In a `S_(N)2` substitution reaction of the type
`R-Br+Cl^(-) overset("DMF")toR-Cl+Br^(+)`
which one of the following has the highest relative rate?

Options

A. `CH_(3)-underset(CH_(3))underset(|)(CH)-CH_(2)Br+Cl^(-)`
B. `CH_(3)-overset(CH_(3))overset(|)underset(CH_(3))underset(|)(C)-CH_(2)Br`
C. `CH_(3)CH_(2)Br`
D. `CH_(3)-CH_(2)-CH_(2)Br`

Explanation

To determine which compound has the highest relative rate in an \( S_N2 \) substitution reaction of the type \( R-Br + Cl^- \overset{DMF}{\rightarrow} R-Cl + Br^+ \), we need to consider the structure of the alkyl halide \( R-Br \) and how steric hindrance affects the reaction rate.

### Step-by-Step Solution:

1. **Understanding the \( S_N2 \) Mechanism**:
- In an \( S_N2 \) reaction, the nucleophile (in this case, \( Cl^- \)) attacks the carbon atom bonded to the leaving group (here, \( Br \)) from the opposite side. This results in a simultaneous bond formation and bond breaking.

2. **Identifying the Role of Steric Hindrance**:
- The rate of an \( S_N2 \) reaction is significantly influenced by steric hindrance. The more crowded the carbon atom is (i.e., the more bulky groups attached to it), the slower the reaction will be. This is because the nucleophile has a harder time approaching the carbon atom.

3. **Comparing Different Alkyl Halides**:
- Let's consider different alkyl halides:
- **Primary Alkyl Halide**: \( CH_3CH_2Br \) (ethyl bromide)
- **Secondary Alkyl Halide**: \( (CH_3)_2CHBr \) (isopropyl bromide)
- **Tertiary Alkyl Halide**: \( (CH_3)_3CBr \) (tert-butyl bromide)
- In these examples, the primary alkyl halide has the least steric hindrance, while the tertiary alkyl halide has the most.

4. **Determining the Relative Rates**:
- The primary alkyl halide will react the fastest in an \( S_N2 \) reaction due to minimal steric hindrance. The secondary will be slower, and the tertiary will be the slowest due to significant steric hindrance.

5. **Conclusion**:
- Therefore, among the options provided, the compound with the least steric hindrance (the primary alkyl halide) will have the highest relative rate in the \( S_N2 \) reaction.

### Final Answer:
The compound with the highest relative rate in the \( S_N2 \) reaction is the primary alkyl halide (e.g., \( CH_3CH_2Br \)).