Alkane + Alkylbenzoate Mixed Solvents. Measurements and

Ind. Eng. Chem. Res. 2005, 44, 7575-7583

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Thermophysical Behavior of n-Alkane + Alkylbenzoate Mixed Solvents. Measurements and Properties Modeling Santiago Aparicio, Rafael Alcalde, Begon ˜ a Garcıá, and Jose´ M. Leal* Universidad de Burgos, Departamento de Quı´mica, 09001 Burgos, Spain

This work contributes to an improved understanding of the structure and dynamics of estercontaining solvents. A broad experimental study has been performed on the thermophysical properties of (C6-C14) n-alkane + (C1-C4) alkylbenzoate binary solvents over the whole composition range and within the 278.15-318.15 K temperature range. The excess and mixing derived properties were analyzed in terms of structural effects and intermolecular forces. To put forward reliable models that may describe the behavior of these complex mixtures, the cubic equations of state by Soave and Peng-Robinson were combined with a set of 10 different mixing rules to correlate the volumetric and viscometric properties. Likewise, to gain a deeper insight into the structure of these solvents the Prigogine-Flory-Patterson and extended real associated solution models were applied to correlate the excess molar volumes. The different contributions to the excess and mixing properties were analyzed as well as their variation with the nature and size of the components involved. Introduction The experimental study and proper interpretation of the thermophysical properties of pure liquids and mixed solvents play a key role in solution chemistry; physical properties and correlation methods may be regarded as the raw material for chemical processes design.1 The modeling of processes is critically dependent on an accurate knowledge of the thermodynamic behavior of the solvents involved; hence, potentially unreliable results arising from such operations may incur either unnecessary costs or fail to achieve production targets.2 Most commercial process simulators are equipped with a set of thermodynamic models and databases; however, these models can neither reliably cover the temperature, pressure, and composition ranges needed nor can they reflect new and innovative compounds. Moreover, to assess the validity for a particular solvent, such models must be probed with experimental data; hence, a combination of experimental data and theoretical modeling provides an optimal approach to study pure and mixed solvents. Benzoic acid esters are a very interesting class of solvents. In addition to their dipolar and hydrophobic nature, the easily polarizable π electron system confers a highly selective ability and converts these solvents into ideally suited compounds for a number of valuable applications. Alkyl benzoates are used to control thickening, flow, and viscosity properties of cellulose ethers, as fragrances and antibacterial agents in cosmetic formulations, as plasticizers to produce poly(vinyl chloride) (PVC) polymers, and as textile dye carriers for the treatment of synthetic fibers, among others. However, despite these applications and promising properties, there is only scant literature available on these solvents.3-6 On the other hand alkanes, the simplest class of organic compounds, contain no functional group, and the smaller alkanes play an important role as model molecules for the behavior of larger compounds. * Corresponding author. E-mail: [email protected].

This work reports a broad study on the (C6-C14) n-alkane + (C1-C4) alkylbenzoate (methylbenzoate (MB), ethylbenzoate (EB), propylbenzoate (PB), and butylbenzoate (BB)) binary mixtures over the 278.15318.15 K temperature range and pursues providing a deeper insight into the intermolecular interactions between alkanes and aromatic esters by extending the database of these systems.7-10 The measured and derived thermophysical properties were treated according to different approaches; the volumetric and viscometric measurements were correlated with the cubic equations of state (EOS) by Soave (SRK)11 and PengRobinson (PR).12 Due to their accuracy, reliability, simplicity, and speed of computation these EOS are widely used in industry and are contained in the modern process simulation packages.13-15 Extending the EOS to multicomponent mixtures requires use of suitable mixing rules. In this work two different approaches were used. A reliable method to extend the pure-fluid equation of state to mixtures, the one-fluid van der Waals model, required seven quadratic and nonquadratic meaningful mixing rules. Another plausible approach consists of combining an EOS with an excess Gibbs energy model; hence, pursuant to the Wong-Sandler model,16 three different mixing rules were also applied. Although cubic EOS are quite valuable tools for correlation and/or prediction purposes, in practice they are not most useful for learning fluid properties at a microscopic level;13 hence, the Prigogine-Flory-Patterson (PFP)17 and modified extended real associated solution (ERAS)18 models were used to correlate the volumetric properties; to obtain meaningful results a global optimization procedure was applied, and thus the parameters and contributions evaluated were analyzed in terms of intermolecular forces. Experimental Section Reagents. Pure solvents, of the highest purity commercially available, were purchased from Fluka and Aldrich. They were degassed with ultrasound and kept

