Hong Life CalculatorsUnderstanding the hydrogen spectrum calculator — concept and principle
When the electron in a hydrogen atom drops from a higher energy level to a lower one, it emits light of one precise wavelength. The Rydberg formula predicts those wavelengths. This calculator takes the lower and upper levels and the nuclear charge and returns the wavelength and the energy of the transition. It is used in chemistry and physics courses, astronomy (H-alpha imaging, spectral classification) and spectroscopy.
How it is calculated
1/λ = R × Z² × (1/n₁² − 1/n₂²)
E = hc/λ = 13.598 eV × Z² × (1/n₁² − 1/n₂²)
n₁ is the lower level and n₂ the higher level (n₂ > n₁, both whole numbers). Z is the nuclear charge: 1 for hydrogen, 2 for He⁺, 3 for Li²⁺, and so on for one-electron ions. The calculator uses R_H = 1.0967758 × 10⁷ m⁻¹, the Rydberg constant corrected for the hydrogen nucleus, and converts 1/λ to nm. The energy is hc/λ from the same R_H, so 13.598 eV replaces the infinite-mass Rydberg energy of 13.6057 eV and the two lines always agree.
The principle
In the Bohr model, and exactly in quantum mechanics for one-electron atoms, the energy of level n is −13.6 eV × Z²/n². A photon emitted in a jump from n₂ to n₁ carries away the energy difference, and λ = hc/ΔE. Because the nucleus is not infinitely heavy, the electron and nucleus orbit their common centre; replacing the electron mass with the reduced mass lowers the Rydberg constant slightly, from R∞ = 1.09737 × 10⁷ m⁻¹ to R_H.
Worked example
- Inputs: n₁ = 2, n₂ = 3, Z = 1.
- 1/2² − 1/3² = 0.25 − 0.11111 = 0.138889.
- 1/λ = 1.0967758 × 10⁷ × 0.138889 = 1.52330 × 10⁶ m⁻¹, so λ = 656.47 nm (H-alpha, red).
- E = 13.598 × 0.138889 = 1.88865095 eV.
Check: hc/λ = 1239.842 eV·nm ÷ 656.47 nm = 1.8887 eV, the same value. With R∞ instead of R_H you would get about 1.8897 eV, 0.05% more.
| Nuclear charge Z | Wavelength (nm) |
|---|---|
| 0.25 | 10,503.51403 |
| 0.5 | 2,625.878507 |
| 0.75 | 1,167.057114 |
| 1 | 656.4696267 |
| 1.25 | 420.1405611 |
| 1.5 | 291.7642785 |
| 2 | 164.1174067 |
How to read the result
Lines with n₁ = 1 form the Lyman series (ultraviolet, Lyman-alpha 121.6 nm); n₁ = 2 is the Balmer series, mostly visible (656, 486, 434, 410 nm); n₁ = 3 is the Paschen series in the infrared. As n₂ grows, lines crowd toward the series limit, about 364.6 nm for Balmer.
Common mistakes
- Putting the larger number in n₁, which gives a negative value.
- Comparing the computed vacuum wavelength with tables in air: H-alpha is 656.28 nm in air.
- Using the formula for multi-electron atoms such as neutral helium, where it does not apply.
Frequently asked questions
Why is H-alpha used in astronomy?
It is the strongest visible hydrogen line, so filters at 656 nm reveal ionised hydrogen in nebulae and the solar chromosphere.
What is the ionisation energy of hydrogen?
Set n₁ = 1 and let n₂ go to infinity: about 13.6 eV, or 91.2 nm.
Can it be used for absorption lines?
Yes. Absorption from n₁ to n₂ occurs at the same wavelength as emission from n₂ to n₁.
Sources: N. Bohr, Philosophical Magazine 26, 1 (1913) · CODATA 2018 recommended values, Rydberg constant (NIST) · NIST Atomic Spectra Database