Improved Prediction of Vapor Pressure for Pure Liquids and Solids

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Improved Prediction of Vapor Pressure for Pure Liquids and Solids from the PR+COSMOSAC Equation of Sate Li-Hsin Wang, Chieh-Ming Hsieh, and Shiang-Tai Lin Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 01 Oct 2015 Downloaded from http://pubs.acs.org on October 4, 2015

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Industrial & Engineering Chemistry Research

Improved Prediction of Vapor Pressure for Pure Liquids and Solids from the PR+COSMOSAC Equation of Sate Li-Hsin Wanga, Chieh-Ming Hsiehb, Shiang-Tai Lina,* a

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan

b

Department of Chemical and Materials Engineering, National Central University, Jhongli 32001, Taiwan

Abstract Vapor pressure of pure substances is a crucial piece of information for many industrial applications. A recently developed PR+COSMOSAC equation of state (EOS) with its molecular interaction parameters determined from quantum mechanical calculations has been shown to provide reasonable prediction accuracy for vapor pressure of almost any chemical species without the issue of missing parameters. In this work, we introduce two modifications to improve its prediction accuracy on the vapor pressure. The overall deviation in pressure for 1124 pure liquids from the modified version is 138%, which is about 1/4 of that from the original model 553%. In particular, the accuracy for the vapor pressure near triple point shows major improvement, with the average error reduced from 1062% to 233%. The sublimation pressure can also be estimated providing that the melting temperature and enthalpy of fusion are available. The average deviation in sublimation pressure from the modified PR+COSMOSAC EOS for 1140 substances is 412%, which is only 1/3 of that from the original model (1249%). This model is capable of providing both the vapor pressure and sublimation pressure over a wide range of conditions (from the critical point to below the triple point). It is particularly useful when experimental data are not available. 1

ACS Paragon Plus Environment


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Keywords:

vapor

pressure,

sublimation

pressure,

solvation

PR+COSMOSAC EOS

∗To

whom correspondence should be addressed: Email [email protected]

2


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free

energy,

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1. Introduction Thermodynamic properties of pure substances and mixtures are important information for the design and development of processes in chemical and pharmaceutical industries.1-3 Among the various thermodynamic properties, saturated pressure (e.g. vapor pressure and sublimation pressure) is of particular importance as it is related to the volatility of a chemical that serves as the basis for leakage monitoring or detection of explosive and poison contents in a luggage. The saturated pressure is also necessary for the calculation the phase equilibrium of a mixture, such as the vapor-liquid equilibrium, the solubility of a drug in solvent or supercritical fluids.4, 5 Numerous thermodynamic models are proposed for the prediction of vapor pressure,6 such as empirical/semi-empirical correlation models,7-10 group-contribution models,11-15 quantitative structure–property relationship (QSPR),16 and artificial neural network (ANN).17, 18 These correlative models usually provide accurate vapor pressure,7-10 but require a few thermophysical properties as input, such as critical temperature, critical pressure, normal boiling temperature, or triple point. Group-contribution models11-15 are usually developed based on the assumption that the property can be obtained from independent contributions of functional groups contained in a chemical. However, some group contribution methods require additional experimental thermophysical properties in order to increase prediction accuracy. For example, the group contribution method proposed by Nannoolal et al.15 is based on Antonie equation; and the three parameters in such equation are determined from group contribution and the normal boiling temperature of the substance of interest. Furthermore, group contribution models usually exhibit larger deviations for substances containing multifunctional groups where proximity effects are important. 3



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QSPR models generally predict thermophysical properties using molecular characters,19 such as topological descriptors (ex: bond angle), electronic descriptors (HOMO level), and molecular parameters (ex: number of rings). A variety of QSPR models have been proposed for vapor pressure prediction with good accuracy.16-18 Since the relationship between chemical structure and the vapor pressure has been found to be nonlinear, the artificial neural network architecture is often used to develop QSPR models. Godavarthy et al.16 used the idea of scaled variable reduced coordinates (SVRC) to generate a universal QSPR model. Yaffe and Cohen17 developed a temperature-dependent ANN model for the vapor pressure of hydrocarbons. Liang and Gallagher18 tested several multilinear regressions and ANN using solely theoretically-derived descriptors. However, the aforementioned group contribution methods and QSPR models are limited to specific properties of interest (e.g., prediction of vapor pressures and heat of vaporizations). Different parameter sets for a group contribution method and QSPR models are necessary when predicting other thermophysical properties. Cubic equations of states (CEOS) are a class of simple and yet powerful models which describe P-V-T (pressure-volume-temperature) relation for pure substances. The Peng-Robison EOS (PR EOS)20 and Soave-Redlich-Kwong EOS (SRK EOS)21 are typical examples of cubic EOS. They provide reasonable accuracy for the vapor pressure for a variety of chemicals. Later modification by Stryjek and Vera, denoted as Peng-Robinson-Stryjek-Vera EOS (PRSV),22 significantly improves the accuracy regarding the temperature dependency of vapor pressure. However, these CEOS models require input of the critical properties and acentric factor of the pure substances. Unfortunately such information is not always available, especially for newly synthesized chemicals, chemically unstable substances in the pure state, and 4