10.1021/ie0502281 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/20/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005

Table 1. Density, G, Dynamic Viscosity, η, and Refractive Index, nD, of the Pure Alkylbenzoates: Measured Values and Comparison with Literature Values compd

278.15 K 288.15 K 298.15 K purity (GC) F (g/cm3) η (mPa s) F (g/cm3) η (mPa s) F (g/cm3) η (mPa s)

MB

99.9

1.102 75a

3.083a

1.092 62a 1.093 34b

2.304a 2.298b

EB

99.9

1.060 21a

3.182a

1.050 59a 1.051 12b

2.441a 2.407b

PB

98.7

1.036 06a

4.077a

3.013a 2.960g

BB

99.1

1.018 59a

4.813a

1.026 95a 1.027 4b 1.026 60g 1.008 38a 1.008 39g

a

3.625a 3.437g

1.083 63a 1.082 62c 1.083 6d 1.041 61a 1.041 38c 1.042d 1.017 91a 1.017 87c 1.017 75g 0.999 84a 1.003 56c 0.999 78g

Values obtained in this work. b From ref 21. c From ref 4.

d

1.970a 1.932d 2.357a 2.311g 2.691a 2.647g

1.514 66a 1.514 57b 1.514 9d 1.503 34a 1.503 48b 1.503 1d 1.498 46a

308.15 K 318.15 K F (g/cm3) η (mPa s) F (g/cm3) η (mPa s) 1.073 99a 1.074 0d 1.072 05e 1.032 25a 1.032 7d

1.504a 1.510d

1.064 28a

1.270a 1.253f

1.633a 1.578d

1.022 88a

1.350a 1.332f

1.009 23a 1.008 83g

1.905a 1.867g

1.000 26a 0.999 81g

1.574a 1.527g

1.495 31a 0.991 25a 0.991 29g

2.208a 2.099g

0.984 37a 0.982 65g

1.819a 1.708g

From ref 22. e From ref 23. f From ref 24. g From ref 5.

out of the light over Fluka 0.3 nm molecular sieves. Their purity was checked by GC (Perkin-Elmer 990 gas chromatograph) and by comparison of the measured thermophysical properties with those obtained from the literature (Table 1). To avoid undesired preferential evaporation, the binary mixtures were prepared by syringing amounts, weighed to ∆m ) (10-5 g with a Mettler AT 261 Delta Range balance, into suitably stoppered bottles, with a stated mole fraction accuracy of (1 × 10-4. The mixtures were completely miscible over the whole composition range. To reach an even data distribution, the sample preparation for each binary system involved 20 experimental points in 0.05 steps in the x1 ∈ [0 1] interval. Instruments and Procedures. The measured thermophysical properties were density, F, dynamic viscosity, η, and refractive index, nD. The molar excess volumes ((10-4 cm3/mol) were deduced from the densities of the pure liquids and mixtures. The densities, F, and speeds of sound, u, were measured with an Anton Paar DSA 5000 digital density meter (0.7 cm3 sample size), equipped with a built-in solid-state thermostat ((0.01 K) and a resident menu program. The density measurements (uncertainty, (1 × 10-6 g/cm3)19,20 were based on the oscillation period of the sample tube. Proper calibration at each temperature was achieved with doubly distilled, deionized water (Milli-Q, Millipore) and n-nonane (Fluka, 99.2%) as standards. The dynamic viscosities, η, were measured with an automated AMV 200 Anton Paar microviscometer, calibrated with doubly distilled and deionized water, hexanoic acid (Aldrich, >99.5%), and butoxyethanol (Aldrich, >99.5%). Viscosity measurements ((0.1% mPa s), based on the rolling ball principle, were evaluated by measuring the shear stress of a steel ball introduced into an inclined, sample-filled glass capillary, inside a thermostatic block ((0.005 K). The stress was monitored by changing the inclination angle within 20°-80° in 5° intervals. Two capillaries of 0.9 mm (0.120 cm3 sample size) and 1.6 mm (0.400 cm3) were used. The calibration constants were evaluated as

k(R) ) ηstand/(Fball - Fstand)t

1.823a 1.825d

nD

(1)

k(R) being the calibration constant at each inclination angle, ηstand the dynamic viscosity of the standard liquid, Fball the density of the ball (7.874 g/cm3), Fstand that of the standard liquid, and t ((0.01 s) the rolling time. An Anton Paar SPV sample changer was used, and the samples were injected with a peristaltic pump. Refractive index ((5 × 10-5) was measured with an automated

