Rydberg Formula Wavelength Calculator

Rydberg formula: 1/λ = R·(1/n₁² − 1/n₂²)

The Rydberg formula tells you the wavelengths of hydrogen's spectral lines whenever an electron jumps between energy levels: 1/λ = R·(1/n₁² − 1/n₂²), where R = 1.097·10⁷ m⁻¹ is the Rydberg constant and n₁ < n₂ are integers. The lines group into families. Lyman (n₁=1) sits in the ultraviolet, Balmer (n₁=2) lands in the visible, and Paschen (n₁=3) is in the infrared. Take the Hα line (n=3 → n=2): it comes out at λ ≈ 656.3 nm, the red glow you see in hydrogen nebulae and discharge lamps. It started as a purely empirical fit in 1888, and only with Bohr's 1913 model did it get derived from first principles.

Applications

You run into this in astrophysics, where absorption and emission lines reveal what stars are made of and the redshift of faraway galaxies. It also shows up in analytical chemistry through atomic emission spectroscopy, in neon and discharge lamps, in calibrating spectrometers, and in LED design that works off semiconductor band gaps.

FAQ

Does the Rydberg formula work for other atoms? It only holds for hydrogen-like ions, the ones with a single electron. Once you have more than one electron, the electron-electron interactions break the simple formula.

What is the Hα line and why is it red? It's the n=3 → n=2 transition in hydrogen, at λ ≈ 656.3 nm, a wavelength that falls in the red. That's why H II regions and solar prominences look the color they do.

Which series is visible to the eye? Just the Balmer series (n₁=2), whose lines Hα, Hβ, Hγ and Hδ sit in the visible range. Lyman is ultraviolet, and Paschen and everything past it is infrared.