Article Thermodynamic Fluid Equations-of-State
Leslie V. Woodcock Department of Physics, University of Algarve, 8005-139 Faro, Portugal; [email protected]
Received: 24 October 2017; Accepted: 30 December 2017; Published: 4 January 2018
Abstract: As experimental measurements of thermodynamic properties have improved in accuracy, to five or six figures, over the decades, cubic equations that are widely used for modern thermodynamic fluid property data banks require ever-increasing numbers of terms with more fitted parameters. Functional forms with continuity for Gibbs density surface ρ(p,T) which accommodate a critical-point singularity are fundamentally inappropriate in the vicinity of the critical temperature (Tc) and pressure (pc) and in the supercritical density mid-range between gas- and liquid-like states. A mesophase, confined within percolation transition loci that bound the gas- and liquid-state by third-order discontinuities in derivatives of the Gibbs energy, has been identified. There is no critical-point singularity at Tc on Gibbs density surface and no continuity of gas and liquid. When appropriate functional forms are used for each state separately, we find that the mesophase pressure functions are linear. The negative and positive deviations, for both gas and liquid states, on either side of the mesophase, are accurately represented by three or four-term virial expansions. All gaseous states require only known virial coefficients, and physical constants belonging to the fluid, i.e., Boyle temperature (TB), critical temperature (Tc), critical pressure (pc) and coexisting densities of gas (ρcG) and liquid (ρcL) along the critical isotherm. A notable finding for simple fluids is that for all gaseous states below TB, the contribution of the fourth virial term is negligible within experimental uncertainty. Use may be made of a symmetry between gas and liquid states in the state function rigidity (dp/dρ)T to specify lower-order liquid-state coefficients. Preliminary results for selected isotherms and isochores are presented for the exemplary fluids, CO2, argon, water and SF6, with focus on the supercritical mesophase and critical region.
Keywords: equation-of-state; liquid-gas criticality; carbon dioxide; argon; water; SF6; virial coefficients
1. Introduction There is a long history and an extensive literature of cubic equations-of-state, going back 140 years, from van der Waals renowned two-term equation for Andrew’s original p-V-T data on carbon dioxide [1], to current research and compilations with hundreds of terms and parameters. As the thermodynamic experimental measurements have improved in accuracy, ever more complex equations have evolved [2]. These equations are used to represent experimental thermodynamic properties for modern data banks such as the NIST compilation for CO2 [2,3]. Over recent decades, ever-increasing numbers of terms and more fitted parameters are required. The Span–Wagner equation for CO2 [2], for example, contains hundreds of terms, each with many fitted parameters, and various adjustable fractional exponents. As even more accurate data become available from future research, these functional forms will need yet more adjustable parameters. The basic reason of this inconvenient modeling practice is the van der Waals gas-liquid continuity hypothesis [1]. Continuous cubic functional forms are inappropriate in the vicinity of Tc and in the supercritical mid-range between gas and liquid phases. A mesophase, confined within percolation loci that bound the existence of gas and liquid phases by higher-order discontinuities, has been identified [4–6]. A simple numerical differentiation of NIST equations-of-state [7] can demonstrate the supercritical mesophase and observe the phase bounds, along any isotherm, of any fluid (e.g., CO2
Entropy 2018, 20, 22; doi:10.3390/e20010022 www.mdpi.com/journal/entropy Entropy 2018, 20, 22 2 of 15
Entropy 2018, 20, 22 2 of 15 Figure1) for any of the 200 fluids in the NIST Thermophysical Property data bank. These boundaries any of the 200 fluids in the NIST Thermophysical Property data bank. These boundaries have been havesmoothed been smoothed over by over the equations-of-state by the equations-of-state used to parameterize used to parameterize the original experimental the original data. experimental data.
CO2 supercritical isotherms 40 750 700 600 35 500 450 30 400 340 25 375 330 p 350 20 320 (MPa) 15 310
10 305 5
0 0 5 10 15 20 density (mol/L)
Figure 1. Supercritical T(K) isotherms for CO2: the green dashed line is the percolation loci PB; the blue Figure 1. Supercritical T(K) isotherms for CO : the green dashed line is the percolation loci PB; the blue dashed line is the percolation loci PA; the green2 and blue solid circles are the maximum experimentally dashed line is the percolation loci PA; the green and blue solid circles are the maximum experimentally observable coexisting gas density, and minimum coexisting liquid density along Tc i.e., 305 K (purple). observable coexisting gas density, and minimum coexisting liquid density along Tc i.e., 305 K (purple). Within the uncertainties, p(ρ,T) is a linear function of ρ in the mesophase region. The origin of Withinthe linearity the uncertainties, is the colloidal p(natureρ,T) isof athe linear supercritical function mesophase of ρ in the [6,8] mesophase with a linear region. combination The origin rule of the linearityfor thermodynamic is the colloidal state nature functions, of the supercriticalsimilar to subcritical mesophase lever [rule6,8] for with the a inhomogeneous linear combination 2-phase rule for region. When appropriate functional forms are used for gaseous and liquid states separately, we find thermodynamic state functions, similar to subcritical lever rule for the inhomogeneous 2-phase region. that all other isothermal thermodynamic state functions in the mesophase are linear functions of When appropriate functional forms are used for gaseous and liquid states separately, we find that all density. In the case of pressure, the negative and positive adjacent deviations for liquid and gas on othereither isothermal side are thermodynamic quadratic. The parameterizations state functions in re thequire mesophase only physical are linearconstants functions belonging of density.to the In the casespecific of pressure, fluid, Boyle the negativetemperature and (T positiveB), critical adjacent temperature deviations (TC), and for coexisting liquid and densities gas on eitheralong the side are quadratic.critical The isotherm, parameterizations and known virial require coefficients only physical b2(T) and constants b3(T) etc. belonging A remarkable to the finding specific is that fluid, for Boyle temperaturethe gas phases (TB), critical below T temperatureB, the coefficient (TC ),b4 andis effectively coexisting zero densities within the along experimental the critical uncertainty. isotherm, and knownThere virial is rigidity coefficients symmetry b2(T) between and b3(T) gas etc. and A liquid remarkable phases findingon either is side that of for the the mesophase gas phases reported below TB, previously [9], that relates the lower coefficients of the liquid equation to properties of the gas in the the coefficient b4 is effectively zero within the experimental uncertainty. There is rigidity symmetry betweenvicinity gas of and the liquid percolation phases loci. on either side of the mesophase reported previously [9], that relates the The objective of this research is to investigate the extent that these findings can be used to describe lower coefficients of the liquid equation to properties of the gas in the vicinity of the percolation loci. experimental thermodynamic properties of a pure fluid over the whole range of equilibrium existence. The objective of this research is to investigate the extent that these findings can be used to describe Here, we investigate equations-of-state for gas, liquid and mesophase separately, each of which, in experimentalan initial thermodynamicapproximation, require properties only ofmeasurable a pure fluid physical over theconstants whole specific range ofto equilibrium particular fluids. existence. Here,Comparisons we investigate with equations-of-stateexperimental data are for made gas, vi liquida the NIST and mesophasethermo-physical separately, databank each [3]. NIST of which, in antabulations initial approximation, reproduce experimental require only data measurable with 100% physicalaccuracy, constantsand therefore, specific provide to particular a foremost fluids. Comparisonscriteria for with testing experimental the alternative data description are made to viavan the der NIST Waals thermo-physical underlying science databank of critical [ 3and]. NIST tabulationssupercritical reproduce behavior experimental of fluids as datadescribed with in 100% references accuracy, [4–9]. and therefore, provide a foremost criteria for testing the alternative description to van der Waals underlying science of critical and supercritical behavior of fluids as described in references [4–9].
