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A generalized correlation for Henry’s Law constants of nonpolar solutes in four polymers
Fluid Phase Equilibria 211 (2003) 241–256
A generalized correlation for Henry’s Law constants of nonpolar solutes in four polymers Shigeki Takishima, Gede Wibawa, Yoshiyuki Sato, Hirokatsu Masuoka∗ Department of Chemical Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan Received 20 February 2003; accepted 28 April 2003
Abstract In this work, Henry’s Law constants of nonpolar solutes in four polymers were obtained by extrapolation of finite concentration vapor–liquid equilibrium (VLE) data using the UNIQUAC equation to infinite dilution condition. The consistency of the results was confirmed by comparing them with infinite dilution data through the linear relationship between logarithm of Henry’s Law constants and inverse of temperatures. This verification provided good method for crosschecking a reliability of finite concentration VLE data with infinite dilution data. Henry’s Law constants were correlated based on both types of data as a function of temperature using the classical van’t Hoff equation. Generalized correlations of the Henry’s Law constants of solutes were proposed for polyisoprene (PI), polyisobutylene (PIB) and poly(n-butyl methacrylate) (PBMA). © 2003 Elsevier B.V. All rights reserved. Keywords: Vapor–liquid equilibria; Activity coefficient; Infinite dilution; Henry’s Law constant; Consistency; Polymer
1. Introduction With the rise in sophistication of polymer technology, polymeric materials have been developed for many applications in aerospace, electronics, automotive, biomedical, and consumer industries. The materials are often processed and produced in solution together with low molecular weight substances such as polymerization solvents, monomers and oligomers. For environment, health and safety reasons, it is necessary to reduce the content of the remaining volatile material in final polymer products. To assess methods for these type of separations, vapor–liquid equilibrium (VLE) data and correlations are required. VLE and solubility data for polymer solutions at both finite concentrations and under conditions of infinite dilution have been measured for many years and are available in [1–3]. Such data are an essential tool for understanding the behavior of mixtures as a basis in developing thermodynamic models. ∗
Corresponding author. Tel.: +81-824-247721; fax: +81-824-247721. E-mail address: [email protected] (H. Masuoka). 0378-3812/03/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-3812(03)00205-X
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To develop prediction methods of VLE, both appropriate models and reliable experimental data are required. Critical evaluation of VLE data can be a substantial task. For low molecular weight substances, thermodynamic consistency tests based on the Gibbs–Duhem equation, such as the method proposed by Herington [4] have been applied widely in checking the reliability of experimental binary VLE data. Unfortunately, such a consistency test for the VLE data of polymer solutions are difficult to apply, because vapor phase does not contain polymer. Newman and Prausnitz [5] tested the reliability of their inverse gas–liquid chromatography (IGC) method by comparing their data with activity coefficients determined by extrapolation of finite concentration VLE data. At that time, VLE data were extremely limited especially in the finite concentration ranges. Kojima et al. [6] evaluated extrapolation methods in calculating infinite dilution data from VLE data for low molecular weight substances by use of thermodynamic models such as the UNIQUAC equation. They concluded that it is possible to obtain reliable estimations of infinite dilution activity coefficients from VLE data of such systems. Evaluations of polymer solutions could not be performed. In our previous work [7], the solubility data of seven nonpolar solvents in four polymers in the temperature range of 293.2–353.2 K were reported along with parameters of the UNIQUAC equation. These parameters can be used to estimate infinite dilution activity coefficients of solutes at the temperature range studied. Therefore, verification of VLE data can be made between finite concentration VLE data and infinite dilution data through thermodynamic relationships and Henry’s Law constants. Much effort has been made in generating infinite dilution data or Henry’s Law constant data for polymer solutions [2,3]. Most data were obtained by the IGC method, which provides a rapid and simple technique for solubility measurement [8,9]. Examples in the literature exist for volatile solutes in polystyrene, polyethylene and copolymers of ethylene and vinyl acetate [5,8,9], and organic solutes in polystyrene [10]. Successful attempts have been made in correlating of Henry’s Law constant of low molecular weight vapors in polymers with temperature. Michaels and Parker [11] showed that the temperature dependence of Henry’s Law constants for gas sorption into solid polymer systems followed Arrhenius behavior. The same result was also found by Durrill and Griskey [12] for gas sorption into molten polymer systems. They also showed that the solubility data could be correlated in terms of a linear relationship between the logarithm of Henry’s Law constants and Lennard–Jones force constants. Since the value of Lennard–Jones force constants are proportional to the critical temperature, the relationship between Henry’s Law constant and critical temperature of solutes can be used to make a generalized correlation as proposed by Stern et al. [13] for Henry’s Law constant of gases and vapors in polyethylene. This approach was found to be applicable for Henry’s Law constant of gases and vapors in molten polystyrene by Stiel and Harnish [10]. A generalized correlation based on the principle of corresponding states was developed by Stiel et al. for nonpolar solutes in low density polyethylene and polyisobutylene (PIB) [14]. Chiu and Chen [15] modified Stiel’s generalized correlation by introducing an additional constant, so the accuracies could be improved. Extension of the correlations of Stiel et al. [14] and Chiu and Chen [15] was developed by Zhong and Masuoka [16] for polar solutes in low density polyethylene and poly(dimethyl siloxane). In this work, the consistency of finite concentration VLE data for polymer solutions was tested by comparing finite concentration VLE data with infinite dilution data through the relationship between Henry’s Law constants and temperatures. Correlations and generalized correlations of Henry’s Law constants are developed. This paper is organized as follows. First, description of the extrapolation method to obtain infinite dilution solubilities from finite concentration VLE data is described. Second, infinite dilution data are summarized. Third, consistency of VLE data are verified with infinite dilution data including the
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correlations obtained from the both types of data. Finally, generalized correlations of Henry’s Law constants are developed. 2. Extrapolation methods Infinite dilution activity coefficients can be obtained by extrapolating finite concentration VLE data to zero concentration of solute using many solution models. In this work, we chose the UNIQUAC equation since the interaction parameters for the solubilities of seven nonpolar solvents in four polymers in the temperature range of 293.2–353.2 K are available [7]. First, the mass fraction activity coefficient of component 1 at infinite dilution, Ω1∞ can be defined by: a1 Ω1∞ = lim (1) w1 →0 w1 Then, the UNIQUAC equation at that condition is written as: ln Ω1∞ = ln Ω1C∞ + ln γ1R∞ where
ln Ω1C∞
r1 M2 = ln r2 M1
rep n1 r1 r1 q2 r1 q2 z rep + 1 − rep − q1 n1 ln +1− q1 r2 q1 r2 n2 r2 2
rep
ln γ1R∞ = q1 n1 (−ln τ21 + 1 − τ12 ) −aij τij = exp RT
(2)
(3) (4) (5)
The mass fraction Henry’s Law constant of solute, H1 is formally defined by: f1 w1 →0 w1
H1 = lim
(6)
where f1 is fugacity of solute in Pa. Therefore, Henry’s Law constant can be calculated from an activity coefficient at the infinitely dilution condition by the following equation: H1 = Ω1∞ f10L
(7)
At low pressure, the fugacity of the pure liquid of component 1, f10L is equal to the vapor pressure in Pa. Another definition of Henry’s Law constant, Kp may be written as: p1 Kp = 0 (8) V1 where p1 is the partial pressure of the solute, in Pa, and V10 the solubility of the solute in molten polymer in m3 (STP)/kg of polymer, and Kp the Henry’s Law constant in Pa kg of polymer/m3 (STP). Therefore, the relationship between Henry’s Law constant in term of H1 and that of Kp is: H1 =
0.022414Kp M1
(9)
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where M1 is the molecular weight of the solute. Henry’s Law constants can be calculated by the UNIQUAC equation with the parameters obtained from our previous work [7] that were presented in term of Kp .
