Ind. Eng. Chem. Res. 2002, 41, 4399-4408
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Thermophysical Behavior of Methylbenzoate + n-Alkanes Mixed Solvents. Application of Cubic Equations of State and Viscosity Models Begon ˜ a Garcı´a, Rafael Alcalde, Santiago Aparicio, and Jose´ M. Leal* Universidad de Burgos, Departamento de Quı´mica, Facultad de Ciencias, 09001 Burgos, Spain
As part of a study on the interactions between aromatic esters and n-alkanes, density, viscosity, and refractive index data were measured for methylbenzoate-n-alkane mixed solvents. From these data, excess molar volumes (V Em), mixing viscosities (∆mixη), excess Gibbs energies of activation of viscous flow (∆G/E m ), and mixing refractive indices (∆mixnD) were deduced. From the excess volumes, the partial molar volumes of the two components were deduced using the intercept method. The cubic equations of state (EOS) proposed by Soave-Redlich-Kwong and Peng-Robinson combined with two simple mixing rules were used to process the excess molar volumes. The change with temperature of the density and viscosity measurements enabled the activation enthalpy (∆Hqm), activation entropy (∆Sqm), Gibbs free energy (∆Gqm), and thermal expansion coefficients (R) to be determined. These results were interpreted in terms of intermolecular interactions and structural effects. The ability of different one-parameter and two-parameter empirical models to predict mixing viscosities was also tested. 1. Introduction The thermophysical properties of solvents have a strong bearing on their practical applications in many settings. Despite the importance of common esters as solvents, these systems have been little investigated up to now.1-3 In particular, aromatic esters are very useful compounds in polymer science and technology.4 Methylbenzoate is a dipolar, hydrophobic, nonprotogenic solvent used in a variety of applications. The presence of an easily polarizable π-electron system confers a great deal of promise on benzoates as selective solvents in the purification of reagents. Petrol, on the other hand, is a mixture of alkanes containing largely isooctane. Alkanes, the simplest class of organic compounds, are extremely unreactive species, because they contain no functional groups. In addition to interest from a fundamental standpoint, alkanes are of industrial importance in the hydrocarbon processing. The manner whereby solute and solvent molecules are associated with one another in a mixed solvent brings about a marked effect on the properties of the resulting liquid. Of importance in this respect is testing the ability of available theoretical models to correlate the viscosity of pure liquids and of binary mixtures and, from these results, predicting the viscosities of multicomponent systems over wide ranges of pressure and temperature. Reliable prediction methods can be valuable in operations such as separation, distillation, and liquid-liquid extraction. The molecular interaction energy governs the distribution of molecules, therefore, a great deal of effort has been devoted to understanding the interactions in mixed solvents. In this work, densities, viscosities, and refractive indices were measured at five different temperatures for the methylbenzoate-n-alkane (hexane, octane, decane, dodecane and tetradecane) mixed solvents. From these measurements, excess molar volumes * Corresponding author. E-mail:
[email protected].
(V Em), mixing viscosities (∆mixη), Gibbs free energies of activation of viscous flow (∆G/E m ), and mixing refractive indices (∆mixnD) were determined. From the excess volumes, the partial excess molar volumes were deduced for the two components using the intercept method, along with the partial molar volumes at infinite dilution. From the effect of temperature on the experimental measurements, the activation thermodynamic properties such as enthalpy ∆Hqm, entropy ∆Sqm, Gibbs free energy ∆Gqm, and thermal expansion coefficients R can be determined. Available prediction models were tested using the data measured for methylbenzoate-n-alkane solvents.5-8 To design theoretical models that enable the prediction of excess molar volumes, the cubic equations of state proposed by Soave-Redlich-Kwong9 (SRK) and Peng-Robinson (PR)10 were applied using two simple mixing rules. These equations require only a few experimental data and are easy to handle for engineering applications. The viscosity-mole fraction data pairs were used to test seven correlating viscosity models with fitting parameters and two prediction models without fitting parameters. 2. Experimental Section Reagents. The reactants methylbenzoate (MB), nhexane, n-octane, n-decane, n-dodecane, and n-tetradecane, of the highest purity available commercially, were used without further purification. The purity was assessed by GC with a Perkin-Elmer 990 gas chromatograph, equipped with a Hewlett-Packard 3390A integrator, and also by comparison of the densities, viscosities, and refractive indices with literature values (Table 1). 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 within ∆m ) 10-5 g with a Mettler AT 261 Delta Range balance, into suitably stoppered bottles. The
