Modeling the Thermodynamic Properties of Saturated Lactones in Nonideal Mixtures with the SAFT-γ Mie Approach

The prediction of the thermodynamic properties of lactones is an important challenge in the flavor, fragrance, and pharmaceutical industries. Here, we develop a predictive model of the phase behavior of binary mixtures of lactones with hydrocarbons, alcohols, ketones, esters, aromatic compounds, water, and carbon dioxide. We extend the group-parameter matrix of the statistical associating fluid theory SAFT-γ Mie group-contribution method by defining a new cyclic ester group, denoted cCOO. The group is composed of two spherical Mie segments and two association electron-donating sites of type e1 that can interact with association electron-accepting sites of type H in other molecules. The model parameters of the new cCOO group interactions (1 like interaction and 17 unlike interactions) are characterized to represent target experimental data of physical properties of pure fluids (vapor pressure, single-phase density, and vaporization enthalpy) and mixtures (vapor–liquid equilibria, liquid–liquid equilibria, solid–liquid equilibria, density, and excess enthalpy). The robustness of the model is assessed by comparing theoretical predictions with experimental data, mainly for oxolan-2-one, 5-methyloxolan-2-one, and oxepan-2-one (also referred to as γ-butyrolactone, γ-valerolactone, and ε-caprolactone, respectively). The calculations are found to be in very good quantitative agreement with experiments. The proposed model allows for accurate predictions of the thermodynamic properties and highly nonideal phase behavior of the systems of interest, such as azeotrope compositions. It can be used to support the development of novel molecules and manufacturing processes.


INTRODUCTION
A lactone is an ester in which the functional group −C(�O)− O− is a part of a cycle.The most common approach for the synthesis of lactones involves the intramolecular esterification of hydroxy acids.Numerous other approaches have been developed to obtain specific lactone structures. 1 Lactones are widely present in fruits, milk, fermentation products, and in many drinks and foods (from plant or animal origin); as such, they are of major interest for the flavor and fragrance industries 1−5 and are relevant to chemical, biological, 6 and pharmaceutical processes. 1For example, oxolan-2-one is present in a large range of food products 2 (dehydrated orange powder, tomato, bread, liquid smoke, popcorn, cocoa, coffee, black tea, wines, beef, etc.) but is also used as an organic solvent and as an intermediate in many syntheses 7 (e.g., for pyrrolidone and derivate compounds).
Knowledge of the thermodynamic properties of lactones can be helpful to synthesize and characterize these molecules and their mixtures and to develop chemical and industrial processes.Oxolan-2-one is, for example, known to be miscible with alcohols, ketones, esters, aromatic compounds, and water, but is not miscible with linear and cyclic aliphatic hydrocarbons. 7−11 Predictive group-contribution approaches can be used in molecular and process design 12−14 to reduce the number of experiments and material use and to study systems over a large range of thermodynamic conditions.
The statistical associating fluid theory (SAFT) 15,16 is a molecular equation of state based on statistical physics at the microscopic scale, providing an accurate description of complex fluids over a large range of thermodynamic conditions.In the original approach, molecules were treated as homonuclear chains of tangentially bonded spherical segments with embedded associating sites to mediate directional interactions that mimic hydrogen bonds. 17,18More-recent versions have been developed to consider spherical segments interacting through various pair potentials, 19,20 e.g., the SAFT-VR version for the square-well potential, 21 soft-SAFT for the Lennard-Jones potential, 22−26 and SAFT-VR Mie, 27 which incorporates the Mie potential (a generalized Lennard-Jones potential) and a third-order high-temperature expansion of the attractive contributions to the free energy.A reference chain fluid is used in PC-SAFT, 28 ePC-SAFT, 29 and PCP-SAFT. 30The PC-SAFT and PCP-SAFT approaches have been used to model 5methyloxolan-2-one, 31,32 and the polymerization of oxepan-2one 33 and of ω-pentadecalactone 34 in a mixture of carbon dioxide and dichloromethane have also been studied with PC-SAFT.These approaches, however, are not group-contribution methods, and as a result the molecular parameters presented are not transferable to other lactones or their mixtures.
The SAFT-VR Mie equation was recast as a groupcontribution approach in the SAFT-γ Mie equation of state 8−11 in which molecules are modeled in terms of their constituent chemical moieties (groups) such that once a group is characterized the thermodynamic properties of molecules and mixtures containing the group can be predicted.The method has been shown to deliver an accurate prediction of a broad range of equilibrium thermodynamic properties, including vapor−liquid equilibria (VLE), liquid−liquid equilibria (LLE), and solid− liquid equilibria (SLE), as well as single-phase and derivative properties. 11Furthermore, the approach can be used to develop SAFT-γ transferable force-field parameters for use in molecular simulation. 35An update of the current capabilities of the SAFT-γ Mie method and the available group interactions can be found in Haslam et al. 11 Recently, the SAFT-γ Mie approach has also been applied to systems of pharmaceutical interest: octanol−water partition coefficients for a range of active pharmaceutical ingredients were predicted by Hutacharoen et al., 36,37 aqueous mixtures of choline and geranate were modeled by Di Lecce et al., 38 solubility predictions were obtained for mefenamic acid in a range of solvents by Febra et al., 39 and pH solubility profiles of aqueous buffered solutions of ibuprofen and ketoprofen were predicted by Wehbe et al. 40 It has also been used to develop accurate models of amines and alkanolamines of interest in the field of carbon capture. 41,42In addition, the predictive capability of SAFT-γ Mie has been tested with the Clapeyron.jltoolkit by Walker et al., 43 and the transferability of the SAFT-γ Mie parameters has been examined by Crespo and Coutinho. 44An alternative treatment, referred to as (structural) s-SAFT-γ Mie, 45−47 has also been proposed to take into account functional group interactions.
Here, we extend the SAFT-γ Mie group-contribution method to represent the family of saturated lactones.The transferability of the new parameters is assessed for a large set of compounds of the lactone family.The parameter estimations and the predictive calculations are carried out by considering vapor-pressure, density, and vaporization-enthalpy data of pure compounds and VLE (including bubble and dew temperature, bubble and dew pressure, and azeotrope composition and temperature), LLE, SLE, density, and molar excess-enthalpy data of binary mixtures.
The different lactones are characterized by their ring size, the presence or absence of side chains, the saturation or unsaturation of the cycle, and the chirality of the atoms of the cycle. 1,3Distinct families of lactones can be defined from their structures; for example, phthalides and coumarins contain an aromatic cycle fused to the lactone cycle.The smallest saturated lactone found in nature is oxolan-2-one, which is a fivemembered cycle composed of three cyclic methylene (cCH 2 ) groups and one lactone (cCOO) group (Figure 1a).The heterocycle results in dipole moments that are larger for lactones than those of free ester chains; for example, the dipole moment of oxolan-2-one is reported as 3.8 D by Longster and Walker 48 and as 4.27 ± 0.03 D in the CRC Handbook of Chemistry and Physics 49 compared with 1.9 D for open-chain esters. 48In addition, the electron pairs of the oxygen atoms in the lactone group can form hydrogen bonds with hydrogen atoms from other molecules. 50n our model, different ring sizes and side chains are accounted for.Chirality, however, cannot be considered with a first-order group-contribution proposition because proximity effects are not taken into account. 11As chiral molecules present near identical thermodynamic properties, there is no need to differentiate between these types of isomers.
In Section 2, we present the SAFT-γ Mie approach and parameter-estimation methods.We detail the models and results for saturated lactones in Section 3 by considering pure fluids (Section 3.1.1)and binary mixtures in a range of solvents: saturated hydrocarbons (Section 3.1.2),primary and secondary The rings of these saturated lactones are modeled with one cCOO group and the corresponding number of cCH 2 groups (in blue and gray, respectively).The linear side chains are modeled with CH 2 (in brown) and CH 3 groups (in green).The carbon participating in the ring and side chain is modeled with a cCH group (in yellow).Association sites are denoted by the smaller red circles, labeled e for electronegative (acceptor) sites.alcohols (Section 3.2), 2-ketones (Section 3.3), esters (Section 3.4), aromatic compounds (Section 3.5), water (Section 3.6), and carbon dioxide (Section 3.7).We conclude and summarize our main findings in Section 4.

