Densities, Dielectric Permittivities, and Dew Points for (Argon +

Article pubs.acs.org/jced

Densities, Dielectric Permittivities, and Dew Points for (Argon + Carbon Dioxide) Mixtures Determined with a Microwave Re‑entrant Cavity Resonator Gergana Tsankova,† Paul L. Stanwix,‡ Eric F. May,‡ and Markus Richter*,† †

Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany Fluid Science and Resources Division, School of Mechanical and Chemical Engineering, The University of Western Australia, Crawley, Western Australia 6009, Australia

‡

ABSTRACT: A microwave re-entrant cavity resonator was used for the determination of dew points, dielectric permittivities, and molar densities of two gravimetrically prepared (argon + carbon dioxide) mixtures, with carbon dioxide mole fractions of 0.9495 and 0.7509. Isochoric dewpoint measurements of a (0.0505 argon + 0.9495 carbon dioxide) mixture were carried out over the temperature range from (257 to 291) K at pressures between (2.4 and 6.0) MPa. The measured dew-point pressures were consistent with the predictions of the recently developed multiparameter equation of state optimized for combustion gases (EOS-CG) within 0.35 %. Measurements of each mixture’s dielectric permittivity in the single-phase gas region over the temperature range from (273.2 to 313.3) K at pressures up to 6.5 MPa were used to determine the mixture molar densities at the same conditions. The method used to determine mixture molar densities from microwave-cavity measurements is based on an inversion of the polarizability mixing rule developed by Harvey and Prausnitz. The microwave-determined mixture densities had relative deviations from values measured for the same mixture with a two-sinker magnetic-suspension densimeter of 0.3 % or less. The new mixture density data agree within 0.37 % of predictions made using the EOS-CG and help identify literature data sets that should receive lower weighting in future model development. range of thermophysical property research. Goodwin et al.7,8 used a re-entrant cavity resonator for the isochoric determination of binary mixture phase boundaries, to measure dielectric polarizabilities of fluids and to determine the dipole moments of gases from dielectric permittivity measurements. May et al.9−12 developed the technique further to enable isothermal and isobaric dew-point measurements of lean hydrocarbon fluid mixtures and to determine liquid volume fractions at vapor−liquid equilibrium. Determinations of singlephase mixture densities from re-entrant cavity measurements have also been described;6,10,12 however, the methods applied were either limited by the use of Oster’s mixing rule for polarizabilities,13 or were only compared with predictions from cubic equations of state. In the present work, we have used a microwave re-entrant cavity resonator, recently described by Tsankova et al.,14 to determine dew points, dielectric permittivities, and molar densities of binary (argon + carbon dioxide) mixtures. The dew points of a gravimetrically prepared (0.0505 argon + 0.9495 carbon dioxide) mixture were measured along isochoric

1. INTRODUCTION Accurate knowledge of the thermodynamic properties of carbon dioxide-rich fluid mixtures is the basis for the design and operation of carbon capture and storage (CCS) systems, which are being investigated to mitigate global CO2 emissions. Key properties, such as density and the phase behavior of fluid mixtures containing carbon dioxide can be predicted with the multiparameter Helmholtz equation of state “EOS-CG” recently developed by Gernert and Span.1 This model, optimized for combustion gases, provides a better description of mixtures relevant to CCS applications than other models commonly used in the industry, such as the GERG-2008 equation of state of Kunz and Wagner2 and the cubic equation of state of Peng and Robinson.3 Nonetheless, the accuracy of all models depends largely on the availability and accuracy of the underlying experimental data. Usually, highly accurate measurements of fluid properties are carried out using experimental devices that are optimized for the measurement of one single property. In many cases such measurements are time and sample consuming, and the apparatus are complex to setup and operate. Marsh and co-workers4−6 helped pioneer the use of the microwave apparatus for rapid property measurements for various fluid mixtures including aggressive or difficult-to-handle systems. In particular, instruments based on microwave reentrant cavities have shown their applicability across a broad © 2017 American Chemical Society

Special Issue: Memorial Issue in Honor of Ken Marsh Received: December 16, 2016 Accepted: April 7, 2017 Published: April 21, 2017 2521

DOI: 10.1021/acs.jced.6b01043 J. Chem. Eng. Data 2017, 62, 2521−2532

Journal of Chemical & Engineering Data

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controller (Eurotherm, UK, model 2416) to obtain a better temperature stability. Three quartz oscillator pressure transducers (Paroscientific, USA, models 223A-101, 2100A-101, and 42K-CE, read-out with Paroscientific, USA, model 735 Intelligent Displays) were used for measuring the pressure. The pressure sensors cover the operational pressure range in three overlapping subranges with maximum pressures of (0.69, 2.7 and 13.8) MPa. The combined expanded uncertainty (k = 1.73) in pressure measurement was 10−4 of the maximum value, as specified by the manufacturer. The key observable in the present work is the resonant frequency of the re-entrant cavity, which has an inverse dependence on the square root of the dielectric permittivity7 and thus the density and composition of a fluid mixture. The fundamental resonant mode of the re-entrant cavity was continuously measured using a vector network analyzer (HP, USA, model 8752C) with a frequency range of (0.3 to 1300) MHz, acquiring the complex transmission coefficient S21. The resonance can be described by a complex frequency response Fα = fα + igα, where fα = Re[Fα] is the resonant frequency and gα = Im[Fα] is the half-width of the resonance, both of which can be determined by fitting the measured transmission coefficients S21 (typically acquired at 401 frequencies spanning an interval fα ± 5gα) to the complex Lorentzian function:

pathways over the temperature range from (257 to 291) K at pressures between (2.4 and 6.0) MPa following the method described and applied previously14 to a (0.2501 argon + 0.7499 carbon dioxide) mixture. To complete the accurate measurements of these mixtures’ properties in the single-phase region, the re-entrant cavity was calibrated for dielectric permittivity measurements with helium and subsequently validated with both argon and carbon dioxide. Dielectric permittivities of the two mixtures were measured at temperatures T = (283.24 and 313.29) K and in the vicinity of the dew-line, as well as at T = (273.20, 283.24 and 293.27) K at pressures up to 6.5 MPa, respectively. Mixture molar densities were then derived from these dielectric permittivity data using mixture polarizabilities estimated from (1) the classical mixing rule of Oster,13 and (2) the state-of-the-art mixing rule of Harvey and Prausnitz.15 The results were compared to highly accurate density data measured with a two-sinker magnetic-suspension densimeter,16 which showed the mixing rule of Harvey and Prausnitz yields density determinations as accurate as 0.3 % or better. Moreover, the microwave-determined mixture densities were compared with predictions made using the fundamental equations of state EOS-CG1 and GERG-20082 as well as with two different implementations of the cubic PR-EOS.3 This work demonstrates that microwave re-entrant cavity resonators are particularly useful for rapid and accurate measurements of high-pressure fluid mixture properties, including densities, over a wide temperature and pressure range. The results produced also establish that the EOS-CG is in general significantly more accurate for these mixtures than other commonly used engineering models, and they help identify literature data sets that should receive reduced weighting in the development of any future models.