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macromolecules that may have been thermally decomposed even before reaching their critical point. The statistical associating fluid theory (SAFT)23, 24 and its extensions such as the PC-SAFT EOS25 are another class of methods that have been shown to provide highly accurate phase behaviors.26-28 Leonhard and coworkers showed that the association parameters in PC-SAFT can be calculated from the binding enthalpies and entropies of gas phase clusters using ab initio calculations.29 More recently Umer et al. demonstrated that all the PC-SAFT parameters of linear alcohols can be determined without input of experimental data.30 Hsieh and Lin proposed an approach utilizing the results from quantum mechanical and COSMO calculations to determined parameters in PR EOS.31,

32

Instead of using the critical properties and acentric factor of pure substances, this approach, denoted as the PR+COSMOSAC EOS, requires the molecular structure as the only input. The PR+COSMOSAC EOS have been applied to predict several different properties for pure substances (such as vapor pressure, liquid density, critical properties, and normal boiling temperature32-34) and phase behavior for mixtures (such as vapor-liquid equilibrium,33 liquid-liquid equilibrium,34,

35

vapor-liquid-liquid

equilibrium,36 and drug solubility in organic solvents at atmospheric condition37, 38 and in the supercritical carbon dioxide39). Furthermore, prediction of partition coefficients37, 40 and phase behavior for systems containing strong association41 are also investigated. Despite these successes, it can be shown that the PR+COSMOSAC EOS provides lowered accuracy in the vapor pressure of pure substances, especially at temperatures far below the normal boiling point. Such data are important for calculating the vapor-liquid equilibrium involving volatile chemicals, and the estimating of chemical potential of sub-cool liquids and the sublimation pressure of solids. 5



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In this study, two modification functions are introduced to improve the accuracy in the prediction of vapor pressures for pure substances. The prediction results from the modified PR+COSMOSAC EOS are greatly improved especially in lower temperature region (the temperature below the normal boiling point of the substance studied). The modified PR+COSMOSAC EOS can be applied to predict the sublimation pressure for a pure substance when its melting temperature and enthalpy of fusion are available. The accuracy of the modified PR+COSMOSAC EOS in the prediction of the sublimation pressure for pure substances and in prediction of binary vapor-liquid equilibrium are also investigated.

2. Theory

2.1 The Original PR+COSMOSAC EOS The Peng-Robinson Equation of state (PR EOS) is proved to be a useful model for describing P-V-T (Pressure-Volume-Temperature) relation of a fluid. =

−

(1)

where a and b are the energy and volume parameters for a substance. Typically, these two parameters are calculated from the critical point and acentric factor of a pure fluid. For mixtures, additional mixing rule is necessary to describe the composition dependence of a and b. Recently, Hsieh and Lin38 proposed a method to obtain a and b on the basis of quantum mechanical solvation free energy calculation and thus these two parameters can be determined without using any experimental data, , =

()

∗ ∑ ∆/ (, )

! = ∑ !

(2) (3) ∗

where CPR is a EOS dependent constant and is -0.623 for the PR EOS, ∆/ 6


the

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charging free energy of solute i dissolved in solvent S, and bi is the molecular cavity volume of molecule i in the COSMO calculation. The charging free energy consists of four energy components: the ideal solvation (is), charge-averaging correction (cc), restoring (res), and dispersion (dsp) contributions: ∗

∗&"'

∗$%" ∆/ , = ∆∗" + ∆∗ + ∆/ , + ∆

()

(4)

The first three terms on the right hand side of Eq. 4 are based on the COSMO-SAC model using results from quantum mechanical and COSMO calculations.42-44 Further details can be found in the literature.33, 38, 45 Since this method combines the PR EOS and the COSMO-SAC model, it is denoted as the PR+COSMOSAC EOS. The last ∗&"'

term on the right hand side of Eq. 4, the dispersion contribution ∆

(), is a

function of temperature and the exposed molecular surface area and molecular structure of molecule i: ∗&"' ∗&"' () + ./0 ( ) ∆∗&"' () = ∑) () *&"',) + +&"',) + ,-

(5)

where Sj is the exposed surface area of atom j of molecule i; Adsp,j and Bdsp,j are the ∗&"' ∗&"' dispersion parameters for atom j; ,- () and ./0 ( ) are empirical correction

to the dispersion free energy for molecules with hydrogen-bonding and ring structure, respectively. ∗&"' ( ) = ,/

1

5

?& () ∑) () *&"',) + +&"',) ∗&"' ∗&"' () + @ >?& () × ./0 ( ) +,1

1

*>?& ( ) = B C − EFGH + 1J

(7b)

D

1

(7a)

1

@ >?& () = BK C − EFGH + 1J

(7c)

D

where a and r are two newly introduced global parameters and ?$ is the critical temperature determined from the original PR+COSMOSAC EOS. Equations 7b and 7c are formulated in a way such that both Amod(T) and Rmod(T) become unity at = ?$ , i.e., the critical temperature from the original PR+COSMOSAC EOS. Therefore, both models provide the same results for the critical point of pure substances. Below the critical temperature ( < ?$ ) both correction factors Amod(T) and Rmod(T) are greater than unity (see Figure S1 in the Supporting Information) and resulting in stronger dispersion interactions (eq. 7a) and reducing the vapor pressure and sublimation pressure). Above the critical temperature ( > ?$ ) the correction factors approach some limiting values that is slightly below unity (e.g., 0.967 for 8


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pentnate). The values of a (in eq. 7b) and r (in eq. 7c) are obtained from the regression of vapor pressures of a few chemicals and all the other parameters are taken from the literature.38 The positive values of a and r enhances the dispersion contribution in the region below the critical temperature from the original PR+COSMOSAC EOS. The values of dispersion parameters are summarized in Table 1.

