The relative reactivity order of following halides towards \(S_N1\) reaction is:

(I) 2-Chloro-2-methylpropane

(II) Chloroethane

(III) Chlorormethane

(IV) Chlorodiphenylmethane

IV > I > II > III

The correct answer is option 1. IV > I > II > III.

Let us explain why the correct reactivity order of the halides towards an \(S_N1\) reaction is IV > I > II > III in detail.

Understanding \(S_N1\) Reactions:

The \(S_N1\) (substitution nucleophilic unimolecular) reaction involves two key steps:

Formation of a carbocation: The halide ion leaves, forming a carbocation intermediate.

Nucleophilic attack: The nucleophile then attacks the carbocation to form the product.

The rate of an \(S_N1\) reaction is determined by the stability of the carbocation intermediate. The more stable the carbocation, the more reactive the halide is in the \(S_N1\) reaction.

Analysis of Each Halide:

IV. Chlorodiphenylmethane:

Carbocation Formed: Benzylic carbocation (a carbocation adjacent to a phenyl group).

Stability: The benzylic carbocation is highly stabilized by resonance with the phenyl rings. The resonance delocalizes the positive charge over the aromatic system, making this carbocation very stable.

Reactivity: Due to this high stability, chlorodiphenylmethane is extremely reactive in \(S_N1\) reactions. Hence, it ranks as the most reactive.

I. 2-Chloro-2-methylpropane:

Carbocation Formed: Tertiary carbocation.

Stability: The tertiary carbocation is stabilized by hyperconjugation and inductive effects from three alkyl groups, making it quite stable, but not as stable as the benzylic carbocation.

Reactivity: The stability of the tertiary carbocation makes 2-chloro-2-methylpropane very reactive in \(S_N1\) reactions, but it is less reactive than chlorodiphenylmethane.

II. Chloroethane:

Carbocation Formed: Primary carbocation (ethyl carbocation).

Stability: The primary carbocation is less stable compared to tertiary and benzylic carbocations due to the absence of significant alkyl group stabilization.

Reactivity: Since primary carbocations are less stable, chloroethane is less reactive in \(S_N1\) reactions compared to the tertiary and benzylic systems.

III. Chloromethane:

Carbocation Formed: Methyl carbocation.

Stability: The methyl carbocation is the least stable because it has no alkyl groups to stabilize the positive charge and is very reactive.

Reactivity: Due to its extreme instability, chloromethane is the least reactive in \(S_N1\) reactions.

Conclusion:

The reactivity order based on the stability of the carbocation intermediates is:

Chlorodiphenylmethane (IV): Most stable carbocation due to resonance stabilization.

2-Chloro-2-methylpropane (I): Stable tertiary carbocation, but less so compared to benzylic.

Chloroethane (II): Less stable primary carbocation.

Chloromethane (III): Least stable methyl carbocation.

Thus, the reactivity order for \(S_N1\) reactions is indeed IV > I > II > III, and option 1 is the correct choice.