Chemistry Reference
In-Depth Information
M
5
k
p
½
2
v
k
p
½
M
=
2
k
t
½
M
=
2
k
t
R
p
(8-59)
5
by substituting for [M
] from
Eq. (8-13)
.
If the polymerization is initiated by thermal homolysis of an initiator,
Eq.
(8-27)
applies, and this expression for [M
]in
Eq. (8-59)
gives
1
=
2
v
k
p
½
M
=
2
ð
fk
d
½
I
k
t
Þ
(8-60)
5
The last two equations show that the average number of monomers converted to
polymer per radical will be inversely proportional to the radical concentration or
the polymerization rate and directly proportional to the monomer concentration.
The number average degree of polymerization (
Section 1.2
), DP
n
, is equal at
any instant to the ratio of the rate of monomer disappearance to the rate at which
completed polymer molecules are produced. That is,
d½
dt
=
M
d½
polymer
DP
n
52
(8-61)
dt
d
½
polymer
M
2
M
2
5
k
tc
½
1
2
k
td
½
(8-62)
dt
because each termination reaction by combination yields one polymer molecule
[?] while each disproportionation produces two macromolecules.
Equations (8-13)
and (8-62)
can be substituted into
Eq. (8-61)
to yield
M
DP
n
5
k
p
½
M
=ð
k
tc
1
2
k
td
Þ½
(8-63)
and with [M
]
5
R
p
/k
p
[M] from
Eq. (8-13)
,
k
p
½
2
DP
n
5
M
=
R
p
ð
k
tc
1
2
k
td
Þ
(8-64)
Also, for initiation by thermal homolysis of an initiator,
Eqs. (8-27) and (8-63)
give
1
=
2
1
=
2
k
p
½
M
ðk
tc
1
k
td
Þ
k
p
½
M
ðk
tc
1
k
td
Þ
DP
n
5
(8-65)
Þ
5
2
R
i
1
=
2
1
=
2
ðk
tc
1
2
k
td
Þðfk
d
½
I
1
ð
k
tc
1
2
k
td
Þ
On c
omparing
Eqs. (8-59) and (8-64)
or
Eqs. (8-60) and (8-65)
, it is evident
that DP
n
5
v
if termination is entirely by disproportionation
(k
tc
5
0). I
n t
he more
common instance where termination is by combination only
k
td
5
0 and DP
n
5
2
v
.
(Recall that for vinyl polymers
M
n
5
M
0
DP
n
(8-66)
where
M
0
is the molecular weight of the monomer.)
The number average degree of polymerization is inversely proportional to the
rate of polymerizati
on a
t given monomer concentration and temperature (
Eq.
8-64
). It follows that DP
n
also varies inversely
wi
th the square root of the rate of
initiation. It is not possible to increase
R
p
and DP
n
simultaneously in free-radical
polymerizations.
