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The equation of state of symmetric extended Lennard-Jones fluids
Physica A 533 (2019) 122090
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
The equation of state of symmetric extended Lennard-Jones fluids ∗
Xiaohong Yang , Weiling Zhu Department of Chemistry and Environmental Engineering, Wuhan Polytechnic University, Hankou, Wuhan, 430023, PR China
graphical
abstract
highlights • • • •
A new model of symmetric extended Lennard-Jones fluid was developed. An extended Lennard-Jones potential function was introduced. The equation of state of symmetric extended Lennard-Jones fluid was derived. The equation described the properties of four simple fluids in acceptable accuracy.
article
info
Article history: Received 22 March 2019 Received in revised form 16 July 2019 Available online 19 July 2019 Keywords: Equation of state Symmetric fluid Extended Lennard-Jones potential Averaged intermolecular potential field
a b s t r a c t This paper developed the conceptual model of symmetric extended Lennard-Jones fluid based on an extended Lennard-Jones potential function. Therein, important relationships between averaged intermolecular potential field (AIPF) and thermodynamic properties were presented, in which the equation of state was included. These formulas can not only predict the p-V-T properties of argon, nitrogen, carbon monoxide and methane that are representatives of spherical, polar and non-polar molecular fluids in acceptable accuracy, but also physically interpret the p-V-T properties from the AIPF. Obtained
∗ Corresponding author. E-mail address: [email protected] (X.H. Yang). https://doi.org/10.1016/j.physa.2019.122090 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
results have theoretical significance and can serve as reference or base in the analysis and computation of the p-V-T properties of simple fluids. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The equation of state (EOS) is of fundamental importance from both the theoretical and practical points of view. It is usually expressed as a relation between state variables. For example, the EOS of ideal gas provides a simple mathematical relationship among its pressure, volume, and temperature. Because a real fluid is much more important and complicated than an ideal gas, its EOS has been studied for more than 100 years. Since van der Waals [1] semi-theoretically introduced the first EOS for real gas by the assumption of a finite volume occupied by the constituent molecules in 1873, numerous investigators have developed EOS based mainly on his ideas, such as the equations from the Redlich–Kwong model [2], Berthelot and modified Berthelot model [3], Peng–Robinson model [4], and so on. These equations principally indicated the early progress. Because of the extreme importance, the research on the EOS has never stopped. In 1986, Baibuz et al. [5] proposed the EOS for high temperature real gas from the concept of covolume. In 1994, Aungier modified the Redlich–Kwong two-parameter equation by employing the acentric factor and the critical point compressibility factor as additional parameters so as to extend its application range to include the critical point [6]. In recent years, efforts have been made on the equations applicable in wide state parameters [7] and more substances [8,9] as well as modified cubic equations [10–14]. Model fluids are idealized imaginary substances that have a well-defined expression for the intermolecular potential, so that the properties of the model fluids are able to be calculated through physical theory with mathematical and/or numerical methods [15–20]. A well-developed model fluid can represent the essential behavior of real fluids and offer data in acceptable accuracy over wide parameter ranges. Furthermore, the model fluids contain physically meaningful parameters, which are directly related to the intermolecular interactions. Therefore, a model fluid is a good reference or base while predicting the thermodynamic properties of pure fluids and their mixtures. Due to the stated benefits, there is always a high need for successful model fluid capable of giving the thermodynamic properties in an analytically straightforward way. In thermodynamics and statistical mechanics there are famous model fluids like the ideal gas, van der Waals fluid, association fluid theories [21], Lennard-Jones model fluid for transport coefficients [22] and equation of state [23] in addition to the just mentioned model fluids in the references [15–20]. Recently, the authors of present paper successfully used a symmetric interaction approximation in calculating viscosity of fluid [24,25]. As a new method, the symmetric interaction approximation is promising to yield novel theoretically significant results in the development of a new model fluid and derivation of the EOS. The aims of this work are to develop a novel model fluid named symmetric extended Lennard-Jones fluids upon the concept of symmetric interaction, and to derive the equation of state based on an extended Lennard-Jones potential function. After that, the availability of the model fluid is to be assessed by applying the derived equations to the most important pure simple fluids: argon, nitrogen, carbon monoxide and methane. In the end, the theoretical and practical significance of the model fluid will be evaluated. 2. Theory of symmetric fluid 2.1. Potential well A symmetric fluid is defined as the fluid whose molecule accepts symmetric force within the fluid from its neighbors around it, and unsymmetric force at boundary from its half neighbor to restrict it inside. By this definition, if a molecule inside receives a force by a neighbor, there is always an opposite force from the molecule at the symmetric position to counteract on it. Then, the total force (F ) on each molecule inside equals zero. If the intermolecular force of the symmetric fluid is represented by a potential field (denoted by u), then F = −∇ u = 04. This equation deduces that the potential field is a constant field independent of spatial coordinates within the symmetric fluid and a step function at the boundary. Such feature, just like a potential well of flat bottom, is expressed by the following equation,
{ u=
w 0
w ithin fluid without fluid
(1)
here, u denotes the intermolecular potential field, w is the amount presenting the depth of the potential well and equals the averaged intermolecular potential field (AIPF) as a result of the constant potential field inside the fluid. Since the symmetric molecule that counteracts at the symmetric position always exists in an ergodic system in thermodynamic equilibrium, the symmetric fluid model is accurate in such system.
