Excitation Energy Of Hydrogen
A hydrogen atom initially in its ground state absorbs a photon and its electron jumps to the second excited state with principal quantum number three. How much energy must the absorbed photon carry for this transition?
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Solution
12.09 eV
The energy a photon must supply equals the difference between the final and initial level energies, where each level is En=−n213.6 eV. The ground state has E1=−13.6 eV and the second excited state has E3=−913.6=−1.51 eV. The required excitation energy is ΔE=E3−E1=−1.51−(−13.6)=12.09 eV. The value 13.6 eV is the full ionisation energy needed to remove the electron entirely (n=1→∞), not to reach n=3. The value 10.2 eV corresponds only to the n=1→2 transition, the first excited state. The value 1.51 eV is merely the magnitude of E3 and is not the energy gap. Only photons whose energy matches an allowed gap can be absorbed; a photon carrying slightly more or less simply passes through the atom without exciting the electron, which is why atomic absorption spectra consist of sharp lines. This applies the NCERT energy-difference rule for photon absorption. A plausibility check confirms the answer lies just below the 13.6 eV ionisation energy, as expected for a transition that nearly, but not quite, frees the electron.
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About This Question
- Subject
- physics
- Chapter
- atoms and nuclei
- Topic
- excitation energy of hydrogen
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
12.09 eV
The energy a photon must supply equals the difference between the final and initial level energies, where each level is En=−n213.6 eV. The ground state has E1=−13.6 eV and the second excited state has E3=−913.6=−1.51 eV. The required excitation energy is ΔE=E3−E1=−1.51−(−13.6)=12.09 eV. The value 13.6 eV is the full ionisation energy needed to remove the electron entirely (n=1→∞), not to reach n=3. The value 10.2 eV corresponds only to the n=1→2 transition, the first excited state. The value 1.51 eV is merely the magnitude of E3 and is not the energy gap. Only photons whose energy matches an allowed gap can be absorbed; a photon carrying slightly more or less simply passes through the atom without exciting the electron, which is why atomic absorption spectra consist of sharp lines. This applies the NCERT energy-difference rule for photon absorption. A plausibility check confirms the answer lies just below the 13.6 eV ionisation energy, as expected for a transition that nearly, but not quite, frees the electron.
This medium difficulty physics question is from the chapter atoms and nuclei, covering the topic of excitation energy of hydrogen. It appeared in the 2025 exam.
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