In hydrogen, the energy difference between the ground state (n=1) and the first excited state (n=2) is 10.2 eV. Which of the following is that difference?

• A. 13.6 eV
• B. 3.4 eV
• C. 0 eV
• D. 10.2 eV

Answer: (D)

In hydrogen, energy levels scale as E_n = -13.6 eV / n^2, so the energy difference between two levels is the difference of their E_n values. For the ground state E1 = -13.6 eV and the first excited state E2 = -3.4 eV. The transition energy is ΔE = E2 − E1 = (-3.4) − (-13.6) = 10.2 eV. This is the energy of the photon involved in moving the electron from n=1 to n=2 (absorbed to climb up, or emitted if the electron drops back down).

The other numbers correspond to different things: 13.6 eV is the ionization energy from the ground state, not the energy difference between those two levels; 3.4 eV is the energy of the n=2 level relative to zero, not the transition energy; and 0 eV would imply no energy change. So the correct energy difference is 10.2 eV.