Karen Silva

Benjamin Katryniok

Marcia Carolina

Araque Marin

Marcia Araque

2-one (2), and 4-Methyl-pentan-2-one (1)-4-Methyl-pentan-2-ol (2) at 101.32 kPa

Keywords: Vapor-liquid equilibrium, Allyl alcohol, 2-hexanol, MIBC, UNIQUAC Model

    Isobaric Vapor-Liquid Equilibrium Data for Six Binary Systems: Prop-2-en-1-ol (1)-Hexan-2-ol (2), Prop-2-en-1-ol (1)-Hexan-2-one (2), Hexan-2-one (1)-Hexan-2-ol (2), Prop-2-en-1-ol (1)-4-Methyl-pentan-2-ol (2), Prop-2-en-1-ol (1)-4-Methyl-pentan-

INTRODUCTION

The Deoxydehydration (DODH) of glycerol is an alternative and sustainable reaction, for the synthesis of allyl alcohol which has recently gained much attention [START_REF] Tazawa | Deoxydehydration with Molecular Hydrogen over Ceria-Supported Rhenium Catalyst with Gold Promoter[END_REF][2][3] [START_REF] Vargas | Deoxydehydration of glycerol to allyl alcohol catalyzed by ceria-supported oxide[END_REF]. Generally, a DODH reaction removes two adjacent hydroxyl groups from vicinal diols to afford the corresponding alkene, in the current case, allyl alcohol from glycerol [5][6]. This reaction is typically catalyzed by rhenium based catalyst and requires a stoichiometric reductant such as H2 or a secondary alcohol [START_REF] Boucher-Jacobs | Catalytic deoxydehydration of glycols with alcohol reductants[END_REF]. The reductant-solvent acts as hydrogen donor and become oxidized to the corresponding ketone and water is also produce. Considering the application of allyl alcohol as a monomer in the chemical industry, it is imperative to isolate it in high purity from the reaction mixture containing the solvent, the co-produced ketone and water. Hence, in order to develop separation and purification strategies for allyl alcohol from the DODH of glycerol, high precision experimental phase equilibrium data are required [START_REF] Duran | Vapor-liquid equilibrium of the ethanol + 3-methyl-1-butanol system at 50.66, 101.33 and 151.99 kPa[END_REF] [START_REF] Moodley | Isobaric Vapor-Liquid Equilibrium Data for Water (1) + 2-Methyl-propan-1-ol (2), 2-Methyl-propan-1-ol (1) + Pyridine (2), and Water (1) + 2-Methyl-propan-1-ol (2) + Pyridine (3) Systems[END_REF].

The Vapor-liquid equilibrium of allyl alcohol/water is already reported in the literature [START_REF] Li | Vapor-Liquid Equilibrium for Binary Systems of Allyl Alcohol + Water and Allyl Alcohol + Benzene at 101.3 kPa[END_REF] and vapor-liquid-liquid equilibrium of solvent/water and ketone/water binary systems are in Aspen database. Thus, this work focuses on the experimental measuring of VLE data for the binary mixtures of Allyl alcohol/2-hexanol, Allyl alcohol/2-hexanone, 2-hexanone/2-hexanol, Allyl alcohol/MIBC, Allyl alcohol/MIBK, and MIBK/MIBC at 101.32 kPa, in order to calculate the adjustable binary interaction parameters from local-composition models [START_REF] Sánchez | Isobaric Vapor Liquid Equilibria for Binary Mixtures of Isoamyl Acetate + Ethyl Acetate at 50 and 100 kPa[END_REF]. In order to validate the accuracy and quality of the VLE data, the Herington [START_REF] Herington | Tests for the consistency of experimental isobaric vapour-liquid equilibrium data[END_REF][13], Van Ness [START_REF] Van Ness | Vapor-liquid equilibrium: Part I. An appraisal of data reduction methods[END_REF] pure component consistency [START_REF] Kang | Quality assessment algorithm for vapor-liquid equilibrium data[END_REF] and Redlich-Kister [START_REF] Kojima | Thermodynamic consistency test of Vapor-liquid equilibrium data[END_REF] total area tests were implemented to verify the thermodynamic consistency of VLE data sets before fitting using the universal quasichemical (UNIQUAC) [START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF], non-random two-liquid (NRTL) [START_REF] Renon | Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures[END_REF] and Wilson [START_REF] Wilson | Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing[END_REF] thermodynamic models.

EXPERIMENTAL SECTION

Materials

All of the experimental chemicals used in this study are listed in Table 1 . The masse fraction reported by the manufacturer was confirmed by gas chromatographic analysis (GC) equipped with a flame ionization detector (FID), using the area method. Furthermore, a comparison between measured boiling temperatures and reported values from literature was carried out. As given in Table 1, close values were noted; the maximum boiling point deviation for Allyl alcohol, 2-hexanol, 2-hexanone, MIBC and MIBK are 0.23 K, 0.27K, 0.58K, 0.1K and 0.17K, respectively. The experimental pressure for the measurement of boiling temperature is 101.32 kPa; standard uncertainties of experimental pressure and temperature are u(Tb) = 0.1K u(P) = 0.1 kPa. b Gas chromatography.

VLE apparatus and procedure

The binary VLE data were determined using an equilibrium apparatus Fischer VLE 602 equipped with a Cottrell circulation pump (Figure 1). Accuracy in measurements of temperature and pressure are ±0.1 K and 0.1 kPa, respectively. The operation procedure is based on the principle of circulation method, to ensure an intimate contact between liquid and vapor phases during the boiling process. At the beginning, the mixed chamber (5) and the side reservoir (1) are loaded with the less volatile pure component, then, different binary mixtures were prepared as needed by controlled addition of the second component to the equilibrium chamber via liquid-phase septum [START_REF] Sánchez | Isobaric Vapor Liquid Equilibria for Binary Mixtures of Isoamyl Acetate + Ethyl Acetate at 50 and 100 kPa[END_REF]. For each equilibrium point, the pressure was fixed and the heating and stirring systems were turned on. The heating supply is adapted so that the reflux flow of the vapor phase is 1-2 drops per second. The apparatus was operated at constant pressure until equilibrium was reached. Equilibrium condition is assumed when temperature variations are less than 0.1 K for at least 30 min at constant pressure. The temperature and pressure variations are monitored in real time by statistical control charts. Once the equilibrium condition attainted, samples of liquid and condensed vapor were taken for gas chromatography analysis. All experiments were carried out at 101.13 kPa of pressure. Between 15 to 22 equilibrium points were generated for each binary system. To verify the performance of the apparatus, vapor pressure data of pure components were also measured, and the coefficients of Antoine extended equation were fitted. Approximately 18 to 20 equilibrium points for each pure component were measured.