SAFT-γ Mie Model and Theory.
In the SAFT-γ Mie approach, 8,51 molecules are modeled as associating heteronuclear chains of fused spherical segments that interact through Mie potentials of variable range, while attractive short-range directional interactions are added by embedding square-well association sites on a given segment.The total Helmholtz free energy A of a fluid of nonionic species is expressed as the sum of four contributions: 8−10,51 where A ideal is the ideal free energy of the mixture, A monomer is the contribution accounting for the Mie segment interactions, A chain is the free energy associated with the formation of chains, and A association is the contribution to the free energy due to association. (Note that the Born and ionic contributions, which appear in the full SAFT-γ Mie free-energy expression, 11 are not included in eq 1 since we do not consider any ionic species in our current work.)8][9][10]51 Following the group-contribution premise, molecules are modeled in terms of functional groups, and it is assumed that the properties of a given molecule (or mixture) can be obtained by accounting for the appropriate contributions of the groups. Thparameters characterizing any given group are treated as transferable to other molecules and mixtures containing the same group.The SAFT-γ Mie representation of four saturated lactones (oxolan-2-one, 5-methyloxolan-2-one, 6-propyloxan-2one, and oxepan-2-one) can be seen in Figure 1.A given group k incorporates a number ν k * of identical spherical segments and a shape factor S k (0 ≤ S k ≤ 1), which is introduced to characterize the contribution of the segment to the overall molecular properties of the molecule.In the simplest case, the interaction between two segments of groups k and l is described through a Mie potential: where r kl is the distance between the centers of the two segments, σ kl is the segment diameter, ε kl is the depth of the potential well, and λ kl r and λ kl a are the repulsive and attractive exponents of the Mie potential, respectively.The prefactor kl is a function of the exponents that ensures that the minimum of potential is −ε kl , and can be expressed as When relevant, hydrogen bonding and strong polar interactions are represented by short-ranged square-well association sites placed on the segments. 52The parameter N ST,k corresponds to the number of association site types in a group k, and n k,a corresponds to the number of sites of a given type a = 1, ..., N ST,k .
The interaction between a site of type a placed on a segment of type k and a site of type b placed on a segment of type l is given by The interactions between groups k and l involve unlike parameters, which can be obtained through combining rules 8 (CR) in the first instance.In our current work, however, ε kl , K kl,ab HB , and ε kl,ab HB are systematically estimated by comparing target calculated and experimental thermophysical properties of pure fluids or mixtures in which the functional group is present, and λ kl r is occasionally estimated for better agreement.
The equilibrium thermodynamic properties of the fluid with c components can be determined from the total Helmholtz free energy A at a temperature T, volume V, and vector number N of molecules, composed of elements N i of compounds i, such that is the total number of molecules in the system.The pressure is obtained from and the chemical potential of a compound i is obtained from These standard relations can be used to determine the fluidphase equilibrium conditions.The solid−liquid solubility of compound i in a given solvent at a given temperature T and pressure P is obtained by solving the equality between the chemical potential of i in the solid phase, assumed pure here, and in the liquid phase: μ i S (T,P,x i S =1) = μ i sat (T,P,x sat ), where x sat is the mole fraction of the saturated solution.The solute mole fraction x i sat (T,P,x sat ) can be calculated as 39 where T i fus is the melting temperature, Δh where Θ is the vector of model parameters, N exp is the total number of experimental points considered in the parameter estimation, N S is the number of systems (pure compounds/ mixtures) used in the estimation, N s P is the number of property types for system s, N s,p D is the number of experimental data points for system s and property p, and w s,p,i is a weight that is used to control the relative importance of data point i for property p of system s.We consider here the same weight for each point (i.e., w s,p,i = 1, for all points).X s,p,i exp is the i th measured value of property p of system s, and X s,p,i calc (Θ) is the corresponding value calculated with SAFT-γ Mie and the parameters Θ.
The percentage absolute average deviation (%AAD) of a property p for a system s, and the absolute average deviation, are used as measures of the accuracy of the theoretical approach.We use the gPROMS 64 tools to perform the parameter estimation and calculations.The open-source toolkits SGTPy 65 and Clapeyron.jl 43can also be used to reproduce the calculations.2 together with the details of the groups used in the SAFT-γ Mie modeling.

