The Thermodynamics of Mixing

The entropy of mixing

Mixing increases the number of ways the molecules can be arranged. For an ideal mixture the result is

Mole fractions are less than one, so their logarithms are negative and the whole expression is positive. The entropy of mixing is always positive, for any composition, which is the single most important fact in this topic.

It is largest at equal mole fractions, since that is where the number of arrangements is greatest — a symmetry worth noting when sketching the curve.

The free energy of mixing

For an ideal solution ΔH_mix is zero, so

Where immiscibility comes from

Real mixtures have a non-zero enthalpy of mixing. In the regular solution model it takes the form

where β measures how unfavourable A–B contacts are relative to A–A and B–B. When unlike interactions are weaker, β is positive and mixing costs energy.

If that cost is large enough, the free energy curve develops two minima separated by a hump. The system then lowers its free energy by splitting into two phases of different composition rather than remaining one — which is exactly what phase separation is.

The role of temperature

The entropy contribution to free energy is multiplied by T, so raising the temperature strengthens it relative to the enthalpy term. Two liquids that separate at low temperature may therefore become fully miscible above a critical temperature.

This gives the familiar phase diagram with a dome: inside it two phases coexist, outside it one. Explaining the dome from the competition between the two terms is the standard question.

Partial molar quantities and chemical potential

In a mixture, a property such as volume is not simply the sum of the pure component values, because each molecule's environment has changed. The partial molar quantity captures how a property changes when a small amount of one component is added at constant composition.

The partial molar Gibbs energy is the chemical potential, and it is the quantity that decides direction: matter flows from higher to lower chemical potential. Every equilibrium condition in solution thermodynamics is a statement that the chemical potential of a component is equal across phases.

Frequently asked questions

Why is the entropy of mixing always positive?

Because mixing increases the number of accessible arrangements. Mole fractions are less than one so their logarithms are negative, and the negative sign in front makes the whole expression positive.

Why do some liquids not mix, if entropy always favours it?

Because the enthalpy of mixing can be sufficiently positive to outweigh the entropy term. Immiscibility is always an enthalpic effect overcoming a favourable entropy.

Why does raising temperature often produce miscibility?

Because the entropy term is multiplied by temperature, so it grows in importance while the enthalpy term does not. Above a critical temperature entropy wins.

What is a regular solution?

One with an ideal entropy of mixing but a non-zero enthalpy of mixing. It is the simplest model capable of describing phase separation.