A 0.001 molal solution of `[Pt(NH_(3))_(4)CI_(4)]` in water had a freezing point depression of `0.0054^(@)C`. If `K_(f)` for water is `1.80`, the correct formulation for the above molecule is

Options

A. `[Pt(NH_3)_4Cl_3]Cl`
B. `[Pt(NH_3)_4Cl_4]`
C. `[Pt(NH_3)_4Cl_2]Cl_2`
D. `[Pt(NH_3)_4Cl]Cl_3`

Explanation

To solve the problem, we need to find the correct formulation of the complex `[Pt(NH₃)₄Cl₄]` based on the given data about the freezing point depression.

### Step-by-Step Solution:

1. **Understand the Given Data**:
- Molality (m) = 0.001 molal
- Freezing point depression (ΔTf) = 0.0054°C
- Cryoscopic constant (Kf) for water = 1.80°C kg/mol

2. **Use the Freezing Point Depression Formula**:
The formula for freezing point depression is given by:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- ΔTf = freezing point depression
- i = Van't Hoff factor (number of particles the solute breaks into)
- Kf = cryoscopic constant
- m = molality of the solution

3. **Substitute the Known Values**:
Plugging in the values we have:
\[ 0.0054 = i \cdot 1.80 \cdot 0.001 \]

4. **Solve for the Van't Hoff Factor (i)**:
Rearranging the equation to solve for i:
\[ i = \frac{0.0054}{1.80 \cdot 0.001} \]
\[ i = \frac{0.0054}{0.0018} = 3 \]

5. **Interpret the Van't Hoff Factor**:
The Van't Hoff factor (i = 3) indicates that the complex ionizes into 3 particles in solution.

6. **Determine the Structure of the Complex**:
Since the complex ionizes into 3 particles, we can deduce the structure:
- The complex must consist of a central metal ion (Pt) surrounded by ligands.
- Given that it has a total of 4 ammonia (NH₃) ligands and 4 chloride (Cl) ligands, we can infer that 2 chloride ions must be outside the coordination sphere to account for the 3 particles (1 complex ion + 2 Cl⁻ ions).

7. **Construct the Correct Formula**:
Therefore, the correct formulation of the complex is:
\[ [Pt(NH₃)₄Cl₂]Cl_2 \]
This indicates that there are 4 ammonia ligands coordinated to platinum and 2 chloride ions outside the coordination sphere.

### Final Answer:
The correct formulation for the complex is:
\[ [Pt(NH₃)₄Cl₂]Cl_2 \]