Applications of transition-state theory

This expression is called the Wigner tunneling correction factor, and is valid for a small degree of tunneling (high and/or broad barrier, hv C Ec ) and, in addition, high temperatures (hv C kBT) are required. Note that the correction factor is larger than one. 

When Eq. (6.44) is compared to the result obtained by a numerical evaluation of the integral in Eq. (6.40) (see Table 6.1), one will find good agreement at high temperatures. At low temperatures, the analytical formula can, however, deviate substantially from the exact result. 

In special cases, say when we want to account for isotope effects in chemical reactions, we just need to consider the changes in the above-mentioned quantities (see, e.g., Problem 6.7). As an example, consider the two exchange reactions 

7This expression is identical to the tunneling correction factor for a parabolic barrier. 

Transition-state theory has been applied to numerous reactions. In order to evaluate the accuracy of the theory in direct comparison to experimental results, highly accurate data for the activated complex is required as basic input to the theory. To that end, we can, e.g., consider the simple reaction D + H2 — HD + H the rate constant at 1000 K (a temperature where tunneling plays a minor role) is 1.78 x 10-12 cm3/(molecule s) (see Problem 6.8) which is in excellent agreement with the experimental value 2.13 x 10-12 cm3/(molecule s) (with an experimental uncertainty of about 25%). 

The notation K is used to emphasize that the scale for K% is based on activities, not simply concentrations. In the gas phase one has available partition functions that can be used to calculate Kt. In solution, however, partition functions are not available. The 

Influence of ionic charges on second-order diffusion-controlled rate constants in aqueous solution at 25 °C 

The TST postulates that the reactants are in equilibrium with the transition state. We can write 

The rate is taken to be proportional to the concentration (not the activity ) of the transition state. 

In the reference state the activity coefficients are, by definition, unity. The reference state may be that in the limit of infinite dilution, but the more conventional reference state is C° = 1 M. With the -y s = 1, 

---

An example application of transition state theory is to simple gas-phase dissociations of a diatomic molecules such as CO... 

The classical approach for discussing adsorption states was through Lennard-Jones potential energy diagrams and for their desorption through the application of transition state theory. The essential assumption of this is that the reactants follow a potential energy surface where the products are separated from the reactants by a transition state. The concentration of the activated complex associated with the transition state is assumed to be in equilibrium... 

Application of Transition State Theory to Traditional and Cavitating Ultrasound Olefin Exchange... 

An example of the application of transition state theory to atmospheric reactions is the reaction of OH with CO. As discussed earlier, this reaction is now believed to proceed by the formation of a radical adduct HOCO, which can decompose back to reactants or go on to form the products H + COz. For complex reactions such as this, transition state theory can be applied to the individual reaction steps, that is, to the steps shown in reaction (15). Figure 5.3 shows schematically the potential energy surface proposed for this reaction (Mozurkewich et al., 1984). The adduct HOCO, corresponding to a well on the potential energy surface, can either decompose back to reactants via the transition state shown as HOCO./ or form products via transition state HOCO,/. ... 

There were several early discussions on the application of transition state theory to activated diffusional transport in crystals [W. Jost (1955)]. The Vineyard treatment [G. Vineyard (1957)] adapts Eyring s concept to the case of vacancy diffusion in a (elemental) crystal and clarifies it by taking into account the many-body features of this diffusion process. 

Arrhenius, 4-factors can also be calculated by direct application of transition-state theory using the expression (24) derived by Herschbach et al. 

The simplest application of transition-state theory is to the collision between two structureless molecules, i.e., two atoms A and B ... 

The expression for the rate constant in Example 6.3 can also be considered as an application of transition-state theory to a bimolecular reaction where the molecules are considered as structureless atoms with the relevant masses. Thus, when we compare this with the proper expression for a bimolecular reaction, we can gain insight into how the internal molecular degrees of freedom affect the rate constant. In Section 4.1.2,... 

During the past four decades the dynamics and mechanisms of electron-transfer processes have been studied via the application of transition-state theory to the kinetics for electrochemical processes. As a result, both the kinetics of the electron-transfer processes (from solid electrode to the solution species) as well as of pre- and post-electron-transfer homogeneous processes can be characterized quantitatively. 

Transition-state theory allows details of molecular structure to be incorporated approximately into rate constant estimation. The critical assumption of transition-state theory is that quasi-equilibrium is established between the reactants and an activated complex, which is a reactive chemical species that is in transition between reactants and products. The application of transition-state theory to the estimation of rate constants can be illustrated by the bimolecular gas-phase reaction... 

Beynon, J.H. and Gilbert, J.R. (1984) Applications of Transition State Theory to Unimolecular Reactions, John Wiley Sons, Inc., New York. 

This assumption is made in nearly every application of transition-state theory. 

The application of transition-state theory to unimolecular reactions is very straightforward. With the subscript 1 identifying the reactant, equation (70) reduces to... 

The most interesting applications of transition state theory have been, perhaps, in solution chemistry, and a number of detailed improvements have been made to bring in some of the effects of the solvent. This has been done mostly within the framework of thermodynamics by introducing in Eq. (I)... 

The competition between intramolecular vibrational relaxation and chemical reaction has been discussed in terms of the applicability of transition state theory to the kinetic analysis [6], If the environment functions mainly as a heat bath to ensure thermalization among the vibrational modes in the excited complex, then transition state theory is a good approximation. On the other hand, when the reaction is too fast for thermalization to occur the rate can depend upon the initial vibronic state. Prompt reaction and prompt intersystem crossing are, by definition, examples of the latter limit. 

APPLICATION OF TRANSITION-STATE THEORY TO ASYMMETRIC INDUCTION... 

Reactions carried out in supercritical fluids have been found to accelerate the reaction rate greatly.- By manipulating the properties of the solvent in which the reaction is taking place, interphase mass transfer limitations can be eliminated. Application of transition states theory discussed in the Professional Reference Shelf on the CD-ROM of Chapter 3 has proven useful in the analysis of these reactions,... 

---