The correct order for the reactivity of following halide towards \(S_N1\) reaction is:

II > III > I > IV

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

The names of the given compounds are:

To determine the correct order of reactivity of the given halides towards the \(S_N1\) reaction, we need to understand the underlying principles that govern the \(S_N1\) mechanism. The \(S_N1\) reaction, or unimolecular nucleophilic substitution, is a two-step process:

Formation of a Carbocation Intermediate: The halide leaves, forming a positively charged carbocation. The stability of this carbocation is crucial because the rate-determining step (slowest step) of the \(S_N1\) reaction is the formation of the carbocation.

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

The reactivity in an \(S_N1\) reaction is influenced primarily by two factors:

Carbocation Stability: The more stable the carbocation, the faster it will form, and therefore, the faster the \(S_N1\) reaction will proceed. Carbocation stability increases in the order:

Tertiary \(> \) Secondary \(> \) Primary

This is due to the inductive effect and hyperconjugation, which help stabilize the positive charge on the carbocation.

Leaving Group Ability: The better the leaving group, the easier it is for the halide to leave and form the carbocation. Leaving group ability increases in the order:

\( \text{F}^- < \text{Cl}^- < \text{Br}^- < \text{I}^- \)

A better leaving group is one that can stabilize the negative charge once it leaves the molecule.

Now, let us analyze each of the given compounds:

I. Chlorocyclohexane :

Carbocation Stability: The carbocation formed after the departure of the chloride ion is a secondary cyclohexyl carbocation. Secondary carbocations are moderately stable but less stable than tertiary carbocations.

Leaving Group: Chlorine (\( \text{Cl}^- \)) is a decent leaving group, but not as good as bromine or iodine.

II. Bromocyclohexane :

Carbocation Stability: Similar to chlorocyclohexane, bromocyclohexane forms a secondary cyclohexyl carbocation. The stability is the same as in chlorocyclohexane.

Leaving Group: Bromine (\( \text{Br}^- \)) is a better leaving group than chlorine, so bromocyclohexane will react faster than chlorocyclohexane.

III. 1-Bromo-1-methylcyclohexane:

Carbocation Stability: Upon the departure of the bromide ion, a tertiary carbocation (1-methylcyclohexyl carbocation) is formed. Tertiary carbocations are significantly more stable than secondary or primary carbocations due to the greater number of alkyl groups that can donate electron density through inductive and hyperconjugation effects.

Leaving Group: Bromine is a good leaving group. Due to the formation of a highly stable tertiary carbocation, 1-bromo-1-methylcyclohexane will have the highest reactivity in the \(S_N1\) reaction.

IV. Iodocyclohexane:

Carbocation Stability: Similar to the previous halides, iodocyclohexane forms a secondary cyclohexyl carbocation, which is moderately stable.

Leaving Group: Iodine (\( \text{I}^- \)) is the best leaving group among the halides mentioned, so iodocyclohexane will react faster than bromocyclohexane and chlorocyclohexane.

Thus the correct Order of Reactivity is III > IV > II > I.