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2.2. Entropy Based on the above zero total force, the molecule in the symmetric fluid can move freely just like the molecule in the ideal gas. The entropy of classical ideal gas is then available for the molecules in the symmetric fluid, which is given below [26],
[ ( 4π m Ek ) 32 ]
3 (2) + Nk 3h2 N 2 here, S presents the entropy of the symmetric fluid, Ek the kinetic energy, N is the total number of molecules restricted in volume V , k is the Boltzmann’s constant, T is absolute temperature, m is the mass of molecule. h is a constant of the dimension of momentum timing distance. As usual, defining the absolute temperature through kinetic energy with the following equation, S Ek , V = Nk ln V
(
)
T =
( ∂ Ek ) ∂S V
(3)
it yields 3
NkT 2 for the symmetric fluid. Ek =
(4)
2.3. Internal energy and parameters In order to derive an expression for the internal energy of the symmetric fluid, it is noticed that the ratio of the 1/3 molecules on boundary to all molecules is approximately 6/N A ∼ 7E − 8 (NA is the Avogadro constant) for a mole symmetric fluid. This small value suggests the error due to the boundary effect is considered so small that it can be ignored. During an adiabatic process, there is no external energy transmitting into the fluid, therefore the internal energy of the symmetric fluid is thus approached as, U = Ek + N w
(5)
here, U denotes the internal energy that belongs a thermodynamic state function. After defining the density (ρ ) as the number of molecules per volume,
ρ=
N
V The equation of state (EOS) is derived through the state function and the Eq. (3) as
(6)
( ∂U ) (7) ∂V S [ ρ ( ∂w ) ] (8) p = ρ kT 1 + kT ∂ρ S When w→0, the Eq. (8) simplifies to the EOS of the ideal gas. By Eq. (8), the compressibility factor (Z ) is expressed as ρ ( ∂w ) Z =1+ (9) kT ∂ρ S p=−
By reviewing the above equations and the relevant derivations, it is fair that all the state functions and parameters of the symmetric fluid can be determined, if w is given. 2.4. Averaged intermolecular potential field With respect to the intermolecular potential field based on the Eulerian approach, it was separated into two terms, u = u2 + um
(10)
here, u2 is the binary potential field that approaches zero in the infinite dilute limit, um stands for the total contributions of many-body potential field that includes triple and higher terms. Averaging Eq. (10) over ensemble that is noted by triangular parenthesis, it gives
⟨ u⟩ = ⟨ u2 ⟩ + ⟨ um ⟩
(11)
This formula offers a relation among the AIPF, averaged binary potential field (ABPF) and averaged many-body potential field (AMPF). Because the triple and higher interacting mechanisms actually transit a molecular event simultaneously to the entire system against the slow binary interacting mechanism, they effectively reveal themselves as the bulk effect of all molecules in system [24,27]. And then, the AMPF does not rely on the intermolecular distance.
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X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
As mentioned above, within the symmetric fluid the depth of the potential well w equals the AIPF. It can also be separated into two terms,
w = w2 + wm
(12)
here, w2 equals the ABPF that approaches zero as the intermolecular distance→∞, wm is identical to the AMPF that merely depends on temperature, when the independent variables of the state function are chosen as the temperature and density.
2.5. Extended Lennard-Jones potential function In order to analytically represent the ABPF, the averaged intermolecular distance (AID, from now on denoted by d) between two molecules is introduced beforehand, which can be measured with either volume or density. For the system of N molecules restricted in volume V , it is averagely believed that the volume is equally shared by the N molecules. If the volume occupied by each molecule is modeled as cube, it yields, V = Nd3
ρ=
N
(13) 1
=
V
(14)
d3
where, ρ is the density of the system. These two equations not only enable us measuring the AID, but also supply a route to define a Lennard-Jones volume and density by the following two formulas, Vσ = N σ 3
ρσ =
(15)
1
(16)
σ3
here, σ is the Lennard-Jones distance. We now introduce an extended Lennard-Jones potential function. The extension involves two aspects: one is extending the function form to cover more potential models; another is the extension of independent variable into the AID or volume or density. The extended function is as follow,
φ = φσ
[( σ )3β d
−θ
( σ )3α ] d
(17)
Obviously, it is extended from the (3β , 3α ) pattern Lennard-Jones binary potential (or Mie potential). Here θ is a parameter independent of the AID, φσ is the Lennard-Jones constant in the unit of energy. This function can cover the potentials of Lennard-Jones’ form, the hard sphere and soft sphere as the parameters take different values. That is
⎧ ⎨θ > 0 ⎩θ ≤ 0 Sphere model
Lennard − Jones′ form { β → ∞ hard shpere others soft sphere
As θ continuously varies from positive to negative values with temperature increasing, the Eq. (17) gives a potential energy surface that continuously transforms the potential curve from the Lennard-Jones’ form into the sphere potential. Fig. 1 shows an example of the potential energy surface when β = 3.8, α = 1.7 and θ = 3 − 0.14T 0.5 . The extended Lennard-Jones potential function enables us constructing a new conceptual model fluid. Comparing with the existing model fluids [16–20], the potential of the new model fluid has the soft sphere repulsive term against the hard sphere repulsions. Therefore, it can approach a real fluid more realistically, especially in the dense fluid where the intermolecular repulsive force may mainly account for the thermodynamic properties of the fluid. Based on the cubic volume model, the Eq. (17) can thus be equivalently represented by the following equations by extending the independent variable through the Eqs. (13) and (14),
[( ρ )β ( ρ )α ] −θ ρσ ρσ [( Vσ )β ( Vσ ) α ] φ = φσ −θ φ = φσ
V
V
(18) (19)
These equations are good approximations to the ABPF and then are called extended Lennard-Jones potential (ELJP). When β = 4, α = 2 and θ = 1, the ELJP reduce to the famous (12, 6) pattern Lennard-Jones potential. Since it is generally believed that the intermolecular potential determines the entire thermodynamic properties, we hope the ELJP with the symmetric fluid theory can add a successful example for this belief.
