Gas pressure in gravitational field from the partition function

[Truecrimson]

Homework Statement

Please see P2 in http://panda.unm.edu/pandaweb/graduate/prelims/SM_S09.pdf

"Starting with $\mathbb{Z} (z_1,z_2)$ above, derive expressions for the gas pressure..."

Homework Equations

The Attempt at a Solution

To find the pressure at the top and the bottom of the gas column, I tried to use $$F=-kT\ln \mathbb{Z}$$, where F is the free energy, and $$P=-\left(\frac{\partial F}{\partial V}\right)_T$$ by writing $h=\frac{V}{A}$ and got a horrible expression. So I don't think that is the way to go for a timed exam. Moreover, I'm not sure how I can get the pressure at a specific height from that.

Now, I'm thinking about starting from the probability $p=\frac{exp[-\beta (\frac{p^2}{2m}+mgz)]}{\mathbb{Z}}$, which directly gives the number of particles at height z. But how can I get the pressure from that?

The second part where I have to calculate the pressure at the top given the pressure at the bottom looks easier though, since I have the pressure at the bottom as a reference point.

 

[Truecrimson]

I think I got it. Integrate $$p=exp[-\beta (\frac{p^2}{2m}+mgz)]$$ along all three momenta, x and y coordinates, and all particle but one. Then I get the probability of a particle to be at height z. From there, I can get the number of particles at height z, and then the force and the pressure.

 

[unchained1978]

It seems like it's easier to just write Z as an integral over 6N dimensional phase space, do the integrals over p and q. All that's left is a function of z1 and z2, then try using the thermodynamic derivatives to give you the pressure.