Phase Equilibrium of Binary Mixtures of Cyclic Ethers + Chlorobutane

9588

J. Phys. Chem. B 2007, 111, 9588-9597

Phase Equilibrium of Binary Mixtures of Cyclic Ethers + Chlorobutane Isomers: Experimental Measurements and SAFT-VR Modeling Beatriz Giner,† Ignacio Gasco´ n,† He´ ctor Artigas,† Carlos Lafuente,*,† and Amparo Galindo‡ Departamento de Quı´mica Orga´ nica y Quı´mica Fı´sica, UniVersidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain, and Department of Chemical Engineering, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom ReceiVed: April 24, 2007; In Final Form: June 7, 2007

The phase equilibria (experimental and modeled) of eight binary mixtures each formed by a cyclic ether (1,3-dioxolane or 1,4-dioxane) and a chlorobutane isomer (1-chlorobutane, 2-chlorobutane, 1-chloro-2methylpropane, or 2-chloro-2-methylpropane) are presented. New experimental vapor-liquid equilibrium data at isothermal conditions (298.15, 313.15, and 328.15 K) has been obtained, and the statistical associating fluid theory for potentials of variable range (SAFT-VR) is used to model the mixtures. The results are discussed in terms of both the molecular characteristics of the pure compounds and the unlike intermolecular interactions present in the mixtures. The SAFT-VR approach is first used together with standard combining rules without adjustable parameters in order to predict the phase behavior at isothermal conditions. Good agreement between experiment and the prediction is found with such a model. Mean absolute deviations for pressures lie between 1 and 3 kPa, while for vapor phase compositions are less than 0.03 in mole fraction. However, a better agreement, can be obtained by introducing one adjustable parameter kij, which modifies the strength of the dispersion interaction between unlike components in the mixtures. This parameter is adjusted so as to model the phase equilibrium of the whole family of mixtures studied here at isothermal and isobaric conditions. We find that a unique unlike parameter kij is valid for all the studied mixtures and it is not temperature or pressure dependent. This unique transferable parameter together with the SAFT-VR approach provide a description of the vapor-liquid equilibrium of the mixtures that is in excellent agreement with the experimental data. In this case, the absolute deviations are of the order of 0.001 in mole fraction for vapor-phase compositions and less than 1 kPa for pressure.

1. Introduction An understanding of the thermodynamic properties and phase behavior of pure substances and their mixtures plays an important role in numerous areas. For instance, accurate phase equilibrium data is crucial to the chemical industry in order to design separation processes such as distillation, extraction, or adsorption.1,2 Thus, the possibility of obtaining accurate thermodynamic information of mixture properties either from models and theories or from experimental measurements is always of interest. Unfortunately, experimental measurements can be expensive and are impractical in extreme conditions. Although great advances are being made (see, for example, ref 3), we are also not yet in a position to predict thermodynamic properties of bulk systems from first principles. We hence rely on experimental data to develop theoretical models, either for use in equations of state or in molecular simulation models. A combination of a few experimental data and reliable theoretical models that can predict the properties of the systems over wide ranges of pressure, temperature, and composition from limited experimental data points in accessible conditions is then ideal. A substantial effort has been made in this area over the last 50 years with the aim of providing a realistic description of the behavior of fluid mixtures from theoretical models, and as a result, quite accurate methods have been developed for describing the thermodynamic behavior of fluids composed of simple molecules, i.e., molecules for which the most relevant inter* Corresponding author. E-mail: [email protected]. † Department of Chemical Engineering, Imperial College London. ‡ Departamento de Quı´mica Orga ´ nica y Quı´mica Fı´sica. Universidad de Zaragoza.

molecular forces are repulsion and dispersion forces or weak electrostatic forces due to dipoles or quadrupoles. There are now numerous theoretical approaches that allow obtaining thermodynamic properties for simple systems: analytic equations of state such as Soave-Redlich-Kwong4,5 or Peng-RobinsonStryjek-Vera;6,7 nonanalytic equations of state like BWR8,9 and Wagner models;10,11 local composition models;12-14 corresponding states methods;15,16 group contribution methods such as UNIFAC17-20 or models based on perturbation theories.21-23 Despite the fact that these methods can be applied easily and are successful at modeling the thermodynamic properties of mixtures of simple fluids, they all rely on the use of arbitrary mixing and/or combining rules; ultimately, the phase behavior of the mixtures is modeled through the use of adjustable mixture parameters, which need to be determined by comparison with experimental data. In most of the cases, the adjustable parameters are found to depend strongly on the temperature or the pressure conditions selected, which reduces the predictive ability of the models, and in addition, it is difficult to assign them a physical meaning. Thus, although these theories are developed as predictive tools, they are more commonly used as correlation methods, which can provide some understanding of the thermodynamic behavior of the systems but have little or no predictive capability. It is now acknowledged that one of the key reasons leading to the limitations of these traditional approaches is their failure to incorporate explicitly molecular detail, both of the shape of the molecules and of the anisotropy of intermolecular interactions such as those leading to hydrogen bonding or complex formation. The statistical associating fluid theory (SAFT) developed by Chapman et al.24,25 in the late 1980s provides the molecular

