Ideal gas inside a sphere under the influence of gravitational field

[Thunderbird88]

Homework Statement

We have an ideal gas consisted of N classical particles, each having mass m. The system is inside a spherical bottle of radius R and is inside the gravitational field of the earth. The system is also in thermal equilibrium with a heat reservoir which has temperature T. I have to find the partition function and the energy of the system.2. The attempt at a solution

The problem I face is finding the energy of a single particle. I'm confused regarding the potential energy. My solution is in the pdf file. Is there something wrong with the spherical coordinates?

 

[vela]

I'm assuming you're placing the origin of the coordinate system in the middle of the sphere and directing the z-axis upward. The potential energy depends on how high the particle is, not how far it is from the origin, so U=mgr isn't correct.