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Applicability of corresponding-states group-contribution methods for the estimation of surface tension of multicomponent liquid mixtures at 298.15 K
Fluid Phase Equilibria 287 (2010) 151–154
Contents lists available at ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Applicability of corresponding-states group-contribution methods for the estimation of surface tension of multicomponent liquid mixtures at 298.15 K Anjali Awasthi, Bhawana S. Tripathi, Aashees Awasthi ∗ Department of Physics, University of Lucknow, Lucknow 226007, India
a r t i c l e
i n f o
Article history: Received 19 August 2009 Received in revised form 1 October 2009 Accepted 7 October 2009 Available online 6 November 2009 Keywords: Corresponding-states group-contribution methods Surface tension Multicomponent liquid mixtures
a b s t r a c t Corresponding-states group-contribution methods (CSGC-ST1 and CSGC-ST2) have been applied to four binary liquid mixtures (propyl acetate + o-xylene, propyl acetate + m-xylene, propyl acetate + p-xylene and propyl acetate + ethyl benzene); two ternary (benzene + cyclohexane + toluene and n-hexane + cyclohexane + benzene) and two quaternary liquid mixtures (pentane + hexane + cyclohexane + benzene and pentane + hexane + benzene + toluene) at 298.15 K. In this work, the CSGC-ST2 method is modified and extended to multicomponent liquid mixtures. The excess magnitudes of surface tension were also calculated and graphs were plotted using Redlich–Kister method. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Surface tension is a thermo-physical phenomenon containing inherent information about the interaction between the molecules and hence can be utilized in studying the behaviour of liquid mixtures. Surface tension plays an important role in heat and mass transfer phenomena at the interface in liquid mixtures. Such studies also prove helpful in cavity theory of molecular association [1–3] and acts as a reliable source in depicting the strength of the solvent in liquid chromatography [4]. Literature survey reveals that experimental data of surface tension of multicomponent liquid systems are scarce. Thus, theoretical modeling for the prediction of surface tension is found to be of great help. Several empirical and thermodynamic methods have been deduced for the computation of surface tension but its application is limited mostly to binary liquid mixtures [5–9]. Brock–Bird [7] and Miller [8] proposed the estimation of surface tension by corresponding-states theorem. Macleod [9] suggested a relation between surface tension, liquid density and vapour density. The Sudgen method [10] requires the determination of the parachor but values of surface tension are found to be highly sensitive to variations in the parachor. Patterson and Rastogi [11] estimated the surface tension of polyatomic liquids using reduced parameters only. Various methods correlate the surface tension with certain physical properties of liquids. However, the accuracy and availability of these properties usually limit the application of these methods.
∗ Corresponding author. Tel.: +91 9415003753. E-mail address: [email protected] (A. Awasthi). 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.10.008
Group-contribution methods have been used for the estimation of transport property of liquid mixtures by several workers. Since the first developments of group-contribution methods by Riedel [12] in 1949 and Lydersen [13] in 1955, a large number of methods have been developed for the estimation of critical property data. Group-contribution methods and the corresponding-states principle have proved to be of great significance in the determination of various physical parameters of compounds. The correspondingstates group-contribution method (CSGC) combines the simplicity of the corresponding-states method with the predictability and flexibility of the group-contribution method. Li et al. [14] proposed CSGC-ST1 and CSGC-ST2 models to evaluate the surface tension of pure components which have been extended to the multicomponent liquid mixtures in the present investigation. In this work, the CSGC-ST1 method is extended to quaternary liquid mixtures and the CSGC-ST2 method is modified and extended to multicomponent liquid mixtures. 2. Prediction of surface tension of liquid mixtures The structural groups were defined in a standardized form and fragmentation of the molecular structures resulted in the evaluation of various critical properties of pure liquids and hence paved way for the estimation of surface tension of liquid mixtures. For the group-contribution predictions, only the molecular structure of the compound is required. Brock and Bird [7] successfully employed the corresponding-states principle to develop a relation for the surface tension of non-polar liquids. Miller on the other hand modified the Brock–Bird relation by introducing a Riedel coefficient [15]. The CSGC-ST1 equation which is similar to the Miller’s modified
152
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
Table 1 Parameters of pure componentsa . Components
Mol. wt.
Tb (K)
20 ◦ C (10−3 N m−1 )
o-Xylene m-Xylene p-Xylene Propyl acetate Ethyl benzene Pentane Hexane Benzene Toluene Cyclohexane
106.168 106.168 106.168 102.134 106.169 72.151 86.178 78.114 92.142 84.162
417.6 412.3 411.5 374.7 409.3 309.2 341.9 353.2 383.8 353.8
30.03 28.63 28.31 24.28 29.04 16.05 18.95 28.88 28.53 24.98
a
Refs. [15,22].
