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Rydberg Equation Calculator

Find the wavelength of a hydrogen emission line.

How it is calculated

The tool uses the Rydberg equation for hydrogen: 1/λ = R × (1/n₁² − 1/n₂²) with R = 1.0973731568×10⁷ m⁻¹, n₁ the lower energy level and n₂ the higher one. Both levels are whole numbers (decimals are cut off), n₁ must be at least 1 and n₂ must be strictly larger than n₁; otherwise no result is shown. The wavelength is λ = 1/(1/λ), converted from metres to nanometres and shown with two decimals, together with the wavenumber 1/λ in m⁻¹ in scientific notation. It describes hydrogen only; ions with one electron need a Z² factor, and heavier atoms need other methods. The value is the wavelength in vacuum.

Worked example

Take the jump between level n₁ = 2 and n₂ = 3. First 1/2² − 1/3² = 0.25 − 0.1111 = 0.1389 (exactly 5/36). Multiplying by R gives a wavenumber of 1.0973731568×10⁷ × 0.1389 ≈ 1.5241 × 10⁶ m⁻¹. Inverting that gives λ ≈ 6.5611×10⁻⁷ m, which the tool displays as 656.11 nm, the red Balmer line of hydrogen.

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Common mistakes

  • Entering the higher level as n₁ and the lower as n₂: the tool needs n₂ greater than n₁ and returns nothing otherwise.
  • Typing a decimal such as 2.5; levels are whole numbers and the fraction is simply dropped.
  • Using the result for helium, lithium or any atom with several electrons; the formula is for hydrogen.
  • Expecting a visible colour for every pair: n₁ = 1 gives ultraviolet lines and n₁ = 3 gives infrared lines.
  • Comparing the vacuum wavelength with a value measured in air without noting the small difference.

About this calculator

This calculator finds the wavelength of a hydrogen spectral line using the Rydberg equation: 1/λ = R · (1/n1² − 1/n2²), where R is the Rydberg constant and n1, n2 are the initial and final principal quantum (energy) levels of the electron. It explains the discrete lines seen in hydrogen's emission and absorption spectra, including the Lyman (UV), Balmer (visible), and Paschen (infrared) series depending on which level the electron ends at.

Chemistry and physics students use it to predict exactly which wavelengths hydrogen will emit or absorb when electrons jump between energy levels, connecting the abstract idea of quantized energy levels to real, measurable spectral lines seen with a spectroscope.

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Comments (1)

  • Editorial team

    Level 1 as the lower level gives the ultraviolet Lyman series and level 2 the visible Balmer series. Try 2 and 3 and you get about 656 nm, the red hydrogen-alpha line. The upper level must be greater than the lower one, and both are whole numbers. The calculation uses the standard Rydberg constant for an infinitely heavy nucleus, so lab values for real hydrogen differ very slightly. Which series are you studying?

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