Technical Notes

In 1873, Johannes Diderik van der Waals defended his doctoral thesis entitled: "On the Continuity of the Liquid and Gaseous States" [1], in which he argued that the properties of a pure component in both the gaseous and liquid states are determined by the location of the critical point. This was only four years after Thomas Andrews had documented the existence of a critical point. As illustrated in Figure 1, van der Waals claimed that gas and liquid coexist at the vapor pressure thanks to a balance between repulsive and attractive molecular forces. The gas phase is dominated by repulsive forces, and the liquid phase by attractive forces. Moving up the vapor pressure curve towards the critical point, the molecular forces in the two phases gradually become more similar, and at the critical point the two phases are identical. At pressures and temperatures higher than at the critical point, a phase separation can no longer occur.

Figure 1 How van der Waals saw the gas and liquid densities develop with temperature and pressure.

 

Van der Waals expressed his findings in the equation shown in Figure 2. It says that the pressure is determined by two opposing molecular interactions, one repulsive and one attractive. The repulsive term is a slight modification of the ideal gas equation, while the attractive term expresses that two molecules will attract each other with a force that increases with the inverse of the squared molar volume, V.

 

Figure 2 Van der Waals equation of state.

 

Figure 3 shows qualitative PV curves at three different temperatures. At a temperature, T, higher than the critical temperature, T_c, the PV curve has a hyperbolic-like shape known from the ideal gas equation. At T_c the PV curve has what van der Waals interpreted as an inflexion point at the critical pressure, P_c. This made it possible to express the parameters a and b as unique functions of T_c and P_c, thus expressing van der Waals' main point, that the properties of a pure component in the entire PT region are determined by the location of the critical point. The vdW equation can be rewritten as a cubic equation in molar volume and is therefore also called the first cubic equation of state. In 1910 van der Waals received the Nobel prize for his work on this equation.

Figure 3 PV isotherms of a pure component at three different temperatures.

 

For many years, the vdW equation was more of an academic discovery than an industrial tool. Industrial interest only took off after Soave in 1972 presented the updated cubic equation shown in Figure 4.

Figure 4 Soave-Redlich-Kwong (SRK) equation.

 

By introducing the mixing rules shown in Figure 4, Soave's equation overcame a major limitation of the van der Waals equation by extending the application from single components to mixtures. The different molecules were represented as one average molecule as sketched in Figure5.

Figure 5 The essence of the mixing rules for cubic equations of state.

 

Soave found that the development of the vapor pressure with temperature was more curved than predicted by the vdW equation and he modified the curvature by making the a-parameter temperature dependent. Figure 6 illustrates Soave’s correction to the vapor pressure curve.

Figure 6 Soave’s modification to the vdW vapor pressure curve.

 

Soave's equation predicted too low liquid densities. In 1976 Peng and Robinson (PR) proposed a modified cubic equation of state [3] that reduced the discrepancy seen with the SRK equation between experimental and calculated liquid densities, but it was not until 1982 satisfactory results for liquid densities could be obtained with a cubic equation of state. As illustrated in Figure 7, Peneloux [4] obtained accurate liquid densities by horizontally shifting the PV curves by a distance, c, to have them match experimental liquid molar volumes.

Figure 7 Volume translation concept by Peneloux et al. [4].

Figure 8 shows how the SRK equation and the Peneloux volume-corrected equation are related.

Figure 8: Relation between SRK equation and Peneloux volume corrected equation.

 

Accurate phase densities are essential, but the most important application of an equation of state is to calculate the amounts and compositions of two or more phases in equilibrium. At equilibrium, all components in each phase will have the same fugacity, f. The left-hand container in Figure 9 shows a system that is not in equilibrium. All components in the gas phase have a higher fugacity than the same components in the liquid phase, and there is a net transport of components from the gas phase to the liquid phase. Fugacity can be seen as a measure of the tendency to escape. In the right-hand container, all components have the same fugacity in both the gas and liquid phases, and there is no net transport across the phase boundary. The fluid system is in equilibrium.

Figure 9 Non-equilibrium system (left-hand) and equilibrium system (right-hand).

 

Carl von Linde, the founder of the Linde chemical company, became familiar with the van der Waals equation as early as 1871/72, which was before van der Waals published the equation, and he tried to use the equation to solve multi-component phase equilibrium problems [5]. He had to give up because the calculations were so cumbersome that he found it almost impossible to solve them manually. The availability of computer power was one of the reasons the use of cubic equations of state took off in the late 1970s.

Even after computers became available, it was unclear how to determine whether a fluid system would remain single-phase or split into two or more phases. Gibbs (father of classical thermodynamics) had already described a procedure for this around 1900 and made it clear that among the possible states, a fluid system will choose the one with the lowest (what was later called) Gibbs energy. However, Gibbs did not have the computational tools needed to make full use of his discovery, and it went almost unnoticed until Professor Michelsen in the early 1980s became aware of it and used it to develop efficient computer algorithms [6].

The rest of the story is well known. Today cubic equations are an indispensable tool in oil and gas production. In addition to classical PVT calculations, this applies to reservoir, pipeline and process simulations.

Cubic equations of state have undergone a great deal of development since 1873, but van der Waals was the first to realize that the location of the critical point of a fluid exerts a decisive influence on the phase behavior, not only near the critical point, but throughout the gas and liquid regions. None of the improvements described above to his original equation have challenged this assumption.