Brock–Bird relation [14,16] has been extended to multicomponent liquid mixtures for the estimation of the surface tension and is represented as;
x P∗ i i ci
mix =
a
101.325
c
b
xi ˛ci − d
1−
i
e xi Tri∗
(1)
i
The Othmer equation can be used to obtain surface tension at 20 ◦ C (i.e. 293.15 K) using a corresponding-states groupcontribution method (CSGC-ST2) and it was used as a Ref. [17]. In the present work, CSGC-ST2 method has been extended successfully to multicomponent liquid mixtures and is represented by the following expression: mix =
xi 20,i
i
∗ g xT i i ∗ ri
1− 1−
(2)
xT i i ri,20
where ∗ Tri,20 =
Tci∗ =
293.15 Tci∗
ATi + BTi
(3)
M j
Tbi
nj Tj + CTi
M j
Propyl acetate (1) + o-xylene (2)
Propyl acetate (1) + m-xylene (2)
Mole frac. (10−3 N m−1 )
Mole frac. (10−3 N m−1 )
x1
Exp.
CSGC-ST1 CSGC-ST2 x1
Exp.
CSGC-ST1 CSGC-ST2
0.0000 0.0998 0.1995 0.2996 0.4002 0.5001 0.6004 0.7002 0.7999 0.9000 1.0000
29.89 28.94 28.14 27.47 26.85 26.27 25.71 25.18 24.69 24.25 23.91 APD
29.57 29.03 28.49 27.95 27.42 26.89 26.36 25.84 25.32 24.81 24.30 1.77
28.47 27.86 27.26 26.70 26.16 25.71 25.30 24.94 24.59 24.25 23.91 APD
28.13 27.74 27.35 26.97 26.58 26.20 25.81 25.44 25.05 24.67 24.30 2.30
29.47 28.90 28.32 27.75 27.17 26.60 26.02 25.45 24.87 24.30 23.72 0.44
0.0000 0.0997 0.1999 0.2986 0.4002 0.4986 0.6004 0.6984 0.8004 0.9007 1.0000
Propyl acetate (1) + p-xylene (2)
xi Tci∗
i
Table 2 Experimentalb and theoretical values of surface tension of various binaries at 298.15 K.
2 + DTi
nj Tj i
M j
3
(4)
0.0000 0.1000 0.1995 0.2981 0.3986 0.4993 0.6002 0.6991 0.7993 0.8999 1.0000 b
28.08 27.56 27.04 26.54 26.05 25.61 25.23 24.89 24.56 24.23 23.91 APD
28.60 28.16 27.73 27.30 26.86 26.43 25.99 25.57 25.15 24.72 24.30 2.56
28.09 27.65 27.21 26.78 26.34 25.91 25.47 25.04 24.59 24.16 23.72 −0.05
Propyl acetate (1) + ethyl benzene (2) 27.77 27.36 26.96 26.56 26.16 25.75 25.34 24.94 24.54 24.13 23.72 −0.16
0.0000 0.1005 0.1994 0.3002 0.4005 0.5000 0.6002 0.6997 0.7998 0.9000 1.0000
28.79 28.15 27.51 26.89 26.35 25.91 25.50 25.10 24.70 24.30 23.91 APD
27.58 27.33 27.06 26.77 26.47 26.14 25.80 25.45 25.08 24.69 24.30 −0.51
28.50 28.02 27.55 27.07 26.59 26.11 25.63 25.16 24.68 24.20 23.72 0.06
Ref. [21].
recognized. That is why normal results show a lowering of surface tension of the mixtures. The n-alkanes and their derivatives form random coils due to internal rotation around C–C bonds, and this tendency increases with an increase in the chain length. In both the ternary and quaternary n-alkane systems, mixtures’ surface layer is enriched by the component of lower surface tension, thereby minimizing the surface tension of the mixture. However, it would not be proper to say that it is the only reason for the discrepancies. Slightly lower surface tension values in ternary and quaternary liq-
nj Tj i
3. Results and discussion The surface tension of the four binary liquid mixtures propyl acetate + o-xylene, propyl acetate + m-xylene, propyl acetate + p-xylene and propyl acetate + ethyl benzene; the two ternary liquid mixtures benzene + cyclohexane + toluene and n-hexane + cyclohexane + benzene and the two quaternary liquid mixtures pentane + hexane + cyclohexane + benzene and pentane + hexane + benzene + toluene have been investigated at 298.15 K. Experimental values of surface tension of these liquid mixtures are taken from literature [18–21]. The properties of the pure components are listed in Table 1 [15,22]. The experimental and calculated (evaluated by the CSGC methods) values of surface tension for binary, ternary and quaternary liquid mixtures are listed in Tables 2–4, respectively. Fig. 1(a–c) shows the excess values of surface tension as a function of mole fraction of the first component for three binary liquid mixtures. From Tables 2–4, it is found that the surface tension decreases gradually with increasing concentration of the first component. It can also be observed that the calculated values of the surface tension using CSGC methods are slightly higher than the experimental values in some cases. Gibbs enrichment of the surface of the mixture by the component with lower surface tension is well
Table 3 Experimentalc and theoretical values of surface tension of ternary liquid mixtures at 298.15 K. Mole frac.