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Fig. 1. Extended Lennard-Jones potential energy surface with β = 3.8, α = 1.7 and θ = 3 − 0.14T 0.5 .
3. Results and discussion 3.1. Internal energy We define a symmetric extended Lennard-Jones fluid (SELJF) as the symmetric fluid whose ABPF equals the value of the ELJP at the AID. It thus has the following equation for the SELJF,
w = φσ
( ρ )α ] ( ) [( ρ )β −θ + wm T ρσ ρσ
(20)
Substituting this equation and Eq. (4) into Eq. (5), the internal energy per mole SELJF is derived as, f
U =
2
RT + RTσ
( ) [( ρ )β ( ρ )α ] −θ + Um T ρσ ρσ
(21)
where, R = 8.314 J/(mol.K) is the ideal gas constant, Tσ = φσ /k, Um = NA wm . NA is the Avogadro constant, f denotes the degree of freedom. As ρ → 0, the internal energy in zero density limit is obtained, U0 =
f 2
( )
RT + Um T
(22)
here, U0 denotes the internal energy in zero density limit, which composes of the kinetic energy and the AMPF. 3.2. The equation of state Because the equation of state (EOS) can both calculate phase equilibria values and p − V − T properties in wide parameter ranges, it has been considered the cornerstone of thermodynamic models [21]. We thus discuss the EOS of the SELJF in this section. Substituting Eq. (20) into Eq. (8), it yields the EOS of a mole SELJF at the constant AMPF, p=
RT { V
1+
Tσ [ ( Vσ )β T
β
V
( )( Vσ )α ]} 2 − αθ + T θ ′ f
V
(23)
Such appearance not only adds a new form for the EOS, but also clearly indicates the influences of the attractive and repulsive power-law terms on the pressure. As Vσ /V →0 (the dilute gas case), this equation also returns to the EOS of ideal gas. For simplicity, we define the reduced temperature (Tr ), density (ρr ), volume (Vr ) and pressure (pr ) with respect to Tσ , ρσ , Vσ and pσ , as usual, Tr =
ρr = Vr = pr =
T Tσ
ρ ρσ V Vσ p pσ
(24) (25) (26) (27)
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X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
Fig. 2. p − V − T surface of SELJF when β = 3.8, α = 1.7, Tσ = 660 K, f = 3 and θ = 3 − 0.14T 0.5 . Table 1 Major properties of the data used. Item
Argon
Nitrogen
Carbon monoxide
Methane
Effective degree of freedom Temperature range (K) Density range (mol/L) Z range No. points
3 90–990 0–50.3 0.00517–12.1 6328
5 70–1000 0–52.0 0.00221–23.1 12 541
5 70–500 0–33.5 0.00120–6.52 6957
8 100–620 0–38.7 0.00151–15.0 4457
where, pσ =
RTσ
(28)
Vσ
The EOS is then represented in the reduced form, pr =
)( 1 )α ]} 1 [ ( 1 )β ( 2 Tr { 1+ β − αθ + T θ ′ Vr Tr Vr f Vr
(29)
When β = 3.8, α = 1.7, Tσ = 660 K, f = 3 and θ = 3 − 0.14T 0.5 , the p − V − T surface of the SELJF was calculated as Fig. 2 by the Eq. (29), which reproduces all the characters of the van der Waals EOS, such as the existence of van der Waals isotherm oscillation, critical point, the liquid, gaseous and supercritical phase regions, as well as phase transition. By the Eq. (29), we could offer an important interpretation for the formation of van der Waals isotherm oscillation. Because the oscillation is analytically linked to the derivative of the ABPF, we can now refer the mechanism for the oscillation occurrence to the intermolecular potential. Despite the SELJF is not a real fluid, its EOS is available to many real pure fluids in acceptable accuracy. This advantage can be certified via the compressibility factor (Z ) as the function of temperature and density that is derived from the Eq. (23) and is analytically equivalent to the EOS, Z =1+
Tσ [ T
β
(
2
βρr − αθ + T θ f
′
)
ρrα
]
(30)
here, we propose a simple relation for the temperature dependence of the θ ,
√ θ =3+λ T
(31)
where λ is a constant coefficient to be fitted. The National Institute of Standards and Technology’s (NIST) open database provides an authoritative compressibility factor data varying with temperature and density for many fluids such as argon, nitrogen, carbon monoxide and methane, which are the representatives for spherical, polar and non-polar molecular fluids with definite amount of the degree of freedom. They are ideal samples for us to certify the validity of the Eq. (30). The major properties of the data used are listed in Table 1, which shows wide parameter ranges of the samples in which the data points almost uniformly distribute. For nitrogen, the Eq. (30) was fitted to the NIST’s data via the Levenberg–Marquardt algorithm and then the coefficients in the Eq. (30) were determined and listed in Table 2. The determination coefficient (R2 ) of the fit was 0.999 closely approaching 1, suggesting a good fit and high precision for the Eq. (30) to describe the NIST’s compressibility factor data of nitrogen. The goodness is visually illustrated in Fig. 3, in which the curves by the Eq. (30) and the coefficients in Table 2 almost meet the NIST’s data points at 150, 300, and 500 K. For all the 12541 data points of nitrogen, the calculated
X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
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Table 2 Fitted coefficients of the four simple fluids. Coefficient
Argon
Nitrogen
Carbon monoxide
Methane
Tσ (K) ρσ (mol/L)
660.6 45.63 1.714 3.821 −0.1439 0.997 3.4
611.9 39.13 1.714 3.941 −0.1740 0.999 3.8
657.3 37.75 1.678 3.607 −0.1633 0.992 5.6
812.8 34.32 1.753 3.893 −0.1428 0.999 5.9
α β λ (K−0.5 )
R2 AAPD (%)
Table 3 Estimated critical temperature and density of the four simple fluids and their absolute percent deviations from experimental data. Parameter
Argon
Nitrogen
Carbon monoxide
Methane
Tc (K) APD (%) ρc (mol/L) APD (%)
169 12 13.8 3
127 0.4 11.4 1.8
136 2 10.6 2.5
187 2 10.2 0.4
Fig. 3. Z factor of nitrogen versus density at 150, 300, 500 K. Curves were calculated via SELJF, symbols denote NIST’s data.