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2. Rigidity and Fluid-State Bounds
Rigidity, (ω)T, is the work required to isothermally and reversibly increase the density of a fluid. With dimensions of a molar energy, this simple state function relates directly to the change in Gibbs energy (G) with density at constant T Entropy 2018, 20, 22 3 of 15
2. Rigidity and Fluid-State Boundsω T = (dp/dρ)T = ρ(dG/dρ)T (1)
The inequalitiesRigidity, that (ω)T distinguish, is the work gasrequired from to liquid isothermally are: and reversibly increase the density of a fluid. With dimensions of a molar energy, this simple state function relates directly to the change in Gibbs energy (G) with density at constant T 2 GAS ρ < ρPB (d ω/dρ)T < 0 (2) ωT = (dp/dρ)T = ρ(dG/dρ)T (1) LIQUID ρ > ρ (d2ω/dρ) > 0 (3) The inequalities that distinguish gas from liquidPA are: T and for the mesophase GAS ρ < ρPB (d2ω/dρ)T < 0 (2)
2 MESOLIQUIDρ PB < ρρ <> ρPA (d(d2ω/dωρ/d)T >ρ 0)T = 0 (3) (4) and for the mesophase The rigidity isotherms shown in Figure2 for CO 2 have been obtained from NIST Thermophysical 2 databank [3]. Inequalities in theMESO derivatives ρPB < ofρ <ω ρPAT (Equations (dω/d (2)ρ)T and= 0 (3)) can thermodynamically (4) define theThe percolation rigidity isotherms loci [9]. shown It follows in Figure that 2 atfor low CO2 densityhave been the obtained percolation from NIST loci mustThermophysical approach the Boyledatabank temperature [3]. Inequalities (TB) by its in definition, the derivatives which of is ω obtainedT (Equations from (2) the and rigidity (3)) can intersection thermodynamically interpolated at zerodefine density. the percolation For CO2 the loci limiting [9]. It follows rigidity that (RT at Blow: where density R isthe the percolation gas constant) loci must along approach the percolation the Boyle temperature (TB) by its definition, which is obtained from the rigidity intersection interpolated at loci is 6.03 kJ/mol and TB is 725 K, i.e., the temperature above which the 2nd-virial coefficient (b2) zero density. For CO2 the limiting rigidity (RTB: where R is the gas constant) along the percolation loci is positive and below which it is negative; at TB, p = ρkT and b2 = 0. In the pressure-density plane, the percolationis 6.03 kJ/mol loci and exhibit TB is 725 maxima K, i.e., the as temperature density decreases above which and approachthe 2nd-virial zero, coefficient i.e., the (b ideal2) is positive gas limit at and below which it is negative; at TB, p = ρkT and b2 = 0. In the pressure-density plane, the percolation low pressures and densities. The same behavior is seen for other atomic (e.g., argon) and molecular loci exhibit maxima as density decreases and approach zero, i.e., the ideal gas limit at low pressures and (e.g.,densities. water) fluids. The same behavior is seen for other atomic (e.g., argon) and molecular (e.g., water) fluids.
Figure 2. Rigidity of fluid phases of CO2 from NIST thermo-physical tables: the loci of gas and Figure 2. Rigidity of fluid phases of CO from NIST thermo-physical tables: the loci of gas and liquid-phase bounds are green (percolation2 line PB) and blue (percolation line PA) respectively; red liquid-phase bounds are green (percolation line PB) and blue (percolation line PA) respectively; red lines are supercritical isotherms, blue lines are subcritical isotherms; the purple isotherm is Tc (305 K):
linesthe are Boyle supercritical temperature isotherms, (TB) = 725 blue K. lines are subcritical isotherms; the purple isotherm is Tc (305 K): the Boyle temperature (TB) = 725 K. It is clear from Equation (1) that ω ≥ 0, i.e., rigidity must always be positive although it can be zero in two-phase coexistence regions. Gibbs energy cannot decrease with pressure when T is constant. From these definitions moreover, not only can there be no “continuity” of gas and liquid but the gas and liquid states are fundamentally different in their thermodynamic description. Rigidity is determined by
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It is clear from Equation (1) that ω ≥ 0, i.e., rigidity must always be positive although it can be zero in two-phase coexistence regions. Gibbs energy cannot decrease with pressure when T is constant. From these definitions moreover, not only can there be no “continuity” of gas and liquid but the gas and liquid states are fundamentally different in their thermodynamic description. Rigidity is determined by number density fluctuations at the molecular level, which have different but complementary statistical origins in each phase, hence the symmetry [9]. There is a distribution of many small clusters in a gas with one large void; there is a distribution of unoccupied pockets in the liquid with one large cluster. The statistical neighborhood properties of unoccupied pockets are the same as occupied sites. The percolation loci and the critical coexisting densities at Tc for CO2 have been obtained by the method previously described in detail for Lennard-Jones fluids [5] and argon [8]. Looking at Figure2, the coexisting densities of the gas ( ρG) and of liquid (ρL) and the meso-density gap vary linearly between the critical temperature (Tc) and Boyle temperature (TB). If the mesophase rigidity is obtainable in terms of TB and TC, we can then represent the density bounds with linear functions and the rigidity in the mesophase as shown in Figure3a,b, and incorporate a linear equation of state for the mesophase that connects with gas and liquid at the percolation bounds PB and PA respectively.
Figure 3. (a) Phase diagram for CO2: data obtained from NIST Thermophysical Properties compilation indicating the percolation loci that bound the supercritical liquid and gas states: the green and blue dashed lines are the gas (PB) and liquid (PA) percolation loci respectively [4–9]; solid red points;
coexisting densities of the gas (ρG) and of liquid (ρL) at Tc are indicated by the arrows; (b) rigidity as a function of temperature in the mesophase.
GAS ρPB(T) = ρG(Tc) [(TB − T)/(TB − TC)] (5)
LIQUID ρPA(T) = ρL(Tc) [(TB − T)/(TB − TC)] (6) Use of original experimental data could result in non-linear Equations (5) and (6) for density state bounds in a more refined analysis. Given these expressions for the temperature dependence of the percolation loci, and the rigidity in the mesophase (Figure3b), in terms of physical constants belonging to the specific fluid, we can now proceed to represent the thermodynamic surfaces using virial expansions. The rigidity in the mesophase varies from kT at TB to zero at Tc. It is linearly dependent on temperature in the near-critical region up about 1.25 T/Tc but there is an increasing small quadratic term as seen in Figure3b. The temperature dependence in the range T c to TB appears to be logarithmic, Entropy 2018, 20, 22 5 of 15 but without a scientific explanation, and within the uncertainty of the data, it is best represented by a simple quadratic 2 ωT = c1T* + c2T* (7) where T* = (T − Tc)/(TB − Tc) and the constants c1 and c2 as given in Figure3b.
