Question

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

(b) a vanadium atom?

Short answer

1. The quantum number that the energy of an electron depends in a hydrogen atom is the principal quantum number n.
2. The quantum number that the energy of an electron depends in a vanadium atom is the principal quantum number n andl .

Step-by-step solution

The given data

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

Understanding the concept of energy

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)

a) Calculation of the energy in the hydrogen atom

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.

b) Calculation of energy in vanadium atom

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 .