(352u) Reformulating and Restructuring the Van Der Waals 1873 Cubic Equation of State: State-of-the-Art Review in Engineering Literature for Progress and Further Applications

One-hundred and ten years after the conception of the Van der Waals theory of cubic equations of state, Robert Reid [1] of MIT for his remarks at the third PPEPPD 1983 Conference (Callaway Gardens, Georgia, USA) queried Chemical Abstracts from 1967-1983 for the literature on cubic equations and reported over 5000 articles have been published that have the word equation of state in the title or abstract and by 1997 Deiters-de Reuck [2] of Cologne, Germany stated over 2000 variations of the Van der Waals cubic equations have been published. Despite those efforts, there is still no fundamental articulation for the procedure of reforming the 1873 equation and which Van der Waals cubic model with same number of parameters is best has no universal answer, however has Martin [3] of University of Michigan, Ann Arbor stated in his 1979 much-quoted review on Van der Waals cubic equations: there is neither obvious choice among the three-, four- and five-parameter model of the 1873 equation. Also, it is noticeable in many of the cubic equations reviewed that the differences in the performance among many variants of cubic equations of state are large (for the same fluid property, e.g. vapor pressure, liquid density, critical volume) in comparison to the Van der Waals 1873 equation. Besides, considering the variants in the parameterization (two-, three-, four-, five-and six parameters) and inaccuracy in the predicted pure substance and mixture properties, it often seems indeed that the estimated parameters of the cubics are more as important as the cubic models’ functionality which in many cases masks the significance of the parameters in the Van der Waals 1873 theory.

While myriad cubic equations in the literature may seems alarming, Van der Waals objective clearly stated in the 1910 Nobel Lecture, “... to determine the relation between p, v and T for a substance,” expected those many cubic equations to come from a universal (or generic) cubic polynomial equation. Consequently, this broad literature review of the philosophy to reform cubic equations reported in the Engineering Research Journals since 1873 to the present shows the optimal way to reformulate the Van der Waals 1873 cubic theory is to reconcile the disagreement between the 1873 PVT equation and the 1880 reduced equation (known universally as the Law of Corresponding States) wherein the 1873 PVT equation is specified by two molecular parameters (a, b) while the reduced equation (which is the 1873 PVT equation scaled by substance critical properties: P_c, V_c, T_c) is unequivocally prescribed by the four Van der Waals molecular parameters (Z_c, Ω_w, Ω_a, Ω_b) that molecular theory has revealed as essentially capturing the size, shape, structure and intermolecular attractive force of the individual pure substances.

Applications of the reformed 1873 equation are demonstrated for volumetric properties (compressibility factor, isothermal compressibility, volume expansivity, second- and third virial coefficients), coexistence gas-liquid densities and high pressure phase equilibria as well as critical properties of binary and ternary systems. The prospect of applying the Van der Waals theory of cubic equations in diversity applications is illustrated for derivative properties (enthalpy, entropy, internal energy, isothermal and isochoric heat capacities, Joule-Thomson coefficient, speed of sound), transport properties (dynamic viscosity, kinematic viscosity, thermal conductivity, thermal diffusivity, Prandtl number, Eucken number, diffusivity (or diffusion coefficient), interfacial (or surface) tension, dielectric constant) and hydrodynamic regime-phase transition of estimating pressure-drop in multiphase fluid flow in pipes and irregular conduits.

This comprehensive review on reformulation of cubic equations of state is a fitting documentary honoring the eightieth birthday of the Sixth Editor of AIChE J., the 1977 Founder and Co-Chair of the Asilomar, Pacific Grove, California Conference of the triennial PPEPPD Conference (which as a preferred reference (for my graduate thermo-class) has been affectionately nickname SS-60 for Storvick-Sandler ACS Symposium Series 60), the 1980 Editor and Co-Chair of the Second PPEPPD Conference in Berlin (West Germany), the Sectional Chair of Phase Equilibria & Properties at the 1995 Annual Meeting of AIChE at Miami Beach, Florida, the overall Program Chairman of the 1998 Annual Meeting of AIChE at Miami Beach, Florida, the Teacher and the Author of Textbooks on Applied Molecular Thermodynamics, Professor Emeritus Stanley (Stan) Sandler of the Chemical and Biomolecular Engineering Department at the University of Delaware. Happy birthday Stan!

[1] Robert Reid (Fluid Phase Equil. (1983), 13, 1-14)

[2] Deiters-de Reuck (Fluid Phase Equil. (1999), 161, 206-219)

[3] J. J. Martin (Ind. Eng. Chem. Fund. 1979, 18(2), 81)