Estimation of the Enthalpy of Vaporization and the Entropy of

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Ind. Eng. Chem. Res. 2005, 44, 8436-8454

Estimation of the Enthalpy of Vaporization and the Entropy of Vaporization for Pure Organic Compounds at 298.15 K and at Normal Boiling Temperature by a Group Contribution Method Zden ˇ ka Kolska´ Department of Chemistry, Institute of Science, J. E. Purkynje University, C ˇ eske´ Mla´ dezˇ e 8, 400 96 U Ä stı´ nad Labem, Czech Republic

Vlastimil Ru ˚ zˇ icˇ ka* Department of Physical Chemistry, Institute of Chemical Technology Prague, Technicka´ 5, 166 28 Prague 6, Czech Republic

Rafiqul Gani CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark

A new group contribution method for estimating the enthalpy of vaporization at 298.15 K (∆HV(298.15 K)) and at the normal boiling temperature (∆HV(Tb)), as well as the entropy of vaporization at the normal boiling temperature (∆SV(Tb)), of pure organic compounds has been developed. Large databases of critically assessed data have been used for group contribution calculations: data for 831 compounds have been used for estimations at 298.15 K, and data for 589 compounds have been used for estimations at the normal boiling temperature. Values obtained by the method developed here have been compared with estimations by the Ducros, Chickos, and Ma and Zhao group contribution methods and by empirical equations by Vetere. A statistical analysis of the regressed data has been also performed, indicating the confidence of the regressed parameters and other related information. The average relative errors (ARE) for the new method are as follows: for ∆HV(298.15 K), 2.2%; for ∆HV(Tb), 2.6%; and for ∆SV(Tb), 1.8%. The error for the enthalpy of vaporization, based on an independent set of various 74 compounds not used for correlation, has been determined to be 2.5%. The new method can also be used for qualitative estimation of the normal boiling temperature. Introduction The enthalpy of vaporization (∆HV, given in units of kJ/mol) is the energy that must be supplied to a unit molar amount of a pure compound to transfer it from the liquid state to the gaseous state at a constant temperature and at the corresponding vapor pressure. Its usage is very widespread. For example, the enthalpy of vaporization is one of the most important parameters for the design and operation of multicomponent and multistage vapor-liquid equilibrium (VLE)-based processes (such as distillation, evaporation, two-phase reactors, etc.), where these values must be known for at least one temperature for the calculations of liquid or vapor enthalpies, using isobaric heat capacity values. The enthalpy of vaporization has also been correlated with the molar volumes to estimate the Hildebrand’s solubility parameters1 and the Hansen’s solubility parameters,2-5 where it serves as an input parameter for the correlation. The enthalpy of vaporization can be used for a controlled extrapolation of vapor pressures, in particular down to the triple point.6,7 In thermochemistry, the standard enthalpy of vaporization is used for conversion of the enthalpy of formation between the liquid state and the ideal gas state.8 These data are * To whom correspondence should be addressed. Tel.: +420220 443 824. Fax: +420 220 445 018. E-mail: [email protected].

usually related to a temperature of 298.15 K. All the points previously discussed indicate the need for determining the enthalpy of vaporization at 298.15 K and at the normal boiling temperature. Entropy characterizes the extent of disorder and shows the direction of a spontaneous process. The entropy of vaporization (∆SV, given in units of J K-1 mol-1) defines these changes in processes involving evaporation and can be calculated from known (experimental) values of the normal boiling temperature (Tb) and the enthalpy of vaporization at Tb. Methods for determining the enthalpy of vaporization can be divided into two main classes:8-10 experimentbased and model-based. Experiment-based methods, such as calorimetry or gas chromatography, provide generally reliable data of good accuracy. However, experimental determinations require expensive equipment, adherence to a good laboratory practice, and, in gas chromatography, also the correct choice of a reference compound. In the case of model-based methods, we can distinguish several groups of methods on the basis of the input information they require.8-10 [Note: In the literature survey, we present only methods published in the last two decades.] Methods based on the Clausius-Clapeyron equation and vapor pressure data are more correlation methods, rather than prediction methods.11-14 Empirical correlations typically require values of the boiling temperature, the molecular mass, the number of C atoms, and sometimes other

10.1021/ie050113x CCC: $30.25 © 2005 American Chemical Society Published on Web 09/27/2005

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8437

molecular characteristics as input information.15-22 Methods based on the tools of statistical thermodynamics23-28 or quantum mechanics29-31 involve complex mathematical calculations and, therefore, are computationally intensive and expensive, especially for large complex multifunctional chemicals. This makes the application range of these methods limited, even though they have a sound theoretical basis. Group contribution methods,32-47 on the other hand, seem to be best-suited for many process-engineering-design/simulation/analysisrelated calculations, because they require only tabulated group contribution values and are computationally simple and inexpensive to apply. They are based on the assumption that a property value of any group has the same contribution in all the compounds in which it appears and that the property value of the compound is only a function of the contributions of all the groups needed for a unique representation of the molecular structure of the compound. Some authors combine more principles, for example, Li et al.,48 who developed an estimation method for the enthalpy of vaporization, as a function of temperature, combining a group contribution approach with a corresponding states theorem. In this work, we have developed group contribution parameters for estimating the enthalpy of vaporization at 298.15 K (∆HV(298.15 K)) and at the normal boiling temperature (∆HV(Tb)), using an extensive database of critically evaluated data. We anticipated that the entropy of vaporization might obey the additivity principle more closely than the enthalpy of vaporization. For this reason, we also included the development of parameters for the entropy of vaporization at the normal boiling temperature (∆SV(Tb)). Parameters for the entropy of vaporization also may be used for estimating the enthalpy of vaporization if a reliable value of the normal boiling temperature (Tb) is known or can be estimated. The entropy of vaporization at 298.15 K is not independent of the enthalpy of vaporization at 298.15 K, because they are mutually related by a constant. For this reason, the development of parameters for the entropy of vaporization at 298.15 K was not performed.

property for a larger set of compounds (compared to the first-level application range), it was unable to improve the reliability of estimation for multifunctional compounds, where groups (or substructures or fragments of the molecule) may be thought to influence the contributions of other groups. To account for these interactions, Marrero and Gani38 introduced a thirdlevel estimation that captured the group interactions through large third-level groups and the addition of further third-level contributions to the first and/or second-level contributions. Through the higher-level contributions, Marrero and Gani38 showed that polyfunctional compounds and large molecules could be adequately represented. Examples of molecular structures that require additional second-level groups are compounds that contain one ring and non-ring chains that have more than four C atoms, whereas multiring compounds with a fused or nonfused aromatic ring or naphthenic rings are examples of molecular structures that require third-level groups. This multilevel molecular representation and property calculation scheme has been shown to enhance the accuracy, reliability, and range of application of the Marrero and Gani method for the following properties: the normal boiling point, the normal melting point, critical properties, the standard heat of formation, the standard heat of fusion, and the standard enthalpy of vaporization. New Model. We have extended this approach for predicting the enthalpy of vaporization and entropy of vaporization by including several new groups, especially at the second level. Also, the third-level contributions have not been used previously for predicting the enthalpy of vaporization, because of the lack of suitable input data. The extension of the number of groups was made possible because of the large databases of recommended data available and resulted in a reduction of the method error. Criteria for the creation of group contributions at the individual levels have been already described.38 The models for the estimation of ∆HV at 298.15 K and at Tb or ∆SV at Tb are given by eqs 1-3:

∆HV(298.15K) )

New Group Contribution Method Theoretical Background and Model Description. The new group contribution method for estimating the enthalpy of vaporization at 298.15 K (∆HV(298.15 K)) and for the estimation of enthalpy of vaporization and entropy of vaporization at the normal boiling temperature Tb (∆HV(Tb) and ∆SV(Tb), respectively) is based on the methodology developed by Marrero and Gani.38 The estimation of properties involves a three-level calculation procedure, covering groups of the first level, the second level, and the third level. In the first (primary) level, contributions from simple groups that allow the representation of a very wide variety of organic compounds are used. Although the application range, in terms of representation of the molecular structures, is large, the accuracy of the estimations suffers at the first level as the size and complexity of the molecular structures increase. To accommodate this deficiency, Marrero and Gani38 (and, prior to that research, Constantinou and Gani37) introduced secondlevel contributions, whose purpose was to add structural information (that the first-level groups could not include) of the molecule as a second-level contribution or correction to the value from the first level. Although the second level increased the reliability of the estimated

n0

h0 +

o0

m0

Mh ,jDh ,j + z ∑ Oh ,kEh ,k ∑ Nh ,iCh ,i + ωj∑ )1 k)1

i)1

0

0

0

0

0

0

(1) ∆HV(Tb) ) nb

hb +

∑

i)1

mb

Nhb,iChb,i + ω

∑

j)1

ob

Mhb,jDhb,j + z

∑ Oh ,kEh ,k

k)1

b

b

(2) ∆SV(Tb) ) nb

sb +

∑

i)1

mb

Nsb,iCsb,i + ω

∑

j)1

ob

Msb,jDsb,j + z

∑ Os ,kEs ,k

k)1

b

b

(3)

In these equations, Nh0,i, Nhb,i, and Nsb,i indicate the number of times the first-level group i appears in the molecular structure and Ch0,i, Chb,i, and Csb,i respectively represent the corresponding contributions to the property. The terms Mh0,j, Mhb,j, and Msb,j indicate the number of times second-level group j appears in the molecular structure, and Dh0,j, Dhb,j, and Dsb,j respectively represent their corresponding second-level contributions to the property. The terms Oh0,k, Ohb,k, and Osb,k indicate the

