Elementary chemical reactions can be classified
as either energy-transfer limited or chemical reaction rate limited (Senkan, 1992). In
energy transfer-limited processes, the observed rate of reaction corresponds to the energy
transfer to or from species either by intermolecular collisions or by radiation, or
intramolecularly due to energy transfer between different degrees of freedom of a chemical
species. All thermally activated unimolecular reactions become energy-transfer limited at
low-density conditions because the reactant can receive the necessary activation energy
only by intermolecular collisions. The reaction then becomes pressure dependent at a given
temperature. An example of such reaction is the thermal decomposition of hydrogen,
(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 Ea 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.
