Correlation of Henry's Constants of Nonpolar and Polar Solutes in

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2826

Ind. Eng. Chem. Res. 2002, 41, 2826-2833

CORRELATIONS Correlation of Henry’s Constants of Nonpolar and Polar Solutes in Molten Polymers Using Connectivity Indices Chongli Zhong,* Chunsheng Yang, and Qunsheng Li Department of Chemical Engineering, P.O. Box 100, Beijing University of Chemical Technology, Beijing 100029, China

New correlations for the estimation of Henry’s constants of nonpolar solutes in molten highdensity polyethylene (HDPE), low-density polyethylene (LDPE), polypropylene (PP), polybutadiene (PB), polyisobutylene (PIB), polystyrene (PS), poly(dimethylsiloxane) (PDMS), poly(vinyl chloride) (PVC), and poly(vinyl acetate) (PVAc), and those of polar solutes in LDPE, PS, PDMS, PVC, and PVAc were proposed. The new correlations only require the connectivity indices of the solutes in the calculation, which are more accurate and easier to apply than the existing correlations based on the physical properties of the solutes. Because connectivity indices can be calculated exactly once the chemical structure of the substance is known, the new correlations are more useful for molecular and process design and development. Introduction Henry’s constants of solutes in molten polymers are important properties because the solubilities of small molecules in polymers are required in the industrial processing of most polymeric materials. Though a lot of experimental data have been measured, similar to other properties, estimation models are still required, particularly for the purpose of process development, design, and optimization. On the other hand, for those solutes whose experimental data are not available, estimation models become the only way to provide us with quantitative numbers, at least crude values. A number of investigations have been carried out on modeling Henry’s constants of solutes in molten polymers. Some are simple correlations between Henry’s constants and the physical properties of solutes,1-6 and others use an equation of state method to correlate or predict Henry’s constants.7,8 However, all of the methods require certain physical properties of the solutes concerned, which may not be available, and the estimation accuracy for Henry’s constants of polar solutes is much less than satisfying. Therefore, a method which can be used to estimate the Henry’s constants of nonpolar and polar solutes in molten polymers based only on the chemical structure of the solutes interested is highly preferable. Connectivity indices, the commonly used molecular structure descriptors, have been widely used to develop correlations for relating the properties of substances to their chemical structures,9-13 and many useful predictive correlations have been successfully proposed. Because connectivity indices can be calculated easily as long as the chemical structure of the substance concerned is known, the correlations only using connectivity * Corresponding author. E-mail: [email protected]. Fax: +86-10-64436781. Tel.: +86-10-64419862.

indices as input parameters are easier to apply and have predictive potential. In this work, we will develop such correlations for Henry’s constants of nonpolar and polar solutes in some industrially important polymers. Connectivity Index Connectivity indices have been widely used as structure descriptors, which contain a large amount of information about the molecule, including the numbers of hydrogen and non-hydrogen atoms bonded to each non-hydrogen atom, the details of the electronic structure of each atom, and the molecular structural features (paths, branches, clusters, and rings).9,10 Details of their definition and the calculation method can be found elsewhere,9,10 and the values of the connectivity indices for the nonpolar and polar solutes used in this work are listed in Tables 1 and 2, respectively. A review on the development of the connectivity index was recently published by Randic´.13 Existing Correlations for Henry’s Constants of Solutes in Molten Polymers Stiel and Harnish3 proposed a generalized correlation for Henry’s constants of nonpolar and slightly polar solutes in molten polystyrene (PS) as follows:

ln(1/Kp) ) -2.338 + 2.706(Tc/T)2

(1)

where T and Tc are the system temperature and solute critical temperature, respectively. Kp is Henry’s constant in atm‚(g of polymer)/cm3 (273.2 K, 1 atm). Stiel et al.4 further proposed the following two correlations for Henry’s constants of nonpolar and slightly polar solutes in molten polyethylene (PE) and polyisobutylene (PIB):

10.1021/ie010966z CCC: $22.00 © 2002 American Chemical Society Published on Web 04/27/2002

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2827 Table 1. Connectivity Indices of the Nonpolar Solutes Used in This Work substance

0χ

0χV

1χ

2χ

2χV

3χ

3χV

5χ CH

ethylene ethane propylene propane 1,3-butadiene isobutane butane isopentane pentane 2-methylbutane 1-hexene 3-hexene hexane 2-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane 1-heptene heptane 2-methylhexane 3-methylhexane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane octane 2-methylheptane 3-methylheptane 2,4-dimethylhexane 2,5-dimethylhexane 3,4-dimethylhexane 2,2,4-trimethylpentane 2,3,4-trimethylpentane nonane 2,2,4-trimethylhexane 2,2,5-trimethylhexane decane undecane dodecane cyclopentane cyclohexane carbon tetrachloride tetraline decalin benzene toluene ethylbenzene m-xylene p-xylene mesitylene