3. Infinite dilution activity coefficient data sources Experimental infinite dilution data sources and number of data points are presented in Table 1. All published data listed in the table were determined from IGC method and were converted to Kp through Eqs. (7)–(9).
4. Consistency of finite concentration with infinite dilution data Infinite dilution activity coefficients of solutes in polymers have been reported to have a linear dependence in terms of the natural logarithm of Henry’s Law constants on inverse of temperatures in the limit of experimental uncertainties [11,12]. This relationship may be applicable to verify the consistency of finite concentration VLE data for polymer solutions with infinite dilution data. In this work, finite concentration VLE data were extrapolated using the UNIQUAC model as described in the earlier section. Parameters of the UNIQUAC equation were reported in our previous work [7] for solubilities of octane, cyclohexane, cyclopentane, benzene, toluene, ethylbenzene and p-xylene in polyisoprene (PI), polyisobutylene and poly(n-butyl methacrylate) (PBMA) in the temperature range of 293.2–353.2 K, and that in poly(vinyl acetate) (PVAc) in the temperature range of 313.2–353.2 K. The parameters allow calculation of Henry’s Law constants over given ranges of temperature. The consistency of the results was verified by comparing them with available infinite dilution data listed in Table 1. For cyclopentane + PI, p-xylene + PIB, cyclopentane + PBMA and cyclopentane + PVAc systems, the consistency of finite concentration VLE data were not evaluated because experimental infinite dilution data were unavailable. For solubilities of the solutes in PI, PIB, PBMA and PVAc, infinite dilution data were compared with the calculated solubilities by the UNIQUAC equation in term of linear dependencies of natural logarithm of Henry’s Law constants on inverse of temperatures and showed good agreement as illustrated in Figs. 1–4, except for the octane + PVAc system at 313.2 and 333.2 K as shown in Fig. 5. This system exhibits a slope change at around 353 K which probably due to the glass transition temperature of PVAc (303.2 K). As observed by Smidsrod and Guillet [17] and Hsiung and Cates [18] using gas chromatography method, discontinuity or changes in slope at several degrees above the glass transition temperature tend to cause non-equilibrium sorption. They also observed that not every solute could exhibit this behavior. The discontinuity of slopes between Henry’s Law constants obtained from extrapolation of finite concentration VLE data and infinite dilution data may be caused by transition between surface adsorption and bulk sorption in the vicinity of glass transition temperature of PVAc. Henry’s Law constants obtained from both finite concentration VLE data and infinite dilution data were correlated using the van’t Hoff equation defined by: 1 b ln (10) =a+ Kp T
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Table 1 Data sources of activity at infinite dilution conditions NPa
Polymer
Solute
T (K)
PI
Octane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Toluene Toluene Ethylbenzene Ethylbenzene Ethylbenzene p-Xylene
298.2–328.2 328.0 238.15 328 298.2–318.2 298.2–328.2 328.15 328 298.2–328.2 328 313.2–328.2 328.15 328.15
3 2 2 1 3 3 1 1 3 1 2 1 1
[19] [20] [21] [20] [22] [19] [21] [20] [19] [20] [19] [21] [21]
PIB
Octane Octane Cyclopentane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Toluene Toluene Ethylbenzene
298.2 298.15 373.15 298.2 313.1–323.1 323.2–423.2 323.2–398.2 298.2 313.2–323.1 323.2–423.2 323.2–398.2 3315.15–373.15 298.2 323.2–423.2 373.15
1 1 1 1 2 5 4 1 2 5 4 2 1 5 1
[23] [24] [25] [23] [26] [27] [28] [23] [26] [27] [28] [29] [23] [27] [25]
PBMA
Octane Octane Octane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Benzene Benzene Toluene Toluene Toluene Toluene
403.2–413.2 343.2–373.2 393.15 373.0–413.0 403.0–413.2 343.2–373.2 393.15 373.0–413.0 393.2–403.2 413.2 343.2 323.2–373.2 348.2–473.2 393.2 373–413 343.2 323.2–373.2 348.2–423.2
2 2 1 3 2 2 1 3 2 1 1 2 9 1 3 1 2 4
[30] [31] [32] [20] [30] [31] [32] [20] [30] [33] [30] [31] [34] [32] [20] [33] [31] [34]
Reference
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Table 1 (Continued ) Polymer
PVAc
NPa
Solute
T (K)
Toluene Ethylbenzene p-Xylene p-Xylene
393.15 373.0–413.0 343.15–373.15 323.15–423.15
1 3 2 5
[32] [20] [31] [34]
Octane Octane Octane Octane Octane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene
353.2–393.2 374.1–433 363.15 373.15 405.15 373.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 374.1–433 398.2–408.2 408.2 393.15 363.15 373.15 353.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 373.2–448.2 374.1–433 373.2–473.2 398.2–418.2 408.2 393.2–423.2 393.15 368.15 405.15 393.15–423.15 373.95–398.15 373.15 398.15 373.15–473.15 353.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 373.2–448.2 374.1–417.2 373.2–473.2 393.2–423.2 393.2 363.15
2 12 1 1 1 2 2 2 3 11 2 1 1 1 2 4 2 2 3 4 12 9 3 1 7 1 1 1 7 2 1 1 5 4 2 2 3 4 12 5 7 1 1
[35] [36] [37] [25] [38] [39] [40] [41] [35] [36] [42] [43] [32] [37] [25] [39] [40] [41] [35] [28] [36] [34] [42] [43] [44] [32] [37] [38] [44] [45] [37] [46] [47] [39] [40] [41] [35] [28] [36] [34] [44] [32] [37]
Reference
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Table 1 (Continued ) Polymer
Total number of data points a
Solute
T (K)
NPa
Reference
Toluene Toluene Toluene Toluene Toluene Ethylbenzene Ethylbenzene Ethylbenzene Ethylbenzene Ethylbenzene p-Xylene
405.15 393.15 373.15–398.15 373.15 373.15–473.15 372.5–417.3 393.2–423.2 405.15 373.2 373.15–473.15 373.15–473.15
1 7 2 1 5 11 7 1 1 5 5
[38] [44] [45] [25] [47] [36] [44] [38] [25] [47] [47]
310
NP: number of data points.