Mole frac.
(10−3 N m−1 )
x1
x2
Exp.
CSGC-ST1
CSGC-ST2
n-Hexane (1) + cyclohexane (2) + benzene (3) 0.0771 0.4315 25.20 0.1269 0.4149 24.60 0.1795 0.3854 24.30 0.2279 0.3501 24.00 0.2812 0.3348 23.60 0.3021 0.3643 23.20 0.3231 0.4851 22.10 0.3448 0.4566 22.90 APD
25.73 25.28 24.84 24.47 23.98 23.67 23.07 22.94 2.16
25.75 25.33 24.94 24.62 24.17 23.85 23.17 23.07 2.64
Benzene (1) + cyclohexane (2) + toluene (3) 0.1331 0.2123 26.80 0.1402 0.2115 26.82 0.1636 0.2297 26.78 0.1854 0.2459 26.72 0.2303 0.2829 26.58 0.2535 0.3018 26.60 0.2763 0.3197 26.64 0.2980 0.3365 26.24 APD
27.23 27.24 27.20 27.17 27.09 27.05 27.01 26.98 1.78
27.22 27.23 27.17 27.12 27.00 26.94 26.88 26.82 1.50
c
Refs. [19,20].
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
153
Table 4 Experimentald and theoretical values of surface tension of quaternary liquid mixtures at 298.15 K. Mole frac.
Mole frac.
Mole frac.
(10−3 N m−1 )
x1
x2
x3
Exp.
CSGC-ST1
CSGC-ST2
Pentane (1) + hexane (2) + cyclohexane (3) + benzene (4) 0.0488 0.1238 0.1831 25.5 25.6 0.0658 0.1078 0.2036 25.1 25.4 0.0813 0.0934 0.2238 24.9 25.3 0.1006 0.0778 0.2430 24.8 25.1 0.1180 0.0629 0.2615 24.7 24.9 0.1243 0.0466 0.2842 24.7 24.9 0.1410 0.1304 0.3129 23.7 23.7 0.1560 0.1262 0.1513 23.9 24.1 0.1285 0.1192 0.5888 22.8 23.0 0.1537 0.0925 0.1685 24.8 24.4 0.1649 0.1013 0.5177 22.9 22.9 0.1368 0.1258 0.1507 24.0 24.3 0.0910 0.1721 0.6137 22.0 22.9 0.0649 0.1378 0.1103 25.5 25.4 0.1810 0.1656 0.2970 23.1 22.8 APD 0.65
25.7 25.5 25.4 25.2 25.1 25.1 23.9 24.4 23.1 24.7 23.1 24.7 23.0 25.6 23.1 1.45
Pentane (1) + hexane (2) + benzene (3) + toluene (4) 0.0943 0.0918 0.4587 25.9 0.1300 0.1373 0.2974 24.6 0.1278 0.1288 0.3589 24.9 0.1450 0.1291 0.3376 24.5 0.1492 0.1384 0.3421 24.4 0.1843 0.1484 0.2711 24.3 0.1823 0.1640 0.3613 24.1 0.1819 0.1601 0.3842 24.0 0.1250 0.1665 0.2455 23.9 0.1691 0.2041 0.2218 23.1 0.1866 0.0826 0.1250 24.3 0.1372 0.1579 0.5548 24.6 0.0660 0.1053 0.7033 25.8 0.0524 0.1434 0.4201 25.5 0.1568 0.0468 0.4582 25.1 APD
26.0 25.1 25.2 25.0 24.9 24.3 24.2 24.3 24.9 23.9 24.9 24.9 26.3 26.0 25.7 1.80
d
25.7 24.6 24.7 24.5 24.3 23.7 23.6 23.7 24.3 23.3 24.1 24.4 26.0 25.8 25.2 −0.30
Ref. [18].
