Boiling-point elevation

Boiling-point elevation is the increase in temperature of the boiling point of a liquid that contains non-volatile solute. It is a colligative property: for dilute solutions of non-electrolytes, the difference in boiling point from that of the pure solvent is proportional to the amount of solute.

Ebullioscopy is the experimental technique that uses boiling-point elevation to measure the molecular weight of compounds. Tonometry is a related technique that directly measures the reduction in vapour pressure of the solution compared to that of the pure solvent. Both techniques are of historical interest only, although their development led to the formulation of Raoult's law.

Description

Solvent	Kb K kg mol−1
Acetic acid	3.07
Acetone	1.71
Aniline	3.22
Benzene	2.53
Carbon disulfide	2.37
Chloroform	3.66
Cyclohexane	2.79
Diethyl ether	1.82
Nitrobenzene	5.26
Tetrachloromethane	4.95
Water	0.51
Data: Kaye & Laby Tables of Physical & Chemical Constants

The boiling-point elevation ΔT_b is proportional to the molality of the solution m: the proportionality constant is called the ebullioscopic constant K_b.

ΔT_b = K_bm

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 boiling-point elevation by

M = K_bw/ΔT_b

The same phenomenon can be described in terms of vapour pressure for any given temperature. If p₀ is the vapour pressure of the pure solvent and p is the vapour pressure of the solution:

(p₀ − p)/p₀ = K′_Rm

where K′_R is a constant that is specific for each solvent and m is the molality of the solution. French physical chemist François-Marie Raoult (1830–1901) investigated this relation in the 1880s, and found in 1887 that a single constant (K_R) for all solvents could be obtained by replacing the molality with the amount fraction of solute x, that is by dividing the molality by the molar mass of the solvent:

(p₀ − p)/p₀ = K_Rx

where K_R ≈ 1 (Raoult found a value of 1.05).

Both of these descriptions only apply for dilute solutions. They also break 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 boild when the solvent in the solution is in equilibrium with pure solvent vapour above the solution, assuming that the solute is involatile, so μ(ℓ) must be equal to μ₀(g), the chemical potential of the pure solvent vapour.

μ₀(g) = μ(ℓ) = μ₀(ℓ) + RTln(1−x)

ln(1−x) = [μ₀(g) − μ₀(ℓ)]/RT = Δ_vapG/RT

From the definition of the Gibbs energy change of vaporization Δ_vapG = Δ_vapH − TΔ_vapS, we can write two equations, the first for the solution and the second for the pure solvent (where T₀ is the boiling point of the pure solvent):

ln(1−x) = Δ_vapH/RT − Δ_vapS/R

ln(1) = Δ_vapH/RT₀ − Δ_vapS/R

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

ln(1−x) = (Δ_vapH/R)[(1/T) − (1/T₀)]

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

x = (Δ_vapH/R)[(1/T₀) − (1/T)]

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

ΔT = (RT20/Δ_vapH)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_vapH)m

K_b = RT20M₀/Δ_vapH

See also

• Freezing-point depression