What exactly constitutes and Equation of State?

[spacelike]

What exactly constitutes an Equation of State?

As the title says I'm not quite sure what I need to get to have an equation of state in thermodynamics.

I think I understand that an equation of state is an equation that relates different state variables of the system such as volume, temperature, pressure, chemical potential, etc...
However, I am not sure how to know exactly which variables to find a relation between.

For example isn't this the equation of state for an ideal gas? $$U=\frac{3}{2}Nk_{b}T$$
So that is a relation between the number of particles, internal energy, and temperature.

We've also done problems where we find from a given Gibbs Free Energy the equation of state of some gas and the final relation is between volume, pressure, and temperature.

So how do I know which variables to get a relation between?
For example we have another problem where given entropy (as a function of energy; with volume constant) find the equation of state.

I didn't ask this in the homework section because it's not a question about my homework.. sure it might help on my homework but I am not asking about a specific problem, my issue is conceptual. It's something I'm confused about in general.

Thanks.

 

[Valerie Park]

An equation of state is an equation that relates two or more thermodynamic properties of a system. The most common equation of state is the Ideal Gas Law, which relates the pressure, temperature, and volume of an ideal gas. Other equations of state can relate pressure, temperature, and volume to other thermodynamic properties such as enthalpy, internal energy, entropy, and chemical potential. The specific variables used in an equation of state depend on the type of system being studied and the form of the equation of state.