Redox balancing using oxidation number changes Steps:

• Identify oxidation changes: Mn decreases from +4 in MnO₂ to +2 in Mn²⁺, gaining 2 e⁻ per Mn atom.
• Identify N change: N decreases from +5 in HNO₃ to +2 in NO (product likely intended despite NO₃⁻ notation), gaining 3 e⁻ per N atom.
• Balance electrons: 3 Mn gain 6 e⁻ (3 × 2), 2 N gain 6 e⁻ (2 × 3), so c = 3, d = 2.
• Confirm ratio c : d = 3 : 2 satisfies overall redox stoichiometry.

Why D is correct:

• Electron balance in redox method requires 3 Mn²⁺ per 2 reduced N species, per oxidation number decrease formula (Δ per atom × atoms = equal e⁻).

Why the others are wrong:

• A: 1:1 assumes equal e⁻ gain per atom, ignoring 2 vs. 3 difference.
• B: 1:2 fits NO₂ product (1 e⁻ for N +4), not dilute HNO₃ yielding NO.
• C: 2:3 reverses the 2 e⁻ (Mn) to 3 e⁻ (N) ratio.

Final answer: D