Let’s see if my brain is up to this topic.

Fundamental to the first IPCC report is that 70% of global warming is caused by carbon dioxide.
The same number made it into the second IPCC report.

a) Prove it.
b) That calculation was done many decades ago, what percentage is it now?

To answer this, I need to integrate the infrared absorption of CO2, after subtracting off the infrared absorption of H2O vapour.

I already know that H2O at atmospheric concentrations is a much stronger absorber of infrared than CO2. At a guess four to five times as strong, but I can check that. CO2 is relatively rigid because of its double bonds and linear shape, H2O is much more flexible because of its single bonds and bent shape. It is this flexibility that makes H2O such a great absorber of infrared radiation.

Calculating the IR absorption from CO2 directly from quantum mechanics would probably be a fiendishly difficult task. Because whereas the line frequencies are easily obtained from quantum mechanics, the line widths are not easy.

That gives the second method of calculation, experimental using measured IR absorption curves. IR absorption of CO2 has been measured as far back as about the 1850s, long before it was known that CO2 was a linear molecule with double bonds.

A good starting place to get some idea of the difficulty of the problem is this paper from 1932. https://tp.physique.usherbrooke.ca/experiences_fichiers/Fourier/References/CO2.pdf

I need to integrate out the curve in Figure 2. And similar curves for the other spectral lines. The centre section has been amplified to better show it on the chart.

It’s no good using the IR absorption spectrum of CO2 in solid form or aqueous solution, those will be different. I don’t know if the infrared emission spectrum of CO2 can be used. If so, this is a chart from a book.

Other books are:
“Infrared Absorption Spectrum of Carbon Dioxide”, 1932
“Infrared Absorption by Carbon Dioxide, Water Vapor, and …”, 1962
“The Infrared Absorption of Carbon Dioxide”, 1961
“Infrared Absorption of Carbon Dioxide and Water Vapor …”, 1969
“The Infrared Absorption Spectrum of Water Vapor and Carbon …”, 1962
“ Determination of Carbon Monoxide and Carbon Dioxide Concentrations at Temperatures Between 295-1250 K Using Fourier Transform Infrared Absorption Spectroscopy”, 1992
etc.

Articles:
“Absorption cross section measurements of carbon dioxide in the wavelength region 118.7–175.5 nm and the temperature dependence”, 1996
“The infrared absorption of carbon dioxide”, 1963

The last of these is subtitled “The infrared absorption of carbon dioxide was calculated over a wide range of path length
and pressure from 1500 to 110,000 cm. The results are presented in extensive tables” which looks promising. Let’s try that.
Nope, have to sign in to the US department of defence to read that one.

Fine. Before I started typing here I already had rough charts of IR absorbance for CO2 and H2O from the book by Seinfeld and Pandis “Atmospheric Chemistry and Physics”, 1998. Let’s look and see where they get their information. The following is a scan of their Page 28.

If I start with the curve in d, subtract off the curves in b and c, that gives the absorption due to CO2 (and minor gases). And multiply that by the curve in a to get the total absorption of IR by a normal amount of CO2.

The relationship between concentration and absorptivity is nonlinear, but if I subtract from 1, multiply by the CO2 concentration ratio and subtract from 1 again. Then multiply by the curve in a, and integrate over wavelength then I could see by how much a change in CO2 concentration affects … Oops. Not quite right. I’d have to use a black body emitter at Earth’s average temperature instead of at the Sun’s temperature to replace Figure a. Then I could get the correct answer. And simply plug the resulting IR absorption from varying CO2 concentrations into the radiation balance equation for Earth to get the global warming due to a rise in CO2 concentration.

OK, so I can do it. This empirical approach is much much much easier than using full quantum chemical calculations.

But can I do it more accurately?

The following chart is from Seinfeld and Pandis Page 32.

This chart comes from the IPCC, but where does the IPCC get its information from?

Fine. Before I started typing here I already had rough charts of IR absorbance for CO2 and H2O from the book by Seinfeld and Pandis “Atmospheric Chemistry and Physics”, 1998. Let’s look and see where they get their information. The following is a scan of their Page 28.

If I start with the curve in d, subtract off the curves in b and c, that gives the absorption due to CO2 (and minor gases). And multiply that by the curve in a to get the total absorption of IR by a normal amount of CO2.

Working straight from Seinfeld and Pandis Page 28. The curves plotted do not exactly inspire confidence. One of the Ozone peaks fails to match up in frequency. Another has an absorption exceeding 100%. The horizontal axis labelling has 6 and 8 microns marked in the wrong place, and the tick marks are a bit out. The absolute best accuracy for the temperature gain from CO2 is +-15% if that. So what I’m going to do is a direct comparison with the IPCC result. If they agree with 30% then that counts as a confirmation of the IPCC computation.

Here is a difference between all atmospheric absorption and that due to O2, O3 and H2O. The difference will be atmospheric absorption due to CO2. The entire atmosphere absorption is in black. That due to O2, O3 and H2O is in blue. Hmm.

mollwollfumble said:

Fine. Before I started typing here I already had rough charts of IR absorbance for CO2 and H2O from the book by Seinfeld and Pandis “Atmospheric Chemistry and Physics”, 1998. Let’s look and see where they get their information. The following is a scan of their Page 28.

If I start with the curve in d, subtract off the curves in b and c, that gives the absorption due to CO2 (and minor gases). And multiply that by the curve in a to get the total absorption of IR by a normal amount of CO2.

