Question

The first excited energy of a He⁺ ion is the same as the ground state energy of hydrogen. Is it always true that one of the energies of any hydrogen-like ion will be the same as the ground state energy of a hydrogen atom?

Answer 1

The energy of hydrogen ion is given by
$E_{\mathrm{n}}=-\frac{(13.6\mathrm{eV})Z^2}{n^2}$
For the first excited state (n = 2), the energy of He⁺ ion (with Z = 2) will be $-$13.6 eV. This is same as the ground state energy of a hydrogen atom.
Similarly, for all the hydrogen like ions, the energy of the (n $-$ 1)th excited state will be same as the ground state energy of a hydrogen atom if Z = n.