# Multiple Triple Points

I was reading Chandler's Introduction to Modern Statistical Mechanics and noticed a strange feature in one of the figures. The phase diagram in the image has two triple points; however, according to the Gibbs phase rule, a one-component system with three coexisting phases should have zero degrees of freedom.

The last paragraph in the image talks about single-component systems, so I assume that is what is shown in the figure. I have two questions about this:

1. Is a discrete variable (only two points in $$(p,T)$$ space) considered a true degree of freedom?
2. If yes to the first question, how are two triple points possible (as in Chandler's Figure 2.4)?

The number of thermodynamic degrees of freedom is the number of independent intensive state variables which can be varied over their domain in the presence of a fixed number of phases.

It is not a triple point or a couple of triple points which represent degrees of freedom. Instead, in a one component system, a triple point corresponds to a situation where one is left with zero degrees of freedom: no intensive variable can be changed (and that's the reason one speaks about triple point). Therefore, there is nothing preventing a complex system to have more than one triple point, as in the example of Chandler's fig. 2.4. In thermodynamics is much more frequent to call this situation as zero-degrees of freedom, more than "discrete degrees of freedom", even though a set of isolated point can always be described as a manyfold of dimension zero.

The reason for the possibility of isolated triple points, while quadruple or higher multiplicity points are impossible, in the case of a one component systems, are all rooted in the proof of the Gibbs' phase rule, i.e. in the counting of the number of variables and the number of equations required to describe phase equilibria.