where a and b were treated as constants over the temperature ranges studied. Since the results of octane + PVAc system at 313.2 and 333.2 K were not in agreement with the infinite dilution data, they were not considered in this regression. The results of the regression are presented in Table 2 for PI, PIB, PBMA and PVAc along with average absolute deviations (AAD) of 1/Kp . Henry’s Law constant correlations of octane and cyclohexane in PVAc gave large AAD of 19.4 and 20.6%, respectively. The large AADs came from the large uncertainties of experimental data caused by the low solubilities of octane and cyclohexane in PVAc as shown in Fig. 5 that the published data obtained from the IGC method showed large variation from author to author with magnitudes within 20%.
Fig. 1. Comparison of Henry’s Law constants of octane in polyisoprene obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
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Fig. 2. Comparison between Henry’s Law constants of toluene in polyisobutylene obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
Fig. 3. Comparison between Henry’s Law constants of benzene in poly(n-butyl methacrylate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
5. Generalized correlation for Henry’s Law constants Generalized correlation for Henry’s Law constants of solutes in polymer solutions were developed based on the principle of corresponding states. Application of the corresponding state principle needs information of critical constants for both solvent and polymer. Since the critical constants for polymer are not available, the generalized correlations were developed only based on critical constants of solutes
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Fig. 4. Comparison between Henry’s Law constants of ethylbenzene in poly(vinyl acetate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
as described by Stern et al. [13], who examined the solubility data for solutes in polyethylene. From the principle of corresponding states, most data can be represented by a linear regression of logarithms of Henry’s Law constants on (Tc /T)2 , where Tc is the critical temperature of the solute. In addition some data might be approximated by straight-line plots of logarithms of Henry’s Law constants versus (Tc /T). Those authors suggested that application of the corresponding states principle must be subjected to stringent tests before validity can be unambiguously ascertained.
Fig. 5. Comparison between Henry’s Law constants of octane in poly(vinyl acetate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
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Table 2 Parameters of the van’t Hoff equation (Eq. (10)) for solutes in nonpolar polymers Solute
AADa (%)
a
b
Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−27.503 – −25.571 −24.018 −26.027 −25.084 −26.696
4941.352 – 3967.211 3483.170 4436.914 4417.964 4996.118
1.0
PIB Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene
−25.691 −24.390 −24.953 −25.167 −25.112 −26.136
4399.694 3283.455 3732.338 3770.435 4059.913 4719.407
4.3 5.1 7.4 9.3 4.6 10.6
PBMA Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−27.153 −25.119 −25.833 −25.843 −26.917 −26.823
4691.687 3665.487 4131.315 4409.450 5064.280 5051.079
6.7 6.1 5.6 2.7 5.9 9.2
PVAc Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−24.755 −24.208 −25.018 −25.145 −25.156 −24.905
3326.620 3016.988 3735.551 3993.397 4184.184 4114.639
19.4 20.6 10.2 10.1 8.6 3.1
PI
1.3 8.8 5.9 4.3 3.6
AAD = (1/NP) |(1/Kp )calcd − (1/Kp )expt |/(1/Kp )expt ; NP, calcd and expt denote number of data points, calculated value and experimental value, respectively. a
There are several notable generalized correlations based on the corresponding states principle proposed for prediction of solubilities of solutes in molten polymers [10,14–16]. Since the generalized correlations of nonpolar solutes proposed by Stiel et al. [14] and Chiu and Chen [15] are convenient for practical use, their methods were tested for the four polymers studied in this work. The correlation proposed by Stiel et al. [14] was based on an analysis that the linear relationship resulting from a plot of ln(1/Kp ) against (Tc /T)2 should give a common intercept with slopes being solute dependent. Stiel et al. [14] found that the slope of each solute was related to its acentric factor, ω of the solute, so the generalized correlation that those authors had the following form:
1 ln Kp
Tc = A + (B + Cω) T
2 (11)
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where Tc is the critical temperature of the solutes. This correlation contains three constants (A, B and C) that should be determined from experimental data. The correlation proposed by Chiu and Chen [15] was based on an analysis that the relationship of a plot of ln(1/Kp ) versus Tc /T should give both slopes and intercepts that are highly correlated to the acentric factor of the solutes. The generalized correlation defined by Chiu and Chen [15] was: 1 Tc ln (12) = A + Bω + (C + Dω) Kp T This correlation contains four constants (A, B, C and D) that should be determined from experimental data. To examine whether either correlation forms of Eq. (11) or Eq. (12) is appropriate for describing the data, plots of logarithms of ln(1/Kp ) versus (Tc /T)2 and Tc /T were prepared for PI, PIB, BMA and PVAc. For PI, a large variation of the intercepts of ln(1/Kp ) at (Tc /T)2 = 0 were predicted for each solutes studied as shown in Fig. 6. Consequently, correlation with Eq. (11) could not be performed. The relationship of intercepts of ln(1/Kp ) at Tc /T = 0 and the slopes of Eq. (12) with ω were found to be close to linear as shown in Fig. 7. Therefore, the generalized correlation in terms of Eq. (12) was made. The following expression was obtained for PI in this work: 1 Tc ln = −21.916 − 13.682ω + (4.501 + 10.409ω) (13) Kp T The AADs for each solute and the overall AAD are presented in Table 3. The overall AAD was calculated by assigning the same weight to each system and was found to be 8.1%. For PIB, the available generalized correlations for Henry’s Law constants of solutes in molten PIB proposed by Stiel et al. [14] and Chiu and Chen [15] were tested. The generalized correlation proposed
Fig. 6. Plot ln(1/Kp ) vs. (Tc /T)2 for solutes in polyisoprene.
