Interpreting Gas-Saturation Vapor-Pressure ... - ACS Publications

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Interpreting Gas-Saturation Vapor-Pressure Measurements Using Virial Coefficients Derived from Molecular Models Shu 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 York 14260-4200, United States ‡ Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, United States S Supporting Information *

ABSTRACT: We calculate virial coefficients of gas mixtures to demonstrate their use for interpreting gas-saturation measurements of the vapor pressure of low-volatility compounds. We obtain coefficients from molecular models, via calculation of Mayer integrals 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 with n-C20H42 are calculated up to third order; cross-coefficients of SF6 with n-C14H30 and all EH-based coefficients are given only to second order. Using these coefficients, we calculate corrections to the vapor pressure of nC20H42 at 323.15 K for all four carrier gases. With the corrections, the derived vapor pressures in He, CO2, and N2 carrier gases are in excellent agreement, resolving most of the variation observed in apparent vapor pressures when gas-phase nonideality is neglected. Results are less satisfactory for SF6 as the carrier gas. We also calculate corrections to vapor-pressure data for n-C14H30 at (283.15, 293.15, 303.15, and 313.15) K in an SF6 carrier gas.

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INTRODUCTION Experimental measurements of vapor pressures of low-volatility substances, such as large organic molecules, can be performed via a gas-saturation technique.1 In this method, the gas (labeled 1) at the temperature of interest T and a convenient (perhaps atmospheric) pressure p is allowed to flow across the pure condensed-phase low-volatility component (labeled 2). The flow is controlled to be slow enough that all of the gas is equilibrated with the condensed phase. At the conclusion of the experiment (which may last for many days), the total amount of the low-volatility solute carried away by the gas is determined, as is the total amount of carrier gas used. This information allows the calculation of the equilibrium vapor mole fraction, y2, which can then be used to infer the vapor pressure, psat 2 (T), of the pure low-volatility compound. A rigorous formula that connects the experimental data to psat 2 can be derived from the standard thermodynamic formalism for vapor−liquid (or vapor−solid) equilibria:2,3 p2sat =

with v2 being the molar volume of pure species 2 in its condensed form and R the gas constant. Finally, allowing for the possibility that some of the carrier gas may be dissolved in the condensed phase, x2 is the species 2 mole fraction in the condensed phase at equilibrium with the gas mixture and γ2 is the 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 − psat 2 )v2/RT]. (2) psat 2 v2/RT ≪ 1: Φ = exp[pv2/RT]. sat (3) The gas of pure species 2 at psat 2 is ideal: ϕ2 = 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

py2 ϕ2 x 2γ2 Φϕ2sat

(1)

p2sat =

Here, ϕ2 is the fugacity coefficient of the low-volatility component in the mixture with the carrier gas, while ϕsat 2 is this quantity as a pure gas at psat 2 ; Φ is the Poynting correction, relating the fugacity of species 2 at p and psat 2 : ⎡ Φ = exp⎢ ⎣

∫p

p

sat 2

⎤ v2 dP ⎥ RT ⎦

x2Φ

=

py2 ϕ2 x 2 exp(pv2 /RT )

(3)

Special Issue: Modeling and Simulation of Real Systems Received: March 13, 2014 Accepted: June 19, 2014

(2) © XXXX American Chemical Society

py2 ϕ2

A

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

Journal of Chemical & Engineering Data

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with the gas-saturation experiment, but this procedure is likely to be cumbersome and inefficient. A much better approach in the current context is the virial equation of state (VEOS),7,8 which nicely combines the molecular and engineering models. The VEOS expresses the compressibility factor Z as a power series in molar density, ρ, as represented in eq 6:

which is appealing because all of the terms on the right-hand side are independent of psat 2 . If we go on and invoke all of the approximations, then p2sat = py2

(4)

Equation 4 requires no input apart from the experimental data, so it is often used in interpreting gas-saturation measurements for psat 2 . However, sometimes it is found that the value obtained this way depends upon the choice of carrier gas,4 indicating that some treatment of nonideality is needed to interpret the experimental 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’s constant or solubility is known or can be approximated for the carrier gas in the condensed phase, then x2 can be estimated. Otherwise, one may need to assume x2 is unity; often this assumption introduces negligible error, but its appropriateness in general depends on the system. The gas-phase fugacity coefficients can be obtained by integration of volumetric data from zero pressure. For the mixture, this requires knowledge of the partial molar volume v̂2: ln φ2 =

