0.002 molal aqueous solution of an ionic compound with molecular formula `Co(NH_(3))_(5)(NO_(2))Cl` freezes at `-0.00744^(@)C`. How many moles of ions does 3 moles of the salt produce on being dissolved in water? [Given `K_(f)` of water=1.86 K kg `mol^(-1)`]

Answer

To solve the problem, we need to determine how many moles of ions are produced when 3 moles of the ionic compound \( \text{Co(NH}_3\text{)}_5\text{(NO}_2\text{)Cl} \) are dissolved in water. We will use the freezing point depression formula and the van 't Hoff factor \( i \).

### Step-by-Step Solution:

1. **Identify the Freezing Point Depression Formula**:
The freezing point depression (\( \Delta T_f \)) is given by the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \( \Delta T_f \) = depression in freezing point
- \( i \) = van 't Hoff factor (number of particles the solute breaks into)
- \( K_f \) = cryoscopic constant of the solvent (water in this case)
- \( m \) = molality of the solution

2. **Calculate \( \Delta T_f \)**:
The freezing point of pure water is 0°C. The solution freezes at -0.00744°C, so:
\[ \Delta T_f = 0 - (-0.00744) = 0.00744 \, \text{°C} \]

3. **Substitute Known Values into the Freezing Point Depression Formula**:
We know:
- \( K_f = 1.86 \, \text{K kg mol}^{-1} \)
- \( m = 0.002 \, \text{mol/kg} \)
Now substituting these values into the formula:
\[ 0.00744 = i \cdot 1.86 \cdot 0.002 \]

4. **Solve for \( i \)**:
Rearranging the equation to solve for \( i \):
\[ i = \frac{0.00744}{1.86 \cdot 0.002} \]
\[ i = \frac{0.00744}{0.00372} \approx 2 \]

5. **Interpret the van 't Hoff Factor \( i \)**:
The van 't Hoff factor \( i = 2 \) indicates that the ionic compound dissociates into 2 particles in solution.

6. **Determine the Number of Ions Produced by 3 Moles of the Salt**:
Since \( i = 2 \), each mole of the compound produces 2 moles of ions. Therefore, for 3 moles of the compound:
\[ \text{Total moles of ions} = 3 \, \text{moles} \times 2 = 6 \, \text{moles of ions} \]

### Final Answer:
The total number of moles of ions produced when 3 moles of the salt are dissolved in water is **6 moles of ions**.