Quantitative Structure-Property Relationships for Prediction of Phase

900

Ind. Eng. Chem. Res. 2010, 49, 900–912

Quantitative Structure-Property Relationships for Prediction of Phase Equilibrium Related Properties Mordechai Shacham,*,† Georgi St. Cholakov,‡ Roumiana P. Stateva,§ and Neima Brauner| Department of Chemical Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa 84105, Israel, Department of Organic Synthesis and Fuels, UniVersity of Chemical Technology and Metallurgy, Sofia 1756, Bulgaria, Institute of Chemical Engineering, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria, School of Engineering, Tel-AViV UniVersity, Tel-AViV 69978, Israel

In this work, novel techniques for predicting vapor pressure and binary interaction coefficients for homologous series are developed based on the previously proposed targeted quantitative structure-property relationship (TQSPR) and QS2PR methods. For predicting vapor pressure variation as a function of temperature, a tworeference compound (TRC) QSPR method is suggested. This method uses two, structurally similar predictive compounds with available vapor pressure data to predict point by point the vapor pressure or the saturation temperature of a target compound. For the target compound, only structural information is required. The two variants of the method were applied to several homologous series. They demonstrate prediction of vapor pressure within experimental uncertainty, depending on the level of similarity between the predictive compounds and the target compound. A targeted QSPR method for prediction of the binary interaction coefficients (kij) in cubic equations of state for a compound with the members of its homologous series is also presented. The coefficients for the Soave-Redlich-Kwong and Peng-Robinson equations, used to test the method, were reproduced within the deviation of those obtained from regressed experimental data. Introduction There is an increased interest in the development and use of quantitative structure-property relationship (QSPR) models. It stems from the remarkable progress achieved in systematic drug discovery and toxicology with quantitative structure-activity relationship (QSAR) models. QSPR models are extensively used for predicting a variety of pure component properties pertaining to chemistry and chemical engineering, environmental engineering and environmental impact assessment, and hazard and operability analysis. In process design, due to their low computational complexity, shortcut calculations allow for fast screening of a large number of design alternatives and preselection of the most favorable candidate structures.1-3 Since these calculations usually are performed prior to accurately measuring the thermodynamic properties in a laboratory experiment, they rely on accurate property predictions. However, the number of the compounds for which measured data are available is at most several thousands, and for many properties, it is much less, while the number of compounds used now by industry, or being of its immediate interest, is estimated at around 100 000. Theoretically possible chemical structures that may eventually interest industry in the future are at least several tens of millions.4 Phaseequilibria related properties that require measurements at various parameters of state and/or mixture compositions (vapor pressure, binary and ternary coefficients, etc.), are even scarcer. In product design, the availability of reliable methods for property prediction (i.e., QSPRs) is also important because fast screening of alternative chemical structures allows for reaching the specification requirements of the market product before the * To whom correspondence should be addressed. E-mail: shacham@ bgu.ac.il. Fax: +972-8-64-72916. † Ben-Gurion University of the Negev. ‡ University of Chemical Technology and Metallurgy. § Bulgarian Academy of Sciences. | Tel-Aviv University.

competition, thus saving time, money, and expert knowledge.5 Moreover, most of the market products are mixtures in which synergism has to be achieved,6,7 so mixing rules are also needed. Completed with “property-property” correlations between different levels of performance estimation,8 the three types of correlations can allow for systematic instead of “trial and error” intuitive development of new products. The choice of the correct method for property estimation, as illustrated by Wakeham et al.,9 can seriously influence process design, so the development of novel, efficient, and accurate property prediction methods is very important. Recently, a new quantitative structure-structure-property-relationship (QS2PR) method10-15 has been developed to complement the existing QSPR methods. It can predict a large number of properties with a single property-property correlation employing the coefficients of the structure-structure correlation of a target compound and the members of its similarity group. Its advantage is that the uncertainty of prediction of an unknown property value can be estimated from the uncertainties of the predictive compounds’ experimental data. The QS2PR method has been proven to give very accurate property predictions10,13-15 for well-represented target compounds in a molecular descriptorphysical property database (i.e., few compounds with known physical properties and of a high level of similarity with the target compounds are available). The “targeted” QSPRs (TQSPRs) are more flexible in terms of the similarity requirements.16 This method is based on preliminary identification of predictive compounds belonging to a similarity group of a particular target compound. This similarity group is then used to tailor a QSPR for a particular property of the similarity group. The TQSPR method has also been proven to have advantages in reliable extrapolation of properties of homologous series.17,18 Prediction of phase-equilibria related properties that require measurements at various parameters of state and/or mixture compositions is a special challenge, when compared to prediction of constant properties. Vapor pressure, for example, is often

10.1021/ie900807j  2010 American Chemical Society Published on Web 12/10/2009

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

represented by the Antoine equation (NIST database), extended Riedel’s equation (DIPPR database), or the Wagner equation.19 The different models that are used for activity coefficient calculations, like the regular solution theory, Wilson, NRTL, and UNIQUAC equations, etc., require different interaction parameters.19-21 The equations that are most widely used for prediction of vapor pressure of a target compound at different temperatures require the use of data, such as normal boiling temperature Tb, critical temperature Tc, and critical pressure Pc (see, for example, the Riedel correlation19). There are several group contribution based methods for predicting vapor pressure. A review of such methods by Dearden22 concluded that “atomic contributions alone do not seem to model vapor pressure well”. Dearden22 also mentions additional works, where vapor pressure was predicted using topological, topostructural, H-bonding, and quantum mechanical descriptors. Multiple linear regression and neural networks were used for deriving the QSPRs.23-25 Fugacity coefficients can be calculated using equations of state, such as the Soave-Redlich-Kwong and the Peng-Robinson equations with mixing rules.20,21,26-28 The pertinent calculations require knowledge of the pure-component constants (Tc, Pc, and acentric factor ω) and the corresponding binary interaction parameters kij. The same parameters of the pure components are needed for the model which is widely used by the petroleum industry to predict binary interaction parameters for the Peng-Robinson equation and mixing rules.31-39 It should be noted that most correlations for estimation of the acentric factor require also the normal boiling point.19 Experimental data for the discussed parameters are available for a limited number of compounds, and for some compounds, the parameters might not be measurable. Various QSPR19,29,30 and QS2PR10,14 methods can predict the pure-compound parameters and acentric factors. Experience shows, though, that large accumulation of deviations in the acentric factor can be observed, especially when combinations of experimental and estimated data are used.19 Still we are not aware of any attempt made so far to predict binary interaction coefficients using relationships between the relevant chemical structures. The above review suggests that recent developments in advanced QSPR and QS2PR methods for property prediction have not penetrated the phase-equilibria calculations, which is a key area in chemical process design and environmental impact assessment. The objective of this work is to advocate, present, and test for selected applications novel targeted-QSPR and QS2PR based techniques that will complement the existing methods for predicting phase-equilibrium related properties. Methodology Brief Description of the QS2PR Approach and Methods. The quantitative structure-structure-property relationship (QS2PR) approach has been described in detail and applied successfully for the prediction of numerous properties of pure components,10-14 so it will be briefly reviewed hereunder. It is assumed that the vector of properties of the target compound t (the dependent variable) is related to a set of m vectors of properties of predictive compounds (independent variables) p1, p2, ..., pm. The following partition of the t and p vectors to subvectors is used: t)

{}

xt ; yt

pi )

{} xi yi

(1)

where xt is an N vector of known properties, and yt is a K vector of unknown properties. Both the N vector xi and the K vector

901

yi contain known properties. In the QS2PR method, the subvectors xt and xi are molecular descriptors, while the subvectors yt and yi contain measured properties with various levels of experimental error. The structure-structure relationship, between xt and the independent variables, x1, x2, ..., xm, is expressed by a linear regression model of the general form: xt ) β1x1 + β2x2... + βmxm + ε

(2)

where the weighing factors β1, β2, ..., βm are the model parameters to be estimated and ε represents independent normal errors of a constant variance. The practical application of eq 2 requires preparation of a bank of potential predictive compounds as a database. The same set of molecular descriptors must be defined for all compounds included in the database, while the span of the molecular descriptors should reflect the difference between the compounds in the database. Having the corresponding molecular descriptors for a target compound, xt, defined as well, a stepwise regression procedure is applied to the database in order to identify the most appropriate predictive compounds that should be included in the structure-structure regression model (eq 2) and to obtain the respective model parameters. Upon identifying the model parameters, the following equation can be used for predicting unknown properties of the target compound: yt ) β1y1 + β2y2... + βmym

(3)

The properties that can be predicted for the target compound are all the properties for which experimental data are available for all the predictive compounds in the structure-structure correlation. The short-cut QS2PR (SC-QS2PR) approach12 stems from the observation that for members of a homologous series, the information required for deriving a structure-structure relation (hence, the property-property relation) can be drastically reduced. The minimum information required for deriving a structure-structure relation for a target compound in terms of m predictive compounds is the availability of m - 1 noncollinear molecular descriptors for both the predictive and the target compounds.12 For example, for m ) 3 the coefficients of the 3 βixi are obtained by the structure-structure relation xt ) ∑i)1 solution of the following system of three linear equations: β1 + β2 + β3 ) 1;