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Fig. 7. Intercepts and slope of Eq. (12) vs. ω of solutes for polyisoprene.
by Stiel et al. [14] was defined by: 2 1 Tc ln = −19.781 + (1.790 + 1.568ω) Kp T
(14)
and the generalized correlation proposed by Chiu and Chen [15] was defined by: 1 Tc ln = −22.234 − 11.8ω + (4.48 + 9.67ω) Kp T
(15)
The AADs for each solutes and overall AAD are presented in Table 4, which shows that the correlation proposed by Chiu and Chen [15] (Eq. (15)) with an overall AAD of 14.8% was better than that proposed by Stiel et al. [14] (Eq. (14)) that had an overall AAD of 18.3%.
Table 3 AAD between experimental and calculated values of 1/Kp for PI Solute
Temperature range (K)
Number of data points
AADa (%) (Eq. (13))
Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2
7 8 12 8 8 5
7.0 4.9 11.5 7.0 6.9 10.7
48
8.1
Overall a
Defined in Table 2.
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Table 4 AAD between experimental and calculated values of 1/Kp for PIB Solute
Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene
Temperature range (K)
293.2–353.2 293.2–373.2 293.2–423.2 293.2–423.2 293.2–423.2 293.2–353.2
Overall a
Number of data points
AADa (%) Eq. (14)
Eq. (15)
6 5 16 18 10 5
12.4 22.9 14.1 21.1 20.2 19.9
13.8 19.8 12.7 18.9 8.6 15.0
60
18.3
14.8
Defined in Table 2.
For PBMA, the analysis showed that both types of correlations were good. For correlations in the forms of Eqs. (11) and (12), the following expressions were proposed for PBMA in this work: 2 1 Tc ln (16) = −20.085 + (2.055 + 0.899ω) Kp T Tc 1 (17) = −23.430 − 9.999ω + (5.491 + 7.676ω) ln Kp T The AADs of Eqs. (16) and (17) for each solute are presented in Table 5. The results show that the correlation of data with Eq. (17) was slightly better than Eq. (16). For PVAc, analysis showed that neither form gave suitable generalized correlations due to scattering of experimental data. For PIB and PBMA, the improvement in AAD shown by correlation of Eq. (12) was not significant as shown in Tables 4 and 5. Hence, the correlation of Eq. (11) should be considered first because of simplicity. In case of Eq. (11) may not be performed, Eq. (12) can be optionally applied. The generalized Table 5 AAD between experimental and calculated values of 1/Kp for PBMA Solute
Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene Overall a
Defined in Table 2.
Temperature range (K)
293.2–403.2 293.2–413.2 293.2–473.2 293.2–423.2 293.2–413.2 293.2–423.2
Number of data points
AADa (%) Eq. (16)
Eq. (17)
9 12 23 15 7 11
17.6 25.7 17.4 20.7 25.2 29.3
26.4 39.3 7.6 11.0 19.8 21.0
77
21.8
18.4
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correlations were also applicable to slightly polar polymer (PBMA), even though the accuracy was not as good as that for nonpolar polymers (PI and PIB). 6. Conclusions Verification of finite concentration vapor–liquid equilibria data of polymer solutions by comparison with infinite dilution data through the linear relationship between logarithms of Henry’s Law constants and inverses of temperature is good method for crosschecking the reliability of finite concentration VLE data and infinite dilution data. The finite concentration solubility data for seven solvents (octane, cyclopentane, cyclohexane, benzene, toluene, ethylbenzene and p-xylene) in four polymers (polyisoprene, polyisobutylene, poly(n-butyl methacrylate) and poly(vinyl acetate) were in good agreement with infinite dilution data when available except for the octane + PVAc system at 313.2 and 333.2 K. Henry’s Law constants were correlated as function of temperature using the classical van’t Hoff equation. Generalized correlations of Henry’s Law constants of solutes could be determined except for PVAc. List of symbols a constant of Eq. (10) ai activity of component i aij interaction parameters of the UNIQUAC equation b constant of Eq. (10) standard state fugacity for the pure solvent (Pa) f0 H mass fraction Henry’s Law constant (Pa) Henry’s Law constant (Pa kg/m3 (STP)) Kp Mi molar mass of repeat unit of component i (kg/mol) rep ni number of repeat unit in component i pi partial pressure (Pa) qi area parameters for repeat unit of component i ri volume parameters for repeat unit of component i T temperature (K) Tc critical temperature (K) Tg glass transition temperature (K) 0 Vi solubility of solute in polymer (m3 (STP)/kg) w mass fraction z coordination number (z = 10) Greek letters γi activity coefficient of component i τ ij defined in the UNIQUAC equation (Eq. (5)) ω acentric factor Ωi mass fraction activity coefficient of component i Subscripts 1 solvent/solute 2 polymer
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A generalized correlation for Henry’s Law constants of nonpolar solutes in four polymers Shigeki Takishima, Gede Wibawa, Yoshiyuki Sato, Hirokatsu Masuoka∗ Department of Chemical Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan Received 20 February 2003; accepted 28 April 2003
Abstract In this work, Henry’s Law constants of nonpolar solutes in four polymers were obtained by extrapolation of finite concentration vapor–liquid equilibrium (VLE) data using the UNIQUAC equation to infinite dilution condition. The consistency of the results was confirmed by comparing them with infinite dilution data through the linear relationship between logarithm of Henry’s Law constants and inverse of temperatures. This verification provided good method for crosschecking a reliability of finite concentration VLE data with infinite dilution data. Henry’s Law constants were correlated based on both types of data as a function of temperature using the classical van’t Hoff equation. Generalized correlations of the Henry’s Law constants of solutes were proposed for polyisoprene (PI), polyisobutylene (PIB) and poly(n-butyl methacrylate) (PBMA). © 2003 Elsevier B.V. All rights reserved. Keywords: Vapor–liquid equilibria; Activity coefficient; Infinite dilution; Henry’s Law constant; Consistency; Polymer
1. Introduction With the rise in sophistication of polymer technology, polymeric materials have been developed for many applications in aerospace, electronics, automotive, biomedical, and consumer industries. The materials are often processed and produced in solution together with low molecular weight substances such as polymerization solvents, monomers and oligomers. For environment, health and safety reasons, it is necessary to reduce the content of the remaining volatile material in final polymer products. To assess methods for these type of separations, vapor–liquid equilibrium (VLE) data and correlations are required. VLE and solubility data for polymer solutions at both finite concentrations and under conditions of infinite dilution have been measured for many years and are available in [1–3]. Such data are an essential tool for understanding the behavior of mixtures as a basis in developing thermodynamic models. ∗
