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Ind. Eng. Chem. Res. 1996, 35, 4386-4388
CORRELATIONS Correlation of the Henry’s Law Constants for Nonpolar Liquids in Molten Polyisobutylene Rex M. H. Chiu Department of Chemical Engineering, National Taipei Institute of Technology, Taipei 10626, Taiwan
B. D. Chen* Department of Chemical Engineering, The University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88, Manchester M60 1QD, U.K.
We present here a correlation for the estimation of solubilities of nonpolar solutes in molten polyisobutylene. Experimental solubility data from the literature are well described by the proposed equation ln(1/Kp) ) A + B/Tr, where Kp is the Henry’s law solubility constant and Tr is the reduced temperature ()T/Tc). The two correlation parameters A and B are generalized by a polynomial expansion of the acentric factor ω and the parachor [P] of the solutes. A total of 115 experimental data points, covering 18 nonpolar substances, is used to determine the proposed correlation. The mean absolute percentage error between experimental and predicted values is 4.4%. This is 2.5% lower than that achieved through use of the presently available generalized method. Introduction Determination of solubility data for a solvent present in a polymer solution is a prerequisite to the rational design of separation process equipment. This is due to the devolatilization of trace amounts of the solvent from polymer solutions which frequently takes place in polymer processing operations. The solubility behavior of a given polymer solution always deviates greatly from Raoult’s law due to the very large difference in molecular weight between the volatile component and the polymer. As a result, theoretical solubilities generally differ significantly from experimental values. In this study, published experimental solubility data determined from the static vapor sorption and gas/liquid chromatographic methods were used to determine an improved correlation for predicting the solubilities of nonpolar solutes, specifically those dissolved in molten polyisobutylene with molecular weights greater than 104 and over a wide range of temperatures (298.2-423.2 K).
T is the absolute temperature. Later investigations of the solubilities of ethylene and other organic solutes in liquid polyethylene by Maloney and Prausnitz (1976) led to a correlation involving the critical temperature and acentric factor (ω) of the solute. The study of Stiel and Harnish (1976) and Stiel et al. (1985) showed that the predictions based on the correlation of Stern et al. (1969) gave a better result than others for nonpolar solutes in both molten polystyrene and low-density polyethylene. An early study for the estimation of the solubilities of nonpolar substances in molten polyisobutylene has been made by Stiel et al. (1985) using an approach similar to that of Stern et al. (1969). Based on a linear relationship between the logarithm of 1/Kp and (Tc/T)2, it was assumed that the intercept at (Tc/T)2 ) 0 was constant for 18 nonpolar solutes. For 18 nonpolar substances, the mean absolute percentage error between the experimental and predicted values is 6.9%. Proposed Correlations
Correlation Methods There are several notable correlations available in the literature for prediction of solubilities of nonpolar solutes in molten or thermally softened polymers. Particular focus has been placed on polyethylene because of its huge worldwide sales. For example, Michaels and Bixler (1961) and later Durrill and Griskey (1969) proposed that the logarithm of the Henry’s law constant is a linear function of either the gas Lennard-Jones force constant or the critical temperature (Tc) of the solute for the solubilities of the solutes in molten or thermally softened polymers. Stern et al. (1969) suggested that the principle of corresponding states could be used to correlate the solubility of both gases and vapors in polyethylene through a linear relationship between the logarithm of the Henry’s law constant and (Tc/T)2, where * Author to whom correspondence should be addressed.
