Equation of State for the Lennard-Jones Truncated and Shifted Model Fluid. Bochum, Germany. Strasse 100, D Paderborn, Germany

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1 Equation of State fo the Lennad-Jones Tuncated and Shifted Model Fluid Monika Thol 1, Gabo Rutkai 2, Roland Span 1*, Jadan Vabec 2, Rolf Lustig 3 1 Themodynamics, Ruh-Univesitaet Bochum, Univesitaetsstasse 150, D Bochum, Gemany 2 Themodynamics and Enegy Technology, Univesitaet Padebon, Wabuge Stasse 100, D Padebon, Gemany 3 Depatment of Chemical and Biomedical Engineeing, Cleveland State Univesity, Cleveland, Ohio 44115, USA Abstact An equation of state is developed fo the Lennad-Jones model fluid, tuncated and shifted at c = 2.5σ. The undelying data set contains themodynamic popeties at about 706 state points including pessue, esidual intenal enegy, fist volume deivative of the esidual intenal enegy, and esidual isochoic heat capacity as a function of tempeatue and density. The equation of state is explicit in tems of the Helmholtz enegy, allowing the detemination of any themodynamic popety by diffeentiation. It is valid fo tempeatues 0.6 < T/T c < 10 and pessues p/p c < 70. High accuacy and good extapolation behavio of the equation of state ae established. Keywods: equation of state; Helmholtz enegy; Lennad-Jones tuncated and shifted; molecula simulation; themodynamic popeties * R.Span@themo.ub.de; Phone: , Fax:

2 1 Intoducion Molecula modeling and simulation have become widely accepted tools in the applied sciences fo detailed undestanding of themophysical pocesses. Molecula models (foce fields) can be used as poweful tools fo themodynamic data pediction. The tuncated and shifted Lennad-Jones (LJTS) potential [1] is one of the most basic and computationally inexpensive molecula models available. Nonetheless, it is still sufficiently ealistic to epesent nonpola spheical molecules. It is defined by u LJTS ( ) = ì u ( ) ( ) LJ - u fo LJ = c c í, (1) î 0 fo > c with u 12 6 éæsö æsö ù ( ) = eêç - ç ú ëè ø è ø, (2) û LJ 4 whee ε and σ ae the enegy and size paametes of the Lennad-Jones (LJ) potential, is the distance between two paticles, and c is a cut-off adius, which is c = 2.5σ hee. Tuncation leads to inexpensive simulation and avoids long-ange coections. The LJ fluid diffes significantly fom the LJTS fluid. Fo example, the citical tempeatue of the Lennad-Jones fluid is about T c,lj = 1.31, wheeas the citical tempeatue of the tuncated and shifted vesion is about T c,ljts = The LJTS has been employed in numeous theoetical studies on phase coexistence [2 13] as well as in applications to noble gases and methane [14], whee e.g. the vapo-liquid suface tension of the noble gases was pedicted within 10 % of the expeimental data. Fo the full Lennad-Jones fluid seveal equations of state wee published, e.g., Nicholas et al. [15], Johnson et al. [16], Kolafa and Nezbeda [17], Mecke et al. [18], and May and Mausbach [19]. To ou knowledge, no fundamental equation of state (FEOS) has been developed pio to this wok fo the LJTS model fluid. Thee is only a geneal appoach how to convet a FEOS fo the full Lennad-Jones model fluid (LJF) into a FEOS fo a cut and shifted potential as published by Johnson et al. [16]. 2 Molecula Simulation The undelying data set was geneated by NVT molecula dynamics using the molecula simulation tool ms2 [20]. Newton s equations of motion wee solved using a fifth-ode Gea 2