SAFT-γ Mie
We introduce here a new cyclic ester group, denoted as cCOO.Ester groups in linear chains can already be modeled with SAFT-γ Mie using the COO group developed in previous work. 8The influence of the heterocycle means, however, that a new cCOO group needs to be characterized for an accurate description of the lactone family.It is worth noting that a cyclic ester group 66 (labeled cy-COO-C) has been introduced in the Modified UNIFAC 67 approach.As in the case of the linear COO group, two association sites of type e are included in our new cCOO group; no like association occurs between cCOO groups, as no e−e bonding is allowed, but these electronegative sites can bond to H sites in other molecules.
We refer to molecules composed only of one cCOO group and a number of cCH 2 groups 51 as ring lactones: three cCH 2 groups for oxolan-2-one (Figure 1a), four cCH 2 groups for oxan-2-one, and five cCH 2 groups for oxepan-2-one (Figure 1d).In addition, we also consider lactones that incorporate an alkyl side chain next to the cCOO group; the intramolecular esterification of hydroxy acids favors the formation of these branches when the OH group of the hydroxy acid is not terminal.Two examples of lactones with side chains can be seen in Figure 1b,c: 5methyloxolan-2-one and 6-propyloxan-2-one, respectively.In order to treat these molecules, the junction between the ring and the side chain is modeled with a cCH group. 41A methyl side chain is composed of a CH 3 group 8 only, and longer linear alkyl chains are modeled by one or several CH 2 groups, 8 with a terminal CH 3 group.
We consider experimental data for 5-alkyloxolan-2-ones (from methyl to hexyl chains) and for 6-alkyloxan-2-ones (from methyl to pentyl chains) to characterize the relevant group interactions.The matrix of group interactions used in current and previous work is shown in Table 3, and the like and unlike group parameter values can be found in Tables 4, 5, and 6 with the corresponding references as appropriate.In order to characterize the group interactions, we use pure-component 58,69−82 and mixture 31,66,83−89  pressures (P vap ), single-phase densities (ρ), and vaporization enthalpies (Δh vap ) for oxolan-2-one, oxan-2-one, and oxepan-2one are compared in Figure 2. As expected, lower vapor pressures and higher vaporization enthalpies are reported for larger ring sizes.For a given temperature, the vapor pressure and liquid-phase density decrease with increasing ring size of the lactone, while the vaporization enthalpy increases with increasing ring size.Good agreement between the calculations and experiments can be seen, although we note that limited data are available for these molecules.The family of 5-alkyloxolan-2-ones corresponds to lactones with five atoms in the lactone ring and an alkyl side chain (in position 5) next to the oxygen of the ring (position 1).Similarly, the family of 6-alkyloxan-2-ones corresponds to lactones with six atoms in the lactone ring and an alkyl side chain (in position 6).The calculated vapor pressures, single-phase densities, and vaporization enthalpies are compared with the corresponding experimental data for pure 5-alkyloxolan-2-ones and 6alkyloxan-2-ones with several side-chain lengths in Figure 3 and Figure 4, respectively.The vapor pressure and liquid-phase density decrease with increasing length of the side chain, while the vaporization enthalpy increases with increasing length of the side chain.Very good quantitative agreement can be seen for alkyllactones although the comparison with experimental data is possible only for a small temperature range, far from the critical temperature.
We report %AAD and AAD in order to assess the performance of our model.It is especially relevant to consider both measures, given the limited data available for comparison and the lowtemperature and pressure nature of the data.The accuracy of the results for pure lactones is summarized in Table 7, where % AADs and AADs can be found for each system and property.The %AADs for the vapor pressure of oxolan-2-one and oxepan-2-one are both smaller than 10% (5.063%, and 7.853%, respectively).The AAD obtained for the vapor pressure of oxolan-2-one (1796 Pa) is, however, much larger than the AAD obtained for oxepan-2-one (5.722 Pa) because of the difference in the temperature ranges (289−478 and 283−353 K, respectively).The largest %AAD shown in Table 7 is for the    69 and diamonds 70 ), oxan-2-one, 71 and oxepan-2-one. 72(b) Temperature−density diagram, with experimental data for oxolan-2-one, 76 oxan-2-one, 79 and oxepan-2-one. 80Saturation densities are represented by the continuous curves, and densities at 1 bar are represented by the dashed curves and lines.(c) Vaporization enthalpy, with experimental data for oxolan-2-one, 69 oxan-2-one, 71 and oxepan-2-one. 72Thermodynamic conditions and the accuracy of the calculations are detailed in Table 7 and in the Zenodo datafile.liquid-phase densities and the vaporization enthalpy of the pure lactones considered are smaller than 5%.In absolute terms, this corresponds to a maximum AAD in liquid-phase densities for oxepan-2-one (51.40 kg/m 3 ) and a maximum AAD in vaporization enthalpy for 6-pentyloxan-2-one (2441 J/mol).
3.1.2.Binary Mixtures of Lactones + Linear/Cyclic Alkanes.We now consider binary mixtures of lactones with cyclohexane or n-alkanes; the groups involved in these mixtures are the same as the groups involved in pure lactones.Specifically, we consider mixtures containing linear n-alkanes from n-hexane to n-octane and from n-undecane to n-nonadecane.Cyclohexane is modeled using six cCH 2 groups; linear alkanes comprise CH 2 and CH 3 groups.As mentioned in the previous section, the experimental data related to these mixtures (summarized in Table 8) are used together with the experimental data for pure lactones (cf.Table 7) to estimate the parameters of the cCOO group and the unlike interactions (specifically for cCOO−CH 2 and cCOO−CH 3 ).
Isobaric temperature−composition phase diagrams for a number of mixtures of lactones (oxolan-2-one or oxepan-2-one) and hydrocarbons (cyclohexane, n-hexane, or n-octane) are shown in Figure 5, in which SAFT-γ Mie calculations and the limited experimental data available can be compared.Specifically, the experimental LLE data for mixtures of oxolan-2-one and cyclohexane, 84 oxolan-2-one and n-hexane, 84,86 and oxepan-2-one and n-hexane 85 can be seen in the figures.Good overall agreement is observed, as indicated by the low AADs summarized in Table 8.The small values of the LLE mole fractions cause some large %AADs, for example, for oxolan-2one + cyclohexane (165.3%),despite a small corresponding AAD (0.04311).Experimental data for LLE 85 and solid− liquid−liquid equilibrium 66 (SLLE) are shown for the mixture of oxepan-2-one and cyclohexane in Figure 5b.The model allows for the correct prediction of the existence and the extent of the LLE region for this system but fails in predicting the composition of the eutectic point (x oxepan-2-one SAFT = 0.084 and x oxepan-2-one exp = 0.847), in part as a consequence of the small difference in the SLLE temperature and the eutectic temperature.The temperature of the eutectic point predicted by SAFTγ Mie with the estimated parameters is T eutectic SAFT = 264 K, while the experimental value 66 is T eutectic exp = 267 K. Several azeotrope data are available for mixtures of oxolan-2one and linear alkanes. 83This allows us to study the influence of the alkane chain length on the azeotrope composition and temperature at 2666.4 Pa.Related experimental data are available for systems containing alkanes from n-undecane (C 11 H 24 ) to n-nonadecane (C 19 H 40 ).SAFT-γ Mie calculations are carried out for n-hexane (C 6 H 14 ) to n-eicosane (C 20 H 42 ).The azeotrope composition x oxolan-2-one az increases with an increase in the alkane length (Figure 6a) and is found to be greater than 0.5 for oxolan-2-one + n-dodecane (C 12 H 26 ) and longer alkanes, both from the experimental data and the predictions.The azeotrope temperature also increases with an increase in the alkane length at a fixed pressure (Figure 6b).The agreement between the experimental data and SAFT-γ Mie The asterisk * indicates that the parameters are characterized in the current work.

Journal of Chemical & Engineering Data
predictions is excellent both for the azeotrope compositions and temperatures, with the corresponding %AADs smaller than 2% for all the mixtures at 2.6664 kPa (cf.Table 8).For 101.3 kPa, only one azeotrope temperature has been reported 83 for the VLE of oxolan-2-one + n-octane (Figures 5e and 6a,b), and the SAFT-γ Mie predictions are found to be in excellent agreement with these data.It also can be seen in Figure 6 that the azeotrope temperature of a given mixture increases significantly with an increase in the pressure, while the compositions remain quite similar.Other %AADs and ADDs associated with the mixtures of lactones and hydrocarbons are summarized in Table 8.

