Science

Bohr Hydrogen Energy Levels & Rydberg Spectral Wavelength Formula

Use a consistent approximate hydrogen Rydberg model for vacuum emission wavelengths, photon energies and shell energies, with positive integer level validation.

Use one mass convention consistently for wavelength and energy

B4 is the approximate hydrogen reduced-mass Rydberg wavenumber 0.01096776 nm^-1. The model gives 1/lambda=R(1/n1²-1/n2²). To keep energy consistent with wavelength, this sheet uses the same R and rounded hc = 1239.84 eV nm for both photon and shell energies.

Using 13.6057 eV for shell energies while using a reduced-mass hydrogen R for wavelength mixes different approximations. Here E_n=-hcR/n² relative to the ionization limit, and photon energy=E_upper-E_lower=hc/lambda.

These are approximate nonrelativistic hydrogen lines, not exact spectroscopy. Fine structure, Lamb shifts and other corrections are omitted. The wavelengths are in vacuum, not air.

For emission require positive integer n2>n1. Lower level 1 labels the Lyman series, 2 the Balmer series and 3 the Paschen series. Balmer includes visible lines but its high-order limit extends into ultraviolet; it is not wholly a visible series.

Vacuum wavelength in nm, D2

The first example is the 3→2 transition.

=IF(COUNT(B2:B4)<>3,"Enter numeric levels and Rydberg constant",IF(OR(B2<1,B3<1,B2<>INT(B2),B3<>INT(B3),B4<=0),"Require positive integer levels and R>0",IF(B3<=B2,"Requires n2 > n1",1/(B4*(1/B2^2-1/B3^2)))))

Photon energy in eV, D3

Uses the same R and hc as the shell model.

=IF(COUNT(B2:B4)<>3,"Enter numeric levels and Rydberg constant",IF(OR(B2<1,B3<1,B2<>INT(B2),B3<>INT(B3),B4<=0),"Require positive integer levels and R>0",IF(B3<=B2,"Requires n2 > n1",1239.84*B4*(1/B2^2-1/B3^2))))

Lower-state energy in eV, D4

Negative energy is relative to the zero-energy ionization limit.

=IF(COUNT(B2:B4)<>3,"Enter numeric levels and Rydberg constant",IF(OR(B2<1,B3<1,B2<>INT(B2),B3<>INT(B3),B4<=0),"Require positive integer levels and R>0",IF(B3<=B2,"Requires n2 > n1",-1239.84*B4/B2^2)))

Upper-state energy in eV, D5

Subtract D4 from D5 to recover the positive photon energy.

=IF(COUNT(B2:B4)<>3,"Enter numeric levels and Rydberg constant",IF(OR(B2<1,B3<1,B2<>INT(B2),B3<>INT(B3),B4<=0),"Require positive integer levels and R>0",IF(B3<=B2,"Requires n2 > n1",-1239.84*B4/B3^2)))

Try the example

Use the original values or change them to test your understanding.