PX284 - H5 - gibbs paradox

distinguishability is key to resolving a paradox posed by gibbs

situation 1

considering the isothermal expansion of a particle:

Δ S = N k_{B} \ln 2

where, $2$ comes from the volume doubling

situation 2!

in this case:

Δ S = 2 N k_{B} \ln 2

situation 3

now, considering the case where both parts of the box have the same particles
before and after appear no different, ie. it is reversible, and from the second law of thermodynamics:

Δ S = 0

why is situation 3 different from situation 2?

because indistinguishability is a fundamental property, not an experimental inadequacy, and this was shown formally by sackur and tetrode