Chemistry Reference
In-Depth Information
or of both kinds that have reacted. Also, N is the total number of moles of mole-
cules (monomer plus polymers of all sizes) when the reaction has proceeded to an
extent p.
Neglecting intramolecular linkages, every time a new linkage is formed the
reaction mixture will contain one less molecule. Therefore, when the number of
molecules has been reduced from N
0
to N moles, the number of linkages which
have been formed is equal to N
0
2
N moles. It takes two functional groups to
form a linkage and so 2(N
0
2
N) moles of functional groups will have been lost
in forming these N
0
2
N moles of linkages.
By definition,
no
:
functional groups used
no
2
ð
N
0
2
N
Þ
N
0
f
av
p
(7-17)
5
functional groups initially
5
:
whenever
1
2
ð
2N
0
2
N
0
pf
av
Þ
N
5
(7-18)
The number average degree of polymerization of the reaction mixture is X
n
;
which is equal to the initial number of monomer units divided by the number of
remaining molecules. That is,
N
0
2
X
n
5
(7-19)
N
0
pf
av
Þ
5
1
2
ð
2N
0
2
2
pf
av
2
This is the Carothers equation. Note that X
n
is the number average degree of
polymerization of the r
ea
ction mixture and not just of the polymer that has been
for
med. Note also that X
n
is not necessarily equal to the degree of polymerization
DP
n
;
defined in Section 1.2 as the average number of rep
ea
ting units per polymer
molecule. In the present contex
t
a single monomer has X
n
5
1
;
a molecule con-
ta
inin
g
two monomers has
X
n
5
2
;
and so on. Thus
for
structure 7-3,
X
n
5
100, since each repeating
unit shown contains the residues of two monomers. The degree of polymerization
derived in this section is generally applicable, even to branched polymers formed
from a mixture of monomers, in which a regular repeating unit cannot be identi-
fied. (Such mixtures are illustrated in
Table 7.1
.) It is unfortunate that these two
somewhat different concepts of degree of polymerization are both current in poly-
mer technology, but the intended meaning is usually clear from the context in
which the term is used.
DP
n
5
100
;
but for 7-4, X
n
5
200 while DP
n
5
H
N
CH
2
C
O
100
7-3