Binary Mixtures of Lactones + Alcohols. 3.2.1. Lactone + Primary Alcohol Mixtures: cCOO−CH 3 OH and cCOO− CH 2 OH Interactions.
Mixtures of lactones and primary alcohols, from methanol to decan-1-ol, are studied in this section.Methanol is represented by a single SAFT-γ Mie molecular group, 51 denoted as CH 3 OH.Longer primary alcohols (ethanol, propan-1-ol, butan-1-ol, etc.) are modeled with a CH 3 group, a number of CH 2 groups appropriate for the length of the alkanol, and a CH 2 OH 36 group (cf.Table 2).The CH 3 OH and CH 2 OH groups include one site of type H, and two e sites (labeled e 1 ) correspond to the lone pairs of the oxygen atom.In mixtures with lactones, an association interaction between the H site of the hydroxyl groups and the e 1 site of the cCOO group is accounted for.The related parameters, characterized here, are , and ; in addition, the unlike dispersion energy parameters cCOO CH OH 3 and cCOO CH OH 2 are also determined (cf.Tables 5 and 6).
The unlike group interaction parameters are estimated from VLE, 92−95 SLE, 66 single-phase density, 96−98 and excess enthalpy 99,100 experimental data.In particular, the corresponding isobaric VLE data 95 are shown in Figure 7a for 5methyloxolan-2-one + methanol and 5-methyloxolan-2-ol + ethanol.No azeotrope is found in these phase diagrams.SLE data (Figure 7b) are available for mixtures of oxepan-2-one + methanol and oxepan-2-one + propan-1-ol; these are used to characterize the interaction parameters with more accuracy.Characteristic nonideal behavior is observed in the experimental density data 97,98 of mixtures of oxolan-2-one and several linear primary alcohols (Figure 7c) at 298.15 K, with a concave shape for the shorter alcohols and a convex shape for the longer alcohols, as a function of mole fraction.The highest densities for x oxolan-2-one < 0.1 are obtained for the longest chains (the longer/ heavier alcohols), while the highest densities for x oxolan-2-one > 0.1 are seen for the smallest chains.An inversion is observed for x oxolan-2-one ≈ 0.1 in the mixtures considered, including the mixture of oxolan-2-one + methanol.Excess molar enthalpy data 99,100 at 298.15 K and calculations are shown in Figure 7d.As can be seen, the excess molar enthalpies of the oxolan-2-one + long primary alcohol mixtures exhibit a similar trend and order of magnitude, with a maximum of approximately 2 kJ/mol at x oxolan-2-one ≈ 0.5.The calculated excess enthalpy for oxolan-2one + methanol presents a lower maximum; we note the limited experimental data available for this system.The asterisk * indicates that the parameters are characterized in the current work.

Journal of Chemical & Engineering Data
Azeotrope experimental data 83 for mixtures of oxolan-2-one and linear primary alcohols are also available (Figure 8) for mixtures containing linear primary alcohols from pentan-1-ol (C 5 H 11 OH) to undecan-1-ol (C 11 H 23 OH) but are not used for the parameter estimation.The azeotrope compositions x oxolan-2-one az at 2666.4 Pa increase with an increase in the alcohol chain length (Figure 8a) and they are higher than 0.5 for oxolan-2-one + octan-1-ol (C 8 H 17 OH) and oxolan-2-one + longer alcohols.As noted earlier, the azeotrope temperatures also increase with an increase in the alkane chain length (Figure 8b) such that the results for the azeotrope compositions and temperatures are qualitatively similar for both oxolan-2-one + linear primary alcohols and oxolan-2-one + linear alkanes (Figure 6).We note, however, that the azeotrope compositions are higher in the mixtures with alcohols than in the mixtures with the alkanes for a given number of carbons in the chain.
Excellent agreement between the experimental data and the SAFT-γ Mie calculation is obtained for the VLE, with %AADs of 0.7520 and 0.1422% for the bubble temperatures and 0.4562 and 0.3411% for the dew temperatures of 5-methyloxolan-2-one + methanol and 5-methyloxolan-2-one + ethanol, respectively.
The solubility of oxepan-2-one (i.e., the SLE curves for T > 200 K in Figure 7b) is determined with accuracy both in methanol (%AAD = 1.702%) and propan-1-ol (%AAD = 4.388%).Densities are also correctly described with SAFT-γ Mie.In particular, the characteristics of the curves shown in Figure 7c, for example, the concave/convex shapes and the intersection of the curves, are in excellent quantitative agreement with the experimental data.The highest %AAD is only 3.438% for oxolan-2-one + octan-1-ol.A concave shape is found for all of the excess enthalpy curves, as shown in Figure 7d.Good agreement is obtained with experiment, 99,100 and all % AADs for excess enthalpy are smaller than 10% except for oxolan-2-one + decan-1-ol (11.06%).The prediction of the excess enthalpy of oxolan-2-one + methanol is significantly different in comparison with the other mixtures containing alcohol, as suggested by the experimental data (the predicted maximum is around 1 kJ/mol only).
Primary alcohols with a branched carbon chain are also considered in order to estimate the unlike interaction between the cCOO group and the CH group. 51Very good accuracy is obtained for the density of oxolan-2-one + 2-methyl-1-propanol (%AAD = 1.367%) and oxolan-2-one +3-methyl-1-butanol (%AAD = 1.384%).The corresponding unlike group parameters are detailed in Table 5, and the association parameters are listed in Table 6.The overall accuracy for all of these systems can be found in Table 9.

