To determine which compound reacts fastest with Lucas reagent, we need to consider the underlying mechanism. Lucas reagent is a mixture of concentrated hydrochloric acid and zinc chloride (\(\text{ZnCl}_2\)) that is used to test for the presence of alcohols, specifically to classify them based on their reactivity. The reactivity order with Lucas reagent is generally tertiary alcohols > secondary alcohols > primary alcohols.
- Lucas test differentiates alcohols based on the ease of formation of their carbocations. Tertiary alcohols form carbocations more readily compared to secondary, and primary alcohols are the least reactive due to the stability of the carbocation.
- The given compounds are:
- Butan-\(1\)-ol: A primary alcohol.
- Butan-\(2\)-ol: A secondary alcohol.
- \(2\)-Methylpropan-\(1\)-ol: A primary alcohol but branched.
- \(2\)-Methylpropan-\(2\)-ol: A tertiary alcohol.
- \(2\)-Methylpropan-\(2\)-ol is a tertiary alcohol and forms the most stable carbocation due to hyperconjugation and the +I effect of alkyl groups. Therefore, it reacts fastest with the Lucas reagent, resulting in turbidity or cloudiness within a short time at room temperature.
Conclusion: The correct answer is \(2\)-methylpropan-\(2-ol\) because it is a tertiary alcohol and reacts fastest with Lucas reagent at room temperature, due to the stable carbocation formation.