In the following pairs of halogen compounds,which compound undergoes faster $S_N1$ reaction?
$(i)$ $2$-chloro-$2$-methylpropane and $3$-chloropentane
(ii) $2$-chloroheptane and $1$-chlorohexane

(N/A) $(i)$ $S_N1$ reaction proceeds via the formation of a carbocation. The alkyl halide $(I)$ is $3^{\circ}$ while $(II)$ is $2^{\circ}$. Therefore,$(I)$ forms a $3^{\circ}$ carbocation while $(II)$ forms a $2^{\circ}$ carbocation. The greater the stability of the carbocation,the faster is the rate of $S_N1$ reaction. Since a $3^{\circ}$ carbocation is more stable than a $2^{\circ}$ carbocation,$(I)$,i.e.,$2$-chloro-$2$-methylpropane,undergoes a faster $S_N1$ reaction than $(II)$,i.e.,$3$-chloropentane.
$(ii)$ The alkyl halide $(I)$ is $2^{\circ}$ while $(II)$ is $1^{\circ}$. $A$ $2^{\circ}$ carbocation is more stable than a $1^{\circ}$ carbocation. Therefore,$(I)$,$2$-chloroheptane,undergoes a faster $S_N1$ reaction than $(II)$,$1$-chlorohexane.