Lactone + Secondary Alcohol Mixtures: cCOO− CHOH and cCH−CHOH Interactions.
We now consider the modeling of lactones and secondary alcohols by incorporating the CHOH group, 11 which is modeled with two sites of type e 1 and one site of type H.The H site of the CHOH group interacts with the e 1 sites of the cCOO group.The unlike group parameters characterized in this section are ε cCOO−CHOH , cCOO CHOH,e H HB 1 and K cCOO CHOH,e H HB 1 (cf.Tables 5 and 6).Experimental VLE data, 95 shown in Figure 9a, are available for 5-methyloxolan-2-one + propan-2-ol at atmospheric pressure for a large range of mole fractions for the bubble temperature (0.00 ≤ x 5-methyloxolan-2-one ≤ 0.77) and a small range for the dew temperature (0.00 ≤ x 5-methyloxolan-2-one ≤ 0.045).Experimental density data can be seen in Figure 9b for oxolan-2-one + propan-2-ol 96 and oxolan-2-one + butan-2-ol. 98The accuracy for the description of mixtures of lactones and secondary alcohols is very good for both VLE and density.The corresponding %AAD is only 0.3007% for the bubble temperature of 5-methyloxolan-2-one + propan-2-ol, and 0.2153% for the related dew temperature.The overall agreement for the density is also very good and includes a crossing of curves at x 5-methyloxolan-2-one ∼ 0.15 seen both from experimental data and the SAFT-γ Mie calculations.
3.3.Mixtures of Lactones + 2-Ketones: cCOO− CH 3 COCH 3 and cCOO−CH 3 CO Interactions.We now consider binary mixtures of lactones + 2-ketones, including acetone.SAFT-γ Mie models for acetone, as well as for 2ketones, have been presented in previous work, 39,51 and the same models are adopted here.Acetone is modeled as a single molecular group 51 denoted by CH 3 COCH 3 , while other 2ketones are modeled by using the CH 3 CO group 39 together with CH 2 and CH 3 groups.No unlike association is considered between the cCOO group and ketone groups; thus, the only parameters that need to be estimated here are cCOO CH COCH Both the VLE of 5-methyloxolan-2-one + acetone and the SLE of oxepan-2-one + pentan-2-one are accurately described with the resulting SAFT-γ Mie model (Figure 10).The corresponding AADs are small, as are the %AADs, which are below 4% for all the available properties (Table 11).Additional VLE diagrams at atmospheric pressure are predicted for binary mixtures of 5-methyloxolan-2-one + butan-2-one, pentan-2-one, and hexan-2-one.No azeotrope is predicted for these systems.The SLE phase diagrams of oxepan-2-one + acetone, + butan-2-one, and + hexan-2-one are also predicted.All of these systems have a eutectic point, as can be seen for oxepan-2-one + pentan-2-one both from experiments and the SAFT calculations.density, 102 and molar excess enthalpy 103 (see Figure 11).In particular, VLE data at atmospheric pressure are available for the mixture of 5-methyloxolan-2-one + ethyl acetate. 101SLE data for oxolan-2-one + diethylbutanedioate 70 are also available, with a eutectic point at x oxolan-2-one ≈ 0.78 and T eutectic ≈ 210 K. Density data at 298.15 K and 101.3 kPa are available for oxolan-2-one + methyl acetate and oxolan-2-one + ethyl acetate 102 for x oxolan-2-one ∈ [0.0, 1.0].Excess molar enthalpy data at 298.15 K and 101.3 kPa are available for oxolan-2-one + methyl acetate 103 for the small range x oxolan-2-one ∈ [0.89, 0.97] with the largest value of only 35.7 J/mol (for x oxolan-2-one = 0.89).
The characterization of the interaction parameters between the cCOO and COO 8 groups in the SAFT-γ Mie modeling is performed by considering the VLE, density, and excess enthalpy data available.SLE data are not used to estimate the parameters here.The resulting agreement between the calculations and the experimental data is excellent for all of the properties.In particular, no azeotrope is found in the VLE of 5-methyloxolan-2-one + ethyl acetate at atmospheric pressure.The density at 298.15 K is larger for oxolan-2-one + methyl acetate than for oxolan-2-one + ethyl acetate, for all values of x oxolan-2-one , which is qualitatively consistent with the experimental findings.The agreement for the excess molar enthalpy of oxolan-2-one + methyl acetate is also very good despite the small values found for this property (quantitatively, we obtain %AAD = 16.86%, which corresponds to an AAD of 3.536 J/mol).
The resulting group parameters are used to make additional predictions of the VLE, SLE, density, and excess molar enthalpy for binary mixtures of lactones and methyl, ethyl, propyl, and pentyl acetates; these are shown in Figure 11.It is of interest to note the "S" shape predicted for the molar excess enthalpy curves shown in Figure 11d, with positive values for lactone-rich phases and negative values for ester-rich phases.For oxolan-2one + pentyl acetate, the predicted excess enthalpy is always  Oxolan-2-one + cyclohexane, with experimental data for the LLE. 84(b) Oxepan-2-one + cyclohexane, with experimental data for the LLE 85 and SLE. 66c) Oxolan-2-one + n-hexane, with experimental data for the LLE. 84,86(d) Oxepan-2-one + n-hexane, with experimental data for LLE. 85(e) Oxolan-2one + n-octane, with experimental data for the azeotrope. 83Thermodynamic conditions and the accuracy of the calculations are detailed in Table 8 and in the Zenodo datafile.) and (b) azeotrope temperatures (T az ) for mixtures of oxolan-2-one + linear alkanes.The number of carbons corresponds to the length of the alkane (six for n-hexane, seven for n-heptane, etc.).Black "×" symbols and blue "+" symbols represent the SAFT-γ Mie calculations.Black squares and blue diamonds represent the experimental points. 83Black points and blue points correspond to results at 2.6664 and 101.3 kPa, respectively.Thermodynamic conditions and the accuracy of the calculations are detailed in Table 8 and in the Zenodo datafile.positive, although highly asymmetric.More experimental data would be useful to confirm or disprove these predictions.
The SLE of oxolan-2-one + diethylbutanedioate for x oxolan-2-one > 0.78 is correctly predicted by using the melting temperature and the enthalpy of fusion of oxolan-2-one 56 (cf.Table 1).The melting temperature of diethylbutanedioate 61 is reported as T diethylbutanedioate fus = 252.55K (cf.Table 1); however, to the best of our knowledge, there is no experimental value of the corresponding enthalpy of fusion.The %AAD for the SLE shown in Figure 11b as a dashed curve for x oxolan-2-one < 0.78 can be minimized by estimating the value of the enthalpy of fusion, which yields Δh diethylbutanedioate fus, estimated = 23 kJ/mol (such that %AAD = 6.238% and AAD = 0.01237).The value of Δh diethylbutanedioate fus, estimated is close to the value obtained from the Joback group-contribution method: 104 Δh diethylbutanedioate fus, Joback = 20.03kJ/mol (we note that both values are higher than the values reported in Table 1 for other compounds).The eutectic composition and temperature are also correctly predicted, as x oxolan-2-one predicted ∼ 0.75, and T eutectic predicted ∼ 211 K with the estimated enthalpy of fusion.Additional information about the accuracy of the SAFT-γ Mie approach for these systems is summarized in Table 12.Selected isobaric thermodynamic properties at atmospheric pressure for lactones + primary alcohols: methanol (black), ethanol (dark gray), propan-1-ol (light gray), butan-1-ol (dark red), pentan-1-ol (red), hexan-1-ol (orange), heptan-1-ol (yellow), octan-1-ol (purple), nonan-1-ol (dark blue), and decan-1-ol (light blue).The curves represent the calculations with SAFT-γ Mie.Experimental data used in the estimation of groupinteraction parameters are represented with filled symbols, and those not used are represented with open symbols.(a) Isobaric vapor−liquid equilibria of 5-methyloxolan-2-one + primary alcohols, with experimental data for methanol 95 and ethanol. 95(b) Isobaric solid−liquid equilibria of oxepan-2-one + primary alcohols, with experimental data for methanol 66 and propan-1-ol. 66(c) Density of oxolan-2-one + linear primary alcohols at 298.15 K, with experimental data for methanol, 97 ethanol, 97 butan-1-ol, 97 and octan-1-ol. 97(d) Excess enthalpy of oxolan-2-one + primary alcohols at 298.15 K, with experimental data for methanol, 100 hexan-1-ol, 96 heptan-1-ol, 96 octan-1-ol, 96 and decan-1-ol. 96Thermodynamic conditions and the accuracy of the calculations are detailed in Table 9 and in the Zenodo datafile.) and (b) azeotrope temperatures (T az ) for mixtures of oxolan-2-one + linear primary alcohols.The number of carbons corresponds to the length of the linear primary alcohol (six for hexan-1-ol, seven for heptan-1-ol, etc.).Black "×" symbols and blue "+" symbols represent the calculations with SAFT-γ Mie.Black squares and blue diamonds represent the experimental points. 83Black and blue points correspond to results at 2.6664 and 101.3 kPa, respectively.Thermodynamic conditions and the accuracy of the calculations are detailed in Table 9 and in the Zenodo datafile.

