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The ratio of energies of the electron in the first excited state to its second excited state in a hydrogen atom is _____.

Asked in GSEB Board July 2015 · Energy levels and ionisation

• (1) 1:4
• (2) 4:9
• (3) 9:4
• (4) 4:1

Step-by-step solution

Why the other options are wrong

• (1) No single slip gives 1:4. Sanity check: the first excited state is n=2 and the second is n=3; since E∝1/(n²), the n=2 level is the larger in size, so the ratio must be greater than 1 — it is 9:4.
• (2) 4:9 is the ratio turned upside down, second excited state to first, which is also what taking the energy to grow as n² gives (2²:3²).
• (4) 4:1 counts the excited states from n=1: taking the first excited state as n=1 and the second as n=2 gives 1/(1²):1/(2²)=4:1. The ground state is n=1, so the first excited state is n=2.

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