Van der Waals supercritical fluid: Exact formulas for special lines

Abstract

In the framework of the van-der-Waals model, analytical expressions for the locus of extrema (ridges) for heat capacity, thermal expansion coefficient, compressibility, density fluctuation, and sound velocity in the supercritical region have been obtained. It was found that the ridges for different thermodynamic values virtually merge into single Widom line only at T < 1.07Tc, P < 1.25Pc and become smeared at T < 2Tc, P < 5Pc, where Tc and Pc are the critical temperature and pressure. The behavior of the Batschinski lines and the pseudo-Gruneisen parameter γ of a van-derWaals fluid were analyzed. In the critical point, the van-der-Waals fluid has γ = 8/3, corresponding to a soft sphere particle system with exponent n = 14.PACS numbers: 64.10.+h, 65.20.De A liquid-gas phase equilibrium curve onto the T, Pplane ends at the critical point. At pressures and temperatures above critical ones (P > P c , T > T c ), the properties of a substance in the isotherms and isobars vary continuously, and it is commonly said that the substance is in its supercritical fluid state, where there is no difference between a liquid and a gas. An anomalous behavior of the majority of characteristics are observed in the vicinity of the critical point. The correlation length for thermodynamic fluctuations diverges at the critical point [1]; one can also observe a critical behavior of the compressibility coefficient β T , thermal expansion coefficient α P , and heat capacity C p : the given properties pass through their maxima under a change of pressure or temperature. Near the critical point, the positions of the maxima of these values in the T, P -plane are close to each other [1]. The same is true for the value of density fluctuations, the speed of sound, thermal conductivity, etc. Thus, in the supercritical region, there is a whole set of the lines of extrema of various thermodynamic values. Each of these lines can be regarded as a "thermodynamical" continuation of the liquid-gas phase equilibrium curve into the supercritical region. The smearing and decreasing (in magnitude) extrema of each of the values form a "ridge" [2][3][4]. A knowledge of the positions of the above "ridges" in the T, P -plane is very important; in particular, it determines a maximum value for such technologically essential characteristics as the dissolving ability of a supercritical fluid, the rate of chemical reactions in a fluid, and others ([2-4] and refs therein). It turned out that the experimentally observed lines of the "ridges" are close to an isochores with a slight decrease in density with increasing temperature [2][3][4]. Most studies on the supercritical region focused on examining the "ridge" for the density fluctuations [2].G. Stanley suggested the name "Widom line" for the line of the maximum of the correlation length isotherms and isobars [5]. Since the lines of the maxima near the critical point merge into one line, the above term was proposed to be used in a wider sense -in reference to the lines of the maxima of all values determined by the ...

Widom line and dynamical crossovers as routes to understand supercritical water

“…The WL of supercritical noble gases, Lennard-Jones and van der Waals fluids was recently studied by experiments and simulations [23][24][25][26][27] . In particular, a recent experimental work has paved the way to a more profound knowledge of hot dense fluids by detecting a crossover in sound velocity of supercritical fluids of noble gases connected to the WL 26 .…”

The generalized scaling laws based on some deductions from the van der Waals equation

“…Finally, we would like to draw attention to the fact that various relations that follow from the WE not only for the ideal lines also appear to be valid, at least qualitatively, for more complex systems. For example, in the supercritical fluid region, the WE "generates" a number of lines that significantly enrich the understanding of this region [23,24]. Among these lines, the line that determines the inversion temperature in the Joule-Thomson process appears to be useful to us.…”

“…Almost 100 years ago, J. Timmermans proposed an empirical relation for the compressibility factor at the critical point that correlates it with the Boyle density (see [4,22]) as (23) The experimental results enabled refinement of the relation [29] by introducing a small correction exponent Z с = (n с /n B ) 1 + α where α = 0.019. Taking into consideration Eq.…”

“…However, the new scaling laws are not necessarily related to the WE, like, for example, the law of the rectilinear diameter of a binodal [3]. They may concern various characteristics of a substance at the critical point and along the phase transition lines [3][4][5][6][7][8][9] or can be related to the universal behavior of some geometric lines on the phase plane [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Some combinations are also possible [21].…”

Dynamical crossover line in supercritical water

“…For some years a so called Widom line, which is a line of supercritical maxima of correlation length and thermodynamic response functions (isobaric heat capacity c P , isothermal compressibility k T , heat expansion coefficion α P ) in fluids, was used to extend a liquid-gas coexistence line into supercritical region1. Later on it was shown that Widom line rapidly splits into a bunch of lines of maxima of different quantities as soon as one goes several dozens percent away from the critical point23. Nevertheless the concept of Widom line is quite important for the description of the anomalous behavior near critical point.…”