10.1021/jp073163j CCC: $37.00 © 2007 American Chemical Society Published on Web 07/21/2007

Equilibrium of Cyclic Ethers + Chlorobutane Isomers bases to address some of the limitations mentioned. The approach stems from the thermodynamic perturbation theory of Wertheim26-29 in which the thermodynamic properties of an associating chain fluid are obtained by following a perturbative approach by using as reference a monomer fluid for which the free energy and contact radial distribution function are known. The original version considered a hard-sphere fluid as the reference monomer system with temperature-dependent diameters and a dispersion term added. Nowadays, the SAFT approach is considered to be one of the most powerful predictive tools for the study of the phase equilibrium of complex fluids. It has been successfully applied for a wide number of chemical compounds, from simple alkanes30-32 to strongly associating compounds,33-35 electrolyte36 or polymer fluids.37-39 These are just but a few examples of the versatility of the approach, and there are now numerous versions and modifications. An excellent review of most SAFT-related advances and systems considered up to 2001 can be found in ref 40. As we mentioned in a recent paper,41 this is a quickly expanding field and it is difficult to list here all the works involving modeling of experimental systems with SAFT approaches. Some of the recent key theoretical developments have been highlighted in our previous work (see ref 41 and references therein for details). In recent years, some of us have been studying the thermodynamic behavior of fluid mixtures formed by hydrocarbons,42,43 haloalkanes,44,45 cyclic ethers,46,47 or alcohols48,49 and have used the experimental data obtained with the aim of contributing to the understanding of fluid-phase behavior and intermolecular interactions.46,50,51 In this paper, a study of the phase equilibrium (experimental and modeled) of binary mixtures containing a cyclic ether (1,3-dioxolane or 1,4-dioxane) and an isomer of chlorobutane (1-chlorobutane, 2-chlorobutane, 1-chloro-2-methylpropane, or 2-chloro-2-methylpropane) at several conditions of temperature and pressure is presented. As we mentioned in our previous work,41 the study of these mixtures is of interest due to the fact that they form new donor-acceptor type interactions in mixtures, which give rise to the appearance of interesting molecular effects. The strength of these new interactions depends on the donor or acceptor ability of each of the chemicals. The acceptor ability of the chloroalkanes studied here is quite similar for all of them. Several studies have, however, highlighted differences in the donor ability of several cyclic ethers. It has been shown that some cyclic monoethers like tetrahydrofuran or tetrahydropyran are stronger donors than the cyclic ethers studied here (1,3-dioxolane and 1,4-dioxane), and based on calorimetric and spectroscopic data, a sequence tetrahydrofuran > tetrahydropyran > 1,3-dioxolane > 1,4dioxane has been presented in terms of their electron donor ability.52-54 In addition to carrying out experimental measurements, we use the well-known Wilson method55 to correlate the measured experimental data and to calculate excess properties of the mixtures of interest and use the SAFT-VR equation to model and predict the phase equilibrium of the mixtures studied. This study is interesting, not only from the point of view of the molecular information, but also because it provides a test of the reliability of the SAFT-VR approach when applied to relatively complex mixtures. This is a double challenge; on one hand, new experimental data of the phase equilibrium of these mixtures is provided and further molecular information has to be extracted from the results. On the other hand, we wondered if the SAFT-VR approach could represent the observed phase behavior and provide physically meaningful models. In a previous work,41 the phase equilibrium of pure cyclic ethers and chloroalkanes was modeled with the SAFT-VR equation and very good agreement between experimental data and calculated results was obtained. Although all of the

