Bohr Hydrogen Energy Level Calculator
Work out the energy levels of hydrogen and hydrogen-like ions, and the wavelength of the light a jump between levels releases.
What it does
Before quantum mechanics, the hydrogen spectrum was a puzzle. Balmer had found a formula that fitted the visible lines with whole numbers, and the puzzle was why an atom should emit light only at a set of fixed wavelengths. Niels Bohr answered it in 1913 with an idea that looks wrong until you see how well it works: the electron in a hydrogen atom is allowed to sit only on certain orbits, and it changes orbit by absorbing or emitting a photon.
OpenStax Chemistry puts the model in one line. The energies are quantised, E n equals minus k over n squared, and inserting that into the energy of a photon gives the Rydberg equation, one over lambda equals the Rydberg constant times one over n1 squared minus one over n2 squared. Bohr calculated the Rydberg constant from his model and got excellent agreement with the measured value, which had been known to high precision already.
The model has a simple extension. OpenStax notes that the Bohr model involves only a single electron, so it also applies to the ions He plus, Li two plus and Be three plus, which differ from hydrogen only in their nuclear charge. Everything scales with Z squared: the energies are deeper, the wavelengths are shorter, and the orbits are tighter.
This page runs those formulas. Choose the species, pick the two levels, and it returns the energy of each level, the wavelength and photon energy of the transition, the radius of the lower orbit and the name of the spectral series the line belongs to. That is enough to work out which colour or which part of the ultraviolet a given jump produces.
Two limits are worth keeping in mind. The Bohr model describes hydrogen-like atoms only, because it ignores the interaction between electrons, so it cannot even get helium right. And the Rydberg constant used here is the one for an infinitely heavy nucleus, so the wavelengths come out a fraction of a picometre away from the measured values for real hydrogen.
How to use
- Pick the species. Hydrogen is the usual case, and the ions behave the same way with the nuclear charge in the formula.
- Enter the lower level the electron falls to, starting at 1 for the ground state.
- Enter the upper level the electron falls from. It has to be above the lower level.
- Read the energy of both levels. They are negative because the electron is bound, and the zero is the electron at rest outside the atom.
- Read the wavelength of the photon released and its energy, then check the series name to see where the line falls in the spectrum.
- Read the orbit radius in the lower level, which grows with the square of the level number.
Formula
- n
- Level number, a positive whole number starting at 1.
- Z
- Nuclear charge of the atom, 1 for hydrogen.
- k
- Constant 2.1798723611e-18 J, the binding energy of a hydrogen electron in the ground state.
- a0
- Bohr radius, 5.29177210903e-11 m.
- R
- Rydberg constant for an infinitely heavy nucleus, 1.0973731568e7 per metre.
For a real hydrogen atom the Rydberg constant is slightly smaller, 10967758.34 per metre, because the nucleus is not infinitely heavy and the electron orbits the centre of mass instead of the nucleus. That difference moves the Balmer alpha line from 656.11 nm to 656.28 nm, which is the value measured in the laboratory. The energies are negative, and a value of zero means the electron has been removed completely, so the ionisation energy from the ground state is 13.6057 eV.
Worked example
Example: hydrogen dropping from level 3 to level 2
Defaults: hydrogen, a lower level of 2 and an upper level of 3.
- Level 2 energy: E = -k / 2 squared = -2.1798723611e-18 / 4 = -5.4496809028e-19 J, which is -3.4014 eV.
- Level 3 energy: E = -k / 3 squared = -2.1798723611e-18 / 9 = -2.4220804012e-19 J, which is -1.5117 eV.
- The energy released is 1.8897 eV, so the photon carries that much energy.
- Wavelength from the Rydberg formula: one over lambda = 1.0973731568e7 x (1/4 - 1/9) = 1.5241294e6 per metre, so lambda = 656.11 nm.
- Orbit radius in level 2: r = 4 x 0.0529177 nm = 0.2117 nm.
