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Supposedly every atom that is in the second period of the periodic table has to obey the octet rule; this means that they have to be surrounded by 8 electrons (or so I was taught). Studying resonance in an exercise, they showed how nitrogen as NO₂ can form 5 bonds, which would mean that NO₂ does not behave according to the rule previously mentioned. I am at a loss; any help would be great.

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No atom has to obey the octet rule. It is a rule. It has no theoretical justification. It works often. But that's all that could be stated. And a lot of compounds are known that do not follow this rule like $O_2, NO, B_2H_6$, etc. And Nitrogen does not form 5 bonds. It cannot. $4$ bonds is the maximum for atoms of the second period (Li, Be, B, C, N, O, F).

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