∫0

v̂ 1⎞ ⎜ 2 − ⎟ dP ⎝ RT P⎠

Z≡

p =1+ ρRT

∞

∑ Bnρn− 1

(6)

n=2

where Bn is the nth-order virial coefficient, which depends on temperature and composition but not density. For use with eq 5, it is more convenient to work with the pressure-series form of the VEOS, which is obtained by reverting the series in eq 6; thus, 2 ⎛ p ⎞ 2 ⎛ p ⎞ ⎟ + (B − B )⎜ ⎟ + ... Z = 1 + B2 ⎜ 3 2 ⎝ ⎝ RT ⎠ RT ⎠

(7)

We denote VEOSn as the VEOS truncated after the nth term, which includes the coefficient Bn (e.g., VEOS3 includes terms to order ρ2 in eq 6 or p2 in eq 7). An important feature of the VEOS is that its composition dependence is given rigorously in terms of a set of well-defined parameters associated with each coefficient Bn.2 In particular, the coefficient Bn for a binary mixture depends on the mole fraction y2 according to

p⎛

(5)

Equation 5 is evaluated with the aid of an analytic equation of state (EOS), expressing the volume of the mixture in terms of temperature, pressure, and composition (mole fraction). The EOS can be developed from experiments and/or from theory. In the former, experimental data are used to obtain the parameters of any of a number of commonly used models, typically a cubic EOS. However, it is difficult to obtain enough data to make a regular practice of fitting the model to mixture properties. It is more common instead to use available data to generate pure-fluid equations of state and then treat the mixture behavior by invoking semiempirical combining rules to define composition-dependent parameters for the EOS model. Mixture equations of state obtained in this way are often unreliable,5 particularly for components that are quite different from each other, which is precisely the case in most applications of the gas-saturation technique. There is reason to expect better results by using molecularbased models to describe the mixture properties. Such models can incorporate the key details that distinguish the behavior of volatile and low-volatility molecules, and thereby provide a better basis for formulating thermodynamic models for their mixtures. The molecular-level combining rules are likely to be much more robust than those formed at the thermodynamic level. In the best case, we would arrive at a treatment that permits calculations from first-principles computational chemistry methods, involving no model fitting, but the present state of the art of ab initio techniques limits these possibilities. Regardless, the aim then is to derive the gas-phase behavior from the molecular models, whatever their source. One avenue is molecular simulation,6 which provides an essentially exact description of the bulk behavior given a detailed molecular model. However, this approach suffers from the same drawback as experiment; namely, it yields “data” rather than an analytical EOS, so its use in interpreting gassaturation results entails model fitting. Alternatively, we could recognize that simulation is capable of providing a value for ϕ2 directly (without eq 5), and we might attempt to interface it

n

Bn =

⎛n⎞

∑ ⎜⎝ k ⎟⎠Bk(n− k)(1 − y2 )k y2n− k

(8)

k=0

The temperature-dependent coefficient Bij depends on the interactions of i molecules of species 1 and j molecules of species 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: ⎛ p ⎞ ⎟ ln φ2 = [2(B11y1 + B02 y2 ) − B2 ]⎜ ⎝ RT ⎠ ⎡3 + ⎢ (B21y12 + 2B12 y1y2 + B03y2 2 ) ⎣2 2 ⎛ 3 ⎞⎤⎛ p ⎟⎞ − 2B2 (B11y1 + B02 y2 ) − ⎜B3 − B2 2 ⎟⎥⎜ ⎝ 2 ⎠⎦⎝ RT ⎠

(9)

A density series can also be derived: ln φ2 = 2ρ(y1B11 + y2 B02 ) +

3 2 2 ρ (y1 B21 + 2y1y2 B12 + y2 2 B03) − ln Z 2

(10)

