On the phase equilibrium of atomic and molecular pure fluids from thermodynamic perturbation theory

Fluid Phase Equilibria 237 (2005) 31–39

On the phase equilibrium of atomic and molecular pure fluids from thermodynamic perturbation theory Osvaldo H. Scalise ∗ Instituto de F´ıisica de L´ıquidos y Sistemas Biol´ogicos (IFLYSIB), UNLP-CONICET-CICPBA, C.C. 565, 1900 La Plata, Argentina Received 16 June 2005; received in revised form 27 July 2005; accepted 3 August 2005 Available online 19 September 2005

Abstract A simplified perturbation theoretical approach is presented for calculating the fluid phase behaviour of both atomic and molecular pure fluids. For atomic Lennard–Jones pure fluids, calculated critical points and vapour–liquid equilibrium coexistence curves show close agreement with computer simulations. For molecular fluids, the approach applied to quadrupolar fluids predicts the behaviour found in computer simulations for fluids with quadrupolar strength of q*2 = 1, 1.5, 2 and 2.5. © 2005 Elsevier B.V. All rights reserved. Keywords: Perturbation theory; Molecular fluids; Lennard–Jones quadrupolar fluid; Phase equilibrium

1. Introduction The study of the phase behaviour of pure fluids is both of fundamental theoretical and practical interests. From the theoretical point of view, they provide information on the relationship between the intermolecular forces and the fluid phase behaviour. On the other hand, from the practical point of view, the knowledge of the fluid phase behaviour is of great value in chemical engineering, physical chemistry and industrial applications. Great efforts have been made for studying the phase behaviour of fluids using different theoretical approaches as integral equation methods, thermodynamic perturbation theory (TPT) [1,2] and computer simulations methods [3]. Although the atomic fluid properties are very well known to date, that is not the case of molecular fluids [1,2]. Different techniques have been applied to theoretically study the properties of such complex fluids in which molecules interact via angular depending forces [1,2], which require a great computational effort. In this work, one of the goals has been directed towards finding a simplified theoretical approach applied to a simple ∗

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molecular model that do not demands much computational effort for calculating the phase behaviour of both atomic and molecular pure fluids. In this note, a simplified TPT approach is presented for studying the properties of both atomic and molecular pure fluids. Mainly, the idea consists in separately taking the radial and electrostatic multipolar forces acting upon molecules as the perturbation potential and, separately considering their contributions to the free energy of the reference system. We consider: (a) for atomic fluids, molecules interactions are via a hard sphere repulsive potential plus an attractive Lennard– Jones tail. (b) for molecular fluids, that molecules interact via a hard sphere repulsive potential plus an attractive Lennard–Jones plus a point multipolar moment placed at the Lennard–Jones site. The radial part of the interactions is then given by the Lennard–Jones potential and the orientation depending interactions are assumed to come from dipolar–dipolar, quadrupolar–quadrupolar, etc. interactions. In this work, we study the case of a quadrupolar fluid, then the multipolar moment placed at the Lennard–Jones site is only just a point quadrupolar moment.

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O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

Usually, in TPT approaches [1,2], the LJ potential is taken as the reference system, which represents the radial interaction of molecules, and the electrostatic multipolar interactions are considered as the total perturbation potential. Here, instead for atomic pure fluids, the LJ potential is considered as the total perturbation potential [4] to the hard-spheres reference system and; for molecular pure fluids, it is assumed to be only a part of the total perturbation potential; the electrostatic multipolar interactions are the other part of the perturbation potential. On implementing the above ideas for atomic and molecular pure fluids, we have applied TPT to the intermolecular potential given in the next section and plotted in Fig. 1. For calculating the Helmholtz free energy (HFE) of the fluid, we consider that perturbation contributions are: (a) for atomic fluids, given by a Pad´e approximant [5] for the LJ function which represent all interactions of molecules and; (b) for molecular fluids, given by a Pad´e approximant that accounts for the LJ interactions, plus another Pad´e approximant that take into consideration the dipole–dipole interactions, plus another Pad´e approximant for the quadrupolar–quadrupolar interactions, etc. Thus, for molecular pure fluids, the presented simplified approach accounts for all contributions to the HFE from radial and different electrostatic multipolar interactions. Such contributions, which separately are taken in different Pad´e type expressions, are added to the HFE of the HS reference system fluid. Since the radial part of the interaction of molecules is assumed to be that of the LJ potential, and just as a test of the approach, in this work, calculations are performed for the atomic LJ pure fluid. For the molecular fluid, a stronger test is chosen to be satisfied by the approach. It is applied to an intermolecular potential that far can be considered as a nearly spherical interaction potential (it is well known [1,2] that TPT gives good results for nearly spherical interaction potentials). The approach then here is applied to the case of the LJ quadrupolar pure fluid. In view of results obtained in this note, the theoretical approach may result useful for calculating the phase behaviour of both atomic and molecular fluids. In this work, we present results for the atomic LJ pure fluid, which show close agreement with computer simulations recently performed for the critical point [6] (see Table 1). Further, for LJ vapour–liquid equilibrium coexistence curves, the approach predicts the behaviour found using modified Benedict–Webb–Rubin (MBWR) equations of state (EOS) with 33 parameters, reported for the LJ pure fluid [7,8] (see Figs. 2–4). More recently. Kolafa and Nezbeda [9] presented an EOS based on a perturbing virial expansion and considered that the size of the reference system of hard-spheres depends on temperature. Using an EOS with 20 parameters, they showed that it performs better than the MBWR type EOS.