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Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005

Table 1. Sources of Input Data

Table 2. Accuracy of Input Data Number of Input Data

Number of Input Data

input data source

for ∆HV(298.15 K)

for ∆HV(Tb)

accuracy class of input data

for ∆HV(298.15 K)

for ∆HV(Tb)

Majer and Svoboda49 DDBST GmbH (ref 50) DDBST GmbH (ref 50)b CAPEC database (ref 51) Osborn and Scott52 CRC (ref 53) NIST (ref 54) Euse´bio et al.55 Ru˚zˇicˇka and Majer6

571 184 (52a) 38 59 5 9 1 2

409 78 (18a) 67 20

A (inaccuracy up 0.25%) B (inaccuracy up 0.5%) C (inaccuracy up 1%) D (inaccuracy up 2%) E (inaccuracy up 5%) unknown inaccuracy

145 263 276 67 0 80

145 84 87 136 52 85

total

831

589

total

831

589

66 16

a Data derived from the temperature dependence of the vapor pressure. b Values were obtained by converting the DDBST data originally at different temperatures to 298.15 K and/or to Tb.

number of times third-level group k appears in the molecular structure, and Eh0,k, Ehb,k, Esb,k respectively represent their corresponding third-level contribution to the property. The subscript “h” indicates the model for enthalpy of vaporization prediction, whereas the subscript “s” indicates the model for entropy of vaporization prediction. The subscripts “0” and “b” are used to distinguish between the temperature of estimation: the subscript 0 represents 298.15 K and the subscript b represents the normal boiling temperature. The variables ω and z are weighting factors that are equal to 0 or 1, depending on whether the second-level and third-level contributions, respectively, are used or not. Thus, for only first-level estimation, ω ) z ) 0; for only first-level and second-level estimation, ω ) 1 and z ) 0; for only first-level and third-level estimation, ω ) 0 and z ) 1; and for first-level, second-level, and thirdlevel estimation, ω ) z ) 1. Terms h0, hb, and sb (denoted generally as x0) are the additional adjustable parameters for ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb), respectively. Database. A large database of critically assessed data on the enthalpy of vaporizationshereafter referenced as the basic databaseswas applied for all group contribution parameter calculations. The pure organic compounds included in the database range in molecular mass from 41 g/mol up to 462 g/mol and cover many families, such as hydrocarbons (saturated, cyclic, unsaturated, aromatic), halogenated hydrocarbons, and compounds that contain O, N, S, and Si atoms. More detailed information about the numbers of compounds and the basic type of individual families is given in Appendix A. Data for 831 organic compounds were used for the 298.15 K determination, whereas the input data for 589 substances were used for the normal boiling temperature correlation, divided into several classes as mentioned previously. Preferably, calorimetrically measured experimental data from the compilation by Majer and Svoboda,49 from the DDBST GmbH database,50 and from the database of CAPEC51 were used. Some other sources6,52-55 were also used, as illustrated in Table 1. Calorimetrical experimental data were supplemented by data derived from vapor pressures, which are denoted in Table 1 in italic type. We preferably selected data with a stated accuracy of better than 1%, when available (as shown in Table 2). If experimental datum of the enthalpy of vaporization at 298.15 K or at Tb was unavailable, we converted an experimental value of ∆HV from the proximity of the respective temperature, using one of the relationships

by Watson,56 by Thiesen57 and its two- and threeparameter extensions,49 or by Xiang.20 The critical temperature (Tc) and the normal boiling temperature Tb, each of which were needed for the conversion of enthalpy of vaporization, were taken from refs 6, 49, 53, 54, and 58-61. Procedure for Estimating the Model Parameters. The following parameters were to be fitted: h0, Ch0,i, Dh0,j, and Eh0,k for ∆HV at 298.15 K; and hb, Chb,i, Dhb,j, Ehb,k and sb, Csb,i, Dsb,j, Esb,k for ∆HV and ∆SV at Tb. The parameter estimation was performed in the following three steps. Step 1. Set ω ) z ) 0 and estimate the adjustable parameters h0, hb, and sb and the first-level contributions Ch0,i, Chb,i, and Csb,i using the entire dataset for ∆HV(298.15 K), for ∆HV(Tb) or for ∆SV(Tb). Note that all compounds in the database had to be fully represented by the first-level groups. Step 2. Set ω ) 1 and z ) 0, keeping h0, hb, and sb and the first-level contributions Ch0,i, Chb,i, and Csb,i from step 1 constant, estimate the second-level contributions Dh0,j, Dhb,j, and Dsb,j. Note that, in the second step, only those compounds that require the second-level groups are used. Step 3. Set ω ) 1 and z ) 1 and keeping h0, hb, and sb the first-level contributions Ch0,i, Chb,i, and Csb,i from step 1 and the second-level contributions Dh0,j, Dhb,j, and Dsb,j from step 2 constant, estimate the third-level contributions Eh0,k, Ehb,k, and Esb,k. Note that, in the third step, only those compounds that require the third-level groups are used. The group contribution parameters are given in Tables 3-5. The statistical significance of groupcontribution values was evaluated using the absolute error (AE), the relative error (RE), the absolute average error (AAE), and the average relative error (ARE), which are given by the following equations:

AE[X]i ) |Xexp - Xest|i RE[X]i )

(4)

|Xexp - Xest|i AE[X]i × 100 ) × 100 (5) Xexp,i Xexp,i

AAE[X] )

ARE[X] )

1

n

1

n

|Xexp - Xest|i ∑ AE[X]i ) n∑ ni ) 1 i)1 1

n

∑

ni)1

RE[X]i )

1

n

∑

AE[X]i

ni)1 Xexp,i

× 100

(6)

(7)

where X represents the estimated value ∆HV(298.15 K) or ∆HV(Tb) or ∆SV(Tb), n is the number of data in the database, and subscripts “exp” and “est” respectively denote the experimental value or the estimated value. Tables 6-8 give the values for the absolute average error AAE and the average relative error ARE for the

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8439 Table 3. List of the First-Level Group Contributions and Their Values for the Enthalpy of Vaporization at 298.15 K and at the Normal Boiling Temperature, as Well as for the Entropy of Vaporization at the Normal Boiling Temperature Entropy of Vaporization (J K-1 mol-1)

Enthalpy of Vaporization (kJ/mol) No.

first-level group, i

x0

additional adjustable parameter

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 27 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 61 62 63 64 65 66 67 68 69 70 71 72

CH3 CH2 CH C CH2dCH CHdCH CH2dC CHdC CdC CH2dCdCH CH#C C#C aCH aC fused with aromatic ring aC fused with nonaromatic subring aC except as above aN in aromatic ring aC-CH3 aC-CH2 aC-CH aC-C aC-CHdCH2 OH aC-OH COOH CH3CO CH2CO CHCO CCO aC-CO CHO aC-CHO CH3COO CH2COO CHCOO CCOO HCOO aC-COO aC-OOC COO except as above CH3O CH2O CH-O C-O aC-O CH2NH2 CHNH2 CNH2 CH3NH CH2NH CHNH CH3N CH2N aC-NH2 aC-NH aC-N NH2 except as above CH2CN CHCN CCN aC-CN CN except as above CH2NO2 CHNO2 CNO2 aC-NO2 NO2 except as above ONO2 CONHCH3 CONHCH2 CONCO NCON

∆HV1i(298.15)a

b

NC1i

9.672 2.266 4.724 4.724 4.809 5.620 8.516 7.741 10.409 11.558 * 6.716 11.063 4.297 5.712 6.190 3.910 9.960 8.121 9.840 9.261 8.843 * 24.930 43.072 28.222 15.707 17.011 17.158 15.889 18.409 13.027 * 20.160 20.587 21.029 18.748 15.714 21.425 23.658 16.457 7.959 8.597 8.278 7.555 -4.968 16.345 13.955 13.171 14.251 12.062 11.151 8.204 6.731 21.137 15.633 17.140 13.582 22.342 22.300 20.881 * 18.099 26.922 * * 20.541 20.593 18.006 48.210 50.953 32.144 29.423

∆HV1i(Tb

)a

b

NC1i

14.876 624 443 152 66 25 27 16 9 2 0 2 4 134 11 14 12 18 52 17 14 11 0 36 5 5 24 11 4 2 7 4 0 15 29 6 7 4 5 2 6 23 83 8 6 4 9 2 1 3 8 1 10 7 6 2 1 5 11 2 1 0 14 2 0 0 4 2 2 2 3 2 2

2.312 2.362 1.430 0.452 4.387 4.066 3.691 5.435 3.870 6.831 5.902 7.786 2.680 4.269 3.223 1.821 7.311 4.470 4.063 3.885 6.561 10.426 16.881 17.013 17.122 10.346 9.882 8.917 5.405 13.570 10.392 14.226 13.259 12.710 10.441 8.658 10.919 15.498 * 11.365 5.878 5.052 4.330 1.687 8.208 10.934 8.193 9.157 9.818 7.232 4.845 4.539 5.334 14.003 * 11.451 9.722 15.899 12.889 7.736 17.626 12.944 19.881 17.299 -24.540 * 16.801 * * * * *

