My crude back-of-the-envelope estimate is about 3.4m.

This is based on a current CO2 concentration of about 400 ppmv (parts per million by volume) - or 4 parts in 10,000 of air. Assuming that CO2 is "well mixed" in the atmosphere, a litre of air contains about 4 x 10^-4 or 0.0004 litres of CO2.

Next, take the hydrostatic relationship:

pressure = density * gravity * height

If the atmosphere had the same density everywhere as at the surface (i.e, about 1.225 kg m^-3), then with surface pressure of 1013 hPa or 1.013^5 Pa, gravity = 9.8 m s^-1, the atmosphere would have a (constant density) height of about 8,438m. Then about 0.0004 of that, or 3.375m, would be CO2.

Since the above is essentially a "volume-based" calculation I don't think you need to worry about the different molecular weights of CO2 and the rest of the air - but you might want to confirm that yourself...

I can confirm that, my not so crude calc, also for 400 ppmv, delivers 3.107 m (total CO2 mass being 3.112E15 kg). By the way, stratospheric Ozone works out to 3.1 mm, tropospheric Ozone to 0.54 mm.

Here is whole picture (excel table does not copy as table but as csv, and needed some rearranging):

component height (if component were to be at STP)

N2 6064.1 m

O2 1626.7 m

A 72.5 m

H2O at 0.4% of atmosphere 49949.6 mm (H2O vapour only, not liquid/droplet H2O)

CO2 3106.6 mm

Ne 141.2 mm

He 40.7 mm

CH4 13.9 mm

Kr 8.9 mm

H2 4.3 mm

N2O 2.6 mm

CO 0.8 mm

Xe 0.7 mm

troposphere O3 0.5 mm

stratosphere polar O3 3.1 mm (=300 Dobson Units)

SO2 7.8 mm

NO2 0.2 mm

iodine I 0.1 mm

So how can the discrepancy between my "crude" estimate of 3.375m and the "precise" calculation of 3.107m be reconciled? Not by directly invoking the different molecular weights of CO2 (44.01g/mol) and mixed air (28.97 g/mol). The ratio between them is about 0.66, while 3.107/3.375 is about 0.92.

Perhaps that 8% or so of a discrepancy is more indirectly due to the different molecular weights, insofar as the heavier CO2 is more concentrated near the surface than above the tropopause, and so the "well-mixed" assumption is only approximate, and maybe only valid for the lower 80% of the atmosphere. That would also explain why my "crude" estimate comes in higher than the more precise one.

Anyway, it's probably safe to say that John Koch's tubes, up to 3.87m long, are plenty long enough for the purposes of measuring CO2 absorption in the atmosphere.

Our answers are within 8% of each other, so essentially confirming each other. I use total amount of CO2 in atmosphere (kg, stated in my response) for 400 ppmv (or 608 ppmm) of the published total dry atmosphere mass (i.e. without H2O), converting to its STP volume (m3) , and dividing by the earth surface (m2) to get height (m). My calc still includes, of course, uncertainties in atm mass and composition, earth surface, etc.

I did have a slight problem with your air density of 1.22 kg/m3 (which is indeed in widespread use, I have seen it used elsewhere) because at STP it is actually close to 1.293 kg/m3, a 6% difference already.

I built up an xls file with geography, atmosphere, ocean, etc data which is handy when issues come up. You are welcome to have a copy if you provide email address. Don't know whether ResGate allows that.

And yes, the 3.87m tube should do. But absorption in pure CO2 will not be a sufficiently correct measure for absorption in atmosphere.

A further point worth making is that the preoccupation with CO2 is all well and good, but CO2 does not cause global warming. Global warming and cooling is caused by cosmic (mainly solar) and earth-bound (mainly oceanic) cyclic processes with a large range of cycle lengths, and these processes all cause increase and decrease in CO2, some short term, others (like ocean outgassing) in long term . Human activity also contributes CO2 but it is neither here nor there.

Humans do other, severe damage to the planet, in the form of surface modification and thus micro-climate modification, solid and liquid pollution of atmosphere and ocean and inland waters, destruction of wildlife habitats and consequent species destruction (Europe lost practically all its wildlife in the last 10000 years). And so forth. But humans do not affect the macro-climate.

Enda William O'Brien and Geert F de Vries,

Thank you for your replies. Between you, you have solved a problem I had with Koch's results. He was experimenting with the absorption of CO2 in tubes of various lengths and at various pressures. He found that the absorption at 300 °C was consistently less than that at 100 °C for the same length and pressure.

The reason was that the density of CO2 (or any other gas) falls as the temperature rises, so the absorption also falls with a rise in temperature. Enda's density of 1.225 kg m-3 is for air at 15 °C : https://en.wikipedia.org/wiki/Density_of_air

whereas Geert's density of 1.293 m-3 is for air at 0 °C i.e STP.

I have roughly translated Koch's paper here: Working Paper Translation of Koch, J., 1901.

and the original German text is available here:

https://www.biodiversitylibrary.org/page/32299447#page/489/mode/1up

Although the concentration of CO2 when Koch was experimenting was only 300 ppm, his discovery that the absorption of CO2 in tubes of only 30 cm changes vary little when the pressure was reduced shows that the absorption of CO2 in the Earth's atmosphere is virtually saturated. See:

Very, F.W. and C.A. (1901) ‘Knut Angstrom on Atmospheric Absorption’, Monthly Weather Review, vol. 1901, no. June, p. 268. http://docs.lib.noaa.gov/rescue/mwr/029/mwr-029-06-0268a.pdf