Last updated: 2026-06-16

Bohr Model

The Bohr model describes a hydrogen-like atom as a single electron orbiting a positive nucleus in one of a set of allowed circular orbits. Each orbit has a fixed energy, and the electron emits or absorbs a photon only when it jumps between orbits. Introduced by Niels Bohr in 1913, it was the first model to explain the discrete spectral lines of hydrogen.

This calculator takes the principal quantum number $n$ and the atomic number $Z$, and returns the energy $E_n$, the orbit radius $r_n$ and the electron speed $v_n$.

How the model works

Bohr postulated that an electron’s angular momentum is quantized, which restricts it to specific orbits. Each allowed orbit has a definite radius, speed and energy. Because the energies are discrete, the photon emitted in a transition carries exactly the energy difference between two levels — producing the sharp lines seen in atomic spectra rather than a continuous rainbow.

Formulas

Here $n = 1, 2, 3, \dots$ is the principal quantum number, $Z$ is the atomic number, $a_0$ is the Bohr radius and $\alpha$ is the fine-structure constant.

Worked example

For ground-state hydrogen, n = 1 and Z = 1.

Energy:

Radius:

Speed:

The electron sits 0.0529 nm from the proton, moving at roughly 0.7% of the speed of light, bound by 13.6 eV.

Energy levels of hydrogen

The levels crowd together as $n$ grows and converge on zero, the ionization threshold. The 13.6 eV gap between the ground state (n = 1) and the free electron is hydrogen’s ionization energy.

Real-world relevance

The Bohr model accounts for the Balmer, Lyman and Paschen spectral series of hydrogen and gives the correct ionization energy. The same formulas, with the appropriate $Z$, describe one-electron ions such as He⁺ and Li²⁺. It remains a standard introduction to atomic structure and to the idea of quantized energy.

Limitation

The Bohr model is exact only for hydrogen-like (single-electron) systems. It does not reproduce the spectra of multi-electron atoms, fine structure or chemical bonding, and its definite orbits conflict with quantum uncertainty. Full quantum mechanics replaces the orbits with orbitals, though Bohr’s energy formula for hydrogen survives intact.