Identify the product $B$ in the following reaction sequence:
$1$-bromo-$4$-chlorobenzene $\xrightarrow[\text{Dry ether}]{Mg (1 \text{ equivalent})} (A)$ $\xrightarrow{HO-CH_2-CH_2-Cl} (B)$

Which alkyl halide from the following pairs would you expect to react more rapidly by an $S_N2$ mechanism? Explain your answer.
$(i)$ $CH_3CH_2CH_2CH_2Br$ or $CH_3CH_2CH(Br)CH_3$
$(ii)$ $CH_3CH_2CH(Br)CH_3$ or $(CH_3)_3CBr$
$(iii)$ $CH_3CH(CH_3)CH_2CH_2Br$ or $CH_3CH_2CH(CH_3)CH_2Br$

Which of the following is product $A$ in the above reaction?

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?

Predict the product for the following reaction:
$\xrightarrow[\Delta]{alc. \ KOH}$ product

The product obtained by the reaction of ethyl bromide with alcoholic silver nitrite is .............. .