Chemistry Reference
In-Depth Information
8.5.6
Measurement of k
d
Dead-end polymerization
is a useful technique for assessing
k
d
.
Equation (8-35)
can be written
ð
ln
½
2
V
0
k
d
M
2
e
2
k
d
t=
2
1
Þ
(8-35a)
2
M
0
5
½
where the parameter
V
0
is defined by
1
=
2
fk
d
k
t
1
=
2
V
0
k
p
½
I
0
(8-55)
A useful feature of
Eq. (8-35a)
is that
k
d
is separated from the other rate con-
stants because it is the only kinetic parameter in the exponential function. If the
starting initiator concentration [I
0
] was insufficient to cause polymerization of all
the monomer, the reaction would reach a dead end (i.e., cease) after a long time
t
N
. The corresponding limiting value of [M] can be used to assess
k
d
if this con-
centration can be determined accurately and if the polymerization does not exhibit
autoacceleration (
Section 8.13.2
) effects at high conversions
[4]
. To avoid these
potential complications, it is convenient to expand the exponential function in
Eq. (8-35a)
in a power series and simplify the resulting expression to
"
#
t
2
t
2
ln
½
M
=½
M
0
k
d
2
k
d
2
V
0
1
2
ð
1
=
2
!
Þ
1
ð
1
=
3
!
Þ
1
?
(8-56)
2
5
t
If the
t
term is much larger than the
t
2
term, the equation may be truncated to
t
ln
½
M
=½
M
0
k
d
4
V
0
2
V
0
2
5
(8-57)
t
A plot of the left-hand side of
Eq. (8-57)
versus time produces a straight line
whose intercept at
t
5
0 is equal to
V
0
and whose slope can be used to calculate
k
d
[5]
.
8.6
Length of the Kinetic Chain and Number Average
Degree of Polymerization of the Polymer
The kinetic chain length
v
is the average number of monomers that react with an
active center from its formation until it is terminated. It is given by the ratio of
the rate of polymerization to the rate of initiation and under steady-state condi-
tions where
R
t
5
R
i
:
M
2
fk
d
½
M
R
p
R
i
5
k
p
½
M
½
k
p
½
M
½
v
(8-58)
5
5
2
I
M
2
k
t
½
from
Eqs. (8-10), (8-13), and (8-23)
. Therefore,
