vessel containing 100 ppmv initially would have 50 ppmv after 23.2 min and 25 ppmv
after 46.4 min.
Example 5.3 The oxidation of ferrous iron(II) to ferric(III) in water follows first-order
kinetics if oxygen is not limiting. A solution of 20 mg/L ferrous iron at pH 8.0 is aerated
for 60 min. The ferric iron that is formed precipitates and can be removed by filtration.
The remaining iron concentration is 4.0 mg/L. What are the rate coefficient and the half-
life?
Answer Rearrange equation (5.22) gives k ¼lnðc=c
0
Þ=t ¼ 0:027 min
1
. Then use
equation (5.23), we have t
1=2
¼ 0:693=0:027 ¼ 25:8 min.
Sometimes a higher-order reaction will behave as if it were first order. For example, if
the reaction is A þ B ) C, the rate law is d½A=dt ¼k½A½B. Now suppose that B is
present far in excess of A. Then as A is depleted, B will decrease only a small amount
proportionally. For a specific example, suppose that the initial concentrations ½A
0
¼ 2:0
and ½B
0
¼ 100: 0. Now as A becomes depleted by 50%, B will only decrease by 1%.
Thus, B can be assumed to have a constant concentration. The concentration of B can
be combined to form a new rate coefficient, k
0
¼ k½B , and the new rate law becomes
first order: d½ A=dt ¼k
0
½A. This assumption is called a pseudo-first-order reaction.
This effect is one reason why a great many reactions can be treated as being first order
even if the underlying reaction mechanism is very complicated.
5.3 ENZYME KINETICS
Enzymes function not only to catalyze reactions but to enable control of those reactions
via sensitivity to environmental conditions and to the presence of cofactors, which can
increase or decrease rates. A detailed treatment of the kinetics of enzyme reactions will
be beneficial for two reasons. First, it will lead to a clearer understanding of the action of
enzymes. Second, the kinetic equations themselves can be useful in modeling biochemical
processes in environmental systems. This can be true even when the process involved is
not a simple enzyme reaction. For example, the first system we describe will be Michaelis–
Menten kinetics. The resulting model is derived rigorously. However, an empirical
equation of the same form, the Monod equation, is used effectively to model substrate
utilization by systems consisting not only of many different enzymes and of substrates,
but of mixed populations of microbial organisms.
Recall from general chemistry that a catalyst is a substance that affects the rate of a
reaction but is not changed by it, and that it acts by lowering the activation energy, E
a
.
The activation energy refers to the height of a barrier between reactants and products of a
reaction. The molecules in a given system have a distribution of energies. If the activation
energy is high, fewer molecules possess enough to surmount the barrier. Lowering the
height of the barrier means that a higher proportion of the molecules has enough energy
to get over it, and the rate at which molecules pass is increased. The effect of activation
energy and temperature on the rate constant of a first-order reaction is given by the
Arrhenius equation:
ln k ¼ B
E
a
RT
ð5:24Þ
90 ENERGY AN D METABOLISM