Assessment and Revision of the MOSCED Parameters for Water

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Assessment and Revision of the MOSCED Parameters for Water: Application to Limiting Activity Coefficients and Binary Liquid-Liquid Equilibrium Pratik Dhakal, and Andrew S Paluch Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04133 • Publication Date (Web): 08 Jan 2018 Downloaded from http://pubs.acs.org on January 8, 2018

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Industrial & Engineering Chemistry Research

Assessment and Revision of the MOSCED Parameters for Water: Application to Limiting Activity Coefficients and Binary Liquid-Liquid Equilibrium Pratik Dhakal and Andrew S. Paluch∗ Department of Chemical, Paper and Biomedical Engineering, Miami University, Oxford, Ohio 45056, USA E-mail: [email protected] Phone: (513) 529-0784. Fax: (513) 529-0761

Abstract MOSCED (MOdified Separation of Cohesive Energy Density) is an attractive method for modeling phase-equilibria because it both can make quantitative predictions and give insight into the system’s underlying molecular-level interactions. While MOSCED predicts satisfactory values of limiting activity coefficients of organics in water, we find that predicted limiting activity coefficients for water (here the solute) in organic solvents shows a much greater disparity with reference data. Moreover, the error is systematic and increases as the value of the limiting activity coefficient increases. After detailed assessment of the MOSCED parameters for water, we find this is not a limitation of the model but results from how the parameters were regressed. As a consequence, in the present study we re-regress MOSCED parameters ∗

To whom correspondence should be addressed

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for water which greatly improves the accuracy with which MOSCED may predict limiting activity coefficients of water in organic solvents, with the average absolute error decreasing from 3056.2% to 63.2%. The improvement is systematic, increasing as the value of the limiting activity coefficient increases. We additionally demonstrate the ability of the revised parameters to make improved binary liquid-liquid equilibrium calculations in aqueous systems.

Introduction The ability to model and predict the phase-behavior of mixtures is of central importance in many physical, environmental, and biological processes. Phase-behavior involving aqueous systems is of particular importance. For example, practically all biochemical reactions take place in an aqueous phase, which may require additional processing, purification, and formulation for a particular application 1 . Water is an inexpensive, eco-friendly solvent capable of solvating a range of ionic, polar inorganic, and organic compounds, causing it to be a popular solvent in industrial processes 1 . Conventional methods to predict the phase-behavior of mixtures, such as the group contribution method UNIFAC 2 and mod-UNIFAC (Dortmund) 3 , often neglect the molecular-level details of the system of interest. However, an understanding of the molecular-level details of the system can greatly improve process intensification 4 . MOSCED (MOdified Separation of Cohesive Energy Density) is a solubility parameter based method to predict limiting (or infinite dilution) activity coefficients 5,6 . As a solubility parameter based method, its use is particularly attractive because in addition to making quantitative predictions, MOSCED may additionally be used to understand the underlying molecular-level driving forces for intuitive solvent selection and formulation. A major improvement of MOSCED over similar solubility parameter-based methods is that it splits the association term, allowing MOSCED to better capture the physics of solvation of associating compounds 7,8 . In the recent 2005 re-parameterization of MOSCED, parameters were regressed for 130 organic solvents using 6441 reference limiting activity coefficients. With these parameters fixed, parameters for water were then regressed. Results were shown for the calculation of the limiting 2


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activity coefficient of organics in water. While the errors were found to be larger than for organic systems, the results were satisfactory 6,9 . In the present study, we find that predicted limiting activity coefficients for water (here the solute) in organic solvents shows a much greater disparity with reference data. We will show how this is not a limitation of the model but results from how the parameters were regressed. For many applications, we are concerned only with solvation in water. The original MOSCED parameters for this case, and can be used to make accurate property predictions 7,8 . However, MOSCED may only predict limiting activity coefficients. To predict composition dependent activity coefficients, as is needed for phase-equilibria predictions, one must parameterize an excess Gibbs free energy model such as the NRTL, UNIQUAC, or Wilson’s equations 2,5,10 . For a binary system, this would require the limiting activity coefficient of each species in the binary pair; that is, we would need to accurately compute the limiting activity coefficient when water is both the solvent and solute. As a result, we re-regress MOSCED parameters for water using reference data in which water is both a solute and solvent. With the proposed revised parameters, the accuracy with which MOSCED can predict limiting activities of water in organic solvents is greatly improved. In addition to limiting activity coefficients, we demonstrate the improved ability of MOSCED to predict mutual solubilities in aqueous systems.