2.3 Vapor-liquid equilibrium (VLE) The criteria for VLE require the equivalence of fugacity for individual compounds in each phase. N̅ = N̅P

(8)

where N̅ and N̅P are the fugacity of component i in vapor and liquid phases, respectively. The fugacity from PR+COSMOSAC EOS can be determined from the charging free energy as following ln

SG (,T,) G T

=

∗DVW

∆0G/U

T

T

G − ln C1 − XH + XT − YZ[

(9)

where Z=PV/RT is the compressibility factor and is the mole fraction of component i in the fluid. The vapor pressure of a pure fluid is obtained from VLE calculations with xi=1.

2.3 Sublimation pressure for pure substance The fugacity of pure solid can be evaluated in two ways. One is to estimate from the fugacity of its pseudo liquid state (NP ) G7 ()Ga () TTGb

N" (, ) = NP (, )exp ` Sf",d

where d , d , and ∆e

∆, b

b

+ Gb C1 − G Hc G

(10)

are the triple point temperature, triple point pressure, and

enthalpy of fusion at triple point temperature of the substance i; g" and gP are the 9



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molar volumes of substance i in the solid and liquid state, respectively. In this study, normal melting temperature > , enthalpy of fusion at normal melting temperature Sf",>

∆e

, and d = > are used to replace the required information at triple point

because most of them are unavailable. g" is assumed to be a constant and estimated to be the volume parameter bi. gP and NP (, ) are determined from the PR+COSMOSAC EOS. The second method for calculating solid fugacity is from the sublimation pressure "f with Poynting correction N" (, ) = hi! ()exp B

G7 (TTGhi!())

J

(11)

Combining Eqs. 10 and 11 and letting P = j ' , the sublimation pressure "f can be predicted by solving the following equation using gP (, ) and NP (, "f ) calculated from the PR+COSMOSAC EOS and > = 1 atm. G7 Ga (,T) TGhi! TGk

hi! = NP (, hi! )exp `

+

lm7,k

,G

Gk

C1 −

Gk

Hc

(12)

The accuracy in sublimation pressure prediction is an indication for the accuracy of fugacity determined in low temperature region.

3. Computational Details

The first step of using the PR+COSMOSAC EOS is to obtain required information from the quantum mechanical (QM) and COSMO calculations.42-44 The quantum mechanical and COSMO calculation results for substances considered in this study are taken from the database developed by Virginia Polytechnic Institute and State University (denoted as the VT-database46, 47) when available. Otherwise, they are generated using the commercial package Dmol348 following the settings in the literature.42 Once the required data are available, the energy parameter a and volume 10


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parameter b in PR+COSMOSAC EOS can be obtained easily according to the computational procedure in literature.33 The computational procedure of the modified PR+COSMOSAC EOS is slightly different from that of the original PR+COSMOSAC EOS and is summarized as follows: 1. Obtaining quantum mechanical and COSMO calculation results from VT-database or commercial package DMol3. 2. Determining

the

critical

temperature

?$

from

the

original

PR+COSMOSAC EOS. The critical volume for PR EOS is calculated from the volume parameter b q

q

g = ! `1 + 4 + 2√2 r + (4 − 2√2)r c

(13)

Then, the critical temperature ?$ is obtained by solving the following equation G

?$ =

EFG (Ds tD s )s

∗

D ∆/ ?$ = ( ) s D

t(D )

(14)

3. Once ?$ is determined, the computational procedure of the modified PR+COSMOSAC EOS is the same as that of the original PR+COSMOSAC EOS, except using the dispersion contribution described in Eqs. 6 and 7.

4. Parameter Optimization

There are a total of 17 universal parameters and three parameters for each element (one for atomic radius and two for the determination of dispersion contribution term) in the modified PR+COSMOSAC EOS. The values of all parameters are taken from previous work,38 except for the two newly introduced parameters a and r (see Eqs 7a to 7c) to improve temperature dependency of atomic 11



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and ring dispersion contribution, respectively. Their values are determined by the regression of experimental sublimation pressures of Type1 and Type3 systems (see Table S1 in the Supporting Information for a complete list of chemicals) and the average logarithmic deviation (ALD) is used as the objective function: 1

ALD-P = / ∑/ }1 ulog1x

TGDyzD

{|8b

TG

u

(15)

where N is the number of experimental data points and superscripts calc and expt indicate the calculated and experimental result of sublimation pressure, respectively. In other words, the vapor pressure of Type1~Type4 systems are not used in the parameterization and are purely predictions. The procedure of parameter optimization for a and r is as following: First, a is obtained by regressing sublimation pressure data of 391 substances without ring structure covering a wide variety of chemicals such as alkanes, alkenes, alcohols, esters, organic acids. Once the value of the parameter a is determined, it is fixed in next parameter optimization step. Second, sublimation pressure data of 295 substances with ring structure (cyclic and aromatic chemicals) is used to determine the value of parameter r. The list of substances used in the parameter optimization is summarized in Supplementary Materials.

