Bohr Model Calculator
Calculate hydrogen energy levels, orbital radius, and photon wavelengths with the Bohr model.
Enter quantum number n, and optionally a transition target level.
In 1913, Niels Bohr proposed that hydrogen electrons occupy fixed circular orbits at specific radii, each with a definite energy. The model was wrong in important ways — it only works for hydrogen and hydrogen-like ions, electrons do not actually follow classical circular paths, and it cannot predict spectral line intensities. But it got the energy levels exactly right.
Energy levels
Eₙ = -13.6 / n² eV
The negative sign means the electron is bound. For n=1 (ground state), E = -13.6 eV. For n=2, E = -3.4 eV. The levels converge toward zero as n increases — at n=∞ the electron is free.
Orbital radius
rₙ = n² × a₀ where a₀ = 0.529 Å (the Bohr radius)
The ground state radius is 0.529 Å. The n=2 orbital is 4× larger (2.116 Å). The n² scaling means higher orbitals grow rapidly.
Orbital velocity
vₙ = 2.187 × 10⁶ / n m/s
The ground state electron moves at about 0.7% of the speed of light. At n=2 it is half that, and so on.
Transitions and spectral lines
When an electron drops from level nᵢ to nᵠ (lower), it emits a photon. The wavelength is given by the Rydberg formula:
1/λ = R_H × (1/nᵠ² - 1/nᵢ²) where R_H = 1.097 × 10⁷ m⁻¹
Transitions to n=1: Lyman series (ultraviolet). Transitions to n=2: Balmer series, which includes the four visible hydrogen lines at 656 nm (red), 486 nm (blue-green), 434 nm (violet), and 410 nm (deep violet). Transitions to n=3: Paschen series (near-infrared).
To calculate a transition: enter n in the first field and the target level in the second. The calculator shows emission or absorption accordingly.
Where the model breaks down
Helium already requires full quantum mechanics to solve properly, because electron-electron repulsion is not in the Bohr model. The model also predicts zero angular momentum for the ground state, which contradicts quantum mechanics. Modern quantum theory replaced it, but the energy eigenvalues it predicts for hydrogen are still exactly correct.