Question No. 16

What is the minimum spatial volume where one could see the greenhouse effect? E.g., would a cube of 1 m3 filled with chlorofluorocarbons suffice?
Answered 1 October 2007.
Question author: anonymous.
Asked 28 September 2007.

The greenhouse effect depends on the so-called optical thickness τ, which is defined as the ratio of the geometric thickness (height) z of the considered area to the free path length l of thermal photons, τ = z/l. Free path length l = 1/, where n is concentration of greenhouse substances (number of molecules in one cubic meter), σ is the absorption cross-section (in square meters) characterizing absorption of thermal photon by one molecule.

In the atmosphere the optical thickness of CO2 is of the order of unity, τ ~ 1. Taking thickness of a uniformly dense atmosphere to be of the order of 10 km, we obtain that free path length is also 10 km (photons travel on average 10 km before they are absorbed by a CO2 molecule). Air density is 2 x 1025 molecules/m3, the relative amount of CO2 is 300 ppm = 3 x 10−4. This means that CO2 concentration is 6 x 1021 molecules/m3 and, hence, σCO2 ~ 2 x 10−26 m2. To achieve optical density τ ~ 1 at z = 1 m, one has to increase CO2 density by 104 times, i.e. up to 6 x 1025 molecules/m3, three times air density, which is technically possible. Absorption cross section of chlorofluorocarbons (CFCs) is about four orders of magnitude larger than that of CO2. Therefore, in order to obtain τ ~ 1 at z = 1 m for CFCs one needs to attain a CFC concentration of 6 x 1023 molecules/m3, i.e. a concentration equal to that of atmospheric CO2. This is technically achievable. Generally, the greater the concentration, the smaller volume one will need to see a preset value of the greenhouse effect.

Further reading:

Gorshkov V.G., Makarieva A.M. (2002) Greenhouse effect dependence on atmospheric concentrations of greenhouse substances and the nature of climate stability on Earth. Atmospheric Chemistry and Physics Discussions, 2, 289-337. Abstract. PDF (0.3 Mb). Screen version (0.5 Mb).