Bohr model: the electron in a one-electron atom is allowed only certain orbits, and each one carries a fixed energy En = -13.6 Z2/n2 eV. This free calculator gives the energy of a level, the radius of its orbit, or the energy, wavelength and frequency of the photon released when the electron drops between two of them — with every step of the working shown.
What Is the Bohr Model Calculator?
The Bohr model calculator is a free online tool built on the formula En = -13.6 · Z2 / n2 eV. Enter the values you already know, in whichever units suit you, and it solves for energy level, transition wavelength or orbit radius and shows every step of the substitution. Hydrogen-like energy levels, transition wavelengths and Bohr orbit radii.
| Symbol | Quantity | Default unit | Also accepts | Example value |
|---|---|---|---|---|
| n | Principal quantum number | (1–20) | — | 3 |
| nf | Final level | (1–20) | — | 2 |
| Z | Atomic number | (1–10) | — | 1 |
How to calculate Bohr model energy levels
Every answer here comes from one equation. Multiply -13.6 eV by the square of the atomic number Z, then divide by the square of the principal quantum number n — En = -13.6 · Z2 / n2. That is the whole model. The 13.6 eV is not arbitrary: it is the energy needed to tear the electron out of ground-state hydrogen, so it is the depth of the deepest well the model has.
There are three things worth asking of it, and the calculator handles them as three modes. Energy level wants only n and Z and returns En in electronvolts and joules. Orbit radius takes the same two numbers through rn = n2a0/Z, where a0 = 5.29177×10-11 m is the Bohr radius. Transition asks for a second level, works out both energies, and takes the size of the gap between them.
That gap is where the light comes from. The photon carries exactly ΔE = |Ei − Ef|, and because λ = 1239.842 / ΔE with the energy in electronvolts, the wavelength drops out in nanometres with no unit juggling at all — 1239.842 eV·nm is simply hc written in the units this problem lives in. If you would rather come at a photon from its frequency, the photon energy calculator runs the same relationship the other way, and the volt to eV converter handles the electronvolt bookkeeping.
Two habits cause most of the wrong answers. The first is treating the levels as evenly spaced: they are not, and the 1/n2 makes the gaps collapse fast, so the jump from n = 2 to n = 1 releases 10.2 eV while the jump from n = 8 to n = 7 releases barely 0.065 eV. The second is raising Z for a neutral atom. Z here counts protons in a nucleus holding one electron, so Z = 2 means He+ and not helium; the moment a second electron joins, it screens the nucleus and this formula stops being true.
Worked example
Take the red line of hydrogen: an electron falling from n = 3 to n = 2 with Z = 1. The two levels are E3 = -13.6/9 = -1.511 eV and E2 = -13.6/4 = -3.400 eV, so the photon carries ΔE = |-1.511 − (-3.400)| = 1.889 eV. Its wavelength is λ = 1239.842 / 1.889 = 656.4 nm, deep red and squarely in the Balmer series — the line named for the lower level of the pair. The n = 3 orbit it started from has radius r3 = 32 × 5.29177×10-11 = 4.763×10-10 m, or 0.4763 nm.
Now strip a helium atom to one electron and run the same drop with Z = 2. Every energy scales by Z2 = 4, so the n = 2 to n = 1 transition that gives 10.2 eV in hydrogen gives 40.8 eV in He+, and the wavelength shortens by the same factor of four to 30.39 nm — out of the visible range entirely and well into the extreme ultraviolet.
Why it matters
The Bohr model is wrong in its picture and right in its numbers, which is a rare and useful combination. Electrons do not travel on circular tracks, but the energies this formula predicts match the measured hydrogen spectrum to better than a part in a thousand, and those spectral lines are how we identify elements in stars, calibrate spectrometers and date the expansion of the universe by their redshift. It is also the first place a physics course meets quantisation as an answer rather than a puzzle: the orbits are fixed because the electron’s de Broglie wavelength has to close on itself around the loop, and the sharp lines in a discharge tube are the direct visible consequence.
Frequently asked questions
Why is the energy negative?
The zero of energy is set at a free electron infinitely far from the nucleus and at rest. An electron bound to the atom has less energy than that, so its energy is negative, and the deeper it sits the more negative it gets. The ground state of hydrogen is -13.6 eV, which is another way of saying it takes 13.6 eV of energy to pull that electron away completely. As n rises the levels climb towards zero from below and the electron gets easier and easier to remove.
What units should I enter?
None. Every input here is a pure count: n is the principal quantum number and Z is the number of protons in the nucleus, and both are dimensionless whole numbers. The units live in the answers instead. Energies come out in electronvolts with the joule value beside them, radii in nanometres with the metre value beside them, wavelengths in nanometres and frequencies in hertz.
How do I find the wavelength of a transition?
Choose the Transition mode, enter the starting level as n and the finishing level as the final level, then read the wavelength from the results. The calculator works out both energies from E = -13.6 Z squared over n squared, takes the size of the difference, and divides 1239.842 by it. That constant is hc expressed in eV nm, so an energy in electronvolts divided into it gives a wavelength straight out in nanometres. For n = 3 down to n = 2 in hydrogen the answer is 656.4 nm, the red line you see in every hydrogen discharge tube.
What does Z mean and when do I change it?
Z is the number of protons in the nucleus. Leave it at 1 for hydrogen. Raise it only for a hydrogen-like ion, meaning an atom stripped down to a single remaining electron: Z = 2 is He+ and Z = 3 is Li2+, not neutral helium or lithium. The Bohr result only holds for one electron, because a second one would screen the nucleus and repel its partner. Within that limit every energy scales as Z squared and every radius shrinks as 1 over Z, so He+ is bound four times as tightly as hydrogen in an orbit half the size.
Why does the calculator refuse when the two levels are the same?
Because there is no transition to describe. If the start and end levels match, the energy difference is exactly zero, and the wavelength formula would divide 1239.842 by zero. Rather than print an infinity, the calculator stops and asks for a different final level. Physically nothing has happened: the electron has not moved, so no photon is emitted and none is absorbed.
References & formula source
- Ling, Sanny & Moebs — University Physics Volume 3 (OpenStax), section on Bohr’s model of the hydrogen atom.
- Eisberg & Resnick — Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, chapter on Bohr’s model of the atom.
- NIST Atomic Spectra Database — hydrogen energy levels and observed transition lines.
- Further reading: Bohr model — Wikipedia