Computational Methods MOSCED The limiting activity coefficient for component 2 in 1 (γ2∞ ) is calculated using MOSCED via the following series of equations 5,6 :

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ln γ2∞ = ln γ2∞,COMB + ln γ2∞,RES ( )aa2 ( )aa2 v2 v2 ∞,COMB ln γ2 = ln +1− v1 v1

[ ] (T ) (T ) (T ) (T ) (T ) 2 2 2 (T ) (α − α )(β − β ) q q (τ − τ ) v 2 2 1 2 2 + 1 ln γ2∞,RES = (λ1 − λ2 )2 + 1 2 1 RT ψ1 ξ1 [( ] )2 (T ) (T ) (T ) aa2 = 0.953 − 0.002314 τ2 + α2 β2 ( ( ( )0.8 )0.8 )0.4 293 K 293 K 293 K (T ) (T ) (T ) αi = αi , βi = βi , τi = τ i T T T

(1)

where i = {1 or 2} (T ) (T )

ψ1 = POL + 0.002629 α1 β1 [ ( )](293 K/T )2 ξ1 = 0.68 (POL − 1) + 3.4 − 2.4 exp −0.002687 (α1 β1 )1.5 [ ( ( )3 )] (T ) 4 POL = q1 1.15 − 1.15 exp −0.002337 τ1 +1 where vi is the (liquid) molar volume, λi , τi , αi and βi are the solubility parameters due to dispersion, polarity, and hydrogen bond acidity and basicity, respectively, where i = {1, 2}, and the induction parameter, qi , reflects the ability of the nonpolar part of a molecule to interact with a polar part. The terms ψ1 and ξ1 are empirical (solvent dependent) asymmetry terms to modify the residual contribution for polar and hydrogen bonding interactions, and aa2 is an empirical (solute dependent) term to modify the size dissimilarity in the combinatorial contribution for polar and hydrogen bonding interactions. These additional empirical terms are not adjustable but are functions of the other parameters (τi , αi , βi and qi ). For all cases aa2 ≤ 0.953, effectively reducing the size dissimilarity and magnitude of the combinatorial contribution, with the value smaller for polar and associating compounds. R is the molar gas constant and T is the absolute temperature. The superscript (T ) is used to indicate temperature dependent parameters, where the temperature dependence is computed using the empirical correlations provided in eq. (1). As suggested by the equations, MOSCED adopts a reference temperature of 293 K (20 ◦ C). An equivalent expression

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for the limiting activity coefficient for component 1 in 2 (γ1∞ ) can be written by switching the subscript indices.

Mutual Solubilities For a binary system in liquid-liquid equilibrium, equilibrium compositions are found by simultaneously solving the iso-activity relation for each component 2 :

II xI1 γ1I = xII 1 γ1

(2)

II xI2 γ2I = xII 2 γ2

where x1 and x2 correspond to the mole fraction of component 1 and 2, respectively, and γ1 and γ2 correspond to the composition dependent activity coefficient of component 1 and 2, respectively, with the superscript indicating the phase (I or II). The composition dependent activity coefficients are modeled using the NRTL equation 2 parameterized using limiting activity coefficients predicted using MOSCED. For the NRTL equation, we have 2 :

[

(

ln γ1 = x22 τ21 [

(

ln γ2 = x21 τ12

G21 x1 + x2 G21 G12 x2 + x1 G12

)2 + )2 +

τ12 G12 (x2 + x1 G12 )2 τ21 G21 (x1 + x2 G21 )2

] (3) ]

where G12 = exp (−α12 τ12 ) and G21 = exp (−α12 τ21 ), and τ12 and τ21 are adjustable parameters. NRTL has a third parameter α12 which is related to the randomness of the mixture; in this work we set α12 = 0.2 so that we are left with just two adjustable parameters. In DECHEMA’s “LiquidLiquid Equilibrium Data Collection,” α12 = 0.2 when modeling all binary systems, motivating the choice here 11 . At infinite dilution, eq. (3) reduces to: 5



ln γ1∞ = τ21 + τ12 G12

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(4)

ln γ2∞ = τ12 + τ21 G21 Calculating ln γ1∞ and ln γ2∞ using MOSCED, eq. (4) provides a system of two equations with two unknowns (τ12 and τ21 ). Solving for τ12 and τ21 , eq. (3) may be used to calculate composition dependent activity coefficients in the solution for eq. (2). While in the present student predictions will be made at a single temperature, in general when modeling temperature-dependent binary liquid-liquid equilibrium, MOSCED may used to calculate ln γ1∞ and ln γ2∞ at each temperature of interest, and then used to compute unique NRTL parameters at each temperature.

