Step-by-step Solution:

1. Identify the given data and the required quantity.

   • Mass of solute (naphthoic acid, C₁₁H₈O₂), w₂ = 20 g.
   • Mass of solvent (benzene), w₁ = 50 g = 0.050 kg.
   • Freezing point depression, $ΔT_f = 2$ K.
   • Molal freezing point depression constant for benzene, $K_f = 1.72$ K kg mol⁻¹.
   • We need to find the van't Hoff factor, i.

2. Calculate the molar mass of the solute (naphthoic acid, C₁₁H₈O₂).

   • The chemical formula is C₁₁H₈O₂.
   • Molar mass M₂ = (11 × ext{Atomic mass of C}) + (8 × ext{Atomic mass of H}) + (2 × ext{Atomic mass of O})
   • M₂ = (11 × 12.01) + (8 × 1.008) + (2 × 16.00)
   • M₂ = 132.11 + 8.064 + 32.00 = 172.174 g/mol. For simplicity, we can use integer masses: M₂ = 11×12 + 8×1 + 2×16 = 132 + 8 + 32 = 172 g/mol.

3. Calculate the theoretical molality (m) of the solution.

   • Molality is defined as the number of moles of solute per kilogram of solvent.
   • m = (moles of solute) / (mass of solvent in kg)
   • Moles of solute n₂ = w₂ / M₂ = 20 g / 172 g/mol ≈ 0.1163 mol.
   • Mass of solvent w₁ = 50 g = 0.050 kg.
   • m = 0.1163 mol / 0.050 kg ≈ 2.326 mol/kg.

4. Use the formula for freezing point depression including the van't Hoff factor.

   • The formula for the depression in freezing point is: $ΔT_f = i × K_f × m$
   • We need to solve for i: $i = \frac{ΔT_f}{K_f × m}$

5. Substitute the known values into the formula to find 'i'.

   • $ΔT_f = 2$ K
   • $K_f = 1.72$ K kg mol⁻¹
   • m = 2.326 mol/kg
   • $i = \frac{2}{1.72 × 2.326}$
   • $i = \frac{2}{4.00072}$
   • $i ≈ 0.4999 ≈ 0.5$

6. Interpret the result and select the correct option.

   • The calculated van't Hoff factor i is approximately 0.5.
   • A van't Hoff factor less than 1 indicates that the solute particles are associating in the solvent. In this case, since i = 0.5, it suggests that naphthoic acid (a carboxylic acid) forms dimers in the non-polar solvent benzene, likely through hydrogen bonding. For complete dimerization, i would be exactly 0.5.
   • Comparing our result with the options:
     • A: 0.5
     • B: 1
     • C: 2
     • D: 3
   • The correct option is A.