Entropy of Mixing - Gibbs' Paradox: "mixing" of Identical Species

Gibbs' Paradox: "mixing" of Identical Species

The molecular species must be distinct for the entropy of mixing to exist. Thus arises the so-called "Gibbs paradox" which states that if the molecular species are identical, there will be no entropy change, because, as defined in proper thermodynamic terms, there is no process and no mixing, yet the slightest detectable distinction between the two will yield a thermodynamically recognized process of mixing and a considerable entropy change, which is just the entropy of mixing. The word "paradox" is used here to refer to the specious appearance that a slightest detectable distinction can lead to a considerably large change. For ideal gases, the entropy of mixing does not depend on the degree of difference between the distinct molecular species, but only on the fact that they are distinct; for non-ideal gases, the entropy of mixing can depend on the degree of difference of the distinct molecular species. The specious "mixing" of identical molecular species is not in thermodynamic terms a mixing at all, because thermodynamics refers to states specified by state variables, not allowing talk of an imaginary labelling of particles. If the molecular species are different, then there is mixing in the thermodynamic sense; but if they are the same then there is no mixing in the thermodynamic sense.

Gibbs himself was clearly aware of the physics here, and he did not say that he saw any paradox in it. The "paradox" is the work of textbook writers.