Measurement and Correlation of the Phase Equilibrium of Aqueous

Article pubs.acs.org/jced

Measurement and Correlation of the Phase Equilibrium of Aqueous Two-Phase Systems Composed of Polyethylene(glycol) 1500 or 4000 + Sodium Sulfite + Water at Different Temperatures Bruno G. Alvarenga,† Luciano S. Virtuoso,*,† Nelson H. T. Lemes,† Luis A. da Silva,† Anderson F. Mesquita,‡ Kelany S. Nascimento,§ Maria C. Hespanhol da Silva,∥ and Luis H. Mendes da Silva∥ †

Grupo de Pesquisa em Química de Colóides, Instituto de Ciências Exatas, Universidade Federal de Alfenas (UNIFAL), Rua Gabriel Monteiro da Silva, 700, 37130-000, Alfenas, MG, Brazil ‡ Grupo de Macromoléculas e Surfactantes, Departamento de Química, Universidade Federal do Espírito Santo (UFES), 29075-910, Vitória, ES, Brazil § Departamento de Engenharia de Pesca, Universidade Federal do Ceará (UFC), 60020-181, Fortaleza, CE, Brazil ∥ Grupo de Química Verde Coloidal e Macromolecular, Departamento de Química, Centro de Ciências Exatas, Universidade Federal de Viçosa, Av. P. H. Rolfs, s/n, 36570-000, Viçosa, MG, Brazil ABSTRACT: The equilibrium behaviors of two-phase liquid−liquid systems composed of poly(ethylene glycol) (PEG) 1500 or 4000 + sodium sulfite + water were experimentally determined at temperatures of (288.15, 298.15, 308.15, and 318.15) K. The effects of the molecular weight of PEG and the temperature on the phase separation were studied. The binodal curves were fitted to an empirical equation that correlates the concentrations of PEG 1500 or 4000 and sodium sulfite, and the coefficients for the different temperatures were estimated. The tie-line compositions were estimated and correlated using the Othmer−Tobias and Bancroft equations, and the parameters are reported. The liquid−liquid equilibrium (LLE) experimental data obtained were wellcorrelated to the activity coefficients of the non-random two-liquid (NRTL) and UNIversal QUAsiChemical (UNIQUAC) models, and the mean deviations were less than 0.36 % and 0.31%, respectively.

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have been reported.8−10 The study of new polymer−salt systems has been used to improve the technique because of the low cost, the low viscosity, and the rapid phase separation of these systems.11 Several studies have combined PEG with different types of salts, for example, sulfates,12 carbonates,13 chlorides,14 and nitrates,15 but no experimental data for PEG + sodium sulfite + water are available in the literature. In this study, experimental data for an ATPS composed of PEG 1500 + sodium sulfite + water and PEG 4000 + sodium sulfite + water [at temperatures of (288.15, 298.15, 308.15, and 318.15) K] were obtained to study the influence of the temperature and the PEG molar mass on the equilibrium data. Experimental phase equilibrium data are required for the development of thermodynamic models. There are three main types of models proposed in the literature; these are either based on the osmotic virial expansion or on lattice theories, such as local composition models. In this study, the nonrandom twoliquid (NRTL)16 and the extended UNIversal QUAsiChemical (UNIQUAC)17 models were used to correlate the experimental

INTRODUCTION Aqueous two-phase systems (ATPS’s) are produced under specific thermodynamic conditions in which a combination of two aqueous solutions of different macromolecules, one polymer and one inorganic salt or two electrolytes, are mixed above a critical concentration and have been applied for over 50 years to the separation, fractionation, and molecular characterization of biological macromolecules and particles. The use of ATPS’s is a useful technique for the separation of biomaterials, including whole cells, membranes, organelles, and other biological particles.1 ATPS’s can be used to simultaneously clarify, concentrate, and partially purify processes into one step. This extraction technique can provide high recovery yields and purities and allows easy scale-up.2 Poly(ethylene glycol) (PEG) has a set of features, such as the fact that it is inexpensive, nontoxic, and biodegradable, which makes it useful for bioseparation applications.3 In addition to bioseparation, several reviews have focused on the applications of PEG in biotechnology and medicine,4,5 as solvents and phasetransfer catalysts (PTCs) in organic synthesis,6 and as alternative separation media in ATPS’s.7 In recent years, a large number of experimental phase equilibrium data sets of systems composed of PEG and a salt © 2014 American Chemical Society

Received: September 17, 2013 Accepted: January 23, 2014 Published: February 5, 2014 382

