Advances in the Correlation of Thermodynamic Properties of Binary

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Ind. Eng. Chem. Res. 2010, 49, 9548–9558

Advances in the Correlation of Thermodynamic Properties of Binary Systems Applied to Methanol Mixtures with Butyl Esters Fernando Espiau, Juan Ortega,* and Eduvigis Penco Laboratorio de Termodina´mica y Fisicoquı´mica de Fluidos, Parque Cientı´fico-Tecnolo´gico, UniVersidad de Las Palmas de Gran Canaria, Canary Islands, Spain

Jaime Wisniak Deparment of Chemical Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa 84105, Israel

This work analyzes the utility of a new model to correlate thermodynamic properties of solutions, the foundations of which have been published in a previous study. The model is applied to a set of experimental data for several properties of binary systems of methanol with four butyl alkanoates (vapor-liquid equilibria at p ) 141.32 kPa and excess enthalpies and volumes at 298.15 and 318.15 K). Vapor-liquid equilibrium data (VLE) indicate that the four binary systems deviate positively from Raoult’s law and do not present azeotrope. Excess enthalpies (hE) are positive for the entire range of compositions and decrease regularly with increasing length of the ester chain, with (∂hE/∂T)p,x > 0. The excess volumes (VE) decrease regularly with the length of the acid chain; they are positive for the binary systems of methanol with butyl (methanoate, ethanoate, and propanoate) and become negative for the system with butyl butanoate, with (∂VE/∂T)p,x > 0. The new model can be used to obtain a satisfactory correlation for Gibbs function gE ) gE(p, T, xi), and for its derivatives. Correlation procedures for the data are described for the stages (x, hE) f [x, gE(T)] for the isobaric data reported here and (x, VE) f [x, gE(p)] for isothermal data reported in the literature. The new method allows a better correlation than the one obtained with the classical models of Wilson, NRTL, and UNIQUAC. We also present a unique correlation of all the properties of the methanol + butyl ethanoate system in the form of an analytical expression ξ(p, T, x, y) ) 0 and conclude that, on the whole, its implementation can be considered an advance in the data treatment of the properties of liquid solutions. 1. Introduction For several years our research team has studied the behavior of ester systems with a second component (alkanols1-3 and alkanes4,5) through their properties in solution. Recently, the former set of compounds has attracted special interest because esters (especially methyl esters) and the first alkanols (especially methanol) constitute the basis of biodiesel, used as alternative fuels.6,7 Our research team has been especially focusing on the study of vapor-liquid equilibria (VLE) under isobaric conditions in systems comprised of methanol and methyl8 and ethyl9 alkanoates. In this work, we extend the study of properties for these systems and report experimental data on the vapor-liquid equilibrium at 141.32 kPa, excess enthalpies (hE) and excess volumes (VE) at 298.15 and 318.15 K, for the binary systems of methanol with the first four alkanoates (methanoate to butanoate), to complete the data published previously.10,11 VLE data for the system methanol + butyl ethanoate have been reported by Beregovykh et al.,12 Patlasov et al.,13 and Resa et al.14 at 101.32 kPa and by Sieg et al.15 at T ) 296.15, 313.15, and 333.15 K. The latter also reported hE data a T ) 293.15 K for the same system. Lo´pez et al.16 have measured excess enthalpies for the system methanol + butyl methanoate at 298.15, and one of us (J.O.)10 has reported excess volumes for the system methanol + butyl ethanoate at 298.15 K. The purpose of this work is to use all the above information to test the utility of a new model for the excess Gibbs function (gE), presented in a previous publication,17 to describe the * To whom correspondence should be addressed. E-mail: jortega@ dip.ulpgc.es.

correlation of several thermodynamic properties of a system at different pressures and temperatures. A successful application of a particular form of the model, applied to isobaric VLE data and excess enthalpies, has been reported previously.4 In the present study, we present an application of the model to the case where isobaric and isothermal data are available simultaneously, and its extension to the case where excess enthalpy and excess volume data are available at two temperatures. We then compare the results obtained with those predicted by other classical models such as Wilson,18 NRTL,19 and UNIQUAC.20 Our model presents an important difference with the latter, in that Gibbs function is dependent on the composition (through the so-called actiVe fraction), pressure, and temperature, in other words gE ) gE[p, T, z(xi)]. From this expression, it is clear that other derived thermodynamic relationships can be used to represent different properties, as can be understood from any textbook on chemical thermodynamics.21 We have previously shown, from a mathematical viewpoint, that this model is sufficiently flexible to be adapted to different situations.17 Specifically, we are going to consider three different applications for the mixtures studied here. Two of these entail the following steps, depending on whether the VLE data are isobaric, [(x, hE) f (x, gE(T)], or isothermic (x, VE) f [x, gE(p)]. For each of these cases, the quantities can be simultaneously treated by [(x, hE(T), gE(T)] for the isobaric case and [(x, VE(p), gE(p)] for the isothermic case. Finally, a global correlation of properties can be carried out, which uses experimental data of all the properties of a study mixture, hence determining the values of the parameters in a single step.

