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Chapter 12 Since solute molality, molar concentration, and mole fraction are proportional to
Multicomponent Phase Equilibrium one another in an ideally dilute solution (Prob. 9.8), dilute-solution colligative prop-
erties can be expressed using any of these composition measures. Equation (12.25)
uses mole fraction, (12.26) uses molality (since n /n M m ), and (12.27) uses
A
B
B
A
molar concentration.
Figure 12.6 plots ß versus solute concentration for aqueous sucrose solutions at
25°C. The dotted line is for an ideally dilute solution.
For solutions that are not ideally dilute, Eq. (12.24) holds. However, a different
(but equivalent) expression for ß in nonideally dilute solutions is often more conven-
ient than (12.24). In 1945, McMillan and Mayer developed a statistical-mechanical
theory for nonelectrolyte solutions (see Hill, chap. 19). They proved that the osmotic
pressure in a nonideally dilute nonelectrolyte two-component solution is given by
3
2
1
ß RT1M r A r A r . . . 2 (12.28)
B
B
3 B
2 B
where M is the solute molar mass and r is the solute mass concentration: r w /V
B B B B
[Eq. (9.2)], where w is the mass of solute B. The quantities A , A , . . . are related to
B 2 3
the solute–solute intermolecular forces in solvent A and are functions of T (and
weakly of P). Note the formal resemblance of (12.28) to the virial equation (8.4) for
Figure 12.6 gases. In the limit of infinite dilution, r → 0 and (12.28) becomes ß RTr /M
B B B
RTw /M V RTn /V c RT, which is the van’t Hoff law.
Osmotic pressure ß of aqueous B B B B
sucrose solutions at 25°C plotted Osmotic pressure is sometimes misunderstood. Consider a 0.01 mol/kg solution
versus sucrose concentration. The of glucose in water at 25°C and 1 atm. When we say the freezing point of this solu-
dotted line is for an ideally dilute tion is 0.02°C, we do not imply that the solution’s temperature is actually 0.02°C.
solution.
The freezing point is that temperature at which the solution would be in equilibrium
with pure solid water at 1 atm. Likewise, when we say that the osmotic pressure of this
solution is 0.24 atm (see Example 12.2), we do not imply that the pressure in the so-
lution is 0.24 atm (or 1.24 atm). Instead, the osmotic pressure is the extra pressure that
would have to be applied to the solution so that, if it were placed in contact with a
membrane permeable to water but not glucose, it would be in membrane equilibrium
with pure water, as in Fig. 12.7.
EXAMPLE 12.2 Osmotic pressure

Find the osmotic pressure at 25°C and 1 atm of a 0.0100 mol/kg solution of glu-
cose (C H O ) in water.
6 12 6
It is a good approximation to consider this dilute nonelectrolyte solution as
ideally dilute. Almost all the contribution to the solution’s mass and volume
3
comes from the water, and the density of water is nearly 1.00 g/cm . Therefore
an amount of this solution that contains 1 kg of water will have a volume very
3
close to 1000 cm 1 L, and the glucose molar concentration is well approxi-
3
mated as 0.0100 mol/dm . (See also Prob. 11.21b.) Substitution in (12.27) gives
1
1
3
3
3
ß c RT 10.0100 mol>dm 2182.06 10 dm atm mol K 21298.1 K2
B
ß 0.245 atm 186 torr
Glucose
P Water P
solution
Figure 12.7
Pure water in equilibrium with Membrane permeable to
water in a glucose solution. water but not glucose