CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS
12.4 COLLIGATIVEPROPERTIES OF ADILUTESOLUTION 381
wherepAis the vapor pressure of the pure solvent at the temperature of the solution.
Thus, approximate expressions for vapor-pressure lowering in the limit of infinite dilu-
tion are
Åp VApAcB and Åp MApAmB (12.4.19)
We see that the lowering in this limit depends on the kind of solvent and the solution com-
position, but not on the kind of solute.
12.4.4 Osmotic pressure
The osmotic pressureis an intensive property of a solution and was defined in Sec.12.2.2.
In a dilute solution of low, the approximation used to derive Eq.12.2.11(that the partial
molar volumeVAof the solvent is constant in the pressure range fromptopC) becomes
valid, and we can write
D
A A
VA
(12.4.20)
In the limit of infinite dilution,A Aapproaches RTlnxA(Eq.12.4.2) andVAbe-
comes the molar volumeVAof the pure solvent. In this limit, Eq.12.4.20becomes
D
RTlnxA
VA
(12.4.21)
from which we obtain the equation
lim
xA! 1
@
@xA
T;p
D
RT
VA
(12.4.22)
The relations in Eq.12.4.7transform Eq.12.4.22into
lim
cB! 0
@
@cB
T;p
DRT (12.4.23)
lim
mB! 0
@
@mB
T;p
D
RTMA
VA
DART (12.4.24)
Equations12.4.23and12.4.24show that the osmotic pressure becomes independent of
the kind of solute as the solution approaches infinite dilution. The integrated forms of these
equations are
DcBRT (12.4.25)
(dilute binary solution)
D
RTMA
VA
mBDART mB (12.4.26)
(dilute binary solution)
Equation12.4.25isvan’t Hoff’s equationfor osmotic pressure. If there is more than one
solute species,cBcan be replaced by
P
i§AciandmBby
P
i§Amiin these expressions.
In Sec.9.6.3, it was stated that=mBis equal to the product ofmand the limiting
value of=mBat infinite dilution, wheremD.A A/=RTMA
P
i§Amiis the
osmotic coefficient. This relation follows directly from Eqs.12.2.11and12.4.26.