# Colligative Properties

Properties of solution that depend on the ratio of no: of solute particles to the total no: of particles in the solution.

1. Relative lowering of vapour pressure:

⇒ Vapour pressure of solvent in solution is less than that of pure solvent.

According to Raoult’s law

PA = XA x P°A

⇒ So the reduction in vapour pressure is given as

Δ PA = P°A – PA

i.e., Δ PA = P°A (1 – XA) = P°AXB

$$\frac{\Delta {{P}_{A}}}{{{P}_{A}}}={{X}_{B}}=\frac{{{n}_{2}}}{{{n}_{1}}+{{n}_{2}}}$$

i.e., therefore, relative lowering of vapour pressure Δ PA/PA is equal to mole fraction of solute XB.

⇒ If two or more non-volatile solutes are present then it is equal to sum of the mole fraction of the solutes

$$\frac{\Delta {{P}_{A}}}{{{P}_{A}}}={{X}_{B}}+{{X}_{C}}+………+{{X}_{N}}$$

For dilute solutions:

$$\frac{\Delta {{P}_{A}}}{{{P}_{A}}}={{X}_{B}}=\frac{{{n}_{2}}}{{{n}_{1}}}$$ (As n₂ << n₁)

$$\frac{\Delta {{P}_{A}}}{{{P}_{A}}}=\frac{{{w}_{B}}\times {{M}_{A}}}{{{M}_{B}}\times {{w}_{A}}}$$

Where, wA = mass of solvent

MA = Molar mass of solvent

WB = mass of solute

MB = Molar mass of solute

2. Elevation of boiling point:

⇒ A liquid boils at a temperature where its vapour pressure becomes equal to atmospheric pressure.

⇒ Since in a solution the vapour pressure of solvent is reduced, the temperature needs to be raised to increase the vapour pressure to atmospheric pressure.

⇒ Therefore, the boiling point of solution is always higher than that of pure solvent.

ΔTb = Tb – T°b

= elevation in boiling point

Tb= boiling point of solution

Tb0 = boiling point of pure solvent

For dilute solutions:

Tb α m

Δ Tb = Kb x m

m = molality

Kb = molal elevation constant (Ebullioscopic Constant). Units: K kg mol-1

$${{K}_{b}}=\frac{R\times {{M}_{1}}\times {{T}_{b}}^{2}}{1000\times {{\Delta }_{vap}}H}$$

R = gas constant

M1 = molar mass of solvent

ΔvapH = enthalpy of vaporization

3. Depression of freezing point

⇒ A solution freezes when its vapour pressure becomes equal to the vapour pressure of pure solid solvent.

⇒ As the vapour pressure of a solvent decreases when a non-volatile solute is added, the freezing point of solvent decreases.

ΔTf = T°f – Tf

ΔTf = depression in freezing point

f = freezing point of pure solvent

Tf = freezing point of solution

For dilute solution:

Δ Tf α m

Δ Tf = Kf x m

Kf = Molal depression constant (Cryoscopic constant). Units: K kg mol¯¹

$${{K}_{f}}=\frac{R\times {{M}_{1}}\times {{T}_{f}}^{2}}{1000\times {{\Delta }_{fus}}H}$$

R = gas constant

M1= molar mass of solvent

ΔfusH = enthalpy of fusion