10.1021/ie101165r  2010 American Chemical Society Published on Web 08/25/2010

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

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Table 1. Physical Properties of Pure Compounds and Comparison with Literature Values F/(kg · m-3)

o Tb,i /K

compound

purity %

exp

lit

methanol

99.8

337.42

337.85a 337.687b

butyl methanoate

97

379.23

379.30a 379.25b,c

>99

399.23

399.211a 399.15b,c

butyl propanoate

99

418.14

butyl butanoate

>99

438.09

butyl ethanoate

a

b

c

nD

exp

lit

exp

lit

298.15f

786.79

1.3266

318.15f 298.15f

767.40 887.64

1.3176 1.3870

1.32652a,b 1.3265c 1.3189d 1.3874a,b,c

318.15f 298.15f

867.74 875.67

1.3773 1.3919

1.3918a,b,c

419.75b,c

318.15f 298.15f

855.42 871.07

1.3821 1.3987

1.3823e 1.4000b,c

438.15b,c

318.15f 298.15f

851.33 864.73

1.3891 1.4040

1.4029b,c

318.15f

846.13

786.64a,b 787.45c 767.43b 886.90a,b 886.89c 866.30b 876.60b 876.06c 856.50b 871.40b 871.88c 852.00b 866.40b 865.75c 849.30b

d

T/K

1.3950

e

Reference 24. Reference 25. Reference 26. Reference 27. Reference 28.

2. Experimental Section E

2.1. Apparatus and Procedures. Excess enthalpies h were measured at (298.15 ( 0.01) and at (318.15 ( 0.01) K in a Calvet microcalorimeter, model MS80D, (SETARAM, Lyon, France). The calorimeter was electrically previously calibrated with a Joule effect, being the uncertainty of experimental results lower than (10 J · mol-1 for hE. This estimation was made previously comparing the measurements obtained for standard mixtures such as indicated in a previous paper.22 The composition of each point presented an uncertainty of (5 × 10-4. Values of VE, over the entire range of compositions for the four binary mixtures were estimated from the densities of the pure substances and the mixtures at temperatures of (298.15 ( 0.01, 318.15 ( 0.01) K. This was done using an Anton Paar digital densimeter DMA-55, with a precision of (0.02 kg · m-3, calibrated according to the procedure described in a previous paper.22 Mean errors in the calculations of the molar fractions of methanol were (10-4 and (2 × 10-9 m3 · mol-1 for the VE. VLE measurements were made using the previously described apparatus and procedure.1,23 Control of the pressure, maintained constant at (141.32 ( 0.02) kPa, was done with a regulator/ calibrator equipment manufactured by Desgranges et Huot, model PPC2. The temperature of both phases in equilibrium was measured with a digital thermometer from Comarks Electronics Ltd., model 6800, calibrated regularly according to ITS-90, and presenting an uncertainty of around (20 mK. Compositions at equilibrium were calculated by applying a quadratic equation of the form: F ) [(F1 - F2)x + F2] + [x(1 - x)(a + bx + cx )] 2

where F, F1, and F2 are the density of the mixture in the sample and the densities of its two pure components, respectively. Constants a, b, and c were calculated previously in an inverse procedure, in other words, using density values of synthetic mixtures of known composition. This indirect procedure can be used to estimate the compositions of the phases with an uncertainty of (2 × 10-3 units in the molar fraction of methanol. Refractive indices of the pure compounds were measured with an Abbe refractometer 320 by Zuzi, with a reading error of (0.0002 units, maintaining the working temperature constant at (298.15 ( 0.01)K with a Heto Birkerod external circulation water bath. 2.2. Materials. Methanol, butyl methanoate, and butyl propanoate were purchased from Fluka, and butyl ethanoate and

butyl butanoate were from Aldrich, the commercial purity of which are shown on the respective Web sites. All were degasified by ultrasound for several hours and then passed through a molecular sieve (Fluka, 0.3 nm) to remove traces of moisture. Then, the most relevant physical properties were o , and determined: boiling point at a pressure of 101.32 kPa, Tb,i density, F, and the refractive index nD, at T ) 298.15 K and at atmospheric pressure. The experimental results recorded in Table 1 show an excellent agreement with data reported in the literature. 3. Correlation Model for Gibbs Excess Function, gE(x, p, T) In a previous work,17 we have presented the development and application to an extensive experimental database of a parametric mathematical model to correlate values obtained for Gibbs function and other properties of multicomponent systems. The expression formulated for a binary system is 2

gE(x1, p, T) ) z1(1 - z1)

∑ g (p, T)z i

i 1

(1)

i)0

where the variation with composition is expressed through the so-called actiVe fraction, zi, by the relationship, z1(x1) )

x1 [x1 + k(1 - x1)]

(2)

for one of the compounds of the solution. The significance of parameter k(p, T) and its influence in the correlation processes was analyzed in the previous work, and several procedures described for its determination.17 An analytical expression was introduced for the gi coefficients of eq 1, which was related to pressure and temperature by the expression: gi(p, T) ) gi1 + gi2p2 + gi3pT +

gi4 + gi5T2 T

(3)

In this way, the general model, given by eq 1, can be adapted for the correlation/estimation of other properties for which a thermodynamic relationship is known. For the case of the excess properties reported in this work, their thermodynamic relationships with the excess Gibbs function are hE ) gE - T

[ ] ∂gE ∂T

x,p

VE )

[ ] ∂gE ∂p

x,T

(4)

