(74s) Molecular Interpretation of Parameters in the Van der Waals Theory of Cubic Equations-of-State

Even though it is universally known that Van der Waals 1873 theory of cubic equations of state predicts inaccurate liquid-phase properties, unreliable for highly polar and hydrogen-bonded fluids, inappropriate for fluid critical point, unsuitable for large components and incapable of accurate virial coefficients, but as yet it is not widely stated that those fluid properties depend on the molecular structure of pure substances which is evidently not reflected by the parameters in the VDW theory. This poster suggested the possibility of including two inspired parameters into the Van der Waals 1873 equation of state. The physically-based empirical parameters that replicate molecular shapes and molecular structure of pure substances are introduced into the reformed Van der Waals (VDW) 1873 equation to reconcile the VDW 1873 theory with the stipulated shapes of the potential energy and length scale of the statistical mechanical potential functions.

By relating two of the empirical parameters (α, β) in the reformed VDW 1873 equation (which has been affectionately named the Lawal-Lake-Silberberg (LLS) cubic equation) to the microscopic force-parameters derived from the statistical mechanics energy potential functions and performing perturbation and other sensitivity analysis; the sensitivity of the derivatives of the structural parameters (P_c, Z_c, T_c) and the sensitivity of the derivatives of second, third and fourth virial coefficients with respect to α show non-linear trend with α-parameter and thus, α replicates structure of pure substances in the configuration of the LLS equation. Similar analysis of the derivatives of the critical volume (V_c) and Boyle temperature (T_B) with respect to β show non-linear trend with β-parameter and thus β reflects shapes of pure substances in the configuration of the LLS equation. Nonetheless, re-constructing the VDW 1873 cubic equation from the microscopic to the macroscopic level has been very challenging because the attraction term of the 1873 theory comes from the VDW intuition, as seen in the Schematic Figure.  However, those empirical parameters reflecting structure and shape of pure substances in the LLS equation resolve the major disagreement between the VDW theory and fluid properties, including fluid critical point.

Since the shape-structure parameters (α, β) occur in the intuitive-part of the VDW theory, we can speculate that instinctive reasoning of Van der Waals is correct but rather Van der Waals forgot to carry his theory of cubic equations of state to logical conclusion before passing away in 1923. We can apply that inference to attest to the widely known but not often said that “some pertinent variables must have been overlooked in the development of cubic equations of state,” which was echoed by Sydney Young (1890) in regard to the analysis of the coexistence gas-liquid volumes and later repeated by Stanley Walas (1985) in regard to the State of The Arts Review of cubic equations of state in his book: Phase Equilibria in Chemical Engineering.  Stanley Walas’ conclusion is “it does appear, however, that experimental data often are more nearly in accord with the principle of corresponding states than with specific EOS,” which reinforced the statement by Guggenheim (1945) that the corresponding states principle “may safely be regarded as the most useful by-product of the Van der Waals 1873 equation of state.” 

The Schematic Figure is, however, contradictory to the widely accepted viewpoint of molecular theory in which repulsive forces are responsible for structure of liquids while the attraction forces play no role whatsoever in the theory of liquids. In contrast, the Schematic Figure illustrates that the attractive pressure term of the VDW equation is responsible for structure of liquids. Also, the Schematic Figure only assists in the understanding of the thermodynamic properties of dense fluids from macroscopic viewpoint not the interpretation of those properties in terms of the forces between the molecules.  The conjecture of the Schematic Figure makes no claim whatsoever to quantify geometrical shape and structure of molecules (i.e. bonding distance and angles) which are in the realm of the applications of quantum chemistry as exemplified by the works of 1998 Nobel Laureates John Pople and Walter Kohn. Furthermore, the Schematic Figure cannot apply at molecular level to compute shapes of molecules because the shapes of Potential Energy Surface resulted from interplay of four basic interaction energies (electrostatic, exchange, induction, and dispersion) as discussed in Chemical Reviews.

While the shape-structure parameters are weighted by molecular size (or molecular volume) in the Schematic Figure, it is the discriminant of the attractive pressure term in the VDW equation that is responsible for accurate saturated liquid densities because Z_c = f (α, β). As typified by the coefficient of the logarithm term and the expression in the ln{} of the fugacity coefficient being depended on the discriminant, as Z_c = f(α, β). Since the Z_c factor solely depends on the limiting critical volume parameter Ω_w (or b/v_c) and they both can be expressed as f (α, β), they should be specified individually in cubic equations of state. In that regard, however, we may have unintentionally addressed the lament of Van der Waals which he stated in the 1910 Nobel Prize Lecture^1-2 as “And here I have come to the weak point in the study of the equation of state. I still wonder whether there is a better way. In fact this question continually obsesses me, I can never free myself from it, and it is with me even in my dreams.” A better way is to use Z_c and Ω_w to specify all the parameters of cubic equations of state because Z_c solely depends on Ω_w and that would fulfil the VDW stated objective²: “to determine the relation between p, v and T for a substance.”