3 pedicto-coecto numeical integato [1]. The tempeatue was kept constant using isokinetic velocity scaling [1]. Themodynamic popeties at about 706 state points include pessue p, esidual intenal enegy u, fist volume deivative of the esidual intenal enegy ( u / v) T, and esidual isochoic heat capacity c v as a function of tempeatue T and density ρ = N/V. At each state point, N = 1372 paticles wee sufficiently equilibated and sampled fo 2 to 5 million poduction time steps of Δtσ -1 (m/ε) -1/2 = 0.001, whee m is the molecula mass. Figue 1 shows the data set in the T, ρ plane. The enegy and size paametes ε and σ of the potential wee used to educe all popeties to dimensionless numbes of ode unity: tempeatue T * = Tk B /ε, density ρ * = ρσ 3, intenal enegy u * = u/ε, enthalpy h * = h/ε, o pessue p * = pσ 3 /ε. Fo bevity, asteisks ae omitted in the following with the undestanding that educed quantities ae used. Results wee validated with simulations of N = 2048 and 3072 paticles and no significant system size dependence was found (cf. Supplementay Mateial C). Statistical uncetainties of all esults wee estimated by a block aveaging method [21]. 3 Equations of State The pesent FEOS is witten in tems of the educed mola Helmholtz enegy α as a function of tempeatue and density. The equation is decomposed into an ideal gas contibution (supescipt o ) and a esidual contibution due to the intemolecula inteactions (supescipt ) A T, o,, o, N a T a T,,, (3) k T k T B B whee k B is Boltzmann s constant, τ = T c / T, and δ = ρ / ρ c with T c = and ρ c = The citical paametes wee detemined duing the fitting pocess. See Section 4.2. As the LJTS model fluid is a classical monatomic model, the isobaic heat capacity of the ideal gas is c o p / k B = 2.5. Integation yields the ideal gas contibution ln 1.5 ln c c. (4) o 1 2 The constants c 1 and c 2 wee abitaily adjusted so that h 0 = 0 and s 0 = 0 at T 0 = 0.8, p 0 = 0.001, and the coesponding density of the ideal gas is ρ 0 = p 0 /T 0. A common fom fo the esidual pat of the Helmholtz enegy consists of polynomial, exponential, Gaussian bell-shaped, and non-analytic tems [22]. Polynomial and exponential tems captue the oveall featues of the FEOS. The Gaussian bell-shaped tems, intoduced 3

4 by Setzmann and Wagne [23] fo methane, impove the epesentation of the popeties in the citical egion. Oiginally, these tems wee used to yield significant contibutions only nea the citical point. Howeve, with modeate paametes, they may also impove the Helmholtz enegy suface ove the entie fluid ange [24]. Non-analytic tems wee used only fo the FEOS of wate [25] and cabon dioxide [26]. These tems allow modeling the isochoic heat capacity and the speed of sound vey close to the citical point. Howeve, they equie vey accuate data in that egion [22] which ae not well accessible by molecula simulation. The functional fom of the pesent FEOS has 21 tems: 6 polynomials, 6 exponentials, and 9 Gaussian s 6 12 di ti di ti li, ni ni exp i 1 i 7 21 i di ti n exp. i i i i i (5) The coelation is valid fo tempeatues 0.64 < T < 11 and pessues p < 6.8, coesponding to 0.6 < T/T c < 10 and p/p c < 70. No accuate tiple point tempeatue T t,ljts of the Lennad-Jones tuncated and shifted model fluid is available in the liteatue. Theefoe, the well-known tiple point tempeatue of the Lennad-Jones fluid, which is about T t,lj = 0.68 [27], has been used as an estimate fo the tiple point tempeatue T t,ljts = 0.56 by naive coesponding states estimation. To ensue that all simulations ae located in the fluid egion, the value was inceased to T = The coefficients, tempeatue and density exponents, as well as the Gaussian bell-shaped paametes ae listed in Table 1. The simultaneous detemination of coefficients and paametes equies a non-linea fit algoithm which was povided by E. W. Lemmon fom the National Institute of Standads and Technology (NIST). Duing the fit, simulation data wee augmented by seveal themodynamic constaints to ensue easonable physical behavio of the FEOS esults. Recent FEOS coelations fo eal substances, e.g. R-125 [28], popane [24], o popylene [29], show that coect FEOS behavio can be enfoced fo popeties and egions whee no data ae available. Special attention was given to the vapo-liquid equilibium (VLE) cuves, ectilinea diamete, heat capacities, speed of sound, Gueneisen coefficient, the phase identifie paamete [30] (which significantly influences heat capacities and speeds of sound), viial coefficients up to the fouth ode, and ideal cuves. Figue 2 shows VLE in the citical egion. The ectilinea diamete fom the pesent FEOS exhibits a sudden bend vey close to the citical point, which is not common fo eal fluids. Molecula simulation does most likely not allow fo a veification of the effect. Howeve, 4