10.1021/ie020008c CCC: $22.00 © 2002 American Chemical Society Published on Web 07/12/2002
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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002
Table 1. Densities, G; Dynamic Viscosities, η; and Refractive Indices, nD, of the Pure Compounds: Measured Values in Comparison with Literature Values 278.15 K
288.15 K
298.15 K
purity (GC)
F × 10-3 (kg m-3)
η × 103 (Pa s)
F × 10-3 (kg m-3)
η × 103 (Pa s)
F × 10-3 (kg m-3)
η × 103 (Pa s)
MB
99.9
1.10275a
3.083a
1.09262a 1.09334b
2.304a 2.298b
1.823a 1.825d
n-hexane
99.5
0.67246a
0.400a
n-octane
99.8
0.71429a
0.670a
0.360a 0.3275b 0.326i 0.588a 0.591i
n-decane
99.4
0.74090a
1.185a
0.66587a 0.66345g 0.66402h 0.70818a 0.70657o 0.70651p 0.73508a 0.73357p
n-dodecane
99.5
0.75966a
2.064a
n-tetradecane
99.5
0.75396a 0.74802q 0.76806a 0.76625q
1.636a 1.637i 2.596a
1.08363a 1.08262c 1.0836d 0.65498a 0.65484b 0.65504j 0.69843a 0.69862b 0.69856j 0.72603a 0.72635b 0.72634j 0.74527a 0.74518b 0.75964a 0.75931j
compound
1.003a 0.990i
0.330a 0.2949i 0.302k 0.518a 0.5151b 0.5092i 0.849a 0.8614b 0.8498i 1.348a 1.3585i 2.031a 2.078i
308.15 K nD 1.51466a 1.51457b 1.5149d 1.37244a 1.37226b 1.39519a 1.39505b 1.40900a 1.40967b 1.41890a 1.41952b 1.42621a
318.15 K
F × 10-3 (kg m-3)
η × 103 (Pa s)
F × 10-3 (kg m-3)
η × 103 (Pa s)
1.07399a 1.0740d 1.07205e 0.64578a 0.6454k 0.6454l 0.69032a 0.6906l 0.69031p 0.71842a 0.7186l 0.71843p 0.73798a 0.7378l 0.75248a 0.75272q
1.504a 1.510d
1.06428a
1.270a 1.253f
0.309a 0.268i 0.274k 0.471a 0.454i 0.453k 0.743a 0.739i 0.720k 1.149a 1.149i 1.709a 1.713i
0.63701a 0.63484m 0.6351n 0.68260a 0.68862m 0.6885n 0.71138a
0.282a 0.245i
0.73118a
0.967a 0.987i 1.407a 1.438i
0.74586a
0.423a 0.408i 0.643a 0.650i
a Values obtained in this work. b Reference 15. c Reference 2. d Reference 16. e Reference 17. f Reference 18. g Reference 19. h Reference 20. i Reference 21. j Reference 22. k Reference 23. l Reference 24. m Reference 25. n Reference 26. o Reference 27. p Reference 28. q Reference 29.
mixtures were completely miscible over the whole composition range. Instruments and Procedures. The molar excess volumes were deduced from the densities, F, of the pure liquids and mixed solvents. Densities were measured with a computer-controlled DMA 58 Anton Paar digital density meter, which requires only a small sample (0.7 cm3). The density measurements were based on the conversion of the period of oscillation of the sample tube into density readings. The instrument is equipped with a solid-state thermostat ((0.01 K) and a menu program. Calibration was achieved with deionized doubly distilled water (Milli-Q, Millipore) and n-nonane (Fluka, 99.2% purity by GC) as reference liquids. Dynamic viscosities, η, were measured with an automated AMV 200 Anton Paar microviscometer calibrated with doubly distilled and deionized water. The viscosity values were based on the rolling ball measuring principle and were determined by measuring the shear stress of a steel ball introduced into an inclined samplefilled glass capillary placed inside a block thermostated by a Julabo F-25 bath. The stress was varied by changing the inclination angle of the capillary within the range 20-80°. The calibration constants were evaluated using the equation
k(R) ) ηstand/(Fball - Fstand)t
(1)
where k(R) is the calibration constant at each inclination angle; ηstand and Fstand represent the dynamic viscosity and density, respectively, of the standard liquid; Fball is the density of the ball; and t represents the rolling time ((0.01 s) of the ball.11-14 The instrument is also wellsuited even for non-Newtonian fluids. The refractive indices, nD, of the pure compounds and of the mixtures were measured using the sodium line of an automatic Leica AR600 refractometer, thermostated by a Julabo F32 MV bath. The prism was equipped with a fitting lid that produces tightness, and preserves the sample from preferential evaporation. The thermostatic time of the samples was small, and the temperature was read at the prism surface. The nD values were measured at 298.15 K only, because the effect of temperature on this property was negligible.
