Radii and energies of the hydrogen atom

Combining the standing-wave condition with Coulomb's law and Newton's second law gives:

• The radius of the n-th orbit: r_n = n² r₁, with r₁ = 0.053 nm (the radius of the first orbit). So r₂ = 4r₁, r₃ = 9r₁, and so on: the orbits get farther apart as n increases.

• The energy of the electron in the n-th level:

  E_n = − 13.6 / n² eV (n = 1, 2, 3, ...)

  The negative sign means the electron is bound to the nucleus. The energy is lowest (most negative) for n = 1. As n increases the energy rises toward zero. At n = ∞ the electron is free (E = 0).

Energies of the first levels:

Ionisation energy of hydrogen: the energy needed to remove the electron from the ground state completely = 13.6 eV (≈ 2.18 × 10⁻¹⁸ J). From the n = 2 level it is only 3.4 eV.