Rydberg Formula
The Rydberg formula predicts the wavelengths of spectral lines emitted by hydrogen and hydrogen-like atoms.
Includes worked examples.
The Formula
The Rydberg formula, developed by Swedish physicist Johannes Rydberg in 1888, predicts the wavelengths of light emitted or absorbed when electrons in a hydrogen atom jump between energy levels. Each jump produces a photon with a very specific wavelength, creating the characteristic spectral lines seen in a hydrogen spectrum.
When an electron falls from a higher energy level (n₂) to a lower one (n₁), it releases a photon of light. The Rydberg formula tells you exactly what wavelength that photon will have. This was one of the first quantitative successes of quantum theory — the formula matched experimental spectral data with remarkable precision.
The Rydberg constant R∞ has a value of 1.097 × 10⁷ m⁻¹ and is one of the most precisely measured constants in physics. The formula was later explained theoretically by Niels Bohr in 1913 using his model of quantized electron orbits.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| λ | Wavelength of emitted or absorbed light | m |
| R∞ | Rydberg constant = 1.097 × 10⁷ | m⁻¹ |
| n₁ | Lower energy level (principal quantum number) | dimensionless |
| n₂ | Upper energy level (must be greater than n₁) | dimensionless |
Example 1
Find the wavelength of light emitted when a hydrogen electron drops from n = 3 to n = 2 (the first line of the Balmer series — visible red light).
Identify: n₁ = 2, n₂ = 3, R∞ = 1.097 × 10⁷ m⁻¹
Apply the formula: 1/λ = 1.097 × 10⁷ × (1/4 − 1/9)
Calculate: 1/λ = 1.097 × 10⁷ × (0.25 − 0.1111) = 1.097 × 10⁷ × 0.1389
1/λ = 1.524 × 10⁶ m⁻¹
λ ≈ 656 nm — this is the famous red H-alpha line visible in hydrogen emission spectra
Example 2
Find the wavelength when a hydrogen electron drops from n = 2 to n = 1 (Lyman series — ultraviolet light).
Identify: n₁ = 1, n₂ = 2, R∞ = 1.097 × 10⁷ m⁻¹
Apply the formula: 1/λ = 1.097 × 10⁷ × (1/1 − 1/4)
1/λ = 1.097 × 10⁷ × 0.75 = 8.228 × 10⁶ m⁻¹
λ ≈ 122 nm — ultraviolet light, invisible to the human eye but important in astrophysics
When to Use It
Use the Rydberg formula when:
- Calculating the wavelength of light emitted or absorbed by hydrogen atoms
- Identifying spectral series — Lyman (n₁=1), Balmer (n₁=2), Paschen (n₁=3)
- Analyzing stellar spectra to determine a star's composition
- Studying atomic structure and quantized energy levels
- Designing spectroscopic instruments and calibrating wavelength standards