3. Results For decades, the thermodynamic properties of mixed solvents have attracted the interest of chemists. The thermodynamic properties of hydrocarbons are of considerable theoretical and practical interest in petrochemistry, organic chemistry, environmental protection, and separation techniques, among others. Excess thermodynamic properties reflect the nonideality of solvents when different liquids are mixed and are attributed to the different shapes and sizes of the components, to the reorientation of the molecules in the mixture, and to the molecular interactions. The excess molar volume is an important property for characterizing mixed solvents and designing handling facilities in chemical processes. The excess molar volumes of the mixtures studied were deduced from the measured densities according to
V Em ) x1M1(F-1 - F1-1) + x2M2(F-1 - F2-1)
(2)
where subscript 1 refers to n-alkane and subscript 2 to methylbenzoate. Mixing viscosities were evaluated from the measured dynamic viscosities using the equation
∆mixη ) η - (x1η1 + x2η2)
(3)
Likewise, mixing refractive indices ∆mixnD were evaluated using an equation similar to eq 3. The Gibbs free energy of activation for viscous flow ∆G/E m represents an interesting mixing property, because it involves density and viscosity data30
∆G/E m ) RT[ln(ηV) - x1 ln(η1V1) - x2 ln(η2V2)]
(4)
where V ) (x1M1 + x2M2)/F is the molar volume of the mixture and Vi is that of pure component i. RedlichKister-type polynomials31 were fitted to the excess and mixing properties, Y E r
E
Ai(2x1 - 1)i ∑ i)1
Y ) x1x2
(5)
where the proper number of Ai fitting coefficients was optimized by an F-test32 and determined by least
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4401
Figure 1. Excess molar volumes, V Em, vs x1 for the x1(n-alkane) + (1 - x1)MB mixtures at 298.15 K. (b) n-hexane, (9) n-octane, (() n-decane, (2) n-dodecane, and (f) n-tetradecane, this work; (- • -) n-heptane data obtained from ref 1.
Figure 3. Mixing refractive indices, ∆mixnD, vs x1 for the x1(nalkane) + (1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1.
with the (∂V E/∂xi)p,T partial derivatives being evaluated from eq 5. Substitution of the values
lim(∂V E/xi)p,T ) 0 xif0
and
limV E ) 0 xif0
into eq 7 enables one to obtain V h ∞i , the partial molar volumes at infinite dilution. Moreover, the partial excess molar volumes at infinite dilution, V h E,∞ i , for the two components were also evaluated using the equation
)V h ∞i - V V h E,∞ i
Figure 2. Mixing viscosities, ∆mixη, vs x1 for the x1(n-alkane) + (1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1.
squares. Figures 1-3 display the isothermal variations of the V Em, ∆mixη, and ∆mixnD values with the alkane chain length across the entire composition range. Partial molar quantities account for the rate of change with concentration of an extensive magnitude and can provide information on binary and higher-order interactions. From the excess molar volumes, the partial molar volumes, V h i, were determined for the two components using the intercept method, i.e.
(8)
and are available in the Supporting Information, along with the V h ∞i values. Plots of V h Ei vs x1 for the ith component are shown in Figure 4a for n-alkanes and in Figure 4b for methylbenzoate. The density and viscosity measurements allow one to evaluate the activation thermodynamic properties enthalpy ∆Hqm, entropy ∆Sqm, and Gibbs free energy ∆Gqm. The activation properties at different compositions can be deduced from the variation of the kinematic Stokes viscosity (ν ) η/F) versus the reciprocal temperature according to
ln(νM) ) ln(Nh) -
∆Sqm ∆Hqm + R RT
(9)
Vi ) Vi + V h Ei
(6)
where h, R, and N are universal constants. Figure 5 shows the effect of composition on ∆Gqm for the different binary mixtures. Finally, the thermal expansion coefficients, R, for each mixture were evaluated from the experimental densities at different temperatures according to
V h Ei ) V E + xj(∂V E/∂xi)p,T
(7)
R ) -F-1(∂F/∂T)p
(10)
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Figure 5. Free molar energies of activation, ∆Gqm, vs x1 for the x1(n-alkane) + (1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1.
Figure 4. Partial excess molar volumes of (a) n-alkanes, V h E1 , and (b) MB, V h E2 , vs x1 for the x1(n-alkane) + (1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1.