S21(f ) =

Af f 2 − Fα2

+ B + C(f − f ) + D(f − f )2 * *

(1)

The complex constants A, B, C, and D, and the complex resonant frequency Fα were treated as fitting parameters, f is the source frequency, and f * is an arbitrary constant frequency near fα. Detailed information about the fitting procedure can be extracted from Mehl.21 The statistical standard uncertainty in the resonant frequency fα determined by the regression was up to 6 Hz. The binary mixtures, investigated in the current work, were prepared gravimetrically in house. Detailed information about the mixture preparation system, the preparation procedure, and the uncertainty analysis of the mixture preparation is described by Schäfer.22

2. APPARATUS DESCRIPTION For the measurements reported in this work, we utilized an apparatus based on the microwave re-entrant cavity resonator built by the National Engineering Laboratory (NEL, East Kilbride, UK),17 with a design adapted from the cavity of Goodwin et al.7 It was originally used for energy measurements, based in part on dielectric permittivity measurements of natural gas.18,19 The refurbished apparatus enables measurements over the temperature range from (243 to 353) K at pressures up to 12 MPa. A detailed description of our measurement system was recently reported by Tsankova et al.,14 with only a brief description included here. The temperature of the cavity was accurately measured using a thermometry chain consisting of a 25 Ω standard platinum resistance thermometer (SPRT, Tinsley, UK, model 5187L), a precision resistance bridge (Isotech, UK, model microK 500) and an external 25 Ω reference resistor (Tinsley, UK, model 5685A). We calibrated the temperature measurement chain along ITS-90 and estimated the combined expanded uncertainty (k = 1.73) in temperature to be 20 mK. (Our assumption, based on the “Guide to the Expression of Uncertainty in Measurement” according to ISO/IEC (2008),20 is that input quantities such as manufacturer specifications and estimates of measurement deviations are described by a symmetric, rectangular probability distribution. Therefore, we assumed a coverage factor for the expanded uncertainty of k = √3 ≈ 1.73, which corresponds to a 91.6 % confidence interval). The temperature of the measuring cell was controlled using a bath thermostat (Grant, UK, model LTC2006-40) specially modified with an extra temperature

3. DETERMINATION OF FLUID MIXTURE DENSITY FROM DIELECTRIC PERMITTIVITY MEASUREMENTS The relationship between molar polarizability P, molar density ρ, and the dielectric permittivity εr is given by the ClausiusMossotti relation, which is valid for fluids consisting of molecules with negligible permanent electric dipole moment: P=

1 ⎛ εr − 1 ⎞ ⎟ ⎜ ρ ⎝ εr + 2 ⎠

(2)

Two different methods were used to calculate the molar polarizability and thus the molar density of each mixture. In the first approach, the molar polarizability was calculated using Oster’s mixing rule,13 which is essentially the mole fraction average of the component polarizabilities at the same pressure and temperature as the mixture, together with the pure fluid polarizability correlations of Schmidt and Moldover:23 2522


Journal of Chemical & Engineering Data εr,i − 1 εr,i + 2

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where v*i is the critical volume of the pure component i. The mixing rule developed by Harvey and Prausnitz15 can then be written as

= Aε ,273ρi (1 + bρi + cρi2 )

⎛ ⎞ ⎛ 273.16 K ⎞ T + A τ ρi ⎜ − 1⎟ + qρi2 ⎜ − 1⎟ ⎝ ⎠ ⎝ 273.16 K ⎠ T

(3)

Pmix =

i

The molar densities ρi of the pure components were calculated with the equation of state of Span and Wagner24 for carbon dioxide and of Tegeler et al. for argon,25 as implemented in ref 26 and ref 27. The coefficients Aε,273, b, c, Aτ and q in eq 3 were taken from ref 23. The second method is based on the mixing rule developed by Harvey and Prausnitz15 combined with the Harvey and Lemmon 28 correlations as implemented in the NIST REFPROP database26 for the molar polarizabilities of the component pure fluids. The correlation of Harvey and Lemmon28 is based on the dielectric virial expansion, as a function of the temperature and the reduced molar density of the pure component i: Pi = Aε +

Aμ T

+ Bε ρr, i + Cρr,Di

∑ xiPi(p , T ) i

(4)

∑ xivi* i

(7)

⎛ ε − 1⎞ ⎛ ρ ∑ xivi* ⎛ ρmix ∑i xivi* ⎞⎞ ⎟⎟⎟⎟ = 0 ρmix − ⎜ r · Pi ⎜⎜T , ⎟/⎜⎜∑ Φ*i · mix i vi* vi* ⎝ εr + 2 ⎠ ⎝ i ⎝ ⎠⎠ (8)

To determine the mixture densities, eq 8 was solved iteratively for ρmix, using the ideal gas density as the initial guess.

4. CALIBRATION AND UNCERTAINTY ANALYSIS 4.1. Calibration of the Microwave Re-entrant Cavity. Before undertaking dielectric permittivity measurements of (argon + carbon dioxide) mixtures, the microwave re-entrant cavity was characterized to establish the response of its resonant frequency to the effects of thermal expansion and pressure distortion. Using a well-established calibration procedure,7,29 the cavity was evacuated and filled with pure helium. Helium was chosen because of its small and accurately known molar polarizability.30 All pure components used in this work (also to prepare the mixtures, described below) are listed in Table 1. Table 1. Provenance and Mole Fraction Purities of the Samples

(5)

where xi and Pi are the mole fraction and the molar polarizability of the pure component i at constant temperature and pressure, respectively. However, Oster’s mixing rule is limited by its use of pure-component molar densities at constant temperature and pressure. In many cases, the temperatures and pressures of interest do not yield the same thermodynamic state for the mixture and its component pure fluids. For this reason, in this work Oster’s rule was only used at temperatures and pressures where pure carbon dioxide is in the vapor phase. 3.2. The Mixing Rule of Harvey and Prausnitz. To avoid the limitation of Oster’s mixing rule, Harvey and Prausnitz15 proposed a corresponding states-type mixing rule based on reduced molar density and temperature. The dimensionless reduced molar mixture density ρr,mix is defined as ρr,mix = ρmix

⎝

ρr,mix ⎞ ⎟ vi* ⎠

with Φ*i = xiv*i /∑jxjv*j , and the reduced molar density of the pure component i is defined as ρr,i = ρr,mix/vi*. The ratio ρr,mix/ vi* gives an equivalent density for the component pure fluid at which the polarizability correlation (eq 4) should be evaluated to estimate its equivalent contribution to the mixture polarizability. This mixing rule is implemented together with the pure component polarizability correlations of Harvey and Lemmon in the software packages REFPROP 9.126 and TREND 3.027 for the calculation of dielectric permittivities of natural gas at a given pressure and temperature via the use of an equation of state for the mixture density. A practical means of inverting this correlation to determine mixture densities from dielectric permittivity measurements has not been previously described. Combining eqs 2, 6 and 7 gives a nonlinear equation for the mixture density:

