Chemistry Reference
In-Depth Information
early stages of the polymerization because the rates of these reactions, (8-16) and
(8-18), are proportional to the square of the total concentration of radicals in the sys-
tem. Eventually the rate of radical generation will be balanced by the rate at which
radicals undergo mutual annihilation, and the concentration of radicals in the system
will reach a steady value. It can be shown that this steady state is reached very early
in the reaction with the usual concentrations of initiator and monomer.
The assumption that the rate of initiation equals the rate of termination is
called the
steady-state assumption
.
It
is equivalent
to the two following
statements:
R
i
5
R
t
at steady state
(8-24)
and
d
½
M
dt
5
0 at steady state
(8-25)
Since the steady state is reached soon after polymerization starts, we can
assume without significant error that
it applies to the whole course of the
polymerization.
Substituting
Eqs. (8-10) and (8-23)
into
Eq. (8-24)
,
M
2
2
fk
d
½
I
5
2
k
t
½
(8-26)
Hence
M
5
½
1
=
2
½
fk
d
½
I
=
k
t
(8-27)
This is an expression for the total concentration of monomer-ended radicals in
terms of experimentally accessible quantities.
The rate of polymerization is taken to be the rate of disappearance of mono-
mer, which is
d
[M]
/dt
. (The concentration of monomer [M] decreases with time
so
d
[M]
/dt
is negative.) The two reactions listed that consume monomer are (8-8)
(initiation) and (8-12) (propagation). Therefore,
d
½
M
=
dt
R
i
1
R
p
(8-28)
2
5
It can be shown that the initiation process accounts for a negligible amount of
monomer if high-molecular-weight polymer is being produced. Then the rate of
polymerization can be taken as equal to the rate of propagation. That is,
2
d½
M
=dt
5
R
p
5
k
p
½
M
½
M
5
ðk
p
=k
t
1
=
2
1
=
2
Þ½
M
ðfk
d
½
I
Þ
(8-29)
The preceding relation is obtained by substituting
Eq. (8-27)
in
Eq. (8-13)
.It
is a differential equation and gives the rate of polymerization in moles of mono-
mer per unit volume per unit time when the monomer concentration is [M] and
the initiator concentration is [I]. Since both these concentrations will decrease as
the reaction proceeds, the amount of polymer formed in a given time is obtained
by integrating
Eq. (8-29)
with respect to time (see
Section 8.3.5
).