Fig. 1. Reduced hard-spheres (HS)-Lennard–Jones (LJ) intermolecular model u* (r* ) = u/ε, vs. reduced intermolecular separation r* = r/σ.

For molecular fluids, the procedure applied to the case of the LJ quadrupolar pure fluid give results for the critical point in good agreement with Gibbs ensemble computer simulations performed recently [10] (see Table 2). For the vapour–liquid equilibrium coexistence curves, the behaviour found in computer simulations for fluids with quadrupolar strength of q*2 = 1, 1.5, 2 and 2.5 is also predicted by the approach. In next section, a brief description of the theoretical approach is given for atomic and molecular pure fluids. In Section 3, results and discussions are presented. Finally, in Section 4, conclusions are presented.

2. Theory In this work, thermodynamic perturbation theory TPT [1,2] is applied to a system of equal N hard-spheres molecules interacting via the LJ potential at a given temperature T, in a volume V. For atomic LJ fluids, it is assumed that the total pair potential between two molecules, say i and j, are separated as follows (see Fig. 1): uij (r) = u0ij (r) + uPij (r),

(1)

with: u0ij (r) = ∞, u0ij (r) = 0,

r≤σ

(2)

r>σ

and uPij (r) = 0, uPij (r)

=

with: u (r) = 4ε ij

r≤σ

uLJ ij (r),


  σ 12 r

(3)

r>σ

−

 σ 6  r

,

(4)

O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

where r is the distance from the center of molecule i, to the center of molecule j. u0ij (r) is the HS potential, which is taken as the fluid reference system. uij (r), the LJ pair potential function is assumed to be, respectively, for atomic pure fluids the total perturbation potential and a part of it for molecular pure fluids. TPT studies [1,2] have considered that the total pair wise additive potential is separated out in an isotropic reference potential and an angular dependent multipolar term that account, respectively, for the interaction of molecules at short, and at intermediate and at long distances [1,2]. Pioneers works of Barker–Henderson (BH) [11] and Week–Chandler–Andersen (WCA) [12] differs on the way the LJ pair potential is separated out for accounting on both repulsive and attractive forces. In both BH and WCA theories, the radial distribution function of the reference fluid, g0 (r), is related to that of a radial distribution function of a fluid of hard-spheres gHS (r) of diameter σ HS . In BH and WCA theories, the value of σ HS is a function of, respectively, only of temperature σ HS (T), and of both density and temperature σ HS (ρ, T). The two theories depend on the way the pair potential is separated out and on the devised procedure for calculating the diameter σ HS value of the HS fluid reference system. Once the value of the function for the diameter of the hard-spheres σ HS is obtained, the radial distribution function of the HS fluid system and the HFE reference system may be calculated [1,2]. Main new idea of this work concerns the assumption that the diameter of the HS reference system is equal to the parameter σ of the Lennard–Jones, hence independent of temperature. 2.1. Atomic pure fluids TPT assumes that the Helmholtz free energy A can be written as [1,2]: A∗ = A∗0 + A∗1 + A∗2 + · · ·

(5)