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Figure 1. (a) Experimental excess molar enthalpies hE, for binaries CH3OH(1) + H2u-1Cu-1COOC4H9 (2) at T ) 298.15 K and comparison with those from the literature: (•) u ) 1, (2) u ) 2; Ortega11 (1) u ) 3, (9) u ) 4; (O) Lo´pez and co-workers16 u ) 1; (∆) Sieg and co-workers15 for u ) 2 at T ) 293.15 K. The curves correspond to eq 5, and coefficients of Tables 2 and 3 for (s) u ) 1, (s s s) u ) 2, (- - -) u ) 3, and ( · · · ) u ) 4. The inset shows the variation of equimolar hE at T ) 298.15 K of the binaries: (() CnH2n+1OH (n ) 3-10) + butyl methanote,16 (•) methanol + butyl methanote, this work, and (O) methanol + butyl methanote.16 (b) Experimental hE at T ) 318.15 K for binaries CH3OH (1) + H2u-1Cu-1COOC4H9 (2). The indications for points and curves are the same used in Figure 1a. Table 2. Coefficients, gij, and Standard Deviations, s, Obtained Using Equation 6 to Correlate hE at T ) (298.15, 318.15) K and p ) 101.32 kPa binary mixture coefficients

methanol (1) + butyl methanoate (2)

methanol (1) + butyl ethanoate (2)

methanol (1) + butyl propanoate (2)

methanol (1) + butyl butanoate (2)

g01 + g02(p/Pa)2 g04 g05 g11 + g12(p/Pa)2 g14 g15 g21 + g22(p/Pa)2 g24 g25 kh s(hE)/(J · mol-1)

-5667 5293 × 102 -0.084 125 7295 × 102 0.089 -2271 -3058 × 102 -6.5 × 10-2 0.913 8

-11129 1977 × 102 -0.012 -7988 3299 × 102 -0.096 -1175 -9414 × 102 -7.1 × 10-2 0.651 9

-8734 6261 × 102 -0.114 2774 -57968 0.084 3976 104 × 102 0.1 × 10-2 0.931 14

-5304 8090 × 102 -0.071 88 -10971 × 102 -2.994 × 10-3 35377 -155 × 102 -0.19 × 10-2 1.267 14

If additional information is required about the final structure of the mixture, eqs 4 can be used to calculate the excess entropy through the expression sE ) -[∂gE/∂T]x,p. Treatment of the experimental data, on the basis of eqs 4, was done in two separate stages as follows: one with the VLE data obtained isobarically and another with the isothermic VLE data. Threedimensional representations of excess Gibbs function can be analyzed with the model, eq 1, in its complete form. 4. Experimental Results and Treatment of Isobaric Data 4.1. Excess Enthalpies. Table S1 (of the Supporting Information) shows the experimental values of hE (J · mol-1) for the four binary mixtures methanol + butyl alkanoate (methanoate to butanoate) obtained at T ) 298.15 and 318.15 K, except for the system methanol + butyl ethanoate at T ) 298.15 K, which have been published previously.11 These values are shown in Figure 1 together with those published in the literature for methanol + butyl methanoate16 at T ) 298.15 K and methanol + butyl ethanoate15 at T ) 293.15 K. Significant differences were observed between the values obtained in this work and those reported by Lo´pez et al.16 This discrepancy has been analyzed by representing the equimolar values of hE obtained by these authors for the alkanol series (propan-1-ol a decan-1-

ol) + butyl methanoate in the inset of Figure 1, together with the equimolar value obtained in this work for the system methanol + butyl methanoate. We can observe that the values measured here present a better regularity in relation to those estimated for other alkanols, because they are represented as a function of alkanol chain length. However, we cannot find any explanation for the discrepancies observed between our values and those of Sieg et al.,15 for methanol + butyl ethanoate, which can only be justified by the relevant differences between the calorimetric techniques used. The values in Table S1 have been correlatedsin relation to canonical variables x, p, and Tsby the following expression, which is obtained by applying the first expression of eqs 4 to the proposed model, eq 1:

∑ (g 2

hE(x1, p, T) ) z1(1 - z1)

i)0

i1

+ gi2p2 + 2

)

gi4 - gi5T2 zi1 T (5)

To obtain this equation, we have had to consider as zero the last term of the expression produced after the derivation, giving

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 3. Coefficients, gij, and Standard Deviations, s, Obtained Using Equation 3 in the Estimation of Isobaric VLE at p ) 141.32 kPa

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a

binary mixture

a

coefficients





g01 + g02(p/Pa)2 g01 g02 g03 g04 g05 g11 + g12(p/Pa)2 g11 g12 g13 g14 g15 g21 + g22(p/Pa)2 g21 g22 g23 g24 g25 kg s(gE/RT) s(γi)

341 -12022 6.191 × 10-7 2.331 × 10-4 5293 × 102 -0.084 -3640 4108 -3.879 × 10-7 -2.230 × 10-4 7295 × 102 0.089 15110 -20656 1.791 × 10-6 -6.781 × 10-5 -3057 × 102 -0.065 0.843 0.013 0.043

-13837 -8264 -2.791 × 10-7 2.182 × 10-4 19771 × 102 -0.012 -30084 15383 -2.277 × 10-6 8.271 × 10-4 3299 × 102 -0.096 17777 -21222 1.953 × 10-6 -3.493 × 10-5 -9414 × 102 -0.071 0.541 0.025 0.043