Sample Analysis

The concentrations of Liquid and condensed vapor samples were determined by gas chromatography using a GC instrument equipped with a column CP-Wax52CB (30m x 250 µm x 0.25 µm) and a flame ionization detector (FID). The initial oven temperature was programmed at 40°C for 2 min, with a heating ramp of 10°C/min, up to 100°C. The injection port temperature was maintained at 250°C using split injection mode adjusted to 140:1 ratio. In each analysis, 1µL of sample was injected. VLE samples of 20 µL were diluted in 980 µL of Tetrahydrofuran (THF). 20 µL of benzene was added in each sample as internal standard. The uncertainty in the concentration measurements was determined by analyzing solutions with known composition, using the calibration curves of the chromatographic technique to this end. Ten calibration standards were prepared gravimetrically per each binary system. A maximum standard uncertainty of ± 0.001 in the mole fraction of each component was estimated.

RESULTS AND DISCUSSION

Vapor pressure correlation of pure components

Despite the availability of vapor pressure data for Allyl alcohol, 2-hexanol, 2-hexanone, MIBC and MIBK in Aspen physical properties database, measurement of the vapor pressure for this components was performed, bearing in mind that this information is fundamental for a proper application of the consistency tests [START_REF] Wisniak | An Assessment of Thermodynamic Consistency Tests for Vapor-Liquid equilibrium data[END_REF] as well as checking the reliability of the equilibrium apparatus used in this work. The experimental vapor pressures of the components are present in Table 2. Numerical data regression was carried out to fit the vapor pressure data with Extended Antoine equation (Eq 1) and the corresponding coefficients are presented in Table 3. A maximum average error of 0.004 kPa between experimental and calculated vapor pressures was observed. As showed in Figure 2, vapor pressure data also agree with the values previously reported (NIST database).

Vapor-liquid equilibrium (VLE) model

VLE experimental data for the binary systems were used to calculate the activity coefficients (𝛾𝛾 𝑖𝑖 ). The VLE relationship at moderate pressure can be expressed as showed in Eq 2 [START_REF] Liu | Vapour-liquid equilibrium measurements and correlation for separating azeotropic mixture (ethyl acetate + n-heptane) by extractive distillation[END_REF]. Due to the fact that the experimental data was obtained at low pressure (101.32 kPa), vapor phase behaviour was considered as ideal gas, and therefore, fugacity coefficient (𝜑𝜑 𝑖𝑖 𝑣𝑣 ) and pointing factor were fitted to one [START_REF] Ma | Isobaric vapour-liquid equilibrium measurements and extractive distillation process for the azeotrope of (N,Ndimethylisopropylamine + acetone)[END_REF]. Thus, VLE expression can be simplified as showed in Eq 3. Where xi and yi are the composition of the liquid and vapor phases, respectively, P is the total system pressure. 𝑃𝑃 𝑖𝑖 𝑠𝑠 is the vapor pressure of pure component i at the system temperature. 𝑃𝑃 𝑖𝑖 𝑠𝑠 was calculated using the Antoine coefficients previously fitted with our experimental data. Experimental VLE data with the corresponding experimental activity coefficients are listed in To evaluate the ideality deviations of the systems under the same composition and pressure conditions, excess Gibbs energy (G E ) was determined by Eq 4 [START_REF] Moodley | Isobaric Vapor-Liquid Equilibrium Data for Water (1) + 2-Methyl-propan-1-ol (2), 2-Methyl-propan-1-ol (1) + Pyridine (2), and Water (1) + 2-Methyl-propan-1-ol (2) + Pyridine (3) Systems[END_REF], where R is the universal gas constant, and T is the system temperature. Taking into account that 𝛾𝛾 1 and 𝐺𝐺 𝐸𝐸 values are above unity and zero (Table 456), respectively. Positive deviations for all binary systems are evidenced. 𝐺𝐺 𝐸𝐸 for the six binaries have been correlated by a sixth order polynomial equation and plotted in Figure 3. The mixtures of Allyl alcohol/2-hexanone and Allyl alcohol/MIBK showed the biggest nonideality from Raoul's law compared to those of the other four mixtures. The largest value of the G E is observed when the mixture have equal mole fraction in allyl alcohol and corresponding ketone. Taking all systems, G E follow the order Allyl alcohol/2-hexanone > Allyl alcohol/MIBK > MIBK/MIBC > 2-hexanone/2-Hexanol > Allyl alcohol/2-Hexanol > Allyl alcohol/MIBC.

𝐺𝐺 𝐸𝐸 = 𝑅𝑅𝑇𝑇 (𝑥𝑥 1 𝐿𝐿𝐿𝐿𝛾𝛾 1 + 𝑥𝑥 2 𝐿𝐿𝐿𝐿𝛾𝛾 2 ) Eq 4

Thermodynamic consistency test

Thermodynamic consistency tests allow the assessment of experimental VLE data on the basis of Gibbs-Duhem equation (Eq 5-6) [START_REF] Kang | Quality assessment algorithm for vapor-liquid equilibrium data[END_REF] [START_REF] Wisniak | An Assessment of Thermodynamic Consistency Tests for Vapor-Liquid equilibrium data[END_REF]. The use of several consistency tests is always recommended to check -in the proper manner -the quality of the data, given that, contradictory consistency results can inform one about an incorrect application of a particular test [START_REF] Wisniak | An Assessment of Thermodynamic Consistency Tests for Vapor-Liquid equilibrium data[END_REF]. In this work, Redlich-Kister total area test, Herington Redlich-Kister total area test (1948) assumed the volume and heat effects of mixing (ε) are negligible [START_REF] Wisniak | The Herington Test for Thermodynamic Consistency[END_REF] (Eq 7 -8). According with Kang et al. [START_REF] Kang | Quality assessment algorithm for vapor-liquid equilibrium data[END_REF], a polynomial relation with the form presented in Eq 9, with n , between 4 and 6, can be used for regressing data 𝐿𝐿𝐿𝐿 � 𝛾𝛾 1 𝛾𝛾 2 �. Nevertheless, due to the absence at the infinite dilution zones, as reported by Duran et al. [START_REF] Duran | Vapor-liquid equilibrium of the ethanol + 3-methyl-1-butanol system at 50.66, 101.33 and 151.99 kPa[END_REF], Margules Activity Model was used in the pure components zones. The corresponding parameters A12 and A21, were obtained by the data regression of Excess Gibbs energy of a binary system [START_REF] Smith | Introduction to Chemical Engineering Thermodynamics, 7th editio[END_REF] (Eq 10), the values were summarized in Table 7. According with Kojima et al [START_REF] Kojima | Thermodynamic consistency test of Vapor-liquid equilibrium data[END_REF], for |A*|< 0.03, the test is passed. In addition, the tool for thermodynamic consistency available in Aspen Plus V9 was used [START_REF]and Aspen Plus V 8.4[END_REF]. This latter is quite similar to Redlich-Kister area test [START_REF] Duran | Vapor-liquid equilibrium of the ethanol + 3-methyl-1-butanol system at 50.66, 101.33 and 151.99 kPa[END_REF][16] [START_REF]and Aspen Plus V 8.4[END_REF]. x 1 As reported Kurihara et al [START_REF] Kurihara | Evaluation of thermodynamic consistency of isobaric and isothermal binary vapor-liquid equilibrium data using the PAI test[END_REF], ε for isobaric systems can be large (as high as 0.04) and neglect it, is not recommended. Herington proposed an semi-empirical approach of this term by use of the total boiling range of the mixture [START_REF] Herington | Tests for the consistency of experimental isobaric vapour-liquid equilibrium data[END_REF] (Eq 11-12), where γ 1 and γ 2 are the activity coefficients which were calculated by Eq 3, and A, B are the areas above and below of the abscissa axis in 𝐿𝐿𝐿𝐿 � 𝛾𝛾 1 𝛾𝛾 2 � vs 𝑥𝑥 1 (area test plot Figure S1). If deviation from experimental errors (|𝐷𝐷 -𝐽𝐽|) are less than 10, the corresponding VLE data set can be judged as thermodynamically consistent.