A∗0 = βA/N is the free energy of the reference system and A∗1 , A∗2 , . . ., etc. are the contribution from the first, second,. . ., etc. perturbation terms to the total HFE of the fluid. Following Zwanzig’s assumption [4], we consider that the perturbing potential to the HS fluid reference system is the LJ potential. Using the so called local compressibility approximation, Barker and Henderson [11] showed that the equation for the total Helmholtz free energy for the pure LJ fluid is written as:  βA ρβ ∗ = A0 + uLJ (r)go (r)4πr 2 dr N 2    ρβ2 ∂[ρ(go (r))] 2 LJ − [u (r)] kT 4πr 2 dr + · · · 4 ∂P (6) where go (r) is the radial distribution function of a system of HS of diameter σ. Thus, the first and second order perturba-

33

tion terms are then, respectively, given by: A∗1 =

8πρ∗ (12) ∗ [I (ρ ) − I (6) (ρ∗ )] T∗

(7)

and A∗2 =

1 −16π ∂ {ρ∗ [I (24) (ρ∗ ) + I (12) (ρ∗ ) ∗ ∗ ∗ T K (ρ, T ) ∂p∗ −2I (18) (ρ∗ )]}.

(8)

The symbols I(n) are the integrals reported by Stell and collaborators in Ref. [5] as Pad´e approximants obtained using a virial series expansion for go (r). They showed that Pad´e results are the best values for I(n) (see Appendix of Ref. [5] for details). β = 1/kB T with kB the Boltzmann’s factor, and K∗ (ρ∗ , T ∗ ) = (1/ε)(∂P (0) /∂ρ), where P(0) is the pressure of the HS reference system [13]. In the above equations, reduced quantities given in terms of the LJ potential parameters are density ρ* = ρσ 3 , temperature T* = kB T/ε, and pressure P* = Pσ 3 /ε with σ = σ HS = σ LJ . Then, Eq. (6) can be written as follows: A∗ = A∗0 + A∗LJ

(9)

where A∗0 is the HFE Carnahan–Starling’s expression [13] and A∗LJ is given by the Pad´e approximant [14]: A∗LJ

=

A∗1



A∗ 1 − 2∗ A1

−1

,

(10)

with A∗1 and A∗2 given, respectively, by Eqs. (7) and (8). 2.1.1. The critical point The critical point of the fluid is calculated by numerically solving the following equations:
∗ ∂P =0 (11) ∂ρ∗ Tc∗ and
2 ∗ ∂ P =0 ∗ ∂ρ2 Tc∗ where P* is the total pressure of the fluid giving by:  ∗ ∗ ∗ ∗ ∗2 ∗ ∂A P =ρ T +ρ T ∂ρ∗ T ∗

(12)

(13)

2.1.2. Phase equilibrium It is well known from thermodynamics [15] that in order for a liquid and its vapor to coexists at a given temperature, say T, they must have the same pressure P and the same chemical potential µ, P(ρg , T ) = P(ρl , T ) = P

(14)

µ(ρg , T ) = µ(ρl , T ) = µ,

(15)

O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

34

where ρg and ρl are the gas and the liquid density, respectively. Solution of these equations gives the equilibrium densities of liquid and vapor phases. An alternative method for obtaining the phase equilibrium is to use the well-known Maxwell equal-area construction [15]. For the case of a pure fluid, however, it is not necessary to calculate the chemical potential to obtain the vapor–liquid equilibrium, since from the Gibbs’s fundamental relation:

q separated by a distance r, is given by: 

3q2 qq uij (r) = 4rij5

1−5(cos2 θi cos2 θj ) − 15cos2 θi cos2 θj + × , 2(cos θi cos θj cos(φi −φj )−4cos θi cos θj )2 (20)

dG = −S dT + V dP + µ dN,

where θ and φ are, respectively, the polar and azimuthal angles with respect to the vector r¯ij . For quadrupolar pure fluids, last term of Eq. (19) is then given by A∗ELEC = A∗q , with:

(16)

and since G/N = µ, by integrating Eq. (16) at constant temperature T, from a saturated liquid state to the saturated vapor state, it follows: Vg G(Vg ) − G(Vl ) =

Vg V dP = P[Vg − Vl ] −

Vl

P dV

(17)

Vl

At equilibrium, the term on the left hand side is zero; the chemical potential must be equal at the vapor and liquid phases. Then,  Vg (s) P [Vg − Vl ] = P dV (18) Vl

The equilibrium values Vl , Vg and P(s) in Eq. (18), for the coexistence vapour–liquid equilibrium at temperature T, are obtained numerically solving Eq. (18).

