Chemistry Reference
In-Depth Information
the propagation rate constant
k
p
for hompolymerization of M
i
under the given
reaction conditions.)
If we now assume that the only significant changes in monomer concentra-
tions result from propagation reactions (i.e., changes in
M
i
from initiation and
transfer reactions are negligible), the rates of monomer disappearance according
to the reaction schemes of (9-2)
½
(9-5) are
M
i
½
M
:
2
½
d
½
M
1
=
dt
k
11
½
M
1
1
k
21
½
M
1
(9-6)
2
5
M
:
2
½
M
:
1
½
d
½
M
2
=
dt
k
22
½
M
2
1
k
12
½
M
2
(9-7)
2
5
Similarly, the time dependence of the concentration of radical M
:
1
is
2
d½
M
:
1
=dt
52
k
12
½
M
:
1
½
M
:
2
½
M
2
1
k
21
½
M
1
(9-8)
M
:
1
M
:
1
=
dt
is negli-
gible compared to the rates of change of concentrations of the reactants. Hence,
setting
d
½
is sufficiently small that
d
½
Under steady-state conditions
M
:
1
=
½
dt
0
5
M
:
1
½
M
:
2
½
k
12
½
M
2
5
k
21
½
M
1
(9-9)
and
M
:
1
=½
M
:
2
5
½
k
21
½
M
1
=
k
12
½
M
2
(9-10)
The relative rates of incorporation of the two monomers into the copolymer at
any instant follow by dividing
Eq. (9-6)
by
Eq. (9-7)
:
d
M
:
1
½
M
:
2
½
½
M
1
k
11
½
M
1
1
k
21
½
M
1
(9-11)
M
2
5
M
:
2
½
M
:
1
½
d
½
k
22
½
M
2
1
k
12
½
M
2
M
:
2
;
Now divide the right-hand side of
Eq. (9-11)
by
½
insert expression (9-10)
M
:
1
=½
M
:
2
;
for
½
and divide through by
k
21
to obtain
d
d½M
2
5
½
½
M
1
M
1
½ð
k
11
=
k
12
Þ½
M
1
1
½
M
2
(9-12)
½
M
2
½ðk
22
=k
21
Þ½
M
2
1
½
M
1
Define the reactivity ratios
r
i
as
r
1
k
11
/k
12
and
r
2
k
22
/k
21
, so that the pre-
ceding equation becomes
d
d½M
2
5
½
½
M
1
M
1
ð
r
1
½
M
1
1
½
M
2
Þ
(9-13)
½
M
2
ðr
2
½
M
2
1
½
M
1
Þ
An equivalent expression can of course be derived in terms of mole fractions
rather than concentrations. If
f
1
and
f
2
are the respective mole fractions of mono-
mers M
1
and M
2
in the reaction feed and
F
1
and
F
2
are the corresponding mole
fractions in the copolymer formed from this mixture, then
r
1
f
1
1
f
1
f
2
r
1
f
1
1
F
1
5
(9-14)
r
2
f
2
2
f
1
f
2
1
and from the definition of mole fraction
