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Bohr Model Energy Calculator

Reviewed by the CalculatorHub team, edited by James Graham. Bohr energy levels from Bohr (1913); Rydberg constant from NIST CODATA 2018. Last verified 15 June 2026.

Niels Bohr's 1913 model of the hydrogen atom introduced quantized electron orbits and correctly predicted the hydrogen spectrum. The energy of the nth orbit is E_n = -13.6 eV / n^2, where the hydrogen ground state ionization energy 13.6 eV is equal to the Rydberg energy (alpha^2 m_e c^2 / 2). When an electron transitions from a higher level n2 to a lower level n1, it emits a photon with energy delta_E = E_n2 - E_n1 = 13.6 * (1/n1^2 - 1/n2^2) eV. This calculator also supports hydrogen-like ions (one electron, atomic number Z) where the energies scale as Z^2. Enter the principal quantum number n and optionally a transition to compute photon wavelength.

H = 1, He(+) = 2, Li(2+) = 3
Must be positive integer, n1 less than n2 for emission
Must be greater than n1
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Bohr model energy formula

E_n = -13.6 * Z^2 / n^2 eV
delta_E = 13.6 * Z^2 * (1/n1^2 - 1/n2^2) eV
lambda = h * c / (delta_E in joules) nm

For hydrogen (Z=1), ground state E1 = -13.6 eV. Ionization energy = 13.6 eV. The Rydberg constant R_H = 1.09678 x 10^7 m^-1 follows from 13.6 eV converted to wavenumbers.

Key hydrogen spectral lines

  • Lyman alpha (n=2 to n=1): 122 nm (UV), delta_E = 10.2 eV.
  • Balmer alpha H-alpha (n=3 to n=2): 656 nm (red visible), delta_E = 1.89 eV.
  • Balmer beta H-beta (n=4 to n=2): 486 nm (blue-green visible), delta_E = 2.55 eV.
  • Paschen alpha (n=4 to n=3): 1,875 nm (near IR), delta_E = 0.66 eV.

Bohr model: frequently asked questions

What is the Bohr model of the atom?

The Bohr model (1913) describes the hydrogen atom as an electron orbiting the nucleus in fixed circular orbits with quantized angular momentum. The energy of orbit n is E = -13.6 eV / n^2, where n = 1, 2, 3... The ground state (n=1) has energy -13.6 eV; the electron is bound. When the electron transitions between orbits, it emits or absorbs a photon with energy equal to the difference in orbit energies.

What is the Rydberg formula?

The Rydberg formula gives the wavelength of photons emitted or absorbed when a hydrogen electron transitions between levels n1 and n2: 1/lambda = R * (1/n1^2 - 1/n2^2), where R = 1.0974 x 10^7 m^-1 is the Rydberg constant. This is derived from the Bohr energy differences and agrees with the Lyman, Balmer, and Paschen spectral series.

What are the main spectral series of hydrogen?

Lyman series: transitions to n=1 (UV, 91 to 122 nm). Balmer series: transitions to n=2 (visible and near UV, 365 to 656 nm; the 656 nm H-alpha line is the red glow of nebulae). Paschen series: transitions to n=3 (near IR). Higher series fall in IR.

Why is the ground state energy negative?

By convention, zero energy is chosen for an electron infinitely far from the nucleus (ionized). Bound states have negative energy because energy must be added (provided) to free the electron. The ground state energy of -13.6 eV means 13.6 eV is needed to ionize hydrogen from its ground state.

Does the Bohr model apply to atoms other than hydrogen?

The Bohr model can be applied to hydrogen-like ions (one electron): He+, Li2+, etc. The formula becomes E = -13.6 * Z^2 / n^2 eV, where Z is the atomic number. He+ ground state energy = -54.4 eV. For multi-electron atoms, the Bohr model fails because electron-electron interactions are not included.

Official sources

Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.