The R values deduced for pure components at the working temperatures are listed in Table 2. Figure 6 shows the effect of composition on the thermal expansion coefficients. 4. Discussion The V Em values were negative for the n-hexane mixtures and became positive at 2:1 alkane/methylbenzoate ratios and higher. This behavior arises from the expansion caused by the fewer holes available in the alkane framework and points to a size effect, the contraction in volume for n-hexane being ascribed to the formation of small heteroaggregates. The smaller size of n-hexane facilitates an easier accommodation into the methylbenzoate structure: n-tetradecane reached a maximum,
Figure 6. Thermal expansion coefficients, R, vs x1 for the x1(nalkane) + (1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1. Table 2. Thermal Expansion Coefficients, r, of Pure n-Alkanes and MB R × 103 (K-1) compound MB n-hexane n-octane n-decane n-dodecane n-tetradecane
278.15 K 288.15 K 298.15 K 308.15 K 318.15 K 0.867 1.353 1.132 1.022 0.940
0.875 1.366 1.142 1.030 0.947 0.897
0.882 1.389 1.157 1.043 0.958 0.907
0.890 1.409 1.171 1.054 0.968 0.916
0.898 1.428 1.184 1.064 0.977 0.924
and n-octane displayed an S-shaped curve with V Em ) 0 at x1 ) 0.6 (Figure 1). The ∆mixη values were negative, with a minimum for n-hexane, and decreased with increasing chain length
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4403
(Figure 2). This effect was ascribed to the dominance of dispersion forces compared to the induced dipole-dipole interactions and reflects the decreased ability of shorter alkanes to form heteroaggregates.33 This behavior is supported by the ability to form dipole-dipole complexes, which increases with increasing polarizability and chain length (Gaussian polarizabilities RS × 1024/ cm3 mol-1: n-heptane, 13.69; n-octane, 15.5; and ndecane, 20.0).34 Only n-hexane gave positive mixing refractive indices, ∆mixnD, across the entire composition range (Figure 3), which reveals a controversial behavior. Alkanes have close to null dipole moments, except hexane (µ298.15 ) 0.085 D), whereas methylbenzoate has a larger one (µ298.15 )1.94 D).15 Hence, the n-hexane-methylbenzoate mixture (V Em < 0) exhibited stronger interactions. The ∆G/E m values were negative for all mixtures across the composition range. The minima for hexane, octane, and decane at x1 ) 0.5 suggest the dominance of dispersion forces. The H Em > 0 data reported for several alkylbenzoate-n-alkane mixtures support an endothermic mixing process resulting from the destruction of order in the pure components;4,5,7,8 if the energy required to disrupt the orientational order in pure alkanes prevails, then an increase in chain length should result in endothermic mixing. These enthalpies increase for higher alkanes. Although an increase in chain length results in larger polarizability (negative contribution to H Em), the repulsive forces (positive contribution to H Em) also increase, with a net positive increase in the excess enthalpy. The partial excess molar volumes, V h Ei (Figure 4a and b) reveal strong interactions at very low concentrations, in the vicinity of the ideal solution limits. Adherence of the V h i/x1 data pairs to Gibbs-Duhem eq 11 showed the partial molar volumes deduced to be consistent within a reasonable 5% difference between the right- and the left-hand sides.
x1(∂V h 1/∂x1)T ) -x2(∂V h 2/∂x1)T
(11)
The sharp V h Ei minima reported for some amidewater mixtures was ascribed to a strong solvation effect;13 for alkane-methylbenzoate mixtures, howvalues ever, such minima are lacking. The alkane V h E,∞ i changed from negative to positive with increasing chain length, indicating a poor interstitial accommodation. This effect is consistent with trapping of lower alkanes in the methylbenzoate network, which yields a smaller volume. In particular, n-hexane led to V h E,∞ MB ) -1.55 3 -1 cm mol , suggesting the formation of weak dipoledipole hexane-methylbenzoate aggregates. The expansion effect arises from three main contributions: (a) The dipole-dipole benzoates structures (µ ) 1.94 D)15 become destroyed upon mixing (dispersion process), giving rise to endothermic contributions,4 because the longer the chain length, the shorter the range of orientational order in pure alkanes.35 (b) The longer the chain length (larger polarizability), the stronger the dipole-dipole interactions. (c) A size-effect appears with increasing chain length. The effect of temperature on the mixing viscosity ∆mixη can be justified in a straightforward manner; an increase in temperature caused a decrease in ∆mixη, with the lowest value corresponding to hexane and the
highest to tetradecane. The thermodynamic activation entropy, ∆Sqm, and enthalpy, ∆Hqm, were deduced according to eq 9. The ∆Hqm values were always positive and decreased with the alkane concentration. The activation entropies were negative and increased with the n-alkane concentration.† The effect of composition on the Gibbs free energies of activation ∆Gqm is shown in Figure 5. The ∆Gqm values exhibited a profile similar to that of the mixing volumes. This permits the conclusion that the formation of heteroaggregates is unlikely. The large ∆Hqm values point to the dominance of dispersion forces and a greater ordering compared to pure liquids. The ∆Sqm values indicate higher ordering for shorter alkanes. Thermal expansion coefficients, R, can be evaluated for pure components and for the mixtures from the effect of temperature on the density measurements according to eq 10 (Table 2). The R values considerably increased with decreasing alkane chain length and varied only very slightly with temperature (Figure 6). At high alkane concentration tetradecane had (roughly) the same R as methylbenzoate (0.9), whereas hexane had a value of 1.4, consistent with the enhanced local density of solvent next to the solute and values derived (see Table III of the the negative V h E,∞ i Supporting Information).