3. Pressure Surfaces p(ρ,T) The pressure for any thermodynamic equilibrium state point can be obtained as a function of density along any isotherm. The gas equation-of-state can be parameterized using the formally exact Mayer virial expansion [10]. The mesophase pressure increases linearly with density the region 0 > T > TB and ρPB(T) > ρ < ρPA(T), and for the “liquid” state ρ > ρPA(T), we can use an empirical expansion. The isothermal equations-of-state for pressure are as follows:
2 GAS p(ρ,T) = kTρ (1 + b2(T)ρ + b3(T)ρ + . . . ) (8)
MESO p(ρ,T) = p(ρPB,T) + ωT (ρ − ρPB) (9) 2 3 LIQUID p(ρ,T) = p(ρPA,T) + [a1(ρ − ρPA) + a2(ρ − PA) + a3(ρ − ρ PA) ...] (10)
For the temperature-density region Tc < T < TB and ρ < ρPA(T), Equations (8)–(10) are based upon the observation that there is a phase transition of the third-order at the mesophase bounds with a discontinuity in the derivative of ω with respect to ρ, i.e., the third derivative of Gibbs energy with respect to ρ at constant T along an isotherm. For temperatures above TB, p(ρPA,T) and ρPA, in Equation (10), are zero, whereupon the coefficients an in Equation (10) become the same as the Mayer virial coefficients bn (T) in Equation (8). For the supercritical fluid at temperatures above TB, Equation (8) up to order b4 is sufficient to reproduce the pressure with five-figure accuracy, i.e., within the margin of the original experimental uncertainty, with a trend-line regression >0.999999, up to temperatures of 1000 K and pressures up to 50 Mpa. For the gas phase, at all temperatures below TB, the 4th virial coefficient is found to be essentially zero, within the uncertainty of the experimental data; all higher terms are negligible. Thus, for the case of CO2, the equation-of-state of the whole “gas” region of Figure3 requires just the second and a third virial coefficients b2(T) and b3(T) within the experimental precision. An overall p(ρ,T) equation of state can be integrated along any isotherm to obtain the Helmoltz energy function A(T) and hence, via the partition function, all other thermodynamic state functions and/or derivatives by long-established procedures [11].
3.1. Supercritical CO2 In order to demonstrate a necessity of three separate equations-of-state, we take, as an example in the first instance, a supercritical isotherm T = 350 K, or T/Tc = 1.15, of CO2. The critical and Boyle temperatures for CO2 according to NIST databank [3] are Tc = 305 K and Tc = 725 K. Using the same method described previously for argon [8], the coexisting densities at Tc are found to be ρc(gas) = 7.771 mol/L and ρc(liq) = 13.46 mol/L. Substituting these physical constant values into equations (5 to 7) for T = 350 K we obtain the gas and liquid state density bounds ρPB(gas) = 7.305 mol/L and ρPA(liq) = 12.02 mol/L, and the rigidity ω(350 K) = 0.318 kT (0.925 kJ/mol). If we try to fit the NIST data for the whole isotherm continuously from 0, just up to only 50 MPa, using a 6-term polynomial for example, we obtain the result:
p = −0.000008ρ6 + 0.000457ρ5 − 0.009373ρ4 + 0.096245ρ3 − 0.596317ρ2 + 3.459264ρ − 0.189239 with R2 = 0.999125 In this typical continuous overall polynomial fit, the lower order coefficients are scientifically meaningless. By contrast, by using virial coefficients and known properties of the mesophase bounds, Entropy 2018, 20, 22 6 of 15 the NIST data for the whole range of three regions can be reproduced with essentially 100% precision when the gas, liquid and meso-states have different equations-of-state. We assume that there is a third-order phase transition, i.e., a discontinuity in the third derivative of Gibbs energy at the gas and liquid-state boundaries in accord with inequalities (2 and 3). The experimental data from the NIST tabulations is reproduced with the same precision as the original data can be obtained, i.e., accurate to 5 figures, as evidenced by the mean-squared regression Entropy 2018, 20, 22 6 of 15 (R2) in the trendline polynomial coefficients as given. Also shown in Figure4 are redefined equations toto givegive thethe virialvirial coefficients coefficients inin EquationsEquations (8)–(10)(8)–(10) asas shownshown onon the the plots. plots. TheThe coefficientcoefficient aa11 inin thethe ω liquid-stateliquid-state expansion expansion is is taken taken to to be be equal equal to to the the value value of of ωTT inin thethe mesophasemesophase toto effecteffect thethe third-orderthird-order discontinuitydiscontinuity betweenbetween thethe mesophasemesophase and and the the liquid liquid state. state. ThereThere isis aa symmetrysymmetry betweenbetween thethe rigidityrigidity ofof gasgas andand liquidliquid onon eithereither sideside ofof the the mesophase mesophase butbut at at this this stage, stage, this this observation observation is is empirical. empirical. AsAs itit presentlypresently lackslacks formalformal theory,theory, wewe assumeassume onlyonly thethe linearlinear coefficientcoefficient inin EquationEquation (10)(10) andand makemake nono assumptionsassumptions regarding regarding higher-order higher-order coefficients coefficients at at this this stage. stage.
(a) (b)(c)
Figure 4. Pressure equations p(ρ) for CO2 at a supercritical temperature (350 K or T/Tc = 1.15) as derived Figure 4. Pressure equations p(ρ) for CO2 at a supercritical temperature (350 K or T/Tc = 1.15) as derivedfrom NIST from Thermophysical NIST Thermophysical Properties Properties compilation: compilation: (a) gas (state;a) gas (b state;) supercritical (b) supercritical meso-phase meso-phase and (c) andliquid. (c) liquid.
