Chemistry Reference
In-Depth Information
however, to dilute the solution, and this flow will continue until sufficient excess
hydrostatic pressure is generated on the solution side to block the net flow of sol-
vent. This excess pressure is the osmotic pressure. At thermodynamic equilibrium
also, the chemical potential of the solvent will be the same on both sides of
the membrane. The relation between this chemical potential, which appears in
Eq. (3-10)
, and the measured osmotic pressure is derived next. Let
μ
0
1
5
chemical potential of the pure solvent in compartment 1 under atmo-
spheric pressure
P
1
. By definition
0
1
5
G
1
.
μ
chemical potential of the solvent in solution (on side 2) under atmo-
spheric pressure
P
1
.
μ
μ
1
5
0
1
chemical potential of the solvent in solution on side 2 under the final
pressure
P
1
1π
5
, where
π
is the osmotic pressure.
The condition for equilibrium is
G
1
5μ
0
1
5μ
1
(3-11)
The chemical potential of solvent at pressure
P
1
1π
is
ð
P
1
1π
@μ
1
@
μ
1
5μ
1
1
dP
(3-12)
P
P
1
T
T
5
The partial molar volume
V
1
of the solvent will
be essentially independent of
P
over the pressure range and will moreover be
essentially equal to the molar volume
V
1
in dilute solutions where osmotic pres-
sure measurements are made. Thus, from
Eq. (3-12)
,
In general
@μ
i
=@
P
V
1
:
ð
P
1
1π
μ
1
5μ
1
1
V
1
dP
(3-13)
P
1
μ
1
5μ
1
1π
V
1
(3-14)
From
Eq. (3-11)
, at equilibrium,
G
1
Þ=
V
1
π52ðμ
1
2
(3-15)
Thus the osmotic pressure
π
is a direct measure of the chemical potential
μ
1
ðμ
1
2
G
1
Þ
of the solvent in the solution. Equating the terms for
in
Eqs. (3-15)
and (3-10)
,
V
1
=
2
M
2
π5
RTc
2
1
=
M
1 ð
Þ
c
2
1
?
(3-16)
In the limit of zero concentration
lim
c
2
-
0
ðπ=c
2
Þ 5RT=M
(3-17)
which is van't Hoff's law of osmotic pressures.
