Chemistry Reference
In-Depth Information
M
o
]
2
, where [
M
o
] represents the initial concentration
of the reactants (of hydroxyl or carboxyl groups in a polyesterification) at time
The constant in the above equation equals 1/[
0.
At the start of the polymerization, there are [M
o
] molecules present. After some progress of the
reaction, there are [M] molecules left; [M
o
]
t ¼
[M] is then the number of molecules that participated
in the formation of polymeric chains. The conversion,
p,
can be written, according to Carothers [
6
], as
p ¼
½
M
o
½
M
½
M
o
or, the concentration of [M] at any given time
t
is
½
M
¼½
M
o
ð
1
pÞ
and the degree of polymerization,
1
DP
¼
1
p
It is important to realize from the above equation that in order to a DP of only 50, it is necessary to
achieve 98% co
nve
rsion (
must equal 0.98).
The value of DP, at any given time,
p
is equal to the ratio of monomer molecules that were present
at the start of the reaction divided by the number of molecules that are still present at that particular
time:
t
¼
½
M
o
½
M
DP
By combining the above expression with Caruthers equation and solving for [
M
], one obtains
½
M
¼½
M
o
ð
1
pÞ
For a second-order rate expression, the above equation can be written as
1
=f½
M
o
ð
1
pÞg
1
½
M
o
¼kt
and by replacing 1
=ð
1
pÞ
with DP, one obtains
DP
¼½
M
o
kt þ
1
Using the above equation, it is possible to calculate from the rate constant (if it is known) and the
concentration of monomers the time required to reach a desired number average molecular weight.
When there is no catalyst present and the carboxylic acid assumes the role of a catalyst itself, then a
third-order rate expression (shown above) must be employed:
3
d
½
M
=
d
t ¼ k½
M
By integrating the third-order rate expression, one obtains:
2
2
1
=½
M
1
=½
M
o
¼
2
kt
