cp organics

Estimation of the Heat Capacity of Organic Liquids as a Function ofTemperature by a Three-Level Group Contribution Method

Zdenka Kolska

Department of Chemistry, Faculty of Science, J. E. Purkinje UniVersity, CeskeMladeze 8,400 96 UÄ stı nad Labem, Czech Republic

Jaromır Kukal

Department of Computing and Control Engineering, Institute of Chemical Technology Prague, Technicka´ 5,166 28 Prague 6, Czech Republic

Milan Za bransky and Vlastimil Ru zicka*

Department of Physical Chemistry, Institute of Chemical Technology Prague, Technicka´ 5,166 28 Prague 6, Czech Republic

A new group contribution method for estimating the heat capacity of pure organic liquids, as a function oftemperature in the range from the melting temperature to the normal boiling temperature, was developed. Alarge set of critically assessed data for 549 compounds was used for group contributions calculation. Valuesobtained using the method developed here were compared with estimations that were determined using theZabranskyand Ruzicka and Chickos group contribution methods. A statistical analysis of the regressed datawas also performed, which indicated the confidence of the regressed parameters and other related information.The average relative errors for the new method are as follows: 1.2% for 549 compounds from the basic setcovering data over the temperature range indicated above, 1.5% for 404 data points at 298.15 K for compoundsfrom the basic dataset, and 2.5% for 149 compounds from an independent test set with data obtained at298.15 K.

Introduction

Isobaric heat capacity of liquid (Cpl ) is an important ther-

modynamic quantity of a pure compound. Its value must beknown for the calculation of an enthalpy difference, which isrequired for the evaluation of heating and cooling duties. Liquidheat capacity also serves as an input parameter, for example, inthe calculation of the temperature dependence of the enthalpyof vaporization, for extrapolation of the vapor pressure and therelated thermal data via their simultaneous correlation, etc. Heatcapacity is one of the directly measured properties. Calori-metrically determined experimental data are known for some2000 organic and inorganic compounds.1,2 A substantial portionof them were determined at only one temperature, mostly at298.15 K, or in a narrow temperature range.

A comparison of available experimental data on liquid heatcapacities and of industrially important compounds clearlyreveals a need for an estimation method to supplement themissing data. As in most engineering applications, the heatcapacity of a liquid is required to calculate the enthalpydifference; an analytical function is desirable to represent thetemperature dependence of the heat capacity. Several estimationmethods were presented in the literature,3-13 some which arebased on a group contribution approach.14-30

In this work, we applied the three-level group contributionmethod that has been reported by Marrero and Gani31 andextended it to the estimation of heat capacity of liquids as afunction of temperature. This is a useful development insofaras the Marrero and Gani method allows the estimation of a pure

compound physical chemical property that is either temperature-independent or required to be measured at a specified temper-ature, primarily 298.15 K.32-35

New Group Contribution Method

Methodology Description. The new group contributionmethod for the estimation of heat capacity of organic liquids isbased on the methodology developed by Constantinou andGani32 and Marrero and Gani.31 The estimation of propertiesinvolves a three-level calculation procedure that covers groupsof the first level for the estimation of simple monofunctionalmolecules, and groups of the second and third levels forimproving the more-complex prediction of compounds.31,35

New Model. We extended the Gani et al. approach forestimating the heat capacity of liquids as a function oftemperature,CP

l (T). The temperature dependence of the groupcontributions was expressed through an empirical polynomialequation.1

The new model for the estimation ofCPl (T) has the form of

the following equations:

with

* To whom all correspondence should be addressed. Fax:+420220445018. E-mail: [email protected].

Cpl (T) ) Cp0

l (T) + ∑I

NiCp1-il (T) + w ∑

j

MjCp2-jl (T) +

z∑k

OkCp3-kl (T) (1)

C p qthlevel-i ,j, or kl (T) ) aq-i, j, or k + bq-i, j, or k( T

100) +

dq-i, j, or k( T100)2

(2)

2075Ind. Eng. Chem. Res.2008,47, 2075-2085

10.1021/ie071228z CCC: $40.75 © 2008 American Chemical SocietyPublished on Web 02/20/2008

In eq 1,Cp1-il (T) is the contribution of the first-level group of

type i, Cp2-j(T) is the contribution of the second-level group oftype j, andCp3-k

l (T) is the contribution of the third-level groupof type k. Ni, Mj, andOk denote the number of occurrences ofthe individual groups (of typei, j, or k, respectively) in acompound.Cp0

l (T) (which could be considered as the contri-bution of the zero-level group) is an additional adjustableparameter. Variablesw and z are weighting factors that areassigned to 0 or 1, depending on whether the second-level andthird-level contributions, respectively, are used or not. In eq 2,aq-i, j, or k, bq-i, j, or k, anddq-i, j, or k are adjustable parameters forthe temperature dependence ofCp0(T), Cp1-i

l (T), Cp2-jl (T), and

Cp3-kl (T).All compounds were described by the same set of groups as

those defined in the paper by Kolska´ et al.35

Database.A large database of critically assessed data onliquid heat capacity1,2 (hereafter referenced as the basic set) wasused for all group contribution parameter calculations. The pureorganic compounds included in the database range in the molarmass 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,and S atoms. A list of names of individual families is given inAppendix A. The basic set used in this work was identical tothat used by Za´branskyand Ruzicka.29

Procedure for Estimation of Model Parameters. Theparameters to be fitted were the following:aq-i, j, or k, bq-i, j, or k,and dq-i, j, or k. For the parameter estimation, we adopted twoapproaches. First, in the so-called hierarchic approach (denotedas new model-H) the parameter calculation was performed asdescribed earlier31,35 in three consecutive steps. Second, in thenonhierarchic approach (denoted as new model-NH), allparameters were calculated in a single step.

Mathematical Background. Determination of adjustableparameters of any group contribution method results in asolution of an overdetermined system of linear equationsTt ) u, whereT ∈Rm×n, t ∈Rn, andu ∈Rm; m is the numberof experimental input data andn is the total number of groupcontributions (n ) n1 + n2 + n3, wheren1, n2, andn3 are thenumbers of contributions in the first, second, and thirdlevels). The traditional least-squares method is based on theminimization of the function SSQ(t) ) |Tt - u|2. The explicitoptimum solution,t* ) (T′T)-1T′u, is not numerically stablefor a large numbern and for ill-posed tasks. The numericaldifficulties can be decreased using the hierarchic approach,which will be demonstrated for three levels of optimization usedin this work. The contribution vector can be decomposed ast) (p,q,s), wherep ∈Rn1, q ∈Rn2 and s ∈Rn3. The adequatedecomposition of the system matrix comes toT ) (P|Q|S) andthe original set of equations can be written asPp + Qq + Ss) u. The three-level hierarchic approach of the least-squaresoptimization is based on the step-by-step minimization of thefunction SSQ*(p,q,s) ) |Pp + Qq + Ss- u|2. Let t* be theoptimum solution of the minimization task SSQ(t) ) min. Then,SSQ(t*) e SSQ*(p*,q*,s*); this inequality follows from thedefinition of the global optimumt* in the theoretical case(“theoretical case” means that the truncation error is absent).However, in a real case, the role of the truncation error andthat of the ill-posed linear system can cause a turnover ofthe inequality SSQ(t*) e SSQ*(p*,q*,s*), when the hierarchicsolution is numerically better than the global one. The globalsolution with a suppressed truncation error was preferred inthis work and is represented by the nonhierarchic approach. Ourprocedure was based on the regularization of the least-

