Gibbs paradox

In statistical mechanics, a semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs, who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the "mixing paradox". If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, in the thermodynamic limit, the paradox is averted. 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. The gas in container #1 is identical in every respect to the gas in container #2 (i.e. in volume, mass, temperature, pressure, etc). There is a certain entropy S associated with each container which depends on the volume of each container. 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. The entropy of the gas in the two-container system can be easily calculated, but if the equation is not extensive, the entropy would not be 2S. In fact, the non-extensive entropy quantity defined and studied by Gibbs would predict additional entropy. Closing the door then reduces the entropy again to S per box, in supposed violation of the Second Law of Thermodynamics. As understood by Gibbs, and reemphasized more recently, this is a misapplication 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 what is defined as ordered, and it would be a mistake to conclude the entropy had not increased.