# For what kind of thermodynamic systems are phase diagrams valid?

I understand that thermodynamic systems can be broadly categorized as open, closed or isolated. Do 2-D pressure vs. temperature type of phase diagrams apply to all three?

Additionally, for closed and isolated systems, it seems that the diagrams don't account for cases where amount of the material present becomes a limiting factor. For instance, I am considering a large evacuated container starting at room temperature with a relatively small amount of water. The system could be considered either closed or isolated. Could there be a scenario where the amount of water present becomes the deciding factor for the final pressure, as opposed to the one defined by the phase diagram at the given temperature? If so, does that mean that phase diagrams implicitly assume the presence of 'enough' material to begin with? Is there a more exact definition of what 'enough' means for defining phase diagrams?

• If it is isolated, $T$ is not exactly well-defined, though we can make do. You must have some way to control the pressure, so there cannot be a case where the "amount of water present becomes the deciding factor for the final pressure"; that can only decide the partial pressure due to water. Phase boundaries themselves are not well-defined until your system is large enough to approximate the thermodynamic limit. These are all standard textbook material. Commented Jul 26 at 11:57
• Can you provide a specific example? Commented Jul 26 at 12:00
• @naturallyInconsistent Why can't T be well defined in an isolated system internally in equilibrium? Commented Jul 26 at 12:26
• @BobD that is the "though we can make do" part. Commented Jul 26 at 15:00
• @naturallyInconsistent Why is it imperative to 'control the pressure' in some way? Is the system that I described not physically viable? I mentioned that the chamber was evacuated, so that the partial pressure of water becomes the total pressure. Yes, I believe that thermodynamic limit was the keyword that I was looking for. Does it have an exact definition or does that depend on context? Commented Jul 27 at 18:59

The conventional phase diagram based on the Clapeyron equation $$\frac{dp}{dT}=\frac{\Delta S }{\Delta V} \tag{1}$$ assumes that the total mass of the system is kept constant $$dn_1+dn_2=0$$ where $$n_1, n_2$$ are the moles of each phase, this is a closed system. If you wish to make the mole numbers variable then you must go back to the full Gibbs-Duhem equation $$SdT-Vdp+nd\mu=0$$ written for each phase and write the 3-variable phase equilibrium as $$\Delta SdT-\Delta Vdp+\Delta nd\mu=0\tag{2}.$$ This gets you a generalization of Clapeyron similar to the variable case as $$\frac{dp}{dT}= \frac{\Delta S}{\Delta V}+\frac{\Delta n}{\Delta V}\frac{d\mu}{dT}=0\tag{3},$$ which is also true but is not as easy to plot.