5 bends in eithe diection wee also seen in low-paamete coelations [31]. Thee, the citical egion was not attempted to be descibed pecisely, so that the bends wee taken as possible atifacts. The pesent coelation, howeve, should descibe the citical egion much bette. Theefoe it is at least not uled out that the bend is coect. The viial coefficients B, C, and D ensue a coect tansition fom the ideal gas into the eal fluid. They ae calculated fom the FEOS as lim B T (6) 0 lim C T (7) lim D T (8) As the limit is ρ 0, they also detemine the ideal cuves, which pove a coect extapolation behavio of the coelation fo high tempeatues, pessues, and densities. See Section 5. Figue 3 is deduced fom the viial equation Z B C D (9) Fo ρ 0, intecepts with the odinate ae B, the slopes of the isothems coespond to C, and D ae the cuvatues. As D can be neglected fo densities ρ < 0.2ρ c, the isothems ae nealy staight lines fo low densities. The absolute values fo B, C, and D ae shown in Fig. 4. The second viial coefficient B passes though zeo at the Boyle tempeatue T BL, eaches a maximum at the Joule-Thomson invesion tempeatue T JT, and appoaches zeo fo high tempeatues [32]. Not only the qualitative but also the quantitative behavio of the viial coefficients is coect. Theefoe, calculated viial coefficients of Shaul [33], Wheatley [33, 34], Hellmann [36], and this wok wee consideed to validate the FEOS. All computed data fo B (Shaul et al. [33], Wheatley [35], and this wok) ae consistent and epesented vey well by the new FEOS. The data fo C of the same authos ae also epoduced well. Only in the citical egion, the FEOS calculations fo C ae too low. Fo D, the behavio is quite simila to C and qualitatively coect. Thee computed data sets of Hellmann [36], Wheatley [35] and this wok agee vey well. Although no data fo D wee used in the fit, the FEOS epesents all of them qualitatively as well as quantitatively. Fo C, thee is an unusual change in cuvatue fo 3 < T < 100. Fo D, a second maximum occus at T 5 in both the FEOS and data fom diect numeical integation. These phenomena appea to exist fo this model fluid, although they ae not common in eal fluids. 5

6 Heat capacities and speed of sound ae illustated in Fig. 5. The esidual isochoic heat capacity of the satuated liquid and vapo phase meet in an absolute maximum at the citical point. Compaed to eal fluids (cabon dioxide [26] o wate [25]), the maximum of the pesent FEOS is less ponounced. Obviously, the citical egion is descibed less accuately than fo cabon dioxide [26] and wate [25], which is a consequence of insufficient data in the citical egion and the neglect of non-analytic tems. Howeve, the citical egion was not the main focus of this wok. The speed of sound should exhibit a steep decease in the citical egion. Again, the effect is not as distinctive as fo wate [25] o cabon dioxide [26]. Nevetheless, the satuated liquid phase of the speed of sound as well as the isobas show a linea tend with negative slope, which indicates pope extapolation behavio. Futhe qualitative physical behavio of othe themodynamic popeties is descibed by Lemmon and Jacobsen [28], Lemmon and Wagne [29], and Lemmon [37]. 4 Simulation Data and Compaison to the Equation of State Table 2 gives an oveview of which popety and how many data points wee published. In this wok all liteatue data wee used fo compaison puposes only. As mentioned befoe, the only available data fom the liteatue ae VLE data (69 state points in total). This data set was extended by nine VLE state points and additional 706 state points in the homogeneous fluid phase (cf. Fig. 1). 4.1 Vapo-Liquid Equilibium Pio wok mainly focused on VLE data. In this wok, a FEOS valid fo the entie fluid egion was developed and used to set up ancillay equations fo vapo pessue, satuated liquid density, and satuated vapo density. Although ancillay equations ae not equied when a full FEOS is available, they ae useful fo estimates in the iteative pocedues to find the satuation states. They should not be used fo calculating pope VLE data. The vapo pessue p v may be epesented by a modified Wagne equation [38], (10) v c ln p T N1 N 2 N3 N 4 N5 pc T whee N 1 = -6.21, N 2 = 1.5, N 3 = -1.92, N 4 = 2.2, N 5 = -4.76, and θ = (1 T/T c ). The values of the citical paametes ae given in section 4.2. Compaison with simulation data and the 6

