Rydberg formula

Meaning: In the Bohr model, hydrogen's electron has quantized energies, En = −13.6 eV/n². A photon is emitted when the electron falls to a lower level and absorbed when it rises, and the photon energy equals the gap between the two levels. The Rydberg formula is that statement written in terms of wavelength. Series are named by the lower level: Lyman (n₁ = 1, ultraviolet), Balmer (n₁ = 2, includes visible lines), Paschen (n₁ = 3, infrared).

Equation: 1/λ = R Z² (1/n₁² − 1/n₂²), where λ is the photon wavelength (m), R = 1.097 × 10⁷ m⁻¹ is the Rydberg constant, Z is the atomic number (1 for hydrogen), and n₁ < n₂ are the lower and upper levels (unitless integers). Equivalently, E = hc/λ.

On the MCAT: Passages give an energy-level diagram or spectrum and ask which transition gives the longest or shortest wavelength, or the highest frequency. Within a series, the smallest gap (n₁ to n₁ + 1) gives the longest wavelength, and n₂ → ∞ gives the shortest. Convert with E = hf and c = fλ.

Common confusion: The formula gives 1/λ, so do not forget to invert. Level spacing shrinks as n rises, so the gap is not proportional to Δn. The formula works for hydrogen and one-electron ions, not for atoms with several electrons.