β1nc1 + β2nc2 + β3nc3 ) nct ; β1ζj1 + β2ζj2 + β3ζj3 ) ζjt

(4)

where nc1, nc2, and nc3 are the numbers of carbon atoms of the predictive compounds and nct is the number of the carbon atoms of the target compound. Since some of the molecular descriptors might have identical values for all members of a homologous series, the requirement that the sum of the parameter values should be one is introduced. The last equation in set 4 must be applied with a molecular descriptor ζj which is well-correlated with the property to be predicted, yP for the group of compounds to which the target and predictive compounds belong (i.e., the homologous series), and not collinear with the number of the carbon atoms. Alternatively, another property (e.g., normal boiling temperature) of the target and predictive compounds, which is well-correlated with yP, can replace the calculated descriptor, ζj. This option is an important practical advantage, because it enables users, without access to libraries of molecular descriptors, to employ the QS2PR technique prediction. The minimum number of predictive compounds required for applying the SC-QS2PR method is two. In this case only, the descriptor identified as well-correlated with the property to be predicted is used, whereby:

902


β1 + β2 ) 1;

β1ζj1 + β2ζj2 ) ζjt

(5)

Solving eq 5 for β1 and β2, the following property-property relation is obtained: ypt )

ζj2 - ζjt

ζjt - ζj1

ζj2 - ζ1

ζj2 - ζj1

yp + j 1

yp2

(6)

The SC-QS2PR method with two predictive compounds is denoted as two reference compound QSPR (TRC-QSPR). This name conveys that only structural information is used for the target compound, and property data of two reference compounds are used for the prediction. The applications of the QS2PR and shortcut QS2PR methods for predicting pure-component properties such as, normal boiling temperature, standard liquid density, melting point, and critical properties, etc. are well-documented.10,11,14 The objective of this research is to extend their use to prediction of properties needed for phase equilibrium calculations, namely vapor pressures of pure compounds and binary interaction parameters. The targeted QSPR method16,40,41 is described hereunder, only in principle. A detailed description is given elsewhere.16 The first stage of the method uses the QS2PR methodology (described above) for identification of a similarity group structurally related to the compound for which properties have to be predicted (the target compound) from a much wider database of compounds. The second stage of the TQSPR method is similar to the typical QSPR technique. The training set is selected from the members of the similarity group for which experimental data for yp (the property to be predicted) are available. The stepwise regression program SROV18 is used for the selection of the independent variables. In each step, it includes in the model one molecular descriptor ζj that reduces the prediction error most strongly. The descriptors are selected to the model in a stepwise manner according to the value of the partial correlation coefficient, |Fyj| between the vector of the property values y, and that of a potential predictive descriptor ζj. The partial correlation coefficient is defined as Fyj ) yj(ζjj)T, where yj and ζjj are row vectors, centered (by subtracting the mean) and normalized to a unit length. Values close to one indicate high correlation between the molecular descriptor and the property. Two criteria for measuring the signal-to-noise ratio in the jth candidate descriptor (TNRj) and in the partial correlation of the jth candidate descriptor with the prediction residual (CNRj) ensure that the selected descriptors contain valuable information and that overfitting is avoided. Additionally, the SROV program provides a procedure for rotation of descriptors, so that eventually a better combination of descriptors might be found. The final model is validated with a selection of (or with all) compounds in the similarity group which are not members of the training set. The application of the TQSPR method for prediction of properties of a homologous series is described in detail elsewhere.18 In brief, the particular homologous series is assumed a priori to be the required similarity group. Then a descriptor collinear with the studied property is identified by subsequently employing different homologous compounds as targets. Sources of Data, Databases, and Software. The experimental data and the descriptors of molecular structure used in this paper were taken from our previously described property and descriptor databases of 326 hydrocarbons and organic oxygen compounds.18,42 The original data for normal boiling temperatures, melting properties, and critical properties, etc.

were collected and updated from the DIPPR43 and the NIST44 databases and journal publications. The “Dragon” program45 was used to calculate 1664 descriptors46 for the compounds in the database from minimized-energy molecular models. The molecular geometry of each molecule was optimized using the CNDO (complete neglect of differential overlap) semiempirical method implemented in the HyperChem package.47 In this work, we have used also data (i.e., for the binary interaction parameters kij in the mixing rule for the mixture energy parameter of the cubic equations of state (CEoS)), which are not experimental, but obtained by regression of experimental data. The main reason for this exception is that binary interaction parameters are essential for modeling and prediction of phase equilibria for mixtures. Hence, we have applied these data only to test our methods and demonstrate their ability to predict for systems with scarce amount of published nonexperimental data for kij. The data in question were taken from the database of the Honeywell UNISIM48 software package, which is used widely by chemical engineers. The software programs used for the selection of similarity groups, collinear descriptors, etc. and for deriving the QS2PRs and TQSPRs were the ones we developed in the MATLAB49 environment on the basis of the SROV program.50 Hereunder, we present and discuss our results on prediction of saturated vapor pressure (as a representative example of the application of the proposed method for predicting temperaturedependent properties) and of binary interaction coefficients (the Peng-Robinson CEoS). We have chosen these examples because they demonstrate very well the specific challenges faced in the prediction of phase equilibria, while we believe that their importance will be well-recognized by the chemical engineering community. Prediction of Pure Compound Vapor (Saturation) Pressures Vapor (or saturation) pressures of pure components are essential in phase equilibrium calculations for nonideal and ideal systems. For nonideal systems, the vapor-liquid equilibrium ratio K (K-value) for the ith component, applying the gamma-phi approach, is given by Ki )

γiL fiL◦ φiVP

(7)

where γiL and fiL° denote the activity coefficient and the standardstate fugacity of the ith component in the liquid phase and φiV is the fugacity coefficients of the ith component in the vapor phase. A value for the pure component saturation pressure Psi is required to calculate fiL° of a pure liquid at the specified temperature and pressure. For ideal systems at low pressures, eq 7 is reduced to Ki ) Psi /P

(8)

where P is the total pressure. Hence, for ideal systems the prediction of Psi is sufficient for phase equilibrium calculations. Several correlations for prediction of vapor pressure are available in the literature (Miller, Riedel, etc.). Most of those correlations require the normal boiling temperature (Tb), the critical temperature (Tc), and pressure (Pc) of the target compound.19 The prediction of these properties, using either the QS2PR or shortcut QS2PR techniques is straightforward,12,42 and therefore, the prediction of vapor pressure via such correlation/models will not be further discussed here.


Pure component vapor pressures are usually available as a function of temperature in an equation of the general form f(Psi , T) ) 0. The most commonly used equations (Antoine, Riedel, and Wagner) are explicit in Psi , but the Antoine equation can also be brought into a form that can be explicitly solved for the temperature. We have applied the property-property correlation to directly predict the parameters of a particular (say, Antoine) vapor pressure equation of a target compound, when the same parameters are available for the predictive compounds, identified by the QS2PR technique and have found that, in general, these parameters do satisfy the property-property correlation. However, this approach has several practical limitations. First, the parameters for the same type of equation must be available for all the predictive compounds and for the same pressure range. Second, as these parameters have been obtained by nonlinear regression of data (in the case of the Antoine equation), or in some cases ill-conditioned linear regression (Riedel’s equation with non-normalized temperatures), the parameters for the target so-obtained may result in deterioration of the vapor pressure predictions. Consequently, a more robust approach was developed. Our approach uses the QS2PR method for predicting the saturation temperature at a specified pressure, and it is based on the QS2PR methods proven ability for high precision prediction of the normal boiling point. Accordingly, the targetcompound saturation temperature Tst corresponding to vapor pressure P should satisfy the property-property correlation (eq 6) when the temperatures corresponding to vapor pressure P of the predictive compounds (Ts1 and Ts2) are substituted on its righthand side (rhs). Thus, the following set of equations is used: Tts )

ζj2 - ζjt

Ts + j 1

ζj2 - ζ1 f(P, T1s) ) 0 and

ζjt - ζj1 ζj2

-

f(P, T3s)

T2s ζj1

(9)

)0

This is a set of three equations with three unknowns (T1s, T2s, which can be solved by simple substitution if, for example, the Antoine equation is used for representing the vapor pressure curve. If more complex equations (such as the Wagner or Riedel equations) are used, numerical (iterative) solution of the implicit equations is required. Generalized Algorithm for Prediction of Vapor (Saturation) Pressures. The algorithm combines the TQSPR and the TRC-QSPR methods and includes the following steps: 1. Identification of the target compound “similarity group”, which is structurally similar to the target compound (i.e., members of homologous series, otherwise use the TQSPR method of Brauner et al.16). It is assumed that data for the normal boiling temperature (Tb) are available for members of the similarity group, and for a few of them (at least two compounds), experimental data and/or models for vapor pressure are available. 2. A stepwise regression program is used to identify a molecular descriptor that is colinear with Tb for the group of predictive compounds. Recalling that the normal boiling temperature (Tb) is the saturation temperature at atmospheric pressure, suggests the use of a descriptor collinear with Tb in the structure-structure correlation of the TRCQSPR. The minimum number of predictive compounds needed in order to establish collinearity between a descriptor and a property is three. Additional predictive compounds, especially if they are located in opposite sides of the target compound (in terms of the selected descriptor) and cover a wide range of property values, can increase Tts),