7 pesent FEOS is pesented in Fig. 6. Except fo the data of Maeschal et al. [8], the ageement is within ± 2.5 %. The satuated liquid density ρ was epesented by the ancillay equation ' N1 N 2 N3 N 4, (11) c whee N 1 = 1.45, N 2 = , N 3 = , and N 4 = Compaison with simulation data and the pesent FEOS is pesented in Fig. 7. Except fo 6 data points out of 77, the ageement is within ± 1 %. The satuated vapo density ρ may be epesented by the ancillay equation " ln N N N N c , (12) whee N 1 = , N 2 = , N 3 = , and N 4 = Compaison with simulation data and the pesent FEOS is pesented in Fig. 8. Again, except fo the data set of Adams and Hendeson [2], Dunikov et al. [5], and Maeschal et al. [8], ageement is within ± 3 %. 4.2 Citical Point Citical paametes fom the liteatue ae given in Table 3. The citical values fo tempeatue and density of Vabec et al. [14] wee taken as a stating point fo setting up the pesent FEOS. As both paametes wee included in the fitting pocess (not constained), thei influence on themodynamic popeties as discussed in Section 3 was monitoed caefully. Special attention was given to the fist and second deivatives of pessue with espect to density, which vanish at the citical point. The citical tempeatue T c = and the citical density ρ c = wee found which happen to be within the uncetainties of the data by Dunikov et al. [5] and Smit [13]. The citical pessue p c = can then be calculated fom the pesent FEOS. 4.3 Homogenous Fluid States In this wok, a compehensive data set fo the homogenous fluid egions was simulated to set up the FEOS which is valid fo a wide ange of tempeatue, density, and pessue. The numeical values of the simulation data ae given in the supplementay mateial B. The accuacy of the FEOS was detemined by elative deviations of all popeties at all state points depicted in Fig. 9 along andomly selected isothems. Compaisons using the complete 7

