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Since the structure of nitrate has both a double and single bond, we have to understand that their is resonance involved. Also, remember that double bonds have shorter bond lengths than single bonds because double bonds are stronger. Resonance affects the strength of bonds because it spreads multiple-bond character over the bonds between atoms so single bonds are strengthened while double bonds become weaker as a result of the bond lengths being averaged.
Due to resonance, the most stable structure of the nitrate will be an average of the bonds nitrate holds which is both a single and double bond. As an overall molecule, nitrate's structure will even out and be an inbetween version of a single and double bond. Therefore, the average of 120 and 140 is 130.
If a structure has multiple bonds then the total molecule will have bonds the average length of all the present bonds. A molecule that has resonance has multiple possible bond placements in a Lewis structure, but this simply translates to having bonds that are averaged out over the different bond lengths. If you have a 120pm and a 140pm set of bonds in a molecule, the electrons will be held an average of those lengths or somewhere around 130pm.
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