dx.doi.org/10.1021/je400843e | J. Chem. Eng. Data 2014, 59, 382−390

Journal of Chemical & Engineering Data

Article

circulated at constant temperature. The temperature was controlled to within ± 0.1 K. The binodal curves were determined by a titration method. A salt solution of known concentration was titrated with the polymer solution or vice versa, until the solution turned turbid. The composition of the mixture was determined by mass using an analytical balance (Shimadzu, AG 220) with a precision of ± 0.0001 g. To determine the tie lines, different ATPS’s were prepared by weighing the appropriate amounts of the components from stock solutions of PEG (50 % (w/w), sodium sulfite (20 %), and water on an analytical balance (Shimadzu, AG 220, with an uncertainty of ± 0.0001 g). In general, 10 g of each system was prepared. After the mixtures were vigorously stirred, the system became turbid and was allowed to settle for (48 to 96) h at the indicated operation temperature of (288.15, 298.15, 308.15, or 318.15) K in a temperature-controlled bath (QUIMIS-BR, with an uncertainty of 0.1 K). The equilibrium state was characterized by the absence of turbidity in both the top and the bottom phases. Aliquots of the top and bottom phases were collected with a syringe for analysis. Construction of Phase Diagrams. The salt concentration was determined by the conductivity (Tecnal, TEC 4 MP, Brazil) of the electrolyte in the range of 10−3 % to 10−2 % (w/w).20 The salt solutions exhibited the same conductivity in water as in the diluted polymer solution (0.1 % to 0.01 % (w/w)). Using this method, the standard deviation of the mass percent of the salt was ± 0.10 %. The polymer was quantified at 298.15 K using a refractometer (Analytik Jena AG Abbe refractometer 09-2001, Germany). Because the refractive index of the phase samples depends on the polymer and salt concentration and it is an additive property, we obtained the PEG concentration by subtracting the salt concentration (obtained by the conductivity)

liquid−liquid equilibrium (LLE) data obtained for the two aqueous (PEG 1500 or 4000 + Na2SO3 + H2O) systems studied. Additionally, the Othmer−Tobias and Bancroft equations18 were used to correlate the tie-line data, and the binodal curves were fitted to two nonlinear equations. This new ATPS has such advantages as easy preparation, low viscosity characteristics, and recyclable utilization, which may find application in partitioning and analysis of biomaterials. In addition, sulfites are widely used as additives in the food industry due to the inhibitory effect on bacteria, yeasts, molds and inhibiting enzymatic as well as nonenzymatic browning reactions during processing and storage which adds interesting features to new ATPS’s.19

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EXPERIMENTAL SECTION Materials. Polyethylene glycol with average molar masses of 1500 g·mol−1 and 4000 g·mol−1 were purchased from Aldrich Table 1. Estimated Values of the a0, a1, and a2 Coefficients for the Different Systems ATPS

a0

a1

a2

PEG 1500 + Na2SO3 + H2O PEG 4000 + Na2SO3 + H2O

1.331 1.331

1.488·10−3 1.478·10−3

1.489·10−3 1.479·10−3

(USA). Analytical grade Na2SO3 (sodium sulfite) was obtained from Synth (Brazil). Milli-Q II water (Millipore, USA) was used to prepare all of the aqueous solutions. Experimental Procedure. The experiments were carried out using a glass vessel with a working volume of 50 cm3 to determine the phase equilibrium (binodal curve). The glass vessel was provided with an external jacket in which water was

Table 2. Equilibrium Data (in Mass Fraction Percent) for the PEG 1500 (w1) + Sodium Sulfite (w2) + Water (w3) System at T = (288.15 to 318.15) K overall tie line

tie line length

w1

w2

top phase w3

1 2 3 4 5

29.87 32.21 34.62 36.48 38.93

9.58 10.22 11.41 12.51 13.33

13.71 14.05 14.45 14.85 15.30

76.71 75.73 74.14 72.64 71.37

1 2 3 4 5

31.85 33.43 35.45 37.20 39.28

9.65 10.38 11.01 11.93 12.84

14.54 14.91 15.41 15.84 16.34

75.81 74.71 73.58 72.23 70.82

1 2 3 4 5

36.45 39.02 41.51 43.46 45.44

10.92 11.74 12.38 13.21 13.85

14.32 14.86 15.47 16.13 16.68

74.76 73.40 72.15 70.66 69.47

1 2 3 4 5

30.91 32.96 36.16 38.95 42.42

9.39 10.18 11.17 12.05 13.04

11.78 12.29 12.74 13.31 13.88

78.83 77.53 76.09 74.64 73.08

w1 288.15 K 28.63 30.48 32.67 34.08 36.20 298.15 K 29.89 31.28 32.85 34.03 36.01 308.15 K 35.40 37.54 39.56 40.91 42.66 318.15 K 30.62 32.19 34.65 37.26 40.48 383

bottom phase

w2

w3

w1

w2

w3

3.73 3.47 3.47 3.38 3.40

67.64 66.05 63.86 62.54 60.40

2.48 2.15 2.01 1.91 1.69

18.17 18.79 19.54 20.59 21.41

79.35 79.06 78.45 77.50 76.90

3.61 3.59 3.58 3.54 3.53

66.50 65.13 63.57 62.43 60.46

1.94 1.88 1.83 1.79 1.77

18.89 19.51 20.73 22.09 22.77

79.17 78.61 77.44 76.12 75.46

3.32 2.97 2.88 2.80 2.77

61.28 59.49 57.56 56.29 54.57

2.04 2.01 1.89 1.77 1.75

18.01 19.11 20.32 21.69 22.55

79.95 78.88 77.79 76.54 75.70

3.44 3.34 2.99 2.94 2.82

65.94 64.47 62.36 59.80 56.70

1.93 1.72 1.31 1.24 1.05

14.95 15.92 16.98 17.76 18.46

83.12 82.36 81.71 81.00 80.49



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from the total solution composition (obtained by the refraction index). For dilute aqueous solutions containing a polymer and a salt, the relationship between the refractive index, nD, and the mass fractions of PEG, w1, and salt, w2, is given by nD = a0 + a1w1 + a 2w2 (1)