8 data set can be found in supplementay mateial. Themodynamic consistency was veified and most of the simulation data wee epesented within thei statistical uncetainty. The uncetainty of density calculated with the pesent FEOS is ±0.2 %. In the extended citical egion (T = 1 to 1.5), the deviations incease to ±1 %. The uncetainty in esidual intenal enegy is ±0.3 %, up to ±0.5 % fo high tempeatues. Deviations fo the esidual isochoic heat capacity and the fist volume deivative of the esidual intenal enegy at constant tempeatue ae less than ±5 %. Fo highe tempeatues, the deviations fo the esidual isochoic heat capacity ae below ±2 %. Deviations of total isochoic heat capacity ae also shown. Ageement is within ±2 % and less than ±0.5 % fo highe tempeatues. Thus, the new FEOS can be classified as a technical equation of state [22]. 5 Extapolation Behavio Although the ange of validity of the FEOS fo the LJTS model fluid is defined by 0.6 < T/T c < 10 and p/p c < 70 (based on the available molecula simulation data), the FEOS can be extended in all diections (highe tempeatues, pessues, densities, and lowe tempeatues) while maintaining a physically easonable behavio. As explained in Section 3, this was achieved by applying pope constaints to the fit, based on expeience fom past FEOS fitting wok on eal substances [24, 27, 28]. Good extapolation behavio is also beneficial within the ange of validity. Poo extapolation usually causes incoect slopes in the validity ange fo popeties such as the heat capacities. The investigation of the extapolation behavio includes fou diffeent aspects: the epesentation of simulation data outside the given ange of validity, the functional fom, the epesentation of ideal cuves, and physically easonable behavio of diffeent themodynamic popeties such as speed of sound o heat capacities. To study the extapolation capability, additional data points wee simulated along the isochoe ρ = 0.5 (T = 15 to 19) and the isothem T = 20 (ρ = 0.1 to 1) outside the ange of validity. Figue 10 shows that fo densities up to ρ = 0.4, the data can be epesented within ±0.01 %. The isochoe ρ = 0.5 was pedicted within ±0.05 %. Fo highe densities, the deviations incease up to ±1.8 %. A easonable behavio can be obseved along the satuated liquid line fo the speed of sound, that is a staight line down to a educed tempeatue of about T/T c = The esidual isochoic heat capacity (Fig. 5) shows an upwad tend in the liquid phase at low tempeatues which is common fo many eal fluids and has been validated expeimentally [24]. Pessue vs. density to exteme conditions is pesented in Fig. 11 on the left. Obviously, extapolation 8

9 is smooth to extemely high tempeatues, pessues, and densities which is contolled by the functional fom of the FEOS. As discussed in detail by Span and Wagne [32], the extapolation to high tempeatues and pessues is mainly influenced by polynomial and exponential tems with powe l i = 1. In the investigated egion, δ = ρ / ρ c eaches a high value, wheeas τ = T c / T is deceasing. Theefoe, the density exponent d i must be high (but not too high to oveestimate the cuvatue) and the tempeatue exponent must be low. The fist polynomial tem of the pesent FEOS with exponents t 1 = 1 and d 1 = 4, which wee found to be an effective combination by Lemmon and Jacobsen [28], accounts fo the coect behavio of the investigated isothems. The density exponent d 1 is high enough to model the inceasing tempeatue at inceasing pessue and density, but avoids uneasonably ponounced cuvatues of isothems. The tempeatue exponent t 1 ensues that intesecting isothems ae avoided and isothems convege towads each othe. Additionally, the coesponding coefficient n 1 has to be positive so that no negative pessues occu. Finally, some ideal cuves wee investigated. Figue 11 shows the Boyle cuve, the Joule- Thomson invesion cuve, the Joule invesion cuve, and the ideal cuve. Shapes ae simila to those of eal fluids [32] without uneasonable inflection points o defomations. Thus these ideal cuves indicate a qualitatively coect extapolation behavio of the FEOS extending to high tempeatues and pessues. 6 Conclusion Based on molecula simulation data, a new FEOS was developed fo the LJTS model fluid. The equation is expessed in tems of the Helmholtz enegy, can be implemented easily in common softwae packages, and can be used to calculate all themodynamic popeties, e.g., density, VLE data, heat capacities, speed of sound, o intenal enegy by diffeentiation only. It is valid fo tempeatues 0.64 < T < 11 and fo pessues p < 6.8, coesponding to 0.6 < T/T c < 10 and p/p c < 70. Uncetainties of the FEOS wee studied by compaison to simulation data. The uncetainty in density is ±0.2 %. In the extended citical egion (T 1 to 1.5), deviations incease to ±1 %. The uncetainty in esidual intenal enegy is ±0.3 % to ±0.5 % fo high tempeatues, and ±1 % fo esidual enthalpy. Deviations fo the esidual isochoic heat capacity and the fist deivative of the esidual intenal enegy with espect to volume at constant tempeatue ae less than ±5 %. Fo highe tempeatues, deviations fo the esidual isochoic heat capacity ae below ±2 %. Refeence values ae given in Table 4 to veify a compute implementation of the FEOS. Additionally, the FEOS is given as a souce code in the supplementay mateial. 9

