Raoult's Law and Vapour Pressure

CHERaoult's Law & Vapour Pressure

Raoult's law links the vapour pressure of a solution to the mole fractions of its components, in both volatile-volatile mixtures and non-volatile-solute solutions. ISC numericals ask for total vapour pressure, vapour composition, and the mass of solute needed to lower vapour pressure to a target value.

Raoult's law (volatile components)

p_A = p_A^\circ x_A, \quad p_B = p_B^\circ x_B, \quad p = p_A + p_B

x_A,x_B

liquid-phase mole fractions,

p_A^\circ,p_B^\circ

pure vapour pressures

Non-volatile solute

p_s = p^\circ \, x_{solvent}

p^\circ

pure-solvent vapour pressure,

x_{solvent}

solvent mole fraction

Total pressure vs liquid mole fraction

p_{total} = p_B^\circ + (p_A^\circ - p_B^\circ)\,x_A

linear in liquid composition

x_A

for an ideal binary mixture

Vapour composition (Dalton's law)

y_A = \dfrac{p_A}{p_{total}}

y_A

mole fraction of

A

in the vapour

Raoult's law (volatile components): $p_A = p_A^\circ x_A$ and $p_B = p_B^\circ x_B$ , with total $p = p_A + p_B$ (mole fractions in the liquid).
Non-volatile solute: $p_s = p^\circ x_{solvent}$ , so vapour pressure is lowered in proportion to the solvent's mole fraction.
Relative lowering equals the solute mole fraction: $\dfrac{p^\circ - p_s}{p^\circ} = x_{solute}$ .
Vapour composition from Dalton's law: mole fraction of $A$ in vapour $y_A = \dfrac{p_A}{p_{total}}$ — the vapour is always richer in the more volatile component.
Mass of solute to reach a target $p_s$ : combine $\dfrac{p^\circ - p_s}{p^\circ} = \dfrac{n_{solute}}{n_{solute}+n_{solvent}}$ and solve for $n_{solute}$ , then $w = n \times M_r$ .
Raoult's law is the limiting form of Henry's law for the solvent (where $K_H = p^\circ$ ).
Raoult's law states that the partial vapour pressure of each volatile component equals its pure vapour pressure times its mole fraction in the liquid phase.
For an ideal solution the total vapour pressure lies on a straight line between $p_A^\circ$ and $p_B^\circ$ as the liquid composition varies, with no maximum or minimum.
Because $y_A/x_A = p_A^\circ/p_{total}$ , the more volatile component is enriched in the vapour — the principle behind fractional distillation.
Vapour-pressure lowering by a non-volatile solute is one of the four colligative properties and depends only on the number of solute particles, so $i$ enters for electrolytes.
Both volatile-volatile mixtures (use both $p_A^\circ x_A$ and $p_B^\circ x_B$ ) and non-volatile-solute solutions (only the solvent contributes to $p_s$ ) follow from the same law.
Trap: $p_A^\circ$ is the vapour pressure of the pure component; use liquid-phase mole fractions for partial pressures and vapour-phase fractions only for composition of the vapour.

Where the marks go

Using vapour-phase mole fractions in $p_A = p_A^\circ x_A$ — partial pressures require liquid-phase mole fractions; $y_A$ is for vapour composition only.
Counting a non-volatile solute as contributing vapour pressure; only the solvent's mole fraction sets $p_s$ .
Writing the relative lowering against $p_s$ rather than $p^\circ$ — the denominator is the pure-solvent vapour pressure.
Assuming the vapour has the same composition as the liquid; it is always richer in the more volatile component except at an azeotrope.
Forgetting the van't Hoff factor when a non-volatile electrolyte lowers the vapour pressure of the solvent.

How the board asks it

Numericaltotal pressure from liquid mole fractions
At $298\,K$ , the vapour pressures of pure benzene and pure toluene are $75\,mmHg$ and $22\,mmHg$ respectively. Calculate the total vapour pressure of a solution containing $2\,mol$ of benzene and $3\,mol$ of toluene.
Numericalmass of non-volatile solute for a target $p_s$
The vapour pressure of pure water at $25^\circ C$ is $23.8\,mmHg$ . Calculate the mass of a non-volatile solute $(M = 60\,g\,mol^{-1})$ that must be dissolved in $100\,g$ of water to lower its vapour pressure to $23.0\,mmHg$ .
Numericalvapour composition from dalton's law
Two volatile liquids $A$ and $B$ have $p_A^\circ = 120\,mmHg$ and $p_B^\circ = 80\,mmHg$ . If the mole fraction of $A$ in the liquid mixture is $0.4$ , calculate the mole fraction of $A$ in the vapour phase.
Define / stateraoult's law for volatile components
State Raoult's law for a solution of two volatile liquids and express it mathematically for the partial vapour pressures $p_A$ and $p_B$ of the two components.
Give reasonsvapour richer in the more volatile component
Account for the fact that the vapour above an ideal solution of two volatile liquids is richer in the more volatile component than the liquid, and explain how this is exploited in fractional distillation.
Give reasonsrelative lowering as a colligative property
Explain why the relative lowering of vapour pressure caused by a non-volatile solute is a colligative property, and state why the van't Hoff factor $i$ must be included for an electrolytic solute.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.