903

significantly the confidence in the validity of the linear relationship. 3. Two or three predictive compounds are selected, which are closest to the target compound (in terms of the selected descriptor value) and preferably located on opposite sides of the target compound. Experimental data and/or models for vapor pressure should be available for the selected predictive compounds. 4. The common vapor pressure range for the predictive and target compounds is determined, according to the data available for the predictive compounds and critical pressure (predicted, if necessary) of the target compound. 5. The vapor pressures variation with temperature of the predictive compounds are represented by appropriate correlations/models (e.g., Antoine, Riedel, or Wagner) in cases where such models are not already available. 6. Application of the TRC-QSPR methods for point-by-point calculation of saturation temperature value for the target compound. This procedure can be simplified if the target and predictive compounds belong to the same homologous series and the Tb value is available for the target compound. In this case, the Tb value can be used instead of a molecular descriptor (step 2 of the procedure is skipped). Prediction of Tts for the 1-Alkene Series with the Antoine Equation. The suggested algorithm will be applied using two predictive compounds and the Antoine equation for this demonstration. The Antoine equation is the following: log(Psi ) ) Ai -

Bi Ci + T

(10)

where Ai, Bi, Ci, are the Antoine equation coefficients for the ith predictive compound, and Pis (bar) is the saturation pressure at temperature T (K). Introducing the log(Pis) values from eq 10 into eq 11 yields log(Pts), where Pts is the saturation pressure (bar) of the target compound at temperature T (K). An alternative form of the Antoine equation, which is explicit in the saturation temperature, Tis, at pressure P is the following: Tis )

Bi - Ci Ai + log(P)

(11)

Introducing the Tis values from eq 11 into eq 6 yields the value of the saturation temperature of the target compound at pressure P. Data for seventeen members of the 1-alkene series, which include the number of carbon atoms (nC), Tb, vapor pressure data in terms of Antoine-equation parameters,44 and the recommended temperature range for its applicability are collected and presented in Appendix A, Table A1 (Appendix A can be found in the Supporting Information (SI)). The corresponding pressure applicability range (obtained by introducing the minimal and maximal temperature values into eq 10) is shown in Table 1. Up to 1-hexadecene, experimental Tb values are available as well as Antoine equation parameters. For higher nC, only a few Tb values and no Antoine-equation parameters are available. Table 1 shows also the values of the descriptor VEv1 (a 2D eigenvalue-based descriptor: eigenvector coefficient sum from van der Waals weighted distance matrix, calculated by the Dragon program) for the 1-alkene series, which can be calculated irrespective of nC. Figure 1 shows a plot of Tb versus the values of the descriptor VEv1 for the 1-alkene series. Observe that the Tb data can be represented by a straight line, Tb ) 1103.7Vev1 - 12.647, with a linear correlation coefficient R2 ) 0.9981. This equation can

904


Table 1. Pressure Ranges and VEv1 Descriptor Values for the 1-Alkene Homologous Series press. range no.

component name

low

high

descriptor VEv1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1-butene 1-pentene 1-hexene 1-heptene 1-octene 1-nonene 1-decene 1-undecene 1-dodecene 1-tridecene 1-tetradecene 1-pentadecene 1-hexadecene 1-heptadecene 1-octadecene 1-nonadecene 1-eicosene

0.0240 0.5369 0.1664 0.0228 0.0637 0.0638 0.0636 0.0640 0.0638 0.0692 0.0692 0.0692 0.0695

1.0908 1.0399 1.0400 0.8786 1.0401 1.0403 1.0389 1.0402 1.0404 1.0104 1.0282 1.0105 1.0109

0.2599 0.2902 0.3177 0.3429 0.3663 0.3884 0.4093 0.4291 0.4480 0.4663 0.4838 0.5007 0.5170 0.5329 0.5483 0.5633 0.5778

be used for predicting Tb for the compounds for which no experimental values are available. In this case, there are many potential predictive compounds for every possible target compound. Our experience has shown that models of the highest accuracy are obtained if the selected predictive compounds, belonging to the similarity group, are the closest to the target compound (in terms of the selected descriptor value) and preferably located on opposite sides of the target compound. Also such models have the widest range of applicability. It is essential however that for the selected predictive compounds experimental data and/or models for vapor pressure variation with temperature are available. Let us select 1-decene and 1-dodecene as predictive compounds and 1-undecene as the target compound. Reliable prediction of vapor pressure can be obtained only if the predictions are carried out in the common temperature applicability range of the Antoine equations for the predictive compounds (for the case of Pts(T)), or the common pressure applicability range of the same equations (for the case of Tts(P)). The vapor pressures vs temperature of the predictive and target compounds are shown in Figure 2. Examining the temperature range of validity of the high and low pressure-range parameters of the predictive compounds shows a rather limited overlap between those ranges. However, there is practically a complete overlap between the pressure ranges of validity. Thus, limiting the prediction range to the range of applicability of the equations

Figure 1. Plot of Tb versus the descriptor VEv1 for the 1-alkene series.

Figure 2. Vapor pressure versus temperature of the predictive and target compounds: (() 1-decene (predictive); (2) 1-undecene (target); (•) 1-dodecene (predictive). Table 2. Prediction of Ts for 1-undecene (target compound) with 1-decene and 1-dodecene Predictive Compounds Pressure

Saturation Temp/K

Deviation1

No.

bar

Antoine Eq.

SC-QS2PR

K

%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.066 0.067 0.068 0.069 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

379.78 380.15 380.51 380.87 381.22 384.55 387.55 390.28 409.57 422.01 431.44 439.12 445.65 451.36 456.45 461.06 465.29

379.65 380.02 380.38 380.74 381.09 384.42 387.42 390.15 409.42 421.85 431.27 438.94 445.46 451.17 456.26 460.87 465.09

0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.14 0.15 0.16 0.17 0.18 0.18 0.19 0.19 0.20 0.20

0.034 0.034 0.034 0.034 0.034 0.034 0.035 0.035 0.037 0.038 0.039 0.040 0.041 0.041 0.042 0.042 0.04

1

s s - TSC-QS2PR | DeViation ) |TAntoine

used provides an important justification for preferring the prediction of Tts(P) over the prediction of Pts(T). Introducing the values of the descriptor VEv1 for the predictive and target compound (ζ1 ) 0.4093; ζ2 ) 0.4663; and ζt ) 0.448) into the structure-structure correlation (eq 5) yields the following parameter values: β1 ) 0.48814 and β2 ) 0.51186. Using these parameter values in the TRC-QSPR property-property relation, values of Tts for 1-undecene are predicted (see Table 2) and the results were compared with those obtained by Antoine equation for 1-undecene (the parameters are given in Table A1). The difference between the TRC-QSPR prediction and the Antoine equation calculation is the deviation of the predicted values. The highest absolute deviation was 0.2 K (0.04%) at P ) 1.0 bar. Considering that the uncertainty in the reported Tb value of 1-undecene is 6.0 K (NIST44), the prediction uncertainty is well below the experimental uncertainty in this case. The good prediction is further verified by Figure 3 showing the predicted vs experimental values for 1-undecene. The use of the two immediate neighbors of 1-undecene in the homologous series as predictive compounds can explain the very low deviation in this case. For a longer-range interpolation we have selected 1-octene (nC ) 8) and 1-tridecene (nC ) 13) as predictive compounds, keeping 1-undecene (nC ) 11) as target compound. This yielded a TRC-QSPR with β1 ) 0.37188 and β2 ) 0.62812. The highest deviation in this case is 1.35 K


Figure 3. Predicted and Experimental Vapor Pressure Data for 1-undecene (see Table A2), •, experimental data; —, predicted data.

(0.3%) at P ) 1 bar. Thus, although there is some deterioration in the prediction accuracy when using a longer range interpolation, the precision is still very good. For testing the extrapolation capabilities of the method, we have selected 1-octene (nC ) 8) and 1-undecene (nC ) 11) as predictive compounds, and 1-tetradecene (nC ) 14) as the target compound. This yielded a TRC-QSPR with β1 ) -0.87029 and β2 ) 1.8703. The highest prediction uncertainty in this case was -3.65 K (0.7%) at P ) 1.0 bar. Thus, extrapolation yields mixed-sign parameter values and expectedly12 results in higher prediction uncertainty than interpolation. Dearden22 provides prediction uncertainty values for current state of the art QSPRs for predicting Tb. For the Alkenes group a QSPR which includes 5 topological descriptors is mentioned. The average prediction uncertainty using this QSPR reported as 2.3%. Thus, with the proposed method the prediction uncertainties are lower than those of state of the art QSPRs even for the case of extrapolation. It can be argued that for close neighbors the corresponding states theorem can provide good estimation of Pts(T), as at Tr ) 1; Pr ) 1 for all compounds. To demonstrate the imprecision associated with the use of this principle at points not so close to Tc we have plotted the values of Pr at Tr ) 0.6 (calculated by the Antoine equation) versus the number of the carbon atoms for part of the 1-alkene series (see Figure 4). It can be seen that there is a considerable difference between the Pr values of even the closest neighbors (for example there is 16.1% difference between the Pr values of 1-undecene and 1-dodecene). Furthermore, the change of Pr with the change of the number of carbon atoms is nonlinear, thus linear interpolation or extrapolation even between close neighbors can be risky. More sophisticated use of reduced properties in the framework of the TRC-QSPR method will be described in the section: “Prediction of for the n-alkanoic Acid Series with the “TRC-QSPR”. Prediction of Tts for the n-alkanoic Acid Series with the Riedel Equation. In this example the Riedel equation: ln(Pis) ) Ai +