J. Phys. Chem. B, Vol. 111, No. 32, 2007 9589 molecules considered are polar, they were modeled using the standard SAFT-VR approach without explicitly taking into account dipolar interactions and treating orientation independent polar interactions effectively as dispersion forces using squarewell potentials of variable range. Furthermore, because interactions between chloroalkane-chloroalkane and cyclic ethercyclic ether molecules are not strong enough to create molecular complexes, we considered all the compounds as nonassociating fluids. The molecules are hence modeled as chains of m tangentially bonded square-well segments of hard-core diameter σ and characterized by a well depth parameter and a range λ. The united atom approach is assumed so that one segment does not correspond to one atom in the molecule. Instead, in each case, an optimal value m for the number of segments representing a molecule is obtained by comparison with experimental data. We find that it is not necessary to treat explicitly the cyclic nature of the ethers, although it is worth noting that the SAFTVR approach as presented here is easily modified to account for the extract contact in cyclic molecules.56,57 In this work, the same pure-component models are used, and are used to study the binary phase behavior of eight mixtures, each containing a cyclic ether and a chlorobutane isomer. In the next section, we present the experimental devices and procedure to obtain the vapor-liquid equilibrium of the studied mixtures. The data are correlated using the Wilson equation, and a discussion the molecular characteristics of pure compounds and the molecular interactions between them is provided. Predictions and models with the SAFT-VR approach are given in Section 3. The calculated results are interpreted in detail, placing special attention in the molecular attributes of the different components of the mixtures and their intermolecular interactions. In Section 4, a summary of the key results is presented. 2. Experimental Section 2.1. Experimental Procedure. To measure the isothermal vapor-liquid equilibrium of the mixtures of interest, 1,3dioxolane, 1-chlorobutane, 2-chlorobutane, and 2-chloro-2methylpropane (99% purity) and 1,4-dioxane (99.9% purity) were obtained from Aldrich, while Fluka provided 1-chloro-2methylpropane (purity greater than 98%). The purity of the chemicals was checked by comparing the measured densities and pressures at a temperature of 298.15 K with those reported in the literature.58-63 No further purification was considered necessary. The experimental vapor-liquid equilibrium data were obtained using an all-glass dynamic recirculating type still that was equipped with a Cottrell pump.64 This is a commercial unit (Labodest model) built by Fischer. The equilibrium temperatures were measured to an accuracy of ( 0.01 K by means of a thermometer (model F25 with a PT100 probe) from Automatic Systems Laboratories, and the pressure in the still was measured with a Digiquartz 735-215A-102 pressure transducer from Paroscientific, equipped with a Digiquartz 735 display unit. The accuracy of the pressure measurements is (0.01% of reading. The compositions of both vapor and liquid phases were determined by densimetric analysis65,66 using an Anton Paar DMA-58 vibrating tube densimeter. The error in the determination of liquid and vapor mole fractions is estimated to be (0.0002. Values of the pressures of pure compounds measured at 298.15, 313.15, and 328.15 K for the pure components are collected in Table 1. 2.2. Experimental Data and Wilson Correlations. The pressure-composition diagrams (P - x1 - y1) for the eight mixtures studied are shown in Figures 1-8. As can be seen in the first three figures, the mixtures formed by 1,3-dioxolane with 1-chlorobutane, 2-chlorobutane, or 1-chloro-2-methylpro-

9590 J. Phys. Chem. B, Vol. 111, No. 32, 2007

Giner et al.

Figure 1. Pressure-composition diagram (P - x1 - y1) for 1,3dioxolane (1) + 1-chlorobutane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

Figure 2. Pressure-composition diagram (P - x1 - y1) for 1,3dioxolane (1) + 2-chlorobutane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

TABLE 1: Experimental Pressures P, Volumes V, and Second Virial Coefficients B at Different Temperatures T Measured in This Work and Compared with Literature Data compound

T/K

Pexp/kPa

V × 106/ B × 106/ m3 mol-1 m3 mol-1

Plit/kPa

1,3-dioxolane

298.15 313.15 328.15

13.535 13.5659 26.830 49.010

69.977 71.238 72.566

-906 -837 -776

1,4-dioxane

298.15 313.15 328.15

4.900 10.170 19.540

4.9558

85.717 87.147 88.620

-1190 -1102 -1022

1-chlorobutane

298.15 313.15 328.15

13.515 13.49960 26.075 46.290

105.11 107.16 109.67

-1722 -1488 -1302

2-chlorobutane

298.15 313.15 328.15

20.905 20.96961 38.540 66.320

106.72 108.89 111.12

-1641 -1426 -1254

1-chloro-2-methyl- 298.15 propane 313.15 328.15

20.350 19.85162

106.26

-1264

37.680 65.025

108.45 110.09

-1165 -1076

110.67

-1140

113.12 115.67

-1047 -966

2-chloro-2-methyl- 298.15 40.130 40.05463 propane 313.15 70.695 328.15 117.310

pane exhibit azeotropic behavior at all of the temperatures considered. It is also observed that increasing temperatures lead to azeotropic compositions richer in the cyclic ether (see details in Table 2). The activity coefficients of the components in the liquid phase were correlated using the Wilson equation.55 Estimation of the two adjustable parameters required in this model was based on minimization of the following objective function FObj in terms of experimental Pexp and calculated Pcal pressure values:67 n

FObj )

∑ i)1

(

)

Pexp - Pcal Pexp

2

Figure 3. Pressure-composition diagram (P - x1 - y1) for 1,3dioxolane (1) + 1-chloro-2-methylpropane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

