Physical Chemistry , 1st ed.

in molarity units varies due to expansion or contraction of the solution’s

volume.

The next colligative property is boiling point elevation.A pure liquid has a

well-defined boiling point at a particular pressure. If a nonvolatile solute were

added, then to some extent those solute molecules would impede the ability of

solvent molecules to escape from the liquid phase, so more energy is required

to make the liquid boil, and the boiling point increases.

Similarly, nonvolatile solvents will make it harder for solvent molecules to

crystallize at their normal melting points because solidification will be im-

peded. Therefore, a lower temperature will be required to freeze the pure sol-

vent. This defines the idea offreezing point depression.A pure liquid will have

its freezing point lowered when a solute is dissolved in it. (This idea is a com-

mon one for anyone who has tried to synthesize a compound in a lab. An im-

pure compound will melt at a lower temperature because of the freezing point

depression of the “solvent.”)

Because the liquid-gas and liquid-solid transitions are equilibria, we can

apply some of the mathematics of equilibrium processes to the changes in

phase transition temperatures. In both cases the argument is the same, but here

we will concentrate on the liquid-solid phase equilibrium and then apply the

final arguments to the liquid-gas phase change.

In some respects, the freezing point depression can be considered in terms

of solubility limits, which we discussed in the previous section. This time, in-

stead of the component of interest being the solute, the component of interest

is the solvent.However, the same arguments and equations apply. By analogy,

we can adapt equation 7.39 and say that

ln xsolvent

(^) f
R

usH

T

1



TM

1

P

 (7.41)

where (^) fusHand TMPrefer to the heat of fusion and melting point of the sol-
vent. If we are considering dilute solutions, then xsolventis very close to 1. Since
xsolvent 1 xsolute, we can substitute to get

ln(1 xsolute) 

(^) f
R

usH

T

1



TM

1

P

 (7.42)

Using a one-term Taylor series expansion of ln (1 x) x,* we substitute

for the logarithm on the left side of the equation and get

xsolute

(^) f
R

usH

T

1



TM

1

P

 (7.43)

where the minus signs have canceled. This equation is rewritten by algebraically

rearranging the temperature terms:

xsolute

(^) f
R
usHT
T
M

P
T



MP

T

 (7.44)

We make one last approximation. Since we are working with dilute solutions,

the temperature of the equilibrium is not much different from the normal

melting point temperature TMP. (Recall that the freezing point and the melt-

ing point are the same temperature and that the phrases “freezing point” and

“melting point” can be used interchangeably.) Therefore, we substitute TMPfor

Tin the denominator of equation 7.44, and define Tfas TMPT: the change

194 CHAPTER 7 Equilibria in Multiple-Component Systems

*The multiterm expansion is ln(1 x) x^12 x^2 ^13 x^3 ^14 x^4   .