† These results reinforce the argument that hydrophobicity implies an unfavorable free energy, leading to the association of the particles.36 Application of Cubic Equations of State. The density of a liquid reflects the heaviness of the molecules in a unit volume. Densities and molar excess volumes are significant quantities for the design of processes and the interpretation of interactions in mixed solvents. The interest on cubic EOS for the correlation and prediction of phase equilibria in mixed solvents has increased in the past few years. Cubic EOS have proved to be useful in the petroleum industry for the prediction of fluidphase thermophysical properties. Different mixing rules have been introduced that allow EOS to describe highly nonideal mixed solvents. In this connection, a recent approach has been proposed that accounts for the effect of density on excess properties of nonideal mixed solvents, while giving zero interaction parameters for the ideal solution limit.37 Although no cubic equation of state can be regarded as the most accurate in predicting all thermophysical solvent properties, because of their simplicity, accuracy, and predictive capability, the Soave-Redlich-Kwong (SRK)9 and PengRobinson (PR)10 cubic EOS were used to predict V Em and V h E,∞ values of the mixtures studied here. These i equations have the advantage of requiring only a little experimental information and were applied using two quite simple mixing rules. The general expression for these cubic equation is38
P)
n2a nRT V - nb (V + δ1nb)(V + δ2nb)
(12)
where n is the number of moles in the mixture, with δ1 ) 1 and δ2 ) 0 for SRK and δ1 ) 1 + x2 and δ2 ) 1 x2 for PR. It would be desirable to find expressions for a and b such that eq 12 yields a mixture volume that matches eq 2 at any concentration, pressure, and temperature. The mixing rules used to calculate the energetic (a) and covolume (b) parameters for the mixture are
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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 r
a)
r
b)
r
∑ ∑xixj(1 - kij)(aiaj)0.5 i)1 j)1
(13)
r
xixj(1 - mij)(bibj)0.5 ∑ ∑ i)1 j)1
(14)
where r represents the number of components. Previously, the classical van der Waals mixing rules for the copressure, along with the nonclassical mixing rules for the covolume factor, were used successfully by different authors in highly nonideal systems, leading to accurate correlations and low deviations in wide intervals of pressure and temperature.39 Hence, these simple rules were chosen instead of others with a deeper theoretical basis, because the complexity of the other rules is not accompanied by better correlations. For the first mixing rule (R1), the kij parameters were fitted by introducing the experimental V Em data, yielding mij ) 0. For the second mixing rule (R2), the mij and kij parameters were also fitted. The fitting parameters were derived using the Marquardt algorithm combined with the NewtonRaphson method for the equation of state. More details about the fitting procedure are provided elsewhere.40 The PR and SRK equations gave accurate V Em and E,∞ V h i results using the R1 and R2 mixing rules. The results deduced by the application of the two equations and mixing rules upon correlation of the experimental V Em data pairs at 298.15 K (Table 3) gave very low deviations for all mixtures. If the same mixing rules are used, then the PR and SRK equations lead to very similar results; the two-parameter mixing rule normally yields better correlations. The V h E,∞ values deduced i with both equations and mixing rules showed low deviations from the experimental data, regardless of the working temperature (Figure 7). The fitting parameters deduced for the mixed solvents studied can be applied Table 3. Values of the Fitting Parameters, K12 and m12, and Standard Deviations, σ, for the SRK and PR Equations of State Applied to the Correlation of Excess 3 -1 Molar Volume, V E m (cm mol ), for the x1(n-alkane) + (1 x1)MB Mixtures at 298.15 K with Two Different Mixing Rules, R1 and R2 SRK R1
PR R2
R1
R2
K12 m12 σ
x1(n-hexane) + (1 - x1)MB 0.0944 0.0805 0.0842 -0.0040 0.0379 0.0189 0.0177
0.0849 -0.0017 0.0176
K12 m12 σ
x1(n-octane) + (1 - x1)MB 0.0567 0.0745 0.0543 -0.0018 0.0217 0.0061 0.0312
0.0842 -0.0003 0.0062
K12 m12 σ
x1(n-decane) + (1 - x1)MB 0.0204 0.1036 0.0230 -0.0130 0.0436 0.0131 0.0452
0.1186 -0.0118 0.0150
K12 m12 σ
x1(n-dodecane) + (1 - x1)MB -0.0196 -0.1704 -0.0135 -0.0667 0.0572 0.0506 0.0569
-0.1605 -0.0660 0.0520
K12 m12 σ
x1(n-tetradecane) + (1 - x1)MB -0.0848 -0.1891 -0.0765 -0.1194 0.0378 0.0072 0.0391
-0.2040 -0.1211 0.0080
Figure 7. Partial excess molar volumes at infinite dilution of (b) h E,∞ n-alkane, V h E,∞ 1 , and (9) MB, V 2 , vs number of carbon atoms, n, in the x1(n-alkane) + (1 - x1)MB mixtures at 298.15 K. Comparison with values obtained from the cubic equations of state by (a) SRK and (b) PR using mixing rule R1 for (O) n-alkane and (0) MB and mixing rule R2 for (x) n-alkane and (!) MB.
