Kinetics transition-state theory

State is that assembly of atoms or moieties that closely resembles the reactant(s), such that only a relatively small reorganization will generate the reactant(s). Analogously, a late transition state more closely resembles the structure of the reaction product(s). See Chemical Kinetics Transition State Theory Potential Energy Surface Hammond Principle Transition Structure... 

This model for the thermal rearrangement of l-phenylbicyclo[2.2.1]hex-ene provides an explanation for the stereochemistry observed in the [1,3] methyl shift. More important, it also brings into question some of the common assumptions of reaction kinetics. Transition state theory is based on the premise that the redistribution of internal kinetic energy is faster than is the progress of a collisionally activated reactant over a potential energy surface to a transition state and then to a product. If this assumption is not valid, then the... 

For chemical kinetics, transition state theory is most useM in the form that starts from reactants in thermal equilibrium. For our purpose we want a more detailed version, that of reactants with a total energy in the range E oE + AE. If we know how to do that, we can and will average over a Boltzmann distribution in E to obtain the thermal results. The first task at hand is to define what is meant by reactants at equilibrium at a total energy within the range (and at given values of any other conserved quantum numbers). It is the foundation of statistical mechanics that equilibrium under such conditions means that all possible quantum states of the reactants are equally probable. ... 

Space limitations preclude a serious treatment of either of these theories of chemical kinetics. Transition-state theory is developing rapidly as a result of the availability of inexpensive computing power and is a component of many graduate courses in chemical kinetics and chemical reaction engineering. The interested student can learn more about transition-state and collision the y from references such as Masel, R. L., Chemical Kinetics and Catalysis, Wiley-Interscience (2001), Benson, S. W., Thermochemical Kinetics, 2nd edition, Wiley-Interscience (1976), and Moelwyn-Hughes, E. A., Physical Chemistry, 2nd revised edition, Pergamon Press (1961). 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

The first two of these we can readily approach with the knowledge gained from the studies of trappmg and sticking of rare-gas atoms, but the long timescales involved in the third process may perhaps more usefiilly be addressed by kinetics and transition state theory [35]. 

Fast transient studies are largely focused on elementary kinetic processes in atoms and molecules, i.e., on unimolecular and bimolecular reactions with first and second order kinetics, respectively (although confonnational heterogeneity in macromolecules may lead to the observation of more complicated unimolecular kinetics). Examples of fast thennally activated unimolecular processes include dissociation reactions in molecules as simple as diatomics, and isomerization and tautomerization reactions in polyatomic molecules. A very rough estimate of the minimum time scale required for an elementary unimolecular reaction may be obtained from the Arrhenius expression for the reaction rate constant, k = A. The quantity /cg T//i from transition state theory provides... 

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 assumptions of transition state theory allow for the derivation of a kinetic rate constant from equilibrium properties of the system. That seems almost too good to be true. In fact, it sometimes is [8,18-21]. Violations of the assumptions of TST do occur. In those cases, a more detailed description of the system dynamics is necessary for the accurate estimate of the kinetic rate constant. Keck [22] first demonstrated how molecular dynamics could be combined with transition state theory to evaluate the reaction rate constant (see also Ref. 17). In this section, an attempt is made to explain the essence of these dynamic corrections to TST. 

The numerical values of AG and A5 depend upon the choice of standard states in solution kinetics the molar concentration scale is usually used. Notice (Eq. 5-43) that in transition state theory the temperature dependence of the rate constant is accounted for principally by the temperature dependence of an equilibrium constant. 

We now carry the argument over to transition state theory. Suppose that in the transition state the bond has been completely broken then the foregoing argument applies. No real transition state will exist with the bond completely broken—this does not occur until the product state—so we are considering a limiting case. With this realization of the very approximate nature of the argument, we make estimates of the maximum kinetic isotope effect. We write the Arrhenius equation for the R-H and R-D reactions... 

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. 

