# Bohr Model Calculator

URL: https://speedycalculator.com/calculators/physics/bohr-model/
Last updated: 2026-10-30

> The Bohr model calculator finds the energy levels of a hydrogen-like atom and the energy, frequency and wavelength of the photon emitted or absorbed when an electron jumps between two levels.

## Inputs

- Atomic number Z
- Initial level n_i
- Final level n_f

## Results

- Initial energy E_i
- Final energy E_f
- Photon energy |ΔE|
- Photon frequency
- Photon wavelength
- Transition

## Formula

```
E_n = −13.6057 eV·Z²/n²,  ΔE = E_f − E_i,  λ = hc/|ΔE|
```

## Explanation

Niels Bohr's 1913 model explained hydrogen's spectral lines: electrons occupy fixed energy levels and emit or absorb light of precise wavelengths when they jump. It works perfectly for one-electron atoms.

### How to use the Bohr Model Calculator

- **Atomic number Z**
- **Initial level n_i**
- **Final level n_f**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the atomic number (1 for hydrogen) and the initial and final principal quantum numbers.

### Inputs explained

FieldWhat to enterAtomic number ZA numberInitial level n_iA numberFinal level n_fA number

### What each result means

- **Initial energy E_i** = -13.605693×[Atomic number Z]^2÷[Initial level n_i]^2
- **Final energy E_f** = -13.605693×[Atomic number Z]^2÷[Final level n_f]^2
- **Photon energy |ΔE|** = abs([Final energy E_f] - [Initial energy E_i])
- **Photon frequency** = [Photon energy |ΔE|]×1.602176634e-19÷6.62607015e-34
- **Photon wavelength** = 6.62607015e-34×299792458÷([Photon energy |ΔE|]×1.602176634e-19)

### Example values

How the main result changes when **Atomic number Z** changes and everything else stays at the example values above:

Atomic number ZPhoton energy |ΔE| (eV)11.88971.252.95261.54.251827.5587

## Worked example

Inputs — Atomic number Z: 1; Initial level n_i: 3; Final level n_f: 2.

Results — Initial energy E_i = -1.511744 eV; Final energy E_f = -3.401423 eV; **Photon energy |ΔE| = 1.88968 eV**; Photon frequency = 456,922,490,348,751 Hz; Photon wavelength = 656.112282 nm; Transition = Emission.

## Key terms

- **Energy level**: Allowed energy of a bound electron.

## FAQ

**What is the ground-state energy of hydrogen?**
−13.6 eV.

**Why is the 3 → 2 line red?**
It emits 656 nm light – the Balmer H-alpha line.

**Does the Bohr model work for helium?**
Only for one-electron ions such as He⁺.

**What replaced the Bohr model?**
Quantum mechanics (Schrödinger's equation).


## Sources

- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)