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# Thermodynamics 0, 1, 2, 3

In teaching the elements of thermodynamics in the graduate course on soft active materials, I have followed this sequence:

- Isolated system: a system capable of no variation. Entropy, S.
- Temperature: a system capable of one independent variation. Entropy is a function of energy, S(U).
- Pressure: a system capable of two independent variations. Entropy is a function of energy and volume, S(U,V).
- Chemical potential: a system capable of three independent variations. Entropy is a function of energy, volume and the number of water molecules, S(U,V,N).

The word “entropy” is the shorthand for the phrase “the logarithm of the number of quantum states”. Thermodynamics stands on a single fundamental postulate: A system isolated for a long time can be in any one of its quantum states with equal probability*.*

We can speak of the number of quantum states only for an isolated system, but we are interested in interactive systems. To study an interactive system in terms of an isolated system, we allow the system to vary quantities such as energy, volume, and the number of water. These quantities share a remarkable property: each is conserved. When the system loses energy, the rest of the world gains the same amount of energy.

Such an open system is characterized by the entropy as a function of the conserved quantities. When these conserved quantities go from one system to another system, the composite of the two systems is an isolated system. The fundamental postulate ascribes the experimental significance to the derivative of the entropy with respect to each of the conserved quantities. Given a system, we can measure its temperature, pressure, and chemical potential of water by equilibrating the system with another system with known temperature, pressure, and chemical potential of water.

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## Comments

## The word "entropy" is short hand for ...

Hi, Zhigang. Hope your class is going well.

To correct you, the word "entropy" is short for "transformational content". What you are talking about is called the Boltzmann-Planck approximation of entropy. You need to clearly state these underlying details, before throwing around loose language, which only adds to the confusion.

The term "transformation content" embodies something equivalent to over 400 years worth of experimentation, mathematical detail, and conceptual understanding, all rooted in Clausius' 1854 theorem of the equivalence of transformations, which itself is rooted in the premise of entropy as an exact differential, which itself is rooted in the older mathematical proof that the absolute temperature is the integrating denominator, a subject pursued in length by Constantin Caratheodory (1908).

You must always keep in mind that entropy is a differential quantity of heat divided by whole number, generally from 1 to 1000. The nature of this quantity of heat is found in the description of quantum electrodynamic interactions.

Jumping into unjustifed discussion of quantum states (a model based on velocity distributions of particles), throws all this by the wayside, and leaves the new student without any fundamental understanding of what he or she is learning.

## Re: The word "entropy" is short hand for ...

Dear Libb: Not entirely sure about your point. As I stated at the end of my notes on isolated systems,

Our presentation above deviates significantly from how the second law was discovered historically. Our presentation, however, is not new. Similar presentations can be found in many textbooks, such as

## Teaching thermodynamics and statistical mechanics

Dear Zhigang,

This may not be best post to make the comments below but I could not locate your old post where you had initiated a discussion on teaching thermodynamics.

I had indicated earlier in the year on iMechanica that I will be teaching a graduate course titled, "Thermodynamics and Statistical Mechanics of Materials". I have a few comments that you and other teachers of Thermodynamics/Stat. Mech may find interesting.

(1) I know a lot more than what I knew when I started. However, I am also quite a bit more confused! I suspect that I will have to teach this course a few times more and spend another couple of years before I can (possibly) reconcile everything in my head.

(2) I think it is vital to integrate statistical mechanics with thermodynamics. I don't believe the latter should be taught separately as it is done quite often in engineering curriculums.

(3) There are several good books...some which you have listed above (---and I did refer to many). However, I don't believe there is a single suitable textbook for teaching this subject as relevant to materials science. The latter has moved quite a bit beyond mere phase diagrams.

(4) I did not get time to go too much into non-equilibrium thermodynamics. However, I did start to read some literature on this. I will definitely try to cover more of it next time I teach this subject. I think I understood equilibrium thermodynamics better once I had a chance to delve into non-equilibrium thermodynamics. So from a pedagogical viewpoint, it may be worthwhile to spend a little time on this.

(5) I came across papers by Terrence Hill on the so-called "nano-thermodynamics". I did not cover it in the course. However, during my readings (---and I am still trying to digest this material) I realized that teaching this (albeit in short dosages) would be useful as well. It may force the student to re-examine the basic postulates of thermodynamics and statistical mechanics which sometimes are taken for granted.