Energy Levels

Let's calculate the energy corresponding to a given orbit in the hydrogen atom according to the Bohr model. The kinetic energy is

KE =

mv² = ke²/(2r_n)

The potential energy from the Coulomb interaction is

PE = -eV = -ke²/r_n

The total energy then becomes

E_n = ke²/(2r_n) - ke²/r_n = - ke²/(2r_n)

Putting in our quantized radii we get

E_n = [(mk²e⁴)/(2

²)] / n² = -E_o/n²

where E_o = 2.2·10^-18 J = 13.6 eV.

The negative sign in the equation for the E_n means that the system has less energy than a free electron and proton or that the electron is bound to the proton, and it takes 13.6 eV of energy to remove the electron from the atom.