Energy of an Hydrogen Atom

Assume that the nucleus has a positive charge Ze ( there are z protons each with positive charge e).

By equating the coulomb force acting between Ze and e to the centripetal acceleration mv²/r, we get r the radius at which the electron revolves.

r = Ze²/4π ε₀v²

From Bohr’s quantization rule,

mvr = nh/2 π
where n is a positive integer

Eliminating v from both the equations we get

r = ε₀h²n²/πmZe²

We get expression for v as

v = Ze²/2 ε₀hn


Hence allowed radii are proportinal to n² and for each value of n = 1,2,3…we allowed orbits.

The smallest radius orbit will have n = 1.



As we have expression for v, we can give an expression for kinetic energy when electron is in nth orbit is

K = ½ mv² = mZ²e⁴/8 ε₀²h²n²

The potential energy of the atom is

V = - Ze²/4π ε₀r = -mZ²e⁴/4ε₀²h²n²

The expression for potential energy is obtained by assuming the potential energy to be zero when the nucleus and the electron are widely separated.

The total energy of the atom is

E = K+V = - mZ²e⁴/8 ε₀²h²n²

When an atom is n^th stationary orbit, it is said to be in the n^th energy state.

In giving an expression for the total energy of the atom, kinetic energy of the electron and potential energy of the electron-nucleus pair are considered. Kinetic energy of the nucleus is assumed to be negligible.

Bohr’s postulates can be used to find the allowed energies of the hydrogen atom when its single electron is in various stationary orbits. The methodology can be used any hydrogen like ions which have only one electron.. Therefore it is valid for He⁺, Li⁺⁺, Be⁺⁺⁺ etc.