2.0000 2.0000 2.7071 2.7071 3.4142 3.5774 3.4142 4.2845 4.1213 4.2845 4.8284 4.8284 4.8284 4.9916 5.2071 5.1547 5.5355 5.5355 5.6987 5.6987 5.9142 5.8618 5.8618 5.9142 6.2426 6.4058 6.4058 6.5689 6.5689 6.5689 6.7845 6.7321 6.9497 7.4916 7.4916 7.6569 8.3640 9.0710 3.5355 4.2426 4.5000 6.8116 6.8116 4.2426 5.1129 5.8200 5.9831 5.9831 6.8534

1.4142

1.0000 1.0000 1.4142 1.4142 1.9142 1.7321 1.9142 2.2701 2.4142 2.2701 2.9142 2.9142 2.9142 2.7701 2.5607 2.6427 3.4142 3.4142 3.2701 3.3081 3.0607 3.1807 3.1259 3.1213 3.9142 3.7701 3.8081 3.6639 3.6259 3.7187 3.4165 3.5534 4.4142 3.9545 3.9165 4.9142 5.4142 5.9142 2.5000 3.0000 2.0000 4.9663 4.9663 3.0000 3.3939 3.9319 3.7877 3.7877 4.1815

0.0000 0.0000 0.7071 0.7071 1.0000 1.7321 1.0000 1.8021 1.3536 1.8021 1.7071 1.7071 1.7071 2.1825 2.9142 2.4880 2.0607 2.0607 2.5361 2.3021 3.3107 2.6295 3.0234 2.8713 2.4142 2.8896 2.6556 3.1430 3.3650 2.7711 4.1586 3.3472 2.7678 4.2782 4.4932 3.1213 3.4749 3.8284 1.7678 2.1213 3.0000 4.0891 4.0891 2.1213 2.7432 2.9123 3.3770 3.3650 4.0226

0.0000 0.0000 0.4083 0.7071 0.4714 1.7321 1.0000 1.8021 1.3536

0.0000 0.0000 0.0000 0.0000 0.5000 0.0000 0.5000 0.8165 0.7071 0.8165 0.9571 0.9571 0.9571 0.8660 1.0607 1.3333 1.2071 1.2071 1.1350 1.4784 1.0000 1.7820 0.9428 1.9142 1.4571 1.3850 1.7474 1.5707 1.3214 2.2593 1.0206 2.1031 1.7071 1.6578 1.4717 1.9571 2.2071 2.4571 1.2500 1.5000 0.0000 3.4663 3.4663 1.5000 1.8939 2.3021 2.1986 2.3045 2.4142

0.0000

0.0000 0.0000

3.4142

4.8284

5.6987

6.2426 6.4058 6.4058 6.5689 6.5689 6.5689 6.7845 6.9497 7.4916 7.6569 9.0710

6.1378 6.8116 3.4641 4.3868 5.0939 5.3094 5.3094 6.2321

1.7071 2.9142 2.0607 2.3021

2.4142 2.8896 2.6556 3.1430 3.3650 2.7711 4.1586 2.7678 4.2782 3.1213 3.8284 1.7678 2.1213 3.8571 2.9758 4.0891 1.1547 1.6547 1.8392 2.1582 2.1547 2.6651

0.5000

2.4571

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2.2606 3.4663 0.6667 0.9405 1.2511 1.1736 1.2182 1.3660

0.1250 0.0000 0.1667 0.1667 0.1250 0.1021 0.1021 0.0833 0.0833 0.0680

3χ

3χV

0.9571

1.4784

1.4571 1.3850 1.7474 1.5707 1.3214 2.2593 1.0206 1.7071 1.6578 1.9571

Table 2. Connectivity Indices of the Polar Solutes Used in This Work substance methyl acetate vinyl acetate ethyl acteate 1-propyl acetate 1-butyl acetate acetone methyl ethyl ketone 2-pentanone 1-propanol 2-propanol 1-butanol 1-pentanol 1-octanol methyl chloride dichloromethane chloroform chlorofluoromethane difluorochloromethane fluorotrichloromethane dichlorodifluoromethane trichlorotrifluoroethane acrylonitrile pyridine fluorobenzene chlorobenzene

0χ

4.9916

3.5774 4.2845 3.5774

2.0000 2.7071 3.5774 2.7071 3.5774 4.5000 4.5000 7.0000 3.4142 5.1129 5.1129

0χV

3.3165 3.6010 4.0236 4.7307 5.4378 2.9083 3.6154 4.3225 2.8614 3.0246 3.5685 4.2756 6.3970 2.1339 2.9749 3.9790 2.2190 2.4672 4.2796 3.5237 5.5356 2.2317 3.3340 3.7647 4.5206

