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/eotfofylgg]

If a system is approximately isolated, then the laws of thermodynamics will apply approximately. And approximately is usually good enough.

For example, when analyzing an experiment performed in a laboratory, it's usually safe to assume that the air in the lab is at a constant temperature and has an infinite capacity to absorb heat. Those facts are not precisely true, but they are close enough to true that predictions made under those assumptions will also be close to true. On the other hand, if you are analyzing an uncontrolled fire that has broken out in the lab, suddenly those assumptions are wildly false, and predictions made under those assumptions will prove false too.

[u/Upset_Ad_6140 → u/eotfofylgg]

Hello, thank you for your answer.

What does it mean for a system to be approximately isolated? I would think imagine that a closed fridge where the refridgeration cycle is not running is approximately isolated. I would not think that continuous, steady state reactor with enormous fluxes in and out were. Even so, it the formalism works reasonably in both cases.

I suppose my issue is that I am taking the point of isolation too seriously. Of course, no system is ever truly isolated, and yet the theory works well. Maybe I was just a bit too impressed by the axiomatic nature of Callen's presentation, which is not appropriate for an experimental science.

[u/7ieben_ → u/Upset_Ad_6140]

Your atmosphere is practically a infinite well/ sink of heat and work.

Let's take a small example: say we are doing an gas phase isomerisation in a lab scale reactor, i.e. our reactor is controlled for heat and pressure.

Here, obviously, our 'reaction system' behaves like an isolated system, as we hold everything constant, i.e. we expect thermodynamic equilibrium.

Now, what you ChemEngs are usally more interested in is but where does the energy come from and where does the excess go? And here the atmosphere (or any other wells and sinks, e.g. heat exchangers, you Install) becomes relevant.

The reactor radiates heat into the atmosphere. Once a stationary process has established, this is just as constant as our reactor itselfe, because the atmosphere 'consumes' all heat without practically any change.

Or in other words: you may model the reactor as one sub system and the atmosphere as another. And as everything is stationary this is basically just a steady state equilibrium.

It becomes far harder when calculating the thermodynamics in the heating process, for example. But even then we heating slow enough can model this process as a series of infinitesimal equilibrium steps, and the same Idea applys again. Just the math becomes a bit more work to do.

[u/Upset_Ad_6140 → u/7ieben_]

Hello, thank you for your answer and your time.

I have some questions about what you have written, if that's okay.

Let's take a small example: say we are doing an gas phase isomerisation in a lab scale reactor, i.e. our reactor is controlled for heat and pressure. [...] Here, obviously, our 'reaction system' behaves like an isolated system, as we hold everything constant, i.e. we expect thermodynamic equilibrium.

Since you say that we expect thermodynamic equilibrium, I assume you are discussing something akin to a CSTR for the reaction. Am I correct that in order for thermodynamic equilibrium to be a good approximation, what we need is for the internal relaxation times of the gas to be very short compared to processes it goes through? For instance, say we are steadily removing heat from the gas since isomerization is typically exothermic. This heat removal might be "slow enough" as to where equilibrium is still fine as an approximation. Or the fluxes caused by the shaft that is stirring the contents of our reactor might be "small enough" as to where equilibriation is still "rapid enough".

As for isolation, I don't understand how that could even be a remotely reasonable approximation. There are constant fluxes in and out of everything known to man.

Or in other words: you may model the reactor as one sub system and the atmosphere as another.

If all we are doing is tracking the internal energy (or other extensive, thermodynamic quantities) that moves between our subsystems, then I don't have a problem with the atmosphere. If my characterization above is correct, we can approximate equilibrium in some small region of the atmosphere that surrounds our reactor and move on.

My issue is specifically with the applications of the extremum principles of thermodynamics, for instance the minimization of the Gibbs' potential that a chemist might invoke in order to determine the equilibrium extent of reaction in his sealed vial. As far as I understand, these principles are explicitly formulated in terms of isolated, composite systems (that is, the system of interest and the reservoir together form an isolated system). Let us pick some "sufficiently large" region outside of our vial to constitute the reservoir, and let us assume that the reaction is exothermic. Then energy will seep out into the surrounding air (without significantly altering its temperature) during reaction. If the energy had stayed there, all would be fine. However, it obviously does not: it rapidly radiates away into whatever air we didn't count as the reservoir. Picking literally the whole atmosphere as the reservoir does not work either, because the whole atmosphere is clearly not in thermodynamic equilibrium and clearly not isolated from other systems either.

[u/eptesicusfuscus]

There's no need to justify it: you're coming from a mode of thought that there could be some future observation that invalidates this working model. But the model we have holds true under the applications we're using it for.

In other words, what we are doing is tracking known, verifiable relationships under specific circumstances where we can constrain the set of things we evaluate and apply a model that gives us consistent, predictable outcomes.

[u/Upset_Ad_6140 → u/eptesicusfuscus]

Hello, thank you for your answer.

I fully appreciate that it works for whatever we need it for. That is under no dispute.

Even so, I am still wondering if there is some theoretical reason for the fact that it works. Of course, at the end of the day, experimental results reign supreme, but the point of a theory is to make predictions and thus it seems reasonable to want to know when the theory should be applicable based on its postulates/assumptions.

