To solve this problem, we can use the equation for the freezing point depression:
ΔTf = Kf * i * molality
where ΔTf is the change in freezing point, Kf is the freezing point depression constant (given as 1.86°C/m), i is the van't Hoff factor, and molality is the number of moles of solute per kilogram of solvent.
We are given that the freezing point depression is 0.53°C and the molality is:
molality = moles of solute / mass of solvent (in kg)
We don't have the mass of solvent, but we can assume that it is 1 kg (1000 g) since the molality is in moles per kilogram of solvent. This means that the mass of solute (K2SO4) is 0.200 * 1000 = 200 g.
The molar mass of K2SO4 is 174.26 g/mol, so the number of moles of solute is:
moles of solute = 200 g / 174.26 g/mol = 1.147 mol
Substituting these values into the freezing point depression equation, we get:
0.53 = 1.86 * i * (1.147/1)
Solving for i, we get:
i = 0.53 / (1.86 * 1.147) = 0.247
Therefore, the van't Hoff factor for this solution is approximately 0.247