1χ

1.0000 1.4142 1.7321 1.4142 1.7321 2.0000 2.0000 3.2500

1χV

2χ

1.3165 1.5522 1.9040 2.4040 2.9040 1.2041 1.7648 2.2648 1.5233 1.4129 2.0233 2.5233 4.0233 1.1339 1.6036 1.9640 1.0690 1.0911 1.8898 1.5119 2.5178 0.9205 1.8497 2.0997 2.4776

1.8021 2.1825 2.1825 2.5361 2.8896 1.7321 1.8021 2.1825 1.0000 1.7321 1.3536 1.7071 2.7678 0.0000

2χV

0.7761

0.9083 1.0556 1.0938

0.0000

1.7321

2.2269

1.0000 2.1213 2.7432 2.7432

1.2956 1.7320

0.8165 0.8660 0.8660 1.1350 1.3850 0.0000 0.8165 0.8660 0.5000 0.0000 0.7071 0.9571 1.7071 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.2500 0.5000 1.5000 1.8939 1.8939

0.2493

0.0000 0.4979 0.0000

0.0000 0.0000

0.7331 0.9851

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For PE

Table 3. Calculated Results of 1/Kp for HDPE

ln(1/Kp) ) -1.561 + (2.057 + 1.438ω)(Tc/T)2 (2) solute

For PIB ln(1/Kp) ) -1.347 + (1.790 + 1.568ω)(Tc/T)2 (3) where ω is the acentric factor of the solute. Recently, Chiu and Chen5 proposed two more accurate correlations for nonpolar solutes in PIB, and one of them is as follows:

ln(1/Kp) ) -3.80 - 11.8ω + (4.48 + 9.67ω)(Tc/T) (4) In our previous work, several generalized correlations for nonpolar solutes in molten polypropylene (PP) and poly(dimethylsiloxane) (PDMS) and for polar solutes in molten low-density PE (LDPE) and PDMS were proposed, and some of them are as follows:6

For nonploar solutes in PP ln(1/Kp) ) -2.316 - 8.885ω + (3.290 + 8.545ω)(Tc/T) (5)

3-methylhexane octane 2-methylheptane 3-methylheptane 2,4-dimethylhexane 2,5-dimethylhexane 3,4-dimethylhexane 2,2,4-trimethylpentane nonane 2,2,4-trimethylhexane decane dodecane toluene ethylbenzene p-xylene m-xylene mesitylene tetralin cis-decalin trans-decalin system average error a

For nonaromatic hydrocarbons in PDMS ln(1/Kp) ) -3.986 - 9.228ω + (4.778 + 7.811ω)(Tc/T) (6) For aromatic hydrocarbons in PDMS ln(1/Kp) ) -3.152 - 13.954ω + (4.025 + 10.758ω)(Tc/T) (7) For polar solutes in LDPE ln(1/Kp) ) -1.448 + [2.062 - 6.326ω + 23.831(ωzc)](Tc/T)2 (8) For polar solutes in PDMS ln(1/Kp) ) -10.913 + 94.546(ωzc) - 410.557(ωzc)2 + 6.219(Tc/T) (9) where zc is the critical compressibility factor of the solute. Development of New Generalized Correlations Based on Connectivity Indices Although the existing correlations can give good estimations for Henry’s constants of solutes in several polymers, the physical properties of the solutes, that is, the critical temperature and acentric factor, and also the critical compressibility factor for polar solutes are required in the calculations. Therefore, they cannot be applied to those solutes whose required parameters are not available. Furthermore, similar correlations are difficult to develop for other polymers. As a result, new correlations requiring only chemical structure information of solutes and applicable to more polymers are necessary. Therefore, we select connectivity indices as structure descriptors to characterize the solutes and develop correlations based only on connectivity index values.

AAD )

1 N

AADa (%)

no. of data temp range (K) points

∑ i

|

1

Kexp P,i

418.55-425.75 418.55-425.45 418.55-425.75 418.55-425.75 418.55-425.75 418.55-425.75 418.55-425.75 418.55-425.75 418.55-425.75 418.55-425.75 418.55-426.45 418.55-426.45 418.55-425.45 418.55-425.45 418.55-425.45 418.55-425.45 418.55-426.45 418.55-426.45 418.55-426.45 418.55-426.45

-

1 Kcal. P,i

|

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

eq 2

this work ref

10.0 3.7 6.6 3.4 9.0 1.1 6.9 3.3 6.7 2.0 8.1 3.0 1.0 6.4 5.3 5.0 4.4 2.3 3.4 3.0 9.5 1.7 4.3 1.7 3.2 4.4 7.8 2.9 3.7 5.5 5.7 1.0 4.0 3.1 14.9 1.3 23.2 13.2 8.6 10.7 7.3 3.9

