The Bohr Radius Calculator computes the orbital radius of an electron in any hydrogen-like atom at any principal quantum number using rₙ = a₀ × n²/Z, where a₀ = 0.529 Å. The Bohr radius is the fundamental unit of atomic length in quantum chemistry — the natural scale of the atom.
0.000000000053
m
0.000000000053
m
52.9177
pm
0.529177
Å
-13.605693
eV
0.00729735
105.8354
pm
0.000000000053
m
0.000000000053
m
52.9177
pm
0.529177
Å
-13.605693
eV
0.00729735
105.8354
pm
The Bohr radius (a₀ = 5.291 × 10⁻¹¹ m = 0.529 Å) is the most probable distance between the proton and electron in a ground-state hydrogen atom — the natural length scale of the atom. Every other orbital radius in the hydrogen series scales from this value: the n=2 orbit is 4 times larger; n=3 is 9 times larger; the orbit expands as n². The Bohr radius calculator computes any orbital radius for hydrogen-like atoms at any quantum level.
rₙ = a₀ × n² / Z
where a₀ = 0.529177 Å = 5.29177 × 10⁻¹¹ m (the Bohr radius constant), n = principal quantum number, Z = atomic number of the nucleus.
Examples for hydrogen (Z=1): r₁ = 0.529 Å; r₂ = 0.529 × 4 = 2.116 Å; r₃ = 0.529 × 9 = 4.762 Å; r₄ = 0.529 × 16 = 8.466 Å. For He⁺ (Z=2, n=1): r₁ = 0.529/2 = 0.265 Å — half the hydrogen radius because the stronger nuclear charge pulls the electron closer. Use this online calculator for any combination. The Bohr model calculator computes energy levels and spectral wavelengths from the same atomic model.
In classical mechanics, the Bohr radius represents the exact orbital radius where the electrostatic attraction equals the centripetal force needed for circular orbital motion. In quantum mechanics, it represents the distance from the nucleus at which the 1s electron probability density is maximum — the most likely location to find the electron. The Bohr radius is a natural unit in atomic physics: measuring distances in units of a₀ (atomic units) simplifies equations by eliminating physical constants. At larger scales: a hydrogen atom (n=1) has radius ≈ 1 Å; a Rydberg atom at n=100 has radius ≈ 0.529 × 10,000 Å = 529 nm — roughly the size of a visible light wavelength.
The Bohr model predicts fixed orbital radii; quantum mechanics predicts probability distributions (orbitals). Despite this conceptual difference: the most probable radius of the quantum mechanical 1s orbital of hydrogen exactly equals the Bohr radius (0.529 Å); the average radius of the 1s orbital is 1.5 × a₀ (slightly larger due to the asymmetric probability distribution); Bohr radii provide accurate estimates for highly excited (Rydberg) states where the quantum number is large and the classical approximation improves. The physical constants calculators provide complementary atomic unit tools.
The orbital radius gives the Bohr model's predicted distance of the electron from the nucleus, displayed in meters, picometers, and angstroms. The energy is the total (kinetic + potential) energy of the orbital, with negative values indicating a bound state. The velocity ratio v/c indicates how relativistic the electron is — when this exceeds ~0.1, relativistic corrections become important and the simple Bohr model loses accuracy.
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The hydrogen ground state has radius a₀ = 52.9 pm, energy -13.6 eV, and electron speed of c/137. This is the fundamental reference point for all atomic physics.
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He⁺ (Z = 2) has half the radius (26.5 pm), four times the binding energy (54.4 eV), and twice the electron velocity compared to hydrogen. This is the simplest ion after hydrogen.
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