3.2. Supercritical Argon 3.2. Supercritical Argon Modern research into fundamental equation-of-state science is more often based upon the Modern research into fundamental equation-of-state science is more often based upon the example example of argon in the first instance because of the relative simplicity of the spherical two-body of argon in the first instance because of the relative simplicity of the spherical two-body pair potential pair potential and the amenability to basic theory. The literature on equation-of-state data to high and the amenability to basic theory. The literature on equation-of-state data to high temperature and temperature and pressure is both extensive and accurate to at least five figures except in the vicinity of pressure is both extensive and accurate to at least five figures except in the vicinity of the critical the critical temperature (Tc) and pressure (pc) [3,12]. For temperatures above the Boyle temperature temperature (Tc) and pressure (pc)[3,12]. For temperatures above the Boyle temperature there appears there appears to be just a single continuous fluid state up to the highest pressures measured, which, in to be just a single continuous fluid state up to the highest pressures measured, which, in the case of the case of argon, is 1000 MPa, according to NIST [3,12]. If there are no discontinuities along isotherms argon, is 1000 MPa, according to NIST [3,12]. If there are no discontinuities along isotherms for T > TB, for T > TB, the Mayer virial expansion is applicable over the whole accessible density range. The the Mayer virial expansion is applicable over the whole accessible density range. The expansion for expansion for the pressure is the pressure is 2 3 4 p =p kT( = ρkT(+ρ b 2+ρ b2+ρ2b +3 ρb3ρ+3 + b 4bρ4ρ4. .……) . . . . ) (11) (11) andand onon differentiationdifferentiation the the rigidity rigidity is is
ω = (dp/dρ)T = kT(1 + 2b2ρ + 3b23ρ2 + 4b43ρ3………) (12) ω = (dp/dρ)T = kT(1 + 2b2ρ + 3b3ρ + 4b4ρ ...... ) (12) where bn are the temperature-dependent virial coefficients with conventional definitions. The NIST data wherefor four bn supercriticalare the temperature-dependent isotherms above and virial including coefficients the withBoyle conventional temperature definitions. (TB) can be The precisely NIST datareproduced for four using supercritical Equation isotherms (12) and abovethe virial and coeffi includingcients thedetermined. Boyle temperature From Stewart (TB )and can Jacobson be precisely [13] reproducedwe obtain T usingB = 414 Equation ± 0.5 for (12)argon, and which the virial is consistent coefficients with determined. the zero value From of the Stewart second and virial Jacobson coefficient [13] weobtained obtain ThereB = 414from± parameterizations0.5 for argon, which of is NIST consistent rigidity with values the zero from value sound of the velocity second compilations. virial coefficient obtainedThe hereequation-of-state from parameterizations isotherms ofare NIST presented rigidity valuesin Figure from 5.sound Also velocitygiven are compilations. the polynomial trend-line parameters up to order 4. In this region, the entire experimental range of pressures, from near zero at ideal gas states at limiting low densities, up to 1000 MPa, can be represented to within the experimental five-figure precision by a three or at most four-term, closed-virial equation. Using Equation (12) we can deduce lower-order values of the virial coefficients of argon as listed in Table 1. The values obtained for b2 and b3 are in agreement with the literature values within the experimental uncertainties, which can be as high as 20% in the case of b3 [9]. For argon at supercritical temperatures, little is presently known experimentally about b4 and higher coefficients.
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The equation-of-state isotherms are presented in Figure5. Also given are the polynomial trend-line parameters up to order 4. In this region, the entire experimental range of pressures, from near zero at ideal gas states at limiting low densities, up to 1000 MPa, can be represented to within the experimental five-figure precision by a three or at most four-term, closed-virial equation. Using Equation (12) we can deduce lower-order values of the virial coefficients of argon as listed in Table1. The values obtained for b2 and b3 are in agreement with the literature values within the experimental uncertainties, which can be as high as 20% in the case of b3 [9]. For argon at supercritical temperatures, little is presently knownEntropy experimentally 2018, 20, 22 about b4 and higher coefficients. 7 of 15
Figure 5. Rigidity isotherms, ω(ρ) for argon at temperatures from 700 K down to TB (414 K) as Figure 5. Rigidity isotherms, ω(ρ) for argon at temperatures from 700 K down to T (414 K) as obtained obtained from NIST Thermophysical Properties compilation up to pressures of 500 MPa;B all of the from NISTEXCEL Thermophysical trendline polynomials Properties of order compilation 4 reproduce th upe NIST to pressures isotherms ofwith 500 total MPa; precision, all of i.e., the a EXCEL trendlineregression polynomials of R2 = of1.000000 order up 4 reproduce to 6-figure precision. the NIST isotherms with total precision, i.e., a regression of R2 = 1.000000 up to 6-figure precision. Table 1. Argon virial coefficients.
T(K) b2 Table(10−3 mol 1. Argon/L) b3 virial(10−6 mol coefficients.2/L2)b4 (10−9 mol3/L3) 700 14.91 0.8191 0.0063 −3 −6 2 2 −9 3 3 T (K) 600 b2 (10 11.45mol/L) 0.9029 b3 (10 mol /L 0.0013) b4 (10 mol /L ) 700500 14.916.130 1.0523 0.8191−0.0082 0.0063 600414 (~TB) 11.450.0081 1.3085 0.9029−0.0248 0.0013 500151 (~Tc) 6.130−86.02 2.48 1.0523 0 −0.0082 414 (~TB)100 0.0081−183.2 0 1.3085 −0.0248 151 (~Tc) −86.02 2.48 0 Lower-order100 virial coefficients,−183.2 as defined by Equation (12), 0 and calculated from the trend-line polynomial coefficients in Figure 5, are summarized in Table 1. All the experimental data from the NIST tabulations are reproduced with maximum precision as evidenced by the mean-squared Lower-orderregression invirial the trendline coefficients, polynomial as defined coefficients by (R Equation2 = 1.000000 (12), to six and decimal calculated places in from every the case); trend-line polynomialthe coefficients coefficients have in Figurebeen redefined5, are summarized to obtain valu ines Table of the1 .virial All theexpansio experimentaln coefficients data defined from thein NIST tabulationsEquation are reproduced(11). These are with in line maximum with previous precision b2(T) and as b evidenced3(T) tabulations by [14] the within mean-squared the uncertainties, regression in the trendlinewhich polynomialare rather wide, coefficients up to a few (R percent2 = 1.000000 for b2(T), to and six up decimal to 25% placesin the case in everyof b3(T). case); the coefficients have been redefined to obtain values of the virial expansion coefficients defined in Equation (11). These 4. Critical Isotherms are in line with previous b2(T) and b3(T) tabulations [14] within the uncertainties, which are rather wide, up4.1. to Argon a few percent for b2(T), and up to 25% in the case of b3(T). The critical temperature of argon according to NIST is 150.87, the value obtained in a revised 4. Criticalanalysis Isotherms of the data of Gilgen et al. is 151.1 ± 0.1. Here we represent the rigidities of the experimental 151 K isotherm from the NIST data tables using standard trend-line polymomials. The coexisting gas 4.1. Argonand liquid densities as obtained from the original experimental data of Gilgen et al. [14,15] are Theρc(gas) critical = 11.89 temperature mol/L and ρc of(liq) argon = 14.95 according mol/L [8]. For to all NIST temperatures is 150.87, between the value Tc and obtained the triple point in a revised (Ttr), the NIST ω(ρ)T isotherms can be reproduced with 100% precision using just the two-parameter analysis of the data of Gilgen et al. is 151.1 ± 0.1. Here we represent the rigidities of the experimental equation. Numerical values of the coefficients b2 and b3 at Tc, obtained from the rigidity isotherms as illustrated in Figure 6a, are included in Table 1. The trend-line polynomials of order 3 reproduce the NIST isotherms over the whole range of equilibrium existence with high accuracy.
