Deriving pressure, density and temperature profile of atmosphere

[voxel]

Homework Statement

Derive the pressure, density and temperature profiles of an adiabatically stratified plane-parallel atmosphere under constant gravitational acceleration g. Assume that the atmosphere consists of an ideal gas of mean molecular weight $$\mu$$.

Given $$\mu$$=14u, g = 9.81m/s^2, z = 8500m, T (@sea level) = 300K, calculate temperature and pressure at the summit.

Homework Equations

Edit: removed the ideal gas law and barometric formula because I think I was on the wrong track with them...

The Attempt at a Solution

I have been able to derive the barometric formula (which doubles as a pressure and density profile) from the ideal gas law, but am stuck in a bit of a circular problem: I need the temperature at the top of the summit to get the pressure, and vice versa. I don't know how to proceed, or maybe I've taken the wrong approach.

Any help would be appreciated!

 

[kuruman]

What does "adiabatically stratified" mean? Is it that pV^γ= const. ?

 

[voxel]

What does "adiabatically stratified" mean?

I interpreted it to mean that the atmosphere can be modeled as planes of thickness dz that are adiabatic.

 

[kuruman]

I interpreted it to mean that the atmosphere can be modeled as planes of thickness dz that are adiabatic.

They "are adiabatic" in what way? Could it be that as z changes, the product pV^γ remains constant? If so you have three equations: barometric, ideal gas and adiabatic condition and three thermodynamic variables. You can eliminate any two variables and find the other in terms of z.

 

[voxel]

I think you're right in that as z changes, the product $$PV^\gamma = const$$.

However, I'm not seeing how I can eliminate P and V to get T(z)..

edit: clarification: I don't see how I can eliminate two of the thermodynamic variables without introducing an unknown constant.

 

[kuruman]

Use the ideal gas law to eliminate the volume in the adiabatic condition to find an expression that says (Some power of p)*(some other power of T) = constant. Find the value of the constant from the initial conditions. Solve for the pressure and replace the expression you get for p in the barometric equation. This will give you an equation with T and z only.