14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14

× Kexp. × 100. P,i

Henry’s constant data for some industrially important polymers were collected from the literature, and they were converted to Kp in atm‚(g of polymer)/cm3 (273.2 K, 1 atm). Detailed conversion and calculation methods were given elsewhere.4 Generalized Correlation for Henry’s Constants of Nonpolar Solutes in Molten High-Density PE (HDPE). HDPE is a very important polymeric material, and Henry’s constant data for 20 nonpolar solutes were collected from the literature.14 Using the data of the systems shown in Table 3, the following correlation was obtained by fitting the data:

ln(1/Kp) ) 4.932 - 1.4330χ - 0.3162χV 12.8475χCH - (6.574 - 3.1781χ - 1.6712χ 300 0.8233χ) (10) T where mχ is the mth-order connectivity index, mχV is the mth-order valence connectivity index, and 5χCH is the chain connectivity index. Parallel calculations using eq 2 proposed by Stiel et al.4 have also been carried out for comparison, and the calculated results are also listed in Table 3. From Table 3, it is obvious that the new correlation proposed gives better correlative accuracy than eq 2. Although it seems that the new correlation is a little more complicated than the existing one, it requires only connectivity indices as input parameters in the calculations, and connectivity indices can be readily calculated once the molecular structure of the substance interested is known. Therefore, the new correlation is easier to apply than the existing one. Generalized Correlations for Henry’s Constants of Nonpolar and Polar Solutes in Molten LDPE. Henry’s constant data of nonpolar solutes in molten LDPE were collected, as shown in Table 4, from the

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2829 Table 4. Calculated Results of 1/Kp for Nonpolar Solutes in LDPE

solute ethylene ethylene butane hexane 3-methylhexane octane octane 2-methylheptane 3-methylheptane 2,4-dimethylhexane 2,5-dimethylhexane 3,4-dimethylhexane 2,2,4-trimethylpentane nonane 2,2,4-trimethylhexane decane dodecane benzene toluene toluene ethylbenzene p-xylene m-xylene mesitylene tetralin cis-decalin system average error

temp range (K)

no. of data points

398.15-523.15 397.15-573.15 397.15-573.15 397.15-573.15 393.15-418.25 393.15-418.25 397.15-573.15 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 397.15-573.15 393.15-418.25 397.15-573.15 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25 393.15-418.25

6 5 5 5 2 2 5 2 2 2 2 2 2 2 2 2 2 5 2 5 2 2 2 2 2 2

AAD (%) eq this 2 work ref 4.2 2.2 4.9 0.9 5.4 4.7 4.2 4.3 1.9 3.0 0.6 4.0 2.9 5.5 6.2 1.8 1.9 4.3 1.3 8.8 2.3 4.8 4.8 2.9 0.9 6.3 2.8 4.7 9.9 4.9 1.4 4.5 7.7 5.6 9.0 3.8 1.1 4.1 8.4 3.6 6.1 11.4 1.0 3.6 4.0 3.7 2.5 3.5 18.7 5.5 3.6 0.3 4.5 4.3

15 2 2 2 14 14 2 14 14 14 14 14 14 14 14 14 14 2 14 2 14 14 14 14 14 14

Table 5. Calculated Results of 1/Kp for Polar Solutes in LDPE

solute

temp range (K)

no. of data points

2-propanol acetone methyl ethyl ketone vinyl acetate vinyl acetate methyl chloride system average error

398.15-523.15 398.15-523.15 398.15-523.15 398.15-523.15 397.15-573.15 398.15-523.15

6 6 6 6 5 6

AAD (%) eq this 8 work 5.0 2.6 9.5 6.8 7.9 1.1 5.4

3.8 3.3 5.5 3.6 3.9 5.0 4.2

ref 15 15 15 15 2 15

literature,2,14,15 and the following correlation was developed by fitting the data.

ln(1/Kp) ) -0.971 - 1.1730χ + 0.7970χV 0.9642χV + (0.152 + 2.0191χ + 1.5912χ + 300 0.3913χV) (11) T From Table 4, where the parallel calculated results using eq 2 are also listed, it is evident that the new correlation, which only requires connectivity indices in the calculations, shows accuracy similar to that of the existing one. However, the new correlation is much easier to apply. Henry’s constant data for polar solutes in LDPE are much less, and the data for five polar solutes were collected from the literature.2,15 Using the data of the systems shown in Table 5, the following correlation was obtained. 0

2

ln(1/Kp) ) -1.489 - 1.11 χ + 1.31 χ + (6.049 + 300 2.3923χ) (12) T Parallel calculations using eq 8 proposed by Zhong and Masuoka6 have also been carried out for compari-

Figure 1. Temperature dependence of Henry’s constants for methyl ethyl ketone, 2-propanol, and methyl chloride in LDPE.

