A newly discovered atom, designated X, exhibits the bound-state energies listed in the table below (all energies relative to the continuum):
| Principal quantum number n | Energy E_n (eV) |
| 1 | −15.0 |
| 2 | −4.80 |
| 3 | −2.10 |
| 4 | −1.20 |
| 5 | −0.75 |
A pulsed discharge excites electrons in a dilute gas of X atoms to the n = 5 level. A spectrometer with a wavelength detection range of 200 nm to 2.5 µm is then used to record the resulting emission spectrum.
(a) Enumerate all possible electronic transitions that can originate from the n = 5 level, whether directly or through cascades, until the atom reaches the ground state. Present your list as ordered pairs (n_i → n_f).
(b) Calculate the photon energy and corresponding wavelength for each transition identified in part (a).
(c) Indicate which of these transitions will produce photons that fall outside the detection range of the spectrometer, and explain why those lines would not be observed.
(d) A colleague claims that the strongest observable line will correspond to the 5 → 2 transition. Evaluate this claim by considering both the transition energy and the number of cascade pathways that can populate the n = 5 → 2 emission. Provide a reasoned argument supporting or refuting the claim.