The earth’s atmosphere can be modeled as an ideal gas! Assume that the atmospher

Question

The earth’s atmosphere can be modeled as an ideal gas! Assume that the atmosphere is only appreciable over a small range of altitudes above sea level when compared to the Earth’s radius. This way, we can assume the gravitational force is constant.

1. Consider a column of air above sea level. (A cylinder, with area A would be a good column). Find the pressure difference dp between air at an altitude of y and y+dy in terms of the molar mass of air MM, g, R, T and p. Hint: use pV = NRT.

2. Solve the differential equation in 1 to obtain an expression for the pressure as a function of altitude.

3. From the result in 2, can you estimate how high is our atmosphere (from sea level to ~ 0.03 atm which is low vacuum.) based on the atmospheric composition? (The atmosphere is ~78% Nitrogen, ~21% Oxygen, ~1% Argon. Ignoring traces and water vapor, the average molar mass is 28.97 g/mol. Use T=300K.

The answer in 3 is wrong. At the altitude given, the temperature is not 300K. The actual model for the atmosphere is very complicated.

Explanation / Answer

1. pv = nrT
   pdv + vdp = nRdT
   assuming dT = 0
   dp = -pdV/V = -P*dy/y
2. P = -Pln(y/yo)

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