lindemarm Theory described above, Henry Eyring, and independently also M.G. Evans and Michael Polanyi, developed around 1935 a theory for the rate of a reaction that is still used, namely the transition state theory. 

The description of molecular adsorption is very similar to that of atoms, provided we account for the molecules internal degrees of freedom. Hence we need to consider how these degrees change in going from the gas phase to the transition state of adsorption. The most general form for the rate constant of adsorption in the transition state theory is... 

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