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The concept of Helmholtz free energy has been giving me a lot of issues and I think the main reason for this is that I do not really have many concrete examples of physical phenomena that I can think about when contemplating the Helmholtz free energy.

I understand that the Helmholtz free energy, $F$, of a system can only decrease in a system at constant temperature and volume (as an aside, is the constant temperature in this statement referring to the temperature of the system, the temp of the surroundings or both?). We can see this by looking at the change in $F$ at constant temperature which is simply $$\Delta F=\Delta U_{sys} -T\Delta S_{sys} $$

Presently, whenever I think about a system with constant volume and temperature, there is only one process I can think of which could lead to $\Delta F <0$. In fact, I can only think of one process at all that can occur under these restrictive conditions. The process I can think of is the spontaneous mixing that will occur if the system is composed of two unmixed gases separated by a thin partition with a hole in the middle of it. With time, the particles will mix and so $\Delta S_{sys}>0$ but $\Delta U_{sys}=0$ and thus $\Delta F<0$. In this scenario, it is forbidden by the second law for the particles to ever unmix and so saying that we require $\Delta F<0$ is equivalent to saying that $\Delta S_{universe}>0$. But this equivalence is limited to processes at constant volume and temperature, of which the only example I can think of is mixing. This makes me feel like $F$ as a concept has extremely limited scope and use which is obviously not the case. So are there any other examples of processes occurring at constant temperature and volume in which I can use to think about $F$ tending to a minimum with time. Also is the temperature in the statement that "$F$ must always decrease in conditions of constant temperature and volume" referring to the temperature of the system, the temperature of the surroundings, the boundary between the two or some combination of the these?

Any help on this issue would be most appreciated!

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From what I narrowed down, your post is asking two questions:

  1. What is the temperature variable within the Helmholtz free energy with respect to?
  2. What other physical examples are there of $\Delta F$ < 0.

What is the temperature variable within the Helmholtz free energy with respect to?

Consider a test-tube housing an ideal gas, submerged within a heat bath:

enter image description here

Our system may be divided into two parts:

  1. The gas within the test-tube.
  2. The heat bath.

Each have their own mechanical and thermodynamic variables, lets list them:

1) The gas within the test-tube

$U_{g},V_{g},T_{g},S_{g},N_{g}$

2) The heat bath

$U_{B},V_{B},T_{B},S_{B},N_{B}$

Let us now direct our attention to the heat bath.

The volume is constant, therefore the internal energy of the bath is:

$dU_{B} = T_{B}dS_{B}-0$

(NOTE: $p_{g}dV_{g} = 0$ because the bath's container is rigid, therefore no work can be done.)

Now consider the total entropy of this system:

$$dS_{sys} = dS_{g} + dS_{B} > 0$$

Solve for the entropy of the heat bath and plug it into our entropy expression:

$$dS_{sys} = dS_{g} + \frac{dU_{B}}{T_{B}} > 0$$

But, the internal energy within our system must be conserved, therefore:

$U_{tot} = U_{g} + U_{B}$

but $dU_{tot} = 0$,

therefore $dU_{B} = -dU_{g}$

Our entropy expression becomes:

$$dS_{g} - \frac{dU_{g}}{T_{B}} > 0$$

Multiply both sides by $-T_{B}$, we get:

$$dU_{g}-T_{B}dS_{g} < 0$$

Or, equivalently, we can call it the "Helmholtz Free energy":

$$dU_{g}-T_{B}dS_{g} = dF < 0$$

What other physical examples are there of $\Delta F$ < 0.

Essentially any closed, constant temperature, constant volume process. It is particularly useful for systems in heat baths. Consider section 3.1 of Matthew Schwartz's Statistical Mechanics, Spring 2019.

https://scholar.harvard.edu/files/schwartz/files/8-freeenergy.pdf

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