-7340 -10210 1.437 × 10-7 4.584 × 10-4 6261 × 102 -0.114 -39334 47312 -4.339 × 10-6 4.251 × 10-4 -579 × 102 0.084 33970 -27749 3.090 × 10-6 -4.331 × 10-4 104 × 102 1.003 × 10-3 0.418 0.030 0.055

31 -10947 5.496 × 10-7 2.256 × 10-4 8090 × 102 -0.071 -10050 10811 -1.045 × 10-6 6.207 × 10-5 -10972 × 102 -2.994 × 10-3 -75939 87600 -8.189 × 10-6 1.803 × 10-3 -155 × 102 -1.865 × 10-3 0.498 0.038 0.070

Coefficients gi4 and gi5 (i ) 0, 1, 2) have been extracted from Table 2.

hE ) gE - Tz1z2

∂z dk(p, T) ∑ ( ∂T )z - T( ∂k )( dT ) × ∂gi

i

2

[(1 - 2z)

2

∑ g z + z(1 - z) ∑ ig z i

i

i-1

]

i

i)0

(6)

i)0

and considering k(p, T) as constant and (dk/dT) ) 0 in eq 6. This assumption has been numerically demonstrated in a previous work,17 where this term was shown to be quantitatively negligible compared to the other two summands. Similarly, the two first summands of eq 5 will be expressed as a single constant (see Table 2), by the grouping: gi1 + gi2p.2 The experimental data were correlated using a least-squares procedure based on an optimization of the following objective function: 2

OF )

l

∑∑ j)1 k)1

[

hE(x1kj, Tj, p ) Cte) - (hE)jk (hE)jk

]

2

(7)

where {[x1kj, (hE)ij; j: 1, 2; k: 1, ..., l]} is the set of experimental points of the hE determined at T1 ) 298.15 K and T2 ) 318.15 K and p ) 101.32 kPa. Table 2 presents the numerical results of this correlation, the values of the parameters in eq 5, and the standard errors obtained for the four binary mixtures methanol + butyl alkanoate (methanoate to butanoate). In Figure 1, the predicted values are compared with the experimental ones and show the goodness of the correlation. 4.2. Vapor - Liquid Equilibria at p ) 141.32 kPa. Study of the behavior of liquid mixtures plays an important role in the analysis of phase equilibria. This is even more important in our present line of research, which aims to apply a generalized model on gE for mixing properties under different experimental conditions. This is why we start by first presenting isobaric VLE data for data treatment. Table S2 (of the Supporting Information) gives the values of the quantities that characterize the equilibrium states at p ) (141.32 ( 0.02) kPa of the liquid and vapor phases: T-x-y for the binary mixtures CH3(OH) + CVH2V+1COOC4H9 (V ) 1-4), determined as indicated previously. These values are compiled in Figure 2, together with the correlation curves obtained as indicated below, and used to

Figure 2. Plots of experimental isobaric VLE values T-x1,y1 at p ) 141.32 kPa for the binaries CH3OH (1) + H2u-1Cu-1COOC4H9 (2): (•) u ) 1, (2) u ) 2, (1) u ) 3, (9) u ) 4, and correlation curves calculated by eq 1 with coefficients in Table 3.

calculate the activity coefficients γi, of the liquid phase, with the following expression ln γi ) ln

( )

(Bii - V°)(p pyi - p°) p i i + δ y2 + p°x RT RT ij j i i

(8)

where δij ) 2Bij - Bii - Bjj. The second virial coefficients of pure compounds, Bii, and of the mixtures, Bij, were calculated using the Tsonopoulos expressions.29 Molar volumes of the pure compounds V°i at the saturation temperature were obtained by the Rackett equation modified by Spencer and Danner,30 with the ZRA coefficients extracted from the work of Reid et al.31 Finally, the vapor pressures were calculated at each equilibrium

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Table 4. Coefficients and Standard Deviations, s, Obtained in the Estimation of VLE and Excess Enthalpies Using Different Models eqs 1 and 5 lack of fit

Wilson

NRTL

UNIQUAC

g0j

g1j

g2j

-8520 7.933 × 10-7 -1.517 × 10-4

-4946 5.714 × 10-7 -1.057 × 10-4 2010 × 102 -1.310 × 10-3 kg ) 0.896 kh ) 0.591 0.013 0.038 10

methanol (1) + butyl methanoate (2) ∆λ12 ) 3142 ∆λ21 ) 1456

s(gE/RT) s(γi) s(hE)

∆g12 ) 9419 ∆g21 ) -7840 R ) -0.05

∆u12 ) 189 ∆u21 ) 2290

0.018 0.189 208

0.025 0.321 62

∆λ12 ) 3480 ∆λ21 ) 967

∆g12 ) 1194 ∆g21 ) 1168 R ) -1.55

∆u12 ) 7 ∆u21 ) 2512

0.015 0.293 185

0.024 0.124 104

0.013 0.295 136

∆λ12 ) 3850 ∆λ21 ) 601

∆g12 ) 1415 ∆g21 ) 1077 R ) -1.39

∆u12 ) -161 ∆u21 ) 2889

0.028 0.378 154

0.032 0.123 79

0.025 0.372 126

∆λ12 ) 4215 ∆λ21 ) 464

∆g12 ) 1643 ∆g21 ) 1171 R ) -0.89

j j j j j

) ) ) ) )

