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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 :

enter image description here

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)

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2 Answers 2

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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.

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  • $\begingroup$ I'm not sure to understand your point on equality of the pressure. Indeed for me the equality of the pressure is between the phases. Thus during all the transformation I must have $P_{liq}=P_{gas}$. But this pressure could move during the transformation ? Like in $1$ I have $P_{liq^1}=P_{gas^1}$ but at the end of the transfo I have $P_{liq^2}=P_{gas^2}$ (always equality of the pressures between phases, but the pressure is not the same as the one at the beginning. I have the same problem for your equality of chemical potentials. $\endgroup$
    – StarBucK
    Jan 6, 2018 at 11:17
  • $\begingroup$ And so if I understand well : the van der Waals model is right enough to describe the pressure of homogenous system. Thus between $V_1$ and $V_2$ it is right (if we assume homogenous system). But in practice in the experiment if we wait long enough (thermo equilibrium) the system won't be homogenous in this zone, thus the hypothesis of the model breakdown. But in practice if we allow a homogenous system it works well everywhere, this is what happen above critical point (we don't have to do any construction on the model to make it work at thermo equilibrium). Am I right ? $\endgroup$
    – StarBucK
    Jan 6, 2018 at 11:26
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    $\begingroup$ 1. As you move on the isotherm past points 1 and 2 the pressure of the homogeneous liquid drops, and the gas pressure rises, so they cannot be in equilibrium. The entire mixed phase consists of $P(gas)=P_2=P(liq)=P_1$, only the volume fractions of liquid and gas change. $\endgroup$
    – Thomas
    Jan 6, 2018 at 14:56
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    $\begingroup$ 2. The van der Waals model refers, by definition, to the homogeneous phase. It correctly describes the fact that there is a regime in which no stable homogeneous phase exists. Because the system is mechanically unstable, it phase separates. The van der Waals model correctly describes the phases it separates into, and the equilibrium condition between them. The vdW model would be wrong if the correct state is not phase separation. That is not the case. $\endgroup$
    – Thomas
    Jan 6, 2018 at 15:00
  • $\begingroup$ Thank you for the point 2, now it is clear. I still have a problem to understand "1". Where do you see that the gas pressure increase ? The only place where the pressure increases is between a and b, and you seem to assume that it correspond to the gas pressure here. But in the model we assume everything homogenous so why do you consider it as the gas pressure ? For me it is juste the pressure we would have in an homogenous state (but I don't know if it is gas or liquid pressure). I don't understand... $\endgroup$
    – StarBucK
    Jan 6, 2018 at 15:38
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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. enter image description here

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).

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