The calculated pressure is obtained by taking into account the nonideality of the vapor phase and the variation of the Gibbs function of the pure compounds with pressure by using 2

Pcal )

[

xiγiPi exp ∑ i)1

]

(Vi - Bii)(P - Pi) - (1 - yi)2 P(δij) RT

(2)

where

(1)

δij ) 2Bij - Bii - Bjj

(3)

Equilibrium of Cyclic Ethers + Chlorobutane Isomers

Figure 4. Pressure-composition diagram (P - x1 - y1) for 1,3dioxolane (1) + 2-chloro-2-methylpropane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

Figure 5. Pressure-composition diagram (P - x1 - y1) for 1,4-dioxane (1) + 1-chlorobutane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

P is the total pressure, T the temperature, and R the gas constant. The compositions of the liquid xi and gas yi, the activity coefficient γi, the pure vapor pressure Pi, and saturated liquid volume Vi of component i are also required. The second virial coefficients of the pure components Bii are estimated from an equation of state6,7 for 1,3-dioxolane, 1,4-dioxane, and 2-chloro2-methylpropane and from the TRC tables68 for the rest of chemicals, while the cross second virial coefficient Bij is calculated using a suitable mixing rule.69 Values of molar volumes and second virial coefficients at 298.15, 313.15, and 328.15 K of pure compounds are collected in Table 1. The vapor-liquid equilibrium data, P - x1 - y1, together with the

J. Phys. Chem. B, Vol. 111, No. 32, 2007 9591

Figure 6. Pressure-composition diagram (P - x1 - y1) for 1,4-dioxane (1) + 2-chlorobutane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

Figure 7. Pressure-composition diagram (P - x1 - y1) for 1,4-dioxane (1) + 1-chloro-2-methylpropane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

activity coefficients and the corresponding excess Gibbs function calculated using the Wilson equation are given in the Supporting Information. Parameters for the activity coefficient correlation along with average deviations in pressure, ∆P, and vapor phase composition, ∆y, are collected in Table 3. The average deviations for pressure are between 0.03 and 0.09 kPa, and the average deviations for vapor composition lie between 0.003 and 0.008. As expected, the Wilson equation provides an excellent correlation of the activity coefficients of these mixtures. Furthermore, the thermodynamic consistency of the experimental results


Giner et al. TABLE 3: Correlation Parameters for the Wilson Equation, λ12 - λ11 and λ21 - λ22, Average Deviation in Pressure, ∆P, and Average Deviation in Vapor Phase Composition ∆y T/K 1,3-dioxolane (1) + 1-chlorobutane (2)

λ12 - λ11/ λ21 - λ22/ Jmol-1 Jmol-1 ∆P/kPa

∆y

298.15 313.15 328.15

953.38 777.17 501.56

823.98 955.91 1370.11

0.028 0.024 0.058

0.0057 0.0038 0.0037

1,3-dioxolane (1) + 298.15 2-chlorobutane (2) 313.15 328.15

1309.06 1418.49 1435.42

362.20 218.46 245.02

0.034 0.058 0.093

0.0040 0.0028 0.0032

1,3-dioxolane (1) + 298.15 1-chloro-2-methyl- 313.15 propane (2) 328.15

928.45 1069.60

764.47 492.23

0.042 0.040

0.0045 0.0039

980.91

726.22

0.035

0.0052

1,3-dioxolane (1) + 2-chloro-2-methylpropane (2)

298.15 313.15

1647.29 1425.09

-124.49 142.05

0.067 0.054

0.0035 0.0021

328.15

1276.21

291.80

0.065

0.0054

1,4-dioxane (1) + 1-chlorobutane (2)

298.15 313.15 328.15

876.85 360.44 451.19

475.02 1190.56 1134.39

0.073 0.061 0.094

0.0073 0.0041 0.0031


Figure 8. Pressure-composition diagram (P - x1 - y1) for 1,4-dioxane (1) + 2-chloro-2-methylpropane (2): experimental data (9, 0) at 298.15 K; (b, O) at 313.15 K; (2, 4) at 328.15 K. SAFT-VR prediction using an additional parameter (solid line).

298.15 313.15 328.15

967.20 926.76 655.03

624.94 582.49 945.84

0.066 0.069 0.077

0.0038 0.0033 0.0025

1,4-dioxane (1) + 1-chloro-2-methylpropane (2)

298.15 313.15

-88.74 39.09

2122.13 1852.00

0.048 0.049

0.0058 0.0061

328.15

-164.84

2150.96

0.069

0.0065

TABLE 2: Measured Pressure, P, and Composition, x1,y1, of the Azeotropic Points


298.15 313.15

20.86 990.09

3498.45 770.94

0.087 0.071

0.0081 0.0052

328.15

1016.64

656.09

0.052

0.0060

T/K

P/kPa

x1,y1

1,3-dioxolane (1) + 1-chlorobutane (2)

