Maxwell construction

The van der Waals gas equation (using reduced variables) can be expanded [1] to

${\displaystyle v_{R}^{3}-\left({\frac {8T_{R}}{p_{R}}}+1\right){\frac {v_{R}^{2}}{3}}+{\frac {9v_{R}}{3p_{R}}}-{\frac {1}{p_{R}}}=0}$

which is of the form

${\displaystyle v_{r}^{3}+a_{1}v_{r}^{2}+a_{2}v_{r}+a_{3}=0}$

To solve this Cubic function one defines several precursor terms :

${\displaystyle a_{1}(T_{R},p_{R})=-{\frac {{\frac {8T_{R}}{p_{R}}}+1}{3}}}$

${\displaystyle a_{2}(p_{R})={\frac {3}{p_{R}}}}$

and

${\displaystyle a_{3}(p_{R})=-{\frac {1}{p_{R}}}}$

so that the following become defined

${\displaystyle \eta =a_{2}-{\frac {a_{1}^{2}}{3}}}$

and

${\displaystyle q=-\left(+{\frac {2}{27}}a_{1}^{3}-{\frac {a_{1}a_{2}}{3}}+a_{3}\right)}$

The first root’s precursor form is:

${\displaystyle z_{1}={\sqrt {\frac {4\eta }{-3}}}\cos \left[{\frac {1}{3}}\cos ^{-1}\left(-{\frac {3q}{2\eta }}{\sqrt {\frac {-3}{\eta }}}\right)\right]}$

leading to

${\displaystyle v_{r_{1}}=z_{1}-{\frac {a_{1}}{3}}}$

and

${\displaystyle v_{r_{2}}={\frac {-z_{1}+{\sqrt {z_{1}^{2}-4(\eta +z_{1}^{2})}}}{2}}-{\frac {a_{1}}{3}}}$

and finally

${\displaystyle v_{r_{3}}={\frac {-z_{1}-{\sqrt {z_{1}^{2}-4(\eta +z_{1}^{2})}}}{2}}-{\frac {a_{1}}{3}}}$

These last four equations depend on two variables, the temperature which is chosen when one determines which isotherm one is working on, and the pressure. One starts with an arbitrary (but reasonable) chosen value and adjusts its values as one solves the equation (below) ultimately obtaining ${\displaystyle p=p_{V}}$ or ${\displaystyle p=p_{\mathrm {eq} }}$ through the Maxwell construction (see below) at that temperature. With these two variables in hand, one can re-substitute the pressure value obtained into the root equations (above) to obtain the three roots.

The Maxwell construction requires solution of the equation (obtained by making the areas under the two loops equal and opposite in value from each other):

${\displaystyle {\frac {8T_{r}}{3}}\ell n\left({\frac {3v_{r_{\ell }}-1}{3v_{r_{g}}-1}}\right)-8T_{r}\left({\frac {v_{r_{\ell }}}{3v_{r_{\ell }}-1}}+{\frac {v_{r_{g}}}{3v_{r_{g}}-1}}\right)+{\frac {6}{v_{r_{\ell }}}}+{\frac {6}{v_{r_{g}}}}=0}$

with a fixed reduced temperature and the solution depending then on the chosen variable reduced pressure, which becomes the reduced vapor pressure. Unfortunately, this equation can not be solved analytically, and requires numerical evaluation. The subscripts in this equation are ${\displaystyle \ell \equiv smallest}$ and ${\displaystyle g\equiv largest}$ have been changed to make clear which two roots of the cubic are to be employed; these roots themselves depend on the equations which precede them (above), and contain the reduced pressure and temperature which are treated as above.

pseudo 3D p-v-T diagram for van der Waals fluid showing some tie lines and isotherms

Maxwell construction for van der Waals fluid

van der Waals coexistence locus of liquid-gas equilibrium i.e., vapor pressure versus molar volumes

By equating the pressures corresponding to the two volumes where there is a discontinuity in the p-v isotherm, one obtains an expression for the temperature independent of pressure, i.e.,

$${\displaystyle T={\frac {{\frac {3}{v_{g}^{2}}}-{\frac {3}{v_{\ell }^{2}}}}{\left({\frac {8}{3v_{g}-1}}-{\frac {8}{3v_{\ell }-1}}\right)}}}$$

which allows elimination of the temperature from the prior equation. Its solution is complicated, but finally becomes

$${\displaystyle v_{g}={\frac {f(d)e^{d}+1}{3}}}$$

and

$${\displaystyle v_{\ell }={\frac {f(d)e^{-d}+1}{3}}}$$

where

$${\displaystyle v_{g}={{1} \over {3}}-{{e^{d}\,\left(-e^{4\,d}+4\,d\,e^{2\,d}+1\right)} \over {6\,\left(d\,e^{3d}-e^{3d}+d\,e^{d}+e^{d}\right)}}}$$

$${\displaystyle v_{\ell }={{1} \over {3}}-{{e^{-d}\,\left(-e^{4\,d}+4\,d\,e^{2\,d}+1\right)} \over {6\,\left(d\,e^{3\,d}-e^{3\,d}+d\,e^{d}+e^{d}\right)}}}$$

The pseudo-3D ${\displaystyle p_{r}-v_{r}-T_{r}}$ diagram for the van der Waals fluid is shown in the accompanying Figure. For details see lectures from University of Connecticut, articles 88, 93, 95 and especially 96.[2]

The Maxwell construction is rarely derived from the condition that the Gibbs free energies of the gas and the liquid must be equal when they coexist. However, it can be shown that this condition is fulfilled. Essentially the same applies to any other thermodynamic system, where ${\displaystyle P}$ and ${\displaystyle V}$ are replaced by a different pair of conjugate variables, e.g. magnetic field and magnetization or chemical potential and number of particles.

The Maxwell construction is related to the common tangent construction[3][4] and the lever rule[5].

• Van der Waals equation
• Phase transition
• Conjugate variables (thermodynamics)

• Maxwell, J. C. (1875). "On the Dynamical Evidence of the Molecular Constitution of Bodies". Nature. 11: 357–359. Bibcode:1875Natur..11..357C. doi:10.1038/011357a0.

• Reichl, L. E. (2009). A Modern Course in Statistical Physics (3rd ed.). New York, NY USA: Wiley-VCH. ISBN 9783527407828.