A hydrogen atom in its first excited state transitions to the ground state, emitting a photon. This photon is completely absorbed by a He ⁺ ion initially in its first excited state, exciting the ion to a higher energy level n=x . The value of x is

Step-by-step solution

For the hydrogen atom ( Z=1 ), the first excited state corresponds to n=2 and the ground state to n=1 . The energy of the emitted photon is: E = 13.6 1^2 ( 1 1^2 - 1 2^2 ) = 13.6 3 4 eV This photon is absorbed by a He ⁺ ion ( Z=2 ). The first excited state of He ⁺ corresponds to n=2 . Let the final state of the He ⁺ ion be n=x . The energy required for this transition is: E = 13.6 Z^2 ( 1 2^2 - 1 x^2 ) Substituting Z=2 and equating the energies: 13.6 3 4 = 13.6 4 ( 1 4 - 1 x^2 ) 3 4 = 4 ( 1 4 - 1 x^2 ) 3 4 = 1 - 4