compressibility factor by SELJF were plotted against the NIST’s data in Fig. 4. It demonstrates that all points narrowly sat along the diagonal line, suggesting an overall accuracy for the SELJF to describe the NIST’s data. The averaged absolute percent deviation (AAPD, defined in Appendix) is 3.8% for the nitrogen. Similar good results were found for the other pure simple fluids in question as indicated by the R2 and AAPD values in Table 2. These results assessed the availability for the SELJF to the most important pure simple fluids, so as to certify the correctness of our theoretical framework and to demonstrate its practical values in its field. With the inflection point method, the Eq. (29) can predict the critical points. For the above-mentioned fluids (argon, nitrogen, carbon monoxide and methane) with the corresponding coefficients listed in Table 2, the critical temperature (Tc ) and density (ρc ) were estimated and presented in Table 3. The absolute percent deviations from experimental data (by NIST Reference Fluid Properties) were listed in the same table. It shows that the estimated critical temperature and density of the four simple fluids were in acceptable accuracy as a whole. 4. Conclusion In this paper, the conceptual model of symmetric extended Lennard-Jones fluid (SELJF) was developed based on an extended Lennard-Jones potential function. Therein, important relationships between microscopic averaged intermolecular potential field (AIPF) and macroscopic thermodynamic properties were obtained via the symmetric interaction approximation, in which the EOS for the model fluid was included. The EOS successfully predicted the corresponding properties of argon, nitrogen, carbon monoxide and methane that are representatives of spherical, polar and non-polar molecular simple fluids. The SELJF not only used the intermolecular potential field representing the intermolecular interaction, which involved the contributions of binary, triple and higher body interactions, avoiding the error and mathematical complexity arising from the pairwise additive potential approximation, but also physically interpreted the relevant thermodynamic properties from the AIPF. The obtained results have both theoretical and practical significance in the following aspects: (1) They enrich the content of thermodynamics and statistical mechanics and give new insights
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X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
Fig. 4. Calculated compressibility factor by SELJF versus NIST’s data for nitrogen.
into the intermolecular interaction. (2) They have captured much of the essential physics of fluids and are close to reality enough to offer a convenient starting point as dealing with actual fluids. (3) They can serve as an accurate reference or base in the computation of relevant real simple fluid properties. Acknowledgment This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix The averaged absolute percent deviation (AAPD) between calculated and the NIST’s data is defined as usual
⏐
AAPD =
⏐
⏐ 1 ∑⏐ ⏐1 − Zcal ⏐ × 100 ⏐ n ZNIST ⏐
here, Zcal and ZNIST are the calculated and the NIST’s compressibility factor data, respectively. n is the number of data points. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
J.D. van der Waals, On the Continuity of the Gaseous and Liquid States (Doctoral dissertation), Universiteit Leiden, 1873. O. Redlich, On the three-parameter representation of the equation of state, Ind. Eng. Chem. Fundam. 14 (3) (1975) 257–260. D. Berthelot, Travaux et Mémoires du Bureau international des Poids et Mesures-Tome XIII, Gauthier-Villars, Paris, 1907. D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. V.F. Baibuz, V.Yu. Zitserman, L.M. Golubushkin, I.G. Malyshev, The covolume and equation of state of high-temperature real gases, J. Eng. Phys. 51 (2) (1986) 955–956. R.H. Aungier, A fast accurate real gas equation of state for fluid dynamic analysis applications, Am. Soc. Mech. Eng. Fluids Eng. Div. 182 (1994) 1–6. A.B. Kaplun, B.I. Kidyarov, A.B. Meshalkin, A.V. Shishkin, Thermal equation of state for real gas in a wide range of state parameters including the critical region, Thermophys. Aeromech. 15 (3) (2008) 359–368. B. Grigor’ev, I. Alexandrov, A. Gerasimov, Generalized equation of state for the cyclic hydrocarbons over a temperature range from the triple point to 700 K with pressures up to 100 MPa, Fluid Phase Equilib. 418 (2016) 15–36. N. Farzi, P. Hosseini, A new equation of state for gaseous, liquid, and supercritical fluids, Fluid Phase Equilib. 409 (2016) 59–71. F. Esmaeilzadeh, F. Samadi, Modification of Esmaeilzadeh–Roshanfekr equation of state to improve volumetric predictions of gas condensate reservoir, Fluid Phase Equilib. 267 (2008) 113–118. I. Polishuk, Generalized cubic equation of state adjusted to the virial coefficients of real gases and its prediction of auxiliary thermodynamic properties, Ind. Eng. Chem. Res. 48 (23) (2009) 10708–10717. S. Hu, W. Dai, A new equation of state for studying the thermodynamic properties of real gases, Phys. Chem. Liq. 53 (1) (2015) 138–145. Luis A. Forero G, J.A. Velásquez J, A generalized cubic equation of state for non-polar and polar substances, Fluid Phase Equilib. 418 (2016) 74–87. M.S. Mahboub, H. Farrokhpour, A generalized perturbation-based equation of state for thermodynamic modelling of fluids over a wide range of density, Phys. Chem. Liq. 56 (6) (2018) 730–750. S. Zhou, J.R. Solana, Thermodynamic properties of diamond and wurtzite model fluids from computer simulation and thermodynamic perturbation theory, Physica A 493 (2018) 342–358. J. Largo, J.R. Solana, Thermodynamic properties of sutherland fluids from an analytical perturbation theory, Int. J. Thermophys. 21 (4) (2000) 899–908.