son, and the calculated results are listed in Table 5. From Table 5 it is obvious that the new correlation proposed gives better accuracy than eq 8, which is also easier to apply. The experimental and calculated Henry’s constants for methyl ethyl ketone, 2-propanol, and methyl chloride in LDPE are shown in Figure 1. The two correlations show similar correlative accuracy with a little different temperature dependence trend. In a comparison of eqs 11 and 12, it seems that a more complicated correlation is required for nonplolar solutes, and both simple and valence connectivity indices are needed to get good correlative accuracy for them. However, the number of systems used to develop the two correlations are different, which may affect the complexity of the correlations proposed. Generalized Correlation for Henry’s Constants of Nonpolar Solutes in Molten PP. PP is an important industrial polymer, and Henry’s constant data for 20 systems were collected from the literature16 to develop the following new correlation based on connectivity indices.

ln(1/Kp) ) -1.944 - 0.3170χ - 0.5551χ + (1.674 + 300 (13) 2.4571χ + 0.4842χ + 0.4233χ) T The calculated results from the new correlation together with those from eq 5 are shown in Table 6. The new correlation shows accuracy similar to that of the existing one; however, the former does not require any physical properties of the solutes in the calculations. Generalized Correlation for Henry’s Constants of Nonpolar Solutes in Molten Polybutadiene (PB). Henry’s constant data for 21 systems were collected from the literature,17 and the following correlation was proposed for PB based on the data of the systems shown in Table 7.

ln(1/Kp) ) -3.847 - 0.3820χ - 0.22χV + (4.744 + 300 (14) 1.5951χ + 0.8152χ + 0.473χ) T The calculated results shown in Table 7 illustrate that a simple correlation like eq 14 can give a good correlation between Henry’s constants and solute molecular

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Table 6. Calculated Results of 1/Kp for PP

solute propylene propane isobutane butane isopentane pentane hexane 2,2-dimethylbutane heptane octane nonane decane dodecane cyclopentane cyclohexane benzene toluene m-xylene ethylbenzene mesitylene system average error

no. of data temp range (K) points 448.2-498.2 448.2-498.2 448.2-498.2 448.2-498.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2 448.2-523.2

3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

AAD (%) eq this 5 work ref 6.9 3.3 5.3 3.2 2.4 4.8 3.3 3.9 3.4 2.8 2.0 3.0 4.0 6.0 7.6 10.9 5.7 3.0 4.7 5.9 4.6

7.2 2.5 4.7 3.3 2.0 6.1 3.5 9.3 4.2 2.7 1.3 1.4 5.9 1.9 3.3 6.0 0.9 6.8 6.5 10.2 4.5

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

Table 7. Calculated Results of 1/Kp for Nonpolar Solutes in PB

solute

temp range (K)

no. of data points

propylene propane 1,3-butadiene isobutane butane isopentane pentane cyclopentane 2,2-dimethylbutane hexane cyclohexane heptane octane nonane decane dodecane benzene toluene ethylbenzene m-xylene mesitylene system average error

423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15 423.15-498.15

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

AAD (%)

ref

5.7 3.4 3.5 5.0 7.9 5.5 9.7 8.6 6.7 1.9 3.4 5.2 2.6 2.2 4.1 5.8 9.4 3.5 8.0 4.3 11.8 5.6

17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17

connectivity indices. A search of the literature shows that no simple correlation like eqs 1-9 exists for this polymer. Generalized Correlation for Henry’s Constants of Nonpolar Solutes in Molten PIB. Based on the Henry’s constant data for the systems shown in Table 8,18-20 the following correlation was developed for PIB:

Table 8. Calculated Results of 1/Kp for Nonpolar Solutes in PIB

solute

no. of data temp range (K) points

pentane hexane hexane 2,3-dimethylbutane cyclohexane cyclohexane heptane 2,2-dimethylpentane octane 2,2,4-trimethylpentane 2,5-dimethylhexane nonane decane benzene toluene system average error

313.16-373.16 313.16-393.16 323.15-398.15 313.16-393.16 313.16-393.16 323.15-398.15 313.16-393.16 313.16-393.16 313.16-413.16 313.16-413.16 313.16-413.16 333.16-433.16 333.16-433.16 323.15-423.15 323.15-423.15

4 5 6 5 5 6 5 5 6 6 6 6 6 5 5

AAD (%) eq 4 2.4 2.4 9.0 1.0 5.1 7.6 1.8 4.2 2.6 3.8 3.0 4.2 4.8 12.4 3.0 4.5

this work ref 5.1 3.9 5.7 2.4 3.3 6.6 3.3 4.2 2.5 4.9 5.8 2.2 4.0 4.1 5.0 4.2

18 18 19 18 18 19 18 18 18 18 18 18 18 20 20

Table 9. Calculated Results of 1/Kp for Nonpolar Solutes in PS

solute

no. of data temp range (K) points

propane isobutene pentane hexane heptane cyclopentane cyclohexane benzene toluene system average error

