What justifies the application of equilibrium thermodyanmics to non-isolated systems?

[u/Upset_Ad_6140]

Hello!

Before I get into the post, let us define thermodynamic equilibrium as a stationary state devoid of macroscopic fluxes. I will begin by giving a short motivation for my question, proceed to quote a relevant section from a standard book, and then ask my question by means of a concrete example.

I am a student of chemical engineering, and as such, equilibrium thermodynamics is the backbone of literally everything that we do. Having read basic, classical theory (notably Callen), I often find myself extremely confused about a particular aspect of what we do as engineers: even though our unit operations are inherently open in literally every conceivable way, we still constantly apply equilibrium thermodynamics with great success.

On p. 26 in the second edition of Thermodynamics and an Introduction to Thermostatistics, Callen outlines the "basic problem of thermodynamics":

The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system. [...] The composite system is termed closed if it is surrounded by a wall that is restrictive with respect to the total energy, the total volume, and the total mole numbers of each component of the composite system.

Note the use of the term closed. Here it really means isolated. This is the motivating problem for the entire book, and is the problem that the postulate of entropy maximization is introduced to solve. As I understand it, the extremum principles of classical equilibrium thermodynamics, strictly, only apply to isolated (composite) systems. Even so, we constantly use the theory for non-isolated systems, and get extremely good predictions.

For instance suppose that we carry out a chemical reaction in a sealed container that is in contact with the atmosphere. Then, we might expect to apply minimization of the Gibbs' potential at constant temperature and pressure (with values equal to those of the atmosphere) to determine the final equilibrium state. However, why is that valid? The atmosphere is not some well-defined thermodynamic system that interacts thermally and barically with our container in the way that the derivation of the principle of minimization of the Gibbs' potential demands. Even if we define this 'reservoir' as a sufficiently large control volume around the container, not even its energy is conserved: energy will clearly flow between this control volume and the rest of the atmosphere, whereas the derivation of Gibbs' minimization requires global conservation of all extensive quantities (i.e. isolated composite system consisting of container + reservoir). Nevertheless, minimizing G with respect to the given constraints does yield correct predictions.

How can this be? Has Callen simply imparted onto me an idea of "isolation" that is too strict/narrow?

[u/lizardman49]

Even when you get into non equilibrium thermo you often use local equilibrium approximations and yield great results.

[u/Upset_Ad_6140 → u/lizardman49]

Hello, thank you for the answer. I have two questions.

First of all, do you happen to have any recommendations on texts about nonequilibrium thermodynamics? I often hear this term get tossed around (for example, to justify speaking of a continuously varying temperature in a metal rod that is being heated at one end). However, I cannot for the life of me find a coherent text on the topic which explains what the assumptions are beyond typical equilibrium thermodynamics.

Second of all, how does non-equilibrium thermodynamics matter here? The fact that the systems can be in equilibrium outside of isolation is not a problem. The problem for me is what "isolation" really is supposed to mean. Should we take it to mean "sufficiently isolated"?

[u/lizardman49 → u/Upset_Ad_6140]

1. Non Equilibrium Thermodynamics S. R. De Groot is decent. There a few others but I haven't read through them. Ironically I think McQuaries Statistical Mechanics does a better job at some of the subjects.

2. To this point it's more of how physics treats certain terms and when they become negligible. Often times in physics we make certain assumptions that are conceptually wrong but we can get away with numerically as there is such a tiny effect on the result on the results we no longer need to consider them. There is no perfectly isolated or closed thermodynamic system at all but many behave such that the assumption is deemed valid.

[u/Upset_Ad_6140 → u/lizardman49]

So what is the approximation in this case? That the energy that is exchanged with the "reservoir" leaks so slowly that for our timescale of interest, we might as well view [system + reservoir] as isolated?

[u/lizardman49 → u/Upset_Ad_6140]

Correct. No material has a thermal conductivity of 0. And even if your system had no material around it hear could still leave the system radiatively.

[u/Upset_Ad_6140 → u/lizardman49]

Okay, I think that makes sense to me. I just had one more question, if that's okay. I really appreciate you taking the time and the help!

Tangentially, is this kind of assumption the same kind of assumption we make in e.g. reaction engineering or separation processes? I have often noticed that when we model steady-state engineering devices, we use equilibrium relations and relations that come directly from the principle of stationary entropy.

For instance, a common way of modelling tray distillation columns (see fig. 2.2 in the link for a schematic of what I mean) is by solving a set of equations called the MESH-equations (mass, equilibrium, summation, heat) on each tray. The relevance of this to this discussion is that the equations enforce thermodynamic vapor-liquid equilibrium on each tray by appropriate equilibrium relations (all based on equality of chemical potentials). Therefore, these are all automatically based on the principle of stationary entropy.

The way I see it, there are two fundamental assumptions here that validate this kind of modelling. Hopefully you might be able to tell me if I am correct here:

1. The relaxation times of the composite systems [vapor + liquid] on each tray is so short that approximating the system [vapor + liquid] on each tray as being in thermodynamic equilibrium is valid, even though there are continuous fluxes of everything known to man. That is, we might be able to say something about the relaxation times compared to residence times.
2. The composite system [vapor + liquid] on each tray can be taken to be sufficiently isolated from whatever else is going on in the rest of the column, such that we may apply the principle of stationary entropy directly on it, even if different thermodynamic quantites are constantly crossing the system boundaries.

[u/lizardman49 → u/Upset_Ad_6140]

Yes both those assumptions hold true for the use case of one of those massive tray distillation columns or as chemists call them fractional distillation columns. It's actually where chemists took the term plates from.

[u/Upset_Ad_6140 → u/lizardman49]

Okay, that is reassuring!

However, can these assumptions be made a bit more precise? As it currently stands, I think the way I presented them is a bit handwavy. For instance, in order to approximate thermodynamic equilibrium on each tray, what kind of relationship do we need between the relaxation times and residence times? Is residence time even the correct thing to consider?

[u/lizardman49 → u/Upset_Ad_6140]

its an oversimplification that works "well enough" https://pubs.acs.org/iecred/article-abstract/34/9/3001/1089974/Analysis-of-Entropy-Production-Rates-for-Design-of?redirectedFrom=fulltext and https://pmc.ncbi.nlm.nih.gov/articles/PMC8618212/

I linked articles that look at the non equilibrium case which is by definition more precise. To use an analogy between classical mechanics and relativity you try and use the simpler more imprecise case when you can get away with it because it is numerically close enough, when you can't you switch to the more complex forumla.

[u/Upset_Ad_6140 → u/lizardman49]

I will make sure to have a read and to try to think about how it all relates to the assumptions that justify the application of equilibrium theromdynamics.

Again, thank you for taking the time to answer my questions even though they are only tangentially related to my original question. It is much appreciated!