Here, the symbol P refers to the molar polarizability, whereas in ref 28 it refers to the electric polarization, which is given by ρP in our notation. The first Aε and second Bε dielectric virial coefficients as well as the empirical parameter C are, in general, temperature-dependent parameters as defined by the equations given by Harvey and Lemmon.28 The exponent parameter D is temperature independent, while the parameter Aμ is used for fluids with electric dipole moments, and was zero for systems considered here. Detailed information concerning the fitting of the correlation, the sources of the pure-component dielectric permittivity data and densities are given in ref 28. We note that the pure Ar and CO2 polarizability correlations of Harvey and Lemmon were anchored to the results of Schmidt and Moldover, so the difference in the two methods considered here is essentially determined by the two mixing rules applied. 3.1. Oster’s Mixing Rule. The conventional way to calculate the mixture molar polarizability Pmix is to use the mixing rule of Oster:13 Pmix(p , T ) =

⎛

∑ Φ*i ·ρr,i ·Pi⎜T ,

chemical name

source

purity/mol fraction

purification method

helium argon carbon dioxide

Air Liquide Air Liquide Air Liquide

0.999999a 0.999990b 0.999995c

none none none

Impurities (stated by supplier): x(H2O) ≤ 0.5·10−6, x(O2) ≤ 0.1· 10−6, x(CmHn) ≤ 0.1·10−6, x(CO2) ≤ 0.1·10−6, x(CO) ≤ 0.1·10−6, x(H2) ≤ 0.1·10−6 bImpurities (stated by supplier): x(H2O) ≤ 2.0·10−6, x(O2) ≤ 2.0·10−6, x(CmHn) ≤ 0.5·10−6, x(CO2) ≤ 0.2·10−6, x(N2) ≤ 5.0·10−6 cImpurities (stated by supplier): x(H2O) ≤ 2.0·10−6, x(O2) ≤ 1.0·10−6, x(CmHn) ≤ 0.1·10−6, x(N2) ≤ 2.0·10−6, x(CO) ≤ 0.5·10−6, x(NOx) ≤ 0.1·10−6 a

The resonant frequencies and half-widths measured at each pressure and temperature were used for the determination of the complex dielectric permittivity εr of the fluid under study using the implicit model developed by Hamelin et al.:29

(6) 2523



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Table 2. Parameters for eq 9, Determined from the Calibration Procedure under Vacuum and with Helium, with Tref = 273.09 K f 00 α /MHz

g00 α /MHz

106 θ/K−1

106 φ/MPa−1

393.8046 ± 0.0002

0.1743 ± 0.0002

−16.89 ± 0.18

1.33 ± 0.19

Figure 1. Results for the molar polarizability P of pure argon (top panels) and carbon dioxide (bottom panels). For the calculation of P we used the pure fluid correlations of Harvey and Lemmon28 and corresponding equations of state.24,25 (a, b) Measured and calculated P versus pressure for argon and carbon dioxide, respectively; (c, d) relative deviations of the measured Pmeas from the calculated Pcalc for argon and carbon dioxide, respectively. Pmeas are indicated by the markers, and Pcalc are indicated by the lines. (a, c) ○, T = 263.14 K; ◇, T = 273.15 K; ▽, T = 302.44 K; ―, T = 263.14 K; − − −, T = 273.15 K; ···, T = 302.44 K. (b, d) ◇, T = 273.21 K; ▽, T = 302.45 K; ×, Tsat = 288.24 K; △, experimental data at saturation pressures between (3.97 and 6.07) MPa reported by May et al.;34 ―, T = 273.21 K; ···, T = 302.45 K; − − −, calculated with the correlation of Harvey and Lemmon28 at saturation pressures from (3.7 to 6.3) MPa.

The pressure distortion coefficient φ was determined by measurements of the resonance frequency fα on a series of isotherms at T = (263.14, 273.09 and 302.46) K as a function of pressure (up to 8 MPa) when the cavity was filled with helium. We determined the value of f 0α at each measured temperature and pressure using values for εr(p, T) calculated with the correlation of Harvey and Lemmon,28 and the density of helium was calculated with the equation by Lemmon32 (as implemented in ref 26). The coefficients f 00 α and φ were derived from a linear least-squares regression of the f 0α values using the previously measured value of θ and by choosing Tref = 273.09 K. The vacuum measurements also revealed that g00 α was independent of temperature, and therefore, its value was fixed. 00 Values of f 00 α gα , θ, and φ for the fundamental mode are summarized in Table 2. To confirm the validity and stability of the calibration coefficients, the re-entrant cavity was evacuated, and the resonant vacuum frequency at T = (283.24 and 293.26) K was measured following a pressure cycle over the measured pressure range. It was found that the frequency shifted by 1.3 kHz (relative change of 3.35·10−6) and 0.5 kHz (relative change of 1.28·10−6), respectively, which is consistent with or smaller than that observed with re-entrant cavities previously.33 The calibration was validated by comparing the measured molar polarizabilities Pmeas of pure argon and pure carbon dioxide with values Pcalc calculated using the correlations of Harvey and Lemmon28 (as implemented in ref 26). Measurements of vapor-phase dielectric permittivities were performed at T = (263.14, 273.15 and 302.44) K for pure argon and at T = (273.21 and 302.45) K and Tsat = 288.24 K (saturation

⎛ f 0 + ig 0 ⎞2 1 + ( −1 + i)(f /f 0 )3/2 ε (2g 0 /f 0 ) r α α α⎟ α α εr = ⎜⎜ α ⎟ 0 0 1 + ( −1 + i)(2gα /f α ) ⎝ fα + igα ⎠ (9)

where the superscript “0” denotes vacuum conditions. For the fluids and frequencies considered in this work, the imaginary part of the dielectric permittivity was not measurably different from zero, so below we treat the dielectric permittivity as a purely real quantity. The temperature and pressure dependence of the vacuum frequency is given by the following equation: f α0 = f α00 (1 + θ(T − Tref ) + φp)

(10)

Here f 00 α is the vacuum frequency at reference temperature Tref, while θ and φ are the thermal expansion and pressure distortion coefficients, respectively, of the cavity’s fundamental mode. The coefficient of thermal expansion θ was measured directly over the temperature range from (255 to 313) K when the cavity was evacuated. The temperature and the resonant frequency were recorded when the system had achieved equilibrium (measured values became constant within preset limits of 0.01 K and 0.01 MHz over 20 min). Linear leastsquares regression of the measured vacuum frequency data was used to compute the thermal expansion coefficient of θ = (−16.89 ± 0.18)·10−6 K−1. This value is in reasonable agreement with the technical specifications for copper− beryllium (the cavity’s material) given by the supplier.31 2524



