# 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:

μLoμHik (m2 kg-1)
50inf 0.18
4050 0.031
20.229.9 1.4
18.220.2 3.5
9.0910.0 0.1
6.25 9.09 0.3
5.0 6.25 1.4
3.85 5.020
1.25 3.8580
1.25 5.8851

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:

μLoμHik (m2 kg-1)
50inf 0.000082
4050 0.000030
29.940 0.037
1618.2 0.00087
15.216 0.00041
13.915.2 0.000017
12.313.9 0.0028
11.412.3 0.052
10.911.436
10.010.932
9.0910.0 0.11
6.25 9.09 0.0026

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:

μLoμHik (m2 kg-1)
16.018.2 3.8

## Chou and Lee 1996

With k in cm-1 atm-1, we first multiply by 100 to get m-1 atm-1. 1 atm of pure O3 would have, as noted above, 1.9953 kg m-2 specific mass, so again we divide by that to find k in m2 kg-1:

μLoμHik (m2 kg-1)
0.1750.225 1527
0.2250.245 9384.1
0.2450.2815132
0.280.295 2147
0.3950.31  355.3
0.310.32   62.6
0.320.4    1.50
0.40.7    2.51

## Essenhigh 2001

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

GasμLoμHik (m2 kg-1)
H2O1.82714
2.53 71.4
58 11
1225  0.86

CO2

1.9

2.1

334
2.62.9 71.00
4.14.5  9.356
1317  0.754

CH4

2.2

2.5

147
3.54.5 18.0
6.57  3.2

## 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 NA* 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:

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

μk (m2 kg-1)
0.830.000015
0.940.000014
1.070.000007
1.280.000007
1.580.000007
20.000007
2.90.0001

## 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.

GasμLoμHik (m2 kg-1)
H2O 6.33  6.5010
6.50  6.6725
6.67  6.8510
10.75 11.75 0.0000050
21.00 25.00 1
25.00 30.00 4
30.00 40.0016
40.00125.0040

O3

8.90

9.35

0.0063
9.35  9.9 0.16
9.9 10.1 0.0063

CO2

14.0

14.5

0.04
14.5 15.5 0.2
15.5 16.0 0.04

## 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:

AbsorberμMWk (m2 kg-1)
CCl2F2 (CFC-12)10.8120.9138  860
CCl3F (CFC-11)11.8137.3684 1900
CCl4 (carbon tetrachloride)12.6152.823 1900
C2Cl4 (perchloroethylene)10.9165.834  430
CHClF2 (CFC-22) 8.945 86.4687 2200
CH3CCl3 (methyl chloroform) 9.174133.4048  270
C2H2 (acetylene)13.9 26.037921000
C2H4 (ethylene)10.5 28.0538 2500
CNH2N+2 (paraffinic carbon) 3.367 30.0696*  310
COS (carbonyl sulfide) 4.866 60.0764 2800
N2O (nitrous oxide) 3.883 44.0128   32
N2O (nitrous oxide) 8.688 44.0128    8.1
*representative value

## 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:

μLoμHik (m2 kg-1)
12130.025
13140.084
14150.13
15160.165
16170.44
17181.72

## Lundt and Kinnunen 1976

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

Gasλ (μ)k (m2 kg-1)
H2O27.97145 ± 8

## 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 μN2 = 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:

μLoμHik (m2 kg-1)
0.1160.1454.3

## Roberts et al. 1976

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

 a = 0.42 m2 kg-1 b = 555.8 m2 kg-1 β = 0.0000787 m k(ν, 296K) = a + b exp(-β ν) k(ν, T) = k(ν, 296K) exp [T0 (1/T - 1/296)] where T0 = 1800 K.

## 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:

λ (μ)k (m2 kg-1)
937,830
1128,640

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:

μLoμHik (m2 kg-1)
0.076- 54,000
0.06610.0796221,000

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:

λPEAK (μ)k (m2 kg-1)
1.437> 230
1.955    5.7
2.013    9.8
2.060    9.8

## 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:

μLoμHik (m2 kg-1)
12130.044
13140.052
14150.063
15160.076
16170.094
17180.129

## References

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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.

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

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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.