Mixtures of Lactones + Aromatic Compounds: cCOO−aCH, cCOO−aCCH 3 , and cCOO−aCCH 2 Interactions.
The aromatic compounds are modeled with specific aromatic groups, denoted as aCH, aCCH 3 , and aCCH 2 .For instance, benzene is composed of six aCH groups. 51One of the aCH groups of benzene is replaced by an aCCH 3 group 37,90 to model toluene or replaced by an aCCH 2 group 51 bonded to a CH 3 group to model ethylbenzene.There is no association between the cCOO group and the aromatic groups, such that only the unlike interaction parameters ε kl and λ kl r need to be characterized (Table 5) to describe binary mixtures of lactones + aromatic compounds.
Bubble-pressure data 105 at 293 and 313 K are used in the characterization of the unlike parameters considering mixtures of benzene + oxolan-2-one, + oxan-2-one, + oxepan-2-one, and + 5-methyloxolan-2-one.The VLE of these mixtures can be seen in Figure 12a for a given temperature; the markedly different vapor pressures of the pure lactones (some hundreds of Pa, as shown in Figures 2−4) and pure benzene 105 (10.03 kPa at 293 K and 24.38 kPa at 313 K) give rise to clear nonideality in the phase diagrams.We also use density data 106 for mixtures of oxolan-2one + benzene, + toluene, and + ethylbenzene to characterize the unlike parameters (cf. Figure 13a).Excess molar enthalphy data, 105,107 shown in Figure 13b−d, represent the most abundant set of data for the mixtures of lactones (oxolan-2one, oxan-2-one, and 5-methyloxolan-2-one) and aromatic compounds (benzene, toluene, and ethylbenzene).The reported values are positive, as well as negative, depending on the mixture and are remarkably small with a maximum value of only 413 J/mol for an equimolar mixture of oxolan-2-one + ethylbenzene.The shape of some of the excess-enthalpy curves is also unusual; for example, the excess enthalpy of oxolan-2-one + benzene has an "M" shape 105 (cf. Figure 13b) with positive values for x oxolan-2-one < 0.21 and x oxolan-2-one > 0.48, negative Isobaric vapor−liquid equilibria of 5-methyloxolan-2-one + secondary alcohols, with experimental data for propan-2-ol. 95(b) Density of oxolan-2-one + secondary alcohols at 303.15 K, with experimental data for propan-2-ol 96 and butan-2-ol. 98Thermodynamic conditions and the accuracy of the calculations are detailed in Table 10 and in the Zenodo datafile.(a) Vapor−liquid equilibria of 5-methyloxolan-2-one + 2-ketones, with experimental data for acetone. 101(b) Solid−liquid equilibria of oxepan-2-one + 2-ketones, with experimental data for pentan-2-one. 66Thermodynamic conditions and accuracy are detailed in Table 11 and in the Zenodo datafile.
values otherwise, and a maximum of about 30 J/mol.SLE data are also available for mixtures of lactones and aromatic compounds 66,108,109 but are not used in the characterization of group interactions.The SAFT-γ Mie calculations using the optimized parameters are also shown in Figures 12 and 13.While the agreement for the bubble-pressure and density curves is excellent, the agreement for the excess enthalpy curves is not always quantitatively accurate.The sign is not correct for two of the nine mixtures considered (for oxan-2-one + toluene and 5-methyloxolan-2one + ethylbenzene), although we note that the highest values obtained from the calculations and the experimental data are always of the order of a few hundred J/mol only.Encouragingly, Figure 11.Isobaric thermodynamic properties of lactones + linear esters at atmospheric pressure: methyl acetate (black), ethyl acetate (dark gray), propyl acetate (light gray), butyl acetate (dark red), pentyl acetate (red), and diethylbutanedioate (green).The curves represent the calculations with SAFT-γ Mie.The dashed curves are determined with Δh diethylbutanedioate fus, estimated , as explained in the text.Experimental data used in the estimation of groupinteraction parameters are represented with filled symbols, and those not used are represented with open symbols.(a) Vapor−liquid equilibria of 5methyloxolan-2-one + esters, with experimental data for ethyl acetate. 101(b) Solid−liquid equilibria of oxolan-2-one + esters, with experimental data for diethylbutanedioate. 70 (c) Density of oxolan-2-one + esters at 298.15 K, with experimental data for methyl acetate 102 and ethyl acetate. 102(d) Excess molar enthalpy of oxolan-2-one + esters at 298.15 K, with experimental data for methyl acetate. 103Thermodynamic conditions and the accuracy of the calculations are detailed in Table 12 and in the Zenodo datafile.the "M" shape of the excess enthalpy curve for oxolan-2-one + benzene is qualitatively obtained with SAFT.The transferability of the parameters characterized is assessed for the prediction of the SLE (Figure 12b).As can be seen in the figure, the eutectic points of oxolan-2-one + benzene and 5-methyloxolan-2-one + benzene are correctly predicted.The corresponding %AADs and AAD for the systems 66,105−109 shown in Figures 12 and 13 (and additional systems 110−113 ) can be found in Table 13.
The VLE 114 and SLE 70 experimental data found for oxolan-2one + water are represented in Figure 14a.No azeotropic or liquid−liquid demixing behavior is seen, but we note the eutectic point in the SLE region.Densities 115−117 at 298.15 K and atmospheric pressure are presented in Figure 14b for binary mixtures of oxolan-2-one + water and 5-methyloxolan-2-one + water, and the excess molar enthalpies 73,118 for the same mixtures are shown in Figure 14c.A highly asymmetric "S" shape can be seen for the two curves, with positive excess enthalpies for x water < 0.9 and small negative values for x water > 0.9.The highest values are about 1 kJ/mol for x water ≈ 0.4 for the two systems.As can be gleaned from the figures, very good agreement is obtained for the bubble temperature of oxolan-2-one + water (%AAD = 0.7953%, and AAD = 2.978 K), the solubility of oxolan-2-one in water (%AAD = 0.3427%, and AAD = 0.003086), and the solubility of water in oxolan-2-one (%AAD = 3.168%, and AAD = 0.01049).The experimental 70 eutectic composition and temperature are also accurately predicted.The calculated densities decrease with an increase in the length of the lactone side chain, and the agreement with experimental data 115−117 is very good for water + oxolan-2-one and reasonable for water +5methyloxolan-2-one.The corresponding %AADs and AADs are found to be small (cf.Table 14).Furthermore, the "S" shape of the excess enthalpy curves is correctly reproduced by the SAFT-γ Mie calculation.The corresponding %AADs are rather large (251.2% for water + oxolan-2-one, and 31.32% for water + 5-methyloxolan-2-one), although we note that these large values are due to the very small enthalpies in the water-rich phase (we have considered all the experimental data 73,118,119 found for the %AAD calculation and not only the points shown in Figure 14c, including data for different temperatures and data at infinite dilution).The corresponding AADs, which are relatively small (43.33 J/mol for water + oxolan-2-one and 129.4 J/mol for water + 5-methyloxolan-2-one), provide a more appropriate measure of the quality of the model and confirm the good agreement seen in Figure 14c.Additional information on the accuracy of the calculations for the systems considered in this section compared with experimental data from the literature 70,73,114−122 can be found in Table 14.
3.7.Mixtures of Lactones + Carbon Dioxide: cCOO− CO 2 Interactions.Carbon dioxide is modeled with the CO 2 molecular group in SAFT-γ Mie. 90 One association site of type   116 and 5-methyloxolan-2-one. 117 (c) Excess molar enthalpy of oxolan-2-one + water at 299.15 K and 5-methyloxolan-2-one + water at 303.15 K, with experimental data for oxolan-2-one + water 118 and 5-methyloxolan-2-one + water. 73−126 although we note that inconsistencies have been reported for some of the data related to these mixtures. 125We use the most recent set of data 125 to estimate the cCOO−CO 2 interaction parameters and present a representative sample of the results in Figure 15.Bubble pressures at 333 K for several binary lactone + carbon dioxide mixtures are shown in Figure 15a, and the influence of temperature is shown in Figure 15b for the bubble pressure of oxepan-2-one + carbon dioxide.For this system, vapor−liquid− liquid equilibrium (VLLE) can be seen at 303 K. 125 At temperatures slightly higher than the critical point of CO 2 (303 K), an upper critical end point signals the end of the three-phase line, and continuous behavior, from VLE at low pressure to LLE at higher pressure, can be seen (in the figure, this behavior is observed at T ≥ 313 K).The calculations are in propyloxolan-2-one, 5-butyloxolan-2-one, 5-pentyloxolan-2-one, and 5-hexyloxolan-2-one (from dark blue to pale blue) at 333 K. Experimental data are represented by triangles for oxolan-2-one + carbon dioxide, 124 diamonds for 5-methyloxolan-2-one + carbon dioxide, 126 and squares for 5ethyloxolan-2-one + carbon dioxide. 125(b) Fluid−phase equilibria of oxepan-2-one + carbon dioxide at 303 K (black), 313 K (dark gray), 323 K (light gray), 333 K (dark red), and 343 K (red).Experimental data are represented by triangles. 125Thermodynamic conditions and the accuracy of the calculations are detailed in Table 15 and in the Zenodo datafile.The dagger symbols indicate that several sets of experimental data can be considered for the %ADD and AAD calculations and are taken together ( †) or separately ( † †).very good agreement with the experimental data, although the critical pressure is overestimated in the case of the oxolan-2-one mixture (Figure 15a).The critical pressures are also slightly overestimated, but the overall agreement in terms of the VLLE (at 303 K) and the high-pressure fluid-phase equilibria (at higher temperatures) shown in Figure 15b is very good.The corresponding %AADs and AADs are detailed in Table 15.When inconsistent sets of data are found, we calculate the %AADs and AADs for the references taken together (denoted as † in the table) or taken separately (denoted as † †).
3.8.Overall Deviations.As a summary of the accuracy of the calculation of the thermodynamic properties and phase equilibria obtained with the SAFT-γ Mie models presented, we collate in Table 16 and Figure 16 the overall deviations of the predicted and calculated data for each property considered.Deviations for temperature, composition, pressure, density, and enthalpy data types are presented in Figure 16 a−e, respectively.As can be seen, the values of the deviations calculated for points not used in parameter estimation (%AAD prediction and AAD prediction ) are found to be similar to those calculated for points used in parameter estimation (%AAD estimation and AAD estimation ), thereby confirming the robustness of the models.Moreover, in the case of bubble and dew temperatures (Figure 16a), as well as for densities for pure fluids and mixtures (Figure 16d), the %AAD prediction values are found to be smaller than those of the %AAD estimation .Azeotrope temperature data are not used in parameter estimation, and we thus report only the corresponding %AAD prediction and AAD prediction ; these deviations are found to be similar to those for bubble and dew temperatures.In terms of the composition deviations, it is important to note that the %AADs for compositions in binary mixtures (i.e., x 1 LLE and x 1 sat ) depend on the choice of compound 1 as a reference.The AADs provide a better metric of performance because these do not depend on the reference compound and they can be compared with the full range of mole fraction values (i.e., from 0 to 1).We find that the AAD estimation and AAD prediction values of the LLE and SLE compositions are small and of similar order of magnitude (Figure 16b).Similarly, in the case of the excess molar enthalpies, the overall %AADs are rather large, while the AADs are reasonably small; this is because of the experimental data values close to zero leading to very large %AADs.In particular, it is interesting to note that %AAD prediction is higher than %AAD estimation for the excess molar enthalpy (470.8% and 181.8%, respectively), while the corresponding AAD prediction is lower than AAD estimation (92.24J/mol and 153.7 J/mol, respectively).