Table 3. Binodal Data (in Mass Fraction Percent) for the PEG 1500 (w1) + Sodium Sulfite (w2) + Water (w3) System at T = (288.15 to 318.15) K 288.15 K

This equation was used for the phase analysis of the poly(propylene glycol) + NaCl + H2O system by Cheluget et al.21 The estimated values of the a0, a1, and a2 coefficients of the present systems are shown in Table 1. The standard deviation of the mass percent of the polymer was on the order of 0.05 %. The water concentration was obtained by subtracting. All of the analytical measurements were performed in triplicate. The tie line lengths (TLL) for the different compositions were calculated according to TLL = [(w1t − w1b)2 + (w2t − w2b)2 ]1/2

wt1,

wb1,

wt2,

(2)

wb2

where and are the top (t) and bottom (b) equilibrium mass fractions of PEG (1) and sodium sulfite (2). The TLLs are expressed in mass fractions.

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RESULTS AND DISCUSSION Effect of Temperature. Tables 2 to 5 show the LLE and the binodal data, which is expressed in mass percent, for the PEG 1500 + sodium sulfite + water and the PEG 4000 + sodium sulfite + water systems in temperatures (288.15 to 318.15) K. The subscripts 1, 2, and 3 in the tables represent the polymer, salt, and water components, respectively. Five tie lines were determined at each temperature. These tie lines were obtained through the linear regression of the corresponding set of overall top-phase and bottom-phase concentrations. An increase in polymer and salt segregation with an increase in global composition and consequently in the TLL is noted. This behavior is in agreement with the reported results for other ATPS’s formed by PEG.2,3,7,21 For all ATPS’s, the temperature has a significant effect on the phase equilibrium compositions. Figures 1 and 2 show the behavior of the PEG 1500 + sodium sulfite + water and the PEG 4000 + sodium sulfite + water systems. As shown in the figures, a reduction in the biphasic area can be observed with decreasing temperature, which indicates that the phase separation process of both systems is endothermic. This behavior is in agreement with the results reported for ATPS’s composed of other salts and PEG with different molar masses.9,22 The effect of the temperature on the phase equilibrium was studied through the application of the slope of the tie line (STL) concept. This value is defined as the ratio of the variations in the polymer concentration in each phase of the system to the variations in the salt concentrations in each phase of the system: STL =

308.15 K

318.15 K

100w1

100w2

100w1

100w2

100w1

100w2

100w1

100w2

39.34 38.23 35.34 33.70 32.00 31.50 29.90 29.30 28.90 28.20 27.95 27.26 26.20 25.58 25.00 23.90 22.35 21.15 20.07 19.23 18.20 17.41 16.57 15.73 15.12 14.20 13.24 10.00 7.50 6.12 4.00 2.48 2.16 2.06 1.96 1.93 1.91 1.92

3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 4.00 4.30 4.45 4.53 4.66 4.80 5.04 5.79 6.29 6.74 7.02 7.53 7.85 8.27 8.75 8.99 9.67 10.19 12.50 14.30 15.73 17.87 20.04 21.01 22.16 23.73 25.42 27.03 28.38

38.20 35.10 33.50 32.10 31.61 30.73 30.04 29.54 27.88 26.89 26.10 25.09 24.23 23.50 22.82 21.80 20.66 19.65 18.95 17.89 17.00 15.73 14.77 12.65 10.09 7.70 6.68 4.99 2.35 2.24 2.16 2.01 1.76 1.70 1.65 1.63 1.58 1.57

3.36 3.37 3.41 3.45 3.51 3.59 3.70 3.72 3.77 3.90 4.08 4.33 4.61 4.78 4.99 5.24 5.54 5.79 5.99 6.14 6.45 6.80 7.38 8.23 10.08 11.78 12.66 14.78 17.87 19.29 20.46 21.31 22.06 23.30 24.06 24.18 27.18 28.87

43.09 41.14 39.89 33.65 31.89 37.87 35.59 30.23 27.77 26.09 24.49 23.26 22.08 21.21 20.34 18.81 18.07 17.75 17.10 15.47 14.97 12.66 9.89 9.00 7.89 7.04 5.33 3.97 3.37 2.22 1.88 1.77 1.72 1.70 1.64 1.36 1.49 1.33

3.01 3.09 3.11 3.14 3.18 3.19 3.21 3.35 3.46 3.79 4.34 4.69 5.07 5.24 5.45 5.82 6.03 6.11 6.26 6.72 6.87 7.56 9.33 9.77 10.56 11.22 13.13 14.99 16.00 17.56 20.55 21.37 23.42 24.15 25.23 25.89 28.06 29.14

37.95 36.87 35.99 35.14 33.21 32.01 30.31 29.50 29.00 28.20 27.64 27.06 26.45 25.87 24.73 23.73 22.80 21.95 21.16 20.42 19.71 19.04 18.44 18.13 17.86 17.59 17.33 17.08 13.00 10.01 5.20 3.08 2.33 1.09 1.07 1.05 1.03 1.01