squares task with the adequate parameterλ > 0. This approachis called the Tikhonov regularization36 or the Levenberg-Marquardt method,37 respectively. The corresponding formulais t+ ) (T′T + λIn)-1T′u. The objective of the regularizationis to obtain SSQ(t+) e SSQ*(p+,q+,s+) < min (SSQ(t*),SSQ*-(p*,q*,s*)). The recommended value of the parameterλ is 10-6.In the calculation of the contribution vectort ) (p,q,s) in thiswork, we used the relative error minimization, so the originalsystem was modified toTrel ) T diag-1(u) and urel ) 1m inthis case.

The group contribution parameters are given in Table 1. Inthis table, we present parameters that have been determined bythe nonhierarchic approach, as well as by the hierarchicapproach. Although it turned out that the nonhierarchic approachis slightly superior (see the next part), for compatibility withcommonly used software packages (e.g., the CAPEC programpackage38-41), we present both sets of parameters.

The statistical significance of group-contribution values wasevaluated using the average relative error (ARE), which is givenby eq 3, and using the median of relative errors (MED):

wheren is the number of data in the database and the suffixes“exp” and “est” denote the experimental and estimated values,respectively. The median is a number that is obtained byseparating the higher half of a sample from the lower half. Themedian of the relative error (RE) values is found by arrangingall of them from the lowest value to the highest value andselecting the one in the middle.

Table 2 gives the ARE and MED values forCpl estimation

over a temperature range. The correlation statistics are shownfor the first level, the second level, and the third level for thehierarchic approach and for a single level for the nonhierarchic(global) approach. The results in Table 2 clearly demonstratethat the nonhierarchic approach is slightly superior, in terms ofthe ARE and MED values for the entire basic dataset. Moredetailed results are shown for individual molecular types inAppendix A.

Figure 1 provides a visualization of the correlation ofexperimental data onCp

l at 298.15 K for compounds from thebasic dataset determined by our model. Note that, except for afew data points, all of the data have been fitted to a high degreeof accuracy over a wide range ofCp

l values, from∼80 J K-1

mol-1 up to ∼2000 J K-1 mol-1.

Performance Analysis and Model Application

The performance of the developed model for estimation ofthe liquid heat capacity was analyzed in terms of comparisonwith other estimation models, as well as in terms of extrapolationfeatures, predictive capability, and consistency.

Comparison with Other Models. Two methods, which wereformulated by Za´branskyand Ruzicka29 and Chickos et al.,24,28

were selected for a comparison of theCpl estimation results.

The former method is representative of a classical Benson’stype second-order group-contribution method that is capable ofestimating the liquid heat capacity over the temperature rangefrom the melting temperature to the normal boiling temperature.The latter method is also based on the Benson’s group ad-ditivity approach of the first-order type, but it is capable ofestimatingCp

l at only one temperature: 298.15 K. The Chickos

ARE[Cpl ] )

1

n∑i)1

n |Cp,expl - Cp,est

l |iCp,exp,i

l× 100 (3)

2076 Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008

Table 1. List of Group Contributions and Their Values for Liquid Heat Capacity ( Cpl , Expressed in Terms of J K-1 mol-1), Determined by the

Nonhierarchic (NH) and Hierarchic (H) Approachesa

New Model, Nonhierarchic-NH New Model, Hierarchic-H

No. group name a b d No.a a b d

0 additional adjustable parameter 105.94 -51.40 7.24 0 89.53 -40.13 5.19

First-Level Group Contributions1 CH3 -10.75 17.70 -1.15 1 1.04 8.29 0.762 CH2 16.19 3.21 0.41 2 19.64 0.78 0.803 CH 50.97 -19.12 3.71 3 12.20 7.90 -1.034 C 53.24 -26.31 4.51 4 25.40 -2.25 0.165 CH2dCH -14.86 47.46 -8.04 5 15.13 9.02 0.996 CHdCH 18.12 40.76 -9.36 6 57.81 -21.62 6.387 CH2dC -13.26 64.86 -13.50 7 35.28 -2.15 2.178 CHdC -0.58 80.98 -21.01 8 46.90 0.39 -1.019 CdC -150.51 186.33 -37.28 9 22.45 33.10 -10.93