in further works for the prediction of volumetric properties in mixtures involving n-alkanes and methylbenzoate. Viscosity Mixture Models. Viscosity is an important bulk property that provides a measure of the internal friction of a fluid and is closely related to selfassociation in liquids. The force required to make a fluid layer move in relation to another layer is called shear, and a Newtonian fluid is one with a constant ratio of the shear stress to the shear rate. Thus, viscosity data are needed in chemical engineering for the evaluation of mass-transport phenomena, and such data provide a valuable insight into the structure and interactions in mixed solvents. To calculate and predict viscosity values of liquid mixtures, the ability of various one-parameter
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4405 Table 4. Values of the Fitting Parameters; Standard Deviations, σ; and Percentage Errors, % E, for the Different Models Tested for the Correlation of Dynamic Viscosity, η (mPa s), for the x1(n-alkane) + (1 - x1)MB Mixtures at 298.15 K Nissan-Grunberg n-hexane + MB n-octane + MB n-decane + MB n-dodecane + MB n-tetradecane + MB
Hind
σ
%E
η12
σ
%E
γ12
γ21
σ
%E
-0.633 -0.783 -0.833 -0.748 -0.521
0.006 0.003 0.008 0.015 0.021
0.685 0.230 0.520 0.926 1.017
0.186 0.410 0.668 1.000 1.449
0.052 0.044 0.035 0.028 0.018
6.940 4.194 2.697 1.814 0.916
-0.689 -0.748 -0.718 -0.541 -0.231
-0.096 0.010 0.058 0.096 0.101
0.002 0.003 0.006 0.006 0.005
0.205 0.234 0.364 0.303 0.216
Wu (A ) 2.45)
Lobe n-hexane + MB n-octane + MB n-decane + MB n-dodecane + MB n-tetradecane + MB
Heric
d
R12
R21
σ
%E
σ
%E
ν12
ν21
σ
%E
0.362 -0.097 -2.161 18.048 3.551
-1.034 -0.749 -0.076 -5.661 -1.619
0.002 0.004 0.008 0.008 0.007
0.158 0.341 0.491 0.435 0.342
0.010 0.010 0.008 0.066 0.192
0.999 0.926 0.555 4.023 9.291
0.565 0.759 1.059 1.514 2.160
0.904 0.991 1.150 1.386 1.729
0.002 0.003 0.006 0.006 0.005
0.212 0.242 0.360 0.304 0.215
Cao (UNIMOD) n-hexane + MB n-octane + MB n-decane + MB n-dodecane + MB n-tetradecane + MB
McAllister
Auslander
GC -UNIMOD
U21 - U11
U12 - U22
σ
%E
B12
A21
B21
σ
%E
σ
%E
-127.505 -145.593 -168.562 -188.390 -96.772
-142.977 -214.534 -215.208 -165.538 -99.502
0.073 0.096 0.120 0.106 0.059
7.897 8.901 8.874 5.974 2.731
0.547 1.777 3.870 3.869 -3.016
0.203 0.605 1.161 0.786 0.540
1.110 0.271 -0.120 -1.423 8.968
0.001 0.001 0.001 0.002 0.004
0.120 0.105 0.105 0.513 0.170
0.071 0.086 0.091 0.103 0.148
4.933 3.905 4.496 5.516 7.191
and two-parameter empirical models available for the prediction of mixing viscosities was tested using the viscosity measurements of the alkane-methylbenzoate mixtures. It should be noted that predictive models do not involve fitting parameters; rather, they use only properties of the pure compounds to predict mixture viscosities. In constrast, correlation models require experimental viscosity data of the mixtures to process the interaction parameters; hence, they need somewhat more detailed experimental information. Therefore, predictive models turn out to be of practical use, even though their results are less accurate. The viscosity-mole fraction data pairs of these mixtures were used to test the empirical one-parameter models proposed by Nissan and Grunberg41 and Hind, McLaughlin, and Ubbelohde;42 the two-parameter models by Heric,43 Lobe,44 McAllister45 (involving three-body interactions), and Cao (UNIMOD);46 and the threeparameter model of Auslander.47 The predictive ability of the GC-UNIMOD48 and Wu49 models was also tested. The measured viscosities and the calculated mixing viscosities were analyzed using the equations discussed below. The Nissan-Grunberg model is based on the Arrhenius equation for the dynamic viscosity of a mixed solvent
ln(η) ) x1 ln(η1) + x2 ln(η2) + x1x2d
(15)
where the parameter d, which is independent of the mixture composition and the characteristics of each system, can be either positive or negative and represents a measure of the intermolecular interactions between unlike molecules. The d values were negative for the mixtures investigated (Table 4), reflecting positive deviations from Raoult’s law. The Hind model provides a useful description of mixed solvents. The η12 parameter involved by analogy with second virial coefficients is independent of the composition, and can be attributed to unlike-pair interactions