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[17] J.P. Sinha, S.K. Sinha, Perturbation theory for thermodynamic properties of a ν -dimensional square-well fluid, Pramana J. Phys. 35 (5) (1990) 473–483. [18] J. Montes-Perez, A. Cruz-Vera, J.N. Herrera, Thermodynamic properties and static structure factor for a Yukawa fluid in the mean spherical approximation, Interdiscip. Sci. Comput. Life Sci. 3 (2011) 243–250. [19] J.N. Herrera, Thermodynamic and structural properties of fluids with a hard-sphere plus multi-Yukawa interaction potential, J. Mol. Liq. 270 (2018) 25–29. [20] B.M. Mishra, S.K. Sinhar, Thermodynamic properties of a two-dimensional square-well fluid, Pramana 23 (1) (1984) 79–90. [21] G.M. Kontogeorgis, I.G. Economou, Equations of state: From the ideas of van der Waals to association theories, J. Supercrit. Fluids 55 (2010) 421–437. [22] K. Meier, A. Laesecke, S. Kabelac, Transport coefficients of the Lennard–Jones model fluid. I. Viscosity, J. Chem. Phys. 121 (8) (2004) 3671–3687. [23] J.K. Johnson, J.A. Zollweg, K.E. Gubbins, The Lennard–Jones equation of state revisited, Mol. Phys. 78 (3) (1993) 591–618. [24] A. Zhang, X.H. Yang, S.X. Zhang, An approach to the averaged intermolecular potential field of methane from viscosity, Chem. Phys. Lett. 682 (2017) 82–86. [25] X.H. Yang, W. Zhu, A new model for the viscosity of liquid short chain (C1–C4) 1-alcohols over density and temperature based on Lennard–Jones potential, J. Mol. Liq. 224 (2016) 234–239. [26] K. Huang, Statistical Mechanics, second ed., John Wiley & Sons, New York, 1987, pp. 151–153. [27] X.H. Yang, W. Zhu, Approaching averaged binary potential field from compressibility factor, J. Mol. Liq. 280 (2019) 367–373.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
The equation of state of symmetric extended Lennard-Jones fluids ∗
Xiaohong Yang , Weiling Zhu Department of Chemistry and Environmental Engineering, Wuhan Polytechnic University, Hankou, Wuhan, 430023, PR China
graphical
abstract
highlights • • • •
A new model of symmetric extended Lennard-Jones fluid was developed. An extended Lennard-Jones potential function was introduced. The equation of state of symmetric extended Lennard-Jones fluid was derived. The equation described the properties of four simple fluids in acceptable accuracy.
article
info
Article history: Received 22 March 2019 Received in revised form 16 July 2019 Available online 19 July 2019 Keywords: Equation of state Symmetric fluid Extended Lennard-Jones potential Averaged intermolecular potential field
a b s t r a c t This paper developed the conceptual model of symmetric extended Lennard-Jones fluid based on an extended Lennard-Jones potential function. Therein, important relationships between averaged intermolecular potential field (AIPF) and thermodynamic properties were presented, in which the equation of state was included. These formulas can not only predict the p-V-T properties of argon, nitrogen, carbon monoxide and methane that are representatives of spherical, polar and non-polar molecular fluids in acceptable accuracy, but also physically interpret the p-V-T properties from the AIPF. Obtained
∗ Corresponding author. E-mail address: [email protected] (X.H. Yang). https://doi.org/10.1016/j.physa.2019.122090 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
results have theoretical significance and can serve as reference or base in the analysis and computation of the p-V-T properties of simple fluids. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The equation of state (EOS) is of fundamental importance from both the theoretical and practical points of view. It is usually expressed as a relation between state variables. For example, the EOS of ideal gas provides a simple mathematical relationship among its pressure, volume, and temperature. Because a real fluid is much more important and complicated than an ideal gas, its EOS has been studied for more than 100 years. Since van der Waals [1] semi-theoretically introduced the first EOS for real gas by the assumption of a finite volume occupied by the constituent molecules in 1873, numerous investigators have developed EOS based mainly on his ideas, such as the equations from the Redlich–Kwong model [2], Berthelot and modified Berthelot model [3], Peng–Robinson model [4], and so on. These equations principally indicated the early progress. Because of the extreme importance, the research on the EOS has never stopped. In 1986, Baibuz et al. [5] proposed the EOS for high temperature real gas from the concept of covolume. In 1994, Aungier modified the Redlich–Kwong two-parameter equation by employing the acentric factor and the critical point compressibility factor as additional parameters so as to extend its application range to include the critical point [6]. In recent years, efforts have been made on the equations applicable in wide state parameters [7] and more substances [8,9] as well as modified cubic equations [10–14]. Model fluids are idealized imaginary substances that have a well-defined expression for the intermolecular potential, so that the properties of the model fluids are able to be calculated through physical theory with mathematical and/or numerical methods [15–20]. A well-developed model fluid can represent the essential behavior of real fluids and offer data in acceptable accuracy over wide parameter ranges. Furthermore, the model fluids contain physically meaningful parameters, which are directly related to the intermolecular interactions. Therefore, a model fluid is a good reference or base while predicting the thermodynamic properties of pure fluids and their mixtures. Due to the stated benefits, there is always a high need for successful model fluid capable of giving the thermodynamic properties in an analytically straightforward way. In thermodynamics and statistical mechanics