Cross Second Virial Coefficients and Dilute Gas Transport Properties

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Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Cross Second Virial Coefficients and Dilute Gas Transport Properties of the (CH4 + C3H8) and (CO2 + C3H8) Systems from Accurate Intermolecular Potential Energy Surfaces Robert Hellmann* Institut für Chemie, Universität Rostock, 18059 Rostock, Germany S Supporting Information *

ABSTRACT: Accurate values for the cross second virial coefficients and dilute gas shear viscosities, thermal conductivities, and binary diffusion coefficients of the (CH4 + C3H8) and (CO2 + C3H8) systems were determined at temperatures from (150 to 1200) K using state-of-the-art computational approaches. The cross second virial coefficients were calculated semiclassically using the Mayer-sampling Monte Carlo method, while the transport properties were computed by means of the classical trajectory approach in conjunction with the kinetic theory of molecular gases. The required intermolecular potential energy surfaces (PESs) for the CH4−C3H8 and CO2−C3H8 interactions are reported in this work, whereas those for the CH4−CH4, CO2−CO2, and C3H8−C3H8 interactions were taken from our previous work on the pure gases. All PESs are based on high-level quantum-chemical ab initio calculations and were fine-tuned to the best experimental data for the second virial and cross second virial coefficients. Overall, the agreement of the calculated thermophysical property values with the few available experimental data is satisfactory.

1. INTRODUCTION The calculation of the thermophysical properties of a fluid requires knowledge of the potential energy surface (PES) that describes the interactions between the individual molecules. In a low-density gas, the thermophysical properties are governed exclusively by binary interactions and thus by the pair PESs. For rare gases and small molecules, accurate pair PESs can be developed by fitting analytical functions to interaction energies calculated using high-level quantum-chemical ab initio methods.1−8 Provided that the pair potential functions are available, it is possible to compute second virial and cross second virial coefficients employing expressions from statistical thermodynamics and transport properties by means of the kinetic theory of gases.5,9−14 Recently, we have calculated the cross second virial coefficients and three transport properties (shear viscosity, thermal conductivity, and binary diffusion coefficient) in the dilute gas limit of the (CH4 + N2),11,13 (CH4 + CO2),14 (CH4 + H2S),14 and (H2S + CO2)14 systems employing intermolecular PESs based on high-level ab initio calculations. Particularly for the transport properties, this investigation improved our knowledge substantially because experimental data are scarce for (CH4 + N2) and (CH4 + CO2) mixtures and nonexistent for the other two. The four studied binary systems are important subsystems of natural gas, whose thermophysical properties are of high practical relevance. In the present work, we extend this investigation to two binary subsystems of natural gas that contain propane (C3H8), namely © XXXX American Chemical Society

(CH4 + C3H8) and (CO2 + C3H8). The required intermolecular PESs are presented in the next section. For the like species interactions, they have been taken from our previous studies on the pure gases,1,4,7 while for the unlike interactions, they have been newly developed. The computational methodologies applied to determine the cross second virial coefficients and the dilute gas shear viscosities, thermal conductivities, and binary diffusion coefficients at temperatures from (150 to 1200) K are summarized in Section 3. The results are presented and discussed in Section 4, and conclusions are given in Section 5.

2. INTERMOLECULAR POTENTIAL ENERGY SURFACES 2.1. CH4−CH4, CO2−CO2, and C3H8−C3H8. While the cross second virial coefficients depend only on the unlike interactions, the transport properties also depend on the like interactions, i.e., in the present case, CH4−CH4, CO2−CO2, and C3H8−C3H8. Highly accurate PESs for the latter were developed previously1,4,7 to investigate the thermophysical properties of pure methane,1,15,16 carbon dioxide,4,17,18 and propane.7 They are based on supermolecular quantum-chemical ab initio calculations at the coupled-cluster level with single, double, and perturbative triple excitations [CCSD(T)]19 or, in the case of the C3H8−C3H8 PES,7 a combination of calculations at the CCSD(T) and Received: October 10, 2017 Accepted: November 27, 2017

A

DOI: 10.1021/acs.jced.7b00886 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Figure 1. Deviations of interaction energies obtained using the fitted analytical CH4−C3H8 and CO2−C3H8 potential functions from the corresponding ab initio calculated values as a function of the latter. The dashed lines indicate relative deviations of ±2%.

number of configurations with small values of R because of excessive overlap of the two monomers, leaving 12 633 CH4− C3H8 configurations and 10 319 CO2−C3H8 configurations. Interaction energies V for all configurations were calculated by means of the counterpoise-corrected supermolecular approach22 at the frozen-core RI-MP2 level of theory with the RI-JK approximation23,24 for the Hartree−Fock self-consistent-field (SCF) part. The aug-cc-pVXZ25 basis sets with X = 4 (Q) and X = 5 were used in these calculations. For both basis set levels, we employed the aug-cc-pV5Z-JKFIT 26 and aug-cc-pV5ZMP2FIT27 auxiliary basis sets. Differences between interaction energies obtained in this manner and those obtained using the much more computationally expensive standard MP2 approach were found to be negligible. The correlation parts of the interaction energies, VRI-MP2 corr, were extrapolated to the complete basis set (CBS) limit employing the widely used twopoint scheme recommended by Halkier et al.,28

resolution of identity second-order Møller−Plesset perturbation theory (RI-MP2)20,21 levels. The monomers were treated as rigid rotors using the zero-point vibrationally averaged geometries. Site−site potential functions with 9 sites for CH4, 7 sites for CO2, and 14 sites for C3H8 were fitted to the calculated interaction energies. All three PESs were fine-tuned to the most accurate experimental data for the second virial coefficients and show excellent performance with respect to the prediction of a variety of thermophysical properties at low densities over wide temperature ranges.1,4,7,15−18 The interested reader is referred to the original papers for the full documentation of these PESs.1,4,7 2.2. CH4−C3H8 and CO2−C3H8. PESs for the unlike interactions CH4−C3H8 and CO2−C3H8 based on high-level ab initio calculations have not yet been reported in the literature. Therefore, we have developed such PESs and present them here. In accordance with our work on the like interactions,1,4,7 the monomers were treated in all ab initio calculations of CH4−C3H8 and CO2−C3H8 interaction energies as rigid rotors using the zero-point vibrationally averaged geometries, which were taken from the previous studies.1,4,7 Each configuration of the molecule pairs can be expressed in internal coordinates using the distance between the centers of mass of the molecules, R, and four (CO2− C3H8) or five (CH4−C3H8) Euler angles. The monomer geometries and the precise definition of the angles are provided in the Supporting Information. A total of 850 and 724 distinct angular orientations were investigated for the CH4−C3H8 and CO2−C3H8 molecule pairs, respectively. In both cases, 17 center-of-mass separations R in the range from (1.6 to 10.0) Å (1 Å = 10−10 m) were considered, resulting in 14 450 (850 × 17) CH4−C3H8 and 12 308 (724 × 17) CO2−C3H8 configurations. However, we discarded a large

