Step 1: Understanding the \(E_1\) Mechanism The \(E_1\) (unimolecular elimination) reaction involves the formation of a carbocation intermediate.
The stability of the carbocation determines the reactivity: the more stable the carbocation, the faster the reaction.
Step 2: Analyzing the Given Compounds The compounds contain halides (\(\text{Cl}\)) attached to different carbon atoms.
The reactivity order depends on the stability of the carbocation formed after the loss of \(\text{Cl}^-\).
Step 3: Determining the Order of Reactivity (A) Primary Carbocation: Least stable, leading to the slowest reaction.
(B) Carbocation Undergoing Rearrangement: Results in a more stable secondary or tertiary carbocation, enhancing reactivity.
(C) Five Hyperconjugating Structures: Provides moderate stability.
(D) Resonance-Stabilized Carbocation: Most stable due to delocalization, increasing reactivity.
Step 4: Final Order of Reactivity The stability of carbocations follows:
\[
D>B>C>A
\]
Final Answer: The correct order is \( D>B>C>A \).