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

Answer

To determine how many isomers of \(C_5H_{11}Br\) show stereochemical inversion when undergoing an \(S_N2\) reaction, we need to follow these steps:

### Step 1: Identify the molecular formula and possible structures
The molecular formula \(C_5H_{11}Br\) indicates that we are dealing with bromopentane. We need to identify all possible structural isomers of this compound.

### Step 2: Draw the structural isomers
1. **Straight-chain isomers**:
- **1-bromopentane**: \(CH_3(CH_2)_3CH_2Br\)
- **2-bromopentane**: \(CH_3CH(CH_2)CH_2Br\)
- **3-bromopentane**: \(CH_3CH_2CH(CH_2)Br\)

2. **Branched-chain isomers**:
- **2-bromobutane**: \(CH_3CH(CH_3)CH_2Br\)
- **3-bromobutane**: \(CH_3CH_2CH(CH_3)Br\)
- **1-bromo-2-methylbutane**: \(CH_3CH(CH_3)CH_2CH_2Br\)
- **2-bromo-2-methylbutane**: \(C(CH_3)_3Br\)

### Step 3: Identify chiral centers
For stereochemical inversion to occur during an \(S_N2\) reaction, the carbon atom bonded to the bromine must be a chiral center. A chiral center is a carbon atom that is attached to four different substituents.

- **1-bromopentane**: No chiral center.
- **2-bromopentane**: Chiral center (attached to \(CH_3\), \(CH_2Br\), \(H\), and \(CH_2CH_3\)).
- **3-bromopentane**: No chiral center.
- **2-bromobutane**: Chiral center (attached to \(CH_3\), \(CH_3\), \(H\), and \(CH_2Br\)).
- **3-bromobutane**: Chiral center (attached to \(CH_3\), \(CH_2Br\), \(H\), and \(CH_2CH_3\)).
- **1-bromo-2-methylbutane**: Chiral center (attached to \(CH_3\), \(CH_2Br\), \(H\), and \(CH(CH_3)\)).
- **2-bromo-2-methylbutane**: No chiral center.

### Step 4: Count the isomers with chiral centers
From our analysis, the following isomers have chiral centers:
- **2-bromopentane**
- **2-bromobutane**
- **3-bromobutane**
- **1-bromo-2-methylbutane**

### Conclusion
The total number of isomers of \(C_5H_{11}Br\) that show stereochemical inversion during an \(S_N2\) reaction is **four**.