CBS −3 VRI ‐ MP2 corr(X ) = V RI ‐ MP2 corr + αX

(1)

The SCF contributions are essentially converged at the X = 5 basis set level and were therefore not extrapolated to the CBS limit. To improve the accuracy of the interaction energies further, we performed counterpoise-corrected supermolecular calculations also at the frozen-core CCSD(T)/aug-cc-pVDZ level for all configurations and added the differences between the CCSD(T)/aug-cc-pVDZ and MP2/aug-cc-pVDZ interaction energies (the latter obtained as a byproduct of the CCSD(T)/ aug-cc-pVDZ calculations) to the RI-MP2/CBS ones. Thus, we obtained an approximation to the frozen-core CCSD(T)/CBS level. For both molecule pairs, the CCSD(T) correction changes the interaction energy typically by only a few percent. B



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to the dispersion part of the analytical site−site PES with only one empirically adjusted parameter was always found to be sufficient to bring the calculated values into agreement with the best experimental data.1,7,11,14 Following this approach, we changed the C6 parameters for the interactions of the site on the C atom of CH4 with the three sites closest to the three C atoms of C3H8 by an amount ΔC6 that was adjusted to the best experimental data for the cross second virial coefficient of the (CH4 + C3H8) system (see Section 4.1). The interactions of the site on the C atom of CO2 and the two sites closest to the O atoms of CO2 with the three sites closest to the C atoms of C3H8 were adjusted in a similar manner. The adjustment procedure increases the maximum well depth of the CH4−C3H8 PES from 559.5 K to 585.6 K and that of the CO2−C3H8 PES from 854.9 K to 875.8 K. Unless otherwise noted, all thermophysical property values reported in this work were obtained using the adjusted versions of the PESs. Tables of all symmetry-distinct minimum structures for both the unadjusted and the adjusted analytical PESs and the corresponding interaction energies can be found in the Supporting Information, which also provides Fortran 90 routines computing the new PESs.

The results of the ab initio calculations for all investigated CH4−C3H8 and CO2−C3H8 configurations are listed in the Supporting Information. The RI-MP2 calculations were performed using ORCA 3.0.3,29 while the CCSD(T) calculations were performed using CFOUR.30 Next, we fitted site−site potential functions to the calculated interaction energies. As in our work on the CH4−CH4,1 CO2− CO2,4 and C3H8−C3H87 PESs, we used 9 sites for CH4, 7 sites for CO2, and 14 sites for C3H8. We also adopted the positions of the sites within the molecule-fixed frames and their partial charges q. The number of symmetry-distinct sites per molecule is three for CH4, four for CO2, and seven for C3H8. This results in 21 and 28 distinct site−site combinations for the CH4−C3H8 and CO2− C3H8 interactions, respectively. Each site−site interaction is represented by a function that depends only on the distance Rij between site i in molecule 1 and site j in molecule 2, qiqj C6ij Vij(R ij) = Aij exp( −αijR ij) − f6 (bij , R ij) 6 + R ij R ij (2) where f6 is a damping function,31 6

f6 (bij , R ij) = 1 − exp( −bijR ij) ∑ k=0

(bijR ij)k k!

3. CALCULATION OF THERMOPHYSICAL PROPERTIES 3.1. Cross Second Virial Coefficients. The classical statistical-mechanical expression for the cross second virial coefficient of two rigid molecules is

(3)

The total interaction energy is obtained as the sum over all site− site interactions between the two molecules,

V=

∑ ∑ Vij(R ij) i

cl =− B12

(4)

j

The parameters A, α, b, and C6 for all distinct site−site combinations (84 and 112 parameters in total for the CH4−C3H8 and CO2−C3H8 PESs, respectively) were optimized in nonlinear least-squares fits to the ab initio calculated interaction energies using a weighting function w given by w=

−3

∫0

∞

⎡ V (R , Ω , Ω ) ⎤ 1 2 exp⎢ − ⎥−1 kBT ⎣ ⎦

dR Ω1, Ω 2

(6)

where NA is Avogadro’s constant, T is the temperature, R is the distance vector between the centers of mass of the two molecules, Ω1 and Ω2 represent their angular orientations, and the angle brackets indicate an average over Ω1 and Ω2. Quantum effects can be accurately taken into account for the considered molecule pairs in a semiclassical manner by replacing the pair potential V by the quadratic Feynman−Hibbs (QFH) effective pair potential,32 a scheme that we also employed in previous studies (see, for example, refs 4, 13, and 18). For the CH4−C3H8 molecule pair, the QFH potential can be written as

exp[a1(R /Å)3 ] [1 + a 2(V /K + a3)2 ]2

NA 2

(5) −6

where a1 = 5 × 10 , a2 = 10 , a3 = 550 for CH4−C3H8 interaction energies, and a3 = 900 for CO2−C3H8 interaction energies. The denominator of this function ensures that the weight of configurations increases with the interaction energy decreasing toward the global minimum (Vmin ≈ −550 K and Vmin ≈ −900 K for the CH4 −C3H 8 and CO 2−C3 H8 PESs, respectively), whereas the numerator ensures an adequate fit quality in the asymptotic regions of the PESs. Note that we quote energies in this work consistently in units of kelvin, i.e., we divide them by Boltzmann’s constant kB but omit kB from the notation for brevity. In Figure 1, the deviations of the fitted interaction energies from the corresponding ab initio calculated values are plotted as a function of the latter up to 8000 K. It can be seen that the relative deviations for the CH4−C3H8 PES are mostly within ±2%, whereas the quality of the fit for the CO2−C3H8 PES is somewhat inferior but still acceptable. Particularly for interactions involving hydrocarbons, the use of rigid monomers in the quantum-chemical ab initio calculations leads to a slight underestimation of the strength of the dispersion interactions, which is most evident in the observed systematic positive deviations of calculated second virial and cross second virial coefficients from most of the experimental data (see refs 1, 7, 11, 14, and references therein). However, a simple correction