298.15 313.15 328.15

15.67 30.19 54.00

0.552 0.556 0.626


298.15 313.15 328.15

21.03 39.06 67.85

0.115 0.170 0.225


298.15 313.15 328.15

20.42 37.98 66.19

0.099 0.145 0.225

was satisfactorily checked for all the mixtures studied using the van Ness method,70 described by Fredenslund et al.17 The results of the thermodynamic consistency test can also be found in the Supporting Information. We find the activity coefficients for all the mixtures studied to be greater than one; i.e., the mixtures present positive deviations from ideality. In general, the activity coefficients of the mixtures formed by 1,4-dioxane are greater than those for mixtures containing 1,3-dioxolane. The excess molar Gibbs functions, GE, which can be easily calculated from the activity coefficients, present positive values for all the mixtures, and they are found to be largest for the case of mixtures containing 1,3-dioxolane, except for those with 2-chloro-2-methylpropane in which case the larger GE is that of the binary mixtures with 1,4-dioxane. In Figures 11 and 12, the excess molar Gibbs functions at 298.15 K are shown. If mixtures are formed by 1,3-dioxolane, all the curves are quite similar and there is not an unambiguous sequence for the values, while for the mixtures containing 1,4dioxane, there is a clear sequence; 1,4-dioxane + 1-chlorobutane < 1,4-dioxane + 1-chloro-2-methylpropane < 1,4-dioxane + 2-chlorobutane < 1,4-dioxane + 2-chloro-2-methylpropane. Let us then pay attention to the molecular factors that determine the behavior of these mixtures. Nonideality of a mixture is commonly attributed on one hand to structural effects,

differences in shape and size of the mixed components, and on the other hand, to the energetic effects, that is, molecular interactions that can be weakened or destroyed or established by mixing. Clearly, the macroscopic properties of a mixture depend directly on the nature of the molecules that form it. It is because of this that the molecular characteristics of the pure compounds and the intermolecular interactions that take place between them should be considered prior to advancing a suitable description of the molecular phenomena that can occur during the mixture process. Because of the presence of two oxygen atoms within the structures of both cyclic ethers, both the molecules of 1,3dioxolane and 1,4-dioxane show relatively strong dipolar and quadrupolar interactions. A number of studies71-75 have pointed out that the ether-ether interaction of 1,3-dioxolane and 1,4dioxane are stronger than those of molecules of other similar cyclic ethers such as tetrahydrofuran or tetrahydropyran, which only have one oxygen atom. It has been suggested74 that the presence of two oxygen atoms strengthens the interaction between adjacent molecules, leading to a tighter-packed structure (cf., the high values of density and viscosity of these compounds compared to those of other cyclic ethers58). With respect of the isomeric chlorobutanes, it is important to note that while all of them are dipolar, the disparity of their structure (from the linear one of the 1-chlorobutane to the globular of the 2-chloro-2methylpropane) can result in important differences in behavior. Once the main characteristics of the pure compounds have been revised, it is useful to consider the mixture. Because of the characteristics of the chemicals studied, new donor-acceptor type interactions between the molecules of the cyclic ethers and those of the chloroalkanes can take place. Several studies have revealed the ability of some of the cyclic ethers as donor substances, confirming again differences between the ethers studied here and other cyclic ethers;54 the donor ability of 1,3-


Figure 9. Temperature-composition diagram (T - x1 - y1) for 1,3dioxolane (1) + 1-chlorobutane (2) (a); + 2-chlorobutane (2) (b); + 1-chloro-2-methylpropane (2) (c); + 1-chloro-2-methylpropane (2) (d): experimental data (9, 0) at 40.0 kPa; (b, O) at 101.3 kPa. SAFTVR prediction using an additional parameter (solid line).

dioxolane and 1,4-dioxane is weaker than that presented by cyclic monoethers. The thermodynamic behavior observed is the result of the superposition of all of the molecular phenomena described. Therefore, it is very useful to have the widest possible set of thermodynamic properties of a given mixture in order to get the broadest view of the phenomena as well as the maximum molecular information possible. In previous papers, we have reported several thermodynamic properties of the mixtures studied here.65,76,77 From these studies, very interesting information relating to the structural and energetic effects that might happen and to the superficial phenomena of these mixtures has been extracted. For instance, the globular structure of 2-methyl2-chloropropane results in mixtures with a considerably different behavior, which must be related to the reorganization of the structure of the mixture due to the interstitial packing of the ethers into the structure of the chloroalkane. Besides, the two opposite energetic effects (weakness of interactions between molecules of pure components and new donor-acceptor interactions) seem to play an important role in these mixtures. Positive activity coefficients and positive values for the excess Gibbs function of the studied mixtures obtained in this paper are a sign of the slight predominance of overall weaklike interactions as compared to the new donor-acceptor interactions. Furthermore, from the excess Gibbs function results, it may be stated that the strength of the cyclic ether-chloroalkane interaction relating to the chlorobutane isomer follows a trend: cyclic ether + 1-chlorobutane > cyclic ether + 1-chloro-2-