10 This wok is a fist step to show that molecula simulation data can be used to set up FEOS coelations fo a wide tempeatue and pessue ange. Wok on a new equation of state fo the Lennad-Jones model is in pogess. Acknowledgement We thank E. W. Lemmon fo his suppot duing the development of the equation of state and G. Guevaa-Caion fo he suppot in caying out molecula simulation wok. This poject was funded by the Deutsche Foschungsgemeinschaft (DFG). Nomenclatue Latin Symbols a Helmholtz enegy c 1, c 2 integation constants of the ideal Helmholtz enegy c v d i h l i N n i N i p c s t * T t i u V X isochoic heat capacity density exponents of the esidual Helmholtz enegy enthalpy density exponents of the exponential tem of the esidual Helmholtz enegy numbe of molecules in the simulation coefficients of the esidual Helmholtz enegy coefficients of the ancillay equations pessue adius cut-off adius entopy time steps tempeatue tempeatue exponents of the esidual Helmholtz enegy potential enegy/intenal enegy volume any themodynamic popety Geek Symbols α β i educed Helmholtz enegy Gaussian bell-shaped paametes 10

11 γ i δ ε ε i η i θ ρ σ τ Gaussian bell-shaped paametes educed density enegy paamete of the molecula model Gaussian bell-shaped paametes Gaussian bell-shaped paametes (1-T/T c ) fo the ancillay equations density size paamete of the molecula model invese educed tempeatue Subscipt c citical LJ Lennad-Jones LJTS Lennad-Jones tuncated and shifted v vapo v isochoic 0 efeence Supescipt o ideal esidual satuated liquid satuated vapo 11

12 Refeences 1. M. P. Allen, D. J. Tildesley, Compute Simulation of Liquids (Claendon, Oxfod, 1987) 2. P. Adams, J. R. Hendeson, Mol. Phys. 73, 1383 (1991) 3. P. J. Camp, M. P. Allen, Mol. Phys. 88, 1459 (1996) 4. L.-J. Chen, J. Chem. Phys. 103, (1995) 5. D. O. Dunikov, S. P. Malyshenko, V. V. Zhakhovskii, J. Chem. Phys. 115, 6623 (2001) 6. M. J. Haye, C. Buin, J. Chem. Phys. 100, 556 (1994) 7. C. D. Holcomb, P. Clancy, J. A. Zollweg, Mol. Phys. 78, 437 (1993) 8. M. Maeschal, R. Lovett, M. Baus, J. Chem. Phys. 106, 645 (1997) 9. M. J. P. Nijmeije, A. F. Bakke, C. Buin, J. H. Sikkenk, J. Chem. Phys. 89, 3789 (1988) 10. M. Rao, D. Levesque, J. Chem. Phys. 65, 3233 (1976) 11. A. Tokhymchuk, J. Alejande, J. Chem. Phys. 111, 8510 (1999) 12. W. Shi, J. K. Johnson, Fluid Phase Equilib , 171 (2001) 13. B. Smit, J. Chem. Phys. 96, 8639 (1992) 14. J. Vabec, G. K. Kedia, G. Fuchs, H. Hasse, Mol. Phys. 104, 1509 (2006) 15. J. J. Nicholas, K. E. Gubbins, W, B. Steett, D. J. Tildesley, Mol. Phys., 37, 1429 (1979) 16. J. K. Johnson, J. A. Zollweg, K. E. Gubbins, Mol. Phys. 78, 591 (1993) 17. J. Kolafa, I. Nezbada, Fluid Phase Equilib., 100, 1 (1994) 18. M. Mecke, A. Mülle, J. Winkelmann, J. Vabec, J. Fische, R. Span, W. Wagne, Int. J. Themophys., 17, 391 (1996) 19. H.-O. May, P. Mausbach, Phys. Rev. E, 85, (2012) 20. S. Deublein, B. Eckl, J. Stoll, S. V. Lishchuk, G. Guevaa-Caion, C. W. Glass, T. Meke, M. Beneuthe, H. Hasse, J. Vabec, Comp. Phys. Comm. 182, 2350 (2011) 21. H. Flyvbjeg, H. G. Petesen, J. Chem. Phys. 91, 461 (1989) 22. R. Span, Multipaamete equations of state (Spinge, Belin, 2000) 23. U. Setzmann, W. Wagne, J. Phys. Chem. Ref. Data 20, 1061 (1991) 24. E. W. Lemmon, M. O. McLinden, W. Wagne, J. Chem. Eng. Data 54, 3141 (2009) 25. W. Wagne, A. J. Puss, J. Phys. Chem. Ref. Data 31, 387 (2002) 26. R. Span, W. Wagne, J. Phys. Chem. Ref. Data 25, 1509 (1996) 27. A. Ahmed, R. J. Sadus, J. Chem. Phys. 131, (2009) 28. E. W. Lemmon, R T. Jacobsen, J. Phys. Chem. Ref. Data 34, 69 (2005) 29. E. W. Lemmon, W. Wagne, to be published 30. G. Venkataathnama, L.R. Oellich, Fluid Phase Equilib. 301, 225 (2011) 12