To use this form, the density is determined for the given pressure and temperature via eq 6 and then inserted into eq 10 to get ϕ2. In the present work, we use only the form given by eq 9. Despite its theoretical advantages, the VEOS sees limited application, for several reasons. At sufficiently high pressure, the truncated series loses accuracy. While it can in principle be improved systematically through the addition of more terms, high-order coefficients are not available for most substances. In addition, the VEOS is inapplicable to condensed phases, so it cannot serve as a global equation of state for fluid behavior. For the present purposes, however, these issues are irrelevant, because we employ the equation only to correct for nonidealB



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gas-phase behavior at rather low pressures, and the low-order VEOS 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 In addition, correlations have been proposed9,10 from corresponding states based on critical properties and perhaps the acentric factor, and some of them can determine the EOS to good accuracy. However, given that it is the measurement of the vapor pressure of the low-volatility component that motivates the analysis, it is unlikely that its critical properties or acentric factor would be available for use in these correlations. Further, even if B2 for the low-volatility component could be characterized 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 not going to be available. Consequently, combining rules must be developed to generate them from the pure-species values, thereby reintroducing the key undesirable feature of cubic EOSs that led us to consider molecular-based approaches to begin with. The principal advantage of virial series is that the virial coefficients can be calculated rigorously from interactions among a small number of molecules, via formulas expressed in terms of cluster integrals.11 The integrals are multidimensional and the computational complexity increases as the order of virial coefficient increases. Previous efforts to use virial coefficients from molecular models to interpret gas-saturation data have been limited to small solute species such as Hg and H2O.12,13 The Mayer-sampling Monte Carlo method (MSMC) has opened up much more opportunity for the direct calculation of virial coefficients from molecular models for systems where such computations were previously prohibitive.14 In the present work, we employ this technique in conjunction with realistic molecular models to evaluate mixture virial coefficients needed to improve the interpretation of gassaturation data. Specifically, we examine four choices of carrier gas that are often used in experiment: helium (He), carbon dioxide (CO2), nitrogen (N2), and sulfur hexafluoride (SF6). We use normal eicosane (n-C20H42) as a prototype for the lowvolatility component and analyze the experimental data for eicosane in these four carrier gases measured by Widegren et al.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, we detail the molecular models used for the species studied here, and we describe the methods used to calculate the virial coefficients from these models. We also describe a corresponding-states correlation that we use for comparison to the molecular-model-based coefficients. We then present our results and provide concluding remarks.

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QFH u12 (r12) = u12 +

⎡ ∂ 2u ℏ2 2 ∂u12 ⎤ ⎢ 122 + ⎥ 24kBT (m /2) ⎣ ∂r12 r12 ∂r12 ⎦ (11)

where ℏ is the reduced Planck constant, kB is the Boltzmann constant, and m is the mass of a He atom. The LJ parameters obtained by fitting B2 for this potential to the second virial coefficient of He are given in Table 1. Table 1. Lennard-Jones Potentials of He and SF6 He SF6

(ϵ/kB)/K

σ/Å

7.8875 178.9

2.5848 6.133

For SF6, we examine several models. Olivet and Vega proposed 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 different potential parameters (“6-site, optimized”), and they also modeled SF6 as seven interaction sites (“7-site”).19 We also consider a simple 1-site model using a single LJ atom for the SF 6 molecule, with parameters determined via fit to experimental 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-unitedatom model (TraPPE-UA).22 TraPPE-UA force fields treat the alkane chain as flexible and include internal degrees of freedom, i.e., a harmonic bond-bending potential and an associated dihedral potential. The carbon and its bonded hydrogens are united into one interaction site represented by a LJ sphere; the methyl (CH3) and methylene groups (CH2) are treated as distinct pseudoatoms. The bonds that connect the united atoms have fixed lengths. For a subset of calculations, we also apply the TraPPE-explicit-hydrogen model (TraPPE-EH)23 to C14H30 and C20H42. In the TraPPE-EH model, hydrogen atoms are modeled as Lennard-Jones interaction sites. For all models, we use Lorentz−Berthelot (LB) combining rules to calculate potential parameters for unlike interaction sites on molecules of different species. We treat alkane−helium interactions classically, using the LJ parameters for He given in Table 1 with the LB combining rules. Mayer-Sampling Monte Carlo. For pairwise-additive models, the virial coefficients can be expressed as integrals of the Mayer function, f ij = exp(−uij/kBT) − 1, where uij is the interaction energy of molecules i and j given the positions of their atoms, which are integrated over to yield the desired coefficient. The virial coefficient Bk(n−k) is determined via such an integral involving k molecules of species 1 and (n − k) molecules of species 2. Corrections to the virial coefficients are required for B3 and higher order coefficients when computed for molecular models having conformational degrees of freedom;24−26 these terms are included in the calculations, where appropriate. Diagrams are frequently used to represent the cluster integrals.11 We use solid points to represent an integral over the coordinates of a molecule and a line connecting two points to represent a Mayer function. Figures 1 and 2 illustrate the diagrams involved in the second and third virial coefficients of a binary mixture. The squares correspond to rigid carrier-gas molecules, and the solid circles correspond to flexible molecules (n-C14H30 or n-C20H42). All of the singly connected diagrams in Figure 2 are flexible corrections