Figure 10. Temperature-composition diagram (T - x1 - y1) for 1,4dioxane (1) + 1-chlorobutane (2) (a); + 2-chlorobutane (2) (b); + 1-chloro-2-methylpropane (2) (c); + 1-chloro-2-methylpropane (2) (d): experimental data (9, 0) at 40.0 kPa; (b, O) at 101.3 kPa. SAFTVR prediction using an additional parameter (solid line).

methylpropane > cyclic ether + 2-chlorobutane > cyclic ether + 2-chloro-2-methylpropane. 3. Modeling the Phase Equilibrium of the Mixtures 3.1. SAFT-VR Equation of State. In this section, we provide only a brief account of the main expressions in the SAFT-VR equation of state for a mixture of nonassociating chain fluids, as this equation has been presented in detail in previous works.30,78 The Helmholtz free energy can be written as the sum of three separate contributions,

A Aideal Amono Achain ) + + NkT NkT NkT NkT

(4)

where N is the number of chain molecules in the mixture, k is the Boltzmann constant, and T is the temperature. In this equation, Aideal is the ideal free energy, Amono the free-energy contribution due to the monomer-monomer square-well interactions, and Achain is the contribution due to the formation chains. The free energy of an ideal mixture is given by:

Aideal NkT

)

(

n

xi ln FiΛi3 ∑ i)1

)

-1

(5)

where n is the total number of components in the mixture, Fi ) Ni/V is the number density, and Λi is the thermal de Broglie wavelength of species i.


Giner et al.

A1 A2 Amono AHS ) + + NkT NkT NkT NkT

(6)

where the residual free energy of the reference hard sphere fluid AHS/NkT is calculated using the expression of Boublı´k and Mansoori et al.81,82 A1/NkT corresponds to the mean attractive energy of the mixture, and the second-order fluctuation term A2/NkT is calculated by using the local compressibility approximation. Details of each of these terms and of the mixing rules can be found in the original works.30,78 In this work, we use mixing rule MX1b as described in ref 78. The contribution to the free energy due to the formation of chains of square-well segments in a binary mixture is written as

Achain NkT

Figure 11. Excess molar Gibbs functions, GE, for the mixtures 1,3dioxolane (1)+ chlorobutane isomers (2) at 298.15 K: 1-chlorobutane, (black line), 2-chlorobutane (red line), 1-chloro-2-methylpropane (green line), 2-chloro-2-methylpropane (blue line).

n

)-

xi (mi - 1) ln yM ∑ ii (σi) i)1

(7)

where yM ii (σi) is obtained from the high-temperature expansion of the pair radial distribution function gM ii (σi), following the original works.30,78 3.2. Prediction of Phase Equilibrium Using the SAFTVR Approach. For two or more phases to be in equilibrium with one another, the pressures, temperatures, and chemical potential of each component must be equal in the coexisting phases. The SAFT-VR parameters relating to the aspect ratio of the model molecules mi, the hard-core diameter of the spherical segments σii, and the depth ii and range λii of the square-well interactions for the pure components obtained previously41 are used to model the phase behavior of the mixtures studied in this work. The unlike size and energy parameters σij and ij are obtained using the Lorentz-Berthelot combining rules;82 i.e.,

σii + σjj 2

(8)

ij ) x(iijj)

(9)

σij ) and

while λij was obtained following

λij )

Figure 12. Excess molar Gibbs functions, GE, for the mixtures 1,4dioxane (1)+ chlorobutane isomers (2) at 298.15 K: 1-chlorobutane, (black line), 2-chlorobutane (red line), 1-chloro-2-methylpropane (green line), 2-chloro-2-methylpropane (blue line).

Following the original SAFT-VR approach,30 the contribution to the free energy due to the monomer-monomer interactions is obtained as high-temperature perturbation expansion up to second order:79,80

σiiλii + σjjλjj σii + σjj

(10)

We first consider a purely predictive model using the combining rules as given above with no further adjustment and calculate the phase behavior of the mixtures. The mean absolute deviations obtained for the pressure and composition for each of the mixtures are shown in Table 4. In general, mean absolute deviation for the pressure increases for increasing temperatures, while the deviations in composition remain constant throughout. The largest deviations are found for the mixtures containing 2-chloro-2-methylpropane. Taking into account that for these calculations no further parameter adjustment has been carried out, i.e., that these are purely predictive calculations, the results can be considered rather satisfactory. The shape of the vaporliquid boundaries for the mixtures which contain 1,4-dioxane are well predicted. Unfortunately, the phase behavior of the systems containing 1,3-dioxolane are not as well described. The azeotrope seen for 1,3-dioxolane + 1-chloropropane is moderately well predicted, but unfortunately the azeotropic behavior