𝐴𝐴 * = ∫ 𝑙𝑙𝐿𝐿 𝛾𝛾 1 𝛾𝛾 2 𝑑𝑑𝑥𝑥 + ∫ 𝜀𝜀 𝑑𝑑𝑥𝑥 1 0 1 0 , 𝜀𝜀 = -� 𝐻𝐻 𝐸𝐸 𝑅𝑅𝑇𝑇 2 � � 𝜕𝜕𝑇𝑇 𝜕𝜕𝑥𝑥 1 � 𝑃𝑃 , 𝜀𝜀 = -� 𝑉𝑉 𝐸𝐸 𝑅𝑅𝑇𝑇 � � 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥 1 � 𝑇𝑇 Eq 7 𝐴𝐴 * = ∫ 𝑙𝑙𝐿𝐿 𝛾𝛾 1 𝛾𝛾 2 𝑑𝑑𝑥𝑥 1 0 Eq 8 𝐿𝐿𝐿𝐿 � 𝛾𝛾 1 𝛾𝛾 2 � = 𝑘𝑘 𝑛𝑛 𝑥𝑥 1 𝑛𝑛 + 𝑘𝑘 𝑛𝑛-1 𝑥𝑥 1 𝑛𝑛-1 + ⋯ + 𝑘𝑘 1 𝑥𝑥 1 + 𝑘𝑘 0 Eq 9 𝐺𝐺 𝐸𝐸 𝑅𝑅𝑇𝑇𝑥𝑥 1 𝑥𝑥 2 2 = 𝐴𝐴 21 𝑥𝑥 1 𝑥𝑥 2 + 𝐴𝐴 12 , ln𝛾𝛾 1 ∞ 𝑥𝑥 1 →0 = 𝐴𝐴 12 , , ln𝛾𝛾 2 ∞ 𝑥𝑥 2 →0 = 𝐴𝐴 21
𝐷𝐷 = 100 � 𝐴𝐴 -𝐵𝐵 𝐴𝐴 + 𝐵𝐵 � = 100 �∫ 𝐿𝐿𝐿𝐿 � 𝛾𝛾 1 𝛾𝛾 2 � 𝑑𝑑𝑥𝑥 1 1 0 � ∫ �𝐿𝐿𝐿𝐿 � 𝛾𝛾 1 𝛾𝛾 2 �� 𝑑𝑑𝑥𝑥 1 1 0 Eq 11 𝐽𝐽 = 50 � ∆𝐻𝐻 𝑚𝑚 ∆𝐺𝐺 𝐸𝐸 � � 𝑇𝑇 𝑚𝑚𝑎𝑎𝑥𝑥 -𝑇𝑇 𝑚𝑚𝑖𝑖𝑛𝑛 𝑇𝑇 𝑚𝑚𝑖𝑖𝑛𝑛 � , � ∆𝐻𝐻 𝑚𝑚 ∆𝐺𝐺 𝑚𝑚 𝐸𝐸 � = 3 Eq 12
Wisniak [START_REF] Wisniak | The Herington Test for Thermodynamic Consistency[END_REF] slightly modified the Herington criteria; ∆𝐻𝐻 𝑚𝑚 data were correlated against ∆𝐺𝐺 𝑚𝑚 𝐸𝐸 from experimental data with good results using the Eq 13. In this work, the data was calculated from experiments, using Eq 3 -4 and plotted in Figure 3. Each data set was correlated by a sixth order polynomial equation and the maximum value of excess free energy of mixing (∆G 𝑚𝑚 𝐸𝐸 ) was calculated by derivation. ∆𝐻𝐻 𝑚𝑚 ∆G 𝑚𝑚 𝐸𝐸 �

Values were summarized in Table 8.

∆H m (J mol -1 ) = -237.02 + 1.3863 ∆𝐺𝐺 𝑚𝑚 𝐸𝐸 (J mol -1 )

Eq 13

The Van Ness method [START_REF] Van Ness | Vapor-liquid equilibrium: Part I. An appraisal of data reduction methods[END_REF] is a suitable method for testing the sensitive to the selected model as well the reliability of VLE values, thought the mean absolute deviation between experimental data and calculated values by a given thermodynamic model [START_REF] Ma | Isobaric vapour-liquid equilibrium measurements and extractive distillation process for the azeotrope of (N,Ndimethylisopropylamine + acetone)[END_REF]. According with the obtained results given in Table 8, all binaries were found consistent with the Redlich Kister area test (|A*|<0.03), Aspen plus tool, Modified Herington test (|𝐷𝐷 -𝐽𝐽| < 10) and Van Ness method (Δy < 1) and ΔT< 1). Despite the fact that the Herington test has been successfully implemented in similar systems containing non-polar compound and alcohols [START_REF] Wisniak | The Herington Test for Thermodynamic Consistency[END_REF][32] [33][34], in this study, from 6 data sets, only 2-hexanone (1)/2-hexanol (2) binary was found consistent |𝐷𝐷 -𝐽𝐽| = 0.99. According to the literature, some researchers have also shown data sets that failed Herington test but passed Redlich-Kister area test [START_REF] Duran | Vapor-liquid equilibrium of the ethanol + 3-methyl-1-butanol system at 50.66, 101.33 and 151.99 kPa[END_REF][35] [START_REF] Resa | Isobaric vapor-liquid equilibria of acetonepropyl ether and isopropyl ether-propyl ether systems. Corroboration of no reverse volatility[END_REF]. In our case, this failure in the Herington test, has been also corrected by leaving out value of ∆𝐻𝐻 𝑚𝑚 ∆𝐺𝐺 𝑚𝑚 𝐸𝐸 � = 3 (Wisniak criterion Eq 13). Additionally to the requirements associated to the Gibbs-Duhem equation, consistency between the end-points of VLE data sets (i.e., mole fraction of 0 and 1) and the vapor pressures of pure components must be applied [START_REF] Kang | Quality assessment algorithm for vapor-liquid equilibrium data[END_REF]. The pure component consistency test is presented in Eq 16 and Eq 17. Where 𝑑𝑑 1 0 and 𝑑𝑑 2 0 are the pure component vapor pressures, 𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑙𝑙𝑒𝑒 is the bubble point pressure. Consistency test is passed if ∆𝑃𝑃 1 0 and ∆𝑃𝑃 2 0 are less than 1. As presented in Table 9, all the experimental data sets pass this criterion.