Leica AR600 refractometer whose temperature is controlled by a Julabo F32 external circulator to (1 × 10-2 K. Calibration was made with doubly distilled water and a standard oil supplied by the manufacturer. Densities and dynamic viscosities for the even (C6C14) n-alkane + (C1-C4) alkylbenzoate binary mixtures were measured over the 278.15-318.15 K temperature range in 10 K steps, however refractive indexes were measured only at 298.15 K since the temperature effect on this property was almost negligible. The experimental values of the measured quantities (over 4000 experimental readings, Tables I and II, Supporting Information) were fitted to the polynomial p

Y)

Ajx1j ∑ j)0

(2)

where Y represents density, dynamic viscosity, and refractive index. The proper number of coefficients was determined according to an F-test procedure and their values deduced with a least-squares algorithm (Tables III-V, Supporting Information). Results and Discussion Nonideal behavior of liquid systems often arises when different components are mixed and can be attributed to the following: (i) the difference in size and shape of the components, (ii) the reorientation of the molecules in the mixture, and (iii) the intermolecular interactions between components. Densities and the derived excess volumes bear relation to the heaviness of the molecules per unit volume, and are relevant quantities to interpret the interactions between the mixture components. The extent to which such mixtures deviate from the ideal behavior can best be reflected by the derived excess and mixing properties instead of the raw experimental measurements. Excess molar volume, VEm, can be defined as

VEm )

M F

-

∑i xi

Mi Fi

(3)

where M stands for the molar mass of either the mixed or pure components. By definition, an excess property requires a reference ideal state, but this is not attainable for nonthermodynamic properties such as dynamic viscosity or refractive index. However, by analogy it is possible to define the so-called mixing property, ∆mixY,

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Figure 1. Molar excess volume, VEm, at 298.15 K for (a) x1 n-alkane + (1 - x1) PB, (b) x1 n-hexane + (1 - x1) alkylbenzoate, (c) x1 n-decane + (1 - x1) alkylbenzoate, and (d) x1 n-tetradecane + (1 - x1) alkylbenzoate. n ) number of carbon atoms of n-alkane; m ) number of alkylic carbons of alkylbenzoate.

which reflects the deviation from the linear additivity and may also be analyzed in terms of intermolecular interactions and structural effects,

∆mixY ) Y -

∑i xiYi

(4)

where Y represents the dynamic viscosity or refractive index. On the other hand partial molar quantities, defined as the rate of change with concentration of extensive functions, have become a useful tool to account for binary and even higher order interactions between components. Partial molar functions represent the rate of change with concentration of extensive thermodynamic mixture properties caused by the addition of an infinitely small amount of one component. The partial molar excess volumes for each component in a mixture, V h Ei , may provide valuable information on the interactions and can be conveniently evaluated with the intercept method. The isobaric thermal expansivities, RP, were evaluated from a proper evaluation (by an F-test) of coefficients by deriving the density vs temperature plot; likewise, the activation properties were calculated according to the Eyring theory,

( )

η ∆Sqm ∆Hqm + ln M ) ln(Nh) F R RT

(5)

where h, R, and N are universal constants, and entropy, ∆Sqm, and enthalpy, ∆Hqm, are activation thermodynamic properties. The effect of the alkane chain length, n, ester chain length, m, and temperature on the structure and intermolecular forces can be inferred from an analysis of the excess and mixing properties. These properties are plotted in Figures 1-8, and the corresponding numerical information can be easily deduced from the fits provided as Supporting Information. Linear n-alkanes have close to null dipole moments, and their structure is dominated primarily by the presence of dispersion forces whose strength increases as the chain length increases.25 The efficient packing observed for short alkanes is distorted for larger alkanes, whose linear shape is less pronounced; hence, the balance between dispersion forces and packing effects determines the properties of n-alkanes; in principle, the dispersion forces dominate, as shown by the increase in density, viscosity, and refractive index with the chain length.

