text: Gibbs considered the following difficulty that arises if the ideal gas entropy is not extensive.  Two containers of an ideal gas sit side-by-side. The gas in container #1 is identical in every respect to the gas in container #2 (i.e. in volume, mass, temperature, pressure, etc). Accordingly, they have the same entropy S. Now a door in the container wall is opened to allow the gas particles to mix between the containers.  No macroscopic changes occur, as the system is in equilibrium. But if the formula for entropy is not extensive, the entropy of the combined system will not be 2S.  In fact, the particular non-extensive entropy quantity considered by Gibbs predicts additional entropy (more than 2S).  Closing the door then reduces the entropy again to S per box, in apparent violation of the Second Law of Thermodynamics.
  As understood by Gibbs, and reemphasized more recently, this is a misuse of Gibbs non-extensive entropy quantity.  If the gas particles are distinguishable, closing the doors will not return the system to its original state – many of the particles will have switched containers.  There is a freedom in defining what is "ordered", and it would be a mistake to conclude that the entropy has not increased.  In particular, Gibbs non-extensive entropy quantity for an ideal gas is not intended for situations where the number of particles changes.
  The paradox is averted by assuming the indistinguishability (at least effective indistinguishability) of the particles in the volume.  This results in the extensive Sackur–Tetrode equation for entropy, as derived next.