Bohr Model Energy Level Calculator
Energy, orbital radius and electron speed for any level of a hydrogen-like atom, plus the photon emitted or absorbed between two levels — wavelength, frequency and spectral series.
How it works
Eₙ = −13.606 eV · Z² ÷ n²; rₙ = a₀·n² ÷ Z; vₙ = 2.19 × 10⁶ m/s · Z ÷ n; λ = hc ÷ (E₂ − E₁) = 1239.84 nm·eV ÷ ΔE
Bohr’s 1913 model quantises the electron’s angular momentum and reproduces the hydrogen spectrum exactly: energies scale as 1/n², radii as n², and transitions between levels emit photons whose wavelengths form the Lyman, Balmer and Paschen series. It also works for any one-electron ion by scaling with Z². It fails for multi-electron atoms and fine structure, but remains the quickest way to get hydrogen’s lines and the right order of magnitude for atomic energies.
Worked example
Hydrogen n = 3 → 2: E₃ − E₂ = −1.511 − (−3.401) = 1.889 eV → 656.3 nm, the red H-α line of the Balmer series.
Frequently asked questions
Why is the energy negative?
Zero is a free electron at rest; bound states lie below it, and the ionisation energy is the positive amount needed to reach zero.
Does the reduced mass matter?
Slightly — using the electron–proton reduced mass shifts hydrogen lines by 0.05%; the Rydberg constant here is the infinite-mass value.
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This calculator is for educational purposes. Double-check important results before acting on them.