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Phenomenological High-Pressure Equation of State for Nitrogen, Methane, Methanol, Carbon Dioxide, and Helium

Abstract

The vapor–liquid phase equilibria, saturation curves and pressure isotherms of nitrogen, methane, methanol, carbon dioxide and helium are modeled with a closed-form non-cubic equation of state (EoS), developed to describe high-pressure and high-density properties of fluids. The EoS is analytic and applicable to pure compounds as well as mixtures in the full temperature range above the melting point (or lambda point in the case of normal fluid helium) and up to the limit density where the pressure singularity occurs. The temperature evolution of the EoS is determined by four temperature-dependent scale factors on which the EoS linearly depends. Above the critical temperature, these scale factors can be regressed from empirical supercritical isotherms. Based on the proposed EoS, the Helmholtz and Gibbs free energies of the mentioned fluids are calculated, also in closed form. In the subcritical regime, a convex Helmholtz free energy is obtained by way of the common tangent construction. The subcritical scale factors of the EoS are inferred from the empirical liquid and vapor saturation densities, so that the EoS is consistent with the common tangent construction and the measured coexistence curve.

Introduction

The purpose of this paper is to discuss phase equilibria, coexistence curves and high-pressure isotherms of nitrogen, methane, methanol, carbon dioxide and helium, based on a phenomenological fluid equation of state (EoS) that is analytic and of closed form. The proposed EoS is non-algebraic, its attractive potential containing a density-dependent exponential. Even though the EoS is non-cubic, it is designed in analogy to cubic equations [1,2,3,4,5,6,7,8]; its wide applicability from classical fluids to extreme quantum fluids like helium is due to the adaptability of the temperature-dependent scale factors of the EoS, which are broken power laws [9], and also because the EoS is detached from microscopic first principles, i.e. from the specific modeling of molecular interactions.

A detailed derivation of the EoS has been given in Ref. [9], centered at the critical point conditions, i.e. vanishing density derivatives of pressure at the critical point, which will not be repeated here. Otherwise, the paper is self-contained, the EoS is defined from scratch, and the main purpose is to demonstrate its applicability by discussing concrete examples of fluids for which accurate saturation data and high-pressure isothermal data are available. The emphasis is on high density and pressure, where cubic equations tend to be particularly inaccurate, as will be illustrated in the figures, and where virial expansions [10, 11] of molecular interaction models usually fail. The applicability of the EoS at high density is due to the fact that it is non-perturbative, avoiding the decomposition into a free exactly solvable model with additive interaction terms that are required to be small for perturbative expansion techniques.

In Sect. 2, we define the EoS, which is parametrized with five temperature-independent substance-specific constants to be regressed from the critical isotherm. These constants also determine the limit density of the EoS, where the pressure becomes singular. The regression of these constants from the empirical critical pressure isotherm is explained in Sect. 3.1.

The temperature evolution of the EoS is determined by four temperature-dependent scale factors. In the supercritical regime, these analytic scale factors are regressed from pressure isotherms, as done in Sects. 3.2 and 3.3. In the subcritical regime above the melting point (or lambda point in the case of normal fluid helium), the scale factors are calculated from the empirical saturation curve, by making use of the common tangent construction ensuring the convexity of the Helmholtz free energy (as a function of volume) in the subcritical temperature interval.

The pressure isotherms of the EoS will be tested with super- and subcritical isothermal data sets of nitrogen, methane, methanol, carbon dioxide and helium. The empirical saturation properties can be implemented in the temperature evolution of the EoS, as will be demonstrated here with these fluids. In the figures depicting selected isotherms and isothermal data sets, we also show the corresponding isotherms of the cubic Peng-Robinson EoS, which serves as a reference equation. We refrain from comparing with more general cubic equations, e.g. EoSs involving volume translations [1, 6] or non-cubic EoSs obtained from molecular interaction models such as SAFT or PC SAFT models [12,13,14], as these equations can be marred by intersecting isotherms and multiple Maxwell loops in the coexistence region [15], the latter also being encountered in multiparameter EoSs [16,17,18,19,20,21,22,23,24,25]. The data sets used for the regression are actually generated by multiparameter EoSs, see below.

