Why do electrons in a cool gas not release photons as they are excited?

Question

When white light is shone on a cool gas the electrons absorb photons of a certain wave lengths and become excited.
Shouldn’t the electrons then return to ground and release photons with the same wavelength as they absorbed meaning that line absorption spectra would be continuous? If the electrons don't return to ground they won't be able to absorb the photons to raise them to $n = 2$.

ProfRob · Answer

Yes, for an isothermal medium in thermal equilibrium, there will be just as many upward as downward radiative transitions. This is known as the principle of detailed balance.
However, if the downward transitions are via spontaneous emission, then the emitted photons go in random directions and only a small fraction will rejoin the original illuminating beam. Hence there would be absorption lines.
Note, that this typical explanation assumes that the mean free path of a photon in the gas is comparable or larger than the extent of the gas. i.e. Most of the original light makes it through. If instead the gas is "optically thick", then photons are absorbed and re-emitted many times before re-emerging and the absorption line would be at least partially infilled by emission.

Why do electrons in a cool gas not release photons as they are excited?

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