An electron in a hydrogen-like ion is in a specific orbit. The energy of the electron in this orbit is equal to the ground state energy of a He ⁺ ion, and the radius of this orbit is equal to the Bohr radius of a hydrogen atom (ground state radius). The atomic number Z of this unknown hydrogen-like ion is :

Step-by-step solution

For a hydrogen-like species, the energy of an electron in the n^ th orbit is proportional to Z^2 n^2 , and the radius of the orbit is proportional to n^2 Z . Let the unknown ion have atomic number Z and the electron be in the n^ th orbit. Given that the energy is equal to the ground state energy of He ⁺ ( Z=2, n=1 ): Z^2 n^2 = 2^2 1^2 = 4 Z = 2n Given that the radius is equal to the Bohr radius of hydrogen ( Z=1, n=1 ): n^2 Z = 1^2 1 = 1 n^2 = Z Substituting Z = 2n into the radius equation: n^2 = 2n Since n 0 , we