Undergrad Request for an example of minimum energy principle (thermodynamics)

[Simobartz]

Summary: Request for an example of minimum energy principle usage

The minimum energy principle states that, for a system at constant entropy, volume and other extensive quantities, the internal energy is minimized at equilibrium. Can you give me an example in which, using this principle, is it possible to determine the equilibrium state of the system?

 

[berkeman]

I can think of a couple of examples offhand. What searching have you been doing so far on this question? Is it for schoolwork?

 

[Simobartz]

Hi, it's for my personal understanding, no schoolwork

I've seen an example (it is written below) in which the principle of maximum entropy is used to determine the state of equilibrium of a system and, since the minimization of the proper potential at equilibrium is correlated to the principle of maximum energy, i thought it would be nice to think at some analogous examples involving thermodynamic potentials. So, examples in which minimization of the proper potentials is used to find the equilibrium state.
To start with I've tried to figure out an example involving internal energy for constant volume and entropy but i have difficulty with the constant entropy require.

The examples in which principle of maximum entropy is used to determine the state of equilibrium of a system: consider the free expansion of a gas in a volume $V$. Will the gas occupy the entire available volume? The answer is, of course, yes.
Indeed we have $dU=TdS-pdV=0$ from which $dS/dV=P/T>0$. So the entropy increases with the volume and since the equilibrium maximize the entropy it also maximize the volume.