∆SV1i(Tb)a

NC1ib

83.861 406 301 123 40 19 7 6 3 3 1 2 3 74 5 3 2 15 32 8 2 1 1 42 7 4 19 9 2 2 2 9 1 17 22 5 2 6 2 0 8 22 53 8 2 2 10 2 2 5 8 1 2 2 5 0 1 3 7 1 2 1 11 2 1 1 0 2 0 0 0 0 0

-0.389 0.657 0.430 0.566 0.712 1.616 1.124 3.853 3.383 1.105 3.010 4.487 0.259 1.529 1.592 4.309 6.857 -0.483 0.606 3.518 7.457 7.338 18.408 16.038 5.579 3.066 3.706 4.507 3.879 7.935 6.203 9.037 8.710 8.292 5.880 6.094 7.972 16.992 * 8.618 1.655 2.546 3.696 0.857 6.396 6.365 6.736 9.810 7.004 2.606 2.393 2.885 4.775 8.539 * 10.792 7.607 3.546 2.831 4.267 13.787 3.994 12.830 10.471 -3.745 * 7.313 * * * * *

406 301 123 40 19 7 6 3 3 1 2 3 74 5 3 2 15 32 8 2 1 1 42 7 4 19 9 2 2 2 9 1 17 22 5 2 6 2 0 8 22 53 8 2 2 10 2 2 5 8 1 2 2 5 0 1 3 7 1 2 1 11 2 1 1 0 2 0 0 0 0 0

8440


Table 3 (Continued) Entropy of Vaporization (J K-1 mol-1)


first-level group, i

∆HV1i(298.15)a

73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

NHCO except as above CH2Cl CHCl CCl CHCl2 CCl3 CH2F CHF2 CF2 CF3 CCl2F HCClF CClF2 aC-Cl aC-F aC-I aC-Br -I except as above -Br except as above -F except as above -Cl except as above OCH2CH2OH -O-OH CH2SH CHSH CSH aC-SH -SH except as above CH3S CH2S CHS CS CO3 C2H3O CH2 (cyclic) CH (cyclic) C (cyclic) CHdCH (cyclic) CHdC (cyclic) CH2dC(cyclic) NH (cyclic) N (cyclic) CHdN (cyclic) O (cyclic) CO (cyclic) S (cyclic) SO2 (cyclic) SiO Si NdN C(cyc)dCHNdO NdN

45.177 13.046 11.487 7.261 17.746 17.913 9.370 8.334 4.075 3.763 11.520 11.811 5.977 10.674 4.378 * 13.381 14.940 10.959 -0.076 5.172 31.438 26.529 17.098 15.155 14.394 * 11.190 14.790 15.324 14.245 14.741 19.949 19.201 4.013 4.075 3.667 9.179 10.070 6.749 14.765 6.576 14.986 6.171 14.837 9.805 * 4.643 0.515 15.017 8.108 16.594 6.809

a

b

NC1i

∆HV1i(Tb)a

b

NC1i

∆SV1i(Tb)a

NC1ib

1 27 9 2 6 3 1 6 19 11 5 13 5 6 9 0 1 14 21 14 12 5 3 12 2 2 0 2 9 17 2 3 1 8 132 73 29 15 11 4 5 6 3 17 11 11 0 6 5 2 3 1 7

* 9.106 5.938 4.650 11.487 12.311 9.072 * 2.293 1.347 9.495 7.659 4.016 6.873 2.894 11.226 9.610 9.941 6.648 -0.735 4.369 21.825 * 10.720 8.568 6.916 11.656 8.177 9.793 9.088 7.059 3.939 * 10.456 2.416 2.263 3.435 5.475 6.000 -4.173 9.021 2.515 7.231 4.655 19.025 7.708 35.865 0.328 0.166 * 4.487 * *

0 18 7 3 4 3 1 0 16 15 3 10 4 9 10 1 1 7 18 14 10 8 0 7 2 2 1 2 8 9 2 2 0 2 65 35 10 13 5 1 4 1 2 15 5 10 1 3 3 0 1 0 0

* 2.889 0.219 2.673 3.469 2.113 9.890 * 0.568 -0.295 4.834 3.595 -0.815 1.563 1.075 0.434 3.110 0.782 0.792 -1.882 0.723 14.703 * 2.721 2.145 1.267 5.121 1.682 2.330 2.600 3.221 -0.087 * 5.731 -0.011 0.971 3.730 1.544 5.859 -14.285 7.095 -6.781 1.472 2.525 20.094 1.116 24.438 2.114 -3.925 * -0.133 * *

0 18 7 3 4 3 1 0 16 15 3 10 4 9 10 1 1 7 18 14 10 8 0 7 2 2 1 2 8 9 2 2 0 2 65 35 10 13 5 1 4 1 2 15 5 10 1 3 3 0 1 0 0

An asterisk symbol (*) denotes that the group value is unavailable. b NC1i ) number of data points used for parameter calculation.

estimation of ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb), respectively. The correlation statistics are shown for the first level and the second level, as well as the third level. NC denotes the number of compounds in the dataset, and NG represents the number of groups determined in the individual correlation levels. More detailed results are shown for individual molecular types in Appendix A. Figures 1-3 provide a visualization of the correlation of all experimental data from the basic database by our model. Note that, except for a few data points, all of the data have been fitted to a high degree of accuracy. Results obtained for all properties are shown in Table 9. The average relative error ARE, the maximum relative error (MRE), the number of input data NC, and

the number of calculated group contributions NG for the individual levels are presented. More-detailed results are presented in Appendix A for all families of compounds. Distribution of the relative errors obtained by our method for all compounds from the database is given in Table 10. The numbers of estimated values within the relative error (RE) range, and their percentage occupation in the database, are presented. 3. Performance Analysis and Model Application The performance of the developed model for estimation of ∆HV(298.15 K) and ∆HV(Tb) was analyzed in terms of comparison with other estimation models, as

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8441 Table 4. List of the Second-Level Group Contributions and Their Values for the Enthalpy of Vaporization at 298.15 K and at the Normal Boiling Temperature, as Well as for the Entropy of Vaporization at the Normal Boiling Temperature Entropy of Vaporization (J K-1 mol-1)


second-level group, j

∆HV2j(298.15)a

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 61 62 63 64 65 66 67 68 69 70 71 72 73

(CH3)2CH (CH3)3C CH(CH3)CH(CH3) CH(CH3)C(CH3)2 CHndCHm-CHpdCHk (k,m,n,p in 0...2) CH3-CHmdCHn (m,n in 0...2) CH2-CHmdCHn (m,n in 0...2) CHp-CHmdCHn (m,n in 0...2; p in 0...1) CH3COCH2 CH3COCH or CH3COC CHCOOH or CCOOH CH3COOCH or CH3COOC CO-O-CO CHOH COH CHm(OH)CHn(OH) (m,n in 0...2) CHm(OH)CHn(NHp) (m,n,p in 0...2) CHm(NH2)CHn(NH2) (m,n in 0...2) CHm(NH)CHn(NH2) (m,n in 1...2) NC-CHn-CHm-CN (n, m in 1...2) HS-CHn-CHm-SH (n, m in 1...2) COO-CHn-CHm-OOC (n, m in 1...2) NC-CHn-COO (n in 1...2) CHm-O-CHndCHp (m,n,p in 0...3) CHmdCHn-F (m,n in 0...2) CHmdCHn-Cl (m,n in 0...2) CHmdCHn-CN (m,n in 0...2) CHndCHm-COO-CHp (m,n,p in 0...3) CHmdCHn-CHO (m,n in 0...2) aC-CHn-NHm (n in 1...2; m in 0...2) aC-CHn-O- (n in 1...2) aC-CHn-OH (n in 1...2) aC-CHn-CN (n in 1...2) aC-CHn-OOC (n in 1...2) aC-CHn-COO (n in 1...2) aC-CH(CH3)2 aC-C(CH3)3 aC-CF3 (CHndC)(cyc)-CHO (n in 0...2) (CHndC)(cyc)--CH3 (n in 0...2) (CHndC)(cyc)-CH2 (n in 0...2) (CHndC)(cyc)-CN (n in 0...2) CH(cyc)-CH3 CH(cyc)-CH2 CH(cyc)-CH CH(cyc)-C CH(cyc)-CHdCHn (n in 1...2) CH(cyc)-CdCHn (n in 1...2) CH(cyc)-OH CH(cyc)-NH2 CH(cyc)-SH CH(cyc)-CN CH(cyc)-CO CH(cyc)-COO C(cyc)-CH3 C(cyc)-CH2 >Ncyc-CH3 AROMRINGs1s2 AROMRINGs1s3 AROMRINGs1s4 AROMRINGs1s2s3 AROMRINGs1s2s4 AROMRINGs1s3s5 AROMRINGs1s2s3s5 PYRIDINEs2 PYRIDINEs3 PYRIDINEs4 PYRIDINEs2s3 PYRIDINEs2s4 PYRIDINEs2s5 PYRIDINEs2s6 PYRIDINEs3s4 PYRIDINEs3s5