1 2 3 4 5

207 3.044 × 10-7 3.414 × 10-6 -2633 × 102 -0.024

-7.879 × 10-4

0.019 0.175 157 methanol (1) + butyl ethanoate (2)


j j j j j

) ) ) ) )

1 2 3 4 5

5291 -4.917 × 10-7 2.009 × 10-4 323 × 102 -0.017

-365 1.024 × 10-7 -4.674 × 10-5 5089 × 102 -0.014

9557 2.454 × 10-8 -7.022 × 10-5 -19409 × 102 -0.014 kg ) 1.522 kh ) 0.613 0.022 0.040 28

-1002 -7.618 × 10-7 2.271 × 10-4 7609 × 102 -4.897 × 10-3

5254 1.812 × 10-7 -2.021 × 10-5 -1246 × 102 -1.201 × 10-3 kg ) 0.483 kh ) 0.543 0.028 0.050 23

-1753 4.494 × 10-8 -2.455 × 10-5

2131 -2.180 × 10-7 8.996 × 10-5 520 × 102 -8.287 × 10-3 kg ) 1.457 kh ) 0.783 0.037 0.068 20



j j j j j

) ) ) ) )

1 2 3 4 5

7034 7.227 × 10-8 -1.250 × 10-5 -8613 × 102 -0.022



0.041 0.443 125

0.034 0.113 104

∆u12 ) -246 ∆u21 ) 3182

j j j j j

) ) ) ) )

1 2 3 4 5

4506 -4.213 × 10-7 3.110 × 10-4 -1470 × 102 -0.051

-0.014

0.038 0.434 98

temperature by Antoine’s equation with the coefficients presented in previous papers (the work of Blanco and Ortega9 for methanol and that of Gonzalez and Ortega2 for the butyl esters). Table S2 (of the Supporting Information) also reports the experimental data and the calculated values of γi and gE/RT. The VLE data were tested for thermodynamic consistency using the global test of Fredenslund et al.,32 and the L-W proposed by Wisniak.33 Both methods showed a positive consistency, with the sole exception of the methanol + butyl butanoate mixture which does not satisfy the test of Wisniak, with a value of D ) 5.3. The above information was then used to treat the isobaric VLE data, considering the following advantages afforded by the flexibility of our method: • Step-to-step correlation: Fitting the hE(x1, p, T) as indicated in section 4.1 permits coefficients gi4 and gi5 of eq 5 to be clearly evaluated, and also the value of the binomial (g01 + g02p2) ) H′ at the working pressure of enthalpies, 101.32 kPa (see Table 2). This corresponds to the first stage of the data treatment. • In a second step, the values obtained for these coefficients are introduced in eq 1 of the Gibbs function, but with the

gi(p, T) according to eq 3. In this way, VLE data can be used to obtain the coefficient (g01 + g02p2) ) G′, but logically with a different value to that of H′, because of the different working pressure in both thermodynamic quantities (101.32 kPa for hE and 141.32 kPa for the VLE data). This stage is continued using a genetic algorithm, based on a least-squares procedure optimizing the following objective function. n

OF )

2

∑∑ j)1 i)1

[

ln γi(xij, p, Tj) - ln γij ln γij

]

2

(9)

where {(Tj,x1j,ln γ1j,ln γ2j); j: 1,...,n} represents the set of experimental VLE data at p ) 141.32 kPa compiled in Table S2 (of the Supporting Information) and ln γi(xi, p, T) will correspond to the expression used to determine the partial molar property corresponding to the Gibbs function, derived from eq 1. In this step, the G′ values are obtained for each binary mixture. As the model proposed is univocal,


9553

Figure 3. (a-d) Representation of experimental values gE/RT (•) and γi (2) vs x1 at p ) 141.32 kPa from Table S2 (of the Supporting Information) and correlation curves (s) using eqs 1 and 5 with coefficients in Tables 3 and 4 for the binaries: CH3OH (1) + H2u-1Cu-1COOC4H9 (2). Comparison with curves by NRTL model (s s) with coefficients in Table 4: for u ) (a) 1, (b) 2, (c) 3, (d) 4. (e-h) Representation of experimental values of hE at 298.15 (black dots) and 318.15 K (red dots) vs x1 for the binaries: CH3OH (1) + H2u-1Cu-1COOC4H9 (2) for u ) (e) 1, (f) 2, (g) 3, (h) 4. The correlation curves (solid black and red lines) correspond to eq 5 with coefficients in Table 2 and (dotted black and red lines) by NRTL with coefficients in Table 4.

9554


although higher overall errors were obtained. The statistical values for this application are presented in Table 4, which also shows the results obtained for the parameters of the proposed model, but in a joint correlation, using the same procedure as that applied in classical models. Although the NRTL model satisfactorily reproduces the activity coefficient and Gibbs energy function, the fits of hE were unsatisfactory, and this excess property even tended to invert with temperature, in relation to experimental values. Figure 3 shows the graphical representation of some properties for the four methanol + butyl ester systems and the corresponding estimations made with the NRTL models and the one proposed here. A comparison of the standard errors and the curves proves that the model used here substantially improves the quality of the joint correlations compared to those obtained with classical models,18-20 hence validating its utility and, at the same time, justifying the presence of a greater number of parameters, but with a more extensive capacity of representation, that the other models do not have.

