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).