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In the emission spectrum, do electrons that jump up to the next shell go to any orbital in that shell? Or if the electron is originally in say the p orbital, does it jump to the next p orbital in the next shell?? Also wouldn’t there be more colors on the emission spectrum if energy is emitted when electrons fall down from an excited orbital down to lower orbital in the same shell? Why does it seem like there are only colors for change in shell and not orbital within a shell?

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Yes there are emission lines for jumps between the orbitals for multi electron lines as seen in the form of doublets( two closely spaced lines) and triplets( three closely spaced lines).

Infact this is one of the reason why Bohr's model failed. Doublets and triplets appear in the spectra of some atoms as very close pairs of lines. Bohr's model cannot say why some energy levels should be very close together. These closely spaced lines corresponds to transition from one orbital to other.

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