uid mixtures may also be due to small contributions from sterichindrance. The average percentage deviations (APD) for the calculated values of surface tension by the CSGC-ST1 and CSGC-ST2 methods in comparison with the experimental values for binary, ternary and quaternary liquid mixtures are presented in Tables 2–4, respectively. The maximum APD values for binary liquid systems are found to be 2.56% (CSGC-ST1) and 0.44% (CSGC-ST2); for ternary liquid systems 2.16% (CSGC-ST1) and 2.64% (CSGC-ST2) and for quaternary liquid systems 0.65% (CSGC-ST1) and 1.80% (CSGCST2), respectively. Thus, it becomes quite obvious that both CSGC methods not only are simple to use but also improved predictive accuracies with overall APD values of 1.51% (CSGC-ST1) and 1.01% (CSGC-ST2) for multicomponent liquid mixtures. The calculated results show that CSGC methods provide a wide range of applicability and their estimation accuracy is significantly superior over conventional corresponding-states methods. The values of excess surface tension ( E ) is evaluated by the relation; E = mix −
xi i
Fig. 1. (a–c) E versus mole fraction (x1 ) at 298.15 K. Explanation: the excess value of surface tension is plotted against mole fraction of the first component of the binary liquid mixture using Redlich–Kister fitting method.
number of coefficients and the model fits the data fairly well with total error being very small. To fit data, Redlich–Kister equation [23] of the following form was used (Table 5): E = x 1 x2
m
ak (2x1 − 1)k
(6)
k=0
(5)
i
Redlich–Kister equation describes the non-ideal behaviour of the system investigated by fitting the experimental data through the thermodynamically consistent correlations. It provides a flexible algebraic expression for representing the excess function of a complex liquid mixture and gives an indication of the apparent complexity of the mixture as accuracy of the experimental data and the fit desired. It offers an increased flexibility using the desired
Table 5 Coefficients (ak ) from Eq. (6) for binary liquid mixtures at 298.15 K. Binary system
Propyl acetate (1) + o-xylene (2) Propyl acetate (1) + m-xylene (2) Propyl acetate (1) + p-xylene (2) Propyl acetate (1) + ethyl benzene (2)
Binary coefficients a0
a1
a2
−0.1868 −2.1288 −0.0976 0.8150
−0.0137 0.2000 −0.0072 −0.0768
−0.0197 −5.3475 −0.0289 −0.0183
154
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
From Fig. 1(a–c), it can be seen that the curves representing these points are asymmetrical in nature. It is interesting to note that a maximum or minimum in excess surface tension values is obtained at almost equimolar concentration of the binary liquid mixtures. A positive deviation is obtained for propyl acetate + ethyl benzene while negative deviations are observed for propyl acetate + o-xylene and propyl acetate + p-xylene over the entire composition range. Positive values of E are due to weak intermolecular interactions while negative values of E are attributed to strong intermolecular interactions between the molecules [24]. Thus, propyl acetate with o-xylene/p-xylene show strong intermolecular associations while propyl acetate + ethyl benzene demonstrates a weak intermolecular interaction between the molecules.
Greek letters surface tension of mixture mix i surface tension of ith component 20, i surface tension at 293.15 K (10−3 N m−1 ) of ith component E excess value of surface tension ˛ci Riedel coefficient (Ref. [25]) of ith component
4. Conclusions
[1] O. Sinanoglu, A. Fernandez, Biophys. Chem. 21 (1985) 157–162. [2] K.A. Connors, J.L. Wright, Anal. Chem. 61 (1989) 194–198. [3] O. Sinanoglu, in: B. Bullman (Ed.), Molecular Associations in Biology, Academic Press, New York, 1968. [4] W.J. Cheong, P.W. Carr, J. Liq. Chromatogr. 10 (1987) 561–581. [5] R. Auerbach, Experiential 4 (1948) 473–474. [6] O. Nomoto, J. Phys. Soc. Japan 8 (1953) 553–560. [7] J.R. Brock, R.B. Bird, AIChE J. 1 (1955) 174–177. [8] D.G. Miller, Ind. Eng. Chem. Fundam. 2 (1963) 78–79. [9] D.B. Macleod, Trans. Faraday Soc. 19 (1923) 38–42. [10] S. Sudgen, J. Chem. Soc. 125 (1924) 1167–1189. [11] D. Patterson, A.K. Rastogi, J. Phys. Chem. 74 (1970) 1067–1071. [12] L. Riedel, Z. Elektrochem. 53 (1949) 222–228. [13] A.L. Lydersen, Univ. Wisconsin Coll. Eng. Exp. Sin. Rept. 3, Madison, WL, April 1955. [14] P. Li, P.S. Ma, J.G. Dai, W. Cao, Fluid Phase Equilib. 118 (1996) 13–26. [15] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw Hill, New York, 1987 (Ch. 12, p. 632). [16] J.D. Pandey, P. Chandra, T. Srivastava, N.K. Soni, A.K. Singh, Fluid Phase Equilib. 273 (2008) 44–51. [17] C.L. Yaws, H.C. Yang, X. Pan, Chem. Eng. 98 (3) (1991) 140–151. [18] J.D. Pandey, R.D. Rai, R.K. Shukla, J. Chem. Soc. Faraday Trans. I 85 (2) (1989) 331–341. [19] R.K. Shukla, A. Kumar, K. Srivastava, N. Singh, J. Solut. Chem. 36 (2007) 1103–1116. [20] J.D. Pandey, N.K. Soni, V.K. Singh, V. Sanguri, Phys. Chem. Liq. 39 (2001) 763–772. [21] J. Deng, H. Tan, Y. Yang, S. Zai, G. Ouyang, Z. Huang, J. Chem. Eng. Data 52 (2007) 1131–1135. [22] J.A. Riddick, W.B. Bunger, T. Sakano, Organic solvents, physical properties and methods of purification, in: Techniques of Chemistry, vol. II, 4th ed., Wiley, New York, 1986. [23] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (1948) 345–350. [24] R.J. Fort, W.R. Moore, Trans. Faraday Soc. 61 (1965) 2102–2111. [25] L. Riedel, Chem. Eng. Technol. 26 (1954) 83–89.
