McKay, Donald. "Front matter" Multimedia Environmental Models ...

Integrating from an initial condition of C O at zero time gives the following
equations:
©2001 CRC Press LLC
ln(C/C O) = –kt or C = C O exp(–kt)
Rate constants have units of frequency or reciprocal time and are therefore not
easily grasped or remembered. A favorite trick question of examiners is to ask a
student to convert a rate constant of 24 h –1 into reciprocal days. The correct answer
is 576 days –1 , so beware of this conversion! It is more convenient to store and
remember half-lives, i.e., the time, t 1/2, which is the time required for C to decrease
to half of C O. This can be related to the rate constant as follows.
When C = 0.5 C O, then t = t 1/2
ln (0.5) = –kt 1/2, therefore, t 1/2 = 0.693/k
For example, an isotope with a half-life of 10 hours has a rate constant, k, of
0.0693 h –1 .
6.3.2 Non-First-Order Kinetics
Unfortunately, there are many situations in which the real reaction rate is not a
first-order reaction. Second-order rate reactions occur when the reaction rate is
dependent on the concentration of two chemicals or reactants. For example, if
A + B Æ D + E
then the rate of the reaction is dependent on the concentration of both A and B.
Therefore, the reaction rate is as follows:
N = Vk C A C B
Reactant “B” is often another chemical, but it could be another environmental
reactant such as a microbial population or solar radiation intensity. Third-order
reaction rates, when the rate of reaction is dependent on the concentration of three
reactants (N = Vk C A C B C C), are very rare and are unlikely to occur under environmental
conditions.
We can often circumvent these complex reaction rate equations by expressing
them in terms of a pseudo first-order rate reaction. The primary assumption is that
the concentration of reactant “B” is effectively constant and will not change appreciably
as the reaction proceeds. Thus, the constant k and concentration of reactant
“B” can be lumped into a new rate constant, k P, and the second-order reaction
becomes a pseudo first-order reaction. Therefore,
N = Vk C AC B