The structure of pure alkylbenzoates has been rather scarcely investigated up to now. Their noticeable dipole moment (µMB,298.15K ) 1.94 D)21 increases with the alkyl chain, giving rise to a pronounced ordering effect due to dipole-dipole interactions; such an effect is reflected by the high density of the pure alkylbenzoates. Although dipole moments increase with the alkyl chain, a direct interaction between dipoles is less likely for the largest alkanes since the less efficient packing causes the separation distance to increase. Hence, geometry factors should also be considered; alkylbenzoates are nearly planar in shape, and this planarity stepwise disappears with increasing the alkyl chain length;10 thereby most efficient packing appears with short chain lengths. Therefore, it is the balance between dipole interactions and geometry effects that is the factor controlling the properties of pure alkylbenzoates. The observed decrease in density and refractive index with increasing alkyl length also supports that geometry effects are prevailing, whereas the observed increase in viscosity is explained by the more compact packing, which makes the viscous flow difficult. The excess and mixing properties of the x1 n-alkane + (1 - x1) alkylbenzoate mixtures reflect the complex behavior of these systems. Figure 1 plots the effect of the n and m parameters on the molar excess volumes; under isothermal conditions, the change observed for a particular ester from negative to positive excess molar volumes with increasing n can, in principle, be ascribed to the geometry effects arising from the planar shape of alkylbenzoates and the linear shape of n-alkanes. The pronounced interstitial accommodation of alkanes into the alkylbenzoate structure (less efficient for large alkanes) justifies the observed variation of molar excess volumes. For a particular alkane, the alkyl chain effect is rather complex; if the excess molar volume is negative, then an increase in the ester chain gives rise to more negative values and, conversely, if the molar excess volume is positive, then a decrease in the property follows (Figure 1a,d). This may be attributed to the less efficient packing for the largest benzoates which enables an efficient interstitial accommodation upon mixing. An additional relevant effect that also should be discussed is the observed weakening of the dipolar ordering of alkylbenzoates when mixed with n-alkanes; the predominant dispersion forces in the mixtures, revealed by the negative mixing viscosities (Figure 2), support that the dipole-dipole interactions become hampered by the introduction of n-alkanes into the ester

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Figure 2. Mixing viscosity, ∆mixη, at 298.15 K for (a) x1 n-alkane + (1 - x1) PB and (b) x1 n-hexane + (1 - x1) alkylbenzoate. n ) number of carbon atoms of n-alkane; m ) number of alkylic carbons of alkylbenzoate.

holes.26 Viscosity data are a valuable tool for learning the interactions and structure of mixed solvents; viscous forces appear when adjacent layers of a fluid move with different speeds; therefore viscosity, defined as the ratio of the tangential stress over the velocity gradient, provides a measure of the extent of interactions. An increase in the alkane chain length is followed by a decrease of mixing viscosities. When n increases, also the n-alkane polarizability increases10 and hence dipole (induced)-dipole (permanent) interactions appear for large alkanes; these interactions do not fully balance the observed weakening of dipole-dipole interactions upon mixing, as the (negative) mixing viscosities reveal, but rather they give rise to a greater mixing viscosity compared to short alkanes. An increase in the m

parameter gives rise to even more negative mixing viscosities (Figure 2b) due to the severe dipolar weakening upon mixing with large alkylbenzoates and the less efficient packing in pure esters. The effects of the n and m parameters on the mixing refractive index (Figure 3) concur with the behavior of molar excess volumes (Figure 1). The efficient packing inherent to negative molar excess volumes caused an increase in the refractive index, giving rise to positive values and vice versa. The partial molar excess volumes evaluated for nalkanes (Figure 4) also point to an efficient packing for small alkanes. These values decreased as the n parameter increased and changed from negative to positive at infinite dilution. In a similar way, these values increased by stepwise increase in m, showing the relevant role of the geometry effect as a critical factor influencing the mixture structure. The values deduced for alkylbenzoate (Figure 4b,d) indicate that an efficient ester introduction into the n-alkane structure is strongly dependent on the similar/dissimilar size of the two components. Because of the greater molecular movement, an increase in temperature favors the packing effect on the excess and mixing properties (Figure 5) and the interstitial accommodation and makes the viscous flow difficult. The assumption of a more efficient packing at high temperatures is confirmed by the observed decrease in the partial molar excess volumes (Figure 6) and by the isobaric thermal expansivity (Figure 7). The (positive) activation enthalpies (Figure 8a,c) decreased with n and increased with m, while the activation entropies were always negative with the same n and m effects. These features discard the formation of stable