408-498 423-498 408-498 423-498 448-498 448 498 423-498 423-473

5 4 5 4 3 1 1 4 3

AAD (%) eq 1 1.2 7.8 6.8 10.7 6.0 8.0 0.6 11.5 13.0 7.3

this work ref 3.2 1.2 2.3 2.8 0.9 7.1 7.3 4.8 3.9 3.7

3 3 3 3 3 3 3 3 3

Table 10. Calculated Results of 1/Kp for Polar Solutes in PS

solute chloromethane dichloromethane chloroform chlorofluoromethane chlorodifluoromethane dichlorodifluoromethane trichlorofluoromethane trichlorotrifluoroethane system average error

temp range (K)

no. of data points

423-448 423-473 448 408-448 408-498 408-498 408-498 423-498

2 3 1 3 5 5 5 4

AAD (%) eq this 1 work 22.6 3.5 46.0 5.0 4.2 7.8 2.9 13.1

3.9 14.3 32.8 2.2 1.9 2.2 6.7 0.8 8.1

ref 3 3 3 3 3 3 3 3

Generalized Correlations for Henry’s Constants of Nonpolar and Polar Solutes in Molten PS. Using the experimental Henry’s constant data3 for the systems shown in Table 9, the following correlation was proposed for Henry’s constants of nonpolar solutes in PS:

300 (15) T

ln(1/Kp) ) -3.379 - 0.4070χ - 0.3752χV + (3.058 + 300 (16) 1.8441χ + 1.02χ + 0.4113χ) T

Parallel calculations have been carried out using eq 4, and the results are shown in Table 8, where the calculated results from the new correlation are also listed. Obviously, the two correlations give similar accuracy; however, the new correlation has the advantage of not requiring any physical properties in the calculation.

The calculated results listed in Table 9 show that the new correlation gives much better accuracy than the existing one, that is, eq 1. Furthermore, Henry’s constant data for eight slightly polar solutes in molten PS were collected from the literature,3 which are shown in Table 10. When the data were fitted, the following correlation was proposed for

ln(1/Kp) ) -1.272 - 1.3990χ + 0.8282χ 8.4645χCH + (1.691 + 3.1341χ + 0.8493χ)

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2831 Table 11. Calculated Results of 1/Kp for Nonpolar Solutes in PDMS

solute pentane pentane 2-methylbutane hexane hexane hexane hexane hexane 2-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane cyclohexane cyclohexane cyclohexane 1-hexene cis-3-hexene trans-3-hexene heptane heptane heptane heptane heptane 2-methylhexane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 1-heptene octane octane octane 2-methylheptane 2,2,4-trimethylpentane 2,2,4-trimethlypentane 2,3,4-trimethylpentane nonane 2,2,5-trimethylhexane decane decane undecane dodecane benzene benzene benzene toluene toluene toluene p-xylene ethylbenzene ethylbenzene system average error

no. of AAD (%) data eq 6 this temp range (K) points or 7 work ref 313.15-353.15 298.15-343.15 298.15 363.15-473.15 313.15-413.15 313.15-353.15 298.15-343.15 303.15-353.15 298.15-343.15 303.15-353.15 303.15-353.15 328.15-453.15 313.15-353.15 303.15-343.15 303.15-353.15 303.15-353.15 303.15-353.15 363.15-473.15 328.15-453.15 313.15-353.15 298.15-343.15 303.15-343.15 298.15-343.15 303.15-353.15 303.15-353.15 303.15-353.15 303.15-353.15 303.15-353.15 363.15-473.15 313.15-353.15 298.15-343.15 298.15-343.15 298.15-343.15 303.15-353.15 303.15-353.15 363.15-473.15 303.15-353.15 363.15-473.15 303.15-353.15 363.15-473.15 363.15-473.15 363.15-473.15 313.15-453.15 298.15-343.15 363.15-473.15 328.15-453.15 298.15-343.15 298.15-343.15 333.15-453.15 298.15-343.15

5 4 1 6 7 5 4 6 4 6 6 8 5 5 6 6 6 6 8 5 4 5 4 6 6 6 6 6 6 5 4 4 4 6 6 6 6 6 6 6 6 6 9 4 6 8 4 4 7 4

6.5 1.9 3.3 17.8 1.5 2.2 4.4 5.7 4.6 8.3 7.4 16.4 11 4.6 8.4 10.2 4.2 14.9 3.7 2.1 7.5 1.6 5.8 3.2 2.7 4.6 3.5 3.2 7.8 4.5 8.3 7.0 7.7 3.2 3.1 3.7 5.5 9.1 4.3 11.8 11.2 9.6 3.2 4.2 12.1 4.9 3.8 3.9 3.3 4.5 6.2