Results and Discussion Original MOSCED Parameters In the 2005 MOSCED parameterization, parameters were regressed for 130 organic solvents using 6441 reference activity coefficients. The root mean squared error (RMSE) for ln γ2∞ was 0.148, and the average absolute error (AAE) for γ2∞ was 10.6% 6,9 , where v u N ( )2 u1 ∑ t ln γ2∞,pred − ln γ2∞,ref RMSE = N i=1

AAE =

1 N

N ∑

∞,pred − γ2∞,ref γ2

i=1

γ2∞,ref

× 100%

(5)

(6)

where the summation is over all N reference values, where the superscripts “pred” and “ref” correspond to the MOSCED prediction and the reference value, respectively. With the MOSCED parameters for the 130 organic solvents fixed, the authors then subsequently regressed MOSCED parameters for water using a separate set of reference limiting activity coefficients involving water. 6


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From ref. 6: The magnitude and range of the infinite dilution activity coefficients for organics in water (10−1 to 1010 ) are much larger than the other organic data. In addition, the variability and discrepancies in experimental data are much larger for aqueous systems than most other organic solvent data due to experimental difficulties. Using the actual molar volume of water (18 mL/mol) in the model resulted in a poor fit of the data and gave unreasonably low values for the activity coefficient of water in organic solvents. Therefore, the molar volume of water was treated as an adjustable parameter and the optimal value was 36 cm3 /mol. The extensive hydrogen bond network present in water could possibly cause water to act with a larger molar volume in solution. With this change, MOSCED is able to correlate the activity coefficients of the organics in water to 41.1% AAE, which is good considering that the uncertainty for nonpolar solutes in water (particularly for hydrocarbon-water mixtures) is close to 50% or even more in some cases. Based on this information, it is unclear if water was parameterized using only values of ln γ2∞ for organics in water, or if values of ln γ2∞ for water in organics was also included. The reference data for water was not included in the supporting information of ref. 6 or in ref. 9. We therefore obtained a set of reference data from “Yaws’ Handbook of Properties for Aqueous Systems” 12 . For the 130 organic solvents for which MOSCED is parameterized, we obtained 88 values of ln γ2∞ for organics in water and 88 values for water in organics mostly at 298.15 K. Additionally, we obtain mutual solubilities at 298.15 K (i.e., binary liquid-liquid equilibrium) for 78 organics in water and for water in 63 organics 13 . Using MOSCED for organics in water, the RMSE for ln γ2∞ was 1.120 and the AAE for γ2∞ was 77.0%. For water in organics, the RMSE for ln γ2∞ was 2.864 and the AAE for γ2∞ was 3056.2%. For the reference set, the values of γ2∞ span the range of 10−2 to 108 for organics in water, and 10−1 to 103 for water in organics. We therefore find the error for water in organics to be significant. If we wish to use limiting activity coefficients predicted using MOSCED for a binary aqueous pair 7



to parameterize an excess Gibbs free energy model for phase-equilibria calculations, we require values of both γ1∞ and γ2∞ , where water is both the solute and solvent. We therefore need to ensure both values are calculated accurately. A parity plot of the MOSCED predictions of ln γ2∞ versus reference values is provided in fig. 1. For the case of organics in water, the data is well correlated with a Pearson correlation coefficient (R2 ) of 0.955, with a slope of the best fit line of 0.946. For the case of water in organics, the data is also well correlated with a Pearson correlation coefficient (R2 ) of 0.934, however, the slope of the best fit line is 1.634. This corresponds to a systematic error in ln γ2∞ that increases as ln γ2∞ increases. Given the ability of MOSCED to too accurately calculate ln γ2∞ for the case of organics in water while exhibiting systematic error for the case of water in organics, we will assume reference data for water in organics was not included in the original MOSCED parameterization of water. Let us consider the impact of this on the parameterization of MOSCED when the volume is treated as an adjustable parameter. Following the work of Abrams and Prausnitz 14 , ln γ2∞ computed using MOSCED (eq. (1)) can be written as the sum of a combinatorial (COMB) and residual (RES) contribution. The combinatorial contribution results from the size dissimilarity of the components, which corresponds to the partial molar excess entropy. For all cases, ln γ2∞,COMB ≤ 0, contributing toward negative deviations from Raoult’s law. The greater the size dissimilarity between the two components, the more negative the value of ln γ2∞,COMB . The residual contribution results from intermolecular interactions; it corresponds to the partial molar excess internal energy. The first two terms in the residual contribution are always greater than or equal to zero, contributing toward positive deviations from Raoult’s law. On the other hand, the third term, which accounts for association, is positive for all cases except when component 1 and 2 prefer to associate with each other rather than with themselves (i.e., when α1 > α2 and β1 < β2 , or when α2 > α1 and β2 < β1 ) 7,8 . If we consider only cases in which water is the solvent (component 1), v1 only appears in the combinatorial term. Of the 130 organic solvents for which MOSCED was parameterized, the smallest molar volume is 40.6 cm3 /mol for methanol. Therefore, the impact of doubling the molar

