Physical Chemistry
Colligative Properties
Four solvent effects that count particles, not identities
Colligative properties depend only on the number of dissolved particles, not their identity. The four classics — vapor-pressure lowering, boiling-point elevation, freezing-point depression, and osmotic pressure — share a single thermodynamic root: solute particles lower solvent chemical potential by an amount proportional to mole fraction. Antifreeze in your car, salt on icy roads, IV saline, and dialysis all run on this physics.
- OriginEntropy of mixing solvent with solute
- Depend onParticle count (mole fraction)
- Independent ofSolute identity
- Water Kf1.86 °C·kg/mol
- Water Kb0.512 °C·kg/mol
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The single idea behind all four
Dissolve any solute in any solvent and four things happen at once: solvent vapor pressure drops, boiling point rises, freezing point falls, and an osmotic pressure develops across any semipermeable membrane separating it from pure solvent. The size of each effect depends on how many particles you dissolved per kilogram of solvent — not on what those particles are. Sugar, urea, glycerol, or sodium-equivalent ions all produce the same shift, particle-for-particle.
The shared cause: mixing raises entropy. Higher entropy means lower Gibbs free energy means lower chemical potential. Solvent in solution has lower escaping tendency than pure solvent — it evaporates less, requires more heat to boil, freezes lower, and pulls in solvent from any pure reservoir across a membrane.
Mathematically, the chemical potential of solvent in solution is:
μ_solvent = μ°_solvent + RT ln(X_solvent)The Xsolvent term is what enters all four colligative equations after the appropriate phase-equilibrium algebra.
The four effects with formulas
| Effect | Formula | Constant for water | Magnitude per molal |
|---|---|---|---|
| Vapor-pressure lowering | ΔP = Xsolute · P° | P°(25°C) = 23.76 mmHg | ~0.43 mmHg/molal |
| Boiling-point elevation | ΔTb = i · Kb · m | Kb = 0.512 °C·kg/mol | 0.512 °C/molal (i=1) |
| Freezing-point depression | ΔTf = i · Kf · m | Kf = 1.86 °C·kg/mol | 1.86 °C/molal (i=1) |
| Osmotic pressure | π = i · M · R · T | RT (25°C) = 24.46 L·atm/mol | 2.45 atm/molar (i=1) |
| Henry's solubility (related) | psolute = KH · Xsolute | System-specific | Linear in solute concentration |
| Activity-coefficient correction | Replace X with γX, m with γ±m | γ → 1 at infinite dilution | ±5–20% at 0.1 m for ions |
m is molality (mol solute per kg solvent), M is molarity (mol solute per litre solution), and i is the van 't Hoff factor — the effective number of particles each formula unit produces in solution.
The van 't Hoff factor
For non-electrolytes (sugar, urea, glycerol), each formula unit dissolves as one particle and i = 1. For ionic compounds, i counts dissociated ions:
| Solute | Dissociation | Ideal i | Actual i (0.1 m) |
|---|---|---|---|
| Sucrose, glucose | None | 1 | ~1.00 |
| NaCl, KCl, NH₄Cl | 2 ions | 2 | ~1.87 |
| MgSO₄ | 2 ions (high charge) | 2 | ~1.30 |
| CaCl₂, MgCl₂ | 3 ions | 3 | ~2.70 |
| K₂SO₄, Na₂SO₄ | 3 ions | 3 | ~2.45 |
| AlCl₃ | 4 ions | 4 | ~3.20 |
| Acetic acid (1 M) | Partial | 1–2 | ~1.013 |
Real i values fall below ideal because of ion pairing — at finite concentration some Na⁺ and Cl⁻ ions transiently associate, reducing the count of free particles. The deviation grows with charge density (MgSO₄ pairs more than NaCl) and concentration. At infinite dilution real i approaches the ideal value.
Worked example: 50% ethylene glycol antifreeze
How much does a 50% by mass ethylene glycol/water mixture lower the freezing point? MW(C2H6O2) = 62.07 g/mol.
Per 1.000 kg solution: 500 g glycol, 500 g water = 0.500 kg water
n_glycol = 500 / 62.07 = 8.06 mol
m = 8.06 mol / 0.500 kg = 16.1 molal
ΔT_f (linear prediction) = 1 × 1.86 × 16.1 = 30°C of depression
Predicted freezing point: 0°C − 30°C = −30°CThe measured freezing point of 50% ethylene glycol/water is ~–37°C — the linear formula underestimates by ~7°C at this concentration. Above 1–2 mol/kg, ideality breaks down and real performance exceeds linear theory. Practical antifreeze tables come from experimental measurement, not extrapolation.
Now a textbook-friendly case: 23 g NaCl in 1.00 kg water (roughly seawater concentration). m = 0.394 mol/kg, i ≈ 1.87, so ΔTf = 1.87 × 1.86 × 0.394 = 1.37°C — within 5% of measured ocean ice formation.
Real-world applications, with numbers
| Application | Solute | Concentration | Effect |
|---|---|---|---|
| Road salt (NaCl) | NaCl | Saturated brine ~6 m | Melts ice down to –21°C (eutectic) |
| Cold-climate de-icer (CaCl₂) | CaCl₂ | Saturated brine | Melts ice down to –51°C (eutectic) |
| Engine antifreeze (50/50) | Ethylene glycol | ~9 m | Freeze protection to –37°C |
| Aircraft de-icing fluid | Propylene glycol + thickener | ~80% v/v | Freeze protection to –50°C, plus shear-thinning film |
| IV normal saline | NaCl | 0.154 M, 308 mOsm/L | Isotonic with plasma (π ≈ 7.7 atm) |
| Seawater | Mixed salts | ~1.1 osmolar | Freezes at –1.9°C; π ≈ 27 atm |
| Ice cream salt-bath churning | NaCl ice slurry | ~3 m brine | Bath at –10°C lets cream freeze without becoming icy |
| Radiator boil-over protection | Ethylene glycol + 1.1 atm cap | ~50% glycol | Boils at ~125°C instead of 100°C |
Phase diagram intuition
On a pressure-vs-temperature phase diagram for water, the triple point sits at 0.01°C, 6.1 mbar. Add a non-volatile solute and the liquid–vapor curve shifts downward (vapor-pressure lowering). The triple point slides down-left; the liquid region expands at both ends of the temperature axis (freezing-point depression at the cold end, boiling-point elevation at the hot). Vapor-pressure lowering, freezing-point depression, and boiling-point elevation are three readings of the same shifted phase diagram, indexed by the same Xsolute.
