91Ó°ÊÓ

The free energy curves at one particular temperature for a two-component system that has three possible solid phases (crystal structures), one of essentially pure A, one of essentially pure B, and one of intermediate composition.

Consider the Gibbs free energy graph below, which shows a system with three solid phases: $\mathrm{Î\pm },\mathrm{β}\text{and}\mathrm{γ}$. One is a pure A substance, one is a pure B substance, and one is a mixture of the two.

First, as shown in the diagram, we draw three tangent lines from left to right: first, the $\mathrm{Î\pm }$phase plus the liquid phase are stable, then the liquid phase is stable, then the $\mathrm{β}$phase plus the liquid phase are stable, then a narrow range of x where only the $\mathrm{β}$phase is stable, and finally only the liquid phase is stable.

The Gibbs free energy is given by

$G=U-TS+PV$

At constant entropy and constant pressure, differentiate the Gibbs free energy to get:

role="math" localid="1647065790638" $dG=dU-SdT+PdV$

By increasing the temperature,

$\frac{∂G}{∂T}=-S$

We can see from this equation that as the temperature rises, the stability ranges of $\mathrm{Î\pm }$and $\mathrm{β}$ vanish. When we lower the temperature, the stability of $\mathrm{γ}$ plus the liquid appears, as seen in the previous figure for large x. As the temperature drops, the liquid's stable range narrows until it vanishes at two locations, which are known as the eutectic points (where all the liquid freezes).