In Sect. 4, we calculate the Helmholtz free energy of the mentioned fluids and perform the common tangent construction, which the EoS has to satisfy in order to be consistent with the empirical saturation properties. In Sect. 5, the equations of the common tangent construction are employed to calculate the subcritical scale factors of the EoS from the measured saturation curves. In this way, an analytic closed-form EoS is obtained, applicable above the melting point (or lambda point in the case of helium) and for densities up to the pressure singularity. In Sects. 35, we specifically derive the EoSs of nitrogen, methane, methanol, carbon dioxide and of normal fluid helium, using isothermal and saturation data sets from Ref. [26]; the latter are synthetic data based on a specific multiparameter reference EoS for the respective fluid inferred from a variety of experimental data sources. (These reference EoSs are quoted in Sect. 4.3 and also in the figures of the pressure isotherms where the data points are depicted.) The emphasis is on the mentioned fluids, but we keep the formalism sufficiently general for wider applicability to other compounds and multi-component mixtures. The conclusions are summarized in Sect. 6.

A Fluid Equation of State Applicable to Pure Compounds and Mixtures at High Pressure

The analytic closed-form EoS P(n,T) (pressure P, molar density n, temperature T) is defined as

P/R=Q(n,T)(1σ(T))(n/nc)2Q0eA(nnc)(nnc+B)
(2.1)

with repulsive potential

Q(n,T)=1(1n/b0)β0[Tn+k=24(1ρk(T))cknk]
(2.2)

A derivation thereof is given in Refs. [9, 27]. R=8.31446 J/(K mol) is the gas constant. The repulsive term Q(n,T) is modeled in analogy to the Carnahan-Starling hard-sphere EoS [28, 29] and depends on five temperature-independent fitting parameters: the amplitude b0 and exponent β0 are positive, and the polynomial coefficients ck=2,3,4 are real. The temperature-dependent scale factors ρk=2,3,4(T) in (2.2) and σ(T) in (2.1) are normalized to zero at the critical temperature ρk(Tc)=σ(Tc)=0. The critical-point constants are denoted by (nc,Tc,Pc) and listed in Table 1 for the fluids studied here, that is nitrogen, methane, carbon dioxide, methanol and helium.

Table 1 Critical constants (temperature Tc, molar density nc, molar volume Vc, pressure Pc) of nitrogen, methane, carbon dioxide (CO2), methanol and helium, cf. Ref. [26]. The melting temperature Tmelt is also listed; in the case of (normal-fluid) helium, the lambda point of 2.17 K is taken as lower edge of the subcritical temperature range. Also recorded are the critical compressibility factor Zc=Pc/(ncTcR), where R=8.31446 J/(K mol) is the gas constant, and the acentric factor ω used in the α function of the cubic PR EoS, cf. (3.2)

We will use the shortcuts Q0, Q,n, Q,n,n for the constants Q0:=Q(nc,Tc)Pc/R and Q,n(nc,Tc), Q,n,n(nc,Tc); a subscript comma followed by n denotes a density derivative. Thus, Q,n and Q,n,n are the first- and second-order density derivatives of the repulsive potential (2.2), taken at the critical point. The EoS (2.1), (2.2) linearly depends on the rho and sigma functions ρk(T), σ(T), which determine the temperature evolution of the EoS, apart from the term linear in temperature in (2.2).

The exponential of the second term (attractive potential of the EoS) in (2.1) is defined by the constants

A=1nc2+Q0Q,n,nQ,n22Q02,B=2ncQ0(ncQ,n2Q0)2Q02nc2Q,n2+nc2Q0Q,n,n
(2.3)

derived from the critical-point conditions, P(nc,Tc)=Pc, P,n(nc,Tc)=0,P,n,n(nc,Tc)=0, which are exactly satisfied by the EoS, cf. Ref. [9]. The attractive term in (2.1) scales n2 at low density and the repulsive term (2.2) is linear in the low-density regime. The positive real exponent β0 in (2.2) is the pressure scaling exponent at the singularity n=b0 (high-density limit). The analytic scale factors ρk=2,3,4(T) and σ(T) in (2.1) and (2.2) will be specified in Sects. 3 and 5 for the super- and subcritical temperature ranges.

