Molecular Radiation and Collisional Lifetime
Posted by Jeff Id on August 17, 2010
DeWitt Payne has kindly completed a nice post explaining how re-absorption of photons emitted by CO2 or other molecules doesn’t appreciably affect the ratio of energy transfer in the atmosphere due to IR emission vs absorption (or molecular collision). It’s another commonly debated topic often improperly cited by critics of the warming effects of CO2. When it’s not expressed correctly, it’s a clue that the individual is less knowledgeable on the science than they may appear. I’m sure everyone will agree that whatever your position on global warming, the basics are critical to making a sound, science based argument.
I’ve done a bit of reformatting for clarity–Jeff
Have you ever seen or heard the statement that CO2 can’t emit radiation in the atmosphere because the decay time for spontaneous emission is long compared to the collisional life? Guess what, that’s completely wrong. A molecule or atom in an excited state has no knowledge of its age. The probability of decay is the same whether the excited state has existed for centuries or picoseconds. The rate of decay of a collection of things in an excited state depends only on the number of things in the excited state, Ni, and the decay constant Kd. The decay constant has units of reciprocal seconds (s-1).
The decay constant is often expressed as the half-life or the amount of time it takes for half the initial number of things to decay. The half-life is equal to ln(2)/Kd.
The difference between radioactive decay and molecular decay in an atmosphere is that 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.
The number of molecules in the excited state depends only on the energy of the excited state and the temperature through the Maxwell-Boltzmann distribution. For the 667.6612 cm-1 CO2 line at 296 K:
Ni/N = exp(-Ei/kT) = exp (-hν/kT) =0.039
What does that actually mean in terms of radiance?
No significant absorption model:
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 (6.022E23*0.002*0.00038*273.2)/(0.0224*296)=1.89E19 CO2 molecules /m2 and 1.89E19*0.039=7.35E17 molecules in the excited state. That gives 1.542*7.35E17=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.
Bulk atmosphere model:
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-08*υ3)/(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.
It’s distinctly possible I’ve made multiple errors here, but if I did, they appear to cancel out.