COMPUTATIONAL METHODS

Molecular Models. The four carrier gases are each characterized via rigid-body molecular models, without consideration of intramolecular interactions. We use the transferable potentials for phase equilibria (TraPPE) for CO2 and N2,16 which are explicit-atom models with point charges. For He, we use a semiclassical Lennard-Jones (LJ) atom, with size and energy parameters fit to the computed second virial coefficient of He over the range (250 to 400) K.17 More specifically, the He−He interaction is given by a quadratic Feynman−Hibbs modification of the LJ potential u12(r): C



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⎤3 ⎡1 vc, ij = ⎢ (vc, i1/3 + vc, j1/3)⎥ , ⎦ ⎣2

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

Tc, ij =

Tc, iTc, j (1 − kij),

1 (p vc, i /Tc, i + pc, j vc, j /Tc, j)Tc, ij/vc, ij , 2 c, i 1 ωij = (ωi + ωj) 2 pc, ij =

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

The characteristic constant kij of some binaries is tabulated by Chueh and Prausnitz.29 For binaries where kij is not available, it is estimated by

(indicated with “C” in the label). These diagrams contain at least one flexible molecule at an articulation point.11 The Mayer-sampling Monte Carlo (MSMC) method is used to evaluate the virial-coefficient integrals.14 We adopt the overlap sampling technique,27 using as a reference the integrals in the corresponding virial coefficient for hard spheres of diameters (4.25, 5.0, and 8.0) Å when calculating pure virial coefficients of CO2, N2 and SF6, respectively, and (σCH3 + 0.5n)

kij = 1 −

(vc, ivc, j)1/2 vc, ij

(14)

Table 2 lists the critical constants and acentric factors of all components for which we used the Tsonopoulos correlation and their sources. Table 2. Critical Constants for the Systems Examined in This Work

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

component CO2 N2 SF6 n-C14H30 n-C20H42

the Lennard-Jones parameter for the CH3 group and equals 3.75 Å for the TraPPE-UA model; n is the number of carbons in the alkane chain: 14 for tetradecane and 20 for eicosane. For the TraPPE-EH model, the diameter of the hard sphere in the reference system is (σC + 0.5n) Å; σC equals 3.3 Å. The choice of a reference system has in principle no effect on the results and can be adjusted within a broad range of values to optimize the efficiency of the calculation. For each virial coefficient, 10 independent simulations with 109 attempted Monte Carlo trials are performed. The sampling of the configurational degrees of freedom for the flexible molecule is performed as described in ref 28, and the flexible corrections are computed as described in ref 26. The second virial coefficients of He and single-site SF6 are computed via direct quadrature method, while third virial coefficients of these gases are computed using Fourier transforms. Corresponding-States Correlation. Tsonopoulos10 proposed a correlation for B2 of pure components, expressed as a function of critical temperature, Tc, critical pressure, pc, and acentric factor, ω. Defining the reduced temperature Tr ≡ T/Tc, B2 =

(13)

a e

Tc/K

pc/bar a

304.1282 126.192b 318.7232c 693g 769d

a

73.773 33.958b 37.5498c 15.7g 10.8d

vc/(cm3/mol)

ω

94.12a 89.41b 196.77c 894g 1284f

0.224e 0.0372e 0.21e 0.644g 0.882f

Reference 42. bReference 43. cReference 21. Reference 20. fReference 45. gReference 2.

d

Reference 44.

