Step 1: Understanding the Question:
The question asks to choose the correct chemical reagent 'A' required to cleanly convert an alkyl chloride ($\text{chloroethane}$, $\mathrm{CH}_3\mathrm{CH}_2\mathrm{Cl}$) into a nitroalkane ($\text{nitroethane}$, $\mathrm{CH}_3\mathrm{CH}_2\mathrm{NO}_2$).
Step 2: Key Formula or Approach:
The nitrite ion ($\mathrm{NO}_2^-$) is an ambident nucleophile that can form a covalent bond through either its nitrogen atom (yielding a nitroalkane, $-\mathrm{NO}_2$) or its oxygen atom (yielding an alkyl nitrite, $-\mathrm{ONO}$). The outcome depends heavily on the ionic or covalent character of the reagent used.
Step 3: Detailed Explanation:
Alkali metal nitrites like $\mathrm{KNO}_2$ or $\mathrm{NaNO}_2$ are highly ionic compounds. In a solution, they dissolve completely to free up the negative nitrite ion. Because the oxygen atom carries a high negative charge density, it acts as the primary nucleophilic center, attacking the alkyl group to yield an alkyl nitrite ($\mathrm{CH}_3\mathrm{CH}_2\mathrm{-O-N=O}$) as the major product.
Conversely, silver nitrite ($\mathrm{AgNO}_2$) is predominantly covalent. The bond between silver and oxygen ($\mathrm{Ag-O}$) is strong and does not dissociate easily. As a result, the lone pair on the less electronegative nitrogen atom is the main available electron pair to carry out the nucleophilic attack on the chloroethane substrate. This clean nitrogen-centered connection forms a nitroalkane:
$$\mathrm{CH}_3\mathrm{CH}_2\mathrm{Cl} + \mathrm{AgNO}_2 \rightarrow \mathrm{CH}_3\mathrm{CH}_2\mathrm{NO}_2 + \mathrm{AgCl}\downarrow$$
Step 4: Final Answer:
The required reagent to produce nitroethane is silver nitrite ($\mathrm{AgNO}_2$), which corresponds to option (B).