The present study proves that the corresponding-states groupcontribution methods can be used to predict surface tension of mixture with input data of pure components only. The CSGCST1 method requires the boiling point of the pure components while CSGC-ST2 method needs the value of surface tension at 20 ◦ C (293.15 K) of the pure compounds only. The present work shows the reliability and wide applicability of corresponding-states groupcontribution methods to predict the surface tension not only for pure liquids but also for multicomponent liquid mixtures with high prediction accuracies. List of symbols xi mole fraction of ith component a–e constants of CSGC-ST1 (Eq. (1)) g constants of CSGC-ST2 (Eq. (2)) ∗ Pci assumed-critical pressure of ith component Tci∗ assumed-critical temperature of ith component Tri∗ reduced temperature of ith component ∗ reduced temperature at 293.15 K of ith component Tri,20 Tbi normal boiling point of ith component T absolute temperature Ti contribution values of ith group to Tc∗ ak Redlich–Kister coefficients j group (shown as subscript in Eq. (4)) AT –DT parameters of Eq. (4): Ref. [14]
Acknowledgement The authors are thankful to the University Grants Commission, New Delhi for providing financial assistance in a Major Research Project: File No. 33-34/2007 (SR). References
Contents lists available at ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Applicability of corresponding-states group-contribution methods for the estimation of surface tension of multicomponent liquid mixtures at 298.15 K Anjali Awasthi, Bhawana S. Tripathi, Aashees Awasthi ∗ Department of Physics, University of Lucknow, Lucknow 226007, India
a r t i c l e
i n f o
Article history: Received 19 August 2009 Received in revised form 1 October 2009 Accepted 7 October 2009 Available online 6 November 2009 Keywords: Corresponding-states group-contribution methods Surface tension Multicomponent liquid mixtures
a b s t r a c t Corresponding-states group-contribution methods (CSGC-ST1 and CSGC-ST2) have been applied to four binary liquid mixtures (propyl acetate + o-xylene, propyl acetate + m-xylene, propyl acetate + p-xylene and propyl acetate + ethyl benzene); two ternary (benzene + cyclohexane + toluene and n-hexane + cyclohexane + benzene) and two quaternary liquid mixtures (pentane + hexane + cyclohexane + benzene and pentane + hexane + benzene + toluene) at 298.15 K. In this work, the CSGC-ST2 method is modified and extended to multicomponent liquid mixtures. The excess magnitudes of surface tension were also calculated and graphs were plotted using Redlich–Kister method. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Surface tension is a thermo-physical phenomenon containing inherent information about the interaction between the molecules and hence can be utilized in studying the behaviour of liquid mixtures. Surface tension plays an important role in heat and mass transfer phenomena at the interface in liquid mixtures. Such studies also prove helpful in cavity theory of molecular association [1–3] and acts as a reliable source in depicting the strength of the solvent in liquid chromatography [4]. Literature survey reveals that experimental data of surface tension of multicomponent liquid systems are scarce. Thus, theoretical modeling for the prediction of surface tension is found to be of great help. Several empirical and thermodynamic methods have been deduced for the computation of surface tension but its application is limited mostly to binary liquid mixtures [5–9]. Brock–Bird [7] and Miller [8] proposed the estimation of surface tension by corresponding-states theorem. Macleod [9] suggested a relation between surface tension, liquid density and vapour density. The Sudgen method [10] requires the determination of the parachor but values of surface tension are found to be highly sensitive to variations in the parachor. Patterson and Rastogi [11] estimated the surface tension of polyatomic liquids using reduced parameters only. Various methods correlate the surface tension with certain physical properties of liquids. However, the accuracy and availability of these properties usually limit the application of these methods.
