On which quantum numbers does the energy of an electron depend in (a) a hydrogen atom and

(b) a vanadium atom?

Atoms are Hydrogen ( Z = 1 ) and Vanadium ( Z = 23 ) .

Using the energy of an electron in the nth Bohr's orbit of an atom, we can calculate the energy of an electron of the required atom that are hydrogen-like species using their atomic number and the principal quantum number of the subshell where the electron is present. But for transition metals that are not hydrogen-like species, the energy of the electron also depends on the azimuthal number irrespective of the dependence of energy on the principal quantum number.

Formula:

The energy of an electron in the nth Bohr's orbit or shell of the atom,

$E=\frac{-13.6Z^2}{n^2}$ ……….(i)

According to the concept and equation (i), we can say that the energy of hydrogen or any other hydrogen-like species depends on the principal quantum number n.

Hence, it depends on the principal quantum number n.

Now, as Vanadium is not hydrogen-like species and is a transition metal, the energy of this atom due to transition depends on both the principal n and azimuthally quantum number l that is, the energy dependence is on the value $\left(n+l\right)$.

Hence, the energy of this atom depends onn and l .