CONCLUSIONS
The thermodynamic properties and phase behavior of small saturated lactones have been modeled with the SAFT-γ Mie group-contribution approach.A total of 86 systems have been considered, which correspond to 13 pure lactones and 73 binary mixtures: 21 with saturated hydrocarbons, 20 with alcohols, 2 with ketones, 4 with esters, 15 with aromatic compounds, 5 with water, and 6 with carbon dioxide.
A new SAFT-γ Mie cCOO group has been introduced, and the relevant like cCOO−cCOO and 17 unlike group interactions have been characterized.The accuracy of the calculations has been assessed by comparison with experimental data graphically with phase diagrams over broad thermodynamic conditions and by calculating the appropriate %AADs and AADs.The overall agreement between the experimental values and the SAFT-γ Mie calculations is found to be very good for all of the mixtures and properties studied.
A number of interesting regular features in the thermodynamic properties are found from the comparison between the available experimental data and the SAFT-γ Mie calculations.The VLE for the mixtures of 5-methyloxolan-2-one and several solvents, 95,101,105,108 including alcohols (Figures 7a and 9a), ketones (Figure 10a), esters (Figure 11a), aromatic compounds (Figure 12a), and carbon dioxide (Figure 15), do not present azeotropes.In the case of the solubility of oxepan-2-one in ketones (Figure 10b) and aromatic compounds (Figure 12b), eutectic points are found.The density of oxolan-2-one mixtures is abundantly documented, [96][97][98]102,106,116 and we have presented results for mixtures with alcohols (Figures 7c and  9b), esters (Figure 11c), aromatic compounds (Figure 13a), and water (Figure 14b).The density is found to be nonideal in most of the cases considered, with a concave or convex shape as a function of mole fraction. The values foud for the excess molar enthalpy, both in the literature 88,89,96,100,103,105,118 and from the SAFT-γ Mie calculations, are generally small as can be seen for the mixtures of oxolan-2-one with hydrocarbons (Table 8),
The parameters characterized in this work are transferable to other lactones, given the group-contribution nature of the SAFT-γ Mie equation of state, such that more complex molecules modeled with the cCOO group can be considered in future work.In particular, the new interaction parameters pave the way for the modeling of a wide range of compounds, for example, unsaturated lactones, ascorbic acid (vitamin C), and active pharmaceutical ingredients (e.g., simvastatin and lovastatin) of current interest.

Figure 1 .
Figure 1.SAFT-γ Mie molecular models of: (a) oxolan-2-one; (b) 5methyloxolan-2-one; (c) 6-propyloxan-2-one; and (d) oxepan-2-one.The rings of these saturated lactones are modeled with one cCOO group and the corresponding number of cCH 2 groups (in blue and gray, respectively).The linear side chains are modeled with CH 2 (in brown) and CH 3 groups (in green).The carbon participating in the ring and side chain is modeled with a cCH group (in yellow).Association sites are denoted by the smaller red circles, labeled e for electronegative (acceptor) sites.
r kl,ab is the distance between the centers of the two sites, ε kl,ab HB is the association energy, and r kl,ab c is the cutoff range of the interaction.Site a is positioned at distance r kk,aa d from the center of segment k, and site b is positioned at distance r ll,bb d from the center of segment l.The range of the association can equivalently be described by the cutoff r kl,ab c or by the bonding volume K kl,ab HB for given values of r kk,aa d and r ll,bb d .

Figure 5 .
Figure 5. Isobaric phase diagrams of ring lactones + hydrocarbons: oxolan-2-one (light blue) and oxepan-2-one (dark red) at atmospheric pressure.The curves represent the calculations with SAFT-γ Mie.The letters V, L and S indicate vapor, liquid, and solid phases, respectively.Experimental data used in the estimation of group-interaction parameters are represented with filled symbols, and those not used are represented with open symbols.(a)Oxolan-2-one + cyclohexane, with experimental data for the LLE.84(b) Oxepan-2-one + cyclohexane, with experimental data for the LLE85 and SLE.66 (c) Oxolan-2-one + n-hexane, with experimental data for the LLE.84,86(d)  Oxepan-2-one + n-hexane, with experimental data for LLE.85 (e) Oxolan-2one + n-octane, with experimental data for the azeotrope.83Thermodynamic conditions and the accuracy of the calculations are detailed in Table8and in the Zenodo datafile.

