Final answer:
In a hydrogen atom energy-level diagram, the longest wavelength photon is emitted by the transition with the smallest energy change, which typically occurs between the closest energy levels.
Step-by-step explanation:
In a hydrogen atom energy-level diagram, the longest wavelength photon corresponds to the transition involving the smallest energy change. The energy change associated with a transition between energy levels is given by the difference in energy between the final and initial states.
According to the Bohr model of the hydrogen atom, the energy difference between levels decreases as the principal quantum number increases, since the energy levels get closer together the farther they are from the nucleus.
Thus, a transition from a higher level to a just slightly lower level (for example, from n=3 to n=2) will have a smaller energy change, and therefore produce a light photon of longer wavelength, than a transition between levels that are further apart (such as n=2 to n=1, where n represents the principal quantum number).
Therefore, among the options provided, the transition with the smallest difference in principal quantum numbers, which indicates the transition occurring between the closest levels, will emit the longest wavelength photon.