What Does the Rydberg Equation Calculate? Hydrogen Line Wavelength Calculator

The Rydberg equation calculates the wavelength of the light a hydrogen atom emits when its electron drops from a higher level n₂ to a lower level n₁ (or absorbs when it goes up). Enter the two levels; you get the wavelength in nanometres (in vacuum), the wavenumber and the photon energy in electronvolts.

Formula

1 ÷ λ = R_H × (1 ÷ n₁² − 1 ÷ n₂²); E = h × c ÷ λ.

• λ: wavelength in vacuum
• R_H = 1.09677583 × 10⁷ m⁻¹, the Rydberg constant for hydrogen (the Rydberg constant R_∞ = 1.0973731568 × 10⁷ m⁻¹ corrected for the electron-to-proton mass ratio)
• n₁ < n₂: whole-number energy levels (n₁ = 1 Lyman, 2 Balmer, 3 Paschen series)
• h = 6.62607015 × 10⁻³⁴ J·s, c = 299,792,458 m/s and e = 1.602176634 × 10⁻¹⁹ C, all exact; 1 eV = e × 1 V

Worked Example

The red Balmer-alpha line of hydrogen, n₂ = 3 to n₁ = 2.

1. 1 ÷ 2² − 1 ÷ 3² = 0.25 − 0.111111 = 0.138889 (exactly 5 ÷ 36)
2. 1 ÷ λ = 1.09677583 × 10⁷ × 5 ÷ 36 = 1,523,299.76 m⁻¹, a wavenumber of 15,233.0 cm⁻¹
3. λ = 1 ÷ 1,523,299.76 m⁻¹ = 6.564696 × 10⁻⁷ m = 656.47 nm
4. E = 6.62607015 × 10⁻³⁴ × 299,792,458 × 1,523,299.76 = 3.02595 × 10⁻¹⁹ J; ÷ 1.602176634 × 10⁻¹⁹ = 1.8887 eV

Assumptions

• The atom is hydrogen (one electron, proton nucleus). For hydrogen-like ions multiply by Z² and use the matching reduced-mass constant; for many-electron atoms the equation does not apply.
• The levels are whole numbers, chosen from the lists; n₂ must be greater than n₁, otherwise the calculator says so instead of giving a result.
• Fine structure and other small corrections are ignored; for Balmer-alpha they amount to about 0.01-0.02 nm.

How to read the result

The result is a vacuum wavelength. Tables of visible lines often list air wavelengths, which are slightly shorter: Balmer-alpha is about 656.47 nm in vacuum and 656.28 nm in air. Larger jumps give shorter wavelengths; all lines ending on n₁ = 1 are ultraviolet.

Limits and common mistakes

• Using R_∞ instead of R_H shifts the wavelength by about 0.05% (0.36 nm for Balmer-alpha).
• Putting the larger level first gives a negative value; the lower level is n₁.

Frequently Asked Questions

What does the Rydberg equation calculate?

The wavelength (or wavenumber) of light emitted or absorbed by a hydrogen atom when its electron moves between two energy levels.

What is the wavelength of the n = 3 to n = 2 transition?

About 656.47 nm in vacuum, the red Balmer-alpha line, with a photon energy of about 1.889 eV.

What is the Rydberg constant for hydrogen?

About 1.09678 × 10^7 m^−1, slightly smaller than R∞ because the proton is not infinitely heavy.

sel n1 | Lower level n1 | 2 (Balmer series)=2, 1 (Lyman series)=1, 3 (Paschen series)=3, 4 (Brackett series)=4, 5 (Pfund series)=5, 6=6
sel n2 | Upper level n2 | 3=3, 4=4, 5=5, 6=6, 7=7, 8=8, 9=9, 10=10, 2=2
in r | Rydberg constant for hydrogen | m^-1 | 10967758.3 | 0
out lam | Wavelength (vacuum) | nm | 1e9 / (r * (1 / n1^2 - 1 / n2^2)) | 2
out wn | Wavenumber | cm^-1 | 10000000 / lam | 1
out ev | Photon energy | eV | 6.62607015e-34 * 299792458 * wn * 100 / 1.602176634e-19 | 4