Bi + Ciln T + DiT Ei T

(12)

is used to represent the vapor pressure data of the predictive compounds. For calculation of Tts(P), the equation is rewritten in an implicit form:

(

f(Tis) ) ln(P) - Ai +

)

Bi + Ci ln T + DiTEi T

(13)

905

Figure 4. Plot of the reduced pressure, Pr versus the number of the carbon s atoms, at Tr ) 0.6, for part of the 1-alkene series, • P(T /Pc; -, linear fit: r ) 0.6) Pr ) -0.0007 nc + 0.0113 (R2 ) 0.9778). Table 3. Pressure Ranges and Descriptor Values for the n-Alcanoic Acid Series press. range component name

low

high

descriptor VEv1

methanoic acid ethanoic acid propanoic acid n-butanoic acid n-pentanoic acid n-hexanoic acid n-heptanoic acid n-octanoic acid n-nonanoic acid n-decanoic acid n-undecanoic acid n-dodecanoic acid n-tridecanoic acid n-tetradecanoic acid n-pentadecanoic acid n-hexadecanoic acid n-heptadecanoic acid n-octadecanoic acid n-eicosanoic acid

2.41 × 103 1.28 × 103 1.31 × 101 1.03 × 101 3.96 × 10-2 3.17 × 10-1 4.65 × 10-2 2.76 × 10-1 4.55 × 10-2 1.45 × 10-1 3.56 × 10-2 7.90 × 10-2 2.22 × 10-2 3.75 × 10-2 1.12 × 10-2 1.83 × 10-2 5.70 × 10-3 7.25 × 10-3 2.82 × 10-3

5.81 × 106 5.74 × 106 4.61 × 106 4.06 × 106 3.63 × 106 3.31 × 106 3.04 × 106 2.78 × 106 2.50 × 106 2.29 × 106 2.10 × 106 1.94 × 106 1.78 × 106 1.64 × 106 1.57 × 106 1.49 × 106 1.37 × 106 1.33 × 106 1.19 × 106

1.728 1.978 2.203 2.41 2.601 2.78 2.948 3.107 3.259 3.404 3.543 3.677 3.806 3.931 4.052 4.169 4.283 4.395 4.609

For each pressure value, this equation has to be solved iteratively to yield Tis. The Tis values are then introduced into the property-property correlation (eqs 6) to obtain the respective Tts value. Vapor pressure data and equations for the first 19 members of the n-alkanoic acid series are available in the DIPPR database34 (Table A3). The data include the number of carbon atoms (nC), the normal boiling temperature (Tb), the Riedel equation parameter values (Ai, Bi, Ci, Di, and Ei), the recommended temperature range of its applicability, and the associated uncertainty in the calculated vapor-pressure values. DIPPR recommends the range between the triple point and the critical temperature as the range of applicability for the Riedel equation (the corresponding vapor pressure ranges are shown in Table 3). The uncertainty associated with the use of this equation varies considerably for the various compounds. For ethanoic acid, the uncertainty is up to 1%, and for most higher nC compounds, it is up to 10% (for eicosanoic acid it is 25%). For the n-alkanoic acid series, Brauner et al.17 have identified the descriptor VEV1 (a 2D eigenvalue-based descriptor, eigenvector coefficient sum from van der Waals weighted distance matrix) as collinear with Tb. The relationship between Tb and the descriptor can be represented by a linear model, Tb )

906


Table 4. Prediction of Ts for n-Octanoic Acid (n-Butanoic and n-Decanoic Acids are Predictive Compounds) saturation temp/K

deviationa

no.

pressure/Pa

Riedel eq

SC-QS2PR

K

%

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

1.0 × 10 3.0 × 102 5.0 × 102 7.0 × 102 1.0 × 103 2.0 × 103 4.0 × 103 6.0 × 103 8.0 × 103 1.0 × 104 2.0 × 104 4.0 × 104 6.0 × 104 8.0 × 104 1.0 × 105 2.0 × 105 4.0 × 105 6.0 × 105 8.0 × 105 1.0 × 106 1.1 × 106 1.2 × 106 1.4 × 106 1.6 × 106 1.8 × 106 2.0 × 106 2.2 × 106

355.16 371.81 380.24 386.07 392.51 405.84 420.42 429.6 436.45 441.96 460.34 480.91 494.15 504.16 512.32 540.11 572.34 593.67 610.06 623.54 629.52 635.08 645.2 654.25 662.43 669.92 676.81

353.54 370.21 378.65 384.49 390.93 404.28 418.86 428.05 434.89 440.4 458.77 479.33 492.55 502.56 510.7 538.47 570.71 592.1 608.59 622.2 628.24 633.88 644.16 653.37 661.73 669.4 676.48

1.62 1.60 1.59 1.58 1.58 1.57 1.56 1.56 1.56 1.56 1.56 1.58 1.60 1.61 1.62 1.64 1.63 1.56 1.47 1.34 1.28 1.20 1.05 0.88 0.70 0.52 0.33

0.46 0.43 0.42 0.41 0.40 0.39 0.37 0.36 0.36 0.35 0.34 0.33 0.32 0.32 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.19 0.16 0.13 0.11 0.08 0.05

a

2

s s - TSC-QS2PR |. Deviation ) |TRiedel

104.56VEv1 + 187.7, with a correlation coefficient value of R2 ) 0.9989. The values of the descriptor VEv1 for the members of the n-alkanoic acid series are shown in Table 3. Let us assume, for example, that n-octanoic acid (no. 8 in Table 3, with a descriptor VEv1 value of 3.107) is the target compound. The closest compounds from opposite sides are the immediate neighbors: n-heptanoic acid (VEv1 ) 2.948) and n-nonanoic acid (VEv1 ) 3.259). In this case, the highest vapor pressure for obtaining Tts estimation is 2.5 MPa (the critical pressure of n-nonanoic acid). Selecting n-butanoic acid (VEv1 ) 2.41) and n-decanoic acid (VEv1 ) 3.404) as predictive compounds yields a model of a more limited pressure range (upper limit of 2.29 MPa). Selecting n-octanoic acid as the target, n-butanoic acid (predictive compound 1), and n-decanoic acid (predictive compound 2), the parameter values for the structure-structure correlation are obtained by introducing the corresponding VEv1 values into eq 5 (β1 ) 0.2988 and β2 ) 0.7012). Validation of the β values can be done by predicting Tb for the target compound. Introducing the β values, together with the Tb values of the predictive compounds (see Table A3) yields an estimate for the Tb of n-octanoic acid Tb/K ) (0.2988 × 436.32 + 0.7012 × 543.15 )) 511.23 K. The experimental value of Tb is 512.85 K (see Table A3), hence the deviation of the predicted value is 0.32%. Table 4 shows the predicted Tts of n-octanoic acid with n-butanoic and n-decanoic acids as predictive compounds, using the above TRC-QSPR parameter values and eq 13 for the predictive compounds. The results shown are for 27 different pressure values within a pressure range from 100 Pa to 2.2 MPa (slightly below the critical pressure of n-decanoic acid). The predicted Tts values were compared with those obtained using the Riedel equation for n-octanoic and the difference between the two is the reported deviation. The highest absolute deviation is 1.64 K at P ) 200 kPa and the highest relative deviation is 0.46% at P ) 100 Pa. A comparison between the predicted

Figure 5. Predicted and experimental vapor pressure data for n-octanoic acid: (-) predicted; (2) Ghiassee and Ambrose;52 (•) Ralston and Pool53 (data listed in Table A4).

values and experimental data reported by Ghiassee and Ambrose52 and Ralston and Pool53 is shown in Figure 5 indicating good agreement. The fit can be further improved by using the two immediate neighbors of n-octanoic acid (n-heptanoic acid and n-nonanoic acid) as predictive compounds. This yields a TRC-QSPR with β1 ) 0.4887 and β2 ) 0.5113. The highest absolute deviation is reduced to 1.1 K at P ) 2.4 MPa, and the highest relative deviation is 0.21% at P ) 100 Pa. Short-range extrapolation yields satisfactory results, as well. Choosing n-pentadecanoic acid (nC ) 15) as a target compound and n-octanoic acid (nC ) 8) and n-tridecanoic acid (nC ) 13) as predictive compounds yields an SC-QS2PR with β1 ) -0.35193 and β2 ) 1.3519. Results for Tts predicted for 23 pressure values (between P ) 100 Pa and P ) 1.4 MPa) indicate that the highest deviation of 2.69 K (0.35%) at P ) 1.4 MPa. However, a longer range extrapolation causes a considerable deterioration of the results. For example, keeping n-pentadecanoic acid as the target compound, but using n-pentanoic acid (nC ) 5) and n-decanoic acid (nC ) 10) as predictive compounds, yields the highest deviation of 5.08 K (1.1%) at P ) 700 Pa. Pertinent Data for the n-Alkane, 1-Alcanol, n-Alkanal, and Alkyl-Benzene Homologous Series. Similar vapor pressure prediction studies were carried out for the n-alkane, 1-alcanol, alkyl-benzene, and n-alkanal homologous series, using the Riedel equation for calculating Tis. The results of these studies are similar to the results obtained for the 1-alkene group. Some of the data required for these predictions are shown in Appendix A (Tables A5, A6, and A7, respectively given in the SI). Results for the use of the TRC-QSPR method for the n-alkanal series are reported elsewhere.54 For the n-alkane series, experimental Tb values43 are available up to n-eicosane (nC ) 20), while for compounds with higher nC values, DIPPR43 reports predicted values. The descriptor VEA1 (a 2D eigenvalue based index: eigenvector coefficient sum from the adjacency matrix) is recommended for n-alkane series up to nC ) 16. For this range of nC, Tb ) 1166.3VEA1 - 48.119, with a correlation coefficient value of R2 ) 0.9993. For nC > 17, the ATS2e descriptor (a 2D autocorrelations descriptor: Broto-Moreau autocorrelation of a topological structureslag 2/weighted by atomic Sanderson electronegativities) is recommended, with Tb ) 175.46ATS2e - 241.99 (R2 ) 1.0). The same descriptor can be used for nC < 17, as well, but with lower precision.


