Minimization of thermodynamic equilibrium

[Simobartz]

Hi, I don't understand what does it mean that at equilibrium the proper thermodynamic potential of the system is minimized.
For example on the book Herbert B. Callen - Thermodynamics and an Introduction to Thermostatistics it is written:

Helmholtz Potential Minimum Principle. The equilibrium value of any unconstrained internal parameter in a system in diathermal contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which $T = T^r$.​

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I tired to figure out an example of this using an ideal gas but I'm not sure i have understood. If i consider a closed system with a pure gas the thermodynamic variables are $P,V,T$. If i put the system in thermal contact with a reservoir at $T^r$ then at equilibrium $T=T^r$. But since the degree of freedom of a pure gas are two, then it is impossible to determine $P$ and $V$. My guess is that if I fix also another external parameter like the volume $V$ then the last degree of freedom, the pressure, will be found by the minimization of the Helmoltz free energy $(dF/dP)_{T,V}$. In this case the minimization of the thermodynamic potential would give me an equation equivalent to the equation of state and this is weird to me.

So, can you please give me an example about a pure gas in which Helmholtz Potential Minimum Principle can be used to determine the equilibrium state?

 

[vanhees71]

The "Helmholtz Potential" is free energy, i.e.,
$$F=U-TS.$$
From the 1st Law,
$$\mathrm{d} U=T \mathrm{d} S-p \mathrm{d} V$$
you get
$$\mathrm{d} F=-S \mathrm{d} T-p \mathrm{d} V.$$
That shows that the Helmholtz free energy is minimized by the equilibrium state for fixed temperature and volume.

I never can remember the names of all these potentials, but what's easy is the mathematical idea behind its definition. You start from the first Law and then take the Legendre transformations from $U$ to the wanted potential such that its "natural variables" (for $U$ it's $S$ and $V$) are the ones kept constant in the process under consideration (i.e., $F$ is the right potential for isothermal-isochoric processes since the natural variables for it are $T$ and $V$).

 

[Simobartz]

Hi, thanks for the reply. Can you give me an example in which Helmholtz free energy is minimized by the equilibrium state for fixed temperature and volume? If possible, can you use an ideal gas?
Sorry for insisting in asking the example but i feel like it is really important in this case to understand what we are speaking about

 

[Simobartz]

I think that if possible it should be easy to give an example. Am i asking something wrong?