Step 1: Understanding the Concept:
The reaction of benzyl halides with KCN in a polar aprotic solvent is typically an \( S_N2 \) mechanism. In benzylic systems, both electron-donating groups (EDGs) and electron-withdrawing groups (EWGs) can influence the rate, but usually, EDGs significantly enhance reactivity if the transition state has significant carbocationic character.
Step 2: Detailed Explanation:
In the transition state for nucleophilic substitution on benzyl halides, the developing positive charge is stabilized by resonance with the aromatic ring.
1. p-amino (b): \( -NH_2 \) is a very strong electron-donating group via the \( +M \) effect, providing maximum stabilization to the transition state.
2. p-hydroxy (a): \( -OH \) is also a strong electron-donating group (\( +M \)), but it is less effective than \( -NH_2 \).
3. m-nitro (d): The nitro group is electron-withdrawing. At the meta position, it acts primarily through the \( -I \) effect.
4. p-nitro (c): At the para position, \( -NO_2 \) exerts both \( -I \) and strong \( -M \) effects, causing the greatest destabilization to a cation-like transition state compared to the meta isomer.
Thus, the reactivity order is: p-amino \(>\) p-hydroxy \(>\) m-nitro \(>\) p-nitro.
Step 3: Final Answer:
The correct order is b \(>\) a \(>\) d \(>\) c.