Parameters and Supercritical Scale Factors of the EoS

Density Dependence of Pressure at the Critical Temperature

The temperature-dependent ρk and σ scale factors of the EoS (2.1), (2.2) vanish at the critical temperature, σ(Tc)=ρk=2,3,4(Tc)=0. The critical isotherms of nitrogen, methane, carbon dioxide, methanol and helium are obtained by four-parameter (b0,ck=2,3,4) regression, since we put β0=1/2 in (2.2) from the outset because of lack of high-pressure data points. The temperature-independent fitting parameters b0,ck are recorded in Table 2. The least-squares functional to be minimized reads

χ2(b0,ck)=i=1N(P(ni;b0,ck)Pi)2Pi2
(3.1)
Table 2 Fitting parameters of the EoSs of nitrogen, methane, carbon dioxide, methanol and helium at the critical temperature Tc. The regressed parameters are the amplitudes b0 and c2,3,4 of the repulsive potential (2.2). The exponent β0 in (2.2) defining the pressure scaling at the critical density is also a fitting parameter but has been put to β0=1/2 for lack of high-pressure data points close to the singularity. The listed parameters b0 and c2,3,4 have been rescaled with nc and Tc to make them dimensionless. The regression of the EoS at Tc is explained in Sect. 3.1. Also recorded are the minimum of the least-squares functional χ2 and the degrees of freedom of the fit (dof: number N of data points (ni,Pi) on the critical isotherm minus number of fitting parameters). The regressed critical isotherms P(n,T=Tc) (cf. (2.1)) are depicted in Figs. 1, 6, 11, 16 and 21

where (ni,Pi)i=1,...,N are data points for the critical isotherm, and P(n;b0,ck) is the EoS (2.1), (2.2) with β0=1/2 and the ρk and σ scale factors put to zero. The regressed isotherms, temperature-dependent scale factors and saturation curves of nitrogen are depicted in Figs. 1, 2, 3, 4, and 5, of methane in Figs. 6, 7, 8, 9, and 10, carbon dioxide in Figs. 11, 12, 13, 14, and 15, methanol in Figs. 16, 17, 18, 19, and 20 and helium in Figs. 21, 22, 23, 24 and 25. The χ2 functionals used for the regression of supercritical isotherms in Sects. 3.2 and for the saturation densities and pressure in Sect. 4.3 are structured analogously to (3.1). The data points depicted in the figures are taken from Ref. [26], see the captions of Figs. 1, 6, 11, 16 and 21 and also Sect. 4.3 where the multiparameter EoSs generating these data are referenced; residual plots of the isotherms are also included in these figures.

Fig. 1
figure 1

Nitrogen pressure isotherms. Data points from Ref. [26]; the multiparameter reference EoS used to generate the data points of the nitrogen isotherms in Ref. [26] is stated in Refs. [16, 17]. Squares/circles/diamonds indicate supercritical/critical/subcritical isothermal data points, respectively. The red solid curves show isotherms of EoS (2.1), (2.2) (with parameters b0,ck,β0 in Table 2). The scale factors ρk(T), σ(T) of the EoS of nitrogen in the supercritical regime T>Tc are defined in (3.3) and Table 4. In the subcritical interval TmeltT<Tc above the melting temperature, the scale factors ρk(T), σ(T) are calculated from the empirical saturation curve (green solid curve), cf. Sect. 5. Depicted are the supercritical isotherms at 2000, 1400, 800, 300 and 200 K, as well as the critical isotherm passing through the critical point at 126.19 K, and also the subcritical 70 K isotherm (red solid curves). The horizontal dashed line crossing the coexistence region connects the vapor and liquid saturation points on the 70 K isotherm, cf. Sect. 4.2. The black diamonds indicate the end points of the liquid–vapor saturation curve at the melting temperature of 63.15 K. Also depicted for comparison are the isotherms of the cubic Peng–Robinson (PR) EoS (blue dashed curves), cf. (3.2). Residuals of selected isotherms are shown in the lower panels, the (ni,Pi) labeling data points

Fig. 2
figure 2

Supercritical scale factors ρk=2,3,4(T), σ(T) of the EoS of nitrogen, cf. Section 3.3. The circles/squares show data points (Tj,ρ2,3,4(Tj)), (Tj,σ(Tj)) from Table 3, at Tc, 200, 300, 800, 1400 and 2000 K, cf. Section 3.2. The solid curves depict the regressed supercritical ρk and σ scale factors (3.3) of the EoS (2.1), (2.2), with fitting parameters in Table 4

Fig. 3
figure 3

Liquid and vapor saturation volumes of nitrogen, cf. Section 4.3. Data points from Ref. [26]. Depicted are the liquid saturation volume V1(T) in (4.6) with regressed parameters in Table 9 (blue curve) and the vapor saturation volume V2(T) in (4.7) with parameters in Table 10 (red curve). The temperature range covers the subcritical interval, TmeltTTc, cf. Table 1. The vertical dashed line indicates the reduced melting temperature, and the diamond depicts the critical point (Tc,Vc). Residuals of the liquid and vapor branches of the regressed saturation volume are shown in the lower panel, the (Ti,Vi) labeling data points