For the 1-alkanol homologous series, the descriptor identified as collinear with the Tb data43 is SPI (a 2D topological descriptor: superpendentic index). For this group, the relationship between Tb and the descriptor can be represented by the model, Tb ) 38.728SPI + 292.07 (R2 ) 0.9993). In the case of the alkyl benzene homologous series, experimental Tb values are available43 up to n-nonylbenzene (nC ) 15). For compounds with higher nC values, DIPPR43 reports predicted values. The descriptor collinear with Tb for the alkylbenzene group was identified as the CIC1 (a 2D information index: complementary information content, neighborhood symmetry of 1-order). For this group the relationship between Tb and the descriptor can be represented by the model, Tb ) 279.43CIC1 - 236.92, with a correlation coefficient value of R2 ) 0.9999. Prediction of PRs for the n-Alkanoic Acid Series with the “TRC-QSPR” Method. The TRC-QSPR method can be used for prediction of the logarithm of the reduced vapor pressure (ln(PRs)) of the target compound at a specified reduced temperature value, instead of predicting the saturation temperature at a particular pressure. Substituting ln(PRs) as the predicted property into eq 6 yields ln(PsR,t) )

ζj2 - ζjt

ζjt - ζj1

ζj2 - ζ1

ζj2 - ζj1

ln(PsR,1) + j

ln(PsR,2)

(14)

s s where PR,1 and PR,2 are the reduced saturation pressures (at a s reduced temperature TR) of the predictive compounds and PR,t is the (predicted) reduced saturation pressure of the target compound at TR. The descriptor ζj used in this case must be collinear with ln(PRs) at a particular TR value. A convenient choice is TR ) 0.7. At this reduced temperature, the acentric factor: ω ) -log(PRs)TR)0.7 - 1, is available for a large number of compounds, and it is collinear with ln(PRs). If ω is used as the descriptor in eq 14, then the TRC-QSPR method reduces to the traditional two reference fluid method, which is discussed in some detail, for example, by Poling et al.19 The two reference fluid method requires the property data ω, Tc, and Pc for the target compound. The critical pressure and temperature are s (TR) value (obtained from eq needed in order to convert the PR,t 14) to a Pts(T) value. In the herein proposed TRC-QSPR method, descriptors collinear with these properties are used in cases where the property value is not available. An additional important enhancement of the traditional method concerns the selection of the predictive compounds. Poling et al.19 mention the following options: (1) Designate the pairs propane and octane and benzene and pentafluorotoluene as reference fluids (predictive compounds), as was suggested by Ambrose and Patel.55 (2) Use any two substances chemically similar to the unknown fluid whose vapor-pressure behavior is well-established. In the proposed technique, the algorithm described in the “generalized algorithm for prediction of vapor (saturation) pressures” section is used. Obviously, for identifying a collinear descriptor for each of the required properties (i.e., ω, Tc, and Pc) these should be available for the compounds included in the corresponding selected similarity group (instead of the Tb values that are required in step 1 of the method when Ts is predicted). To demonstrate the use of the “two-reference fluid” method and the TRC-QSPR method, the reduced vapor pressure of n-octanoic acid is predicted with two sets of predictive compounds. In the first set, following the recommendation of Ambrose and Patel,55 propane (with acentric factor ω1 ) 0.152, DIPPR43) and octane (ω2 ) 0.3978) are used as predictive

907

s Figure 6. Prediction of ln(PR,t ) for n-octanoic acid, in the temperature range, 0.4 e TR e 1, using the two reference fluid method: (- - -) propane; (••••) octane; (- • -) target compound (Riedel equation); (s) target compound (predicted).

compounds, with n-octanoic acid (ωt ) 0.773) as the target compound. Additional pertinent data for the predictive compounds, propane and n-octane, are shown in Table A3 (see the SI). The Riedel equation was used for calculating the vapor pressure of the predictive compounds. The common range of applicability of the equations for the predictive compounds and the target compound in terms of reduced temperature is 0.4 e s ) for the target and the predictive TR e 1. The plot of ln(PR,t -1 compounds versus TR , within the range of validity of the Riedel equations, is shown in Figure 6. The target compound s ) was predicted using eq 14, and compared with the values ln(PR,t obtained by the Riedel equation (parameters provided in Table A3). In this case, the prediction represents extrapolation, as the s s s ) is well below the curves of ln(PR,1 ) and ln(PR,2 ). curve of ln(PR,t The difference between the calculated and predicted values of s ) (denoted ε) is clearly distinguishable and increases with ln(PR,t -1 s ) is shown as “case 1” in TR . The variation of ε versus ln(PR,t Figure 7. The smallest deviation is ε ) -0.00088 for TR ) 0.95, and the largest one is ε ) -0.58 for TR ) 0.4. To improve the precision of the prediction we next applied the two-reference fluid method for the same target compound (n-octanoic acid), but using predictive compounds, which are similar to the target, namely n-butanoic acid (ω1 ) 0.675, from DIPPR43) and n-decanoic acid (ω2 ) 0.813). The prediction error for this case is shown as case 2 in Figure 7. The maximal (absolute) error is also at TR ) 0.4; however, it is reduced to ε ) -0.14. To apply the TRC-QSPR method to the same system, we identified a descriptor which is collinear with the acentric factor for the n-alkanoic acid homologous series (MWC03, a 2D walk and path count descriptor: molecular walk count of order 3). s ) while using in eq 14 the descriptor values Predicting ln(PR,t instead of the acentric factors yields prediction error as shown in case 3 of Figure 7. The maximal (absolute) deviation is further reduced to ε ) -0.048 (at TR ) 0.87). Thus, in this case, the TRC-QSPR method yields more than an order of magnitude reduction of the prediction error in comparison to the original two-reference fluid method. In order to compare the methods that predict Ts with the methods that predict ln(PRs), the error propagation formula is used to relate the uncertainty values of δTs and δ[ln(PRs)]. If

908


aij ) (aiiajj)0.5(1 - kij)

s Figure 7. Prediction uncertainty versus ln(PR,t ) for n-octanoic acid using the two-reference fluid and the TRC-QSPR methods: (••••) case 1, propane and octane predictive compounds, ω used in eq 14; (s) case 2, n-butanoic and n-decanoic acids predictive compounds, ω used in eq 14; (- • • -) case 3, n-butanoic and n-decanoic acids predictive compounds, descriptor MWC03 is used in eq 14.

the Riedel equation is used for calculating the vapor pressure, the following error conversion formula applies: δ[ln(PRs)]

(

)

d[ln(PRs)] Bi Ci δT ) - 2 + + EiDiT Ei-1 δT ) dT T T (15)

We have compared the prediction uncertainties of Ts versus the prediction of ln(PRs) using the above formula and found that the uncertainty levels obtained in the two cases are similar. Prediction of Binary Interaction Parameters for Equation of State Models If the equilibrium liquid and vapor phases are both modeled by an EoS (phi-phi approach), then the K-value is expressed as follows: Ki ) φiL /φiV

(16)

To calculate the fugacity coefficients of the components in the liquid (φiL) and vapor (φiV) phases, usually either the Soave-Redlich-Kwong (SRK) or Peng-Robinson (PR) cubic EoS with mixing rules (e.g., one-fluid van der Waals mixing rule) are employed. In the EoS, the properties of the pure compounds required are the critical temperature (Tc) and pressure (Pc) and the acentric factor (ω). The prediction of these properties by applying the QS2PR, the shortcut QS2PR or the TQSPR techniques is straightforward. However, the mixing rules require binary interaction parameters. These can be predicted with TQSPR with the modifications illustrated below. The application, for example, of either the SRK or PR cubic EoS to mixtures requires the use of mixing rules for the mixture energy parameter amix (which accounts for interactions between the species in the mixture) and for the covolume parameter bmix (which accounts for the excluded volume of the species of the mixture): amix )

∑ ∑xxa ; i j ij

i

j

bmix )

∑xb

i ii

(17)

i

The cross coefficient aij is related to the corresponding purecomponent parameters by the following combining rule:

