Why is the octet rule true? How can electrons even stay around nuclei with lesser number of protons?
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Simply put: Nobel gases have completely full shells, so they are stable and do not interact much with other elements. That is why other elements 'mimic' nobel gases. They want to obtain the stable configuration. As for why the octet rule is true, you might want to check @Andreas Hagen's answer to this question: What is the quantum mechanical explanation of the octet rule?
PNS
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By octet is meant $4$ orbitals, each filled with $2$ electrons (one of spin $+1/2$, one of spin $-1/2$), so $8$ electrons in total. That is the electron configuration of the outer shells of all the noble gases (except $\mathbf{He}$)
As we know empirically as well as from Quantum Mechanical theory such configurations are very stable (they are of very low energy).
Electropositive elements (like the alkali and earth alkali metals) can achieve an octet structure by shedding their outer (valence) electrons.
By contast, electronegative elements (like the halogens and chalcogens) can achieve an octet structure by acquiring additional electrons into their outer (valence) electrons.
By doing so these elements acquire the stable octet structure of the noble gases.
Gert
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