Colligative Properties

Colligative properties are physical properties of solutions that depend only on the number of dissolved solute particles, not on their chemical identity. The four colligative properties are vapour-pressure lowering, boiling-point elevation, freezing-point depression, and osmotic pressure.

The underlying cause is the same for all four: dissolved solute particles reduce the tendency of solvent molecules to escape into the vapour phase. Because these effects depend on particle count, electrolytes that dissociate into multiple ions produce a larger colligative effect than the same molality of a nonelectrolyte.

Each colligative-property equation uses a different concentration unit:

• Boiling-point elevation and freezing-point depression use molality (m) — moles of solute per kilogram of solvent. Molality is independent of temperature, which makes it the natural choice when the solution is being heated to its boiling point or cooled to its freezing point.
• Osmotic pressure uses molarity (M) — moles of solute per liter of solution — because the equation Π = iMRT is derived from the same gas-law-style framework that uses volume.
• Raoult’s law uses mole fraction (χ) — the fraction of total moles that the solvent represents.

Converting between molarity, molality, mass percent, and mole fraction (using density to bridge volume and mass) is the subject of T9 LO 9.3 — see T9: Solutions and Concentration for the four-way conversion workflow. This topic assumes you arrive with the right concentration unit for the equation you’re using.

Molality of a saturated solution. Solubility data are usually reported as grams of solute per 100 g of water at a given temperature. Convert that directly to molality by computing moles of solute (mass ÷ molar mass) and dividing by the mass of water in kilograms. Without a measured density of the saturated solution, molarity cannot be inferred from solubility data alone — molality is the natural concentration unit for these problems.

Effective particle concentration. When a strong electrolyte dissolves, the relevant particle count is the sum of every ion released, not the formula-unit count. Each ion concentration equals the formula-unit concentration multiplied by the subscript of that ion in the formula. For 0.250 M CaCl₂: [Ca²⁺] = 0.250 M, [Cl⁻] = 2 × 0.250 = 0.500 M, total dissolved-particle concentration = 0.750 M. For 0.500 m Al₂(SO₄)₃: [Al³⁺] = 1.00 m, [SO₄²⁻] = 1.50 m, total = 2.50 m. This total is the input to every colligative-property equation below, encapsulated by the van’t Hoff factor i.

Raoult’s law states that the vapour pressure of a solvent above a solution equals the mole fraction of the solvent times the vapour pressure of the pure solvent:

P_solution = χ_solvent · P°_solvent

• χ_solvent = mole fraction of the solvent = mol solvent / total mol
• P°_solvent = vapour pressure of the pure solvent at that temperature

Because adding a nonvolatile solute reduces χ_solvent below 1, the solution’s vapour pressure is always lower than that of the pure solvent. The magnitude of the lowering is ΔP = χ_solute · P°_solvent.

This vapour-pressure lowering is the root cause of boiling-point elevation and freezing-point depression — all colligative properties trace back to the reduced tendency of solvent molecules to escape into the gas phase.

Adding a nonvolatile solute to a solvent raises its boiling point. Because the solute lowers the vapour pressure, a higher temperature is needed for the vapour pressure to reach atmospheric pressure (the condition for boiling).

The equation: ΔT_b = i · K_b · m

• ΔT_b = boiling-point elevation (°C)
• i = van’t Hoff factor (number of particles per formula unit)
• K_b = ebullioscopic constant of the solvent (for water, 0.512 °C/m)
• m = molality (mol solute / kg solvent)

The new boiling point = normal boiling point + ΔT_b. For example, a 1.00 m aqueous glucose solution (i = 1) boils at 100.00 + (1)(0.512)(1.00) = 100.51 °C. The small size of K_b means boiling-point elevation is a relatively subtle effect in dilute solutions.

Adding a solute lowers the freezing point of a solution. Solute particles disrupt the formation of the ordered crystal lattice that defines the solid phase, so a lower temperature is required for the solvent to freeze.

The equation: ΔT_f = i · K_f · m

• ΔT_f = freezing-point depression (°C)
• K_f = cryoscopic constant of the solvent (for water, 1.86 °C/m)
• m = molality; i = van’t Hoff factor

The new freezing point = normal freezing point − ΔT_f. This principle explains why salt (NaCl) is spread on icy roads — the dissolved ions depress the freezing point of water well below 0 °C. A 0.50 m aqueous glucose solution gives ΔT_f = (1)(1.86)(0.50) = 0.93 °C, so it freezes at −0.93 °C. Note that K_f for water is roughly four times larger than K_b, making freezing-point depression easier to measure experimentally.

