To determine which statement is false, we will analyze each option provided in the question step by step.
### Step 1: Analyze Option 1
**Statement:** Two different solutions of sucrose of the same molality prepared in different solvents will have the same depression in freezing point.
**Analysis:**
- The depression in freezing point (\( \Delta T_f \)) is given by the formula:
\[
\Delta T_f = K_f \cdot m \cdot i
\]
where \( K_f \) is the cryoscopic constant of the solvent, \( m \) is the molality, and \( i \) is the van't Hoff factor.
- For sucrose, the van't Hoff factor \( i = 1 \) because it does not dissociate in solution.
- While the molality \( m \) is the same for both solutions, the cryoscopic constant \( K_f \) will differ for different solvents.
- Therefore, the depression in freezing point will not be the same for both solutions since \( K_f \) varies with the solvent.
**Conclusion:** This statement is FALSE.
### Step 2: Analyze Option 2
**Statement:** The osmotic pressure of a solution is given by \( \pi = CRT \).
**Analysis:**
- The formula for osmotic pressure is indeed given by:
\[
\pi = iCRT
\]
where \( i \) is the van't Hoff factor. However, the statement simplifies to \( \pi = CRT \) which is technically correct if \( i = 1 \) (non-electrolyte solutions).
- Since this statement holds true for non-electrolytes, we consider it correct.
**Conclusion:** This statement is TRUE.
### Step 3: Analyze Option 3
**Statement:** Decreasing order of osmotic pressure for 0.01 M aqueous solutions of barium chloride, potassium chloride, acetic acid, and sucrose.
**Analysis:**
- The osmotic pressure is calculated as:
\[
\pi = iCRT
\]
- For barium chloride (BaCl2), \( i = 3 \) (dissociates into Ba²⁺ and 2 Cl⁻).
- For potassium chloride (KCl), \( i = 2 \) (dissociates into K⁺ and Cl⁻).
- For acetic acid (CH3COOH), \( i \) is less than 2 due to partial dissociation (typically around 1.2).
- For sucrose, \( i = 1 \) (non-electrolyte).
- Thus, the order of osmotic pressure from highest to lowest is: BaCl2 > KCl > CH3COOH > Sucrose.
**Conclusion:** This statement is TRUE.
### Step 4: Analyze Option 4
**Statement:** According to Raoult's law, the vapor pressure exerted by a volatile component of a solution is directly proportional to its mole fraction in the solution.
**Analysis:**
- Raoult's law states that the vapor pressure of a component in a solution is directly proportional to its mole fraction:
\[
P_i = X_i \cdot P_i^0
\]
where \( P_i \) is the vapor pressure of the component in the solution, \( X_i \) is the mole fraction, and \( P_i^0 \) is the vapor pressure of the pure component.
- This statement is a correct representation of Raoult's law.
**Conclusion:** This statement is TRUE.
### Final Conclusion
After analyzing all the options, we conclude that the FALSE statement is:
**Option 1:** Two different solutions of sucrose of the same molality prepared in different solvents will have the same depression in freezing point.
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