6.0 11.0 9.0 4.1 6.9 3.0 6.5 4.5 5.6 3.0 6.5 2.7 3.9 10.5 4.5 6.6 5.6 7.3 6.2 3.2 6.2 7.4 5.5 3.3 4.6 5.6 7.7 4.1 4.2 3.9 7.6 5.1 1.7 9.5 10.6 3.7 14.1 6.1 11.8 6.1 5.4 7.5 5.6 8.4 11.9 4.5 4.0 11.1 12.8 11.8 6.6

21 22 22 23 24 21 22 25 22 25 25 24 21 25 25 25 25 23 24 21 23 25 22 25 25 25 25 25 23 21 22 22 22 25 25 23 25 23 25 23 23 23 24 22 23 24 22 22 24 22

polar solutes in this work.

ln(1/Kp) ) -3.327 - 1.0721χV + 0.1673χ + (3.526 300 (17) 3.8660χ + 2.7790χV + 5.4971χ) T The calculated results listed in Table 10 show that the new correlation gives better correlative accuracy than eq 1. Generalized Correlations for Henry’s Constants of Nonpolar and Polar Solutes in Molten PDMS. A lot of Henry’s constant data are available for nonpolar solutes in molten PDMS, and the systems collected from the literature21-25 are shown in Table 11. Using the

Figure 2. Temperature dependence of Henry’s constants for benzene, pentane, octane, and 2-methylheptane in PDMS.

database, the following correlation was developed.

ln(1/Kp) ) -2.176 - 0.7960χ - 5.0895χCH + (2.856 + 300 (18) 2.4081χ + 0.5082χ + 0.543χ) T The calculated results are shown in Table 11, where parallel calculations using our previously proposed correlations, that is, eq 6 for nonaromatic hydrocarbons and eq 7 for aromatic hydrocarbons, are also reported. Though the accuracy is comparable, our previous work shows that two different correlations have to be used for aromatic and nonaromatic hydrocarbons separately to get reasonable correlative accuracy. It is interesting to see that if connectivity indices are adopted as input parameters, one correlation is enough, which illustrates, to some extent, the advantage of using connectivity indices as structure descriptors to characterize solutes. The experimental and calculated Henry’s constants for benzene, pentane, octane, and 2-methylheptane in PDMS are shown in Figure 2. All of the correlations show good correlative accuracy; however, two different correlations were used for aromatic and nonaromatic hydrocarbons in the existing correlations. Henry’s constant data for the systems shown in Table 12 were collected for polar solutes from the literature.23,26,27 Using the database, the following correlation was developed for polar solutes.

ln(1/Kp) ) -2.898 - 0.9360χV + 0.4112χ + (5.443 + 300 2.6081χV + 0.6013χ) (19) T The calculated results are shown in Table 12, where parallel calculations using the existing correlation, that is, eq 9, are also reported. Obviously, the new correlation shows much better correlative accuracy. Generalized Correlations for Henry’s Constants of Nonpolar and Polar Solutes in Molten Poly(vinyl chloride) (PVC). Equations 20 and 21 were proposed for Henry’s constants of nonpolar and polar solutes in PVC based on the experimental data of the

2832

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002

Table 12. Calculated Results of 1/Kp for Polar Solutes in PDMS

solute methyl acetate ethyl acetate ethyl acetate 1-propyl acetate 1-butyl acetate acetone methyl ethyl ketone 2-pentanone 1-propanol 1-butanol 1-butanol 1-pentanol 1-octanol pyridine chlorobenzene system average error

no. of data temp range (K) points 373.15 363.15-473.15 373.15 373.15 373.15 373.15 373.15 363.15-473.15 373.15 363.15-473.15 373.15 373.15 363.15-473.15 363.15-473.15 433.15-468.15

1 6 1 1 1 1 1 6 1 6 1 1 6 6 3

AADa (%) eq this 9 work ref 18.9 1.2 7.3 3.7 10.3 7.6 16.9 5.0 21.4 3.4 4.3 14.8 52.3 4.7 6.1 11.9

9.7 1.4 7.6 7.1 6.4 11.0 9.9 1.7 10.9 3.3 1.1 4.8 1.7 5.8 4.1 5.8

26 23 26 26 26 26 26 23 26 23 26 26 23 23 27

Table 13. Calculated Reuslts of 1/Kp for Nonpolar Solutes in PVC

solute

temp range (K)

no. of data points

nonane decane benzene toluene ethylbenzene p-xylene carbon tetrachloride system average error

388.15-423.15 388.15-423.15 388.15-423.15 388.15-423.15 388.15-423.15 388.15-423.15 388.15-408.15

