Preferential Solvation in Ternary Solutions Containing

13478

J. Phys. Chem. B 2003, 107, 13478-13486

Preferential Solvation in Ternary Solutions Containing Methylbenzoate. A Kirkwood-Buff Fluctuation Theory Study Begon˜ a Garcıá, Santiago Aparicio, Rafael Alcalde, and Jose´ M. Leal* Departamento de Quı´mica, UniVersidad de Burgos, 09001 Burgos, Spain ReceiVed: January 30, 2003; In Final Form: July 1, 2003

Density, dynamic viscosity, and refractive index of the hexane/cyclohexane/methylbenzoate ternary solvent and its binary constituents were measured at 298.15 K over the whole composition range. From the experimental measurements, excess and mixing properties were evaluated and correlated with composition using different models. The structure and interactions in the ternary solvent were analyzed starting from the experimental properties by application of the Kirkwood-Buff fluctuation theory of solutions along with the UNIFAC group contribution method for predicting activity coefficients. The local composition and excess (or deficit) number of molecules aggregated around a central one were determined using the Kirkwood-Buff integrals. The properties of the ternary system were used to test the predictive ability of several models. The volumetric properties of the binary and ternary systems were correlated and predicted with the cubic equations of state by Soave and Peng-Robinson combined with two simple mixing rules.

Introduction Grouping of solvents into classes often is based on the nature of the intermolecular forces because the manner whereby solute and solvent molecules are associated with one another brings about a marked effect on the resulting properties. After the introduction of the concept of ionization power of solvent,1 much work has been devoted to solvent effects on rate and equilibrium processes.2 Because of the close connection between liquid structure and macroscopic properties, determination of volumetric, electrical, and rheological properties is a valuable tool to learn the liquid state.3 On the other hand, the obtaining of reliable measurements of solvent properties over a wide range of composition, pressure, and temperature often is not feasible; hence, prediction and correlation methods constitute a valuable option to overcome such difficulties.4,5 This work contributes to the study of the structure and interactions of ester-containing ternary mixtures. These solvents are widely used in polymer science and syntheses operations; however, the scarce literature existing makes convenient a systematic research on their properties.6 Density, dynamic viscosity, and refractive index data of the methylbenzoate/ cyclohexane/n-hexane ternary solvent and its constituent nhexane/cyclohexane, n-hexane/methylbenzoate, and cyclohexane/ methylbenzoate binary components are reported over the full composition range.7,8 From these measurements, the excess and mixing properties were evaluated and interpreted on the basis of interactions and geometry factors. The volumetric properties of the binary systems were correlated using the cubic equations of state by Soave9 (SRK) and Peng-Robinson10 (PR) combined with two simple mixing rules. On the basis of the properties of the binary components, the same properties of the ternary solvent were approached using several models. The statistical thermodynamic theory by Kirkwood-Buff (KB)11 provides a proper link between macroscopic and microscopic properties and was used to interpret intermolecular interactions and preferential * Corresponding author. E-mail: [email protected]. Tel: +34 947 258 819. Fax: +34 947 258 831.

solvation effects.12 This approach describes properly deviations from ideality, provides a reliable description of the solvation effect, and can be extended to different types of particles, not only spherical ones.13 Also it enables evaluation of the local composition of the components and has become popular for binary and multicomponent mixtures.14 The activity coefficients were evaluated using the group contribution UNIFAC III model.15 The excess molar volumes were applied by deriving expressions for the Kirkwood-Buff integrals and the corresponding excess number of molecules around a central one.16 Experimental Section Reagents. The reactants, methylbenzoate (MB), hexane (hex), and cyclohexane (cyc), of the highest purity commercially available were used without further purification. The purity was assessed by gas chromatography (GC) with a Perkin-Elmer 990 gas chromatograph, equipped with a Hewlett-Packard 3390A integrator and also by comparing the densities, viscosities, and refractive indices with literature values (Table S1, Supporting Information, and refs 17-31 contained therein). The liquids were degassed with ultrasound for several days before use and kept out of the light over Fluka Union Carbide 0.4 nm molecular sieves. To prevent the samples from preferential evaporation, the mixtures were prepared by syringing amounts, weighed to ∆m ) 10-5 g with a Mettler AT 261 Delta Range balance, into suitably stoppered bottles. The mixtures were completely miscible over the whole composition range. Instruments and Procedures. The molar excess volumes ((10-4 cm3 mol-1) were deduced from the densities of the pure liquids and mixtures. The densities, F, were measured with a computer-controlled DMA 58 Anton Paar digital density meter ((5 × 10-6 g cm-3) by introducing the samples into a U-shaped oscillating tube the natural frequency of which is influenced by the mass of the sample. The proper calibration was achieved at all working temperatures ((0.01 K) with deionized doubly distilled water (Milli-Q, Millipore) and n-nonane (Fluka, 99.2% purity by GC) as reference liquids; the apparatus takes care of