3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.69 3.71 3.77 3.87 3.97 4.18 4.34 4.49 4.63 4.76 5.04 5.22 5.33 5.55 5.66 5.70 5.89 6.01 6.12 7.60 9.23 12.50 14.88 17.19 18.85 20.09 21.90 22.90 24.31

increase in the temperature decreases the attraction between the PEG and the water molecules, which results in an increase in the PEG concentration in the top phase and a decrease in the salt concentration in the lower phase.22,23 An increase in the TLL was observed with each temperature analyzed because of an increase in the global composition. Similar results have been reported in the literature.24 A similar effect was observed for the PEG 4000 + sodium sulfite + water system. Figure 2b shows the effect of temperature on the displacement of the biphasic area in the PEG 4000 + sodium sulfite + water system. As the temperature was increased from (288.15 to 318.15) K, an increase in the biphasic region was observed. Similar results were obtained by other researchers with other PEG−salt systems.24,25 It is worth mentioning that, in aqueous solutions, sulfite exists in the form of different sulfur species and their relative abundance in solution depends on the concentration and the pH. In recent work, Müller et al.26 have shown the pH-dependent sulfite

w1t − w1b w2t − w2b

298.15 K

(3)

Table 6 shows the STL values for the ATPS containing PEG 1500 or 4000 + sodium sulfite + water at different temperatures. According to the results, an increase in the temperature promotes an increase in the STL (Figure 2a) for all ATPS’s. A possible explanation for this change is related with increased hydrophobicity of the PEG polymer molecules because of the increase in temperature and spontaneous diffusion of water molecules from the top phase to the bottom phase, resulting in an increase in the polymer composition in the upper phase and a reduction in the salt composition in the lower phase. In other words, an 384



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Table 4. Equilibrium Data (in Mass Fraction Percent) for the PEG 4000 (w1) + Sodium Sulfite (w2) + Water (w3) System at T = (288.15 to 318.15) K overall tie line

tie line length

w1

w2

top phase w3

1 2 3 4 5

26.28 29.90 34.39 37.68 40.27

13.90 16.17 18.48 19.67 20.56

8.78 9.56 10.35 11.01 11.72

77.32 74.27 71.17 69.32 67.72

1 2 3 4 5

27.64 30.78 33.33 35.33 38.59

14.81 16.39 17.90 19.36 20.75

7.90 8.25 8.70 9.12 9.49

77.29 75.36 73.40 71.52 69.76

1 2 3 4 5

25.27 30.14 35.55 37.93 41.16

14.81 16.60 17.90 19.40 20.80

7.12 7.62 8.12 8.69 9.20

78.07 75.78 73.98 71.91 70.00

1 2 3 4 5

26.29 30.43 34.28 36.55 38.44

13.81 15.28 16.84 18.24 19.79

6.28 6.79 7.28 7.77 8.27

79.92 77.93 75.87 73.99 71.93

w1 288.15 K 27.28 30.21 33.84 36.27 38.45 298.15 K 30.71 32.73 34.56 36.23 39.07 308.15 K 28.27 32.33 36.68 38.56 41.50 318.15 K 26.23 29.67 33.15 34.93 36.14

w3

w1

w2

w3

2.71 2.55 2.21 1.90 1.79

70.01 67.24 63.95 61.83 59.76

3.69 3.61 3.56 3.43 3.59

14.29 16.20 18.52 20.36 21.95

82.01 80.19 77.91 76.21 74.46

2.56 2.30 2.20 2.15 1.66

66.73 64.97 63.24 61.62 59.27

4.68 3.93 3.59 3.49 3.37

11.88 13.15 14.52 15.41 16.31

83.44 82.91 81.89 81.10 80.33

2.10 1.62 1.56 1.36 1.23

69.63 66.05 61.76 60.08 57.27

4.67 3.99 3.09 3.03 3.01

11.10 11.85 13.19 14.65 15.82

84.23 84.16 83.73 82.32 81.17

2.41 1.89 1.59 1.36 1.21

71.36 68.45 65.25 63.71 62.65

1.14 1.06 0.91 0.78 0.71

10.29 12.26 13.24 14.42 16.15

88.57 86.67 85.85 84.80 83.14

⎛ wb ⎞ ⎛ wt ⎞ ln⎜⎜ 3b ⎟⎟ = ln k 2 + r ln⎜ 3t ⎟ ⎝ w1 ⎠ ⎝ w2 ⎠

species distribution in a solution with a total sulfite concentration of 0.02 M and verified that for pH < 1 the sulfur dioxide (SO2) concentration dominates; between pH 1.5 and pH 6, the sum of the two bisulfite isomers (HSO3−, SO3H−) dominates; and in the alkaline pH range SO32− dominates, and there are minor amounts of NaSO3−. The amount of S2O52‑ is very small in all pH ranges. In this work, it was found that the pH of the upper and lower stages, at different temperatures of the ATPS’s in study, always remained between 8 and 10. Effect of PEG Molecular Weight. The effect of PEG molecular weight on phase separation was evaluated. Figure 3 shows the binodal curves at 298.15 K for the two ATPS’s composed of sodium sulfite and PEGs 1500 and 4000 g mol−1. As shown in this figure, the binodal curve for the PEG + sodium sulfite ATPS at 298.15 K moves closer to the origin with increasing molecular weight, thus requiring lower concentrations for phase separation. This effect may be caused by an increase in the hydrophobic character of a PEG polymer with a higher molecular weight compared with a PEG polymer with a lower molecular weight. This increase in the hydrophobicity will increase the incompatibility between the system components and will thus demand lower concentrations for phase separation. Similar results were obtained by other authors for other PEG− salt systems.27 Correlation of the Tie Lines. The Othmer−Tobias (eq 4) and Bancroft (eq 5) equations were used to correlate the tie-line compositions:19,28 ⎛ 1 − wb ⎞ ⎛ 1 − w1t ⎞ 2 ⎜⎜ ⎟⎟ k n ln⎜ ln ln = + ⎟ 1 t b ⎝ w1 ⎠ ⎝ w2 ⎠

