On the Gibbs paradox: What does indistinguishability really mean?

The Gibbs paradox concerns the entropy of mixing of ideal gases: although the entropy change of mixing samples of different gases is unequal to the entropy change of mixing samples of the same gas, it is not clear within the framework of classical thermodynamics how the identity of the particles effects the analysis. An explicit examination of the concept of indistinguishability can clarify some of the implications of the quantum-mechanical resolution of the Gibbs paradox.