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What Is the Rydberg Equation?
The Rydberg equation predicts the wavelengths of spectral lines produced when electrons in hydrogen-like atoms transition between energy levels. Originally derived empirically by Johannes Rydberg in 1888, it was later explained theoretically by Niels Bohr's atomic model. The equation accurately predicts the emission and absorption spectra of hydrogen and hydrogen-like ions.
When an electron drops from a higher energy level (n2) to a lower one (n1), it emits a photon whose wavelength is given by the Rydberg formula. When it absorbs a photon, it jumps from a lower to a higher level. The specific wavelengths produced form discrete spectral lines, which was one of the key observations leading to quantum mechanics.
The Rydberg Formula
Where λ is the wavelength, R_H is the Rydberg constant (1.0974 x 10^7 m^-1), Z is the atomic number, n1 is the lower energy level, and n2 is the upper energy level (n2 > n1).
Hydrogen Spectral Series
| Series | n1 | n2 Range | Region |
|---|---|---|---|
| Lyman | 1 | 2, 3, 4... | Ultraviolet |
| Balmer | 2 | 3, 4, 5... | Visible / Near UV |
| Paschen | 3 | 4, 5, 6... | Infrared |
| Brackett | 4 | 5, 6, 7... | Infrared |
| Pfund | 5 | 6, 7, 8... | Far infrared |
Key Hydrogen Transitions
| Transition | Wavelength (nm) | Name |
|---|---|---|
| 3 → 2 | 656.3 (red) | H-alpha |
| 4 → 2 | 486.1 (cyan) | H-beta |
| 5 → 2 | 434.0 (blue) | H-gamma |
| 6 → 2 | 410.2 (violet) | H-delta |
Frequently Asked Questions
Does the Rydberg equation work for all atoms?
The basic Rydberg equation works only for hydrogen-like species (one electron): H, He+, Li2+, etc. For multi-electron atoms, electron-electron repulsion complicates the energy levels and the simple formula no longer applies. The Z^2 factor accounts for the nuclear charge in hydrogen-like ions.
What is the Rydberg constant?
The Rydberg constant R_H = 1.0973731568539 x 10^7 m^-1 is one of the most precisely measured constants in physics. It can be derived from fundamental constants: R = m_e * e^4 / (8 * epsilon_0^2 * h^3 * c), where m_e is electron mass, e is elementary charge, and h is Planck's constant.
Why are only certain wavelengths emitted?
Energy levels in atoms are quantized -- electrons can only occupy specific energy states. The energy difference between two levels determines the photon's wavelength. Since only discrete energy differences are possible, only specific wavelengths are emitted, creating the characteristic line spectrum.