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151 K isotherm from the NIST data tables using standard trend-line polymomials. The coexisting gas and liquid densities as obtained from the original experimental data of Gilgen et al. [14,15] are ρc(gas) = 11.89 mol/L and ρc(liq) = 14.95 mol/L [8]. For all temperatures between Tc and the triple point (Ttr), the NIST ω(ρ)T isotherms can be reproduced with 100% precision using just the two-parameter equation. Numerical values of the coefficients b2 and b3 at Tc, obtained from the rigidity isotherms as illustrated in Figure6a, are included in Table1. The trend-line polynomials of order 3 reproduce the
NIST isothermsEntropy 2018 over, 20, 22 the whole range of equilibrium existence with high accuracy. 8 of 15 These trend-line equations for ωT(ρ) can be integrated to obtain pressure equations, and rewritten as: These trend-line equations for ωT(ρ) can be integrated to obtain pressure equations, and rewritten as: 2 3 Gas ρ < ρc(gas) p = kT (ρ + b2 ρ + b3 ρ ) (13) Gas ρ < ρc(gas) p = kT (ρ + b2 ρ2 + b3 ρ3) (13) Meso ρc(gas) < ρ < ρc(liq) p = pc (14) Meso ρc(gas) < ρ < ρc(liq) p = pc 2 3 (14) Liquid ρ > ρc(liq) p = p(ρcl,T) + kT [a (ρ − ρcl) + a2(ρ − ρcl) + a3(ρ − ρcl) ] (15) Liquid ρ > ρc(liq) p = p(ρcl,T) + kT [a (ρ − ρcl) + a2(ρ − ρcl)2 + a3(ρ − ρcl)3] (15) where ρcl is the critical coexisting liquid density ρc(liq). For argon, the critical pressure in Equation (14) where ρcl is the critical coexisting liquid density ρc(liq). For argon, the critical pressure in Equation (14) pc pc = 4.949 MPa [4]. Thus, we have an equation-of-state for argon along the whole critical isotherm, = 4.949 MPa [4]. Thus, we have an equation-of-state for argon along whole of the critical isotherm given with no divergent singularity. The coefficients in Equations (13) and (15), are obtainable from the the coefficients in Figure 6a,b for Equation (15), obtainable, for example, using the data in Figure 6b, rigidity data plotted in Figure6a,b, for example, by dividing the coefficients a by nkT. by dividing the coefficients an by nkT. n
(a) (b)
Figure 6. Rigidity isotherms, ω(ρ), for argon along Tc from NIST Thermophysical Properties compilation Figure 6. Rigidity isotherms, ω(ρ), for argon along T from NIST Thermophysical Properties [3]: (a) gas phase from zero to maximum gas density at criticalc pressure 4.949 MPa (b) liquid phase a compilationfrom minimum [3]: ( ) gasliquid phase density from from zeropc (4.949 to MPa) maximum to 326 MPa. gas The density rigidity at along critical the critical pressure divide 4.949 is MPa (b) liquidequal phase to zero. from minimum liquid density from pc (4.949 MPa) to 326 MPa. The rigidity along the critical divide is equal to zero. 4.2. Steam and Water 4.2. Steam andThe Water evidence for supercritical percolation transitions bounding the existence of water and steam in the supercritical region, and the parameters describing the mesophase, have all been the Thesubject evidence of a previous for supercritical article with percolation an extended transitions debate [16]. boundingFor the present the existencepurposes, we of wateronly need and steam in the supercriticalvalues of some region, physical and constants the parameters that are given describing as an ancillary the mesophase, data list by NIST have [3]. all been the subject of a previous articleFigure with 7 shows an extended how the debatecritical point [16]. in For the the T-p present plane is purposes, thermodynamically we only needdefined values by an of some physicalintersection constants of that two are percolation given as loci, an ancillaryusing the exam dataple list of by water. NIST The [3 ].coexisting density curves are Figuretaken7 fromshows the how experimental the critical measurements point in theof the T-p IAPWS plane International is thermodynamically Steam Tables defined[17]. The by an supercritical percolation transition points PA and PB intersect at Tc, and continue to define the intersection of two percolation loci, using the example of water. The coexisting density curves are taken metastable limit of existence loci, usually referred to as spinodals, of the subcritical gas and liquid from thewithin experimental the two-phase measurements region. Values of of the the IAPWScritical physical International constants Steam for water Tables are: [T17c =]. 647.1 The K; supercritical pc = percolation22.05transition MPa; critical points steam PAdensity and ρ PBcG = intersect13.02 mol/L; at critical Tc, and water continue density to ρc defineL = 20.61 the mol/L metastable [3,16]. limit of existence loci,Also usually plotted referredin Figure 7 to are as the spinodals, experimentally of the ob subcriticalserved spinodals gas and[18] that liquid bound within the regions the two-phase of metastable existence within the two-phase coexistence region at subcritical temperatures. At Tc, the region. Values of the critical physical constants for water are: Tc = 647.1 K; pc = 22.05 MPa; critical percolation loci, PA and PB in Figure 7, cross the critical coexistence line at its extremities to become steam density ρcG = 13.02 mol/L; critical water density ρcL = 20.61 mol/L [3,16]. the subcritical boundary of the metastable compressed gas (p > psat) and metastable expanded liquid Also plotted in Figure7 are the experimentally observed spinodals [ 18] that bound the regions state (p < psat) stability limits respectively. This behavior was also shown for liquid argon [8]. It is of metastableconsistent existence with a phenomenological within the two-phase definition coexistenceof PA and PB regionwhen both at liquid subcritical and gas temperatures. have the same At Tc, values of the rigidity (ωT) on the same isotherm whereupon (d2p/dρ2)T = 0 at both PA and PB but at
Entropy 2018, 20, 22 9 of 15 the percolation loci, PA and PB in Figure7, cross the critical coexistence line at its extremities to become the subcritical boundary of the metastable compressed gas (p > psat) and metastable expanded liquid state (p < psat) stability limits respectively. This behavior was also shown for liquid argon [8]. It is consistent with a phenomenological definition of PA and PB when both liquid and gas have the same 2 values of theEntropy rigidity 2018, 20, 22 (ω T) on the same isotherm whereupon (d2p/dρ )T = 0 at both PA9 andof 15 PB but at different pressures. At these instability points, there is also zero surface tension and consequently no Entropydifferent 2018, 20 pressures., 22 At these instability points, there is also zero surface tension and consequently no9 of 15 barrier to spontaneous nucleation of steam from water at PA, or water from steam at PB, for example. barrier to spontaneous nucleation of steam from water at PA, or water from steam at PB, for example. different pressures. At these instability points, there is also zero surface tension and consequently no barrier to spontaneous nucleation of steam from water at PA, or water from steam at PB, for example.
Figure 7. A phase diagram of water and steam in the p-T plane: the red and blue data points and Figure 7. Adashed phase lines diagram are the percolation of water loci and state steam bounds in PA the (water) p-T and plane: PB (steam) the red respectively. and blue data points and dashed lines are the percolation loci state bounds PA (water) and PB (steam) respectively. FigureThe 7.equations-of-state A phase diagram for of steamwater and watersteam alongin the thep-T criticalplane: theisotherm red and are blue shown data in points Figure and 8a,b respectively.dashed lines In are both the cases percolation a quartic loci equation state bounds is requ PAired (water) to fit andthe data PB (steam) with the respectively. five-figure precision of The equations-of-state for steam and water along the critical isotherm are shown in Figure8a,b the NIST tables. The fourth virial coefficient for steam at Tc, however, still appears to be quite small in respectively.itsThe contribution In bothequations-of-state cases right a up quartic to forthe steamnear equation critical and waterdensity. is required along We alsothe to notecritical fit thethat isotherm datathe liquid with are water theshown critical five-figure in Figureisothermprecision 8a,b of the NISTrespectively. tables.extends Thefrom In fourthbotha critical cases virial pressure a quartic coefficient of equation 22.05–1000 for is steamrequMPa iredand at tocan T fitc ,be the however, reproduced data with still withthe appears five-figure six-figure to precision be quite of small in by the quartic polynomial. its contributionthe NIST righttables. upThe to fourth the nearvirial criticalcoefficient density. for steam We at also Tc, however, note that still the appears liquid to water be quite critical small in isotherm extendsits from contribution a critical right pressure up to the of near 22.05–1000 critical density. MPa andWe also can note be reproducedthat the liquid with water six-figure critical isotherm precision by extends from a critical pressure of 22.05–1000 MPa and can be reproduced with six-figure precision the quartic polynomial. by the quartic polynomial.