10 CH2CH2dCdCH 34.08 9.23 1.39 10 37.77 7.91 1.4911 CH2CH2dCdC 50.34 3.06 1.38 11 43.17 10.62 -0.3912 CHtC 25.94 -0.53 3.95 12 27.11 0.04 3.7013 CtC 4.16 28.24 -4.93 13 -3.00 35.78 -6.7014 aCH -1.28 8.17 -0.43 14 -2.09 8.58 -0.4715 aC fused with an aromatic ring 384.23 -151.04 15.62 15 -4.91 8.60 -0.9616 aC fused with a nonaromatic subring 539.17 -291.04 41.64 16 -3.73 10.46 -1.5117 aC, except as above 474.70 -258.65 37.43 17 -9.24 12.84 -1.5918 aN in an aromatic ring 10.79 5.74 -0.98 18 19.83 -0.12 0.0519 aC-CH3 20.23 9.63 -0.43 19 14.90 11.61 -0.3320 aC-CH2 49.18 -6.09 1.15 20 37.15 4.17 -0.6121 aC-CH 96.82 -30.00 3.37 21 97.94 -37.93 5.6622 aC-C 52.70 -15.10 1.17 22 88.73 -28.81 2.6023 aC-CHdCH2 -0.29 38.36 -5.08 23 20.15 25.04 -2.8624 aC-CHdCH -134.98 110.23 -17.46 24 237.57 -66.05 6.4525 aC-CdCH2 -62.49 82.00 -13.56 25 -30.89 44.81 -5.2026 aC-CtCH 58.32 -2.92 1.88 26 78.77 -16.25 4.1027 OH 13.08 -17.97 9.02 27 -99.86 71.89 -7.7028 aC-OH 178.51 -34.86 1.98 28 163.82 -29.49 1.8329 COOH 12.58 20.70 -0.21 29 -2.21 30.79 -1.9830 aC-COOH -738.97 370.11 -39.06 30 -739.60 363.53 -37.2231 CH3CO 127.88 -47.29 10.19 31 69.38 -2.63 1.9132 CH2CO 65.01 15.15 -3.53 32 52.07 26.42 -5.8133 CHCO 67.08 10.77 -3.23 33 79.70 -2.77 -0.9734 CCO -23.93 32.22 -4.45 34 -40.63 68.57 -11.0735 aC-CO 103.77 -17.30 1.21 35 112.43 -21.22 1.5336 CHO 31.10 11.01 -2.20 36 32.92 10.22 -1.9637 aC-CHO 26.21 18.40 -2.50 37 46.66 5.07 -0.2838 CH3COO 58.31 -4.75 5.65 38 34.25 35.65 -5.1739 CH2COO 91.18 23.69 -9.10 39 74.76 13.70 -2.9940 CHCOO 7.84 20.47 -2.85 40 -1.46 52.01 -7.9141 HCOO -40.23 40.10 -5.38 41 -52.52 58.54 -6.9242 aC-COO 87.72 -6.65 0.79 42 72.73 3.36 -0.8743 COO, except as above -3.40 43.61 -7.28 43 12.98 38.74 -7.6844 CH3O 7.24 25.20 -2.53 44 21.08 14.60 -0.7145 CH2O 42.67 9.68 -1.36 45 48.82 4.92 -0.4046 CH-O 54.36 3.90 -1.66 46 66.98 -9.64 0.6047 aC-O 40.33 23.30 -5.76 47 116.47 -23.43 1.3248 CH2NH2 48.21 27.05 -4.95 48 83.30 3.17 -1.2449 CHNH2 34.31 39.40 -7.34 49 61.60 15.06 -2.8350 CNH2 66.17 11.97 -2.34 50 220.92 -94.89 15.4851 CH3NH 14.59 44.77 -8.21 51 84.91 -12.05 3.1152 CH2NH 221.83 -45.82 0.63 52 209.44 -69.77 8.7453 CH3N -46.47 75.91 -13.72 53 -5.20 42.02 -7.0854 CH2N 78.76 8.23 -5.69 54 -391.63 253.86 -37.1355 aC-NH2 133.67 -25.76 2.28 55 124.73 -23.07 2.4356 aC-NH 90.55 -16.34 2.06 56 156.25 -58.04 8.3757 aC-N 41.79 8.14 -2.78 57 38.67 13.64 -4.3858 NH2, except as above 51.85 -3.48 0.44 58 41.12 6.56 -1.1659 CH2CN 36.93 7.45 1.85 59 21.25 25.29 -2.7960 CCN 47.87 11.05 -1.77 60 24.68 36.07 -7.4661 aC-CN 21.97 20.75 -3.50 61 36.89 10.70 -1.7562 CN, except as above 9.56 -3.63 2.19 62 5.54 19.92 -2.5963 aC-NCO 28.13 24.91 -3.35 63 44.07 14.42 -1.5364 CH2NO2 88.87 -13.06 4.00 64 93.49 -14.92 4.1465 aC-NO2 380.42 -171.77 23.67 65 380.97 -174.45 24.4766 ONO2 -99.91 132.09 -23.00 66 -103.93 135.83 -23.9067 CH2Cl 10.00 21.75 -2.18 67 17.62 17.23 -1.4768 CHCl 93.83 -23.16 3.38 68 87.64 -18.64 2.5569 CHCl2 36.39 18.82 -1.98 69 31.35 24.59 -3.3070 CCl2 43.52 24.65 -4.95 70 36.42 32.16 -6.7171 CCl3 57.68 16.51 -1.30 71 49.56 23.46 -2.67

Ind. Eng. Chem. Res., Vol. 47, No. 6, 20082077

Table 1. (Continued)



72 CH2F 6.30 12.33 1.27 72 48.23 -18.42 6.9473 CHF2 8.74 25.00 -2.39 73 33.50 3.27 2.3674 CF2 12.93 19.38 -2.34 74 35.62 -0.02 1.6875 CF3 -29.59 55.77 -6.90 75 -23.33 47.21 -4.6976 CCl2F 35.62 22.90 -2.58 76 41.45 15.75 -0.7777 HCClF 14.09 22.86 -1.30 77 31.20 12.21 0.3278 CClF2 -10.02 45.46 -5.93 78 32.33 7.80 2.2079 aC-Cl 18.32 11.79 -1.69 79 36.94 -2.11 0.9280 aC-F 8.20 14.34 -2.09 80 12.20 8.78 -0.5481 aC-I 15.62 25.52 -5.08 81 36.07 12.20 -2.8682 aC-Br 6.08 22.28 -3.60 82 29.22 4.63 -0.2683 -I, except as above -95.22 85.97 -14.69 83 -87.17 82.44 -14.3884 -Br, except as above -21.39 32.75 -5.29 84 -10.87 24.80 -3.7385 -F, except as above -11.58 12.82 -1.08 85 -12.79 12.63 -0.8386 -Cl, except as above 39.99 -5.52 -0.14 86 23.19 -15.79 6.0287 OCH2CH2OH -5.75 59.46 -4.83 87 -36.22 80.39 -8.3588 CH2SH 31.55 20.87 -3.14 88 38.20 16.82 -2.5089 CHSH 54.58 10.27 -2.48 89 51.69 11.05 -2.6290 CSH 90.39 -18.41 2.69 90 57.67 9.27 -2.8991 aC-SH 40.68 15.13 -2.91 91 61.12 1.81 -0.6992 -SH, except as above -14.27 21.96 -3.68 92 43.47 -4.34 0.9693 CH3S 45.30 7.02 -0.08 93 51.47 3.49 0.4194 CH2S 53.18 8.70 -2.35 94 34.28 23.43 -4.9695 CHS 54.06 22.96 -6.19 95 66.68 9.42 -3.9496 aC-S- 99.88 -18.58 1.85 96 120.51 -27.86 2.3697 CO3 130.69 -29.16 4.45 97 126.26 -22.66 2.7898 C2H3O 49.63 -16.59 4.07 98 40.12 5.99 1.5399 CH2 (cyclic) -5.86 11.43 -0.54 99 0.60 7.28 0.12

100 CH (cyclic) 104.05 -54.34 8.72 100 22.63 -0.97 -0.13101 C (cyclic) 18.20 6.76 -2.66 101 83.25 -38.77 5.10102 CHdCH (cyclic) 18.76 0.52 2.28 102 19.00 -0.72 2.63103 CHdC (cyclic) 49.82 -27.08 5.26 103 56.86 -21.05 4.49104 CH2dC (cyclic) 43.14 -15.85 5.44 104 40.16 -14.67 5.51105 NH (cyclic) -209.01 172.27 -28.22 105 47.34 12.58 -3.89106 O (cyclic) 123.35 -82.77 16.64 106 -2.31 15.12 -2.72107 CO (cyclic) 33.03 8.24 -1.69 107 9.62 20.64 -3.18108 S (cyclic) -2.80 24.33 -3.96 108 33.15 -4.87 1.14109 -O-, except as above -30.04 24.29 -4.13 109 -130.06 108.77 -18.70110 -S-, except as above -48.86 56.79 -9.06 110 -26.06 34.45 -4.55

