Phase Stability and Chemical Potential

It is a consequence of the Second Law that at equilibrium, the chemical potential of a substance must be the same throughout, regardless of the number of phases present.

(If chemical potentials were not equal throughout, it would be possible for the system to reduce its Gibbs energy by transfer of substance from the area of higher chemical potential to the area of lower chemical potential. This process would be spontaneous, so the system cannot be at equilibrium under these conditions.)

At low temperatures, the solid phase of a substance usually has the lowest chemical potential (providing the pressure is not too low) and so is the most stable phase. However, the chemical potentials of the different phases change with temperature in different ways. Thus as the temperature is raised, it is possible for the chemical potential of another phase (liquid or vapour) to fall below that of the solid. This will then become the most stable phase, and a phase transition should be observed (as long as it is not prevented on kinetic grounds):

The above diagram illustrates the temperature dependence of the chemical potential of the solid, liquid and vapour phases of a substance. Note that in practice the lines would actually be curved. The phase with the lowest chemical potential at any temperature is the most stable phase at that temperature.

We have derived the relationship for the temperature dependence of the Gibbs energy as (δG / δT) = -S. Since the chemical potential of a pure substance is equal to its molar Gibbs energy, it follows that:

Since every substance has a positive molar entropy, the chemical potential of a pure substance always decreases as the temperature is raised. Since liquids usually have a higher molar entropy than solids, and gases have a higher molar entropy than either, the slope of a graph of μ against T is steepest for gases and shows the least slope for solids.

Similarly, from the pressure dependence of the Gibbs energy, (δG / δp)  =  V , we obtain

Usually, the molar volume of a liquid is greater than the molar volume of the substance in its solid phase. In this case, increasing the pressure increases the chemical potential of the liquid more than the solid, and so the melting point is raised. i.e. the liquid will condense to a solid at a higher temperature than it did at the lower pressure. This may perhaps be more easily understood by thinking of a higher pressure compacting the sample together, and favouring the denser phase (usually the solid) over the less dense one (usually the liquid).

In the unusual case where the molar volume of the solid is greater than the liquid (eg water) increasing the pressure increases the chemical potential of the solid more than the liquid. This lowers the melting point.

We also need to consider the effect of the external (applied pressure) upon the vapour pressure of a condensed phase. When pressure is applied to a condensed phase (eg by an inert pressurising gas, or by the action of a gas-permeable piston upon the condensed phase) the vapour pressure rises. (This can be crudely viewed as the extra pressure forcing particles out of the condensed phase as gas. Thus the number of particles present in the gas phase, and hence its pressure, increases.) The quantitative relationship is given by the following equation:

where p is the vapour pressure when an additional pressure ΔP is applied, and p* is the vapour pressure in the absence of any additional applied pressure.