(a) (b)
Figure 8. Rigidity (ω) for steam and water along the critical isotherm from NIST Thermophysical
Properties compilation [4]: (a) steam from zero to ρcG at Tc (rigidity = 0); (b) liquid phase from minimum
liquid density (ρcL) at pc = 22.05 MPa (ω = 0) to a pressure of 1000 MPa (ω ~ 100 kJ/mol). (a) (b) It is clear from the above that the equations-of-state for subcritical isotherms will require a Figure 8. Rigidity (ω) for steam and water along the critical isotherm from NIST Thermophysical Figurescientific 8. Rigidity knowledge (ω) for of the steam forms and for the water extension along of the the criticalpercolation isotherm loci into from the subcritical NIST Thermophysical two-phase regionProperties on the compilation density surface. [4]: (a) Atsteam present from wezero do to notρcG athave Tc (rigidity this information. = 0); (b) liquid It would phase befrom rather minimum unwise ρ Propertiesliquid compilation density (ρcL) at [4 p]:c = ( a22.05) steam MPa ( fromω = 0) zeroto a pressure to cG ofat 1000 Tc MPa(rigidity (ω ~ 100 = 0);kJ/mol). (b) liquid phase from
minimum liquid density (ρcL) at pc = 22.05 MPa (ω = 0) to a pressure of 1000 MPa (ω ~100 kJ/mol). It is clear from the above that the equations-of-state for subcritical isotherms will require a scientific knowledge of the forms for the extension of the percolation loci into the subcritical two-phase region on the density surface. At present we do not have this information. It would be rather unwise
Entropy 2018, 20, 22 10 of 15
It is clear from the above that the equations-of-state for subcritical isotherms will require a scientific knowledge of the forms for the extension of the percolation loci into the subcritical two-phase region on the density surface. At present we do not have this information. It would be rather unwise to make any assumptions based only upon the known supercritical behavior of the density loci of PA and PB and rather scanty subcritical spinodals. In the following section, therefore, we will limit the presentEntropy extension 2018, 20 of, 22 the above to an analysis of the 100 K isotherm of liquid argon, to demonstrate10 of 15 the applicability of the simple equation-of-state forms Equations (8)–(10) to subcritical gas and liquid state isotherms.to make any assumptions based only upon the known supercritical behavior of the density loci of PA and PB and rather scanty subcritical spinodals. In the following section, therefore, we will limit the 5. Subcriticalpresent Isothermsextension of the above to an analysis of the 100 K isotherm of liquid argon, to demonstrate the applicability of the simple equation-of-state forms Equations (8)–(10) to subcritical gas and liquid state isotherms. Subcritical Argon 5. Subcritical Isotherms The 100 K (T/Tc ~2/3) isotherm data for the rigidity from the NIST compilation of gaseous and liquid argonSubcritical are re-plottedArgon in Figure9. The gas pressure up to the condensation line can be represented with just theThe two-term 100 K (T/T equationc ~ 2/3) isotherm with the data value for the of rigidity b2 given from in the Table NIST1; bcompilation3 is essentially of gaseous zero. and Thus the accurateliquid equation argon are for re-plotted argon gas in atFigure lower 9. The temperatures gas pressure up is simplyto the condensation line can be represented with just the two-term equation with the value of b2 given in Table 1; b3 is essentially zero. Thus the accurate equation for argon gas atp lowergas (100 temperatures K) = kTρ is(1 simply + b2 ρ) (16)
pgas (100 K) = kTρ (1 + b2 ρ) (16) −3 The value obtained from the linear fit of ωT in Figure8a, is 183.2 × 10 mol/L compares The value obtained from the linear fit of ωT in Figure 8a, is 183.2 × 10−3 mol/L compares favorably − 3 favorablywith with an experimental an experimental value − value187 (cm3/mol)187 (cmobtained/mol) in 1958 obtained [13] in 1958 [13].
(a) (b)
Figure 9. Rigidity ω(ρ) for argon along the sub-critical isotherm T = 100 K from NIST [4]: (a) gas Figure 9. Rigidity ω(ρ) for argon along the sub-critical isotherm T = 100 K from NIST [4]: (a) gas phase from zero to maximum equilibrium gas coexistence density (0.42 mol/L): (b) liquid phase from phase fromminimum zero toliquid maximum equilibrium equilibrium coexistence gas density coexistence (32.9 mol/L) density at the (0.42coexistence mol/L): pressure (b) liquid (MPa) phaseto a from minimumpressure liquid of equilibrium100 MPa. coexistence density (32.9 mol/L) at the coexistence pressure (MPa) to a pressure of 100 MPa. The corresponding pressure data for liquid argon at 100 K up to freezing (62 MPa) may not be continuous in all its derivatives, and hence the equation-of-state may not be as simple as it looks if Thethere corresponding is a higher-order pressure discontinuity. data for The liquid liquid argon isotherm at 100 appears K up to to comprise freezing two (62 linear MPa) parts may not be continuouscontaining in all an its intersection derivatives, at an and intermediate hence the density equation-of-state of around 35.0 mol/L. may This not beis consistent as simple with as a it looks if thereprevious is a higher-order conjecture [19] discontinuity. that as the crystallization The liquid point isotherm is approached, appears the toliquid comprise itself may two become linear parts containinganother an intersectionkind of mesophase at an as intermediate large clusters densityof ordered of arrangements around 35.0 of mol/L. atoms, probably This is consistent bcc in the with a previouscase conjecture of argon, begin [19] thatto stabilize. as the This crystallization could be a feature point of is the approached, equilibrium “liquid” the liquid state itself structure may in become the near triple-point density region. One can regard the data in Figure 9b as a fragment of evidence anotherfor kind this ofconjecture. mesophase We will as large not consider clusters this of intriguing ordered arrangementspossibility further of at atoms, this stage. probably We note, bcc in the case of argon,however, begin that if to this stabilize. is indeed Thisthe underlying could be scie a featurence, any of equation-of-state the equilibrium for the “liquid” equilibrium state ‘pure structure in the nearliquid’ triple-point may terminate density before region. the One equilibrium can regard crystallization the data in density Figure 9isb reached as a fragment along subcritical of evidence for this conjecture.isotherm, Webelow will around not consider 115 K in the this case intriguing of argon. possibility further at this stage. We note, however, that if this is indeed the underlying science, any equation-of-state for the equilibrium ‘pure liquid’ may
Entropy 2018, 20, 22 11 of 15 terminate before the equilibrium crystallization density is reached along subcritical isotherm, below around 115 K in the case of argon.
