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Entropy 2020, 22, 437 22 of 26 not have access to our present understanding of cluster percolation, his intuitive conclusions and background science are essentially correct, in relating divergence of cluster sizes to percolation transitions or 3rd order phase transitions [67]. Unfortunately, by the time the 2nd Edition of Mayer and Mayer’s “Statistical Mechanics” was published, 20 years later, Joseph Mayer was no longer active in research, and the concept of universality had been acclaimed as established physics [10–13]. The original chapter on “Condensation and the Critical Region” is missing from the 2nd Edition of Mayer and Mayer. The theory of van der Waals and his renowned equation-of-state survived for about 50 years until, increasingly, there was evidence that it could not account for the experimental data unless it was generalised to include more and more terms with many more adjustable parameters. Whilst modern equations-of-state with countless terms and parameters [4,68] can reproduce experimental p-V-T data to high precision, they tell us virtually nothing about the underlying physical science. The ultimate test of any scientific theory is its ability to account for experimental observations. From the foregoing review, we see that when the van der Waals hypothesis and the concept of universality is abandoned, the pressure equation-of-state, and hence also the thermodynamic state functions, can be expressed simply in terms of a few physical constants belonging to the fluid, and coefficients in virial expansions [28]. Surface tension plays a central role in the thermodynamic stabilisation of colloidal equilibrium states. Computer experiments on a model Lennard-Jones fluid show that it goes to zero at a finite density difference at Tc [25,27]. Surface tension, nevertheless, is the clue to understanding the thermodynamic states both above and below the critical divide. It is positive and well-defined below Tc, and we have two-phase coexistence with liquid condensation that minimizes the interfacial area and hence also total Gibbs energy. At Tc, the surface tension is zero, but it cannot be thermodynamically well-defined above Tc as the two states are not coexisting phases, but colloidally dispersed states that do not exist independently of each other, as the precursor states to the percolation transitions are dispersed hetero-phase fluctuations [69]. As temperature increases above Tc within the mesophase, the pressure difference between liquid and gas states along an isotherm at the percolation bounds increases. Equilibrium is obtained if both the liquid and gas pure phases, both inter-disperse to percolate the phase volume, with an interfacial surface tension that balances any Gibbs energy difference between gas and liquid at the same T,p state point. Thus, both pure gas and pure liquid states coexist at the same T,p states with uniform chemical potential throughout at equilibrium in the mesophase. Looking at the experimental data, (Figures 9–11) it appears that the mesophase may extend all the way to low density at the Boyle temperature, narrowing the density gap as it does so. Comparing critical point hypothesis with experiment establishes a scientific truth. The data described as “experiment” in reference [4], i.e., as obtained from NIST tabulations, are based upon the TSW-equation-of-state [68] with hypothetical “crossover equations” [65] near Tc. In the region of the critical divide and the immediate supercritical region, data points are either fabricated by hypothetical scaling equations, or distorted by the continuity hypothesis implicit in the equations-of-state used. These multiparameter equations tell us nothing about the underlying science of critical and supercritical state bounds; they produce hypothetical near-critical and supercritical mesophase data that have never been experimentally measured. Figure 16 shows how the scientific malpractice of assuming the van der Waals and universal scaling hypotheses to be established scientific truth, leads to the spurious experimental p-V-T data [4,19] in the vicinity of Tc. Finally, it only takes one experimental property to disagree with theory for the basic hypothesis to be wrong. The present equation-of-state analysis with confirmation of a critical divide would appear to suffice. However, there will always be those who cite the “logarithmic divergence of the isochoric heat capacity on both sides of Tc” as discovered by the historic measurements of the 1960s and confirmed by the space shuttle microgravity experiment. It seems rather unusual, if these measurements were such an important fundamental cornerstone of the theory of critical-point universality [10,11,53], thatPDF Image | Supercritical Fluid Gaseous and Liquid States
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