Why osmometry wins for measurement
For molar-mass determination of polymers or proteins, the four colligative properties are not equal. For a 0.001 molal solution of a 100,000 g/mol protein: vapor-pressure lowering predicts ΔP = 0.000018 mmHg (unmeasurable); boiling-point elevation ΔTb = 0.0005°C (marginal); freezing-point depression ΔTf = 0.0019°C (borderline); but osmotic pressure π = 18 mmHg (strong signal). This is why membrane osmometry became the gold-standard method for polymer and protein number-average molar mass — the same concentration that gives an unmeasurable temperature shift gives a generous pressure reading.
Variants and refinements
- Eutectic mixtures. The NaCl/water system has a eutectic at 23 wt%, –21°C, where saturated brine and ice freeze together. Below that road salt stops working.
- Cryoscopy in non-aqueous solvents. Camphor has Kf = 40°C·kg/mol, ~22× larger than water. Rast's method exploits this for organic MW determination by mixing 1% of unknown into molten camphor.
- Ebullioscopic constants. Kb values: water 0.512, benzene 2.53, chloroform 3.63, naphthalene 5.8 °C·kg/mol. Higher Kb = more sensitive boiling-point measurement.
- Activity correction at high m. Replace m with γ±m via Debye–Hückel or Pitzer equations. At 1 m NaCl, γ± ≈ 0.66; ignoring activity overestimates colligative effects by ~30%.
Pitfalls and common mistakes
- Forgetting i for ionic solutes. 1 m NaCl behaves like ~1.87 m of single particles. Plain m gives the wrong answer by nearly a factor of two.
- Confusing molality and molarity. Molality (mol/kg solvent) is temperature-independent and used for boiling/freezing equations. Molarity (mol/L solution) varies with temperature and is used for osmotic pressure. Mixing them leaks errors of a few percent.
- Linear extrapolation past 1 mol/kg. ΔTf = Kf·m·i is a limiting law. At antifreeze concentrations, real depressions exceed linear predictions by factors of 1.2 to 2. Always consult experimental tables for high-concentration design.
- Ignoring volatile solutes. Boiling-point elevation requires the solute to have negligible vapor pressure. Ethanol in water doesn't depress vapor pressure linearly because both species evaporate.
- Using ΔTf for solubility predictions. Freezing-point depression measures particles in the liquid only — it cannot tell you whether the solute is partially solid (saturated). Always cross-check against the solubility limit.
Frequently asked questions
Why doesn't the identity of the solute matter?
Colligative effects depend on the entropy of mixing — adding particles increases solvent disorder by an amount that depends only on particle count, not identity. The chemical potential of the solvent drops by the same amount whether you dissolve sugar, urea, or NaCl-equivalent moles of particles. Identity matters for the magnitude only through the van 't Hoff factor i, which counts particles per formula unit.
How much ethylene glycol is in real antifreeze?
A 50/50 by volume ethylene glycol/water mix protects to about –37°C and raises the boiling point to ~108°C at 1 atm. That is roughly 9 molal in glycol, predicting 9 × 1.86 = 16.7°C of freezing-point depression — the experimental value (~37°C) far exceeds the linear prediction because at high concentrations the simple equation breaks down. Most modern coolants run 50–60% glycol for both freeze protection and corrosion inhibition.
Why does salt melt ice?
Salt dissolves in the thin water film on the ice surface, depressing the freezing point of that brine below the ambient temperature. With melting now thermodynamically favoured, the ice underneath supplies more water, more salt dissolves, and a melting front propagates. NaCl works to about –9°C; CaCl₂ down to about –25°C because its three-ion dissociation gives a higher van 't Hoff factor and CaCl₂ also releases heat as it dissolves.
Why does seawater freeze around –1.9°C, not 0°C?
Average seawater is 35 g/kg salts (~1.1 osmolar of dissociated ions). Multiplying by Kf = 1.86°C/molal gives ΔTf ≈ 1.9°C of depression — almost exactly observed. Polar oceans stay liquid down to that limit; below it sea ice forms but expels brine, so the residual seawater is even saltier and freezes at even lower temperature, driving thermohaline circulation.
Which colligative property is most sensitive?
Osmotic pressure by far. A 0.001 m solution gives only 0.0019°C of freezing-point depression and 0.000018 mmHg of vapor-pressure lowering — both unmeasurable in practice. The same solution exerts 18 mmHg of osmotic pressure, easily measured. That sensitivity is why osmometry is the standard method for biological osmolarity and polymer molar mass.
Do colligative properties work for non-electrolytes only?
No — they apply to any dissolved species, but you must include the van 't Hoff factor i for ionic compounds. NaCl gives i ≈ 2 (Na⁺ + Cl⁻), CaCl₂ gives i ≈ 3 (Ca²⁺ + 2 Cl⁻). At high concentrations real i values fall below ideal because of ion pairing. Weak electrolytes have i between 1 and 2 depending on degree of dissociation.