I am trying to understand the liq-gas transition with van der Waals model. And I am very confused by everything. Here is what I understood and I hope you will correct me.
I consider the Free energy $F(V,T,N)$ and the Gibbs energy $G(P,T,N)$.
Those functions are thermodynamic potential. It means that their values should be minimum when the system is at thermodynamic equilibrium.
To have a stable system, thus a system that can have thermodynamic equilibrium, the free energy must be concav and the gibbs must be convex. If it is not possible then no equilibrium is possible in the region
Do you agree with this paragraph? Am I right by saying that if these thermodynamic quantities don't have the nice convexity properties then it physically mean that the system is not stable and thus doing a phase transition. In fact I am a little confused by this, I will explain with an example just below.
Here is the curve showing $P(V)$ for the van der Waals model for $T<T_c$.
And now the associated free energy :
Question 2: In this curve of the Free energy in function of $V$ we then understand that it is not possible to have thermodynamic equilibrium for the red part (because the curve is not concav here). But then, what happens if I fix $V$ to such a value? The system won't be in equilibrium but as $T,V,N$ are fixed what changes then (bc everything is fixed)? Does this mean by contradiction that, thus, I can't fix all those values in this area?
When we do maxwell construction, we try to take the concav envelop of $F$ to make it "respect" the convexity property.
And on the graph of $P(V)$ it then become the "area" property that will give the final value of the pressure in function of volume:
But I would like to really understand the point of maxwell construction. What does the curve mean if we don't do it ? Like the pressure in function of volume. Are they physically "wrong" if we don't to this trick ? I am not sure how I should interpret the maxwell construction. Should I understand this as:
(1): I have a model of $P(V)$ that works very well excepted in a specific zone. Then I locally modify my model by using the maxwell construction (that has physical justification of course). At the end I have a good result.
(2): The maxwell construction is more a "change of point of view". Like the Pressure-Volume curve without doing it are not physically wrong but as we have a phase transition occuring here the pressure when two phase are in contact doesn't have the same meaning. Then we have to change the pressure curve.
My last question is probably not very clear but I hope you will understand what I mean.