Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

If we multiply the vdW EOS (expression 7.11a in Module 7) by ${\tilde{v}}^{2}$ and expand the factorized product by applying the distributive law, the result is the vdW EOS expressed in terms of molar volume, as follows:

$${\tilde{v}}^{3}-\left(b+\frac{RT}{P}\right){\tilde{v}}^{2}+\left(\frac{a}{P}\right)\tilde{v}-\frac{ab}{P}=0$$ (9.1)

Note that equation (9.1) is a third order polynomial in $\tilde{v}$
i.e., it is *cubic* in molar volume. Additionally, we can substitute the definition of compressibility factor Z,

$$Z=\frac{P\tilde{v}}{RT}$$ (9.2)

into equation (9.1) and obtain a different cubic polynomial in Z, as shown:

$${Z}^{3}-\left(1+\frac{bP}{RT}\right){Z}^{2}+\left(\frac{aP}{{R}^{2}{T}^{2}}\right)Z-\frac{ab{P}^{2}}{{\left(RT\right)}^{3}}=0$$ (9.3)

As we see, vdW EOS is referred to as *cubic* because it is a polynomial of order 3 in molar volume (and hence in compressibility factor Z). In general, any equation of state that is cubic in volume (and Z) and explicit in pressure (equation 7.11b) is regarded as a *cubic equation of state*. vdW EOS is a cubic EOS, and all the transformations and modifications that it has undergone during the more than one hundred years since its publication are also *cubic EOS;* or better, they are in-the-van-der-Waals-spirit EOS or of-the-van-der-Waals-family EOS.