Chemistry Reference
In-Depth Information
integrated form of the copolymer equation. The intersection point that corre-
sponds to the “best” values of
r
1
and
r
2
is selected imprecisely and subjectively
by this technique. Each experiment yields a straight line, and each such line
can intersect one line from every other experiment. Thus,
n
experiments yield
(
n
(
n
1)
unreliable intersections. Various attempts to eliminate subjectivity and reject
dubious data on a rational basis have not been successful.
Alternatively
[6]
, the simple copolymer equation can be solved in a linear
graphical manner by substituting
x
1)/2 intersections and even one “wild” experiment produces (
n
2
2
[M
1
]/[M
2
],
z
d
[M
1
]/[M
2
],
so that
5
5
Eq. (9-13)
becomes
z
x
ð
1
r
1
x
Þ=ð
r
2
1
x
Þ
(9-33)
5
1
Equation (9-33)
can be linearized in the alternative forms
G
x
ð
z
1
Þ=
z
r
1
H
r
2
(9-34)
5
2
5
2
and
G
=
H
r
2
=
H
r
1
52
1
(9-35)
x
2
/z
. Linear least squares fits to
Eq. (9-34)
or
(9-35)
yield one reactiv-
ity ratio as the intercept and the other as the slope of the plotted line. The experi-
mental data are, however, unequally weighted by these equations and the values
obtained at low [M
2
]in
Eq. (9-34)
or low [M
1
]in
Eq. (9-35)
have the greatest
influence on the slope of a line corresponding to these equations.
Equations (9-34)
and (9-35)
are not symmetrical in
r
1
and
r
2
. The same set of experimental data can
yield different
r
1
,
r
2
sets depending on which monomer is indexed as M
1
and
which is M
2
. This procedure (the Fineman
where
H
5
Ross method) has been widely used,
because it is simple and can be treated graphically or by a linear least squares
regression on the data. The latter procedure is statistically unsound, however.
Equation (9-13)
is not linear in
r
1
and
r
2
. Its transformation into the linear form of
Eq. (9-34)
or
(9-35)
is algebraically correct, but the error structure in the copol-
ymer composition (
F
1
and
F
2
) is also changed, so that the errors in
G
and
H
do
not have constant variance and zero mean. This means that
Eqs. (9-34) and (9-35)
do not meet the statistical requirements for linear least squares computations
[7]
.
More recent linear graphical methods are invariant to the inversion of mono-
mer indexes. One procedure
[8]
uses the form
H
1
=
2
H
1
=
2
z
1
=
2
z
2
1
=
2
r
2
=
r
1
=
(9-36)
Another method
[9]
involves recasting the copolymer equation in the form
5
2
1
η
5
ðr
1
1
r
2
Þ=αÞξ
2
r
2
=α
(9-37)
or
η
5
r
1
ξ
2
ðr
2
=αÞð
1
2
ξÞ
(9-38)
