22 g of benzoic acid (molar mass 122 g mol^-1) is dissolved in 50 g of benzene (Kf=5.12 K kg mol^-1). Benzoic acid associates only as a dimer…

Molality of benzoic acid $=(1.22/122)/0.050=0.20$ mol/kg. Without association the depression would be $5.12\times0.20=1.024$ K, so the van't Hoff factor is $i=0.64/1.024=0.625$. For dimerisation, $i=1-\alpha/2$, so $\alpha=0.75$. At equilibrium $m_M=0.20(1-0.75)=0.050$ and $m_D=0.20(0.75)/2=0.075$ mol/kg. Therefore $K=m_D/m_M^2=0.075/(0.050)^2=30$ kg/mol. The key step is recognising that each dimer removes two solute particles from the solution, so the effective particle count falls by $\alpha/2$ of the formal moles, not by $\alpha$. Using $\alpha=1$ would give $i=0.5$, which is the limiting value for complete dimerisation, and the observed 0.625 lies between that and 1, confirming partial association.