bottom phase

w2

(5)

in which k1, n, k2, and r are the fitting parameters whose values were determined according to a least-squares criterion. All other variables are defined above. These equations have been widely used to correlate the tie line compositions of some similar ATPS with significant success. The fitting parameters of the eqs 4 and 5 together with the root-mean-square deviation (σ) and the linear coefficient of determination (R2) are shown in Table 7. As can be seen, these equations can be satisfactorily used to correlate the tie-line data of the investigated systems. Five tie lines were experimentally determined at each temperature; therefore, if we assume that degrees of freedom are equal to 3 and the critical value of Pearson’s correlation coefficient (Rc) is equal to 0.878 with a significance level of 0.05,29 one may conclude that there is an acceptable correlation between the tie-line experimental data and equation predictions. These results are in excellent agreement with both systems studied (with PEG 1500 or PEG 4000) and both equation models (eqs 4 and 5). With knowledge of the parameters of such models, it will be possible to determine the tie line for any given point of the binodal curve. Binodal Fitting. The binodal data for the PEG 1500 and the PEG 4000 two-phase systems are shown in Tables 3 and 5 and Figure 1. The binodal curves were fitted to two models, eqs 6 and 7, one with exponential term and the other with polynomial dependence:30 w1 = a exp(bw20.5 − cw23)

(4)

and

(6)

and 385



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a, b, and c in both equations were determined using the Simplex method31 with the objective function defined by sum of squared difference between experimental and calculated values. Based on the final value of the objective function after optimization step, which was less than 3 % as shown in Table 8, we can observe that both eqs 6 and 7 provide a good description of the binodal data for both ATPS, with PEG 1500 or PEG 4000. Thermodynamic Modeling. The theoretical models are a useful tool when equilibrium conditions cannot be studied experimentally, which is of great use in technological applications. For example, activity coefficient models, such as NRTL or UNIQUAC, are often used to find new tie lines from other experimental LLE data. The estimation of the model parameters is usually performed using the procedure described in the manuscript published by Stragevitch and d’Á vila.29 The previous procedure consists of the minimization of an objective function using the Simplex method31 and several direct calculations of the equilibrium curve. The LLE curve is determined through the optimization of a nonlinear equilibrium system: xti γti − xbi γbi = 0, in which i represent the three components of ternary mixture. However, a different method for the determination of the equilibrium molar fractions in both liquid phases will be discussed in this study. The direct solution to the problem is one step of the inverse problem, which can be achieved with the minimizing objective function by using some optimization strategies, such as simplex, neural network, or genetic algorithm.31 The Rachford−Rice equation32 can be used as an easy method to determine the molar fractions in both the liquid phases of an equilibrated system. This equation is a more robust and efficient method when compared with nonlinear equation systems obtained by equilibrium condition between the phases.29 The inverse procedure used here is shown below. The NRTL model for the activity coefficient, of the component i in the phase α, of ternary systems is expressed as follows17

Table 5. Binodal Data (in Mass Fraction Percent) for the PEG 4000 (w1) + Sodium Sulfite (w2) + Water (w3) System at T = (288.15 to 318.15) K 288.15 K

298.15 K

308.15 K

318.15 K

100w1

100w2

100w1

100w2

100w1

100w2

100w1

100w2

42.27 34.18 30.64 29.73 26.94 26.20 25.50 24.78 23.46 22.90 22.32 21.77 20.74 19.78 19.34 18.89 16.79 11.56 7.63 5.32 4.91 4.68 4.45 4.43 4.36 4.28 4.24 4.17 4.12 4.06

1.77 2.37 2.57 2.62 2.76 2.86 2.97 3.11 3.39 3.47 3.60 3.71 3.93 4.17 4.26 4.38 5.00 7.10 8.77 10.08 10.72 10.98 16.20 12.55 15.10 14.21 17.21 19.20 21.09 22.89

45.85 42.24 40.48 38.73 37.31 35.88 34.49 32.65 31.04 29.96 28.99 27.79 26.95 26.19 25.26 24.65 24.05 23.47 22.27 15.06 10.93 6.87 4.79 4.42 4.22 4.21 4.01 3.99 3.37 3.12

1.41 1.60 1.65 1.74 1.81 1.96 2.14 2.25 2.35 2.46 2.55 2.66 2.69 2.76 2.96 3.17 3.21 3.26 3.47 5.40 6.44 7.90 10.08 10.72 10.98 11.36 11.84 12.55 14.21 15.10

