Enter the atomic number (Z), initial energy level (n₁), and final energy level (n₂) to calculate the wavelength of light emitted or absorbed during an electron transition using the Rydberg Equation. You also get the corresponding frequency, photon energy, and the spectral series name for hydrogen-like atoms.
Wavelength (λ)
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Wavenumber (1/λ)
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Frequency (ν)
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Photon Energy (E)
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Photon Energy (E)
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Spectral Series
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Electromagnetic Region
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The Rydberg equation is a formula used to predict the wavelength of light emitted or absorbed when an electron in a hydrogen-like atom transitions between energy levels. It is written as 1/λ = R·Z²·(1/n₂² − 1/n₁²), where R is the Rydberg constant (1.0974 × 10⁷ m⁻¹), Z is the atomic number, and n₁ and n₂ are the higher and lower principal quantum numbers respectively.
First calculate the wavelength λ using the Rydberg equation. Then use the relationship ν = c / λ, where c is the speed of light (≈ 3 × 10⁸ m/s) and ν is the frequency in Hz. For example, a wavelength of 656 nm gives a frequency of about 4.57 × 10¹⁴ Hz (457 THz).
The original Rydberg formula was derived for hydrogen, but it can be extended to any hydrogen-like atom — that is, any atom or ion with only one electron. Examples include He⁺ (Z=2), Li²⁺ (Z=3), and Be³⁺ (Z=4). For these atoms, the equation uses a modified Rydberg constant multiplied by Z².
The Rydberg constant (R∞) has a value of approximately 1.0974 × 10⁷ m⁻¹ (or 10,973,731.568 m⁻¹). For hydrogen specifically (Z=1), the constant remains R∞. For other hydrogen-like atoms, the effective constant becomes R∞·Z².
When an electron in hydrogen drops from n₁ = 4 to n₂ = 2, the emitted wavelength is approximately 486.1 nm. This falls in the visible blue-green region of the electromagnetic spectrum and is part of the Balmer series, known as the Hβ line.
Hydrogen has several spectral series named after their discoverers. The Lyman series (n₂ = 1) falls in the ultraviolet range. The Balmer series (n₂ = 2) is partially visible. The Paschen series (n₂ = 3), Brackett series (n₂ = 4), and Pfund series (n₂ = 5) all lie in the infrared region.
Once you have the wavelength λ, the photon energy is calculated using E = hc/λ, where h is Planck's constant (6.626 × 10⁻³⁴ J·s) and c is the speed of light. The result in joules can be converted to electron-volts (eV) by dividing by 1.602 × 10⁻¹⁹.
In the Rydberg equation, n₁ represents the higher energy level and n₂ the lower one. The electron emits a photon as it falls from a higher to a lower energy state. If n₁ were equal to or less than n₂, the equation would yield a zero or negative value, which has no physical meaning for an emission transition.