Chemistry Reference
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given by
M
/r
P
, and [
]
M
yields the hydrodynamic volume of one mole of isolated
polymer coils so that F
o
becomes:
M
r
P
½
=
1
½
F
o
¼
M
¼
(27)
r
P
Upon the expansion of the logarithm in (
17
) up to the second term (which suffices in
view of the low F
o
values typical for the present systems), we obtain the following
expression for l:
1
2
þ
½
r
P
N
l
¼
(28)
Relating the intrinsic viscosity to
N
by means of the Kuhn Mark Houwink relation:
K
N
N
a
½¼
(29)
the intramolecular interaction parameter becomes:
1
2
þ
ð
1
a
Þ
l
¼
k
N
(30)
where k
K
N
r
P
.
The insertion of (
30
) into (
22
) and employing (
15
) enables the rationalization of
the experimental finding that the
A
2
values for the solutions of a given polymer of
different chain length do not exclusively decrease with rising
M
in good solvents,
but might also increase. The resulting equation reads:
¼
zk
r
2
P
V
N
A
2
þ
ð
1
a
Þ
A
2
¼
(31)
where
A
2
is the limiting value of
A
2
for infinite molar mass of the polymer. The
reason for an anomalous molecular weight dependence of the second osmotic virial
coefficient lies in the sign of z, which is positive in most cases, but may also become
negative under special conditions. For theta systems,
A
2
¼
0, irrespective of
M
, and
z becomes zero. One consequence of the present experimentally verified consider-
ation concerns the way that
A
2
(
M
) should be evaluated. Equation (
31
) requires plots
of
A
2
as a function of
M
(1
a
)
, instead of the usual double logarithmic plots, and
does not in contrast to the traditional evaluation automatically yield zero second
osmotic virial coefficient in the limit of infinitely long chains.
Another helpful consequence of (
30
) lies in the fact that its second term is almost
always negligible (with respect to 1/2) for polymers of sufficient molar mass. This
feature allows the merging of the parameters z and l into their product zl, and the
