Calculation of Energy of an Electron:

The total energy, E of the electron is the sum of kinetic energy and potential energy.

Kinetic energy of the electron = ½ mv²

Potential energy = -KZe² /r

Total energy = 1/2 mv² – KZe² /r                               … (4)

From equation (1) we know that

mv²= KZe² /r

∴ ½ mv² = KZe² /2r

Substituting this in equation (4)

Total energy (E) =KZe² /2r – KZe² /r  = -KZe² /2r

Substituting for r, gives us

E = 2π²mZ²e'K²/n²h²      where n = 1, 2, 3………

This expression shows that only certain energies are allowed to the electron. Since this energy expression consist of so many fundamental constant, we are giving you the following simplified expressions.

E = –21.8 x 10^–12 x z²n²  erg per atom 

 = –21.8 x10^–19 x z²n²  J per atom

 = –13.6 x z²n²  eV per atom          

(1eV = 3.83 x 10^–23 kcal 

1eV = 1.602 x 10^–12 erg 

1eV = 1.602 x10^–19J) 

E = –313.6 x z²n² kcal / mole (1 cal = 4.18 J)

The energies are negative since the energy of the electron in the atom is less than the energy of a free electron (i.e., the electron is at infinite distance from the nucleus) which is taken as zero. The lowest energy level of the atom corresponds to n=1, and as the quantum number increases, E becomes less negative.

When n = ∞, E = 0, which corresponds to an ionized atom i.e., the electron and nucleus are infinitely separated.

H —→ H⁺+ e^– (ionization).