The vertical axis is free energy, showing AGO for the net conversion of A to P, and AG, the activation free energy for each of the kinetic steps. According to Eyring s transition state theory (Chapter 7), AG is given by... 

Transition state theory has been useful in providing a rationale for the so-called kinetic isotope effect. The kinetic isotope effect is used by enzy-mologists to probe various aspects of mechanism. Importantly, measured kinetic isotope effects have also been used to monitor if non-classical behaviour is a feature of enzyme-catalysed hydrogen transfer reactions. The kinetic isotope effect arises because of the differential reactivity of, for example, a C-H (protium), a C-D (deuterium) and a C-T (tritium) bond. 

Kinetics on the level of individual molecules is often referred to as reaction dynamics. Subtle details are taken into account, such as the effect of the orientation of molecules in a collision that may result in a reaction, and the distribution of energy over a molecule s various degrees of freedom. This is the fundamental level of study needed if we want to link reactivity to quantum mechanics, which is really what rules the game at this fundamental level. This is the domain of molecular beam experiments, laser spectroscopy, ah initio theoretical chemistry and transition state theory. It is at this level that we can learn what determines whether a chemical reaction is feasible. 

Equation (12) also contains a pre-exponential factor. In Section 3.8.4 we treated desorption kinetics in terms of transition state theory (Figure 3.14 summarizes the situations we may encounter). If the transition state of a desorbing molecule resembles the chemisorbed state, we expect pre-exponential factors on the order of ek T/h = 10 s . However, if the molecule is adsorbed in an immobilized state but desorbs via a mobile precursor, the pre-exponential factors may be two to three orders of magnitude higher than the standard value of 10 s . ... 

The interpretation of phenomenological electron-transfer kinetics in terms of fundamental models based on transition state theory [1,3-6,10] has been hindered by our primitive understanding of the interfacial structure and potential distribution across ITIES. The structure of ITIES was initially studied by electrochemical and thermodynamic analyses, and more recently by computer simulations and interfacial spectroscopy. Classical electrochemical analysis based on differential capacitance and surface tension measurements has been extensively discussed in the literature [11-18]. The picture that emerged from... 

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]. 

This chapter treats the descriptions of the molecular events that lead to the kinetic phenomena that one observes in the laboratory. These events are referred to as the mechanism of the reaction. The chapter begins with definitions of the various terms that are basic to the concept of reaction mechanisms, indicates how elementary events may be combined to yield a description that is consistent with observed macroscopic phenomena, and discusses some of the techniques that may be used to elucidate the mechanism of a reaction. Finally, two basic molecular theories of chemical kinetics are discussed—the kinetic theory of gases and the transition state theory. The determination of a reaction mechanism is a much more complex problem than that of obtaining an accurate rate expression, and the well-educated chemical engineer should have a knowledge of and an appreciation for some of the techniques used in such studies. 

The transition state theory provides a useful framework for correlating kinetic data and for codifying useful generalizations about the dynamic behavior of chemical systems. This theory is also known as the activated complex theory, the theory of absolute reaction rates, and Eyring s theory. This section introduces chemical engineers to the terminology, the basic aspects, and the limitations of the theory. 

The thermodynamic formulation of the transition state theory is useful in considerations of reactions in solution when one is examining a particular class of reactions and wants to extrapolate kinetic data obtained for one reactant system to a second system in which the same function groups are thought to participate (see Section 7.4). For further discussion of the predictive applications of this approach and its limitations, consult the books by Benson (59) and Laidler (60). Laidler s kinetics text (61) and the classic by Glasstone, Laidler, and Eyring (54) contain additional useful background material. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

As noted in Chapter 16, transition state theory does not require that kinetic rate laws take a linear form, although most kinetic studies have assumed that they do. The rate law for reaction of a mineral A can be expressed in the general nonlinear form,... 

Lasaga, A. C., 1981b, Transition state theory. In A. C. Lasaga and R. J. Kirkpatrick (eds.), Kinetics of Geochemical Processes. Mineralogical Society of America, Washington DC, pp. 135-169. 

---