Thread by @zhigangsuo

#thermofact
Thermodynamics of phase transition

1/10 This thread focuses on a simple case, which illustrates the thermodynamic method.

Make a system of one species of molecules.
Neglect change in volume.

Thus, the system has a single independent variable: energy.

2/10 Consider a system of H2O molecules and two phases, ice and water. Change in volume is small, and is neglected in this model.

Water is a phase, which can be in many states. The same is true for ice.

Water and ice can equilibrate in mixtures.

3/10 Let u be the average energy per molecule, and s be the average entropy per molecule.

Draw a u-s plane.

On the u-s plane, each pure state corresponds to a point, and all pure states correspond to a set of points.

4/10 Two pure states, (uA, sA) and (uB, sB), form a mixture. The average energy and entropy (u,s) is a linear combination:

(u,s) = yA (uA, sA) + yB (uB, sB)

yA = fraction of molecules state A
yB = fraction of molecules state B

The mixture is a point on the segment.

5/10 Similarly, a mixture of three pure states is a point in the triangle connecting the three pure states.

6/10 One can similarly locate a mixture of any number of pure states.

Each mixture corresponds to a point on the u-s plane. Points of all mixtures are in a polygon, called the convex hull.en.wikipedia.org/wiki/Convex_hu…

7/10 The system is an isolated system if energy u is fixed, corresponding to a vertical line on the (u,s) plane.

The isolated system equilibrates by maximizes entropy s, corresponding to the interaction between the vertical line and the upper boundary of the convex hull.

8/10 As u changes, the system equilibrates in states corresponding to the upper boundary of the convex hull.

The system can equilibrium as either a pure state, or a mixture of two pure states.

The system cannot equilibrate as a mixture of three or more pure states.

9/10 All pure states of a phase form a smooth curve.

On the u-s plane are a curve for ice (A), and a curve for water (B).

For the two curves of pure states, the convex hull consists of three pieces: part of curve A, part of curve B, and a tangent common to both curves.

10/10 Define temperature T by 1/T = ds(u)/du.

In a mixture of two pure states, the temperature of the two states are equal.

Approximate experimental data for ice and water are given in the T-s and T-u planes.

Writing this thread focussed me on essentials. I have updated my class notes under the heading "Thermodynamics of phase transition".

Rule of mixture implies convex hull.

An isolated system conserves energy and maximizes entropy.
docs.google.com/document/d/12j…