26
The kinetics of complex reactions
Solutions to exercises
Discussion questions
E26.1(b)
In the analysis of stepwise polymerization, the rate constant for the second-order condensation is
assumed to be independent of the chain length and to remain constant throughout the reaction. It
follows, then, that the degree of polymerization is given by
h
n
i=
1
+
kt
[A]
0
Therefore, the average molar mas can be controlled by adjusting the initial concentration of monomer
and the length of time that the polymerization is allowed to proceed.
Chain polymerization is a complicated radical chain mechanism involving initiation, propagation,
and termination steps (see Section 26.4 for the details of this mechanism). The derivation of the
overall rate equation utilizes the steady state approximation and leads to the following expression for
the average number of monomer units in the polymer chain:
h
n
2
k
[M][I]
−
1
/
2
,
where
k
=
1
/
2
k
P
(f k
i
k
t
)
−
1
/
2
, with
k
P
,
k
i
, and
k
t
, being the rate constants for the propagation, initiation,
and termination steps, and
f
is the fraction of radicals that successfully initiate a chain. We see that
the average molar mass of the polymer is directly proportional to the monomer concentration, and
inversely proportional to the square root of the initiator concentration and to the rate constant for
initiation. Therefore, the slower the initiation of the chain, the higher the average molar mass of the
polymer.
E26.2(b)
Refer to eqns 26.26 and 26.27, which are the analogues of the Michaelis–Menten and Lineweaver–
Burk equations (26.21 and 26.22), as well as to Fig. 26.12. There are three major modes of inhibition
that give rise to distinctly different kinetic behaviour (Fig. 26.12). In competitive inhibition the
inhibitor binds only to the active site of the enzyme and thereby inhibits the attachment of the
substrate. This condition corresponds to
α>
1 and
α
0
=
1 (because ESI does not form). The slope of
the Lineweaver–Burk plot increases by a factor of
α
relative to the slope for data on the uninhibited
enzyme (
α
=
α
0
=
1). The
y
-intercept does not change as a result of competitive inhibition. In
uncompetitive inhibition, the inhibitor binds to a site of the enzyme that is removed from the active
site, but only if the substrate is already present. The inhibition occurs because ESI reduces the
concentration of ES, the active type of the complex. In this case
α
=
1 (because EI does not form)
and
α
0
>
1. The
y
- intercept of the Lineweaver–Burk plot increases by a factor of
α
0
relative to the
y
-intercept for data on the uninhibited enzyme, but the slope does not change. In non-competitive
inhibition, the inhibitor binds to a site other than the active site, and its presence reduces the ability
of the substrate to bind to the active site. Inhibition occurs at both the E and ES sites. This condition
corresponds to
1 and
α
0
>
1. Both the slope and
y
-intercept of the Lineweaver–Burk plot increase
upon addition of the inhibitor. Figure 26.12c shows the special case of
K
I
=
K
I
0
and
α
=
α
0
, which
results in intersection of the lines at the
x
-axis.