A complex is represented as `CoCl_(3) . XNH_(3)`. Its `0.1` molal solution in aqueous solution shows `Delta T_(f) = 0.558^(circ). (K_(f)` for `H_(2)O` is `1.86 K "molality"^(-1))` Assuming `100%` ionisation of complex and co-ordination number of `Co` as six, calculate formula of complex.

Answer

To solve the problem step by step, we will use the information given in the question and apply the relevant formulas for colligative properties.

### Step 1: Understand the given data
We are given:
- The complex is represented as \( \text{CoCl}_3 \cdot X\text{NH}_3 \).
- The molality of the solution is \( 0.1 \, \text{mol/kg} \).
- The depression in freezing point \( \Delta T_f = 0.558^\circ C \).
- The freezing point depression constant \( K_f \) for water is \( 1.86 \, \text{K} \cdot \text{molality}^{-1} \).
- The complex undergoes 100% ionization.
- The coordination number of Co is 6.

### Step 2: Use the formula for depression in freezing point
The formula for depression in freezing point is given by:
\[ \Delta T_f = K_f \cdot m \cdot i \]
Where:
- \( \Delta T_f \) is the depression in freezing point.
- \( K_f \) is the freezing point depression constant.
- \( m \) is the molality of the solution.
- \( i \) is the van 't Hoff factor (number of particles the solute breaks into).

### Step 3: Rearrange the formula to find \( i \)
We can rearrange the formula to find \( i \):
\[ i = \frac{\Delta T_f}{K_f \cdot m} \]

### Step 4: Substitute the values into the equation
Substituting the known values:
\[ i = \frac{0.558}{1.86 \cdot 0.1} \]
Calculating this gives:
\[ i = \frac{0.558}{0.186} \approx 3 \]

### Step 5: Interpret the value of \( i \)
The value \( i = 3 \) indicates that the complex dissociates into three particles in solution.

### Step 6: Identify the components of the complex
The complex \( \text{CoCl}_3 \cdot X\text{NH}_3 \) will dissociate as follows:
- \( \text{Co}^{3+} \) (1 particle)
- \( 3 \text{Cl}^- \) (3 particles)

If the complex is \( \text{Co(NH}_3)_x\text{Cl}_3 \), then it will dissociate into:
- \( \text{Co(NH}_3)_x^{3+} \) (1 particle)
- \( 3 \text{Cl}^- \) (3 particles)

### Step 7: Determine the coordination number
Since the coordination number of Co is 6, we have:
- 3 Cl ions contribute 3 to the coordination number.
- Therefore, \( x \) (the number of NH3 ligands) must be 3 to make a total of 6.

### Step 8: Write the final formula of the complex
Thus, the formula of the complex is:
\[ \text{Co(NH}_3)_3\text{Cl}_3 \]

### Summary of the Solution
The formula of the complex is \( \text{Co(NH}_3)_3\text{Cl}_3 \).

---