Arrange the following alkyl halides in the increasing order of their reactivity towards nucleophilic substitution reaction:
$(I)$ $CH_3CH_2CH(Cl)CH_3$
$(II)$ $CH_2=CHCH(Br)CH_3$
$(III)$ $CH_2=CHCH(Cl)CH_3$
$(IV)$ $CH_3CH=C(Br)CH_3$

$CH_3-CH_2-Cl \xrightarrow{alc. KOH} A$,the product is

Identify $A$ and $B$ in the following reactions:
$A \xrightarrow[\Delta]{aq. NaOH} C_2H_5OH \xleftarrow{AgOH} B$

Identify the product obtained when $2-\text{chloro}-2-\text{methylbutane}$ is reacted with sodium metal in the presence of dry ether.

Among the given halides,which one will give the same product in both $S_{N^1}$ and $S_{N^2}$ reactions?
$(I) \ CH_3-CH(CH_3)-CH_2-CH(Br)-CH_3$
$(II) \ \text{2-methylcyclohexyl chloride}$
$(III) \ \text{Chlorocyclohexane}$
$(IV) \ CH_3-CH(Br)-CH_2-CH_3$

Compound $A$ with molecular formula $C_4H_9Br$ is treated with aqueous $KOH$ solution. The rate of this reaction depends upon the concentration of compound $A$ only. When another optically active isomer $B$ of this compound was treated with aqueous $KOH$ solution,the rate of reaction was found to be dependent on the concentration of both the compound and $KOH$.
$(i)$ Write down the structural formula of both compounds $A$ and $B$.
$(ii)$ Out of these two compounds,which one will be converted to the product with inverted configuration?