45.46 42.24 40.64 39.15 37.18 35.94 34.71 33.68 32.55 31.59 30.60 29.71 28.87 28.08 27.29 26.51 25.81 25.12 24.46 18.75 15.00 11.56 5.58 3.57 3.29 3.12 2.79 2.27 2.02 1.91

1.12 1.22 1.35 1.47 1.60 1.70 1.83 1.88 2.03 2.10 2.23 2.33 2.42 2.50 2.61 2.73 2.83 2.94 3.05 3.48 4.49 5.64 7.43 9.58 10.10 11.16 11.97 14.28 15.18 15.85

47.58 45.63 43.03 40.63 38.55 37.20 35.45 34.30 32.21 31.17 30.27 28.63 27.84 26.42 24.55 23.96 18.75 15.11 10.39 4.90 1.74 1.48 1.36 1.13 0.69 0.64 0.41 0.41 0.25 0.13

0.91 1.01 1.14 1.29 1.39 1.50 1.58 1.67 1.84 1.98 2.05 2.18 2.27 2.40 2.57 2.64 3.43 4.39 5.37 7.29 8.25 8.49 8.69 8.93 9.44 9.66 10.24 13.57 11.03 12.47

ln γiα =

∑j τjiGjixj ∑j Gjixj

+

⎡ Gx ⎛ ∑k τkjGkjxk ⎞⎤ ji j ⎢ ⎜ ⎟⎥ τ − ∑j ⎢ ⎜ ij ∑k Gkjxk ⎟⎠⎥⎦ ⎣ ∑k Gkjxk ⎝ (8)

when molar fractions xi of the i component are used. For this model

τij =

Δgij (9)

RT

and Gij = exp( −αijτij)

in which Δgij is an parameter that characterizes the enthalpic contribution in the mixture, αij is related with local composition assumption, and R = 1.987 cal·(mol·K)−1 represents the gas constant. The dependence of the data on the temperature can improve if Δgij is a function of the temperature T, such as Δgij = Aij + BijT in which Aij and Bij are empirical parameters. For a ternary system, there are 15 adjustable parameters; these parameters can be estimated from experimental LLE data. The NRTL model is restricted to weak electrolyte systems in which long-range interaction contribution can be considered to be negligible. In this case, all ionic contributions to the activity coefficient were incorporated into the parameter related to the

Figure 1. Binodal experimental data for the ○, PEG 1500 + sodium sulfite + water system and □, the 4000 + sodium sulfite + water system at a temperature of 298.15 K. The binodal fitting is indicated by the filled symbols and dotted curves.

w1 = a + bw20.5 + cw2

(10)

(7)

in which a, b, and c are fitting parameters, listed in Table 8. All the other variables are defined earlier in this article. The parameters 386



Article

Figure 2. (a) Binodal curves at ▲, 288.15 K and △, 318.15 K, and tie lines at ■, 288.15 K and □, 318.15 K for the PEG 1500 + sodium sulfite + water system. (b) Binodal curves at ▲, 288.15 K and △, 318.15 K, and tie lines at ●, 288.15 K and ○, 318.15 K for the PEG 4000 + sodium sulfite + water system.

Table 6. STL Values for Aqueous Systems Composed of PEG 1500 or PEG 4000 and Sodium Sulfite T/K tie line

288.15

298.15

308.15

PEG 1500 + Sodium Sulfite + Water −1.81 −1.83 −2.27 −1.85 −1.85 −2.20 −1.91 −1.81 −2.16 −1.87 −1.74 −2.07 −1.92 −1.78 −2.07 PEG 4000 + Sodium Sulfite + Water −2.04 −2.79 −2.62 −1.95 −2.65 −2.77 −1.86 −2.51 −2.89 −1.78 −2.47 −2.67 −1.73 −2.44 −2.64

1 2 3 4 5 1 2 3 4 5

318.15 −2.49 −2.42 −2.38 −2.43 −2.52 −3.18 −2.76 −2.77 −2.61 −2.37

salt component. This approach has been used in similar systems showing excellent agreement.32 For the UNIQUAC model, the activity coefficient are given by the sum of two terms: combinatorial and residual contributions,18 ln γiα = ln γiC + ln γi R

Figure 3. Effect of the molecular weight of PEG. Phase diagrams for the ATPS’s formed with water + sodium sulfite and either □, PEG 1500 or ●, PEG 4000 at 298.15 K.

and the surface area fraction of the component i, respectively, and are given by the following expressions q xi θi = n i ∑ j = 1 qjxj (15)

(11)

where each term can be written as ln γiC = ln

ϕi xi

+

ϕ n θ z qi ln i + Ii − i ∑ Ijxj xi j = 1 2 ϕi

⎡ n ln γi = qi⎢1 − ln(∑ τijθj) − ⎢⎣ j=1 R

n

∑ j

⎤ ⎥ n ∑k = 1 τkjθk ⎥⎦

(12)

and ϕi =

τijθj

(13)

rx i i n ∑ j = 1 rjxj

(16)

and the interaction parameter τij can be written as τij = exp(−Aij/ RT). In the UNIQUAC model, 12 parameters (r1, r2, r3, q1, q2, q3, A12, A21, A13, A31, A23, A32) must be determined, and coordination number z was fixed to value 10 during calculations. In this study,

and Ij = z/2(ri − qi) − (ri − 1); ri and qi are molecular parameters of a pure component, representing volume and surface area, respectively. The parameters θi and ϕi are the volume fraction 387