there are famous model fluids like the ideal gas, van der Waals fluid, association fluid theories [21], Lennard-Jones model fluid for transport coefficients [22] and equation of state [23] in addition to the just mentioned model fluids in the references [15–20]. Recently, the authors of present paper successfully used a symmetric interaction approximation in calculating viscosity of fluid [24,25]. As a new method, the symmetric interaction approximation is promising to yield novel theoretically significant results in the development of a new model fluid and derivation of the EOS. The aims of this work are to develop a novel model fluid named symmetric extended Lennard-Jones fluids upon the concept of symmetric interaction, and to derive the equation of state based on an extended Lennard-Jones potential function. After that, the availability of the model fluid is to be assessed by applying the derived equations to the most important pure simple fluids: argon, nitrogen, carbon monoxide and methane. In the end, the theoretical and practical significance of the model fluid will be evaluated. 2. Theory of symmetric fluid 2.1. Potential well A symmetric fluid is defined as the fluid whose molecule accepts symmetric force within the fluid from its neighbors around it, and unsymmetric force at boundary from its half neighbor to restrict it inside. By this definition, if a molecule inside receives a force by a neighbor, there is always an opposite force from the molecule at the symmetric position to counteract on it. Then, the total force (F ) on each molecule inside equals zero. If the intermolecular force of the symmetric fluid is represented by a potential field (denoted by u), then F = −∇ u = 04. This equation deduces that the potential field is a constant field independent of spatial coordinates within the symmetric fluid and a step function at the boundary. Such feature, just like a potential well of flat bottom, is expressed by the following equation,
{ u=
w 0
w ithin fluid without fluid
(1)
here, u denotes the intermolecular potential field, w is the amount presenting the depth of the potential well and equals the averaged intermolecular potential field (AIPF) as a result of the constant potential field inside the fluid. Since the symmetric molecule that counteracts at the symmetric position always exists in an ergodic system in thermodynamic equilibrium, the symmetric fluid model is accurate in such system.
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2.2. Entropy Based on the above zero total force, the molecule in the symmetric fluid can move freely just like the molecule in the ideal gas. The entropy of classical ideal gas is then available for the molecules in the symmetric fluid, which is given below [26],
[ ( 4π m Ek ) 32 ]
3 (2) + Nk 3h2 N 2 here, S presents the entropy of the symmetric fluid, Ek the kinetic energy, N is the total number of molecules restricted in volume V , k is the Boltzmann’s constant, T is absolute temperature, m is the mass of molecule. h is a constant of the dimension of momentum timing distance. As usual, defining the absolute temperature through kinetic energy with the following equation, S Ek , V = Nk ln V
(
)
T =
( ∂ Ek ) ∂S V
(3)
it yields 3
NkT 2 for the symmetric fluid. Ek =
(4)
2.3. Internal energy and parameters In order to derive an expression for the internal energy of the symmetric fluid, it is noticed that the ratio of the 1/3 molecules on boundary to all molecules is approximately 6/N A ∼ 7E − 8 (NA is the Avogadro constant) for a mole symmetric fluid. This small value suggests the error due to the boundary effect is considered so small that it can be ignored. During an adiabatic process, there is no external energy transmitting into the fluid, therefore the internal energy of the symmetric fluid is thus approached as, U = Ek + N w
(5)
here, U denotes the internal energy that belongs a thermodynamic state function. After defining the density (ρ ) as the number of molecules per volume,
ρ=
N
V The equation of state (EOS) is derived through the state function and the Eq. (3) as
(6)
( ∂U ) (7) ∂V S [ ρ ( ∂w ) ] (8) p = ρ kT 1 + kT ∂ρ S When w→0, the Eq. (8) simplifies to the EOS of the ideal gas. By Eq. (8), the compressibility factor (Z ) is expressed as ρ ( ∂w ) Z =1+ (9) kT ∂ρ S p=−
By reviewing the above equations and the relevant derivations, it is fair that all the state functions and parameters of the symmetric fluid can be determined, if w is given. 2.4. Averaged intermolecular potential field With respect to the intermolecular potential field based on the Eulerian approach, it was separated into two terms, u = u2 + um
(10)
here, u2 is the binary potential field that approaches zero in the infinite dilute limit, um stands for the total contributions of many-body potential field that includes triple and higher terms. Averaging Eq. (10) over ensemble that is noted by triangular parenthesis, it gives
⟨ u⟩ = ⟨ u2 ⟩ + ⟨ um ⟩
(11)
This formula offers a relation among the AIPF, averaged binary potential field (ABPF) and averaged many-body potential field (AMPF). Because the triple and higher interacting mechanisms actually transit a molecular event simultaneously to the entire system against the slow binary interacting mechanism, they effectively reveal themselves as the bulk effect of all molecules in system [24,27]. And then, the AMPF does not rely on the intermolecular distance.
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As mentioned above, within the symmetric fluid the depth of the potential well w equals the AIPF. It can also be separated into two terms,
w = w2 + wm
(12)
here, w2 equals the ABPF that approaches zero as the intermolecular distance→∞, wm is identical to the AMPF that merely depends on temperature, when the independent variables of the state function are chosen as the temperature and density.
