Step 1: Understanding the \( S_N1 \) Mechanism.
The \( S_N1 \) reaction follows a two-step process:
1. Formation of a carbocation (the rate-determining step).
2. Nucleophilic attack on the carbocation.
The rate of the \( S_N1 \) reaction is influenced by the stability of the carbocation. Tertiary carbocations are more stable due to greater alkyl group substitution, which provides enhanced electron donation through hyperconjugation and inductive effects. This increased stability facilitates faster carbocation formation, thereby accelerating the reaction rate.
When comparing two halogen compounds, the one with the tertiary halogenated carbon (more substituted carbon) will form a more stable carbocation and therefore undergo the \( S_N1 \) reaction at a faster rate.