Chemistry Reference
In-Depth Information
N
a
2
1
4a
1
1
y
3
a
y
2
1
5
(7-31c)
4
ð
Þ
1
2
a
y
5
1
From Eqs. (2-6) and
(7-24)
:
X
X
p
i
2
1
X
n
5
x
i
i
5
i
ð
1
2
p
Þ
(7-32)
i
i
Terms in p alone can be removed from the summation since this calculation is
at fixed degree of conversion. Then
N
1
2
p
1
ip
i
2
1
X
n
5
ð
1
p
Þ
(7-33)
2
5
2
5
1
p
ð
1
p
Þ
2
2
i
5
1
(invoking the series sum of
Eq. 7-31a
). Of course, this calculation gives the same
result as the Carothers equation for step-growth polymerization with f
av
5
2
(
Eq. 7-20
).
The reader is reminded of Eq. (1-1), which reads as follows in the current
context:
M
n
5
X
n
M
0
5
M
0
=ð
1
2
p
Þ
(7-34)
where M
0
is the formula weight of the repeating unit which is the residue of a sin-
gle monomer in this case.
The weight average degree of polymerization X
w
can be derived similarly by
using the series summation of
Eq. (7-31b)
. From Eqs. (2-13) and
(7-26)
,
X
w
5
P
i
iw
i
5
P
i
2
2
p
i
2
1
ð
1
2
p
Þ
P
i
2
p
i
2
1
2
5
ð
1
p
Þ
ð
1
p
Þ
1
p
2
1
1
(7-35)
2
5
ð
1
p
Þ
2
5
3
1
p
ð
1
p
Þ
2
2
The breadth of the number distribution in equilibrium step-growth polymeriza-
tion of linear polymers is indicated by
X
w
=
X
n
5
M
w
=
M
n
5
1
p
(7-36)
1
from
Eqs. (7-34) and (7-35)
. In standard statistical terms, since M
w
5
ð
M
n
from the last equation, then the variance of the number distribution of molecular
weights is
1
p
Þ
1
M
n
5
s
n
5
2
M
w
M
n
2
p
ð
M
n
Þ
(7-37)
(cf. Eq. 2-33) and the standard deviation of the number distribution is
p
1
=
2
M
n
s
n
5
(7-38)