Figure 3. Mixing refractive index, ∆mixnD, at 298.15 K for (a) x1 n-alkane + (1 - x1) PB, (b) x1 n-hexane + (1 - x1) alkylbenzoate, and (c) x1 n-tetradecane + (1 - x1) alkylbenzoate. n ) number of carbon atoms of n-alkane; m ) number of alkylic carbons of alkylbenzoate.

Figure 4. Partial molar excess volume, V h Ei , at 298.15 K: for x1 n-alkane + (1 - x1) PB, (a) n-alkane partial molar volume and (b) PB partial molar volume; for x1 n-hexane + (1 - x1) alkylbenzoate, (c) n-alkane partial molar volume and (d) alkylbenzoate partial molar volume. n ) number of n-alkane carbon atoms; m ) number of alkyl carbons of alkylbenzoate.


for the correlation of the experimental molar excess volumes. A detailed literature survey shows the huge number of mixing rules suggested for multicomponent mixtures.14,27,28 Among all such proposals two main model types are remarkable: (i) those based on the van der Waals one-fluid mixing model and (ii) those which combine an EOS with an excess Gibbs energy model (Wong-Sandler approach). In this work several mixing rules corresponding to both approaches were applied. The van der Waals type mixing rules calculate the mixture covolume, b, according to eqs 6 and 7, with one correlation parameter for binary mixtures, δ12: Figure 5. Temperature effect on (a) molar excess volume, VEm, and (b) mixing viscosity, ∆mixη, for x1 n-hexane + (1 - x1) PB.

∑i ∑j xixjbij

(6)

1 (b + bjj)(1 - δij) 2 ii

(7)

b) bij )

The mixture copressure parameter, a, was calculated according to eq 8, the cross-coefficients

a)

Figure 6. Temperature effect on (a) partial molar excess volume h E2 , for x1 n-hexane + (1 - x1) PB. of n-hexane, V h E1 , and (b) PB, V

Figure 7. Isobaric thermal expansivity, RP, for (a) x1 n-alkane + (1 - x1) PB at 298.15 K and (b) x1 n-hexane + (1 - x1) PB at different temperatures. n ) number of n-alkane carbon atoms.

heteroaggregates and point to the prevalence of the geometry effects. So far, the existing literature data of excess enthalpies confirm a strong endothermic mixing for all alkanes and alkylbenzoates investigated;6 the observed increase of the excess enthalpies with n and decrease with m indicate that the disruption of the dipolar ordering of alkylbenzoates, not fully balanced by the formation of new heteroassociations, is the main contributing energy factor to the mixing process and support that the geometry effects arising from the size and shape of the molecules involved exerting control of the mixture structure, confirming the above conclusions. Property Modeling Cubic Equations of State. The cubic equations of state by Soave11 and Peng- Robinson12 were applied

∑i ∑j xixjaij

(8)

aij being defined on the basis of the corresponding pure component parameters according to different combining rules. The proper number of binary parameters depends on the particular rule and was obtained by fitting the EOS to the measured quantities. The simple quadratic one-parameter form (R1),27 Panagiotopoulos-Reid (two parameters, R2),29 Adachi-Sugie (two parameters, R3),30 Sandoval (two parameters, R4),31 SchwartzentruberRenon (three parameters, R5),32 and Stryjek-Vera van Laar (two parameters, R6)33 rules were applied in this work. The above nonquadratic mixing rules (R2-R6) are prone to the so-called Michelsen-Kistenmacher syndrome;34 that is, they may yield distinct results if the mole fraction of one component is arbitrarily split into two composite mole fractions; although this effect is particularly crucial for mixtures containing two or more very similar components,27 these rules have been applied successfully to binary mixtures. The MathiasKlotz-Prausnitz mixing rule (three parameters, R7)35 does not suffer from the Michelsen-Kistenmacher syndrome and can be applied safely to systems with similar components. The main shortcoming of the van der Waals one-fluid mixing rules is that they are applicable to moderate nonideal systems only. Among the different modern approaches available to reliably describe nonideal liquid mixtures, that by Wong and Sandler16 appears most appropriate. The mixing rule developed by Wong and Sandler for two-parameter cubic EOS combines an EOS and an excess Gibbs energy model. This densityindependent rule produces the desired EOS behavior at both low and high densities;27 this mixing rule is consistent with the statistical mechanical requirements and was applied together with the Wilson (three parameters, WS-Wilson, R8),36 NRTL (three parameters, WS-NRTL, R9),37 and UNIQUAC (three parameters, WS-UNIQUAC, R10)38 models. The pure component parameters required in the application of cubic EOS are reported as Supporting Information (Table VI, refs 39-44 contained therein). The correlation was performed according to a leastsquares procedure; for simplicity, the fitting parameters