Figure 4. Excess molar volumes VE, at T ) 298.15 K (closed symbols) obtained in this work and the literature and the corresponding correlation curves by eq 12 (s), with coefficients in Table 5 for the binaries: CH3OH (1) + H2u-1Cu-1COOC4H9 (2) (•) for u ) 1, Ortega,10 (2) u ) 2, (1) u ) 3, and (9) u ) 4. The open symbols and dashed-lines correspond to the values at T ) 318.15 K.

the following system of equations should be resolved for each system studied: gi1 + gi2(101320)2 ) H′ gi1 + gi2(141320)2 ) G′

(10)

Another advantage of this procedure, in relation to the flexibility of use of the proposed model, is that it permits a single solution to be obtained for correlation of the different quantities (which are thermodynamically related) and determined under different conditions of pressure. The results obtained for coefficients gi1 and gi2 are shown in Table 3. The model obtained can now be used to reproduce the experimental data of each of the systems studied, together with the corresponding fitting curves for T-x-y. Figure 2 shows the quality of the estimations made with the model. In an attempt to verify the utility of the proposed model and the fitting procedure mentioned previously, a simultaneous correlation was carried out (since none of them permit a stepby-step procedure) of VLE data and hE with other previously known models, in particular Wilson,18 NRTL,19 and UNIQUAC.20 In these three cases, optimization was done through a single objective function made by combining eqs 7 and 9,

5. Experimental Results and Treatment of Isothermic Data 5.1. Excess Volumes. The VE data obtained for this work for the binary systems methanol + butyl esters were determined at temperatures of T ) 298.15 and 318.15 K and atmospheric pressure. Values determined for the four systems are presented in Table S3 (of the Supporting Information), except for those obtained at 298.15 K for methanol + butyl methanoate that were reported previously.10 The (x, VE) points are depicted in Figure 4 and show volume contractions that decrease with increasing acid chain length of the butyl alkanoate, reaching positive values along the whole range of compositions for the methanol + butyl butanoate systems. The effect of increasing the temperature seems to be significant in these mixtures, with values for the slope (∂VE/∂T)p,x > 0. The curves in Figure 4 represent the correlation of volumes using an equation that results from applying the second ratio of eqs 4 to the model of eq 1. Hence, VE ) z(1 - z)

∑ ( ∂p ) ∂gi

zi +

T,x

[(1 - 2z)

( dkdz )( dk(p,dp T) ) ×

2

∑

2

gizi + z(1 - z)

i)0

∑ ig z i

i-1

]

(11)

i)0

Assuming the second summand in eq 11 to be negligible (owing to the independence of parameter k from pressure and temperature) and introducing the derivative of the coefficients given by eq 3 yields the following parametric equation:

Table 5. Coefficients, gij, and Standard Deviations, s, Obtained Using Equation 12 to Correlate WE at T ) 298.15 and 318.15 K binary mixture coefficients





g02 g03 g12 g13 g22 g23 kv 109s(VE)/(m3 · mol-1)

-2.016 × 10-11 1.304 × 10-8 7.185 × 10-12 -6.165 × 10-9 4.630 × 10-13 -3.915 × 10-10 0.562 2

-3.586 × 10-11 2.395 × 10-8 4.291 × 10-11 -2.995 × 10-8 -5.371 × 10-12 4.474 × 10-9 0.688 2

-2.768 × 10-11 1.873 × 10-8 -5.023 × 10-12 3.209 × 10-9 4.069 × 10-11 -2.594 × 10-8 1.101 2

-2.594 × 10-11 1.773 × 10-8 1.850 × 10-11 -1.287 × 10-8 1.340 × 10-11 -7.418 × 10-9 1.349 3

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 6. Coefficients, gij, and Standard Deviations, s, Obtained Using Equation 1 in the Estimation of Isothermic VLE at T ) 296.5, 313.15, or 333.15 Ka binary mixture coefficients


g01 g02 g03 g04 g05 g11 g12 g13 g14 g15 g21 g22 g23 g24 g25 kg s(gE/RT) s(γi)

5640 -3.586 × 10-11 2.395 × 10-8 -4401 × 102 -6.339 × 10-3 -1239 4.291 × 10-11 -2.995 × 10-8 -10011 4.731 × 10-4 1179 -5.371 × 10-12 4.474 × 10-9 -8342 × 102 1.570 × 10-2 1.223 0.020 0.082

2

∑ (2g

i2p

+ gi3T)zi1

Table 7. Coefficients, gij, and Standard Deviations, s, Obtained Using Equations 1, 5, and 12 in the Simultaneous Correlation of Isothermic VLE at T ) 296.5, 313.15, or 333.15 K, Isobaric VLE at p ) 141.32 kPa, Excess Molar Enthalpies, hE, and Excess Molar Volumes, WE, at T ) 298.15 or 318.15 K binary mixture

a Coefficients gi2 and gi3 (i ) 0, 1, 2) have been extracted from Table 5.

