What this calculator does
The Bohr model gives exact energy levels for any atom with a single electron. For hydrogen the ground state sits at −13.606 eV with a radius of 0.0529 nm, the Bohr radius, and both figures fall directly out of the model.
Energies scale with the square of atomic number and inversely with the square of the level number, so a singly ionised helium ion has levels four times deeper than hydrogen's while its orbits are half the size.
The formula
Energy is −13.606 eV times atomic number squared divided by level number squared. Radius is the Bohr radius times level number squared divided by atomic number. The ionisation energy from a level is simply the magnitude of its energy.
| Term | Meaning |
|---|---|
| Principal quantum number (n) | The energy level, starting at 1 for the ground state. |
| Bohr radius | 0.0529 nm, the ground-state orbital radius of hydrogen. |
| Ionisation energy | The energy needed to remove the electron entirely from that level. |
The inputs explained
| Field | What to enter |
|---|---|
| Atomic number (Z) | The atomic number. Hydrogen is 1, singly ionised helium is 2. The model only applies with one electron present. |
| Principal quantum number (n) | The principal quantum number of the level, a positive whole number. |
When to use it
Finding an ionisation energy
The energy needed to free the electron from a given level is the magnitude of that level's energy.
Understanding spectral lines
The difference between two levels gives the photon energy emitted in a transition between them.
Comparing hydrogen-like ions
Changing atomic number shows how much more tightly a larger nuclear charge binds the same single electron.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do energy and radius change with the level?
Successive energy levels of the hydrogen atom.
| Level (n) | Energy of level n (En) | Orbital radius (rn) |
|---|---|---|
| n = 1 | -13.606 eV | 0.0529 nm |
| n = 2 | -3.401 eV | 0.2117 nm |
| n = 3 | -1.512 eV | 0.4763 nm |
| n = 4 | -0.8504 eV | 0.8467 nm |
Questions
Why are the energies negative?
Because zero is defined as the electron being free and at rest, infinitely far away. A bound electron has less energy than that, so its value is negative, and the magnitude is the energy needed to liberate it.
Does the Bohr model actually work?
For one-electron systems it gives the correct energies, which is why it survives in teaching. It fails for atoms with more than one electron and does not describe orbital shapes correctly, which required full quantum mechanics.
What is special about 13.6 eV?
It is the ionisation energy of hydrogen from its ground state, one of the most frequently quoted numbers in atomic physics. It emerges from the fundamental constants rather than being measured and inserted.
Why do levels crowd together at high n?
Because energy depends on one over n squared, so the gaps shrink rapidly. The levels converge toward zero, which is the ionisation limit, and that crowding is visible as the series limit in spectra.
For the wavelengths emitted between these levels, see the Rydberg equation calculator. For the photon energies involved, see the photon energy calculator.