Fig. 4
figure 4

Saturation pressure of nitrogen. Data points from Ref. [26]. Depicted is the regressed saturation pressure Psat(T) in (4.8), with parameters in Table 11 (solid red curve). The vertical dashed line indicates the reduced melting temperature, and the diamond depicts the critical point (Tc,Pc). The blue dashed curve shows the saturation pressure P(n1(T),T) calculated via EoS (2.1), (2.2) (with subcritical scale factors ρk(T), σ(T) in (5.7)), where n1(T)=1/V1(T) is the liquid saturation density, cf. Sects. 4.2, 4.3 and Fig. 3. Psat(T) and P(n1(T),T) very nearly coincide, which is a consistency check of the subcritical EoS, cf. Section 5. Residuals of the regressed and calculated (from EoS) saturation pressure are depicted in the lower panels, the (Ti,Pi) labeling data points

Fig. 5
figure 5

Subcritical scale factors of the EoS of nitrogen. The red and green curves show the scale factors ρ2(T) and σ(T) of EoS (2.1), (2.2) in the subcritical interval Tmelt<T<Tc, cf. (5.7). (The scale factors ρ3(T) and ρ4(T) of the EoS have been put to zero in this interval.) The vertical dashed line indicates the reduced melting temperature. The depicted data points were calculated from empirical saturation data in Fig. 3 by making use of the common tangent construction to the free energy, cf. Sects. 4.2 and 5

Fig. 6
figure 6

Methane pressure isotherms. Data points from Ref. [26]; see also Ref. [19] for the multiparameter reference EoS used to generate the data. The red solid curves show isotherms of EoS (2.1), (2.2) (with parameters b0,ck,β0 in Table 2). The scale factors ρk(T), σ(T) of the EoS of methane in the supercritical regime T>Tc are defined in (3.3), (3.4) and Table 5. In the subcritical temperature interval TmeltT<Tc, the scale factors are calculated from the empirical saturation curve (green solid curve), cf. Sects. 4.2, 4.3 and 5. Depicted are the supercritical isotherms at 600, 400, 300 and 220 K as well as the critical isotherm at 190.56 K and also the subcritical 100 K isotherm (red solid curves). The horizontal dashed line connects the vapor and liquid saturation points on the 100 K isotherm, cf. Section 4.2. The black diamonds indicate the end points of the liquid–vapor saturation curve at the melting temperature of 90.69 K. Also depicted for comparison are the isotherms of the cubic PR EoS (blue dashed curves), cf. (3.2) and Table 1. Residuals of selected isotherms are shown in the lower panels

Table 3 Parameters of selected supercritical pressure isotherms of nitrogen, methane, carbon dioxide, methanol and helium. The isotherms are defined by EoS P(n,T) in (2.1), (2.2) at a fixed supercritical temperature T=Tj>Tc, cf. Section 3.2. The temperature-independent parameters β0,b0,ck of the EoS are taken from the least-squares fit of the critical isotherm, cf. Table 2. The rho and sigma functions ρ2,3,4(Tj), σ(Tj) at the listed temperatures are fitting parameters regressed from the respective experimental Tj isotherm, based on data sets from Ref. [26]. (See also the captions of Figs. 1, 6, 11, 16, 21 and Sect. 4.3 for multiparameter reference EoSs generating the data sets.) Also recorded are the minimum of the least-squares functional χ2 and the degrees of freedom (dof) of each fit, cf. the caption of Table 2
Table 4 Fitting parameters of the supercritical ρk and σ scale factors of the EoS of nitrogen, cf. Section 3.3. Recorded are the parameters of the scale factors ρk(T), k=2,3,4, and σ(T) in (3.3), regressed from supercritical data sets (Tj,ρ2(Tj)), (Tj,ρ3(Tj)), (Tj,ρ4(Tj)) and (Tj,σ(Tj)), respectively, listed in Table 3. The analytic structure of these scale factors is (ρk(T),σ(T))=b(T/Tc)ηαβ((T/Tc)α1)β, with amplitude b and positive exponents α, β and real exponent η recorded in this table for nitrogen. χ2 is the minimum of the least-squares functional. The data points and regressed scale factors of nitrogen are depicted in Fig. 2
Table 5 Fitting parameters of the supercritical ρk and σ scale factors of the EoS of methane. The analytic scale factors are broken power laws, (ρ2(T),σ(T))=b(T/Tc)ηαβ((T/Tc)α1)β, cf. (3.3), and (ρ3(T),ρ4(T))=b(T/Tc)η1αβ((T/Tc)α1)β(T/Tc1c), cf. (3.4), with real amplitude b, positive amplitude c, positive exponents α, β and real exponent η recorded in this table for methane. The least-squares regression is based on supercritical data sets (Tj,ρ2(Tj)), (Tj,ρ3(Tj)), (Tj,ρ4(Tj)) and (Tj,σ(Tj)) in Table 3. Also listed is the minimum χ2 of the least-squares functional. The data points and the regressed analytic ρk and σ functions of methane are depicted in Fig. 7
Fig. 7
figure 7

