Converting Absorption Coefficients--A Tour through the Math

(c) 2014 by Barton Paul Levenson

An earlier web page of mine listed absorption coefficients of interest for atmosphere modeling. It used all units as given in the original sources, with a few copying mistakes on my part-- sorry about that! Here I step through the math needed to convert everything to a standard unit. I target the measure I find most congenial, the mass absorption coefficient in m² kg^-1.

Bartko and Hanel 1968

This source uses cm^-1 NTP for CO₂ and O₃, but cm² g^-1 for H₂O. The H₂O conversion is therefore easy: multiply by 1/10,000 m² cm^-2 and 1,000 g kg^-1, or a factor of 0.1:

NTP, "Normal temperature and pressure," is defined as 293.15 K and 101,325 Pa. Our first step for CO₂ and O₃ would then be to convert from cm^-1 to m^-1 by multiplying by 100. We now have m^-1, and how much CO₂ is there in one meter at NTP? The ideal gas law is

P V = n R T

We'll use a volume of 1 m³, reducing this to P = n R T. Solving for n:

n = P / R T = 101325 / (8314.4621 x 293.15) = 0.041571 kmol. One kmol of CO₂ has a mass of 44.0098 kg, so we're talking a specific mass of 1.8295 kg m^-2.

τ = β s = k SM

k = β s / SM

and since s = 1, k = β / SM, finally giving us, for CO₂:

Unfortunately, this gives the lowest figure for the 15-micron absorption band, which should be one of the strongest. Very unlikely, but the error seems to be in the original paper, which itself gives the lowest absorption coefficient for that wavenumber domain.

Lastly, using similar logic, the specific mass for ozone is 1.9953 kg m^-2 (MW = 47.9982), giving the ozone absorption coefficient as:

Chou and Lee 1996

Essenhigh 2001

Using the same reasoning, for STP rather than NTP, and the appropriate molecular weights, Essenhigh's figures become

Griffith et al. 2003

per km per amagat is an odd one. Per km to per meter involves just multiplying by 1/1,000, but what's an amagat? It's a measure of "number density:"

η = n / n₀

where η is the number density in amagat (agt) and n₀ is "Loschmidt's number," the number density of an ideal gas at STP (P = 101325 Pa, T = 273.15 K). The numerical value of n₀ is 2.6867805 x 10²⁵ m^-3.

1 m³ of pure methane at STP would have a number density of 0.044615 kilomoles. 1 kmol is 1000 N_A* or 6.0221413 x 10²⁶, or to put it another way, the Loschmidt number is 0.044615 kmol--not surprising, since n₀ is defined at STP. In other words, we'd have exactly 1 amagat of methane at STP. The specific mass would be 0.71575 kg m^-2. We therefore find:

*N_A is Avogadro's number, 6.0221413 x 10²³.

Hanel et al. 1963

This is an easy conversion: multiply by 1/10,000 m² cm^-2 times 1,000 g kg^-1, or a net factor of 0.1.

Hanst et al. 1975

My first absorbers web page incorrectly gave the absorption coefficient units from this source as cm^-1, whereas the original paper actually says cm^-1 atm^-1. There are a bunch of different absorbers here, so I summarize the findings below:

Kondratiev and Niilisk 1960

This requires some logic. Units are cm^-1 of precipitable water. What the heck does that mean?

Precipitable water is the depth of liquid if the water vapor in the air rained out. A 1 cm depth of water covering a square meter would have a volume of 0.01 m, and given liquid water density at 1000 kg m^-3, this would contain 10 kg. So 1 precipitable cm would be 10 kg to the square meter, or 0.1 m² kg^-1. To find absorption coefficient in m² kg^-1, then, we multiply precipitable cm^-1 by 0.1:

Lundt and Kinnunen 1976

This is an easy conversion. Multiply by 1,000 g kg^-1 to find m² kg^-1:

Okabe 1978

This source gives the absorption cross-section of molecular nitrogen (N₂) as σ = 2 x 10^-21 cm² at wavelengths in the far-ultraviolet of 0.116-0.145 μ. This is an area of 2 x 10^-25 m². One molecule of nitrogen has a mass of μ_N2 = 28.0134 times 1 atomic mass unit, 1.660538921 x 10^-27 kg, or 4.65173 x 10^-26 kg. The cross-section for a whole kilogram, then, would be σ divided by this mass:

Roberts et al. 1976

This is another easy conversion. Change cgs units to mks.