∆𝑃𝑃 1 0 = � 𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑙𝑙𝑒𝑒 (𝑥𝑥 1 → 1) -𝑑𝑑 1 0 𝑑𝑑 1 0 � Eq 16 ∆𝑃𝑃 2 0 = � 𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑙𝑙𝑒𝑒 (𝑥𝑥 1 → 0) -𝑑𝑑 2 0 𝑑𝑑 2 0 � Eq 17

VLE data correlation

The measured data for all binary systems were correlated by three thermodynamic models, UNIQUAC, NRTL and Wilson, using Aspen plus V11 regression tool. The corresponding interaction parameters were obtained by minimizing the maximum likelihood objective function (OF), defined by Eq 18. This OF consist on the sum of differences of measured and predicted variables [START_REF] Sánchez | Isobaric Vapor Liquid Equilibria for Binary Mixtures of Isoamyl Acetate + Ethyl Acetate at 50 and 100 kPa[END_REF], where P and T are the equilibrium pressure and temperature, respectively; xi and yi the liquid and vapor mole fraction, respectively; N and 𝜎𝜎 are the number of data point and the standard deviation, respectively. According to Renon and Prausnitz [START_REF] Renon | Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures[END_REF], the non-randomness parameter (𝛼𝛼 𝑖𝑖𝑖𝑖 ) was set at 0.3. Concerning the UNIQUAC model, volume (r) and area (q) parameters for each component are given in Table 10. Eq 18 Table 10. Structural parameters r and q for UNIQUAC model a

𝑂𝑂𝑂𝑂 = �
To evaluate the difference between the experimental data and the calculated results, the root-meansquare deviation (RMSD) of equilibrium temperature and composition was calculated, according to equations 19 and 20, respectively. The corresponding binary coefficients of the three thermodynamic models for the six systems and those of RMSD are given in Table 10. For all binary systems, one can observe that RMSD (T) and RMSD (y) values are less than 0.24 K and 0.0089, respectively, which indicates that all binary systems have been adjusted successfully by the three thermodynamic models. The quality of regression could be also confirmed through the correlation of experimental and calculated activity coefficients at 101.kPa, since these last coefficients are determined on the basis of all parameters involves in the objective function (Eq 18). The experimental data and regression results obtained by UNIQUAC, NRTL and Wilson, for all binary systems are shown in Figure 4, 5 and 6. As shown, Txy diagrams have been adjusted successfully in all composition range unlike experimental activity coefficients in the infinite dilution zones. The latter is caused by high uncertainties in the GC characterization at high (>0.9) and low (<0.1) mole fractions [START_REF] Wisniak | An Assessment of Thermodynamic Consistency Tests for Vapor-Liquid equilibrium data[END_REF]. considering that activity coefficients values at low pressures depends of K-values, system pressure and equilibrium temperature (Eq 3), this difference is possibly due to a slight shift in these parameters.

𝑅𝑅𝑀𝑀𝑅𝑅𝐷𝐷(𝑇𝑇

CONCLUSION

Isobaric vapor-liquid equilibrium (VLE) data for Allyl alcohol/2-hexanol, Allyl alcohol/2hexanone, 2-hexanone/2-hexanol, Allyl alcohol/MIBC, Allyl alcohol/MIBK, and MIBK/MIBC systems were measured at 101.32 kPa. Non azeotrope was formed in these six binary systems and positive deviations from Raoul's law were found in all systems, following the order Allyl alcohol/2-hexanone > Allyl alcohol/MIBK > MIBK/MIBC > 2-hexanone/2-Hexanol >Allyl alcohol/2-Hexanol > Allyl alcohol/MIBC. Where Allyl alcohol/2-hexanone and Allyl alcohol/MIBK showed the biggest non-ideality compared to those of the other four mixtures. All measured VLE data sets passed the thermodynamic consistency with Herington (Wisniak criterion), Van Ness, pure component consistency and Redlich-Kister total area tests. Furthermore, all systems were successfully fitted by UNIQUAC, NRTL and Wilson thermodynamic models, and the corresponding binary coefficients were correlated. MIBK/MIBC system was compared with existing binary coefficients in Aspen data based (NISTV90 NIST-IG and NISTV90 NIST-RK), using UNIQUAC model.

Figure 1 .

 1 Figure 1. Apparatus Fisher VLE 602 used in this work. 1: Reservoir with immersed rod heater. 2: Condenser. 3: Vapor-phase temperature sensor. 4: Liquid-phase temperature sensor. 5: Mixer chamber. 6: Coolant water connection. 7: Pressure control line. 8: Liquid sampling port. 9: Vapor sampling port. 10: Condenser. 11: Liquid-phase septum. 12: Vapor-phase septum.

  test, Modified Herington test by Wisniak and Van Ness test were employed.

Figure 4 .

 4 Figure 4. Phase diagram Txy (left) and plot of Lnɣ1, Lnɣ2 and G E /RTx1x2 vs x1 at 101.13 kPa (right) for Allyl Alcohol/2-Hexanol and Allyl Alcohol/MIBC binaries. (□) Experimental data (○) Experimental Lnɣ1, Lnɣ2 (•) Experimental GE/RTx1x2. Green solid line, red dotted line and blue points correspond to UNIQUAC, NRTL and Wilson adjustment.

Figure 5 .Figure 6 .

 56 Figure 5. Phase diagram Txy (left) and plot of Lnɣ1, Lnɣ2 and GE/RTx1x2 vs x1 at 101.13 kPa (right) for Allyl Alcohol/2-Hexanone and Allyl Alcohol/MIBK binaries. (□) Experimental data (○) Experimental Lnɣ1, Lnɣ2 (•) Experimental GE/RTx1x2. Green solid line, red dotted line and blue points correspond to UNIQUAC, NRTL and Wilson adjustment

Figure 7 .

 7 Figure 7. Diagram (x-y) (left) and plot of Lnɣ1, Lnɣ2 and G E /RTx1x2 vs x1 at 101.13 kPa (right) for MIBK/MIBC system. Green solid line, red dotted line and blue points correspond to simulated equilibrium using UNIQUAC parameters from this work and Aspen data based, respectively.

Figure 7

 7 presents the phase equilibrium diagram x-y at 101.32 kPa and the correlation of the activity coefficients. As observed, this work agrees with the vapor-liquid distribution ratios (K-values), using UNIQUAC parameters from the Aspen data based. Nevertheless, a deviation in the activity coefficients has been identified. Hence,

Table 1 .

