How to use it
- Enter Z: 1 for hydrogen, 2 for He⁺, 3 for Li²⁺ – any atom stripped to one electron.
- Enter the starting and final levels. Falling to a lower level emits a photon; rising absorbs one.
- Read the photon wavelength, energy and frequency, the series, and the electron’s energy, radius and speed on the starting level.
Formulas
- Level energy: Eₙ = −13.605693 eV × Z² ÷ n²
- Orbit radius: rₙ = a₀ n² ÷ Z, a₀ = 52.9177 pm
- Electron speed: vₙ = α c Z ÷ n, α ≈ 1/137.036
- Rydberg formula: 1/λ = R∞ Z² (1/n₁² − 1/n₂²), R∞ = 10,973,731.568 1/m
- Photon energy: E = E_upper − E_lower = h c ÷ λ
Worked example
Hydrogen, electron falls from n = 3 to n = 2:
- 1/λ = 10,973,731.568 × (1/4 − 1/9) = 10,973,731.568 × 5/36 → λ = 656.112 nm (656.47 nm with the proton’s mass), emission, Balmer series
- E = 13.605693 × 5/36 = 1.8897 eV, frequency 456.92 THz
- On n = 3: E₃ = −1.5117 eV, r₃ = 9 × 52.9177 = 476.26 pm, v₃ = 729.2 km/s
He⁺ (Z = 2) from n = 2 to n = 1: λ = 121.502 ÷ 4 = 30.376 nm, 40.8171 eV.
Tips
- Levels are negative because the electron is bound; 0 eV means free. The ionization energy is the size of the level energy.
- Lyman lines are ultraviolet, Balmer lines include the visible ones, Paschen and higher are infrared.
- Vacuum wavelengths. Tables measured in air list slightly shorter values, for H-alpha 656.28 nm.