-0.066 -0.022 1.082 1.920 * 0.040 -0.208 0.219 1.440 1.443 * -1.965 * 0.656 -0.405 -4.381 * 2.618 0.178 * 0.812 1.674 * 0.985 * -0.506 1.088 * * 0.398 -0.412 * * * 0.758 0.445 0.833 -1.971 * 3.216 * -4.900 0.112 -0.529 0.736 1.565 0.222 -2.416 3.070 -3.722 0.000 -1.516 1.930 0.336 -1.355 1.196 0.599 -0.832 -0.073 0.473 -2.297 1.245 0.192 0.110 -2.462 -0.392 -0.142 -1.036 -1.286 * -1.286 1.764 0.714

b

NC2j

∆HV2j(Tb

109 63 10 4 0 34 44 16 11 3 0 3 0 7 3 3 0 1 1 0 1 6 0 2 0 3 5 0 0 5 2 0 0 0 2 6 6 1 0 7 0 1 12 22 2 1 2 2 2 1 2 4 1 3 6 3 3 9 8 10 5 5 6 2 1 1 1 1 1 0 1 1 1

-0.143 0.092 1.171 1.126 -0.419 0.191 0.419 * -0.091 0.808 4.280 -1.832 0.903 -0.877 -2.140 -1.157 1.438 1.235 -0.351 1.826 1.614 * 4.345 0.091 0.556 -0.502 -0.341 0.269 -1.355 * * 1.198 0.756 -2.028 * -0.049 * 1.270 1.801 0.715 3.351 * -0.212 0.031 0.843 0.307 -0.751 * 1.266 -2.803 0.000 0.091 1.752 -0.668 -0.351 -1.411 0.000 0.894 0.256 0.517 * -0.419 -0.241 * -1.206 -0.026 0.134 -0.086 -0.636 -0.486 -0.659 0.824 0.294

)a

b

NC2j

∆SV2j(Tb)a

NC2jb

97 37 6 3 2 12 13 0 8 3 1 2 3 9 4 3 2 1 1 1 1 0 1 3 1 4 7 3 1 0 0 1 1 2 0 2 0 1 1 3 1 0 13 7 1 1 1 0 3 1 2 2 1 2 4 1 1 10 5 8 0 3 1 0 1 1 1 1 1 1 1 1 1

-0.061 -0.091 0.430 -0.639 -1.406 -0.160 0.754 * -0.735 1.181 11.590 -1.606 1.952 1.526 1.589 -7.223 -1.501 0.768 -2.282 -1.004 1.189 * 7.906 -1.430 1.153 -0.625 0.567 1.313 -3.966 * * 1.191 1.098 -2.873 * -0.156 * 0.570 -0.658 -1.705 8.086 * -0.552 -1.252 -0.398 -0.864 -3.764 * 7.931 -3.611 0.000 -1.630 0.754 -2.174 0.867 -4.181 0.000 1.849 0.405 0.612 * -0.301 -0.768 * -1.431 -1.768 -1.642 -0.566 -1.257 -0.827 -0.821 -2.115 -1.875

97 37 6 3 2 12 13 0 8 3 1 2 3 9 4 3 2 1 1 1 1 0 1 3 1 4 7 3 1 0 0 1 1 2 0 2 0 1 1 3 1 0 13 7 1 1 1 0 3 1 2 2 1 2 4 1 1 10 5 8 0 3 1 0 1 1 1 1 1 1 1 1 1

8442


Table 4 (Continued) Entropy of Vaporization (J K-1 mol-1)


second-level group, j

∆HV2j(298.15)a

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

AROMRINGs1s2s3s4s5 CH2SCOCH3 CHSCOCH3 CSCOCH3 OCH2COO ClCH2COO (Cl2)CHCOO (Cl3)CCOO CH2OCHO CH2COOCH2 CCOOCH3 CCOOCH2 CHCOOCH2 CH3COOCH2 OCH2CH2O CH2COOCH3 OCH2O OCH2O CH2SSCH2 CSSC CF2O (CH2OCH2)(cyc) (CH2OCH)(cyc) (CHdCHOCHdCH)(cyc) (COOCH2)(cyc) (COOCd)(cyc) (CH2NHCH2)(cyc) (dCHNHCHd)(cyc) (CH3)3SiaCaNaC (CH2SCH2)(cyc) (CH2SCHd)(cyc) (dCHSCHd)(cyc) (dCHSCd)(cyc) (3 F) (5 F) (perfluoro) (CH2)3Si(O)3Si-

0.372 -3.636 -1.830 -3.965 6.140 * 4.665 0.867 -0.233 -1.319 1.280 -0.841 -2.354 -0.622 1.276 -0.248 0.104 1.292 -0.956 1.567 -0.020 -0.056 1.175 * * 1.448 -3.294 2.544 * -2.119 3.666 * -5.061 -5.338 -0.203 0.037 0.033 * -0.144

a

b

NC2j

∆HV2j(Tb)a

b

NC2j

∆SV2j(Tb)a

NC2jb

1 3 1 1 2 0 2 2 3 14 2 5 5 11 16 14 6 2 4 1 12 12 6 0 0 1 3 1 0 3 4 0 4 1 14 10 10 0 8

0.194 -2.323 -1.043 -0.210 2.428 4.878 2.913 * 0.095 -0.395 0.544 -0.728 -0.899 0.228 0.876 -0.598 -0.094 2.932 -0.096 * 0.819 0.307 -0.979 -3.382 -1.427 0.000 -1.403 3.901 0.795 -1.047 2.032 0.347 -2.055 -3.187 -1.270 -0.068 0.203 -0.999 -0.495

1 2 1 1 2 2 1 0 5 15 1 1 3 12 11 6 4 1 1 0 9 8 3 1 1 1 3 1 2 3 4 1 3 1 20 9 9 1 2

-0.116 -1.156 0.057 3.322 2.074 5.215 3.624 * -0.013 0.030 1.001 -0.819 -1.334 0.138 0.967 -1.790 1.273 3.236 -0.316 * 0.037 0.902 -1.052 -0.475 -13.548 0.000 -1.004 2.111 2.053 0.059 2.285 -0.207 0.041 -2.394 -0.570 0.202 0.116 -3.251 -0.941

1 2 1 1 2 2 1 0 5 15 1 1 3 12 11 6 4 1 1 0 9 8 3 1 1 1 3 1 2 3 4 1 3 1 20 9 9 1 2

An asterisk symbol (*) denotes that the group value is unavailable. b NC2j ) number of data points used for parameter calculation.

well as in terms of extrapolation features, predictive capability, and consistency. Comparison with Other Models. Two methods, formulated by Ducros and co-workers44-47 and by Chickos et al.,34 were selected for a comparison of ∆HV(298.15 K) results. The former method is a representative of a classical group contribution method, and the latter is a type of recent and more-general purpose additive method. Both methods were developed for estimation at 298.15 K only. The method devised by Ducros and co-workers44-47 applies the relation ∆HV ) k νi(∆HV)i, where νi is the number of groups of type Σi)1 i in the molecule with the enthalpy of vaporization value (∆HV)i. The method devised by Ducros and coworkers44-47 lacks groups for compounds that contain F and I atoms in the molecule, and, for polycyclic and some miscellaneous compounds (for example, heterocyclic nitrogen or sulfur compounds, or compounds with different atoms in the molecule, such as nitrogen and oxygen, a halogen and oxygen, or sulfur and oxygen). Therefore, it was possible to predict values for only 526 compounds from our basic database using this method. Chickos et al.34 proposed four relationships for different organic compounds. All of them start from the basic form ∆HV ) 4.69(nC - nQ) + 1.3nQ + 3.0, where nC and nQ denote the total number of C atoms and number of quaternary C atoms in the molecule, respec-

tively. This relation works well for hydrocarbons. For other families of compounds, this relation is extended by including terms to account for branching and hybridization, for the presence of metal atoms, and for the presence of rings. Two equations by Vetere18 (eqs 6 and 7 in the original paper) and a group contribution method by Ma and Zhao62 were chosen for a comparison of ∆HV(Tb) and ∆SV(Tb) results. Vetere18 presented parameters of two empirical equations for estimating ∆HV(Tb) for different families of compounds, including hydrocarbons and CCl4, alcohols, and other polar compounds. The relationships are universal and enable the estimation for all 589 compounds from our basic database. The results of the comparison are presented in Table 11, where NC represents the number of compounds included in the comparison and ARE is the average relative error of the model. The method by Ma and Zhao62 was proposed to estimate the entropy of vaporization at normal boiling temperature using the following equation: ∆SV(Tb) ) n νi∆SV(Tb)i, where νi is the number of groups of A + Σi)1 type i in the molecule with an entropy of vaporization value of ∆SV(Tb)i. This method lacks group contributions for nitriles connected to cyclic and aromatic rings, for benzenethiols, for compounds that contain Si atoms in the molecule, for bromine that is connected to aromatic

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8443 Table 5. List of the Third-Level Group Contributions and Their Values for Enthalpy of Vaporization at 298.15 K (∆HV(298.15 K)) and at the Normal Boiling Temperature (∆HV(Tb)), as Well as the Entropy of Vaporization at the Normal Boiling Temperature (∆SV(Tb))a Entropy of Vaporization (J K-1 mol-1)

Enthalphy of Vaporization (kJ/mol) ∆HV3k(298.15)a NC3kb ∆HV3k(Tb)a NC3kb

No.