VQFH(T ) = V +

⎡ ℏ2 ⎢ 1 ⎛ ∂ 2V ∂ 2V ∂ 2V ⎞ ⎟ ⎜ 2 + 2 + 24kBT ⎢⎣ μ ⎝ ∂x ∂y ∂z 2 ⎠

+

⎛ ⎞ 1 ⎜ ∂ 2V ∂ 2V ∂ 2V ⎟ 1 ∂ 2V + + + I1 ⎜⎝ ∂ψ1,2a I2, a ∂ψ2,2a ∂ψ1,2b ∂ψ1,2c ⎟⎠

+

⎤ 1 ∂ 2V 1 ∂ 2V ⎥ + I2, b ∂ψ2,2b I2, c ∂ψ2,2c ⎥⎦

(7)

where ℏ is Planck’s constant divided by 2π; μ is the reduced mass of the molecule pair; x, y, and z are the Cartesian components of R; I1 is the moment of inertia of molecule 1 (CH4); I2,a, I2,b, and I2,c are the principal moments of inertia of molecule 2 (C3H8); and the angles ψi,a, ψi,b, and ψi,c correspond to rotations around the principal axes of molecule i. The expression for the CO2− C3H8 molecule pair is similar, the only difference being that there is no contribution from the rotation around the molecular axis of CO2. C



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numbers. For each molecule pair, up to 2 × 106 trajectories were calculated at each total energy value. Below 300 K, the number of trajectories had to be gradually reduced down to only 200 000 at 60 K because the computational effort required to calculate trajectories with the desired accuracy grows dramatically with decreasing energy. A weighted integration over the total energy yielded temperature-dependent generalized cross sections in the center-of-mass frame41 for temperatures from (150 to 1200) K. Finally, the center-of-mass cross sections were converted to the laboratory frame cross sections38,41 needed to determine the transport properties. The generalized cross sections for binary collisions between two CH4, two CO2, and two C3H8 molecules were obtained in a similar way; the details of these calculations have been published previously.4,7,11 The relative standard uncertainty of the calculated transport property values for the two investigated mixtures due to the Monte Carlo integration scheme employed in the classical trajectory approach is estimated (based on uncertainty estimates provided by TRAJECT for the individual cross sections, see ref 40 for details) to be smaller than 0.15% for viscosity and the binary diffusion coefficient and 0.3% for thermal conductivity for all temperatures and compositions. The neglect of third- and higher-order contributions to the thermal conductivity is estimated to introduce errors of at most (0.1−0.2)%. Uncertainties due to the neglect of quantum effects, the numerical integration of Hamilton’s equations, the numerical integration over the total energy, and the neglect of fourth- and higher-order contributions to the viscosity and binary diffusion coefficient should be negligibly small.

The cross second virial coefficients were calculated for a large number of temperatures in the range from (150 to 1200) K by means of the Mayer-sampling Monte Carlo (MSMC) approach of Singh and Kofke33 implemented in an in-house software code that was also used in several previous studies (see, for example, refs 4, 11, 14, and 18). Two hard spheres with a diameter of 5 Å were utilized as the reference system. The results for all temperatures were obtained simultaneously by performing multitemperature simulations33,34 of 2 × 1010 trial moves with a sampling temperature of 150 K. To avoid unphysical negative interaction energies at very small intermolecular separations R, both potentials were set to infinity when R was smaller than 2.0 Å or when any of the site−site distances Rij were smaller than 1.2 Å. In each trial move, one of the two molecules was displaced and rotated. Maximum step sizes were adjusted in short equilibration runs to achieve a 50% acceptance rate. The second derivatives needed in eq 7 were evaluated analytically. For both molecule pairs, the calculated cross second virial coefficients from eight independent simulation runs were averaged. The standard uncertainty of the averages due to the Monte Carlo integration is smaller than 0.02 cm3·mol−1 at all temperatures and hence negligible. 3.2. Dilute Gas Transport Properties. The transport properties of a low-density gas mixture are accessible through the kinetic theory of molecular gases.9,11,13,14,35−41 For each transport property, a system of linear equations, whose coefficients are related to so-called generalized cross sections, has to be solved. The generalized cross sections are determined by binary collisions in the gas and are thus directly linked to the intermolecular PESs. The approaches employed in this work for the computation of the shear viscosity η in the third-order approximation, the thermal conductivity λ (under steady-state conditions, see ref 13 for details) in the second-order approximation, and the product of molar density and binary diffusion coefficient, ρmD, in the third-order approximation from the generalized cross sections are the same as in our previous work.11,13,14 Therefore, they are not repeated here. The approach for the thermal conductivity13 requires knowledge of the vibrational contributions to the ideal gas heat capacities of the pure gases, which were obtained from the current reference correlations of the isochoric ideal gas heat capacities42−44 by subtracting the translational and classical rotational contributions. The basic methodology for the calculation of the generalized cross sections is the same as in our previous studies on transport properties of dilute molecular gases and gas mixtures (see, for example, refs 7, 11, 14, 45, and references therein). The generalized cross sections for binary collisions between CH4 and C3H8 and between CO2 and C3H8 were calculated within the rigid-rotor approximation by means of the highly efficient and reliable classical trajectory method employing an extended version of the TRAJECT software code.11,40,41 The collision trajectories were obtained by integrating Hamilton’s equations from pre- to postcollisional values with a very large initial and final separation of 500 Å to avoid any cutoff effects. The integration accuracy was chosen such that the relative drift in the total energy between the initial and final states was typically in the range from 10−9 to 10−6, the maximum tolerated value being 10−4. Total-energy-dependent generalized cross sections in the center-of-mass frame, which can be formulated as integrals over the initial states of the trajectories, were calculated at 37 values of the total energy in the range from (60 to 30 000) K by means of a simple Monte Carlo integration scheme using quasi-random