13 31. R. Lustig, G. Rutkai, J. Vabec, to be published (2014) 32. R. Span, W. Wagne, Int. J. Themophys. 18, 1415 (1997) 33. K. R. S. Shaul, A. J. Schultz, D. A. Kofke, Collect. Czech. Chem. Commun. 75, 447 (2010) 34. R. J. Wheatley, Phys. Rev. Lett. 110, (2013) 35. R. J. Wheatley, pivate communication (2013) 36. R. Hellmann, pivate communication (2013) 37. E. W. Lemmon, 18 th Symp. Themophys. Pop., USA, (2012) 38. W. Wagne, Fotsch.-Be. VDI, Düsseldof, VDI-Velag, 3 (1974) 13

14 List of Tables Table 1 Paametes and coefficients of the esidual pat of the FEOS coelation, Eq. 5. Table 2 VLE simulation data fo the LJTS model fluid with c = 2.5σ fom the liteatue. Table 3 Citical paametes fo the LJTS model fluid with c = 2.5σ fom the liteatue. Table 4 Calculated values of popeties fo algoithm veification. 14

15 Tables Table 1 i n i t i d i l i η i β i γ i ε i

16 Table 2 Autho Yea Popety No. of data points a Refeence Adams & Hendeson 1991 p v / ρ / ρ 5 [2] Camp & Allen 1996 ρ / ρ 6 [3] Chen 1995 ρ / ρ 1 [4] Dunikov et al ρ / ρ 17 [5] Haye & Buin 1993 ρ / ρ 6 [6] Holcomb et al ρ / ρ 1 [7] Maeschal et al p v / ρ / ρ 7 [8] Nijmeije et al ρ / ρ 5 [9] Rao & Levesque 1976 ρ / ρ 1 [10] This wok 2014 ρ / ρ 9 - Tokhymchuk & Alejande 1999 p v / ρ / ρ 10 [11] Vabec et al p v / ρ / ρ 15 [14] a The numbe of data points is the same fo each popety Table 3 Autho Yea Refeence T c ρ c p c Dunikov et al [5] 1.085(5) 0.317(3) 0.097(8) Haye and Buin 1993 [6] 1.078(2) - - Shi and Johnson 2001 [12] (2) (5) - Smit 1992 [13] 1.085(5) 0.317(6) - This wok Tokhymchuk & Alejande 1999 [11] Vabec et al [14] Table 4 T p ρ u c v w a

