VAN’T HOFF’S LAW OF FREEZING-POINT DEPRESSION AND BOILING-POINT ELEVATION

15.5 VAN’T HOFF’S LAW OF FREEZING-POINT DEPRESSION AND BOILING-POINT ELEVATION

Let us consider a pure solid phase, such as ice, in equilibrium with a pure liquid phase, such as water, at some specified temperature and pressure. If the two phases are in equilibrium

m † 1 ;s ¼m † 1 (15 :59) in which m † 1,s represents the chemical potential of the pure solid and m 1 † represents the

chemical potential of the pure liquid. If solute is added to the system, and if it dis- solves only in the liquid phase, then the chemical potential of the liquid solvent will be decreased:

m 1 ,m † 1

To reestablish equilibrium, m 1,s † must be decreased also. This decrease in m can be accomplished by decreasing the temperature. The chemical potential of the liquid solvent is decreased by the drop in temperature as well as by the addition of solute. Equilibrium is reestablished if

dm † 1 ;s ¼ dm 1 (15 :60) As the chemical potential of the solid phase depends only on the temperature,

whereas that of the solvent in the solution depends on both temperature and concen- tration of added solute, the total differentials of Equation (15.60) can be expressed in terms of the appropriate partial derivatives, as follows:

2 (15 :61) T P

dT ¼

@T P ,X 2 @ ln X 2 P ,T

351 From Equation (9.24),

15.5 VAN’T HOFF’S LAW OF FREEZING-POINT AND BOILING-OINT

and from Equation (15.21) and Equation (15.37)

@m 1 RT

d ln X 2 dX 2 (15 :63)

@ ln X 2 P ,T

If we substitute from Equations (15.62) and (15.63) into Equation (15.61), we obtain

If we express Equation (15.64) as a limiting law, which is consistent with the obser- vation that Henry’s law is valid only in very dilute solutions,

At equilibrium at constant temperature and pressure [Equations (8.9) and (8.10)]

DH m DS m ¼ T

so that

in which D f H m1 is the molar enthalpy of fusion of pure solid component 1 to pure, supercooled, liquid component 1. This result is analogous to that in Equation (14.65), to which Equation (15.66) is a limiting-law equivalent. Whether we choose to use the

DILUTE SOLUTIONS OF NONELECTROLYTES

equations to describe the temperature dependence of the solubility of a component or the concentration dependence of the freezing point is a matter of point of view.

If D f H m1 is assumed to be constant in the small temperature range from the freezing point of the pure solvent T 0 to the freezing point of solution T, Equation (15.66) is integrated as

If T 0 ffi T, another approximation is

The preceding expression, like the other laws of the dilute solution, is a limiting law. It is expressed more accurately as

By a similar set of arguments, it can be demonstrated that the boiling point elevation for dilute solutions containing a nonvolatile solute is given by the expression

in which D v H m1 is the molar enthalpy of vaporization of pure liquid component 1. According to Equation (15.6), in a solution sufficiently dilute that the limiting form of Equation (15.70) applies,

¼n 2 w 1

where w 1 is the mass of solvent, which is usually expressed in kilograms. But n 2 /w 1 is equal to m 2 , so that in dilute solution,

X 2 ¼m 2 M 1 (15 :72)

353 If we substitute for X 2 from Equation (15.72) in Equation (15.70), which is written for

EXERCISES

dilute solutions, then

¼K f m 2 (15 :73) where K f is the molal freezing point depression constant of the solvent, which is equal

to RT 2 M 1 /D f H m1 . As the laws of dilute solution are limiting laws, they may not provide an adequate approximation at finite concentrations. For a more satisfactory treatment of solutions of finite concentrations, for which deviations from the limiting laws become appreci- able, the use of new functions, the activity function and excess thermodynamic functions, is described in the following chapters.