As pointed out in the Introduction, without knowledge of the critical properties of both species, as well as the acentric factor (which is determined from the vapor pressure), these correlations cannot be employed. Hence, they will most likely not be helpful in application to interpreting gas-saturation measurements of psat 2 . Nevertheless, it is of interest to examine them in this study, as they provide a standard to gauge our predictions of mixture properties from the molecular models, and the necessary data are available for the prototype lowvolatility components.

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RESULTS AND DISCUSSION Pure-Component Virial Coefficients. First we compare our calculated pure-component virial coefficients to those available from experiment and correlation. This comparison can be used to characterize the accuracy of the molecular models. For interpretation of the gas-saturation data of n-C20H42, our interest is for the single temperature of 323.15 K, but it is worthwhile to examine the behavior of the virial coefficients over a broader range of temperature. Inaccuracies in a model that cause a discrepancy in the virial coefficient at one temperature can be manifested at a different temperatureone perhaps showing no discrepancy in the virial coefficientwhen the model is used for a different purpose, such as for computing a 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 them well. The comparison for B3 is not as good, which can be ascribed to (1) compounding of the approximation of the two-

RTc (0) [F (Tr) + ωF (1)(Tr)], pc

F (0)(Tr) = 0.1445 − 0.330Tr −1 − 0.1385Tr −2 − 0.0121Tr −3 − 0.000607Tr −8 , F (1)(Tr) = 0.0637 + 0.331Tr −2 − 0.423Tr −3 − 0.008Tr −8 (12)

We apply the Tsonopoulos correlation to cross second virial coefficients B11 with the approach of Chueh and Prausnitz.29 The cross critical constants are calculated from critical properties of each component with the combining rules: D



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true test of the TraPPE model potentials, which were fit to properties over a broad range of state conditions, including the liquid. Given that they are not optimized for low-density behavior, they perform rather well. For CO2, the comparison shows that the TraPPE B2 is slightly higher (less negative) than values from experiment over all temperatures, while the deviation for B3 is more considerable. At the temperature of interest, the error in B2 is less than 5 %. The performance for N2 is about the same as that observed for CO2. The difference between the TraPPE B2 and experiment at the temperature of interest appears worse in fractional terms, but this is only because 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),

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

body He−He interaction with a simple LJ form and (2) neglect of nonadditive (three-body) interactions. Comparisons with experimental virials for CO2 and N2 are given in Figures 4 and 5, respectively. These comparisons are a

Figure 6. Second virial coefficient B2 of SF6 from (200 to 500) K: red line with squares, 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); black line, Guder and Wagner.20,21

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

Figure 4. Second virial coefficient B2 and third virial coefficient B3 of CO2 from (200 to 500) K: red lines with squares, B2 (solid line) and B3 (dashed line) from TraPPE model; black line with circles, B2 from correlation (eq 12); black lines, B2 (solid line) and B3 (dashed line) from Span and Wagner.20,42

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

models are slightly larger than the reference B2, especially at low temperature, B3 values from these two models agree with reference data well and behave much better than the single-site LJ model. The multisite models were formulated by fitting to a broad range of experimental data, including the liquid phase, so one should expect them to capture the three-body behavior better than the model that is fit to just B2, while not performing as well in describing B2 itself.

Figure 5. Second virial coefficient B2 and third virial coefficient B3 of N2 from (200 to 500) K: red lines with squares, B2 (solid line) and B3 (dashed line) from TraPPE model; black line with circles, B2 from correlation (eq 12); black lines, B2 (solid line) and B3 (dashed line) from Span et al.20,43

E



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

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 are CO2/C20H42, and dashed lines are N2/C20H42.

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 for C20H42.

