What is the ratio of the energy required to excite an electron in a He ⁺ ion from its ground state to its first excited state, to the ionization energy of a hydrogen atom?

Step-by-step solution

The energy difference between two Bohr orbits n₁ and n₂ for a hydrogen-like species is given by: E = 13.6 Z^2 ( 1 n₁^2 - 1 n₂^2 ) eV For the excitation of a He ⁺ ion ( Z = 2 ) from the ground state ( n₁ = 1 ) to the first excited state ( n₂ = 2 ): E_ He ⁺ = 13.6 2^2 ( 1 1^2 - 1 2^2 ) = 13.6 4 3 4 = 13.6 3 eV The ionization energy of a hydrogen atom ( Z = 1 ) corresponds to the transition from n₁ = 1 to n₂ = : E_ ion = 13.6 1^2 ( 1 1^2 - 1 ) = 13.6 1 eV The required ratio is: E_ He ⁺ E_ ion = 13.6 3 13.6 1 = 3 Answe