Chemistry Reference
In-Depth Information
3.1.4
Virial Equations and Virial Coefficients
The real solutions used to study the characteristics of macromolecular solutes are
rarely ideal even at the highest dilutions that can be used in practice. The expres-
sions derived earlier for ideal solutions are therefore invalid in the experimental
range. It is useful, however, to retain the form of the ideal equations and express
the deviation of real solutions in terms of empirical parameters. Thus the usual
practice in micromolecular thermodynamics is to retain
Eq. (3-10)
but substitute
fictitious concentrations, called
activities
, for the experimental solute concentra-
tions. In polymer science, on the other hand, the measured concentrations are
taken as accurate and deviations from ideality are expressed in the coefficients of
the concentration terms.
For example, the osmotic pressure of an ideal solution is given by
Eq. (3-16)
as
2
G
1
Þ
RTV
1
2
M
2
V
1
Þ
2
ðμ
1
2
c
2
V
1
5
c
2
5
RT
M
1
RT
ð
c
2
1
?
c
2
1
(3-23)
3
M
3
The osmotic data of a real solution are then expressed in a parallel form as
A
3
c
2
π=
c
5
RT
½
1
=
M
n
1
A
2
c
1
1
?
(3-24)
where
A
2
and
A
3
, the second and third virial coefficients, would be determined in
the final analysis by the fitting of corresponding
and
c
2
data to
Eq. (3-23)
. (In a
two-component solution the subscript 2, which refers to solute, is often deleted.)
Unfortunately, there is no uniformity in the exact form of the virial equations
used in polymer science. Alternatives to
Eq. (3-24)
include
π
1Γ
3
C
2
π=
c
5 ðπ=
c
Þ
c5
0
½
1
1Γ
2
c
1
?
(3-25)
and
Cc
2
π=
c
5 ð
RT
=
M
n
Þ 1
Bc
1
1
?
(3-26)
The three forms are equivalent if
B
5
RTA
2
5 ð
RT
=
M
ÞΓ
2
(3-27)
Authors may report virial coefficients without specifying the equation to
which they apply, and this can usually be deduced only by inspecting the units of
the virial coefficient. Thus,
A
2
has units of mol cm
3
/g
2
if
M
is a gram-mole with
units of g/mol. If
M
is in g, however, then
A
2
is in cm
3
/g. The units of
Γ
2
and
B
may depend on the particular units chosen for
R
,
c
2
, and
M
.
In polymer science, the ideal form of the thermodynamic equations is pre-
served and the nonideality of polymer solutions is incorporated in the virial coef-
ficients. At low concentrations, the effects of the
c
2
terms in any of the equations
will be very small, and the data are expected to be linear with intercepts that
