Transition state theory for unimolecular reactions

As is implied by the name, a unimolecular reaction is one in which a single molecule of reactant decomposes or rearranges to give rise to product molecules. Ordinary thermal reactions can be modeled by a process which considers the reactant to be in thermal equilibrium with a transition state which then decomposes (rearranges) to give products. One can theoretically describe the process and its isotope effects using transition state theory. For unimolecular reactions, on the other hand, while there is still a transition state, it is not in thermal equilibrium with the reactant except for systems at high pressure. Consequently, a more elaborate theoretical framework is required to understand unimolecular reactions and their isotope effects. 

In Chapter 7 we turn to the other basic type of elementary reaction, i.e., uni-molecular reactions, and discuss detailed reaction dynamics as well as transition-state theory for unimolecular reactions. In this chapter we also touch upon the question of the atomic-level detection and control of molecular dynamics. In the final chapter dealing with gas-phase reactions, Chapter 8, we consider unimolecular as well as bimolecular reactions and summarize the insights obtained concerning the microscopic interpretation of the Arrhenius parameters, i.e., the pre-exponential factor and the activation energy of the Arrhenius equation. 

Detailed Cross-sections and Rates.—The RRKM version of transition-state theory for unimolecular reactions, as developed 25 years ago and sununarized in its useful practical form in recent books, has continued to find wide applications in unimolecular rate theory. As has been pointed out by Marcus in the 1973 Faraday Discussion on molecular beams, it is both a weakness and a strength of transition-state theory that it does not make very detailed statements on specific cross-sections and rates. With such information becoming accessible experimentally, more detailed statistical dynamical theories were to come. We have now four such detailed statistical approaches ... 

G. Transition state theory for unimolecular reactions. In the high-pressure limit one can assume that the energy-rich species A has reached thermal equi-libriiun. (a) Verify the TST result for the rate of unimolecular dissociation k(T) = (ytBr/A)(gV0exp(— o)wheregis the partition function forAand 0 is the partition function for the transition state, (b) This result looks just like the TST expression for the bimolecular thermal reaction rate constant. But this cannot be. A imimolecular reaction rate constant has different dimensions from a bimolecidar one. Resolve this dilemma, (c) The thermal dissociation of ethane. 

Within the framework of transition-state (RRKM) theory for unimolecular reactions, one can obtain a microscopic interpretation of the activation energy that is analogous to the one presented above (Problem 8.4). 

RRKM theory, developed from RRK theory by Marcus and others [20-23], is the most commonly applied theory for microcanonical rate coefficients, and is essentially the formulation of transition state theory for isolated molecules. An isolated molecule has two important conserved quantities, constants of the motion , namely its energy and its angular momentum. The RRKM rate coefficient for a unimolecular reaction may depend on both of these. For the sake... 

A change in pressure of only 15 bar increased the rate constant by an order of magnitude, because the density and thus the solvent strength increased significantly. This solvent effect can be explained in an alternative but more complex manner by using transition state theory. For a unimolecular reaction, A = A products, the activation volume may be expressed as... 

The induction time for the initial endothermic bond breaking reaction can be calculated using the high pressure, high temperature transition state theory. Experimental unimolecular gas phase reaction rates under low temperature (<1000K) shock conditions obey the usual Arrhenius law ... 

Within the Eyring transition state theory for a unimolecular gas-phase reaction SiN [SiN] —> SiN with the activated complex [S1n] the rate constant k is then related to the activation free enthalpy of the... 

Equation (3.24) thus gives the prediction from transition-state theory for the rate of a reaction in terms appropriate for a supercritical fluid. The rate is seen to depend on (1) the pressure (except for unimolecular reactions and as a consequence of defining the rate coefficient in terms of mole fractions) and some universal constants, (2) the equilibrium constant for the activated-complex formation in an ideal gas and (3) a ratio of fugacity coefficients, which express the effect of the supercritical medium. 

TRANSITION-STATE THEORY FOR REACTIONS ON SURFACES 10.4.1 Unimolecular reactions... 

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... 

It is worthwhile to first review several elementary concepts of reaction rates and transition state theory, since deviations from such classical behavior often signal tunneling in reactions. For a simple unimolecular reaction. A—>B, the rate of decrease of reactant concentration (equal to rate of product formation) can be described by the first-order rate equation (Eq. 10.1). 

For a temperature of 1000 K, calculate the pre-exponential factor in the specific reaction rate constant for (a) any simple bimolecular reaction and (b) any simple unimolecular decomposition reaction following transition state theory. 

Let us consider in more detail the concept of a free energy barrier. Transition state theory also uses the idea that there is such a barrier in the reaction path. What is special about TST is that it ascribes certain properties to the species at the top of the barrier, the activated complex. According to TST for a unimolecular reaction,... 

At high temperatures and low pressures, the unimolecular reactions of interest may not be at their high-pressure limits, and observed rates may become influenced by rates of energy transfer. Under these conditions, the rate constant for unimolecular decomposition becomes pressure- (density)-dependent, and the canonical transition state theory would no longer be applicable. We shall discuss energy transfer limitations in detail later. 

Consider the simple unimolecular reaction of Eq. (15.3), where the objective is to compute the forward rate constant. Transition-state theory supposes that the nature of the activated complex. A, is such that it represents a population of molecules in equilibrium with one another, and also in equilibrium with the reactant, A. That population partitions between an irreversible forward reaction to produce B, with an associated rate constant k, and deactivation back to A, with a (reverse) rate constant of kdeact- The rate at which molecules of A are activated to A is kact- This situation is illustrated schematically in Figure 15.1. Using the usual first-order kinetic equations for the rate at which B is produced, we see that... 

Elementary reactions are initiated by molecular collisions in the gas phase. Many aspects of these collisions determine the magnitude of the rate constant, including the energy distributions of the collision partners, bond strengths, and internal barriers to reaction. Section 10.1 discusses the distribution of energies in collisions, and derives the molecular collision frequency. Both factors lead to a simple collision-theory expression for the reaction rate constant k, which is derived in Section 10.2. Transition-state theory is derived in Section 10.3. The Lindemann theory of the pressure-dependence observed in unimolecular reactions was introduced in Chapter 9. Section 10.4 extends the treatment of unimolecular reactions to more modem theories that accurately characterize their pressure and temperature dependencies. Analogous pressure effects are seen in a class of bimolecular reactions called chemical activation reactions, which are discussed in Section 10.5. 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

R. A. Marcus It certainly is a good point that transition state theory, and hence RRKM, provides an upper bound to the reactive flux (apart from nuclear tunneling) as Wigner has noted. Steve Klippenstein [1] in recent papers has explored the question of the best reaction coordinate, e.g., in the case of a unimolecular reaction ABC — AB + C, where A, B, C can be any combination of atoms and groups, whether the BC distance is the best choice for defining the transition state, or the distance between C and the center of mass of AB, or some other combination. The best combination is the one which yields the minimum flux. In recent articles Steve Klippenstein has provided a method of determining the best (in coordinate space) transition state [1]. 

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