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Entropy 2020, 22, 437 3 of 26 Alongside Andrew’s original data for CO2, we show in Figure 2b the same 6 isotherms taken from a modern experimental p-ρ-T thermodynamic fluid property databank [4]. Also shown are the coexistence curves and the law of rectilinear diameters (LRD) extended into the supercritical region [5]. A foremost observation is that, when the density is on a linear scale, there is a linear region to the supercritical isotherms centred upon the extended LRD. The slope ωT = (∂p/∂ρ)T, i.e., the thermodynamic state function rigidity, decreases with T at this midpoint, approaching zero at the critical temperature (Tc = 29.9 ◦C). Note also that (∂p/∂ρ)T decreases with density for gas isotherms and increases with density for liquid. A distinction between subcritical coexisting gas and liquid phases evidently extends to the supercritical isotherms. This fundamental difference between gas and liquid has not received the observation and attention it deserves. The practice of plotting pressure-volume, instead of the natural intensive variable density, can obscure to some degree the linear region of the supercritical fluid phase. 1 ∂V In addition, the commonly used compressibility −V ∂p T, a mechanical property used by van der Waals, and by the science community since, is rather uninformative. By contrast, its reciprocal, rigidity ωT = (∂p/∂ρ)T, with dimensions of molar energy, is more amenable to molecular interpretation. Indeed, it is the thermodynamic state function defined as the work to increase the density of the fluid. Rigidity is directly proportional to the reciprocal molecular number density fluctuations. Had Andrews published his data with a linear density scale, early indications of non-continuity between gas and liquid might have attracted more attention. Moreover, in 150 years since Andrews’ discovery of a critical temperature (Tc), nobody has reported a direct observation of a liquid-gas critical density. This hypothetical property has only ever been defined and obtained by an a priori assumption of its existence by application of LRD [5], which seems to be illogical cyclic reasoning. Soon after van der Waals suggested his 2-parameter equation-of-state, it was found to be inadequate for the p-V-T data in the critical, near-critical and supercritical region. Empirically parameterized cubic equations-of-state, in order to be accurate in the critical region, have required ever-increasing numbers of adjustable parameters to reproduce the thermodynamic state functions with 5- or 6-figure experimental numerical accuracy. Current research and modern compilations require 25+ parameters [4]. This presents a typical case of Bayesian adjustment of the hypothesis, which may be an indicator of faulty reasoning. Thus, this progressive inadequacy of cubic equations of state is per se indicative that the hypothesis of supercritical continuity is fundamentally incorrect. There are many other indicators from experimental sources that have been available all along but have been ignored. In sections II to IV below, we review a selection of the salient research reports that have provided compelling experimental evidence for the non-existence of van der Waals critical volume point. Computer experiments by molecular dynamics (MD) simulations and Monte Carlo (MC) methods became a research tool in the latter part of last century. Computer experiments have certain advantages over real laboratory experiments, at least the absence of impurities and the absence of gravity. The evidence from computer experiments relies on the usage of simple model molecular Hamiltonians, ranging from sticky spheres, square-well models, including the van der Waals mean-field limit, to Lennard–Jones fluids. All of these model classical fluids, without exception, show an unequivocal connection between percolation transitions that characterize the limits of gas and liquid phases, and criticality. The percolation transition that bounds the gas phase (designated PB) occurs at the density whereupon the size of bonded molecular clusters suddenly diverges to length scales of the order of the size of the system. The percolation transition that bounds the liquid state (PA) is when available volume pockets in the liquid, large enough to accommodate an additional molecule, also suddenly diverge. The critical point (pc, Tc), on Gibbs p-T surface, is the intersection of the two percolation loci, which bound the existence of pure gas and liquid states [6–8]. At this juncture, the two states, liquid and gas, have the same T, p and chemical potential, but slightly different densities, by about 20%. Considering this body of empirical evidence, we ask how could the erroneous theory of liquid-gas criticality, which is so clearly inconsistent with the experimental literature, have survived for so long? One reason is the misguided generalization of Onsager’s two-dimensional (2D) Ising model solution [9].PDF Image | Supercritical Fluid Gaseous and Liquid States
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