17 List of Figues Fig. 1 Simulation data set used in the pesent FEOS development. Fou independent themodynamic popeties wee measued at each state point: pessue, esidual intenal enegy, fist volume deivative of the esidual intenal enegy, and esidual isochoic heat capacity Fig. 2 VLE in the citical egion. Dashed line: satuated vapo density ρ, solid line: satuated liquid density ρ, dashed-dotted line: ectilinea diamete (ρ + ρ )/2 Fig. 3 (Z 1)/ρ as a function of ρ along isothems (T = ) Fig. 4 Second, thid, and fouth viial coefficients (B, C, and D) as functions of tempeatue. The dashed line indicates the citical tempeatue. The dashed-dotted line indicates the Boyle tempeatue (T BL = 2.784), and the dotted line the Joule-Thomson invesion tempeatue (T JT = 20.09) Fig. 5 Typical plots of esidual isochoic heat capacity c v (left) and speed of sound w (ight) as functions of tempeatue along isobas (p = ) Fig. 6 Compaison with vapo pessue data fom the liteatue. The FEOS is epesented by the zeo line, the ancillay equation by the solid line, and eo bas ae simulation uncetainties (if given by the authos) Fig. 7 Compaison with satuated liquid density data fom the liteatue. The FEOS is epesented by the zeo line, the ancillay equation by the solid line, eo bas ae simulation uncetainties (if given by the authos) Fig. 8 Compaison with satuated vapo density data fom the liteatue. The FEOS is epesented by the zeo line, the ancillay equation by the solid line, and eo bas ae simulation uncetainties (if given by the authos) Fig. 9 Compaison of the pesent molecula simulation data with the FEOS as a function of density along selected isothems Fig. 10 Compaison of additional molecula simulation data with the extapolated FEOS at isothem T = 20 (left) and isochoe ρ = 0.5 (ight). These data wee not used fo the development of the FEOS 17

18 Fig. 11 Pessue vesus density diagam along isothems (left) and ideal cuves (ight). p v : vapo pessue cuve; ID: Ideal cuve 0, BL: Boyle cuve 2 0 2, JTI: Joule-Thomson invesion cuve , JI: Joule invesion cuve

19 Figues Fig. 1 Fig. 2 19

20 Fig. 3 20

21 Fig. 4 21

22 Fig. 5 Fig. 6 22

23 Fig. 7 23

24 Fig. 8 24

25 Fig. 9 25

26 Fig. 9 continued 26

27 Fig. 10 Fig

28 Supplementay A Simulation esults fo the Lennad-Jones tuncated and shifted model fluid fo vaious state points. N is the numbe of paticles, T is the tempeatue, ρ is the density, u is the esidual intenal enegy (potential enegy aising fom intemolecula inteactions), p is the pessue, and c v is the esidual isochoic heat capacity. Popeties ae given in Lennad- Jones educed units. Numbes in backets denote the uncetainty in the last digit fo the peceding values. N = 1372 N = 2048 N = 3072 T = 1.0 ρ = 0.04 u (2) (9) (7) p (1) (2) (2) c v (3) (17) (17) T = 10.0 ρ = 0.04 u (4) (3) (13) p (1) (16) (3) c v (2) (7) (8) T = 6.0 ρ = 0.5 u (2) (4) (4) p (5) (12) (10) c v (5) (15) (19) T = 0.64 ρ = 0.82 u (2) (2) (2) p (9) (10) (10) c v (9) (13) (13) T = 1.0 ρ = 0.88 u (2) (3) (3) p (2) (2) (2) c v (11) (14) (11) T = 1.09 ρ = 0.34 u (1) (2) (2) p (1) (3) (2) c v 1.81 (5) 1.56 (10) 1.80 (16)

29 Supplementay B p ρ T u h ( u / v) T c v p u h ( u / v) T c v

30 p ρ T u h ( u / v) T c v p u h ( u / v) T c v

31 p ρ T u h ( u / v) T c v p u h ( u / v) T c v

32 p ρ T u h ( u / v) T c v p u h ( u / v) T c v

33 p ρ T u h ( u / v) T c v p u h ( u / v) T c v