Hydrogen-like Ions
A singly ionised helium atom, He+, behaves as a one-electron system with nuclear charge twice that of hydrogen. What energy is required to excite its electron from the ground state to the first excited state?
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Solution
40.8 eV
For a hydrogen-like ion of nuclear charge Z, the energy levels scale as En=−n213.6Z2 eV, so the entire spectrum is stretched by a factor of Z2. For He+, Z=2, hence Z2=4. The ground state is E1=−13.6×4=−54.4 eV and the first excited state is E2=−413.6×4=−13.6 eV. The excitation energy is ΔE=E2−E1=−13.6−(−54.4)=40.8 eV. The value 10.2 eV is the corresponding n=1→2 gap for ordinary hydrogen and ignores the Z2=4 scaling. The value 54.4 eV is the full ionisation energy of He+, not the first excitation energy. The value 30.6 eV results from an incorrect arithmetic combination of the levels. Because of this Z2 enhancement, hydrogen-like ions emit much shorter-wavelength, more energetic radiation than neutral hydrogen for the corresponding transition. This uses the NCERT generalisation of the Bohr formula to hydrogen-like species. A plausibility check confirms the He+ excitation energy is four times the hydrogen value of 10.2 eV, exactly as the Z2 dependence predicts.
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About This Question
- Subject
- physics
- Chapter
- atoms and nuclei
- Topic
- hydrogen-like ions
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
40.8 eV
For a hydrogen-like ion of nuclear charge Z, the energy levels scale as En=−n213.6Z2 eV, so the entire spectrum is stretched by a factor of Z2. For He+, Z=2, hence Z2=4. The ground state is E1=−13.6×4=−54.4 eV and the first excited state is E2=−413.6×4=−13.6 eV. The excitation energy is ΔE=E2−E1=−13.6−(−54.4)=40.8 eV. The value 10.2 eV is the corresponding n=1→2 gap for ordinary hydrogen and ignores the Z2=4 scaling. The value 54.4 eV is the full ionisation energy of He+, not the first excitation energy. The value 30.6 eV results from an incorrect arithmetic combination of the levels. Because of this Z2 enhancement, hydrogen-like ions emit much shorter-wavelength, more energetic radiation than neutral hydrogen for the corresponding transition. This uses the NCERT generalisation of the Bohr formula to hydrogen-like species. A plausibility check confirms the He+ excitation energy is four times the hydrogen value of 10.2 eV, exactly as the Z2 dependence predicts.
This hard difficulty physics question is from the chapter atoms and nuclei, covering the topic of hydrogen-like ions. It appeared in the 2025 exam.
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