Transition State Theory for Gas-Phase Reactions

Variational Transition State Theory for Gas-Phase Reactions 127... 

VARIATIONAL TRANSITION STATE THEORY FOR GAS-PHASE REACTIONS... 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

Intramolecular hydrogen transfer is another important class of chemical reactions that has been widely studied using transition state theory. Unimolecular gas-phase reactions are most often treated using RRKM theory [60], which combines a microcanonical transition state theory treatment of the unimolecular reaction step with models for energy redistribution within the molecule. In this presentation we will focus on the unimolecular reaction step and assume that energy redistribution is rapid, which is equivalent to the high-pressure limit of RRKM theory. 

The Arrhenius A factors for the propagation reactions are low and of the order one would expect from any of the transition-state theories for a bimolecular reaction between two large molecules (Table XII.2). The activation energies Ep for propagation are also low and of the order observed for similar addition reactions in the gas phase of radicals to a double bond. The values of At are in the range to be expected for diffusion-controlled reactions (Sec. XV.2) except for vinyl chloride, which must certainly be in error. As pointed out earlier in discussing diffusion-controlled reactions, it is expected that the activation energies will be of the order of a... 

Theoretical predictions, based on AMI MO theory, for gas-phase elimination reactions of 3-chloropropanoic and 2-chlorobutanoic acids are consistent with experimental results four-, five-, and six-membered transition states have been discussed. ... 

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. 

Transition state theory was also developed as a means of rationalizing rate constants for gas phase reactions and their temperature dependence. It is most directly applied to bimolecular reactions and is based on three fundamental postulates for reactions in solution ... 

The simplest model that accounts for the Arrhenius expression is the collision theory of gas-phase reaction rates, in which it is supposed that reaction occurs when two reactant molecules collide with at least a minimum kinetic energy (which is identified with the activation energy. Figure 2). A more sophisticated theory is the activated complex theory (also known as the transition state theory), in which it is supposed that the reactants encounter each other, form a loosened cluster of atoms, then decompose into products. 

Equations (1.3-14) and (1.3-15) thus give the prediction from transition-state theory for the rate of a reaction in terms appropriate for an SCF. The rate is seen to depend on (i) the pressure, the temperature and some universal constants (ii) the equilibrium constant for the activated-complex formation in an ideal gas and (iii) a ratio of fugacity coefficients, which express the effect of the supercritical medium. Equation (1.3-15) can therefore be used to calcu-late the rate coefficient, if Kp is known from the gas-phase reaction or calculated from statistical mechanics, and the ratio (0a 0b/0cO estimated from an equation of state. Such calculations are rare an early example is the modeling of the dimerization of pure chlorotrifluoroethene = 105.8 °C) to 1,2-dichlor-ohexafluorocyclobutane (Scheme 1.3-2) and comparison with experimental results at 120 °C, 135 °C and 150 °C and at pressures up to 100 bar [15]. 

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... 

Collision state theory is useful for gas-phase reactions of simple atoms and molecules, but it cannot adequately predict reaction rates for more complex molecules or molecules in solution. Another approach, called transition-state theory (or activated-complex theory), was developed by Henry Eyring and others in the 1930s. Because it is applicable to a wide range of reactions, transition-state theory has become the major theoretical tool in the prediction of chemical kinetics. 

Part IV again is a theoretical one, in which different approaches for the calculation of state-specific and thermal rate data are described. The article by A.F. Wagner presents a new approach to describe the influence of hindered rotations on recombination/dissociation kinetics in the framework of transition state theory. In the papers by D.C. Clary and G. Nyman an approximate quantum mechanical method is described and used to calculate thermal rate coefficients for gas phase reactions of interest in atmospheric chemistry which involve polyatomic molecules. Finally, different approaches to describe vibrational relaxation of diatoms in thermal collisions are discussed by E,E, Nikitin,... 

Transition state theory proposes that every reaction (or every step in an overall reaction) goes through its own transition state. Fignre 16.17 presents reaction energy diagrams for two gas-phase reactions. In each case, the structure of the transition state is predicted from the orientations of the reactant atoms that must become bonded in the product. 

For gas phase reactions, the reaction can be viewed as a scattering event and k T) can be calculated from scattering properties. However, a more intuitive approach is to calculate the thermal rate constant directly from a simulation of the dynamics in the vicinity of the reaction barrier. This approach has a number of advantages it decreases the numerical effort compared to a full scattering calculation, enables an equivalent treatment of reactions in gas and condensed phase, and results in a very intuitive interpretation based on ideas adopted from transition state theory. The present article intends to... 

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]. 

Transition state theory is presented with an emphasis on solution reactions and the Marcus approach. Indeed, to allow for this, I have largely eliminated the small amount of material on gas-phase reactions that appeared in the First Edition. Several treatments have been expanded, including linear free-energy relations, NMR line broadening, and pulse radiolytic and flash photolytic methods for picosecond and femtosecond transients. 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

Statistical rate theories have been used to calculate rate constants for gas-phase Sn2 reactions.1,7 For a SN2 reaction like Cl" + CH3Clb, which has a central barrier higher than the reactant asymptotic limit (see Figure 1), transition state theory (TST) assumes that the crossing of the central barrier is rate-limiting. Thus, the TST expression for the SN2 rate constant is simply,... 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

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