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Figure 8. Activation molar enthalpy, ∆Hqm, and entropy, ∆Sqm, for (a and b) x1 n-alkane + (1 - x1) PB and (c and d) x1 n-hexane + (1 x1) alkylbenzoate. n ) number of carbon atoms of n-alkane; m ) number of alkylic carbons of alkylbenzoate.

Figure 9. Total absolute average percentage deviation, AAD, for the correlation of (a) molar excess volume and (b) dynamic viscosity, with the SRK and PR EOS and R1-R10 mixing rules, for x1 n-alkane + (1 - x1) alkylbenzoate over the 278.15318.15 K temperature range.

Figure 10. (a) Molar excess volume, VEm, (b) dynamic viscosity, η, and (c) mixing dynamic viscosity, ∆mixη, for x1 n-alkane + (1 - x1) PB at 298.15 K. (b) Experimental values and (;) values obtained with PR EOS and WSNRTL mixing rule R9.

are not reported here and only the average absolute percentage deviation, AAD (eq 9), is provided as Supporting Information (Table VII). In Figure 9 the total AAD, eq 10, is reported,

AAD )

100 N

∑ | N

|

YE(calc) - YE(expt)

AAD )

YE(expt) AADM ∑ M M

(9)

(10)

where N stands for the number of experimental data pairs for each binary system (usually 20) and M for the number of binary systems (96 systems studied). The

correlations obtained with both EOS are nearly similar (Figure 9a). As a rule, good results were obtained with most mixing rules tested, although the Wong-Sandler model gives the best results for the most complex systems such as those with an S-shaped molar excess volume profile (mixtures with n-octane and n-decane). As an example Figure 10a shows a comparison between some experimental values and those correlated with the PR + WS-NRTL model; the values and the shapes of the curves are reproduced fairly well despite the pronounced nonideal behavior of these systems. To reliably determine dynamic viscosities according to the Lee model,45 the correlating EOS parameters deduced from the molar excess volumes were used together with an additional parameter;45 in order to calculate the excess Gibbs energy, this model combines


a proper EOS with Eyring’s absolute rate theory of viscosity. On the basis of the results deduced by application of cubic EOS to molar excess volumes, the PR EOS combined with the R1, R6, and R9 mixing rules were used to correlate the viscosities measured. Only the additional parameter by the Lee model45 was correlated (for each system at every working temperature) with the experimental dynamic viscosities measured. The AAD values are reported as Supporting Information (Table VII) and the averaged AAD values shown in Figure 9. As a rule this model provides highquality correlations and rather low deviations for both the dynamic viscosity and the mixing property (Figure 10b,c). In summary, it is remarkable the fair ability of cubic EOS to concurrently correlate volumetric and viscometric properties in these strongly nonideal systems with only scant parameters. The Wong-Sandler mixing rule together with the NRTL and UNIQUAC equations appear to be ideally suited. Prigogine-Flory-Patterson Model. The PFP17 theory has been widely applied to interpret the excess thermodynamic properties of different types of multicomponent mixtures, including polar components. This model has proved to be particularly useful in analyzing the influence on liquid mixture behavior of the size, shape, and momentum of the components. The PFP model was applied to the molar excess volumes of the n-alkane + alkylbenzoate binary systems at 298.15 K to deduce molecular level information that otherwise cannot be inferred from the EOS treatment. According to the model the molar excess volume of a binary system can be split into three different contributions: E E effect VEm ) VEfree volume + Vinteractional + Vp*

(11)

