Check option A: Raoult's law
Raoult’s law for an ideal solution states that the partial vapour pressure of a component is $p_i = x_i p_i^0$ where $x_i$ is the mole fraction of the component in the solution.
Hence, vapour pressure is proportional to mole fraction.
Therefore, A is true.

Check option B: Freezing point depression
Freezing point depression is given by $\Delta T_f = iK_f m$ For sucrose, $i=1$ since it is a non-electrolyte. So, $\Delta T_f = K_f m$ Even if the molality $m$ is same, the cryoscopic constant $K_f$ depends on the solvent. Different solvents generally have different $K_f$ values. Therefore, the freezing point depression will not be the same in different solvents.
Hence, B is false.

Check option C: Order of osmotic pressure
Osmotic pressure is $\pi = iMRT$ For equal molarity at the same temperature, osmotic pressure depends on van’t Hoff factor $i$.

• $BaCl_2 \rightarrow Ba^{2+} + 2Cl^-$, so $i \approx 3$
• $KCl \rightarrow K^+ + Cl^-$, so $i \approx 2$
• $CH_3COOH$ is a weak electrolyte, so $1 < i < 2$
• sucrose is non-electrolyte, so $i=1$

Therefore, $BaCl_2 > KCl > CH_3COOH > \text{sucrose}$ So, C is true.

Check option D: Formula for osmotic pressure
The osmotic pressure relation is $\pi = CRT$ If concentration is expressed in molarity $M$, then $\pi = MRT$ for non-electrolytes, and more generally for electrolytes, $\pi = iMRT$ Since the given statement matches the standard expression for dilute solutions, D is true.

Conclusion
The false statement is: $\boxed{B}$

Comparison with stored answer
Stored correct answer: B
Derived answer: B
Therefore, they agree.