Second-Level Group Contributions111 (CH3)2CH 2.30 -6.45 1.55 1 19.48 -16.15 3.01112 (CH3)3C -4.29 8.08 -2.02 2 -23.06 20.55 -4.46113 CH(CH3)CH(CH3) -35.67 24.12 -3.81 3 -13.18 12.25 -2.46114 CH(CH3)C(CH3)2 -70.53 50.14 -9.22 4 -44.90 32.11 -6.66115 C(CH3)2C(CH3)2 -54.47 39.79 -6.90 5 -37.80 26.83 -5.56116 CHndCHm-CHpdCHk (k, m, n, p ) 0, ..., 2) 42.52 -64.77 16.23 6 -8.21 7.32 -1.29117 CH3-CHmdCHn (m, n ) 0, ..., 2) 22.93 -33.28 8.05 7 1.13 -0.82 0.12118 CH2-CHmdCHn (m, n ) 0, ..., 2) 22.37 -33.23 7.89 8 -3.40 2.39 -0.53119 CHp-CHmdCHn (m, n ) 0, ..., 2;p ) 0, ,.., 1) -0.34 -15.08 5.06 9 -16.74 15.08 -2.98120 CHCHO or CCHO -364.47 252.44 -42.14 10 -339.11 233.99 -39.34121 CH3COCH2 -71.15 54.13 -9.95 11 -9.11 7.88 -1.43122 CH3COCH or CH3COC -84.87 63.96 -12.01 12 -10.48 8.76 -2.21123 CHCOOH or CCOOH -127.32 62.69 -7.40 13 -107.39 47.19 -4.55124 CH3COOCH or CH3COOC -338.60 142.49 -16.75 14 -114.97 -44.39 20.91125 CHOH -393.77 284.02 -47.66 15 -124.66 78.06 -10.03126 COH -356.37 249.59 -37.58 16 -216.13 138.02 -17.23127 CHm(OH)CHn(OH) (m, n ) 0, ..., 2) -169.45 150.35 -31.31 17 94.71 -54.40 6.36128 CHm(OH)CHn(NHp) (m, n, p ) 0, ..., 2) -194.13 102.90 -13.55 18 -72.71 35.13 -4.05129 CHm(NH2)CHn(NH2) (m, n ) 0, ..., 2) 36.85 -43.85 9.13 19 -136.35 73.44 -9.81130 NC-CHn-CHm-CN (n, m ) 1, ..., 2) -46.52 39.70 -9.47 20 -12.58 0.87 0.67131 COO-CHn-CHm-OOC (n, m ) 1, ..., 2) 167.72 -57.46 4.29 21 91.78 -2.29 -5.77132 OOC-CHm-CHm-COO (n, m ) 1, ..., 2) -178.24 125.41 -20.24 22 -108.97 78.24 -12.39133 CHm-O-CHndCHp (m, n, p ) 0, ..., 3) 45.65 -52.96 11.86 23 7.25 -6.77 1.24134 CHmdCHn-F (m, n ) 0, ..., 2) 43.05 -39.85 7.54 24 6.94 -8.08 2.28135 CHmdCHn-Cl (m, n ) 0, ..., 2) -32.11 -6.64 5.20 25 -18.07 16.79 -3.81136 CHmdCHn-CN (m, n ) 0, ..., 2) 9.56 -3.63 2.19 26 0.00 0.00 0.00137 CHndCHm-COO-CHp (m, n, p ) 0, ..., 3) 63.70 -55.53 10.43 27 2.36 -0.38 -0.23138 CHmdCHn-CHO (m, n ) 0, ..., 2) 163.18 -105.60 17.22 28 148.10 -83.06 11.52139 CHmdCHn-COOH (m, n ) 0, ..., 2) 850.61 -562.94 92.16 29 948.72 -610.36 97.75140 aC-CHn-X (n ) 1, ..., 2), where X is a halogen 57.09 -45.85 9.65 30 106.37 -59.16 7.48141 aC-CHn-O- (n ) 1, ..., 2) -60.08 48.57 -8.26 31 -1.05 0.67 -0.11142 aC-CHn-OH (n ) 1, ..., 2) -233.61 171.75 -28.90 32 -88.19 58.31 -8.20143 aC-CH(CH3)2 5.51 -12.78 3.67 33 -11.49 9.08 -1.55144 aC-C(CH3)3 43.76 -23.35 4.39 34 4.95 -3.79 1.12


Table 1. (Continued)



145 aC-CF3 8.94 8.25 -3.23 35 27.80 -21.80 3.92146 (CHndC)(cyc)-CH3 (n ) 0, ..., 2) 49.82 -27.08 5.26 36 -1.77 1.60 -0.27147 CH(cyc)-CH3 -60.66 37.50 -5.87 37 -4.22 2.53 -0.37148 CH(cyc)-CH2 -50.85 31.05 -4.97 38 11.24 -9.71 1.87149 CH(cyc)-CH -151.74 87.41 -13.13 39 -84.48 40.25 -5.04150 CH(cyc)-C -35.95 20.41 -3.20 40 -69.43 24.49 -1.83151 CH(cyc)-CHdCHn (n ) 1, ..., 2) 1.49 -22.73 6.84 41 31.37 -22.71 4.13152 CH(cyc)-CdCHn (n ) 1, ..., 2) -33.33 7.74 0.06 42 39.30 -25.73 3.69153 CH(cyc)-Cl -78.46 40.50 -4.38 43 85.90 -63.30 11.85154 CH(cyc)-OH -173.40 96.39 -11.09 44 8.85 -39.69 13.58155 CH(cyc)-NH2 -198.45 149.07 -26.01 45 -116.79 91.89 -16.34156 CH(cyc)-SH -14.27 21.96 -3.68 46 0.00 0.00 0.00157 CH(cyc)-S- -45.91 27.08 -4.31 47 -2.87 1.00 -0.06158 C(cyc)-CH3 24.35 -20.88 4.24 48 4.78 -4.24 0.83159 AROMRINGs1s2 -5.82 1.12 0.26 49 8.43 -4.57 0.54160 AROMRINGs1s3 -49.93 25.88 -2.94 50 -15.93 7.90 -0.89161 AROMRINGs1s4 -32.01 14.84 -1.30 51 -12.07 4.91 -0.38162 AROMRINGs1s2s3 22.01 -14.75 2.98 52 32.27 -17.71 2.50163 AROMRINGs1s2s4 -15.05 7.75 -0.47 53 -18.74 15.33 -2.80164 AROMRINGs1s3s5 -60.51 29.24 -3.03 54 -26.33 11.40 -1.31165 AROMRINGs1s2s3s4 -70.85 47.97 -7.66 55 -19.78 13.51 -2.12166 AROMRINGs1s2s3s5 -32.41 20.35 -2.69 56 18.66 -14.12 2.85167 AROMRINGs1s2s4s5 -76.59 53.81 -8.73 57 -25.54 19.35 -3.19168 PYRIDINEs2 -4.29 0.74 0.25 58 11.65 -8.29 1.30169 PYRIDINEs3 -11.90 6.64 -0.83 59 4.04 -2.39 0.22170 PYRIDINEs4 -21.96 12.13 -1.52 60 -6.03 3.10 -0.47171 PYRIDINEs2s3 -9.50 6.43 -0.45 61 10.95 -4.16 0.46172 PYRIDINEs2s4 -12.26 4.65 -0.06 62 8.19 -5.94 0.86173 PYRIDINEs2s5 -15.61 5.79 -0.11 63 4.84 -4.80 0.81174 PYRIDINEs2s6 -95.43 62.60 -10.84 64 -99.60 63.93 -10.67175 PYRIDINEs3s4 -21.29 15.56 -1.96 65 -0.83 4.97 -1.04176 PYRIDINEs3s5 -16.32 7.33 -0.54 66 4.13 -3.26 0.38177 AROMRINGs1s2s3s4s5 73.70 -55.26 10.65 67 -11.43 8.84 -1.32178 CH2OCHO -40.23 40.10 -5.38 68 -21.70 17.54 -3.23179 CH2COOCH2 16.41 -37.05 11.15 69 18.74 -14.29 2.41180 CCOOCH2 -23.93 32.22 -4.45 70 23.03 -20.53 4.46181 CHCOOCH2 7.84 20.47 -2.85 71 -19.92 16.43 -3.06182 CH3COOCH2 -21.27 34.02 -9.40 72 -3.46 -0.73 0.36183 OCH2CH2O 23.69 -11.32 0.99 73 37.76 -20.87 2.79184 CH2COOCH3 21.12 -35.22 10.86 74 30.37 -17.66 2.99185 OCH2O 19.75 -20.97 4.28 75 5.03 -8.63 2.29186 CH2SSCH2 11.00 -15.56 4.70 76 42.25 -38.27 8.39187 (CH2OCH2)(cyc) -126.93 100.97 -20.41 77 -1.41 2.28 -0.66188 (CH2OCH)(cyc) 49.63 -16.59 4.07 78 -8.92 7.78 -1.56189 (CHdCHOCHdCH)(cyc) -151.24 110.58 -20.98 79 -9.66 3.91 -0.27190 (COOCH2)(cyc) -174.57 123.71 -22.84 80 -24.78 12.17 -1.54191 (CH2NHCH2)(cyc) 271.87 -166.98 24.72 81 18.78 -10.30 1.12192 aCaNaC 61.37 -47.23 9.64 82 85.98 -59.16 10.39193 (CH2SCH2)(cyc) 47.26 -37.50 6.61 83 3.29 -3.60 0.95194 (dCHSCHd)(cyc) -21.48 12.32 -2.63 84 -47.55 35.65 -6.83195 (dCHSCHd)(cyc) -78.10 55.00 -10.33 85 -65.12 48.87 -9.35196 (3 F) 58.27 -51.18 10.74 86 9.25 -6.59 1.10197 (5 F) 20.27 -29.74 8.20 87 -9.12 5.13 -0.71198 (perFlouro) 35.87 -45.37 12.02 88 0.99 -1.12 0.39d