4. RESULTS AND DISCUSSION 4.1. Cross Second Virial Coefficients. Table 1 lists the calculated semiclassical values for the cross second virial coefficients of both molecule pairs at 46 temperatures and estimates of their combined expanded uncertainties (coverage factor k = 2 or approximately 95% confidence level), which are discussed below. In Figure 2, the calculated values for the cross second virial coefficient of the CH4−C3H8 molecule pair are compared with most of the available experimental data 46−50 and two experimentally based correlations.50,51 We reanalyzed the data of Dantzler et al.46 using our own, more accurate values for the pure-component virial coefficients,1,7 which resulted in changes of up to 4.6 cm3·mol−1. The error bars shown in this figure (and all following figures) correspond to those given by the respective authors except for the reanalyzed data of Dantzler et al., for which the uncertainties were also reassessed. Particularly the data close to room temperature of Wormald et al.,47 Jaeschke et al.,48 and Trusler et al.49 were used as reference for the empirical adjustment of the PES described in Section 2.2. The adjustment changes the cross second virial coefficient by −50.1 cm3·mol−1 at 150 K, −13.1 cm3·mol−1 at 300 K, and −2.5 cm3·mol−1 at 1200 K and, as can be seen in the figure, improves the agreement with the experimental data and the correlations substantially. The figure also shows the classical values obtained using the adjusted potential function. They differ from the corresponding semiclassical values by −12.7 cm3·mol−1 at 150 K, −1.5 cm3·mol−1 at 300 K, and −0.09 cm3·mol−1 at 1200 K. On the basis of the comparison with the experimental data, the combined expanded (k = 2) uncertainty of the semiclassical values obtained using the adjusted CH4−C3H8 PES is estimated to be one-half of the absolute value of the difference between the D



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Table 1. Calculated Semiclassical Values for the Cross Second Virial Coefficients, BQFH 12 , of the CH4−C3H8 and CO2−C3H8 Molecule Pairs and Their Combined Expanded (k = 2) Uncertainties, U(BQFH 12 ), as a Function of Temperature T CH4−C3H8

CO2−C3H8

T

BQFH 12

(K)

(cm ·mol )

(cm ·mol )

(cm ·mol )

(cm3·mol−1)

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 420 440 460 480 500 520 540 560 580 600 650 700 750 800 850 900 950 1000 1100 1200

−536.3 −467.7 −412.3 −366.8 −328.7 −296.4 −268.7 −244.7 −223.7 −205.2 −188.8 −174.1 −160.9 −149.0 −138.2 −128.4 −119.4 −111.1 −103.5 −96.43 −89.91 −83.84 −78.19 −72.92 −67.99 −63.36 −54.93 −47.44 −40.75 −34.74 −29.31 −24.38 −19.89 −15.79 −12.02 −8.56 −1.00 5.29 10.59 15.11 19.01 22.39 25.35 27.96 32.33 35.83

25.0 21.5 18.7 16.5 14.7 13.3 12.1 11.1 10.2 9.5 8.8 8.3 7.8 7.3 6.9 6.6 6.2 6.0 5.7 5.4 5.2 5.0 4.8 4.7 4.5 4.3 4.1 3.8 3.6 3.4 3.3 3.1 3.0 2.8 2.7 2.6 2.4 2.2 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

−818.9 −697.9 −604.1 −529.6 −469.0 −418.9 −376.9 −341.2 −310.4 −283.7 −260.3 −239.6 −221.2 −204.8 −190.0 −176.6 −164.5 −153.4 −143.2 −133.9 −125.2 −117.3 −109.9 −103.0 −96.62 −90.63 −79.77 −70.18 −61.65 −54.02 −47.15 −40.94 −35.30 −30.16 −25.45 −21.12 −11.71 −3.91 2.65 8.23 13.04 17.21 20.86 24.08 29.47 33.80

34.4 28.2 23.7 20.3 17.7 15.6 13.9 12.5 11.4 10.4 9.6 8.9 8.2 7.7 7.2 6.8 6.4 6.1 5.8 5.5 5.3 5.0 4.8 4.6 4.4 4.3 4.0 3.7 3.5 3.3 3.1 3.0 2.8 2.7 2.6 2.5 2.2 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

3

U(BQFH 12 ) −1

3

−1

BQFH 12 3

U(BQFH 12 ) −1

Figure 2. Deviations of experimental data, two experimentally based correlations, and calculated values for the cross second virial coefficient of the CH4−C3H8 molecule pair from values calculated semiclassically using the CH4−C3H8 PES of the present work as a function of temperature: ■, Dantzler et al.,46 reanalyzed; ○, Wormald et al.;47 ●, Jaeschke et al.;48 □, Trusler et al.;49 ×, Richter and McLinden;50 − − −, correlation of Dymond et al.;51 , correlation of Richter and McLinden;50 − · −, semiclassical result for the unadjusted CH4− C3H8 PES; − · · −, classical result for the adjusted CH4−C3H8 PES; ······, ± U(BQFH BQFH 12 12 ) with k = 2. Error bars for the data of ref 50 were omitted for clarity.

Figure 3. Deviations of experimental data and calculated values for the cross second virial coefficient of the CO2−C3H8 molecule pair from values calculated semiclassically using the CO2−C3H8 PES of the present work as a function of temperature: ●, Jaeschke et al.;48 ○, McElroy et al.;52 □, Feng et al.;53 ■, Feng et al.,53 reanalyzed; − · −, semiclassical result for the unadjusted CO2−C3H8 PES; − · · −, classical ± U(BQFH result for the adjusted CO2−C3H8 PES; ······, BQFH 12 12 ) with k = 2.

values resulting from the unadjusted and adjusted PESs or 2 cm3· mol−1, whichever is larger. Figure 2 shows that this uncertainty estimate is consistent with most of the data. We note that chemical reactions occurring in the investigated gas mixtures at very high temperatures are not accounted for in our uncertainty estimates for the cross second virial coefficients and the transport properties.

Figure 3 shows the comparison between the calculated values for the cross second virial coefficient of the CO2−C3H8 molecule pair and the experimental data of Jaeschke et al.,48 McElroy et al.,52 and Feng et al.53 Also depicted in the figure are values for the cross second virial coefficient that we extracted from the data E



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Table 2. Coefficients of the Function Fitted to the Semiclassical Values for the Cross Second Virial Coefficient (Eq 8) molecule pair

b0

b0.5

b1

b2

b3

b4

b5

b6

CH4−C3H8 CO2−C3H8

4.1732 × 101 4.7281 × 101

2.2061 × 102 2.2291 × 102

−8.4189 × 102 −9.3045 × 102

1.7654 × 102 7.6145 × 101

−1.1864 × 103 −1.4490 × 103

8.2542 × 102 4.0499 × 102

−6.5951 × 102 −2.3437 × 102

−9.2895 × 102

Table 3. Calculated Values for the Shear Viscosities η (in μPa·s) of (CH4 + C3H8) and (CO2 + C3H8) Mixtures in the Dilute Gas Limit as a Function of Mole Fraction x and Temperature Ta CH4 (1) + C3H8 (2)

CO2 (1) + C3H8 (2)

T (K)

x1 = 0.2

x1 = 0.4

x1 = 0.6

x1 = 0.8

x1 = 0.2

x1 = 0.4

x1 = 0.6

x1 = 0.8

150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 550 600 650 700 750 800 900 1000 1100 1200