10.1021/jp034255l CCC: $25.00 © 2003 American Chemical Society Published on Web 11/08/2003

Preferential Solvation in Ternary Solutions

J. Phys. Chem. B, Vol. 107, No. 48, 2003 13479

the electrical control of sample temperature and calls for recalibration if the control is missed. Dynamic viscosities, η, were measured with an automated AMV 200 Anton Paar microviscometer (0.120 cm3 sample volume). Viscosity measurements are based on the shear stress of a rolling ball introduced into an inclined, sample-filled glass capillary placed inside a thermostated block ((0.005 K) by measuring the time ((0.01 s) needed for the ball to roll a fixed distance between two sensors. The stress was varied by changing the inclination angle of the capillary in 5° intervals. Viscosities were determined as an average of 28 readings (seven inclination angles, four readings each), the estimated error being (0.005 mPa‚s. Careful calibration was achieved at every inclination angle and working temperature as a function of water as the standard liquid and the density of the ball (7.874 g cm-3). The refractive indices, nD, were measured using the sodium line of an automatic Leica AR600 refractometer. A Julabo F32MV thermostat ((0.01 K) guarantees an accuracy better than (5 × 10-5. Small sample volumes were put on a prism equipped with a lid that produces tightness to avoid preferential evaporation of the sample; the thermostatic time of the samples was small, and the temperature was read at the prism surface. The nD values were taken every two minutes as an average of 150 readings, the measurements being reproducible to (10-5. Results and Discussion Table S2 (Supporting Information) lists density, viscosity, and refractive index measurements for the x hexane/(1 - x) cyclohexane mixed solvent as a function of composition, along with the derived excess and mixing properties; the properties for the other two binary constituents, x hexane/(1 - x) MB and x cyclohexane/(1 - x) MB, were reported elsewhere.7,8 Table S3 lists same properties (Supporting Information) for the x1 hexane/x2 cyclohexane/(1 - x - x2) MB ternary system. Excess molar volume, VEm, mixing viscosity, ∆mixη, and mixing refractive index, ∆mixnD, were evaluated from the experimental measurements using the classical expressions8

VEm )

M F

Mi

c

-

xi ∑ i)1 F

(1) i

c

∆mixη ) η -

xiηi ∑ i)1

(2)

c

∆mixnD ) nD -

xinD,i ∑ i)1

(3)

where c stands for the number of components of the mixture, M and Mi are the molar mass of the mixture and of the pure components, F and Fi represent the density of the mixture and of pure components, η and ηi are the corresponding dynamic viscosities, and nD and nD,i are the refractive indices. The Redlich-Kister polynomials (eq 4) were fitted to excess and mixing properties YE k

YE ) x(1 - x)

Ai(2x - 1)i ∑ i)1

(4)

by least squares using the Marquardt algorithm, the optimum number of Ai fitting coefficients being optimized by an F-test. The coefficients for the x hexane/(1 - x) cyclohexane mixed solvent are listed in Table 1. Figure 1 plots the excess and

Figure 1. Excess molar volume, VEm (a), mixing viscosity, ∆mixη (b), and mixing refractive index, ∆mixnD (c), at 298.15 K for the binary mixtures x hexane/(1 - x) MB (b), x cyclohexane/(1 - x) MB (9), and x hexane/(1 - x) cyclohexane (2).

TABLE 1: Ai Coefficients of Eq 4 and Standard Deviation, σ, for the x Hexane + (1 - x) Cyclohexane Binary Mixture at 298.15 K, Excess Molar Volume, VEm (cm3 mol-1), Mixing Viscosity, ∆mixη (mPa‚s), and Mixing Refractive Index, ∆mixnD property VEm ∆mixη ∆mixnD

A0

A1

A2

A3

σ

0.7149 -0.2487 -0.0391 -0.1801 0.0007 -0.572 0.378 -0.213 0.002 -0.011 07 0.002 75 -0.000 86 -0.001 09 0.000 08

mixing properties for the hexane/MB, cyclohexane/MB, and hexane/cyclohexane binary components. The YETER excess and mixing properties for the hexane/cyclohexane/MB ternary solvent were correlated with composition using the Cibulka equation (eq 5) and the Redlich-Kister coefficients deduced from eq 4 for the binary constituents: E YTER ) YEBIN + x1x2(1 - x1 - x2)(B0 + B1x1 + B2x2) (5)

where the YEBIN term involves the sum extended to the three binary components, and YEij can be evaluated for each property

13480 J. Phys. Chem. B, Vol. 107, No. 48, 2003

Garcıá et al.