third-level group, k

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

OH-(CHn)m-OH (m > 2, n in 0...2) NH2-(CHn)m-NH2 (m > 2; n in 0...2)) SH-(CHn)m-SH (m > 2; n in 0...2) aC-aC (different rings) aC-CHn(cyc) (different rings) (n in 0...1) aC-CHn(cyc) (fused rings) (n in 0...1) CH(cyc)-CH(cyc)(different rings) CH multiring C multiring aC-CHm-aC (different rings) (m in 0...2) aC-CO-aC (different rings) aC-CHm-CO-aC (different rings) (m in 0...2) aC-CO(cyc) (fused rings) aC-S(cyc) (fused rings) aC-O-aC (different rings) AROM.FUSED[2] AROM.FUSED[2]s1 AROM.FUSED[2]s2 AROM.FUSED[2]s1s4 AROM.FUSED[2]s1s3 PYRIDINE.FUSED[2] PYRIDINE.FUSED[2-iso] a

-0.128 * 1.368 * 1.891 0.279 0.588 -0.191 0.027 -0.194 * * 0.746 * * -0.615 0.128 1.051 -0.392 -0.582 1.149 *

1 0 3 0 2 7 2 18 14 7 0 0 3 0 0 6 3 2 2 2 2 0

1.933 1.744 * -0.167 * 1.046 * 0.473 -0.220 * 0.029 0.146 * 0.275 -0.176 -0.825 1.783 * * * 1.043 0.343

3 1 0 1 0 2 0 6 5 0 1 1 0 1 1 8 2 0 0 0 1 1

∆SV3k(Tb)a

NC3kb

-2.653 1.686 * -0.648 * 1.034 * 0.566 -0.138 * 0.456 -0.313 * 0.074 -0.143 -0.522 4.103 * * * -4.901 -0.171

3 1 0 1 0 2 0 6 5 0 1 1 0 1 1 8 2 0 0 0 1 1

An asterisk symbol (*) denotes that the group value is unavailable. b NC3k number of data points used for parameter calculation.

Table 6. Results for ∆HV(298.15 K) Estimation estimation level

NC

NG

AAE (kJ/mol)

ARE (%)

116

1.3

2.8

486 486 compounds after the first level only

91

0.8 1.3

1.8 2.8

third third

55 55 compounds after the first and second levels only

15

1.1 1.4

2.1 2.5

all levels

831

222

1.0

2.2

first

831

second second

Table 7. Results for ∆HV(Tb) Estimation estimation level

NC

NG

AAE (kJ/mol)

ARE (%)

first

589

111

1.2

3.2

second second


100

0.9 1.2

2.5 3.4

third third

23 23 compounds after the first and second levels

14

0.8 1.3

1.9 2.7

all levels

589

225

0.9

2.6

NG

AAE (J K-1 mol-1)

ARE (%)

Table 8. Results for ∆SV(Tb) Estimation estimation level

NC

first

589

111

2.1

2.2

second second


100

1.8 2.3

1.9 2.4

third third

23 23 compounds after the first and second levels only

14

1.9 2.5

1.9 2.5

all levels

589

225

1.7

1.8

rings, and for some other miscellaneous compounds that contain halogene and double bonds. Also, application of this method for heterocyclic compounds that contain nitrogen is unclear, because the value for the nitrogen group in the aromatic ring is missing. Because of this lack of groups, the method was applicable to only 549 compounds of our basic database. We also assembled data for the enthalpy of vaporization for 74 compounds that were not used for parameter calculation. The ARE of the estimated values for the enthalpy of vaporization (66 values at 298.15 K, 8 values

at Tb) for this independent set, hereafter called the test set, is 2.5%. We also tested three methods that were developed for estimating the enthalpy of vaporization at a full temperature range: two first-order group contribution methods, by Basarˇova´ and Svoboda42 and by Tu and Liu,43 and a corresponding states group contribution method by Li et al.48 All three methods require, in addition to group contribution parameters, some other input data, such as the critical temperature42,43 and normal boiling temperature and the enthalpy of vapor-

8444


Figure 1. Comparison of estimated and experimental values for the enthalpy of vaporization at 298.15 K (∆HV(298.15 K)).

Figure 3. Comparison of estimated and experimental values for the entropy of vaporization at the normal boiling temperature Tb (∆SV(Tb)). Table 9. Results for Estimations of the Enthalpy of Vaporization and the Entropy of Vaporization NG

∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb)

ARE (%)

MRE (%)

2.2 2.6 1.8

12.6 16.5 17.4

NC

first levela

second level

third level

831 589 589

115 + 1 110 + 1 110 + 1

91 100 100

15 14 14

a The term “+1” in the fifth column means one adjustable additional parameter x0 (h0, hb, and sb in eqs 1-3), which is calculated in the first-level correlation for all properties.

Table 10. Distribution of Relative Errors (RE) Obtained for All Compounds from the Basic Database

Figure 2. Comparison of estimated and experimental values for the enthalpy of vaporization at the normal boiling temperature Tb (∆HV(Tb)).

ization at normal boiling temperature48 and lack group contribution parameters for some families of compounds (e.g., carboxylic acids, multiring aromatic compounds, some heterocyclic compounds). A comparison with experimental data on C1-C25 compounds selected mainly from the DDBST GmbH database,50 which contains data for 121 compounds at 298.15 K and 144 compounds at normal boiling temperature, gave the results that are presented in Table 12. All three methods42,43,48 work fine for monosubstituted compounds but often fail for morecomplex multisubstituted compounds. Extrapolation Features. Figures 4-7 show the extrapolation feature of the developed model (denoted by the subscript “our model”) for n-alkanes and 1-alkanols C3-C38 (note that our basic database contained data for n-alkanes from C5 to C24 and for 1-alkanols from C2 to C14 for ∆HV(298.15 K) and data for n-alkanes from C4 to C20 and for 1-alkanols from C2 to C14 for ∆HV(Tb)). In these figures, ∆HV(298.15 K) (Figure 4), ∆HV(Tb) (Figure 5), ∆SV(Tb) (Figure 6), and Tb ) ∆HV(Tb)/∆SV(Tb) (Figure 7) were plotted as a function of the number of C atoms (nC). The experimental data (denoted by the subscript ”experim.”) used for parameter calculation are also

RE

∆HV(298.15 K)

∆HV(Tb)

∆SV(Tb)

up to 1% 1%-2% 2%-5% 5%-10% over 10%

299 (36.0%) 203 (24.4%) 240 (28.9%) 80 (9.6%) 9 (1.1%)

196 (33.3%) 112 (19.0%) 191 (32.4%) 71 (12.1%) 19 (3.2%)

284 (48.2%) 125 (21.2%) 131 (22.3%) 37 (6.3%) 12 (2.0%)

Table 11. Results of the Comparison of Enthalpy of Vaporization at 298.15 K (∆HV(298.15 K)) and at the Normal Boiling Temperature (∆HV(Tb)), as Well as the Entropy of Vaporization at the Normal Boiling Temperature Ducros Ma 18 our and coChickos Vetere and model workers44-47 et al.34 eq 6 eq 7 Zhao62 ∆HV(298.15 K) NC ARE (%) ∆HV(Tb) NC ARE (%) ∆SV(Tb) NC ARE (%)

831 2.2

526 3.1

800 4.7

589 2.6

589 589 4.6 3.4

549 2.5

589 1.8

589 589 4.6 3.4

549 2.5

plotted. In addition, some values calculated by our model outside the range of experimental data (denoted by the subscript “extrapol.”) are shown. Figure 4 shows that the model extrapolates for n-alkanes and for 1-alkanols safely along the carbon number scale and predicts the properties with reasonable accuracy for ∆HV(298.15 K). In the same way, we also plotted nine recommended data points from Steele and Chirico63 of ∆HV(298.15 K) for the family of 1-alkenes of C5-C16 and determined that the dependence was similar to that

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8445 Table 12. Results of the Comparison of the Enthalpy of Vaporization at 298.15 K (∆HV(298.15 K)) and at the Normal Boiling Temperature (∆HV(Tb))

∆HV(298.15 K) NC ARE (%) ∆HV(Tb) NC ARE (%)

our model

Basarˇova´ and Svoboda42

Tu and Liu43

121 2.8

78 4.4

74 6.3

70 3.6

144 2.9

105 5.6

104 7.3

not applicable

Li et al.48

observed for n-alkanes. In Figures 5-7, the data of n-alkanes and 1-alkanols are plotted for ∆HV(Tb), ∆SV(Tb) and Tb. It can be observed, particularly from Figure 7, that the extrapolated data have a tendency to deviate from experimental values above a carbon number of 20. This is not surprising, because the enthalpy of vaporization, as well as the normal boiling temperature, are not linearly dependent on the number of C atoms in the molecule. The intermolecular forces do not increase linearly with the molecular size. We also tested a suitability of the developed model for estimation of the normal boiling temperature using an independent set of 203 organic compounds that were

Figure 4. Comparison of experimental, predicted, and extrapolated ∆HV(298.15 K) values for n-alkanes and 1-alkanols.

Figure 5. Comparison of experimental, predicted, and extrapolated ∆HV(Tb) values for n-alkanes and 1-alkanols.

not used for the calculation of group contributions for ∆HV(Tb) and ∆SV(Tb). The average absolute deviation (AAE) between the experimental normal boiling temperature and that calculated as a ratio of the estimated enthalpy of vaporization and the estimated entropy of vaporization was 7.8 K. A graphical comparison is shown in Figure 8. Analysis of Results of RegressionsHistograms. Figures 9 and 10 show histograms of the enthalpies of vaporization at the two temperatures for experimental values and values calculated by various models. These histograms illustrate the following results. First, they show that distributions of all datasets are not normal, and, therefore, it is necessary to use nonparametric methods for a statistical comparison of regression results obtained in this work. Second, it is obvious that the agreement between values obtained by our model and the experimental data is good, because our model is able to determine almost the same number of ∆HV(298.15 K) values as the number of data points in the basic database. Furthermore, the histograms also illustrate that other models tested in this work are unable to describe all compounds from the basic database, because of a lack of groups required for some families of compounds or due to inaccuracy of the particular model. The latter is

Figure 6. Comparison of experimental, predicted, and extrapolated ∆SV(Tb) values for n-alkanes and 1-alkanols.

