Option 1 (Benzoic acid dimer) — observed molar mass route:
Step 1: Association changes the effective particle count, so first find the experimental (observed) molar mass from the depression data using \(M_{obs} = \dfrac{K_f \times w_2 \times 1000}{\Delta T_f \times w_1}\), where \(w_2\) is solute mass and \(w_1\) is solvent mass in grams.
Step 2: \(M_{obs} = \dfrac{4.9 \times 2 \times 1000}{1.62 \times 25} = \dfrac{9800}{40.5} = 241.98\) g mol\(^{-1}\).
Step 3: The van't Hoff factor is the ratio of the true molar mass to the observed molar mass: \(i = \dfrac{M_{normal}}{M_{obs}} = \dfrac{122}{241.98} = 0.504\).
Step 4: Because two molecules combine into one dimer, \(i = 1 - \dfrac{\alpha}{2}\), giving \(\alpha = 2(1 - 0.504) = 0.992\).
Step 5: \[\boxed{\text{Percentage association} = 99.2\%}\] The observed molar mass being nearly double 122 confirms almost complete dimerisation.
Option 2 (Protein) — moles-of-particles route:
Step 1: First find the number of moles of protein directly from \(n = \dfrac{\pi V}{RT}\).
Step 2: \(n = \dfrac{(2.57\times10^{-3}\text{ bar})(0.200\text{ L})}{(0.083\text{ L bar K}^{-1}\text{mol}^{-1})(300\text{ K})} = \dfrac{5.14\times10^{-4}}{24.9} = 2.064\times10^{-5}\) mol.
Step 3: Molar mass is total mass divided by moles: \(M = \dfrac{w}{n} = \dfrac{1.26}{2.064\times10^{-5}}\).
Step 4: \[\boxed{M \approx 61039\text{ g mol}^{-1}}\] a large value typical of macromolecules such as proteins.