E Figure 2. Excess molar volume, Vm,TER (a), and ternary contribution E E to excess molar volume, Vm,TER - Vm,BIN (b), at 298.15 K for the x1 hexane/x2 cyclohexane/(1 - x1 - x2) MB ternary system.

TABLE 2: Bi Coefficients of Eq 5 and Standard Deviation, σ, for the x1 Hexane + x2 Cyclohexane + (1 - x1 - x2) MB Ternary Mixture at 298.15 K, Excess Molar Volume, E (cm3 mol-1), Mixing Viscosity, ∆mixηTER (mPa‚s), and Vm,TER Mixing Refractive Index, ∆mixηD,TER property E Vm,TER ∆mixηTER

∆mixnD,TER

B0

B1

B2

σ

2.7925 -0.614 -0.1382

-13.0009 1.527 0.1242

-9.0608 0.805 -0.1579

0.0397 0.005 0.00120

from eq 6:

YEBIN ) YE12 + YE13 + YE23

(6)

with YEij given by eq 4 and subscripts 1, 2, and 3 referring to hexane, cyclohexane, and MB, respectively. The Bi parameters deduced by least squares are listed in Table 2. The so-called “ternary contributions” YETER - YEBIN, which represent the difference between the value measured for the ternary property and that predicted from the binary contributions, provide information on the ternary interactions and were deduced from eqs 5 and 6. Figures 2-4 plot the excess and mixing properties for the ternary system, along with the ternary contributions, and Table 3 lists the maxima or minima or both displayed. Excess and mixing properties allow one to draw information on the structure and interactions of mixed solvents. Hexane is a linear, nonpolar molecule, MB is a planar, polar molecule (µ298.15K ) 1.9 D),18 and cyclohexane, globular in shape, often is regarded as an order destroyer of other solvents.32 Linear hexane favors an efficient package and yielded negative hexane/ MB excess molar volumes with a minimum at equimolar

Figure 3. Mixing viscosity, ∆mixηΤΕR (a), and ternary contribution to mixing viscosity, ∆mixηTER - ∆mixηTER (b), at 298.15 K for the x1 hexane/x2 cyclohexane/(1 - x1 - x2) MB ternary system.

composition. The cyclohexane breaking ability of differently shaped solvents is accompanied by an expansion effect (positive VEm, Figure 1a), and as a consequence, a dilution of the MB dipoles in cyclohexane/MB follows. Negative mixing viscosities reflect the prevalence of dispersion forces.33 The ∆mixη values reported in Figure 1b were all negative for the three binary components; the efficient packing observed in hexane/MB leads to negative ∆mixη values, whereas the very different shape of cyclohexane leads to a decrease in ∆mixη. Mixing refractive indices normally display a behavior opposite to VEm; contractive mixtures such as hexane/MB give rise to positive ∆mixnD, and the opposite is observed for hexane/ cyclohexane; consistently, the ∆mixnD values were positive for hexane/MB and negative for hexane/cyclohexane (Figure 1c). However, cyclohexane/MB, which displays an expansive character, also yielded positive mixing refractive indices; the dipolar dilution caused by the positive VEm becomes partly balanced by the cyclohexane contribution (µ298.15K ) 0.3 D)18 to the polarity of the resulting mixture. The very different shape of the three components causes noticeable effects on the behavior of the ternary mixture. Excess molar volumes, VEm,TER, were positive at high cyclohexane concentration and negative at low concentration (Figure 2a). The ternary contribution VETER - VEBIN, predominantly positive, gave a minimum (Table 3) that represents some 200% relative to VEm,TER at the same composition (Figure 2b); this feature reflects the important ternary effects observed in this system. Mixing viscosities, ∆mixηTER, were negative over the whole composition range (Figure 3a); these values involve low positive and negative ∆mixηTER - ∆mixηBIN ternary contributions (Figure