(17a)

where kij ) 0, when i ) j. The usual procedure is to obtain kij by fitting to available experimental vapor-liquid equilibrium data. Predicted values can be obtained by the PPR78 model,31-39 which requires data for Tc, Pc, and acentric factor, if available. When TQSPR is employed to predict directly from chemical structure kij for a system consisting of a given compound (say compound i) and members of a homologous series (say compounds j), the corresponding kij for at least several members of the series with compound i have to be available. We have tested the applicability of the TQSPR method16 to homologous series following the algorithm described in detail elsewhere.18 For the present task, this algorithm requires the available members of homologous series to be consecutively chosen as targets until a descriptor collinear with the dependence between kij and chemical structure is identified from single descriptor linear relationships, developed for each binary system consisting of the given compound and a member of the homologous series. In most cases,18 the descriptor sought is found among the descriptors suggested by the SROV program as having the highest correlation coefficient of partial correlation with the property, |Fyj|. In a limited number of cases, the collinear descriptor might not be suggested by the program. Some of the reasons for this inefficiency are discussed elsewhere.42 In such cases, the descriptor is sought among those selected for the different binary systems. After the collinear descriptor is identified, its linear correlation with the interaction coefficients of the systems might be used for prediction of unavailable values of the binary coefficients for systems of interest, consisting of the given compound and a member of the homologous series. Hereunder, this method is demonstrated by calculating kij for the SRK EoS of “propane-n-alkane” systems, and for the PR EoS of “n-dodecane-n-alkane” systems. For the SRK EoS, following the described procedure, the descriptor Xu was identified as the descriptor collinear with the binary interaction parameters of the studied systems. The Xu index is described as a topological descriptor calculated from the adjacency and distance matrices accounting for molecular size and branching.46 It was second among the 10 descriptors suggested by the SROV program as having the highest correlation coefficient of partial correlation with the property, |Fyj| for the target n-nonane. The following relationship kij ) -0.0087542 + 0.0023293Xu

(18)

where the coefficients of the equation are calculated from the kij values of nine compounds (training set), was identified as the most structurally related to the target n-nonane. Equation 18 was used to predict the binary interaction coefficients, presented in Table 5. The average absolute deviation, AAD, for all studied systems was calculated according to AAD )

N

|kUNISIM - kcalculated | ij ij

j)1

kUNISIM ij

( )∑ 1 N

(19)

where kUNISIM are the binary interaction parameters, taken from ij are the values of the Honeywell UNISIM48 databank; kcalculated ij the binary interaction parameters, calculated with eq 18; and N is the number of available data points. Table 5 displays a relatively large deviation from the Honeywell UNISIM48 data for the first four members of the homologous series. However, without these compounds, the deviations are the following: AAD ) 0.0007176 and AAD/% ) 1.32, which is


909

Table 5. Prediction of the Binary Interaction Parameters of the Propane-n-Alkane Systems for the SRK CEoS compounda

kij48

kij, predicted

absolute deviation

absolute deviation/%

descriptor Xu

C2 n-C4 n-C5 n-C6 n-C7 n-C8 n-C9 n-C10 n-C11 n-C12 n-C13 n-C14 n-C15 n-C16 n-C17 n-C18 n-C19 n-C20 n-C21 n-C22 n-C23 n-C24 n-C25 n-C26 n-C27 n-C28 n-C29 n-C30 n-C32 n-C35 n-C36 n-C40 n-C44 n-C60 AAD AAD/%

0.0018529 0.0015385 0.0030508 0.0031659 0.0082870 0.0112210 0.0140420 0.0169510 0.0197760 0.0223090 0.0254440 0.0277260 0.0299600 0.0325760 0.0353680 0.0380110 0.0404140 0.0427550 0.0448390 0.0469330 0.0489270 0.0508810 0.0527420 0.0545580 0.0563050 0.0579860 0.0596120 0.0611960 0.0653387

-0.0087542 -0.0013912 0.0019443 0.0051495 0.0082428 0.0112410 0.0141550 0.0169920 0.0197610 0.0224680 0.0251190 0.0277160 0.0302670 0.0327730 0.0352350 0.0376600 0.0400470 0.0424000 0.0447200 0.0470100 0.0492690 0.0515010 0.0537070 0.0558850 0.0580410 0.0601730 0.0622830 0.0643730 0.0684890 0.0745190 0.0764940 0.0842370 0.0917420 0.1198600

0.0106070 0.0029297 0.0011065 0.0019836 0.0000442 0.0002040 0.0011300 0.0004100 0.0000150 0.0001590 0.0003250 0.0000100 0.0003070 0.0001970 0.0001330 0.0003510 0.0003670 0.0003500 0.0001190 0.0000770 0.0003420 0.0006200 0.0009650 0.0013270 0.0017360 0.0021870 0.0026710 0.0031770

572.47 190.43 36.27 62.66 0.53 0.18 0.80 0.24 0.07 0.71 1.28 0.04 1.02 0.60 0.37 0.92 0.91 0.83 0.27 0.16 0.70 1.22 1.83 2.43 3.08 3.77 4.48 5.19

0.000 3.161 4.593 5.969 7.297 8.584 9.835 11.053 12.242 13.404 14.542 15.657 16.752 17.828 18.885 19.926 20.951 21.961 22.957 23.940 24.910 25.868 26.815 27.750 28.676 29.591 30.497 31.394 33.161 35.750 36.598 39.922 43.144 55.215

a

0.0012089 31.91

Short notation, i.e. C2 is ethane.

Figure 8. Prediction of binary interaction coefficients, kij, of propane and the homologous series of n-alkanes for the Soave-Redlich-Kwong CEoS: (- • -) data from the UNISIM48 databank; (••O••) data predicted with eq 18.

compatible with the deviations obtained by regression of experimental data.56 Figure 8 might partially explain the larger relative deviations for the first four members of the homologous series. As seen, the data from the Honeywell UNISIM48 data bank, especially for ethane and butane, as expected, do not follow the general trend within the homologous series, so the linear model cannot reflect their behavior. Moreover, the values of Xu are zeroes for both methane and ethane. This problemszero values and/

or lack of definition of some descriptors for low molecular compoundsshas been discussed in detail elsewhere.42 The nearly linear dependence of the binary coefficients upon the number of carbon atoms of n-alkanes is typical for systems in which the constant component of the systems (i.e., propane in the above discussion) is in the beginning or the end of the homologous series. For such systems, single collinear descriptors can be identified, and relationships, like eq 18, can be used for estimation of missing data. Figure 8 suggests eventual systematic increase of the deviations for extrapolations at high number of C atoms. In the UNISIM48 databank for the SRK CEoS, there are no data for the system propane-n-C32. However, the comparison with the value for the same system for the Peng-Robinson CEoS (0.0653387) indicates a lower relative deviation for propane-n-C32 (4.82%) than for the propane-nC30 system in Table 5. Thus, it might be expected that, for the longer chain n-alkanes used in most industrial processes (even in petroleum processing), the extrapolated values would remain within those of regressed experimental data for the binary coefficients. When the constant component is around the middle of the series, the typical dependences are in a form similar to the one shown in Figure 9 for the PR CEoS binary coefficients of the “n-dodecane-n-alkanes” systems. As seen from Figure 9, the studied dependence is not asymptotic, as those for most propertiessnumber of carbon atoms relationships. Still, the TQSPR method can be useful, but in such cases, two descriptors have to be used. The data predicted, shown in the figure, have been obtained with two

910


nonzero row elements in a reduced distance matrix, where the rows correspond to all non-hydrogen atoms and the columns to only the terminal atoms. In order to avoid too large numbers, in the Dragon program, the product sums are substituted by logarithm sums. VEA2 is an average eigenvector coefficient sum, calculated from the adjacency matrix. Since the relationship in Figure 9 is complex, the kij values for 17 compounds (training set), neighbors of the target n-decane, suggested by the program, have been used to determine the coefficients of eq 20. The descriptors have been identified by the SROV program rotation procedure as the couple best describing the available data. Table 6 presents in more detail the results from the modeling of the studied systems.

Figure 9. Prediction of binary interaction coefficients, kij, of n-dodecane and the homologous series of n-alkanes for the Peng-Robinson CEoS: (- • -) data from the UNISIM48 databank; (••O••) data predicted with eq 20.

descriptors, identified by the TQSPR program, by the relationship, shown below: ) -0.00691980 + 0.0045504SPI + 0.15746VEA2 kPREoS ij (20) 46

where the superpendentic index (SPI) is a topological index, calculated as the square root of the sum of the products of the

The value of SPI is zero for ethane (and for methane), which, as above, might explain partially the relatively large deviations for this compound. As a whole, eq 19 describes the studied dependence with deviations within those of data regressed from experimental data56 and can be used for estimation of missing data for binary interaction coefficients. The results obtained indicate that for n-alkanes with carbon atoms numbers approximately more than 18, it is also possible to develop by the TQSPR method a singe descriptor linear relationship, which would provide even more reliable extrapolated data. Finally, we would like to point out that all three chosen descriptors are of the 2D type which supports the findings of a previous work42 that with the present state of chemical structure

Table 6. Prediction of the Binary Interaction Parameters of the n-Dodecane-n-Alkane Systems for the PR CEoS descriptors

a

compounda

kij48

kij predicted

absolute deviation

absolute deviation/%

SPI

VEA2

C2 n-C3 n-C4 n-C5 n-C6 n-C7 n-C8 n-C9 n-C10 n-C11 n-C13 n-C14 n-C15 n-C16 n-C17 n-C18 n-C19 n-C20 n-C21 n-C22 n-C23 n-C24 n-C25 n-C26 n-C27 n-C28 n-C29 n-C30 n-C32 n-C35 n-C36 n-C40 n-C44 n-C60 AAD AAD/%

