# Gibbs phase rule and degrees of freedom at the triple point / triple line

The Gibbs phase rule tells me that at a substance's triple point, where there are 3 phases in equilibrium, there should be 0 degrees of freedom. Based on my understanding, that means there should be 0 intensive properties that can be varied.

When you look at a P-T phase diagram, the triple point is actually a point so the 0 DF makes sense. But looking at a 3D phase diagram (like the one here https://commons.wikimedia.org/wiki/File:PvT_3D_plot_-_single_component.png), I see that specific volume can vary arbitrarily along the triple line. Why is that not considered a degree of freedom?

An analogous question holds for the phase transition "lines" with 1 DF (which are planes in 3D). Again, the volume can be varied arbitrarily while pressure and temperature are held constant. Why do these areas have 1 DF and not 2?

Because specific volume or specific dipole moment, or specific "anything" are not really intensive parameters. What is a "true" intensive parameter for which the Gibbs rule holds, and what is only a "specific anything" intensive and there Gibbs does not hold, can be seen only in their relationship in the energy variation equation of state: $dU=TdS-pdV+\mu dN$. The true intensives are the partial derivatives of the internal energy with respect to the extensives, e.g.$\frac{\partial U}{\partial V}=-p$, and this realtionship carries over to the "specific variables" that is $\frac{\partial u}{\partial v}=-p$ where $u=\frac{dU}{dm}$ and $v=\frac{dV}{dm}$.
• I think what is meant by "The state of a simple compressible system is completely specified by two independent, intensive properties" is a definition of a simple system, one that is homogeneous and whose state can be described by $p$ and $T$ at any point. A two-phase system in equilibrium, say, liquid and vapor, is not homogeneous even though it has uniform $p$ and $T$ throughout but has two distinct specific volumes $v_{vapor} >> v_{liquid}$. – hyportnex Jul 8 '15 at 11:30