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temperature at p = 5.10 MPa) for pure carbon dioxide. The relative combined expanded uncertainty (k = 2) in the dielectric permittivity measurements for argon and carbon dioxide in the vapor and in the liquid phase was estimated to be Uc(εr,Ar)/εr,Ar = 15·10−6, Uc(εr,CO2,vap)/εr,CO2,vap = 38·10−6, and Uc(εr,CO2,liq)/ εr,CO2,liq = 900·10−6, respectively. Details of the uncertainty analysis are given in section 4.2. When combined with the EOS uncertainty for these pure fluids, this corresponds to a relative uncertainty of 0.19 % for the polarizability of Ar, 0.09 % for the polarizability of CO2 in the vapor phase, and 0.17 % for the saturated liquid CO2 polarizability. The measured and calculated molar polarizabilities are shown in Figure 1 for argon (top panel, a) and carbon dioxide (bottom panel, b). At low pressures (e.g., up to 2 MPa) where the measurement’s signal-to-noise is reduced and the pressure transducer’s uncertainty is increased, the measured data for argon show deviations from the calculated values of between (−0.24 and 0.08) %, which approximately corresponds to the uncertainty of the measured data. At higher pressures, our experimental data are in excellent agreement with the predicted values. The data for carbon dioxide agree very well with the calculated values, with relative deviations of the measured molar polarizabilities from the calculated values between (−0.05 and 0.03) % at higher pressures, where the signal-to-noise ratio of the measurements is large. The agreement is within 0.12 % at low pressures, where the relative combined standard uncertainty in the measured molar polarizability was estimated to be 0.19 %. Furthermore, the reproducibility of the molar polarizability measurements of carbon dioxide improves with increasing pressure; for example, at T = 302.45 K and low pressures, the reproducibility of the measured polarizabilities was approximately 0.05 %. In contrast, the reproducibility at high pressures was conservatively estimated to be 0.005 %. Liquid-phase dielectric permittivities were validated with pure carbon dioxide at a saturation temperature Tsat = 288.24 K, as shown in Figure 1 (bottom panel, b) and are also compared with the data reported by May et al.34 for saturation pressures between (4.6 to 6.7) MPa. The relative deviations of the measured molar polarizabilities in the liquid phase of carbon dioxide from the calculated values range between −(0.20 and 0.29) %, which is slightly larger than the relative combined expanded uncertainty in the measured values. However, our results are in very good agreement with those presented by May et al.,34 which together exhibit a deviation systematically increasing with pressure from values calculated with the model of Harvey and Lemmon of up to −0.3 %. 4.2. Uncertainty Analysis of the Measured Dielectric Permittivities. The measurement uncertainty in the dielectric permittivity was determined by application of the “Guide to the Expression of Uncertainty in Measurement” according to ISO/ IEC (2008): see ref 20. The combined standard uncertainty in the experimental dielectric permittivity u c (ε r ) can be determined by

Here, u(εr) is the standard uncertainty in the dielectric permittivity measurement. The partial derivatives (∂εr/∂T)p and (∂εr /∂p) T represent the dependence of dielectric permittivity on temperature and pressure, and they were calculated from a sensitivity analysis with the experimental data. Standard uncertainties in temperature and pressure measurement are represented by u(T) and u(p). The uncertainty that arises from the reproducibility of the dielectric permittivity measurements, u(εr,repro), was estimated individually at each measured state point. When conducting measurements on mixtures, the uncertainty in the mixture composition needs to be taken into accountsee the last term in eq 11, for which u(Mmix) is the uncertainty in the molar mass of the gas mixture. The approach reported by Richter and McLinden35 cannot easily be transferred to this work because while the uncertainty of the mixture preparation can be determined, correcting the uncertainty for sorption effects36 in this re-entrant cavity system is not trivial. As the surface area and quality (gold plating) of the measuring cell of the present system is similar to that of the two-sinker densimeter used by Ben Souissi et al.16 and by Yang et al.37 for density measurements on exactly the same mixtures, we assume that the expanded uncertainties (k = 2) in the mixture molar masses are also 0.0032 g·mol −1 and 0.0054 g·mol−1, respectively. We note that the dominant contribution to permittivity measurements made with reentrant cavity resonators is usually associated with the stability of the cavity’s dimensions under the cycling of pressure and/or temperature. This contribution manifests itself in u(εr,repro), and for the measurements reported in this work, this term was approximately 14 % of the combined uncertainty for the pure substances and (14 to 31) % for the mixtures. Nevertheless, for the mixtures, the largest uncertainty contribution originated from the mixture composition, which was between (65 and 78) % of the combined uncertainty. Considering eq 10, the standard uncertainty u(εr) in the dielectric permittivity measurement arises from the individual standard uncertainties of the resonant frequency u(f 0α) and the half-width u(g0α) corresponding to the (deformed) “evacuated” cavity at equivalent (T, p) conditions, and of the resonant frequency u(fα) and the half-width u(gα) when the cavity was filled with a fluid. Thus, the standard uncertainty u(εr) in the dielectric permittivity measurement can be calculated as follows: 2 ⎧⎡⎛ ⎡⎛ ⎤2 ⎪ ∂εr ⎞ 0 ⎤ ∂εr ⎞ 0 ⎥ ⎢ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ u(εr) = ⎨ ⎜ 0 ⎟u(f α ) + ⎜ 0 ⎟u(gα ) ⎢⎣⎝ ∂g ⎠ ⎥⎦ ⎥⎦ ⎪⎢⎣⎝ ∂f α ⎠ α ⎩

⎤2 ⎫ ⎡⎛ ⎤ 2 ⎡⎛ ⎪ ∂εr ⎞ ∂εr ⎞ ⎢ ⎢ ⎥ ⎟⎟u(g )⎥ ⎬ ⎟⎟u(f ) + ⎜⎜ + ⎜⎜ α α ⎢⎣⎝ ∂gα ⎠ ⎥⎦ ⎪ ⎢⎣⎝ ∂fα ⎠ ⎥⎦ ⎭

1/2

(12)

The uncertainty analysis procedure for the vacuum resonant frequency u( f α0 ) determined from the helium calibration measurements was as follows:. a. measurement of the resonant frequency ( f He α ) when the cavity resonator was filled with helium at reference temperature Tref as the pressure was cycled from approximately (0.5 to 8.0) MPa b. calculation of the dielectric permittivity of helium εHe r,calc at the measured temperatures and pressure using ref 26 He c. calculation of the vacuum frequency (f 0,He α ) using f α He and εr,calc

⎡ ⎡⎛ ⎞ ⎤ 2 ⎡⎛ ⎞ ⎤2 ∂ ε ∂ ε ⎢ 2 r r uc(εr) = ⎢u(εr) + ⎢⎜ ⎟ u(T )⎥ + ⎢⎜ ⎟ u(p)⎥ ⎢⎣⎝ ∂T ⎠ p ⎥⎦ ⎢⎣⎝ ∂p ⎠ ⎥⎦ T ⎣ 1/2 2⎤ ⎤ ⎡ ε Δ ⎥ + u(εr,repro)2 + ⎢ r u(M mix )⎥ ⎥ ⎦ ⎣ ΔM ⎦

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dioxide, as well as details concerning the uncertainty analysis of the experimental dew points are described in Tsankova et al.14 The dew points in the present work were measured over the temperature range from (257 to 291) K at pressures from (2.4 to 6.0) MPa. The estimated combined expanded uncertainty (k = 2) in dew-point temperature and pressure ranged from (0.024 to 0.04) K and from (0.06 to 0.024) MPa, respectively (as indicated in Table 3). The phase boundary of the mixture

d. computation of the vacuum frequency at reference temperature f 00 α and the pressure distortion coefficient φ from the linear least-squares regression of the measured data e. calculation of the vacuum resonant frequency f 00 α using eq 9 According to the calibration procedure described above, the determination of the standard uncertainty in the vacuum resonant frequency u( f 0α) (see eq 12) should take into account the standard deviations of the regressions for the vacuum frequency at the reference temperature u( f 00 α ), the pressure distortion coefficient u(φ), and thermal expansion coefficient u(θ).

