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.