Calculate orbital radius, energy levels, electron velocity, and photon wavelengths for hydrogen-like atoms using the Bohr model of the atom.
The Bohr model, proposed by Niels Bohr in 1913, was the first quantum model of the atom to successfully explain the discrete emission spectrum of hydrogen. In this model, electrons orbit the nucleus at fixed radii determined by their principal quantum number n, with each orbit corresponding to a specific quantized energy level. Although superseded by full quantum mechanics, the Bohr model remains remarkably accurate for hydrogen-like (single-electron) ions and is still widely taught as a foundational concept in physics and chemistry.
This calculator lets you explore the Bohr model for any hydrogen-like ion by specifying the atomic number Z and the principal quantum number n. You can compute the orbital radius (which scales as n²/Z times the Bohr radius a₀ ≈ 0.529 Å), the total energy of the electron (E = −13.6 Z²/n² eV), the electron's orbital velocity, angular momentum, and de Broglie wavelength. The transition mode lets you calculate the energy and wavelength of photons emitted or absorbed when an electron jumps between two shells, covering the Lyman, Balmer, Paschen, and higher spectral series.
Use the presets to quickly load common systems such as the hydrogen atom, He⁺, or Li²⁺ ions, or enter custom values to explore exotic high-Z ions and Rydberg states at large n.
Understanding the Bohr model is essential for students learning about atomic structure, spectroscopy, and the historical development of quantum theory. This calculator provides instant results for orbital radii, energies, velocities, and photon transitions — saving time on repetitive calculations in physics and chemistry courses. The shell table and energy level diagram give a visual overview that deepens intuition about how quantum numbers shape atomic properties.
Bohr Model Equations (hydrogen-like ion with atomic number Z): • Orbital radius: rₙ = a₀ · n² / Z, where a₀ = 0.529 Å (Bohr radius) • Energy: Eₙ = −13.6 · Z² / n² eV • Velocity: vₙ = 2.188 × 10⁶ · Z / n m/s • Transition photon energy: ΔE = 13.6 · Z² · (1/n_f² − 1/n_i²) eV • Photon wavelength: λ = hc / ΔE
Result: −3.4 eV, radius 2.116 Å
For hydrogen (Z=1) in the n=2 shell, the energy is −13.6/4 = −3.4 eV, and the orbital radius is 4 × 0.529 Å = 2.116 Å.
Niels Bohr introduced his atomic model in 1913 to resolve a critical failure of classical physics: according to Maxwell's equations, an orbiting electron should continuously radiate energy and spiral into the nucleus within nanoseconds. Bohr's revolutionary postulate was that electrons occupy stationary orbits where they do not radiate, and only emit or absorb energy when jumping between these orbits. This bold hypothesis correctly predicted the wavelengths of hydrogen's spectral lines to remarkable accuracy.
While the Bohr model works perfectly for hydrogen-like systems, it fails for multi-electron atoms, cannot explain fine structure or the Zeeman effect, and treats the electron as a classical particle on a fixed orbit rather than a probabilistic wave. The Schrödinger equation (1926) replaced the Bohr model with a full wave-mechanical treatment, where the "orbit" becomes an orbital — a probability distribution in three-dimensional space. Nevertheless, the Bohr model's energy formula Eₙ = −13.6 Z²/n² eV remains exactly correct for one-electron systems and is still used as a first approximation in many contexts.
Despite its limitations, the Bohr model framework is used in plasma physics (ionization balance calculations), astrophysics (recombination lines in nebulae), and laser physics (stimulated emission between Bohr levels). Rydberg atoms — atoms excited to Bohr orbits with n > 50 — are central to modern quantum computing and quantum simulation experiments, where their enormous dipole moments enable strong long-range interactions.
The Bohr model is a semi-classical atomic model where electrons orbit the nucleus in discrete circular orbits with quantized angular momentum L = nħ. It accurately predicts the energy levels and spectral lines of hydrogen-like atoms.
No. The Bohr model is exact only for one-electron systems (H, He⁺, Li²⁺, etc.). For multi-electron atoms, electron-electron repulsion makes the problem much more complex and requires full quantum mechanical treatment.
The Bohr radius a₀ ≈ 0.529 Å (5.29 × 10⁻¹¹ m) is the most probable distance of the electron from the nucleus in the ground state of hydrogen. It serves as the natural length scale for atomic physics.
Transitions to n=1 form the Lyman series (UV), to n=2 the Balmer series (visible), to n=3 the Paschen series (IR), and higher series (Brackett, Pfund) lie further in the infrared. Use this as a practical reminder before finalizing the result.
In the ground state of hydrogen, the electron moves at about 2.19 × 10⁶ m/s, roughly 0.73% of the speed of light. Velocity increases with Z and decreases with n.
A Rydberg atom has one or more electrons in a very high principal quantum number (n >> 1), giving the atom an enormous radius (scales as n²) and making it extremely sensitive to electric fields. Such atoms are used in quantum computing research.