Table 3. Experimental Dew-Point Temperatures Tdew and Dew-Point Pressures pdew for the (0.0505 Argon + 0.9495 Carbon Dioxide) Mixturea. The Number Following the Symbol ± Is the Numerical Value of the Expanded Uncertainty (k = 2)

⎧ ⎤2 ⎡⎛ ∂f He ⎞ ⎪ He ⎥ u(f α0 ) = ⎨u(fαHe )2 + u(fαHe,fit )2 + ⎢⎜⎜ α ⎟⎟u(εr,calc ) + u(f α00 )2 ⎥⎦ ⎢⎣⎝ ∂εr ⎠ ⎪ ⎩ 1/2 ⎤2 ⎫ ⎤2 ⎡⎛ ∂f He ⎞ ⎡⎛ ∂f He ⎞ ⎪ α α ⎟u(θ)⎥ ⎬ ⎟u(φ)⎥ + ⎢⎜⎜ + ⎢⎜⎜ ⎥⎦ ⎪ ⎥⎦ ⎢⎣⎝ ∂θ ⎟⎠ ⎢⎣⎝ ∂φ ⎟⎠ ⎭ (13)

Tdew/K 291.134 288.008 284.395 280.935 277.242 277.145 272.696 268.485 263.577 257.544

Here u( f He α ) is the absolute measurement uncertainty in frequency originating from the vector network analyzer; according to the manufacturer’s specifications, the relative He uncertainty in the frequency measurement is u(f He α )/f α = 10· −6 He 10 , and u( f α,fit) is the standard deviation of the frequency fit He (see eq 1), and u(εr,calc ) is the standard uncertainty of the dielectric permittivity calculated with REFPROP,26 see section 4.1. For the determination of the standard uncertainty in the vacuum half-width u(g0α), an equation similar to eq 12 was used but the last three terms were negligible as the vacuum halfwidth is essentially independent of temperature and pressure. The standard uncertainty in the resonant frequency u( fα) and the half-width u(gα), when the cavity resonator was filled with the fluid under study depends only on the absolute uncertainty of the vector network analyzer indicated by the subscript “meas” and on the standard deviation of fitting the measured complex transmission coefficient indicated by the subscript “fit”. This can be expressed with the following equations: u(fα ) = [u(fα ,meas )2 + u(fα ,fit )2 ]1/2

± ± ± ± ± ± ± ± ± ±

0.038 0.039 0.029 0.038 0.034 0.027 0.040 0.037 0.026 0.024

pdew/MPa 6.0077 5.5472 5.0565 4.6228 4.1946 4.1837 3.7119 3.3049 2.8749 2.4060

± ± ± ± ± ± ± ± ± ±

0.0235 0.0202 0.0133 0.0158 0.0126 0.0103 0.0125 0.0102 0.0071 0.0057

a The expanded uncertainty (k = 2) in composition is 0.0004 mole fraction carbon dioxide.

calculated with the EOS-CG and the location of the measured dew points is shown in Figure 2 (panel a). The experimental dew-point temperatures and pressures with their corresponding uncertainty are listed in Table 3 and plotted in Figure 3 relative

(14)

and u(gα ) = [u(gα ,meas)2 + u(gα ,fit )2 ]1/2

(15)

5. MIXTURE RESULTS AND DISCUSSION 5.1. Dew-Point Results. Dew points of a (0.0505 argon + 0.9495 carbon dioxide) mixture were measured with respect to temperature decrements along an isochoric pathway. In general, the dew-point estimation is based on detecting the change in the derivative of the vapor phase permittivity with temperature that occurs when the dew line has been crossed. In a mixture, the slope discontinuity in dielectric permittivity is caused by the slope discontinuities in density and composition. Monitoring the resonance frequency as the temperature is changed allows these slope discontinuities to be detected upon the onset of the phase transition. Detailed information concerning the measurement procedure and validation of the re-entrant cavity resonator for phase envelope determinations with pure carbon

Figure 2. p,T-diagrams and measurement conditions of (a) (0.0505 argon +0.9495 carbon dioxide) mixture; (b) (0.2491 argon +0.7509 carbon dioxide) mixture: ―, phase envelope calculated with the EOS-CG by Gernert and Span;1 ○, dew points measured in the present work. Conditions at which dielectric permittivities and molar densities were measured: ◇, T = 273.20 K; ×, T = 283.24 K; △, T = 293.27 K; □, T = 313.29 K; +, in the vicinity of the phase boundary, see section 5.2. 2526



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temperature range from (252 to 294) K, at pressures from (2 to 7) MPa, and with vapor phase carbon dioxide mole fractions, yCO2, from (0.45 to 0.99), and have relative deviations from the EOS-CG ranging between (−1.2 and 7.8) %. The values reported by Coquelet et al.41 with yCO2 = (0.94 and 0.97) and Sarashina et al.43 with yCO2 = (0.90 and 0.94) are in very good agreement with our new experimental data. The data reported by Ahmad et al.40 with yCO2 = (0.95 and 0.98) show a large scatter, with relative deviations from (−0.3 to 7.8) %. Sarashina et al.43 and Tsankova et al.14 reported data with a composition farther away from our current composition (yCO2 = (0.45 to 0.85) and yCO2 = 0.75, respectively), but those data nevertheless show similarly small deviations from the EOS-CG. 5.2. Experimental Results for the Dielectric Permittivity of a (0.05 Argon + 0.95 Carbon Dioxide) and a (0.25 Argon + 0.75 Carbon Dioxide) Mixture. The measured dielectric permittivities of the two (argon + carbon dioxide) mixtures with carbon dioxide mole fractions of 0.9495 and 0.7509 are listed in Table 4. The measurements of the (0.0505 argon + 0.9495 carbon dioxide) mixture were performed along two isotherms at T = (283.24 and 313.29) K. At T = 283.24 K, the measurements were performed by varying the pressure from (1.0 to 4.5) MPa, while the measurements at T = 313.29 K were interspersed with the dewpoint measurements for this mixture: the apparatus was refilled for each isochoric measurement at T = 313.29 K, with the pressure selected according to the target dew point. This starting temperature was chosen to ensure the mixture did not enter its two-phase region when the sample was transferred into the evacuated apparatus at constant enthalpy. The dielectric permittivities were measured at these “filling points” before the temperature of the system was decreased to approximately 2.5 K above the estimated dew-point temperature (which represented the starting point for the dew-point measurements14). The relative combined expanded uncertainty (k = 2) in the dielectric permittivity measurements at T = (283.24 and 313.29) K and at temperatures in the vicinity of the dew line was estimated to be 0.028 %. Dielectric permittivities of the (0.2491 argon + 0.7509 carbon dioxide) mixture were measured along three isotherms T = (273.20, 283.24 and 293.27) K as the pressure was cycled from (0.5 to 6.5) MPa, with an estimated relative combined expanded uncertainty (k = 2) in the measured dielectric permittivities of 0.018 %. In Figure 4, the measured dielectric permittivities have been combined with molar densities calculated with the EOS-CG by Gernert and Span1 (as implemented in ref 27) according to eq 2 to calculate the molar polarizabilities of the mixtures. We estimated a relative expanded uncertainty (k = 2) in the experimental molar polarizabilities of approximately 0.7 %, increasing to 1.1 % for data in the vicinity of the dew line. This increase near the dew point is in part due to the increased uncertainty of the densities calculated with the EOS-CG. The relative deviations of the measured molar polarizability Pmix,meas of the (0.0505 argon + 0.9495 carbon dioxide) mixture and the (0.2491 argon + 0.7509 carbon dioxide) mixture from calculated values Pmix,calc,27 as shown in Figure 4 (c and d), range from (−0.19 to 0.27) % and from (−0.37 and 0.24) %, respectively. In general, the measured molar polarizabilities are in good agreement with the theoretical prediction, with the relative deviations increasing with increasing pressure and slightly increasing with decreasing temperature. For the data

