Thermodynamic state having 2 degrees of freedom (i.e., for properties)

[swampwiz]

TL;DR

What is the underlying reason that thermodynamic state has 2 degrees of freedom - i.e., that any 2 properties (saturated state excepted) completely determines the state?

I'm trying to delve into the reason why this is so. It seems that there are 5 fundamental properties:

P - Pressure
V - Volume (specific)
T - Temperature
S - Entropy (specific)
U - Internal Energy

(Yes, there are other types of energy, but they are fully determinable from these 5 - e.g., Enthalpy: H = U + PV)

Since there are 5 such possible domain variables, having a net of 2 DOF means that there must be 3 constraint equations. The 1st Law of Thermodynamics provides 1 of the equations:

dU = δQ - δW = T dS - P dV

so what are the other 2?

I can see for an ideal gas that the Ideal Gas law provides another:

p V = R T

but even for this model, there must be yet another.

 

[Master1022]

Do you know about "Gibbs' Phase Rule"? That might help to answer your question

 

[swampwiz]

Do you know about "Gibbs' Phase Rule"? That might help to answer your question

So it seems that because the the 1st Law of Thermodynamics involves only 2 true-extensive properties T & P (i.e., specific Volume or Entropy is not a true-extensive property), the topology is as per a 2-D domain, with the result that phases are regions within that domain, and thus a maximum of 3 phases can exist at single points, with there being 0 DOF of those true-extensive properties there (e.g., the triple point has no DOF - it is a singular point), and such that along points at which there are 2 phases (i.e., saturation paths), there is 1 DOF of true-extensive properties (e.g., saturated water liquid/vapor has a specific temperature for a given pressure and vice-versa), and thus 2 DOF when inside a solitary phase.

However, this idea can be simplified such that the number of true-extensive properties is the number of DOF outside of a saturation state. But this doesn't explain why intensive properties are not part of the DOF. I think there is a deeper reason here.