Question

Vapour pressure of pure water at 298 K is $23.8$ mmHg. 50 g of urea is dissolved in 850 g of water. Calculate the vapour pressure of water for this solution at its relative lowering.

Answer 1

Hint: The vapour pressure of a solvent is the pressure exerted by the vapour of the solvent on the walls of the container and the addition of solute to the solvent reduces this vapour pressure. This relative lowering of the vapour pressure solution is dependent on the number of the solute molecules.
Formula used:
$\dfrac{{{{\text{P}}^{\text{0}}} - {\text{ P}}}}{{{{\text{P}}^{\text{0}}}}}{\text{ = }}\dfrac{{{{\text{w}}_{\text{2}}} \times {{\text{M}}_{\text{1}}}}}{{{{\text{M}}_{\text{2}}} \times {{\text{w}}_1}}}$

Complete step by step answer:
The relative lowering of vapour pressure is a colligative property of the solution that is mathematically represented as:
$\dfrac{{{{\text{P}}^{\text{0}}} - {\text{ P}}}}{{{{\text{P}}^{\text{0}}}}}{\text{ = }}\dfrac{{{{\text{w}}_{\text{2}}} \times {{\text{M}}_{\text{1}}}}}{{{{\text{M}}_{\text{2}}} \times {{\text{w}}_1}}}$
Here, ${{\text{w}}_2}$ = 50 g, Molecular weight of urea, ${{\text{M}}_2} = 60.06$ , ${{\text{w}}_1}$ = 850 g, ${{\text{M}}_1}$ = 18 ${\text{g/mol}}$ and the vapour pressure of the pure solvent ${{\text{P}}^0}$ = $23.8$ mmHg
Putting the values of the variable in the equation, we get,
$\dfrac{{23.8 - {\text{P}}}}{{23.8}}{\text{ = }}\dfrac{{50 \times 18}}{{60.6 \times 850}}$
$ \Rightarrow 23.8 - {\text{P = }}\dfrac{{{\text{21420}}}}{{51510}} = 0.415$
$ \Rightarrow {\text{ P = }}23.8 - 0.415$ = $23.385$ mmHg
Solving this we get, the final vapour pressure of the solution is $23.385$mm of Hg.
The relative lowering of vapour pressure is thus, $\dfrac{{{P^0} - P}}{{{P^0}}} = \dfrac{{\Delta P}}{{{P^0}}}$ = $\dfrac{{0.415}}{{23.8}}$ = \[0.0174\]
As this is a ratio of two quantities so the relative lowering has no unit.

Note:
The colligative properties of the solution depend on the number of the solute molecules present in the solution and not on the type or the nature of the solute.
Besides the lowering of vapour pressure, there are three other colligative properties namely, elevation in boiling point of the solvent, depression in freezing point of the solvent and osmotic pressure.