 1 Experimental components information.

						Tb [K] a	Analysis
	Component	CAS	Suppliers	Mass fraction	This work	Lit	method
	Allyl Alcohol	107-18-6	Sigma-Aldrich	≥ 0,990	369.98	369.75 [10]	GC b
						370.07 [20]
						369.94 [21]
	2-hexanol	626-93-7	Sigma-Aldrich	≥0,990	412.77	413.04	GC b
						[22][23]
	2-hexanone	591-78-6	Acros Organics	≥0,980	400.12	400.7[23]	GC b
	4-methyl-2-pentanol (MIBC)	108-11-2	Sigma-Aldrich	≥ 0,980	404.75	404.85[23]	GC b
	4-methyl-2-pentanone (MIBK)	108-10-1	Sigma-Aldrich	≥ 0.995	388.83	389[23]	GC b
	a

Table 2 . Measured vapor pressure data for all components Pressure (kPa) Temperature (K)

 2

			Allyl alcohol 2-hexanol 2-hexanone MIBC	MIBK
		101.32	369.98	412.77	400.12	404.75	388.83
		95	368.22	410.51	397.64	402.78	386.61
		90	366.78	408.88	395.78	401.14	384.78
		85	365.27	407.15	393.9	399.45	382.87
		80	363.69	405.28	391.87	397.66	380.88
		75	362.01	403.41	389.82	395.78	378.76
		70	360.26	401.41	387.53	393.82	376.53
		65	358.39	399.28	385	391.73	374.18
		60	356.38	397.16	382.49	389.5	371.68
		55	354.26	394.65	379.79	387.14	368.98
		50	351.96	391.98	376.8	384.6	366.11
		45	349.43	389.21	373.75	381.84	363
		40	346.68	386.15	370.49	378.83	359.58
		35	343.63	382.61	366.59	375.5	356
		30	340.2	378.87	362.37	371.76	351.57
		25	336.23	374.64	357.44	367.48	346.71
		20	331.54	369.48	351.06	362.46	341
		a u(T) =0.036K and u(P)=0.004 kPa
	𝐿𝐿𝐿𝐿 � 𝑃𝑃 𝑖𝑖 𝑠𝑠 � 𝑘𝑘𝑃𝑃𝑘𝑘	� = 𝐶𝐶 1 +	𝐶𝐶 2 𝑇𝑇 + 𝐶𝐶 3	+ 𝐶𝐶 4 𝑇𝑇 + 𝐶𝐶 5 𝐿𝐿𝐿𝐿(𝑇𝑇) + 𝐶𝐶 6 𝑇𝑇 𝐶𝐶 7 𝑓𝑓𝑓𝑓𝑓𝑓 𝐶𝐶 8 ≤ 𝑇𝑇 ≤ 𝐶𝐶 9	Eq 1

Table 3 .

 3 Coefficients of the extended Antoine equation

	Component	C1	C2	C3	C4	C5	C6	C7	C8/K	C9/K
	Allyl Alcohol	77.91	-7702.23	-10.25	-0.0011	-8.72	1.83E-16	5.78	144.15	545.1
	2-hexanol	109.31	-8191.02	-47.58	-0.0032	-13.65	4.32E-06	2.09	223	585.3
	2-hexanone	100.13	-9625.98	26.73	0.0062	-12.64	5.26E-06	1.82	217.35	587.61
	4

-methyl-2-pentanol (MIBC)

	66.27	-4160.32	-98.28	0.0095	-8.65	8.07E-16	4.82	183	574.4
	4

-methyl-2-pentanone (MIBK)

	72.26	-8325.25	36.24	0.0075	-8.52	-1.63E-16	5.82	189.15	574.6
	Figure 2. Vapor pressure for 2-hexanol, MIBC, 2-hexanone, MIBK and Allyl alcohol. The continuous blue line denotes the
	Antoine equation with regressed parameters. (•) represents experimental data and (○) reported data from the NIST database.

Table 4 - 6 .

 46 𝑃𝑃𝑦𝑦 𝑖𝑖 𝜑𝜑 𝑖𝑖 𝑣𝑣 = 𝑃𝑃 𝑖𝑖 𝑠𝑠 𝜑𝜑 𝑖𝑖 𝑠𝑠 𝛾𝛾 𝑖𝑖 𝑥𝑥 𝑖𝑖 exp � 𝑉𝑉 𝑖𝑖 𝑙𝑙 𝑑𝑑𝑑𝑑 (𝑖𝑖 = 1.2.3. … . 𝑁𝑁)

		𝑃𝑃
			Eq 2
		𝑃𝑃 𝑖𝑖 𝑠𝑠
	𝛾𝛾 𝑖𝑖 =	𝑦𝑦 𝑖𝑖 𝑃𝑃 𝑥𝑥 𝑖𝑖 𝑃𝑃 𝑖𝑖 𝑠𝑠	Eq 3

Table 4 .

 4 Isobaric Vapor-liquid equilibrium data for Allyl alcohol/2-hexanol and Allyl alcohol/MIBC systems at 101.32 kPa

			Allyl alcohol (1)/2-hexanol (2)				Allyl alcohol (1)/MIBC (2)
	T(K)	𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐			𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐	� 𝐺𝐺 𝐸𝐸 (𝑎𝑎)
	412.62	0	0		1.0003			404.74	0	0		1.0002
	412.02	0.0060	0.0247	1.0467	1.0003	0.0454	1.789	403.51	0.0181	0.0570	1.0290	0.9999	0.0287		1.537
	410.84	0.0183	0.0727	1.0451	0.9998	0.0443	2.212	402.15	0.0386	0.1170	1.0321	1.0001	0.0315		4.363
	408.62	0.0422	0.1576	1.0482	1.0001	0.0470	7.210	401.9	0.0426	0.1280	1.0309	1.0000	0.0304		4.363
	406.88	0.0622	0.2207	1.0484	1.0004	0.0468	11.236	399.76	0.0768	0.2170	1.0350	1.0003	0.0342		9.584
	404.09	0.0965	0.3158	1.0508	1.0006	0.0490	17.865	398.03	0.1063	0.2850	1.0361	1.0006	0.0349		14.099
	400.69	0.1425	0.4207	1.0510	1.0026	0.0472	30.980	395.82	0.1465	0.3670	1.0374	1.0008	0.0359		19.959
	396.98	0.1989	0.5246	1.0532	1.0033	0.0485	42.636	393.51	0.1917	0.4466	1.0380	1.0017	0.0356		27.801
	394.72	0.2369	0.5826	1.0544	1.0030	0.0499	48.707	390.77	0.2506	0.5340	1.0368	1.0032	0.0329		37.196
	391.78	0.2910	0.6514	1.0542	1.0044	0.0484	60.063	389.45	0.2808	0.5734	1.0371	1.0038	0.0326		42.028
	389.74	0.3320	0.6955	1.0540	1.0051	0.0475	67.466	387.47	0.3295	0.6296	1.0357	1.0053	0.0297		48.698
	386.16	0.4130	0.7657	1.0495	1.0094	0.0390	81.718	385.00	0.3961	0.6947	1.0321	1.0089	0.0227		57.146
	383.2	0.4890	0.8180	1.0460	1.0119	0.0332	89.235	383.93	0.4268	0.7213	1.0310	1.0104	0.0201		60.521
	381.79	0.5290	0.8410	1.0430	1.0147	0.0274	92.511	381.38	0.5058	0.7807	1.0269	1.0170	0.0097		69.022
	379.92	0.5860	0.8691	1.0376	1.0253	0.0119	101.042	380.34	0.5405	0.8035	1.0251	1.0206	0.0044		71.921
	377.70	0.6610	0.9010	1.0303	1.0378	-0.0073	101.523	378.47	0.6076	0.8429	1.0207	1.0285	-0.0076	73.928
	376.8	0.6940	0.9130	1.0264	1.0491	-0.0219	102.564	377.05	0.6639	0.8718	1.0155	1.0371	-0.0210	70.376
	374.29	0.7940	0.9460	1.0163	1.0758	-0.0569	86.846	375.65	0.7228	0.8990	1.0107	1.0482	-0.0365	64.817
	373.00	0.8510	0.9620	1.0101	1.1064	-0.0911	73.310	374.41	0.7772	0.9219	1.0075	1.0608	-0.0516	59.029
	371.87	0.9040	0.9760	1.0051	1.1392	-0.1252	52.905	373.35	0.8257	0.9408	1.0053	1.0737	-0.0658	52.136
	370.79	0.9570	0.9890	1.0008	1.2223	-0.1999	28.998	371.99	0.8919	0.9647	1.0026	1.0924	-0.0858	36.601
	369.91	1	1	1.0003				370.94	0.9459	0.9827	1.0006	1.1180	-0.1109	20.330
								369.89	1	1	1.0010

