Reference Values for the Second Virial Coefficient and Three Dilute

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

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Reference Values for the Second Virial Coefficient and Three Dilute Gas Transport Properties of Ethane from a State-of-the-Art Intermolecular Potential Energy Surface Robert Hellmann* Institut für Chemie, Universität Rostock, 18059 Rostock, Germany S Supporting Information *

ABSTRACT: The second virial coefficient and the dilute gas shear viscosity, thermal conductivity, and self-diffusion coefficient of ethane (C2H6, R-170) were determined with high accuracy at temperatures from (90 to 1200) K using advanced computational approaches. The second virial coefficient was calculated semiclassically by means of the Mayer-sampling Monte Carlo technique, while the transport properties were obtained using the classical kinetic theory of polyatomic gases. The required intermolecular potential energy surface was developed as part of this work. It is based on high-level quantum-chemical ab initio calculations and was fine-tuned to reliable experimental data for the second virial coefficient. The computed thermophysical property values are in excellent agreement with the best available experimental data and are recommended as reference values. Correlations based entirely on the calculated values are proposed for practical applications in the low-density gas phase.

1. INTRODUCTION The calculation of the thermophysical properties of a pure fluid or a fluid mixture requires that the potential energy surface (PES) describing the interactions between the molecules is known. In dilute gases, the thermophysical properties are governed by binary interactions only and hence by the pair PESs. For rare gases and simple molecules, such as methane,1 hydrogen sulfide,2 carbon dioxide,3 propane,4 and xenon,5 accurate pair PESs can be constructed by fitting suitable analytical functions to interaction energies determined using high-level quantum-chemical ab initio approaches. If the pair potential functions are available, it is possible to calculate the second virial coefficient (or the cross second virial coefficient in the case of unlike interactions) utilizing expressions from statistical thermodynamics and the transport properties employing the kinetic theory of gases.6−11 We previously investigated the second virial coefficients and dilute gas transport properties of pure alkanes for the first and third member of the series, i.e., for methane (CH4) and propane (C3H8).1,4,12,13 In the present work, we extend this study to the second alkane, ethane (C2H6). It is a minor component of natural gas and used mainly as a feedstock for ethylene production by steam cracking. It is also used as a lowtemperature refrigerant (R-170) with an ozone depletion potential of zero and a low global warming potential. The only intermolecular PES based on high-level ab initio calculations that was developed specifically for ethane (i.e., not optimized for transferability of PES parameters between different alkanes at the expense of accuracy) is that of Rowley et al.14 Their eight-center site−site potential function is based on counterpoise-corrected15 supermolecular calculations for © XXXX American Chemical Society

523 configurations of the two molecules using second-order Møller−Plesset perturbation theory (MP2) and employing the basis set 6-311+G(2df,2pd). This PES was published already in 2001 and is outdated by today’s standards. Therefore, we developed a new state-of-the-art pair PES for ethane, which is presented in Section 2. In Section 3, the computational methodologies used to determine the second virial coefficient and the dilute gas shear viscosity, thermal conductivity, and selfdiffusion coefficient at temperatures from (90 to 1200) K are summarized. The results are compared with experimental data and discussed in Section 4. New correlations for all four properties, which are based solely on the calculated values, are presented in Section 5, followed by conclusions in Section 6.

2. INTERMOLECULAR POTENTIAL ENERGY SURFACE 2.1. Monomer Geometry. The PES of the ethane molecule pair is 42-dimensional if all inter- and intramolecular degrees of freedom are considered. In contrast, treating the ethane molecules as rigid rotors results in a PES that is only sixdimensional. While it might be argued that at least the hindered internal rotation has to be taken into account, the results of our study of the second virial coefficient and dilute gas transport properties of propane4 indicate that this should not be necessary. The PES for the propane molecule pair yielded values for the investigated properties that are in excellent agreement with the best experimental data despite freezing all Received: December 8, 2017 Accepted: January 3, 2018

A

DOI: 10.1021/acs.jced.7b01069 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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To increase the accuracy of the interaction energies further, we also performed counterpoise-corrected supermolecular calculations at the frozen-core CCSD(T)/aug-cc-pVDZ and CCSD(T)/aug-cc-pVTZ levels for all configurations. The differences between the CCSD(T) and MP2 interaction energies (the latter obtained as a byproduct of the CCSD(T) calculations) were then extrapolated to the CBS limit in the same way as the VRI‑MP2corr values and added to the RI-MP2/ CBS interaction energies. Thus, an approximation to the frozen-core CCSD(T)/CBS level was obtained. For about 69% of the configurations, the CCSD(T) correction influences the interaction energy by less than 5%. The results of the ab initio calculations for all investigated configurations are tabulated in the Supporting Information. All RI-MP2 and CCSD(T) calculations reported in this work were carried out using ORCA 3.0.328 and CFOUR,29 respectively. 2.3. Analytical Potential Function. A site−site potential function with 11 sites per molecule was fitted to the calculated interaction energies. Each site−site interaction is represented by