E TABLE 3: Coordinates for the Maxima and Minima of the Ternary Contributions to Excess Molar Volume (Vm,TER E , cm-3 mol-1), Mixing Viscosity, (∆mixηTER - ∆mixηBIN, mPa‚s), and Mixing Refractive Index (∆mixnD,TER - ∆mixnD,BIN) for Vm,BIN the x1 Hexane + x2 Cyclohexane + (1 - x1 - x2) MB Ternary Mixture at 298.15 K E E (Vm,TER - Vm,BIN )

position

amount

position

-0.1978

x1 ) 0.5345 x2 ) 0.2814 x1 ) 0.1299 x2 ) 0.2094

maximum minimum

x1 ) 0.4371 x2 ) 0.3391

∆mixηTER - ∆mixηBIN

∆mixnD,TER - ∆mixnD,BIN

amount

position

amount

x1 ) 0.2551 x2 ) 0.4334

-0.0060

0.0119 -0.0044

plication to a wide range of different particles is feasible over the whole composition range. This model is a powerful tool to draw structural information from macroscopic properties of liquids;16 the statistical mechanics formalism involved links macroscopic effects such as isothermal compressibility, chemical potential, and partial molar volumes to microscopic properties using spatial integrals and radial distribution functions. The pioneering work by Ben-Naim enables application of the model to binary mixtures and has been applied to a wide range of mixed solvents to interpret the solvent structure.14 A convenient link between molecular and macroscopic properties is provided by the so-called Kirkwood-Buffs integrals (KBI), or fluctuation integrals, Gij, defined as

Gij )

Figure 4. Mixing refractive index, ∆mixnD,TER (a), and ternary contribution to mixing refractive index, ∆mixnD,TER - ∆mixnD,BIN (b), at 298.15 K for the x1 hexane + x2 cyclohexane + (1 - x1 - x2) MB ternary system.

3b), geometry and shape features being those that primarily determine the ternary properties. Mixing refractive indices, ∆mixnD,TER, were predominantly positive (Figure 4a) with negative ternary contributions ∆mixnD,TER - ∆mixnD,BIN, the minimum of which represents some 600% of the property at same composition (Figure 4b); this feature is in reasonably good agreement with the highly negative ternary contributions to excess molar volumes and reflects the prevalence of the geometric effects on the intermolecular interactions. Kirkwood-Buff Theory Theory. The macroscopic effects observed in liquids can be attributed to several different microscopic properties, and it is not always feasible to trace back which microscopic property is the origin for a particular observed behavior. The KirkwoodBuff theory of solutions links thermodynamic quantities with molecular distribution functions.11 This model involves an spatial pair correlation function and considers no assumptions on the additivity of the total potential energy; hence, universal ap-

∫0∞(gij - 1)4πr2 dr

(7)

where gij, the radial distribution function of species i around a central molecule j, describes the solution structure at the macroscopic level, r being the distance between the i-j centers. To apply the Kirkwood-Buff formalism to study the interactions and local composition in ternary mixtures, the need for a proper reference level has been emphasized.12 The model considers the symmetrical ideal system as a convenient reference level; hence, the KBIs for this level must be subtracted from the KBIs of the mixture to provide a proper description of the mixture structure. The KBIs used for binary and ternary mixtures fulfilled the aforementioned criteria. Another relevant parameter in the application of the Kirkwood-Buff theory is the excess number of molecules around a central one, nij, which can be evaluated from the KBIs according to

) nij ) ci(Gij - GIDEAL ij

(8)

Source of Data. Accurate data pairs of three macroscopic properties are needed for a proper evaluation of KBIs: isothermal compressibility, activity coefficient, and partial molar volume. Evaluation of partial molar volumes for binary and ternary solvents can be properly achieved by application of the intercept method to excess molar volumes.7,8 On the other hand, the contribution of isothermal compressibility of the ternary mixture to KBIs was negligible in this work (ideal behavior),16 so the values for this property were deduced from those of pure components available in the literature.18 Evaluation of activity coefficients, normally from data of vapor-liquid equilibria, is costly and time-consuming. Full characterization of a ternary system requires weeks of continuous work for collecting reliable readings;34 instead, application of group contribution methods may enable one to obtain accurate databases of phase equilibria and related mixture properties. In particular, the UNIFAC method has been shown to provide accurate predictions. The original UNIFAC method suffered from some weakness to predict activity coefficients at infinite dilution.35 In fact, the average error of activity coefficients predicted by this method ranged some 15%;36 however, this can be substantially reduced


Figure 5. Kirkwood-Buff integrals, Gij (a), excess or deficit of molecules around a central hexane molecule (b), and excess or deficit of molecules around a central cyclohexane molecule (c) in the x hexane + (1 - x) cyclohexane binary mixture at 298.15 K.

to less than 1/2 for systems with low deviations from ideality, as was the case in this work.37 Moreover, the modified UNIFAC (Dortmund) model used in this work noticeably has improved the results of a set of mixture properties (activity coefficients and heat capacity, among others). For instance, the infinite ∞ dilution (xmethanol f 0) integral Gmethanol-water calculated according to our procedure is -39.1 cm3 mol-1, whereas the value by Ruckenstein38 is -37.5 cm3 mol-1; hence, the temperaturedependent group parameters can be obtained by a suitable fitting over a sufficiently large temperature range. The activity coefficients and their concentration dependence needed to evaluate the chemical potential, experimentally not accessible, were calculated using the UNIFAC III group contribution method,