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volume of water from 18 to 36 cm3 /mol is to decrease the size dissimilarity in the combinatorial term, reducing its negative contribution to ln γ2∞ . Additionally, because the molar volume of water is smaller than the other organic solvents, the magnitude of the combinatorial term for water is larger than for an organic solvent with the same solute. Next consider the case of water as the solute (component 2). The molar volume of water, v2 , now appears in both the combinatorial and residual term. In the combinatorial term, doubling the molar volume again decreases the size dissimilarity in the combinatorial term, reducing its negative contribution to ln γ2∞ . However, in the residual term, v2 serves as a scale factor for our differences in intermolecular interactions. By doubling the molar volume, we are doubling the residual term. This would explain why the slope of the best fit line in the parity plot of the reference values of ln γ2∞ vs MOSCED predictions is 1.7 times greater for the case of water in organics than for organics in water. To further investigate this behavior, we kept all MOSCED parameters for water the same, including the molar volume used in the combinatorial term, but decreased the molar volume used in the residual term (i.e., the scale factor) to the actual molar volume of water of 18 cm3 /mol. With this set of parameters the MOSCED predictions for ln γ2∞ for organics in water is unchanged. However, for water in organics, the RMSE for ln γ2∞ decreases to 2.090 and the AAE for γ2∞ decreases to 74.1%. This is a significant improvement. This set of MOSCED predictions is slightly less correlated with the reference values of ln γ2∞ , with R2 decreasing to 0.865, and the slope of the best fit line decreases to 0.716. Use of the factor of 2 “overcorrects” the error. Using the original MOSCED parameters for water, the results for mutual solubility predictions, where MOSCED is used to parameterize the NRTL equation, are similar to that for limiting activity coefficients with more spread in the data for water in organics. A parity plot of the mutual solubilities predicted using MOSCED versus reference values is provided in fig. 2. Within fig. 2, a vertical dotted line is plotted at a mole fraction solubility of 0.01. As suggested by Sherman et al. 15 , when γ2∞ > 100 it may be reasonable to approximate γ2∞ = 1/xI2 or ln γ2∞ = − ln xI2 . We find that most of the data satisfies this condition. The choice of using the NRTL equation there-

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fore should not influence the overall results. The choice of excess Gibbs free energy model when predicting mutual solubilities with MOSCED may be important to evaluate in the future 2,16,17 . For organics in water, the RMSE for ln xI2 was 1.260 and the AAE for xI2 was 443.2%. The best fit line for the MOSCED predictions versus reference ln xI2 values resulted in an R2 value of 0.950 II with a slope of 0.930. For water in organics, the RMSE for ln xII 1 was 3.192 and the AAE for x1

was 105.9%. The best fit line for the MOSCED predictions versus reference ln xII 1 values results in an R2 value of 0.853 with a slope of 1.392. For organics in water, the mole fraction solubility range is 11 orders of magnitude (10−2 to 10−13 ), and for water in organics the range is 3 orders of magnitude (10−1 to 10−4 ). Additionally, of all of the systems studies, MOSCED (used to parameterize NRTL) predicts that water/1-butanol and water/2-methyl-1-propanol are miscible. Considering the homologous series of linear alcohols, 1-butanol is the smallest alcohol that is partially miscible with water.

Revised MOSCED Parameters As a result of the aforementioned shortcomings when using MOSCED to predict limiting activity coefficients for water in organics, and the necessity to predict accurate values of limiting activity coefficients when water is both a solvent and solute to parameterize an excess Gibbs free energy model to make phase equilibria calculation, here we will re-regress MOSCED parameters. As we have shown, it may be possible to tune the molar volume of water in the combinatorial and residual term separately to improve accuracy. However, we will use a single molar volume for simplicity. We also treat the molar volume as an adjustable parameter as in the original MOSCED parameterization. However, we include limiting activity coefficients for water in organics in addition to organics in water. We took as our reference set the data from “Yaws’ Handbook of Properties for Aqueous Systems” 12 for 88 values of limiting activity coefficients for organics in water and 88 values for water in organics mostly at 298.15 K. Parameters were regressed by minimizing the objective function (OBJ):

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OBJ =

176 ( ∑

∞,ref ∞,MOSCED ln γ2,i − ln γ2,i

)2 (7)