internal degrees of freedom, including the hindered internal rotations. The torsional barriers are also of comparable magnitude in both molecules with values of about 1490 K for ethane16 and about 1660 K for propane.16 (Note that we quote energies consistently in units of kelvin in this work, i.e., we divide them by Boltzmann’s constant kB but omit kB from the notation for brevity.) Therefore, we applied the rigid-rotor approximation also in the present work. The monomer geometry used in all further calculations was determined fully ab initio in multiple steps. First, to obtain an accurate equilibrium structure, a geometry optimization was performed at the all-electron coupled-cluster level with single, double, and perturbative triple excitations [CCSD(T)]17 employing the cc-pwCV5Z18 basis set. Then, a geometry optimization was also performed at the all-electron CCSD(T) level using the smaller cc-pwCVTZ18 basis set, followed by a cubic force field calculation to obtain a zero-point vibrationally averaged geometry. Finally, the differences in the Cartesian coordinates of the atoms between the vibrationally averaged structure and the equilibrium structure at the all-electron CCSD(T)/cc-pwCVTZ level were added to the Cartesian coordinates of the all-electron CCSD(T)/cc-pwCV5Z equilibrium structure. Thus, the final geometry represents an approximation to the zero-point vibrationally averaged structure at the all-electron CCSD(T)/cc-pwCV5Z level. It is characterized by a C−C distance of 1.5333 Å, a C−H distance of 1.0952 Å, and a C−C−H angle of 111.35°, which is in good agreement with results from rovibrational spectroscopy of 1.5335(5) Å, 1.0955(5) Å, and 111.10(5)°, respectively.19 2.2. Calculation of Interaction Energies. Each configuration of two rigid ethane molecules can be expressed in internal coordinates by means of the distance between the centers of mass of the two molecules, R, and five Euler angles, whose precise definition is provided in the Supporting Information. A total of 249 distinct angular orientations was investigated. The main grid consists of 123 angular configurations, for which 27 center-of-mass separations R in the range from (1.75 to 12.0) Å were considered. For the additional 126 angular configurations, the number of considered center-of-mass separations was reduced to six in the range from (2.0 to 8.0) Å. This results in a total of 4077 configurations. However, we discarded many configurations with small R values because of excessive overlap of the two molecules, leaving 3456 configurations. The interaction energies V for all configurations were computed using the counterpoise-corrected supermolecular approach at the frozen-core resolution of identity MP2 (RIMP2)20,21 level applying the RI-JK approximation22,23 for the Hartree−Fock self-consistent-field (SCF) part. The aug-ccpVXZ24 basis sets with X = 4 (Q) and X = 5 were used in these calculations. For both basis set levels, the aug-cc-pV5Z-JKFIT25 and aug-cc-pV5Z-MP2FIT26 auxiliary basis sets were employed. Differences between interaction energies obtained in this way and those obtained using the standard MP2 method were found to be insignificant. The correlation parts of the interaction energies, VRI‑MP2corr, were extrapolated to the complete basis set (CBS) limit using the popular two-point scheme recommended by Halkier et al.,27 CBS −3 VRI ‐ MP2corr(X ) = V RI ‐ MP2corr + αX

Vij(R ij) = Aij exp( −αijR ij) − f6 (bij , R ij) − f8 (bij , R ij)

C8ij R ij8

+

C6ij R ij6

qiqj R ij

(2)

where Rij is the separation between site i in molecule 1 and site j in molecule 2, and the damping functions f n are given by30 n

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

(bijR ij)k k!

(3)

The total interaction potential is obtained as the sum of all site−site contributions, 11

V=

11

∑ ∑ Vij(R ij) i=1 j=1

(4)

The 11 sites were arranged such that there are four distinct types of sites, resulting in 10 different types of site−site combinations. Note that if we were to use only the eight atoms in each molecule as interaction sites, we would have only two types of sites and thus only three types of site−site combinations. This would be insufficient for an accurate fit unless the site−site interactions are made anisotropic. The parameters A, α, b, C6, and C8 for the 10 distinct site− site combinations, the partial charges q for all types of sites except the site in the center of mass (whose partial charge was set to zero), and the positions of the sites within the moleculefixed frame (constrained to the D3d symmetry of ethane) were optimized in a nonlinear least-squares fit to the 3456 ab initio calculated interaction energies, which were weighted by w=

exp[0.005(R /Å)3 ] [1 + 10−6(V /K + 650)2 ]2

(5)

The denominator of this function ensures that the weight of configurations increases with the interaction energy decreasing toward its most negative values (V > −650 K for all calculated interaction energies), whereas the numerator ensures an adequate fit quality in the asymptotic regions of the PES. The site charges were constrained to yield neutral molecules with a quadrupole moment equal to that resulting from an ab

(1)

The SCF contributions were not extrapolated to the CBS limit because they are essentially converged at the X = 5 basis set level. B



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initio calculation at the frozen-core CCSD(T)/aug-cc-pV5Z level of −0.51272 au. Furthermore, the C6ij coefficients were constrained such that the isotropic part of the C6 dispersion coefficient of ethane resulting from our model, C6iso = ∑i 11= 1∑j 11= 1C6ij, is equal to an ab initio value for this quantity, initio Cab = 358.9 au. This value was extracted from interaction 6iso energies computed at R values between (20 and 30) Å for four different angular orientations at the frozen-core CCSD(T)/augcc-pVTZ level. The four orientations were chosen such that a weighted average of their interaction energies, Vave, for a given R value contains only the isotropic part of the C6 dispersion contribution as well as contributions proportional to R−8, R−9, etc. (for the similar case of two centrosymmetric linear initio molecules, see, for example, ref 31). Then, we have Cab = 6iso 6 limR→∞VaveR . Further details of the procedure are provided in the Supporting Information. We note that the same approach initio using the smaller aug-cc-pVDZ basis set yields a Cab value 6iso that differs by less than 0.1% from that determined using the aug-cc-pVTZ basis set, indicating that the value for the latter basis set is close to that for the CBS limit. The final auxiliary condition is related to the C8ij coefficients. They were constrained such that the isotropic part of the C8 dispersion coefficient of ethane for the site−site model, C8iso = ∑i =11 1∑j =11 1C8ij, is equal to 21821 au, which is the value initio resulting from multiplying Cab by an ab initio value for the 6iso 32 ratio C8iso/C6iso. Figure 1 shows the optimized positions of the 11 interaction sites, while Figure 2 shows the deviations of the fitted

Figure 2. Deviations of interaction energies computed using the fitted analytical potential function from the corresponding ab initio values as a function of the latter. The dashed red lines indicate relative deviations of ±2%.

6.4% larger than our ab initio value for rigid monomers. In our study of the interactions of two methane molecules,1 we found that the respective DOSD value for C6 is 5.3% larger than the ab initio value for rigid methane, which we determined in a similar way as that for ethane. However, for all PESs involving hydrocarbons that we have developed so far,1,4,8,11,33 a simple correction to the dispersion part with only one empirically adjusted parameter was always sufficient to bring the calculated values and the best experimental data into satisfying agreement.1,4,8,11,33 In the present work, we employed a correction scheme similar to that previously used for the PES of the methane molecule pair,1

Figure 1. Visualization of the optimized positions of the 11 interaction sites (red spheres) in the ethane molecule.

2

interaction energies from the corresponding ab initio values as a function of the latter up to 8000 K. It can be seen that the relative deviations are mostly within ±2%. In our previous studies of intermolecular PESs involving hydrocarbons,1,4,8,11,33 we observed systematic positive deviations of the ab initio calculated values for the second virial and cross second virial coefficients from most of the experimental data. This is mainly due to the use of rigid monomers, resulting in the neglect of important vibrational contributions to the dispersion coefficients, see ref 34 and references therein. The use of vibrationally averaged monomer geometries in the ab initio calculations only partly accounts for these contributions. For the interaction of two ethane molecules, the reference value for C6iso from the dipole oscillator strength distribution (DOSD), CDOSD = 381.8 au,35 which includes vibrational 6iso contributions and has a stated relative uncertainty of only 1%, is

Vadj = Vunadj +

2

∑ ∑ ΔVij(R ij) (6)

i=1 j=1

with ΔVij(R ij) = −f6 (bcorr , R ij)

ΔC6iso 4R ij6

− f8 (bcorr , R ij)

ΔC8iso 4R ij8 (7)

where Vadj is the adjusted analytical site−site PES, Vunadj is the unadjusted one, sites 1 and 2 are those closest to the two ab initio carbon atoms (see Figure 1), ΔC6iso = CDOSD , and the 6iso − C6iso value of ΔC8iso was chosen such that the resulting relative correction for C8iso is identical to that for C6iso (i.e., +6.4%). The damping parameter bcorr was adjusted such that the calculated values for the second virial coefficient agree with the best experimental data (see Section 4.1). The adjustment procedure C



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3.2. Dilute Gas Transport Properties. The kinetic theory expression for the shear viscosity η of a dilute polyatomic gas in the first-order approximation is6

increases the maximum well depth of the PES from 748.4 K to 769.1 K. Unless otherwise noted, all thermophysical property values reported in this work were obtained using Vadj. A Fortran 90 routine computing the PES as well as tables of the symmetry-distinct minimum structures for both Vadj and Vunadj and the corresponding interaction energies are provided in the Supporting Information.

NA 2

∫0

∞

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

where ⟨v⟩= 4(kBT/πm) is the average relative thermal speed and σ(2000) denotes a temperature-dependent generalized cross section, which is determined by the binary collisions in the gas and is thus directly linked to the intermolecular PES. The expressions for higher-order approximations contain further generalized cross sections.3,6 If we make the reasonable assumptions that vibrationally inelastic and resonant collisions are rare and that the vibrational motion does not significantly influence the trajectories, we can approximate the thermal conductivity λ in the dilute gas limit as the sum of a rigid-rotor part λrr and a vibrational part λvib,9,41,42

dR Ω1, Ω 2

(8)

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. The mass and moments of inertia of ethane are large enough to justify a semiclassical treatment of quantum effects by means of a modification of the pair potential known as the quadratic Feynman−Hibbs (QFH) effective pair potential.36 We employed this scheme already in previous studies; see, for example, refs 4, 33, and 37. For the ethane molecule pair, the QFH potential is given by VQFH = V + 1 + 2

λ = λrr + λ vib

(11)

In the first-order approximation, the rigid-rotor part is given by λrr =

(1) (1) (1) (1) 5kB2T S11 − rS21 − rS12 + r 2S22 2m⟨v⟩ S(1)

(1)

where S

⎡ ∂ 2V ⎞ ℏ2 ⎢ 1 ⎛ ∂ 2V ∂ 2V ⎟ ⎜ 2 + 2 + 12kBT ⎢⎣ m ⎝ ∂x ∂y ∂z 2 ⎠ ⎛ 2 2 2 ⎞⎤ ∑ ⎜⎜ 1 ∂ V2 + 1 ∂ V2 + 1 ∂ V2 ⎟⎟⎥⎥ I⊥ ∂ψi , b I⊥ ∂ψi , c ⎠⎦ i = 1 ⎝ I ∂ψi , a

(10) 1/2

3. CALCULATION OF THERMOPHYSICAL PROPERTIES 3.1. Second Virial Coefficient. The classical second virial coefficient for rigid molecules is given as B2cl = −

kBT ⟨v⟩σ(2000)

η=

is a determinant of rigid-rotor cross sections, ⎛1010 ⎞ ⎟ σ(1010)rr σ ⎜ ⎝1001⎠rr

S(1) =

2

(12)

⎛1001⎞ ⎟ σ(1001)rr σ⎜ ⎝1010 ⎠rr

(13)

S(1) ij

and are its minors. The dimensionless parameter r is defined as

(9)

⎛ 2 Crot ⎞1/2 r=⎜ ⎟ ⎝ 5 kB ⎠

where ℏ is Planck’s constant divided by 2π; m is the molecular mass; x, y, and z are the Cartesian components of R; I∥ and I⊥ are the moments of inertia parallel and perpendicular to the molecular figure axis, respectively; and the angles ψi,a, ψi,b, and ψi,c correspond to rotations around the principal axes of molecule i, with axis a being the figure axis. We computed the second virial coefficient at a large number of temperatures in the range from (90 to 1200) K by means of the Mayer-sampling Monte Carlo (MSMC) approach of Singh and Kofke38 implemented in an in-house software code that we also used in many previous studies; see, for example, refs 4, 33, 37, 39, and 40. Hard spheres with a diameter of 5 Å were employed as the reference system. The results at all temperatures were obtained simultaneously from multitemperature simulations,38,39 with the temperature governing the sampling distribution being 120 K. Unphysical negative interaction energies at very small intermolecular separations R were avoided by placing hard spheres with a diameter of 1 Å on all interaction sites. In each MC trial move, one of the molecules was displaced and rotated. Maximum step sizes for the moves were adjusted in short equilibration runs such that an acceptance rate of 50% was achieved. The second derivatives appearing in eq 9 were computed analytically. Calculated values for the second virial coefficient from 16 independent simulation runs of 2 × 1010 trial moves each were averaged. The standard uncertainty of the averages due to the MC integration decreases with temperature from 0.07 cm3·mol−1 at 90 K to only 0.003 cm3·mol−1 at 1200 K.

(14)

where Crot is the contribution of the rotational degrees of freedom to the ideal gas heat capacity. The vibrational contribution to the thermal conductivity is given by9,41,42 λ vib = NACvib ρm Dself

(15)

Here, Cvib is the contribution of the vibrational degrees of freedom to the ideal gas heat capacity, ρm is the molar density, and Dself is the self-diffusion coefficient. In the first-order approximation, the product of the latter two quantities is given as6 ρm Dself =

kBT NAm⟨v⟩σ ′(1000)

(16)

where σ′(1000) denotes the so-called self-part of the cross section σ(1000). An important quantity in the kinetic theory of gases is the dimensionless parameter A*, which is defined as6,7 A* =

5 σ(2000) 5 NAmρm Dself = η 6 σ ′(1000) 6

(17)

where ρmDself and η correspond to the first-order approximations of these quantities. For unlike interactions in gas mixtures, A* is analogously defined in terms of the binary diffusion coefficient and the so-called interaction viscosity.6,8 D



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Table 1. Classically Calculated Second Virial Coefficient, Bcl2 , and Semiclassically Calculated Second Virial Coefficient, BQFH , of Ethane and the Combined Expanded (k = 2) 2 Uncertainty of the Latter, U(BQFH ), as a Function of 2 Temperature T

Because A* as a function of an appropriately defined reduced temperature is only weakly dependent on the intermolecular PES, its main practical application is the estimation of diffusion coefficients from experimental viscosity data. In this work, we computed the shear viscosity in the thirdorder approximation3 and the thermal conductivity and the selfdiffusion coefficient in the second-order approximation.9 Values of Cvib for ethane were derived from the reference formulation of the ideal gas isochoric heat capacity43 by subtracting the translational (Ctr = 3kB/2) and classical rotational (Crot = 3kB/ 2) contributions. Thus, the contribution of the hindered internal rotation is part of Cvib. We determined the generalized cross sections needed for all three transport properties using the same basic methodology as in our previous investigations of transport properties of dilute molecular gases; see, for example, refs 3, 4, 44, and references therein. All generalized cross sections were calculated within the rigid-rotor approximation by means of the highly efficient and reliable classical trajectory method using an extended version of the TRAJECT code.45,46 The collision trajectories were obtained by integrating Hamilton’s equations from pre- to postcollisional values. To avoid any cutoff effects, the initial and final separation was set to 500 Å. The accuracy of the integration 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 with a maximum tolerated value of 10−4. Total-energy-dependent generalized cross sections in the center-of-mass frame, which are 13-dimensional integrals over the initial states of the trajectories, were computed at 37 values of the total energy from (60 to 30 000) K by means of a simple Monte Carlo integration scheme employing quasi-random numbers. Up to 4 × 106 collision trajectories were generated at each total energy value. As the computational effort required to determine a trajectory with the desired accuracy grows dramatically with decreasing energy, the number of trajectories had to be gradually reduced at energies below 300 K down to only 400 000 at 60 K. A weighted integration over the total energy then yielded temperature-dependent generalized cross sections in the center-of-mass frame46 for the temperature range from (90 to 1200) K. Finally, the center-of-mass cross sections were converted to laboratory frame cross sections,46,47 which are the ones appearing in the expressions for the transport properties. The relative standard uncertainty of the computed transport property values due to the Monte Carlo integration scheme used in the classical trajectory approach is estimated (based on uncertainty estimates generated by TRAJECT for the generalized cross sections, see ref 45 for details) to be smaller than 0.1% for shear viscosity and self-diffusion and 0.2% for thermal conductivity at all temperatures. The neglect of thirdand higher-order contributions to the thermal conductivity and the self-diffusion coefficient should not result in errors of more than (0.1−0.2)%. Uncertainties due to the neglect of fourthand higher-order contributions to the shear viscosity as well as due to the classical treatment of the collision dynamics, the numerical integration of Hamilton’s equations, and the numerical integration over the total energy should be negligibly small.

T (K)

Bcl2 (cm3·mol−1)

BQFH (cm3·mol−1) 2

U(BQFH ) (cm3·mol−1) 2

90 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 900 1000 1100 1200

−3948 −2638 −1912 −1466 −1171 −964.1 −812.3 −696.8 −606.6 −534.2 −475.1 −425.9 −349.0 −291.8 −247.7 −212.7 −184.2 −160.7 −140.9 −124.1 −109.6 −96.96 −85.88 −76.08 −67.36 −59.55 −52.52 −37.67 −25.81 −16.14 −8.12 −1.36 4.39 13.65 20.73 26.30 30.78

−3631 −2478 −1821 −1409 −1133 −937.5 −792.8 −682.1 −595.1 −525.0 −467.6 −419.7 −344.6 −288.5 −245.1 −210.6 −182.5 −159.3 −139.7 −123.1 −108.7 −96.17 −85.18 −75.45 −66.79 −59.04 −52.05 −37.29 −25.49 −15.86 −7.88 −1.16 4.57 13.79 20.85 26.40 30.86

60 33 21 14 10 7.8 6.2 5.0 4.2 3.6 3.1 2.7 2.1 1.8 1.5 1.3 1.1 1.0 0.9 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

atures. In Figure 3, the semiclassical values are compared with several experimental data sets,48−56 values resulting from the reference equation of state (EOS) of Bücker and Wagner,43 and the classical values, whereas the comparison with the semiclassical values resulting from the unadjusted PES, BQFH 2,unadj, is shown in Figure 4. The data of Funke et al.55 were derived from extremely accurate pρT measurements with a dual-sinker densimeter. We used their data close to room temperature as reference for the adjustment of the PES described in Section 2.3. Apart from only two data points, all of the experimental data depicted in Figure 3, which cover temperatures from (220 to 623) K, agree with the calculated values within ±1 cm3· mol−1. This is remarkable considering that the adjustment lowers the second virial coefficient quite substantially (e.g., by 16.2 cm3·mol−1 at 300 K). A similar observation was previously made for propane.4 It is also interesting to note that the increasing deviations of the data of Douslin and Harrison49 from the calculated values above about 500 K are not

4. RESULTS AND DISCUSSION 4.1. Second Virial Coefficient. Table 1 lists the classically calculated values for the second virial coefficient, Bcl2 , and the semiclassically calculated ones, BQFH , at 37 selected temper2 E



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Figure 3. Deviations of experimental data,48−56 of values derived from the current reference EOS,43 and of calculated values for the second virial coefficient of ethane from values calculated semiclassically using the new intermolecular PES: ○, Michels et al.;48 ●, Douslin and Harrison;49 □, Katayama et al.;50 ■, Mansoorian et al.;51 ◇, Jaeschke;52 ◆, Bell et al.;53 △, Estrada-Alexanders and Trusler;54 ▲, Funke et al.;55 ▽, Cristancho et ± U(BQFH ) with k = 2. al.;56 , EOS of Bücker and Wagner;43 −·−, classical result; ······, BQFH 2 2

mainly on the comparison with the experimental data, is given by QFH U (B2QFH ) = max(a1|B2QFH − B2,unadj | + a 2|B2QFH − B2cl | , a3)

(18)

where a1 = 0.06, a2 = 0.1 (accounting for the fact that the semiclassical QFH approach is not exact) and a3 = 0.8 cm3· mol−1. The resulting uncertainty values are provided in Table 1 and are also depicted in Figure 3. 4.2. Dilute Gas Transport Properties. The calculated values for the shear viscosity, thermal conductivity, and product of the molar density and the self-diffusion coefficient at 33 selected temperatures are listed in Table 2. Figure 5 shows the comparison of the calculated viscosity values with highly accurate experimental data,57−60 a recommended reference value at 298.15 K by Berg and Moldover,61 the zero-density part of the reference correlation of Vogel et al.,60 and values obtained from the unadjusted PES. The figure shows that the calculated values agree extremely well with all experimental data. In the case of the recommended value of Berg and Moldover, which has a relative standard uncertainty of only 0.033%, and the data of Vogel et al.,60 the agreement is essentially perfect. In contrast, the values obtained from the unadjusted PES differ significantly from all experimental data included in the comparison. We stress again that only experimental data for the second virial coefficient were used in the adjustment procedure. The zero-density part of the correlation of Vogel et al.60 is valid at temperatures from the triple point, Tt = 90.368 K,43 to 675 K with stated expanded (k = 2) uncertainties of 0.5% between (290 and 625) K, 1% between (212 and 290) K, and 6% below 212 K and above 625 K. The relative deviations from the calculated values are within ±0.5% above 150 K but increase to −6.7% at Tt.

Figure 4. Calculated values and experimental data48−56 for the second virial coefficient of ethane: − − −, semiclassical values resulting from the unadjusted PES; , semiclassical values resulting from the adjusted PES. The symbols represent the experimental data and have the same meaning as those in Figure 3.

corroborated by the values resulting from the reference EOS, whose upper limit of validity is 675 K. Our estimate for the combined expanded uncertainty (coverage factor k = 2 or approximately 95% confidence level) of the semiclassical values, U(BQFH ), which is based 2 F



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Table 2. Calculated Shear Viscosity η, Thermal Conductivity λ, and Product of the Molar Density and the Self-Diffusion Coefficient, ρmDself, of Ethane in the Dilute Gas Limit as a Function of Temperature Ta T (K) 90 105 120 135 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 650 700 750 800 850 900 950 1000 1100 1200

106 × η

103 × λ

104 × ρmDself

−1

−1

(Pa·s)

(W·m ·K )

(mol·m−1·s−1)

3.050 3.462 3.879 4.304 4.739 5.480 6.237 7.003 7.770 8.533 9.287 10.03 10.76 11.48 12.18 12.87 13.54 14.20 14.84 15.47 16.08 16.68 17.27 18.41 19.51 20.58 21.60 22.60 23.56 24.50 25.42 27.19 28.88

4.357 5.100 5.893 6.741 7.652 9.326 11.23 13.39 15.82 18.53 21.51 24.74 28.20 31.87 35.71 39.70 43.81 48.02 52.32 56.69 61.11 65.58 70.08 79.16 88.31 97.49 106.7 115.8 124.9 133.9 142.8 160.4 177.4

1.161 1.356 1.559 1.769 1.986 2.356 2.733 3.111 3.487 3.857 4.220 4.575 4.921 5.258 5.586 5.906 6.218 6.521 6.817 7.106 7.387 7.663 7.932 8.455 8.957 9.441 9.909 10.36 10.80 11.23 11.65 12.46 13.24

Figure 5. Relative deviations of experimental data,57−60 a recommended reference value,61 a correlation,60 and calculated values for the dilute gas shear viscosity of ethane from values calculated using the new intermolecular PES: ◇, Iwasaki and Takahashi;57 ▲, Hunter and Smith;58 □, Wilhelm et al.;59 ◆, Vogel et al.;60 ○, recommended value at 298.15 K by Berg and Moldover;61 , correlation of Vogel et al.;60 − − −, values resulting from the unadjusted PES; ······, ηcalc ± U(ηcalc) with k = 2.

Carmichael et al.,66 Yakush et al.,68 and Roder and Nieto de Castro.72 Some remarks need to be made regarding the four data points of Millat et al.,73 which were measured by means of the transient hot-wire (THW) technique and have stated uncertainties of at most 0.6%. They differ from the computed values by +1.5% at 308 K, +2.0% at 332 K, +4.2% at 380 K, and +6.3% at 426 K. We found similarly increasing deviations with temperature for THW data from the same laboratory for methane, carbon dioxide, and nitrous oxide from our respective calculated values and several other data sets.3,13,76 We also found such behavior for THW data from a different laboratory for the mixtures (CH4 + N2) and (CH4 + CO2).10,11 While there is now significant circumstantial evidence that THW measurements on dilute gases at temperatures above room temperature are often burdened with large systematic errors, the cause of the problem has not yet been identified.77 The zero-density correlation of Hendl et al.74 for temperatures from (225 to 725) K, which was adopted by Vesovic et al. for their thermal conductivity correlation of the entire fluid region,78 has a stated relative uncertainty of 2% between (300 and 500) K, increasing to 3% at the lowest and highest temperatures. Its deviations from the calculated values vary from −4.9% at 225 K to +4.8% at 725 K. It can be seen in Figure 6 that the correlation constitutes a compromise between several inconsistent data sets including that of Millat et al.73 Therefore, Hendl et al. stated that their correlation is “provisional” and that additional high-precision measurements, particularly at higher temperatures, are needed.74 However, no new measurements have been reported since then, and the correlation has therefore never been updated.

a

The relative combined expanded (k = 2) uncertainties Ur are Ur(η) = 0.3%, Ur(λ) = 2.0%, and Ur(ρmDself) = 1.0% at temperatures from (250 to 700) K, and Ur(η) = 1.0%, Ur(λ) = 3.0%, and Ur(ρmDself) = 2.0% at all other temperatures.

In Figure 6, the calculated thermal conductivity values are compared with most of the available experimental data,62−73 the correlations of Hendl et al.74 and Friend et al.,75 and the values for the unadjusted PES. It can be seen that many of the data sets are not mutually consistent within their claimed uncertainties (if uncertainties are provided at all), making it difficult to assess the quality of the calculated values. However, it is noteworthy to point out that the calculated values agree remarkably well with the data of Mann and Dickins,62 which were determined using the steady-state hot-wire technique. The average relative deviation from the calculated values is only −0.1%. This might be coincidental to a certain degree, but the data of Mann and Dickins for methane and propane from the same paper agree also very well with the respective calculated values4,13 with average relative deviations from the latter of +0.5% and −0.1%, respectively. Our calculated values also agree well with the data of Vines and Bennett,63 Lambert et al.,64 and Le Neindre,67 and at least partly also with the data of G



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Figure 7. Relative deviations of experimental data79 and calculated values for the dilute gas limit of the product of the molar density and the self-diffusion coefficient of ethane from values calculated using the new intermolecular PES: ●, Mueller and Cahill;79 − − −, values resulting from the unadjusted PES; ······, ρmDself,calc ± U(ρmDself,calc) with k = 2.

Figure 6. Relative deviations of experimental data,62−73 two correlations,74,75 and calculated values for the dilute gas thermal conductivity of ethane from values calculated using the new intermolecular PES: ○, Mann and Dickins;62 ◇, Vines and Bennett;63 ◆, Lambert et al.;64 ★, Leng and Comings;65 +, Carmichael et al.;66 ●, Le Neindre;67 ■, Yakush et al.;68 □, Fleeter and Kestin;69 ▼, Prasad and Venart;70 ▲, Zheng et al.;71 △, Roder and Nieto de Castro;72 ▽, Millat et al.;73 , correlation of Hendl et al.;74 −·−, correlation of Friend et al.;75 − − −, values resulting from the unadjusted PES; ······, λcalc ± U(λcalc) with k = 2.

calculated values. For the other two properties, the data situation does not allow to assess whether the same holds true, and our uncertainty estimates are therefore much more conservative. The reason that the estimate of the relative uncertainty is largest for the thermal conductivity is that of the three transport properties studied, it should be the only one appreciably affected by allowing for binary collisions that are inelastic with respect to the hindered internal rotation. At room temperature, such collisions were found to occur on average about once every 16 collisions.81 Finally, we examine the behavior of the dimensionless parameter A*. It is plotted as a function of temperature for CH4, C2H6, C3H8, and the unlike interaction CH4−C3H8 in Figure 8. The curves for CH4, C3H8, and CH4−C3H8 were computed from generalized cross sections obtained in previous studies.4,8,33 As expected, A*(T) changes in a systematic

The reference correlation of Friend et al.75 for temperatures from (90 to 600) K was published in the same year as the correlation of Hendl et al.74 Its stated uncertainty in the dilute gas limit is 10% below 200 K, 3% from (200 to 350) K, and 4% above 350 K. The deviations from the calculated values are as large as −33.8% at 90 K and +4.7% at 600 K. Figure 7 shows the comparison of the calculated values for the product of the molar density and the self-diffusion coefficient with the only available experimental data set79 and the results for the unadjusted PES. Mueller and Cahill79 derived their data from measurements of the binary diffusion coefficient of ethane and ethane-d1 using the approach developed by Winn.80 They did not provide estimates of the uncertainty of their data, but the agreement with the calculated values can be considered as satisfactory when taking into account that it appears to be very difficult to obtain self-diffusion coefficients with good precision using the approach of Winn, see the comparison in ref 12 of our calculated values for methane with the respective data by Winn.80 Estimates of the relative combined expanded (k = 2) uncertainties of the calculated values for the three transport properties are provided in the footnote of Table 2 and are also depicted in Figures 5−7. The largest source of uncertainty is the treatment of the ethane molecule as a rigid rotor in all stages of the calculation. For the shear viscosity, the comparison with the experimental data showed that this shortcoming of our approach is mostly compensated by the adjustment of the PES to experimental data for the second virial coefficient, thus allowing for very small uncertainties to be assigned to the

Figure 8. Dimensionless parameter A* as a function of temperature: , CH4; − − −, C2H6; −·−, C3H8; ······, CH4−C3H8. H



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manner from methane to ethane to propane, and the curve for the interaction between methane and propane is close to that for ethane.

and appears to extrapolate reasonably to temperatures below 90 K and above 1200 K. The uncertainty of the correlation corresponds to that of the calculated values given in Table 1. The new correlations for the three transport properties in the dilute gas limit are based exclusively on the calculated values of the present work at 145 temperatures from (90 to 1200) K and are of the form

5. CORRELATIONS A correlation for the second virial coefficient was obtained by fitting an expression of the form ∑i 6= 1 bi/(T*)ki with T* = T/ (100 K) to the semiclassical values at 116 temperatures from (90 to 1200) K. The exponents ki are positive integer multiples of 1/2, whose optimal values were found along with the coefficients bi using the symbolic regression software Eureqa (version 1.24.0).82 The resulting correlation is given by B2QFH 3 −1

cm ·mol

= b1 +

b2 1/2

+

b3 b4 + T* (T *)3 b6

(T *) b5 + + (T *)6 (T *)21/2

106 × η T̅ 1/2 = Pa·s Sη(T̅ )

(20)

Cp0 T̅ 1/2 103 × λ = kB Sλ(T̅ ) W·m−1·K−1

(21)

104 × ρm Dself mol ·m−1·s−1

(19)

bi

hi

li

di

44.8446 227.152 −946.805 −1136.01 −622.656 −44.2151

0.79330 262.946 13.8366 1339.77 −322.242

−0.16481 10.6848 1115.78 170.862

1.30405 13.7494 245.838 −10587.2 119559

(22)

where T̅ = T/K, and is the isobaric ideal gas heat capacity (in J·K−1) as given as a function of temperature by Bücker and Wagner.43 If η and ρmDself were obtained from the first-order kinetic theory (see eqs 10 and 16), the functions Sη(T̅ ) and SD(T̅ ) would be proportional to the cross sections σ(2000) and σ′(1000), respectively. Furthermore, apart from a constant factor, the right-hand side of eq 21 would be identical to the simplified thermal conductivity expression of Thijsse et al.6,74,83,84 if the function Sλ(T̅ ) were to be replaced by the cross section σ(10E).6,74,83,84 The detailed functional forms of Sη(T̅ ), Sλ(T̅ ), and SD(T̅ ) were obtained again by symbolic regression using Eureqa.

Table 3. Coefficients of Eqs 19 and 23−25 i

T̅ 1/2 SD(T̅ )

C0p

where the coefficients b1 to b6 are given in Table 3. Equation 19 reproduces the calculated values to within ±0.02 cm3·mol−1

1 2 3 4 5 6

=

Figure 9. Correlations for the dilute gas shear viscosity (left panels), thermal conductivity (middle panels), and product of the molar density and the self-diffusion coefficient (right panels) of ethane. Solid lines represent the correlations of the present work, dotted lines the viscosity correlation of Vogel et al.60 and the thermal conductivity correlation of Hendl et al.,74 and dashed lines the thermal conductivity correlation of Friend et al.75 The correlations are colored in blue within the stated temperature ranges of validity and in red outside these ranges. I



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Following Laesecke and Muzny,85,86 who correlated the dilute gas viscosities of CO2 and CH4 obtained by us from kinetic theory3,12 in a similar way, T̅ was restricted to appear solely in integer powers of T̅ 1/6. In addition, we allowed only constants, exponential functions, and the operators addition, subtraction, multiplication, division, and negation to occur. For each transport coefficient, we obtained several solutions that reproduce the calculated values sufficiently well. The functions that represent the best compromise with respect to goodness of fit, mathematical complexity, and extrapolation behavior are Sη(T̅ ) = h1 + h2exp( −T̅ 1/3) + + h5T̅ exp( − 2T̅

Sλ(T̅ ) = l1 +

l2 T̅

SD(T̅ ) = d1 +

1/6

d2 T̅

1/3

+

experimental data is excellent not only for the second virial coefficient, where this is expected to some extent due to the adjustment of the PES to the best experimental data for this quantity, but also for the shear viscosity, for which the deviations from the most accurate data are of the order of only ±(0.1 to 0.2)% over a wide temperature range. In the case of the thermal conductivity, which is a quantity that is generally much harder to measure accurately than the shear viscosity, several of the partly contradictory data sets agree satisfactorily with the calculated values. The only available data set for the self-diffusion coefficient is also in good agreement with the values obtained from kinetic theory. For all four properties, the calculated values are recommended as reference data. Practical correlations based entirely on these values were also developed. The new correlations for the transport properties in the dilute gas limit should be incorporated into any future reference formulations covering the entire fluid region.

h3 + h4 exp( −T̅ 1/3) T̅ 1/2

1/3

)

(23)

l3 l4 + exp(T̅ 1/3)

(24)

■

+ T̅ 1/6[d3exp( −T̅ 1/3)

+ d4 exp( − 2T̅ 1/3) + d5T̅ 1/6exp(− 3T̅ 1/3)]

ASSOCIATED CONTENT

S Supporting Information *

(25)

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b01069. Details of the internal coordinates, results of the ab initio calculations for all investigated configurations, detailed description of the procedure employed to obtain the isotropic part of the C6 dispersion coefficient, minimum structures of the ethane molecule pair, and a Fortran 90 routine computing the new analytical potential function (ZIP)

where the coefficients h1 to h5, l1 to l4, and d1 to d5 are listed in Table 3. The correlations reproduce the calculated values for η, λ, and ρmDself to within ±0.02%, ±0.07%, and ±0.01%, respectively. The relative uncertainties correspond to those given in the footnote of Table 2. Figure 9 illustrates the extrapolation behavior of the new correlations for all three transport properties as well as of the viscosity correlation of Vogel et al.60 and the thermal conductivity correlations of Hendl et al.74 and Friend et al.75 The extrapolation behavior of the latter three correlations is unsatisfactory except for the correlation of Vogel et al. at low temperatures. The new correlations, on the other hand, extrapolate in a reasonable manner down to zero kelvin and up to several thousand kelvin (although, of course, real ethane would decompose instantly at such high temperatures).

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Robert Hellmann: 0000-0003-3121-6827 Funding

6. CONCLUSIONS Highly accurate values for the second virial coefficient and three traditional transport properties (shear viscosity, thermal conductivity, and self-diffusion coefficient) in the dilute gas limit were determined for ethane at temperatures from (90 to 1200) K. For this purpose, a state-of-the-art intermolecular PES for interactions between two rigid ethane molecules was developed. It is based on quantum-chemical ab initio calculations at the RI-MP220,21 and CCSD(T)17 levels of theory for 3456 mutual configurations and is represented analytically by a site−site potential function with 11 sites per molecule. To account for deficiencies mainly due to treating ethane as a rigid rotor, a physically motivated correction term was added to the PES. This term contains a single adjustable parameter, which was chosen such that agreement with the best experimental data for the second virial coefficient was achieved. The maximum well depth of the PES is increased from 748.4 K to 769.1 K by the inclusion of the correction term. The second virial coefficient was determined by performing MSMC 38 simulations, in which quantum effects were accounted for semiclassically,36 while the three transport properties were computed using the classical trajectory approach in conjunction with the kinetic theory of polyatomic gases.3,6,9,45,46 The agreement between calculated values and

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

The author declares no competing financial interest.

■

ACKNOWLEDGMENTS The author wishes to thank Prof. Eckhard Vogel, Dr. Sebastian Herrmann, and Dr. Arno Laesecke for helpful discussions on viscosity correlations.

■

REFERENCES

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Reference Values for the Second Virial Coefficient and Three Dilute

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