Bohr Model Calculator
Enter an atomic number and one or two principal quantum numbers to instantly get the orbital radius, electron velocity, binding energy, and angular momentum for any hydrogen-like atom or ion. Add a second (lower) quantum number to calculate the transition: photon wavelength, frequency, energy released, and the spectral series it belongs to. Works for hydrogen (Z=1), singly ionised helium He+ (Z=2), doubly ionised lithium Li2+ (Z=3), and any other one-electron ion.
What the Bohr model is
The Bohr model, proposed by Niels Bohr in 1913, describes a hydrogen-like atom as a tiny nucleus orbited by one electron moving in fixed, quantised circular paths called shells or orbits. Unlike the earlier Rutherford model, Bohr required that the electron only occupy orbits where its angular momentum is an integer multiple of h-bar (reduced Planck constant). This single postulate explains why atoms emit and absorb light only at specific wavelengths: an electron dropping from a higher shell (n1) to a lower shell (n2) releases exactly the energy difference as a photon, while absorbing a photon of the same energy causes the reverse jump. The model reproduces the Rydberg formula and predicts the hydrogen spectrum to high accuracy, making it the first quantum mechanical model of the atom.
Bohr model formulas used in this calculator
Four quantities define each orbital level. The orbital radius scales as r = n² × a₀ / Z, where a₀ = 52.9177 pm is the Bohr radius. The electron velocity is v = αc × Z/n, where α ≈ 1/137 is the fine-structure constant, giving 2187.7 km/s for hydrogen in n=1. The binding energy is E = -13.606 × Z²/n² eV (negative because the electron is bound). The quantised angular momentum is L = n × h-bar. For a transition from n_high to n_low, the photon energy is the difference in binding energies: ΔE = 13.606 × Z² × (1/n_low² - 1/n_high²) eV, and the wavelength follows from λ = hc/ΔE.
Hydrogen-like ions and the Z² scaling
The Bohr model applies exactly to any atom or ion with only one electron: hydrogen H (Z=1), singly ionised helium He+ (Z=2), doubly ionised lithium Li2+ (Z=3), and so on. Because energy scales as Z², the ground-state binding energy of He+ is 4 × 13.6 = 54.4 eV, and all spectral lines shift to shorter wavelengths by the same factor. Orbital radius scales as 1/Z, so the electron in He+ orbits roughly half as far from the nucleus as in hydrogen. This Z² scaling is a powerful way to predict the spectra of stripped ions seen in stellar atmospheres and hot plasmas.
Where the Bohr model works and where it fails
The Bohr model correctly predicts the energy levels and emission wavelengths of one-electron species. It also correctly accounts for the quantisation of angular momentum and gave physicists the first glimpse of wave-particle duality before de Broglie and Schrodinger formalised quantum mechanics. However, it fails for multi-electron atoms because it ignores electron-electron repulsion and spin. It also cannot explain fine structure (small splittings in spectral lines), the relative intensities of spectral lines, or the behaviour of atoms in magnetic and electric fields (Zeeman and Stark effects). Modern quantum mechanics, using the Schrodinger equation and quantum numbers n, l, m_l, and m_s, replaces the Bohr model for all but the simplest one-electron systems.
Hydrogen (Z=1) spectral series
| Series | n_final | Wavelength range | Region | Example line |
|---|---|---|---|---|
| Lyman | 1 | 91.2 - 121.6 nm | Ultraviolet | Ly-α: 121.6 nm (2→1) |
| Balmer | 2 | 364.6 - 656.3 nm | Visible / near-UV | H-α: 656.3 nm (3→2) |
| Paschen | 3 | 820.4 - 1875 nm | Near-infrared | P-α: 1875 nm (4→3) |
| Brackett | 4 | 1458 - 4051 nm | Infrared | Br-α: 4051 nm (5→4) |
| Pfund | 5 | 2279 - 7460 nm | Mid-infrared | Pf-α: 7460 nm (6→5) |
Key emission lines of atomic hydrogen, grouped by the series named after their discoverers. The final level (n_final) determines which series a transition belongs to.
Frequently asked questions
What is the Bohr model of the atom?
The Bohr model (1913) depicts the atom as a dense nucleus surrounded by electrons moving in fixed circular orbits at specific energies. The key postulate is that only orbits where the electron angular momentum equals an integer multiple of h-bar are allowed. This quantisation means electrons cannot spiral into the nucleus and explains why atoms emit light only at discrete wavelengths when electrons jump between levels.
How do I calculate the wavelength of a transition?
Enter the higher quantum number in n1 and the lower quantum number in n2, along with the atomic number Z. The calculator computes the photon energy using the Rydberg formula: ΔE = 13.606 × Z² × (1/n2² - 1/n1²) eV. It then converts to wavelength via λ = hc/ΔE. For example, the hydrogen Balmer H-alpha line has n1=3, n2=2, giving ΔE = 1.889 eV and λ = 656.3 nm (red light).
What is the Bohr radius?
The Bohr radius (symbol a₀) is the most probable distance between the proton and electron in a hydrogen atom in its ground state. Its value is approximately 52.9177 pm (picometres), or about 0.529 angstroms. For any hydrogen-like ion in quantum level n, the orbit radius is r = n² × a₀ / Z. This means the second orbit in hydrogen (n=2) has radius 4 × 52.9 = 211.7 pm.
What are the hydrogen spectral series?
The spectral series group hydrogen transitions by the lower level (n_final) the electron lands on. The Lyman series (n_final=1) lies in the ultraviolet. The Balmer series (n_final=2) includes visible lines: H-alpha at 656 nm (red), H-beta at 486 nm (blue-green), and others. The Paschen series (n_final=3) is in the near-infrared. Higher series - Brackett, Pfund, and beyond - fall progressively deeper into the infrared.
Does the Bohr model work for helium or other atoms?
It works only for one-electron (hydrogen-like) species. These include hydrogen (H), singly ionised helium (He+), doubly ionised lithium (Li2+), and any other atom stripped to a single electron. For neutral helium or heavier atoms with multiple electrons, electron-electron repulsion makes the Bohr model inaccurate, and the full quantum-mechanical treatment is needed.
Why are energy values negative in the Bohr model?
The convention is to set the reference energy at zero for an electron infinitely far from the nucleus (n = infinity). Any bound electron has lower energy than a free electron, so all bound-state energies are negative. The binding energy for hydrogen in n=1 is -13.606 eV, meaning you need to supply at least 13.606 eV to remove the electron completely (ionisation). Higher levels are less negative (less tightly bound): n=2 gives -3.401 eV, n=3 gives -1.512 eV, and so on.