In $S_N2$ substitution reaction of the type $\text{R-Br} + \text{Cl}^- \xrightarrow{\text{DMF}} \text{R-Cl} + \text{Br}^-$, which one of the following has the highest relative rate?

The given reaction proceeds via an $S_N2$ mechanism. The rate of an $S_N2$ reaction is highly dependent on the steric hindrance (spatial arrangement) around the electrophilic carbon atom. As the steric bulk around the $\alpha$-carbon increases, the approach of the incoming nucleophile ($\text{Cl}^-$) is hindered, leading to a lower reaction rate. Among the given primary alkyl bromides, bromoethane ($\text{CH}_3\text{CH}_2\text{Br}$) has the least steric hindrance. Branching at the $\beta$-carbon (as seen in the other options: propyl, isobutyl, and neopentyl bromides) increases steric resistance, making the approach of the nucleophile difficult. Thus, $\text{CH}_3\text{CH}_2\text{Br}$ has the highest relative rate.