Atmospheric Transmission Beer's Law

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3/31/10Optical depth problemThe optical depth of the stratospheric aerosol layer is typically about 1 x 10 -4 forwavelengths of 1 µm. Following the eruption of Mt. Pinatubo volcano in June1991, the optical depth at 1 µm increased to 1 x 10 -2 . How much more or lessinfrared radiation was transmitted through this layer? What do you think theconsequences of this change might be?Pinatubo eruption, June 1991 Space Shuttle view of aerosol veil, August 1991From the restatement of Beer's Law, we know that I/I 0 = exp (-τ), where the ratioI/I 0 represents the fraction of light transmitted through the medium.We can make use of an approximation: for small values of x, e -x ≈ (1 - x). Bothvalues for the optical depth are quite small, so we can approximate:Typical: I/I 0 = e -(0.0001) ≈ (1 - 0.0001) ≈ 1Optical depth problemPost-volcano: I/I 0 = e -(0.01) ≈ (1 - 0.01) ≈ 0.99The amount of light transmitted through the aerosol layer decreased followingthe volcanic eruption because the optical depth increased. Approximately 1%less light was transmitted. That missing 1% had to go somewhere - either bescattered by the particles or absorbed by them. In this case, it was likelyabsorbed.Evidence has shown that the enhanced aerosol layer following the Pinatuboeruption was responsible for slightly increasing the temperature of the lowerstratosphere..10
3/31/10More Beer’s LawAlso known as the Beer-Lambert Law or the Beer-Bouguer-Lambert Law, orother combinations…I 0βdIt = II 0= e −βdThe law states that there is a logarithmic dependence between the transmission (ortransmissivity), t, of light through a substance and the product of the absorptioncoefficient (β) and the distance the light travels through the material, or the pathlength (d).€More Beer’s LawFor a gaseous absorber, the absorption coefficient (β) is written as theproduct of an absorption cross-section (σ, cm 2 ) and the number density ofabsorbers (molecules cm -3 ):I 0σ, NdIt = II 0= e −σNd(For liquids the absorption coefficient is the product of the molar absorptivity of theabsorber and the concentration of the absorber)€11
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Atmospheric Transmission Beer's Law

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