Chemistry Reference
In-Depth Information
the reaction mixture is iM
0
N
i
. The total weight of all molecules equals N
0
M
0
(neglecting unreacted ends). Hence the weight fraction w
i
of i-mers is
2
p
i
2
1
N
i
ð
iM
0
Þ
N
0
M
0
5
N
0
ð
1
p
Þ
ð
iM
0
Þ
2
w
i
5
N
0
M
0
(7-26)
2
p
i
2
1
i
ð
1
p
Þ
5
2
This is the differential weight distribution function.
Figure 7.5
a shows the number distribution of degrees of polymerization in a
linear (f
2) step-growth polymerization for several extents of reaction p. This is
a plot of
Eq. (7-22)
. The corresponding weight distribution function (
Eq. 7-26
)is
plotted in
Fig. 7.5b
. The former figure shows that monomer is the most prevalent
species on a mole basis at any extent of reaction. The corresponding weight frac-
tion of monomer is negligible, however, at any p
5
0.99 [w
1
/x
1
5
(1
p) from
$
2
Eqs. (7-22) and (7-26)
].
The weight distribution in
Fig. 7.5b
exhibits a peak at a particular degree of
polymerization, i, for any given extent of reaction. This peak moves to higher i
and flattens out as p increases. The value of i at which this peak occurs is
obtained from the derivative of w
i
with respect to i:
2
p
i
2
1
dw
i
=
di
5
ð
1
p
Þ
ð
Þð
1
i ln p
Þ
(7-27)
2
1
When dw
i
/di
0, the curve has a maximum and i
1/ln p. A series expansion
5
5
of
1)
as p approaches 1. This is the value of X
n
in linear step-growth polymerizations
(
Eq. 7-20
). Thus, as a first approximation, the peak in the weight distribution of
high conversion linear step-growth polymers is located at M
n
of the polymer if
the synthesis was carried out under conditions where interchange reactions and
molecular weight equilibration could occur.
It is also easy to calculate how much polymer has been made at a given
degree of conversion. Defining polymer as any species with i
1/ln p gives p /(p
1) as the first te
rm
and this fraction approaches 1/(p
2
2
2
2, the weight frac-
tion of polymer must be equal to the total weight fraction of all species minus the
weight fraction of monomer. That is,
N
$
N
w
i
5
w
i
2
w
1
(7-28)
i
5
2
i
5
1
Substituting for w
1
from
Eq. (7-26)
and setting
P
w
5
1
w
i
5
1
;
N
2
p
0
p
2
w
i
5
1
1
ð
1
p
Þ
2p
(7-29)
2
2
5
2
i
5
2
Similarly from
Eq. (7-24)
, the mole fraction of the reaction mixture that is a
polymer at any degree of conversion p is