Tsonopoulos correlation,10 but the use of the log scale in the figure perhaps makes the agreement look better than it is. Values of B2 for C14H30 from TraPPE-UA are consistently less negative than the correlation, while the comparison for C20H42 shows some variation with temperaturethe TraPPE-UA values are more negative than correlation at lower temperatures and less negative for temperatures above 325 K. Both the TraPPE-UA model and the correlation are being used outside of the range of models or conditions used for their fitting, so it is not possible to determine what gives rise to the discrepancy between them. There are no data or correlations for the third virial coefficient for these alkanes, so we do not include B3 in the 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 any mixture considered here are available, so we compare our results just with B11 from correlation. For the mixtures with He, even correlation cannot provide a meaningful basis for comparison, because the temperature of interest is much higher than the critical temperature of He. Hence, we show B11 only for CO2/C20H42 (Figure 9), N2/C20H42 (Figure 9), and SF6/C20H42 (Figure 10). B11 tends to adopt behavior that is qualitatively between that of the pure components. The B11 curve of CO2/C20H42 shows the best agreement with correlation, especially at temperatures higher than 350 K. B11 results of N2/C20H42 display consistent negative deviation, whereas B11 results of SF6/C20H42 (Figure 10) are generally less negative than those from correlation and are more negative for temperatures higher than about 280 K. As with the purecomponent virials, the B11 values calculated from the corresponding-states correlation represent extrapolation outside the range in which the correlation was fitted, so quantitative conclusions cannot be drawn from these comparisons. However, the qualitative agreement suggests that our molecular-model results are physically reasonable. Gas-Saturation Data Analysis. We turn now to calculation of the saturation vapor pressure of n-C20H42 at 323.15 K. The pressure under which the gas-saturation experiments were conducted is p = 83.2 kPa, and mole fractions y2 of C20H42 in each carrier gas measured and reported

Figure 10. Second cross virial coefficient B11 of SF6/C20H42 from (200 to 500) K: black triangles, B11 of SF6 (triangle up, 1-site LJ potential; triangle down, 7-site model)/C20H42 from the TraPPE-EH model at 323.15 K. All lines are from the TraPPE-UA model: red line with squares, 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).

by Widegren et al.4 are given in Table 3. The solubilities of the carrier gas in the liquid phase (x1; also given in Table 3) are Table 3. Reported4 Mole Fractions (y2) of n-C20H42 in Each Carrier Gas, And Estimated Mole Fraction (x1) of Carrier Gas Dissolved in the Liquida

a

carrier gas

y2 × 107

x1

He CO2 N2 SF6

5.454 5.702 5.564 6.825

0.0003 0.010 0.0012 0.006

All data are at 323.15 K.

estimated based on experimental Henry’s constants or solubility data;30−35 in some cases extrapolation in temperature and/or solvent carbon number was required. Since this solubility provides 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 procedure detailed in the Introduction. The results for various F



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Table 4. Saturation Vapor Pressure of n-C20H42 in Different Carrier Gases for TraPPE-UA and TraPPE-EH Models at 323.15 Ka −2 psat Pa) 2 /(10

py2ϕ2/Φx2 carrier gas He CO2 N2

TraPPE model

py2 (id)

py2/Φx2

VEOS2

UA EH UA EH UA EH

4.538

4.488

−

4.744

4.738

4.629

4.582

4.522 4.527 4.479 4.481 4.483 4.481

UA EH UA EH UA UA

5.678

5.648

4.889 4.585 5.000 5.038 4.978 4.988

5.067

VEOS2/C

4.480 4.514

VEOS3

ϕ2

py2/psat 2

4.527 − 4.479 − 4.483 −

1.0087 1.0087 0.9454 0.9457 0.9782 0.9778

1.0024 1.0024 1.0593 1.0587 1.0327 1.0331

4.885 − 4.997 − 4.945 4.956

0.865 0.812 0.885 0.892 0.876 0.877

1.162 1.238 1.136 1.127 1.148 1.146

SF6 1-site 1-site 7-site 7-site 6-site 6-site, optimized

Values are determined using the indicated approximate form of eq 1; “id” denotes the idealized treatment given by eq 4. VEOSn includes ϕ2 employing the truncated virial equation using coefficients determined from the molecular models, and VEOS2/C is the same but using the correlation. For the TraPPE-UA model, the tabulated value of ϕ2 is based on VEOS3 and is, for the assumed molecular model, exact to the digits given. The last column is the enhancement factor, given as the ratio of psat 2 from eq 4 divided by that value based on VEOS3. For TraPPE-EH, all of the tabulated values are based on VEOS2. a

