Physical Chemistry Third Edition

6.7 Colligative Properties 293

We assume that the solution is sufficiently dilute that the solvent obeys the ideal solution

equation:

μ∗ 1 (liq)+RTln(x 1 )μ∗ 1 (solid) (6.7-2)

wherex 1 is the mole fraction of the solvent in the solution. By the Gibbs phase rule

there are two independent intensive variables. If the pressure and temperature are

chosen to be independent variables the mole fraction of the solvent in the liquid phase

is a dependent variable.

Our strategy is to differentiate Eq. (6.7-2), then to apply thermodynamic relations

to it and then to integrate. We divide byTand then differentiate with respect toTat

constantP:

(

∂(μ∗ 1 (liq)/T)

∂T

)

P

+R

(

∂ln(x 1 )

∂T

)

P



(

∂μ∗ 1 (solid)/T

∂T

)

P

(6.7-3)

By the use of thermodynamic relations

(

∂

(

μ∗ 1 /T

)

∂T

)

P



T

(

∂μ∗ 1 /∂T

)

P−μ

∗

1

T^2



−TSm,1∗ −μ∗ 1

T^2

−

Hm,1∗

T^2

(6.7-4)

Use of this identity in Eq. (6.7-3) gives

R

(

∂ln(x 1 )

∂T

)

P



H

∗(liq)

m,1 −H

∗(solid)

m,1

T^2



∆fusHm,1∗

T^2

(6.7-5)

where∆fusHm,1∗ is the enthalpy change of fusion of the pure solvent.

The equilibrium temperature will be lower than the freezing temperature of the pure

solvent. We multiply Eq. (6.7-5) bydTand integrate both sides of Eq. (6.7-5) from the

normal melting temperature of the pure solventTm,1to some lower temperatureT′.

R

∫T′

Tm,1

(

∂ln(x 1 )

∂T

)

P

dT

∫T′

Tm,1

∆fusHm,1∗

T^2

dT (6.7-6)

To a good approximation,∆fusHm,1∗ is constant over a small range of temperature. To

this approximation

Rlnx 1 (T′)−∆fusHm,1∗

[

1

T′

−

1

Tm,1

]

(6.7-7)

wherex 1 (T′) is the mole fraction of the solvent in the solution that is at equilibrium

with the pure solvent at temperatureT′, and where we have used the fact thatx 1 1at

temperatureTm,1. Our system (pure solid solvent plus liquid solution) corresponds to

one of the curves in a solid–liquid phase diagram such as Figure 6.20, so that Eq. (6.7-7)

is the equation for this curve in the case that there is no appreciable solid solubility and

the liquid solvent acts ideally. For dilute solutions, Eq. (6.7-7) is simplified by using

the first term of a Taylor series

ln(x 1 )ln(1−x 2 )−x 2 −

x 22

2

−···≈−x 2 (6.7-8)