Corresponding author. Tel.: +81-824-247721; fax: +81-824-247721. E-mail address: [email protected] (H. Masuoka). 0378-3812/03/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-3812(03)00205-X
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To develop prediction methods of VLE, both appropriate models and reliable experimental data are required. Critical evaluation of VLE data can be a substantial task. For low molecular weight substances, thermodynamic consistency tests based on the Gibbs–Duhem equation, such as the method proposed by Herington [4] have been applied widely in checking the reliability of experimental binary VLE data. Unfortunately, such a consistency test for the VLE data of polymer solutions are difficult to apply, because vapor phase does not contain polymer. Newman and Prausnitz [5] tested the reliability of their inverse gas–liquid chromatography (IGC) method by comparing their data with activity coefficients determined by extrapolation of finite concentration VLE data. At that time, VLE data were extremely limited especially in the finite concentration ranges. Kojima et al. [6] evaluated extrapolation methods in calculating infinite dilution data from VLE data for low molecular weight substances by use of thermodynamic models such as the UNIQUAC equation. They concluded that it is possible to obtain reliable estimations of infinite dilution activity coefficients from VLE data of such systems. Evaluations of polymer solutions could not be performed. In our previous work [7], the solubility data of seven nonpolar solvents in four polymers in the temperature range of 293.2–353.2 K were reported along with parameters of the UNIQUAC equation. These parameters can be used to estimate infinite dilution activity coefficients of solutes at the temperature range studied. Therefore, verification of VLE data can be made between finite concentration VLE data and infinite dilution data through thermodynamic relationships and Henry’s Law constants. Much effort has been made in generating infinite dilution data or Henry’s Law constant data for polymer solutions [2,3]. Most data were obtained by the IGC method, which provides a rapid and simple technique for solubility measurement [8,9]. Examples in the literature exist for volatile solutes in polystyrene, polyethylene and copolymers of ethylene and vinyl acetate [5,8,9], and organic solutes in polystyrene [10]. Successful attempts have been made in correlating of Henry’s Law constant of low molecular weight vapors in polymers with temperature. Michaels and Parker [11] showed that the temperature dependence of Henry’s Law constants for gas sorption into solid polymer systems followed Arrhenius behavior. The same result was also found by Durrill and Griskey [12] for gas sorption into molten polymer systems. They also showed that the solubility data could be correlated in terms of a linear relationship between the logarithm of Henry’s Law constants and Lennard–Jones force constants. Since the value of Lennard–Jones force constants are proportional to the critical temperature, the relationship between Henry’s Law constant and critical temperature of solutes can be used to make a generalized correlation as proposed by Stern et al. [13] for Henry’s Law constant of gases and vapors in polyethylene. This approach was found to be applicable for Henry’s Law constant of gases and vapors in molten polystyrene by Stiel and Harnish [10]. A generalized correlation based on the principle of corresponding states was developed by Stiel et al. for nonpolar solutes in low density polyethylene and polyisobutylene (PIB) [14]. Chiu and Chen [15] modified Stiel’s generalized correlation by introducing an additional constant, so the accuracies could be improved. Extension of the correlations of Stiel et al. [14] and Chiu and Chen [15] was developed by Zhong and Masuoka [16] for polar solutes in low density polyethylene and poly(dimethyl siloxane). In this work, the consistency of finite concentration VLE data for polymer solutions was tested by comparing finite concentration VLE data with infinite dilution data through the relationship between Henry’s Law constants and temperatures. Correlations and generalized correlations of Henry’s Law constants are developed. This paper is organized as follows. First, description of the extrapolation method to obtain infinite dilution solubilities from finite concentration VLE data is described. Second, infinite dilution data are summarized. Third, consistency of VLE data are verified with infinite dilution data including the
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correlations obtained from the both types of data. Finally, generalized correlations of Henry’s Law constants are developed. 2. Extrapolation methods Infinite dilution activity coefficients can be obtained by extrapolating finite concentration VLE data to zero concentration of solute using many solution models. In this work, we chose the UNIQUAC equation since the interaction parameters for the solubilities of seven nonpolar solvents in four polymers in the temperature range of 293.2–353.2 K are available [7]. First, the mass fraction activity coefficient of component 1 at infinite dilution, Ω1∞ can be defined by: a1 Ω1∞ = lim (1) w1 →0 w1 Then, the UNIQUAC equation at that condition is written as: ln Ω1∞ = ln Ω1C∞ + ln γ1R∞ where
ln Ω1C∞
r1 M2 = ln r2 M1
rep n1 r1 r1 q2 r1 q2 z rep + 1 − rep − q1 n1 ln +1− q1 r2 q1 r2 n2 r2 2
rep
ln γ1R∞ = q1 n1 (−ln τ21 + 1 − τ12 ) −aij τij = exp RT
(2)
(3) (4) (5)
The mass fraction Henry’s Law constant of solute, H1 is formally defined by: f1 w1 →0 w1
H1 = lim
(6)
where f1 is fugacity of solute in Pa. Therefore, Henry’s Law constant can be calculated from an activity coefficient at the infinitely dilution condition by the following equation: H1 = Ω1∞ f10L
(7)
At low pressure, the fugacity of the pure liquid of component 1, f10L is equal to the vapor pressure in Pa. Another definition of Henry’s Law constant, Kp may be written as: p1 Kp = 0 (8) V1 where p1 is the partial pressure of the solute, in Pa, and V10 the solubility of the solute in molten polymer in m3 (STP)/kg of polymer, and Kp the Henry’s Law constant in Pa kg of polymer/m3 (STP). Therefore, the relationship between Henry’s Law constant in term of H1 and that of Kp is: H1 =
0.022414Kp M1
(9)
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where M1 is the molecular weight of the solute. Henry’s Law constants can be calculated by the UNIQUAC equation with the parameters obtained from our previous work [7] that were presented in term of Kp .