2.5. Extended Lennard-Jones potential function In order to analytically represent the ABPF, the averaged intermolecular distance (AID, from now on denoted by d) between two molecules is introduced beforehand, which can be measured with either volume or density. For the system of N molecules restricted in volume V , it is averagely believed that the volume is equally shared by the N molecules. If the volume occupied by each molecule is modeled as cube, it yields, V = Nd3
ρ=
N
(13) 1
=
V
(14)
d3
where, ρ is the density of the system. These two equations not only enable us measuring the AID, but also supply a route to define a Lennard-Jones volume and density by the following two formulas, Vσ = N σ 3
ρσ =
(15)
1
(16)
σ3
here, σ is the Lennard-Jones distance. We now introduce an extended Lennard-Jones potential function. The extension involves two aspects: one is extending the function form to cover more potential models; another is the extension of independent variable into the AID or volume or density. The extended function is as follow,
φ = φσ
[( σ )3β d
−θ
( σ )3α ] d
(17)
Obviously, it is extended from the (3β , 3α ) pattern Lennard-Jones binary potential (or Mie potential). Here θ is a parameter independent of the AID, φσ is the Lennard-Jones constant in the unit of energy. This function can cover the potentials of Lennard-Jones’ form, the hard sphere and soft sphere as the parameters take different values. That is
⎧ ⎨θ > 0 ⎩θ ≤ 0 Sphere model
Lennard − Jones′ form { β → ∞ hard shpere others soft sphere
As θ continuously varies from positive to negative values with temperature increasing, the Eq. (17) gives a potential energy surface that continuously transforms the potential curve from the Lennard-Jones’ form into the sphere potential. Fig. 1 shows an example of the potential energy surface when β = 3.8, α = 1.7 and θ = 3 − 0.14T 0.5 . The extended Lennard-Jones potential function enables us constructing a new conceptual model fluid. Comparing with the existing model fluids [16–20], the potential of the new model fluid has the soft sphere repulsive term against the hard sphere repulsions. Therefore, it can approach a real fluid more realistically, especially in the dense fluid where the intermolecular repulsive force may mainly account for the thermodynamic properties of the fluid. Based on the cubic volume model, the Eq. (17) can thus be equivalently represented by the following equations by extending the independent variable through the Eqs. (13) and (14),
[( ρ )β ( ρ )α ] −θ ρσ ρσ [( Vσ )β ( Vσ ) α ] φ = φσ −θ φ = φσ
V
V
(18) (19)
These equations are good approximations to the ABPF and then are called extended Lennard-Jones potential (ELJP). When β = 4, α = 2 and θ = 1, the ELJP reduce to the famous (12, 6) pattern Lennard-Jones potential. Since it is generally believed that the intermolecular potential determines the entire thermodynamic properties, we hope the ELJP with the symmetric fluid theory can add a successful example for this belief.
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Fig. 1. Extended Lennard-Jones potential energy surface with β = 3.8, α = 1.7 and θ = 3 − 0.14T 0.5 .
3. Results and discussion 3.1. Internal energy We define a symmetric extended Lennard-Jones fluid (SELJF) as the symmetric fluid whose ABPF equals the value of the ELJP at the AID. It thus has the following equation for the SELJF,
w = φσ
( ρ )α ] ( ) [( ρ )β −θ + wm T ρσ ρσ
(20)
Substituting this equation and Eq. (4) into Eq. (5), the internal energy per mole SELJF is derived as, f
U =
2
RT + RTσ
( ) [( ρ )β ( ρ )α ] −θ + Um T ρσ ρσ
(21)
where, R = 8.314 J/(mol.K) is the ideal gas constant, Tσ = φσ /k, Um = NA wm . NA is the Avogadro constant, f denotes the degree of freedom. As ρ → 0, the internal energy in zero density limit is obtained, U0 =
f 2
( )
RT + Um T
(22)
here, U0 denotes the internal energy in zero density limit, which composes of the kinetic energy and the AMPF. 3.2. The equation of state Because the equation of state (EOS) can both calculate phase equilibria values and p − V − T properties in wide parameter ranges, it has been considered the cornerstone of thermodynamic models [21]. We thus discuss the EOS of the SELJF in this section. Substituting Eq. (20) into Eq. (8), it yields the EOS of a mole SELJF at the constant AMPF, p=
RT { V
1+
Tσ [ ( Vσ )β T
β
V
( )( Vσ )α ]} 2 − αθ + T θ ′ f
V
(23)
Such appearance not only adds a new form for the EOS, but also clearly indicates the influences of the attractive and repulsive power-law terms on the pressure. As Vσ /V →0 (the dilute gas case), this equation also returns to the EOS of ideal gas. For simplicity, we define the reduced temperature (Tr ), density (ρr ), volume (Vr ) and pressure (pr ) with respect to Tσ , ρσ , Vσ and pσ , as usual, Tr =
ρr = Vr = pr =
T Tσ
ρ ρσ V Vσ p pσ
(24) (25) (26) (27)
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Fig. 2. p − V − T surface of SELJF when β = 3.8, α = 1.7, Tσ = 660 K, f = 3 and θ = 3 − 0.14T 0.5 . Table 1 Major properties of the data used. Item
Argon
Nitrogen
Carbon monoxide
Methane
Effective degree of freedom Temperature range (K) Density range (mol/L) Z range No. points
3 90–990 0–50.3 0.00517–12.1 6328
5 70–1000 0–52.0 0.00221–23.1 12 541
5 70–500 0–33.5 0.00120–6.52 6957
8 100–620 0–38.7 0.00151–15.0 4457
where, pσ =
RTσ
(28)
Vσ
The EOS is then represented in the reduced form, pr =
)( 1 )α ]} 1 [ ( 1 )β ( 2 Tr { 1+ β − αθ + T θ ′ Vr Tr Vr f Vr
(29)
When β = 3.8, α = 1.7, Tσ = 660 K, f = 3 and θ = 3 − 0.14T 0.5 , the p − V − T surface of the SELJF was calculated as Fig. 2 by the Eq. (29), which reproduces all the characters of the van der Waals EOS, such as the existence of van der Waals isotherm oscillation, critical point, the liquid, gaseous and supercritical phase regions, as well as phase transition. By the Eq. (29), we could offer an important interpretation for the formation of van der Waals isotherm oscillation. Because the oscillation is analytically linked to the derivative of the ABPF, we can now refer the mechanism for the oscillation occurrence to the intermolecular potential. Despite the SELJF is not a real fluid, its EOS is available to many real pure fluids in acceptable accuracy. This advantage can be certified via the compressibility factor (Z ) as the function of temperature and density that is derived from the Eq. (23) and is analytically equivalent to the EOS, Z =1+
