Hydrogen Emission Spectrum & Rydberg Formula
The Rydberg equation calculator helps you compute the wavelength, frequency, and energy of light emitted (or absorbed) when an electron transitions between energy levels in a hydrogen-like atom. A hydrogen-like atom has only one electron — examples include H, He⁺, Li²⁺, and Be³⁺.
The Rydberg Formula
The wavelength λ of the emitted photon is given by:
1/λ = R·Z²·(1/n₁² − 1/n₂²)
where:
- λ — wavelength of emitted light (in vacuum)
- R — Rydberg constant ≈ 1.0974 × 10⁷ m⁻¹
- Z — atomic number (Z = 1 for hydrogen)
- n₁ — principal quantum number of the lower (final) state (n₁ ≥ 1)
- n₂ — principal quantum number of the upper (initial) state (n₂ > n₁)
From the wavelength, the calculator also derives:
- Frequency: ν = c / λ (in THz)
- Photon energy: E = h·ν (in eV and J)
- Wavenumber: 1/λ (in m⁻¹ or cm⁻¹)
Hydrogen Emission Spectrum — Spectral Series
According to the Bohr model, electrons orbit the nucleus only at discrete energy levels. When an electron drops from a higher level (n₂) to a lower level (n₁), it releases a photon with a specific wavelength. The collection of all possible transitions forms the hydrogen emission spectrum, organised into named series:
| Series | Final level n₁ | Initial level n₂ | Spectrum region |
|---|---|---|---|
| Lyman | 1 | ≥ 2 | Ultraviolet |
| Balmer | 2 | ≥ 3 | Visible / UV |
| Paschen | 3 | ≥ 4 | Near-Infrared |
| Brackett | 4 | ≥ 5 | Infrared |
| Pfund | 5 | ≥ 6 | Mid-Infrared |
| Humphreys | 6 | ≥ 7 | Far-Infrared |
The Balmer series is the most familiar — its lines fall in the visible range and give hydrogen its characteristic red (656 nm), blue-green (486 nm), violet (434 nm), and deep violet (410 nm) colours.
Hydrogen-Like Atoms
The formula works for any ion with a single electron. For such an ion with atomic number Z, the Rydberg constant is effectively multiplied by Z². This means the spectral lines shift to shorter wavelengths (higher energy) as Z increases:
- H (Z = 1) — standard hydrogen spectrum
- He⁺ (Z = 2) — wavelengths four times shorter than hydrogen for the same transition
- Li²⁺ (Z = 3) — nine times shorter
- Be³⁺ (Z = 4) — sixteen times shorter
Spectroscopy Applications
The Rydberg equation is fundamental to atomic spectroscopy — the technique of identifying chemical elements from their characteristic emission or absorption lines. Each element emits a unique set of wavelengths (its spectral fingerprint), which can be measured with a spectrometer. Applications include:
- Identifying elements in stellar atmospheres (stellar spectroscopy)
- Chemical analysis in laboratories
- Plasma physics and fusion research
- Laser physics and quantum optics
- Verification of quantum mechanical models
Units: Metric vs. American
This calculator supports two unit systems:
- Metric: wavelength in nm (nanometres), energy in eV and J (Joules), frequency in THz, wavenumber in m⁻¹
- American: wavelength in Å (Ångströms, 1 Å = 0.1 nm), energy in eV, frequency in THz, wavenumber in cm⁻¹ (the preferred spectroscopic unit in the USA)
Frequently Asked Questions
What is the Rydberg constant?
The Rydberg constant R ≈ 1.0973731568508 × 10⁷ m⁻¹ is a physical constant derived from other fundamental constants: R = m_e·e⁴ / (8·ε₀²·h³·c). It was originally determined empirically by Johannes Rydberg in 1888 to fit the hydrogen spectral lines measured by Balmer.
Why must n₂ be greater than n₁?
The formula describes emission — light released when an electron falls from a higher energy level (n₂) to a lower one (n₁). If n₂ ≤ n₁, the term 1/n₁² − 1/n₂² would be negative or zero, which is physically meaningless for emission. For absorption, the roles swap, but the wavelength is identical.
What wavelengths does the Balmer series produce?
The first four Balmer lines (n₂ = 3, 4, 5, 6 → n₁ = 2) are:
- Hα: 656.3 nm (red) — n=3→2
- Hβ: 486.1 nm (blue-green) — n=4→2
- Hγ: 434.0 nm (violet) — n=5→2
- Hδ: 410.2 nm (deep violet) — n=6→2
Can I use this calculator for multi-electron atoms?
No. The Rydberg formula is exact only for hydrogen-like (one-electron) species. For atoms with multiple electrons, electron–electron repulsion complicates the energy levels and a more sophisticated quantum mechanical treatment is required.
What is the series limit?
As n₂ → ∞, the photon energy approaches the ionisation limit for that series. For the Lyman series (n₁ = 1), the series limit is at λ ≈ 91.2 nm — photons with shorter wavelengths ionise hydrogen from its ground state.