∗ Corresponding author. Tel.: +91 9415003753. E-mail address: [email protected] (A. Awasthi). 0378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2009.10.008
Group-contribution methods have been used for the estimation of transport property of liquid mixtures by several workers. Since the first developments of group-contribution methods by Riedel [12] in 1949 and Lydersen [13] in 1955, a large number of methods have been developed for the estimation of critical property data. Group-contribution methods and the corresponding-states principle have proved to be of great significance in the determination of various physical parameters of compounds. The correspondingstates group-contribution method (CSGC) combines the simplicity of the corresponding-states method with the predictability and flexibility of the group-contribution method. Li et al. [14] proposed CSGC-ST1 and CSGC-ST2 models to evaluate the surface tension of pure components which have been extended to the multicomponent liquid mixtures in the present investigation. In this work, the CSGC-ST1 method is extended to quaternary liquid mixtures and the CSGC-ST2 method is modified and extended to multicomponent liquid mixtures. 2. Prediction of surface tension of liquid mixtures The structural groups were defined in a standardized form and fragmentation of the molecular structures resulted in the evaluation of various critical properties of pure liquids and hence paved way for the estimation of surface tension of liquid mixtures. For the group-contribution predictions, only the molecular structure of the compound is required. Brock and Bird [7] successfully employed the corresponding-states principle to develop a relation for the surface tension of non-polar liquids. Miller on the other hand modified the Brock–Bird relation by introducing a Riedel coefficient [15]. The CSGC-ST1 equation which is similar to the Miller’s modified
152
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
Table 1 Parameters of pure componentsa . Components
Mol. wt.
Tb (K)
20 ◦ C (10−3 N m−1 )
o-Xylene m-Xylene p-Xylene Propyl acetate Ethyl benzene Pentane Hexane Benzene Toluene Cyclohexane
106.168 106.168 106.168 102.134 106.169 72.151 86.178 78.114 92.142 84.162
417.6 412.3 411.5 374.7 409.3 309.2 341.9 353.2 383.8 353.8
30.03 28.63 28.31 24.28 29.04 16.05 18.95 28.88 28.53 24.98
a
Refs. [15,22].
Brock–Bird relation [14,16] has been extended to multicomponent liquid mixtures for the estimation of the surface tension and is represented as;
x P∗ i i ci
mix =
a
101.325
c
b
xi ˛ci − d
1−
i
e xi Tri∗
(1)
i
The Othmer equation can be used to obtain surface tension at 20 ◦ C (i.e. 293.15 K) using a corresponding-states groupcontribution method (CSGC-ST2) and it was used as a Ref. [17]. In the present work, CSGC-ST2 method has been extended successfully to multicomponent liquid mixtures and is represented by the following expression: mix =
xi 20,i
i
∗ g xT i i ∗ ri
1− 1−
(2)
xT i i ri,20
where ∗ Tri,20 =
Tci∗ =
293.15 Tci∗
ATi + BTi
(3)
M j
Tbi
nj Tj + CTi
M j
Propyl acetate (1) + o-xylene (2)
Propyl acetate (1) + m-xylene (2)
Mole frac. (10−3 N m−1 )
Mole frac. (10−3 N m−1 )
x1
Exp.
CSGC-ST1 CSGC-ST2 x1
Exp.
