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Reaction Kinetics & Chemical Reaction Models

2. Transition State Theory of Unimolecular/Bimolecular Reactions

A chemical reaction is presumably a continuous process involving a gradual transition from reactants to products. It has been found extremely helpful, however, to consider the arrangement of atoms at an intermediate stage of reaction as though it were an actual molecule. This intermediate structure is the transition state, and its energy content corresponds to the top of the reaction energy barrier along the reaction coordinate. The rate coefficients according to TST will be given here without proof, they correspond to a thermodynamic approach where the reaction rate is given in terms of thermodynamic functions. One of the main assumptions of TST is that the process is chemical rate limited (Laidler, 1987).

Bimolecular Reactions

In a successful bimolecular collision, part of the kinetic energy of the fast-moving reactant molecules is used to provide the energy of activation and thus to produce the high-energy molecular arrangement of the transition state. TST applied to the reaction,

 in which AB^¡  is the transition state structure, leads to the following expression,

(28)

the units are cm³/mol-sec. TST shows a T² dependence on temperature, and the change in entropy leading to the transition state is needed. There is now quantum chemistry software available that make it possible to estimate the properties of the transition state, this will be discussed more fully later. Unfortunately, properties of the transition state cannot yet be tested experimentally, thus the uncertainty in the calculations for the transition state would not be well known. The best approach is to be most familiar with the particular quantum chemistry package that is to be used and its limitations in general.


Unimolecular Reactions

In unimolecular reactions, the necessary energy for the reaction may accumulate in the molecule as the result of intermolecular collisions, photon activation, or as the result of unimolecular chemical activation. Once energy is imparted to the molecule, it is rapidly distributed amongst its vibrational and rotational energy levels with the energized molecule taking many configurations. If one of these configurations corresponds to the localization of enough energy along the reaction coordinate, then the reaction occurs.

The application of TST theory to the process below,

leads to the following expression ( the units are sec^-1 )

(30)

TST predicts a first order temperature dependence for the rate coefficient. As with bimolecular reactions, the entropy change leading to the transition state will be required, and quantum chemistry methods may be used for this.

Lindemann’s Approach to Unimolecular Reactions

No discussion on chemical kinetic theory would be complete without Lindemann’s theory of unimolecular reactions which attempts to explain the pressure dependence of unimolecular reactions. The overall unimolecular reaction is given below,

At a given temperature, and for high pressure conditions, the rate of decomposition of A is first order in its concentration, but a low enough pressures, the rate becomes pressure dependent, i.e., the process is energy transfer limited. The dependence of k_uni on pressure is shown in Figure 1.

 In the mechanism developed by Lindemann (Laidler, 1987), the decomposition of reactant A occurs according to the following two step scheme,

In reaction (32), molecules of A are energized by collision with a second body M. Reaction (33) describes the process by which the energized molecules of A* turns into the final product. The results of this approach will be given below without a proof, Laidler (1987), Garginer (1972) present a full discussion of Lindemann’s Theory.  

In the high pressure limit the unimolecular rate coefficient takes the form,

The high-pressure limit does not show a dependence on pressure, in the other hand, the low pressure coefficient is dependent on pressure through the term [M], as it is found experimentally. Estimates of the high-pressure limit rate coefficient k^oo_uni can be made using TST, quantum chemistry can be used to estimate the properties of the transition state. In equation (35), in order to calculate the low-pressure coefficient, k₁ is expressed as follows,

Z_A*M is the molar collisional frequency between energized A* molecules and M (see Equation 25), f(E_o) is the fraction of molecules with energies higher than E_o and can be activated according to reaction (32), this term may be given in terms of the Boltzmann distribution function P(E) or

b is a collisional efficiency that accounts for the fact that not every collision between an activated A* molecule and M results in deactivation of A* back to A.

 For thermodynamic conditions where unimolecular reactions fall in a regime that is between high and low pressure or the fall-off regime, software is available that can make estimates of the coefficient based on the constants given in equations (34) and (35): Kee et al. (1993), Stewart et al (1989). The reader is referred to these references for more details.

Treatment of Energy Transfer Limited Reactions.