Chemistry Reference
In-Depth Information
1
þ r
DP
¼
þ r
rp
2
2
The molecular weight of the product can be controlled by precise stoichiometry of the polymeri-
zation reaction. This can be done by simply quenching the reaction mixture at a specified time when
the desired molecular weight is achieved.
Flory derived statistical methods for relating the molecular weight distribution to the degree of
conversion [
1
,
3
]. In these polymerizations, each reaction step links two monomer molecules
together. This means that the number of mers in the polymer backbone is always larger by one
than the number of each kind of functional groups, A or B. If there are
x
monomers in a chain, then the
number of functional groups that have reacted is
x
1. The functional groups that are unreacted
p
remain at the ends of the chains. If we designate
as the extent of the reaction or the degree of
p
x
1
, where
x
conversion, as above, then the probability that
1 of A or B has reacted is
p ¼ðN
o
NÞ=N
o
and the probability of finding an unreacted functional group is
p
1. The probability of finding a
polymer molecule that contains
x
monomer units and an unreacted functional group A or B is
p
x
1
ð
1
pÞ
. At a given time
t
, the number of molecules present in the reaction mixture is
N
.
The fraction that contains
x
units can be designated as
N
x
and can be defined as:
N
x
¼ Np
x
1
ð
pÞ
1
The Carothers equation defines
N
/
N
o
¼
1
p
. The above expression for
N
x
can, therefore, be
written as:
2
p
x
1
N
x
¼ N
o
ð
1
pÞ
where
N
o
is, of course, the number of monomer units that are present at the start of the reaction.
To determine the molecular weight distribution of the polymeric species that form at any given degree
of conversion, it is desirable to express the weight average and number average molecular weights by
terms, like
p
.BydefiningM
o
as the mass of the repeating unit, the number average molecular weight is:
M
n
¼
M
o
DP
¼
M
o
ð
1
=
1
pÞ
and the weight average molecular weight is:
X
M
w
¼
w
x
xM
o
where
w
x
is the weight fraction of molecules containing
x
monomer units. It is equal to
xN
x
/
N
o
and
2
p
x
1
. The weight average
that can be written, based on the above equation for
N
x
,as
w
x
¼ xð
1
pÞ
molecular weight can now be expressed as:
2
X
2
p
x
1
M
w
¼
M
o
ð
pÞ
x
1
It can be shown that summation,
P
x
3
. Based on that, the weight average
p
x
1
2
¼ð
1
þ pÞ=ð
1
pÞ
molecular weight is:
M
w
¼
M
o
ð
1
þ pÞ=ð
1
pÞ
