# Thermodynamics

A cloud moving across the ocean at 2000 m (height) encounters a mountain range. As it rises to 3500 m, it undergoes adiabatic expansion from p=0.802 to 0.602 atm. If the initial temperature of the cloud is 288 K, will it rain or snow on the mountains? (Assume air to be an ideal gas with Cpm = 28.86 J/mol*K)

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

This is a case of adiabatic expansion. For adiabatic expansion, following equations must be satisfied :

PVÎ³ = Constant where P, V are the pressure and volume and Î³ = Cp/Cv

Or P1V1Î³ = P2V2Î³

Or (P1/ P2) = (V2/V1)Î³ .......(1)

And assuming air to be an ideal gas : PV = nRT

Or P1V1 = nRT1 and P2V2 = nRT2

Or (V2/V1) = (P1/P2)(T2/T1) ......(2)

Substituting from (2) in (1) : (P1/ P2) = (P1/P2)Î³(T2/T1)Î³

Or (P1/ P2)1-Î³ = (T2/T1)Î³

Or (P1/ P2)(1-Î³)/Î³ = (T2/T1) ........(3)

For air Cp = 28.86 J/mole.K

Also we know that Cp - Cv = R or 28.86 - Cv = 8.31 or Cv = 20.55 J/mole.K

Hence, Î³ = Cp/Cv = 28.86/20.55 = 1.4

Substituting P1 = 0.802 atm, P2 = 0.602 atm, T1 = 288 K and Î³ = 1.4 in (3) we get :

(0.802/0.602)(1-1.4)/1.4 = T2/288

Or 1.332-0.286 = T2/288

Or 1.3320.286 = 288/T2

Taking log on both the sides : 0.286 log101.332 = log10288 - log10T2

Or log10T2 = 2.424 or T2 = 265 K (or - 8OC)

As the temperature falls to -8OC (below the freezing point of water), it will snow.

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