η ) x12η1 + x22η2 + 2x1x2η12
(16)
The predictive Heric equation for kinematic viscosities
ln(ν) ) x1 ln(ν1) + x2 ln(ν2) + x1 ln(M1) + x2 ln(M2) ln(x1M1 + x2M2) + x1x2[γ12 + γ21(x1 - x2)] (17) was fitted to the (ν, x1) data pairs, with the γ12 and γ21 parameters determined by least squares. The Lobe equation involves the volume fractions of the components, f1 and f2, and was also tested by fitting the parameters R12 and R21 to the (ν, x1) data pairs
[
ν ) f1ν1 exp f2R12 ln
( )] ν2 ν1
( )]
[
+ f2ν2 exp f1R21 ln
ν2 ν1
(18)
The McAllister model, based on Eyring’s absolute reaction rates theory, involves three-body interactions and leads to the cubic semiemprirical eq 19 for kinematic viscosities
ln(ν) ) x13 ln(ν1) + x23 ln(ν2) + 3x12x2 ln(ν12) + M2 + 3x22x1 ln(ν21) - ln x1 + x2 M1 M2 2 1 2M2 + 3x22x1 ln + + 3x12x2 ln + 3 3M1 3 3M1 M2 x23 ln (19) M1
(
)
(
(
)
)
( )
where ν12 and ν21 represent the interaction parameters between unlike molecules. Equation 19 provides reliable results if the r2/r1 ratio of radii of the components is 1.5 or less. The Cao model, based on Eyring’s absolute reaction rates theory, statistical thermodynamics, and the corresponding-states principle, involves the concept of a local composition parameter and predicts viscosities as a function of temperature and composition
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ln(νM) )
∑ i)1
r
∑ i)1
φi ln(νiMi) + 2
φi ln
-
φi n iqi
r
∑ i)1
() xi
φ1
r
θij ln(τij) ∑ j)1
ri
(20)
where the qi parameter is evaluated from the surface area of the UNIFAC group50 and the local composition parameter θij from the partition function of the mixture. The τij interaction parameter
(
τij ) exp -
)
z Uji - Uii 2 RT
(21)
involves the Uij potential interaction energy between sites i and j and can be deduced from the viscosities of pure components. The semiempirical Auslander model was initially applied to mixtures in which the components do not have widely differing viscosities, such as oil and hydrocarbon mixtures; it involves a less complex threeparameter equation and gives very low deviations
η)
η1x1(x1 + B12x2) + η2[A21x2(B21x1 + x2)] x1(x1 + B12x2) + (A21x2)(B21x1 + x2)
(22)
The fitting parameters were deduced from the experimental data by nonlinear least squares. The group parameters needed to apply these models were obtained from the UNIFAC model by Fredenslund.50 These models were tested with fairly good agreement for the solvent studied. In particular, the low deviations yielded by the Nissan-Grunberg model are remarkable; however, the Hind and Cao (UNIMOD) models gave somewhat higher deviations. The GC-UNIMOD and Wu predictive models led to very close results, with the percentage of deviation being lower than 20% and even lower than 10% for the heptane, octane, and decane mixtures. Table 4 summarizes the results obtained. The correlating models gave better results, but they only enable the fitting of experimental viscosity data. The predictive models have the advantage of requiring only very few measured data. Figure 8 shows the very good agreement between the kinematic viscosities measured at 298.15 K and those calculated with the Wu model. 5. Concluding Remarks Aromatic esters manifest a specific behavior due to self-aggregation. The structural changes and the appearance of dispersion forces and further heteroassociations by mixing with alkanes makes these systems particularly interesting. Mixing of alkanes with methylbenzoate causes the benzoate structure to be broken to an extent that depends on chain length. The excess Gibbs energy of viscous flow, thermal expansion coefficients, and activation properties point to a complex mixture behavior, with a balance between interactions of opposite sign and a slight dominance of dispersion forces. For shorter alkanes, accommodation of the two components in the network of the cosolvent is favored by increasing temperature. The cubic SRK and PR cubic EOS involving two mixing rules provided good predictions of the volumetric properties of these systems. The interaction parameters deduced can be used further to predict these properties in multicomponent mixtures.
