An electron in a hydrogen atom transitions from a higher energy level to the first excited state, emitting a photon. This photon is incident on a photosensitive metal surface having a work function of 1.0 eV , ejecting photoelectrons with a maximum kinetic energy of 1.55 eV . Which spectral series does this transition belong to, and what is the principal quantum number of the higher energy state ( n₂ )?

Step-by-step solution

According to the photoelectric equation, the energy of the incident photon is: E = + K_ max = 1.0 eV + 1.55 eV = 2.55 eV The transition occurs to the first excited state, which corresponds to the principal quantum number n₁ = 2 . The energy of a photon emitted during a transition in a hydrogen atom is given by: E = 13.6 ( 1 n₁^2 - 1 n₂^2 ) eV Substituting the known values: 2.55 = 13.6 ( 1 2^2 - 1 n₂^2 ) 2.55 13.6 = 1 4 - 1 n₂^2 3 16 = 1 4 - 1 n₂^2 1 n₂^2 = 1 4 - 3 16 = 1 16 n₂^2 = 16 n₂ = 4 Since the transition is