5. Results and Discussions

The modified PR+COSMOSAC EOS is validated using vapor pressures of 1125 substances and sublimation pressures of 1140 substances. All experimental data for vapor pressures and sublimation pressures of pure substances are taken from the DIPPR database.49 To facilitate the analysis, the pure substances are categorized into four types according to two conditions: (1) ring structures (e.g., cyclic and aromatic compounds) and (2) hydrogen bonding. Type1 substances are those without any ring 12


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structure and no hydrogen bonding interactions, e.g. n-pentane, acetone, diethyl ether. Type2 substances do not have any ring structure but contain hydrogen bonding interactions, e.g. 1-pentanol, acetic acid, n-butylamine. Type3 include substances with ring structure but without hydrogen bonds, e.g. benzene, cyclohexane, 1,4-dioxane. Type4 are substances with ring structure and capable of forming hydrogen bonds, e.g. phenol, cyclohexanol, benzylamine. Furthermore, the VLE of 1118 binary systems are used to examine the predictive accuracy of the modified PR+COSMOSAC EOS without using any experimental data. The experimental VLE data are retrieved from the DECHEMA Chemistry Data Series.50,

51

The binary VLE systems are simply

categorized into nonhydrogen bonding (nhb) and hydrogen bonding (hb) systems.

5.1 Prediction of vapor pressures for pure substances Figure 1 illustrates the prediction of vapor pressures for all systems predicted from original PR+COSMOSAC EOS and the modified PR+COSMOSAC EOS. The systematic over prediction of saturated pressures for all 4 types of species can be clearly seen. The overestimation, particularly at low temperatures (Figure 1a to 1d), implies that the intermolecular interactions, measured by parameter a in the PR EOS (eq. 1), are underestimated. Since this parameter is dominated by the charging free energy (eq. 2), the modified model, with the enhanced dispersion interactions using functions Amod(T) and Rmod(T), effectively corrects such systematic inaccuracy. The average logrithmetic deviation (ALD-P) is used to measure the accuracy of EOS in describing the vapor pressures of pure substances. Table 2 summarizes the deviation from the original and modified PR+COSMOSAC EOS for 1125 substances which cover a wide variety of chemicals. The calculation results from PR EOS are used as a baseline reference for comparison. At high temperature region (from normal boiling temperature to critical temperature of every substance), the accuracy in vapor 13



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pressure prediction from the modified PR+COSMOSAC EOS shows a slight improvement in comparison with that from the original PR+COSMOSAC EOS. This is because the modification functions Amod(T) and Rmod(T) only provide small contribution to the dispersion contribution at high temperature region. At low temperature region (from near triple point temperature to normal boiling temperature of every substance), the overall deviations in ALD-P is 0.321 (233.08%) from the modified PR+COSMOSAC EOS, compared with 0.391 (1061.80%) from the original PR+COSMOSAC EOS. The overall deviation is greatly reduced by 0.07 (or 1/4 in terms of percentage error) in the low temperature region. As listed in Table 2, it is worth to mention that ALD-P for types 1,3 and 4 compounds shows improvement both in the high and low temperature range. Although the ALD-P from the modified PR+COSMOSAC EOS is larger than that from PR EOS, the PR EOS requires experimental critical temperature, critical pressure, and acentric factor for every substance as input while the modified PR+COSMOSAC uses the molecular structure as the only input. The success of the modified model implies that the temperature dependence of the dispersion interaction in the original model (in the form of AT+B, see eq. 5) may not be suitable at low temperatures. The modification effectively introduces a 1/T temperature dependence to the original model, i.e., AT+B+C/T. Note that the value of B is about -200 J/mol/Å2 and that of A is about 0.2 J/mol/Å2 (see Table 1). Therefore, the original model has a moderate temperature correction by parameter A, which diminishes at low temperatures. The introduced modification enhances the dispersion contributions (more negative) at low temperatures. The small changes in the calculated vapor pressure from the modified PR+COSMOSAC for Type2 (hydrogen bonding) species are a result of two facts: (1) the correction factors Amod(T) of Type2 species are generally small, and (2) the 14


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charging free energy is dominated by electrostatic term and not the dispersion term for Type2 chemicals. Take properties of hexanol (Type2) and hexane (Type1) at the triple point (Tt) as an example. The correction factor Amod(T= Tt) for hexanol is 1.04, while that for hexane is 1.06 (see Table S2 in the Supporting Information). More importantly, the charging free energy of hexane (-16.98 RTt) is dominated by the dispersion energy (-17.15 RTt); while for hexanol the dispersion energy (-15.51 RTt) is only about 70% of the charging free energy (-21.39 RTt). As a result, there is a 1 RTt reduction in the charging free energy for hexane but the reduction is only 0.57 RTt in the case of hexanol. Therefore, the correction is, in general, less important for Type2 species. Figure 2 illustrates four examples of temperature dependence of vapor pressure from the original and modified PR+COSMOSAC EOS. The semi-empirical function used in the original PR+COSMOSAC EOS provides acceptable prediction results for vapor pressure at high temperature range, but a larger deviation is observed near the triple point temperature. The deviation between experimental data and predicted results becomes larger with decreasing temperature from the critical point to triple point. The modified PR+COSMOSAC EOS provides better prediction results in whole temperature range, indicating that the heat of vaporization are better described. It is also useful to compare the proposed method to a similar approach based on PC-SAFT (see Table 3). Umer et al.30 recently developed a computational approach that allows to determine the PC-SAFT parameters for linear alcohols and thus predict their vapor pressure and liquid molar volumes. It can be seen that the PR+COSMOSAC model is slightly more accurate in the vapor pressure but is less accurate in the liquid density.

