Question

Prove that the relative lowering of the vapour pressure of a solution is equal to the mole fraction of non-volatile solute present in the solution.

Answer 1

Step 1 Raoult’s law:

Raoult’s law states that the partial vapour pressure of any component in the solution is directly proportional to its mole fraction.

Mathematically, Raoult’s law equation can be expressed as,

${\mathrm{P}}_{\mathrm{solution}}={\mathrm{{\rm X}}}_{\mathrm{solvent}}{\mathrm{P}}_{\mathrm{solvent}}^0$

Where,

${\mathrm{P}}_{\mathrm{solution}}$ is the vapour pressure of the solution
${\mathrm{{\rm X}}}_{\mathrm{solvent}}$ is the mole fraction of the solvent
$\mathrm{P}{°}_{\mathrm{solvent}}$is the vapour pressure of the pure solvent

Step 2 Relative lowering of vapour pressure

The relative lowering of vapour pressure is the ratio of lowering the vapour pressure of the solvent in a solution to the vapour pressure of the pure solvent.

According to Raoult’s law,

If the vapour pressure of the solution is $\mathrm{P}$ and the partial vapour pressure of the solvent is ${\mathrm{P}}_{\mathrm{A}}$ then $\mathrm{P}={\mathrm{P}}_{\mathrm{A}}$

That is,

${\mathrm{P}}_{\mathrm{A}}={\mathrm{P}}_{\mathrm{A}}^0{\mathrm{X}}_{\mathrm{A}}$ …..(1)

Where, ${\mathrm{P}}_{\mathrm{A}}^0$ is the vapour pressure of pure solvent and ${\mathrm{x}}_{\mathrm{A}}$ and ${\mathrm{x}}_{\mathrm{B}}$ are the mole fractions of solvent and solute respectively.

$$\begin{gathered} x_A+x_B=1 \\ {x}_A=(1-x_B) \end{gathered}$$

Substitute this in equation (1)

$$\begin{gathered} \begin{array}{r}\mathrm{P}={\mathrm{P}}_{\mathrm{A}}^0\left(1-{\mathrm{x}}_{\mathrm{B}}\right)\\ \mathrm{P}={\mathrm{P}}_{\mathrm{A}}^0-{\mathrm{P}}_{\mathrm{A}}^0\cdot {\mathrm{x}}_{\mathrm{B}}\end{array} \\ P-{\mathrm{P}}_{\mathrm{A}}^0=-{\mathrm{P}}_{\mathrm{A}}^0\cdot {\mathrm{x}}_{\mathrm{B}} \\ \frac{{\mathrm{P}}_{\mathrm{A}}^0-\mathrm{P}}{{\mathrm{P}}_{\mathrm{A}}^0}={\mathrm{x}}_{\mathrm{B}} \end{gathered}$$

Where, $\frac{{\mathrm{P}}_{\mathrm{A}}^0-\mathrm{P}}{{\mathrm{P}}_{\mathrm{A}}^0}$ is the relative lowering of vapour pressure.

Therefore, the relative lowering of the vapour pressure of a solution is equal to the mole fraction of non-volatile solute present in the solution.