Raoult's Law

What Raoult's law says

A pure liquid in a closed flask reaches equilibrium with its vapor at a definite pressure that depends only on temperature — water at 25°C exerts P° = 23.76 mmHg, ethanol exerts 59 mmHg, mercury 0.0019 mmHg. Now dissolve a non-volatile solute, say sugar, in the water. The vapor pressure drops. Add more sugar and it drops further. Raoult quantified the drop:

P_solvent = X_solvent · P°_solvent

where X_solvent is the mole fraction of solvent (the number of solvent molecules divided by the total number of molecules). If X_solvent = 0.9, vapor pressure drops to 90% of the pure value. The lowering is proportional to the mole fraction of the solute alone — independent of what the solute is. That is the colligative property at work.

For a mixture of two volatile liquids A and B, both contribute to the total vapor pressure:

P_total = P_A + P_B = X_A · P°_A + X_B · P°_B

This linear relationship between vapor pressure and composition is the diagnostic feature of an ideal solution. Plotting P_total versus X_A in an ideal mixture gives a straight line from P°_B at X_A = 0 to P°_A at X_A = 1.

The microscopic picture

At a pure liquid's surface, every escaping molecule is solvent. Escape rate balanced by return sets P°. In solution, only X_solvent of surface positions hold solvent; the rest are solute. Escape rate falls in direct proportion to X_solvent, and equilibrium vapor pressure follows.

This works exactly when solute occupies the surface as readily as solvent and has no preferential interactions. Real solutions deviate when those assumptions fail: water–ethanol has hydrogen-bond networks that don't combine cleanly; chloroform–acetone has unusually strong A–B attraction; benzene–toluene barely deviates because the molecules are nearly identical electronically.

Worked example: vapor pressure over a sugar solution

Calculate the vapor pressure of water above a solution of 50.0 g sucrose (C₁₂H₂₂O₁₁, MW = 342.30 g/mol) in 250.0 g water at 25°C, where P°_water = 23.76 mmHg.

n_sucrose = 50.0 / 342.30 = 0.146 mol
n_water = 250.0 / 18.02 = 13.87 mol
X_water = 13.87 / (13.87 + 0.146) = 0.9896
P_water = 0.9896 × 23.76 = 23.51 mmHg

The 0.25 mmHg lowering looks tiny, but it shifts the boiling point measurably — from 100.000°C for pure water to 100.30°C for this solution. Same logic, applied to a 1.0 m sucrose solution, gives the standard textbook K_b = 0.512°C/molal for water.

Now a binary volatile mixture: a flask holds 0.4 mol benzene (P° = 95.1 mmHg) and 0.6 mol toluene (P° = 28.4 mmHg) at 25°C, both volatile.

P_benzene = 0.4 × 95.1 = 38.0 mmHg
P_toluene = 0.6 × 28.4 = 17.0 mmHg
P_total = 55.0 mmHg
Mole fraction in vapor: Y_benzene = 38.0 / 55.0 = 0.69

Vapor is enriched in benzene (0.69) versus liquid (0.40). That enrichment is the entire mechanism of fractional distillation: every theoretical plate concentrates the more volatile component until the column produces nearly pure benzene at the top.

Ideal vs non-ideal solutions

	Ideal (Raoult)	Positive deviation	Negative deviation	Azeotrope
ΔHmix	0	>0 (endothermic)	<0 (exothermic)	Either sign
ΔVmix	0	>0 (volume expands)	<0 (volume contracts)	Either sign
A–B interactions	= A–A and B–B	Weaker than pure	Stronger than pure	Either
Vapor pressure	Linear in X	Above ideal line	Below ideal line	Maximum or minimum vs X
Boiling-point curve	Smooth	Minimum (low BP)	Maximum (high BP)	Pinch point at extremum
Examples	Benzene/toluene, hexane/heptane	Ethanol/water, acetone/CS2	Acetone/chloroform, HCl/water	95.6% EtOH, 20.2% HCl
Distillation outcome	Pure components possible	Limited by min-BP azeotrope	Limited by max-BP azeotrope	Cannot purify past the pinch

Positive deviation: A and B "like each other less" than their own kind — disrupting cohesion, raising vapor pressure, lowering the boiling point. Negative deviation: A and B form a stronger association than either pure liquid (chloroform donating H to acetone's carbonyl, for example), depressing vapor pressure and raising boiling point.