Third-Level Group Contributions199 HOOC-(CHn)m-COOH (m > 2, n ) 0, ..., 2) 47.68 35.65 -11.76 1 54.95 25.28 -9.02200 OH-(CHn)m-OH (m > 2, n ) 0, ..., 2) -776.13 474.81 -74.60 2 -578.38 311.30 -43.20201 NC-(CHn)m-CN (m > 2) -31.39 29.87 -8.09 3 12.93 -14.66 2.86202 aC-(CHndCHm)cyc (fused rings) (n, m ) 0, ..., 1) -361.00 234.70 -37.84 4 7.34 2.93 -1.73203 aCsaC (different rings) -949.70 542.18 -78.94 5 16.65 -4.81 0.32204 aC-CHn(cyc) (fused rings) (n ) 0, ..., 1) -352.63 223.53 -34.99 6 -5.73 4.19 -0.64205 aC-(CHn)m-aC (different rings) (m > 1; n ) 0, ..., 2) -162.91 94.57 -13.04 7 -114.37 58.67 -7.13206 CH(cyc)-CH(cyc) (different rings) -16.53 32.62 -7.48 8 108.59 -48.77 6.03207 CH multiring -82.22 50.32 -8.17 9 -14.06 5.01 -0.48208 aC-CHm-aC (different rings) (m ) 0, ..., 2) -328.63 177.13 -24.66 10 19.44 -12.71 1.93209 aC-(CHmdCHn)-aC (different rings) (m, n ) 0, ..., 2) -134.98 110.23 -17.46 11 0.54 -0.21 0.02210 aC-S-aC (different rings) -509.98 283.05 -41.17 12 -22.17 5.46 -0.28211 aC-O-aC (different rings) -476.00 250.62 -34.67 13 -43.71 10.48 -0.34212 aC-CHn-O-CHm-aC (different rings) (n, m ) 0, ..., 2) -30.04 24.29 -4.13 14 -0.01 0.00 0.00213 AROM.FUSED[2] -392.10 164.35 -17.68 15 -8.67 4.97 -0.69214 AROM.FUSED[2]s1 -392.02 161.30 -16.51 16 30.93 -16.04 2.22215 AROM.FUSED[2]s2 -426.19 177.11 -18.53 17 -20.53 8.67 -0.94216 AROM.FUSED[2]s2s3 -363.01 152.00 -15.61 18 33.70 -10.53 0.72217 AROM.FUSED[2]s1s4 -344.79 133.58 -12.83 19 17.88 -10.67 1.15218 AROM.FUSED[2]s1s3 -402.80 165.45 -17.16 20 -16.41 6.87 -1.02


method cannot be used for some more-complex compounds, incontrast to the Za´branskyand Ruzicka method. The averagerelative error of the new model for the basic dataset of 549compounds over the temperature range was 1.2%; the error bythe Zabranskyand Ruzicka method for the same dataset wasalso 1.2%. For a comparison of heat capacity estimation at298.15 K, we used both methods (Za´branskyand Ruzicka andChickos) and evaluated them for 404 compounds from the basicset and also for 149 additional compounds that were not usedin the parameter calculation. The latter set of 149 compoundsis hereafter called the test set. Because of the unavailability of

some groups in the Chickos et al. method,24,28 the number ofheat capacity values estimated by this method is lower. Resultsof this comparison are presented in Table 3 and in Appendix A.

It turned out (see Tables 2 and 3) that the new model issuperior, in terms of the ARE and MED values for the entirebasic set, as well as for the test set. Furthermore, the use of thenonhierarchic approach in the new model yields slightly betterresults.

