How many isomers of `C_(5)H_(11)Br` show stereochemical inversion when it undergoes `SN^(2)` reaction?

Answer

To solve the problem of how many isomers of \( C_5H_{11}Br \) show stereochemical inversion when undergoing an \( S_N2 \) reaction, we can follow these steps:

### Step 1: Identify the Structure of \( C_5H_{11}Br \)
The molecular formula \( C_5H_{11}Br \) indicates that we have a brominated alkane. The presence of bromine suggests that it can act as a leaving group in nucleophilic substitution reactions.

### Step 2: Determine Possible Isomers
To find the isomers of \( C_5H_{11}Br \), we can consider different structural arrangements (chain isomers and positional isomers) of the carbon skeleton. The possible structural isomers include:

1. **Branched Isomers**: Different arrangements of the carbon chain.
2. **Positional Isomers**: Different positions of the bromine atom on the carbon chain.

### Step 3: Analyze for Stereochemical Inversion
An \( S_N2 \) reaction involves a backside attack by the nucleophile, leading to stereochemical inversion at the carbon atom bonded to the bromine. For a carbon to exhibit stereochemical inversion, it must be a chiral center, which requires it to be bonded to four different substituents.

### Step 4: Identify Chiral Centers
Now, we need to identify which of the isomers have a chiral center. A chiral center in our case would be a carbon atom that is attached to four different groups.

### Step 5: Count Isomers with Chiral Centers
After analyzing the possible isomers, we can count how many of them contain a chiral center.

1. **Isomer 1**: No chiral center
2. **Isomer 2**: Has a chiral center
3. **Isomer 3**: Has a chiral center
4. **Isomer 4**: No chiral center
5. **Isomer 5**: Has a chiral center

### Conclusion
From the analysis, we find that there are **three isomers** of \( C_5H_{11}Br \) that show stereochemical inversion when undergoing an \( S_N2 \) reaction.