Question

The entropy of two liquids always increases when they are mixed. Yet, immiscible solutions exist. Explain this using state variables.

Answer 1

For binary mixtures the entropy of random mixing can be considered as a function of the mole fraction of one component.

For all possible mixtures,, so that and are both negative and the entropy of mixing is positive and favors mixing of the pure components.

Also the curvature of as a function of is given by the second derivative

This curvature is negative for all possible mixtures, so that mixing two solutions to form a solution of intermediate composition also increases the entropy of the system. Random mixing therefore always favors miscibility and opposes phase separation.