Supercritical scale factors ρk=2,3,4(T), σ(T) of the EoS of methane, cf. Section 3.3. The circles/squares show data points (Tj,ρ2,3,4(Tj)), (Tj,σ(Tj)) from Table 3, at Tc, 220, 250, 300, 400, 500 and 600 K, cf. Section 3.2. The solid curves depict the regressed supercritical rho and sigma functions ρ2(T), σ(T) in (3.3) and ρ3,4(T) in (3.4), with fitting parameters in Table 5

Fig. 8
figure 8

Liquid and vapor saturation volumes of methane, cf. Section 4.3. Data points from Ref. [26]. Depicted are the liquid saturation volume V1(T) in (4.6) with parameters in Table 9 (blue curve) and the vapor saturation volume V2(T) in (4.7) with parameters in Table 10 (red curve). The temperature range covers the subcritical interval, from the melting point at 90.69 K to the critical temperature at 190.56 K, cf. Table 1. The diamond indicates the critical point (Tc,Vc). Residuals of the regressed liquid and vapor saturation volumes are shown in the lower panel

Fig. 9
figure 9

Saturation pressure of methane. Data points from Ref. [26]. Depicted is the regressed saturation pressure Psat(T) in (4.8) with parameters in Table 11 (solid red curve). The vertical dashed line indicates the reduced temperature of the melting point, and the diamond depicts the critical point (Tc,Pc), cf. Table 1. The blue dashed curve shows the saturation pressure P(n1(T),T) calculated via EoS (2.1), (2.2) (with subcritical scale factors ρk(T), σ(T) in (5.7)), where n1(T)=1/V1(T) is the liquid saturation density, cf. Sects. 4.2, 4.3 and Fig. 8. Psat(T) and P(n1(T),T) very nearly coincide, which is a consistency check of the subcritical EoS, cf. Sect. 5. Residuals of the regressed and calculated (from EoS) saturation pressure are shown in the lower panels

Fig. 10
figure 10

Subcritical scale factors of the EoS of methane. The red and green curves show the scale factors ρ2(T) and σ(T) of EoS (2.1), (2.2) in the subcritical interval Tmelt<T<Tc, calculated in (5.7). (The scale factors ρ3(T) and ρ4(T) of the EoS have been put to zero in this interval.) The vertical dashed line indicates the melting temperature, cf. Table 1. The depicted data points were obtained from empirical saturation data in Fig. 8 by making use of the double-tangent construction to the free energy, cf. Sects. 4.2 and 5

In Figs. 1, 6, 11, 16 and 21, the critical pressure isotherm of EoS (2.1), (2.2) and selected super- and subcritical isotherms discussed in the subsequent sections are compared with the universal cubic Peng- Robinson (PR) EoS, cf. e.g. Refs. [1,2,3,4,5,6,7,8]

P/R=nT1n/pα(T)an21+n/b+n2/c
(3.2)

where p=p^nc, b=b^nc, c=c^nc2, a=(Tc/nc)a^, and p^=3.9514, a^=1.4874, b^=1.9757, c^=15.6133 are universal constants. The scale factor α(T) (alpha function [30,31,32,33]) of the PR EoS (3.2) reads α(T)=(1+Ω(ω)(1T/Tc))2, with Ω(ω)=0.3919+1.4996ω0.2721ω2+0.1063ω3, cf. Ref. [34], where ω denotes the acentric factor of the fluid. The critical point parameters (nc,Tc,Pc) and ω are recorded in Table 1 for nitrogen, methane, carbon dioxide, methanol and helium.

