This from a blog on climate change;
Using Kirchhoff’s law, absorptivity of a material = emissivity of a material for the same wavelength and direction. For diffuse surfaces – and for gases – direction does not affect these material properties, so they are only a function of wavelength. (And for all intents and purposes, absorptance is the same term as absorptivity, and transmittance is the same term as transmissivity).
Emission of radiation at any given wavelength for a blackbody (a perfect emitter) is given by Planck’s law, which is usually annotated as Bλ(T), where T = temperature.
The absorptivity of a gas, aλ = 1-tλ =emissivity of a gas, ελ.
For a very small change in monochromatic radiation due to emission:
dIλ = ελBλ(T) .ds [10a]
dIλ = nσBλ(T) .ds [10b]
So if now combine emission and absorption, equations 1 & 10:
dI/ds = nσ.(Bλ(T) – Iλ) 
If we combine this with our definition of optical thickness, from equation :
dIλ/dτ = Iλ – Bλ(T) 
which is also known as Schwarzschild’s Equation – and is the fundamental description of changes in radiation as it passes through an absorbing (and non-scattering) atmosphere.
It says, in not so easy to understand English:
The change in monochromatic radiation with respect to optical density is equal to the difference between the intensity of the radiation and the Planck (blackbody) function at the atmospheric temperatureSorry it’s not clearer in English.
In more vernacular and less precise terms:
As radiation travels through the atmosphere, the intensity increases if the Planck blackbody emission is greater than the incoming radiation and reduces if the Planck blackbody emission is less than the incoming radiation
When will this madness end?