5.2 Prediction of sublimation pressures for pure substances 15



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With the melting temperature and enthalpy of fusion at melting temperature, sublimation pressures of pure substances can also be estimated from the cubic equation of states through Eq. 10. Figure 3 compares the prediction of sublimation pressures from original PR+COSMOSAC EOS and the modified PR+COSMOSAC EOS with experiment for 1140 compounds containing a wide variety of chemicals in a wide temperature region. Here again, the systematic inaccuracy can be solved for all types of systems by the newly introduced function in this study. Table 4 summarizes the prediction accuracy from different methods. The PR EOS provides the lowest overall deviation among these three approaches and thus serves as a baseline for the comparison. The overall deviation from the modified PR+COSMOSAC EOS (ALD-P = 0.71 or 412%) is 0.5 smaller than that from the original PR+COSMOSAC EOS (ALD-P = 1.13 or 1249%), i.e. a reduction of 41% in overall deviation. Similarly, the accuracy for the prediction of sublimation pressure is improved for all types of substances considered in this study. This improvement indicates that the systematic error in prediction of phase equilibrium for sub-cooled liquid is corrected significantly. Figure 4 illustrates the temperature dependency of sublimation pressure prediction from the original and modified PR+COSMOSAC EOS for four substances: n-pentane, 1-propanol, α-methylstyrene, and 2-pyprolidone. The range of sublimation pressure of these four substances is from 10-16 Pa ~ 1 Pa. The modified PR+COSMOSAC EOS shows improved temperature dependency over the original model.

5.3 Prediction of VLE for binary system The AARD-P and average absolute deviation in vapor phase composition (AAD-y1) are used to evaluate the accuracy in prediction of binary VLE, 1

AARD-P = / ∑/ }1

{|8b

~TGDyzD TG

{|8b TG

~

× 100% 16


(16)

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1

%'d AAD-y1 = ∑/ }11, − 1, × 100% /

(17)

where N is number of data point and the superscripts calc and expt indicate results from calculation and experiment, respectively. Table 5 summarizes the overall deviation for 1118 binary VLE systems from the original and modified PR+COSMOSAC EOS and modified UNIFAC (a group contribution method).52 In the case of VLE for binary mixtures, the calculation results from the modified UNIFAC are regarded as a baseline reference for comparison. As listed in Table 5, the modified UNIFAC provides great prediction results and has the lowest deviation, 2.62% in AARD-P and 1.28% in AAD-y1, among these three approaches. The overall deviation from the modified PR+COSMOSAC EOS is 32.3% in AARD-P and 8.99% in AAD-y1. The AARD-P from the modified PR+COSMOSAC EOS is lower than that from the original PR+COSMOSAC EOS (39.96%), while the AAD-y1 from the modified PR+COSMOSAC EOS is similar to that form the original PR+COSMOSAC EOS (8.61%). This can be attributed to the modified dispersion contribution, which significantly improves the accuracy in vapor pressure prediction for pure substances, i.e. the endpoints of binary VLE phase diagram. This phenomenon can be observed in phase diagram of binary VLE for n-hexane + ethanol and 1-butanol + 2-butanol (Fig. 5). Although the modified UNIFAC provides best prediction accuracy, it should be noted that the experimental vapor pressures of relevant pure substances are required because the modified UNIFAC is an activity coefficient model. Such a group contribution based method is not applicable should any of the necessary parameters is missing. This is why fewer systems are investigated for the modified UNIFAC. Furthermore, different properties may require different sets of parameter, for example Pharma Mod. UNIFAC is developed for the solubility of active pharmaceutical ingredients in organic solvents

53

. It has been shown that the accuracy in VLE 17



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prediction from PR+COSMOSAC EOS for mixtures can be improved significantly when the same amount of experimental data are used 33.

5.4 Application of PR+COSMOSAC EOS when other methods are not applicable Anthraquinone and benzocaine are important chemicals in pharmaceutical industry and are used as a case study to demonstrate the capability of the modified PR+COSMOSAC EOS. Typical CEOS, such as PR EOS, cannot be applied to estimate the vapor and sublimation pressures for both substances because the experimental critical properties are not available in literature. Figure 6 shows the results of sublimation pressure prediction from the proposed method for both substances. The experimental data of sublimation pressure for anthraquinone are available in the temperature range of 346 K ~ 471 K

54-56

and these data covering a

wide temperature range may be sufficient for the design and optimization of relevant processes. However, in some cases, few or no experimental data are available for some compounds, such as benzocaine. In these two cases, the proposed model can be used to predict the sublimation pressure for a border temperature range with inputs of molecular structure, melting temperature, and enthalpy of fusion. As mentioned above, the modified PR+COSMOSAC EOS can also be applied to predict phase behaviors of mixtures without any further efforts, such as VLE, partition coefficient, and solubility in organic solvents. Figure 7 is an example of binary VLE (1,4-benzoquinone and benzocaine) for which the modified UNIFAC model is not applicable because experimental vapor pressures for both substances and the values of necessary pair interaction parameters are unavailable. Although the modified PR+COSMOSAC EOS can be applied to estimate several different thermodynamic properties for pure substances and phase behavior for mixtures with a single set parameter, its accuracy is not always the best. As presented in this work, the modified 18


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UNIFAC model can provide better binary VLE prediction if all required information is available.

