The possible numerof structural and stereo isomers for the complex `[MBr_(2)Cl_(2)]SO_(2)` is

Answer

To determine the possible number of structural and stereo isomers for the complex \([MBr_2Cl_2]SO_4\), we can follow these steps:

### Step 1: Identify the Type of Complex
The complex is of the form \([MA_2B_2]SO_4\), where \(M\) is the metal, \(A\) represents the bromine (Br) ligands, and \(B\) represents the chlorine (Cl) ligands. The sulfate ion \((SO_4^{2-})\) acts as a counter ion.

### Step 2: Determine Structural Isomers
Structural isomers can arise from different arrangements of the ligands. Here, we can have:

1. **Ionization Isomers**: These occur when the counter ion (sulfate) can be exchanged with one of the ligands.
- One possible ionization isomer is \([MBr_2SO_4]\) with both Cl ions as counter ions.
- Another possible ionization isomer is \([MCl_2SO_4]\) with both Br ions as counter ions.
- A third possibility is \([MBrClSO_4]\) where one Br and one Cl are exchanged with the sulfate.

Thus, we have **3 structural isomers** from ionization.

### Step 3: Determine Geometrical Isomers
For the complex \([MBr_2Cl_2]^{2-}\), we can also have geometrical isomers due to the arrangement of the ligands around the metal center.

1. **Cis Isomer**: Both Br ligands are adjacent to each other, and both Cl ligands are adjacent to each other.
2. **Trans Isomer**: The Br ligands are opposite each other, and the Cl ligands are opposite each other.

Thus, we have **2 geometrical isomers** (cis and trans).

### Step 4: Total Isomers
Now, we can combine the structural and geometrical isomers:
- Structural isomers: 3
- Geometrical isomers: 2

Therefore, the total number of isomers is:
\[ \text{Total Isomers} = \text{Structural Isomers} + \text{Geometrical Isomers} = 3 + 2 = 5 \]

### Final Answer
The possible number of structural and stereo isomers for the complex \([MBr_2Cl_2]SO_4\) is **5**.

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