4.509 5.172 5.844 6.524 7.210 7.895 8.578 9.256 9.927 10.59 11.24 11.88 12.51 13.13 13.74 14.92 16.05 17.14 18.20 19.21 20.20 22.08 23.85 25.54 27.15

4.732 5.448 6.171 6.897 7.623 8.345 9.060 9.765 10.46 11.14 11.81 12.47 13.12 13.75 14.37 15.57 16.72 17.83 18.90 19.93 20.93 22.85 24.65 26.38 28.03

5.008 5.788 6.571 7.350 8.123 8.886 9.636 10.37 11.09 11.80 12.49 13.16 13.82 14.47 15.10 16.32 17.50 18.63 19.72 20.77 21.79 23.74 25.59 27.35 29.05

5.359 6.218 7.072 7.913 8.739 9.546 10.33 11.10 11.85 12.58 13.29 13.98 14.66 15.32 15.96 17.21 18.41 19.56 20.67 21.74 22.78 24.77 26.67 28.49 30.24

4.861 5.580 6.312 7.055 7.805 8.555 9.303 10.05 10.78 11.50 12.22 12.92 13.61 14.28 14.94 16.23 17.46 18.65 19.80 20.90 21.97 24.00 25.92 27.75 29.49

5.454 6.282 7.124 7.976 8.832 9.686 10.54 11.37 12.20 13.02 13.82 14.60 15.37 16.12 16.86 18.29 19.66 20.97 22.24 23.46 24.64 26.90 29.02 31.05 32.98

6.115 7.059 8.017 8.985 9.954 10.92 11.88 12.82 13.75 14.66 15.56 16.43 17.29 18.13 18.95 20.55 22.07 23.54 24.95 26.31 27.63 30.14 32.51 34.76 36.92

6.855 7.921 9.002 10.09 11.18 12.26 13.33 14.39 15.43 16.45 17.45 18.43 19.40 20.34 21.26 23.04 24.75 26.39 27.97 29.50 30.97 33.79 36.45 38.99 41.41

a The listed viscosity values should be scaled using eqs 9 and 10 to obtain the recommended values. The relative combined expanded (k = 2) uncertainty of the scaled values is 1.0% for 300 ⩽ T/K ⩽ 700, 1.5% for 200 ⩽ T/K < 300 and 700 < T/K ⩽ 1200, and 2.5% for 150 ⩽ T/K < 200. See refs 7 and 14 for the calculated viscosities of the pure gases.

The coefficients of the following analytical function were fitted to the semiclassically calculated values for the cross second virial coefficients of the two molecule pairs:

of Feng et al.53 for the second virial coefficient of a mixture with xCO2 = 0.5158 using our own values for the pure-component virial coefficients.4,7 The reanalyzed data of Feng et al. were used as reference for the adjustment of the PES (see Section 2.2) and agree very well with the calculated values except for the datum at the lowest temperature. The calculated values are also consistent with the data of McElroy et al.,52 whereas the data of Jaeschke et al.48 differ more significantly. However, the uncertainties given by Jaeschke et al. are much higher than those for their data for the CH4−C3H8 molecule pair. Therefore, we performed the adjustment without considering the data of Jaeschke et al. The adjustment changes the cross second virial coefficient by −51.6 cm3·mol−1 at 150 K, −10.2 cm3·mol−1 at 300 K, and −1.7 cm3· mol−1 at 1200 K. The classical values obtained using the adjusted PES, which are also shown in the figure, deviate from the corresponding semiclassical values by −13.5 cm3·mol−1 at 150 K, −1.1 cm3·mol−1 at 300 K, and −0.06 cm3·mol−1 at 1200 K. The combined expanded (k = 2) uncertainty of the semiclassical values obtained using the adjusted CO2−C3H8 PES is estimated to be two-thirds of the absolute value of the difference between the values resulting from the unadjusted and adjusted PESs or 2 cm3·mol−1, whichever is larger.

QFH B12

cm 3·mol−1

= b0 +

b0.5 T*

6

+

∑ i=1

bi (T *)i

(8)

where T* = T/(100 K), and the coefficients b are given in Table 2. Equation 8 reproduces the calculated values for both systems to within ±0.03 cm3·mol−1. 4.2. Dilute Gas Transport Properties. The calculated values for the viscosities, thermal conductivities, and products of the molar densities and the binary diffusion coefficients are listed for different compositions of the two systems at 25 temperatures from (150 to 1200) K in Tables 3−5. In Figures 4 and 5, the calculated values for the dilute gas shear viscosities of the (CH4 + C3H8) and (CO2 + C3H8) systems are compared with the available experimental data for the mixtures54−56,59 and with the experimental data for the pure components of Vogel57,60 and Vogel and Herrmann.58 The latter data sets, which extend from ambient temperature up to more than 600 K, are of reference quality with relative standard uncertainties of at most 0.3%. Their relative deviations from the F



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Table 4. Calculated Values for the Thermal Conductivities λ (in mW·m−1·K−1) of (CH4 + C3H8) and (CO2 + C3H8) Mixtures in the Dilute Gas Limit as a Function of Mole Fraction x and Temperature Ta CH4 (1) + C3H8 (2)

CO2 (1) + C3H8 (2)

T (K)

x1 = 0.2

x1 = 0.4

x1 = 0.6

x1 = 0.8

x1 = 0.2

x1 = 0.4

x1 = 0.6

x1 = 0.8

150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 550 600 650 700 750 800 900 1000 1100 1200

7.041 8.623 10.40 12.40 14.65 17.16 19.93 22.94 26.17 29.60 33.21 36.95 40.82 44.78 48.83 57.09 65.52 74.05 82.62 91.22 99.80 116.8 133.6 150.0 165.9

8.487 10.27 12.23 14.39 16.78 19.42 22.31 25.44 28.79 32.35 36.07 39.94 43.94 48.05 52.24 60.83 69.60 78.51 87.49 96.52 105.6 123.6 141.3 158.8 175.8

10.33 12.37 14.56 16.92 19.48 22.28 25.31 28.58 32.07 35.75 39.62 43.64 47.80 52.07 56.44 65.40 74.59 83.94 93.41 103.0 112.5 131.7 150.7 169.5 187.8

12.74 15.13 17.61 20.22 23.00 25.98 29.18 32.61 36.26 40.11 44.14 48.33 52.67 57.14 61.71 71.11 80.80 90.70 100.8 110.9 121.2 141.8 162.3 182.6 202.7