Article

Table 7. Values of Othmer−Tobias and Bancroft Constants, Root-Mean-Square Deviation, and Regression Coefficients Othmer−Tobias T

n

k1

R2

288.15 298.15 308.15 318.15

1.6252 1.0605 1.0381 1.6728

0.2125 0.4955 0.3749 0.1277

0.9821 0.9702 0.9895 0.9602

288.15 298.15 308.15 318.15

0.9774 0.9472 1.3473 0.9511

0.4625 0.3434 0.1452 0.3542

0.9997 0.9656 0.9563 0.9574

Bancroft σt1a

σt2

r

System: PEG 1500 + Sodium Sulfite + Water 0.1579 0.3528 0.5781 0.2645 0.3729 0.8988 0.1713 0.2614 0.9535 0.2621 0.7057 0.5538 System: PEG 4000 + Sodium Sulfite + Water 0.0492 0.0725 1.0461 0.3217 0.5443 1.0102 0.3527 0.9595 0.6858 0.4468 0.7807 1.0260

k2

R2

σb3

σt3

2.6787 2.0534 2.6452 3.5515

0.9800 0.9696 0.9906 0.9585

0.7887 1.1992 0.7627 1.4517

1.1244 1.0780 0.5882 2.0068

2.1455 3.1461 4.2258 3.0819

0.9996 0.9713 0.9475 0.9556

0.2991 1.7523 2.7416 3.0996

0.2358 1.3227 3.0943 2.2949

a α σi

= 100(∑ni=1(wαi (exp) − wαi (cal))2/n)1/2, where wαi represents the mass fraction of composition i in phase α and n represents the number of bimodal data points.

Table 8. Values of the Parameters of Equations 3 and 4, Root-Mean-Square Deviation, and Correlation Coefficients eq 3

a

T

a

b

288.15 298.15 308.15 318.15

0.4340 0.4472 0.4727 0.3333

−4.1261 −4.8849 −5.3033 −4.2680

288.15 298.15 308.15 318.15

0.6727 0.3321 0.2909 0.1319

−6.9198 −5.0410 −5.1023 −3.0714

eq 4 σa

c

b

c

σa

−1.0498 −1.1724 −1.2144 −0.8750

0.8099 1.0356 1.0820 0.7345

1.1201 1.3031 1.5756 0.7497

−1.1973 −0.6760 −0.6341 −0.2898

1.0668 0.5290 0.5076 0.1792

2.2462 0.6292 0.8387 0.6556

a

System: PEG 1500 + Sodium Sulfite + Water −3.2661 1.1029 0.3726 7.5394 1.1749 0.3688 10.6366 1.2678 0.3726 4.4771 0.6953 0.2946 System: PEG 4000 + Sodium Sulfite + Water 17.2881 2.1040 0.3598 4.2770 0.5052 0.2343 5.4729 0.6529 0.2152 −5.8030 0.6204 0.1251

cal 2 1/2 σ = 100(∑ni=1(wexp 1,i − w1,i ) /n) .

Table 9. Fitted NRTL Parameters, with Aij in cal·mol−1 and Bij in cal·(mol·K)−1

Table 11. Equilibrium Condition ATPS’s with PEG 1500 at T = 288.15 K and Using the UNIQUAC Model

i

j

Aij

Aji

Bij

Bji

αij

tie line

γt1xt1/γb1xb1

γt2xt2/γb2xb2

γt3xt3/γb3xb3

PEG1500 PEG4000 H2O PEG1500 PEG4000

Na2SO3 Na2SO3 Na2SO3 H2O H2O

−254.5 5911 −19.58 −46.26 1747

254.3 −216.9 171.5 −8.827 −2789

−0.7563 −1.881 −6.360 −8.630 −10.75

7.442 6744 14.89 20.77 25.98

0.49 0.50 0.32 0.42 0.50

1 2 3 4 5

1.2149 1.1535 1.0661 1.0452 0.9370

1.0686 1.0706 1.0379 1.0356 1.0023

0.9988 0.9989 0.9995 0.9994 1.0000

Table 10. Fitted UNIQUAC Parameters

Table 12. Comparison of Experimental Results and Predictions for TLL and STL at T = 298.15 K

parameters

ATPS (1)a

ATPS (2)b

r1 r2 r3 q1 q2 q3 A12 A21 A13 A31 A23 A32

1.0613 16.10 0.7935 1.116 1.346 0.3597 −3115 2734 −1180 11168 1356 −2687

15.54

tie line

TLLexp

TLLmodel

error %

STLexp

STLmodel

error %

12.19

1 2 3 4 5

31.8 33.4 35.4 37.2 39.3

30.5 33.8 34.7 36.9 39.5

4.1 0.99 2.1 0.68 0.69

−1.82 −1.85 −1.81 −1.74 −1.78

−1.71 −1.87 −1.78 −1.76 −1.83

6.2 1.5 1.4 1.1 2.9

−3732 −4655 10534 −3556

the structural ri and qi and interaction parameters Aij have been adjusted to reproduce LLE data, and thus the UNIQUAC model has become purely empirical. In this article, the NRTL and UNIQUAC were used to correlate the data from the tie lines for the studied system. The procedure began by choosing the initial conditions for the model parameters by examining similar systems in the literature. Thereafter, the model was used to determine the activity coefficients for a given overall composition. Then, the objective function,

a

ATPS (1): PEG 1500 (1) + Na2SO3 (2) + water (3). bATPS (2): PEG 4000 (1) + Na2SO3 (2) + water (3).