S0888-5885(96)00290-4 CCC: $12.00
Herein we propose a correlation to describe the temperature dependence of the solubility of nonpolar liquids in molten polyisobutylene
ln(1/Kp) ) A + B(Tc/T)
(1)
where A and B are correlation parameters for a given system. Gas-liquid chromatographic (GLC) studies of solubility in polyisobutylene were made by Newman and Prausnitz (1972, 1973, 1974) for pentane, cyclohexane, benzene, and toluene for 298.2-423.2 K. A similar technique has been used by Leung and Eichinger (1974a,b) for pentane, hexane, heptane, nonane, cyclohexane, and benzene for 298.2-338.2 K and by Hammers and De Ligny (1971) for 11 nonpolar organic compounds for 313.2-433.2 K. Lichtenthaler et al. (1974) conducted the solubility studies for hexane, © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4387
cyclohexane and benzene for 323.2-398.2 K with both capillary and packed chromatographic columns in gasliquid chromatography. Low-pressure vapor sorption experiments at finite concentrations were performed by Prager et al. (1953) for butane, isobutane, pentane, isopentane, and neopentane for 298.2-319.7 K and by Baker et al. (1962) for pentane for 298.2-328.2 K. The experimental results from the finite concentration studies were extrapolated to infinite dilution of the solute to obtain the values of 1/Kp used in the present study. It has been pointed out by numerous authors (Hammers and De Ligny, 1971; Leung and Eichinger, 1974a,b; Lichtenthaler et al., 1974) that gas-liquid partition chromatography shows much promise as a technique for rapid measurement of polymer-solvent interactions in highly concentrated polymer solutions, whereas conventional methods (e.g., vapor sorption) usually are not capable of giving a direct measurement. However, a comparison between the limiting activity coefficients obtained directly by GLC and by extrapolation of data from concentration regions accessible by conventional methods has been made by several authors (Hammers and De Ligny, 1971; Newman and Prausnitz, 1972, 1973; Leung and Eichinger, 1974a,b; Lichtenthaler et al., 1974), indicating that good agreement was obtained between these two methods. In the present study, published data from both GLC and vapor sorption measurements were used to determine the proposed correlation. The experimental solubilities of gases in molten polymers reported in the literature as mentioned above appear to be correlatable by Henry’s law
X ) Pg/Kp
(2)
where X is the concentration of gas in the melt, cm3 (273.2K, 1 atm) g-1, Kp is Henry’s law constant, g atm cm-3 (273.2 K, 1 atm), and Pg is partial pressure of the dissolved gas, atm. Different solubility forms have been modified to yield a consistent form of the Henry’s law constant in terms of 1/Kp. Detailed calculation procedures for 1/Kp are given elsewhere (Stiel et al., 1985). For 18 different nonpolar liquids including alkanes, isomeric alkanes, cycloalkanes, aromatics, and tetrachloromethane, the correlations suggested by eq 1 are best represented by a single curve with the average regression coefficient (r2) of 0.999. Values of intercept A and slope B are listed in Table 1, together with the critical temperature Tc, the parachor [P], and the acentric factor ω of the solute. Both the slopes and intercepts are highly related to the acentric factor ω of the solutes. The regression lines are described by
A ) -(3.80 ( 0.33) - (11.8 ( 0.9)ω
(3)
and
Table 1. Values of Solute Properties and Regressed Parameters, A and B substance
T c ,a K
ωa
[P]b
A
B
butane pentane hexane heptane octane nonane decane isobutane neopentane isopentane 2,3-dimethylbutane 2,2-dimethylpentane 2,2,4-trimethylpentane 2,5-dimethylhexane cyclohexane benzene toluene tetrachloromethane
425.2 469.7 507.5 540.3 568.8 594.6 617.7 408.2 433.8 460.4 500.0 520.5 544.0 550.0 553.5 562.2 591.8 556.4
0.193 0.251 0.296 0.351 0.394 0.444 0.490 0.176 0.197 0.227 0.247 0.289 0.303 0.352 0.213 0.212 0.257 0.194
190.3 231.0 270.7 310.8 251.0 390.5 431.0 191.0c 231.0c 229.0c 266.2 306.5 343.8 347.0 241.7 206.1 246.2 219.8
-5.99 -6.63 -6.70 -7.98 -8.50 -9.34 -9.64 -4.49 -6.52 -7.90 -6.65 -7.14 -7.11 -8.06 -6.28 -6.31 -6.62 -6.84
6.29 6.80 7.01 7.92 8.33 8.96 9.24 5.10 6.57 7.59 6.82 7.20 7.21 7.94 6.59 6.43 6.84 6.79
a Reid et al., 1987. b Quayle, 1953. c Estimated by the method of Quayle (Quayle, 1953).