The interactional contribution, which is proportional to the only correlation parameter, χ12, accounts for the intermolecular forces in the mixture; if this contribution is positive, then it entails absence of H-bonding or other specific interactions. The free volume contribution arises from the difference in the expansion degree of the two components The last term, the so-called pressure contribution, depends on the difference of internal pressure and reduced volume of the components.40 The pure component parameters required by the PFP model are reported as Supporting Information (Table VI). The χ12 interaction parameters for each binary system were evaluated from the molar excess volumes using a global minimization procedure with a simulated annealing algorithm. Classical methods such as that by Levemberg-Marquardt only are able to give correlation parameters by simple minimization procedures provided they have local minima; instead, use of the PFP algorithm is preferred, since it yields reliable, physically meaningful parameters. Bearing this in mind, no initial assumptions on the parameters and no prior restrictions are required; hence, the initial algorithm inputs that might condition the final results are avoided. This optimization procedure has been successfully applied to other thermodynamic and phase equilibria calculations.46 The PFP model interaction parameters and AAD values are reported as Supporting Information (Table IX). Most binary systems gave good correlations, as revealed by the low AAD values with just one-parameter correlation. Only those systems displaying complex S-shaped molar excess volumes (n-octane and n-decane)

Figure 11. PFP contributions to the equimolar molar excess E volume, Vm,0.5 , for (a) x1 n-alkane + (1 - x1) PB and (b) x1 n-hexane + (1 - x1) alkylbenzoate at 298.15 K. n ) number of n-alkane carbon atoms; m ) number of alkyl carbons of alkylbenzoate.

gave rise to higher deviations, not surprisingly for a oneparameter model. Figure 11 plots the n and m effects on the three PFP contributions to molar excess volumes. The interaction parameters, always positive (Table IX, Supporting Information), follow the usual trend for systems where specific interactions are lacking. The interactional contribution slightly decreased with n and m, a feature attributable to both the increasing alkane polarizabilities as the chain length increases and the increasing ester dipole moment. The free volume and pressure contributions were negative throughout. The free volume contribution, which represents a measure of the molecular fitting, is particularly important for short n-alkanes; it reached nearly the same value as that of the interactional contribution for n-hexane (Figure 11a), but tends to zero when n increases. This effect stems from the less efficient interstitial accommodation of large alkanes into the ester structure, the larger size and the less linearity making such fitting difficult. The free volume contribution decreased to more negative values with increasing m (Figure 11b) due to the less compact structure of the large esters, which enables a more efficient fitting. The pressure contribution, ascribable to the differences in the properties of pure components, approached zero with increasing n because the difference between the properties of both molecules are shortened and hence decreased in absolute value with m. In summary, pronounced specific interactions that might balance the dipolar ordering disruption upon mixing are lacking in the systems investigated, and according to the PFP analysis their structure is controlled mainly by geometry factors. Extended Real Associated Solution Model. The ERAS theory, originally developed by Heintz,47 combines the real association solution model with Flory’s equation of state. A recent modification of the ERAS model was applied in this work for the correlation of excess molar volumes;18 the molar excess volumes can be split into two main contributions:

VEm ) VEphys + VEchem

(12)

The physical contribution, VEphys, stems from the nonpolar van der Waals effects, while the chemical contribution, VEchem, is produced by the chemical interactions between components and, in particular, by H-bonding and specific interactions.

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Figure 12. Experimental and correlated ERAS (a) molar excess volume, VEm, and (b) molar excess enthalpy, HEm, for x1 n-hexane + (1 - x1) EB at 298.15 K. Symbols: (a) (b) Experimental values from this work; (b) (b) experimental values from Grolier et al.3

