Calculate spectral line wavelengths, photon energies, and wavenumbers using the Rydberg equation for hydrogen and hydrogen-like ions across all spectral series.
The Rydberg equation is one of the most important formulas in atomic physics, predicting the wavelengths of all spectral lines emitted by hydrogen and hydrogen-like ions. Originally discovered empirically by Johannes Rydberg in 1888, the equation was later derived from Bohr's atomic model and confirmed by quantum mechanics. It relates the wavelength of a photon emitted during an electronic transition to the principal quantum numbers of the initial and final states.
For hydrogen-like atoms with atomic number Z, the generalized Rydberg equation is 1/λ = R∞·Z²·(1/n_f² − 1/n_i²), where R∞ = 1.0974 × 10⁷ m⁻¹ is the Rydberg constant, n_i is the initial (higher) level, and n_f is the final (lower) level. The spectral lines naturally group into series named after their discoverers: Lyman (n_f = 1, UV), Balmer (n_f = 2, visible), Paschen (n_f = 3, IR), and so on. Each series converges to a series limit as n_i → ∞.
This calculator computes the wavelength, energy, frequency, and wavenumber for any transition, generates full series tables with color indicators for visible lines, and compares all six named spectral series at a glance. It supports hydrogen-like ions (He⁺, Li²⁺, etc.) where the Rydberg formula remains exact.
The Rydberg equation is a cornerstone of atomic spectroscopy and a staple of physics and chemistry courses at all levels. This calculator saves time on repetitive spectral line calculations, provides visual confirmation for visible lines, and offers series tables that are invaluable for lab report preparation and exam review. The hydrogen-like ion support makes it useful for advanced topics like astrophysical plasma diagnostics.
Rydberg Equation: • 1/λ = R∞ · Z² · (1/n_f² − 1/n_i²) • R∞ = 1.0974 × 10⁷ m⁻¹ (Rydberg constant) • Photon energy: E = hc/λ = 13.6 · Z² · (1/n_f² − 1/n_i²) eV • Wavenumber: ṽ = 1/λ (in cm⁻¹) • Series limit: λ_min = 1/(R∞ · Z² / n_f²) as n_i → ∞
Result: λ = 656.3 nm (H-α, red visible light)
The transition from n=3 to n=2 in hydrogen gives the H-alpha line at 656.3 nm — the bright red line in the Balmer series, used extensively in astronomical observations.
Before quantum mechanics existed, Rydberg and others empirically discovered mathematical formulas that accurately predicted hydrogen spectral lines. Johann Balmer found the pattern for visible lines in 1885, and Rydberg generalized it in 1888. Their purely empirical discovery was later explained by Bohr's 1913 atomic model, which showed that the Rydberg constant encodes fundamental constants: R∞ = mₑe⁴/(8ε₀²h³c). This was one of the earliest and most spectacular confirmations of quantum theory.
Spectral analysis using the Rydberg equation is the primary tool for determining the chemical composition, temperature, and motion of celestial objects. The hydrogen 21 cm line (hyperfine transition), Lyman-alpha forest in quasar spectra, and Balmer absorption lines in stellar atmospheres all rely on precise knowledge of hydrogen energy levels. Redshifted Rydberg lines reveal the expansion rate of the universe and the distances to remote galaxies.
The hydrogen atom is the simplest atom and therefore the most precisely calculable in quantum electrodynamics (QED). Measurements of hydrogen spectral lines now test QED to extraordinary precision. The discrepancy between the measured proton charge radius from hydrogen spectroscopy and from muonic hydrogen (the "proton radius puzzle") drove years of experimental and theoretical work, illustrating how the humble Rydberg equation continues to push the frontiers of fundamental physics.
The Rydberg constant R∞ = 1.0974 × 10⁷ m⁻¹ is one of the most precisely measured physical constants. It relates energy levels in hydrogen-like atoms to spectral line wavelengths and is fundamental to atomic physics.
Lyman (n_f=1, UV), Balmer (n_f=2, visible/near-UV), Paschen (n_f=3, near-IR), Brackett (n_f=4, IR), Pfund (n_f=5, far-IR), and Humphreys (n_f=6, far-IR). The Balmer series is the most famous because its lines are visible to the eye.
H-alpha (Hα) is the first Balmer line (n=3→2) at 656.3 nm. It appears as a bright red emission line and is one of the most important lines in astronomy — it is used to detect hydrogen gas in nebulae, stars, and galaxies.
It works exactly only for hydrogen-like (one-electron) atoms — H, He⁺, Li²⁺, etc. For multi-electron atoms, electron screening modifies the energy levels, and more complex models are needed.
As nᵢ approaches infinity, the spectral lines crowd together and converge to a minimum wavelength called the series limit. Beyond this limit, the photon has enough energy to ionize the atom.
The nuclear charge Z affects the Coulomb attraction between the nucleus and electron. Since energy scales as Z², the wavelengths scale as 1/Z². He⁺ lines are 4× shorter (4× more energetic) than hydrogen's.