Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.4 COLLIGATIVEPROPERTIES OF ADILUTESOLUTION 378

In the limit of infinite dilution, whennBis much smaller thannA, 1 xAapproaches the
valuenB=nA. Then, using expressions in Eq.9.1.14on page 225 , we obtain the relations

dxADd.1xA/Dd.nB=nA/

DVAdcB

DMAdmB (12.4.7)

(binary solution at

infinite dilution)

which transform Eq.12.4.5into the following:^6

lim

cB! 0



@Tf

@cB



p

D

VAR.Tf/^2

Åfus,AH

lim

mB! 0



@Tf

@mB



p

D

MAR.Tf/^2

Åfus,AH

(12.4.8)

We can apply these equations to a nonelectrolyte solute by settingequal to 1.
AscBormBapproaches zero,TfapproachesTf. The freezing-point depression (a
negative quantity) isÅTfDTfTf. In the range of molalities of a dilute solution in which
.@Tf=@mB/pis given by the expression on the right side of Eq.12.4.8, we can write

ÅTfD

MAR.Tf/^2

Åfus,AH

mB (12.4.9)

Themolal freezing-point depression constantor cryoscopic constant,Kf, is defined
for a binary solution by

KfdefD  lim

mB! 0

ÅTf

mB

(12.4.10)

and, from Eq.12.4.9, has a value given by

KfD

MAR.Tf/^2

Åfus,AH

(12.4.11)

The value ofKfcalculated from this formula depends only on the kind of solvent and the
pressure. For H 2 O at 1 bar, the calculated value isKbD1:860K kg mol^1 (Prob. 12. 4 ).
In the dilute binary solution, we have the relation

ÅTfDKfmB (12.4.12)

(dilute binary solution)

This relation is useful for estimating the molality of a dilute nonelectrolyte solution (D 1 )
from a measurement of the freezing point. The relation is of little utility for an electrolyte
solute, because at any electrolyte molality that is high enough to give a measurable depres-
sion of the freezing point, the mean ionic activity coefficient deviates greatly from unity and
the relation is not accurate.

(^6) A small dependence ofV
AonThas been ignored.