+ A(2)∗ + A(3)∗ + ··· A∗q = A(1)∗ q q q

(21)

(i)∗

where Aq are the i-order perturbation contribution terms from quadrupolar interactions. Taken up to third order perturbation terms, and by assuming that the above series can be summed up using a Pad´e approximant [5], the contribution to the HFE from quadrupolar interactions is then given by:

 ∗(3) −1 A q A∗q = A∗(2) 1 − ∗(2) ; A∗(1) = 0, [1] (22) q q Aq with: =− A∗(2) q

14πρ∗ (2) ∗ I (ρ ) × q∗4 T ∗2 2

(23)

and 2.2. Molecular pure fluids For molecular pure fluids, not only radial but electrostatic multipolar interaction forces do act upon molecules. For radial interaction, it is assumed, as for atomic pure fluids, that such interactions are represented by the LJ Pad´e approximant by the above given Eq. (10). For the intermolecular forces depending on orientation, such as dipole–dipole, quadrupole–quadrupole, etc., they are separately considered in different Pad´e approximants. As induced forces are neglected, and since LJ interaction are not coupled with multipolar electrostatic interactions, the total HFE for the molecular pure fluid reads as follows: A∗ = A∗0 + A∗LJ + A∗ELEC .

(19)

Last term on the right hand side refers to dipolar–dipolar, quadrupolar–quadrupolar, etc. contributions to the HFE from multipolar electrostatic interactions. Thus, the total HFE given by Eq. (19) accounts for all contributions coming from the radial and electrostatic multipolar interactions. In this work, and in order to test the theoretical approach for molecular pure fluids, we consider that case of LJ quadrupolar pure fluids. Here, it is considered that case of symmetric cylindrical distributions of charge, then q, the quadrupole moment, is an scalar magnitude and the quadrupole–quadrupole interaction between two molecules say, i and j, with quadrupole moments

= A∗(3) q

144π ρ∗ (15) ∗ π2 ρ∗2 ∗6 I (ρ ) × q + I3 (ρ∗ ) × q∗6 2 245 T ∗3 800 T ∗3 (24)

I2 (ρ∗ ) are given in Refs. [5,16] and the I3 (ρ* ) is given by the following expression [14]: (n)

6.75 + 5.9346498ρ∗ − 0.5483549ρ∗2 I3 (ρ∗ ) =

− 2.3178757ρ∗3 1.0 − 1.536875ρ∗ + 0.8886999ρ∗2

(25)

− 0.1949384ρ∗3 with q* = q/(εσ 5 )1/2 is the reduced quadrupole moment of strength, q.

3. Results and discussion 3.1. Atomic pure fluids In this work, we consider that two representative hardspheres molecules of an atomic pure fluid interact, until they make contact with each other, via the LJ potential. The proposed molecular model is then that of the HS plus the LJ potential (see Fig. 1). TPT is applied to the given molecular model, and the LJ is considered, after Zwanzig [4], as the perturbation potential. Further, it is assumed that the LJ

O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

35

Table 1 Critical point parameters for the pure Lennard–Jones fluid ρ*

T*

P*

0.304a 0.3197b 0.316c 0.316d 0.35e 0.310f 0.278g

1.316a 1.1876b 1.3226c 1.3120d 1.308e 1.313f 1.308g

0.1279d 0.142e 0.13f 0.136g

a b c d e f g

Ref. [20]. Ref. [21]. Ref. [22]. Ref. [6]. Ref. [7]. Ref. [8]. This work.

