Van der Waals model for liquid gas phase transition : Understanding Maxwell construction

Question

I have a question on the context of Maxwell construction, spinodal lines. In this pdf https://www.uam.es/personal_pdi/ciencias/evelasco/master/tema_III.pdf they first compute the Van der Waals model that give them the curves $P(T,V)$.

And then they are able to compute the free energy $F$ and the Gibbs energy $G$.

They have thus the curves in black :

But then they remark that between a and b the compressibility $\kappa_T=-\frac{1}{V}\frac{\partial V}{\partial P}$ in negative, which reflects an instability. This instability is also reflected in the concavity of the Free energy as $\frac{\partial^2 F}{\partial V^2}=\frac{1}{V \kappa_T}$

They use this remark to say that we can't have thermodynamic equilibrium here.

Thus, we do a Maxwell construction that will correct the behavior between a and b. And in practice it will modify the curve between 1 and 2.

The Maxwell construction is from what I understood, based on the fact that :

• we want the same pressure at the beginning (point 1) and at the end (point 2) of the phase transition (eq 3.38 of the article) : $\frac{\partial F}{\partial V}_1 = \frac{\partial F}{\partial V}_2$
• we want this pressure to be constant during all the phase transition (eq 3.39) : $\frac{\partial F}{\partial V}_1 = \frac{\partial F}{\partial V}_2=P_2=\frac{F_1-F_2}{V_1-V_2}$

First question :

How can we understand well the argument behind the Maxwell construction ? Is it because the Van der Waals model is wrong during the phase transition, but good elsewhere. Then we have to locally correct it using the experimental knowledge (or at least an external knowledge) that the pressure is constant during the phase transition ? Thus it can be understood as a "correction" we do on the model.

If I am right with what is above, then I don't understand the equation (3.40).

Indeed, they want to compute $F_1-F_2$ to be able to know $P_{coex}=P_1=P_2$ by using (3.39). But, to compute this difference they use the pressure given by the Van der Waals model.

Second question

How can we use the pressure law given by the Van Der waals model in the phase transition zone to compute $F_1-F_2$ if this model is precisely wrong in this zone ? (this assume I was right with the guess in my first question, else the question is not relevant anymore)

Answer 1

The van der Waals model is not wrong (it's a model, but it is not qualitatively wrong). It correctly describes the pressure of a homogeneous system. However, there are regions where the homogeneous phase is unstable, and phase separates into two phase. The two phases are a high density "liquid" phase, and a low density "gas" phase.

Phase equilibrium implies $P_1=P_2$ (mechanical equilibrium) and $T_1=T_2$, $\mu_1=\mu_2$ (thermodynamic equilibrium). At the endpoints of the mixed phase the isotherms must be on the homogeneous van der Waals isotherms. Also, the high and low density endpoint must be at the same $T,\mu,P$. This is what the Maxwell construction ensures. Obviously $P=const$ on a horizontal line in the $P,V$ diagram. At constant $T$ (isotherm) we have $$ d\mu = v dP $$ where $v=1/n$ is the specific volume. Then $\int d\mu=0$ gives the equal area rule.

Answer 2

This is a late answer, but hopefully, adding helpful information.

How can we understand well the argument behind the Maxwell construction?

Besides the argument based on the experimental finding that below the critical point (the horizontal inflection point of an isotherm), coexisting equilibrium states must have the same temperature, pressure, and chemical potential, there is a compelling theoretical argument from basic thermodynamics.

Indeed, the unstable part of the van der Waals loop, with a negative value of the isothermal compressibility, is signaling the presence of a region where the Helmholtz free energy is not convex as a function of the volume. Remember that thermodynamic potentials' convexity/concavity properties are directly related to thermodynamic stability and the minimum property of thermodynamic potentials. Thus, it is a fundamental property we would like to find in every theoretical model.

Van der Waals's free energy dependency on the volume, as obtained from the equation of state, fails in this respect. A simple way of restoring convexity is to substitute the non-everywhere convex free energy with the so-called convex envelope. It amounts to replacing the region around the concave intruder with a linear part joining the two points having a common tangent, as schematically represented in the following figure.

Of course, a linear region of the Helmholtz free energy as a function of the volume implies a constant pressure in the same volume range.

To summarize, Maxwell's construction is a way to restore the right convexity of the underlying free energy.

Notice that the presence of the linear region in the free energy immediately justifies the result of eq. (3.39).