VE(x1, p, T) ) z1(1 - z1)

9555

(12)

coefficients


g01 g02 g03 g04 g05 g11 g12 g13 g14 g15 g21 g22 g23 g24 g25 kg kh kv s(gE/RT) s(γi) s(hE) 109s(VE)/(m3 · mol-1)

2311 -1.689 × 10-11 1.132 × 10-8 -194 × 102 -1.697 × 10-3 5508 -2.265 × 10-11 1.440 × 10-8 -5144 × 102 -2.990 × 10-2 4322 3.942 × 10-11 -2.639 × 10-8 -9151 × 102 2.999 × 10-3 0.656 0.668 0.424 0.022 0.099 28 3

i)0

where now T is constant (for this work corresponds with 298.15 or 318.15 K depending on the case). The fit is carried out by applying a least-squares procedure using the following objective function: 2

OF )

l

∑∑ j)1 k)1

[

VE(x1kj, p, Tj) - (VE)jk E

(V )jk

]

Here, an analogous procedure can be followed to that used for isobaric equilibria. After a first step (with the previously described correlation of the VE), isothermic VLE data are correlated, now using the following objective function 3

2

OF )

(13)

where {(x1kj,(VE)jk); j: 1, 2; k: 1, ..., l} represents the set of experimental points determined at T1 ) 298.15 K and T2 ) 318.15 K and p ) 101.32 kPa. Table 5 reports the resulting values of the parameters of eq 11, for each of the four binary mixtures, together with the associated standard errors.

n

2

∑∑∑ j)1 k)1 i)1

[

ln γi(x1jk, pjk, Tj) - ln γijk ln γijk

]

2

(14)

where {(x1jk, pjk, Tj, ln γ1jk, ln γ2jk); j: 1, 2, 3; k: 1, ..., n} represents the set of experimental data for isothermic VLE at the temperatures Tj ) 296.15, 313.15, or 333.15 K reported by

Figure 5. (a) Representation of experimental (from ref 15) values of gE/RT and γi vs x1 and correlation curves using from eq 1 with coefficients in Tables 5 and 6 of isothermic VLE data at T ) 296.5 (•, O, s), 313.15 (2, ∆, s s), and 333.15 K (1, 3, - - -) for the system CH3OH (1) + -H3CCOOC4H9 (2). (b) Representation of experimental VE data and correlation curves by eq 12 and coefficients in Table 5 for the VE vs x1 at T ) 298.15 (•, s) and 318.15 K (9, s) for the system CH3OH (1) + H3CCOOC4H9 (2).

9556


Figure 6. Representation, for the binary mixture CH3OH (1) + H3CCOOC4H9 (2) of experimental values and correlated curves using eqs 1, 5, and 12 and coefficients of Table 7, by a simultaneous correlation procedure of the quantities gE/RT and γi vs x1 for (a and b) isobaric VLE data at p ) 141.32 kPa and (c and d) isothermic VLE data at T ) 296.5 (•, s), 313.15 (1, s s), and 333.15 K (2, - - -). Open symbols correspond, respectively, to the projections of the quantities mentioned above on the x1-gE/RT and x1-γi planes. (e and f) Representation of experimental and correlated curves in the simultaneous procedure indicated for hE and VE at T ) 298.15 (•, s) and 318.15 K (9, s) as a function of x1.

Sieg et al.15 for the binary system methanol + butyl ethanoate at T ) 296.15, 313.15, and 333.15 K, the only ones reported in the literature for the group of mixtures studied. The numerical results of the correlation are given in Table 6, while Figure 5 shows the curves calculated, together with the experimental data. The model proposed is also a good instrument to use to correlate equilibrium properties obtained

isothermically. In this case, a comparison has not been made correlating the data with those obtained with classical models, as was done for isobaric VLE data, since those ones did not include pressure as a regressor variable. In other words, it is not possible to use one of eqs 4 to obtain an expression for excess volumes from the one corresponding to excess Gibbs energy function.


9557

(1) isobaric, with the data obtained from this work, (2) isothermic, with VLE data from the literature, and (3) a treatment combining the two previous ones correlated in a single procedure. Compared with standard correlation methods,18-20 the new model proposed contains more parameters, but has a greater capacity to represent all the data corresponding to the different properties of a single solution. It was, therefore, necessary to use stochastic optimization techniques, such as genetic algorithms, to resolve optimization problems with several objective functions. In future works these results will be compared with those obtained using a technique of multiobjective optimization.34 Of the classical models considered here, NRTL was the one that presented the best behavior. However, shown in Figure 7 is a new system able to simultaneously represent all the variables, reflecting the superior capacity for representation of the model proposed before and its potential as a tool to correlate the different thermodynamic quantities of a system. Figure 7. Three-dimensional representation of experimental data (•) in this work and from the literature (isobaric13 and isothermic15) and correlation curves using eq 1 for the binary system methanol + butyl ethanoate.

Acknowledgment

6. Overall Treatment of Isobaric + Isothermic Data

The authors (F.E., J.O., E.P.) gratefully acknowledge the financial support received from Ministerio de Ciencia e Innovacioń (Spain) for the Project CTQ2009-12482.

