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Gibbs's paradox is a paradox in thermodynamics about the entropy of mixing, and it has to do with the nature of distinguishability and indistinguishability. Thinking deeply about this paradox is a good way to understand the limitations of the classical framing and logical structure of thermodynamics, and when I teach thermodynamics I always encourage my students to think about this paradox. The description of the paradox, as I understand it, is as follows. I have a box divided in two by a barrier. In the left-hand half of the box I have one gas, and in the right-hand half I have another gas. The entropy of mixing is defined is the change in entropy if I remove the barrier and allow the gases to intermix. We consider two extreme cases: in the first case, the two gases are so different that in fact there is a membrane one can produce that lets through one type of gas and not the other. In that case, one can show that the entropy of mixing (times the temperature at which the whole thing takes place) is equal to the minimum work I need to do to separate back out the two gases using that membrane. That quantity is the same no matter what two gases are involved. The second case is the one where I actually have the same type of gas on the two sides of the barrier. In that case, the entropy of mixing is zero -- I can recover the original state without doing any work by just putting the barrier back in. True, if you look closely, you might notice that not every gas molecule that started in the left-hand side ended up in the left-hand side when I put the barrier back in, but the quantities of gas on the two sides are the same as when I started and that means that thermodynamically the states are the same. Now, the paradox comes from considering mixing two gases that are only marginally different. Say they have the same chemical properties but are made of different nuclear isotopes; or say they are the same chemically and nuclearly, but I've hung labels on the molecules of one gas that say A and labels that say B on the molecules of the other; or say that the scientists of today have not yet found a way to distinguish, isolate, or separate the two gases, but the scientists in the future will. Is the entropy of mixing the same as for identical gases, is it the same as for unrelated gases, or is it some intermediate value? In the third scenario above, would the entropy change when the two gases are discovered to be distinguishable?

It looks like there are two ways to go about this from the point of view of formal thermodynamics, neither of which would yield a contradiction in the theory. One is to say that whenever the two gases are even slightly different, then the entropy of mixing is the same as for completely different gases. That is the natural route to take, but it irked a lot of the people in Gibbs's time, who called this a discontinuity in the nature of the entropy of mixing -- the entropy, which they understood to be a measurable physical quantity varies discontinuously as one changes the "character" of the gas constituents "continuously". The second option is to take a more subjective approach and say that whenever we have no way of separating or distinguishing the two gases then there is no entropy gain from mixing them. That introduces no difficulty in the theory because whenever there is no reversible process by which one can convert one state to the other, then there is no way of calculating the change in the entropy between the two states. When we discover a way of distinguishing the two gases, we will have to revise our notion of the entropy of mixing these two gases.

What good is the entropy then, if it's subjective like so? Wasn't the whole point in introducing entropy that it was a measurable, calculable, physical quantity with laws and formulae it obeyed? Well, that's the beauty of Gibbs's whole description of thermodynamics in terms of statistics. In a meta-theory, like Gibbs showed thermodynamics to be -- where the whole basis of the theory is that we cannot in fact measure everything accurately and we have to lump stuff together at some scale using statistics -- the things which you calculate are intrinsically linked to which things you can measure or not. It is no surprise then that things which are in fact not measurable do not receive a well-defined value from the theory. This is a bit different from many other fields of physics where any thought-up experiment, realistically measurable or not, can be put through the machinery of the theory and a well-defined answer should come out.

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