Internal energy and gravitation

[hokhani]

TL;DR

Is gravitational potential included in the internal energy?

Suppose that we just put a cylinder of an ideal gas in a higher place. Does it's internal energy increase?

 

[A.T.]

Suppose that we just put a cylinder of an ideal gas in a higher place. Does it's internal energy increase?

Internal energy of the cylinder? No.

Internal energy of the cylinder+earth system? Yes.

 

[hokhani]

Internal energy of the cylinder? No.

Internal energy of the cylinder+earth system? Yes.

Thanks, but as I read in some references, internal energy is equal to the sum of the average kinetic and potential energies of all molecules. So, at higher places the potential energy of particles increases and hence, the internal energy must increase. Otherwise I think this definition of the internal energy is not exact! Isn't it?

 

[A.T.]

Thanks, but as I read in some references, internal energy is equal to the sum of the average kinetic and potential energies of all molecules.

This is obviously not specific enough. For example, kinetic energy is frame dependent, so you always have to provide the reference frame.

A common definition of internal energy considers:
- kinetic energy due to motion of system parts relative the center of mass of the system
- potential energy due to mutual interactions between system parts

So, at higher places the potential energy of particles increases and hence, the internal energy must increase.

If your system is just the cylinder of gas, then no, because the gravitational interaction with the Earth is not internal to the cylinder. The increased gravitational potential energy is the internal energy of the Earth-cylinder-system.

Note that assigning all the gravitational potential energy to just one of the gravitationaly interacting bodies (the much less massive one), is an approximation that simplifies the math, and happens to give the right numerical results in some computations. But you should always keep in mind, that gravitational potential energy is the energy of the system that includes all the gravitationally interacting bodies.

Otherwise I think this definition of the internal energy is not exact!

You would have to provide those references and full quotes, in order to judge them.