Bohr Hydrogen Spectrum
Choose hydrogen energy levels and connect emission or absorption to photon energy, wavelength and orbital scale.
By Riano · Published · Updated · Report a correction
Bohr Hydrogen Spectrum
The Bohr model of the hydrogen atom provides an educational framework for understanding discrete energy levels and spectral lines. In this model, the electron is confined to circular orbits with specific quantized energies. The energy of the electron in the n-th level is given by En = -13.6 / n2 eV, where n is a positive integer (the principal quantum number). The radius of the orbit is rn = a0n2, where a0 is the Bohr radius.
When an electron transitions from an initial level ni to a final level nf, the atom's energy changes by ΔE = Ef - Ei. If nf < ni, the atom loses energy and a photon is emitted. If nf > ni, the atom gains energy by absorbing a photon. The photon energy is exactly equal to the magnitude of the atomic energy change. The corresponding photon wavelength λ is given by λ = hc / Ephoton.
Predicting a Paschen Series Transition
Consider a transition from the initial level ni = 4 to the final level nf = 3. We use the model constants hc = 1239.841984 eV·nm and a0 = 0.05291772109 nm.
- Calculate the initial energy: E4 = -13.6 / 42 = -13.6 / 16 = -0.85 eV.
- Calculate the final energy: E3 = -13.6 / 32 = -13.6 / 9 ≈ -1.51111111111 eV.
- Find the atomic energy change: ΔE = E3 - E4 ≈ -1.51111111111 - (-0.85) = -0.66111111111 eV.
- Since the final energy is lower than the initial energy, this is an Emission transition. The emitted photon has an energy of roughly 0.66111111111 eV.
- The emitted wavelength is λ = hc / 0.66111111111 ≈ 1239.841984 / 0.66111111111 ≈ 1875.3912363 nm (in the infrared Paschen series).
- The initial orbital radius is r4 = 0.05291772109 × 16 ≈ 0.84668353744 nm, and the final radius is r3 = 0.05291772109 × 9 ≈ 0.47625948981 nm.
Compare spectral transitions
Open the Bohr Hydrogen Spectrum module and use its labeled controls to explore the model.
- Set Initial level ni to 3 and Final level nf to 2. This represents the first line of the Balmer series (H-alpha). The module calculates the photon energy as approximately 1.88888888889 eV and the wavelength as 656.386932706 nm, characteristic of visible red light.
- Set Initial level ni to 1 and Final level nf to 3. The atom transitions from a lower to a higher energy state, meaning it undergoes an Absorption process, capturing a photon of roughly 12.0888888889 eV.
- Try setting both levels equal, e.g., Initial level ni to 1 and Final level nf to 1. The energy change is exactly 0 eV, and the wavelength is correctly marked as
Unavailable: no transition. - Drag the visual display to explore the drawn orbits up to n=6, and use Reset view to restore the centered framing.
Model Limits
The Bohr model is an early semi-classical approximation that successfully predicts the gross spectral lines of hydrogen but has significant limitations. It treats the electron as a localized particle moving in definitive macroscopic circular orbits, which contradicts modern quantum mechanical descriptions involving probability density wavefunctions. The stated educational constants omit the finite-nuclear-mass correction that slightly alters real hydrogen and deuterium spectra; the electron is not assigned infinite mass. Furthermore, it does not incorporate relativistic fine structure, Lamb shift, or hyperfine splitting. The displayed radii illustrate the historic classical model dimensions and are not measurable trajectories.
Source material
OpenStax University Physics Volume 3: Bohr's Model of the Hydrogen Atom details the energy levels, the calculation of transition photon energies, and discusses the limitations of this educational model.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Bohr Hydrogen Spectrum module.
Try Bohr Hydrogen Spectrum on Riano →