Figure 8. Experimental dynamic viscosities, η, vs x1 for the x1(n-alkane) +(1 - x1)MB mixtures at 298.15 K. Symbols as in Figure 1. (- - -) Values predicted by the Wu model with A ) 2.45 for the different mixtures.
The viscosity models tested show a good ability to fit the experimental data, and the predictive GC-UNIMOD and Wu models gave reasonably good results considering that they have no fitting parameters. Nomenclature F ) density η ) dynamic viscosity ν ) kinematic viscosity nD ) refractive index Mi ) molar mass of the ith component M ) molar mass of the mixture xi ) mole fraction of the ith component fi ) volume fraction of the ith component V ) molar volume V Em ) molar excess volume V h i ) partial molar volume of the ith component V h Ei ) partial molar excess volume of the ith component V h ∞i ) partial molar volume at infinite dilution of the ith component V h E,∞ ) partial excess molar volume at infinite dilution of i the ith component H Em ) excess molar enthalpy ∆mixη ) mixing viscosity ∆mixnD ) mixing refractive index ∆G/E m ) activation of viscous flow free energy ∆Hqm ) activation enthalpy ∆Sqm ) activation entropy ∆Gqm ) activation Gibbs free energy R ) thermal expansion coefficient µ ) dipole moment RS ) Gaussian polarizability Ai ) coefficients in eq 5 P ) pressure T ) absolute temperature R ) universal gas constant a ) mixture copressure parameter in eq 12 b ) mixture covolume parameter in eq 12 ai ) copressure parameter of the ith component
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4407 bi ) energetic parameter of the ith component δ1 ) parameter in eq 12 δ2 ) parameter in eq 12 kij ) parameter in eq 13 mij ) parameter in eq 13 d ) Nissan-Grunberg parameter in eq 15 η12 ) Hind parameter in eq 16 γ12, γ21 ) Heric parameters in eq 17 R12, R21 ) Lobe parameters in eq 18 ν12, ν21 ) McAllister parameter in eq 19 φi ) average area fraction in the Cao model, eq 20 ni ) constants of the Cao model in eq 20 qi, ri ) van der Waals properties of compound i in eq 20 θij ) local composition parameter in eq 20 Uji, Uii ) Cao interaction potential energy parameters, eq 21 B12, B21, A21 ) Auslander parameters in eq 22 R1 ) mixing rule 1 R2 ) mixing rule 2 Errors xi ) (10-5 F/(g cm-3) ) (10-5 nD ) (10-5 V Em/(cm3 mol-1) ) (10-4 ∆mixη/(mPa s) ) (10-3 -1 ∆G/E m /(J mol ) ) (0.04 ∆nD ) ((4 × 10-5) V h i/(cm3 mol-1) ) (0.01 V h Ei /(cm3 mol-1) ) (10-3 R/K-1 ) (10-6 ∆Hqm/(J mol-1) (0.1 ∆Sqm/(J mol-1 K-1) (0.01 ∆Gqm/(J mol-1) (0.1
Acknowledgment The financial support of Junta de Castilla y Leo´n, Spain, Project BU05/99, is gratefully acknowledged. Supporting Information Available: The measured densities, viscosities and refractive indices (Table I), Redlich-Kister fitting coefficients (Table II), partial molar volumes at inifinite dilution (Table III), activation properties for all mixtures and temperatures (Table IV) and free molar Gibbs energies of activation of viscous flow (Figure I) are provided in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Grolier, J. P. E.; Ballet, D.; Viallard, A. Thermodynamics of ester-containing mixtures. Excess enthalpies and excess volumes for alkyl acetates and alkyl benzoates + alkanes, + benzene, + toluene, and + ethylbenzene. J. Chem. Thermodyn. 1974, 6, 895. (2) Dusart, O.; Piekarski, C.; Piekarski, S.; Viallard, A. Excess volumes of binary mixtures n-heptane + esters and n-heptane + ketones in homologous series. J. Chim. Phys. 1976, 73, 837. (3) Jayaprakash, P. R.; Rambabu, K.; Ramachandran, D.; Nagaswara, M.; Rambabu, C. Densities, adiabatic compresibility, freelength, viscosity and excess volumes of dimethyl sulfoxide with some aromatic esters at 303.15-318.15 K. Phys. Chem. Liq. 1997, 34, 51. (4) Ortega, J.; Postigo, M. Experimental values and analysis of mixing heats of the binary mixtures formed by alkyl benzoates and n-alkanes. Fluid Phase Equilib. 1995, 108, 121. (5) Garcı´a, B.; Ortega, J. C. Excess viscosity, excess volume, and excess free energy of activation at 283, 293, 303, 313, and 323 K for mixtures of acetonitrile and alkyl benzoates. J. Chem. Eng. Data 1988, 33, 200.
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Received for review January 3, 2002 Revised manuscript received May 29, 2002 Accepted June 4, 2002 IE020008C