6.016 7.437 9.059 10.91 13.00 15.34 17.90 20.68 23.65 26.79 30.06 33.46 36.94 40.50 44.12 51.49 58.95 66.46 73.98 81.49 88.95 103.7 118.1 132.1 145.7

6.167 7.585 9.198 11.02 13.06 15.31 17.75 20.38 23.16 26.07 29.10 32.22 35.41 38.65 41.94 48.61 55.34 62.09 68.83 75.53 82.19 95.30 108.1 120.5 132.5

6.327 7.731 9.320 11.10 13.06 15.20 17.50 19.94 22.49 25.15 27.89 30.69 33.55 36.44 39.36 45.25 51.17 57.08 62.96 68.79 74.57 85.90 96.91 107.6 117.9

6.493 7.867 9.412 11.12 12.98 14.98 17.09 19.30 21.59 23.95 26.35 28.80 31.27 33.76 36.27 41.29 46.30 51.28 56.21 61.09 65.89 75.29 84.36 93.12 101.6

a

The thermal conductivity values for the (CO2 + C3H8) system should be scaled using eq 11 to obtain the recommended values. The relative combined expanded (k = 2) uncertainty of the listed values for the (CH4 + C3H8) system and of the scaled values for the (CO2 + C3H8) system is 3.0% for 300 ⩽ T/K ⩽ 700, 3.5% for 200 ⩽ T/K < 300 and 700 < T/K ⩽ 1200, and 4.5% for 150 ⩽ T/K < 200. See refs 7 and 14 for the calculated thermal conductivities of the pure gases.

where ηcalc and ηrec are the calculated and recommended values (which are also shown in the figures), respectively. We have already suggested such a scaling procedure in our work on the (CH4 + N2),11 (CH4 + CO2),14 (CH4 + H2S),14 and (H2S + CO2)14 systems. Figures 4 and 5 also depict the viscosities at 300 K resulting from using the unadjusted PESs for the unlike interactions. They differ from those obtained using the adjusted PESs by at most +0.7% and +0.5% for the (CH4 + C3H8) and (CO2 + C3H8) systems, respectively. Figures 6 and 7 show the comparison of the computed values for the dilute gas thermal conductivities with the experimental data of Smith et al.62 for the (CH4 + C3H8) system and of Cheung et al.63 at about 370 K for both systems (data of Cheung et al. are also available at higher temperatures but only for the pure components). Additionally, the comparison includes the highly accurate experimental data of Haarman64 for pure CO2, which have a stated uncertainty of only 0.3% and deviate on average by +1.1% from the calculated values. For the other pure gases, data of similar accuracy are not available over a wider temperature range. While there is considerable disagreement between the calculated values and the data of Smith et al.62 for the (CH4 + C3H8) system, with deviations from the calculated values as large as −8% for pure methane, the agreement with the data of Cheung et al.63 is quite satisfactory, particularly for the (CO2 + C3H8) system. On the basis of the average relative deviations of the reference data of Haarman64 from the computed values, we propose a scaling similar to that for the shear viscosity values as

computed values are only weakly dependent on temperature and are on average −0.45% for methane, +0.55% for carbon dioxide, and +0.30% for propane. The relative deviations of the experimental data for the mixtures54−56,59 from the calculated values are mostly positive and exhibit no clear mole fraction dependence. The pure-component data reported alongside the mixture data in three of the papers54,55,59 also exhibit mostly positive relative deviations, which are of similar magnitude as those for the mixtures. Note that the claimed uncertainty of the data of Abe et al. for the two systems is only 0.3%.55,59 However, it has been shown61 that the viscometer used by Abe et al. suffered from a design flaw that resulted in viscosity values that are always systematically too high above room temperature by up to about 1%.3,5,11,14,15,45,57,58 This is consistent with the observed deviations. To obtain viscosity values with the highest possible accuracy for the two mixtures, we propose a simple scaling of the calculated viscosity values by a temperature-independent factor that depends linearly on the mole fraction and reduces the average relative deviations from the reference data of Vogel57,60 and Vogel and Herrmann58 for the pure components to zero. Thus, we have ηrec = ηcalc(0.9955xCH4 + 1.0030xC3H8)

(9)

and ηrec = ηcalc(1.0055xCO2 + 1.0030xC3H8)

(10) G



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Table 5. Calculated Values for the Products of the Molar Densities and the Binary Diffusion Coefficients, ρmD (in 10−4 mol·m−1·s−1), of the (CH4 + C3H8) and (CO2 + C3H8) Systems in the Dilute Gas Limit as a Function of Mole Fraction x and Temperature Ta CH4 (1) + C3H8 (2)

CO2 (1) + C3H8 (2)

T (K)

x1 → 0

x1 = 0.5

x1 → 1

x1 → 0

x1 = 0.5

x1 → 1

150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 550 600 650 700 750 800 900 1000 1100 1200

2.507 2.957 3.407 3.851 4.287 4.714 5.129 5.534 5.928 6.312 6.685 7.048 7.403 7.749 8.086 8.739 9.364 9.966 10.55 11.11 11.65 12.69 13.68 14.63 15.54

2.506 2.957 3.407 3.852 4.288 4.714 5.130 5.534 5.927 6.310 6.682 7.045 7.398 7.742 8.078 8.728 9.349 9.947 10.52 11.08 11.62 12.65 13.64 14.58 15.48

2.505 2.957 3.407 3.852 4.289 4.714 5.129 5.533 5.925 6.305 6.675 7.035 7.385 7.727 8.060 8.702 9.316 9.907 10.47 11.02 11.56 12.57 13.54 14.47 15.36

1.671 1.985 2.306 2.628 2.948 3.265 3.575 3.878 4.174 4.462 4.742 5.014 5.279 5.537 5.787 6.269 6.728 7.166 7.587 7.991 8.381 9.123 9.824 10.49 11.13

1.672 1.985 2.306 2.628 2.948 3.264 3.574 3.877 4.173 4.460 4.740 5.012 5.276 5.533 5.783 6.265 6.723 7.160 7.580 7.983 8.372 9.113 9.813 10.48 11.12

1.672 1.986 2.306 2.628 2.948 3.264 3.574 3.876 4.171 4.458 4.738 5.009 5.273 5.529 5.779 6.259 6.716 7.152 7.571 7.973 8.361 9.101 9.799 10.46 11.10

