Chemistry Reference
In-Depth Information
where
η ðηÞ
5
G
=ðα
1
H
Þ
(9-39)
ξð
x
i
Þ
5
H
=ðα
1
H
Þ
(9-40)
and
α
(alpha) is defined from
1
=
2
α
5
ð
H
max
H
min
Þ
(9-41)
with
H
max
the maximum and
H
min
the lowest values of ([M
1
]/[M
2
])
2
d
[M]
2
/d
[M]
1
in the series of measurements.
Other procedures
[10,11]
rely on computerized adjustments of
r
1
and
r
2
to
minimize the absolute values of deviations between predicted and measured
copolymer compositions.
The last four methods cited
[8
11]
are statistically inexact, in that they cannot
give good estimates of the reactivity ratio values, but they can provide good esti-
mates of
r
1
and
r
2
if the copolymerization experiments are suitably designed.
The copolymerizations should not be carried out over a random range of feed
compositions, but the available effort should be devoted to replications of copoly-
merizations at two monomer feeds and
f
0
1
and
f
v
1
given by
f
0
1
5
r
1
Þ
2
=ð
2
(9-42)
1
and
r
2
ð
r
2
Þ
(9-43)
where
r
1
and
r
2
are approximated values of the reactivity ratios
[12]
. (A conve-
nient method for approximating reactivity ratios is given in
Section 9.11
.)
The most generally useful methods and the only statistically correct proce-
dures for calculating reactivity ratios from binary copolymerization data involve
nonlinear least squares analysis of the data or application of the “error in variables
(EVM)” method. Effective use of either procedure requires more iterations than
can be performed by manual calculations. An efficient computer program for non-
linear least squares estimates of reactivity ratios has been published by Tidwell
and Mortimer
[13]
. The EVM procedure has been reported by O'Driscoll and
Reilly
[14]
.
Another important recent contribution is the provision of a good measurement
of the precision of estimated reactivity ratios. The calculation of independent
standard deviations for each reactivity ratio obtained by linear least squares fitting
to linear forms of the differential copolymer equations is invalid, because the two
reactivity ratios are not statistically independent. Information about the precision
of reactivity ratios that are determined jointly is properly conveyed by specifica-
tion of joint confidence limits within which the true values can be assumed to
coexist. This is represented as a closed curve in a plot of
r
1
and
r
2
. Standard sta-
tistical techniques for such computations are impossible or too cumbersome for
application to binary copolymerization data in the usual absence of estimates of
f
v
1
5
2
1