Figure 7. Comparison of experimental, predicted, and extrapolated normal boiling temperature (Tb) values for n-alkanes and 1-alkanols.

8446


Figure 8. Comparison of experimental and predicted Tb values for 203 organic compounds.

Figure 9. Histogram of the ∆HV(298.15 K) values.

Figure 10. Histogram of the ∆HV(Tb) values.

observed in Figure 10; eq 6 in the work by Vetere18 is clearly inferior to other methods examined. The occurrence of data calculated by this equation in the range of 30-35 kJ/mol is lower than that for experimental data, even though the Vetere method was applied to the same number of compounds. An opposite effect is observed in the adjacent interval of 35-40 kJ/mol.

Analysis of Results of RegressionsStatistical Approach. For an exact analysis of all obtained results, we applied some statistical tools. For all estimated properties, ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb), the following datasets were analyzed: experimental data, values determined by our method, and values determined by all other methods examined (Ducros and Chickos for ∆HV(298.15 K), and Vetere and Ma and Zhao for ∆HV(Tb) and ∆SV(Tb)). Also, relative errors for all data obtained by all methods were compared between each other for each estimated property. Initially, we needed to determine whether the distribution of the individual dataset was normal or different. Normality (or normal distribution of the set) is one of the fundamental presumptions for the applied statistical test for the future statistical comparison.64,65 When the dataset originates from a normal distribution, we can use parametric statistical tests, whereas in all other cases, it is necessary to use only the nonparametric statistical tests. The test of normality was made by the W-test by Shapiro and Wilks.66 This tests the assumption that the data follow a normal distribution, rather than a different distribution. The Shapiro-Wilks test is based on comparing the quantiles of the fitted normal distribution to the quantiles of the data. For all tested datasets, this test resulted in rejection of the hypothesis of the normal distribution on the 90%, 95%, and 99% confidence level. Therefore, nonparametric methods for other statistical comparisons were considered. To compare experimental data and individual sets of estimated values, we applied the Mann-Whitney W-test (also known as the Wilcoxon test)67 for a comparison of medians and the test by Kolmogorov-Smirnov,67 to determine if a statistically significant difference exists between the distribution of the two samples (the term “sample” in this work means the set of data) at a given confidence level. The same comparative tests were applied for comparison of the relative errors obtained in this work. The test by MannWhitney compares the medians of the two samples. This test is constructed by combining the two samples, sorting the data from the smallest to the largest value, and then comparing the average ranks of the two samples of combined data. It tests the hypothesis that the distribution of two independent samples from populations is the same and if there is or is not a statistically significant difference between the medians of samples at the chosen confidence level (mostly at the 95% confidence level). The Kolmogorov-Smirnov statistics calculates the observed cumulative distribution for the two samples and the maximum positive, maximum negative, and absolute differences. If the distance is large enough, it indicates that a statistically significant difference exists between the two distributions at a given confidence level. This calculation is a nonparametric method that tests the null hypothesis that two samples are from the same distribution. As a result, we obtained the box-and-whisker plots, which are presented in Figures 11 and 12. An explanation of the box-and-whisker plots is given in Figure 13. This diagram creates a plot for each dataset (sample), which is divided into four equal areas of frequency (quartiles). The plot shows the most extreme values in the data (maximum and minimum values), the lower and upper quartiles, and the median. A box encloses the middle 50%, where the median is drawn as a vertical line inside the box. Horizontal lines that are known as whiskers extend from each end of the box. The left (or


Figure 11. Comparison of results for ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb).

Figure 12. Comparison of relative errors for ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb).

Figure 13. Explanation of the box-and-whisker plot.

lower) whisker is drawn from the lower quartile to the smallest point within 1.5 interquartile ranges from the lower quartile. The other whisker is drawn from the

upper quartile to the largest point within 1.5 interquartile ranges from the upper quartile. The rectangular portion of the plot extends from the lower quartile to the upper quartile, covering the center half of each sample. The center lines within each box show the location of the sample medians. The plus signs represent the location of the sample means. The whiskers extend from the box to the minimum and maximum values in each sample, except for any outside or far-outside points (the distant values in Figure 13), which are plotted separately. Outside points lie more than 1.5 times the interquartile range above or below the box; they are shown as small squares. Far-outside points lie more than 3.0 times the interquartile range above or below the box; they are shown as small squares with plus signs through them. For a comparison of models, this figure

8448


shows the following: the more similar the box-andwhisker plots of compared sets, the better the agreement between compared data. Especially for median and mean values, lower and upper quartile should be at the same position. The statistical tests for all applied models confirmed that the models are suitable for estimation of the properties being examined, because no significant difference exists between the medians obtained by the individual models. The plots in Figure 11 clearly show how the individual models are able to describe the experimental data. It is evident that our model shows the best agreement with the experimental data, in comparison with other models examined (including the Ducros, Chickos, and Ma and Zhao models, as well as eqs 6 and 7 in the work by Vetere18) for all estimated properties. The medians and means of our model lie at the same values as the medians and means of the experimental data, whereas the medians and means of other models differ from the experimental data. It is worth emphasizing that the plots only illustrate how different methods describe experimental data. The medians and means have no physical meaning, with the exception of the median of ∆SV(Tb). Its value of 88 J K-1 mol-1 is identical to the Trouton rule8 value. Plots in Figure 12 show the results for comparison of obtained relative errors RE. For all properties, the median and mean values of the relative error of our model are smaller than medians and means of the relative error obtained by other models. In these plots, the medians and means provide information about which method gives the best results. It can also be observed that eq 6 in the work by Vetere18 yields significantly diverse RE values, in comparison with our model and also in comparison with the second Vetere’s equation (eq 7). In this figure, the maximum relative errors MRE of the individual models can also be observed. The application of the developed model for estimation of the ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb) values for 1,1,4,7-tetramethylindane, octadecafluorodecahydrotrans-naphthalene, camphor, and hexane-1,6-diol is illustrated in Appendix B. 4. Conclusions A new group contribution method for estimating the enthalpy of vaporization at 298.15 K (∆HV(298.15 K)) and at the normal boiling temperature Tb (∆HV(Tb)), as well as for estimating the entropy of vaporization at the normal boiling temperaturee (∆SV(Tb)) was developed, compared with some other group contribution and empirical methods, and analyzed in terms of predictive capability, consistency, and extrapolation features. The new models seem to perform better than other similar models. The group parameter tables for the new models will permit an easy implementation, as well as a fast evaluation, of the enthalpy of vaporization and entropy of vaporization. Because the enthalpy of vaporization is an important parameter for many property correlations, a reliable estimated value for it, especially when it is difficult to measure, contributes to the fast and accurate analysis of process-product alternatives. Current and future work involves the development of a connectivity-index-based method for the generation of the unavailable group contribution, using the same database that was used for the development of the new

models. The addition of this feature will significantly increase the model application range without the need for too much additional experimental data. It was demonstrated that the entropy of vaporization at the normal boiling temperature adheres more closely to the additivity principle than the enthalpy of vaporization. Therefore, group contribution parameters for the entropy of vaporization should be used preferably also for estimating the enthalpy of vaporization if a reliable value for the normal boiling temperature is available or can be estimated. A ratio of the estimated enthalpy of vaporization and the estimated entropy of vaporization at the normal boiling temperature can serve as a qualitative estimate of the normal boiling temperature. Thirty-nine new groups from the second-level family have been proposed and included into the model for the estimation of the previously mentioned properties. Because of this extension, the model is now able to apply these new second-level groups for description of more compounds. The original model was capable of describing 387 compounds of the input database with the second-level groups, whereas our model is able to encompass 486 compounds with these second-level groups. It results in improving and refining all predictions. This model is now also able to apply 15 third-level groups for the estimation of 55 compounds for the 298.15 K temperature and 14 thirdlevel groups for prediction of 23 substances at Tb, in comparison with the earlier work, where no third-level groups were applied to estimate the enthalpy of vaporization. Appendix A This appendix gives a detailed summary of the estimation methods used for a variety of families of compounds. Appendix B This appendix gives examples of the use of our model for estimating the enthalpy of vaporization at 298.15 K, as well as for estimating the enthalpy of vaporization and the entropy of vaporization at the normal boiling temperature, and it gives their comparison with experimental data and with values obtained using other methods. The first example involves estimations for 1,1,4,7tetramethylindane, C13H18 (CAS No. 1078-04-2).

A comparison of the ∆HV(298.15 K) values from Table B1 gave the following: ∆HV(298.15 K)OM ) 60.35 kJ/ mol and RE ) 1.7%; ∆HV(298.15 K)D ) 62.30 kJ/mol and RE ) 1.5%; and ∆HV(298.15 K)CH ) 60.58 kJ/mol and RE ) 1.3%.