Pressure Isotherms of the EoS Above the Critical Temperature

The rho and sigma functions ρ2,3,4(T) and σ(T) of the EoS (2.1), (2.2) can be treated as four independent fitting parameters at a fixed supercritical temperature T=Tj>Tc. The temperature-independent parameters (b0,ck=2,3,4,β0) of the EoS have already been determined from the critical isotherm, cf. Section 3.1, so that ρ2,3,4(Tj) and σ(Tj) remain the only fitting parameters. In Table 3, we list ρ2,3,4(Tj) and σ(Tj) at selected supercritical temperatures Tj>Tc, for nitrogen, methane, carbon dioxide, methanol and helium. The regressed supercritical isotherms of these fluids are depicted in Figs. 1, 6, 11, 16 and 21, together with their residual plots. The blue dashed curves in these figures show the corresponding isotherms (at the same supercritical temperatures Tj) of the cubic PR EoS.

Analytic Temperature Evolution of the Supercritical EoS

Above the critical temperature, T>Tc, the ρk and σ scale factors of the EoS (2.1), (2.2) can be specified as broken power-law density [35,36,37],

σ(T;b,α,β,η)=b(T/Tc)ηαβ((T/Tc)α1)β
(3.3)

and the same power law is used for the rho functions ρk(T;b,α,β,η) (with different fitting parameters b, α, β, η, of course). If there is a sign change of ρk(T) or σ(T), as it happens in the case of methane and carbon dioxide, cf. Table 3, we use a slightly more general broken power law by adding a third factor,

ρk(T;b,c,α,β,η)=b(T/Tc)η1αβ((T/Tc)α1)β(T/Tc1c)
(3.4)

The fitting parameters are the real amplitude b, the positive exponents α, β and the real exponent η. In (3.4), the positive amplitude c in the third factor is an additional fitting parameter determining the temperature at the sign change of ρk(T).

The regression is based on data sets (Tj,ρ2(Tj)), (Tj,ρ3(Tj)), (Tj,ρ4(Tj))(Tj,σ(Tj)) recorded in Table 3. The regressed parameters (b,α,β,η) defining the supercritical scale factors of nitrogen, methanol and helium and the parameters (b,c,α,β,η) of methane and carbon dioxide are listed in Tables 4, 5, 6, 7 and 8, and the least-squares fits of the broken power laws (3.3) and (3.4) are depicted in Figs. 2, 7, 12, 17 and 22 for these fluids.

Fig. 11
figure 11

Carbon dioxide pressure isotherms. Data points from Ref. [26]; see also Ref. [20] for the multiparameter reference EoS used to generate the data. The red solid curves show isotherms of EoS (2.1), (2.2) (with parameters b0,ck,β0 for carbon dioxide in Table 2). The scale factors ρk(T), σ(T) of the EoS of carbon dioxide in the supercritical regime T>Tc are defined in (3.3), (3.4) and Table 6. In the subcritical temperature interval TmeltT<Tc, the scale factors ρk(T), σ(T) are calculated from the empirical saturation curve (green solid curve), cf. Sects. 4.2, 4.3 and 5. The red solid curves show the supercritical isotherms at 1100, 600 and 400 K, as well as the critical isotherm passing through the critical point at 304.13 K and also the subcritical 250 K isotherm. The horizontal dashed line connects the vapor and liquid saturation points on the 250 K isotherm, cf. Section 4.2. The endpoints of the liquid–vapor saturation curve at the melting temperature of 216.6 K are depicted as black diamonds. Also shown for comparison are the isotherms of the cubic Peng–Robinson EoS (blue dashed curves), cf. (3.2) and Table 1. Residuals of selected isotherms are shown in the lower panels

Table 6 Fitting parameters of the supercritical rho and sigma scale factors of the EoS of carbon dioxide (CO2). The scale factors are broken power laws (ρ2(T),σ(T))=b(T/Tc)ηαβ((T/Tc)α1)β, cf. (3.3), and (ρ3(T),ρ4(T))=b(T/Tc)η1αβ((T/Tc)α1)β(T/Tc1c), cf. (3.4), with real amplitude b, positive amplitude c, positive exponents α, β and real exponent η recorded in this table for carbon dioxide. The least-squares regression is based on supercritical data sets (Tj,ρ2(Tj)), (Tj,ρ3(Tj)), (Tj,ρ4(Tj)) and (Tj,σ(Tj)) in Table 3. The data points and regressed analytic functions are depicted in Fig. 12. Also listed is the minimum χ2 of the least-squares functional (which is structured analogously to χ2 in (3.1))
Fig. 12
figure 12

Supercritical scale factors ρk=2,3,4(T),