Thermodynamics and Chemistry

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
ÅpVApAcB and ÅpMApAmB (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

AA

VA

(12.4.20)

In the limit of infinite dilution,AAapproachesRTlnxA(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.AA/=RTMA

P

i§Amiis the

osmotic coefficient. This relation follows directly from Eqs.12.2.11and12.4.26.