Explanation:

The Lucas reagent is a mixture of concentrated hydrochloric acid (HCl) and zinc chloride (ZnCl2). It is used to test for the presence of primary, secondary, and tertiary alcohols.

The reaction of alcohols with Lucas reagent involves the substitution of the hydroxyl group (-OH) with a chloride ion (-Cl). The rate of this reaction depends on the stability of the carbocation intermediate formed during the reaction. Tertiary carbocations are more stable than secondary carbocations, which are more stable than primary carbocations.

Comparing the given options:

a) Butan-1-ol: This is a primary alcohol. The carbocation intermediate formed during the reaction is a primary carbocation. Primary carbocations are the least stable among the three types. Therefore, the reaction is expected to be slow.

b) Butan-2-ol: This is a secondary alcohol. The carbocation intermediate formed during the reaction is a secondary carbocation. Secondary carbocations are more stable than primary carbocations but less stable than tertiary carbocations. Therefore, the reaction is expected to be faster than butan-1-ol but slower than 2-methylpropan-2-ol.

c) 2-Methylpropan-1-ol: This is a primary alcohol. Similar to butan-1-ol, the carbocation intermediate formed during the reaction is a primary carbocation. Therefore, the reaction is expected to be slow.

d) 2-Methylpropan-2-ol: This is a tertiary alcohol. The carbocation intermediate formed during the reaction is a tertiary carbocation. Tertiary carbocations are the most stable among the three types. Therefore, the reaction is expected to be the fastest among the given options.

Conclusion:

Based on the stability of the carbocation intermediates formed during the reaction with Lucas reagent, the compound that reacts fastest at room temperature is 2-methylpropan-2-ol (option D).