Colligative properties

Four properties of dilute solutions depend only on the number of solute particles, not on what they are: vapour-pressure lowering, boiling-point elevation, freezing-point depression and osmotic pressure. Each can be used to find a molar mass.

Properties that depend on the number of particles

A non-volatile solute lowers the vapour pressure of the solvent by a fraction equal to the solute’s mole fraction. That fraction depends on how many solute particles there are, not on what kind they are. Four measurable properties follow from this.

They are called colligative properties, from the Latin for “bound together”. In a dilute solution, each depends only on the number of solute particles. One mole of glucose and one mole of urea, each dissolved in 1 kg of water, lower its freezing point by the same amount.

Because each property counts particles, measuring it gives the number of moles of solute. If the mass of solute is known, its molar mass follows.

1. Relative lowering of vapour pressure

By Raoult’s law, the relative lowering of vapour pressure equals the mole fraction of the solute:

(p₁° − p₁) / p₁° = x₂

Measuring the vapour pressure of the pure solvent and of the solution gives x₂. From x₂ and the masses of solute and solvent, the molar mass of the solute can be calculated.

2. Elevation of boiling point

A liquid boils when its vapour pressure equals the external pressure. A solution has a lower vapour pressure than the pure solvent at every temperature. So it must be heated to a higher temperature before its vapour pressure reaches the external pressure. Its boiling point is higher. For a dilute solution, the rise ΔT_b is proportional to the molality m:

ΔT_b = K_b m

K_b is the boiling point elevation constant, a property of the solvent. For water, K_b = 0.52 K kg mol⁻¹. A solution of 1 mol of glucose in 1 kg of water boils at about 100.52°C.

3. Depression of freezing point

A solution freezes when its vapour pressure equals the vapour pressure of the solid solvent. Because the solution’s vapour pressure is lowered, this happens at a lower temperature. The freezing point of the solution is lower than that of the pure solvent:

ΔT_f = K_f m

K_f is the freezing point depression constant. For water, K_f = 1.86 K kg mol⁻¹. Salt spread on an icy road dissolves in the thin layer of water on the ice. The solution has a freezing point below 0°C, so if the road is at, say, −5°C, the ice melts. Ethylene glycol in a car’s radiator lowers the freezing point of the coolant in the same way.

4. Osmotic pressure

A semipermeable membrane lets solvent molecules pass but not solute particles. Cell membranes and parchment are examples. If a solution and pure solvent are separated by such a membrane, solvent flows through the membrane into the solution. This net flow of solvent, from lower to higher solute concentration, is osmosis.

The flow is caused by the difference in solvent concentration across the membrane: more solvent molecules strike the membrane per second from the pure-solvent side. Applying extra pressure to the solution increases the rate at which solvent molecules are pushed back through. The osmotic pressure, Π, is the extra pressure that must be applied to the solution to stop the net flow. For a dilute solution:

Π = C R T

C is the molar concentration of solute, R is the gas constant, 0.0821 L atm K⁻¹ mol⁻¹, and T is the temperature in kelvin. A 0.010 M solution at 300 K has Π = 0.010 × 0.0821 × 300 ≈ 0.25 atm. That pressure supports a column of water about 2.5 m high, which is easy to measure. The same solution would lower the freezing point of water by only about 0.019 K.

Osmosis in everyday situations

• Raw mango pieces placed in strong salt solution lose water by osmosis and shrink, because the solution outside has a higher solute concentration than the fruit’s cells.
• Wilted flowers placed in fresh water take in water by osmosis and become firm again.
• Red blood cells in pure water take in water and can burst. In concentrated salt solution they lose water and shrink. Fluid injected into a vein is therefore isotonic with blood: it has the same osmotic pressure. For sodium chloride, that is a 0.9% (mass/volume) solution.
• Meat salted heavily, or fruit kept in thick sugar syrup, resists spoiling. Bacteria on the surface lose water by osmosis and cannot grow.