Energy comes in fixed amounts
An electron in an atom can only have certain energies. When it falls from a higher level to a lower one, the energy it loses leaves as one photon. The photon’s frequency f and wavelength λ are fixed by the size of the drop:
h is the Planck constant, 6.626 × 10⁻³⁴ J s, and c is the speed of light, 3.00 × 10⁸ m s⁻¹.
Take sodium’s brightest line, 589.0 nm:
That is the energy of one photon. Multiply by the Avogadro constant and you get the energy of a mole of photons, about 203 kJ mol⁻¹: similar in size to the energy of a chemical bond. In sodium, this is the drop of the outer electron from the 3p level back to 3s.
Hydrogen, the cleanest case
Hydrogen has one electron, so its spectrum is simple enough to predict exactly. Its four visible lines are all drops that end on level n = 2:
| From level | Wavelength in air | Colour |
|---|---|---|
| 3 | 656.3 nm | red |
| 4 | 486.1 nm | blue-green |
| 5 | 434.0 nm | violet |
| 6 | 410.2 nm | violet |
Drops that end on n = 1 give out more energy, in the ultraviolet, where you can’t see them. As the starting level gets higher, the levels crowd together, so the lines in each series crowd together too, until they merge at a limit. The energy at that limit, for the series that ends on n = 1, is the energy needed to remove the electron completely: the ionisation energy of hydrogen, 13.6 eV, or 1312 kJ mol⁻¹.
If you work out these wavelengths with the Rydberg formula, you will get 656.5 nm and not 656.3 nm for the red line. The formula gives wavelengths in a vacuum. Light travels very slightly slower in air, which shortens its wavelength by about 0.03%, and the measured values in tables like this one are in air.
What a flame test really shows
Lithium’s crimson is its strong 670.8 nm line. Potassium’s lilac is a mix of deep red lines near 770 nm, which the eye barely sees, and a violet line near 404 nm. Calcium and copper are more complicated: a flame is cool by the standards of a spectroscope, and in it these metals form small molecules, such as CaOH and CuCl, whose bands of light add to the atomic lines. That is one reason instruments, not eyes, are used for real analysis.
Flame tests and spectroscopy are done in a lab with a teacher or technician. Don’t heat or burn chemicals at home.