- The lower level is 2, so the line belongs to the Balmer series, visible light.
- The tool shows 656.11 nm, 1.8897 eV, -3.4014 eV, -1.5117 eV, 0.2117 nm and the Balmer series, visible light.
Reference table
| Transition | Wavelength | Photon energy | Series |
|---|---|---|---|
| Level 2 to 1 | 121.50 nm | 10.2043 eV | Lyman, ultraviolet |
| Level 3 to 1 | 102.52 nm | 12.0939 eV | Lyman, ultraviolet |
| Level 3 to 2 | 656.11 nm | 1.8897 eV | Balmer, visible |
| Level 4 to 2 | 486.01 nm | 2.5511 eV | Balmer, visible |
| Level 4 to 3 | 1874.61 nm | 0.6614 eV | Paschen, infrared |
| Level 5 to 4 | 4050.08 nm | 0.3061 eV | Brackett, infrared |
Limits
- The model covers one electron atoms and ions only. It cannot describe helium or anything with more electrons.
- Fine structure, spin and the Lamb shift are not included, so the numbers are the simple textbook values.
- The Rydberg constant used assumes an infinitely heavy nucleus, so real hydrogen lines sit about 0.17 nm away for the Balmer series.
- Only transitions between two levels are covered, so the page does not list the relative strength of each line.
Common mistakes
- Using the upper level number for the lower level. The energy is released when the electron falls from the higher level to the lower one, and the series is named after the lower level.
- Reading the energies as positive. They are negative because the electron is bound, and zero is the free electron.
- Expecting the energy to scale with n. It scales with one over n squared, so the levels get closer together as n grows.
- Forgetting that the orbit radius also scales with n squared, so level 100 is a large orbit with almost no binding energy.
- Applying the model to helium or lithium metal. Those need a real quantum treatment with electron interactions.
Try it yourself
Quick checks. Answers sit tucked under each question.
What wavelength does a jump from level 3 to level 2 give in hydrogen?
656.11 nm. One over lambda equals 1.0973731568e7 x (1/4 - 1/9), which gives 1.5241294e6 per metre, so lambda = 656.11 nm.
What is the energy of level 2 in hydrogen?
-3.4014 eV, which is -k over n squared with k = 2.1798723611e-18 J and n = 2.
How large is the level 3 orbit of hydrogen compared with level 2?
It is 9 over 4, or 2.25 times larger, because the radius goes with n squared. Level 2 sits at 0.2117 nm and level 3 at 0.4763 nm.
Questions
What does the Bohr model say?
That the electron in a hydrogen-like atom can only occupy certain orbits, and that it moves between them by absorbing or emitting a photon whose energy is the difference between the levels.
What is the Rydberg equation?
One over lambda equals R times one over n1 squared minus one over n2 squared, with R the Rydberg constant. It gives the wavelength of every line of the hydrogen spectrum from the two level numbers alone.
Which series is which?
The series is named after the level the electron falls to. Level 1 is Lyman in the ultraviolet, level 2 is Balmer in the visible, level 3 is Paschen in the infrared, level 4 Brackett and level 5 Pfund.
Why are the level energies negative?
Because the electron is bound and needs energy to escape. Zero is the energy of a free electron at rest, so the binding energy of the ground state of hydrogen is 13.6057 eV.
Does the model work for helium?
No. It handles a single electron, so it works for hydrogen and the ions He plus, Li two plus and Be three plus. Helium with two electrons falls outside the model.
What is the Bohr radius?
The radius of the ground state orbit of hydrogen, 5.29177210903e-11 m. Every other radius is that value multiplied by n squared and divided by the nuclear charge.
Sources
- OpenStax Chemistry 2e, 6.2: the Bohr model and the Rydberg equation
- OpenStax College Physics 2e, 30.3: Bohr's theory of the hydrogen atom
- Chemistry LibreTexts: the Bohr model and quantised energy levels