Question

An aqueous solution is prepared by dissolving $0.1 \,mol$ of an ionic salt in $1.8\, kg$ of water at $35^{\circ} C$. The salt remains $90 \%$ dissociated in the solution. The vapour pressure of the solution is $59.724\, mm$ of Hg Vapor pressure of water at $35^{\circ} C$ is $60.000\, mm$ of Hg. The number of ions present per formula unit of the ionic salt is_____

Answer 1

Correct Answer: 5

Let P⁰ be the vapour pressure of pure water and P_s be the vapour pressure of the solution.
Given:

• Moles of ionic salt (solute), \( n_{\text{solute}} \) = 0.1 mol
• Mass of water (solvent) = 1.8 kg = 1800 g
• Molar mass of water (H₂O) = 18 g/mol
• Moles of water, \( n_{\text{solvent}} \) = Mass of water / Molar mass of water = 1800 g / 18 g/mol = 100 mol
• Vapour pressure of pure water, P⁰ = 60.000 mm of Hg
• Vapour pressure of the solution, P_s = 59.724 mm of Hg
• Degree of dissociation, α = 90% = 0.90

According to Raoult's law, the relative lowering of vapour pressure is given by: $$ \frac{P^0 - P_s}{P^0} = X_{\text{solute, effective}} $$ where \( X_{\text{solute, effective}} \) is the mole fraction of all solute particles in the solution.
If 'i' is the Van't Hoff factor, then the effective number of moles of solute is \( i \cdot n_{\text{solute}} \).
So, $$ X_{\text{solute, effective}} = \frac{i \cdot n_{\text{solute}}}{i \cdot n_{\text{solute}} + n_{\text{solvent}}} $$ Substituting the given values: $$ \frac{60.000 - 59.724}{60.000} = \frac{i \cdot 0.1}{i \cdot 0.1 + 100} $$ $$ \frac{0.276}{60.000} = \frac{0.1 i}{0.1 i + 100} $$ $$ 0.0046 = \frac{0.1 i}{0.1 i + 100} $$ Now, solve for i: $$ 0.0046 (0.1 i + 100) = 0.1 i $$ $$ 0.00046 i + 0.46 = 0.1 i $$ $$ 0.46 = 0.1 i - 0.00046 i $$ $$ 0.46 = (0.1 - 0.00046) i $$ $$ 0.46 = 0.09954 i $$ $$ i = \frac{0.46}{0.09954} \approx 4.621257 $$

The Van't Hoff factor 'i' is related to the degree of dissociation (α) and the number of ions produced per formula unit (n) by the formula: $$ i = 1 + (n - 1)\alpha $$ We have i ≈ 4.621257 and α = 0.90. $$ 4.621257 = 1 + (n - 1) \times 0.90 $$ $$ 4.621257 - 1 = (n - 1) \times 0.90 $$ $$ 3.621257 = (n - 1) \times 0.90 $$ $$ n - 1 = \frac{3.621257}{0.90} $$ $$ n - 1 \approx 4.023618 $$ $$ n \approx 4.023618 + 1 $$ $$ n \approx 5.023618 $$ Since the number of ions present per formula unit must be an integer, we round n to the nearest integer. $$ n = 5 $$

The final answer is \({5}\)