The reactivity of alcohols with Lucas reagent is ordered as: Tertiary ($ \text{3}^\circ $) > Secondary ($ \text{2}^\circ $) > Primary ($ \text{1}^\circ $).
This reactivity order is attributed to the stability of the carbocation intermediate formed. The reaction mechanism involves alcohol protonation by $ \text{HCl} $, followed by water elimination to generate a carbocation, which then swiftly reacts with a chloride ion. Tertiary carbocations exhibit the highest stability due to hyperconjugation and inductive effects from three alkyl groups, resulting in the fastest reaction rates. Secondary carbocations are less stable, and primary carbocations are the least stable, often requiring elevated temperatures or extended reaction times.
Analysis of Given Alcohols:
(1) $ \text{CH}_3 - \text{CH}_2 - \text{CH}_2 - \text{CH}_2\text{OH} $: This is a primary alcohol (the hydroxyl-bearing carbon is bonded to one other carbon). Primary alcohols typically show no reaction or very slow reaction with Lucas reagent at room temperature, with no immediate turbidity observed.
(2) $ \text{CH}_3 - \text{CH}_2 - \underset{\underset{\text{CH}_3}{|}}{\text{CH}} - \text{OH} $: This is a secondary alcohol (the hydroxyl-bearing carbon is bonded to two other carbons). Secondary alcohols react with Lucas reagent, producing turbidity within 5 to 10 minutes.
(3) $ \text{CH}_3 - \underset{\underset{\text{CH}_3}{|}}{\text{CH}} - \text{CH}_2\text{OH} $: This is also a primary alcohol (the hydroxyl-bearing carbon is bonded to one other carbon). Consistent with (1), this alcohol does not react instantaneously with Lucas reagent at room temperature.
(4) $ \underset{\underset{\text{CH}_3}{|}}{\overset{\overset{\text{CH}_3}{|}}{\text{CH}_3 - \text{C} - \text{OH}}} $: This is a tertiary alcohol (the hydroxyl-bearing carbon is bonded to three other carbons). Tertiary alcohols react almost instantly with Lucas reagent, producing immediate turbidity due to the formation of a stable tertiary carbocation.
Conclusion: Alcohol (4) reacts instantaneously with Lucas reagent.
Final Answer: The alcohol that reacts instantaneously with Lucas reagent is: $ \boxed{(4)} $