i=1

the squared difference between the reference (ref) value of ln γ2∞ and that predicting using MOSCED, where the summation is over all 176 reference data points. This is the same objective function as used in the original MOSCED parameterization 6,9 . During the regression, we set q = 1, and treated v, λ, τ , α, and β as adjustable parameters. The objective function was minimized using the differential evolution global optimization method 18 as implemented in GNU Octave 19 . The “original” and “revised” MOSCED parameters for water regressed in the presented study are provided in table 1. Similar to the original MOSCED parameterization, we obtain an optimal molar volume greater than the actual molar volume of water, 18 cm3 /mol. However, the optimal molar volume here is less than the value of 36 cm3 /mol found previously. This again has the effect of decreasing the size dissimilarity in the combinatorial term, decreasing its magnitude, albeit to a lesser extent. The most important effect of the smaller molar volume is on the residual contribution when water is a solute. Using MOSCED with our new water parameters for organics in water, the RMSE for ln γ2∞ was 1.090 and the AAE for γ2∞ was 144.8%. For water in organics, the RMSE for ln γ2∞ was 0.771 and the AAE for γ2∞ was 63.2%. For the case of organics in water, we find that RMSE decreases slightly, while AAE increases. We acknowledge that RMSE is what is minimized in our objective function. For the case of water in organics, we obtain a substantial decrease in both RMSE and AAE. The largest difference is for alkanes, alkenes, aromatics, and chlorinated hydrocarbons. For the 22 alkanes included in the reference set, the AAE for γ2∞ decreased from 6278.7% to 64.1% using the revised water parameters. For the 9 alkenes and aromatics included in the reference set (1-pentene, 1-hexene, 1-octene, benzene, toluene, p-xylene, ethylbenzene, isopropylbenzene, and butylbenzene), the AAE for γ2∞ decreased from 6003.0% to 45.1% using the revised water parameters. For the 7 chlorinated hydrocarbons (chloroform, carbon tetrachloride, 1,1-dichloroethane, 1,2-dichloroethane, 1,1,1-trichloroethane, trichloroethylene, and 1-chlorobutane) the AAE for γ2∞ 11



decreased from 7109.1% to 77.2% using the revised water parameters. These correspond to highly non-ideal cases where γ2∞ spans the range of 97.57 to 2602. Even for moderately non-ideal case of water in ethers (diethyl-ether, dipropyl-ether, dibutyl-ether, diisopropyl-ether, and methyl-tertbutyl-ether), where γ2∞ spans the range of 16.28 to 34.16, the improvement is appreciable with the AAE for γ2∞ decreasing from 1817.1% to 96.7.1% using the revised water parameters. As already pointed out, using the original water parameters for water in organics, the error in ln γ2∞ is systematic and increases as ln γ2∞ increases. Here we find that the improvement in the calculation of ln γ2∞ is systematic and increases as ln γ2∞ increases. On the other hand, for the case of water alcohols (methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 2-butanol, 2-methyl-2-propanol, 2-methyl-1-propanol, 1-pentanol, 1-hexanol, and 1-octanol), where γ2∞ spans the range of 1.148 to 4.83, the change in the AAE for γ2∞ is much smaller, increasing slightly from 28.2% to 57.6% using the revised water parameters. A parity plot of the MOSCED predictions of ln γ2∞ versus reference values is shown in fig. 1. For the case of organics in water, the data is well correlated with a Pearson correlation coefficient (R2 ) of 0.953, with a slope of the best fit line of 0.899. For the case of water in organics, the data is also well correlated with a Pearson correlation coefficient (R2 ) of 0.931, with a slope of the best fit line of 1.026. As a comparison, we were able to compute ln γ2∞ for 58 organics in water and for water in 64 of the organic solvents in the reference set using mod-UNIFAC (Dortmund) 3,20–24 with all calculations performed using CHEMCAD 25 . For this reduced set using mod-UNIFAC (Dortmund), the RMSE for ln γ2∞ was 3.144 and the AAE for γ2∞ was 93.9% for organics in water, and the RMSE for ln γ2∞ was 1.993 and the AAE for γ2∞ was 105.3% for water in organics. For the parity plot of the mod-UNIFAC (Dortmund) predictions of ln γ2∞ versus reference values, for the case of organics in water we have an R2 value of 0.782 with a slope of 0.592, and for the case of water in organics we have an R2 value of 0.725 with a slope of 0.495. The mod-UNIFAC (Dortmund) predictions are included in our parity plot in fig. 1. For this limited test set, MOSCED with our revised water parameters out-performs both MOSCED with the original parameters for water and mod-UNIFAC

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(Dortmund). Next, let us use the revised MOSCED parameters for water to make mutual solubility predictions, where MOSCED is used to parameterize the NRTL equation. A parity plot of the mutual solubilities predicted using MOSCED versus reference values is provided in fig. 2. For organics in water, the RMSE for ln xI2 was 1.414 and the AAE for xI2 was 733.1%. The best fit line for the reference ln xI2 versus the MOSCED prediction resulted in an R2 value of 0.921 with a slope of II 0.987. For water in organics, the RMSE for ln xII 1 was 1.097 and the AAE for x1 was 147.4%. 2 The best fit line for the reference ln xII 1 versus the MOSCED prediction results in an R value of

