To determine the maximum bond angle at nitrogen, we need to analyze the hybridization and molecular geometry of nitrogen in each species.

Step 1: Analyze ${\mathrm{NO}}_2^{-}$ (Nitrite ion)

• Nitrogen has 5 valence electrons. Oxygen has 6 valence electrons. Total valence electrons = $5+2(6)+1=18$.
• Lewis structure: N is central. One N=O bond, one N-O bond, and a lone pair on N.
• Steric number = 2 (bonded atoms) + 1 (lone pair) = 3.
• Hybridization = ${\mathrm{sp}}^2$.
• Geometry: Bent.
• Bond angle: Approximately $120^{\circ }$ (due to lone pair repulsion, it's slightly less, around $115^{\circ }$).

Step 2: Analyze ${\mathrm{NO}}_2^{+}$ (Nitronium ion)

• Nitrogen has 5 valence electrons. Oxygen has 6 valence electrons. Total valence electrons = $5+2(6)-1=16$.
• Lewis structure: N is central, with two N=O double bonds. No lone pairs on N.
• Steric number = 2 (bonded atoms) + 0 (lone pairs) = 2.
• Hybridization = $\mathrm{sp}$.
• Geometry: Linear.
• Bond angle: $180^{\circ }$.

Step 3: Analyze ${\mathrm{NO}}_3^{-}$ (Nitrate ion)

• Nitrogen has 5 valence electrons. Oxygen has 6 valence electrons. Total valence electrons = $5+3(6)+1=24$.
• Lewis structure: N is central. One N=O bond, two N-O single bonds (resonance structures exist). No lone pairs on N.
• Steric number = 3 (bonded atoms) + 0 (lone pairs) = 3.
• Hybridization = ${\mathrm{sp}}^2$.
• Geometry: Trigonal planar.
• Bond angle: $120^{\circ }$.

Step 4: Analyze ${\mathrm{NO}}_2$ (Nitrogen dioxide)

• Nitrogen has 5 valence electrons. Oxygen has 6 valence electrons. Total valence electrons = $5+2(6)=17$.
• Lewis structure: N is central. One N=O bond, one N-O bond, and one unpaired electron on N.
• Steric number = 2 (bonded atoms) + 1 (unpaired electron acts like a half-lone pair) = 3.
• Hybridization = ${\mathrm{sp}}^2$.
• Geometry: Bent.
• Bond angle: Approximately $134^{\circ }$ (the unpaired electron causes less repulsion than a lone pair, leading to a larger angle than ${\mathrm{NO}}_2^{-}$).

Comparing the bond angles:

• ${\mathrm{NO}}_2^{-}$: $\approx 115^{\circ }$
• ${\mathrm{NO}}_2^{+}$: $180^{\circ }$
• ${\mathrm{NO}}_3^{-}$: $120^{\circ }$
• ${\mathrm{NO}}_2$: $\approx 134^{\circ }$

The maximum bond angle is $180^{\circ }$ in ${\mathrm{NO}}_2^{+}$.

Correct Answer: (B)