0.000001
m
656.0962
nm
1.8912
eV
1,524,166.6667
m^-1
0.000001
m
656.0962
nm
1.8912
eV
1,524,166.6667
m^-1
The Rydberg Equation Calculator computes the wavelength of light emitted or absorbed when an electron in a hydrogen-like atom transitions between two energy levels. The Rydberg formula, developed empirically by Johannes Rydberg in 1888 and later derived from first principles by Niels Bohr, is one of the most successful equations in spectroscopy.
When an electron in a hydrogen atom transitions from a higher energy level n2 to a lower energy level n1, it emits a photon whose wavelength satisfies 1/lambda = R * (1/n1^2 - 1/n2^2), where R = 1.0974 x 10^7 m^-1 is the Rydberg constant. The reverse process (absorption) follows the same formula. Transitions to n1 = 1 form the Lyman series (ultraviolet), to n1 = 2 form the Balmer series (visible), to n1 = 3 the Paschen series (infrared), and so on.
The visible hydrogen spectrum — four colored lines known since 1885 from Balmer's work — has historically been one of the most important testing grounds for atomic theory. The fact that Bohr's model derived the Rydberg constant from fundamental constants (electron mass, electron charge, Planck's constant, and the speed of light) was a triumph that confirmed quantum theory.
For hydrogen-like ions with atomic number Z (He+, Li2+, etc.), the formula generalizes to 1/lambda = R * Z^2 * (1/n1^2 - 1/n2^2). This calculator supports any atomic number Z, allowing computation for hydrogen (Z=1), singly ionized helium (Z=2), doubly ionized lithium (Z=3), and so on.
The Rydberg equation is fundamental to stellar spectroscopy (identifying elements in stars from their absorption lines), laser design (tuning transitions), astronomy (21 cm hydrogen line), and quantum chemistry education. Modern derivations from Schrodinger's equation reproduce the Rydberg constant to extraordinary precision, confirming quantum mechanics.
1/lambda = R * Z^2 * (1/n1^2 - 1/n2^2), where R = 1.0974 x 10^7 m^-1 is the Rydberg constant, Z is the atomic number, n1 is the lower level, and n2 is the upper level (n2 > n1). Wavelength is 1 divided by the wavenumber. Photon energy is E = h*c/lambda, converted to eV by dividing by 1.602 x 10^-19.
Transitions to n1=1 (Lyman series) produce UV photons (91-122 nm). Transitions to n1=2 (Balmer series) produce visible and near-UV photons (364-656 nm): H-alpha at 656 nm (red), H-beta at 486 nm (blue-green). Larger n differences give higher-energy photons. For Z > 1 the photon energy scales as Z^2.
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The H-alpha line at 656.3 nm gives hydrogen its characteristic red color in nebulae and is the most prominent line in the visible hydrogen spectrum.
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Lyman-alpha at 121.6 nm (UV) is the most intense hydrogen line in the solar spectrum and is important in studying the intergalactic medium.
R = 1.0974 x 10^7 m^-1 is a fundamental physical constant related to the energy levels of hydrogen. It can be derived from electron mass, charge, Planck's constant, and speed of light.
Lyman (n1=1, UV), Balmer (n1=2, visible/near-UV), Paschen (n1=3, near-IR), Brackett (n1=4, mid-IR), Pfund (n1=5, far-IR).
The wavenumber 1/lambda is proportional to photon energy (E = h*c/lambda = h*c * wavenumber). Rydberg found empirically that wavenumbers of spectral lines fit simple integer arithmetic.
Not directly. The Rydberg formula applies exactly only to hydrogen-like (one-electron) systems. Multi-electron atoms require quantum defect corrections or full quantum mechanical treatment.
It corresponds to a transition from n1=1 to n2=infinity: E = R*h*c = 13.6 eV, the ionization energy of hydrogen. Wavelength approaches 91.2 nm (the Lyman series limit).
Electrons in atoms can only occupy discrete energy levels. When an electron drops from a higher to a lower level, it emits a photon of exactly the energy difference. These discrete energies produce discrete (line) spectra.
Astronomers identify elements in stars by matching observed spectral lines to Rydberg formula predictions. Redshift of these lines also reveals a star's or galaxy's velocity relative to Earth.
Yes. The same formula applies for absorption (electron jumping from n1 to n2). Stars show dark Fraunhofer absorption lines where cooler outer gas absorbs photons from the hot interior.
Niels Bohr (1913) proposed that electrons orbit the nucleus only in allowed circular orbits with quantized angular momentum. This model derived the Rydberg formula from first principles for hydrogen.
For singly ionized helium (He+, Z=2), use Z=2 in the formula. All wavelengths are 4 times shorter (energies 4 times higher) than for hydrogen at the same n values.
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