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The electron absorbs energy and moves to a higher energy level. It then produces a line emission spectrum when it moves back down to a lower energy level (goes from an excited state to the ground state) and releases energy. The energy released is negative and should equal to the amount of energy put in because energy is conserved.
The change in energy of the electron is negative because the electron is moving from an excited state back down to its ground state. However, the energy released by the electron is equal to the energy of the photon. Therefore, the change in electron energy is negative while the change in the emitted photon's energy is positive.
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