3. Infinite dilution activity coefficient data sources Experimental infinite dilution data sources and number of data points are presented in Table 1. All published data listed in the table were determined from IGC method and were converted to Kp through Eqs. (7)–(9).
4. Consistency of finite concentration with infinite dilution data Infinite dilution activity coefficients of solutes in polymers have been reported to have a linear dependence in terms of the natural logarithm of Henry’s Law constants on inverse of temperatures in the limit of experimental uncertainties [11,12]. This relationship may be applicable to verify the consistency of finite concentration VLE data for polymer solutions with infinite dilution data. In this work, finite concentration VLE data were extrapolated using the UNIQUAC model as described in the earlier section. Parameters of the UNIQUAC equation were reported in our previous work [7] for solubilities of octane, cyclohexane, cyclopentane, benzene, toluene, ethylbenzene and p-xylene in polyisoprene (PI), polyisobutylene and poly(n-butyl methacrylate) (PBMA) in the temperature range of 293.2–353.2 K, and that in poly(vinyl acetate) (PVAc) in the temperature range of 313.2–353.2 K. The parameters allow calculation of Henry’s Law constants over given ranges of temperature. The consistency of the results was verified by comparing them with available infinite dilution data listed in Table 1. For cyclopentane + PI, p-xylene + PIB, cyclopentane + PBMA and cyclopentane + PVAc systems, the consistency of finite concentration VLE data were not evaluated because experimental infinite dilution data were unavailable. For solubilities of the solutes in PI, PIB, PBMA and PVAc, infinite dilution data were compared with the calculated solubilities by the UNIQUAC equation in term of linear dependencies of natural logarithm of Henry’s Law constants on inverse of temperatures and showed good agreement as illustrated in Figs. 1–4, except for the octane + PVAc system at 313.2 and 333.2 K as shown in Fig. 5. This system exhibits a slope change at around 353 K which probably due to the glass transition temperature of PVAc (303.2 K). As observed by Smidsrod and Guillet [17] and Hsiung and Cates [18] using gas chromatography method, discontinuity or changes in slope at several degrees above the glass transition temperature tend to cause non-equilibrium sorption. They also observed that not every solute could exhibit this behavior. The discontinuity of slopes between Henry’s Law constants obtained from extrapolation of finite concentration VLE data and infinite dilution data may be caused by transition between surface adsorption and bulk sorption in the vicinity of glass transition temperature of PVAc. Henry’s Law constants obtained from both finite concentration VLE data and infinite dilution data were correlated using the van’t Hoff equation defined by: 1 b ln (10) =a+ Kp T
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Table 1 Data sources of activity at infinite dilution conditions NPa
Polymer
Solute
T (K)
PI
Octane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Toluene Toluene Ethylbenzene Ethylbenzene Ethylbenzene p-Xylene
298.2–328.2 328.0 238.15 328 298.2–318.2 298.2–328.2 328.15 328 298.2–328.2 328 313.2–328.2 328.15 328.15
3 2 2 1 3 3 1 1 3 1 2 1 1
[19] [20] [21] [20] [22] [19] [21] [20] [19] [20] [19] [21] [21]
PIB
Octane Octane Cyclopentane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Toluene Toluene Ethylbenzene
298.2 298.15 373.15 298.2 313.1–323.1 323.2–423.2 323.2–398.2 298.2 313.2–323.1 323.2–423.2 323.2–398.2 3315.15–373.15 298.2 323.2–423.2 373.15
1 1 1 1 2 5 4 1 2 5 4 2 1 5 1
[23] [24] [25] [23] [26] [27] [28] [23] [26] [27] [28] [29] [23] [27] [25]
PBMA
Octane Octane Octane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Benzene Benzene Toluene Toluene Toluene Toluene
403.2–413.2 343.2–373.2 393.15 373.0–413.0 403.0–413.2 343.2–373.2 393.15 373.0–413.0 393.2–403.2 413.2 343.2 323.2–373.2 348.2–473.2 393.2 373–413 343.2 323.2–373.2 348.2–423.2
2 2 1 3 2 2 1 3 2 1 1 2 9 1 3 1 2 4
[30] [31] [32] [20] [30] [31] [32] [20] [30] [33] [30] [31] [34] [32] [20] [33] [31] [34]
Reference
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Table 1 (Continued ) Polymer
PVAc
NPa
Solute
T (K)
Toluene Ethylbenzene p-Xylene p-Xylene
393.15 373.0–413.0 343.15–373.15 323.15–423.15
1 3 2 5
[32] [20] [31] [34]
Octane Octane Octane Octane Octane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Benzene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene
353.2–393.2 374.1–433 363.15 373.15 405.15 373.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 374.1–433 398.2–408.2 408.2 393.15 363.15 373.15 353.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 373.2–448.2 374.1–433 373.2–473.2 398.2–418.2 408.2 393.2–423.2 393.15 368.15 405.15 393.15–423.15 373.95–398.15 373.15 398.15 373.15–473.15 353.2–393.2 398.2–413.2 398.2–413.2 353.2–393.2 373.2–448.2 374.1–417.2 373.2–473.2 393.2–423.2 393.2 363.15
2 12 1 1 1 2 2 2 3 11 2 1 1 1 2 4 2 2 3 4 12 9 3 1 7 1 1 1 7 2 1 1 5 4 2 2 3 4 12 5 7 1 1
[35] [36] [37] [25] [38] [39] [40] [41] [35] [36] [42] [43] [32] [37] [25] [39] [40] [41] [35] [28] [36] [34] [42] [43] [44] [32] [37] [38] [44] [45] [37] [46] [47] [39] [40] [41] [35] [28] [36] [34] [44] [32] [37]
Reference
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Table 1 (Continued ) Polymer
Total number of data points a
Solute
T (K)
NPa
Reference
Toluene Toluene Toluene Toluene Toluene Ethylbenzene Ethylbenzene Ethylbenzene Ethylbenzene Ethylbenzene p-Xylene
405.15 393.15 373.15–398.15 373.15 373.15–473.15 372.5–417.3 393.2–423.2 405.15 373.2 373.15–473.15 373.15–473.15
1 7 2 1 5 11 7 1 1 5 5
[38] [44] [45] [25] [47] [36] [44] [38] [25] [47] [47]
310
NP: number of data points.
