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.

 

[webplodder]

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?

Right, so imagine we're chatting about physics over tea. Gravitational potential? Not lumped in with internal energy at all. Internal energy's just the buzz inside the gas - its heat, its motion. Raising the whole thing up... well, that just gives the cylinder extra height on the shelf, not extra jiggle in the molecules.

 

[Herman Trivilino]

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?

Here's the problem. Textbooks and instructors often refer to the gravitational potential energy of an object, such as a ball or in your case a cylinder or a molecule. But this is a sloppy practice. These objects do not possess gravitational potential energy. It's the object-Earth system that possesses the potential energy.

So, as @A.T. pointed out in Post #2 the internal energy of the cylinder does not increase.

Potential energy is a property of a system. When you increase the height of the cylinder you increase the potential energy of the cylinder-Earth system.

 

[hokhani]

Here's the problem. Textbooks and instructors often refer to the gravitational potential energy of an object, such as a ball or in your case a cylinder or a molecule. But this is a sloppy practice. These objects do not possess gravitational potential energy. It's the object-Earth system that possesses the potential energy.

So, as @A.T. pointed out in Post #2 the internal energy of the cylinder does not increase.

Potential energy is a property of a system. When you increase the height of the cylinder you increase the potential energy of the cylinder-Earth system.

Usually in elementary physics text book, dealing with energy concept is in such a way that we first choose a reference frame for kinetic energy and an origin for potential energy, then study the system dynamics. But as far as I understood, it seems that for potential energy we must have another point in mind that potential energy for one object is meaningless. Also potential energy of every system is only due to the interactions between all the parts of the system and the interactions with outside is not included.

 

[Andrew Mason]

The internal energy is a thermodynamic property of a system and it is a function of its thermodynamic state. If the work done on the system is done by some means other than a thermodynamic process, (e.g. lifting a volume of gas in a fixed volume container in Earth's gravitational field) there is no change in its thermodynamic state and, therefore, no change in its internal energy.

The internal energy of a system - the sum of the average kinetic and average potential energies of the molecules or particles that make up a system - can be determined from the first law of thermodynamics and measurement of the heat into, and thermodynamic work done by, the system. We then use temperature (which is the average translational kinetic energy of the molecules/particles of the system relative to the centre of mass of the system) and the degrees of freedom of the particles/molecules in the system to determine average kinetic energy of the molecules/particles.

Average potential energy of the molecules depends on the relative positions and forces between molecules. It is easy to understand the concept of average potential energy of the molecules but impossible to measure directly. It is calculated or inferred from the total internal energy and total average kinetic energy.

[Note: Distribution of internal energy between translational, rotational, vibrational kinetic energy and potential energy modes often involves rules of quantum mechanics that can result in some modes being entirely or partially 'frozen out' below certain temperatures.]

 

[Herman Trivilino]

Usually in elementary physics text book, dealing with energy concept is in such a way that we first choose a reference frame for kinetic energy and an origin for potential energy, then study the system dynamics. But as far as I understood, it seems that for potential energy we must have another point in mind that potential energy for one object is meaningless.

Potential energy of one object is indeed zero. I agree with that but don't understand what you mean by an origin or by "another point". Perhaps you could give us an example?

Also potential energy of every system is only due to the interactions between all the parts of the system and the interactions with outside is not included.

Right. That's consistent with the internal energy of a system including only interactions that are internal to the system. So in your example of a cylinder of gas, you wouldn't include the gravitational interaction that the cylinder has with planet Earth, because Earth is external to the cylinder.

 

[A.T.]

in mind that potential energy for one object is meaningless.

When a mass M and a much smaller mass m, are released to fall towards eachother, then in the common center of mass frame, the graviational potential energy of M & m will be almost completely converted into kinetic energy of m. And since M barely accelerates, the rest frame of M is almost indetical to the inertial common center of mass frame where energy conservation applies.

This is what allows the shortcut, of assigning all the graviational potential energy of M & m to m, and then using energy conservation.

 

[hokhani]

but I don't understand what you mean by an origin or by "another point". Perhaps you could give us an example?

The potential by its own is not physicaly meaningful and it is the potential difference which is important. We can choose any point as the origin of potential, i.e, the point at which the potential is zero.
By "another point" I meant kind of "another fact".

 

[Herman Trivilino]

By "another point" I meant kind of "another fact".

You need at least two objects (usually modeled as particles) interacting with each other (via a conservative force) to have potential energy. So that's a two-object system for which that potential energy is at least part of the internal energy of that system.

Therefore, consider a system consisting of only one of those two objects and that potential energy of the two-object system is not part of the internal energy of this one-object system.

Does that not resolve the issue raised in your OP?

 

[gyorgiy]

The crucial word in this context is "internal". If you include interactions with external objects, it no longer differs from total energy.

However, there are things that will change the internal energy, even if the container is totally rigid and a perfect insulator.

First, you are moving the container. This involves acceleration and deceleration (however small) each of which will create some minor acoustic disturbance that will eventually be converted to heat.
Similarly, the reduction in gravitational field in finite time will change the distribution of the gas, and the acoustic energy involved will be converted to heat.

 

[hokhani]

Does that not resolve the issue raised in your OP?

Yes. Thanks all. Your comments resolved the ambiguity about internal energy and potential energy.

 

[Chestermiller]

The more general form of the 1st law of thermodynamics is $$\Delta U+\Delta (PE)+\Delta (KE)=Q-W$$where PE is the gravitational potential energy and KE is the non-random kinetic energy.