Garcıá et al. which consists of a broad database of group parameters and provided accurate results.15 Results and Analysis. Evaluation of KBI and nij values for the ternary mixture and the three binary components provides valuable information. The ternary mixture was treated as a pseudobinary mixture, that is, as a mixture with constant mole fraction for one component. Two pseudobinary mixtures were chosen in principle: hexane/MB at constant 0.1 cyclohexane mole fraction and cyclohexane/MB at constant 0.1 hexane mole fraction. The KBI data for hexane/cyclohexane, small and negative, suggest a negligible affinity between the binary components (Figure 5).12 The nhex-hex and ncyc-cyc values reported, positive and small, displayed a maximum; however, the nhex-MB and ncyc-MB values, small and negative, displayed a minimum at the same composition, which reveals a solvation effect by like molecules. The modest KBI values deduced point to a short solvation radius, probably not exceeding that of the first coordination sphere around the central molecule,39 a feature consistent with the behavior observed for the excess and mixing properties. The large KBI values deduced for hexane/MB were positive with a maximum for Gii and negative with a minimum for Gij (Figure 6). The lack of affinity between hexane and MB, revealed by the negative Ghex-MB values, was confirmed by the large positive nii values and the negative nij values; hence, the hexane molecules are surrounded by hexane molecules, while the MB molecules become solvated by MB molecules, the solvation radii being high, as inferred from the KBI values. This effect can be explained considering the different shape of both molecules (linear hexane and planar MB) and the observed preferential solvation between like molecules. Cyclohexane/MB gave positive Gii and negative Gij KBIs (Figure 7) with maxima and minima somewhat less pronounced; the positive nii and negative nij values with maxima and minima some 10 times lower than those for hexane/MB clearly point to a solvation effect with somewhat shorter solvation radii. These findings are in good agreement with the behavior of the properties deduced for these systems. Figures 8 and 9 show the results for the ternary mixture provided by the model. Figure 8 plots the nii and nij values for the pseudobinary hexane/MB mixture at constant 0.1 cyclohexane mole fraction; the excess (or deficit) number of molecules aggregated around a central hexane molecule are plotted in Figure 8a, around a central cyclohexane in Figure 8b, and around a central MB in Figure 8c. Figures 8a and 6b reveal the decrease in number of hexane molecules around a central one upon addition of cyclohexane; likewise, Figures 8c and 6c reveal the decrease in the number of MB molecules surrounding a central MB upon increasing the cyclohexane content. Likewise, unlike solvation decreased upon addition of cyclohexane to hexane/ MB. Only cyclohexane, with positive ncyc-hex values, shows certain solvation ability around a central hexane molecule (Figure 8a). Addition of cyclohexane to hexane/MB causes a decrease both in the hexane content around a central hexane molecule and in the MB content around a central MB molecule. Only unlike solvation of cyclohexane around a central hexane and of hexane around a central cyclohexane occurs, in contrast with the observed solvation in hexane/cyclohexane, where there is not a clear solvation trend between both molecules (Figure 5b,c). Figure 9 plots nii and nij for the pseudobinary cyclohexane/ MB mixture at constant 0.1 hexane mole fraction. Figure 9a shows the excess or deficit of molecules around a central hexane,


Figure 6. Kirkwood-Buff integrals, Gij (a), excess or deficit of molecules around a central hexane molecule (b), and excess or deficit of molecules around a central MB molecule (c) in the x hexane + (1 - x) MB binary mixture at 298.15 K.

Figure 9b around a central cyclohexane, and Figure 9c around a central MB. Figures 9b and 7b reveal the increase in thecyclohexane solvation around a central cyclohexane upon addition of hexane; unlike MB solvation around a central hexane becomes unlikely when the ternary mixture is formed. Also, the remarkable excess of hexane around a central cyclohexane molecule points to a certain unlike solvation between these components. The same conclusion can be arrived at by comparing Figures 7c and 9c: addition of hexane favors like solvation of MB around a central MB molecule, whereas unlike cyclohexane solvation around a central MB is not feasible because of the observed deficit of cyclohexane around the central MB. Finally, Figure 9a shows that hexane becomes solvated preferentially by cyclohexane; therefore, unlike solvation becomes very important.


Figure 7. Kirkwood-Buff integrals, Gij (a), excess or deficit of molecules around a central cyclohexane molecule (b), and excess or deficit of molecules around a central MB molecule (c) in the x cyclohexane + (1 - x) MB binary mixture at 298.15 K.