where a and b were treated as constants over the temperature ranges studied. Since the results of octane + PVAc system at 313.2 and 333.2 K were not in agreement with the infinite dilution data, they were not considered in this regression. The results of the regression are presented in Table 2 for PI, PIB, PBMA and PVAc along with average absolute deviations (AAD) of 1/Kp . Henry’s Law constant correlations of octane and cyclohexane in PVAc gave large AAD of 19.4 and 20.6%, respectively. The large AADs came from the large uncertainties of experimental data caused by the low solubilities of octane and cyclohexane in PVAc as shown in Fig. 5 that the published data obtained from the IGC method showed large variation from author to author with magnitudes within 20%.
Fig. 1. Comparison of Henry’s Law constants of octane in polyisoprene obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
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Fig. 2. Comparison between Henry’s Law constants of toluene in polyisobutylene obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
Fig. 3. Comparison between Henry’s Law constants of benzene in poly(n-butyl methacrylate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
5. Generalized correlation for Henry’s Law constants Generalized correlation for Henry’s Law constants of solutes in polymer solutions were developed based on the principle of corresponding states. Application of the corresponding state principle needs information of critical constants for both solvent and polymer. Since the critical constants for polymer are not available, the generalized correlations were developed only based on critical constants of solutes
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Fig. 4. Comparison between Henry’s Law constants of ethylbenzene in poly(vinyl acetate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
as described by Stern et al. [13], who examined the solubility data for solutes in polyethylene. From the principle of corresponding states, most data can be represented by a linear regression of logarithms of Henry’s Law constants on (Tc /T)2 , where Tc is the critical temperature of the solute. In addition some data might be approximated by straight-line plots of logarithms of Henry’s Law constants versus (Tc /T). Those authors suggested that application of the corresponding states principle must be subjected to stringent tests before validity can be unambiguously ascertained.
Fig. 5. Comparison between Henry’s Law constants of octane in poly(vinyl acetate) obtained from extrapolation with the UNIQUAC equation and those from published IGC data.
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Table 2 Parameters of the van’t Hoff equation (Eq. (10)) for solutes in nonpolar polymers Solute
AADa (%)
a
b
Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−27.503 – −25.571 −24.018 −26.027 −25.084 −26.696
4941.352 – 3967.211 3483.170 4436.914 4417.964 4996.118
1.0
PIB Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene
−25.691 −24.390 −24.953 −25.167 −25.112 −26.136
4399.694 3283.455 3732.338 3770.435 4059.913 4719.407
4.3 5.1 7.4 9.3 4.6 10.6
PBMA Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−27.153 −25.119 −25.833 −25.843 −26.917 −26.823
4691.687 3665.487 4131.315 4409.450 5064.280 5051.079
6.7 6.1 5.6 2.7 5.9 9.2
PVAc Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
−24.755 −24.208 −25.018 −25.145 −25.156 −24.905
3326.620 3016.988 3735.551 3993.397 4184.184 4114.639
19.4 20.6 10.2 10.1 8.6 3.1
PI
1.3 8.8 5.9 4.3 3.6
AAD = (1/NP) |(1/Kp )calcd − (1/Kp )expt |/(1/Kp )expt ; NP, calcd and expt denote number of data points, calculated value and experimental value, respectively. a
There are several notable generalized correlations based on the corresponding states principle proposed for prediction of solubilities of solutes in molten polymers [10,14–16]. Since the generalized correlations of nonpolar solutes proposed by Stiel et al. [14] and Chiu and Chen [15] are convenient for practical use, their methods were tested for the four polymers studied in this work. The correlation proposed by Stiel et al. [14] was based on an analysis that the linear relationship resulting from a plot of ln(1/Kp ) against (Tc /T)2 should give a common intercept with slopes being solute dependent. Stiel et al. [14] found that the slope of each solute was related to its acentric factor, ω of the solute, so the generalized correlation that those authors had the following form:
1 ln Kp
Tc = A + (B + Cω) T
2 (11)
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where Tc is the critical temperature of the solutes. This correlation contains three constants (A, B and C) that should be determined from experimental data. The correlation proposed by Chiu and Chen [15] was based on an analysis that the relationship of a plot of ln(1/Kp ) versus Tc /T should give both slopes and intercepts that are highly correlated to the acentric factor of the solutes. The generalized correlation defined by Chiu and Chen [15] was: 1 Tc ln (12) = A + Bω + (C + Dω) Kp T This correlation contains four constants (A, B, C and D) that should be determined from experimental data. To examine whether either correlation forms of Eq. (11) or Eq. (12) is appropriate for describing the data, plots of logarithms of ln(1/Kp ) versus (Tc /T)2 and Tc /T were prepared for PI, PIB, BMA and PVAc. For PI, a large variation of the intercepts of ln(1/Kp ) at (Tc /T)2 = 0 were predicted for each solutes studied as shown in Fig. 6. Consequently, correlation with Eq. (11) could not be performed. The relationship of intercepts of ln(1/Kp ) at Tc /T = 0 and the slopes of Eq. (12) with ω were found to be close to linear as shown in Fig. 7. Therefore, the generalized correlation in terms of Eq. (12) was made. The following expression was obtained for PI in this work: 1 Tc ln = −21.916 − 13.682ω + (4.501 + 10.409ω) (13) Kp T The AADs for each solute and the overall AAD are presented in Table 3. The overall AAD was calculated by assigning the same weight to each system and was found to be 8.1%. For PIB, the available generalized correlations for Henry’s Law constants of solutes in molten PIB proposed by Stiel et al. [14] and Chiu and Chen [15] were tested. The generalized correlation proposed
Fig. 6. Plot ln(1/Kp ) vs. (Tc /T)2 for solutes in polyisoprene.
