How do I calculate predicted curves (or surfaces) on a substance's $p$-$T$ (or $p$-$V$-$T$) phase diagram?

Question

I just saw the announcement "Breakthrough in melting point prediction: over 100-year-old physics problem solved by Queen Mary Professor", which says that

Professor Trachenko ... demonstrates that melting lines [in a substance's phase diagram] can be described by a simple parabolic equation. This not only offers a practical tool for predicting melting points but also reveals a surprising universality across different material types. This universality comes from observing that parameters in the parabolic equation are governed by fundamental physical constants such as the Planck constant and electron mass and charge.

Skimming the announcement and the paper to which it links, I learned that the curves on a substance's phase diagram can be calculated from known physical properties of the substance and from fundamental physical constants. In my high-school physics classes (long ago, now) I was taught that the curves were entirely found by experimental testing. The ability to calculate predicted shapes for these curves from first principles intrigues me, but my physics and math classes concluded many years ago, my degree is in the arts, and I'm struggling to process the details of the paper and the other resources I've found online. So my question is twofold:

1. What is the smallest set of physical properties of a substance required to calculate the sublimation/boiling curve/surface, the melting curve/surface, the triple point, and the critical point of said substance's p–T or p–v–T phase diagram?
2. Using those properties and whatever fundamental physical constants are necessary, how do I find explicit or implicit equations for the sublimation/boiling curve/surface and the melting curve/surface that can be plotted in software like GeoGebra?

Answer 1

Enthalpy (as a function of temperature and pressure), entropy (as a function of temperature and pressure) and surface energy (as a function of temperature and pressure) should be sufficient. This gives you the Gibbs free energy, which determines the equilibrium phase at any temperature and pressure condition, and tells you when the surface energy drops to zero, which defines the critical point. The triple point corresponds to three phase boundaries meeting.

Alternatively, you could know the heat capacity (i.e., the entropy rate of change with temperature), the specific volume (i.e., the enthalpy rate of change with pressure), and the surface energy. Or combinations of these.

See the Clausius–Clayperon relation for obtaining the slope of the phase-change coexistence line from this information.

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