Reduction Reaction:
Ag⁺ + 1e⁻ → Ag ------ (1)
Oxidation Reaction:
NO₂⁻ → NO₃⁻ + 2e⁻
Balance Oxygen atoms, by adding H₂O to side having less number of O atoms, and H⁺ to side having greater number of O atoms.
NO₂⁻ + H₂O → NO₃⁻ + H⁺ + 2e⁻
Balance the H⁺'s,
NO₂⁻ + H₂O → NO₃⁻ + 2 H⁺ + 2e⁻ ------ (2)
Multiply eq. 1 by 2 to balance the number of electrons in overall reaction.
2 Ag⁺ + 2e⁻ → 2 Ag ----- (3)
Compare eq. 2 and 3,
NO₂⁻ + H₂O → NO₃⁻ + 2 H⁺ + 2e⁻
2 Ag⁺ + 2e⁻ → 2 Ag
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NO₂⁻ + H₂O + 2 Ag⁺ → NO₃⁻ + 2 Ag + 2 H⁺
