Illustration of The Problem
Gibbs himself considered the following problem that arises if the ideal gas entropy is not extensive. Two identical containers of an ideal gas sit side-by-side. There is a certain amount of entropy ("S") associated with each container of gas, and this depends on the volume of each container. Now a door in the container walls is opened to allow the gas particles to mix between the containers. No macroscopic changes occur, as the system is in equilibrium. The entropy of the gas in the two-container system could be immediately calculated, but if the equation is not extensive, the entropy would not be 2*S. In fact, Gibbs' non-extensive entropy equation would predict additional entropy. Closing the door then reduces the entropy again to 2*S, in supposed violation of the Second Law of Thermodynamics.
As understood by Gibbs, and reemphasized more recently, this is a misuse of the entropy equation. 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 what is defined as ordered, and it would be a mistake to conclude the entropy had not increased. In particular, Gibbs' non-extensive entropy equation of an ideal gas was not intended for varying numbers of particles.
The paradox is averted by concluding 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.
Read more about this topic: Gibbs Paradox
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