Rydberg Equation Calculator — Hydrogen Spectral Lines
Calculate the exact wavelength of light emitted or absorbed when a hydrogen electron transitions between two energy levels — enter the lower (n₁) and upper (n₂) principal quantum numbers.
Visible — Balmer series (n₁ = 2)
- 1
1 ÷ n₁²
1 ÷ 2² = 0.25 - 2
1 ÷ n₂²
1 ÷ 3² = 0.111111 - 3
Inverse wavelength 1/λ = R∞ × (1/n₁² − 1/n₂²)
1.097 × 10⁷ × (0.25 − 0.111111) = 1,524,129 m⁻¹R∞ = 1.0974 × 10⁷ m⁻¹ (Rydberg constant, NIST 2018 CODATA). - 4
Wavelength λ = 10⁹ ÷ (1/λ)
10⁹ ÷ 1,524,129 = 656.11
How does this calculator work?
The Rydberg equation 1/λ = R∞(1/n₁² − 1/n₂²) gives the wavelength of a photon emitted (or absorbed) by a hydrogen atom transitioning between principal quantum levels n₁ and n₂. The Rydberg constant R∞ = 1.097 × 10⁷ m⁻¹; photon energy E = hc/λ. Named series: Lyman (n₁=1, UV), Balmer (n₁=2, partly visible), Paschen (n₁=3, IR). The equation is exact for hydrogen; hydrogen-like ions scale by Z².
Formula
How this is calculated
The Rydberg equation, 1/λ = R∞ × (1/n₁² − 1/n₂²), predicts the wavelength of light produced when a hydrogen atom's single electron moves between two principal quantum number shells. n₁ is the lower (final) level and n₂ is the higher (initial) level; emission occurs when the electron falls from n₂ to n₁, releasing a photon of exactly that wavelength. Absorption is the reverse — a photon of that wavelength is absorbed when the electron is promoted from n₁ to n₂. The Rydberg constant R∞ = 1.0973731568539 × 10⁷ m⁻¹ is one of the most precisely measured constants in physics (NIST 2018 CODATA).
Grouped by n₁, the lines form named series: n₁ = 1 is the Lyman series (all ultraviolet), n₁ = 2 is the Balmer series (partly visible — the familiar red Hα at 656 nm, blue-green Hβ at 486 nm, and violet lines), n₁ = 3 is the Paschen series (near infrared), n₁ = 4 is Brackett, and n₁ = 5 is Pfund. The photon energy E = hc/λ (using Planck's constant h and the speed of light c) is expressed in electron-volts for convenience.
The Rydberg equation applies exactly only to hydrogen (one proton, one electron). For hydrogen-like ions (He⁺, Li²⁺) the constant scales by Z² (atomic number squared). Multi-electron atoms require quantum-mechanical perturbation theory and do not follow the simple Rydberg formula. This calculator uses the infinite nuclear mass approximation; reduced-mass corrections shift wavelengths by less than 0.05% for hydrogen.
Frequently asked questions
R∞ = 1.0973731568539 × 10⁷ m⁻¹ is derived from fundamental constants: R∞ = mₑe⁴/(8ε₀²h³c), where mₑ is electron mass, e is elementary charge, ε₀ is permittivity of free space, h is Planck's constant and c is the speed of light. Empirically, it was discovered by Johannes Rydberg in 1888 by fitting hydrogen spectral data — quantum mechanics later explained why.
The Balmer series (n₁ = 2) includes all transitions that end at the second energy level. The first four lines fall in the visible spectrum: Hα (n₂=3, 656 nm, red), Hβ (n₂=4, 486 nm, blue-green), Hγ (n₂=5, 434 nm, violet) and Hδ (n₂=6, 410 nm, violet). Higher n₂ lines converge on the series limit at 365 nm (UV). The Balmer series is the reason hydrogen nebulae glow red (Hα dominates).
Directly, no — the Rydberg equation is derived for one-electron (hydrogen-like) atoms. For hydrogen-like ions with atomic number Z, multiply R∞ by Z² (e.g. He⁺ uses 4R∞). For multi-electron atoms, electron screening shifts energy levels and the Rydberg formula fails; spectroscopy tables or quantum-mechanical calculations are required.
Also known as
TG we-Calculate Editorial Team. (2026). Rydberg Equation Calculator — Hydrogen Spectral Lines [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rydberg-equation-calculator
TG we-Calculate Editorial Team. "Rydberg Equation Calculator — Hydrogen Spectral Lines." TG we-Calculate. 2026. https://we-calculate.com/calculator/rydberg-equation-calculator.
TG we-Calculate Editorial Team, "Rydberg Equation Calculator — Hydrogen Spectral Lines," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rydberg-equation-calculator
@misc{wecalculate_rydberg_equation_calculator, title = {Rydberg Equation Calculator — Hydrogen Spectral Lines}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rydberg-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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