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8449 Table A1. Overview of Average Relative Errors of the Individual Estimation Methods for Various Families of Organic Compoundsa D family of organic compounds aliphatic saturated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) aliphatic unsaturated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) cyclic saturated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) cyclic unsaturated and aromatic hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) hydrocarbons (total) ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) fluorinated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) chlorinated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) brominated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) iodinated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous halogenated hydrocarbons ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) halogenated hydrocarbons (total) ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) ethers ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) alcohols and phenols ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) aldehydes and ketones ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) acids ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) esters ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) heterocyclic O-compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous O-compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb)

CH

OM

VET6

VET7

M&Z

NC

ARE

NC

ARE

NC

ARE

NC

ARE

NC

ARE

NC

ARE

62

2.1

59

3.4

59 59 59

1.5 2.3 0.8

59 59

5.8 5.8

59 59

1.7 1.7

59 59

2.9 2.3

62 17 17

1.9 3.2 1.2

17 17

4.1 4.1

17 17

1.8 1.8

17 17

3.7 1.4

48 27 27

2.3 1.9 1.2

27 27

7.5 7.5

27 27

1.7 1.7

25 25

3.2 1.1

67 19 19

2.2 1.7 1.6

19 19

6.9 6.9

19 19

2.1 2.1

18 18

2.8 2.5

236 122 122

2.0 2.2 1.1

122 122

6.1 6.1

122 122

1.8 1.8

119 119

1.9 2.0

19 20 20

1.2 2.4 1.0

20 20

5.1 5.1

20 20

1.7 1.7

20 20

3.4 1.8

38 31 31

2.4 2.3 1.5

31 31

1.9 1.9

31 31

2.1 2.1

31 31

3.1 1.4

20 13 13

1.7 3.5 0.8

13 13

1.8 1.8

13 13

1.0 1.0

12 12

3.2 0.8

14 8 8

1.1 0.8 0.3

8 8

1.1 1.1

8 8

2.0 2.0

7 7

3.1 0.2

12 16 16

3.4 5.3 1.9

16 16

5.8 5.8

16 16

1.9 1.9

15 15

3.4 2.1

103 88 88

2.0 2.9 1.3

88 88

3.3 3.3

88 88

1.8 1.8

85 85

1.4 1.4

81 66 66

2.3 1.7 1.7

66 66

2.2 2.2

66 66

2.2 2.2

66 66

3.1 2.1

38 44 44

2.1 4.1 4.4

44 44

8.7 8.7

44 44

7.7 7.7

44 44

2.2 5.6

44 33 33

2.3 3.3 2.3

33 33

4.0 4.0

33 33

3.9 3.9

29 29

2.8 3.1

4 4 4

8.7 5.6 5.6

4 4

9.5 9.5

4 4

9.5 9.5

4 4

2.6 9.2

55 48 48

1.7 3.3 3.0

48 49

3.9 3.9

48 49

3.6 3.6

48 48

2.7 3.1

13 14 14

3.2 2.1 1.5

14 14

4.3 4.3

14 14

4.6 4.6

10 10

3.4 0.9

27 16 16

2.4 5.3 5.1

16 16

8.8 8.8

16 16

9.2 9.2

16 16

2.3 5.0

59

34

48

203

0

27

20

0

0

47

76

28

32

2

37

9

11

2.5

2.0

3.0

2.4

n.a.

4.9

3.4

n.a.

n.a.

4.3

2.7

1.5

2.6

44.4

3.3

10.8

12.3

62

48

67

236

19

38

20

14

12

103

81

38

44

4

55

13

19

3.4

4.1

4.6

3.9

8.0

3.2

2.9

2.6

4.2

4.0

3.3

4.5

4.9

26.4

5.2

7.5

8.6

8450


Table A1 (Continued) D family of organic compounds compounds with oxygen atom (total) ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) amines ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) nitriles ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) heterocyclic N-compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous N-compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) compounds with nitrogen atom (total) ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) sulfides ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) thiols ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) heterocyclic S-compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) compounds with sulfur atom (total) ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with halogen and oxygen ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with oxygen and nitrogen ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with oxygen and sulfur ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with nitrogen and sulfur ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with nitrogen and halogen ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) miscellaneous compounds with silicon or oxygen and silicon ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb) total for all compounds ∆HV(298.15 K) ∆HV(Tb) ∆SV(Tb)

a

CH

OM

VET6

VET7

M&Z

NC

ARE

NC

ARE

NC

ARE

NC

ARE

NC

ARE

NC

ARE

195

3.9

254

5.1

262 225 225

2.3 3.1 2.9

225 225

4.8 4.8

225 225

4.6 4.6

217 217

3.5 3.5

40 32 32

2.9 1.9 1.4

32 32

2.6 2.6

32 32

2.9 2.9

31 31

3.0 1.7

28 20 20

2.3 2.5 1.5

20 20

5.6 5.6

20 20

5.3 5.3

17 17

2.9 1.8

23 18 18

1.6 0.7 0.8

18 18

2.2 2.2

18 18

2.3 2.3

3 3

2.8 2.9

13 3 3

2.5 3.3 1.3

3 3

6.0 6.0

3 3

6.3 6.3

3 3

2.9 1.2

104 73 73

2.4 1.9 1.3

73 73

3.5 3.5

73 73

3.5 3.5

69 69

3.2 3.2

26 17 17

2.2 1.7 0.8

17 17

1.5 1.5

17 17

1.3 1.3

17 17

3.0 1.1

17 14 14

2.1 1.1 0.6

14 14

1.5 1.5

14 14

1.3 1.3

13 13

3.2 0.9

9 9 9

2.0 0.8 0.6

9 9

1.7 1.7

9 9

1.3 1.3

9 9

2.9 2.0

52 40 40

2.2 1.3 0.7

40 40

1.6 1.6

40 40

1.3 1.3

39 39

1.2 1.2

23 17 17

2.6 4.5 3.5

17 17

7.6 7.6

17 17

5.6 5.6

17 17

3.0 3.2

34 11 11

2.9 0.9 1.1

11 11

8.3 8.3

11 11

8.7 8.7

11 11

2.4 5.4

5 5 5

0.6 0.1 0.1

5 5

3.3 3.3

5 5

3.9 3.9

4 4

2.8 1.2

1 1 1

2.5 2.1 2.4

1 1

3.5 3.5

1 1

4.0 4.0

1 1

2.8 4.9

0 2 2

n.a. 7.8 3.8

2 2

4.3 4.3

2 2

1.4 1.4

2 2

4.1 2.5

11 5 5

3.1 1.6 0.4

5 5

6.1 6.1

5 5

6.0 6.0

0 0

n.a. n.a.

831 589 589

2.2 2.6 1.8

589 589

4.6 4.6

589 589

3.4 3.4

549 549

2.5 2.5

26

14

0

0

40

24

17

0

41

0

0

0

0

0

0

526

2.9

1.2

n.a.

n.a.

2.3

1.2

2.5

n.a.

1.8

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

3.1

40

28

23

1

92

26

17

9

52

23

26

5

0

0

9

800

6.8

3.6

6.6

9.1

5.8

3.7

3.7

3.8

3.7

3.5

6.6

2.3

n.a.

n.a.

18.4

4.7

The term “n.a.” denotes that the method is not applicable for this family of compounds.

Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8451 Table B1. Example 1: Estimation for 1,1,4,7-Tetramethylindane ∆HV(298.15 K) NG

contribution

∆HV(Tb) NG

∆SV(Tb)

contribution

NG

contribution

2.312 2.680 3.223 4.470 2.416 3.435 14.876 48.51 kJ/mol

2 2 2 2 2 1 1

-0.389 0.259 1.592 -0.483 -0.011 3.730 83.861 89.53 J K-1 mol-1

-0.351 47.81 kJ/mol

2

0.867 91.27 J K-1 mol-1

1.046 * 49.90 kJ/mol

2 1

1.034 * 93.33 J K-1 mol-1

First-Level Estimation first-level group number 1 13 15 18 107 109 x0 Xest Xexp RE (%)

2 2 2 2 2 1 1

2.266 4.297 6.190 8.121 4.013 3.667 9.672 63.11 kJ/mol 61.37 kJ/mol 2.8

2 2 2 2 2 1 1

Second-Level Estimation second-level group number 55 Xest Xexp RE (%)

2

-1.355 60.40 kJ/mol 61.37 kJ/mol 1.6

2

Third-Level Estimation third-level group number 6 19 Xest Xexp RE (%)

2 1

0.279 -0.615 60.346 kJ/mol 61.37 kJ/mol 1.7

2 1

Table B2. Example 2: Estimation for (octadecafluorodecahydro-)trans-Naphthalene ∆HV(298.15 K) NG

contribution

∆HV(Tb) NG

∆SV(Tb)

contribution

NG

contribution

-0.735 3.435 14.876 36.00 kJ/mol 35.88 kJ/mol 0.3

18 10 1

-1.882 3.730 83.861 87.29 J K-1 mol-1 86.50 J K-1 mol-1 0.9

0.203 36.20 kJ/mol 35.88 kJ/mol 0.9

1

0.116 87.40 J K-1 mol-1 86.50 J K-1 mol-1 1.0

-0.220 35.76 kJ/mol 35.88 kJ/mol 0.3

2

-0.138 87.13 J K-1 mol-1 86.50 J K-1 mol-1 0.7

First-Level Estimation first-level group number 92 109 x0 Xest Xexp RE (%)

18 10 1

-0.076 3.667 9.672 44.97 kJ/mol 45.40 kJ/mol 0.9

18 10 1


1

0.033 45.01 kJ/mol 45.40 kJ/mol 0.9

1

Third-Level Estimation third-level group number 9 Xest Xexp RE (%)

2

0.027 45.06 kJ/mol 45.40 kJ/mol 0.8

The second example involves estimations for transnaphthalene (octadecafluorodecahydro-, C10F18) (CAS No. 60433-12-7).

