A cloud mass moving across the ocean at an altitude of 2000 m encounters a coast

Question

A cloud mass moving across the ocean at an altitude of 2000 m encounters a coastal mountain range. As it rises to a height of 3500 m to pass over the mountains, it undergoes an adiabatic expansion. The pressure at 2000 and 3500 m is 0.802 and 0.602 atm, respectively. If the initial temperature of the cloud mass is 288 K, what is the cloud temperature as it passes over the mountains? Assume that CP for air is 28.86J?K?1 ?mol?1 and the air obeys the ideal gas law. If you are on the mountain, should you expect rain or snow? PLEASE EXPLAIN WITH FULL DETAIL TO BE SELECTED AS BEST ANSWER.

Explanation / Answer

look at a one mole system to work with because they gave cp as 28.86joules/(kelvin-mol).
use the top one r in the table which is 8.3 because it matches the units of cp. now use this to get cv: R=cp-cv. so cv=cp-R=28.86-8.3=20.86. now get gamma.
gamma=cp/cv=1.38. find your initial volume. v=nrt/p.
this time use the second r in the table because we will be working with volumes in liters and pressures in atms. r=0.08205. v=(1x r x 288)/(0.802) = 29.46 liters
ln (p/pi)= (-1)(gamma)ln(v/vi), i means initial
ln(0.602/0.802)=(-1)(1.38)ln(v/29.46)
-0.287=(-1.38)ln(v/29.46)
0.208=ln(v/29.46) take the e of both sides
1.23x29.46=final volume in liters=36.27 liters.
now use pv=nrt to find final temp in kelvin. use the second r in the table at the wiki link. pv=nrt so t=pv/(nr)=(.602x36.27)/0.08205 because n is one. t final = 266 degrees kelvin. change kelvin to centigrade and its about 13 degrees below zero. it will snow. too cold to rain

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