## 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. 3–5, 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

with repulsive potential

A derivation thereof is given in Refs. [9, 27]. $R=8.31446\text{J/(K mol)}$ 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 ${b}_{0}$ and exponent ${\beta}_{0}$ are positive, and the polynomial coefficients ${c}_{k=2,3,4}$ are real. The temperature-dependent scale factors ${\rho}_{k=2,3,4}(T)$ in (2.2) and $\sigma (T)$ in (2.1) are normalized to zero at the critical temperature ${\rho}_{k}({T}_{\text{c}})=\sigma ({T}_{\text{c}})=0$. The critical-point constants are denoted by $({n}_{\text{c}},{T}_{\text{c}},{P}_{\text{c}})$ and listed in Table 1 for the fluids studied here, that is nitrogen, methane, carbon dioxide, methanol and helium.

We will use the shortcuts ${Q}_{0}$, ${Q}_{,n}$, ${Q}_{,n,n}$ for the constants ${Q}_{0}:=Q({n}_{\text{c}},{T}_{\text{c}})-{P}_{\text{c}}/R$ and ${Q}_{,n}({n}_{\text{c}},{T}_{\text{c}})$, ${Q}_{,n,n}({n}_{\text{c}},{T}_{\text{c}})$; 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 ${\rho}_{k}(T)$, $\sigma (T)$, 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

derived from the critical-point conditions, $P({n}_{\text{c}},{T}_{\text{c}})={P}_{\text{c}}$, ${P}_{,n}({n}_{\text{c}},{T}_{\text{c}})=0,$${P}_{,n,n}({n}_{\text{c}},{T}_{\text{c}})=0$, which are exactly satisfied by the EoS, cf. Ref. [9]. The attractive term in (2.1) scales $\propto {n}^{2}$ at low density and the repulsive term (2.2) is linear in the low-density regime. The positive real exponent ${\beta}_{0}$ in (2.2) is the pressure scaling exponent at the singularity $n={b}_{0}$ (high-density limit). The analytic scale factors ${\rho}_{k=2,3,4}(T)$ and $\sigma (T)$ 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 ${\rho}_{k}$ and $\sigma $ scale factors of the EoS (2.1), (2.2) vanish at the critical temperature, $\sigma ({T}_{\text{c}})={\rho}_{k=2,3,4}({T}_{\text{c}})=0$. The critical isotherms of nitrogen, methane, carbon dioxide, methanol and helium are obtained by four-parameter $({b}_{0},{c}_{k=2,3,4})$ regression, since we put ${\beta}_{0}=1/2$ in (2.2) from the outset because of lack of high-pressure data points. The temperature-independent fitting parameters ${b}_{0},\phantom{\rule{thinmathspace}{0ex}}{c}_{k}$ are recorded in Table 2. The least-squares functional to be minimized reads

where $({n}_{i},{P}_{i}{)}_{i=1,...,N}$ are data points for the critical isotherm, and $P(n;{b}_{0},{c}_{k})$ is the EoS (2.1), (2.2) with ${\beta}_{0}=1/2$ and the ${\rho}_{k}$ and $\sigma $ 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 ${\chi}^{2}$ 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.

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]

where $p=\hat{p}{n}_{\text{c}}$, $b=\hat{b}{n}_{\text{c}}$, $c=\hat{c}{n}_{\text{c}}^{2}$, $a=({T}_{\text{c}}/{n}_{\text{c}})\hat{a}$, and $\hat{p}=3.9514$, $\hat{a}=1.4874$, $\hat{b}=1.9757$, $\hat{c}=-15.6133$ are universal constants. The scale factor $\alpha (T)$ (alpha function [30,31,32,33]) of the PR EoS (3.2) reads $\alpha (T)=(1+\mathrm{\Omega}(\omega )(1-\sqrt{T/{T}_{c}}){)}^{2}$, with $\mathrm{\Omega}(\omega )=0.3919+1.4996\omega -0.2721{\omega}^{2}+0.1063{\omega}^{3}$, cf. Ref. [34], where $\omega $ denotes the acentric factor of the fluid. The critical point parameters (${n}_{\text{c}}$,${T}_{\text{c}}$,${P}_{\text{c}}$) and $\omega $ 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 ${\rho}_{2,3,4}(T)$ and $\sigma (T)$ of the EoS (2.1), (2.2) can be treated as four independent fitting parameters at a fixed supercritical temperature $T={T}_{j}>{T}_{\text{c}}$. The temperature-independent parameters $({b}_{0},{c}_{k=2,3,4},{\beta}_{0})$ of the EoS have already been determined from the critical isotherm, cf. Section 3.1, so that ${\rho}_{2,3,4}({T}_{j})$ and $\sigma ({T}_{j})$ remain the only fitting parameters. In Table 3, we list ${\rho}_{2,3,4}({T}_{j})$ and $\sigma ({T}_{j})$ at selected supercritical temperatures ${T}_{j}>{T}_{\text{c}}$, 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 ${T}_{j}$) of the cubic PR EoS.

### Analytic Temperature Evolution of the Supercritical EoS

Above the critical temperature, $T>{T}_{\text{c}}$, the ${\rho}_{k}$ and $\sigma $ scale factors of the EoS (2.1), (2.2) can be specified as broken power-law density [35,36,37],

and the same power law is used for the rho functions ${\rho}_{k}(T;b,\alpha ,\beta ,\eta )$ (with different fitting parameters $b$, $\alpha $, $\beta $, $\eta $, of course). If there is a sign change of ${\rho}_{k}(T)$ or $\sigma (T)$, 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,

The fitting parameters are the real amplitude $b$, the positive exponents $\alpha $, $\beta $ and the real exponent $\eta $. In (3.4), the positive amplitude $c$ in the third factor is an additional fitting parameter determining the temperature at the sign change of ${\rho}_{k}(T)$.

The regression is based on data sets $({T}_{j},{\rho}_{2}({T}_{j}))$, $({T}_{j},{\rho}_{3}({T}_{j}))$, $({T}_{j},{\rho}_{4}({T}_{j}))$, $({T}_{j},\sigma ({T}_{j}))$ recorded in Table 3. The regressed parameters $(b,\alpha ,\beta ,\eta )$ defining the supercritical scale factors of nitrogen, methanol and helium and the parameters $(b,c,\alpha ,\beta ,\eta )$ 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.