0.828 with a slope of 0.928. Using the revised MOSCED parameters for water, we now properly predict that water/1-butanol and water/2-methyl-1-propanol are partially miscible. As compared to using the original MOSCED parameters for water, we find that with the revised parameters the RMSE of predictions for organics in water increases slightly, while the RMSE for water in organics decreases greatly. For ln γ2∞ , we found the RMSE decreased with the revised water parameters for both organics in water and water in organics. The set of systems for which mutual solubility calculations were performed is a subset of the set of systems for which limiting activity coefficients are available, and the likely cause of this observation. The change is expected to be most drastic for water in organics as this data was excluded from the original parameterization. Additionally, we notice a much larger change in AAE as compared to RMSE. However, this is an artifact of how the errors are computed. Consider the solubility of water in organics. In fig. 1 we see that in general, using the original MOSCED parameters ln γ2∞ is over-predicted. As seen in fig. 2, using the original MOSCED parameters the solubility is therefore under-predicted. When the solubility is under-predicted, the maximum AAE is 100%. When the solubility is overpredicted, the AAE may exceed 100%. Let’s consider the specific case of the solubility of water in trichloroethylene. The reference solubility is 2.0458 × 10−3 mol fracs 12 . Using the original MOSCED parameters, the solubility is under-predicted by two order of magnitude, 1.3315 × 10−5 mol fracs. This results in an absolute error of 99.3%, while the squared difference of log solubili-

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ties (which contributes to RMSE) is 25.3. Using the revised MOSCED parameters, the solubility is over-predicted by one order of magnitude, 1.8131 × 10−2 mol fracs. This results in a much larger absolute error of 786.3%, while the squared difference of log solubilities is 4.8. MOSCED parameters were regressed by minimizing the squared difference of log limiting activity coefficients, which are related to log solubilities. As a comparison, we computed mutual solubilities for organics in water using mod-UNIFAC (Dortmund) with all calculations performed using CHEMCAD. For this set using mod-UNIFAC (Dortmund), the RMSE was 2.792 and the AAE was 3117.0%. Excluded from the calculation of RMSE and AAE are 1,1,1-trichloroethane, trichloroethylene, iodomethane, diiodomethane, iodoethane, and benzonitrile for which mod-UNIFAC (Dortmund) predicts the components are miscible with water. Nonane, decane, dodecane, tetradecane, hexadecane, butylbenzene, and ethyl benzoate were also excluded because the predicted solubility was less than the precision used by CHEMCAD (resulting in a value of 0). We additionally computed mutual solubilities for water in organics. Diethyl phthalate was excluded because CHEMCAD did not contain the necessary groups, despite mod-UNIFAC (Dortmund) parameters being available 26 . For this set using modUNIFAC (Dortmund), the RMSE was 1.871 and the AAE was 662.8%. Excluded from the calculation of RMSE and AAE are 1,1,1-trichloroethane, trichloroethylene, and benzonitrile, for which mod-UNIFAC (Dortmund) predicts the components are miscible with water. The mod-UNIFAC (Dortmund) predictions are shown alongside the MOSCED predictions in fig. 2. Having found that the accuracy with which γ2∞ is computed for water in alkanes is greatly improved using the revised MOSCED parameters for water, we next look at the ability of MOSCED to predict the solubility of the homologous series of linear alkanes in water at 298.15 K and 1 bar. The solubility and molecular conformations of linear alkanes in water was recently investigated using state-of-the-art molecular simulations in ref. 27, which offers an excellent comparison. Additionally, we recently developed a group-contribution method (GC-MOSCED) to predict MOSCED parameters 8,28 , which we will test here too. At these conditions, ethane, propane, and butane are gases. For these systems, the solute solubility (x2 ) was computed using Henry’s law 2 :

14


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x2 =

y2 p y2 p = ∞ sat H2,1 γ2 p2

(8)

where y2 is the vapor phase mole fraction of the solute (component 2), p is the total pressure of 1 bar, H2,1 is the Henry’s law constant for the solute in water, and psat 2 is the saturation (or vapor) pressure of the solute at 298.15 K. We assume that y2 = 1 and take psat 2 at 298.15 K from ref. 29. For pentane up to hexadecane, which are liquid, the values correspond to the (mutual) solubility in water as computed above. In order to accurately parameterize the NRTL equation, we need accurate limiting activity coefficients for both water in the alkanes and for the alkanes in water. As shown in fig. 3, over the studied range we find that predictions made using MOSCED and GC-MOSCED are in excellent agreement. Furthermore, the MOSCED and GC-MOSCED predictions are in excellent qualitative and quantitative agreement with the state-of-the-art molecular simulation results. In all cases, the correct trend is captured, with all three sets of predictions slightly overestimating the solubility.

