Why is it not possible to derive Cv from equation of state?

[Frank0]

Assume you have no knowledge about the microscopic information, all you know about the system of pure material is it's equation of state and number of particles. Why can't we get Cv=(∂U/∂T)_V from here(if this is wrong how do we solve it?)? Why can we obtained some other quantities just from equation of state(Cp-Cv, (∂U/∂V)_T for example)? Is there any fundamental difference between these different types of thermodynamic quantities?

 

[Andrew Mason]

Assume you have no knowledge about the microscopic information, all you know about the system of pure material is it's equation of state and number of particles. Why can't we get Cv=(∂U/∂T)_V from here(if this is wrong how do we solve it?)? Why can we obtained some other quantities just from equation of state(Cp-Cv, (∂U/∂V)_T for example)? Is there any fundamental difference between these different types of thermodynamic quantities?

A simple equation of state would be: PV=nRT

You would be able to determine that Cp(T)-Cv(T) = R. But it is not enough to allow you to determine Cv or Cp. To determine Cp or Cv you first would need information about the degrees of freedom of the molecules (monatomic, diatomic, polyatomic). If it is not monatomic, one would also need to have experimental data as to how Cv and Cp varied with temperature.

AM