Investigation of critical lines and global phase behavior of unequal size of molecules in binary gas–liquid mixtures in the combined pressure–temperature–concentration planes around the van Laar point

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Highlights

  • We combined the modified Tompa model with the van der Waals equation.

  • The critical lines for a binary gas–liquid mixture were studied.

  • We focused on the critical lines at the van Laar point and its vicinity.

  • We found that our P, T plots correspond to the type II, type III, type IV behaviors.

Abstract

We investigate critical curves and global phase behavior of unequal size of molecules in binary gas–liquid mixtures at the van Laar point and its vicinity. The van Laar point is only point at which the mathematical double point curve is stable, and also the intersection of the tricritical point and the double critical end point. The critical line structure is displayed for various combinations of the chain length and system parameters in the reduced pressure (P) temperature (T) plane, as is usually done with experimental results and temperature–concentration (T,x) plane. The P,T diagrams are discussed in accordance with the Scott and van Konynenburg binary phase diagram classification. We found that our P,T plots correspond to the type II, type III, type IV phase diagram behaviors and they are in good agreement with the theoretical and experimental studies. It is also found that the critical lines and phase behavior are extremely sensitive to small modifications in the system parameters.

Introduction

This paper constitutes the continuation of the treatment of previously published work (and referred to as paper I in the following)  [1]. In paper I, we introduced a new model that combines the modified Tompa model with the van der Waals equation. We calculate the critical lines as a function of x1 and x2 that the density of type 1 molecules and the density of type 2 molecules for various values of the system parameters and study critical lines for an unequal size of molecules in a binary gas–liquid mixture around the van Laar point in the density–density plane. We also investigated the connectivity of critical lines at the van Laar point and its vicinity and discuss these connections according to Scott and van Konynenburg classifications  [2]. Furthermore, we discussed in detail, how one can regain the van Laar point that disappears in case of a mixture of molecules of unequal size. Finally, we find that the critical lines and phase behavior are extremely sensitive to small modifications in the system parameters. We should also mention that the van Laar point is coined by Meijer  [3] at which the mathematical double point is stable and it is the intersection of the tricritical point and the double critical end point.

In this paper, we investigate the critical lines and global phase behavior of unequal size of molecules in binary gas–liquid mixtures at the van Laar point and its vicinity. The critical line structure is displayed for various combinations of the size differences factors and system parameters in the combined PTx planes. We also discuss the PT diagrams in accordance with the Scott and van Konynenburg classifications  [2]. This type of calculation was originally introduced by Meijer  [3]. He studied the van der Waals equation of state around the van Laar point in the (x1x2) and (PT) planes  [3]. Pelt and Loos also investigated the connectivity of the critical lines of binary mixtures around the van Laar point in the (Tx) planes  [4]. Finally, we also show the variation of the critical points of the pure components 1 and 2 to size differences ratios in our current study.