4 4 4 4 4 4 3

AAD (%)

ref

3.4 4.5 0.9 4.1 0.9 1.9 2.6 2.6

28 28 28 28 28 28 28

solute

temp range (K)

vinyl acetate acetone methyl ethyl ketone chloroform acrylonitrile fluorobenzene chlorobenzene system average error

388.15-408.15 388.15-408.15 388.15-423.15 388.15-408.15 388.15-423.15 388.15-423.15 388.15-423.15

3 3 4 3 4 4 4

solute

temp range (K)

no. of data points

ethylene ethane nonane decane benzene toluene ethylbenzene p-xylene carbon tetrachloride system average error

398.15-473.15 398.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15

4 4 5 5 5 5 5 5 5

AAD (%)

ref

8.5 8.6 8.9 7.0 5.3 5.4 7.6 5.7 7.7 7.2

29 29 28 28 28 28 28 28 28

Table 16. Calculated Results of 1/Kp for Polar Solutes in PVAc

solute

temp range (K)

no. of data points

vinyl acetate vinyl acetate acetone acetone methyl ethyl ketone methyl ethyl ketone isopropyl alcohol methyl chloride chloroform fluorobenzene chlorobenzene system average error

373.15-473.15 398.15-473.15 373.15-473.15 398.15-473.15 373.15-473.15 398.15-473.15 398.15-473.15 398.15-473.15 373.15-473.15 373.15-473.15 373.15-473.15

5 4 5 4 5 4 4 4 5 5 5

AAD (%)

ref

13.6 8.1 34.7 4.2 14.2 19.3 15.9 13.9 7.7 2.3 23.2 14.3

28 29 28 29 28 29 29 29 28 28 28

nonpolar and polar solutes, respectively.

Table 14. Calculated Results of 1/Kp for Polar Solutes in PVC no. of data points

Table 15. Calculated Results of 1/Kp for Nonpolar Solutes in PVAc

For nonpolar solutes ln(1/Kp) ) -2.809 - 0.8560χ + 0.642χV +

AAD (%)

ref

5.3 5.3 12.7 6.7 9.2 9.8 4.4 7.6

28 28 28 28 28 28 28

systems28 shown in Tables 13 and 14.

ln(1/Kp) ) -1.364 - 1.10χ + 0.9492χV + (3.369 + 300 (20) 0.8481χ + 0.3782χ + 2.1353χ) T ln(1/Kp) ) 7.66 - 3.260χ + 0.6312χ - (2.265 300 6.970χV + 9.5791χV - 4.8683χ) (21) T From Tables 13 and 14, it is obvious that the two correlations work well for this polymer, while a search of the literature shows that no simple correlations similar to eqs 1-9 exist for this polymer. Generalized Correlations for Henry’s Constants of Nonpolar and Polar Solutes in Molten Poly(vinyl acetate) (PVAc). The last industrially important polymer considered in this work is PVAc. Henry’s constant data for nonpolar and polar solutes were collected from the literature,28,29 and the following correlations were developed based on these data for

6.8055χCH + (3.394 + 1.1321χ + 1.0192χ + 300 (22) 0.8613χ) T For polar solutes ln(1/Kp) ) -7.129 + 0.92χV + (11.822 + 1.2070χV 300 3.0181χV + 2.8053χV) (23) T The correlative results from eqs 22 and 23 are shown in Tables 15 and 16, respectively. The correlations give reasonable accuracy, while no similar simple correlations can be found for this polymer from the literature. Conclusions New correlations for Henry’s constants of nonpolar and polar solutes in molten polymers were developed in this work. The calculated results show that the proposed correlations based on connectivity indices are not only more accurate than those correlations using the physical properties of the solutes as input parameters but also can be applied to more polymers. Furthermore, the correlations using connectivity indices as input parameters are easier to apply because connectivity indices can be calculated accurately as long as the molecular structure of the substance concerned is known. Therefore, they are more useful, particularly for the purposes of molecular and process design.

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2833

Acknowledgment Financial support of the Ministry of Education of China (Contract 99013) and the Natural Science Foundation of China (Contract 20106001) is greatly appreciated. Nomenclature AAD ) average absolute deviation defined in footnote a of Table 3 Kp ) Henry’s constant, atm‚(g of polymer)/cm3 (273.2 K, 1 atm) N ) number of data points T ) temperature, K Tc ) critical temperature, K zc ) critical compressibility factor Greek Letters ) mth-order connectivity index ) mth-order valence connectivity index 5χ CH ) chain connectivity index ω ) acentric factor mχ

mχV

Subscripts c ) critical value i ) data point i Superscripts cal ) calculated value exp ) experimental value

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Received for review November 30, 2001 Revised manuscript received March 12, 2002 Accepted March 14, 2002 IE010966Z