How will you show that depression in freezing point is a colligative property?

Explanation

Freezing point of a liquid is the temperature at which both solid and liquid have same vapour pressure. Addition of a non-volatile solute decreases the vapour pressure of solution. Now solution will freeze at low temperature. Depression in freezing point `(DeltaT_(f))` i.e. `DeltaT_(f)=T_(f)^(@)-T_(f)`
The curve AB, BC represents pure solvent. In this at point B, solid starts apppearing and `T_(f)^(@)` is the temperature at this point where solid and liquid are in equilibrium. Below point B solvent get frozen.
At an any temperature the vapour pressure of solution is less than that of the solvent the curve for the solution lies below that of solvent. It will follow the path DE. At point E the solid appears. Hence E represent the freezing point `T_(f)` of the solution.
Depression in freezing point `DeltaT_(f)` is directly proportional to the decreases in vapour pressure. For dilute solutions, decrease in vapour pressure is directly proportional to molality of solution. Hence `DeltaT_(f)alpha` m i.e. `DeltaT_(f)` is a colligative property.