TABLE 4: Mean Absolute Deviations Obtained for the Pressure, ∆P, and Composition, ∆y, between Experimental and Predicted Values with the SAFT-VR Approach T/K

∆P/kPa

∆y


298.15 313.15 328.15

1.09 2.26 3.49

0.0313 0.0278 0.0237


298.15 313.15 328.15

1.30 2.78 5.07


298.15 313.15 328.15


mixture

TABLE 5: Mean Absolute Deviations Obtained for the Pressure, ∆P, and Composition, ∆y, between Experimental and Predicted Values with the SAFT-VR Approach Using an Additional Parameter T/K

∆P/kPa

∆y


0.0304 0.0294 0.0296

298.15 313.15 328.15

0.10 0.15 0.30

0.0036 0.0016 0.0029


1.08 2.09 3.84

0.0258 0.0235 0.0243

298.15 313.15 328.15

0.14 0.28 0.98

0.0042 0.0034 0.0044


298.15 313.15 328.15

1.11 3.16 6.66

0.0194 0.0209 0.0217

298.15 313.15 328.15

0.56 0.73 0.71

0.0212 0.0172 0.0111



298.15 313.15 328.15

0.55 1.20 2.07

0.0229 0.0246 0.0240

298.15 313.15 328.15

1.16 0.55 1.37

0.0112 0.0083 0.0083



298.15 313.15 328.15

0.81 1.48 2.50

0.0241 0.0224 0.0221

298.15 313.15 328.15

0.29 0.39 0.56

0.0117 0.0075 0.0047



298.15 313.15 328.15

0.87 1.51 2.00

0.0235 0.0211 0.0197

298.15 313.15 328.15

0.34 0.54 0.82

0.0065 0.0070 0.0054



298.15 313.15 328.15

1.67 2.93 5.87

0.0348 0.0224 0.0234

298.15 313.15 328.15

0.36 0.60 1.33

0.0084 0.0073 0.0064


298.15 313.15 328.15

0.95 0.45 1.41

0.0145 0.0041 0.0064

of the mixtures of 1,3-dioxolane with 2-chlorobutane or 1-chloro-2-methylpropane is not predicted. To improve the description of the phase equilibrium of the mixtures, an unlike adjustable parameter kij ) 0.0165 is introduced to correct the unlike dispersion interaction; i.e., ij ) (1 - kij)(i‚j)1/2. We find that just one parameter is enough, and that it can be transferred for all the mixtures and conditions of interest here. Using the kij correcting factor, new mean absolute deviations for the pressure and composition for each of the mixtures are shown in the Table 5. In addition, the SAFTVR calculations using this additional parameter are shown in Figures 1-8 together with the experimental data. In view of these results, we can conclude that using the kij parameter, the modeling of vapor-liquid equilibrium of the studied mixtures is considerably improved and a good agreement between experimental and predicted data has been obtained. Mean absolute deviations both for pressure and composition have diminished to a large extent. It is worth mentioning that the largest deviations are observed for mixtures containing 2-chloro2-methylpropane and for the predictions at 328.15 K. At this point, it is important to recall the fact that the proposed unlike parameter is valid for all the studied mixtures and it is not temperature dependent. It is also of interest to consider the validity of the model in isobaric conditions. With the aim to answer this question, we have calculated the vapor-liquid equilibrium of these mixtures for which isobaric data at pressures of 40.0 and 101.3 kPa was obtained experimentally in previous works.44,77 We show the SAFT-VR calculations using the same unlike parameter kij ) 0.0165 together with the experimental data in Figures 9 and 10. In Table 6, the mean absolute deviations for each of the mixtures are presented. The results reveal a very good agreement with experimental data, including those mixtures which present azeotropic behavior. Results have been compared to those obtained previously47 with the original UNIFAC model,83-85 and we can confirm that the SAFT-VR approach with the additional parameter provides a considerably better representation of the isobaric VLE for the studied mixtures

mixture

TABLE 6: Mean Absolute Deviations Obtained for the Temperature, ∆T, and Composition, ∆y, between Experimental and Predicted Values with the SAFT-VR Approach Using an Additional Parameter P/kPa

