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JEE Advanced 2020 Paper 2 · Chemistry · Solutions, Solid State and Surface Chemistry · medium

Benzoic acid (molar mass 122 g mol^-1) associates into dimers in benzene. When 6.1 g of benzoic acid is dissolved in 100 g of benzene (taking Kf=5.12…

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Benzoic acid (molar mass 122 g mol−1122\ \mathrm{g\,mol^{-1}}) associates into dimers in benzene. When 6.1 g6.1\ \mathrm{g} of benzoic acid is dissolved in 100 g100\ \mathrm{g} of benzene (taking Kf=5.12 K kg mol−1K_f=5.12\ \mathrm{K\,kg\,mol^{-1}}), the observed depression in freezing point is 1.38 K1.38\ \mathrm{K}. The percentage of benzoic acid molecules that are present as dimers is ______. (Give the answer correct to two decimal places.)
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Answer: 92.19

Molality is m=6.1/1220.100 kg=0.500 mol kg−1m=\dfrac{6.1/122}{0.100\ \mathrm{kg}}=0.500\ \mathrm{mol\,kg^{-1}}. If the observed depression is ΔTf=i Kfm\Delta T_f=i\,K_fm, then i=1.385.12×0.500=0.5391i=\dfrac{1.38}{5.12\times0.500}=0.5391. For association 2A⇌A22\mathrm{A}\rightleftharpoons \mathrm{A_2} with a fraction α\alpha of the molecules dimerised, 11 mol of monomer gives (1−α)+α/2=1−α/2(1-\alpha)+\alpha/2=1-\alpha/2 mol of particles, so i=1−α/2i=1-\alpha/2. Therefore α=2(1−i)=2(1−0.5391)=0.9219\alpha=2(1-i)=2(1-0.5391)=0.9219. The fraction of molecules present as dimers is thus 92.19%92.19\%. The value of ii is below one because the number of solute particles falls on association, which is the opposite of the dissociation case where i>1i>1. Had the acid not associated, ii would be 11 and the depression would be 2.56 K2.56\ \mathrm{K}; the observed value, a little over half of that, shows that the molecules are almost completely present as hydrogen-bonded dimers, which is the reason the apparent molar mass in benzene is nearly double 122122.
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