Entropy of mixing

When two different ideal gases, initially separated and at the same temperature and pressure, are allowed to interdiffuse and fill a common volume, the entropy of the system rises by ΔS_mix = −R(n_A ln x_A + n_B ln x_B), where x_A and x_B are the mole fractions. The quantity is always positive, and for equal amounts it reduces to (n_A + n_B)R ln 2.

What makes the entropy of mixing remarkable is that it occurs with no heat added, no work done, and no change in temperature. The increase reflects only the larger number of microscopic arrangements available once each species can roam the whole volume. It is one of the clearest demonstrations that entropy is fundamentally about counting configurations rather than tracking heat.

The result applies only to distinguishable species; mixing a gas with more of itself produces no entropy change, a subtlety known as the Gibbs paradox that pointed toward the quantum indistinguishability of identical particles.

The entropy of mixing was worked out by Gibbs in the 1870s; the paradox it raised for identical gases became an early hint of the deep role of particle indistinguishability later clarified by quantum mechanics.