Essentials of Computational Chemistry

520 15 ADIABATIC REACTION DYNAMICS
15.1.1 Unimolecular Reactions
The simplest unimolecular reaction may be expressed in equilibrium form as
A k1
−−−⇀ ↽−−− B (15.3)
k−1
where A and B are isomeric, e.g., conformationally or constitutionally. The unimolecular
rate constants above and below the equilibrium arrows are associated with the forward and
reverse steps in the equilibrium process. Thus, the rate at which A is converted into B is
k1[A] while the rate at which B is converted into A is k−1[B]. These rate constants truly are
‘constants’, i.e., they are independent of time.
Note that at equilibrium, the rate at which A is converted into B must be exactly equal to
the rate at which B is converted into A, i.e., the system is stationary with respect to reactant
and product concentrations. Thus
This may be rearranged to yield
k1[A]eq = k−1[B]eq
k1
= [B]eq
[A]eq
k−1
= Keq
(15.4)
(15.5)
where Keq is the equilibrium constant for Eq. (15.3). So, it is a straightforward task to
measure the ratio of the elementary rate constants, but how is any one measured individually?
If we consider the system perturbed from equilibrium – let us suppose that there is an
excess of A – then the rate of return to equilibrium may be expressed either as the rate of
disappearance of A, i.e., −d[A]/dt, or as the rate of appearance of B, i.e., d[B]/dt. Using
the first choice, we may write
− d[A]
dt = k1[A] − k−1[B] (15.6)
If the second term on the r.h.s. can be ignored, either because k−1 ≪ k1 (a so-called ‘irreversible’
reaction), or because we start with [A] ≫ [B] and only observe the system over a
time frame where that relationship continues to hold, then we may rearrange Eq. (15.6) to
give the first-order rate expression
− d[A]
[A] = k1dt (15.7)
where ‘first order’ implies that the sum of the exponents for concentration terms on the r.h.s.
of the general rate expression written in the form of Eq. (15.2) is one. Integration of both