Final answer:
When atoms are excited to the second excited state and then relax back to lower energy levels, three distinct spectral lines will be observed, corresponding to the transitions from n=3 to n=2, n=3 to n=1, and potentially from n=2 to n=1.
Step-by-step explanation:
If a large number of atoms are excited to the second excited state (which we can call the state with quantum number n = 3), the emitted light observed will consist of a number of spectral lines corresponding to the transitions between excited states and the ground state. According to the rules governing electron transitions, the electron can fall back from the second excited state to lower energy levels in a stepwise fashion, releasing photons with specific energy differences as emitted radiation at discrete wavelengths. To calculate the number of possible spectral lines, we consider the following transitions: from n=3 to n=2, n=3 to n=1, and also from n=2 to n=1, if the electron first goes from n=3 to n=2 and then to n=1.
Using Bohr's model of the hydrogen atom, the electron in the excited state can take multiple pathways to reach the ground state. Each transition results in the emission of a photon with a specific wavelength, contributing to the overall emission spectrum. To illustrate, the transition from n=3 to n=2 is one spectral line, from n=3 to n=1 is another spectral line, and the transition previously made from n=2 to n=1 (if it occurs after the initial drop from n=3) would be the third spectral line. Consequently, we will observe three distinct spectral lines in the emission spectrum of these excited atoms.