Atmospheric Transmission Beer's Law

3/31/10Mass extinction coefficient• The density of the absorber is given by:ρ = M HAWhere M is the mass of absorber in the solution, H is the depth of thesolution, and A is the horizontal area of the ‘container’.• The transmittance is then:€t = exp(−τ ) = exp(−β eH)⎡⎡ ⎛⎛ M ⎞⎞ ⎤⎤= exp⎢⎢−k e ⎜⎜ ⎟⎟ H⎣⎣ ⎝⎝ HA⎠⎠⎥⎥⎦⎦= exp ⎡⎡−k ⎛⎛⎢⎢ e⎜⎜⎣⎣ ⎝⎝⇒ τ = k e• So the transmittance does not depend on H, only M and A.MAMA⎞⎞ ⎤⎤⎟⎟⎠⎠⎥⎥⎦⎦€Extinction cross-section• The dimensions of k e are area per unit mass, or extinction cross-section perunit mass• The extinction cross-section is used when considering molecules of anabsorbing gas, droplets of water in a cloud, or soot particles in a smokeplume.• We use the number density (N; m -3 ) to characterize the particleconcentration.• To link the extinction coefficient and N we can write:• Where the constant of proportionality σ e has dimensions of area and is theextinction cross-section; or the effective area of the particle ‘blocking’transmission of radiation€β e= σ eN15