Extrapolation Features.Figures 2-5 show the extrapolationcapability of the developed model, in terms of theCp

l predic-tion at 298.15 K forn-alkanes C4-C30, 1-alkanols C2-C19,1-iodoalkanes C2-C7, and methyl esters ofn-alkanoic acids C3-C16. Note that our datasets contained the following data: forn-alkanes, C5-C20 (the basic dataset) and C4-C30 (the testdataset); for 1-alkanols, C2-C14 (the basic dataset) and C11-C19 (the test dataset); for 1-iodoalkanes, C2-C3 (the basicdataset) and C4-C7 (the test dataset); and for methyl esters of

Table 1. (continued)


No. group name a b d No.a a b

219 AROM.FUSED[3] -304.97 122.21 -12.63 21 -27.44 9.10 -0.68220 AROM.FUSED[4a] -738.46 301.95 -31.23 22 37.17 -16.16 1.74221 AROM.FUSED[4p] -762.03 309.79 -31.93 23 24.07 -15.28 2.12222 PYRIDINE.FUSED[2] -394.59 160.60 -16.45 24 13.29 -7.59 0.96223 PYRIDINE.FUSED[2-iso] -375.09 150.09 -15.03 25 32.72 -18.06 2.38

a For the hierarchic approach, serial numbers were assigned for the first-level, second-level, and third-level group contributions independently.

Table 2. Results for Estimation of Liquid Heat Capacity as aFunction of Temperature Obtained by the Present Model (eq 1) forthe Basic Dataset (549 Compounds)

ARE (%) MED (%) NGRa

New Model, Hierarchic-Hfirst level 1.9 1.2 110+ 1d

second level 1.6b 1.0 88third level 1.5c 0.9 25

New Model, Nonhierarchic-NHfirst + second+ third levels 1.2 0.8 223+ 1d

a NGR is the number of group contributions determined at individualestimation levels for the hierarchic approach (new model (H)) or the numberof all group contributions calculated for the nonhierarchic approach (newmodel (NH)).b Includes all compounds represented by the first-level andthe second-level group contributions.c Includes all compounds representedby the first-level, second-level, and third-level group contributions.d Thedigit “1” denotes the additional adjustable parameter.

Figure 1. Comparison of the experimental data and the estimated andpredicted values for liquid heat capacity at 298.15 K. For an explanationof terms “estimated” and “predicted”, see the section entitled “ExtrapolationFeatures”.

Table 3. Results for Comparison ofCpl at 298.15 K Obtained by the

New Models (Nonhierarchic Approach (New Model-NH) andHierarchic Approach (New Model-H) and by the Methods ofZabransky and Ruzicka and Chickos et al. for the Basic Datasetand the Test Dataset of Compounds

method NC ARE (%) MED (%)

Basic Datasetnew model-H 404 1.6 1.1new model-NH 404 1.5 1.0Zabranskyand Ruzicka (ZR) 404 1.8 1.1Chickos et al. (CH) 399 3.9 3.0

Test Datasetnew model-H 149 2.7 2.1new model-NH 149 2.5 2.0Zabranskyand Ruzicka (ZR) 149 3.0 2.0Chickos et al. (CH) 147 4.2 3.1

Figure 2. Comparison of the experimental, estimated, and predicted heatcapacities at 298.15 K forn-alkanes. For an explanation of terms “estimated”and “predicted”, see the section entitled “Extrapolation Features”.


n-alkanoic acids, C4 (the basic data set) and C3-C16 (the testdataset). It is shown that the model predicts this property withreasonable accuracy along the carbon number scale. Experi-mental data and the estimated and predicted values for some

members of the homologous series are also plotted. In thesefigures, as well as throughout the entire paper, the term“estimated” is used when the heat capacity estimation isperformed for a compound that has a carbon number withinthe range of compounds from the basic dataset, whereas theterm “predicted” is used when the property estimation isperformed beyond the carbon number range of the basic datasetcompounds. Thus, the predicted values characterize the ex-trapolation capability of the developed model.

Analysis of Results of Regression: Statistical Approach.For an exact analysis of all obtained results, we applied somestatistical tools that have been described earlier in detail.35 First,we tested that obtained data comes from a normal distribution.After rejecting this hypothesis, we continued with the compari-son by nonparametric methods of mathematical statistics. Wecompared the experimental data and the estimated valuesobtained forCp

l at 298.15 K using the following methods: ournew model, the nonhierarchic approach (new model-NH); ournew model, the hierarchic approach (new model-H); themethod by Za´branskyand Ruzicka29 (ZR); and the method byChickos24,28 (CH).

All of the procedures and tests used have been alreadyexplained.35 The results are presented in so-called “box-and-whiskers” plots and are shown in Figures 6 and 7. Anexplanation of the box-and-whiskers plot is given in Figure 8.This diagram creates a plot for each dataset (sample), which isdivided into four equal areas of frequency (quartiles). The plotshows the most extreme values in the data (maximum andminimum values), the lower and upper quartiles, and the median.A box encloses the middle 50% values. The median of valuesis drawn as a vertical line inside the box. Horizontal lines, knownas whiskers, extend from each end of the box. The left (or lower)

Figure 3. Comparison of the experimental, estimated, and predicted heatcapacities at 298.15 K for 1-alkanols.

Figure 4. Comparison of experimental, estimated, and predicted heatcapacities at 298.15 K for 1-iodoalkanes.

Figure 5. Comparison of experimental, estimated, and predicted heatcapacities at 298.15 K for methyl esters ofn-alkanoic acids (MEAA).

Figure 6. Comparison of the results for liquid heat capacity estimation at298.15 K. Legend: New model-NH, this work, nonhierarchic approach;New model-H, this work, hierarchic approach; ZR, the Za´branskyandRuzicka method; and CH, the Chickos et al. method.

Figure 7. Comparison of relative errors of liquid heat capacity estimationat 298.15 K. Legend: New model-NH, this work, nonhierarchic approach;New model-H, this work, hierarchic approach; ZR, the Za´branskyandRuzicka method; and CH, the Chickos et al. method.


whisker is drawn from the lower quartile to the smallest pointwithin 1.5 interquartile ranges from the lower quartile. The otherwhisker is drawn from the upper quartile to the largest pointwithin 1.5 interquartile ranges from the upper quartile. Therectangular portion of the box extends from the lower quartileto the upper quartile, covering the center half of each sample.The centerlines within each box show the location of the samplemedians. The cross symbols (+) in the box near the medianrepresent the location of the sample means. The whiskers extendfrom the box to the minimum and maximum values in eachsample, except for any outside or far-outside points (distant andextreme values in Figure 8), which are plotted separately.Outside points (distant value) lie more than 1.5 times theinterquartile range above or below the box; they are shown assmall squares. Far-outside points (extreme value) lie more than3.0 times the interquartile range above or below the box; theyare shown as small squares with crosses through them. Forcomparison of the models, this figure shows the followingrelation: the more similar the box-and-whisker plots of com-pared sets, the better the agreement between compared data.Especially, the median and mean values, and the lower andupper quartile, should be located at the same position. Thestatistical tests for all applied models confirmed whether themodels are suitable or unsuitable for estimation of the propertiesbeing examined, because there is (or is not) a significantdifference between the medians obtained by the individualmodels.

The plots in Figure 6 clearly show how the individual modelsare able to describe the experimental data. It is evident that bothof our new models, as well as the model by Za´branskyandRuzicka, show good agreement with experimental data.

