Bohr Model Calculator

The Bohr radius a₀ = 5.29177210903×10⁻¹¹ m is the most probable electron–nucleus distance in hydrogen's ground state. This calculator gives r₁ = 52.9465 pm (0.052947 nm) for hydrogen n=1, including the reduced-mass correction; the textbook infinite-nuclear-mass value is 52.9177 pm. Radius scales as rₙ = n²a₀/Z, so n=2 sits at 211.7862 pm and n=3 at 476.5189 pm.

E₁ = −13.5983 eV for hydrogen (n=1), including the reduced-mass correction — the infinite-mass Bohr value is −13.6057 eV. The measured ionization energy of hydrogen is 13.59844 eV, which is exactly why this calculator applies the correction by default instead of the textbook value.

Eₙ = −13.6/n² eV is the handy approximation. Full-precision values for hydrogen: E₂ = −3.39957 eV, E₃ = −1.510921 eV, E₄ = −0.849893 eV, E₅ = −0.543931 eV, E₆ = −0.377730 eV. For hydrogen-like ions multiply by Z²: He⁺ ground state −54.4153 eV, Li²⁺ ground state −122.4417 eV.

121.5684 nm (ultraviolet), from the n=2→1 transition with ΔE = 10.19872 eV. You will often see 121.567 nm quoted — that comes from a rounded Rydberg constant; 121.5684 nm is the full-precision CODATA 2018 value with reduced-mass correction. Without the correction the Bohr model gives 121.5023 nm.

656.4696 nm — the famous red line of the Balmer series (n=3→2), with photon energy ΔE = 1.88865 eV. Its siblings are H-beta at 486.2738 nm (n=4→2, blue-green) and H-gamma at 434.1730 nm (n=5→2, violet).

Each series is named by its lower level n₁: Lyman (n₁=1, ultraviolet), Balmer (n₁=2, visible), Paschen (n₁=3, infrared), Brackett (n₁=4), Pfund (n₁=5), Humphreys (n₁=6). Example: Paschen-alpha (n=4→3) is 1875.6274 nm, deep in the infrared; Lyman-beta (n=3→1) is 102.5734 nm.

Yes — it works for any one-electron (hydrogen-like) ion: scale energies by Z² and radii by 1/Z. He⁺ ground state is −54.4153 eV and its 2→1 line sits at 30.3797 nm; Li²⁺ ground state is −122.4417 eV. It fails for multi-electron atoms because the inner electrons screen the nuclear charge.

Muonium (μ⁺e⁻) is an exotic hydrogen-like atom: a positive muon orbited by an electron, studied in precision spectroscopy. Because the muon is only ~207× heavier than the electron, μ/mₑ = 0.995187, so its ground state is −13.54021 eV — about 0.5% less bound than hydrogen — and its Lyman-alpha line is at 122.0899 nm instead of 121.5684 nm.

For hydrogen's ground state v₁ = 2.187691×10⁶ m/s — about 0.73% of the speed of light (c/137.036, the fine-structure constant at work). One orbit takes T₁ = 1.5207×10⁻¹⁶ s. Speed scales as Z/n.

Real nuclei are not infinitely heavy — nucleus and electron orbit their common center of mass. Including the reduced mass μ shifts every wavelength slightly: Lyman-alpha computes as 121.5023 nm without it and 121.5684 nm with it, matching measurement. The calculator applies it by default for H, He⁺, Li²⁺ and muonium (uncheck the box to see pure textbook values).

Three honest limits: (1) multi-electron atoms — screening breaks the Z² scaling (it predicts helium's first ionization as 54.4 eV vs the measured 24.58 eV); (2) fine structure and the Lamb shift, which split lines the model says are single; (3) electrons do not travel in circular orbits — quantum mechanics replaced them with probability clouds in 1926. Use this tool for hydrogen-like spectra, not for chemistry.

Enter the wavelength in the “Identify λ” tab. Example: 486.27 nm identifies as hydrogen 4→2 (H-beta, Balmer series) — predicted 486.2738 nm, a 0.0008% difference. The tool ranks the five closest hydrogen transitions so you can judge ambiguous cases.