Why mixing two identical gases shouldn't create new entropy

Science · Published 18 October 2026 Swipe more like this →

In short: Imagine two identical containers of gas. If you open a door between them, nothing seems to change macroscopically. Yet, early classical physics predicted that the entropy of the system would increase—and then magically decrease if you closed the door again. This contradiction is the Gibbs paradox.

Proposed by Josiah Willard Gibbs in 1874-1875, the paradox arises from a flaw in semi-classical derivations of entropy. In classical mechanics, particles are treated as distinguishable entities. If you calculate the entropy of an ideal gas using this assumption, the resulting formula is not 'extensive,' meaning it doesn't scale proportionally with the amount of substance. This leads to a mathematical absurdity: if you mix two identical gases and then re-seal the container, the math suggests the entropy returns to its original value, seemingly violating the second law of thermodynamics.

The resolution lies in the fundamental nature of matter. The paradox is averted by assuming that gas particles are indistinguishable. In quantum mechanics, if you swap two identical particles, the state of the system remains unchanged. By accounting for this indistinguishability, the entropy formula becomes extensive. This transition from classical 'labeled' particles to indistinguishable ones leads to the Sackur–Tetrode equation, which correctly predicts that the entropy of a system remains consistent regardless of whether the particles have swapped positions.

This isn't just a pedantic math problem; it highlights a profound shift in how we understand reality. It demonstrates that 'order' is a matter of definition. If particles are distinguishable, closing the door doesn't return the system to its original state because many particles have moved. However, if they are indistinguishable, the state is identical to the start. The paradox serves as a historical bridge, forcing physicists to adopt quantum principles—like the fact that we cannot specify a particle's position and momentum with infinite accuracy—to maintain the consistency of thermodynamics.

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