� 𝐺𝐺 𝐸𝐸 (𝑎𝑎) T(K)

Table 5

 5

	T(K)	𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐	� 𝐺𝐺 𝐸𝐸 (𝑎𝑎) T(K)	𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐	� 𝐺𝐺 𝐸𝐸 (𝑎𝑎)
	399.90	0.0000	0.0000		1.0001				388.84	0.0000	0.0000		1.0000
	398.93	0.0120	0.0390	1.2214	1.0004	0.1996		9.205	388.01	0.0152	0.0388	1.3592	0.9997	0.3072	14.079
	397.31	0.0330	0.1020	1.2216	1.0012	0.1990		25.519	387.10	0.0330	0.0807	1.3418	0.9998	0.2942	30.609
	395.65	0.0560	0.1650	1.2267	1.0012	0.2031		41.389	385.17	0.0750	0.1659	1.2943	1.0036	0.2544	72.486
	392.69	0.1010	0.2700	1.2227	1.0037	0.1973		77.191	383.41	0.1190	0.2418	1.2615	1.0092	0.2232	113.854
	390.87	0.1320	0.3320	1.2198	1.0050	0.1937		99.187	381.91	0.1619	0.3142	1.2680	1.0038	0.2336	132.065
	388.70	0.1730	0.4020	1.2095	1.0089	0.1813		130.090	380.47	0.2091	0.3786	1.2429	1.0068	0.2107	160.820
	386.63	0.2170	0.4670	1.1994	1.0125	0.1693		158.190	379.09	0.2596	0.4389	1.2174	1.0130	0.1838	191.138
	384.73	0.2630	0.5250	1.1854	1.0174	0.1528		183.702	377.92	0.3097	0.4955	1.2001	1.0129	0.1696	205.224
	382.45	0.3260	0.5910	1.1628	1.0296	0.1217		218.965	376.85	0.3608	0.5429	1.1719	1.0246	0.1343	228.039
	380.21	0.3990	0.6530	1.1336	1.0527	0.0740		255.730	375.98	0.4070	0.5842	1.1528	1.0324	0.1103	240.064
	378.20	0.4770	0.7100	1.1056	1.0793	0.0240		276.010	375.02	0.4643	0.6263	1.1210	1.0587	0.0572	260.579
	376.40	0.5590	0.7620	1.0786	1.1146	-0.0327	282.111	374.22	0.5177	0.6618	1.0932	1.0915	0.0016	274.909
	375.16	0.6240	0.7990	1.0588	1.1504	-0.0830	275.566	373.47	0.5737	0.7056	1.0805	1.1009	-0.0187	265.246
	374.04	0.6900	0.8350	1.0416	1.1892	-0.1325	254.436	372.84	0.6256	0.7382	1.0605	1.1374	-0.0700	263.402
	373.17	0.7460	0.8640	1.0286	1.2318	-0.1803	229.487	372.30	0.6774	0.7733	1.0463	1.1631	-0.1058	245.714
	372.36	0.8020	0.8930	1.0183	1.2778	-0.2270	195.251	371.79	0.7266	0.8076	1.0378	1.1841	-0.1319	226.037
	371.79	0.8440	0.9150	1.0122	1.3135	-0.2605	163.192	371.42	0.7692	0.8351	1.0274	1.2166	-0.1690	204.015
	371.09	0.8990	0.9440	1.0058	1.3689	-0.3082	113.867	371.05	0.8188	0.8667	1.0154	1.2678	-0.2220	171.138
	370.82	0.9210	0.9560	1.0041	1.3879	-0.3236	91.507	370.69	0.8705	0.9020	1.0072	1.3195	-0.2701	129.796
	370.41	0.9560	0.9750	1.0016	1.4358	-0.3602	53.581	370.15	0.9520	0.9613	1.0012	1.4307	-0.3570	56.280
	369.91	1.0000	1.0000	1.0003					369.90	1.0000	1.0000	1.0006

. Isobaric Vapor-liquid equilibrium data for Allyl alcohol/2-hexanone and Allyl alcohol/MIBK systems at 101.32 kPa Allyl alcohol (1)/2-hexanone (2) Allyl alcohol (1)/MIBK (2)

Table 6 .

