Hydrogen Spectrum Calculator Formula

Understand the math behind the hydrogen spectrum calculator. Each variable explained with a worked example.

Use the Hydrogen Spectrum Calculator

Formulas Used

Wavelength

wavelength = 1 / (1.0974e7 * (1/pow(n_lower, 2) - 1/pow(n_upper, 2)))

Wavelength (nm)

wavelength_nm = 1 / (1.0974e7 * (1/pow(n_lower, 2) - 1/pow(n_upper, 2))) * 1e9

Photon Energy

photon_energy = 13.6 * (1/pow(n_lower, 2) - 1/pow(n_upper, 2))

Variables

VariableDescriptionDefault
n_lowerLower Level (n1)2
n_upperUpper Level (n2)3

How It Works

Hydrogen Spectral Lines (Rydberg Formula)

When an electron transitions between energy levels in hydrogen, it emits or absorbs a photon with a specific wavelength.

Formula

1/lambda = R_H * (1/n1^2 - 1/n2^2)

  • *R_H* = 1.0974 x 10^7 m^-1 (Rydberg constant)
  • *n1* = lower energy level
  • *n2* = upper energy level (n2 > n1)

    • Series: Lyman (n1=1, UV), Balmer (n1=2, visible), Paschen (n1=3, IR).

    Worked Example

    H-alpha line: n1 = 2 to n2 = 3 (Balmer series).

    n_lower = 2n_upper = 3
    1. 011/lambda = R * (1/4 - 1/9)
    2. 021/4 - 1/9 = 5/36 = 0.1389
    3. 031/lambda = 1.0974e7 * 0.1389 = 1.524e6 m^-1
    4. 04lambda = 6.563e-7 m = 656.3 nm (red)

    Ready to run the numbers?

    Open Hydrogen Spectrum Calculator