# B Question about a system in thermodynamic equilibrium

1. Feb 13, 2019

### hgandh

What are the precise conditions for thermodynamic equilibrium? I know that a system in thermodynamic equilibrium must have constant temperature and that there can be no net macroscopic flow of energy or matter. However, is it possible for there to be a system in equilibrium that has a spatially varying energy density? For example, a system of particles treated as independent oscillators where the energy of the oscillators depends on its position in space. Can such a system be in equilibrium or must there be a mechanism that pushes it towards uniform energy density?

2. Feb 13, 2019

### Andrew Mason

For thermodynamic equilibrium to exist there can be no internal or external macroscopic energy flows ie temperature is uniform and any changes in thermodynamic properties must occur over very long time intervals. Independent oscillators whose energy depends upon their position within the volume do not meet these criteria. Energy distribution being position dependent necessarily means that there will be directional energy flows. For example, a living cell has independent energy oscillators (mitochondria or chloroplasts) that are positioned throughout the cytoplasm of the cell. These provide energy flows within the cells that are highly directed and allow the cell to perform various function i.e. not thermodynamic equilibrium.

AM

3. Feb 14, 2019 at 6:33 AM

### Staff: Mentor

@hgandh, could you clarify what you mean by energy density?

4. Feb 14, 2019 at 6:37 AM

### Staff: Mentor

I don't agree. If something has a position dependent heat capacity (e.g., layered materials), then wouldn't the energy content depend on the local heat capacity at uniform temperature?

5. Feb 14, 2019 at 7:23 AM

### Andy Resnick

'Thermodynamic equilibrium' requires 2 systems, not one- it's a unique definition- and simply states that the two objects are in equilibrium if they have the same temperature. Regarding the possibility of a 'spatially varying energy density', sure- an isothermal column of homogeneous fluid (with gravity present) has a spatially-varying potential energy density and can be in thermal equilibrium with its container and surroundings.

6. Feb 14, 2019 at 7:27 AM

### Staff: Mentor

There is also the concept of internal equilibrium. Two parts of what one would call a unique system can be out of equilibrium with each other.

7. Feb 14, 2019 at 7:32 AM

### Andy Resnick

If the two subsystems are out of thermal equilibrium with respect to each other, what prevents the flow of thermal energy?

8. Feb 14, 2019 at 7:34 AM

### Staff: Mentor

You would get a flow of energy in the direction to restore thermal equilibrium.

9. Feb 14, 2019 at 9:31 AM

### Andy Resnick

Right, that's my point. My original point is that thermal equilibrium requires *at least 2* systems (which can be subsystems of a larger system), that is in contrast with say, mechanical equilibrium which is a statement about a single system.

10. Feb 14, 2019 at 9:49 AM

### Andrew Mason

You are correct. It is difference in average translational kinetic energy per molecule that is determines whether there will be a flow of energy within a body or system. By posing the question as the OP did: "For example, a system of particles treated as independent oscillators where the energy of the oscillators depends on its position in space" I incorrectly assumed that meant that the average translational kinetic energy per particle in the oscillators depended on position within the body. I should have said "If molecular translational kinetic energy distribution is position dependent, this necessarily means that there will be directional energy flows".

AM