2

A comparison of the ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb) values from Table B2 gave the following: ∆HV(298.15 K)OM ) 45.06 kJ/mol and RE ) 0.8%; ∆HV(298.15 K)D, cannot be used, because of unavailability of groups for this compound; ∆HV(298.15 K)CH ) 54.50 kJ/mol and RE ) 20.0%; ∆HV(Tb)OM ) 35.76 kJ/mol and RE ) 0.3%; ∆HV(Tb)VET6 ) 34.18 kJ/mol and RE ) 4.7%; ∆HV(Tb)VET7 ) 35.92 kJ/mol and RE ) 0.1%; ∆HV(Tb) M&Z ) 31.93 kJ/mol and RE ) 11.0%; ∆SV(Tb)OM ) 87.125 J K-1 mol-1 and RE ) 0.7%; ∆SV(Tb)VET6 ) 82.39 J K-1 mol-1 and RE ) 4.7%; ∆SV(Tb)VET7 ) 86.60 J K-1 mol-1 and RE ) 0.1%; and ∆SV(Tb)M&Z ) 76.99 J K-1 mol-1 and RE ) 11.0%.

8452


Table B3. Example 3: Estimation for Camphor ∆HV(298.15 K) NG

contribution

∆HV(Tb) NG

∆SV(Tb)

contribution

NG

contribution

2.312 2.416 2.263 3.435 19.025 14.876 57.22 kJ/mol 59.50 kJ/mol 3.8

3 3 1 2 1 1

-0.389 -0.011 2.263 3.435 20.094 83.861 111.89 J K-1 mol-1 123.37 J K-1 mol-1 9.3

-0.351 56.52 kJ/mol 59.50 kJ/mol 5.0

2

0.867 113.62 J K-1 mol-1 123.37 J K-1 mol-1 7.9

0.473 -0.220 56.77 kJ/mol 59.50 kJ/mol 4.6

1 1

0.566 -0.138 114.05 J K-1 mol-1 123.37 J K-1 mol-1 7.6

First-Level Estimation first-level group number 1 107 108 109 117 x0 Xest Xexp RE (%)

3 3 1 2 1 1

2.266 4.013 4.075 3.667 14.837 9.672 54.76 kJ/mol

3 3 1 2 1 1


2

-1.355 52.05 kJ/mol

2

Third-Level Estimation third-level group number 8 9 Xest Xexp RE (%)

1 1

-0.191 0.270 52.12 kJ/mol

1 1

Table B4. Example 4: Estimation for Hexane-1,6-diol ∆HV(298.15 K) NG

∆HV(Tb)

contribution

NG

∆SV(Tb)

contribution

NG

contribution

2.362 16.881 14.876 62.81 kJ/mol 66.18 kJ/mol 5.1

6 2 1

0.657 18.408 83.861 124.62 J K-1 mol-1 126.48 J K-1 mol-1 1.5

1.933 64.74 kJ/mol 66.18 kJ/mol 2.2

1

-2.653 121.97 J K-1 mol-1 126.48 J K-1 mol-1 3.6

First-Level Estimation first-level group number 2 23 x0 Xest Xexp RE (%)

6 2 1

4.724 24.930 9.672 87.88 kJ/mol 90.99 kJ/mol 3.4

6 2 1

Second-Level Estimationa Third-Level Estimation third-level group number 1 Xest Xexp RE (%) a

1

-0.128 87.75 kJ/mol 90.99 kJ/mol 3.6

1

There was no occurrence of any second-level group.

The third example involves estimations for camphor (bicyclo-2,2,1-heptan-2-one, 1,7,7-trimethyl-, C10H16O) (CAS No. 76-22-2).

A comparison of the ∆HV(Tb) and ∆SV(Tb) values from Table B3 gave the following: ∆HV(Tb)OM ) 56.77 kJ/ mol and RE ) 4.6%; ∆HV(Tb)VET6 ) 43.19 kJ/mol and RE ) 27.4%; ∆HV(Tb)VET7 ) 42.85 kJ/mol and RE ) 28.0%; ∆SV(Tb)OM ) 114.05 J K-1 mol-1 and RE ) 7.6%; ∆SV(Tb)VET6 ) 89.56 J K-1 mol-1 and RE ) 27.4%; and ∆SV(Tb)VET7 ) 88.84 J K-1 mol-1 and RE ) 28.0%.

The fourth example involves estimations for hexane1,6-diol, C6H14O2 (CAS No. 629-11-8).

A comparison of the ∆HV(298.15 K), ∆HV(Tb), and ∆SV(Tb) values from Table B4 gave the following: ∆HV(298.15 K)OM ) 87.75 kJ/mol and RE ) 3.6%; ∆HV(298.15 K)D: 87.36 kJ/mol and RE ) 4.0%; ∆HV(298.15 K)CH ) 90.94 kJ/mol and RE ) 0.5%; ∆HV(Tb)OM ) 64.72 kJ/mol and RE ) 2.2%; ∆HV(Tb)VET6 ) 43.96 kJ/mol and RE ) 33.6%; ∆HV(Tb)VET7 ) 64.03 kJ/mol and RE ) 3.2%; ∆HV(Tb)M&Z ) 67.57 kJ/mol and RE ) 1.5%; ∆SV(Tb)OM ) 121.97 J K-1 mol-1 and RE ) 3.6%; ∆SV(Tb)VET6 ) 84.02 J K-1 mol-1 and RE ) 33.6%; ∆SV(Tb)VET7 ) 122.37 J K-1 mol-1 and RE ) 3.2%; and ∆SV(Tb) M&Z ) 129.14 J K-1 mol-1 and RE ) 2.1%.


Acknowledgment This work was conducted within the framework of Grant Nos. CB MSM 223400008 and MSM 6046137307 and Internal Grant No. 320911031701 of Faculty of Education of J. E. Purkynje University in U Ä stı´ nad Labem (Czech Republic). We are thankful to the following individuals: Prof. Ju¨rgen Gmehling (University Oldenburg and DDBST GmbH Dortmund Data Bank), for providing an extensive database of experimental data on enthalpy of vaporization; Prof. Jirˇ´ı Cihla´rˇ (Department of Mathematics of J. E. Purkynje University U Ä stı´ nad Labem), for his advice in performing statistical analyses; and Jorge Marrero-Morejon (CAPEC, Department of Chemical Engineering of Technical University of Denmark), for his cooperation. Nomenclature ∆HV ) enthalpy of vaporization (kJ/mol) ∆SV ) entropy of vaporization (J K-1 mol-1) X ) ∆HV or ∆SV ∆HV(298.15 K) ) enthalpy of vaporization at 298.15 K (kJ/ mol) ∆HV(Tb) ) enthalpy of vaporization at the normal boiling temperature (kJ/mol) ∆SV(Tb) ) entropy of vaporization at the normal boiling temperature (J K-1 mol-1) x0 ) general expression of additional adjustable parameter in eqs 1-3 h0, hb, sb ) adjustable additional parameters in eqs 1-3 Ch0,i, Chb,i, Csb,i ) first-level group contribution of type i in eqs 1-3 Dh0,j, Dhb,j, Dsb,j ) second-level group contribution of type j in eqs 1-3 Eh0,k, Ehb,k, Esb,k ) third-level group contributions of type k in eqs 1-3 Nh0,i, Nhb,i, Nsb,i ) number of occurrences of the first-level groups in eqs 1-3 Mh0,j, Mhb,j, Msb,j ) number of occurrences of the secondlevel groups in eqs 1-3 Oh0,k, Ohb,k, Osb,k ) number of occurrences of the third-level groups in eqs 1-3 ω, z ) parameters in eqs 1-3 νi ) number of groups of type i in the molecule for the methods by Ducros and Ma and Zhao Tb ) normal boiling temperature (K) Tc ) critical temperature (K) NC ) number of compounds NG ) number of group contributions AE ) absolute error (kJ/mol for ∆HV; J K-1 mol-1 for ∆SV) RE ) relative error (%) AAE ) average absolute error (kJ/mol for ∆HV; J K-1 mol-1 for ∆SV) ARE )average relative error (%) MRE )maximum relative error (%) Subscripts exp ) experimental datum est ) estimated value 1i ) first-level groups in eqs 1 and 2 and in Table 3 2j ) second-level groups in eqs 1 and 2 and in Table 4 3k ) third-level groups in eqs 1 and 2 and in Table 5 Indexes in Appendix A and in Appendix B OM ) value obtained by our model D ) value obtained by the Ducros method CH ) value obtained by the Chickos method

VET6, VET7 ) value obtained by the Vetere equations (eqs 6 or 7, respectively, in his referenced work) M&Z ) value obtained by the Ma and Zhao method

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Received for review January 28, 2005 Revised manuscript received July 28, 2005 Accepted August 15, 2005 IE050113X

Estimation of the Enthalpy of Vaporization and the Entropy of

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