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It's not just P, S, and Cl that can exist with expanded valence shells. I believe that any element which has its valence electrons in the 3rd principal level or higher can accommodate more than 8 electrons, as it gains access to d-orbitals which first exist at the 3rd principal level and can be filled to form additional bonds or hold additional lone pairs for atoms of such elements
I also read in the book that the size of the atom is another main factor that determines whether or not an atom can expand its valence shell or not. A potassium atom is capable of having five chlorine atoms bond to it, but a nitrogen atom is much smaller in size and cannot form this many bonds.
When it comes to expanding the shells, it means that the electrons that should've gone to the new shell instead goes to the orbital's d-shell. For example, if the element is 3p5 and they have to accommodate 3 more electrons, they would bring them to 3d shell.
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