Conclusion In the 2005 MOSCED parameterization, parameters were regressed for 130 organic solvents using 6441 reference limiting activity coefficients. The root mean squared error (RMSE) for ln γ2∞ was 0.148, and the average absolute error (AAE) for γ2∞ was 10.6% 6,9 . With the MOSCED parameters for the 130 organic solvents fixed, the authors then regressed MOSCED parameters for water using reference limiting (or infinite dilution) activity coefficients. Using these parameters, satisfactory values of ln γ2∞ for organics in water may be predicted. However, we observe large systematic errors when making predictions for water in organics. This is not a limitation of the model but results from how the parameters were regressed. During the 2005 MOSCED parameterization for water, the molar volume of water was treated as an adjustable parameter. When water is a solvent, the molar volume only effects the combinatorial term. However, when water is a solute, the molar volume effects both the combinatorial 15



and the residual term. If MOSCED is parameterized using only data wherein water is a solvent, systematic errors would result when making calculations for the case where water is a solute. We therefore regressed revised MOSCED parameters for water using a reference set of limiting activity coefficients of organic solvents in water and water in organic solvents. Using this revised set of MOSCED parameters, the accuracy of MOSCED to predict limiting activity coefficients of water in organic solvents improved greatly; the AAE for γ2∞ for water in organic solvents decreased from 3056.2% to 63.2%. The improvement was systematic, increasing as the value of the limiting activity coefficient increases. The ability of the revised parameters to make improved mutual solubility calculations was additionally demonstrated.

Acknowledgement Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund (56896-UNI6) for support of this research.

Supporting Information Available Tabulation of reference and MOSCED predicted limiting activity coefficients for organics in water and water in organics using both the original and revised MOSCED parameters for water. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Table 1: The “original” 6 and “revised” MOSCED parameters for water regressed in the present study, where v has units of cm3 /mol, q is dimensionless, and λ, τ , αi , and βi have units of MPa1/2 or (J/cm3 )1/2 . original revised

v 36.0 26.6

λ 10.58 6.53

τ 10.48 14.49

17

q 1.00 1.00

α 52.78 45.34


β 15.86 12.81


Figure 1: Parity plot of ln γ2∞ predicted using MOSCED and mod-UNIFAC (Dortmund) versus reference values from ref 12. The top pane is for organics in water, and the bottom pane is for water in organics. With MOSCED, predictions are made using both the “original” and “revised” parameters for water.

20 15

mod-UNIFAC original revised

10 5 0 -5 -5

organics in water 0

10

5

15

20

15

∞

ln γ2 MOSCED

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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water in organics 10 5 0 -5 -5

0

ln

∞ γ2

5

10

Reference

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30

organics in water mod-UNIFAC original 20 revised 10

I

−ln(x2 / mol frac) MOSCED

Figure 2: Parity plot of mutual solubilities predicted using MOSCED and mod-UNIFAC (Dortmund) versus reference values from ref 12. The top pane is for organics in water (xI2 ), and the bottom pane is for water in organics (xII 1 ). With MOSCED, predictions are made using both the “original” and “revised” parameters for water. The vertical dotted line corresponds to a mole fraction solubility of 0.01 and is drawn as a reference.

0 0

10 I −ln(x2

12

20

30

/ mol frac) Reference

water in organics

8 4

II

−ln(x1 / mol frac) MOSCED

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 0

4

8

II

12

−ln(x1 / mol frac) Reference 19



Figure 3: Equilibrium solubility (xI2 ) of the homologous series of linear alkanes in water at 298.15 K and 1 bar as a function of the carbon chain number. Reference values from ref. 12 are provided along with prediction made using state-of-the-art molecular simulations 27 , GC-MOSCED, and MOSCED. The GC-MOSCED and MOSCED calculations both use the revised water parameters from the present study.

-4

linear alkanes in water Reference Simulation GC-MOSCED MOSCED

-6

-8

I

log10(x2 / mol frac)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-10

-12

-14

3

6

9

12

Carbon number 20


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References 1. Weingärtner, H.; Teermann, I.; Borchers, U.; Balsaa, P.; Lutze, H. V.; Schmidt, T. C.; Frank, E. U.; Wiegand, G.; Dahmen, N.; Schwedt, G.; Frimmel, F. H.; Gordalla, B. C. Ullmann’s Encyclopedia of Industrial Chemistry; John Wiley and Sons, Inc. 2. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluidphase Equilibria, 2nd ed.; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986. 3. Weidlich, U.; Gmehling, J. A Modified UNIFAC Model. 1. Prediction of VLE, hE , and γ ∞ . Ind. Eng. Chem. Res. 1987, 26, 1372–1381. 4. Freund, H.; Sundmacher, K. Ullmann’s Encyclopedia of Industrial Chemistry; John Wiley and Sons, Inc. 5. Thomas, E. R.; Eckert, C. A. Prediction of limiting activity coefficients by a modified separation of cohesive energy density model and UNIFAC. Ind. Eng. Chem. Proc. Des. Dev. 1984, 23, 194–209. 6. Lazzaroni, M. J.; Bush, D.; Eckert, C. A.; Frank, T. C.; Gupta, S.; Olson, J. D. Revision of MOSCED Parameters and Extension to Solid Solubility Calculations. Ind. Eng. Chem. Res. 2005, 44, 4075–4083. 7. Dhakal, P.; Roese, S. N.; Stalcup, E. M.; Paluch, A. S. Application of MOSCED to Predict Hydration Free Energies, Henry’s Constants, Octanol/Water Partition Coefficients, and Isobaric Azeotropic Vapor-Liquid Equilibrium. J. Chem. Eng. Data 2017, accepted. 8. A Better Solution Ahead: Welcome to MOSCED. https://sites.google.com/ view/mosced, (accessed December 1, 2017). 9. Lazzaroni, M. J. Optimizing Solvent Selection for Separation and Reaction. Ph.D. thesis, Georgia Institute of Technology, 2004.

