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?

Question

Philoid · Accepted Answer

We know that energy E of a hydrogen or hydrogen-like species is given by,

Now, for the ground state energy of hydrogen(Z=1), Z/n= 1. Thus, E=-13.6 eV.

Also, for the first excited energy of helium ion(Z=2), Z/n= 1. Thus, E=-13.6 eV.

Thus, we have seen that in all hydrogen like species where Z=n, E=-13.6 eV.

So, for the second excited state of lithium ion(Z=2), for the third excited state of beryllium ion(Z=3), so on, we have E=-13.6 eV.

Hence, there will always be an energy level of an hydrogen like species which will be same as the ground state energy of hydrogen.

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?

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?