6. Energy and Heat Capacities
6.1. Argon Critical Isotherm and Isochores The only real evidence for a van der Waals-like singularity and a critical point, with continuity of gas and liquid [1] and universal scaling properties [20], is evidently in historic heat capacity measurements [21,22]. Here we focus briefly at the equation-of-state for the internal energy U(ρ)T for the critical isotherm of argon close to Tc (151 K). The NIST data with the trend-line parameters are shown in Figure9. The foremost observation is that along this isotherm the gas phase is represented by a quadratic requiring two coefficients whereas the liquid phase is linear up to a density around 35 mol/L. The criticalEntropy 2018,divide 20, 22 is linear by definition in accord with the lever rule or11 proportionality.of 15 The internal6. Energy energy and forHeat all Capacities temperatures below Tc in the two-phase region is defined to be linear by the lever rule since the physical state is of two-phase coexistence. Above the critical divide the three 6.1. Argon Critical Isotherm and Isochores regions shown in Figure 10, however, look much the same, so we can identify two higher-order phase The only real evidence for a van der Waals-like singularity and a critical point, with continuity transitions boundingof gas and the liquid mesophase. [1] and universal There scaling is no properti evidencees [20], of is divergentevidently in historic behavior heat capacity of slope of U(T)ρ, i.e., measurements [21,22]. Here we focus briefly at the equation-of-state for the internal energy U(ρ)T for in the isochoric heat capacity Cv. Isotherms from Tc to 2Tc all show a constant value of (dU/dT)ρ in the critical isotherm of argon close to Tc (151 K). The NIST data with the trend-line parameters are the supercriticalshown region, in Figure as 9. seen The foremost in Figure observation 11. Along is that along the this critical isotherm divide, the gas phase the isochoricis represented heat capacity is defined by the leverby a quadratic rule: requiring for T ≤ twoTc coefficients whereas the liquid phase is linear up to a density around 35 mol/L. The critical divide is linear by definition in accord with the lever rule or proportionality. The internal energy for all temperatures below Tc in the two-phase region is defined to be linear by the lever rule sinceCv the= physical (dU/dT) stateV is =of Xtwo- Cphasev(liq) coexistence. + (1 − AboveX) Cv the(gas) critical divide the three (17) regions shown in Figure 10, however, look much the same, so we can identify two higher-order phase where X is moletransitions fraction bounding of liquid. the mesophase. There is no evidence of divergent behavior of slope of U(T)ρ, i.e., in the isochoric heat capacity Cv. Isotherms from Tc to 2Tc all show a constant value of (dU/dT)ρ in Uhlenbeckthe [ 20supercritical] appears region, to as have seen in been Figure the 11. Along first the to critical recognize divide, the what isochoric he heat described capacity is as “surprising Russian results”defined [21,22 by ]the as lever being rule: supportivefor T ≤ Tc of a universal picture of critical phenomena in which the isochoric heat capacity Cv is believedCv = to(dU/dT) divergeV = X Cv logarithmically(liq) + (1 − X) Cv(gas) on both sides of(17) the critical point. When the proceedingswhere X is mole of thefraction Washington of liquid. Conference were published one-year after the event [20], Uhlenbeck [20] appears to have been the first to recognize what he described as “surprising the articles by Rowlinson,Russian results” and[21,22] by as Fisher,being supportive in particular, of a universal hailed picture the of critical Russian phenomena discovery in which asthe a breakthrough (referenced by Rowlinsonisochoric heat capacity though Cv is curiously believed to notdiverge referenced logarithmically by on Fisher), both sides with of the the critical implicit point. proclamation that the van derWhen Waals the criticalproceedings point, of the Washingt with continuityon Conference of were liquid published and one-year gas, had after become the event [20], established the scientific articles by Rowlinson, and by Fisher, in particular, hailed the Russian discovery as a breakthrough truth, and universal(referenced scaling by Rowlinson properties though similar curiously to not a refere 2D-Isingnced by modelFisher), with lattice the implicit gas would proclamation describe gas-liquid and indeed allthat similar the van “critical der Waals point critical singularities”.point, with continuity In of fact, liquid an and abundance gas, had become of experimentalestablished evidence scientific truth, and universal scaling properties similar to a 2D-Ising model lattice gas would describe against this conjecturegas-liquid and was indeed already all similar there ‘critical in point the singularities’. literature In [23 fact,] and an abundance not supportive of experimental of van der Waals hypothesis or universality.evidence against this conjecture was already there in the literature [23] and not supportive of van der Waals hypothesis or universality.
Figure 10. Internal energy U(ρ) for argon along the isotherm (T = 151 K) using data from NIST [4]. Figure 10. Internal energy U(ρ) for argon along the isotherm (T = 151 K) using data from NIST [4].
Entropy 2018, 20, 22 12 of 15
Entropy 2018, 20, 22 12 of 15
0.6 0.4 0.2 GAS 0 -0.2 2-PHASE -0.4 -0.6 -0.8 LIQUID
Internal Energy (KJ/mol) -1 145 146 147 148 149 150 151 152 153 154 155 156 T(K) (a) (b)
Figure 11. (a) Internal energy isochores, U(T)ρ for argon for a temperature range from near triple Figure 11. (a) Internal energy isochores, U(T)ρ for argon for a temperature range from near triple point point to supercritical from NIST [4] showing an expanded window picture of the near critical to supercritical from NIST [4] showing an expanded window picture of the near critical isochores to isochores to the right. (b) Magnified data in critical region showing a weak discontinuity in (dU/dT)v, the right. (b) Magnified data in critical region showing a weak discontinuity in (dU/dT) , i.e., Cv, at T i.e., Cv, at Tc but clearly no evidence of logarithmic divergence on either side of Tc. v c but clearly no evidence of logarithmic divergence on either side of Tc. Despite using complex equations-of-state that could accommodate the concept of a critical Despitedensity at using the node complex of a parabola, equations-of-state the NIST thermody that couldnamic accommodate data bank is thestill conceptvalid as “experimental” of a critical density at theevidence node of that a parabola, the 1965 Washington the NIST thermodynamic proclamation of ‘a data universality bank is theory’ still valid was as fundamentally “experimental” incorrect. evidence It was based upon a misinterpretation of the isochoric heat capacity Cv for T < Tc, and spurious results that the 1965 Washington proclamation of ‘a universality theory’ was fundamentally incorrect. It was at, or millikelvins greater than, Tc [24,25]. Along any isochore within the critical divide there can be no based upon a misinterpretation of the isochoric heat capacity Cv for T < Tc, and spurious results at, divergence of the isochoric heat capacity. For T > Tc a divergence of Cv would be a contradiction of or millikelvinslaws of thermodynamics. greater than, If T cone[24 could,25]. Alongreversibly any add isochore heat (Q withinrev) to a classical the critical Gibbs divide fluid theresystem can that be no divergencedoes no of work the isochoricwith no increase heat capacity. in the temperat For T >ure, Tc ait divergencewould be imply, of Cv forwould example, be a that contradiction Qrev i.e., of lawsenthalpy, of thermodynamics. or (Qrev/T) i.e., Ifentropy, one could are not reversibly state functions add heat[26]. (Qrev) to a classical Gibbs fluid system that does no work with no increase in the temperature, it would be imply, for example, that Qrev i.e., 6.2. SF6 near Critical Isochors enthalpy, or (Qrev/T) i.e., entropy, are not state functions [26]. Sulphur hexafluoride has a special importance in this scientific debate; its critical temperature 6.2. SFand6 near pressure Critical are Isochors amenable to experiments at near ambient conditions. It has been the subject of a Sulphurmicrogravity hexafluoride experiment hasaboard a special the NASA importance Space Shuttle in this [27] scientific that appeared debate; to confirm its critical the erroneous temperature result obtained some 25 years earlier by Voronel and his colleagues [24,25]. We have obtained the and pressure are amenable to experiments at near ambient conditions. It has been the subject of a original published data points of the space shuttle experiments (see acknowledgements) and have microgravity experiment aboard the NASA Space Shuttle [27] that appeared to confirm the erroneous plotted them alongside the U(T)V equation-of-state from the NIST data bank parameterized as a simple result obtained some 25 years earlier by Voronel and his colleagues [24,25]. We have obtained the quadratic for T < Tc and linear for T > Tc equations in Figure 12a. The equations for U(T) at the same originaldensity published as the space data shuttle points μg of experiment the space can shuttle be differentiated experiments to give (see the acknowledgements) isochoric heat capacities and Cv have plottedin themFigure alongside 12b. In contrast the U(T) to Vbothequation-of-state the historic Russian from results the NIST for dataargon, bank and parameterizedthe space shuttle as “C a simplev” quadraticvalues for also T
Entropy 2018, 20, 22 13 of 15 any isochore along the critical divide. In the case of the SF6 microgravity experiments [22], if there was no stirring, there must be some other reason for the discrepancy seen in Figure 12, that is was no stirring, there must be some other reason for the discrepancy seen in Figure 12, that is presently presentlyinexplicable. inexplicable.