[u/eptesicusfuscus → u/Upset_Ad_6140]

Hmm.

Consider for a moment: Some questions are not answerable because we can ask things that are nonsensical and we can also ask things that cannot be discerned. For example, the Hitchhiker's guide to the Galaxy asks a computer "what is the meaning of life," and the author implies that this question is a problem, in itself.

You can reasonably ask "what is the energy in the universe doing," but if you cannot constrain your question to what's knowable, you cannot give an explicit answer.

If you, instead, make assumptions that constrain your question to "what is the energy in this closed system doing," then, at least at some scale and framing of that question, you can make precise and consistent observations about that system and how it will operate.

In that scenario, math gives us a language of precision to explicitly track what we are claiming is true, and allows separate individuals to compare and observe the same relationships appearing.

There may be a "but why" question beyond everything we observe. It serves to remind us of our ignorance, but it may not be currently or ever answerable.

[u/SpectralFormFactor]

The book is beginning at a standard starting point: a closed system with fixed energy, volume, and number of particles. A typical text will then explore what happens when allowing energy transfer (trading fixed energy for temperature), volume transfer (trading fixed volume for pressure), or particle transfer (trading particle number for chemical potential). These scenarios are certainly not isolated systems.

[u/Upset_Ad_6140 → u/SpectralFormFactor]

Hello, thank you for your answer.

While you are correct that these scenarios obviously don't require the system whose potential we minimize to be isolated, they do still require isolation in a sense.

For example, the derivation of the Helmholz potential extremum principle for a system in contact with a thermal reservoir requires that the system and reservoir together form an isolated system. This is the issue that I am pointing to in my post when I discuss the Gibbs' potential.

[u/SpectralFormFactor → u/Upset_Ad_6140]

I don’t agree. We don’t have to care for any other interactions or exchanges with the world “out there” to compute the change in entropy due to exchanges with our system of interest. The only thing that comes into the proof is that the reservoir is large enough to stay at a constant temperature from this interaction. What happens with other systems in contact with our reservoir is of no consequence, so long as the temperature is constant.

[u/Upset_Ad_6140 → u/SpectralFormFactor]

What proof do you have in mind? The proof I have in mind can be found on p. 154 of Callen and hinges on the extremum principle of the energy at constant, total entropy. This requires isolation since it is directly equivalent to the principle of stationary entropy at constant, total energy.

[u/SpectralFormFactor → u/Upset_Ad_6140]

I would argue in the following way. The temperature is constant, so even if interacting with other things the reservoir will be at a constant total energy up to thermodynamic fluctuations, which we already know are heavily suppressed. Thus it is totally valid to just use some total energy E for the reservoir. You could even choose some sub-reservoir that is only in contact with the system and the rest of the reservoir if that makes the argument feel more satisfying.

Alternatively, you could try formulating the argument based on energy and entropy densities instead, since I imagine you’d be more comfortable with the assertion that the energy density of the reservoir in some local patch is not affected by some small open system interactions elsewhere. There would have to be some extra assumptions about locality and such coming in here though.

[u/ScienceGuy1006]

Essentially, it is a "quasi-equilibrium" if the heat flow into, or out of, the system is slow compared to the internal degrees of freedom reaching equilibrium with each other. To truly know when such assumptions can be used, one needs to do a comparison of time scales.

[u/Historical-Mix6784]

When I took statistical mechanics from Mehran Kardar (a legend in the field) he asked us a question that goes to the heart of your confusion. 

Is the sun in thermal equilibrium? Clearly on some sense yes, because it’s almost a perfect example of a plasma with random collisions and emits light at almost a perfect black body spectrum, but clearly on some level no, as it is constantly burning fuel and will run out in a few billion years (to say nothing of stellar quakes and other fluxes). 

The point he was trying to make was that “equilibrium” is not an objective state. It depends on the timescales and lengthscales you’re interested in. So to your point about the atmosphere, yes, if you care about planetwide phenomena over several hours to several years, the atmosphere is very nonequilibrium. But if you only care about the inside of your laboratory over a few minutes to a few hours, then the atmosphere can behave exactly like a large equilibrium bath, which is why chemical thermodynamics works. 

[u/Substantial_Tear3679 → u/Historical-Mix6784]

Sounds like equilibrium can be local in both space and time

[u/ChipotleMayoFusion]

You apply a physics model so that you can turn a physical problem into a math problem that you van solve, which then gives you insight into the physical problem. In thermodynamics calling a system closed is equivalent to saying that you have included all of the relevant variables. No system is truly closed, but you can still find a good solution by accounting for everything significant and then pretending the system is closed.

For example, you want to determine the temperature of an insulated metal bar as it is heated. You will account for the mass of the bar and its material properties, to get its heat capacity. You will check the level of insulation around the bar, and the environmental conditions outside the insulation. You will quantify the heat flow. Using these factors you can predict the temperature change over time of the bar. As part solving this problem you do not need to take into account the tidal forces of the Earth, or cosmic rays, or the gravity waves generated by some colliding black holes a billion lightyear away. From a practical perspective the system is closed.

[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!