Figure 4. Relative deviations of experimental data and calculated values for the dilute gas shear viscosity of the (CH4 + C3H8) system from values calculated using the CH4−C3H8 PES of the present work, the CH4− CH4 PES of ref 1, and the C3H8−C3H8 PES of ref 7 as a function of methane mole fraction: ○, Giddings et al.,54 283 K; □, Giddings et al.,54 (310 and 311) K; △, Giddings et al.,54 344 K; ▽, Giddings et al.,54 378 K; ◇, Giddings et al.,54 (411 and 413) K; ●, Abe et al.,55 298 K; ■, Abe et al.,55 333 K; ▲, Abe et al.,55 373 K; ▼, Abe et al.,55 418 K; ◆, Abe et al.,55 468 K; ⊕, Locke et al.,56 280 K; ⊞, Locke et al.,56 298 K; ×, Vogel57 and Vogel and Herrmann,58 (292 to 682) K for methane and (297 to 625) K for propane; − · −, result for the unadjusted CH4−C3H8 PES at 300 K; ······, recommended values resulting from eq 9. Error bars for the data of refs 55, 57, and 58 were omitted for clarity.

a

The relative combined expanded (k = 2) uncertainty is 2.0% for 300 ⩽ T/K ⩽ 700, 2.5% for 200 ⩽ T/K < 300 and 700 < T/K ⩽ 1200, and 3.5% for 150 ⩽ T/K < 200.

we have done previously for the (CH4 + CO2) and (H2S + CO2) systems,14 λrec = λcalc(1.011xCO2 + xC3H8)

(11)

Figures 6 and 7 also show the thermal conductivities at 300 K resulting from using the unadjusted PESs for the unlike interactions. The relative deviations from the values obtained using the adjusted PESs are close to those for the shear viscosities and thus much smaller than the deviations of most of the experimental data points. Finally, Figures 8 and 9 show the comparison of the calculated values for the product of molar density and binary diffusion coefficient in the dilute gas limit for both systems with the few available experimental data.65−67 The binary diffusion coefficient is determined almost entirely by the unlike interactions and, as can be seen in Table 5, its product with molar density is only very weakly dependent on composition. Therefore, the two figures depict the deviations as a function of temperature instead of mole fraction. Apart from one data point, the agreement with the data of Gotoh et al.65 for the (CH4 + C3H8) system is satisfactory. The single datum of Kugler et al.67 for the (CO2 + C3H8) system is also consistent with the calculated value considering the rather large uncertainty given by the authors. Unfortunately, Arora et al.66 provided only smoothed values of their data for the (CH4 + C3H8) system in the form of a polynomial in temperature without uncertainty estimates, but the agreement with the

Figure 5. Relative deviations of experimental data and calculated values for the dilute gas shear viscosity of the (CO2 + C3H8) system from values calculated using the CO2−C3H8 PES of the present work, the CO2− CO2 PES of ref 4, and the C3H8−C3H8 PES of ref 7 as a function of carbon dioxide mole fraction: ●, Abe et al.,59 298 K; ■, Abe et al.,59 333 K; ▲, Abe et al.,59 373 K; ▼, Abe et al.,59 418 K; ◆, Abe et al.,59 468 K; ×, Vogel60 and Vogel and Herrmann,58 (298 to 683) K for carbon dioxide and (297 to 625) K for propane; − · −, result for the unadjusted CO2−C3H8 PES at 300 K; ······, recommended values resulting from eq 10. Error bars for the data of refs 58−60 were omitted for clarity.

H



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Figure 8. Relative deviations of experimental data and calculated values for the binary diffusion coefficient of the (CH4 + C3H8) system in the dilute gas phase from values calculated at the same molar densities using the CH4−C3H8 PES of the present work, the CH4−CH4 PES of ref 1, and the C3H8−C3H8 PES of ref 7 as a function of temperature: ●, Gotoh et al.,65 xCH4 = 0.5; , Arora et al.,66 xCH4 = 0.2, smoothed values; − · −, result for the unadjusted CH4−C3H8 PES with xCH4 = 0.5.

Figure 6. Relative deviations of experimental data and calculated values for the dilute gas thermal conductivity of the (CH4 + C3H8) system from values calculated using the CH4−C3H8 PES of the present work, the CH4−CH4 PES of ref 1, and the C3H8−C3H8 PES of ref 7 as a function of methane mole fraction: ○, Smith et al.,62 323 K; □, Smith et al.,62 348 K; △, Smith et al.,62 373 K; ▽, Smith et al.,62 398 K; ◇, Smith et al.,62 423 K; ●, Cheung et al.,63 (366 to 373) K; − · −, result for the unadjusted CH4−C3H8 PES at 300 K.

Figure 9. Relative deviations of the only existing experimental datum and calculated values for the binary diffusion coefficient of the (CO2 + C3H8) system in the dilute gas phase from values calculated at the same molar densities using the CO2−C3H8 PES of the present work, the CO2−CO2 PES of ref 4, and the C3H8−C3H8 PES of ref 7 as a function of temperature: ●, Kugler et al.,67 xCO2 = 0.5; − · −, result for the unadjusted CO2−C3H8 PES with xCO2 = 0.5.

Figure 7. Relative deviations of experimental data and calculated values for the dilute gas thermal conductivity of the (CO2 + C3H8) system from values calculated using the CO2−C3H8 PES of the present work, the CO2−CO2 PES of ref 4, and the C3H8−C3H8 PES of ref 7 as a function of carbon dioxide mole fraction: ●, Cheung et al.,63 (368 to 376) K; ×, Haarman,64 (328 to 468) K; − · −, result for the unadjusted CO2−C3H8 PES at 300 K; ······, recommended values resulting from eq 11. Error bars for the data of ref 64 were omitted for clarity.

strong evidence that the excellent agreement for the (CH4 + C3H8) system is not coincidental. The results for the binary diffusion coefficients of equimolar mixtures obtained using the unadjusted PESs for the unlike interactions are also shown in Figures 8 and 9. At 300 K, they deviate from the results for the adjusted PESs by +1.6% and +1.2% for the (CH4 + C3H8) and (CO2 + C3H8) systems, respectively. The larger relative deviations, compared with those

calculated values is excellent with deviations of at most 0.3%. In this context, it is worth mentioning that Jäger and Bich8 found similarly good agreement between highly accurate ab initio calculated values for the binary diffusion coefficient of the (Kr + He) system with data reported by Arora et al.,68 thus providing I



Article

Funding

for viscosity and thermal conductivity, are due to the fact that, as already mentioned above, the binary diffusion coefficient depends essentially only on the unlike interactions. Conservative estimates for the relative combined expanded (k = 2) uncertainties of the calculated transport property values, which are based on our experience, are given in the footnotes of Tables 3−5. If a scaling is recommended, the estimates refer to the scaled values. The main source of uncertainty in the calculated transport property values is most likely the treatment of the molecules as rigid rotors in all stages of the calculation.

