Calculate the vapour pressure of a solution containing \( 61 \, g \) of benzoic acid (molar mass \( = 122 \, g \, mol^{-1} \)) dissolved in \( 500 \, g \) of benzene when the vapour pressure of pure benzene at this temperature of experiment is \( 66 \, torr \). Assume complete dimerization of benzoic acid in benzene.
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The van't Hoff factor for dimerization is always \( 0.5 \) if association is complete.
Solution and Explanation
• Vapor pressure lowering is a colligative property. According to Raoult's Law for a non-volatile solute: \[ \frac{P^\circ - P_s}{P^\circ} = \frac{i \cdot n_B}{n_A + i \cdot n_B} \approx \frac{i \cdot n_B}{n_A} \text{ (for dilute solutions)} \]
• When a solute undergoes association (like dimerization), the number of effective particles decreases.
• The van't Hoff factor (\( i \)) for dimerization is calculated as \( i = 1 + (\frac{1}{n} - 1)\alpha \). For complete dimerization, \( \alpha = 1 \) and \( n = 2 \).
Step 1: Calculate the number of moles of solute and solvent
Mass of benzoic acid (\( W_B \)) \( = 61 \, g \)
Molar mass of benzoic acid (\( M_B \)) \( = 122 \, g \, mol^{-1} \)
Moles of benzoic acid (\( n_B \)) \( = \frac{61}{122} = 0.5 \, mol \)
Mass of benzene (\( W_A \)) \( = 500 \, g \)
Molar mass of benzene (\( C_6H_6 \), \( M_A \)) \( = 78 \, g \, mol^{-1} \)
Moles of benzene (\( n_A \)) \( = \frac{500}{78} \approx 6.41 \, mol \)
Step 2: Determine the van't Hoff factor (\( i \))
Benzoic acid dimersize in benzene as: \( 2C_6H_5COOH \rightarrow (C_6H_5COOH)_2 \)
For complete dimerization (\( \alpha = 1 \)): \[ i = 1 + \left(\frac{1}{2} - 1\right)(1) = 1 - 0.5 = 0.5 \]
Step 3: Apply Raoult's Law formula
Let the vapor pressure of the solution be \( P_s \).
Pure vapor pressure (\( P^\circ \)) \( = 66 \, torr \)
\[ \frac{66 - P_s}{66} = \frac{0.5 \times 0.5}{6.41 + (0.5 \times 0.5)} \] \[ \frac{66 - P_s}{66} = \frac{0.25}{6.41 + 0.25} = \frac{0.25}{6.66} \approx 0.03753 \]
Step 4: Calculate the final vapor pressure
\[ 66 - P_s = 66 \times 0.03753 \approx 2.477 \, torr \] \[ P_s = 66 - 2.477 = 63.523 \, torr \] The final answer is \( 63.52 \, torr \).
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