 6 Isobaric Vapor-liquid equilibrium data for 2-Hexanone/2-hexanol and MIBK/MIBC systems at 101.32 kPa

			2-Hexanone (1)/2-hexanol (2)					MIBK (1)/MIBC (2)
	T(K)	𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐	� 𝐺𝐺 𝐸𝐸 (𝑎𝑎)	T(K)	𝒙𝒙 𝟏𝟏	𝒚𝒚 𝟏𝟏	𝜸𝜸 𝟏𝟏	𝜸𝜸 𝟐𝟐	𝑳𝑳𝑳𝑳 � 𝜸𝜸 𝟏𝟏 𝜸𝜸 𝟐𝟐	� 𝐺𝐺 𝐸𝐸 (𝑎𝑎)
	412.62	0.0000	0.0000		1.0003			404.75	0.0000	0.0000		0.9999
	412.46	0.0079	0.0129	1.1497	1.0003	0.1392	4.731	404.33	0.0179	0.0311	1.1364	1.0001	0.1278	7.957
	411.87	0.0388	0.0610	1.1268	1.0007	0.1187	18.034	403.89	0.0370	0.0631	1.1299	1.0005	0.1217	16.634
	411.04	0.0851	0.1281	1.1036	1.0023	0.0963	35.890	403	0.0771	0.1273	1.1196	1.0012	0.1118	32.827
	409.92	0.1509	0.2165	1.0843	1.0060	0.0749	58.956	402.19	0.1150	0.1841	1.1088	1.0025	0.1007	47.217
	409.16	0.2010	0.2774	1.0648	1.0105	0.0523	71.399	401.31	0.1575	0.2450	1.1027	1.0034	0.0943	60.894
	407.96	0.2820	0.3722	1.0523	1.0159	0.0352	87.314	400.53	0.1966	0.2980	1.0969	1.0042	0.0883	71.749
	407.03	0.3490	0.4449	1.0428	1.0215	0.0206	96.298	399.7	0.2410	0.3525	1.0821	1.0082	0.0708	83.669
	406.36	0.4010	0.4976	1.0341	1.0273	0.0066	99.789	398.67	0.2964	0.4190	1.0751	1.0104	0.0620	95.377
	405.75	0.4510	0.5449	1.0240	1.0360	-0.0116	101.701	397.88	0.3400	0.4682	1.0699	1.0129	0.0547	103.874
	404.82	0.5256	0.6172	1.0214	1.0403	-0.0183	100.627	397.05	0.3894	0.5193	1.0597	1.0182	0.0400	110.827
	404.20	0.5790	0.6647	1.0161	1.0484	-0.0313	97.920	396.95	0.3970	0.5253	1.0543	1.0216	0.0314	111.866
	403.35	0.6555	0.7313	1.0113	1.0567	-0.0439	88.430	396.25	0.4370	0.5666	1.0530	1.0234	0.0285	117.343
	402.66	0.7189	0.7837	1.0077	1.0670	-0.0571	79.523	396.15	0.4450	0.5724	1.0475	1.0279	0.0190	118.276
	401.92	0.7903	0.8408	1.0043	1.0799	-0.0726	65.153	395.25	0.5050	0.6262	1.0351	1.0394	-0.0042	120.146
	401.53	0.8299	0.8716	1.0025	1.0880	-0.0819	54.774	394.4	0.5586	0.6757	1.0338	1.0418	-0.0077	120.047
	400.79	0.9060	0.9299	1.0006	1.1029	-0.0973	32.491	393.65	0.6120	0.7192	1.0255	1.0536	-0.0271	116.671
	400.37	0.9500	0.9630	1.0002	1.1091	-0.1033	17.998	392.85	0.6700	0.7654	1.0194	1.0647	-0.0435	109.563
	399.89	1.0000	1.0000	1.0004				392.1	0.7258	0.8085	1.0152	1.0742	-0.0566	99.611
	399.89	1.0000	1.0000	1.0003				391.3	0.7874	0.8532	1.0100	1.0929	-0.0789	87.063
	399.89	1.0000	1.0000	1.0003				390.68	0.8380	0.8892	1.0067	1.1070	-0.0950	71.534
	399.89	1.0000	1.0000	1.0003				389.99	0.8968	0.9296	1.0029	1.1320	-0.1211	49.974
	399.89	1.0000	1.0000	1.0003				389.51	0.9387	0.9582	1.0013	1.1515	-0.1397	31.917
	399.89	1.0000	1.0000	1.0003				388.87	1.0000	1.0000	0.9991

E ) of six binaries at 101.32 kPa. (○) Allyl alcohol/2-hexanol (•) Allyl alcohol/MIBC (Δ) Allyl alcohol/2-hexanone ( ) Allyl alcohol/MIBK (□) 2-hexanone/2-Hexanol ( ) MIBK/MIBC, (-) Corresponding polynomial adjustment.

Eq 10 Table 7 .

 107 Margules parameters obtained by G E /RTX1X2 2 linear regression

	System	Parameters	Linear regression
		A12	A21	R 2
	Allyl Alcohol (1)/2-hexanol(2)	0.030	0.2224	0.9986
	Allyl Alcohol (1)/MIBC(2)	0.0337 0.1341	0.9999
	Allyl Alcohol (1)/2-hexanone(2) 0.2661 0.4213	0.9999
	Allyl Alcohol (1) /MIBK(2)	0.2788 0.4039	0.9996
	2-Hexanone (1)/2-hexanol(2)	0.1362 0.1124	0.9996
	MIBK (1)/MIBC(2)	0.1223 0.1731	0.9997

  It is defined by Eq 14 and Eq 15, where N mean the number of experimental points, 𝑃𝑃 𝑖𝑖 𝑒𝑒𝑥𝑥𝜕𝜕 and y i exp the experimental pressures and mole fraction in vapor phase, 𝑃𝑃 𝑖𝑖 𝑐𝑐𝑎𝑎𝑙𝑙 and y i cal the calculated values that obtained by UNIQUAC, NRTL, and WILSON thermodynamic models. As to the Van Ness method, VLE data can be considered thermodynamically consistent if ∆y and ∆𝑃𝑃 values are less than 1.

	∆y =	1 N	N i=1 � 100 �y i exp -y i cal �		Eq 14
	∆𝑃𝑃 =	1 𝑁𝑁	� 100 � 𝑃𝑃 𝑖𝑖 𝑁𝑁 𝑒𝑒𝑥𝑥𝜕𝜕 -𝑃𝑃 𝑖𝑖 𝑐𝑐𝑎𝑎𝑙𝑙 𝑖𝑖=1 𝑃𝑃 𝑖𝑖 𝑒𝑒𝑥𝑥𝜕𝜕	�	Eq 15

Table 8 .

 8 Results of Herington test and Van Ness Method for thermodynamic consistency check UNIQUAC model, b NRTL model, c WILSON model, d J value corrected with

	System	� ∆𝑯𝑯 𝒎𝒎 ∆𝑮𝑮 𝑬𝑬 �		Herington		Wisniak (Modified Herington)	Van Ness	Redlich-Kister total area test and Aspen tool
			D	J	|D -J|	D	J d	|D -J|	∆𝑦𝑦	∆𝑷𝑷 � 𝑘𝑘𝑃𝑃𝑘𝑘	|A*|<	Result
											0.03
	Allyl Alcohol	0.94	12.24	66.21	53.96	12.24	20.86	8.6	0.259 b 0.225 a	0.027 b 0.024 a	0.007	Passed
	(1)/2-hexanol (2)								0.182 c	0.045 c
	Allyl Alcohol	1.89	24.88	54.04	29.16	24.88	34.09	9.21	0.043 a	0.054 a	0.010	Passed
	(1)/MIBC(2)								0.044 b	0.060 b

a

Table 9 .

 9 Pure component consistency test

	System	100∆𝑃𝑃 1 0 100∆𝑃𝑃 2 0
	Allyl alcohol (1)/ 2-hexanol(2)	0.028	0.025
	Allyl alcohol (1)/ MIBC(2)	0.101	0.023
	Allyl alcohol (1)/ 2-hexanone(2)	0.028	0.011
	Allyl alcohol (1)/ MIBK(2)	0.064	0.002
	2-Hexanone (1)/ 2-hexanol(2)	0.039	0.025
	MIBK (1)/ MIBC(2)	0.088	0.010

Table 11 .

 11 Binary interaction coefficients, root-mean-square (RMSD) deviations in the equilibrium temperature and vapor phase mole fraction for all binary systems using NRTL, Wilson and UNIQUAC models.