∆T/K

∆y


40.0 101.3

0.22 0.11

0.0043 0.0026


40.0 101.3

0.12 0.16

0.0033 0.0014


40.0 101.3

0.79 0.71

0.0151 0.0144


40.0 101.3

0.72 0.22

0.0069 0.0064


40.0 101.3

1.07 1.40

0.0133 0.0120


40.0 101.3

0.40 0.69

0.0082 0.0056


40.0 101.3

1.40 1.47

0.0148 0.0117


40.0 101.3

0.62 0.62

0.0035 0.0077

mixture

than the UNIFAC model. This suggests that the SAFT-VR approach using an additional kij parameter is adequate to obtain the vapor-liquid equilibrium of this type of mixtures for a relatively wide range of temperature and pressure conditions. The kij parameter corrects the Lorentz-Berthelot mixing rule; a positive value suggests that the actual interactions between the compounds of the mixture are weaker than the combining rule a priori predicts. Taking into account the molecular phenomena described in the previous section based on the experimental values found for both the activity coefficients and the excess Gibbs functions of the mixtures studied, the value of kij ) 0.0165 may be associated to the energetic contributions due to the interactions weakened between the pure components of the mixture and the new donor-acceptor interactions

9596 J. Phys. Chem. B, Vol. 111, No. 32, 2007 established. Although care should be taken to place too much physical insight into the role played by the adjustable unlike parameter, the fact that a unique value is valid for all the mixtures studied, it is possible to argue that the extent of the weakening of the cyclic diether-cyclic diether and chlorobutane-chlorobutane interactions due to the mixing process and the formation of new interactions is reflected in this value. 4. Conclusions We have studied the phase equilibrium of binary mixtures formed by a cyclic ether (1,3-dioxolane or 1,4-dioxane) and isomeric chlorobutanes (1-chlorobutane, 2-chlorobutane, 1-chloro2-methylpropane, or 2-chloro-2-methylpropane). We have obtained the experimental isothermal vapor-liquid equilibrium at several temperatures and have used the well-known Wilson method to correlate the data and obtain activity coefficients and excess functions. The results obtained have been analyzed in terms of molecular phenomena that take place during the mixture process. The SAFT-VR approach has been used in order to predict the phase equilibrium of the mixtures, and although good overall agreement is obtained, the approach fails to predict the azeotropic behavior exhibited by some of the mixtures. A unique additional unlike parameter related to the energetic parameter, ij, has been introduced to model all the mixtures. The unlike parameter is found to be temperature and pressure independent, and valid for all the mixtures of interest, providing an accurate description of the phase behavior of the mixtures. In this way, we provide further demonstration of the validity of the SAFTVR approach for the study of phase behavior of complex mixtures. Acknowledgment. We are grateful for the financial assistance from the Ministerio de Educacioń y Ciencia and Fondos FEDER (BQU 2003-01765). We are also indebted to D.G.A. and the Universidad de Zaragoza for financial support. Supporting Information Available: Vapor-liquid equilibrium data, P - x1 - y1, together with the activity coefficients and the corresponding excess Gibbs function calculated using the Wilson equation; results of the thermodynamic consistency test. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) McGlashan, M. L. Pure Appl. Chem. 1985, 57, 89. (2) Schneider, G. M. Pure Appl. Chem. 1991, 63, 1313. (3) McGrath, M. J.; Siepmann, J. I.; Kuo, I.-F. W.; Mundy, C. J.; J. VandeVondele, J.; Hutter, J.; Mohamed, F.; Krack, M. J. Phys. Chem. A 2006, 110, 640. (4) Redlich, O.; Kwong, J. S. Chem. ReV. 1949, 44, 233. (5) Soave, G. Chem. Eng. Sci. 1972, 27, 1197. (6) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. (7) Stryjek, R.; Vera, J. H. Can. J. Chem. Eng. 1986, 64, 820. (8) Benedict, M.; Webb, G. B.; Rubin, L. C. J. Chem. Phys. 1940, 8, 334. (9) Benedict, M.; Webb, G. B.; Rubin, L. C. J. Chem. Phys. 1942, 10, 747. (10) Setzmann, U.; Wagner, W. Int. J. Thermophys. 1989, 10, 1103. (11) Setzmann, U.; Wagner, W. J. Phys. Chem. Ref. Data 1991, 20, 1061. (12) Kim, C. H.; Vimalchand, P.; Donohue, M. D.; Sandler, S. I. AIChE J. 1986, 32, 1726. (13) Chen, C. C.; Evans, L. B. AIChE J. 1986, 32, 444. (14) Vera, J. H.; Sayegh, S. G.; Ratcliff, G. A. Fluid Phase Equilib. 1977, 1, 113. (15) Hakala, R. J. Phys. Chem. 1967, 71, 1880. (16) Leland, T. L.; Chappelear, P. S. Ind. Eng. Chem. 1968, 60, 15. (17) Fredeslund, A.; Gmehling, J.; Ramusen, P. Vapour-Liquid Equilibrium Using UNIFAC; Elsevier: New York, 1977.

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Phase Equilibrium of Binary Mixtures of Cyclic Ethers + Chlorobutane

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