Rydberg equation calculator

What is the Rydberg equation?

It gives the wavelengths of the spectral lines of hydrogen: 1/λ = R (1/n₁² − 1/n₂²), where n₁ is the lower energy level and n₂ the upper one. For hydrogen-like ions with one electron (He⁺, Li²⁺…) multiply by Z², the nuclear charge. The photon energy is E = hc/λ (OpenStax Chemistry 2e and University Physics).

Example: the red H-alpha line

An electron falling from n = 3 to n = 2 in hydrogen emits the first Balmer line. With the hydrogen Rydberg constant this gives 656.47 nm in vacuum, or 656.28 nm in air — the value listed in the NIST Atomic Spectra Database. The other visible Balmer lines come out at 486.1, 434.0 and 410.2 nm in air.

What are the Lyman, Balmer and Paschen series?

Lines grouped by the lower level: n₁ = 1 is the Lyman series (ultraviolet, starting at 121.6 nm), n₁ = 2 the Balmer series (the visible lines), n₁ = 3 Paschen, 4 Brackett, 5 Pfund and 6 Humphreys (OpenStax University Physics, Volume 3).

Which Rydberg constant should I use?

R∞ = 10,973,731.568157 m⁻¹ (CODATA 2022) assumes an infinitely heavy nucleus. Real hydrogen’s nucleus is a proton, so its constant is slightly smaller: R_H = R∞ ÷ (1 + mₑ/mₚ) ≈ 10,967,758 m⁻¹, using the CODATA proton-to-electron mass ratio 1836.152673426. R_H matches measured hydrogen lines much better; textbooks often use R∞, which gives 656.11 nm for H-alpha.

Example: a He⁺ ion

OpenStax Chemistry asks for the photon from n = 5 to n = 3 in He⁺ (Z = 2) and gets 6.198 × 10⁻¹⁹ J and 3.205 × 10⁻⁷ m. With Z = 2 and R∞ this calculator gives 6.201 × 10⁻¹⁹ J and 320.4 nm — the small difference comes from the rounded constants in the book.

How much energy does the Lyman-alpha line carry?

The jump from n = 2 to n = 1 releases a photon of 10.20 eV with a wavelength of 121.6 nm in vacuum, in the ultraviolet. This is the largest-wavelength line of the Lyman series, and the series approaches a limit near 91.2 nm as the upper level goes to infinity.

Reading the equation

The Rydberg equation, 1/λ = R Z² (1/n₁² − 1/n₂²), gives the wavenumber of the photon emitted when an electron drops from the upper level n₂ to the lower level n₁, or absorbed in the reverse jump. Z is the nuclear charge and applies only to atoms with a single electron, such as H, He⁺ and Li²⁺. R is the Rydberg constant, in inverse meters, so the wavelength is the inverse of the result. The energy of the photon follows from E = hc/λ, and in electronvolts from dividing by the elementary charge.

Worked examples for hydrogen

With R_H = 10,967,758 m⁻¹: Lyman-alpha (2 to 1) is 121.6 nm, Balmer H-beta (4 to 2) is 486.3 nm and Paschen-alpha (4 to 3) is 1,876 nm in the infrared. For the He⁺ ion (Z = 2) the 3 to 2 line is 164.1 nm, four times the wavenumber of the same hydrogen line in this model. Using R∞ = 10,973,732 m⁻¹ instead of R_H shifts the hydrogen 3 to 2 line from 656.47 nm to 656.11 nm, a change of 0.05%.

Common mistakes

Mixing up the levels is harmless to the result, since the equation uses the difference of the two terms, but the series name depends on the lower level, so the transition 4 to 2 is Balmer and not Brackett. Forgetting that the wavelength is the reciprocal of the sum, not the sum itself, is the most common arithmetic slip. A second is the Z² factor: it applies to one-electron ions, not to atoms with several electrons, where the equation fails. A third is comparing a vacuum wavelength with a table value measured in air, which differ by a factor of about 1.0003.

Sources: OpenStax University Physics Volume 3, section 6.4 (The Bohr Model of the Hydrogen Atom), and OpenStax Chemistry 2e on atomic structure; the Rydberg constant R∞ and the proton-to-electron mass ratio from CODATA 2022 (NIST); air wavelengths from the NIST Atomic Spectra Database. Related: photoelectric effect calculator, wavelength calculator, de Broglie wavelength calculator and electron configuration calculator.