Azeotropes: where distillation gives up

An azeotrope is a fixed-composition mixture that boils at constant temperature and produces vapor of the same composition as the liquid. Distill it and the distillate is identical to the pot — no separation possible.

System	Azeotrope composition	Type	BP (°C)
Ethanol / water	95.6 wt% ethanol	Min-BP (positive)	78.15
2-Propanol / water	87.7 wt% 2-propanol	Min-BP	80.4
HCl / water	20.2 wt% HCl	Max-BP (negative)	108.6
HNO₃ / water	68 wt% HNO₃	Max-BP	122
Benzene / water	91.1 wt% benzene	Heteroazeotrope	69.4
Acetone / chloroform	34 wt% chloroform	Max-BP	64.5

Anhydrous ethanol cannot be made from fermentation broths by distillation alone — past 95.6% the column is useless. Plants dehydrate over 3 Å zeolite molecular sieves, use entrainment distillation with cyclohexane (forming a ternary heteroazeotrope), or run pressure-swing distillation, which shifts azeotrope composition with pressure to drive purity past 99.9%.

Fractional distillation and Raoult's law

A fractional distillation column is a tall tube packed with beads or trays. Vapor rising from the boiler partially condenses on cold packing, re-evaporates, and condenses again — each cycle is a "theoretical plate" that re-enriches vapor in the more volatile component per Raoult's law. Relative volatility α = P°_A/P°_B measures separation power: α = 2.5 gives 2.5× enrichment per plate, so going from 50/50 to 99.9/0.1 takes about 7.5 plates plus reflux losses. Industrial columns range from 20 plates (alcohol, aromatic separation) to 175+ plates for xylene isomers, where α ~ 1.03 forces massive plate counts.

Vapor-pressure lowering with non-volatile solutes

When the solute has effectively zero vapor pressure — sugar, salt, polymers, electrolytes — the total vapor pressure above the solution simplifies to:

P_total = X_solvent · P°_solvent
ΔP / P°_solvent = 1 − X_solvent = X_solute

The relative lowering equals the mole fraction of solute. This is the colligative core: vapor-pressure lowering (Raoult), boiling-point elevation, freezing-point depression, and osmotic pressure are all consequences of the same X_solute dependence.

Real-world: 35 g/L seawater (~0.5 osmolar dissociated ions vs ~55 mol/L water) lowers water's vapor pressure by ~1%, raises boiling point ~0.5°C, depresses freezing point ~1.9°C — keeping polar oceans liquid down to that temperature.

Variants and refinements

• Modified Raoult's law (activity coefficients). For non-ideal solutions, replace X_i with γ_iX_i. γ > 1 = positive deviation; γ < 1 = negative. UNIFAC and NRTL models predict γ in simulation packages like Aspen.
• Henry's law as Raoult's complement. At infinite dilution the solute follows P = K_H·X_solute. The two laws describe opposite ends of the same vapor pressure curve.
• Reactive distillation. When components react (acetic acid + n-butanol → butyl acetate + water), reaction couples to phase equilibrium. Reactive columns run reaction and separation simultaneously, slashing esterification capex.
• Real gas effects (fugacity). Above a few bar, vapor is no longer ideal. Replace pressure with fugacity for high-pressure flash calculations in petrochemical plants.

Pitfalls and common mistakes

• Using mass fraction instead of mole fraction. Raoult's law requires X_i = n_i/n_total, not w_i. Mass fraction misses the molecular-weight bookkeeping that drives colligative effects.
• Ignoring dissociation of electrolytes. A 0.1 m NaCl solution behaves like a 0.2 m solution of single particles. Multiply mole fraction of solute by van 't Hoff factor i for ionic species, just as in osmotic pressure or freezing-point depression.
• Assuming ideality far from infinite dilution. Above ~10 mol% solute, real solutions diverge significantly from Raoult. Use modified Raoult's with activity coefficients or experimental P-x-y data.
• Forgetting the temperature dependence of P°. Pure-component vapor pressures rise sharply with temperature (Clausius–Clapeyron). Re-evaluate at the actual operating temperature, not just 25°C.
• Mistaking total pressure for partial pressure. Each component's partial pressure follows Raoult; the system pressure is their sum. In dynamic distillation the system also has flow and condenser duty terms.