Chemistry Reference
In-Depth Information
and similarly
P
22
N
n
j
5
1
P
21
1
1
P
21
n
j
P
n
22
5
N
2
5
(9-58)
P
22
5
1
2
Every M
1
sequence is joined to two M
2
sequences in the interior of the copol-
ymer. The number of sequences of each type cannot differ by more than one,
therefore. For long polymer molecules, which will contain a large number of
sequences of each type, this difference is negligible and the number of M
1
sequences effectively equals the number of M
2
sequences. Then the ratio of M
1
units to M
2
units in the copolymer,
d
[M
1
]
/d
[M
2
] in our previous notation, must
equal the ratio of the respective average sequence lengths. That is to say,
d
M
1
d½
M
2
5
½
N
1
N
2
5
1
P
12
1
=P
21
=
(9-59)
Substituting
Eqs. (9-56) and (9-58)
into
Eq. (9-59)
and rearranging, we obtain
the simple copolymer
Eq.
(9-14)
without specific reference to steady-state
approximations.
Since considerations of sequence distributions can be used to derive the simple
copolymer equation, it is not surprising that measured values of triad distributions
in binary copolymers [by
1
Hor
13
C NMR analyses] can be inserted into the
copolymer equation to calculate reactivity ratios
[19]
.
If one of the reactivity ratios, say,
r
2
, is zero, then
P
22
5
0 and
Eq. (9-54)
becomes
0
for
n
2
6¼
1
P
n
j
2
1
22
N
ð
M
2
;
n
j
Þ
5
ð
1
P
22
Þ
5
(9-60)
2
1
for
n
2
5
1
Thus, all M
2
units are present as isolated sequences.
When one reactivity ratio, say,
r
1
, is greater than 1, the average sequence
length of M
1
units increases, since
N
1
5
1
=P
12
5
r
1
ð½
M
1
=½M
2
Þ
1
1
(9-61)
Calculations of sequence lengths as a function of the
r
1
r
2
product and the
monomer feed concentration ratio are available in tabular form
[20]
. The calcula-
tions above refer, strictly speaking, to infinitely long copolymer molecules. There
is still the possibility that different copolymer molecules of finite length will vary
in composition even though the overall composition will be as calculated. This
likelihood has been analyzed
[21]
, and it is expected theoretically that the varia-
tion of composition about the mean value calculated from the simple copolymer
equation will be quite narrow and will shrink as the mean molecular weight of
the copolymer increases. If the reaction medium is homogeneous (this is not
always true), this conclusion indicates that most practical copolymerizations from
a given feed yield a product in which all macromolecules have close to the same
composition at any instant.
