SCHLAEFLI'S PHASE RULE

The Gibbs phase rule is P - C + F = 2, where the variables P, C and F are represented by integers. The thermodynamic system in question, in the phase rule, is typically a chemical system comprised of an integer number of phases P, and an integer number of components C, and an integer number of degrees of freedom F. And thus P, C and F are in thermodynamic equilibrium in an isolated chemical system, in which pressure, given by "p", and temperature, given by "T", and volume, given by "V", comprise the 3 "thermodynamic" degrees of freedom within the chemical system. In the Euler formula for the innumerable polyhedra we have V - E + F = 2, where V is an integer number of vertices, and E is an integer number of edges, and F is an integer number of faces. The Gibbs phase rule and the Euler formula for the polyhedra are therefore both of the same general form, and in both formulas the variables are strictly integers. Earlier it has been shown that the Euler formula for the polyhedra (V - E + F = 2) can be transformed, using the identities nF = 2E and pV = 2E, (where "n" is identified as the averaged polygon size over the faces in a given polyhedron, and "p" is identified as the average number of edges meeting at the vertices of a given polyhedron) into an alternative polyhedron formula that was originally proposed by the German mathematician Schlaefli in the 19th century. Schlaefli's formula has earlier been shown to lend itself to a mapping construction, for all the various polyhedra, in which the mapping space is defined by ordered pairs of the form (n, p). Here we suggest an analogous type of mapping scheme, based upon the Gibbs phase rule, that is defined by the analogous variables "a" and "b" as in the following equations: aP = 2C and bF = 2C. One could calculate the variables in the ordered pair (a, b), for all possible chemical systems in thermodynamic equilibrium, and which are thus governed by the Gibbs phase rule, and simply plot them in "a-b" space thus, where "a" is identified as "components-per unit-phases and "b" is identified as "components-per unit-degrees of freedom". This mapping of chemical systems in "a-b" space may likely provide insight into the identities and relationships of any given isolated chemical system to any other isolated chemical system...