0.0336810 0.0223094 0.0147631 0.0097199 0.0062583 0.0038676 0.0022121 0.0011884 0.0005406 0.0002107 0.0002116 0.0004092 0.0006928 0.0011260 0.0016955 0.0023261 0.0029693 0.0036543 0.0043093 0.0050076 0.0057073 0.0064248 0.0071353 0.0078537 0.0085662 0.0092715 0.0099710 0.0106688 0.0125636

0.0421260 0.0257520 0.0160990 0.0101390 0.0062904 0.0038345 0.0021823 0.0011991 0.0005928 0.0002149 0.0001990 0.0004128 0.0007339 0.0011670 0.0017120 0.0023781 0.0028503 0.0036010 0.0043245 0.0050161 0.0056803 0.0064748 0.0072510 0.0081574 0.0088836 0.0097490 0.0105920 0.0114210 0.0131770 0.0156170 0.0165080 0.0199310 0.0233200 0.0366430

0.0084450 0.0034426 0.0013359 0.0004191 0.0000321 0.0000331 0.0000298 0.0000107 0.0000522 0.0000042 0.0000126 0.0000036 0.0000411 0.0000410 0.0000165 0.0000520 0.0001190 0.0000533 0.0000152 0.0000085 0.0000270 0.0000505 0.0001157 0.0003037 0.0003174 0.0004775 0.0006210 0.0007522 0.0006134

25.07 15.43 9.05 4.31 0.51 0.86 1.35 0.90 9.65 1.98 5.94 0.87 5.93 3.64 0.97 2.23 4.01 1.46 0.35 0.17 0.47 0.78 1.62 3.87 3.71 5.15 6.23 7.05 4.88

0 1.177 1.893 2.521 3.094 3.627 4.129 4.605 5.06 5.496 6.323 6.716 7.098 7.47 7.832 8.186 8.532 8.87 9.202 9.527 9.846 10.159 10.468 10.771 11.069 11.363 11.652 11.938 12.497 13.31 13.575 14.604 15.591 19.211

0.707 0.569 0.487 0.431 0.39 0.359 0.334 0.314 0.297 0.282 0.258 0.248 0.239 0.231 0.224 0.218 0.211 0.206 0.201 0.196 0.191 0.187 0.183 0.18 0.176 0.173 0.17 0.167 0.162 0.154 0.152 0.144 0.137 0.117

Short notation, i.e. C2 is ethane.

0.00060158 4.43


description, for homologous series 3D descriptors eventually introduce more error than useful structural information. Conclusions and Future Work We have demonstrated above the potential of our new methods to predict temperature-dependent properties and interaction parameters for phase equilibria calculations, when applying EoSs. Our preliminary results show that our methods can be used also for predicting other important parameters (e. g., activity coefficients for nonideal binary systems). The new QSPR-based methods for predicting temperaturedependent variation of vapor pressure of chemical compounds are based on the identification of potential predictive compounds, which are structurally similar to the target compound (a similarity group) and for which data for a vapor pressure related property (e.g., normal boiling temperature or acentric factor) are available. A molecular descriptor, which is collinear with this property for the members of the similarity group, is used to develop simple structure-structure relations (TRC-QSPR). These relations are then applied for predicting of reduced vapor pressure or saturation temperatures of the target compound in the reduced temperature or pressure range, where valid vapor pressure data exist for two selected predictive compounds. The methods can be considered as refinement of the wellestablished two reference fluids method, which has been in use for over 30 years now. The proposed methods have several advantages over the traditional predictive methods, namely: • Only structural information (no measured property values) are needed for the target compound. • Predictive compounds similar to the target are selected in a systematic manner. • The temperature-vapor pressure relationships of the predictive compounds are used only in their valid range of applicability. • The two methods provide flexibility with regard to the use of normal boiling point or acentric factor, depending on the accuracy of data available. • It is possible to predict either saturation temperature or vapor pressure giving more flexibility regarding the range and uncertainty of the predictions. The new methods were applied to the n-alkane, 1-alkene, n-alkanoic acid, n-alcohol, alkyl-benzene, and n-alkanal homologous series. It was demonstrated that the methods enable the prediction of vapor pressure within experimental uncertainty, depending on the level of similarity between the predictive compounds and the target compound. In cases of longer range interpolation or extrapolation, some degradation of the prediction is noticed; however, even in those cases, the proposed method is superior to the traditional prediction techniques. A targeted QSPR method for prediction of the binary interaction coefficients (kij) in cubic equations of state for a compound with the members of its homologous series is also presented. The coefficients for the Soave-Redlich-Kwong and Peng-Robinson equations, used to test the method, were reproduced within the deviation of those obtained from regressed experimental data.55 In this work, we have studied the application of our methods, choosing as examples particular homologous series. Given the fact that chemical structures in most homologous series differ only by the number of carbon atoms, we believe that our methods are applicable for systems with any homologous series, similar to the one studied in this work. Estimation of data for homologous series is important by itself both for theory and industrial application.18 The developed method for predicting vapor pressure and relationships for the

911

binary interaction coefficients (kij) in cubic equations of state provide reliable data for such series, even when extrapolation within practical ranges is required. In our future work, we intend to study the applicability of the SC-QS2PR and the TQSPR based methods for prediction of phase-equilibrium related properties in systems which contain components that are not members of a homologous series. Supporting Information Available: (1) Vapor pressure related data from the NIST44 database for the 1-alkene homologues series, (2) experimental vapor pressure data for 1-undecene, (3) vapor pressure related data from the DIPPR43 database for the n-alkanoic acid homologues series and reference fluids, (4) experimental vapor pressure data for n-octanoic acid, (5) normal boiling temperatures44 and collinear descriptors for the n-alkane homologous series, (6) normal boiling temperatures44 and collinear descriptor for the 1-alkanol homologous series, and (7) normal boiling temperatures44 and collinear descriptor for the alkyl-benzene homologous series. This material is available free of charge via the Internet at http:// pubs.acs.org. Literature Cited (1) Bru¨ggemann, S.; Marquardt, W. Rapid Screening of Design Alternatives for Nonideal Multiproduct Distillation Processes. Comput. Chem. Eng. 2005, 29, 165–179. (2) Bru¨ggemann, S.; Marquardt, W. Shortcut Methods for Non-ideal Multi-component Distillation: 3. Extractive Distillation Columns. AIChE J. 2004, 50, 1129-1149. (3) Lee, J. W.; Bru¨ggemann, S.; Marquardt, W. : Shortcut Method for Kinetically Controlled Reactive Distillation Systems. AIChE J. 2003, 49, 1471–1487. (4) Horwath, A. L. Molecular Design; Elsevier: Amsterdam, 1992. (5) Cholakov, G. Some Approaches to Computer Aided Design of Lubricants. Oxid. Commun. 1994, 4, 303. (6) Stanulov, K.; Harhara, N.; Cholakov, G. An Opportunity for Partial Replacement of Phosphates and Dithiophosphates in EP Packages with Boron Containing Additives. Trib. Int. 1998, 31, 257–263. (7) Cholakov, G.; Stanulov, K.; Devenski, P.; Iontchev, H. Quantitative Estimation and Prediction of Tribological Performance of Pure Additive Compounds through Computer Modeling. Wear 1998, 216, 194–201. (8) Cholakov, G.; Rowe, J. Lubricating Properties of Grinding Fluids: II. Comparison of Fluids in Four Ball Tribometer Tests. Wear 1992, 155, 331. (9) Wakeham, W. A.; Cholakov, G., St.; Stateva, R. P. Consequences of Property Errors on the Design of Distillation Columns. Fluid Phase Equilib. 2001, 185, 1–12. (10) Brauner, N.; Shacham, M.; Cholakov, G., St.; Stateva, R. P. Property Prediction by Similarity of Molecular Structures - Practical Application and Consistency Analysis. Chem. Eng. Sci. 2005, 60, 5458–5471. (11) Cholakov, G., St.; Stateva, R. P.; Shacham, M.; Brauner, N. Consistency Analysis of Pure Component Property Data Based on StructureStructure Correlations. 7th World Congress of Chemical Engineering, Glasgow, Scotland, July 10-14, 2005. (12) Cholakov, G., St.; Stateva, R. P.; Shacham, M.; Brauner, N. Prediction of Properties in Homologous Series with a Short-Cut QS2PR Method. AIChE J. 2007, 53, 150–159. (13) Shacham, M.; Brauner, N.; Cholakov, G., St.; Stateva R. P. Combining Stepwise Regression with Outlier Detection for Identification of Collinear Groups. 7th World Congress of Chemical Engineering, Glasgow, Scotland, July 10-14, 2005. (14) Shacham, M.; Brauner, N.; Cholakov, G., St.; Stateva, R. P. Property Prediction by Correlations Based on Similarity of Molecular Structures. AIChE J. 2004, 50, 2481–2492. (15) Shacham, M.; Brauner, N.; Cholakov, G., St.; Stateva, R. P. A Unified Correlation For Prediction of Pure Component Properties Based on Similarity of Molecular Descriptors of Various Compounds. Proceedings of the 6th International Conference on Foundations of Computer Aided Process Design, Princeton, NJ, July 11-16, 2004; Floudas, C. A., Agrawal, R., Eds.; p 391-394. (16) Brauner, N.; Stateva, R. P.; Cholakov, G., St.; Shacham, M. A. Structurally “Targeted” QSPR Method for Property Prediction. Ind. Eng. Chem. Res. 2006, 45, 8430–8437.

