Mixing paradox

In thermodynamics and statistical mechanics, the mixing paradox involves the calculation of the entropy of mixing of two thermodynamic systems before and after their contents are mixed. It was first considered by Willard Gibbs (Gibbs, 1961).

Suppose we have a box divided in half by a movable partition. On both sides of the box we have an ideal gas "A" at a particular temperature and pressure. Each side of the box has the same entropy S. If we now remove the partition, the entropy of the gas inside the entire box will be just what it was before we removed the partition, namely 2S. This follows from the extensive nature of the entropy.

Now lets suppose that one side is an ideal "A" gas and the other side is an ideal "B" gas. They are again at equal temperature and pressure. If we now remove the partition, and wait until the two gases are mixed, we will find that the resulting total entropy is greater than 2S, the extra entropy being termed the "entropy of mixing". It must be stressed that the entropy of mixing is completely independent of the nature of the difference between the two gases. The entropy of mixing is just $(N_{A}+N_{B})k\ln(2)$ where $N_{A}$ and $N_{B}$ are the total number of A and B particles respectively.

The paradox is the discontinuous nature of the entropy of mixing. It can be shown that the entropy of mixing multiplied by the temperature is equal to the amount of work one must do in order to restore the original conditions: gas A on one side, gas B on the other. If the gases are the same, no work is needed, but with the tiniest difference between the two, the work needed jumps to a large value, and furthermore it is the same as if the difference between the two gases was great.

The quantum resolution

The paradox is resolved by quantum mechanics by realizing that if the two gases are composed of indistinguishable particles, they obey different statistics than if they are distingushable. Since the distinction between the particles is discontinuous, so is the entropy of mixing.

Gibbs resolution

Gibbs himself posed a solution to the problem. The crux of his resolution is the fact that if one develops a classical theory based on the idea that the two different types of gas are indistinguishable, and one never carries out any measurement which detects this fact, then the theory will have no internal inconsistencies. In other words, if we have two gases A and B and we have not yet discovered that they are different, then assuming they are the same will cause us no theoretical problems. If ever we perform an experiment with these gases that yields incorrect results, we will certainly have discovered a method of detecting their difference.

This insight suggests that the idea of thermodynamic state and entropy are somewhat subjective. Suppose that the two different gases are separated by a partition, but that we cannot detect the difference between them. We remove the partition. How much work does it take to restore the original thermodyamic state? None - simply reinsert the partition. The fact that the different gases have mixed does not yield a detectable change in the state of the gas, if by state we mean a unique set of values for all parameters that we have availiable to us to distinguish states. The minute we become able to distinguish the difference, at that moment the amount of work necessary to recover the original macroscopic configuration becomes non-zero, and the amount of work does not depend on the magnitude of the difference.

References

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• "The Gibbs Paradox" (PDF). November 8. {{cite web}}: Check date values in: |date= and |year= / |date= mismatch (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Format= ignored (|format= suggested) (help); Unknown parameter |PublishYear= ignored (help)