This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG), Grant HE 6155/2-1. Notes

The author declares no competing financial interest.

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(1) Hellmann, R.; Bich, E.; Vogel, E. Ab initio intermolecular potential energy surface and second pressure virial coefficients of methane. J. Chem. Phys. 2008, 128, 214303. (2) Hellmann, R.; Bich, E.; Vogel, E.; Vesovic, V. Ab initio intermolecular potential energy surface and thermophysical properties of hydrogen sulfide. Phys. Chem. Chem. Phys. 2011, 13, 13749−13758. (3) Hellmann, R. Ab initio potential energy surface for the nitrogen molecule pair and thermophysical properties of nitrogen gas. Mol. Phys. 2013, 111, 387−401. (4) Hellmann, R. Ab initio potential energy surface for the carbon dioxide molecule pair and thermophysical properties of dilute carbon dioxide gas. Chem. Phys. Lett. 2014, 613, 133−138. (5) Jäger, B.; Hellmann, R.; Bich, E.; Vogel, E. State-of-the-art ab initio potential energy curve for the krypton atom pair and thermophysical properties of dilute krypton gas. J. Chem. Phys. 2016, 144, 114304. (6) Garberoglio, G.; Jankowski, P.; Szalewicz, K.; Harvey, A. H. Alldimensional H2−CO potential: Validation with fully quantum second virial coefficients. J. Chem. Phys. 2017, 146, 054304. (7) Hellmann, R. Intermolecular potential energy surface and thermophysical properties of propane. J. Chem. Phys. 2017, 146, 114304. (8) Jäger, B.; Bich, E. Thermophysical properties of krypton-helium gas mixtures from ab initio pair potentials. J. Chem. Phys. 2017, 146, 214302. (9) McCourt, F. R. W.; Beenakker, J. J. M.; Köhler, W. E.; Kušcě r, I. Nonequilibrium Phenomena in Polyatomic Gases; Clarendon Press: Oxford, 1990; Vol. I: Dilute Gases. (10) Bich, E.; Mehl, J. B.; Hellmann, R.; Vesovic, V. In Experimental Thermodynamics Volume IX: Advances in Transport Properties of Fluids; Assael, M. J., Goodwin, A. R. H., Vesovic, V., Wakeham, W. A., Eds.; The Royal Society of Chemistry: Cambridge, 2014; Chapter 7, pp 226−252. (11) Hellmann, R.; Bich, E.; Vogel, E.; Vesovic, V. Intermolecular potential energy surface and thermophysical properties of the CH4−N2 system. J. Chem. Phys. 2014, 141, 224301. (12) Hellmann, R.; Bich, E. An improved kinetic theory approach for calculating the thermal conductivity of polyatomic gases. Mol. Phys. 2015, 113, 176−183. (13) Hellmann, R.; Bich, E.; Vesovic, V. Calculation of the thermal conductivity of low-density CH4−N2 gas mixtures using an improved kinetic theory approach. J. Chem. Phys. 2016, 144, 134301. (14) Hellmann, R.; Bich, E.; Vesovic, V. Cross second virial coefficients and dilute gas transport properties of the (CH4 + CO2), (CH4 + H2S), and (H2S + CO2) systems from accurate intermolecular potential energy surfaces. J. Chem. Thermodyn. 2016, 102, 429−441. (15) Hellmann, R.; Bich, E.; Vogel, E.; Dickinson, A. S.; Vesovic, V. Calculation of the transport and relaxation properties of methane. I. Shear viscosity, viscomagnetic effects, and self-diffusion. J. Chem. Phys. 2008, 129, 064302. (16) Hellmann, R.; Bich, E.; Vogel, E.; Dickinson, A. S.; Vesovic, V. Calculation of the transport and relaxation properties of methane. II. Thermal conductivity, thermomagnetic effects, volume viscosity, and nuclear-spin relaxation. J. Chem. Phys. 2009, 130, 124309. (17) Hellmann, R.; Vesovic, V. Influence of a magnetic field on the viscosity of a dilute gas consisting of linear molecules. J. Chem. Phys. 2015, 143, 214303. (18) Hellmann, R. Nonadditive three-body potential and third to eighth virial coefficients of carbon dioxide. J. Chem. Phys. 2017, 146, 054302. (19) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479−483. (20) Weigend, F.; Häser, M. RI-MP2: first derivatives and global consistency. Theor. Chem. Acc. 1997, 97, 331−340.

5. CONCLUSIONS The cross second virial coefficients and the dilute gas shear viscosities, thermal conductivities, and binary diffusion coefficients of the (CH4 + C3H8) and (CO2 + C3H8) systems, which are important binary subsystems of natural gas, were calculated at temperatures from (150 to 1200) K. Highly accurate intermolecular PESs for the CH4−C3H8 and CO2−C3H8 interactions were developed for this purpose. They are based on quantum-chemical ab initio calculations at the RI-MP220,21 and CCSD(T)19 levels of theory for thousands of mutual configurations of the molecules and are represented analytically using site−site potential functions. Both PESs were fine-tuned to the best experimental data for the cross second virial coefficients. The PESs for the CH4−CH4, CO2−CO2, and C3H8−C3H8 interactions, which are needed only for the computation of the transport properties, were taken from our previous work on the thermophysical properties of the pure gases.1,4,7 The cross second virial coefficients were determined by performing MSMC33 simulations, in which quantum effects were accounted for in a semiclassical manner,32 while the dilute gas transport properties were calculated using the classical trajectory method in conjunction with the kinetic theory of molecular gases.9,11,13,14,40,41 These approaches are now well established. The agreement between the calculated values for the investigated thermophysical properties of the two mixtures and the few available experimental data is overall satisfactory. Tables of calculated values for the cross second virial coefficients and the three transport properties, for which in some cases a small empirical adjustment in the form of a scaling factor is recommended, are provided together with conservative uncertainty estimates. These values are not only more accurate than most of the existing experimental data but also span a much wider temperature range.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00886. Details of the internal coordinates, detailed results of the ab initio calculations for all investigated configurations, and minimum structures of the CH 4 −C 3 H 8 and CO2−C3H8 molecule pairs as well as Fortran 90 routines computing the new analytical potential functions (ZIP)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Robert Hellmann: 0000-0003-3121-6827 J



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Cross Second Virial Coefficients and Dilute Gas Transport Properties

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