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Fig. 7. Intercepts and slope of Eq. (12) vs. ω of solutes for polyisoprene.
by Stiel et al. [14] was defined by: 2 1 Tc ln = −19.781 + (1.790 + 1.568ω) Kp T
(14)
and the generalized correlation proposed by Chiu and Chen [15] was defined by: 1 Tc ln = −22.234 − 11.8ω + (4.48 + 9.67ω) Kp T
(15)
The AADs for each solutes and overall AAD are presented in Table 4, which shows that the correlation proposed by Chiu and Chen [15] (Eq. (15)) with an overall AAD of 14.8% was better than that proposed by Stiel et al. [14] (Eq. (14)) that had an overall AAD of 18.3%.
Table 3 AAD between experimental and calculated values of 1/Kp for PI Solute
Temperature range (K)
Number of data points
AADa (%) (Eq. (13))
Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene
293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2 293.2–353.2
7 8 12 8 8 5
7.0 4.9 11.5 7.0 6.9 10.7
48
8.1
Overall a
Defined in Table 2.
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Table 4 AAD between experimental and calculated values of 1/Kp for PIB Solute
Octane Cyclopentane Cyclohexane Benzene Toluene Ethylbenzene
Temperature range (K)
293.2–353.2 293.2–373.2 293.2–423.2 293.2–423.2 293.2–423.2 293.2–353.2
Overall a
Number of data points
AADa (%) Eq. (14)
Eq. (15)
6 5 16 18 10 5
12.4 22.9 14.1 21.1 20.2 19.9
13.8 19.8 12.7 18.9 8.6 15.0
60
18.3
14.8
Defined in Table 2.
For PBMA, the analysis showed that both types of correlations were good. For correlations in the forms of Eqs. (11) and (12), the following expressions were proposed for PBMA in this work: 2 1 Tc ln (16) = −20.085 + (2.055 + 0.899ω) Kp T Tc 1 (17) = −23.430 − 9.999ω + (5.491 + 7.676ω) ln Kp T The AADs of Eqs. (16) and (17) for each solute are presented in Table 5. The results show that the correlation of data with Eq. (17) was slightly better than Eq. (16). For PVAc, analysis showed that neither form gave suitable generalized correlations due to scattering of experimental data. For PIB and PBMA, the improvement in AAD shown by correlation of Eq. (12) was not significant as shown in Tables 4 and 5. Hence, the correlation of Eq. (11) should be considered first because of simplicity. In case of Eq. (11) may not be performed, Eq. (12) can be optionally applied. The generalized Table 5 AAD between experimental and calculated values of 1/Kp for PBMA Solute
Octane Cyclohexane Benzene Toluene Ethylbenzene p-Xylene Overall a
Defined in Table 2.
Temperature range (K)
293.2–403.2 293.2–413.2 293.2–473.2 293.2–423.2 293.2–413.2 293.2–423.2
Number of data points
AADa (%) Eq. (16)
Eq. (17)
9 12 23 15 7 11
17.6 25.7 17.4 20.7 25.2 29.3
26.4 39.3 7.6 11.0 19.8 21.0
77
21.8
18.4
254
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correlations were also applicable to slightly polar polymer (PBMA), even though the accuracy was not as good as that for nonpolar polymers (PI and PIB). 6. Conclusions Verification of finite concentration vapor–liquid equilibria data of polymer solutions by comparison with infinite dilution data through the linear relationship between logarithms of Henry’s Law constants and inverses of temperature is good method for crosschecking the reliability of finite concentration VLE data and infinite dilution data. The finite concentration solubility data for seven solvents (octane, cyclopentane, cyclohexane, benzene, toluene, ethylbenzene and p-xylene) in four polymers (polyisoprene, polyisobutylene, poly(n-butyl methacrylate) and poly(vinyl acetate) were in good agreement with infinite dilution data when available except for the octane + PVAc system at 313.2 and 333.2 K. Henry’s Law constants were correlated as function of temperature using the classical van’t Hoff equation. Generalized correlations of Henry’s Law constants of solutes could be determined except for PVAc. List of symbols a constant of Eq. (10) ai activity of component i aij interaction parameters of the UNIQUAC equation b constant of Eq. (10) standard state fugacity for the pure solvent (Pa) f0 H mass fraction Henry’s Law constant (Pa) Henry’s Law constant (Pa kg/m3 (STP)) Kp Mi molar mass of repeat unit of component i (kg/mol) rep ni number of repeat unit in component i pi partial pressure (Pa) qi area parameters for repeat unit of component i ri volume parameters for repeat unit of component i T temperature (K) Tc critical temperature (K) Tg glass transition temperature (K) 0 Vi solubility of solute in polymer (m3 (STP)/kg) w mass fraction z coordination number (z = 10) Greek letters γi activity coefficient of component i τ ij defined in the UNIQUAC equation (Eq. (5)) ω acentric factor Ωi mass fraction activity coefficient of component i Subscripts 1 solvent/solute 2 polymer
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