Gibbs Free Energy and Equation of State

[electricspit]

I'm wondering why the Gibbs function is related to the equation of state as follows (supposedly):

$$V=(\frac{∂G}{∂P})_T$$

I found a thread on here that mentions this relationship, but doesn't explain it at all. Any help understanding this would be appreciated, this is my first introduction to the Gibbs and Helmholtz functions and I'm trying to understand as fully as I can.

 

[Jano L.]

It is a consequence of the definition G:

$$
G = U - TS + PV.
$$

Since

$$
dU = TdS - PdV,
$$

it follows that

$$
dG = -SdT + VdP
$$

and thus the volume V is given by

$$
V = \left(\frac{\partial G}{\partial P}\right)_T.
$$

 

[electricspit]

Awesome thank you, just what I was looking for!

So then a state function itself is just any relation that relates 2 or more extensive variables?

 

[hilbert2]

So then a state function itself is just any relation that relates 2 or more extensive variables?

If you have a simple system ("simple" here means that the system can do work only by volume expansion) that consists of one pure substance, giving the values of three independent thermodynamic variables fixes the values of all other variables. An equation of state is an equation that can be used to solve a fourth variable when three are known. A familiar example is the ideal gas equation PV=nRT, but other variables like entropy or free energy can also appear in an equation of state.