6. Conclusion

The unique feature of PR+COSMOSAC EOS is its capability of providing thermodynamic properties of almost any fluids, pure or mixture, without input of any experimental data and species dependent parameters. It has been shown to provide reasonable prediction accuracy for a variety of phase behaviors within milliseconds on modern computers. The focus of the present work is to improve the prediction accuracy of saturated vapor pressure from the PR+COSMOSAC EOS, particularly in the low temperature regions. By analyzing the vapor pressure data of 1125 chemicals, it is found that the PR+COSMOSAC EOS, in general, overestimates the vapor pressure as a result of underestimating the intermolecular interactions at temperatures below the normal boiling point. In this work, we propose to introduce two modification functions to the dispersion contribution to correct for the systematic underestimation of molecular interactions in the original model. The modification was so designed such that the correction diminishes at high temperatures. The accuracy of the modified PR+COSMOSAC EOS in prediction of saturated pressure is improved significantly, especially near and below the triple point, while the performance near the critical point is not changed. This allows one to apply the method for the prediction of phase equilibrium containing solid phase, such as drug solubility in organic and supercritical solvents. The PR+COSMOSAC EOS developed in this study may provide useful information in early stage of drug discovery where usually very few experiment data is available. 19



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

Acknowledgments This research was partially supported by the National Science Council of Taiwan (NSC 101-2628-E-002-014-MY3, MOST 103-2218-E-008-003-MY2) and Ministry of Education of Taiwan (NTU-CDP-104R7876). The computational resources from the National Center for High-Performance Computing of Taiwan and the Computing and Information Networking Center of the National Taiwan University are acknowledged.

Supporting Information available The complete list of chemicals used in the parameterization and the temperature dependence of the correction factors are provided in the Supporting Information. This information is available free of charge via the Internet at http://pubs.acs.org/.

20


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Correlations from Droplet Evaporation Data. Chem Eng Sci 1984, 39, 1043-1049. (10) An, H.; Yang, W. M. A new generalized correlation for accurate vapor pressure prediction. Chem. Phys. Lett. 2012, 543, 188-192. (11) Ruzicka, V. Estimation of vapor pressures by a group-contribution method. Ind. Eng. Chem. Fundam. 1983, 22, 266-267. (12) Tu, C. H. Group-contribution method for the estimation of vapor-pressures. Fluid Phase Equilib. 1994, 99, 105-120. (13) Pankow, J. F.; Asher, W. E. SIMPOL.1: a simple group contribution method for predicting vapor pressures and enthalpies of vaporization of multifunctional organic compounds. Atmos. Chem. Phys. 2008, 8, 2773-2796. (14) Edwards, D. R.; Prausnitz, J. M. Estimation of vapor pressures of heavy liquid hydrocarbons containing nitrogen or sulfur by a group-contribution method. Ind. Eng. Chem. Fundam. 1981, 20, 280-283. (15) Nannoolal, Y.; Rarey, J.; Ramjugernath, D. Estimation of pure component 21



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analyzing their physical significance and predicting properties. J Phys Chem B 2008, 112, 5693-5701. (30) Umer, M.; Albers, K.; Sadowski, G.; Leonhard, K. PC-SAFT parameters from ab initio calculations. Fluid Phase Equilibr 2014, 362, 41-50. (31) Lin, S.-T.; Hsieh, C.-M.; Lee, M.-T. Solvation and chemical engineering thermodynamics. J. Chin. Inst. Chem. Eng. 2007, 38, 467-476. (32) Hsieh, C.-M.; Lin, S.-T. Determination of cubic equation of state parameters for pure fluids from first principle solvation calculations. AIChE J. 2008, 54, 2174-2181. (33) Hsieh, C.-M.; Lin, S.-T. First-principles predictions of vapor-liquid equilibria for pure and mixture fluids from the combined use of cubic equations of state and solvation calculations. Ind. Eng. Chem. Res. 2009, 48, 3197-3205. (34) Hsieh, C.-M.; Lin, S.-T. Prediction of liquid-liquid equilibrium from the Peng-Robinson plus COSMOSAC equation of state. Chem. Eng. Sci. 2010, 65, 1955-1963. (35) Lin, S.-T.; Wang, L.-H.; Chen, W.-L.; Lai, P.-K.; Hsieh, C.-M. Prediction of miscibility gaps in water/ether mixtures using COSMO-SAC model. Fluid Phase Equilib. 2011, 310, 19-24. (36) Hsieh, C.-M.; Lin, S.-T. First-principles prediction of vapor-liquid-liquid equilibrium from the PR plus COSMOSAC equation of state. Ind. Eng. Chem. Res. 2011, 50, 1496-1503. (37) Hsieh, C.-M.; Wang, S.; Lin, S.-T.; Sandler, S. I. A predictive model for the solubility and octanol-water partition coefficient of pharmaceuticals. J. Chem. Eng. Data 2011, 56, 936-945. (38) Hsieh, C.-M.; Lin, S.-T. First-principles prediction of phase equilibria using the PR+COSMOSAC equation of state. Asia-Pac. J. Chem. Eng. 2012, 7, S1-S10. (39) Wang, L.-H.; Lin, S.-T. A predictive method for the solubility of drug in supercritical carbon dioxide. J. Supercrit. Fluids 2014, 85, 81-88. (40) Hsieh, C.-M.; Lin, S.-T. Prediction of 1-octanol-water partition coefficient and infinite dilution activity coefficient in water from the PR plus COSMOSAC model. Fluid Phase Equilib. 2009, 285, 8-14. (41) Chen, W.-L.; Hsu, C.-C.; Lin, S.-T. Prediction of phase behaviors of acetic acid containing fluids. Fluid Phase Equilib. 2013, 353, 61-68. (42) Lin, S.-T.; Sandler, S. I. A priori phase equilibrium prediction from a segment contribution solvation model. Ind. Eng. Chem. Res. 2002, 41, 899-913. (43) Klamt, A.; Schuurmann, G. COSMO - A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient. J. Chem. Soc.-Perkin Trans. 2 1993, 799-805. 23