We should also mention that the critical lines and global phase behavior for a wide variety of binary fluid mixtures are studied extensively. An early attempt to study the critical lines and phase equilibria of binary mixtures was made by van Konynenburg and Scott  [2]. They classified almost all known kinds of fluid phase equilibria using the van der Waals equation of state with the van der Waals mixing rules. They also suggested a classification of binary phase diagrams, i.e., Fig. 1, which was taken from a paper by Peters et al.  [5] illustrates the six main classes of the binary phase diagrams suggested by van Konynenburg and Scott  [2]. A detailed discussion of the experimentally observed phase diagrams can be found in several reviews, e.g., in Ref.  [6]. Furman et al.  [7] made similar studies of lattice–gas mixtures and the resultant lattice–gas equation was extended by Furman and Griffiths  [8] to the van der Waals lattice–gas equation of state. They discovered new types of phase behaviors and also introduced another classification procedure. Mazur and co-workers  [9] and Boshkov and co-workers  [10] investigated the global phase diagrams for binary mixtures using the Ree equation of state. Meijer and co-workers  [3], [11], [12], [13], [14], [15], [16], [17] studied the global phase behavior of the binary gas–liquid systems in the x1x2, PT and Tx planes, using the three-state lattice–gas  [18] and the van der Waals lattice–gas models. Deiters and Pegg  [19] made a systematic investigation of the phase behavior in binary fluid mixtures using the Redlich–Kwong equation of state. Kraska and Deiters  [20] performed similar calculations using the Carnahan–Starling–Redlich–Kwong equation of state. van Pelt and co-workers  [21] used the simplified-perturbed hard-chain theory in order to study critical lines, phase equilibria, and some thermodynamic properties, as well as transition states in the global phase diagrams of the binary fluid mixtures. Nezbeda and co-workers  [22], regardless of any equation of state, gave a general description of types of phase equilibrium phenomena in the global phase diagrams of binary mixtures, and investigated the global phase diagrams of binary fluid mixtures based on the model and real binary fluid mixtures system by using Lorentz–Berthelot  [23] and Non-Lorentz–Berthelot  [24] mixtures of attractive hard spheres. They sketched the PT diagrams of the critical lines and their topology in different regions of the global phase diagram. Scott  [25] made a general evaluation of van der Waals-like equation of state for global phase diagrams. It is worthwhile to mention that a new nomenclature for phase diagram classes of binary fluid mixtures has been proposed by Bolz  [26] in which his classification scheme is a combination of the classification of van Konynenburg and Scott and the classification according to Furman and co-workers. Bluma and Deiters  [27] discussed the topological classes of the ternary phase diagrams and their relationship to the classes of the binary subsystems by using van der Waals equation of state. Napari et al.  [28] applied the density-functional theory to study gas–liquid phase behavior and nucleation in binary mixtures consisting of Lennard-Jones atoms with hard-spherical cores (monomers) and bonded Lennard-Jones atoms of two (dimers) or three (trimers) hard spheres. Wang et al.  [29] investigated closed-loop phase transition and global phase behavior of binary mixtures by using Guggenheim equation. They illustrated the critical lines in the reduced pressure–temperature and pressure-packing fraction planes. Quiñones-Cisneros  [30] studied how the barotropic behavior is linked to the transition from type II to type III phase behavior in asymmetric systems, by using PC-SAFT model. Wang and Sadus  [31] reported how the global phase diagram of binary mixtures could be represented by pure components of different critical properties, which is a proxy for components of different size, by using Carnahan–Starling–van der Waals equation of state in conjunction with the one-fluid model and the Lorentz–Berthelot combining rules for unlike interactions. They presented the results in reduced temperature–volume planes as well as temperature–pressure planes. Mejia et al.  [32] predicted subcritical phase and interface behaviors in type-I and type V equal-size Lennard-Jones mixtures by using density gradient theory and molecular-dynamics simulations. Mejia and Segura  [33] analyzed interface properties and wetting transitions for mixtures that belong to the shield region. Cismondi and Michelsen  [34] presented a general strategy for global phase equilibrium calculations, i.e. for calculations of critical lines; critical end points and liquid–liquid–vapor equilibrium in binary mixtures, and the results were given pressure–temperature planes and temperature–molar fraction planes. We should also mention that the phase behavior has its formal foundation on Gibbs theory in which Baker et al.  [35] presented a sort of very basic modern revision of Gibbs theory. Köfinger et al.  [36] studied the phase behavior of symmetrical binary fluid mixture for the chemical potentials of the two species; in case the equal and the unequal situation. Their investigations are based on the mean spherical approximation and grand canonical Monte Carlo simulations. The critical lines are expressed in temperature–pressure-chemical potential variation space. Woywod and Schoen  [37] calculated the complete phase diagrams of binary fluid mixtures composed of molecules of equal size by employing mean-field lattice density functional theory. They focused on the topology of phase diagrams in the space spanned by the thermodynamic fields temperature-mean chemical potentials-incremental chemical potentials of pure mixture components 1 and 2. Patel and Sunol  [38] developed a systematic methodology for automatic generation of global phase equilibrium diagram. Their algorithm is not only able to generate all six major types of phase diagrams but also can handle different types of solid–fluid topography resulting from the presence of solid phase. The results were demonstrated in temperature–pressure planes as well as mole fraction–temperature planes. Gençaslan et al.  [39] investigated the closed critical loops in the phase diagrams for a binary gas–liquid mixture using a modified lattice–gas model. Scalise and Henderson  [40] investigated the fluid phase behavior of some binary Yukawa mixtures in the reduced pressure–temperature–mole fraction surface as well as reduced pressure–temperature planes. Gençaslan  [41] studied the global phase behavior of polymer mixtures in the shield region. A near-exhaustive classification scheme of fluid phase equilibria in binary systems starting van Konynenburg and Scott was presented by Privat and Jaubert  [42]. They describe the transitions between the various types of systems in that paper.

The outline of this paper is as follows. In Section  2, the free energy and equation of state for the model are given. The calculations of critical lines and the van Laar point are presented in Section  3. The numerical calculations of critical lines at the van Laar point and its around are presented in the combined PTx planes. Finally, the discussions of the results are given in the last section.

Section snippets

The free energy and equation of state for the model

In order to investigate the critical lines and global phase behaviors of a mixture, a free energy expression is needed. The free energy of the model can be described within the molecular field approximation in terms of the probability of a site. Let the number of sites occupied by solvent molecules be n0 and let there be two types of species labeled 1 and 2, and assume that the number of species 1 molecules is r1n1, and the number of species 2 molecules is r2n2. The total number of lattice

Calculations of critical lines and the van Laar point

In order to calculate the critical lines and investigate phase behaviors around the van Laar point, one needs to find the expressions for the critical line around the van Laar point and its coordinates.

Discussions of the results

In order to display the connectivity of the critical lines at and around van Laar point in the combined PTx plane, the interaction parameters w1=1, w2=2.884549+Δw, and w12=2(1k)w1w2 are used. The results are given in three groups of figures and a table. Meanwhile, each of the results we found, we compare the recent theoretical and experimental results.

Fig. 2 displays the topology of the critical lines for the various σ ​parameters and Δw=0. As seen in the figure, for the each case of σ, we

Acknowledgment

This work was supported by the Research Fund of Erciyes University ​Grant No. FBA–11–3451.

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