Bohr Model Calculator

When Niels Bohr proposed his atomic model in 1913, he solved a problem that had stumped classical physics for decades: why do excited hydrogen atoms emit light at specific discrete wavelengths rather than a continuous spectrum? His answer — electrons orbit the nucleus only at specific allowed energy levels, and emit or absorb photons when jumping between them — was a revolutionary step toward quantum mechanics. The Bohr model calculator applies his equations to any hydrogen-like transition.

Bohr Model Energy Level Formula

The energy of electron level n in a hydrogen-like atom:

Eₙ = −13.6 eV × Z² / n²

where Z = atomic number (Z = 1 for hydrogen), n = principal quantum number (1, 2, 3, ...). The energy is negative because electrons are bound — n = 1 (ground state) has the lowest (most negative) energy; higher n means higher (less negative) energy and larger orbital.

Transition energy: ΔE = 13.6 eV × Z² × (1/n₁² − 1/n₂²)

If ΔE > 0: photon is emitted (emission); if ΔE < 0: photon is absorbed. Photon wavelength: λ = hc/ΔE = 1240 eV·nm / ΔE. Use this online calculator to compute any hydrogen transition wavelength.

The Hydrogen Spectral Series

Named groups of hydrogen emission lines by the final quantum number (n₁):

• Lyman series (n₁ = 1): transitions to ground state; UV light (91–122 nm)
• Balmer series (n₁ = 2): transitions to n = 2; visible light (365–656 nm); the red Hα line (656 nm, n=3→2) is the most prominent hydrogen emission line in the universe
• Paschen series (n₁ = 3): near-infrared (820 nm – 1875 nm)
• Brackett series (n₁ = 4): mid-infrared (1460 nm – 4050 nm)
• Pfund series (n₁ = 5): far-infrared

The Bohr radius calculator and atomic physics calculators provide complementary quantum chemistry tools.

Bohr Radius: The Scale of Electron Orbits

The Bohr radius (a₀ = 5.29 × 10⁻¹¹ m = 0.529 Å) is the most probable distance between the proton and electron in ground-state hydrogen. Orbital radius at level n: rₙ = a₀ × n² / Z. At n = 1: r₁ = 0.529 Å (for hydrogen). At n = 2: r₂ = 4 × 0.529 = 2.116 Å — four times larger. At n = 3: r₃ = 9 × 0.529 = 4.76 Å. The orbital radius scales as n² — electron orbits expand quadratically with quantum number.

Limitations of the Bohr Model

The Bohr model works exactly for hydrogen and hydrogen-like ions (He⁺, Li²⁺) but fails for multi-electron atoms. It cannot explain: fine structure (spin-orbit coupling); the Zeeman effect (splitting in magnetic fields); the relative intensities of spectral lines; or molecular bonding. The quantum mechanical model (Schrödinger equation, electron probability clouds) supersedes Bohr for accuracy, but Bohr's energy level formula remains exact for hydrogen and is foundational for understanding spectroscopy, atomic structure, and quantum mechanics pedagogy.