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
dxAD d.1 xA/D d.nB=nA/
D VAdcB
D MAdmB (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) isÅTfDTf Tf. 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
ÅTfD KfmB (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.