Chemistry Reference
In-Depth Information
and, by substituting for [
M
] the Carothers equation and then rearranging the resultant equation, one
obtains:
1
1
2
2
¼
2
kt
2
½
M
o
ð
1
pÞ
½
M
o
this can also be written as:
2
2
1
=ð
1
pÞ
¼
2
kt½
M
o
þ
1
or
DP
2
2
¼
2
kt½
M
o
þ
1
The above equation shows that without a catalyst the molecular weight increases more gradually.
It can be deduced from the above discussion that a high stoichiometric balance is essential for
attaining high molecular weight. This means that any presence of a monofunctional impurity has a
strong limiting effect on the molecular weight of the product. The impurity blocks one end of the
chain by reacting with it. This is useful, however, when it is required to limit the DP of the product.
For instance, small quantities of acetic acid are sometimes added to preparations of some polyamides
to limit their molecular weight.
In polymerizations of monomers with the same functional groups on each molecule, like A
——
A
and B
——
B (i.e., a diamine and a diacid), the number of functional groups present can be designated
as
o
o
o
o
B
represent the number of functional
groups present at the start of the reaction. They are twice the number of A
——
A and B
——
B
molecules that are present. If the number
N
A
for A type and
N
B
for B type. These numbers
N
A
and
N
o
o
N
B
is slightly larger than
N
A
, then we have a
stoichiometric
imbalance
in the reaction mixture. This imbalance is designated as
r
.
o
o
B
r ¼ N
A
=N
as less than or equal to unity, so, in the above, B groups are
present in excess.) The total number of monomers at the start of the reaction are
(It is common to define the ratio
r
o
o
ðN
A
þ N
B
Þ=
2or
o
N
A
ð
2.
The extent of the reaction,
1
þð
1
=rÞÞ=
can be defined as the portion of the functional groups A that reacted at
any given time. The portion of the functional groups B that reacted at the same time can be designated
by
p,
,
respectively. The total number of unreacted A groups in the reaction mixture would then be
N
rp
. The unreacted portions of A and B groups can then be designated as 1
p
and 1
rp
o
o
unreacted B groups. The total number
of chain ends on the polymer molecules is the sum of the unreacted A and B groups. Because each
polymer molecule
A
ð
1
pÞ
. This reaction mixture also contains
N
B
ð
1
rpÞ
has
two
chain
ends,
the
total
number
of
chain
ends
is
then
o
o
½N
2.
The number average degree of polymerization is equal to the total number of A
——
A and B
——
B
molecules present at the start of the reaction divided by the number of polymer molecules at the end.
This can be represented as follows:
A
ð
1
pÞþN
B
ð
1
rpÞ=
A
ð
N
1
þ
1
=rÞ=
2
DP
n
¼
o
B
½N
A
ð
1
pÞþN
ð
1
rpÞ=
2
o
o
the expression can be reduced (since
r ¼ N
A
=N
B
)to