distance parameter coincides with the diameter of the hardspheres. This assumption, as shown above, introduces a great simplification on calculations. The critical point and the fluid phase behaviour of the pure Lennard–Jones fluid have been the subject of both theoretical, [17] and computer simulations studies [18]. Recently, in a reformulation of WCA first order perturbation theory, BenAmotz and Stell [19] have reported the critical values ρc∗ = 0.32 and Tc∗ = 1.29 for the atomic LJ pure fluid. In this work, the theoretical procedure is applied to the LJ pure fluid, just in order for testing whether the approach is useful for describing the properties of that fluid. Using the equations given in Section 2, calculations are performed for the critical point and for the fluid phase behaviour of the LJ atomic pure fluid. Table 1 presents results for the critical parameters of the LJ fluid reported in given references and from this work (last row) obtained using the equations given in Section 2. As shown, theory predicts results in close agreement with computer simulation, for the critical parameters. The vapour–liquid equilibrium coexistence curves results (see Figs. 2–4) also are in agreement with results obtained using MBWR EOS with 33 parameters [7,8]. For the vapour branch, and up to the critical point, the agreement is good. For the liquid branch, however, the agreement is not so good. The vapour–liquid curve is very sensitive to both attractions and repulsions, particularly to the attractive A∗1 and A∗2 terms. Here, the fact that the saturated liquid densities are underestimates by the theory is due to two different things: (1) the original BH theory (with the original BH temperature dependent diameter) usually under-estimate slightly the saturated liquid densities (see Fig. 6 in Ref. [11]). Great influences on the calculated vapour–liquid equilibrium are also due to the radial distribution function of the hard sphere reference system gHS (r), used for calculating the integrals I(n) in Ref. [5]; and, (2) the fact of using the hard-sphere diameter instead of the BH diameter contributes to under-estimate even more the saturated liquid densities. The reason is that the BH

Fig. 2. Coexistence curves for the atomic Lennard–Jones pure fluid. The dotted line is the Nicolas et al. MBWR EOS [7]. The dash-dotted-dotted line is the Johnson et al. new MBWR EOS [8]. The solid line and (䊉) are, respectively, the theoretical coexistence curve and critical point; () is the GEMC computer simulation critical point [6]; (
) and () are, respectively, the critical points from Refs. [7,8].

Fig. 3. Saturated liquid and vapour densities as a function of the vapour pressure for the pure Lennard–Jones fluid. Captions are as given in Fig. 1.

Fig. 4. Vapor pressure curves for the pure Lennard–Jones fluid. Captions are as given in Fig. 1.

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O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

diameter is lower than the hard sphere diameter at high temperature, so there are less repulsions, hence higher liquid densities and critical temperatures. Fig. 3 presents the saturated liquid and vapour densities as a function of the vapour pressure for the atomic LJ pure fluid. As noted, for the vapor branch and up the critical point, results are in agreement with MBWR EOS results [7,8]. For the liquid branch, however, the agreement is not so good because of the above reasons. It is interesting to note, however, that the shape of vapour–liquid equilibrium coexistence curves is described by such an oversimplified molecular model. Fig. 4 shows results for reduced vapor pressure curve for the atomic LJ pure fluid. As noted, results are in good agreement with MBWR EOS results [7,8] for all values of temperature and up to the critical point. In view of results presented in Figs. 2–4, and taking into consideration that the approximations made for getting the HFE are not few as shown, still sufficient information is

available on the HFE given in last section for predicting the properties of atomic pure fluids. Finally, it is interesting to note, however, the fact that for atomic LJ pure fluids, the oversimplified theoretical scheme presented in this note can predict results in close agreement with those obtained using theoretical approaches that require much more computational effort. 3.2. Molecular fluids A second goal of our work has been to investigate whether the phase behaviour of fluids with molecules interacting via orientation depending forces, i.e. molecular fluids, can be described using the equations given in Section 2. Quadrupolar fluids has been the subject of theoretical and computer simulations studies (see Ref. [23] and references therein). It has been shown [1,2] that TPT predicts good results for molecules that are not much different from the fluid reference system. Since the quadrupolar fluid is quite different from the

Fig. 5. Vapour–liquid equilibrium results for the Lennard–Jones quadrupolar pure fluid with quadrupolar strengths: (a) q*2 = 1.0; (b) q*2 = 1.5; (c) q*2 = 2.0; and (d) q*2 = 2.5. Note that in (a) () is the predicted critical point (CP) for quadrupolar strength of q*2 = 1.0; (
) the CP computer simulation value and () the vapour–liquid coexistence simulation data [10]. (b) (䊉) is the CP for quadrupolar strength of q*2 = 1.5; () the CP computer simulation value and () the vapour–liquid coexistence simulation data [10]. (c) () is the CP for quadrupolar strength of q*2 = 2; () the CP computer simulation value and () the vapour–liquid coexistence simulation data [10]. (d) ( ) is the CP for quadrupolar strength of q*2 = 2.5; () the CP computer simulation value and ( ) the vapour–liquid coexistence simulation data [10].