The virial-corrected results for He, CO2, and N2 carrier gases are in excellent mutual agreement, with scatter on the order of 1 %, compared to roughly 5 % scatter when the data are interpreted with ideal-gas assumptions. For He, the Poynting correction and the effect of the vapor-phase fugacity coefficient roughly cancel each other, leaving a corrected vapor pressure close to that resulting from ideal-gas assumptions. We also examined alternative LJ parameters of He from the literature.37 These variants changed the computed psat 2 of C20H42 by less than 0.1 %, which is not significant in this context, so we do not report them here. For CO2 and N2, failure to consider gas nonideality would lead to errors of several percent in the derived vapor pressures. The results for these carrier gases are insensitive to the choice of the UA or EA formulation of the TraPPE alkane potential. The picture is somewhat different for SF6. It is clearly the most nonideal case, but the calculated correction for nonideality using the TraPPE-UA model is only about 60 % of the magnitude needed to bring the corrected vapor pressures into agreement with the other data. Addition of the third virial coefficient does not improve the agreement significantly, nor does use of more sophisticated models for the SF6 molecule. However, the use of TraPPE-EH for C20H42 with the simple 1site model for SF6 does yield a significant further reduction in psat 2 , almost to within the range of variation of the other data. Unfortunately, when we employ TraPPE-EH with the explicitatom (7-site) model for SF6, the improvement is lost, and we recover a vapor pressure not much different from that given by TraPPE-UA. It seems then that the improvement shown by TraPPE-EH + 1-site SF6 is fortuitous. The general discrepancy exhibited by the SF6 data could have two possible explanations. Either there was some systematic problem with the experiments that used SF6 as the carrier gas or else the molecular model does not correctly represent the mixture. The TraPPE-EH model yields significantly better second virial coefficients for alkanes than the TraPPE-UA model,23 so we had some hope that switching to it might yield a consistent improvement. The fault may lie with the use of the

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

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

Assuming accurate data from experiment, if all factors in eq 1 are properly handled (or can be safely ignored), then the vapor pressure from the gas-saturation procedure should give a result that is independent of the choice of carrier gas. Clearly this is not the case if only the idealized eq 4 is applied. The Poynting correction is practically the same for all carrier gases (using a constant molar volume36 of C20H42, Φ = 1.011, which lowers all sat psat 2 by about 1 %), so it does not remove the discrepancy in p2 across them. In contrast, the solubilities of the gases in C20H42 are significantly different (Table 3), and the correction for this raises the derived psat 2 by 0−1 %. At 323.15 K, the second virial coefficients are negative (except B11 of He/C20H42), so VEOS2 lowers psat 2 (except for He). The effect of the third-order virial coefficient is completely negligible, except for the SF6 carrier gas, where it lowers psat 2 by about half a percent. G



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Table 5. Saturation Vapor Pressure of n-C14H30 in SF6 for TraPPE-UA and TraPPE-EH Models Using the Indicated Approximate Form of Equation 1a psat 2 /Pa py2ϕ2/Φx2 T/K

TraPPE model

py2 (id)

py2/Φx2

VEOS2

VEOS2/C

ϕ2

py2/psat 2

283.15

UA EH UA EH UA EH UA EH

0.421

0.420

0.371

1.192

1.189

3.181

3.176

7.995

7.975

0.364 0.342 1.047 0.990 2.831 2.695 7.189 6.883

0.867 0.813 0.880 0.832 0.891 0.849 0.901 0.863

1.155 1.231 1.139 1.204 1.124 1.180 1.112 1.162

293.15 303.15 313.15

1.067 2.888 7.336

“id” denotes the idealized treatment given by eq 4. VEOS2 includes ϕ2 employing the second-order virial equation with the 1-site SF6 model, and VEOS2/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 the digits given. The last column is the enhancement factor, based on psat 2 from VEOS2.

