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 $v=\frac{dV}{dm}$.

  • $\begingroup$ Thanks for this. So for these "not true" intensive parameters like specific volume etc., I'm guessing the 2nd state postulate also doesn't have to hold? For example volume in the 3D phase diagram is not a function of p and T. $\endgroup$ – agalick Jul 7 '15 at 23:59
  • $\begingroup$ 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 $v_{vapor} >> v_{liquid}$. $\endgroup$ – hyportnex Jul 8 '15 at 11:30

The specific volume may vary, but the material still will be at triple equilibrium. In other words, specific volume is not an intensive variable that determines whether or not the material is at its triple point. Notice in the 3-D phase diagram that the triple line is at one pressure and one temperature. It is the boundary between liquid + gas and solid + gas.

  • $\begingroup$ "The specific volume may vary, but the material still will be at triple equilibrium" -> This is the exact idea that had me thinking this would count as a degree of freedom. $\endgroup$ – agalick Jul 7 '15 at 23:54
  • $\begingroup$ @agalick: The specific volume is not free to vary at the triple line. The specific volume is a constraint on the system at the triple line, rather than a variable. However, I think I see what you are getting at: If specific volume were to change, the system would move from equilibrium. But the point is that the specific volume for that material is not free to change at the triple line. It's specific to that particular material. As user31748 points out, the specific volume is not a partial derivative in the equation of state. $\endgroup$ – Ernie Jul 8 '15 at 10:42

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