O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

37

Table 2 Critical point parameters for the Lennard–Jones quadrupolar fluid q*2

T*

ρ*

P*

0

1.31a 1.31b

0.31a 0.28b

0.14b

1

1.60a 1.59b

0.34a 0.31b

0.19b

1.5

1.89a

0.36a

1.84b

0.32b

0.24b

2.25a 2.12b

0.38a 0.34b

0.29b

2.62a 2.40b

0.41a 0.346b

0.35b

2.0 2.5 a b

Ref. [10]. This work.

hard-sphere reference fluid, it is certainly a good example to test if the theoretical scheme is accurate for predicting the phase behaviour of LJ quadrupolar pure fluids. Using the equations given in the above section, calculations are performed for the critical point and the fluid phase behaviour of the LJ quadrupolar pure fluid. Table 2 presents results for the critical parameters obtained using the equations given in Section 2 and computer simulations for the LJ quadrupolar pure fluid [10]. As shown, theory predicts results in close agreement with computer simulations for the critical parameters of quadrupolar fluids with different quadrupolar strength. Fig. 5 presents results for the critical point and for the vapour–liquid equilibrium coexistence curves for LJ quadrupolar pure fluids. As shown in Fig. 5(a)–(d), results for quadrupolar fluids with different quadrupolar strengths present a close agreement with computer simulations [10] on the vapour branch as well as for the critical point. For the liquid branch, however, the agreement is not so good. Fig. 6 shows the vapour–liquid equilibrium coexistence curves for quadrupolar fluids with quadrupolar strengths q*2 = 1.0 and q*2 = 1.5. As shown, the critical temperature increases with quadrupolar strength. Same behaviour is found for higher values of quadrupolar strengths. As shown in Fig. 7, the vapour–liquid equilibrium coexistence curves are shifted towards the higher values temperature region as quadrupolar strength increases. Thus, the behaviour found on the coexistence curves and shown in Figs. 6 and 7 is as that found in computer simulations for LJ quadrupolar pure fluids with different quadrupolar strengths [10]. To summarize, we have presented a simplified TPT approach that: (a) for pure atomic fluids, considers that contributions to the total free energy comes only from radial interactions which are considered as the total perturbation potential; and (b) for molecular pure fluids, that contributions to the total free energy come from both the radial and electrostatic multipolar intermolecular forces. Such contributions separately enter in different Pad´e type expressions, which are added to the HFE of the reference system.

Fig. 6. Vapour–liquid equilibrium results for the Lennard–Jones quadrupolar fluid with quadrupolar strengths q*2 = 1.0 and q*2 = 1.5. Captions are as in Fig. 5.

Fig. 7. Vapour–liquid equilibrium results for the Lennard–Jones quadrupolar fluid with quadrupolar strengths q*2 = 2.0 and q*2 = 2.5. Captions are as in Fig. 5.

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O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39

At present, we are investigating whether the procedure is useful when it is applied for calculating thermodynamic properties of more complex fluids. Results will be reported on completion. 4. Conclusion In this article, a simplified perturbation theoretical scheme is applied to the molecular model given in Section 2, to investigate the fluid phase behaviour of both atomic and molecular pure fluids. It consists in considering that contributions to the Helmholtz free energy of the reference system come: (a) for atomic pure fluids only from radial interactions; and (b) for molecular pure fluids from both radial and multipolar electrostatic interaction of molecules. These contributions enter separately in Pad´e type expressions that are summed to the HFE of the reference system. Results for the critical point and vapour–liquid equilibrium coexistence curves for the pure LJ fluid are, respectively, in agreement with computer simulations [6] and present the behaviour found in two works using MBWR EOS with 33 parameters for the LJ pure fluid [7,8] (see Table 1 and Figs. 2–4). For molecular pure fluids, the approach applied to the LJ quadrupolar pure fluid give results in agreement for the critical point and vapour–liquid equilibrium coexistence curves, predicting the behaviour found in computer simulations [10]; as the quadrupole moment strength of molecules increases, critical values of both temperature and density also increases (see Table 2 and Figs. 6 and 7). It must be mentioned, however, that in general equations of state over-predict the critical temperature (as they do not satisfy the scaling laws around the critical point). Hence, the good predictions of the critical temperature (quantitatively) by the presented theory is also due to a cancellation of errors (tendency to over-predict the critical temperature Tc because of scaling laws, cancelled by the tendency to under-predict Tc because the assumption for the Barker–Henderson diameter equal to the hard sphere diameter). Acknowledgements I gratefully acknowledge the Reviewer’s useful comments. Support of this work given by CICPBA and CONICET and discussions with Drs. M. Silbert and G. Zarragoicoechea are acknowledged. O.H. Scalise is member of Carrera del Investigador Cient´ıfico, Comisi´on de Investigaciones Cient´ıficas de la Provincia de Buenos Aires, (CICPBA). List of symbols A reduced Helmholtz free energy of hard-spheres g(r) radial distribution function G Gibbs free energy I(n) integrals related to hard-spheres pair distribution function

