Helmholtz free energy

Defined as

Its physical meaning:

• $U$ = internal energy (total microscopic energy). Also denoted as $⟨E⟩$.
• $TS$ = “energy unavailable to do work” because it’s bound up in disorder.
• $F$ = maximum useful work you can get from the system at fixed $T$ and $V$.
  From $S=k\beta U+k\ln Z$ and $\beta =1/(kT)$ (see partition function), we get:

So:

The term $TS$ represents the portion of a system's internal energy that is "locked up" in its microscopic disorder, making it unavailable to perform coordinated, useful work. Energy is only useful for work (like pushing a piston) when it's directed.

Let's use the example of a 2D box with a piston and two balls. To make a fair comparison, we'll keep the total internal energy ($U$) and temperature ($T$) the same in both scenarios. The only thing we'll change is the arrangement of the motion, which changes the entropy ($S$).

Scenario 1: Low Entropy (Organized Motion)

Imagine the two balls are moving in perfect unison, horizontally, straight towards the piston.

• Entropy ($S$): Low. There is essentially only one way for the system to be arranged to achieve this state: both balls moving perfectly together. This is a highly ordered, low-entropy state.
• Work Extraction: When these two balls strike the piston, they transfer all their kinetic energy into a single, coordinated push. This exerts the maximum possible force, moving the piston a significant distance. We can extract a large amount of useful work from this collision.
• Free Energy ($F$): High. Because the entropy ($S$) is very low, the "unavailable energy" term $TS$ is close to zero. Therefore, the Helmholtz free energy $F=U-TS$ is very close to the total internal energy $U$. Almost all the energy is "free" to do work.

Scenario 2: High Entropy (Chaotic Motion)

Now, imagine the two balls have the same total kinetic energy ($U$), but they are moving randomly in all directions—bouncing off the walls and each other.

• Entropy ($S$): High. There are countless ways the balls can move to have this same total energy. One could be moving up-left while the other moves down-right; they could be spinning; etc. The system is highly disordered, so the entropy is high.
• Work Extraction: What happens at the piston?
  • A ball might hit the piston, giving it a tiny push.
  • Another ball might hit the top wall, contributing no work to the piston.
  • The two balls will likely hit the piston at different times and at different angles, resulting in weak, uncoordinated taps instead of a single strong push.
    A significant portion of the system's energy is tied up in motion that doesn't contribute to pushing the piston (e.g., vertical movements). This energy is "unavailable."
• Free Energy ($F$): Low. Because the entropy ($S$) is high, the "unavailable energy" term $TS$ is large. Therefore, the Helmholtz free energy $F=U-TS$ is much smaller than the total internal energy $U$. Very little of the system's total energy is available to do useful work.

## Conclusion 💡

By comparing these two scenarios with the same total energy ($U$), you can see the physical meaning of $TS$. It's the energy that, due to the random, disorganized nature of the system's microscopic components (high entropy), cannot be harnessed into a single, useful output.

The Helmholtz free energy, $F$, is what's left over: the portion of the internal energy that is organized enough to be extracted as work under conditions of constant temperature and volume.