388


Journal of Chemical & Engineering Data n

F′ =

∑ i

(ki − 1)xi =0 (ki − 1)θ + 1

Article

δw = 100 F ″ /2nml (17)

in which the degree of agreement between the experimental data and that calculated by the thermodynamic model is measured. Using the NRTL model, the results obtained for PEG 1500 and PEG 4000 systems are 0.54 % and 1.28 %, respectively. For UNIQUAC model, the mean deviation values were 0.36 % for the PEG 1500 system and 1.68 % for the PEG 4000 system. In both models, the mean deviation is below the value considered in the literature for adequate agreement (3 %).32 Thus, the NRTL and UNIQUAC models are well-fitted with the experimental data indicating its feasibility for describing the behavior of other equilibrium systems.

in which 0 ≤ θ ≤ 1, ki = and and are the activity coefficients in the top and bottom phases, respectively, for each component. Equation 17 was minimized using the Newton− Raphson procedure with respect to the parameter θ, which is the mole fraction of phases. With a known value of θ, we can use eqs 18 and 19 to calculate the mole fractions of the components in both phases. yti /ybi ,

xit =

kixi (ki − 1)θ + 1

xib =

xi (ki − 1)θ + 1

yti

ybi

■

(18)

CONCLUSION Experimental ATPS data were acquired for a PEG 1500 + sodium sulfite + water system and a PEG 4000 + sodium sulfite + water system at (288.15, 298.15, 308.15, and 318.15) K. In the evaluation of data, it was observed that the effect of the temperature on the phase equilibrium promotes an increase in the STL. This result indicates that the PEG molecules become more hydrophobic with increasing temperature. An increase in the molar mass of the polymer increased the hydrophobic character of the phase. This effect will result in an increase in the segregation between the system components, which will decrease the concentrations of salt and water that are required for phase separation. To avoid errors, the experimental equilibrium data were correlated using the Othmer−Tobias correlation. The results are very satisfactory and show the consistency of the experiments that were conducted. The data were correlated with the NRTL and UNIQUAC models to determine the activity coefficient. The results of both thermodynamics models are very satisfactory.

(19)

To compare the calculated data with the experimental data, the molar fractions are transformed into mass fractions: M(Na2SO3) = 126 g·mol−1, M(H2O) = 18 g·mol−1 and M(PEG) = (1500 or 4000) g·mol−1. Therefore, the direct step can be solved using the Rachford−Rice equation. This step represents a direct method, in which phase composition in equilibrium curve can be obtained knowing the parameters of the thermodynamic model. The retrieval of the parameter model from the experimental data represents an example of an inverse problem, which is another part of our computational program. Finally, the concentrations of the phase components obtained using the thermodynamic model were then compared with the experimental data. Subsequently optimal parameters were found by minimizing the objective function (eq 20) via the Simplex method: n

2

F″ =

m

l

■

∑ ∑ ∑ ∑ [(wit(exp) − wit(cal))2 PEG T

+

tie line

(wib(exp)

wti

−

i

wib(cal))2 ]

(21)

AUTHOR INFORMATION

Corresponding Author

(20)

*Tel.: +55-35-3299-1260. Fax: +55-35 3299 1384. E-mail: [email protected].

wbi

in which and represent the experimental and calculated mass fractions of each component phase, in the top and bottom phases. In the above equation summations ∑2PEG, ∑nT, ∑mtie line, and ∑il represent the contribution of both systems at temperatures of 288.15 K, 308.15 K, and 318.15 K, at all tie lines, and all components, respectively. This way, the fit was obtained by imposing the same value for the same parameter in both systems. Using the Simplex method to solve this problem is common in literature, and it provides good results when compared with other methods. All parameters of the NRTL and UNIQUAC model for the studied system obtained using this procedure are listed in Tables 9 and 10, respectively. The use of Rachford-Rice equation, although polymers are present in ATPS, was adequate to find equilibrium condition γαi xαi = γβi xαi in the end, as shown in Table 11. Similar results can be found in all temperature data sets. The quality of fit of the UNIQUAC model and optimization method was also evaluated by prediction of liquid−liquid equilibrium curve at a new temperature of 298.15 K, that is different from those that were used to optimize (288.15, 308.15, and 318.15) K. This prediction was then compared with experimental data at 298.15 K, as shown in Table 12. The comparison of experimental results and predictions for TLL and STL shows that the model may be used in determining equilibrium curve at different temperatures. The mean deviation δw was calculated to quantitatively evaluate the quality of fit of the thermodynamic model:

Funding

We gratefully acknowledge the Fundaçaõ de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) for financially supporting this project. Notes

The authors declare no competing financial interest.

■

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Measurement and Correlation of the Phase Equilibrium of Aqueous

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