Plots in Figure 7 show the results for a comparison of REvalues obtained by the individual methods. The median andmeans of the relative errors obtained by our new model aresmaller than those obtained by other models. The new modelsthat have been presented in this paper provide the lowestestimation error. The method reported by Za´bransky andRuzicka yields estimation errors that are similar to the errorsobtained by our models. The use of the Chickos method resultsin significantly diverse relative errors, in comparison with othermethods.

Conclusions

A new group-contribution method for estimating the liquidheat capacity as a function of temperature over the range fromthe melting temperature up to the normal boiling temperaturehas been developed. The method is based on a novel three-level group-contribution model that contains a larger set of groupcontributions, compared to a classical Benson-type method, and,therefore, is capable of being applied for prediction of the heatcapacity of more-complex molecular structures, including poly-functional and multiring compounds. Group-contribution pa-

rameters were calculated via two approaches: a hierarchical onethat has been used exclusively so far and entails evaluation ofparameters in three consecutive steps, and a nonhierarchical onein which parameters were evaluated in a single step.

The new method was compared with two other group-contribution methods, for predictions both over a temperaturerange and at a single temperature (298.15 K). The new methodwas determined to provide liquid heat capacity values that werein good agreement with experimental data and, therefore, shouldbe suitable for applications in engineering calculations. Thenonhierarchical approach was proven to be superior to thehierarchical approach, in terms of the relative deviation fromexperimental data. The group-contribution parameters that havebeen developed by the hierarchical approach will be applied inexisting computer-aided systems for the design, simulation, andanalysis of chemical processes.

Appendix A: Error Estimation Methods

Table A1 gives a detailed summary of the estimation methodsfor a variety of families of compounds. NC denotes the numberof compounds in the individual family, ARE is the averagerelative error, and MED is the median of relative errors.

Appendix B: Example Application of the New Model

Table B1 gives an example of using our model to estimatethe heat capacity of liquid 1,3,5-trimethylbenzene and itscomparison with experimental data and with values obtainedby other methods.

Appendix C: New Model, First Version

In our first attempt, we adopted the methodology developedby Constantinou and Gani32 and by Marrero and Gani.31 Weextended their approach for predicting the heat capacity ofliquids and modified the method to estimate heat capacity atsome selected temperatures (298.15 K and several reducedtemperatures (Tr ) 0.3, 0.4, 0.5, 0.6, whereTr ) T/Tc andTc is the critical temperature). The results are presented in TableC1.

The model for predicting the heat capacity at an individualtemperature was given as follows:

whereCpl represents the liquid heat capacity at an individual

temperature (298.15 K or a specified reduced temperature(Tr ) 0.3, 0.4, 0.5, or 0.6,),Cp0

l is the adjustable parameter,Ci is the first-level group contribution of typei, Dj is the second-level group contribution of typej, andEk is the third-level groupcontributions of the typek. Ni, Mj, andOk denote the numberof occurrences of individual group contributions. Para-metersω and z are assigned values of unity and zero, de-pending on the usage of the individual level group contribu-tions.31

Although the results obtained for heat capacity at individualtemperatures were satisfactory, the analysis of the obtainedvalues for individual groups revealed that group values did notfollow the expected monotonic increase with temperature.Therefore, we developed a new version of the model definedby eqs 1 and 2.

Figure 8. Explanation of the box-and-whisker plot.

Cpl ) Cp0

l + ∑i)1

n

NiCi + ω ∑j)1

m

MjDj + z∑k)1

0

OkEk (C1)


Table A1. Overview of Average Relative Error (ARE) and Median of Relative Errors (MED) Values of the Individual Estimation Methods forVarious Families of Organic Compounds for the Estimation ofCp

l at 298.15 K and over a Temperature Rangea

New Model-NH New Model-H Zabranskyand Ruzicka, ZR Chickos et al., CH

family of organic compounds NCARE(%)

MED(%) NC

ARE(%)

MED(%) NC

ARE(%)

MED(%) NC

ARE(%)

MED(%)

Hydrocarbonsaliphatic saturated hydrocarbons

at 298.15 K 21 1.5 1.3 21 2.0 1.3 21 1.7 1.3 21 1.9 1.9over a temperature range 24 1.5 1.5 24 2.0 2.0 24 1.0 1.0

aliphatic unsaturated hydrocarbonsat 298.15 K 26 1.5 1.3 26 1.7 1.3 26 2.0 0.6 26 4.1 3.2over a temperature range 25 0.9 1.0 25 1.0 1.0 25 0.5 0.5

cyclic saturated hydrocarbonsat 298.15 K 35 1.5 0.8 35 1.4 0.6 35 1.8 0.9 35 4.2 3.1over a temperature range 43 1.0 0.8 35 1.1 1.0 43 1.0 0.8

cyclic unsaturated and aromatic hydrocarbonsat 298.15 K 41 1.1 0.7 41 1.2 0.8 41 1.3 0.9 41 4.0 3.0over a temperature range 65 0.6 0.5 65 0.9 0.7 65 0.8 0.6

totalat 298.15 K 123 1.4 1.0 123 1.5 1.1 123 1.7 0.9 123 3.7 2.7over a temperature range 157 0.9 0.7 157 1.1 1.0 157 0.8 0.7

Oxygen-Containing Compoundsethers


aldehydes, ketonesat 298.15 K 20 1.3 0.9 20 1.2 1.0 20 2.1 1.1 20 2.7 2.6over a temperature range 30 1.3 0.9 30 1.4 1.0 30 1.6 1.0

alcohols, phenolsat 298.15 K 27 3.5 3.0 27 3.3 2.4 27 4.0 3.9 27 6.3 3.7over a temperature range 41 3.7 3.3 41 4.4 4.4 41 2.8 2.5

acidsat 298.15 K 9 1.3 1.1 9 1.2 0.7 9 1.5 0.6 9 4.0 4.7over a temperature range 21 1.2 1.0 21 1.6 1.1 21 1.6 1.3

estersat 298.15 K 27 1.5 1.2 27 1.5 1.2 27 2.8 2.0 25 3.8 3.3over a temperature range 48 1.0 0.8 48 1.0 0.9 48 1.5 1.4

heterocyclic O-compoundsat 298.15 K 8 2.6 2.3 8 2.7 2.3 8 2.0 1.6 8 6.0 4.7over a temperature range 9 1.6 1.0 8 1.7 1.2 9 1.5 1.6

miscellaneous O-compoundsat 298.15 K 6 1.3 1.3 6 1.8 1.8 6 2.8 2.4 6 4.8 4.3over a temperature range 11 1.2 1.0 6 1.3 1.3 11 1.8 1.9


Halogenated Compoundschlorinated hydrocarbons


brominated hydrocarbonsat 298.15 K 7 1.3 0.7 7 1.6 0.8 7 1.1 0.6 7 2.9 3.1over a temperature range 7 1.6 1.2 7 2.0 1.4 7 1.1 0.7

iodinated hydrocarbonsat 298.15 K 4 0.7 0.7 4 0.5 0.5 4 0.5 0.3 4 2.0 1.8over a temperature range 4 0.6 0.8 4 0.7 0.9 4 0.5 0.7

fluorinated hydrocarbonsat 298.15 K 14 0.5 0.4 14 0.9 0.7 14 1.4 0.8 14 3.2 2.7over a temperature range 21 0.8 0.7 21 1.1 1.0 21 1.4 1.1

miscellaneous halogenated compoundsat 298.15 K 12 1.4 1.5 12 1.3 1.1 12 1.6 1.2 12 4.5 4.8over a temperature range 19 1.0 1.1 19 1.2 1.2 19 1.1 1.2