K−1 kB N P P(s) q* P* rij S T T* u* V Vg Vl

reference system compressibility Boltzmann constant total number of molecules in system pressure saturation pressure reduced quadrupole moment, q* = q/(εσ 5 )1/2 reduced pressure, P* = Pσ 3 /ε distance between the centers of molecules i and j entropy absolute temperature reduced temperature, T* = kB T/ε reduced intermolecular potential energy, u* = u/ε volume saturated vapour equilibrium volume saturated liquid equilibrium volume

Greek letters φi azimuthal angle of quadrupole vector of molecule i

Helmholtz free energy contribution term ε Lennard–Jones energy parameter σ HS diameter of hard-spheres σ Lennard–Jones distance parameter µ chemical potential ρ number density ρg saturated vapour density ρl saturated liquid density θi polar angle of quadrupole vector of molecule i Superscripts * denotes reduced with potential parameters (i) i-order perturbation term 0 reference system P perturbation potential qq quadrupole–quadrupole interaction LJ Lennard–Jones potential Subscripts c critical ELEC electrostatics interactions HS hard-spheres i,j molecules i and j LJ Lennard–Jones potential q notation for quadrupole

References [1] C.G. Gray, K.E. Gubbins, Theory of Molecular Fluids: Fundamentals, vol. 1, Clarendon Press, Oxford, 1984. [2] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic, London, 1986. [3] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1987; D.A. Frenkel, B. Smit, Understanding Molecular Simulation. From Algorithms to Applications, Academic Press, 1996. [4] R.W. Zwanzig, J. Chem. Phys. 22 (1954) 1420.

O.H. Scalise / Fluid Phase Equilibria 237 (2005) 31–39 [5] G. Stell, J.C. Rasaiah, H. Narang, Mol. Phys. 27 (1974) 1393–1414. [6] J.J. Potoff, A.Z. Panagiotopoulos, J. Chem. Phys. 109 (1998) 10914–10920. [7] J.J. Nicolas, K.E. Gubbins, W.B. Streett, D.J. Tildesley, Mol. Phys. 37 (1979) 1429–1454. [8] J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78 (1993) 591–618. [9] J. Kolafa, I. Nezbeda, Fluid Phase Equilibria 100 (1994) 1. [10] B. Smit, C.P. William, J. Phys.: Condens. Mater. 2 (1990) 4281–4288. [11] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (1967) 4714–4721. [12] J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54 (1971) 5237–5247. [13] N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (1969) 635–636.

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[14] O.H. Scalise, G.J. Zarragoicoechea, A.E. Rodriguez, R.D. Gianotti, J. Chem. Phys. 86 (1987) 6432–6437. [15] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, second ed., John Wiley & Sons, 1985. [16] J.C. Rasaiah, B. Larsen, G. Stell, J. Chem. Phys. 63 (1975) 722. [17] See Ref. [8] and references therein and Ref. [9] and references therein. [18] See Ref. [3] and Ref. [6] and references therein. [19] D. Ben-Amotz, G. Stell, J. Phys. Chem. 108 (2004) 6877–6882. [20] B. Smith, J. Chem. Phys. 96 (1992) 8639–8640. [21] N.B. Wilding, Phys. Rev. E 52 (1995) 602–611. [22] J.M. Caillol, J. Chem. Phys. 109 (1998) 4885–4893. [23] A.L. Benavides, Y. Guevara, A.F. Estrada-Alexander, J. Chem. Thermodyn. 32 (2000) 945–961.