## Lecture 18## Kinetics- simple rate equations |

The Arrhenius equation is based on the empirical observation that rates
of reactions increase with temperature. This is accounted for by an increase
in rate consant with temperature, which led to the concept of an activation
barrier in the reaction pathway. The height of barrier is given by the
activation energy, E_{act}.

where k_{¥} is the rate constant
when Eact is zero, often called the pre-exponential
factor. In collisional ractions, k_{¥}
is the diffusion limited rate constant (see diffusion note above).

where r_{m} and r_{n}
are the radii of reacting species, D_{m} and
D_{n} are diffusion coefficients, N_{o}
is the Avogadro number, and the division by 1000 gives units of cm^{2}.s^{-1}

**Eyring, Randall-Wilkins theory:**

Consider the reaction below, which proceeds through an activation barrier
populated by the species MN^{#}:

k_{1}
k_{2}
**M + N <==> M--N <==> MN ^{# }---->
P**

k

The probability for forming MN^{#}, the
intermediate at the top of the activation barrier, is determined by the
equilibrium constant for formation (K^{#}),
and hence the free-energy:

The rate of formation of products is given by:

The rate constant k_{2} is considered to occur in the first
vibrational transition, and decay of MN^{# }is assumed to go with
equal probability forward or back, so that:

where *k*_{B} is the Boltzmann constant,
*h* is the Planck constant, and *k*_{B}T/*h*
has a value of ~10^{13} s^{-1}. Substituting:

Since we can also express the rate in the conventional form:

and since K^{#} = exp[-DG^{act}/RT]

We introduce a second factor, k, to account
for differences between adiabatic and non-adiabatic processes (see Lecture
19), and convert to molecular units by substituting *k*_{B}
for R, to get:

More generally, we can consider that the limiting step of a reaction may be the collisional process, in which case the overall rate will be determined by the rate of formation of M--N through the second order process above. Then:

where B is either collision frequency (the diffusion limited rate constant,
also called A above, 10^{11 }M^{-1}.s^{-1}) if
liquid phase collisions are involved, or B is vibrational frequency (*k*_{B}T
/ *h *(~10^{13} s^{-1})) if the reaction is intramolecular
or monomolecular (first order), or some mixed term if a collisional process
proceeds via an intramolecular complex.

If we equate E^{act} of the Arrhenius equation with DH^{act},
then:

©Copyright 1996, Antony Crofts, University of Illinois at Urbana-Champaign, a-crofts@uiuc.edu