A `0.002 M` aqueous solution of an ionic compound `[Co(NH_(3))_(5)(NO_(2))] Cl` freezes at `-0.00732^(@)C`. Find the number of moles of ions which `1` mole of ionic compound produces of being dissolved in water. `(K_(f) = -1.86^(@)C//m)`.

Options

A. 2
B. 3
C. 4
D. 1

Explanation

To solve the problem, we will use the freezing point depression formula and the concept of van't Hoff factor (i). Here are the steps to find the number of moles of ions produced by one mole of the ionic compound when dissolved in water.

### Step 1: Understand the given data
- The molarity of the solution (C) = 0.002 M
- The freezing point of the solution (Tf) = -0.00732 °C
- The freezing point depression constant (Kf) = -1.86 °C/m

### Step 2: Calculate the change in freezing point (ΔTf)
The change in freezing point (ΔTf) can be calculated as:
\[ \Delta Tf = 0 - (-0.00732) = 0.00732 \, °C \]

### Step 3: Use the freezing point depression formula
The freezing point depression formula is given by:
\[ \Delta Tf = Kf \times m \times i \]
Where:
- ΔTf = change in freezing point
- Kf = freezing point depression constant
- m = molality of the solution
- i = van't Hoff factor (number of particles produced per formula unit)

### Step 4: Assume molarity is approximately equal to molality
For very dilute solutions, we can assume that molarity (C) is approximately equal to molality (m). Therefore:
\[ m \approx 0.002 \, m \]

### Step 5: Rearrange the formula to solve for i
Rearranging the formula to find the van't Hoff factor (i):
\[ i = \frac{\Delta Tf}{Kf \times m} \]

### Step 6: Substitute the values into the equation
Substituting the known values into the equation:
\[ i = \frac{0.00732}{1.86 \times 0.002} \]

### Step 7: Calculate the value of i
Calculating the denominator:
\[ 1.86 \times 0.002 = 0.00372 \]
Now substituting back into the equation for i:
\[ i = \frac{0.00732}{0.00372} \approx 1.968 \]

### Step 8: Round the value of i
Since the van't Hoff factor (i) must be a whole number, we round 1.968 to approximately 2.

### Conclusion
This means that one mole of the ionic compound \([Co(NH_3)_5(NO_2)]Cl\) produces approximately 2 moles of ions when dissolved in water.

### Final Answer
The number of moles of ions produced by 1 mole of the ionic compound is **2**.

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