Table 2. Evaluation of the Henry’s Law Constants Correlation average deviation,a %
solute
no. of data points ref
butane 3 pentane 3 pentane 4 pentane 3 pentane 3 pentane 3 hexane 3 hexane 6 hexane 5 heptane 5 heptane 3 octane 5 nonane 4 nonane 1 decane 4 isobutane 3 neopentane 3 isopentane 3 2,3-dimethyl5 butane 2,2-dimethyl5 pentane 2,2,4-trimethyl6 pentane 2,5-dimethyl5 hexane cyclohexane 1 cyclohexane 6 cyclohexane 5 benzene 3 benzene 5 toluene 5 tetrachloro5 methane average (115)
b c d e b f e g c c e c c h c b b b c
this work ∆T, K
eqs 1, 3, eqs 1, Stiel et al. and 4 5, and 6 method
298.2-320.2 313.2-353.2 298.2-328.2 298.2-338.2 298.2-320.2 298.2-323.2 298.2-338.2 323.2-398.2 313.2-393.2 313.2-393.2 298.2-338.2 313.2-393.2 333.2-393.2 298.2 333.2-393.2 298.2-320.2 298.2-320.2 298.2-320.2 313.2-393.2
1.54 2.26 5.49 7.52 2.07 2.95 7.26 8.85 2.99 1.73 5.98 1.73 1.94 8.45 3.12 4.69 15.03 5.06 1.21
7.22 2.45 4.46 6.87 0.79 2.69 5.57 7.56 1.91 3.29 3.32 2.27 3.42 3.93 1.79 5.53 7.92 4.02 1.34
4.26 5.60 2.65 5.62 1.54 3.04 8.88 6.57 9.39 7.38 7.04 5.71 4.84 0.92 4.53 1.31 9.86 4.38 6.17
c 313.2-393.2
4.26
4.28
9.13
c 313.2-393.2
6.94
2.28
10.39
c 313.2-393.2
2.70
3.14
7.36
10.98 7.54 5.57 13.50 12.58 2.80 7.90
8.38 9.71 7.18 6.19 6.95 2.69 4.32
3.18 11.41 8.37 22.28 10.15 3.54 2.81
5.4
4.4
6.9
h g c e i i c
298.2 323.2-373.2 313.2-393.2 298.2-338.2 323.2-423.2 323.2-423.2 313.2-393.2
∑|(exptl. value - calcd. value)/exptl. value|(100/no. of data points). b Prager et al., 1953. c Hammers and De Ligny, 1971. d Baker et al., 1962. e Leung and Eichinger, 1974b. f Newman and Prausnitz, 1972. g Lichtenthaler et al., 1974. h Leung and Eichinger, 1974a. i Newman and Prausnitz, 1973. a
B ) (4.48 ( 0.18) + (9.67 ( 0.51)ω
(4)
with regression coefficients of 0.92 and 0.96, respectively. Equations 1, 3, and 4 yield the first proposed approach for predicting the solubilities of nonpolar solutes in molten polyisobutylene. From 115 data points covering 18 substances, predicted results give a 5.4% mean absolute percentage error from the experimental values, whereas the corresponding value for the method of Stiel et al. is 6.9%. The mean absolute percentage error for
each individual compound is given in Table 2, together with the number of experimental solubilities used and the temperature range (∆T) studied. Attempts have been made to improve the accuracy of the proposed approach through the introduction of a second solute property, the parachor [P], for the correlations of the values of A and B. The combination of
4388 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
these two parameters in a polynomial expansion is intended to separate polarity effects from those due to molecular size and shape. This can be extended to include treatment of polar materials (Chiu et al., 1988). Hence, a relatively simple generalized expression is obtained which can be widely used for estimating the physical properties of both polar and nonpolar materials. The intercept A and slope B were then used to correlate the polynomial coefficients with the parachor and acentric factor of the solutes. For 18 nonpolar liquids, the results lead to the following two equations:
A ) -3.535 - 3.610 × 10-2[P] + 2.127 × 10ω + 1.569 × 10-2[P]ω + 5.100 × 10-5[P]2 5.818 × 10ω2 (5) and
