Considering acetic acid dissociates in water, its dissociation constant is \(6.25 \times 10^{-5}\). If \(5…

Considering acetic acid dissociates in water, its dissociation constant is \(6.25 \times 10^{-5}\). If \(5 \mathrm{~mL}\) of acetic acid is dissolved in 1 litre water, the solution will freeze at \(-x \times 10^{-2}{ }^{\circ} \mathrm{C}\), provided pure water freezes at \(0^{\circ} \mathrm{C}\). \(x=\) ______ . (Nearest integer) Given : $\begin{aligned} & \left(K_f\right)_{\text{{water}}}=1.86 \, \text{{K kg mol}}^{-1} \\ & \text{{density of acetic acid is }} 1.2 \, \text{{g mol}}^{-1} \\ & \text{{molar mass of water }}=18 \, \text{{g mol}}^{-1} \\ & \text{{molar mass of acetic acid}}=60 \, \text{{g mol}}^{-1} \\ & \text{{density of water }}=1 \, \text{{g cm}}^{-3} \end{aligned}$ Acetic acid dissociates as $CH_3COOH \rightleftharpoons CH_3COO^{-}+H^{+}$

Solution

$\begin{aligned} & \text { Mass of } \mathrm{CH}_3 \mathrm{COOH}=\mathrm{V} \times \mathrm{d} \\ & =5 \mathrm{ml} \times 1.2 \mathrm{~g} / \mathrm{ml} \\ & =6 \mathrm{gm} \\ & \mathrm{n}_{\mathrm{CH}_3 \mathrm{COOH}}=\frac{6}{60}=0.1 \mathrm{~mol} \\ & \mathrm{~m}_{\mathrm{CH}_3 \mathrm{COOH}} \approx \mathrm{M}_{\mathrm{CH}_3 \mathrm{COOH}}=\frac{0.1}{1}=0.1 \mathrm{M} \\ & \mathrm{CH}_3 \mathrm{COOH} \rightleftharpoons \mathrm{CH}_3 \mathrm{COO}^{-}+\mathrm{H}^{+}\end{aligned}$ C $\begin{aligned} & \mathrm{C}-\mathrm{C} \alpha \quad \mathrm{C} \alpha \quad \mathrm{C} \alpha \\ & \mathrm{K}_{\mathrm{a}}=\frac{\mathrm{C} \alpha^2}{1-\alpha} \\ & 1-\alpha \approx 1 \Rightarrow \mathrm{K}_{\mathrm{a}}=\mathrm{C} \alpha^2 \\ & \alpha=\sqrt{\frac{\mathrm{Ka}}{\mathrm{C}}}=\sqrt{\frac{6.25 \times 10^{-5}}{0.1}}=25 \times 10^{-3}\end{aligned}$ $\begin{aligned} & \text { V.f. }(\mathrm{i})=1+\alpha(\mathrm{n}-1)=1+\alpha(2-1)=1+\alpha \\ & =1+25 \times 10^{-3}=1.025 \\ & \Delta \mathrm{T}_{\mathrm{f}}=\mathrm{i} \mathrm{K}_{\mathrm{f}} \mathrm{m} \\ & =(1.025)(1.86)(0.1) \\ & =0.19 \\ & =19 \times 10^{-2}\end{aligned}$

Asked in: JEE Main 2024 (05 Apr Shift 2)

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