Figure 3. Relative deviations of experimental dew-point pressures pexp for the (0.0505 argon + 0.9495 carbon dioxide) mixture from dewpoint pressures pEOS‑CG calculated with the EOS-CG by Gernert and Span1 (zero line); (a) relative deviations of the data measured in this work plotted versus temperature T; (b) relative deviations of these data and those in the literature plotted versus vapor mole fraction carbon dioxide yCO2; ○, dew-point pressures measured in this work; − − −, dew-point pressures calculated with the GERG-2008 equation of state of Kunz and Wagner;2 −·− , dew-point pressures calculated with the PR-EOS3 as implemented in ref 38; ···, dew-point pressures calculated with an advanced implementation of PR-EOS as implemented in ref 39: △, Tsankova et al.;14 □, Sarashina et al.;43 ◇, Ahmad et al.;40 ×, Coquelet et al.;41 +, Kaminishi et al.42 The uncertainty of the EOS-CG (zero line) in dew-point pressure is about 1.0 % as reported in ref 1.

to values calculated with the EOS-CG by Gernert and Span1 (as implemented in ref 27). The relative deviations range from −(0.27 to 0.35) %, which is clearly within the uncertainty of about 1.0 % reported for this equation. Figure 3 also shows the relative deviations in the dew-point pressures calculated with the GERG-2008 equation of state of Kunz and Wagner2 (as implemented in ref 26) from those calculated with the EOSCG; these range between (0.07 and 0.35) %. The GERG-2008 does not employ a binary specific departure function for the (argon + carbon dioxide) system, whereas the EOS-CG does. Moreover, in Figure 3, the relative deviations of dew-point pressures calculated with two different implementations of the cubic Peng−Robinson equation of state (PR-EOS)3 from EOSCG are shown. The relative deviations of the values calculated with the PR-EOS (as implemented in ref 38) and with the advanced PR-EOS (as implemented in ref 39) from values calculated with the EOS-CG are as large as 2.3 % and −0.6 %, respectively. In general, the relative deviations of the dew-point pressures calculated with GERG-2008 equation and with the advanced PR-EOS from values computed with the EOS-CG are consistent with the dew-point pressures measured in the present work. The relative deviations of the dew-point pressures determined in this work and of other experimental VLE-data for the (argon + carbon dioxide) mixture found in the literature14,40−43 are compared with the EOS-CG in Figure 3 (b, bottom panel). The values plotted on the figure are in the 2527



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Table 4. Experimental Data for the Dielectric Permittivity εr,mixa, and Mixture Density ρHP mix, Determined with the Mixing Rule of Harvey and Prausnitz for Both (Argon + Carbon Dioxide) Mixturesb T/K 313.300 313.303 313.299 313.296 313.295 313.299 313.290 313.262 313.265 313.262 313.277 283.244 283.240 283.234 283.234 283.240 283.235 283.235 283.229 292.669 289.873 283.443 279.734 271.396 266.153 260.141 255.103 294.268 279.728 273.211 273.200 273.199 273.203 273.204 273.200

p/MPa

εr,mix

−3 ρHP mix/ mol·dm

(0.0505 Argon + 0.9495 Carbon Dioxide) 7.0271 1.09523 6.5500 1.08408 6.0284 1.07333 5.5664 1.06488 5.0830 1.05688 4.5507 1.04884 4.0896 1.04247 3.5921 1.03607 3.0520 1.02963 2.6026 1.02461 7.0836 1.09670 1.0299 1.01008 2.0061 1.02120 4.0077 1.05273 4.4939 1.06407 3.9737 1.05203 2.9982 1.03482 2.0115 1.02127 0.9694 1.00945 6.0863 1.10382 5.6320 1.09119 4.7108 1.06985 4.2661 1.06129 3.3656 1.04566 2.9195 1.03887 2.4469 1.03197 2.0689 1.02661 6.4192 1.11591 4.2678 1.06132 (0.2491 Argon + 0.7509 Carbon Dioxide) 0.5066 1.0045 1.0016 1.0091 2.0043 1.0195 3.0087 1.0314 4.0290 1.0459 4.4870 1.0535

T/K

4.1832 3.7140 3.2569 2.8945 2.5486 2.1983 1.9183 1.6356 1.3484 1.1236 4.2444 0.4640 0.9684 2.3612 2.8499 2.3308 1.5765 0.9715 0.4351 4.5286 4.0000 3.0969 2.7293 2.0509 1.7525 1.4473 1.2090 5.0256 2.7307

273.202 273.201 273.201 283.245 283.250 283.240 283.242 283.240 283.242 283.235 283.233 283.235 283.239 283.239 283.232 293.268 293.272 293.273 293.276 293.271 293.277 293.272 293.271 293.273

p/MPa

εr,mix

−3 ρHP mix/ mol·dm

(0.2491 Argon + 0.7509 Carbon Dioxide) 0.4548 1.0040 0.9627 1.0088 2.0148 1.0196 1.0089 1.0088 2.0122 1.0186 2.9933 1.0294 2.9928 1.0294 3.9956 1.0420 4.9918 1.0568 5.9835 1.0748 6.4913 1.0858 5.9870 1.0749 4.0064 1.0421 2.0020 1.0185 0.4928 1.0042 1.0071 1.0085 2.0057 1.0177 3.0062 1.0279 3.9996 1.0393 4.9934 1.0523 5.9946 1.0674 6.4992 1.0761 5.8476 1.0651 4.0066 1.0394

0.2043 0.4439 0.9841 0.4471 0.9349 1.4660 1.4657 2.0792 2.7874 3.6357 4.1487 3.6383 2.0856 0.9293 0.2134 0.4297 0.8900 1.3960 1.9514 2.5758 3.2953 3.7021 3.1830 1.9553

a The relative expanded combined uncertainty (k = 2) in the dielectric permittivity measurements for the (0.0505 argon + 0.9495 carbon dioxide) mixture was 2.8·10−4; for the (0.2491 argon + 0.7509 carbon dioxide) mixture it was 1.8·10 −4 . The combined expanded uncertainties (k = 1.73) in temperature and pressure measurement were 20 mK and 10−4·pmax, respectively. The value of pmax corresponds to the maximum pressure of the utilized pressure transducer (pmax = 0.69, 3.45, and 13.8 MPa), which depends on the investigated pressure range. bThe expanded uncertainty (k = 2) in the composition (from gravimetrical preparation) for the (0.0505 argon + 0.9495 carbon dioxide) and (0.2491 argon + 0.7509 carbon dioxide) mixture is 0.0004 mole fraction carbon dioxide.

