Essentials of Computational Chemistry

15.1 REACTION KINETICS AND RATE CONSTANTS 521
sides from time 0 to time τ leads to
[A]0
ln = k1τ (15.8)
[A]τ
Thus, experimentally, one plots the logarithm of the concentration ratio against time under
conditions where Eq. (15.7) holds in order to determine k1. The reverse rate constant k−1
may either be determined analogously, or from Eq. (15.5) once k1 is known.
Note that for a first-order reaction, the time required for the reactant concentration to drop
by some constant factor is a simple function of the rate constant. Thus, for instance, the half-
life τ1/2, which is the time required such that [A]τ1/2
= 1
2 [A]0 for any starting concentration,
may be determined from Eq. (15.8) to be k −1
1 ln 2.
Fragmentation is another possible unimolecular reaction. A fragmentation reaction may
be expressed as
A k1
−−−⇀ ↽−−− B + C (15.9)
k−1
(in principle, fragmentations involving more than two products all of which are produced
simultaneously are possible, but examples are very rare). The rate for disappearance of A
when in excess of its equilibrium value is
− d[A]
dt = k1[A] − k−1[B][C] (15.10)
The only difference from an experimental viewpoint between Eqs. (15.6) and (15.10) is that
Eqs. (15.7) and (15.8) can now be made to apply by ensuring that either one (or both) of B
and C have vanishingly small concentrations over the course of the rate measurement.
15.1.2 Bimolecular Reactions
The opposite of a fragmentation reaction is a condensation reaction, i.e.,
A + B k1
−−−⇀ ↽−−− C (15.11)
Note that in practice the abstract species A and B may themselves already be molecules
or supermolecules formed from prior condensations, but simple probability arguments make
condensation reactions simultaneously involving more than two species impossible under
most sets of experimental conditions. The rate law associated with eq. 15.11 is
k−1
− d[A]
dt = k1[A][B] − k−1[C] (15.12)
where k1 is a second-order rate constant because it multiplies a set of concentrations whose
exponents sum to 2. The simplest evaluation of k1 proceeds by arranging for a vanishingly