The general model proposed according to eq 1, of the form gE ) gE(p, T, xi), also allows treating together the two case studies discussed in previous sections. In other words, in just one step, the data obtained for the activity coefficients, isobaric and/or isothermic VLE, can be correlated; the VE values using eqs 11 and 12 and the hE values using eqs 5 and 6. We propose that this be carried out by extending the previous cases, using a least-squares procedure to carry out the correlation implemented in a genetic algorithm, optimizing an objective function containing four summands with the eqs 7, 9, 13, and 14 to obtain parameters. Of the four binary systems studied here, the mixture methanol + butyl ethanoate is the only one for which sufficient γi (isobaric and isothermic VLE), VE, and hE data are available to analyze the utility of the new model, to verify the fitting procedure described for this general situation, and also for more widespread applications. Table 7 shows the values for the parameters of eqs 1, 5, and 12 for the best fits, together with the standard errors. Figure 6 offers a qualitative evaluation and illustrates the good behavior of the model, not only in relation to the correlation of hE and VE reflected in graphs e and f of this figure, but also for the excess Gibbs energy function and the activity coefficients, as recorded in the 3D graphs a and b and c and d, for the correlation of isobaric and isothermal VLE, respectively. To provide more complete information, projections of the experimental points and the curves fitted in planes gE/RT-x1 and γi-x1 have also been added. However, for the same reasons as those given for the isothermic case, here it was not possible either to compare the behavior of the model used with that of classical models.18-20 7. Conclusions This work presents the experimental isobaric VLE data at p ) 141.32 kPa, and the quantities of hE and VE determined at the temperatures of T ) 298.15 and 318.15 K for four binary mixtures comprised of methanol + butyl alkanoate (methanoate to butanoate). A previously described parametric model has been used, defined for the excess Gibbs energy function, of the form gE ) gE(p, T, xi), to derive expressions for other thermodynamic properties of solutions, such as VE and hE. The corresponding expressions have been applied to three experimental situations:

Supporting Information Available: Tables containing the experimental values of excess enthalpies, excess volumes, and vapor-liquid equilibrium. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Ortega, J.; Penã, J. A.; De Alfonso, C. Isobaric Vapor-Liquid Equilibria of Ethyl Acetate+Ethanol Mixtures at 760 ( 0.5 mmHg. J. Chem. Eng. Data 1986, 31, 339–342. (2) Gonza´lez, E.; Ortega, J. Densities and Isobaric Vapor-Liquid Equilibria of Butyl Esters (Methanoate to Butanoate) with Etanol aty 101.32 kPa. J. Chem. Eng. Data 1995, 40, 1178–1183. (3) Ortega, J.; Espiau, F.; Postigo, M. A. Isobaric Vapor-Liquid Equilibria and Excess Quantities for Binary Mixtures o fan Ethyl Ester+tertButanol and a New Approach to VLE Data Processing. J. Chem. Eng. Data 2003, 48, 916–924. (4) Ortega, J.; Espiau, F.; Vreekamp, R.; Tojo, J. Modeling and Experimental Evaluation of Thermodynamic Properties for Binary Mixtures of Dialkylcarbonate and Alkanes Using a Parametric Model. Ind. Eng. Chem. Res. 2007, 46, 7353–7366. (5) Blanco, A. M.; Ortega, J. Experimental study of miscibility, density and isobaric vapor-liquid equilibrium values for mixtures of methanol in hydrocarbons (C5, C6). Fluid Phase Equilib. 1996, 122, 207–222. (6) Oliveira, M. B.; Miguel, S. I.; Queimada, A. J.; Coutinho, J. A. P. Phase Equilibria of Ester + Alcohol Systems and Their Description with the Cubic-Plus-Association Equation of State. Ind. Eng. Chem. Res. 2010, 49, 3452–3458. (7) Jothiramalingam, R.; Wang, M. K. Review of Recent Developments in Solid Acid, Base, and Enzyme Catalysis (Heterogeneous) for Biodiesel Production via Transesterification. Ind. Eng. Chem. Res. 2009, 48, 6162– 6172. (8) Blanco, A. M.; Ortega, J. Isobaric Vapor-Liquid Equilibria of Methanol + Methyl Ethanoate, + Methyl Propanoate, + Methyl Butanoate at 141.32 kPa. J. Chem. Eng. Data 1996, 41, 566–570. (9) Blanco, A. M.; Ortega, J. Densities and Vapor-Liquid Equilibrium Values for Binary Mixtures Composed of Methanol + an Ethyl Ester at 141.32 kPa with Application of and Extended Correlation Equation for Isobaric VLE Data. J. Chem. Eng. Data 1998, 43, 638–645. (10) Ortega, J. Excess Molar Volumes of Binary Mixtures of Butyl Formate with Normal Alcohols at 298.15 K. J. Chem. Eng. Data 1985, 30, 465–467. (11) Ortega, J. Excess enthalpy of alcohols + esters. Int. Data Ser., Sel. Data Mixtures Ser. A 1995, 3, 154–183. (12) Beregovykh, V. V.; Timofeev, V. S.; Luk’yanova, R. N.; Yakushev, V. M.; Serafimov, L. A. Liquid-vapor equilibrium in systems formed by water and butyl acetate with homologs of monohydric C1-C5 saturated alcohols. Uch. Zap. Mosk. Inst. Tonkoi Khim. Tekhnol. 1971, 1 (3), 38–44.

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ReceiVed for reView May 27, 2010 ReVised manuscript receiVed August 6, 2010 Accepted August 6, 2010 IE101165R

Advances in the Correlation of Thermodynamic Properties of Binary

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