(a) (b)
Figure 12. (a) Internal energy isochores, U(T)ρ for SF6 for a temperature range from near triple point Figure 12. (a) Internal energy isochores, U(T)ρ for SF6 for a temperature range from near triple point to supercritical from NIST Thermophysical Properties compilation [4]; (b) Isochoric heat capacities to supercritical from NIST Thermophysical Properties compilation [4]; (b) Isochoric heat capacities calculated from the equations in Figure 1a, i.e., (dU/dT)ρ alongside the original space shuttle heat ρ calculatedcapacity from data thepoints. equations in Figure1a, i.e., (dU/dT) alongside the original space shuttle heat capacity data points. 7. Conclusions 7. ConclusionsFrom the foregoing analysis, we conclude that the pressure equation-of-state, and hence all otherFrom thermodynamic the foregoing analysis,state functions, we conclude can be expre that thessed pressure simply in equation-of-state, terms of a few physical and henceconstants all other thermodynamicbelonging to state the fluid, functions, and coefficients can be expressed in virial simply expansions. in terms The of van a few der physical Waals equation-of-state constants belonging survived for about 50 years until, increasingly, there was evidence that it could not account for the to the fluid, and coefficients in virial expansions. The van der Waals equation-of-state survived for experimental data unless it was extended to include more and more terms and adjustable parameters. aboutModern 50 years equations-of-state until, increasingly, with a there large was number evidence of terms that and it couldadjustable not accountparameters for can the reproduce experimental dataexperimental unless it was p-V-T extended data to to high include precision more for and chemical more engineering terms and applications, adjustable parameters.but they may Modernno equations-of-statelonger reflect an with underlying a large numberphysical ofscience terms in and the adjustablespirit of van parameters der Waals. can reproduce experimental p-V-T dataWe to have high found precision that the for equation-of-state chemical engineering for the low applications, density gas phases, but they in mayall three no longertemperature reflect an underlyingregions physicalT > TB, TB science>T > Tc, and in the T < spirit Tc, can of all van be represented der Waals. by the first two or three terms of a Mayer virialWe have expansion. found For that T the> TB equation-of-statea single virial equation for thecan lowrepresent density the whole gas phases, isotherms in all upthree to the temperaturehighest recorded pressures, which in the case of argon is 100 MPa. For supercritical temperatures below the regions T > TB,TB >T > Tc, and T < Tc, can all be represented by the first two or three terms of a Mayer Boyle temperature the gas phase can be represented by a term virial expansion to within the accuracy virial expansion. For T > TB a single virial equation can represent the whole isotherms up to the highest that the original pressures reported by NIST [3]. The virial equations-of-state are fundamental to all recorded pressures, which in the case of argon is 100 MPa. For supercritical temperatures below the atomic and molecular fluids and appear to be accurate up to second order, i.e., to quadratic terms for Boyle temperature the gas phase can be represented by a term virial expansion to within the accuracy all temperatures below TB. We will report detailed comparisons and virial coefficients for the case that theCO2 original, and also pressures some preliminary reported comparisons by NIST [3 ].with The experimental virial equations-of-state data and NIST are equations, fundamental for the to all atomicexemplary and molecular fluid argon, fluids in due and course. appear to be accurate up to second order, i.e., to quadratic terms for all temperaturesAt this stage below in the T Bdevelopment. We will report of science-based detailed alternative comparisons forms and of the virial equation-of-state coefficients we for have the case CO2,not and attempted also some to parameterise preliminary the comparisons second and third with vi experimentalrial coefficients as data functions and NIST of temperature. equations, We for the exemplarynote, however, fluid argon, there inis dueevidence course. that b2(T) may be non-analytic at the Boyle temperature. It can be representedAt this stage by in two the different development series expansions, of science-based at least in alternative the case of a forms simple of Lennard–Jones the equation-of-state model, we haveabove not attempted and below to TB parameterise [29,30]. There theare secondalso some and indications third virial that coefficients b3(T) exhibits as a functions maximum of at temperature.or near Tc. Also, we find that for simple fluids, e.g., argon, below TB, the coefficient b4(T) appears to become We note, however, there is evidence that b (T) may be non-analytic at the Boyle temperature. It can be vanishingly small. A more accurate determination2 of lower-order virial coefficients from original representedthermodynamic by two data different ought series now to expansions, become a research at least priority. in the case of a simple Lennard–Jones model, above and below TB [29,30]. There are also some indications that b3(T) exhibits a maximum at or near Tc. Also,Acknowledgments: we find that We for wish simple to thank fluids, Richard e.g., Sadus argon, of belowSwinburn TBe ,University the coefficient for providing b4(T) appearsthe original to heat become vanishinglycapacity data small. points A from more the accuratespace shuttle determination experiments [27] of re-plotted lower-order in Figure virial 12b. coefficients from original thermodynamic data ought now to become a research priority. Entropy 2018, 20, 22 14 of 15
Acknowledgments: We wish to thank Richard Sadus of Swinburne University for providing the original heat capacity data points from the space shuttle experiments [27] re-plotted in Figure 12b. Conflicts of Interest: The author declares no conflict of interest.
References
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