Bohr Radius Calculator

Bohr radius: rₙ = a₀·n²/Z

In the atomic model Bohr proposed back in 1913, the radius of the n-th allowed electron orbit comes out to rₙ = a₀·n²/Z. Here a₀ = 5.29·10⁻¹¹ m is the Bohr radius, which is hydrogen's ground-state orbit, n is the principal quantum number, and Z the atomic number. Consider hydrogen, where Z=1. At n=1 you get r₁ ≈ 0.529 Å. Move up to n=2 and r₂ ≈ 2.12 Å, which is four times larger. The model accounted for hydrogen's discrete spectrum. Full quantum mechanics later replaced it (Schrödinger, 1926), swapping the fixed orbits for probability clouds, but it remains one of the easiest ways to picture how small an atom really is.

Applications

You'll run into it in introductory chemistry and physics courses, in rough estimates of atomic size (~10⁻¹⁰ m), and when comparing hydrogen-like ions such as He⁺ and Li²⁺. Computational chemistry relies on it too, where it serves as the baseline scale, the atomic unit of length.

FAQ

Why does the radius grow as n²? When you take Bohr's quantization condition (angular momentum = n·ħ) and balance it against the Coulomb force, the algebra produces an n² dependence. That's why higher-energy orbits end up so far from the nucleus.

Is the Bohr model still valid? Only for hydrogen-like atoms, and only as a rough mental picture. Modern quantum mechanics treats electrons as orbitals, meaning probability distributions rather than classical orbits.

What is the Bohr radius a₀ exactly? 5.29177·10⁻¹¹ m. The number falls out of fundamental constants like ℏ, the electron mass, the elementary charge and the Coulomb constant, and it also serves as the atomic unit of length.