Modeling Thermophysical Properties Semiempirical Models. Although thermophysical properties of binary mixtures abound in the literature, such properties measured for ternary systems have been rather scarcely investigated up to now. Experimental measurements of thermophysical properties of mixed solvents, even with modern instrumentation, become progressively more expensive and timeconsuming upon addition of extra components.40 A plausible objective of solution thermodynamics consists of developing predictive models for multicomponent solvents. In this work, several models were used to predict excess and mixing properties of the hexane/cyclohexane/MB ternary solvent from those of the binary constituents; these models can be classified as (i) symmetric, if prediction is independent of the ordering of the


Garcıá et al.

Figure 8. Excess or deficit of molecules (a) around a central hexane molecule, (b) around a central cyclohexane molecule, and (c) around a central MB molecule in the pseudobinary x′ hexane + (1 - x′) MB mixture at 298.15 K with xcyclohexane ) 0.1.

Figure 9. Excess or deficit of molecules (a) around a central hexane molecule, (b) around a central cyclohexane molecule, and (c) around a central MB molecule in the seudobinary x′ cyclohexane + (1 - x′) MB mixture at 298.15 K with xhexane ) 0.1.

components in the mixture, and (ii) asymmetric, if predictions depend on the arbitrary numbering of the components. Therefore, symmetry involves the contribution of the three binary components to the ternary mixture, all contributing equally, while asymmetry indicates different contribution of any of the components. The symmetric models by Jacob-Fitzner,41 Ko¨hler,42 and Colinet43 and asymmetric models by Tsao-Smith,44 Toop,45 and Scatchard46 were used to predict properties of the ternary solvent from data of the binary constituents (Table 4). Most models yielded accurate results and low deviations for the properties analyzed. The three symmetric equations yielded close predictions. The asymmetric models yielded deviations that depend on the arbitrary numbering of the solvents used; except for

excess molar volume, the best predictions were obtained when cyclohexane was designed as component 1. Predictions obtained with both symmetric and asymmetric methods were quite similar; hence, the quoted properties can be predicted advantageously with the more simple symmetric models. Cubic Equations of State. Densities and excess volumes are relevant quantities to interpret the interactions in mixed solvents because they are related to the heaviness of the molecules in a unit volume. The main purpose of using equations of state (EOS) is attaining a proper representation of the mixed solvents properties. In the past few years, the interest related to cubic EOS for the correlation or prediction or both of excess molar volumes, excess enthalpies, vapor-liquid equilibria, and other properties has increased. The main feature of these equations



TABLE 4: Standard Deviations, σ, for the Predictions Obtained for the x1 Hexane + x2 Cyclohexane + (1 - x1 - x2) MB Ternary Solvent at 298.15 K model

E σ(Vm,TER ) (cm3 mol-1)

σ(∆mixηTER) (mPa‚s)

σ(∆mixnD,TER)

Jacob-Fitzner Ko¨hler Colinet Tsao-Smith Toop Scatchard

0.0994 0.1043 0.1042 0.0750b 0.0718b 0.0700b

0.007 0.009 0.009 0.056b 0.008b 0.010b

0.0036 0.0037 0.0037 0.0050b 0.0039b 0.0039b

0.1776a 0.0951a 0.0908a

0.1764c 0.1522c 0.1519c

0.065a 0.030a 0.030a

0.017c 0.011c 0.012c

0.0054a 0.0039a 0.0039a

0.0028c 0.0032c 0.0032 c

a For asymmetric equations, the component 1 is n-hexane. b For asymmetric equations the component 1 is cyclohexane. c For asymmetric equations, the component 1 is MB.

TABLE 5: k12 and m12 Parameters and Standard Deviations, σ, Obtained by Application of Soave (SRK) and Peng-Robinson (PR) Cubic Equations of State with the R1 and R2 Mixing Rules for the Correlation (Binary Mixtures) and Prediction (Ternary Mixture) of Excess Molar Volume (cm3 mol-1) at 298.15 Ka SRK

PR

R1

R2

R1

R2

mixture

k12

σ

k12

m12

σ

k12

σ

k12

m12

σ

hexane + cyclohexane hexane + MB cyclohexane + MB hexane + cyclohexane + MB

0.0139 0.0944 0.1260

0.0302 0.0379 0.0592 0.1621

-0.0642 0.0805 0.0920

-0.0202 -0.0040 -0.0193

0.0048 0.0189 0.0075 0.2057

0.0152 0.0842 0.1337

0.0297 0.0177 0.0508 0.1521

-0.0716 0.0849 0.1009

-0.0207 -0.0017 -0.0190

0.0052 0.0176 0.0082 0.2033

a

Parameters for hexane + MB are taken from ref 7, and those for cyclohexane + MB are taken from ref 8.

