Chemistry · General chemistry II · Concept

Colligative properties of solutions

Calculate freezing-point depression, boiling-point elevation, vapor pressure with nonvolatile or volatile solutes, and osmotic pressure, and use each to find a molar mass or a van ’t Hoff factor.

Why only the number of particles matters

A dissolved solute dilutes the solvent. Fewer solvent molecules at the surface lowers the vapor pressure, and that one effect raises the boiling point, lowers the freezing point and drives osmosis. In dilute solutions these changes depend on how many solute particles there are, not what they are. That is what colligative means.

Ionic solutes count every ion. The van ’t Hoff factor i is the number of particles per formula unit: ideally 1 for glucose, 2 for NaCl and 3 for CaCl₂. Measured values are a little smaller because some ions pair up.

Which concentration each property uses
Property	Equation	Concentration	Why
Vapor pressure	P = χ_solvent P°	Mole fraction χ	Counts particles at the surface
Freezing point	ΔTf = i Kf m	Molality m (mol/kg solvent)	Mass of solvent doesn’t change with temperature
Boiling point	ΔTb = i Kb m	Molality m	Same as freezing point
Osmotic pressure	Π = i M R T	Molarity M (mol/L solution)	Like a gas pressure: amount per volume

Vapor pressure with a nonvolatile solute

Raoult’s law: the solvent’s vapor pressure is its mole fraction times the pure-solvent vapor pressure at the same temperature. The lowering itself is proportional to the solute mole fraction, counting particles, so multiply the moles of an ionic solute by i.

P_{solution} = χ_{solvent} P_{solvent}^{\circ}, Δ P = χ_{solute} P_{solvent}^{\circ}

Vapor pressure when both components are volatile

When the solute also evaporates, each component contributes its own Raoult’s-law pressure, and Dalton’s law adds them. The vapor is richer in the more volatile component than the liquid is, which is why distillation separates liquids. Real mixtures deviate: if the two kinds of molecules attract each other more strongly than they attract themselves, the total pressure falls below the ideal line. If they attract each other less strongly, it rises above.

P_{total} = χ_{A} P_{A}^{\circ} + χ_{B} P_{B}^{\circ}, y_{A} = \frac{χ_{A} P_{A}^{\circ}}{P_{total}}

Freezing-point depression and boiling-point elevation

Both shifts are proportional to molality, with a constant that belongs to the solvent: for water, Kf = 1.86 °C·kg/mol and Kb = 0.512 °C·kg/mol. The freezing point goes down and the boiling point goes up. Molality uses kilograms of solvent, not litres of solution.

Δ T_{f} = i K_{f} m, Δ T_{b} = i K_{b} m, m = \frac{n_{solute}}{kg solvent}

Osmotic pressure

Osmotic pressure is the pressure needed to stop solvent flowing through a semipermeable membrane into the more concentrated solution. It has the form of the ideal gas law, with molarity in place of n/V and T in kelvin. Even dilute solutions give pressures you can measure, which makes osmotic pressure the best way to find the molar mass of large molecules such as proteins.

Π = i M R T, R = 0.08206 L atm {mol}^{- 1} K^{- 1}

Working backwards: molar mass and i

Each equation can be run in reverse. From a measured ΔTf or Π, find moles of solute particles, then divide the sample mass by the moles of solute to get grams per mole. For a known solute, divide the measured shift by the shift expected with i = 1 to get the measured van ’t Hoff factor.

M = \frac{mass of solute}{n_{solute}}, i_{measured} = \frac{Δ T_{measured}}{K m}

Common mistakes

Using molarity in ΔT = iKm, or molality in Π = iMRT.
Forgetting i for ionic solutes, or assuming the ideal i is exact.
Subtracting ΔTb from the boiling point, or adding ΔTf to the freezing point.
Using °C in Π = iMRT.
Using mass fraction instead of mole fraction in Raoult’s law.
Assuming only the solvent evaporates when the solute is volatile.

Key terms

Colligative properties: Properties of a solution that depend on how many solute particles are dissolved, not what they are: vapor-pressure lowering, boiling-point elevation, freezing-point depression and osmotic pressure. Real solutions deviate at higher concentrations.
Molality: Moles of solute per kilogram of solvent (mol/kg). Unlike molarity, it uses the solvent’s mass rather than the solution’s volume, so it doesn’t change with temperature.
Van’t Hoff factor: The number of particles a solute gives per formula unit when it dissolves: 1 for sugar, about 2 for NaCl and about 3 for CaCl₂. Measured values are often a little lower because ions pair up.
Raoult’s law: For an ideal solution, each volatile component’s vapor pressure equals its mole fraction in the liquid times the vapor pressure of the pure component at that temperature: P = χP°.
Vapor pressure: The pressure of a vapor in equilibrium with its liquid or solid at a given temperature. It rises with temperature; a gas collected over water includes water’s vapor pressure.
Freezing-point depression: The drop in a solvent’s freezing point caused by dissolved particles: ΔTf = i·Kf·m, where the constant Kf depends on the solvent.
Boiling-point elevation: The rise in a solvent’s boiling point caused by a dissolved nonvolatile solute: ΔTb = i·Kb·m. Use the solvent’s boiling constant Kb, not its freezing constant.
Osmotic pressure: The pressure needed to stop solvent from flowing through a semipermeable membrane into a solution. For a dilute solution, π = iMRT.
Mole fraction: mole fraction of A = (moles of A) ÷ (total moles of everything in the mixture). It has no units, and all the mole fractions in a mixture add up to 1.
Ideal solution: A solution that follows Raoult’s law at every composition because attractions between unlike particles equal those between like ones. Mixing it releases or absorbs no heat.

Work through an example

25.0 g of ethylene glycol, C₂H₆O₂, is dissolved in 100.0 g of water. Find the freezing and boiling points of the solution. For water, Kf = 1.86 °C·kg/mol and Kb = 0.512 °C·kg/mol.

Freezing and boiling points of antifreeze →

Vapor pressure of a glucose solution →

Vapor pressure of a benzene–toluene mixture →

Molar mass of a protein from osmotic pressure →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Tro, Chemistry: A Molecular Approach, 4th ed., §13.5 Expressing Solution Concentration, pp. 585–592 (molality, p. 588)
Tro, Chemistry: A Molecular Approach, 4th ed., §13.6 Colligative Properties: Vapor Pressure Lowering, Freezing Point Depression, Boiling Point Elevation, and Osmotic Pressure, pp. 593–605 (volatile solutes, p. 597; osmotic pressure, pp. 603–604)
Tro, Chemistry: A Molecular Approach, 4th ed., §13.7 Colligative Properties of Strong Electrolyte Solutions, pp. 605–607
OpenStax Chemistry 2e — Colligative properties

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