Bohr Model Energy Level Calculator
Find the quantized energy levels, orbital radii and transition photon energy or wavelength of a hydrogen-like atom using the Bohr model.
Wavelength: 656.471 nm
- 1
Energy at initial level n_i
−13.6 × 1 ÷ 3² = -1.511 eV - 2
Energy at final level n_f
−13.6 × 1 ÷ 2² = -3.4 eV - 3
Energy difference ΔE
13.6 × 1 × (1/2² − 1/3²) = 1.889 eVNegative ΔE means a photon is emitted; positive means absorbed. - 4
Photon energy absorbed |ΔE|
|1.889| = 1.889
How does this calculator work?
The Bohr model gives a hydrogen-like atom energy E_n = −13.6·Z²/n² eV and radius r_n = 0.529·n²/Z Å. A jump from n_i to n_f changes energy by ΔE = 13.6·Z²·(1/n_f² − 1/n_i²) eV; the emitted or absorbed photon carries |ΔE| with wavelength λ = 1240/|ΔE| nm.
Formula
How this is calculated
The Bohr model treats a single electron orbiting a nucleus of charge +Z. Its bound energy at principal quantum number n is E_n = −13.6·Z²/n² electronvolts, where 13.6 eV is the Rydberg energy. Energies are negative because the electron is bound; they rise toward zero as n grows. The orbital radius scales as r_n = 0.529·n²/Z ångström, with 0.529 Å being the Bohr radius for hydrogen's ground state.
A transition from an initial level n_i to a final level n_f changes the energy by ΔE = 13.6·Z²·(1/n_f² − 1/n_i²) eV. When n_f < n_i the electron drops, ΔE is negative and a photon is emitted; when n_f > n_i energy must be absorbed. The photon's energy equals |ΔE|, and its wavelength follows λ = 1240/|ΔE| nm from the relation E = hc/λ with hc ≈ 1240 eV·nm.
Inputs are the atomic number Z and the two integer levels. The model is exact only for one-electron systems (H, He+, Li2+, ...); for multi-electron atoms it is an approximation that ignores electron–electron repulsion, spin–orbit coupling and relativistic effects. Quantum numbers must be positive integers, so entries are rounded, and identical levels give no transition.
Frequently asked questions
A negative energy means the electron is bound to the nucleus. Zero energy corresponds to a free electron, so deeper (more negative) levels are more tightly bound; the n=1 ground state is the lowest.
If the electron falls to a lower level (n_f < n_i), ΔE is negative and a photon carrying |ΔE| of energy is emitted. If it jumps to a higher level (n_f > n_i), that energy must be absorbed instead.
It is accurate only for hydrogen-like (single-electron) species such as H, He+ and Li2+. For multi-electron atoms it ignores electron repulsion and finer effects, so it gives only rough estimates.
Also known as
TG we-Calculate Editorial Team. (2026). Bohr Model Energy Level Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/bohr-model-energy-calculator
TG we-Calculate Editorial Team. "Bohr Model Energy Level Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/bohr-model-energy-calculator.
TG we-Calculate Editorial Team, "Bohr Model Energy Level Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/bohr-model-energy-calculator
@misc{wecalculate_bohr_model_energy_calculator, title = {Bohr Model Energy Level Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/bohr-model-energy-calculator}}, year = {2026}, note = {TG we-Calculate} }
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