If the freezing point of a `0.01` molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is `-0.0558^(@)C`, the number of chloride (s) in the coordination sphere of the complex if `[K_(f)` of water `=1.86 K kg mol^(-1)]`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
- The molality (m) of the solution is `0.01 mol/kg`.
- The freezing point depression (ΔTf) is `-0.0558°C`.
- The freezing point constant (Kf) of water is `1.86 K kg mol^(-1)`.

### Step 2: Calculate the freezing point depression
The formula for freezing point depression is given by:

\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- \(\Delta T_f\) = freezing point depression
- \(i\) = van 't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = molal freezing point depression constant
- \(m\) = molality of the solution

### Step 3: Rearranging the formula to find \(i\)
We can rearrange the formula to solve for \(i\):

\[ i = \frac{\Delta T_f}{K_f \cdot m} \]

### Step 4: Substitute the values into the equation
Substituting the known values into the equation:

\[ i = \frac{-0.0558}{1.86 \cdot 0.01} \]

### Step 5: Calculate \(i\)
Now, we perform the calculation:

\[ i = \frac{-0.0558}{0.0186} \approx -3 \]

Since we are interested in the absolute value of \(i\):

\[ i \approx 3 \]

### Step 6: Determine the number of chloride ions
The van 't Hoff factor \(i\) indicates the number of particles the solute dissociates into. For a cobalt(III) chloride-ammonia complex, we can assume the complex has the following form:

\[ \text{[Co(NH}_3\text{)}_x\text{Cl}_y]^{3+} \]

Where \(x\) is the number of ammonia ligands and \(y\) is the number of chloride ions.

Given that \(i = 3\), we can deduce that the complex dissociates into 3 ions. Since cobalt(III) has a +3 charge, and ammonia is neutral, the total charge contributed by the chloride ions must balance the +3 charge of the cobalt ion.

### Step 7: Conclusion
If we assume that there are \(y\) chloride ions, the dissociation can be represented as:

\[ \text{[Co(NH}_3\text{)}_x\text{Cl}_y]^{3+} \rightarrow \text{Co}^{3+} + y \text{Cl}^- \]

To satisfy the charge balance, we can have \(y = 3\) (since the cobalt ion is +3 and the chlorides are -1 each). Therefore, the number of chloride ions in the coordination sphere of the complex is:

\[ \text{Number of Cl}^- = 3 \]

### Final Answer
The number of chloride ions in the coordination sphere of the complex is **3**.

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