Rydberg Formula Calculator

The Rydberg formula predicts every spectral line of hydrogen with remarkable precision. It was discovered empirically before quantum mechanics explained why it works.

🌈 Spectroscopy📐 1/λ = R(1/n₁² - 1/n₂²)⚛️ Atomic Physics
Lower level n₁
Upper level n₂
Ion charge Z (1 = hydrogen)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Wavenumber1/λRH(1/n1² − 1/n2²)m⁻¹
Rydberg ConstantRH1.097 × 10⁷ m⁻¹m⁻¹
Wavelengthλ1/(wavenumber)m
For Ion (charge Z)—1/λ = RHZ²(1/n1²−1/n2²)m⁻¹

Step-by-Step Examples

Example 1
Balmer H-alpha

n1=2, n2=3, Z=1.

  • 1/lambda = 1.097e7*(1/4 - 1/9) = 1.524e6 m^-1
  • lambda = 656.3 nm
✓ 656.3 nm — red H-alpha line
Example 2
Lyman Alpha

n1=1, n2=2.

  • 1/lambda = 1.097e7*(1 - 0.25) = 8.228e6 m^-1
  • lambda = 121.6 nm
✓ 121.6 nm — UV Lyman-alpha
Example 3
He+ Ion

Helium ion He+ (Z=2), n1=2, n2=4.

  • 1/lambda = 1.097e7*4*(1/4 - 1/16) = 1.097e7*4*0.1875
  • lambda = 164 nm
✓ 164 nm for He+ Balmer analog (Z=2)

Real-World Applications

🔭
Stellar Spectroscopy
Hydrogen lines identify stellar composition, temperature, and radial velocity via Doppler shift.
🌀
Cosmological Redshift
Lyman-alpha emission from distant galaxies measures their recession velocity and distance.
🧪
Atomic Theory History
Rydberg formula (1888) preceded quantum mechanics. Bohr explained WHY it works in 1913.
🔬
Plasma Physics
Rydberg formula applied to hot plasma identifies which ions are present from emission spectra.

Common Mistakes to Avoid

⚠️
n2 must be greater than n1

For emission: electron falls from n2 to n1 (n2 > n1). For absorption: n2 > n1 too (absorbs photon to go up).

⚠️
Z=1 for hydrogen only

Multiply R by Z^2 for hydrogen-like ions. He+ has Z=2, Li2+ has Z=3.

⚠️
Rydberg formula gives vacuum wavelengths

Wavelengths in air are slightly shorter due to refractive index. Laboratory measurements correct for this.

Frequently Asked Questions

Who discovered the Rydberg formula? ▾
Johannes Rydberg in 1888, working purely empirically from spectral data. He did not know why it worked. Balmer had earlier found the visible hydrogen series formula.
Why does it work? ▾
Bohr model (1913) showed E_n = -13.6/n^2 eV. Transition energy DE = hf = hc/lambda. The Rydberg constant R = m_e*e^4/(8*epsilon_0^2*h^3*c).
How precisely does it predict wavelengths? ▾
To better than 0.01% for hydrogen. More precise hydrogen spectroscopy requires quantum electrodynamics (QED) corrections.
What is the Rydberg constant value? ▾
R_H = 1.09677581e7 m^-1 (for infinite nuclear mass). Real hydrogen uses reduced mass, giving R_H = 1.09678e7 m^-1.
What is the series limit? ▾
As n2 -> infinity, wavelength approaches series limit: lambda = n1^2/R. For Balmer: 364.6 nm (UV), for Lyman: 91.2 nm.

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Formula Explorer connections

Interpretation: This relationship connects quantized energy, wavelength, probability, nuclear mass or radioactive change. Assumption: Use the correct particle, quantum state, nuclide and energy units. Idealized potentials, nonrelativistic motion or single decay channels may be assumed.

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