CSGC-ST1 CSGC-ST2
0.0000 0.0998 0.1995 0.2996 0.4002 0.5001 0.6004 0.7002 0.7999 0.9000 1.0000
29.89 28.94 28.14 27.47 26.85 26.27 25.71 25.18 24.69 24.25 23.91 APD
29.57 29.03 28.49 27.95 27.42 26.89 26.36 25.84 25.32 24.81 24.30 1.77
28.47 27.86 27.26 26.70 26.16 25.71 25.30 24.94 24.59 24.25 23.91 APD
28.13 27.74 27.35 26.97 26.58 26.20 25.81 25.44 25.05 24.67 24.30 2.30
29.47 28.90 28.32 27.75 27.17 26.60 26.02 25.45 24.87 24.30 23.72 0.44
0.0000 0.0997 0.1999 0.2986 0.4002 0.4986 0.6004 0.6984 0.8004 0.9007 1.0000
Propyl acetate (1) + p-xylene (2)
xi Tci∗
i
Table 2 Experimentalb and theoretical values of surface tension of various binaries at 298.15 K.
2 + DTi
nj Tj i
M j
3
(4)
0.0000 0.1000 0.1995 0.2981 0.3986 0.4993 0.6002 0.6991 0.7993 0.8999 1.0000 b
28.08 27.56 27.04 26.54 26.05 25.61 25.23 24.89 24.56 24.23 23.91 APD
28.60 28.16 27.73 27.30 26.86 26.43 25.99 25.57 25.15 24.72 24.30 2.56
28.09 27.65 27.21 26.78 26.34 25.91 25.47 25.04 24.59 24.16 23.72 −0.05
Propyl acetate (1) + ethyl benzene (2) 27.77 27.36 26.96 26.56 26.16 25.75 25.34 24.94 24.54 24.13 23.72 −0.16
0.0000 0.1005 0.1994 0.3002 0.4005 0.5000 0.6002 0.6997 0.7998 0.9000 1.0000
28.79 28.15 27.51 26.89 26.35 25.91 25.50 25.10 24.70 24.30 23.91 APD
27.58 27.33 27.06 26.77 26.47 26.14 25.80 25.45 25.08 24.69 24.30 −0.51
28.50 28.02 27.55 27.07 26.59 26.11 25.63 25.16 24.68 24.20 23.72 0.06
Ref. [21].
recognized. That is why normal results show a lowering of surface tension of the mixtures. The n-alkanes and their derivatives form random coils due to internal rotation around C–C bonds, and this tendency increases with an increase in the chain length. In both the ternary and quaternary n-alkane systems, mixtures’ surface layer is enriched by the component of lower surface tension, thereby minimizing the surface tension of the mixture. However, it would not be proper to say that it is the only reason for the discrepancies. Slightly lower surface tension values in ternary and quaternary liq-
nj Tj i
3. Results and discussion The surface tension of the four binary liquid mixtures propyl acetate + o-xylene, propyl acetate + m-xylene, propyl acetate + p-xylene and propyl acetate + ethyl benzene; the two ternary liquid mixtures benzene + cyclohexane + toluene and n-hexane + cyclohexane + benzene and the two quaternary liquid mixtures pentane + hexane + cyclohexane + benzene and pentane + hexane + benzene + toluene have been investigated at 298.15 K. Experimental values of surface tension of these liquid mixtures are taken from literature [18–21]. The properties of the pure components are listed in Table 1 [15,22]. The experimental and calculated (evaluated by the CSGC methods) values of surface tension for binary, ternary and quaternary liquid mixtures are listed in Tables 2–4, respectively. Fig. 1(a–c) shows the excess values of surface tension as a function of mole fraction of the first component for three binary liquid mixtures. From Tables 2–4, it is found that the surface tension decreases gradually with increasing concentration of the first component. It can also be observed that the calculated values of the surface tension using CSGC methods are slightly higher than the experimental values in some cases. Gibbs enrichment of the surface of the mixture by the component with lower surface tension is well
Table 3 Experimentalc and theoretical values of surface tension of ternary liquid mixtures at 298.15 K. Mole frac.
Mole frac.
(10−3 N m−1 )
x1
x2
Exp.
CSGC-ST1
CSGC-ST2
n-Hexane (1) + cyclohexane (2) + benzene (3) 0.0771 0.4315 25.20 0.1269 0.4149 24.60 0.1795 0.3854 24.30 0.2279 0.3501 24.00 0.2812 0.3348 23.60 0.3021 0.3643 23.20 0.3231 0.4851 22.10 0.3448 0.4566 22.90 APD
25.73 25.28 24.84 24.47 23.98 23.67 23.07 22.94 2.16
25.75 25.33 24.94 24.62 24.17 23.85 23.17 23.07 2.64
Benzene (1) + cyclohexane (2) + toluene (3) 0.1331 0.2123 26.80 0.1402 0.2115 26.82 0.1636 0.2297 26.78 0.1854 0.2459 26.72 0.2303 0.2829 26.58 0.2535 0.3018 26.60 0.2763 0.3197 26.64 0.2980 0.3365 26.24 APD
27.23 27.24 27.20 27.17 27.09 27.05 27.01 26.98 1.78
27.22 27.23 27.17 27.12 27.00 26.94 26.88 26.82 1.50
c
Refs. [19,20].