is that they require introduction of proper mixing rules to allow EOS to describe nonideal mixed solvents and determine the mixture properties;47 an approach has recently been proposed that accounts for the effect of the density on the excess properties of nonideal mixed solvents while giving zero interaction parameters for the ideal solution limit.48 Cubic equations of state are widely used for chemical engineering purposes; because of their simplicity, ease of handling, little information required, and accurate results provided, the cubic equations of state proposed by Soave9 (SRK) and Peng-Robinson10 (PR) were used to correlate the excess molar volumes of the binary mixtures and to predict those of the ternary mixture using the general eq 9

P)

nRT n2a V - nb (V + δ1nb)(V + δ2nb)

(9)

with δ1 ) 1 and δ2 ) 0 for SRK and δ1 ) 1 + x2 and δ2 ) 1 - x2 for PR, n being the mole number in the mixture. The two mixing rules used to calculate the copressure, a, and covolume, b, are shown in eqs 10 and 11. c

c

a)

xixj(1 - kij)(aiaj)0.5 ∑ ∑ i)1 j)1

b)

∑ ∑xixj(1 - mij)(bibj)0.5 i)1 j)1

c

(10)

c

(11)

where c is the number of components. For the first mixing rule (R1), the kij parameters were fitted introducing the experimental VEm data, and the values mij ) 0 were deduced. For the second mixing rule (R2), the mij and kij parameters were also fitted. The fitting parameters were determined using the Marquardt algorithm combined with the Newton-Raphson method used to solve the equation of state. The results obtained for the binary and ternary mixtures are reported in Table 5. The fitting parameters of the binary components were used to predict the excess molar volume in the ternary solvent. SRK and PR conduce to similar correlations when applied with the same mixing rule to the binary mixtures; both mixing rules yielded

low deviations, but these were still lower with the biparametric mixing rule. However, the predictions were not fully satisfactory using these equations with binary interaction parameters. The high ternary contribution to the excess molar volume (Figure 2b) indicates that ternary effects are important. Conclusions The Kirkwood-Buff theory of solutions was applied to the hexane/cyclohexane/MB ternary system. Like-solvation is prevailing in the binary components, while there is a deficit of unlike molecules; this effect is remarkable for the MB binary components, whereas preferential solvation was not observed for hexane/cyclohexane. MB also showed preferential like solvation in the ternary mixture; this effect decreased with increasing cyclohexane concentration. Cyclohexane becomes solvated mainly by cyclohexane, but also a noticeable unlike hexane solvation around a central cyclohexane was observed, this effect increasing with increasing the hexane content. Hexane showed a rather more complex behavior with some MB deficit around a central hexane. Cyclohexane solvates hexane very efficiently at low cyclohexane concentrations. MB molecules do not solvate hexane and cyclohexane; instead MB tends to become surrounded by MB molecules. In contrast, hexane and cyclohexane can be solvated by one another to an extent that depends on the relative concentrations. Acknowledgment. The finantial 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: Table S1 containing experimental and literature properties of the pure solvents, Table S2 containing experimental measurement for hexane/cyclohexane mixture, and Table S3 containing experimental measurements for hexane/cyclohexane/MB ternary mixture. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Reichardt, C. SolVents and SolVent Effects in Organic Chemistry; VCH: Weinheim, Germany, 1988; Chapters 5 and 7.