	𝑖𝑖 ) = �	∑ (𝑇𝑇 𝑖𝑖 𝑒𝑒𝑥𝑥𝜕𝜕 -𝑇𝑇 𝑖𝑖 𝑐𝑐𝑎𝑎𝑙𝑙 𝑁𝑁 𝑖𝑖=1 𝑁𝑁	) 2	�	0.5	Eq 19
	𝑅𝑅𝑀𝑀𝑅𝑅𝐷𝐷(𝑦𝑦 𝑖𝑖 ) = �	∑ (𝑦𝑦 𝑖𝑖 𝑒𝑒𝑥𝑥𝜕𝜕 -𝑦𝑦 𝑖𝑖 𝑐𝑐𝑎𝑎𝑙𝑙 𝑁𝑁 𝑖𝑖=1 𝑁𝑁	) 2	�	0.5	Eq 20

Univ. Lille, CNRS, Centrale Lille, ENSCL, Univ. Artois, UMR 8181 -UCCS -Unité de Catalyse et Chimie du Solide, F-59000 Lille, France

Supporting information

Isobaric Vapor-liquid equilibrium data for six binary systems: Allyl alcohol/2hexanol, Allyl alcohol/2-hexanone, 2-hexanone/2-hexanol, Allyl alcohol/MIBC, Allyl alcohol/MIBK, and MIBK/MIBC at 101.32 kPa.

Karen Silva 1 , Marcia Araque 1 , Benjamin Katryniok 1

Supporting Information contains: 1 pages, and 1 figure

x 1


Davide Fissore
email: davide.fissore@inria.frenrico.tassi@inria.fr

Enrico Tassi

A new Type-Class solver for Coq in Elpi

Type classes: from programming to interactive proving

Type classes [START_REF] Wadler | How to make ad-hoc polymorphism less ad hoc[END_REF] were first introduced in the Haskell language as a way to implement ad-hoc polymorphism, i.e. symbol overloading. The Haskell compiler synthesizes the appropriate implementation for the overloaded symbol by looking at the types of its arguments and by composing user-provided implementations for atomic types and type formers. We call the software component in charge of this synthesis the type class solver.

Type classes soon were introduced in Coq version 8.2, 15 years ago, by Sozeau and Oury [START_REF] Sozeau | First-class type classes[END_REF]. Given the expressiveness of the language, Coq type classes not only relate a type to some operations, but also give them a specification. For example the Eq type class below links a type T to a boolean function eqb which is proved to be correct.

Class Eq T := { eqb: T -> T -> bool; eqP : forall x y, eqb x y = true <-> x = y }.

In the light of that type classes can play the role of interfaces and their solver plays the role of a pervasive form of automation, a major difference of use case w.r.t. Haskell. In particular by organizing theories around interfaces the application of a generic theorem to a specific example is automatically justified by the type class solver which proves that the example implements the required interfaces. Most modern Coq libraries are built around type classes, making their solver a key component of Coq.

Weaknesses of the current type class solver By using Type classes one quickly realizes that Coq's solver is implemented by calling the eauto tactic which in turn iterates the apply tactic using a dedicated hint database containing known type class implementations.

The main advantage of the current implementation is that it reuses powerful existing features that are continuously maintained and updated. There are, however, several downsides, severe enough to justify the study of a new implementation.

First and foremost, the apply tactic is not designed with automation in mind: when users apply lemmas they expect them to work, hence the apply tactic tries very hard to unify the conclusion of the lemma with the goal, and failing to do so can take a considerable amount of time. On the other hand automatic solvers are not driven by the educated guess of the user but they rather try to explore all possibilities by brute-force. Hence most of the attempts lead to a dead end and failing quickly is crucial. Over the years the type class solver got plenty of settings to tweak its behavior such as: preventing the unfolding of constants; avoiding goals which are too unconstrained; cutting search branches that match undesired patterns; etc. We count more than ten of these.

Even so, power users are still demanding features and extra degrees of configurability. One can use Hint Extern to locally take over the search, but cannot change the overall searching procedure, nor easily instrument it or debug it and fix it as we are used to do with any computer code we are not happy about. Our diagnosis is that the solver lacks of a proper "language" to describe how the search is done, a language power users could take full advantage of.

A new solver written in the Elpi Logic Programming language

Elpi [START_REF] Dunchev | ELPI: fast, embeddable, λProlog interpreter[END_REF] is a dialect of λProlog enriched with constraints. It is a higher order logic programming language well suited to manipulate syntax trees with binders and holes (i.e. lack of information to be inferred), such as Coq terms during type class resolution. Elpi comes in the Coq-Elpi plugin as an interpreter, and Coq can call a dedicated Elpi program whenever a type class needs to be solved. In the code block below we show, side by side, the Coq code declaring two type-class instances for the Eq class and the corresponding Elpi code (curly braces quote Coq syntax within Elpi, and lp:X escapes back, letting one bind a subterm to the variable X):

Instance eqBool : Eq bool := { ... }. Instance eqProd `{Eq A, Eq B}: Eq (A * B) := { ... }.

tc {{ Eq bool }} {{ eqBool }}. tc {{ Eq (lp:A * lp:B) }} {{ eqProd lp:SA lp:SB }} :tc {{ Eq lp:A }} SA, tc {{ Eq lp:B }} SB.

The instance for bool corresponds to a fact, a rule with no premises, while the instance for the pair (A * B) requires an implementation of Eq for A and B and hence corresponds to a rule with two premises. Whenever Coq needs to solve a type class we rephrase the problem as an Elpi query. For example solving Eq on (bool * bool) amounts to run the Elpi query "tc {{ Eq (bool * bool) }} Solution". The program above succeeds by chaining the two rules and answers "Solution = {{ eqProd eqBool eqBool }}".

We developed an automatic translator generating the Elpi code on the right starting from the Coq code on the left, but power users are free to write custom rules by hand, and even use different solvers for different type classes if they feel the need for it. Finally Elpi comes with a trace browser which allows to interactively navigate the steps made by the interpreter, substantially simplifying the task of debugging the type class solver.

The complexity of type classes search The Elpi's solver allows power users to have a full control on the search strategy. For example they can program a non-backtracking search for specific type classes, or pre-process the proof context by forward rewriting. In some cases this is sufficient to change the complexity class of type class search, especially in case of failure. One of such cases is a hierarchy of type classes featuring a tower of diamonds as in [START_REF] Selsam | Tabled typeclass resolution[END_REF]. Synthetic benchmark We are currently testing our implementation on the stdpp [START_REF] Krebbers | [END_REF] library to compare Coq and Elpi's solvers performances. The plot on the right shows the time taken to solve a given problem whose size is the number of nodes in the term driving the search (i.e. the type for which we search the Eq class). On small values Coq is faster even if any solver takes a negligible amount of time. For bigger problems our solver is about 2.5 times faster, showing that a more principled search strategy can be competitive even when run by an interpreter: our solver does not call apply, but rather uses the simpler and more efficient unification of Elpi.

Future work Currently, our solver is not in a stable and complete, but results are promising since we already cover about the 73.54% of stdpp type-class problems; we aim to cover the remaining part soon. We also plan to make Elpi's solver work with the iris library and the setoid rewriting one. Both use type class search as a form of automation and go to great lengths to optimize the search strategy, proving an even better test bench for our approach.

Finally, whilst reasonably efficient Elpi's runtime could benefit from forms of memoization, building on the work of Pientka [START_REF] Pientka | Tabling for higher-order logic programming[END_REF] and De Moura [START_REF] Selsam | Tabled typeclass resolution[END_REF].