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The length of the bonds is in part based on the distance between two atoms. Atoms that form double bonds with one another have a stinger attraction between them which means they are pulled closer together than if they were connected by a single bond. Thus, the distance between them is smaller which means that double bonds are shorter than single bonds.
Correct, the lewis structure is just a rough model and Dr. Lavelle did say that the actual length is 1.24 angstrom. The double bonds and triple bonds contain more electrons, thereby exerting a stronger force on the nuclei of the atoms and this pulls the atoms closer together hence shorter bond lengths.
In general, yes. The NO3 is a special case of resonance as its double bond is not on one of the bonds, but rather each bond is a bit stronger than a single bond and a bit weaker than a double bond.
I spoke with an LA after class about this specific topic. He explained that the electrons of the NO3- ion are evenly distributed amongst the three bonds. Thus, in reality, there is no distinguishable difference between the double/single bonds. This is the reasoning behind the experimental 1.24-angstrom value that Dr. Lavelle provided. I hope this add-on helps :)
Additionally, in resonance structures, the bond lengths are equal due to the delocalization of electrons between the atoms. The molecule, as a result, has a lower energy, and this helps to stabilize the molecule.
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