Gibbs paradox

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Originally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances, ^{[1] [2]} the Gibbs paradox (Gibbs' paradox or Gibbs's paradox) applies to thermodynamics. It involves the discontinuous nature of the entropy of mixing. This discontinuous nature is paradoxical to the continuous nature of entropy itself with respect to equilibrium and irreversibility in thermodynamic systems.

Suppose we have a box divided in half by a movable partition. On one side of the box is an ideal gas A, and on the other side is an ideal gas B at the same temperature and pressure. When the partition is removed, the two gases mix, and the entropy of the system increases because there is a larger degree of uncertainty in the position of the particles. 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 given a tiniest difference between the two, the work needed jumps to a large value, and furthermore it is the same value as when the difference between the two gases is great. Entropy of mixing of liquids, solids and solutions can be calculated in a similar fashion and Gibbs paradox can be applied to liquids, solids and solutions in condensed phases as well as the gaseous phase.