For molecular decay, the number of molecules in the excited state is constant at constant temperature and pressure. CO2 molecules are continually being raised to the excited state and the excited states are lowered back to the ground state by inelastic collisions with other molecules. In inelastic collisions, kinetic energy is converted to vibrational energy and back. Most molecular collisions are elastic and total kinetic energy is preserved. **Only about 1 in 10,000 collisions is inelastic at Earth surface temperature and pressure. Since the mean time between collisions is about 1 ns under those conditions, that means the expected lifetime of a CO2 molecule excited to the 15 micrometer vibrational excited state is on the order of 1-10 microseconds. This also means that only about 1 in 10,000 excited molecules decays by emission of radiation rather than collision. For a system to be in local thermal equilibrium it is necessary for this ratio to be very small.

The decay constant for a molecular line is the Einstein A21 coefficient. The value of A21 for any ghg molecular transition can be found in the HITRAN database. The database can be searched using the extract data tab in the line browser feature of SpectralCalc ( http://www.spectralcalc.com/spectral_browser/db_data.php ). For the most intense CO2 line at 667.6612 cm-1, the A21 coefficient is 1.542 s-1 or a half life of 0.45 s.

Let’s take a very thin layer of gas so that self absorption can be neglected. If we use surface atmospheric conditions with a CO2 volume mixing ratio (VMR) of 0.00038, the transmittance at the line peak according to SpectralCalc is 0.992 for a layer 2 mm thick. The absorptance is 1-0.992 or 0.008. For a surface area of 1 m2, that’s a volume of 0.002 m3. At STP (1013 mbar and 273.2 K) there are 0.0224 m3/mole and 6.022E23 molecules/mole. Correcting for the temperature difference between 296 and 273.2 and the VMR, there are (6.022E230.0020.00038273.2)/(0.0224296)=1.89E19 CO2 molecules /m2 and 1.89E190.039=7.35E17 molecules in the excited state. That gives 1.5427.35E17=7.13E18 photons/sec. The photon energy is hν=1.33E-20 and a radiance (ignoring layer thickness) of 7.13E18*1.33E-20/4π= 1.20E-03 W m-2 sr-1.

Now let’s take the same layer of gas and calculate the radiance using the Planck equation. For an emissivity of 0.008, a frequency in cm-1 and radiance in W m-2 sr-1:

I(υ,T)= ε(1.191427E-08υ3)/(exp(1.438775*ν/T)-1)= 1.15E-03 W m-2 sr-1.

1.20E-03 W m-2 sr-1 approximately equates to 1.15E-03 W m-2 sr-1.