Step 1: Understanding the \(E_1\) Mechanism
The \(E_1\) (unimolecular elimination) reaction proceeds via a carbocation intermediate.
Carbocation stability directly correlates with reactivity; higher stability leads to a faster reaction rate.
Step 2: Analyzing the Given Compounds
The compounds feature halides (\(\text{Cl}\)) bonded to various carbon atoms.
The reactivity sequence is determined by the stability of the carbocation formed upon halide dissociation.
Step 3: Determining the Order of Reactivity
(A) Primary Carbocation: Exhibits lowest stability, resulting in the slowest reaction.
(B) Carbocation Undergoing Rearrangement: Rearrangement yields a more stable secondary or tertiary carbocation, thus increasing reactivity.
(C) Five Hyperconjugating Structures: Contributes moderate carbocation stability.
(D) Resonance-Stabilized Carbocation: Achieves maximum stability through electron delocalization, thereby enhancing reactivity.
Step 4: Final Order of Reactivity
The order of carbocation stability is:\[D>B>C>A\]Final Answer: The correct order is \( D>B>C>A \).