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10. Schreiber, L. B.; Eckert, C. A. Use of Infinite Dilution Activity Coefficients with Wilson’s Equation. Ind. Eng. Chem. Process Des. Develop. 1971, 10, 572–576. 11. Sørensen, J. M., Arlt, W., Eds. Liquid-Liquid Equilibrium Data Collection, Part 1: Binary Systems; DECHEMA: Frankfurt am Main, Germany, 1979. 12. Yaws, C. L. Yaws’ Handbook of Properties for Aqueous Systems; Knovel, 2012. 13. The mutual solubility data is from Chapter 1 “Solubility of Organic Compounds in Water” and Chapter 2 “Solubility of Water in Organic Compounds”. We included only data in which the organic was liquid at the temperature of interest. For the set of organic solvents that could be modeled with MOSCED, we therefore excluded propane and butane which are a gas. 14. Abrams, D. S.; Prausnitz, J. M. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy or partly or completely miscible systems. AIChE J. 1975, 21, 116–128. 15. Sherman, S. R.; Trampe, D. B.; Bush, D. M.; Schiller, M.; Eckert, C. A.; Dallas, A. J.; Li, J.; Carr, P. W. Compilation and Correlation of Limiting Activity Coefficients of Nonelectrolytes in Water. Ind. Eng. Chem. Res. 1996, 35, 1044–1058. 16. Lobien, G. M.; Prausnitz, J. M. Infinite-Dilution Activity Coefficients from Differential Equiliometry. Ind. Eng. Chem. Fundam. 1982, 21, 109–113. 17. Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth Publishers: Stoneham, MA, 1985. 18. Storn, R.; Price, K. Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. J. Global. Optim. 1997, 11, 341–359. 19. Eaton, J. W.; Bateman, D.; Hauberg, S. GNU Octave version 3.0.1 manual: a high-level interactive language for numerical computations; CreateSpace Independent Publishing Platform, 2009; ISBN 1441413006. 22


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20. Gmehling, J.; Li, J.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178–193. 21. Gmehling, J.; Lohmann, J.; Jakob, A.; Li, J.; Joh, R. A Modified UNIFAC (Dortmund) Model. 3. Revision and Extension. Ind. Eng. Chem. Res. 1998, 37, 4876–4882. 22. Gmehling, J.; Wittig, R.; Lohmann, J.; Joh, R. A Modified UNIFAC (Dortmund) Model. 4. Revision and Extension. Ind. Eng. Chem. Res. 2002, 41, 1678–1688. 23. Jakob, A.; Grensemann, H.; Lohmann, J.; Gmehling, J. Further Development of Modified UNIFAC (Dortmund): Revision and Extension 5. Ind. Eng. Chem. Res. 2006, 45, 7924–7933. 24. Constantinescu, D.; Gmehling, J. Further Development of Modified UNIFAC (Dortmund): Revision and Extension 6. J. Chem. Eng. Data 2016, 61, 2738–2748. 25. CHEMCAD 7.1.0.9402. http://www.chemstations.com. 26. Confirmed using the “Group Assignment” tool online provided by DDBST at http://www. ddbst.com/unifacga.html, mod-UNIFAC groups exist for diethyl phthalate. However, these groups were not available in CHEMCAD. 27. Ferguson, A. L.; Debenedetti, P. G.; Panagiotopoulos, A. Z. Solubility and Molecular Conformations of n-Alkane Chains in Water. J. Phys. Chem. B 2009, 113, 6405–6414. 28. Dhakal, P.; Roese, S. N.; Stalcup, E. M.; Paluch, A. S. GC-MOSCED: A Group Contribution Method for Predicting MOSCED Parameters with Application to Limiting Activity Coefficients in Water and Octanol/Water Partition Coefficients. Fluid Phase Equilib. 2017, DOI: 10.1016/j.fluid.2017.11.024. 29. NIST Chemistry WebBook, Standard Reference Database 69, Thermophysical Properties of Fluid Systems. http://webbook.nist.gov/chemistry/fluid/, (accessed August 28, 2017).

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Graphical TOC Entry 20

original parameters revised parameters 10

10

∞

ln γ2 MOSCED

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 0 organics in water 0

10

20

ln

∞ γ2

water in organics 0

Reference

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Assessment and Revision of the MOSCED Parameters for Water

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