About the Bohr Model
Niels Bohr's 1913 model of the atom treats an electron as orbiting the nucleus only in fixed, quantized energy levels labeled by a principal quantum number n = 1, 2, 3, ... An electron can jump between two levels by absorbing a photon (moving up) or emitting a photon (moving down), and the photon's energy exactly equals the difference between the two levels' energies. This calculator applies that model to a hydrogen-like atom — a nucleus of atomic number Z with a single orbiting electron, such as hydrogen (Z=1), He+ (Z=2), or Li2+ (Z=3).
Understanding the formula
The energy of an electron in level n of a hydrogen-like atom is E(n) = -13.6 eV × Z² / n², where 13.6 eV is the Rydberg energy (hydrogen's ionization energy) and the minus sign shows the electron is bound to the nucleus. For a transition between an initial level n1 and a final level n2, the photon energy is ΔE = |E(n2) - E(n1)|. That energy converts to a wavelength via λ = h·c / ΔE (Planck's constant h and the speed of light c) and to a frequency via f = ΔE / h = c / λ.
Reading the direction and series
- If n1 is greater than n2, the electron falls to a lower level and a photon is emitted; if n1 is less than n2, a photon must be absorbed to push the electron up.
- Transitions that end on n=1 form the Lyman series (ultraviolet), those ending on n=2 form the Balmer series (visible and near-UV, including the famous red Hα line), and those ending on n=3 form the Paschen series (infrared).
- Always include units in your answer — a wavelength without nm or a frequency without Hz is not meaningful.
Knowing the limits
The Bohr model is exact only for one-electron systems (hydrogen and hydrogen-like ions); it does not correctly predict energy levels for multi-electron atoms because it ignores electron-electron repulsion and orbital shape (s, p, d, f) effects captured by full quantum mechanics. It also omits fine-structure and relativistic corrections, which shift real spectral lines by a small amount. For hydrogen and hydrogen-like ions, though, the energy levels it predicts match measured spectra closely.