Figure 13 .
Figure 13.Isobaric thermodynamic properties of binary mixtures of lactone + aromatic compound: benzene (black), toluene (dark gray), and ethylbenzene (light gray) at atmospheric pressure.The curves represent calculations with SAFT-γ Mie.Experimental data used in the estimation of group-interaction parameters are represented with filled symbols, and those not used are represented with open symbols.(a) Density of oxolan-2-one + aromatic compounds at 293.15 K, with experimental data for benzene,106 toluene,106 and ethylbenzene.106(b) Excess molar enthalpy of oxolan-2-one + aromatic compounds at 293.15 K, with experimental data for benzene,105 toluene,105 and ethylbenzene.105(c) Excess enthalpy of oxan-2-one + aromatic compounds at 293.15 K, with experimental data for benzene,107 toluene,107 and ethylbenzene.107(d) Excess enthalpy of 5-methyloxolan-2one + aromatic compounds at 293.15 K, with experimental data for benzene,107 toluene,107 and ethylbenzene.107Thermodynamic conditions and the accuracy of the calculations are detailed in Table13and in the Zenodo datafile.

Figure 14 .
Figure 14.Isobaric thermodynamic properties of lactones + water at atmospheric pressure: oxolan-2-one (light blue) and 5-methyloxolan-2-one (dark blue).The curves represent the calculations with SAFT-γ Mie.Experimental data used in the estimation of group-interaction parameters are represented with filled symbols, and those not used are represented with open symbols.(a) Isobaric vapor−liquid equilibrium and solid−liquid equilibrium of oxolan-2-one + water, with experimental data for the bubble temperature114 and solid−liquid equilibrium.70(b) Density of lactones + water at 298.15 K, with experimental data for oxolan-2-one116 and 5-methyloxolan-2-one.117 (c) Excess molar enthalpy of oxolan-2-one + water at 299.15 K and 5-methyloxolan-2-one + water at 303.15 K, with experimental data for oxolan-2-one + water118 and 5-methyloxolan-2-one + water.73Thermodynamic conditions and the accuracy of the calculations are detailed in Table14and in the Zenodo datafile.

2.2. Parameter Estimation. The
parameters characterizing the groups are determined by minimizing the objective function data of relevant systems.The new group interactions required to model pure lactones and mixtures of lactones and linear alkanes are cCOO−cCOO, cCOO−cCH 2 , cCOO−cCH, cCOO−CH 3 , and cCOO−CH 2 .The results for pure lactones are detailed in the current section; the results for mixtures of lactones and linear alkanes are detailed in the next section.The experimental and calculated vapor

Table 2 .
List of the Saturated Lactones and Solvents Considered in Our Work, Together with the SAFT-γ Mie Group Representation

Table 3 .
Group Interactions Used to Model Saturated Lactones in a Range of Solvents With the SAFT-γ Mie Approach a

Table 4 .
SAFT-γ Mie Group Like Parameters of the Groups Considered in Our Current Work (Excluding Association) a aThe asterisk * in the ref column indicates that the cCOO group is characterized in the current work.

Table 5 .
Unlike Group Parameters (Excluding Association) for Use with the SAFT-γ Mie Approach a

Table 6 .
Group Association Parameters for Use with the SAFT-γ Mie Approach a 101at atmospheric pressure for 5-methyloxolan-2-one + acetone for a small range of x 5-methyloxolan-2-one values only and SLE data 66 of oxepan-2-one + pentan-2-one with a eutectic point at 194.46 K.As can be seen in Table11, most of the experimental data available are used to characterize the parameters.
3 3 and cCOO CH CO 3 .Limited experimental data are available, which include VLE data

Table 7 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Vapor Pressures P vap (T), Densities ρ(T,P), and Vaporization Enthalpies Δh vap (T) for Pure Lactones, Where N s,p D Is the Number of Experimental Data Points Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p

Table 8 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of the Azeotrope Composition x oxolan-2-one az and Temperature T az , Liquid−Liquid Equilibrium Compositions x 1 LLE , Solubilities x 1 sat , and Excess Molar Enthalpies Δh mix for Binary Mixtures of Lactones and Hydrocarbons, Where N s,p D Is the Number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p

Table 9 .
Overview of the of Accuracy SAFT-γ Mie in the Calculation of Azeotrope Compositions x oxolan-2-one az and Temperatures T az , Bubble Temperatures T bub , Dew Temperatures T dew , Bubble Pressures P bub , Solubilities x 1 sat , Densities ρ, and Molar Excess Enthalpies Δh mix for Mixtures of Lactones and Primary Alcohols, Where N s,p D Is the Number of Experimental Data Used in the the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p

Table 10 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Temperatures T bub , Dew Temperatures T dew , and Densities ρ for Mixtures of Lactones and Secondary Alcohols, Where N s,p D Is the Number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s pand AAD s p for System s and Property p

Table 11 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Temperatures T bub , Dew Temperatures T dew , and Solubilities x 1 sat for Binary Mixtures of Lactones + 2-Ketones, Where N s,p D Is the Number of Experimental Data Used in Parameter Estimation, and N s,p D,total is the Number of Experimental Data Used to Calculate %AAD s,p and AAD s,p for System s and Property p

Table 12 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Temperatures T bub , Dew Temperatures T dew , Solubilities x 1 sat , Densities ρ, and Excess Molar Enthalpies Δh mix for Binary Mixtures of Lactones + Esters, Where N s,p D Is the Number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p and in the Zenodo datafile.sites of type e 1 (one for each lone pair of electrons of the oxygen atom) and two sites of type H (corresponding to the hydrogen atoms).The H 2 O−H 2 O e 1 −H interactions are incorporated into the pure-water model.In mixtures with lactones, hydrogen bonding interactions between the H sites in the H 2 O group and the e 1 sites of the cCOO group are also accounted for.

Table 13 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Pressures P bub , Solubilities x 1 sat , Densities ρ, and Excess Enthalpies Δh mix for Binary Mixtures of Lactones + Aromatic Compounds, Where N s,p D Is the Number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p

Table 14 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Temperatures T bub , Dew Temperatures T dew , and Excess Molar Enthalpies Δh mix for Binary Mixtures of Lactones + Water, Where N s,p D Is the number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p

Table 15 .
Overview of the Accuracy of SAFT-γ Mie in the Calculation of Bubble Pressures P bub , Dew Pressures P dew , and Densities ρ for Mixtures of Lactones and Carbon Dioxide, Where N s,p D Is the Number of Experimental Data Used in the Parameter Estimation, and N s,p D,total Is the Number of Experimental Data Used to Calculate %AAD s p and AAD s p for System s and Property p a

Table 16 .
Overview of the Accuracy of SAFT-γ Mie Calculations for the Pure Lactones and the Binary Mixtures That Contain Lactones Considered in Our Work a estimation is the number of experimental data used in parameter estimation for property p. N p prediction is the number of experimental data not used in parameter estimation (prediction only) for property p.We calculated the corresponding %AAD estimation , %AAD prediction , AAD estimation , and AAD prediction .

Data Availability Statement Data
underlying this article can be accessed on Zenodo at DOI: 10.5281/zenodo.8268756and used under the Creative Commons Attribution license.