If 1m solution of benzoic acid in benzene has a freezing point depression of `3.84 ^(@)C` . `(K_(f)=5.12^(@)C mol^(-1)`kg) and boiling point elevation of `2.53^(@)C(K_(b)` =`2.53^(@)C "mol"^(-1) kg)`, then select the correct statement/s :lt brgtstatement I : there are dimar formation when under =going freezing
Statement II : there are no change when undergoing boiling
Statement III : reverse of I and II ltbr. Statement IV : dimer formation in freezing and boiling state

Options

A. I, II
B. II, III
C. III, I
D. only I

Answer

To solve the problem, we need to analyze the freezing point depression and boiling point elevation of a 1m solution of benzoic acid in benzene. We will use the formulas for freezing point depression and boiling point elevation to determine the van 't Hoff factor (I) and draw conclusions about dimer formation.

### Step-by-Step Solution:

1. **Understanding Freezing Point Depression**:
The formula for freezing point depression is given by:
\[ \Delta T_f = i \cdot m \cdot K_f \]
Where:
- \(\Delta T_f\) = freezing point depression
- \(i\) = van 't Hoff factor
- \(m\) = molality of the solution
- \(K_f\) = freezing point depression constant

2. **Substituting Values for Freezing Point Depression**:
Given:
- \(\Delta T_f = 3.84^\circ C\)
- \(m = 1 \, \text{mol/kg}\)
- \(K_f = 5.12 \, \text{C mol}^{-1} \text{kg}\)

Substituting the values into the formula:
\[ 3.84 = i \cdot 1 \cdot 5.12 \]

3. **Calculating the Van 't Hoff Factor (I)**:
Rearranging the equation to solve for \(i\):
\[ i = \frac{3.84}{5.12} = 0.75 \]

Since \(i < 1\), this indicates that dimer formation is occurring because the expected value of \(i\) for a non-dissociating solute is 1.

4. **Understanding Boiling Point Elevation**:
The formula for boiling point elevation is given by:
\[ \Delta T_b = i \cdot m \cdot K_b \]
Where:
- \(\Delta T_b\) = boiling point elevation
- \(K_b\) = boiling point elevation constant

5. **Substituting Values for Boiling Point Elevation**:
Given:
- \(\Delta T_b = 2.53^\circ C\)
- \(m = 1 \, \text{mol/kg}\)
- \(K_b = 2.53 \, \text{C mol}^{-1} \text{kg}\)

Substituting the values into the formula:
\[ 2.53 = i \cdot 1 \cdot 2.53 \]

6. **Calculating the Van 't Hoff Factor (I) for Boiling Point**:
Rearranging the equation to solve for \(i\):
\[ i = \frac{2.53}{2.53} = 1 \]

Since \(i = 1\), this indicates that there is no dimer formation during boiling.

7. **Conclusion**:
- **Statement I**: There is dimer formation when undergoing freezing (True).
- **Statement II**: There is no change when undergoing boiling (True).
- **Statement III**: Reverse of I and II (False).
- **Statement IV**: Dimer formation in freezing and boiling state (False).

### Final Answer:
The correct statements are:
- Statement I: True
- Statement II: True
- Statement III: False
- Statement IV: False