An unknown alcohol is treated with the "Lucas reagent" to determine whether the alcohol is primary, secondary or tertiary. Which alcohol reacts fastest and by what mechanism:

First we recall the basic facts about the Lucas test. The Lucas reagent is a mixture of concentrated hydrochloric acid and anhydrous zinc chloride, written in equations as $$ZnCl_2 + conc\,HCl$$. It converts an alcohol $$R{-}OH$$ into the corresponding alkyl chloride $$R{-}Cl$$: $$R{-}OH + HCl \rightarrow R{-}Cl + H_2O$$, with $$ZnCl_2$$ acting as a Lewis-acid catalyst. The turbidity (cloudiness) that appears comes from insoluble alkyl chloride droplets. The time taken for this turbidity tells us the class of the alcohol.

Now we state the empirical observation that governs the test:

$$\text{Rate of reaction: } \text{tertiary} \; > \; \text{secondary} \; > \; \text{primary}.$$

To justify this order we analyse the stepwise mechanism. The first and rate-determining step is the formation of a carbocation via protonation followed by departure of water. Writing every step:

1. Protonation of the alcohol oxygen by concentrated $$HCl$$ gives the oxonium ion:
$$R{-}OH + H^+ \rightarrow R{-}OH_2^+.$$

2. The zinc chloride coordinates with the lone pairs on oxygen, pulling electron density away and making the -OH2+ group an even better leaving group:
$$(R{-}OH_2^+)\,ZnCl_2 \rightarrow R{-}OH_2^+\;ZnCl_2.$$

3. Departure of water produces the carbocation:
$$R{-}OH_2^+ \rightarrow R^+ + H_2O.$$

4. Finally chloride ion attacks the carbocation to give the alkyl chloride:
$$R^+ + Cl^- \rightarrow R{-}Cl.$$

This sequence—slow ionisation to a carbocation followed by fast nucleophilic attack—is the classical $$S_N1$$ mechanism (substitution, nucleophilic, unimolecular). In an $$S_N1$$ process the stability of the carbocation controls the rate: more stable carbocation ⇒ faster reaction.

Carbocation stability follows the rule

$$3^\circ \; > \; 2^\circ \; > \; 1^\circ,$$

because additional alkyl groups donate electron density by both hyperconjugation and the inductive effect, dispersing the positive charge.

Therefore a tertiary alcohol, which forms a highly stabilised tertiary carbocation, loses water almost instantly; the solution becomes cloudy within a few seconds. A secondary alcohol forms a less stable carbocation, so the cloudiness appears more slowly (often in a few minutes). A primary alcohol forms an unstable primary carbocation, so the $$S_N1$$ path is extremely slow; instead it can react by an $$S_N2$$ path only on prolonged heating, which is why primary alcohols usually give no turbidity at room temperature.

Putting everything together, the fastest reacting alcohol in the Lucas test is the tertiary alcohol and it proceeds through the $$S_N1$$ mechanism.

Hence, the correct answer is Option D.