Related Formula
Rate of SN1 ∝ Stability of intermediate carbocation$$\text{Rate of } S_N1 \propto \text{Stability of intermediate carbocation}$$
Core Logic
Evaluate the stability of the carbocations formed upon dissociation of the bromide ion:
Structure (I) and (II) form carbocations at bridgehead positions of rigid bicyclic systems. According to Bredt's rule, these are highly unstable because they cannot adopt the necessary planar sp²$sp^2$ geometry. Among them, (I) has an additional methyl group which provides slight +I stabilization compared to (II). So, (II) is even less stable than (I).
Structure (III) forms a tertiary carbocation, but it's not a rigid bridgehead preventing planarity, so it's significantly more stable than I and II.
Structure (IV) is a highly stable trityl-like carbocation where the positive charge is stabilized by extensive resonance with the adjacent phenyl ring(s).
Step 1: Final Ordering
Stability order: (II) < (I) < (III) < (IV)$(II) < (I) < (III) < (IV)$. Thus, the SN1$S_{N}1$ rate follows the same order.
Pattern Recognition
Bridgehead halides effectively do not undergo SN1$S_N1$ (or SN2$S_N2$) reactions due to Bredt's rule. Resonance stabilized carbocations always dominate.
Chapter Mix
Class 12 Chemistry: Haloalkanes and Haloarenes