912


(17) Brauner, N.; Cholakov, G., St.; Kahrs, O.; Stateva, R. P.; Shacham, M. Linear QSPRs for Predicting Pure Compound Properties in Homologous Series. AIChE J. 2008, 54, 978–990. (18) Cholakov, G., St.; Stateva, R. P.; Brauner, N.; Shacham, M. Estimation of Properties of Homologous Series with Targeted Quantitative Structure-Property Relationships. J. Chem. Eng. Data 2008, 53, 2510–2520. (19) Poling, B. E.; Prausnitz, J. M.; O’Connel, J. P. Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (20) Prausnitz, J. M.; Tavares, F. W. Thermodynamics of Fluid-Phase Equilibria for Standard Chemical Engineering Operations. AIChE J. 2004, 50, 739–761. (21) Seader, J. D.; Henley, E. J. Separation Process Principles; John Wiley & Sons: New York, 1998. (22) Dearden, J. C. Quantitative Structure-Property Relationships for Prediction of Boiling Point, Vapor Pressure, and Melting Point. EnViron. Toxicol. Chem. 2003, 22, 1696–1709. (23) Goll, E. S.; Jurs, P. C. Prediction of Vapor Pressures of Hydrocarbons and Halohydrocarbons from Molecular Structure with a Computational Neural Network Model. J. Chem. Inf. Comput. Sci. 1999, 39, 1081– 1089. (24) Rajer-Kanduc, K.; Zupan, J.; Majcen, N. Separation of Data on the Training and Test Set for Modelling: a Case Study for Modelling of Five Colour Properties of a White Pigment. Chemom. Intell. Lab. Syst. 2003, 65, 221–229. (25) Li, Q.; Chen, X.; Zhide Hu, Z. Quantitative Structure-Property Relationship Studies for Estimating Boiling Points of Alcohols Using Calculated Molecular Descriptors with Radial Basis Function Neural Networks. Chemom. Intell. Lab. Syst. 2004, 72, 93–100. (26) Chen, C. C.; Mathias, P. M. Applied Thermodynamics for Process Modeling. AIChE J. 2002, 48, 194–200. (27) Orbey, H.; Sandler, S. I. Modeling Vapor-Liquid Equilibria. Cubic Equation of State and Their Mixing Rules; Cambridge University Press: New York, 1998. (28) Wong, D. S. H.; Sandler, S. I. A Theoretically Correct Mixing Rule for Cubic Equations of State. AIChE J. 1992, 38, 671. (29) Cholakov, G., St; Wakeham, W. A.; Stateva, R. P. Estimation of Normal Boiling Temperature of Industrially Important Hydrocarbons from Descriptors of Molecular Structure. Fluid. Phase Equil. 1999, 163, 21–42. (30) Wakeham, W. A.; Cholakov, G., St.; Stateva, R. P. Liquid Density and Critical Properties of Hydrocarbons Estimated from Molecular Structure. J. Chem. Eng. Data 2002, 47, 559–570. (31) Jaubert, J.-N.; Mutelet, F. VLE Predictions with the Peng-Robinson Equation of State and Temperature Dependent kij Calculated Through a Group Contribution Method. Fluid Phase Equilib. 2004, 224, 285–304. (32) Jaubert, J.-N.; Vitu, St.; Mutelet, F.; Corriou, J.-P. Extension of the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method) to systems containing aromatic compounds. Fluid Phase Equilib. 2005, 237, 193–211. (33) Mutelet, F.; Vitu, St.; Privat, R.; Jaubert, J.-N. Solubility of CO2 in branched alkanes in order to extend the PPR78 model (predictive 1978, Peng-Robinson EOS) with temperature-dependent kij calculated through a group contribution method) to such systems. Fluid Phase Equilib. 2005, 238, 157–168. (34) Vitu, St; Jaubert, J.-N.; Mutelet, F. Extension of the PPR78 model (Predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method) to systems containing naphthenic compounds. Fluid Phase Equilib. 2006, 243, 9–28. (35) Vitu, St; Privat, R.; Jaubert, J.-N.; Mutelet, F. Predicting the phase equilibria of CO2 + hydrocarbon systems with the PPR78 model (PR EOS and kij calculated through a group contribution method). J. Supercrit. Fluids 2008, 45, 1–26. (36) Privat, R.; Jaubert, J.-N.; Mutelet, F. Addition of the Nitrogen Group to the PPR78 Model (Predictive 1978, Peng Robinson EOS with Temperature-Dependent kij Calculated through a Group Contribution Method). Ind. Eng. Chem. Res. 2008, 47, 2033–2048.

(37) Privat, R.; Jaubert, J.-N.; Mutelet, F. Addition of the sulfhydryl group (-SH) to the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method). J. Chem. Thermodyn. 2008, 40, 1331–1341. (38) Privat, R.; Jaubert, J.-N.; Mutelet, F. Use of the PPR78 Model To Predict New Equilibrium Data of Binary Systems Involving Hydrocarbons and Nitrogen. Comparison with Other GCEOS. Ind. Eng. Chem. Res. 2008, 47, 7483–7489. (39) Privat, R.; Mutelet, F.; Jaubert, J.-N. Addition of the Hydrogen Sulfide Group to the PPR78 Model (Predictive 1978, Peng-Robinson Equation of State with Temperature Dependent kij Calculated through a Group Contribution Method). Ind. Eng. Chem. Res. 2008, 47, 10041–10052. (40) Shacham, M.; Kahrs, O.; Cholakov, G., St.; Stateva, R. P.; Marquardt, W.; Brauner, N. The Role of the Dominant Descriptor in Targeted Quantitative Structure Property Relationships. Chem. Eng. Sci. 2007, 62, 6222–6233. (41) Kahrs, O.; Brauner, N.; Cholakov, G., St.; Stateva, R. P.; Marquardt, W.; Shacham, M. Analysis and Refinement of the Targeted QSPR Method. Comp. Chem. Eng. 2008, 32, 1397–1410. (42) Paster, I.; Shacham, M.; Brauner, N. Investigation of the Relationships between Molecular Structure, Molecular Descriptors, and Physical Properties. Ind. Eng. Chem. Res. 2009, 48, 9723–9734. (43) Rowley, R. L.; Wilding, W. V.; Oscarson, J. L.; Yang, Y.; Zundel, N. A. DIPPR Data Compilation of Pure Chemical Properties Design Institute for Physical Properties; Brigham Young University: Provo, Utah, 2006; http://dippr.byu.edu (accessed Jan 2009). (44) ChemistryWebBook, NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology (NIST): Gaithersburg, MD, June 2005; http://webbook.nist. gov (accessed Jan 2009). (45) Dragon, version 5.4; http://www.talete.mi.it (accessed Jan 2009). (46) Todeschini, R.; Consonni, V.; Mauri, A.; Pavan, M. DRAGON user manual; Talete SRL: Milano, Italy, 2006. (47) HyperChem program, version 7.01; http://www.hyper.com/ (accessed Jan 2009). (48) Honeywell UniSim. http://hpsweb.honeywell.com/Cultures/en-US/ default.htm (accessed Jan 2009). (49) MATLAB, version 7.04. (R14); The MathWorks Inc.: Natick, MA, 2005. (50) Shacham, M.; Brauner, N. The SROV Program for Data Analysis and Regression Model Identification. Comp. Chem. Eng. 2003, 27, 701– 714. (51) Shacham, M.; Brauner, N.; Shore, H.; Benson-Karhi, D. Predicting Temperature-Dependent Properties by Correlations Based on Similarity of Molecular Structures - Application to Liquid Density. Ind. Eng. Chem. Res. 2008, 47, 4496–4504. (52) Ghiassee, N. B.; Ambrose, D. Vapor Pressures and Critical Temperatures and Critical Pressures of Some Alkanoic Acids: C1 to C10. J. Chem. Thermodyn. 1987, 19, 505. (53) Ralston, A. W.; Pool, W. O. Boiling Points of n-Alkyl Acids. Ind. Eng. Chem. 1942, 34, 1104. (54) Brauner, N.; Shacham, M. Prediction of Saturated Vapor Pressure with Simple Structure Property- Property Relationships, AIChE’s 2009 National Annual Meeting, Nashville, TN, Nov 8-13, 2009; paper 180ab. (55) Ambrose, D.; Patel, N. C. The Correlation and Estimation of VaporPressures 4. Extrapolation of Vapor-Pressures and Estimation of Critical Pressures by the Principle of Corresponding States Using 2 Reference Fluids with Non-Spherical Molecules. J. Chem. Thermodyn. 1984, 16, 459–468. (56) Rodrigues, W. L.; Mattedi, S.; Abreu, J. C. N. Experimental VaporLiquid Equilibria Data for Binary Mixtures of Xylene Isomers. Braz. J. Chem. Eng. 2005, 22, 453–462.

ReceiVed for reView May 18, 2009 ReVised manuscript receiVed October 4, 2009 Accepted November 16, 2009 IE900807J