Describe the reactions of alcohols with $(a)$ $HX$ and Lucas test,$(b)$ $PX_3$,and $(c)$ dehydration reactions.

(N/A) Reactions involving the cleavage of $C-O$ bond in alcohols:
$(a)$ Reaction with $HX$ and Lucas test:
$(i)$ Alcohols react with hydrogen halides $(HX)$ to form alkyl halides: $R-OH + HX \rightarrow R-X + H_2O$.
The reactivity order of $HX$ with alcohols is $3^{\circ} > 2^{\circ} > 1^{\circ}$.
$(ii)$ Lucas test: The reaction of alcohols with concentrated $HCl$ in the presence of anhydrous $ZnCl_2$ is known as the Lucas test.
$3^{\circ}$ alcohols react immediately to form turbidity.
$2^{\circ}$ alcohols form turbidity within $5-10$ minutes.
$1^{\circ}$ alcohols do not react at room temperature.
$(b)$ Reaction with $PX_3$:
Alcohols react with phosphorus trihalides ($PX_3$,where $X = Cl, Br$) to form alkyl halides: $3R-OH + PX_3 \rightarrow 3R-X + H_3PO_3$.
$(c)$ Dehydration reactions:
Alcohols undergo dehydration in the presence of acid catalysts (like conc. $H_2SO_4$ or $H_3PO_4$) to form alkenes.
The ease of dehydration follows the order: $3^{\circ} > 2^{\circ} > 1^{\circ}$.