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(44) Barone, V.; Cossi, M. Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model. J. Phys. Chem. A 1998, 102, 1995-2001. (45) Hsieh, C.-M.; Sandler, S. I.; Lin, S.-T. Improvements of COSMO-SAC for vapor-liquid and liquid-liquid equilibrium predictions. Fluid Phase Equilib. 2010, 297, 90-97. (46) Mullins, E.; Liu, Y. A.; Ghaderi, A.; Fast, S. D. Sigma profile database for predicting solid solubility in pure and mixed solvent mixtures for organic pharmacological compounds with COSMO-based thermodynamic methods. Ind. Eng. Chem. Res. 2008, 47, 1707-1725. (47) Mullins, E.; Oldland, R.; Liu, Y. A.; Wang, S.; Sandler, S. I.; Chen, C.-C.; Zwolak, M.; Seavey, K. C. Sigma-profile database for using COSMO-based thermodynamic methods. Ind. Eng. Chem. Res. 2006, 45, 4389-4415. (48) Huron, M.-J.; Vidal, J. New mixing rules in simple equations of state for representing vapor-liquid-equilibria of strongly non-ideal mixtures. Fluid Phase Equilib. 1979, 3, 255-271. (49) Lin, S. T.; Hsieh, C. M. Efficient and accurate solvation energy calculation from polarizable continuum models. J. Chem. Phys. 2006, 125. (50) Gmehling, J.; Onken, U.; Arlt, W.; Grenzheuser, P.; Weidlich, U.; Kolbe, B.; Rarey, J. Vapor-Liquid Equilibrium Data Collection. DECHEMA: Frankfurt, 1982-2002; Vol. I. (51) Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection Dechema: Frankfurt, 1977. (52) 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. (53) Hahnenkamp, I.; Graubner, G.; Gmehling, J. Measurement and prediction of solubilities of active pharmaceutical ingredients. Int J Pharmaceut 2010, 388, 73-81. (54) Bardi, G.; Gigli, R.; Malaspina, L.; Piacente, V. Vapor pressure and sublimation enthalpy of anthraquinone and of 1,5- and 1,8-dihydroxyanthraquinones. J. Chem. Eng. Data 1973, 18, 126-130. (55) Goldfarb, J. L.; Suuberg, E. M. Vapor pressures and sublimation enthalpies of seven heteroatomic aromatic hydrocarbons measured using the Knudsen effusion technique. J Chem Thermodyn 2010, 42, 781-786. (56) Monte, M. J. S.; Sousa, C. A. D.; Fonseca, J. M. S.; Santos, L. Thermodynamic study on the sublimation of anthracene-like compounds. J. Chem. Eng. Data 2010, 55, 5264-5270. 24


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Table 1. The values of parameters for dispersion contribution in the PR+COSMOSAC EOS Parameter

Value

Adsp,HB (J/mol)

-465876.8150

Bdsp,HB (K)

-429.5556

Cdsp,HB (K)

-141.8436

Adsp,RING (J/mol/K)

-0.9181

Bdsp,RING (J/mol) a (-)

-365.0667 15.76

r (-)

254.75 Atom Specific Parameters

atom type

Ri (Å)

Adsp,i (J/mol/K/Å2)

Bdsp,i (J/mol/Å2)

H

1.30

0.1694

-191.4602

C

2.00

0.1694

-191.4602

N

1.83

0.4045

-207.9411

O

1.72

0.2701

-178.0767

F Cl Br I

1.72 2.05 2.16 2.32

0.1806 0.1566 0.0759 0.1902

-125.7842 -201.7754 -204.3810 -275.5677

S P

2.16 2.12

0.2604 3.903

-299.5098 -2395.547

25



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Page 26 of 36

Table 2. Comparison of deviations in vapor pressure prediction from the original and modified PR+COSMOSAC EOS and PR EOS High temperature regiona PR+COSMOSACmod

PR+COSMOSAC n Type1

c

Type2

d

Type3

e

Type4

f

Overall

b

ALD-P

n

523

0.103

193

b

PR b

ALD-P

n

ALD-P

523

0.106

523

0.016

0.193

193

0.195

193

0.055

290

0.133

290

0.153

290

0.024

118

0.150

118

0.144

118

0.026

1124

0.131(43.63%)

1124

0.137(42.46%)

1124

0.026(7.56%)

Low temperature regiona PR+COSMOSACmod

PR+COSMOSAC n

b

ALD-P

n

b

PR

ALD-P %

n

b

ALD-P

Type1c

524

0.348

524

0.280

524

0.090

Type2

d

193

0.476

193

0.477

193

0.230

Type3

e

290

0.430

290

0.307

290

0.109

Type4

f

118

0.344

118

0.284

118

0.102

1125

0.321 (233.08%)

Overall

1125 0.391(1061.80%) a.

1125 0.120(53.65%)

High temperature: from normal boiling temperature to critical temperature of

each species; low temperature: near triple point temperature to normal boiling temperature. b. Number of substances considered. c. Substances without any ring structure and no hydrogen bonding interactions. d.

Substances without any ring structure and having hydrogen bonding interactions. Substances with ring structure and no hydrogen bonding interactions. f. Substances with ring structure and having hydrogen bonding interactions. e.