0.2282 0.4624 0.9779 1.5660 2.2666 2.6295

the gravimetrically determined mixture composition; the composition uncertainty in this work corresponds to a polarizability uncertainty for these mixtures of 0.06 % at a maximum. 5.3. Experimental Results for the Density of the (0.05 Argon + 0.95 Carbon Dioxide) and a (0.25 Argon + 0.75 Carbon Dioxide) Mixtures. In Figure 5 we plot the relative deviations of the experimental mixture densities determined using the two methods described in section 3 from values calculated with the EOS-CG1 for the (0.0505 argon + 0.9495 carbon dioxide) mixture (panel a) and the (0.2491 argon + 0.7509 carbon dioxide) mixture (panel b), respectively. The blue symbols represent the relative deviations for the mixture densities determined with the mixing rule of Oster, and the green symbols represent those using the mixing rule of Harvey and Prausnitz. The relative deviations in the molar densities observed with the mixing rule of Oster are systematically low, ranging between −(0.82 and 0.04) %. In contrast, the mixing rule of Harvey and Prausnitz provided a better prediction of the molar densities, with relative deviations of the determined values (except for the points observed in the vicinity of the

measured in the vicinity of the phase boundary of the (0.0505 argon + 0.9495 carbon dioxide) mixture, however, relative deviations of up to 0.92 % were observed, increasing with pressure and temperature. Possible reasons for such large deviations include (a) sorption effects near to the phase boundary (e.g., see refs 35 and 36) or (b) inconsistencies in the mixture densities computed with the EOS-CG.44 Estimates of the zero density molar polarizabilities for the two mixtures were computed by extrapolation of the measured polarizabilities and are plotted in Figure 4 (a and b). In the ideal gas limit, the mixture molar polarizabilities should have a value equal to the mole fraction average of the component polarizabilities, and thus calculation of this quantity provides a check on the mixture composition, at least within the uncertainty of the polarizability measurement which increases as density decreases. As can be seen from Figure 4 (c and d), as the measured mixture polarizabilities approach the zero density condition, their relative deviations from the predicted correlations are consistent within ±0.3 %. This establishes that for these measurements the uncertainty of the molar polarizability determination was significantly larger than that of 2528



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Figure 4. Results for the molar polarizability Pmix of both investigated mixtures. For the calculation of Pmix we used Harvey and Prausnitz’s mixing rule,15 combined with the pure fluid correlations of Harvey and Lemmon28 and the EOS-CG of Gernert and Span.1 (a,b) measured and calculated Pmix versus pressure for the (0.0505 argon + 0.9495 carbon dioxide) and (0.2491 argon + 0.7509 carbon dioxide) mixtures, respectively; (c,d) relative deviations of the measured Pmix,meas from the calculated Pmix,calc for the (0.0505 argon + 0.9495 carbon dioxide) and (0.2491 argon + 0.7509 carbon dioxide) mixtures, respectively. Pmix,meas are indicated by the markers, and Pmix,calc are indicated by the lines: ◇, T = 273.20 K; ×, T = 283.24 K; △, T = 293.27 K; □, T = 313.29 K; +, temperatures in the vicinity of the phase boundary, see Figure 2; ○, zero density polarizability; − − − ,T = 283.24 K; (a): ―, T = 313.29 K; ···, temperatures in the vicinity of the phase boundary; (b) ―, T = 273.20 K; ···, T = 293.27 K.

phase boundary) from the EOS-CG1 in the range (−0.19 to 0.26) % for the (0.0505 argon + 0.9495 carbon dioxide) mixture and (−0.37 to 0.11) % for the (0.2491 argon + 0.7509 carbon dioxide) mixture; for both binaries, the deviations obtained with the Harvey and Prausnitz mixing rule are clearly within the uncertainty of 1.0 % reported by the authors of the EOS-CG. The relative deviations of molar densities determined in the vicinity of the dew line for the (0.0505 argon + 0.9495 carbon dioxide) mixture from the values calculated with EOSCG1 are between (0.16 and 0.91) %, which is still within the EOS-CG uncertainty for the calculation of vapor phase densities (1.0 %). The mixture densities ρHP mix determined in this work using the model of Harvey and Prausnitz are listed in Table 4. To investigate the likely uncertainty of mixture densities derived from dielectric permittivities using the polarizability mixing rule of Harvey and Prausnitz, the values derived in this work for the (0.2491 argon + 0.7509 carbon dioxide) mixture were compared with the highly accurate two-sinker magneticsuspension densimeter measurements of exactly the same mixture conducted by Ben Souissi et al.16 Figure 5 also shows a comparison of the densities determined in this work at T = (273.20, 283.24, and 293.27) K and the density data reported by Ben Souissi et al.16 at T = (273.15, 283.15, and 293.15) K, which have a relative combined expanded uncertainty (k = 2) of at most 0.03 %. For clarity, the uncertainties of the data of Ben Souissi et al.16 are only shown for the T = 293.15 K isotherm (the uncertainties for the other isotherms are similar). Generally, the deviations of the mixture densities determined in this work from the values reported by Ben Souissi et al.16 increase with pressure, with the largest relative deviation being approximately 0.3 % at 6.5 MPa. On the basis of this comparison, we estimated a relative combined expanded uncertainty (k = 1.73) in our mixture density of

uc(ρmix(εmix,T))/ρmix = 0.3 %. We note that the density measurements utilizing the two-sinker densimeter (as, e.g., reported by Ben Souissi et al.16 and Yang et al.37) required at least 1 week per isotherm and several liters of sample (due to thorough flushing of the measuring cell). In contrast, the microwave measurements required only 1 day per isotherm and less than a liter of sample. Figure 6 shows a comparison of our mixture densities with other experimental (p, ρ, T, x) data for the (0.0505 argon +0.9495 carbon dioxide) mixture and the (0.2491 argon + 0.7509 carbon dioxide) mixture with comparable data found in the literature.16,37,43,45,46 The literature data considered are in a temperature range within ±10 K of the present work, at pressures up to 8 MPa and have a composition range within ±0.05 CO2 mole fraction of the current measurements. The experimental mixture densities reported by Yang et al.37 (twosinker magnetic suspension densimeter) for a (0.0505 argon + 0.9495 carbon dioxide) mixture at T = (273.15, 283.15, 293.15, 308.15 and 323.15) K and pressures up to 9.0 MPa agree with the EOS-CG within 0.18 %. The experimental data for a (0.2491 argon + 0.7509 carbon dioxide) mixture plotted in Figure 6 (panel b), reported by Ben Souissi et al.16 (two-sinker magnetic suspension densimeter) at T = (273.15, 283.15, and 293.15) K and pressures up to 9.0 MPa as well as by Wegge46 (single-sinker magnetic suspension densimeter) T = (273.15 and 283.15) K with pressures up to 20.0 MPa, are in excellent agreement with the EOS-CG1. In contrast, the data reported by Sarashina et al.43 (glass capillary cell) for a (0.248 argon + 0.752 carbon dioxide) mixture show large deviations from the equation of state of up to 1.26 %. The agreement of the data reported by Mantovani et al.45 (vibrating tube densimeter) for a (0.0308 argon + 0.9692 carbon dioxide) mixture at T = (303.22 and 323.18) K and pressures up to 20.0 MPa with the EOS2529



