An isomerr of the complex `CoBrCl_(2)(en)_(2)(H_(2)O)`, on reaction with concentrated `H_(2)SO_(4)` (dehydrating agent), suffers no loss in weight and on reaction with `AgNO_(3)` solution it gives only white precipitate, which is soluble in `NH_(3)` solution.
Q. The correct formula of the complex is:

Options

A. `[CoBrH_(2)O(en)_(2)]Cl_(2)`
B. `[CoCl(en)_(2)H_(2)O]BrCl`
C. `[CoBrCl(en)_(2)]Cl*H_(2)O`
D. `[CoCl_(2)(en)_(2)]Br*H_(2)O`

Explanation

To solve the problem, we need to analyze the information given about the complex \( \text{CoBrCl}_2(\text{en})_2(\text{H}_2\text{O}) \) and its reactions with concentrated \( \text{H}_2\text{SO}_4 \) and \( \text{AgNO}_3 \).

### Step-by-Step Solution:

1. **Identify the Coordination Complex**:
The complex given is \( \text{CoBrCl}_2(\text{en})_2(\text{H}_2\text{O}) \). Here, \( \text{Co} \) is cobalt, \( \text{Br} \) is bromine, \( \text{Cl} \) is chlorine, \( \text{en} \) is ethylenediamine (a bidentate ligand), and \( \text{H}_2\text{O} \) is water.

2. **Reaction with Concentrated \( \text{H}_2\text{SO}_4 \)**:
The complex suffers no loss in weight when reacted with concentrated \( \text{H}_2\text{SO}_4 \). This indicates that all water molecules in the complex are coordinated (i.e., part of the coordination sphere) and not free. Therefore, the water molecule must be inside the coordination sphere.

3. **Reaction with \( \text{AgNO}_3 \)**:
When the complex reacts with \( \text{AgNO}_3 \), it produces only a white precipitate that is soluble in \( \text{NH}_3 \). The white precipitate is likely \( \text{AgCl} \), which indicates that the chloride ions are outside the coordination sphere. If bromide ions were outside, we would expect a yellow precipitate of \( \text{AgBr} \), which is not observed.

4. **Determine the Position of Ligands**:
Since \( \text{Cl}^- \) ions are outside the coordination sphere (as evidenced by the formation of \( \text{AgCl} \)), and since the complex contains two \( \text{en} \) ligands and one \( \text{H}_2\text{O} \) molecule, we can conclude that the bromine must be part of the coordination sphere along with the cobalt and the ethylenediamine ligands.

5. **Construct the Correct Formula**:
From the above deductions, we can construct the correct formula for the complex. The coordination sphere consists of \( \text{Co} \), \( \text{Br} \), \( \text{en} \) ligands, and \( \text{H}_2\text{O} \), while the \( \text{Cl} \) ions are outside. Therefore, the correct formula of the complex is:
\[ \text{[Co(en)_2Br(H}_2\text{O)]Cl}_2 \]

### Final Answer:
The correct formula of the complex is \( \text{[Co(en)_2Br(H}_2\text{O)]Cl}_2 \).