If a system in thermodynamic equilibrium contains P phases and C components, then the phase rule states that the number of degrees of freedom is given by F = C − P + 2. Degrees of freedom F represents the number of intensive variables (such as pressure, temperature, and composition) that can be varied arbitrarily over some finite range without changing the number of phases. This simulation considers only the simplest cases of one- and two-component systems. Omitted are such phenomena as multiple crystal structures, solid solutions, partial miscibility of liquids, and azeotrope formation. You can drag the locator over various regions of the phase diagrams. For one-component systems, this selects values for the pressure and temperature. For two-component systems, this selects the temperature and composition. The two-component phase diagram should actually be three-dimensional, with pressure providing an additional degree of freedom.

Following is a compact derivation of the phase rule. At equilibrium, the chemical potential of each component is equal across every phase boundary, as are the temperature and pressure (taking account of hydrostatic effects, if necessary):

This gives a total of

variables — the C chemical potentials (or compositions) plus temperature and pressure. However, these variables must conform to an equation of state in each of the P phases, which removes one degree of freedom per phase. This leaves

arbitrary intensive variables — the Gibbs phase rule:

Remarkably, Gibbs' phase rule is isomorphic with Euler's formula, relating the number of vertices, edges, and faces of a simply-connected polyhedron:

C = number of components  |  P = number of phases  |  F = degrees of freedom  |  \(\mu_i\) = chemical potential of component i  |  V, E, F (Euler) = vertices, edges, faces

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It is a JavaScript/HTML5 implementation of a Mathematica simulation by S. M. Blinder. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.