The purpose of radiative transfer equations is to show that carbon dioxide blocks some radiation from leaving the atmosphere. The blocking concepts has a lot of surrounding realities attached to it, while there is no such thing.

It starts with the concept of absorption spectra. Absorption means absorption; it doesn't mean blocking or trapping. But it is interpreted to mean blocking and trapping.

Absorption spectra are approximately the inverse of transmission spectra. That means the wavelengths that are absorbed do not show up as being transmitted in those wavelengths.

So incompetents retranslate the absence of transmission as blocking or trapping and blocking or trapping as accumulation of heat.

The problem with those assumptions is that nothing blocks of traps heat. The energy that is absorbed heats the molecule, which is interpreted by incompetents as meaning the energy heats the atmosphere. What they miss is that every molecule in existence above absolute zero is emitting radiation in proportion to its temperature. It means heat is not trapped by a heated molecule.

The closest concept to trapping heat would be an insulating substance. Insulators have their own characteristics which are never evaluated for the atmosphere, because there is no such thing as an insulating material in the atmosphere.

That means the heated molecule is radiating its energy away as fast as it is absorbed. That's why the atmosphere cools about 20°C on clear nights.

So the net effect is that the radiant energy absorbed by a molecule at one wavelength is re-emitted at another wavelength.

The wavelengths absorbed by so-called greenhouse gases are narrow bands called fingerprint radiation. That type of absorption results from internal bonds stretching, rotating and vibrating.

That activity heats the whole molecule. The heat of whole molecules is emitted as wider bands and longer wavelengths, because the frequency of vibration is slower for large molecules than it is for groups attacked to the molecule.

Re-emitting absorbed radiation at wider bandwidth and longer wavelengths is not blocking or trapping. The incompetents omit the re-emission.

So radiative transfer equations have the purpose of determining how much absorbed radiation never gets out of the atmosphere. There is no such thing. Heat gets radiated at whatever wavelength a molecule vibrates at—always, without exception, everywhere—no blocking or tapping physically possible.