Working straight from Seinfeld and Pandis Page 28. The curves plotted do not exactly inspire confidence. One of the Ozone peaks fails to match up in frequency. Another has an absorption exceeding 100%. The horizontal axis labelling has 6 and 8 microns marked in the wrong place, and the tick marks are a bit out. The absolute best accuracy for the temperature gain from CO2 is +-15% if that. So what I’m going to do is a direct comparison with the IPCC result. If they agree with 30% then that counts as a confirmation of the IPCC computation.

Here is a difference between all atmospheric absorption and that due to O2, O3 and H2O. The difference will be atmospheric absorption due to CO2. The entire atmosphere absorption is in black. That due to O2, O3 and H2O is in blue. Hmm.

That approximation is not good enough. The peak of the 188 K blackbody spectrum is to the very right of that graph, near 16 microns, and the CO2 absorption there is very iffy. Here’s the blackbody spectrum.

I need gaseous absorption information for CO2, H2O and other gases al longer wavelengths.

mollwollfumble said:

mollwollfumble said:

Fine. Before I started typing here I already had rough charts of IR absorbance for CO2 and H2O from the book by Seinfeld and Pandis “Atmospheric Chemistry and Physics”, 1998. Let’s look and see where they get their information. The following is a scan of their Page 28.

If I start with the curve in d, subtract off the curves in b and c, that gives the absorption due to CO2 (and minor gases). And multiply that by the curve in a to get the total absorption of IR by a normal amount of CO2.

Working straight from Seinfeld and Pandis Page 28. The curves plotted do not exactly inspire confidence. One of the Ozone peaks fails to match up in frequency. Another has an absorption exceeding 100%. The horizontal axis labelling has 6 and 8 microns marked in the wrong place, and the tick marks are a bit out. The absolute best accuracy for the temperature gain from CO2 is +-15% if that. So what I’m going to do is a direct comparison with the IPCC result. If they agree with 30% then that counts as a confirmation of the IPCC computation.

Here is a difference between all atmospheric absorption and that due to O2, O3 and H2O. The difference will be atmospheric absorption due to CO2. The entire atmosphere absorption is in black. That due to O2, O3 and H2O is in blue. Hmm.

That approximation is not good enough. The peak of the 188 K blackbody spectrum is to the very right of that graph, near 16 microns, and the CO2 absorption there is very iffy. Here’s the blackbody spectrum.

I need gaseous absorption information for CO2, H2O and other gases al longer wavelengths.

Very important to note that the H2O absorption near the peak of the 288 K blackbody curve (15 to 20 microns wavelength) is not zero. This gives us a nonzero absorption of radiation when there is 0% CO2. This is vital to understand if comparing 100% CO2 to 200% CO2. In the image below we see the same thing from another source. Has the IPCC got it right? Probably, but it’s worth checking.

Here’s a good one from Wikipedia. The title reads “Absorption spectrum (attenuation coefficient vs. wavelength) of liquid water (red), atmospheric water vapor (green) and ice (blue line) between 667 nm and 200 μm. The plot for vapor is a transformation of data Synthetic spectrum for gas mixture ‘Pure H2O’ (296K, 1 atm) retrieved from Hitran on the Web Information System.”. It is the water vapour that interests us here, and it is quite significant in the wavelength range of 288 K blackbody radiation from 5 to 100 microns.

The attenuation coefficient is “per metre” so needs to be integrated over the height of water vapour in the atmosphere, which in turn is the concentration of water vapour in the atmosphere (constant in the troposphere, low in the stratosphere) times the integrated density of the atmosphere.

Is there anything equivalent for carbon dioxide? Not obviously on wikipedia. Here’s one from quora.

mollwollfumble said:

CO2 is relatively rigid because of its double bonds and linear shape, H2O is much more flexible because of its single bonds and bent shape.

I have considered the IR-absorption spectra of CO₂ and H₂O in a Chemistry III subject about Molecular Symmetry and Group Theory. Without going into much technical detail, CO₂ has 4 vibrational modes: 2 of these are a degenerate pair of bending modes and are IR-active but with the same energies due to the degeneracy; 1 is a symmetric stretching mode which is IR-inactive due to the unchanging electric dipole moment; and 1 is an asymmetric stretching mode which is IR-active. H₂O has 3 vibrational modes: a bending mode; a symmetric stretching mode; and an asymmetric stretching mode; all of which are IR-active. It is worth mentioning that although both CO₂ and H₂O are triatomic molecules, CO₂ has one more vibrational mode due to it being a linear molecule (unlike H₂O). That is, both molecules have 9 dimensions of molecular motion (3 atoms, each independently with 3 dimensions of motion), 3 of these dimensions are translations of the molecule as a whole, 3 of these dimensions are rotations of the molecule as a whole in the case of non-linear H₂O, but only 2 of these dimensions are rotations of the molecule as a whole in the case of linear CO₂. This leaves 3 of these dimensions as vibrations in the case of H₂O, but 4 as vibrations in the case of CO₂.

Symmetry won’t tell you what the frequencies of the vibrational modes are, nor their absorbances. It will tell you which modes are IR-active or IR-inactive, as well as which modes are degenerate. However, each mode does correspond to several peaks in the spectra. Firstly, there are peaks corresponding to the fundamental frequency of the vibrational mode, and additional peaks corresponding to the higher harmonics of that mode. Secondly, and more significantly to your question, each main peak has a series of side peaks corresponding to the different rotational states of the molecule. In the gas phase, these show as more-or-less discrete peaks, whereas in the liquid or solution phase, they show as broadness of the main peak.

I could also do the IR spectrum of methane, but unlike carbon dioxide and water, which can be done by inspection, methane is too complicated to be examined without the full force of group theory.