Now that we have a reasonably complete structure of thermodynamics, we can tackle more complicated problems. In the following section we introduce the concepts of local and global stability, and show how local stability puts restrictions on second-order derivatives. In Section 4.2 we see that application of local stability criteria to the proposed fundamental relations leads to predictions of spinodal curves, which indicate when substances will spontaneously change state. In Section 4.2.2 the application of global stability to a van der Waals fluid leads to predictions of vapor saturation curves and liquid saturation curves. These curves are sometimes called binodal curves (or coexistence curves), and can be predicted from PVT equations of state alone. We then show how thermodynamic diagrams useful for refrigeration, or power-cycle design, can be constructed from PVT relations in Section 4.4. Section 4.4.2 shows generically how one can make predictions of differences in thermodynamic quantities from any of the equations of state shown in Appendix B or from experimental data.
What happens if you take gaseous nitrogen and compress it, keeping the internal energy constant by removing heat? Eventually, the nitrogen begins to condense in the container, and you have a mixture of liquid and gaseous nitrogen. If you isolate this system and wait for a long time, then you see that the system is indeed at a stable equilibrium. From Section 2.8 we know that these two phases have the same temperature and pressure. How can that be? Why is some of the nitrogen happy to stay gaseous, while the rest saw fit to condense into a liquid? The two phases have the same temperature and pressure, yet they have different densities.