Although the separation between physical and chemical contributions could be regarded as a simplified and artificial approach to liquid mixtures structure, not well justified by the modern theories of intermolecular forces,48 in fact it provides a useful treatment for nonideal solutions. The ERAS model usually is applied to systems with at least one strongly associated component; also it has proved to be useful for systems, like those studied in this work, with one component (alkylbenzoate) showing pronounced dipolar intermolecular interactions. The molar excess volumes of the n-alkane + alkylbenzoate binary systems were analyzed at 298.15 K according to this model; also the optimization was applied according to the above global minimization procedure. The literature values of pure component parameters required by the ERAS model (Supporting Information, Table VI) were not available for alkylbenzoates; instead, they were evaluated by simultaneous correlation of experimental molar excess volumes and literature molar excess enthalpy data for selected alkylbenzoate + inert solvent binary systems (Figure 12). The calculated ERAS parameters and the deviations are reported as Supporting Information (Table X). Since no significant cross-association is established between the ester and the alkane, only the mixture Flory parameter, XAB, was fitted, and thus no cross-parameters such as the complexation constant or complexation molar volume are obtained. The agreement between the experimental and calculated molar excess volumes is good, the greater deviations being obtained for the S-shaped systems. Figure 13 plots the n and m effects on the physical and chemical contributions. The physical contribution to molar excess volumes is prevailing. The physical contribution arises from the geometry and weak van der Waals effects and controls the amount and profile of the property; this approached zero with increasing n because of the less favorable fitting effect observed with large alkanes. The (small) chemical contribution, always positive, denotes the absence of specific heteroassociations and increased with n due to the greater alkane polarizability. An increase in m remained the physical contribution unaffected, but approached zero the chemical contribution due to the greater dipole moment of large esters. Concluding Remarks Three main factors govern the structure of n-alkane + alkylbenzoate mixed solvents: (i) the disruption of

Figure 13. ERAS contributions to the equimolar molar excess E volume, Vm,0.5 , for (a) x1 n-alkane + (1 - x1) PB and (b) x1 n-hexane + (1 - x1) alkylbenzoate at 298.15 K. n ) number of n-alkane carbon atoms; m ) number of alkyl carbons of alkylbenzoate.

the ester dipolar structure; (ii) the efficient alkane fitting into the ester structure, which is weakened for the large alkanes; (iii) the weak heteroassociation between both components by induced dipole-dipole interactions and van der Waals forces. These conclusions are supported by the physically meaningful analysis carried out with the PFP and modified ERAS models. The study reported on the correlation ability of SRK and PR EOS with different mixing rules demonstrates that these models are quite adequate to calculate thermophysical properties in these strongly nonideal systems. Acknowledgment The financial support by Junta de Castilla y Leoń, Project BU10/03, and Ministerio de Ciencia y Tecnologıá, Project PPQ2002-02150, Spain, is gratefully acknowledged. Supporting Information Available: Experimental values of measured quantities (Tables I and II), polynomial fitting coefficients of density (Table III), dynamic viscosity (Table IV), and refractive index (Table V), pure solvents parameters (Table VI), AAD values for cubic equations of state (Tables VII and VIII), parameters and deviations of the PFP (Table IX) and ERAS (Table X) models (PDF). This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Cox, K. R. Physical property needs in industry. Fluid Phase Equilib. 1993, 82, 15. (2) Dohrn, R.; Pfohl, O. Thermophysical propertiessIndustrial directions. Fluid Phase Equilib. 2002, 194-197, 15. (3) Grolier, J. P. E.; Ballet, D.; Viallard, A. M. Thermodynamics of ester-containing mixtures. Excess enthalpies and excess volumes for alkyl acetates and alkylbenzoates + alkanes, + benzene, + toluene, and + ethylbenzene. J. Chem. Thermodyn. 1974, 6, 895. (4) Dusart, O.; Piekarski, C.; Viallard, A. Excess volumes of binary mixtures n-heptane + esters and n-heptane + ketones in homologous series. J. Chim. Phys. 1976, 73, 837. (5) Blanco, A. M.; Ortega, J.; Garcıá, B.; Leal, J. M. Studies on densities and viscosities of binary mixtures of alkylbenzoates in n-heptane. Thermochim. Acta 1993, 222, 127. (6) Garcıá, B.; Miranda, M. J.; Leal J. M.; Ortega, J.; Matos J. S. Densities and viscosities of mixing for the binary system of methyl benzoate with n-nonane at different temperatures. Thermochim. Acta 1991, 186, 285. (7) Garcıá, B.; Alcalde, R.; Aparicio, S.; Leal, J. M. Thermophysical behavior of methylbenzoate + n-alkanes mixed solvents. Application of cubic equations of state and viscosity models. Ind. Eng. Chem. Res. 2002, 41, 4399.

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Received for review February 22, 2005 Revised manuscript received July 5, 2005 Accepted July 18, 2005 IE0502281

Alkane + Alkylbenzoate Mixed Solvents. Measurements and

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