Final answer:
A hydrogen atom is excited to a higher energy level when struck by a 12.5 eV electron beam. The number of lines emitted as it returns to the ground state depends on the principal quantum number reached after excitation, calculated using the difference in energy levels.
Step-by-step explanation:
When a hydrogen atom in the ground state is excited by an electron beam of 12.5 eV, it transitions to an excited energy state. The ground state of a hydrogen atom has an energy of E1 = -13.6 eV. Applying a 12.5 eV energy beam raises the energy level of the electron however, only certain energy levels are allowed according to the Bohr model of the hydrogen atom.
In the Bohr model, the energy levels are described by the formula E_n = -13.6 eV / n^2, where n is the principal quantum number. Since the electron starts at the ground state (n=1), it will be excited to a higher state where the difference in energy corresponds to the energy of the electron beam.
To find the state to which the electron is excited, we solve E_n = E1 + 12.5 eV for n. Then, the number of spectral lines emitted when the electron transitions back to the ground state is given by the formula N = n(n - 1) / 2, because the electron can transition between any two energy levels, resulting in the emission of a unique photon for each transition.