Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.4 COLLIGATIVEPROPERTIES OF ADILUTESOLUTION 380

12.4.2 Boiling-point elevation

We can apply Eq.12.3.6to the boiling pointTbof a dilute binary solution. The pure phase of
A in equilibrium with the solution is now a gas instead of a solid.^7 Following the procedure
of Sec.12.4.1, we obtain

lim

mB! 0



@Tb

@mB



p

D

MAR.Tb/^2

Åvap,AH

(12.4.13)

whereÅvap,AHis the molar enthalpy of vaporization of pure solvent at its boiling pointTb.
Themolal boiling-point elevation constantor ebullioscopic constant,Kb, is defined
for a binary solution by

KbdefD lim

mB! 0

ÅTb

mB

(12.4.14)

whereÅTbDTbTbis the boiling-point elevation. Accordingly,Kbhas a value given by

KbD

MAR.Tb/^2

Åvap,AH

(12.4.15)

For the boiling point of a dilute solution, the analogy of Eq.12.4.12is

ÅTbDKbmB (12.4.16)

(dilute binary solution)

SinceKfhas a larger value thanKb(becauseÅfus,AHis smaller thanÅvap,AH), the mea-
surement of freezing-point depression is more useful than that of boiling-point elevation for
estimating the molality of a dilute solution.

12.4.3 Vapor-pressure lowering

In a binary two-phase system in which a solution of volatile solvent A and nonvolatile solute
B is in equilibrium with gaseous A, the vapor pressure of the solution is equal to the system
pressurep.
Equation12.3.7on page 375 gives the general dependence ofponxAfor a binary
liquid mixture in equilibrium with pure gaseous A. In this equation,Åsol,AV is the molar
differential volume change for the dissolution of the gas in the solution. In the limit of
infinite dilution,Åsol,AVbecomesÅvap,AV, the molar volume change for the vaporization
of pure solvent. We also apply the limiting expressions of Eqs.12.4.4and12.4.7. The result
is

lim

cB! 0



@p

@cB



T

D

VART

Åvap,AV

lim

mB! 0



@p

@mB



T

D

MART

Åvap,AV

(12.4.17)

If we neglect the molar volume of the liquid solvent compared to that of the gas, and
assume the gas is ideal, then we can replaceÅvap,AV in the expressions above byVA(g)D
RT=pAand obtain

lim

cB! 0



@p

@cB



T

VApA lim

mB! 0



@p

@mB



T

MApA (12.4.18)

(^7) We must assume the solute is nonvolatile or has negligible partial pressure in the gas phase.