Gibbs paradox

In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the 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. 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, 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. 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 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 2S, 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. In particular, Gibbs' non-extensive entropy quantity for 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.

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