# How can internal energy be expressed as a function of any two of $p, v, T$?

In the book of Irey, Theormodynamics, the author states that (while talking about single phase substances)

For a simple compressible media, we may choose our measurable independent variables any two of $$p, v, T$$.The traditional choice for internal energy is temperature and specific volume, $$u = u(T,v)$$.

However, this statement is as if it just falls from the sky; he does not provide any argument why $$u$$ can be expressed as a function of any two of those variable, nor does he give any argument about the relationship of one of those variables to another two.

Question:

I'm looking for an explanation about the concerns that I raised above.

Edit:

As @SolarMike pointed out, the author explicitly consider gaseous substances in the above comment; however, later he also defines $$Z = \frac{ pv}{RT } = Z (p, T) ,$$ i.e knowing $$p,T$$ allows you to calculate $$Z$$, and then you can find $$v$$, but he still does not give any argument why $$Z = Z (p,T)$$. As far as I can see, Charles's law, and the other two law accompanying it are for ideal gases, but we are not woking with ideal gases, yet.

• Just like Ohm's law, knowing two defines the third... – Solar Mike Jun 3 at 12:47
• @SolarMike I got it, but why and how ? what is that relationship, i.e if I give any two of them what is the third one in terms of the first two ? – onurcanbektas Jun 3 at 12:49
• So what relates p, v & T - have you looked at Boyles or Charles laws and the universal gas constant? – Solar Mike Jun 3 at 12:52
• @SolarMike The author talks about a single phase substances in general, not just gases. As far as I know, those law valid only for gases. – onurcanbektas Jun 3 at 12:55
• Is not the statement "compressible media" - I'm assuming "compressile" was your typing... – Solar Mike Jun 3 at 12:57

The component that you're missing is that the various state variables of a system are related by the equation of state for the system, which reduces the number of degrees of freedom by one. For ideal gases, this is the ideal gas law; for real gases, something like the van der Waals equation serves as an equation of state. For solids and liquids, there are various forms of the equation of state that vary in the kind of behavior they're meant to model best; for a short sampling of them, see here: https://en.wikipedia.org/wiki/Equation_of_state#Equations_of_state_for_solids_and_liquids.

• I see, a constraint exists between the generalised coordinates, but what is it ? depends on the systems, and it might not turn out to be as simple as the ideal gas law. – onurcanbektas Jun 3 at 13:17
• @onurcanbektas There's a different "true" equation of state for every state of matter, and even then, there are many different models for the equations of state of different materials depending on which macroscopic properties are important to model. – probably_someone Jun 3 at 13:23