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Figure 6. Relative deviations of experimental mixture densities ρ from the densities ρEOS‑CG calculated with EOS-CG1 (zero line). (a) Results for the (0.0505 argon + 0.9495 carbon dioxide) mixture. Density values determined in this work: ×, T = 283.24 K; □, T = 313.29 K. Density values reported by other authors plotted for comparison. Yang et al.:37 ◊, T = 273.15 K; ▽, T = 283.15 K; △, T = 293.15 K; ○, T = 308.15 K; +, T = 323.15 K. Mantovani et al.45 for (0.0308 argon + 0.9692 carbon dioxide) mixture: ―, T = 303.22 K; |, T = 323.18 K. (b) Results for the (0.2491 argon + 0.7509 carbon dioxide) mixture. Density values determined in this work: +, T = 273.20 K; ○, T = 283.24 K; □, T = 293.27 K. Density values reported by other authors plotted for comparison. Ben Souissi et al.:16 ◇, T = 273.15 K; ▽, T = 283.15 K; △, T = 293.15 K; Wegge:46 ―, T = 273.15 K; |, T = 283.15 K. Sarashina et al.43 for (0.248 argon + 0.752 carbon dioxide) mixture: ×, T = 288.15 K. The uncertainty of the EOS-CG (zero line) in density is 1.0 % as reported in ref 1.

Figure 5. Relative deviations of the determined mixture densities ρ from the densities ρEOS‑CG calculated with EOS-CG1 (zero line); the densities observed with mixing rule of Oster13 are indicated by the blue symbols, the densities observed with mixing rule of Harvey and Prausnitz15 are indicated by the green symbols; (a) results for the (0.0505 argon + 0.9495 carbon dioxide) mixture; (b) results for the (0.2491 argon + 0.7509 carbon dioxide) mixture: ◇, T = 273.20 K; ×, T = 283.24 K; △, T = 293.27 K; □, T = 313.29 K; +, temperatures in the vicinity of the phase boundary, see Table 4. Experimental densities reported by Ben Souissi et al.16 are indicated by the gray symbols: ◇, T = 273.15 K; ∗, T = 283.15 K; △, T = 293.15 K; for clarity, error bars for the relative combined expanded uncertainty (k = 2) in the mixture density are only illustrated for the measurements at T = 293.15 K. The uncertainty of the EOS-CG (zero line) in density is 1.0 % as reported in ref 1.

CG1 is also rather poor, with relative deviations between (−0.95 and 0.05) % that exhibit no systematic trend.

pressures between −(0.27 and 0.35) % from the EOS-CG were observed, which is within the uncertainty of about 1.0 % given by the authors of the EOS. Furthermore, we compared our results with those calculated with GERG-2008 equations of state of Kunz and Wagner,2,26 as well as with two different implementations of the cubic PR-EOS.3,38,39 Our results are in good agreement with the GERG-20082,26 equation and advanced PR-EOS.3,39 However, relative deviations up to 2 % from PR-EOS3,38 were observed. In general, the dew-points determined in this work are consistent with the literature for the investigated (argon + carbon dioxide) system; however, they have significantly less scatter. Dielectric permittivities of the gravimetrically prepared binary (argon + carbon dioxide) mixtures with carbon dioxide mole fractions of 0.9495 and 0.7509 were measured at temperatures T = (283.24 and 313.29) K and in the vicinity of the dew-point curve, as well as at T = (273.20, 283.24, and 293.27) K with pressures up to 6.5 MPa, respectively. The relative combined expanded uncertainty (k = 2) in the dielectric permittivity measurements was estimated to be 0.028 % for the mixture with a carbon dioxide mole fraction of 0.9495 and 0.018 % for the mixture with a carbon dioxide mole fraction of 0.7509. Importantly, we demonstrated how the measured dielectric permittivities could be used to determine mixture molar densities using the polarizability mixing rule of Harvey and Prausnitz.15 By comparison of our data with those measured by

6. CONCLUSIONS A microwave re-entrant cavity resonator apparatus, recently described by Tsankova et al.,14 has been used for dew-point, dielectric permittivity, and density determinations of two (argon + carbon dioxide) mixtures. The calibration of the reentrant cavity for dielectric permittivity determination included measurements in vacuum and with helium. Validation measurements were carried out with pure argon and both gaseous and liquid carbon dioxide. Except at the lowest pressures where the experimental uncertainty increases, the pure fluid dielectric permittivity measurements when combined with densities from the reference EOS for each pure fluid24,25 produced molar polarizabilities that deviate from the pure fluid correlations of Harvey and Lemmon28 by less than 0.1 %. For liquid CO2, the relative deviations from the reference correlation increased to −0.29 % but are in an excellent agreement (0.03 %) with those measured by May et al.34 using a very different technique (cross capacitor) with entirely different systematic errors. Dew points of the binary (0.0505 argon + 0.9495 carbon dioxide) mixture over the temperature range from (257 to 291) K at pressures from (2.4 to 6.0) MPa were measured with estimated relative combined expanded uncertainties (k = 2) of between (0.024 and 0.040) K and (0.060 and 0.024) MPa, respectively. Relative deviations of the measured dew-point 2530



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Ben Souissi et al.16 with a two-sinker densimeter, we estimated a relative combined expanded uncertainty (k = 1.73) in our mixture density of 0.3 %. The results were compared with the equation of state EOS-CG developed by Gernert and Span, with relative deviations between (−0.19 to 0.26) % for the (0.0505 argon + 0.9495 carbon dioxide) mixture and (−0.37 to 0.11) % for the (0.2491 argon +0.7509 carbon dioxide), which is clearly within the uncertainty of 1.0 % reported by the authors of the equation of state. The densities determined in the vicinity of the dew line agreed within 0.91 %, which is still within the EOS-CG uncertainty for the calculation of vapor phase densities (1.0 %). Our results show that for mixtures of industrial interest, dielectric permittivity measurements can deliver densities with sufficiently small uncertainties so that they can be used to improve reference equations of state for mixtures. However, such an approach requires accurate purecomponent polarizabilities and equations of state over wide temperature and pressure ranges.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +49-234−32-26395. ORCID

Eric F. May: 0000-0001-5472-6921 Markus Richter: 0000-0001-8120-5646 Funding

Gergana Tsankova’s work at Ruhr-University Bochum was supported by the “Forschungsschule für Energieeffiziente Produktion und Logistik” funded by the Ministry of Education, Science and Research of the German State of North-Rhine Westphalia. Markus Richter thanks the institute of advanced studies of The University of Western Australia for supporting his research fellowship. Paul Stanwix is supported by an ARC DECRA fellowship (DE140101094). The authors acknowledge the German Academic Exchange Service (DAAD) for funding the international collaboration between the two groups at RuhrUniversity Bochum and The University of Western Australia. Moreover, this work was supported by the Ruhr-University Research School PLUS, funded by Gemany’s Excellence Initiative [DFG GSC 98/3]. Notes

The authors declare no competing financial interest.

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