Tσ [ T
β
(
2
βρr − αθ + T θ f
′
)
ρrα
]
(30)
here, we propose a simple relation for the temperature dependence of the θ ,
√ θ =3+λ T
(31)
where λ is a constant coefficient to be fitted. The National Institute of Standards and Technology’s (NIST) open database provides an authoritative compressibility factor data varying with temperature and density for many fluids such as argon, nitrogen, carbon monoxide and methane, which are the representatives for spherical, polar and non-polar molecular fluids with definite amount of the degree of freedom. They are ideal samples for us to certify the validity of the Eq. (30). The major properties of the data used are listed in Table 1, which shows wide parameter ranges of the samples in which the data points almost uniformly distribute. For nitrogen, the Eq. (30) was fitted to the NIST’s data via the Levenberg–Marquardt algorithm and then the coefficients in the Eq. (30) were determined and listed in Table 2. The determination coefficient (R2 ) of the fit was 0.999 closely approaching 1, suggesting a good fit and high precision for the Eq. (30) to describe the NIST’s compressibility factor data of nitrogen. The goodness is visually illustrated in Fig. 3, in which the curves by the Eq. (30) and the coefficients in Table 2 almost meet the NIST’s data points at 150, 300, and 500 K. For all the 12541 data points of nitrogen, the calculated
X.H. Yang and W. Zhu / Physica A 533 (2019) 122090
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Table 2 Fitted coefficients of the four simple fluids. Coefficient
Argon
Nitrogen
Carbon monoxide
Methane
Tσ (K) ρσ (mol/L)
660.6 45.63 1.714 3.821 −0.1439 0.997 3.4
611.9 39.13 1.714 3.941 −0.1740 0.999 3.8
657.3 37.75 1.678 3.607 −0.1633 0.992 5.6
812.8 34.32 1.753 3.893 −0.1428 0.999 5.9
α β λ (K−0.5 )
R2 AAPD (%)
Table 3 Estimated critical temperature and density of the four simple fluids and their absolute percent deviations from experimental data. Parameter
Argon
Nitrogen
Carbon monoxide
Methane
Tc (K) APD (%) ρc (mol/L) APD (%)
169 12 13.8 3
127 0.4 11.4 1.8
136 2 10.6 2.5
187 2 10.2 0.4
Fig. 3. Z factor of nitrogen versus density at 150, 300, 500 K. Curves were calculated via SELJF, symbols denote NIST’s data.
compressibility factor by SELJF were plotted against the NIST’s data in Fig. 4. It demonstrates that all points narrowly sat along the diagonal line, suggesting an overall accuracy for the SELJF to describe the NIST’s data. The averaged absolute percent deviation (AAPD, defined in Appendix) is 3.8% for the nitrogen. Similar good results were found for the other pure simple fluids in question as indicated by the R2 and AAPD values in Table 2. These results assessed the availability for the SELJF to the most important pure simple fluids, so as to certify the correctness of our theoretical framework and to demonstrate its practical values in its field. With the inflection point method, the Eq. (29) can predict the critical points. For the above-mentioned fluids (argon, nitrogen, carbon monoxide and methane) with the corresponding coefficients listed in Table 2, the critical temperature (Tc ) and density (ρc ) were estimated and presented in Table 3. The absolute percent deviations from experimental data (by NIST Reference Fluid Properties) were listed in the same table. It shows that the estimated critical temperature and density of the four simple fluids were in acceptable accuracy as a whole. 4. Conclusion In this paper, the conceptual model of symmetric extended Lennard-Jones fluid (SELJF) was developed based on an extended Lennard-Jones potential function. Therein, important relationships between microscopic averaged intermolecular potential field (AIPF) and macroscopic thermodynamic properties were obtained via the symmetric interaction approximation, in which the EOS for the model fluid was included. The EOS successfully predicted the corresponding properties of argon, nitrogen, carbon monoxide and methane that are representatives of spherical, polar and non-polar molecular simple fluids. The SELJF not only used the intermolecular potential field representing the intermolecular interaction, which involved the contributions of binary, triple and higher body interactions, avoiding the error and mathematical complexity arising from the pairwise additive potential approximation, but also physically interpreted the relevant thermodynamic properties from the AIPF. The obtained results have both theoretical and practical significance in the following aspects: (1) They enrich the content of thermodynamics and statistical mechanics and give new insights
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Fig. 4. Calculated compressibility factor by SELJF versus NIST’s data for nitrogen.
into the intermolecular interaction. (2) They have captured much of the essential physics of fluids and are close to reality enough to offer a convenient starting point as dealing with actual fluids. (3) They can serve as an accurate reference or base in the computation of relevant real simple fluid properties. Acknowledgment This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix The averaged absolute percent deviation (AAPD) between calculated and the NIST’s data is defined as usual
⏐
AAPD =
⏐
⏐ 1 ∑⏐ ⏐1 − Zcal ⏐ × 100 ⏐ n ZNIST ⏐
here, Zcal and ZNIST are the calculated and the NIST’s compressibility factor data, respectively. n is the number of data points. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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