13486 J. Phys. Chem. B, Vol. 107, No. 48, 2003 (2) Grunwald, E.; Winstein, S. J. Am. Chem. Soc. 1948, 70, 846. (3) Praustnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice-Hall: New York, 1986; Chapter 4. (4) Tohidi, B.; Todd, A. C.; Danesh, A.; Burgess, R. W.; Gozalpour, F. Int. J. Thermophys. 2001, 22, 1661. (5) Reid, C. R.; Praustnitz, J. M.; Poling, B. E. The properties of Gases and Liquids; McGraw-Hill: New York, 1988, Chapter 1. (6) Garcıá, B.; Ortega, J. C. J. Chem. Eng. Data 1988, 33, 200. (7) Garcıá, B.; Alcalde, R.; Aparicio, S.; Leal, J. M. Ind. Eng. Chem. Res. 2002, 41, 4399. (8) Garcıá, B.; Alcalde, R.; Aparicio, S.; Leal, J. M. Phys. Chem. Chem. Phys. 2002, 4, 5840. (9) Soave, G. Chem. Eng. Sci. 1972, 27, 1197. (10) Peng, Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. (11) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (12) Matteoli, E. J. Phys. Chem. B 1997, 101, 9800. (13) Zielkiewicz, J. J. Chem. Soc., Faraday Trans. 1998, 94, 1713. (14) Ben-Naim, A. J. Chem. Phys. 1977, 67, 4884. (15) Gmehling, J.; Li, J.; Schiller, M. Ind. Eng. Chem. Res. 1993, 32, 178. (16) Ruckenstein, E.; Shulgin, I. Fluid Phase Equilib. 2001, 180, 281. (17) Aminabhavi, T. M.; Raikar, S. K. Collect. Czech. Commun. 1993, 58, 1761. (18) Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic SolVents, Physical Properties and Methods of Purification; Wiley: New York, 1986. (19) Matilla, A. D.; Tardajos, GG.; Junquera, E.; Aicart, E. J. Solution Chem. 1991, 20, 805. (20) Ortega, J.; Legido, J. L. Fluid Phase Equilib. 1993, 86, 251. (21) Ewing, M. B.; Marsh, K. N. J. Chem. Thermodyn. 1977, 9, 357. (22) El-Banna, M. M.; El-Batouti, M. M. Can. J. Chem. 1998, 76, 1860. (23) Ambrose, D.; Tsonopoulos, C. J. Chem. Eng. Data. 1995, 40, 531. (24) Dymond, J. M.; Oye, H. A. J. Phys. Chem. Ref. Data. 1994, 23, 41. (25) Aralaguppi, M. J.; Jadar, C. V.; Aminabhavi, T. M. J. Chem. Eng. Data. 1999, 44, 435.

Garcıá et al. (26) Dymond, J. H.; Young, K. J. Int. J. Thermophys. 1980, 1, 331. (27) Petrino, P. J.; Gaston-Bonhomme, Y. H.; Chevalier, J. L. E. J. Chem. Eng. Data. 1995, 40, 136. (28) Decominges, B. E.; Pin˜eiro, M. M.; Mosteiro, L.; Iglesias, T. P.; Legido, J. L.; Andrade, M. I. P. J. Chem. Eng. Data. 2001, 46, 1206. (29) Jimeńez, E.; Casas, H.; Segade, L.; Franjo, C. J. Chem. Eng. Data. 2000, 45, 862. (30) TRC Thermodynamic Tables; Thermodynamic Research Center, Texas A&M University: College Station, TX, 1994. (31) Iglesias, M.; Pin˜eiro, M. M.; Marino, G.; Orge, B. B.; Domıńguez, M.; Tojo, J. Fluid Phase Equilib. 1999, 154, 123. (32) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures; Butterworths: London, 1982; Chapter 5. (33) Fort, R.; Moore, W. R. Trans. Faraday Soc. 1966, 62, 1112. (34) Deiters, U.K.; Hloucha, M. In Chemical Thermodynamics; Letcher, T. M., Ed.; Blackwell Science: Oxford, U.K., 1999; Chapter 15. (35) Letcher, T. M.; Whitehead, P. G. J. Chem. Thermodyn. 1997, 29, 1261. (36) Gmehling, J. Fluid Phase Equilib. 1998, 144, 37. (37) Voutsas, E. C.; Tassios, D. P. Ind. Eng. Chem. Res. 1996, 35, 1438. (38) Ruckenstein, E.; Sulgin, I. Fluid Phase Equilib. 2001, 180, 281. (39) Zielkiewicz, J. Phys. Chem. Chem. Phys. 2000, 2, 2925. (40) Acree, W. A. Thermodynamic Properties of Nonelectrolyte Solutions; Academic Press: Orlando, FL, 1984; Chapter 4. (41) Jacob, K. T.; Fitzner, K. Thermochim. Acta 1977, 18, 197. (42) Ko¨hler, F. Monatsh. Chem. 1960, 91, 738. (43) Colinet, C. Ph.D. Thesis, University of Grenoble, France, 1967. (44) Tsao, C. C.; Smith, J. M. Chem. Eng. Prog. Symp. Ser. 1953, 49, 107. (45) Toop, G. W. Trans. TMS-AIME 1965, 223, 850. (46) Scatchard, G.; Ticknor, L. B.; Goates, J. R.; McCartney, E. R. J. Am. Chem. Soc. 1952, 74, 3721. (47) Iglesias, M.; Pineiro, M. M.; Marino, G.; Orge, B.; Tojo, J. Fluid Phase Equilib. 1999, 154, 123. (48) Zabaloy, M. S.; Brignole, E. A.; Vera, J. H. A. Fluid Phase Equilib. 1999, 245, 158.

Preferential Solvation in Ternary Solutions Containing

Recommend Documents