(23)

The energy for the reaction to occur is generated by collision with a second body M.

Chemical rate limited processes, in the other hand, correspond to chemical reactions occurring under conditions in which the statistical distribution of molecular energies obey the Maxwell-Boltzman form, i.e., the fraction of molecules that have an energy, E, or larger is proportional to e^-E/RT. The rates of intermolecular collisions are very rapid and all species are in equilibrium with the gas mixture.

Table 1 depicts several theories that can be applied to estimate rate coefficients in order of increasing complexity. In the simplest approach, the rate coefficient of a bimolecular reaction is simple the collision frequency between the molecules. To improve upon this approximation, the collision frequency needs to be corrected to account for the fact that only those collisions with energies above the activation energy of the reaction will result in a net reaction. Also, a steric factor has to be included, since only collisions taking place in a given spatial arrangement will lead to a net reaction.

The next level of complexity is Transition State Theory (TST) of both unimolecular and bimolecular reactions. In TST, the rate coefficients include an activation energy factor, and an entropy factor to account for steric factors. TST only applies to chemical rate limited processes. The Lindemann approach to unimolecular reactions would fall within this level of complexity.

Finally, the most complex theories involve the quantum mechanical treatment of energy transfer limited processes such as thermal activation and unimolecular/bimolecular chemical activation. By chemical activation, in the case of a bimolecular reaction for example, is meant that as the result of a bimolecular reaction an intermediate species is formed possessing excess energy over the ground state that can more easily lead to some final product by decomposition,

These quantum theories account for the dependence of the overall rate coefficients on the excess vibrational energy of the molecular species.

In reaction modeling rate coefficients are normally expressed in the modified Arrhenius form,

(24)

A is the collision frequency factor, T is the temperature (the exponent n accounts for non-Arrhenius behavior to fit experimental data) and E_a is the activation energy. Non-Arrhenius behavior is most obvious in reactions that have little activation energies with the pre-exponential factor determining the temperature dependence.

The discussion above has established an approach for building more complex reaction mechanisms, and the theoretical foundations for the estimation of reaction rate coefficients. The most important consideration always is the chemistry included in the mechanism. Assembling the elementary reactions composing the mechanism is followed then by the best assessment for the mathematical expression giving the rate coefficients of each reaction. The procedure to follow based on the author’s experience is discussed below.