Carbon Dioxide Saturation and Global Warming

Vincent J. Curtis

16 Sept 23

In two previous postings, Beer’s Law and Climate Change (16 Nov 21) and The Molecular Mechanics of Global Warming (16 Nov 21), I explained in a qualitative way how carbon dioxide contributes to warming of the atmosphere, and also that there is limit to how much infra-red radiation CO₂ can absorb in its contribute to atmospheric warming.  The important conclusion was that adding more carbon dioxide to the atmosphere above which is already present will contribute nothing more to atmospheric warming.  Those qualitative explanations will now be demonstrated quantitatively.

In 1998, Heinz Hug determined the extinction coefficient of the bending mode of CO₂ at the frequency of 667 cm^-1, the famous 15 micron absorption peak.  The other principle CO₂ absorption peak, asymmetric stretch, occurs at 2390 cm^-1; but this is not significant to the global warming effect.

The Beer-Lambert law is:

            log₁₀ (I/I₀) = -Ɛ_ƛCd      where Ɛ_ƛ is the extinction coefficient at wavelength ƛ

                                                C is concentration

                                                d is distance

Hug determined Ɛ for CO₂ at the frequency of 667 cm^-1.  At a concentration of 357 ppmv CO₂ in atmospheric air containing 2.6 % water vapor, the extinction coefficient was found to be 20.2 m²/mole.  This enables us to calculate the amount of radiation absorbed over a distance of 10 m in the atmosphere.

We convert 357 ppmv into moles/m³ as follows:

357 ppmv = 0.357 L/1000 L of air.  At STP, 1 mole of gas occupies 22.414 L volume; hence, .357 L = 357/22.414 = 0.0159 moles.  And since 1000 L = 1 m³ we get:

357 ppmv = 0.0159 moles/m³.

Substituting:  log (I/I₀) = - (20.2 * 0.0159 * 10)  m²/mol * mol/m³ * m

                       log (I/I₀) = --3.21

                      Exponentiating:   I/I₀ = 0.0006

 This means that 99.94 % of the radiation is absorbed in the passage through 10 m of air.  Increasing the concentration of carbon dioxide merely shortens the distance in which 99.94 % of the radiation is absorbed.  If, for instance, concentration were doubled, the distance would be halved.  The earth’s lower atmosphere is effectively opaque to IR radiation at the 667 cm^-1 wave number already.  A higher concentration of carbon dioxide in the atmosphere won’t make it any more opaque.

All of this radiation is converted to heat.  This is demonstrated as follows:

Let N_i(t) be the number of molecules of CO₂ in its excited vibrational state at time t.

Let N₀ be the number of molecules of CO₂ in its excited vibrational state at time t=0.

The decay constant, K_d ,of CO₂ returning to its ground state is 1.54/s, which is known from the HITRAN database.

The half-life of the excited state is calculated as follows:

            N_i(t)/N₀ = ½ = exp - (K_d * t_1/2)   where exp = e, the base of the natural logarithms.

            ln (1/2) = - K_d * t_1/2   or    t_1/2 = ln (2)/1.54 = 0.45 s.

The half-life of a vibrationally excited CO₂ molecule is roughly half a second.

Thermal quenching is the process in which a vibrationally excited CO₂ molecule is reduced to the ground vibrational state by a collision with another molecule: the vibrational energy of the CO₂ being converted into kinetic energy in the colliding molecule, i.e. a higher speed of the other molecule exiting the collision.  We have data for that also.

For nitrogen:    K_N2 = 5.5 * 10^-15  cm³/molecule-sec

For oxygen:      K_O2 = 3.1 * 10^-15  cm³/molecule-sec

Averaging for atmospheric composition:  K_air = 5 * 10^-15 cm³/molecule-sec.

Since there are approximately 2.5 * 10¹⁹ molecules per cm³ of air, multiplying K_air with N, the number of molecules, we get:

            5 * 10^-13 * 2.5 * 10¹⁹  =  12.75 * 10⁴  quenching collisions per second.

Hence, vibrationally excited CO2 undergoes roughly 100,000 no-radiative deexcitation collisions per second, or about 50,000 over the half-life of a radiative deexcitation.  A non-radiative thermal deexcitation is 50,000 times faster, and hence far more probable, than re-radiation.  Thus, thermalization effectively kills back-radiation, and hence ends any further potential greenhouse effect of more CO₂.  Because the CO₂ already present in the atmosphere is so quickly returned to the vibrational ground state, it is ready to receive another quanta of IR radiation much faster than if had to wait half a second to de-excite by re-radiation.  This makes the CO₂ already present in the atmosphere so efficient at absorbing IR radiation and converting it to heat.

So, what happens with all this heat transferred to nitrogen and oxygen?  By convection and thermal diffusion, this heat is carried to the upper atmosphere, where, by a process of thermal excitation, ground state infra-red active molecules are excited into a higher vibrational state, and radiatively return to the ground state.  This process becomes more favored because, at the low pressures of the upper atmosphere, thermal quenching is far less probable.

-30-

Heinz Hug, 10 Aug 12 eike-kilma-energie.eu

HITRAN data base: spectra-calc.com/ spectra browser/db data.php

Siddles, Wilson, Simpson, Chemical Physics 184 (1994) 779-91