Varanasi and Nemtchinov 1994

For this source we use the cross-section logic. The mass of one molecule of CFC-12 is 2.00782 x 10^-25 kg, so the absorption coefficients become:

A pretty powerful greenhouse gas!

Weissler et al. 1952

This source uses pure reciprocal cm, which we first change to reciprocal m, for β = 68,000 and 276,000 m^-1 at 0.076 and 0.0661-0.0796 μ, respectively.

A meter of nitrogen N₂ at STP holds 1.24982 kg of nitrogen. The absorption coefficients therefore become:

Normally innocuous nitrogen is dead black in the far ultraviolet!

Werbe-fuentes et al. 2008

This team measured CO₂ absorption coefficients at peak wavelengths in the near infrared. Their absorption coefficients are in the rare units of m² mol^-1, where 1 mole of particles is Avogadro's number, 6.0221413 x 10²³. A mole of CO₂ has a mass of 0.0440098 kg, so these figures become:

Yamamoto 1953

This source uses the same units as Kondratiev and Niilisk 1960, for water vapor under the same conditions. We therefore apply the same conversion factor:

References

Bartko F, Hanel RA 1968. Non-gray equilibrium temperature distributions above the clouds of Venus. ApJ 151, 365-378.

Chou M-D, Lee K-T 1996. Parameterizations for the Absorption of Solar Radiation by Water Vapor and Ozone. J. Atm. Sci. 53, 1203-1208.

Essenhigh 2001. In Box: Robert Essenhigh Replies. Chemical Innovation 31, 62-64 (Nov. 2001).

Griffith CA, Owen T, Geballe TR, Rayner J, Rannou P 2003. Evidence for the Exposure of Water Ice on Titan's Surface. Science 300, 628-630.

Hanel RA, Bandeen WR, Conrath BJ 1963. The Infrared Horizon of the Planet Earth. J. Atmos. Sci. 20, 73-86.

Hanst PL, Spiller LL, Watts DM, Spence JW, Miller MF 1975. Infrared Measurement of Fluorocarbons, Carbon Tetrachloride, Carbonyl Sulfide, And Other Atmospheric Trace Gases. J. Air Pollution Control Assn. 25, 1220-1226.

Kondratiev KY, Niilisk HI 1960. On the question of carbon dioxide heat radiation in the atmosphere. Pure and Appl. Geophys. 46, 216-230.

Lundt PB, Kinnunen, L 1976. An application of a water vapour laser. J. Physics E: Scientific Instruments 9, 528-529.

McDade I 2007. EATS 4230 & ESS 5230, Remote Sensing of the Atmosphere, Assignment #2, Distributed Sunday 18th February 2007 - due Friday 2nd March 2007. http://www.yorku.ca/mcdade/eats4230/ Assignment%202%202007.pdf, accessed 3/05/2007.

Okabe H 1978. Photochemistry of Small Molecules. NY: Wiley-Interscience.

Roberts E, Selby JEA, Biberman IM 1976. Infrared Continuum Absorption by Atmospheric Water Vapor in the 8-12 μm Window. Appl. Opt. 15, 2085-2090.

Varanasi P, Nemtchinov V 1994. Thermal infrared absorption coefficients of CFC-12 at atmospheric conditions. J. Quant. Spectroscopy Rad. Transfer 51, 679-687.

Weissler GL, Lee PO, Mohr EI 1952. Absolute Absorption Coefficients of Nitrogen in the Vacuum Ultraviolet. JOSA 42, 84-90.

Werbe-fuentes J, Moody M, Korol O, Kading T 2008. Carbon dioxide absorption in the near infrared. http://jvarekamp.web.wesleyan.edu/public_htmlA/ public_htmlA/CO2/FP-1.pdf, accessed 12/03/2014.

Yamamoto G 1953. Radiative equilibrium of Earth's atmosphere (I). The grey case. Sci. Rept. Tohuku Univ., Ser. 5, Geophys. 5, No. 2.