Emission Spectra

Electrons in an atom can't sit at just any energy. They're stuck on specific rungs of a ladder, each with a fixed energy value. When an electron drops from a higher rung to a lower one, it has to shed exactly the energy difference between those two rungs, and it does that by emitting a single photon. That photon's energy fixes its wavelength through E = 1240/lambda (with E in eV and lambda in nm), so each possible jump produces one exact color of light, not a smear of colors. A hot hydrogen gas has electrons making many different jumps at once, but only a handful of those land in the visible range, which is why the tube glows with just a few sharp lines instead of a full rainbow.

For hydrogen specifically, the allowed energies follow E_n = -13.6 eV / n^2. A transition from a higher level n1 down to a lower level n2 releases a photon with energy E = 13.6 eV x (1/n2² - 1/n1²). Plug that into E = 1240/lambda and you get the wavelength directly. Every element has its own unique ladder of energy levels, set by its own electron structure and nuclear charge, so the set of possible jumps — and therefore the set of emitted wavelengths — is different for every element. That's why spectral lines work as a fingerprint: no two elements share the same pattern.

This connects directly to absorption spectra and to the idea of quantized energy from earlier in the unit: the same ladder of levels that lets an atom emit specific colors also means it can only absorb those same specific colors. Heating a gas hotter populates higher energy levels, which opens up more possible downward jumps and adds more lines to the spectrum, including ones further into the ultraviolet or infrared.