Thermodynamic Equilibrium — Why the need for so many potentials?
It’s a widely known statement that our physical world is constantly striving to become more chaotic; or in more concrete terms — an isolated system strives towards its entropy maximum. By putting it that way we are ascribing some sort of need to nature. This is akin to the teleological physics practiced by ancient and medieval philosophers; where siphons worked because nature abhors a vacuum. Therefore, it is more accurate to say, that the division of energy, volume and other macroscopic parameters within a system in such a way that the amount of accessible microstates is maximized is, by definition, the most likely. And so the entropy being proportional to that number of microstates or configurations is most likely to be maximized. Furthermore, it’s possible to show that the highest entropy is the most likely by a lot, meaning the relative width of the probability distribution is negligible. This is known as the law of large numbers and, as the name suggests, becomes increasingly true for bigger systems.
That distinction becomes much more clear through the equations deriving the principle of maximizing entropy. The important thing moving forward is that the entropy as a function of energy and other parameters, the volume in our case, is maximized at thermodynamic equilibrium.
So why do we need thermodynamic Potentials? Why the Helmholtz free energy F or the Gibbs free energy G?
Because not all systems can be isolated. Yes, we can predict how a system will end up as long as we know the exact energy E and volume V that system has at its disposal. But most real life scenarios aren’t like that. The only truly isolated system is the universe as a whole. Instead, systems usually exchange energy freely. In that case, the known variable is the temperature T. Sometimes, even volume is exchanged leaving us merely with the pressure information P. How can we predict how the system will behave in those scenarios? How can we tell if a chemical reaction will take place or if it is unfavorable? This is precisely where the thermodynamic potentials come in. They aren’t functions of energy and volume. Instead the natural variables of the Helmholtz free energy are T and V and the natural variables of the Gibbs free energy are T and P.
So how do we put them to use?
We imagine our system A to be much smaller than another adjacent system B. The equilibrium of the combined system A + B is as usual at the Entropy maximum. And the energy and volume of the combined system is constant.
Heat Exchange
Let’s first look at the case, where the two Systems can exchange heat freely. Maximizing entropy yields the following results; where we use a Taylor expansion and the fact that system A is much smaller than system B.
As we can see, the condition of maximizing the entropy of the combined system, turns into a condition of minimizing the Helmholtz free energy of our System A.
Volume Exchange
If we now additionally allow volume exchange and follow the same steps the condition of maximizing the entropy of the combined system, turns into a condition of minimizing the Gibbs free energy of our System A.
As we can see thermodynamic potentials aren’t just a fun exercise in Legendre transformations but are essential to understanding thermodynamic equilibria in closed and open systems.
References
[1] Fließbach, T.: Statistische Physik: Lehrbuch zur Theoretischen Physik IV. 5. Auflage, Spektrum Akademischer Verlag, Heidelberg 2010. ISBN 978–3–8274–2527–0
