Freezing-point depression

Freezing-point depression is the decrease in temperature of the freezing point of a solution compared to that of the pure solvent. It is a colligative property: for dilute solutions of non-electrolytes, the difference in freezing point is proportional to the amount of solute.

Cryoscopy is the experimental technique that uses freezing-point depression to measure the molecular weight of compounds: it is of historical interest only.

Description

Solvent	Kf K kg mol−1
Acetic acid	3.90
Acetone	2.40
Aniline	5.87
Benzene	5.12
Camphor	40
Carbon disulfide	3.8
Chloroform	4.90
Cyclohexane	20.1
Diethyl ether	1.79
Naphthalene	6.94
Nitrobenzene	6.90
Phenol	7.27
Pyridine	4.75
Tetrachloromethane	30
Water	1.86
Data: Kaye & Laby Tables of Physical & Chemical Constants

The freezing-point depression ΔT_f is proportional to the molality of the solution m: the proportionality constant is called the cryoscopic constant K_f.

ΔT_f = K_fm

The molality of the solution is the amount of solute divided by the mass of solvent. For dilute solutions, where the mass of the solution can be approximated by the mass of solvent, the molality is equal to the mass fraction w of solute divided by its molar mass M. Hence, the molar mass of a solute can be calculated from the freezing-point depression by

M = K_fw/ΔT_f

This description only applies for dilute solutions. It also breaks down if the solute dissociates (e.g. electrolytes) or associates (e.g. acetic acid in non-polar solvents) in solution.

Derivation

We assume ideal solution behaviour, so that the chemical potential of the solvent in the solution μ(ℓ) is given by

μ(ℓ) = μ₀(ℓ) + RTln(1−x)

where μ₀(ℓ) is the chemical potential of the pure solvent and x is the amount fraction of solute (so that 1−x is the amount fraction of solvent). The solution freezes when the solvent in the solution is in equilibrium with pure solid solvent, so μ(ℓ) must be equal to μ₀(s), the chemical potential of the pure solid solvent.

μ₀(s) = μ(ℓ) = μ₀(ℓ) + RTln(1−x)

ln(1−x) = [μ₀(s) − μ₀(ℓ)]/RT = Δ_fusG/RT

From the definition of the Gibbs energy change of fusion Δ_fusG = Δ_fusH − TΔ_fusS, we can write two equations, the first for the solution and the second for the pure solvent (where T₀ is the freezing point of the pure solvent):

ln(1−x) = Δ_fusH/RT − Δ_fusS/R

ln(1) = Δ_fusH/RT₀ − Δ_fusS/R

The difference of the two equations, remembering that ln(1) = 0, is

ln(1−x) = (Δ_fusH/R)[(1/T) − (1/T₀)]

Approximating ln(1−x) ≈ −x (for x ≪ 1, i.e. dilute solution) gives

x = (Δ_fusH/R)[(1/T₀) − (1/T)]

As T ≈ T₀ (small depression of the freezing point), we can approximate (1/T₀) − (1/T) ≈ ΔT/T20, so

ΔT = (RT20/Δ_fusH)x

The amount fraction of solute is equal to the molality of the solution m multiplied by the molar mass of the sovent M₀, so

ΔT = (RT20M₀/Δ_fusH)m

K_f = RT20M₀/Δ_fusH

This derivation shows why freezing-point depression is proportional to the amount fraction of solute and hence, for dilute solutions, to the molality. Nevertheless, cryoscopic constants K_f are usually treated as empirical constants to be determined experimentally rather than being calculated from enthalpies of fusion.

See also

• Boiling-point elevation

Notes and references

Further reading

• Getman, Frederick H. François-Marie Raoult, Master Cryoscopist. J. Chem. Educ. 1936, 13 (4), 153–55. DOI: 10.1021/ed013p153.