Nitrogen-Containing Compoundsamines


nitrilesat 298.15 K 7 1.0 0.2 7 1.1 0.2 7 2.3 0.3 6 4.8 4.3over a temperature range 8 0.8 0.5 8 0.9 0.7 8 0.6 0.2

heterocyclic N-compoundsat 298.15 K 20 0.6 0.1 20 0.7 0.1 20 0.9 0.8 20 2.3 2.1over a temperature range 20 0.6 0.4 20 0.7 0.5 20 1.0 0.8

miscellaneous N-compoundsat 298.15 K 5 3.0 2.1 5 3.2 2.1 5 1.6 0.5 5 5.8 7.7over a temperature range 6 2.6 1.7 6 3.5 3.0 6 0.7 0.5


Nomenclature

λ ) regularization parameterAAE ) average absolute error (J K-1 mol-1)ARE ) average relative error (%)Cp

l ) isobaric heat capacity of pure liquid (J K-1 mol-1)Cp

l (298.15 K)) heat capacity at 298.15 K (J K-1 mol-1)Cp

l (Tr ) X) ) heat capacity at reduced temperatureTr ) X,where X) 0.3, 0.4, 0.5, 0.6 (J

K-1 mol-1)m ) total number of experimental data pointsMED ) median of relative errors (%)n ) total number of group contributionsn1, n2, n3 ) number of the first-level, second-level, and third-

level group contributionsNC ) number of compoundsNGR ) number of group contributionsSSQ) the sum of squares functionSSQ* ) decomposed form of SSQT ) temperature (K)Tc ) critical temperature (K)Tr ) reduced temperature;Tr ) T/Tc

1m ) unit vector of sizemdiag ) function for creating a diagonal matrixIn ) identity matrix of sizenp, q, s ) contribution vectors of the first-level, the second-

level, and the third-level group contributionsP, Q, S) structural matrices of the first-level, the second-level,

and the third-level group contributionst*, p*, q*, s* ) least-squares estimatest+, p+, q+, s+ ) regularized estimatest ) total contribution vectorT ) total structural matrixTrel ) relative structural matrixu ) experimental data vector (individualCp

l values)

Table B1. Example of Using the New Model for1,3,5-Trimethylbenzene

groupnumber of

occurrences a b d

additional adjustable parametercp°(T) 1 105.94 -51.40 7.24aCH 3 -1.28 8.17 -0.43aC-CH3 3 20.23 9.63-0.43AROMRINGs1s3s5 1 -60.51 29.24 -3.03

Cpl (298.15 K)exp ) 208.37 J K-1 mol-1

(value taken from Za´branskyet al.1,2)Cp

l (298.15 K)New_model_NH) 209.8 J K-1 mol-1

RE ) 0.7%; RE) (|Cp,expl - Cp,est

l |)/(Cp,expl ) × 100

Cpl (298.15 K)ZR ) 211.3 J K-1 mol-1

RE ) 1.4%Cp

l (298.15 K)CH ) 216.0 J K-1 mol-1

RE ) 3.7%

Table C1. Results for Estimation of Heat Capacity (eq C1) atSelected Temperatures

Cpl

number ofcompounds,

NC

number ofgroup

contributions,NGR

averageabsolute

error, AAEa

(J K-1 mol-1)

averagerelativeerror,

AREb (%)

Cpl (298.15 K) 678 122+ 1c + 65d 4.2 1.7

Cpl (Tr ) 0.3) 66 21+ 1c + 10d 2.2 1.5

Cpl (Tr ) 0.4) 283 87+ 1c + 37d 6.2 2.7

Cpl (Tr ) 0.5) 384 89+ 1c + 48d + 14e 6.8 2.6

Cpl (Tr ) 0.6) 269 83+ 1c + 38d + 13e 6.3 2.5

a AAE ) (1/n)∑i)1n |Cp,exp

l - Cp,estl |i. b See eq 3.c Number of group

contributions of the first level plus 1 adjustable additional parameter.d Number of group contributions of the second level.e Number of groupcontributions of the third level.

Table A1. (continued)

New Model-NH New Model-H Zabranskyand Ruzicka, ZR Chickos et al., CH

family of organic compounds NCARE(%)

MED(%) NC

ARE(%)

MED(%) NC

ARE(%)

MED(%) NC

ARE(%)

MED(%)


Sulfur-Containing Compoundssulfides


thiolsat 298.15 K 17 1.1 1.0 17 0.9 0.6 17 0.8 0.6 17 3.0 2.8over a temperature range 17 1.0 1.0 17 0.9 0.7 17 0.8 0.8

heterocyclic S-compoundsat 298.15 K 6 1.3 0.8 6 1.6 1.5 6 0.5 0.4 6 2.9 2.4over a temperature range 9 1.2 1.1 9 1.1 0.4 9 0.6 0.4


Miscellaneous Compounds with a Mixture of Oxygen, Nitrogen, or Halogencompounds with halogen and oxygen


compounds with nitrogen and oxygenat 298.15 K 13 1.6 1.4 13 2.5 2.3 13 2.2 1.6 13 4.7 5.6over a temperature range 22 1.5 1.4 22 2.0 1.6 22 1.9 1.6

compounds with halogen, oxygen, and nitrogenat 298.15 K 1 4.0 4.0 1 3.9 2.9 1 2.7 2.7 1 15.2 15.2over a temperature range 1 2.2 2.2 1 2.1 2.1 1 1.5 1.5

overall total forT ) 298.15 K 404 1.5 1.0 404 1.6 1.1 404 1.9 1.1 399 3.9 3.0overall total for a temperature range 549 1.2 0.8 549 1.5 1.1 549 1.2 0.8

a For each family of compounds the first line gives values of the number of compounds (NC), the average relative error (ARE), and the median of relativeerrors (MED) of the individual estimation methods for estimation at 298.15 K and the second line gives the same values estimated over a temperature range.


urel ) relative experimental data vector (estimatedCpl values

divided by the experimental ones)

Subscripts

i, j, k ) first-, second-, third-level group contribution of typei,j, k

Acknowledgment

This work was conducted within the IUPAC (Project No.2004-010-3-100), the Institutional Research Plans (Nos. MSM6046137307 and MSM 6046137306), and the Grant Agency ofthe Academy of Sciences of the Czech Republic (under GrantNo. IAA 400720710). The authors acknowledge the use theICAS (Version 9.0) software provided by R. Gani, especiallyits ProPred tool.

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ReceiVed for reView September 11, 2007ReVised manuscript receiVed December 14, 2007

AcceptedDecember 17, 2007

IE071228Z


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