Osmosis is the net flow of solvent through a semipermeable membrane from a region of lower solute concentration to higher solute concentration. The minimum pressure needed to halt this flow is the osmotic pressure (Π).

The equation: Π = iMRT

• Π = osmotic pressure (atm)
• i = van’t Hoff factor
• M = molarity of the solution (mol/L)
• R = 0.08206 L·atm/(mol·K)
• T = temperature in kelvins

Osmotic pressure is extremely sensitive to solute concentration, making it the preferred colligative property for determining molar masses of large molecules such as proteins, where ΔT_b or ΔT_f would be immeasurably small. In biology, solutions with equal osmotic pressure are called isotonic; a solution with higher Π is hypertonic; lower is hypotonic. Intravenous (IV) fluids must be isotonic with blood to prevent cell damage.

The van’t Hoff factor (i) accounts for the dissociation of electrolytes into multiple particles, which amplifies every colligative effect:

• Nonelectrolytes (e.g., glucose, sucrose): i = 1 — molecules remain intact.
• Strong electrolytes: i equals the total number of ions produced. NaCl → Na⁺ + Cl⁻, so i = 2. CaCl₂ → Ca²⁺ + 2 Cl⁻, so i = 3.
• Weak electrolytes: 1 < i < theoretical maximum, because dissociation is incomplete.

In practice, measured i values for strong electrolytes are slightly less than theoretical due to ion pairing — oppositely charged ions can temporarily associate in solution, reducing the effective particle count. This deviation grows at higher concentrations. For example, 0.10 m NaCl ideally gives ΔT_f = (2)(1.86)(0.10) = 0.37 °C, but the measured value is typically about 0.35 °C.

Because colligative-property equations link a measurable physical change to the amount of dissolved solute, they can be rearranged to find an unknown molar mass. The general strategy is:

1. Measure the colligative effect (ΔT_b, ΔT_f, or Π).
2. Use the appropriate equation to calculate molality (or molarity for Π).
3. From molality and the known mass of solute and solvent, calculate moles of solute.
4. Divide the mass of solute (in grams) by the moles to get the molar mass.

Freezing-point depression is commonly used for small molecules because K_f is large enough to give measurable temperature changes. For macromolecules such as proteins or polymers, osmotic pressure is preferred — even very dilute solutions generate measurable Π values, allowing accurate molar-mass determinations where ΔT methods would fail.

From molar mass to molecular formula. Once colligative measurement gives the molar mass, a separate step combines it with percent composition to identify the molecular formula. The workflow: (1) percent composition → mole ratio of each element per 100 g → empirical formula and its empirical molar mass; (2) molecular molar mass ÷ empirical molar mass = n (a small whole number); (3) multiply empirical subscripts by n. Worked example: an organic compound with composition 93.46% C / 6.54% H gives the empirical formula C₆H₅ (empirical MW 77.10 g/mol). If the colligative-property data yields a molar mass of 154.2 g/mol, then n = 154.2 / 77.10 = 2.00, so the molecular formula is (C₆H₅)₂ = C₁₂H₁₀ (biphenyl). For pure elements like sulfur, the molar mass is divided by the atomic mass to recover the allotrope subscript n (e.g., MW ≈ 256 g/mol ÷ 32.07 g/mol = 8, identifying S₈). The empirical-formula-to-molecular-formula derivation itself is the subject of T8: The Mole and Chemical Formulas — this topic uses it as the closing step in colligative-property molar-mass problems that ask for the molecular formula.

A reliable workflow for colligative property calculations:

1. Determine the solute type: molecular (i = 1) or electrolyte (i = number of ions produced per formula unit).
2. Calculate molality (not molarity) using moles of solute per kilogram of solvent.
3. Select the correct equation: ΔT_b = iK_bm for boiling-point elevation, ΔT_f = iK_fm for freezing-point depression, Π = iMRT for osmotic pressure.
4. Apply the van ’t Hoff factor. For strong electrolytes, i equals the number of ions per formula unit (NaCl → i = 2, CaCl₂ → i = 3).
5. Check the direction: boiling points go up, freezing points go down. If your ΔT has the wrong sign relative to the pure solvent, recheck.

Common mistakes: using molarity instead of molality for boiling-point and freezing-point equations, forgetting the van ’t Hoff factor for ionic solutes (this can double or triple the expected effect), treating a weak electrolyte as though it fully dissociates, and confusing the new boiling/freezing point with the ΔT value itself.