B ) 3.817 + 2.776 × 10-2[P] - 1.273 × 10ω 1.952 × 10-3[P]ω - 4.545 × 10-5[P]2 + 3.614 × 10ω2 (6) with average deviations of 2.2 and 1.2%, respectively. Equation 1, in conjunction with eqs 5 and 6, constitutes the second proposed approach for the estimation of solubilities of nonpolar solutes in molten polyisobutylene. From 115 experimental solubility points covering 18 solutes, calculated results give a 4.4% mean absolute percentage error from experimental data. This is 2.5% lower than that achieved from the approach of Stiel et al. (1985). The mean absolute percentage error for each substance is given in Table 2. The predicted results reveal that both the two proposed approaches generally offer considerable improvement in accuracy and reliability over the correlation of Stiel et al. (1985), in particular the approach using a combination of the acentric factor and parachor of the solute. Concepts similar to those described above would also be useful for other solute-polymer systems. Acknowledgment We would like to thank Professor John Garside for his advice in this work. Nomenclature A, B ) parameters defined in eq 1 Kp ) Henry’s law constant, g atm cm-3 (273.2 K, 1 atm) Pg ) partial pressure of the dissolved gas, atm [P] ) parachor of solute T ) absolute temperature, K Tc ) critical temperature, K Tr ) reduced temperature ∆T ) range of temperatures studied X ) concentration of gas in the melt, cm3 (273.2 K, 1 atm) g-1 ω ) acentric factor of solute
Literature Cited Baker, C. H.; Brown, W. B.; Gee, G.; Rowlinson, J. S.; Stubley, D.; Yeadon, R. E. A Study of the Thermodynamic Properties and Phase Equilibria of Solutions of Polyisobutene in n-Pentane. Polymer 1962, 3, 215. Chiu, R. M. H.; Chen, B. D.; Chien, W. N. Correlation of Liquid Vapour Pressures with An Acentric Factor and Parachor. CIChE J. 1988, 19, 131. Durrill, P. L.; Griskey, R. G. Diffusion and Solution of Gases into Thermally Softened or Molten Polymers: Part 2. Relation of Diffusivities and Solubilities with Temperature, Pressure and Structural Characteristics. AIChE J. 1969, 15, 106. Hammers, W. E.; De Ligny, C. L. A Gas-Chromatographic Investigation of the Thermodynamics of Solutions of Some Normal and Branched Alkanes, Cyclohexane, Benzene and Tetrachloromethane in Polyisobutylene. Recl. Trav. Chim. 1971, 90, 912. Leung, Y.-K.; Eichinger, B. E. Gas-Liquid Chromatography on Polymers. I. Polyisobutylene Hydrocarbons at 25°. J. Phys. Chem. 1974a, 78, 60. Leung, Y.-K.; Eichinger, B. E. Gas-Liquid Chromatography on Polymers. II. Temperature Coefficients for PolyisobutyleneHydrocarbons. Macromolecules 1974b, 7, 685. Lichtenthaler, R. N.; Liu, D. D.; Prausnitz, J. M. Polymer-Solvent Interactions from Gas-Liquid Chromatography with Capillary Columns. Macromolecules 1974, 7, 565. Maloney, D. P.; Prausnitz, J. M. Solubilities of Ethylene and Other Organic Solutes in Liquid, Low-Density Polyethylene in the Range 124-300 °C. AIChE J. 1976, 22, 74. Michaels, A. S.; Bixler, H. J. Solubility of Gases in Polyethylene. J. Polym. Sci. 1961, 50, 393. Newman, R. D.; Prausnitz, J. M. Polymer-Solvent Interactions from Gas-Liquid Partition Chromatography. J. Phys. Chem. 1972, 76, 1492. Newman, R. D.; Prausnitz, J. M. Thermodynamics of Concentrated Polymer Solutions Containing Polyethylene, Polyisobutylene, and Copolymers of Ethylene with Vinyl Acetate and Propylene, AIChE J. 1973, 19, 704; 1974, 20, 206. Prager, S.; Bagley, E.; Long, F. A. Equilibrium Sorption Data for Polyisobutylene Hydrocarbon Systems. J. Am.Chem. Soc. 1953, 75, 2742. Quayle, O. R. The Parachors of Organic Compounds, Chem. Rev. 1953, 53, 439. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987; Appendix A. Stern, S. A.; Mullhaupt, J. T.; Gareis, P. J. The Effect of Pressure on the Permeation of Gases and Vapours through Polyethylene. Usefulness of the Corresponding States Principle. AIChE J. 1969, 15, 64. Stiel, L. I.; Harnish, D. F. Solubility of Gases and Liquids in Molten Polystyrene. AIChE J. 1976, 22, 117. Stiel, L. I.; Chang, D. K.; Chu, H. H.; Han, C. D. The Solubility of Gases and Volatile Liquids in Polyethylene and Polyisobutylene at Elevated Temperatures. J. Appl. Polym. Sci. 1985, 30, 1145.
Received for review May 24, 1996 Revised manuscript received July 18, 1996 Accepted July 18, 1996X IE960290L
X Abstract published in Advance ACS Abstracts, September 15, 1996.
