Rydberg Formula Calculator
Enter lower n₁, upper n₂ and nuclear charge Z to get the photon wavelength, frequency, energy and spectral series. H: n₁=2, n₂=3 → 656 nm (Balmer, red).
Input Data
Results
At a glance:In 1888 Swedish physicist Johannes Rydberg, analysing spectral lines of several elements, proposed an empirical formula predicting the emission/absorption wavelengths of atoms — especially one-electron (hydrogen-like) systems: 1/λ=R_M·Z²·(1/n₁²−1/n₂²). R_M≈1.09678e7 m⁻¹ (corrected for nuclear mass) is the atomic Rydberg constant. The n₁ series carry historical names: Lyman (n₁=1, UV), Balmer (n₁=2, visible), Paschen (n₁=3, IR), Brackett (n₁=4, IR), Pfund (n₁=5, IR), Humphreys (n₁=6, far IR). The formula was later derived from first principles by Bohr's 1913 atomic model and became a key precursor to quantum mechanics. With R_M (reduced-mass correction) R_M=R_∞/(1+m_e/M), R_∞=1.0973731568160e7 m⁻¹ and M the nuclear mass; photon frequency f=c/λ (Hz); photon energy E=h·f=h·c/λ (h=6.62607015e-34 J·s; 1 eV=1.602176634e-19 J).
Formula
Rydberg (wavenumber): 1/λ = R_M·Z²·(1/n₁² − 1/n₂²), n₂>n₁≥1
Reduced mass: R_M = R_∞/(1 + m_e/M), R_∞=1.0973731568160e7 m⁻¹
Frequency: f = c/λ (Hz)
Photon energy: E = h·f = h·c/λ (h=6.62607015e-34 J·s)
$$\frac{1}{\lambda} = R_M\,Z^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad R_M = \frac{R_\infty}{1 + \dfrac{m_e}{M}}$$How to Use
- Enter lower level n₁ (1 Lyman, 2 Balmer, 3 Paschen…).
- Enter upper level n₂ (> n₁) and nuclear charge Z (1 H, 2 He⁺…).
- The tool outputs λ (nm, m), f (THz), E (eV) and spectral region.
Case Studies
Hydrogen Balmer red line (Hα)
H: Z=1, n₁=2, n₂=3.
1/λ = R·(1/4 − 1/9) = R·(5/36).
λ ≈ 656 nm — the red Hα line, visible and used in astronomy.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.