26


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Table 3. Comparison of prediction of vapor pressure from PR+COSMOSAC and PC-SAFT for linear alcohols PC-SAFTa

PR+COSMOSAC AARD-Pb

AARD-Vl

AARD-P

AARD-Vl

Methanol Ethanol

19.8% 18.5%

15.4% 16.2%

8.7% 42.8%

9.5% 4.8%

1-Propanol

38.4%

15.3%

49.2%

3.8%

1-Butanol 1-Hexanol 1-Decanol

56.0% 39.8% 18.8%

16.0% 15.7% 19.9%

53.0% 48.0% 65.8%

3.0% 2.0% 8.9%

overal

31.9%

16.4%

44.6%

5.3%

a

Results of PC-SAFT are taken from ref.

b

AARD-P =

1

∑/ / }1

1

AARD-v = / ∑/ }1

{|8b

~TGDyzD TG

{|8b

TG

{|8b

~jGDyzD jG

{|8b jG

~

~

30

× 100%

× 100%

Table 4. Comparison of deviations in sublimation pressure prediction from the original and modified PR+COSMOSAC EOS and PR EOS PR+COSMOSAC nsys

a

PR+COSMOSACmod

ALD-P

nsys

a

PR

ALD-P

nsys

a

ALD-P

Type1b

528

1.23

528

0.75

528

0.40

Type2

c

203

0.97

203

0.96

203

0.73

Type3

d

290

1.25

290

0.53

290

0.36

Type4

e

119

0.72

119

0.57

120

0.34

1140

1.13 (1249%)

1140

0.71 (412%)

1140

0.44 (175%)

Total a. b.

Number of substances considered. Substances without any ring structure and no hydrogen bonding interactions.

c.

Substances without any ring structure and having hydrogen bonding interactions. Substances with ring structure and no hydrogen bonding interactions. e. Substances with ring structure and having hydrogen bonding interactions. d.

27



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Page 28 of 36

Table 5. Comparison of deviations in prediction of binary VLE from the original and modified PR+COSMOSAC EOS and the modified UNIFAC52 PR+COSMOSAC na

AARD-P AAD-y1 (%) (%)

PR+COSMOSACmod na


Modified UNIFAC na


nhb

559

46.37

7.72

559

36.44

8.62

482

1.82

0.89

hb

559

33.56

9.50

559

28.17

9.35

549

3.32

1.62

overall

1118

39.96

8.61

1118

32.30

8.99

1031

2.62

1.28

a.

Number of systems considered.

28


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Fig. 1. Comparison of vapor pressures for (a) type1 (b) type2 (c) type3 (d) type4 systems from original PR+COSMOSAC EOS and the (e) type1 (f) type2 (g) type3 (h) type4 systems from modified PR+COSMOSAC. (blue symbols: high temperature range, red symbols: low temperature range)

29


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log10(Pvap/Pa)


Page 30 of 36

8

8

7

6

6

4

5

2

4

0

3

-2

2

-4

1

-6

0

-8 0.008

0

0.002

0.004 1/T (K)

0.006

Fig. 2. Comparison of vapor pressures from experiments (symbols), the original PR+COSMOSAC EOS (dashed lines), and the modified PR+COSMOSAC EOS (solid lines) for n-pentane (squares), 1-propanol (triangles),  -methylstyrene (circles), and 2-pyprolidone (diamonds).

30


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Fig. 3. Comparison of sublimation pressures for (a) type1 (b) type2 (c) type3 (d) type4 systems from PR+COSMOSAC EOS and the (e) type1 (f) type2 (g) type3 (h) type4 from modified PR+COSMOSAC.

31



0 -2 log10(Psub /Pa)

-4 -6 -8 -10 -12 -14 -16 0.006

0.008

0.01

0.012

1/T (K) 2 1 log10(Psub /Pa)

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

Page 32 of 36

0 -1 -2 -3 0.003

0.004

0.005

1/T (K) Fig. 4. Comparison of sublimation pressures from experiments (symbols), the original PR+COSMOSAC EOS (dashed lines), and the modified PR+COSMOSAC EOS (solid lines) for n-pentane (squares), 1-propanol (triangles), α-methylstyrene (circles), and 2-pyprolidone (diamonds). 32


Page 33 of 36

P (kPa)

(a)

x1 , y1 (b)

P (kPa)

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


x1 , y1 Fig 5. Comparison of binary VLE from experiment (squares), the original PR+COSMOSAC EOS (dashed lines), and the modified PR+COSMOSAC EOS (solid lines) for (a) n-hexane (1) + ethanol (2) at 263.15 K and (b) 1-butanol (1) + 2-butanol (2) at 313.15 K.50 33


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

log10(Psub /Pa)


benzocaine

anthraquinone

T (K) Fig. 6. The prediction of sublimation pressure from the modified PR+COSMOSAC EOS for anthraquinone (solid line) and benzocaine (dash line). The experimental sublimation pressures for anthraquinone are taken from Goldfarb and Suuberg55 (circles), Bardi et al.54 (squares), and Monte et al.56 (triangles).

34


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P (kPa)

Page 35 of 36

x1 , y1 Fig. 7. VLE prediction from the modified PR+COSMOSAC EOS for binary mixture of 1,4-benzoquinone (1) + benzocaine (2) at 500 K.

35



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For Table of Content Graphics Only

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Improved Prediction of Vapor Pressure for Pure Liquids and Solids

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