Chemistry Reference
In-Depth Information
It is useful to express
Eq. (3-1)
in terms of the solute mole fraction
x
2
. For the
arbitrary variable
y
in general,
N
K5
1
ð2
1
Þ
K
ð
y
2
1
Þ
K1
1
ln
y5
;
0
#y#
2
(3-3)
K
and since, for a two-component solution,
x
1
5
2
x
2
1
1
2
x
2
2
1
3
x
2
?
(3-4)
ln
x
1
5
ln
ð
1
2x
2
Þ52x
2
2
Thus, the solvent chemical potential
μ
1
follows from
Eqs. (3-1) and (3-4)
as
1
G
1
1
μ5μ
1
RT
ln
ð
1
2
x
2
Þ 5
RT
ln
ð
1
2
x
2
Þ
2
4
3
5
(3-5)
1
2
x
2
1
1
3
x
2
1
?
G
1
2
5
RT x
2
1
In dilute solution, the total number of moles of all species in unit volume will
approach
n
1
, the molar concentration of the solvent. Then the mole fraction
x
i
of
any component
i
can be expressed as
x
i
5n
i
X
n
i
-
n
i
n
1
(3-6)
as the solution behavior approaches ideality.
If the molar and weight concentrations of the solute are
n
2
and
c
2
, respec-
tively, then
c
2
5
n
2
M
(3-7)
and
Mn
1
(3-8)
If the molar volume of pure solvent is
V
1
, with the same volume unit as is
used to express the concentrations
c
i
and
n
i
(e.g., liters), then
n
1
5
x
2
5
c
2
=
V
1
and
1
=
c
2
V
1
=
x
2
5
M
(3-9)
Substituting in
Eq. (3-5)
,
c
2
1 ð
c
2
1
?
G
1
52
RTV
1
c
2
=
V
1
=
2
M
2
V
1
Þ
2
3
M
3
μ
1
2
M
1
=
(3-10)
Equation (3-10)
is the key to the application of colligative properties to poly-
mer molecular weights. We started with
Eq. (3-1)
, which defined an ideal solution
in terms of the mole fractions of the components.
Equation (3-10)
, which follows
simple arithmetic expresses the difference in chemical potential of the solvent in
the solution and in the pure state in terms of the mass concentrations of the solute.
