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Homework 3 Solutions

For the parameters given the mean molecular weight is

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Unformatted text preview: κλ(0.03 kg/m3)(0.3 m) = 2/3 => κλ = 74 m2/kg Not surprisingly, clay is a lot more opaque than water! (3) Looking through gaseous water [25 pts] (a) [10 pt] Water vapor in the air can also absorb starlight, although in this case its density is not constant with height. Show that atmospheric pressure scales vertically as P (z ) = P (0)e−z/H where H= kT gµmH is the scale height. Compute this scale height assuming standard temperature and pressure (STP): T = 0 ºC = 273 K and P = 100 kPa = 1 bar; and an atmosphere composed (by volume) of 77% N2, 21% O2, 1% H2O and 1% Ar. Consider a slab of gas with area A and vertical width dz, so that the mass of this slab m = ρAdz. The balance of forces  ­ air pressure acting on the top and bottom of the slab, and the slab’s own weight  ­ keep the slab in place only if Ftop − Fbottom + g ρAdz = 0 dP ⇒ = −g ρ dz where we’ve used P = F/A. Using the ideal gas law, this right side be written in terms of pressure: dP P µmH dP gµmH = −g ⇒ =− dz dz kT P kT With H as defined above, we get the pressure equation as shown. For the parameters given, the mean molecular weight is µ = (0.77)(28) + (0.21)(3...
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