$A$ halide $C_5H_{11}Br$ on treatment with alc. $KOH$ gives $2$-pentene only. The halide will be

$A$ primary alkyl halide $(a)$ with molecular formula $C_{4}H_{9}Br$ reacts with alcoholic $KOH$ to give compound $(b)$. Compound $(b)$ reacts with $HBr$ to give compound $(c)$,which is an isomer of $(a)$. When $(a)$ reacts with sodium metal,it gives compound $(d)$ with molecular formula $C_{8}H_{18}$,which is different from the compound formed when $n$-butyl bromide reacts with sodium. Identify the structural formula of $(a)$ and write the equations for all the reactions.

For the reactions given below:
$(I)$ Cyclohexyl chloride $\rightarrow$ Cyclohexyl cation + $Cl^-$,$\Delta H_1^{\circ}$
$(II)$ Cyclohex$-2-$enyl chloride $\rightarrow$ Cyclohex$-2-$enyl cation + $Cl^-$,$\Delta H_2^{\circ}$
$(III)$ Benzyl chloride $\rightarrow$ Benzyl cation + $Cl^-$,$\Delta H_3^{\circ}$
$(IV)$ Chlorobenzene $\rightarrow$ Phenyl cation + $Cl^-$,$\Delta H_4^{\circ}$
The correct decreasing order of enthalpies of reaction for producing the carbocation is:

$A$ compound $X$ on heating with alcoholic $AgNO_3$ gives a white precipitate. Oxidation of $X$ gives an acid with the formula $C_8H_6O_4$,which easily forms a cyclic anhydride on heating. The compound $X$ is

The correct order of the rate of reaction of the following reactants with a nucleophile by $S_{N}1$ mechanism is:
(Given: Structure $I$ and $II$ are rigid)

The decreasing order of reactivity towards $S_N 1$ reaction for the following compounds is: