Interpreting Gas-Saturation Vapor-Pressure Measurements UsingVirial Coefficients Derived from Molecular ModelsShu Yang,† Andrew J. Schultz,† David A. Kofke,*,† and Allan H. Harvey‡

†Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York14260-4200, United States‡Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado80305, United States

*S Supporting Information

ABSTRACT: We calculate virial coefficients of gas mixtures to demonstrate their use forinterpreting gas-saturation measurements of the vapor pressure of low-volatilitycompounds. We obtain coefficients from molecular models, via calculation of Mayerintegrals that rigorously connect the models and the coefficients. We examine He, CO2, N2,and SF6 as carrier gases, and n-C14H30 and n-C20H42 as prototype low-volatility compounds,considering both united-atom (UA) and explicit-hydrogen (EH) alkane models for them.Both the pure virial coefficients of every species and the cross-coefficients of each gas withn-C20H42 are calculated up to third order; cross-coefficients of SF6 with n-C14H30 and allEH-based coefficients are given only to second order. Using these coefficients, we calculate corrections to the vapor pressure of n-C20H42 at 323.15 K for all four carrier gases. With the corrections, the derived vapor pressures in He, CO2, and N2 carrier gasesare in excellent agreement, resolving most of the variation observed in apparent vapor pressures when gas-phase nonideality isneglected. Results are less satisfactory for SF6 as the carrier gas. We also calculate corrections to vapor-pressure data for n-C14H30at (283.15, 293.15, 303.15, and 313.15) K in an SF6 carrier gas.

■ INTRODUCTIONExperimental measurements of vapor pressures of low-volatilitysubstances, such as large organic molecules, can be performedvia a gas-saturation technique.1 In this method, the gas (labeled1) at the temperature of interest T and a convenient (perhapsatmospheric) pressure p is allowed to flow across the purecondensed-phase low-volatility component (labeled 2). Theflow is controlled to be slow enough that all of the gas isequilibrated with the condensed phase. At the conclusion of theexperiment (which may last for many days), the total amount ofthe low-volatility solute carried away by the gas is determined,as is the total amount of carrier gas used. This informationallows the calculation of the equilibrium vapor mole fraction, y2,which can then be used to infer the vapor pressure, p2

sat(T), ofthe pure low-volatility compound. A rigorous formula thatconnects the experimental data to p2

sat can be derived from thestandard thermodynamic formalism for vapor−liquid (orvapor−solid) equilibria:2,3

ϕγ ϕ

=Φ

ppy

x2sat 2 2

2 2 2sat

(1)

Here, ϕ2 is the fugacity coefficient of the low-volatilitycomponent in the mixture with the carrier gas, while ϕ2

sat isthis quantity as a pure gas at p2

sat; Φ is the Poynting correction,relating the fugacity of species 2 at p and p2

sat:

∫Φ =⎡⎣⎢

⎤⎦⎥

vRT

Pexp dp

p2

2sat

(2)

with v2 being the molar volume of pure species 2 in itscondensed form and R the gas constant. Finally, allowing forthe possibility that some of the carrier gas may be dissolved inthe condensed phase, x2 is the species 2 mole fraction in thecondensed phase at equilibrium with the gas mixture and γ2 isthe corresponding activity coefficient.At this point, several approximations of roughly increasing

severity may be introduced:(1) The condensed phase is incompressible (v2 is constant):

Φ = exp[(p − p2sat)v2/RT].

(2) p2satv2/RT ≪ 1: Φ = exp[pv2/RT].

(3) The gas of pure species 2 at p2sat is ideal: ϕ2

sat = 1.(4) Species 2 in the equilibrated condensed phase at p

behaves ideally because x2 ≈ 1: γ2 = 1.(5) The carrier gas has a negligible solubility in the

condensed phase: x2 = 1.(6) pv2/RT ≪ 1: Φ = 1.(7) The gas mixture is an ideal gas: ϕ2 = 1.If we invoke the first four approximations, then eq 1 reduces

to

ϕ ϕ=

Φ=p

py

x

py

x pv RTexp( / )2sat 2 2

2

2 2

2 2 (3)

Special Issue: Modeling and Simulation of Real Systems

Received: March 13, 2014Accepted: June 19, 2014

Article

pubs.acs.org/jced

© XXXX American Chemical Society A dx.doi.org/10.1021/je500245f | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

which is appealing because all of the terms on the right-handside are independent of p2

sat. If we go on and invoke all of theapproximations, then

=p py2sat

2 (4)

Equation 4 requires no input apart from the experimental data,so it is often used in interpreting gas-saturation measurementsfor p2

sat. However, sometimes it is found that the value obtainedthis way depends upon the choice of carrier gas,4 indicating thatsome treatment of nonideality is needed to interpret theexperimental data properly.The molar volume v2 is relatively easy to obtain, so in general

it is not difficult to apply the Poynting correction. If a Henry’sconstant or solubility is known or can be approximated for thecarrier gas in the condensed phase, then x2 can be estimated.Otherwise, one may need to assume x2 is unity; often thisassumption introduces negligible error, but its appropriatenessin general depends on the system.The gas-phase fugacity coefficients can be obtained by

integration of volumetric data from zero pressure. For themixture, this requires knowledge of the partial molar volume v2:

∫φ =

−⎛⎝⎜

⎞⎠⎟

vRT P

Pln1

dp

2 0

2

(5)

Equation 5 is evaluated with the aid of an analytic equation ofstate (EOS), expressing the volume of the mixture in terms oftemperature, pressure, and composition (mole fraction). TheEOS can be developed from experiments and/or from theory.In the former, experimental data are used to obtain theparameters of any of a number of commonly used models,typically a cubic EOS. However, it is difficult to obtain enoughdata to make a regular practice of fitting the model to mixtureproperties. It is more common instead to use available data togenerate pure-fluid equations of state and then treat the mixturebehavior by invoking semiempirical combining rules to definecomposition-dependent parameters for the EOS model.Mixture equations of state obtained in this way are oftenunreliable,5 particularly for components that are quite differentfrom each other, which is precisely the case in most applicationsof the gas-saturation technique.There is reason to expect better results by using molecular-

based models to describe the mixture properties. Such modelscan incorporate the key details that distinguish the behavior ofvolatile and low-volatility molecules, and thereby provide abetter basis for formulating thermodynamic models for theirmixtures. The molecular-level combining rules are likely to bemuch more robust than those formed at the thermodynamiclevel. In the best case, we would arrive at a treatment thatpermits calculations from first-principles computational chem-istry methods, involving no model fitting, but the present stateof the art of ab initio techniques limits these possibilities.Regardless, the aim then is to derive the gas-phase behaviorfrom the molecular models, whatever their source.One avenue is molecular simulation,6 which provides an

essentially exact description of the bulk behavior given adetailed molecular model. However, this approach suffers fromthe same drawback as experiment; namely, it yields “data”rather than an analytical EOS, so its use in interpreting gas-saturation results entails model fitting. Alternatively, we couldrecognize that simulation is capable of providing a value for ϕ2directly (without eq 5), and we might attempt to interface it

with the gas-saturation experiment, but this procedure is likelyto be cumbersome and inefficient.A much better approach in the current context is the virial

equation of state (VEOS),7,8 which nicely combines themolecular and engineering models. The VEOS expresses thecompressibility factor Z as a power series in molar density, ρ, asrepresented in eq 6:

∑ρ

ρ≡ = +=

∞−Z

pRT

B1n

nn

2

1

(6)

where Bn is the nth-order virial coefficient, which depends ontemperature and composition but not density. For use with eq5, it is more convenient to work with the pressure-series formof the VEOS, which is obtained by reverting the series in eq 6;thus,

= + + − +⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠Z B

pRT

B Bp

RT1 ( ) ...2 3 2

22

(7)

We denote VEOSn as the VEOS truncated after the nth term,which includes the coefficient Bn (e.g., VEOS3 includes termsto order ρ2 in eq 6 or p2 in eq 7). An important feature of theVEOS is that its composition dependence is given rigorously interms of a set of well-defined parameters associated with eachcoefficient Bn.

2 In particular, the coefficient Bn for a binarymixture depends on the mole fraction y2 according to

∑= −=

−−⎜ ⎟⎛

⎝⎞⎠B

nk

B y y(1 )nk

n

k n kk n k

0( ) 2 2

(8)

The temperature-dependent coefficient Bij depends on theinteractions of i molecules of species 1 and j molecules ofspecies 2, as detailed in the next section.Combining eqs 5, 7, and 8, one obtains an expression for the

fugacity coefficient, including terms up to p2:

φ = + −

+ + +

− + − −

⎜ ⎟

⎜ ⎟⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠⎤⎦⎥⎛⎝

⎞⎠

B y B y Bp

RT

B y B y y B y

B B y B y B Bp

RT

ln [2( ) ]

32

( 2 )

2 ( )32

2 11 1 02 2 2

21 12

12 1 2 03 22

2 11 1 02 2 3 22

2

(9)

A density series can also be derived:

φ ρ

ρ

= +

+ + + −

y B y B

y B y y B y B Z

ln 2 ( )32

( 2 ) ln

2 1 11 2 02

21

221 1 2 12 2

203 (10)

To use this form, the density is determined for the givenpressure and temperature via eq 6 and then inserted into eq 10to get ϕ2. In the present work, we use only the form given by eq9.Despite its theoretical advantages, the VEOS sees limited

application, for several reasons. At sufficiently high pressure, thetruncated series loses accuracy. While it can in principle beimproved systematically through the addition of more terms,high-order coefficients are not available for most substances. Inaddition, the VEOS is inapplicable to condensed phases, so itcannot serve as a global equation of state for fluid behavior. Forthe present purposes, however, these issues are irrelevant,because we employ the equation only to correct for nonideal-

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gas-phase behavior at rather low pressures, and the low-orderVEOS is perfectly suited for this task.Values of B2 and B3 obtained from regression of experimental

data are tabulated for pure substances and mixtures.7 Inaddition, correlations have been proposed9,10 from correspond-ing states based on critical properties and perhaps the acentricfactor, and some of them can determine the EOS to goodaccuracy. However, given that it is the measurement of thevapor pressure of the low-volatility component that motivatesthe analysis, it is unlikely that its critical properties or acentricfactor would be available for use in these correlations. Further,even if B2 for the low-volatility component could becharacterized by a correlation, it is the cross coefficients (i.e.,B11, B12, B21, etc.) that are needed by the gas-saturation method,and direct knowledge of these quantities is almost certainly notgoing to be available. Consequently, combining rules must bedeveloped to generate them from the pure-species values,thereby reintroducing the key undesirable feature of cubicEOSs that led us to consider molecular-based approaches tobegin with.The principal advantage of virial series is that the virial

coefficients can be calculated rigorously from interactionsamong a small number of molecules, via formulas expressed interms of cluster integrals.11 The integrals are multidimensionaland the computational complexity increases as the order ofvirial coefficient increases. Previous efforts to use virialcoefficients from molecular models to interpret gas-saturationdata have been limited to small solute species such as Hg andH2O.

12,13 The Mayer-sampling Monte Carlo method (MSMC)has opened up much more opportunity for the directcalculation of virial coefficients from molecular models forsystems where such computations were previously prohib-itive.14 In the present work, we employ this technique inconjunction with realistic molecular models to evaluate mixturevirial coefficients needed to improve the interpretation of gas-saturation data. Specifically, we examine four choices of carriergas that are often used in experiment: helium (He), carbondioxide (CO2), nitrogen (N2), and sulfur hexafluoride (SF6).We use normal eicosane (n-C20H42) as a prototype for the low-volatility component and analyze the experimental data foreicosane in these four carrier gases measured by Widegren etal.4 Additionally, we analyze some data for normal tetradecane(n-C14H30) in SF6 reported for four temperatures.15

This work is organized as follows. In the next section, wedetail the molecular models used for the species studied here,and we describe the methods used to calculate the virialcoefficients from these models. We also describe a correspond-ing-states correlation that we use for comparison to themolecular-model-based coefficients. We then present ourresults and provide concluding remarks.

■ COMPUTATIONAL METHODS

Molecular Models. The four carrier gases are eachcharacterized via rigid-body molecular models, withoutconsideration of intramolecular interactions. We use thetransferable potentials for phase equilibria (TraPPE) for CO2and N2,

16 which are explicit-atom models with point charges.For He, we use a semiclassical Lennard-Jones (LJ) atom, withsize and energy parameters fit to the computed second virialcoefficient of He over the range (250 to 400) K.17 Morespecifically, the He−He interaction is given by a quadraticFeynman−Hibbs modification of the LJ potential u12(r):

= + ℏ ∂∂

+∂∂

⎡⎣⎢

⎤⎦⎥u r u

k T mu

r rur

( )24 ( /2)

212QFH

12 12

2

B

212

122

12

12

12

(11)

where ℏ is the reduced Planck constant, kB is the Boltzmannconstant, and m is the mass of a He atom. The LJ parametersobtained by fitting B2 for this potential to the second virialcoefficient of He are given in Table 1.

For SF6, we examine several models. Olivet and Vegaproposed a flexible six-site model18 (denoted here as “6-site”);we expect the flexibility to have little effect in our application,so we consider this model using a rigid-body implementation.Dellis and Samios adjusted the 6-site model, using differentpotential parameters (“6-site, optimized”), and they alsomodeled SF6 as seven interaction sites (“7-site”).19 We alsoconsider a simple 1-site model using a single LJ atom for theSF6 molecule, with parameters determined via fit toexperimental second virial values from (240 to 440) K.20,21

These parameters are included in Table 1.For n-C14H30 and n-C20H42, we employ the TraPPE-united-

atom model (TraPPE-UA).22 TraPPE-UA force fields treat thealkane chain as flexible and include internal degrees of freedom,i.e., a harmonic bond-bending potential and an associateddihedral potential. The carbon and its bonded hydrogens areunited into one interaction site represented by a LJ sphere; themethyl (CH3) and methylene groups (CH2) are treated asdistinct pseudoatoms. The bonds that connect the united atomshave fixed lengths. For a subset of calculations, we also applythe TraPPE-explicit-hydrogen model (TraPPE-EH)23 toC14H30 and C20H42. In the TraPPE-EH model, hydrogenatoms are modeled as Lennard-Jones interaction sites. For allmodels, we use Lorentz−Berthelot (LB) combining rules tocalculate potential parameters for unlike interaction sites onmolecules of different species. We treat alkane−heliuminteractions classically, using the LJ parameters for He givenin Table 1 with the LB combining rules.

Mayer-Sampling Monte Carlo. For pairwise-additivemodels, the virial coefficients can be expressed as integrals ofthe Mayer function, f ij = exp(−uij/kBT) − 1, where uij is theinteraction energy of molecules i and j given the positions oftheir atoms, which are integrated over to yield the desiredcoefficient. The virial coefficient Bk(n−k) is determined via suchan integral involving k molecules of species 1 and (n − k)molecules of species 2. Corrections to the virial coefficients arerequired for B3 and higher order coefficients when computedfor molecular models having conformational degrees offreedom;24−26 these terms are included in the calculations,where appropriate. Diagrams are frequently used to representthe cluster integrals.11 We use solid points to represent anintegral over the coordinates of a molecule and a lineconnecting two points to represent a Mayer function. Figures1 and 2 illustrate the diagrams involved in the second and thirdvirial coefficients of a binary mixture. The squares correspondto rigid carrier-gas molecules, and the solid circles correspondto flexible molecules (n-C14H30 or n-C20H42). All of the singlyconnected diagrams in Figure 2 are flexible corrections

Table 1. Lennard-Jones Potentials of He and SF6

(ϵ/kB)/K σ/Å

He 7.8875 2.5848SF6 178.9 6.133

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(indicated with “C” in the label). These diagrams contain atleast one flexible molecule at an articulation point.11

The Mayer-sampling Monte Carlo (MSMC) method is usedto evaluate the virial-coefficient integrals.14 We adopt theoverlap sampling technique,27 using as a reference the integralsin the corresponding virial coefficient for hard spheres ofdiameters (4.25, 5.0, and 8.0) Å when calculating pure virialcoefficients of CO2, N2 and SF6, respectively, and (σCH3

+ 0.5n)

Å when calculating virial coefficients of C14H30 or C20H42 andall cross virial coefficients for the TraPPE-UA model; σCH3

is

the Lennard-Jones parameter for the CH3 group and equals3.75 Å for the TraPPE-UA model; n is the number of carbonsin the alkane chain: 14 for tetradecane and 20 for eicosane. Forthe TraPPE-EH model, the diameter of the hard sphere in thereference system is (σC + 0.5n) Å; σC equals 3.3 Å. The choiceof a reference system has in principle no effect on the resultsand can be adjusted within a broad range of values to optimizethe efficiency of the calculation. For each virial coefficient, 10independent simulations with 109 attempted Monte Carlo trialsare performed. The sampling of the configurational degrees offreedom for the flexible molecule is performed as described inref 28, and the flexible corrections are computed as described inref 26. The second virial coefficients of He and single-site SF6are computed via direct quadrature method, while third virialcoefficients of these gases are computed using Fouriertransforms.Corresponding-States Correlation. Tsonopoulos10 pro-

posed a correlation for B2 of pure components, expressed as afunction of critical temperature, Tc, critical pressure, pc, andacentric factor, ω. Defining the reduced temperature Tr ≡ T/Tc,

ω= +

= − −− −

= + − −

− −

− −

− − −

BRT

pF T F T

F T T TT T

F T T T T

[ ( ) ( )],

( ) 0.1445 0.330 0.13850.0121 0.000607 ,

( ) 0.0637 0.331 0.423 0.008

2c

c

(0)r

(1)r

(0)r r

1r

2

r3

r8

(1)r r

2r

3r

8

(12)

We apply the Tsonopoulos correlation to cross second virialcoefficients B11 with the approach of Chueh and Prausnitz.29

The cross critical constants are calculated from criticalproperties of each component with the combining rules:

ω ω ω

= +

= −

= +

= +

⎡⎣⎢

⎤⎦⎥v v v

T T T k

p p v T p v T T v

12

( ) ,

(1 ),

12

( / / ) / ,

12

( )

ij i j

ij i j ij

ij i i i j j j ij ij

ij i j

c, c,1/3

c,1/3

3

c, c, c,

c, c, c, c, c, c, c, c, c,

(13)

The characteristic constant kij of some binaries is tabulatedby Chueh and Prausnitz.29 For binaries where kij is notavailable, it is estimated by

= −kv v

v1

( )ij

i j

ij

c, c,1/2

c, (14)

Table 2 lists the critical constants and acentric factors of allcomponents for which we used the Tsonopoulos correlationand their sources.

As pointed out in the Introduction, without knowledge of thecritical properties of both species, as well as the acentric factor(which is determined from the vapor pressure), thesecorrelations cannot be employed. Hence, they will most likelynot be helpful in application to interpreting gas-saturationmeasurements of p2

sat. Nevertheless, it is of interest to examinethem in this study, as they provide a standard to gauge ourpredictions of mixture properties from the molecular models,and the necessary data are available for the prototype low-volatility components.

■ RESULTS AND DISCUSSIONPure-Component Virial Coefficients. First we compare

our calculated pure-component virial coefficients to thoseavailable from experiment and correlation. This comparison canbe used to characterize the accuracy of the molecular models.For interpretation of the gas-saturation data of n-C20H42, ourinterest is for the single temperature of 323.15 K, but it isworthwhile to examine the behavior of the virial coefficientsover a broader range of temperature. Inaccuracies in a modelthat cause a discrepancy in the virial coefficient at onetemperature can be manifested at a different temperatureoneperhaps showing no discrepancy in the virial coefficientwhenthe model is used for a different purpose, such as for computinga mixture cross virial coefficient.Results for helium are shown in Figure 3. The model was fit

to these B2 data from (250 to 400) K, and it describes themwell. The comparison for B3 is not as good, which can beascribed to (1) compounding of the approximation of the two-

Figure 1. Cluster integrals for binary mixture with one flexiblemolecule (circle) at second order.

Figure 2. Cluster integrals for binary mixture with one flexiblemolecule (circle) at third order.

Table 2. Critical Constants for the Systems Examined inThis Work

component Tc/K pc/bar vc/(cm3/mol) ω

CO2 304.1282a 73.773a 94.12a 0.224e

N2 126.192b 33.958b 89.41b 0.0372e

SF6 318.7232c 37.5498c 196.77c 0.21e

n-C14H30 693g 15.7g 894g 0.644g

n-C20H42 769d 10.8d 1284f 0.882f

aReference 42. bReference 43. cReference 21. dReference 44.eReference 20. fReference 45. gReference 2.

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body He−He interaction with a simple LJ form and (2) neglectof nonadditive (three-body) interactions.Comparisons with experimental virials for CO2 and N2 are

given in Figures 4 and 5, respectively. These comparisons are a

true test of the TraPPE model potentials, which were fit toproperties over a broad range of state conditions, including theliquid. Given that they are not optimized for low-densitybehavior, they perform rather well. For CO2, the comparisonshows that the TraPPE B2 is slightly higher (less negative) thanvalues from experiment over all temperatures, while thedeviation for B3 is more considerable. At the temperature ofinterest, the error in B2 is less than 5 %. The performance forN2 is about the same as that observed for CO2. The differencebetween the TraPPE B2 and experiment at the temperature ofinterest appears worse in fractional terms, but this is onlybecause it is near the Boyle temperature for N2.For SF6, as with He, the 1-site LJ model provides a good fit

of the B2 over the range of temperatures considered (Figure 6),

while the behavior of B3 is again not described as well (Figure7). In contrast, although B2 values from the 6- and 7-site

models are slightly larger than the reference B2, especially atlow temperature, B3 values from these two models agree withreference data well and behave much better than the single-siteLJ model. The multisite models were formulated by fitting to abroad range of experimental data, including the liquid phase, soone should expect them to capture the three-body behaviorbetter than the model that is fit to just B2, while not performingas well in describing B2 itself.

Figure 3. Second virial coefficient B2 and third virial coefficient B3 ofHe from (200 to 500) K: red lines with squares, B2 (solid line) and B3(dashed line) for semiclassical LJ potential with parameters given inTable 1; black lines, B2 from an accurate ab initio potential17 (solidline) and B3 from REFPROP20 (dashed line); black triangles, B3 fromaccurate ab initio two-body and three-body potentials.46

Figure 4. Second virial coefficient B2 and third virial coefficient B3 ofCO2 from (200 to 500) K: red lines with squares, B2 (solid line) andB3 (dashed line) from TraPPE model; black line with circles, B2 fromcorrelation (eq 12); black lines, B2 (solid line) and B3 (dashed line)from Span and Wagner.20,42

Figure 5. Second virial coefficient B2 and third virial coefficient B3 ofN2 from (200 to 500) K: red lines with squares, B2 (solid line) and B3(dashed line) from TraPPE model; black line with circles, B2 fromcorrelation (eq 12); black lines, B2 (solid line) and B3 (dashed line)from Span et al.20,43

Figure 6. Second virial coefficient B2 of SF6 from (200 to 500) K: redline with squares, 1-site LJ potential; green line with triangles, 7-sitemodel; green line with diamonds, 6-site model; black line with circles,correlation (eq 12); black line, Guder and Wagner.20,21

Figure 7. Third virial coefficient B3 of SF6 from (200 to 500) K: reddashed line with squares, 1-site LJ potential; green dashed line withtriangles, 7-site model; green dashed line with diamonds, 6-site model;black dashed line, Guder and Wagner.20,21

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Values of B2 for n-C14H30 and n-C20H42 from the TraPPE-UAmodel, shown in Figure 8, agree fairly well with values from the

Tsonopoulos correlation,10 but the use of the log scale in thefigure perhaps makes the agreement look better than it is.Values of B2 for C14H30 from TraPPE-UA are consistently lessnegative than the correlation, while the comparison for C20H42shows some variation with temperaturethe TraPPE-UAvalues are more negative than correlation at lower temperaturesand less negative for temperatures above 325 K. Both theTraPPE-UA model and the correlation are being used outsideof the range of models or conditions used for their fitting, so itis not possible to determine what gives rise to the discrepancybetween them. There are no data or correlations for the thirdvirial coefficient for these alkanes, so we do not include B3 inthe figure.Our B2 and B3 results are tabulated in the Supporting

Information.Mixture Virial Coefficients. We consider now the cross

second virial coefficients. No experimental data for B11 of anymixture considered here are available, so we compare ourresults just with B11 from correlation. For the mixtures with He,even correlation cannot provide a meaningful basis forcomparison, because the temperature of interest is muchhigher than the critical temperature of He. Hence, we show B11only for CO2/C20H42 (Figure 9), N2/C20H42 (Figure 9), andSF6/C20H42 (Figure 10). B11 tends to adopt behavior that isqualitatively between that of the pure components. The B11curve of CO2/C20H42 shows the best agreement withcorrelation, especially at temperatures higher than 350 K. B11results of N2/C20H42 display consistent negative deviation,whereas B11 results of SF6/C20H42 (Figure 10) are generally lessnegative than those from correlation and are more negative fortemperatures higher than about 280 K. As with the pure-component virials, the B11 values calculated from thecorresponding-states correlation represent extrapolation out-side the range in which the correlation was fitted, soquantitative conclusions cannot be drawn from thesecomparisons. However, the qualitative agreement suggeststhat our molecular-model results are physically reasonable.Gas-Saturation Data Analysis. We turn now to

calculation of the saturation vapor pressure of n-C20H42 at323.15 K. The pressure under which the gas-saturationexperiments were conducted is p = 83.2 kPa, and molefractions y2 of C20H42 in each carrier gas measured and reported

by Widegren et al.4 are given in Table 3. The solubilities of thecarrier gas in the liquid phase (x1; also given in Table 3) are

estimated based on experimental Henry’s constants or solubilitydata;30−35 in some cases extrapolation in temperature and/orsolvent carbon number was required. Since this solubilityprovides only a small correction to the derived vapor pressure,these rough estimates are adequate for our purposes.We use the virial coefficients obtained above to determine

the saturation vapor pressure of C20H42 following the proceduredetailed in the Introduction. The results for various

Figure 8. Second virial coefficient B2 of n-C14H30 and n-C20H42 from(200 to 500) K: red lines with squares, TraPPE model; black lines,correlation (eq 12). Dashed lines are for C14H30, and solid lines are forC20H42.

Figure 9. Second cross virial coefficient B11 of CO2/C20H42 and N2/C20H42 from (200 to 500) K: red triangle, B11 of CO2/C20H42 and N2/C20H42 from TraPPE-EH model at 323.15 K; red lines with squares,TraPPE-UA model; black lines, correlation (eq 12). Solid lines areCO2/C20H42, and dashed lines are N2/C20H42.

Figure 10. Second cross virial coefficient B11 of SF6/C20H42 from (200to 500) K: black triangles, B11 of SF6 (triangle up, 1-site LJ potential;triangle down, 7-site model)/C20H42 from the TraPPE-EH model at323.15 K. All lines are from the TraPPE-UA model: red line withsquares, 1-site LJ potential; green line with triangles, 7-site model;green line with diamonds, 6-site model; black line with circles,correlation (eq 12).

Table 3. Reported4 Mole Fractions (y2) of n-C20H42 in EachCarrier Gas, And Estimated Mole Fraction (x1) of CarrierGas Dissolved in the Liquida

carrier gas y2 × 107 x1

He 5.454 0.0003CO2 5.702 0.010N2 5.564 0.0012SF6 6.825 0.006

aAll data are at 323.15 K.

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approximations to eq 1 are listed in Table 4, and key results areillustrated graphically in Figure 11.

Assuming accurate data from experiment, if all factors in eq 1are properly handled (or can be safely ignored), then the vaporpressure from the gas-saturation procedure should give a resultthat is independent of the choice of carrier gas. Clearly this isnot the case if only the idealized eq 4 is applied. The Poyntingcorrection is practically the same for all carrier gases (using aconstant molar volume36 of C20H42, Φ = 1.011, which lowers allp2sat by about 1 %), so it does not remove the discrepancy in p2

sat

across them. In contrast, the solubilities of the gases in C20H42are significantly different (Table 3), and the correction for thisraises the derived p2

sat by 0−1 %. At 323.15 K, the second virialcoefficients are negative (except B11 of He/C20H42), so VEOS2lowers p2

sat (except for He). The effect of the third-order virialcoefficient is completely negligible, except for the SF6 carriergas, where it lowers p2

sat by about half a percent.

The virial-corrected results for He, CO2, and N2 carrier gasesare in excellent mutual agreement, with scatter on the order of1 %, compared to roughly 5 % scatter when the data areinterpreted with ideal-gas assumptions. For He, the Poyntingcorrection and the effect of the vapor-phase fugacity coefficientroughly cancel each other, leaving a corrected vapor pressureclose to that resulting from ideal-gas assumptions. We alsoexamined alternative LJ parameters of He from the literature.37

These variants changed the computed p2sat of C20H42 by less

than 0.1 %, which is not significant in this context, so we do notreport them here. For CO2 and N2, failure to consider gasnonideality would lead to errors of several percent in thederived vapor pressures. The results for these carrier gases areinsensitive to the choice of the UA or EA formulation of theTraPPE alkane potential.The picture is somewhat different for SF6. It is clearly the

most nonideal case, but the calculated correction for non-ideality using the TraPPE-UA model is only about 60 % of themagnitude needed to bring the corrected vapor pressures intoagreement with the other data. Addition of the third virialcoefficient does not improve the agreement significantly, nordoes use of more sophisticated models for the SF6 molecule.However, the use of TraPPE-EH for C20H42 with the simple 1-site model for SF6 does yield a significant further reduction inp2sat, almost to within the range of variation of the other data.Unfortunately, when we employ TraPPE-EH with the explicit-atom (7-site) model for SF6, the improvement is lost, and werecover a vapor pressure not much different from that given byTraPPE-UA. It seems then that the improvement shown byTraPPE-EH + 1-site SF6 is fortuitous.The general discrepancy exhibited by the SF6 data could have

two possible explanations. Either there was some systematicproblem with the experiments that used SF6 as the carrier gasor else the molecular model does not correctly represent themixture. The TraPPE-EH model yields significantly bettersecond virial coefficients for alkanes than the TraPPE-UAmodel,23 so we had some hope that switching to it might yield aconsistent improvement. The fault may lie with the use of the

Table 4. Saturation Vapor Pressure of n-C20H42 in Different Carrier Gases for TraPPE-UA and TraPPE-EH Models at 323.15Ka

p2sat/(10−2 Pa)

py2ϕ2/Φx2

carrier gas TraPPE model py2 (id) py2/Φx2 VEOS2 VEOS2/C VEOS3 ϕ2 py2/p2sat

He UA 4.538 4.488 4.522 − 4.527 1.0087 1.0024EH 4.527 − 1.0087 1.0024

CO2 UA 4.744 4.738 4.479 4.480 4.479 0.9454 1.0593EH 4.481 − 0.9457 1.0587

N2 UA 4.629 4.582 4.483 4.514 4.483 0.9782 1.0327EH 4.481 − 0.9778 1.0331

SF61-site UA 5.678 5.648 4.889 5.067 4.885 0.865 1.1621-site EH 4.585 − 0.812 1.2387-site UA 5.000 4.997 0.885 1.1367-site EH 5.038 − 0.892 1.1276-site UA 4.978 4.945 0.876 1.1486-site, optimized UA 4.988 4.956 0.877 1.146

aValues are determined using the indicated approximate form of eq 1; “id” denotes the idealized treatment given by eq 4. VEOSn includes ϕ2employing the truncated virial equation using coefficients determined from the molecular models, and VEOS2/C is the same but using thecorrelation. For the TraPPE-UA model, the tabulated value of ϕ2 is based on VEOS3 and is, for the assumed molecular model, exact to the digitsgiven. The last column is the enhancement factor, given as the ratio of p2

sat from eq 4 divided by that value based on VEOS3. For TraPPE-EH, all ofthe tabulated values are based on VEOS2.

Figure 11. Saturation vapor pressure of n-C20H42 from eq 4 (labeled“Ideal”) and from VEOS2, modeling C20H42 with TraPPE-UA andTraPPE-EH as labeled, in different carrier gases (from left to rightwithin each set: He, CO2, N2, and SF6(1-site) and SF6(7-site)).

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standard combing rules used to obtain the unlike-pair potentialfrom the SF6 and n-C20H42 pair potentials. It has been observedthat interactions between fluorocarbons and hydrocarbons arenot represented well by simple combining rules or correspond-ing-states treatments.38−40 What is true for fluorocarbons mightalso be true for SF6.Finally, we estimate the vapor pressure of n-C14H30 from data

with the SF6 carrier gas using TraPPE-UA and TraPPE-EHmodels. Measurements at four temperatures were performed byWidegren and Bruno.15 The estimated mole fraction of C14H30in the condensed phase is estimated from experimental Henry’sconstants or solubility data.34,35 The molar volume data arefrom Valencia et al.41 We employed the 1-site model for SF6and estimate p2

sat with VEOS truncated at second order. Asshown in Table 5, the results for the TraPPE-UA model fromVEOS2 agree well with those from correlation, and they arelower than those from the ideal case. The experimental p2

sat

reported by Widegren and Bruno are based on the ideal-gasassumption and are without the Poynting or solubilitycorrections. The apparent measured vapor pressures are 5−10% above the corrected values. This demonstrates theimportance of including gas nonideality. The calculated p2

sat

for the TraPPE-EH model with the 1-site SF6 model are evenlower than those from the TraPPE-UA model, by aboutanother 5 %. However, given the discrepant results mentionedabove for SF6 with C20H42, our corrected p2

sat for C14H30 withSF6 as the carrier gas probably still contains significantinaccuracy.

■ CONCLUSION

We have demonstrated how virial coefficients computed frommolecular models can be applied to improve estimates of thevapor pressure of low-volatility compounds measured in gas-saturation experiments. For the experiments analyzed here, wefind that the third-order virial term can be neglected, but thatthe second-order virial term produces a correction of severalpercent. In other situations these outcomes could differ,depending on the pressure and temperature of the experimentand the nature of the molecular species involved. We note twotrends that will be true in general: (1) the vapor nonidealitycorrections for experiments near sea level will be somewhatlarger than for the experiments analyzed here, which wereperformed at lower pressure due to the facility’s location at ca.1700 m elevation; (2) as is evident in Table 5, the vapor

nonideality effect will be larger at lower temperatures. This isbecause B11 becomes increasingly negative at lower temper-atures for mixtures of gases with large organic molecules.The Poynting correction and solubility of the carrier gas in

the solute can also be significant at the percent level. ThePoynting correction is simple to compute from the molarvolume of the solvent (which can be reasonably estimated if itis not known), so there is no reason not to include it. Thesolubility of the gas in the liquid may not be known, so tominimize the corresponding uncertainty, it may be preferable tochoose a less soluble gas such as nitrogen or (especially)helium.Our results for vapor nonideality also suggest some factors

that experimentalists can consider in choosing the carrier gasfor a gas-saturation experiment. The nonideality correction willin general be smallest for helium, and if high accuracy is notrequired it will often be reasonable to neglect it. Helium may,however, have disadvantages, both because of its propensity toleak and because its low molar mass makes it more difficult toaccurately measure gravimetrically. The nonideality correctionwill be relatively small for nitrogen, and it is likely that argonwould have similar characteristics. SF6 is perhaps the worstpossible choice, both because of its large deviation from ideal-gas behavior and the difficulty of molecular models to representits interactions with hydrocarbon molecules. Of course if one isgoing to apply a nonideality correction as we havedemonstrated here, a key consideration would be theavailability of an accurate pair potential for the interaction ofthe solute with the carrier gas.For the prototype systems studied here, we could turn to

correlation to provide some verification of the virial-seriescalculations, but in most cases one will not have the dataneeded to implement the correlation, in which case one mustturn to virial coefficients computed from a molecular model. Ingeneral, even when correlations are available, it is reasonable toexpect that virial coefficients computed from a good molecularmodel will be more reliable, particularly when used to evaluatemixture cross-coefficients.As molecular modeling continues to advance, it will become

increasingly practical to turn to these methods as tools forinterpreting experimental data, as shown in the gas-saturationanalysis demonstrated here. Further improvements in theaccuracy and transferability of molecular models will certainlyenhance these prospects. Even more, the increasing practicalityof ab initio methods to quantify intermolecular interactions will

Table 5. Saturation Vapor Pressure of n-C14H30 in SF6 for TraPPE-UA and TraPPE-EH Models Using the IndicatedApproximate Form of Equation 1a

p2sat/Pa

py2ϕ2/Φx2

T/K TraPPE model py2 (id) py2/Φx2 VEOS2 VEOS2/C ϕ2 py2/p2sat

283.15 UA 0.421 0.420 0.364 0.371 0.867 1.155EH 0.342 0.813 1.231

293.15 UA 1.192 1.189 1.047 1.067 0.880 1.139EH 0.990 0.832 1.204

303.15 UA 3.181 3.176 2.831 2.888 0.891 1.124EH 2.695 0.849 1.180

313.15 UA 7.995 7.975 7.189 7.336 0.901 1.112EH 6.883 0.863 1.162

a“id” denotes the idealized treatment given by eq 4. VEOS2 includes ϕ2 employing the second-order virial equation with the 1-site SF6 model, andVEOS2/C is the same but using the correlation. Tabulated values of ϕ2 are based on VEOS2 and are, for the assumed molecular model, exact to thedigits given. The last column is the enhancement factor, based on p2

sat from VEOS2.

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make it possible to characterize the interactions of differentspecies, without requiring experiment or some formulationbased on combining the pure-species behaviors. In all of thesecases, virial coefficients provide a very convenient route fromthe molecular behavior to the bulk properties.

■ ASSOCIATED CONTENT*S Supporting InformationTable of virial coefficients calculated in this work. This materialis available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: kofke@buffalo.edu.FundingThis work is supported by the U.S. National ScienceFoundation (Grant No. CHE-1027963).NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Jason Widegren for helpful discussions and formaking his data available prior to publication.

■ REFERENCES(1) Delle Site, A. The Vapor Pressure of Environmentally SignificantOrganic Chemicals: A Review of Methods and Data at AmbientTemperature. J. Phys. Chem. Ref. Data 1997, 26, 157−193.(2) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. MolecularThermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall:Upper Saddle River, NJ, USA, 1999.(3) Dahm, K. D.; Visco, D. P. Fundamentals of Chemical EngineeringThermodynamics; Cengage Learning: Stamford, CT, USA, 2014.(4) Widegren, J. A.; Harvey, A. H.; McLinden, M. O.; Bruno, T. J.Vapor Pressure Measurements by the Gas Saturation Method: TheInfluence of the Carrier Gas. J. Chem. Eng. Data 2014, manuscript inpreparation.(5) Cullen, E. J.; Kobe, K. A. Benedict equation of state: Applicationto vapor−liquid equilibria. AIChE J. 1955, 1, 452−455.(6) Frenkel, D.; Smit, B. Understanding Molecular Simulation: FromAlgorithms to Applications; Academic Press: San Diego, CA, USA, 2001.(7) Dymond, J. D.; Marsh, K. N.; Wilhoit, R. C. Virial Coefficients ofPure Gases and Mixtures; Springer-Verlag: New York, 2002.(8) Carruth, G. F.; Kobayashi, R. Vapor pressure of normal paraffinsethane through n-decane from their triple points to about 10 mmmercury. J. Chem. Eng. Data 1973, 18, 115−126.(9) McGlashan, M. L.; Potter, D. J. B. An Apparatus for theMeasurement of the Second Virial Coefficients of Vapours; TheSecond Virial Coefficients of Some n-Alkanes and of Some Mixtures ofn-Alkanes. Proc. R. Soc. London, Ser. A 1962, 267, 478−500.(10) Tsonopoulos, C. An empirical correlation of second virialcoefficients. AIChE J. 1974, 20, 263−272.(11) Hansen, J.; McDonald, I. Theory of Simple Liquids, 3rd ed.;Academic Press: San Diego, CA, USA, 2006.(12) Bielska, K.; Havey, D. K.; Scace, D. E.; Lisak, D.; Harvey, A. H.;Hodges, J. T. High-accuracy measurements of the vapor pressure of icereferenced to the triple point. Geophys. Res. Lett. 2013, 40, 6303−6307.(13) Harvey, A. H. Estimation of the Enhancement Factor forMercury in Air. Int. J. Thermophys. 2010, 31, 297−307.(14) Singh, J. K.; Kofke, D. A. Mayer Sampling: Calculation ofCluster Integrals using Free-Energy Perturbation Methods. Phys. Rev.Lett. 2004, 92, No. 220601.(15) Widegren, J. A.; Bruno, T. J. Gas Saturation Vapor PressureMeasurements of Mononitrotoluene Isomers from (283.15 to 313.15)K. J. Chem. Eng. Data 2010, 55, 159−164.

(16) Potoff, J. J.; Siepmann, J. I. Vapor−liquid equilibria of mixturescontaining alkanes, carbon dioxide, and nitrogen. AIChE J. 2001, 47,1676−1682.(17) Cencek, W.; Przybytek, M.; Komasa, J.; Mehl, J. B.; Jeziorski, B.;Szalewicz, K. Effects of adiabatic, relativistic, and quantum electro-dynamics interactions on the pair potential and thermophysicalproperties of helium. J. Chem. Phys. 2012, 136, No. 224303.(18) Olivet, A.; Vega, L. F. Optimized molecular force field for sulfurhexafluoride simulations. J. Chem. Phys. 2007, 126, No. 144502.(19) Dellis, D.; Samios, J. Molecular force field investigation forsulfur hexafluoride: A computer simulation study. Fluid Phase Equilib.2010, 291, 81−89.(20) Lemmon, E. W.; McLinden, M. O.; Huber, M. L. NIST ReferenceFluid Thermodynamic and Transport PropertiesREFPROP, NISTStandard Reference Database 23, Version 9.1; National Institute ofStandards and Technology: Gaithersburg, MD, USA, 2013.(21) Guder, C.; Wagner, W. A Reference Equation of State for theThermodynamic Properties of Sulfur Hexafluoride (SF6) for Temper-atures from the Melting Line to 625 K and Pressures up to 150 MPa. J.Phys. Chem. Ref. Data 2009, 38, 33−94.(22) Martin, M. G.; Siepmann, J. I. Transferable Potentials for PhaseEquilibria. 1. United-Atom Description of n-Alkanes. J. Phys. Chem. B1998, 102, 2569−2577.(23) Chen, B.; Siepmann, J. I. Transferable Potentials for PhaseEquilibria. 3. Explicit-Hydrogen Description of Normal Alkanes. J.Phys. Chem. B 1999, 103, 5370−5379.(24) Caracciolo, S.; Mognetti, B. M.; Pelissetto, A. Virial coefficientsand osmotic pressure in polymer solutions in good-solvent conditions.J. Chem. Phys. 2006, 125, No. 094903.(25) Caracciolo, S.; Mognetti, B.; Pelissetto, A. Third virial coefficientfor 4-arm and 6-arm star polymers. Macromol. Theory Simul. 2008, 17,67−72.(26) Shaul, K. R. S.; Schultz, A. J.; Kofke, D. A. Mayer-samplingMonte Carlo calculations of uniquely flexible contributions to virialcoefficients. J. Chem. Phys. 2011, 135, No. 124101.(27) Benjamin, K. M.; Schultz, A. J.; Kofke, D. A. Gas-PhaseMolecular Clustering of TIP4P and SPC/E Water Models fromHigher-Order Virial Coefficients. Ind. Eng. Chem. Res. 2006, 45, 5566−5573.(28) Schultz, A. J.; Kofke, D. A. Virial coefficients of model alkanes. J.Chem. Phys. 2010, 133, No. 104101.(29) Chueh, P. L.; Prausnitz, J. M. Third virial coefficients ofnonpolar gases and their mixtures. AIChE J. 1967, 13, 896−902.(30) Gasem, K. A. M.; Robinson, R. L., Jr. Solubilities of carbondioxide in heavy normal paraffins (C20−C44) at pressures to 9.6 MPaand temperatures from 323 to 423 K. J. Chem. Eng. Data 1985, 30,53−56.(31) Schwarz, B. J.; Prausnitz, J. M. Solubilities of methane, ethane,and carbon dioxide in heavy fossil-fuel fractions. Ind. Eng. Chem. Res.1987, 26, 2360−2366.(32) Huang, S. H.; Lin, H. M.; Chao, K. C. Solubility of carbondioxide, methane, and ethane in n-eicosane. J. Chem. Eng. Data 1988,33, 145−147.(33) Tong, J.; Gao, W.; Robinson, R. L., Jr.; Gasem, K. A. M.Solubilities of Nitrogen in Heavy Normal Paraffins from 323 to 423 Kat Pressures to 18.0 MPa. J. Chem. Eng. Data 1999, 44, 784−787.(34) Hesse, J. J.; Battino, R.; Scharlin, P.; Wilhelm, E. Solubility ofGases in Liquids. 20. Solubility of He, Ne, Ar, Kr, N2, O2, CH4, CF4and SF6 in n-Alkanes n-ClH2l+2 (6 ≤ l ≤ 16) at 298.15 K. J. Chem. Eng.Data 1996, 41, 195−201.(35) Wilcox, R. J.; Battino, R. Solubilities of gases in liquids II. Thesolubilities of He, Ne, Ar, Kr, O2, N2, CO, CO2, CH4, CF4, and SF6 inn-octane, 1-octanol, n-decane, and 1-decanol. J. Chem. Thermodyn.1978, 10, 817−822.(36) Dutour, S.; Daridon, J.; Lagourette, B. Speed of sound, density,and compressibilities of liquid eicosane and docosane at varioustemperatures and pressures. High Temp. − High Pressures 2001, 33,371−378.

Journal of Chemical & Engineering Data Article

dx.doi.org/10.1021/je500245f | J. Chem. Eng. Data XXXX, XXX, XXX−XXXI

(37) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties ofGases and Liquids, 5th ed.; McGraw-Hill Professional: New York,2000.(38) Siebert, E. M. D.; Knobler, C. M. Interaction virial coefficients inhydrocarbon−fluorocarbon mixtures. J. Phys. Chem. 1971, 75, 3863−3870.(39) Lo Nostro, P.; Scalise, L.; Baglioni, P. Phase Separation inBinary Mixtures Containing Linear Perfluoroalkanes. J. Chem. Eng.Data 2005, 50, 1148−1152.(40) Pratas de Melo, M. J.; Dias, A. M. A.; Blesic, M.; Rebledo, L. P.N.; Vega, L. F.; Coutinho, J. A. P.; Marrucho, I. M. Liquid-liquidequilibrium of (perfluoroalkane + alkane) binary mixtures. Fluid PhaseEquilib. 2006, 242, 210−219.(41) Valencia, J. L.; Gonzalez-Salgado, D.; Troncoso, J.; Peleteiro, J.;Carballo, E.; Romaní, L. Thermophysical Characterization of LiquidsUsing Precise Density and Isobaric Heat Capacity Measurements as aFunction of Pressure. J. Chem. Eng. Data 2009, 54, 904−915.(42) Span, R.; Wagner, W. A New Equation of State for CarbonDioxide Covering the Fluid Region from the Triple-Point Temper-ature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data1996, 25, 1509−1596.(43) Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.;Yokozeki, A. A Reference Equation of State for the ThermodynamicProperties of Nitrogen for Temperatures from 63.151 to 1000 K andPressures to 2200 MPa. J. Phys. Chem. Ref. Data 2000, 29, 1361−1433.(44) Nikitin, E. D.; Pavlov, P. A.; Popov, A. P. Vapour-liquid criticaltemperatures and pressures of normal alkanes with from 19 to 36carbon atoms, naphthalene and m-terphenyl determined by the pulse-heating technique. Fluid Phase Equilib. 1997, 141, 155−164.(45) Frenkel, M.; Chirico, R. D.; Diky, V.; Kroenlein, K.; Muzny, C.D.; Kazakov, A. F.; Magee, J. W.; Abdulagatov, I. M.; Lemmon, E. W.NIST ThermoData Engine, NIST Standard Reference Database 103b,version 8; National Institute of Standards and Technology:Gaithersburg, MD, USA, 2013.(46) Shaul, K. R. S.; Schultz, A. J.; Kofke, D. A. Path-integral Mayer-sampling calculations of the quantum Boltzmann contribution to virialcoefficients of helium-4. J. Chem. Phys. 2012, 137, No. 184101.

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