What does the Bohr model calculator do?

The Bohr model calculator lets you choose a hydrogen-like ion and principal quantum numbers n and n'. It then computes the orbit radius, electron energy, and—if a transition is defined—the photon wavelength, frequency, and whether the photon is absorbed or emitted.

For which atoms is the Bohr model valid?

The Bohr model is quantitatively valid only for hydrogen-like systems: atoms or ions with a single electron, such as H, He+, Li2+, Be3+, etc. For multi-electron atoms, it gives only a rough qualitative picture of shells and energy levels.

How do you calculate the energy levels in the Bohr model?

In the Bohr model, the energy of level n in a hydrogen-like ion is E_n = -13.6 eV × Z^2 / n^2, where Z is the atomic number. The calculator uses this formula and converts the result to joules if needed.

How do you get the photon wavelength from a Bohr transition?

First compute the energy difference ΔE = E_n' - E_n between the final and initial levels. Then use E = h c / λ to get λ = h c / |ΔE|. The calculator does this automatically and reports the wavelength in nanometers and the corresponding frequency.

Bohr Model Calculator

Compute orbit radius, electron energy, and photon wavelength/frequency-distribution.html for hydrogen-like atoms using the Bohr model.

Bohr Model Interactive Calculator

Atom / ion (hydrogen-like)

Nuclear charge Z

Positive integer, e.g. 1 for H, 2 for He⁺, 3 for Li²⁺.

Initial level n

Final level n′ (optional)

Leave blank to compute only a single level.

Transition type

Bohr model: key formulas

The Bohr model describes a single electron moving in circular orbits around a nucleus with charge \(+Ze\). It works quantitatively for hydrogen-like ions (one electron only).

Orbit radius

\[ r_n = \frac{n^2}{Z} a_0 \]

where \(n = 1,2,3,\dots\) is the principal quantum number, \(Z\) is the atomic number, and \(a_0 \approx 5.29177 \times 10^{-11}\,\text{m}\) is the Bohr radius.

Energy levels

\[ E_n = -\frac{Z^2}{n^2} \, 13.6\ \text{eV} \]

Negative energy means the electron is bound. As \(n \to \infty\), \(E_n \to 0\) and the electron becomes free.

Photon energy, wavelength and frequency

\[ \Delta E = E_{n'} - E_n \]

\[ |\Delta E| = h\nu = \frac{hc}{\lambda} \]

If \(\Delta E < 0\), a photon is emitted (emission line). If \(\Delta E > 0\), a photon is absorbed (absorption line).

Spectral series (hydrogen)

Lyman series: final level \(n' = 1\) (ultraviolet).
Balmer series: final level \(n' = 2\) (visible and near-UV).
Paschen series: final level \(n' = 3\) (infrared).
Brackett, Pfund, Humphreys: \(n' = 4,5,6\) (infrared).

How to use the Bohr model calculator

Select a hydrogen-like ion (or choose “Custom Z” and enter any positive integer Z).
Enter the initial level \(n\). For the ground state, use \(n = 1\).
Optionally enter a final level \(n'\) to define a transition.
Click Calculate to see orbit radius, energy, and photon properties.

Worked example: Balmer-α line of hydrogen

Consider an electron dropping from \(n = 3\) to \(n' = 2\) in hydrogen (\(Z = 1\)).

\(E_3 = -13.6 / 3^2 \approx -1.51\ \text{eV}\)
\(E_2 = -13.6 / 2^2 = -3.40\ \text{eV}\)
\(\Delta E = E_2 - E_3 \approx -1.89\ \text{eV}\) (photon emitted)
\(\lambda = hc / |\Delta E| \approx 656\ \text{nm}\) (red light, Balmer-α)

Enter \(Z = 1\), \(n = 3\), \(n' = 2\) in the calculator to reproduce this result.

Limitations of the Bohr model

Accurate only for one-electron systems (H, He⁺, Li²⁺, …).
Assumes circular orbits and ignores electron spin and relativistic effects.
Replaced by full quantum mechanics (Schrödinger equation) for multi-electron atoms.