Deriving pressure, density and temperature profile of atmosphere

voxel

1. The problem statement, all variables and given/known data
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 [tex]\mu[/tex].

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


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


3. 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

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

kuruman

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 [tex]PV^\gamma = const[/tex].

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.