a

standard combing rules used to obtain the unlike-pair potential from the SF6 and n-C20H42 pair potentials. It has been observed that interactions between fluorocarbons and hydrocarbons are not represented well by simple combining rules or corresponding-states treatments.38−40 What is true for fluorocarbons might also 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-EH models. Measurements at four temperatures were performed by Widegren and Bruno.15 The estimated mole fraction of C14H30 in the condensed phase is estimated from experimental Henry’s constants or solubility data.34,35 The molar volume data are from Valencia et al.41 We employed the 1-site model for SF6 and estimate psat 2 with VEOS truncated at second order. As shown in Table 5, the results for the TraPPE-UA model from VEOS2 agree well with those from correlation, and they are lower than those from the ideal case. The experimental psat 2 reported by Widegren and Bruno are based on the ideal-gas assumption and are without the Poynting or solubility corrections. The apparent measured vapor pressures are 5−10 % above the corrected values. This demonstrates the importance of including gas nonideality. The calculated psat 2 for the TraPPE-EH model with the 1-site SF6 model are even lower than those from the TraPPE-UA model, by about another 5 %. However, given the discrepant results mentioned above for SF6 with C20H42, our corrected psat 2 for C14H30 with SF6 as the carrier gas probably still contains significant inaccuracy.

nonideality effect will be larger at lower temperatures. This is because B11 becomes increasingly negative at lower temperatures 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. The Poynting correction is simple to compute from the molar volume of the solvent (which can be reasonably estimated if it is not known), so there is no reason not to include it. The solubility of the gas in the liquid may not be known, so to minimize the corresponding uncertainty, it may be preferable to choose 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 gas for a gas-saturation experiment. The nonideality correction will in general be smallest for helium, and if high accuracy is not required it will often be reasonable to neglect it. Helium may, however, have disadvantages, both because of its propensity to leak and because its low molar mass makes it more difficult to accurately measure gravimetrically. The nonideality correction will be relatively small for nitrogen, and it is likely that argon would have similar characteristics. SF6 is perhaps the worst possible choice, both because of its large deviation from idealgas behavior and the difficulty of molecular models to represent its interactions with hydrocarbon molecules. Of course if one is going to apply a nonideality correction as we have demonstrated here, a key consideration would be the availability of an accurate pair potential for the interaction of the solute with the carrier gas. For the prototype systems studied here, we could turn to correlation to provide some verification of the virial-series calculations, but in most cases one will not have the data needed to implement the correlation, in which case one must turn to virial coefficients computed from a molecular model. In general, even when correlations are available, it is reasonable to expect that virial coefficients computed from a good molecular model will be more reliable, particularly when used to evaluate mixture cross-coefficients. As molecular modeling continues to advance, it will become increasingly practical to turn to these methods as tools for interpreting experimental data, as shown in the gas-saturation analysis demonstrated here. Further improvements in the accuracy and transferability of molecular models will certainly enhance these prospects. Even more, the increasing practicality of ab initio methods to quantify intermolecular interactions will

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CONCLUSION We have demonstrated how virial coefficients computed from molecular models can be applied to improve estimates of the vapor pressure of low-volatility compounds measured in gassaturation experiments. For the experiments analyzed here, we find that the third-order virial term can be neglected, but that the second-order virial term produces a correction of several percent. In other situations these outcomes could differ, depending on the pressure and temperature of the experiment and the nature of the molecular species involved. We note two trends that will be true in general: (1) the vapor nonideality corrections for experiments near sea level will be somewhat larger than for the experiments analyzed here, which were performed at lower pressure due to the facility’s location at ca. 1700 m elevation; (2) as is evident in Table 5, the vapor H



Article

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make it possible to characterize the interactions of different species, without requiring experiment or some formulation based on combining the pure-species behaviors. In all of these cases, virial coefficients provide a very convenient route from the molecular behavior to the bulk properties.

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ASSOCIATED CONTENT

S Supporting Information *

Table of virial coefficients calculated in this work. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: kofke@buffalo.edu. Funding

This work is supported by the U.S. National Science Foundation (Grant No. CHE-1027963). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Jason Widegren for helpful discussions and for making his data available prior to publication. REFERENCES

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