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
153
Table 4 Experimentald and theoretical values of surface tension of quaternary liquid mixtures at 298.15 K. Mole frac.
Mole frac.
Mole frac.
(10−3 N m−1 )
x1
x2
x3
Exp.
CSGC-ST1
CSGC-ST2
Pentane (1) + hexane (2) + cyclohexane (3) + benzene (4) 0.0488 0.1238 0.1831 25.5 25.6 0.0658 0.1078 0.2036 25.1 25.4 0.0813 0.0934 0.2238 24.9 25.3 0.1006 0.0778 0.2430 24.8 25.1 0.1180 0.0629 0.2615 24.7 24.9 0.1243 0.0466 0.2842 24.7 24.9 0.1410 0.1304 0.3129 23.7 23.7 0.1560 0.1262 0.1513 23.9 24.1 0.1285 0.1192 0.5888 22.8 23.0 0.1537 0.0925 0.1685 24.8 24.4 0.1649 0.1013 0.5177 22.9 22.9 0.1368 0.1258 0.1507 24.0 24.3 0.0910 0.1721 0.6137 22.0 22.9 0.0649 0.1378 0.1103 25.5 25.4 0.1810 0.1656 0.2970 23.1 22.8 APD 0.65
25.7 25.5 25.4 25.2 25.1 25.1 23.9 24.4 23.1 24.7 23.1 24.7 23.0 25.6 23.1 1.45
Pentane (1) + hexane (2) + benzene (3) + toluene (4) 0.0943 0.0918 0.4587 25.9 0.1300 0.1373 0.2974 24.6 0.1278 0.1288 0.3589 24.9 0.1450 0.1291 0.3376 24.5 0.1492 0.1384 0.3421 24.4 0.1843 0.1484 0.2711 24.3 0.1823 0.1640 0.3613 24.1 0.1819 0.1601 0.3842 24.0 0.1250 0.1665 0.2455 23.9 0.1691 0.2041 0.2218 23.1 0.1866 0.0826 0.1250 24.3 0.1372 0.1579 0.5548 24.6 0.0660 0.1053 0.7033 25.8 0.0524 0.1434 0.4201 25.5 0.1568 0.0468 0.4582 25.1 APD
26.0 25.1 25.2 25.0 24.9 24.3 24.2 24.3 24.9 23.9 24.9 24.9 26.3 26.0 25.7 1.80
d
25.7 24.6 24.7 24.5 24.3 23.7 23.6 23.7 24.3 23.3 24.1 24.4 26.0 25.8 25.2 −0.30
Ref. [18].
uid mixtures may also be due to small contributions from sterichindrance. The average percentage deviations (APD) for the calculated values of surface tension by the CSGC-ST1 and CSGC-ST2 methods in comparison with the experimental values for binary, ternary and quaternary liquid mixtures are presented in Tables 2–4, respectively. The maximum APD values for binary liquid systems are found to be 2.56% (CSGC-ST1) and 0.44% (CSGC-ST2); for ternary liquid systems 2.16% (CSGC-ST1) and 2.64% (CSGC-ST2) and for quaternary liquid systems 0.65% (CSGC-ST1) and 1.80% (CSGCST2), respectively. Thus, it becomes quite obvious that both CSGC methods not only are simple to use but also improved predictive accuracies with overall APD values of 1.51% (CSGC-ST1) and 1.01% (CSGC-ST2) for multicomponent liquid mixtures. The calculated results show that CSGC methods provide a wide range of applicability and their estimation accuracy is significantly superior over conventional corresponding-states methods. The values of excess surface tension ( E ) is evaluated by the relation; E = mix −
xi i
Fig. 1. (a–c) E versus mole fraction (x1 ) at 298.15 K. Explanation: the excess value of surface tension is plotted against mole fraction of the first component of the binary liquid mixture using Redlich–Kister fitting method.
number of coefficients and the model fits the data fairly well with total error being very small. To fit data, Redlich–Kister equation [23] of the following form was used (Table 5): E = x 1 x2
m
ak (2x1 − 1)k
(6)
k=0
(5)
i
Redlich–Kister equation describes the non-ideal behaviour of the system investigated by fitting the experimental data through the thermodynamically consistent correlations. It provides a flexible algebraic expression for representing the excess function of a complex liquid mixture and gives an indication of the apparent complexity of the mixture as accuracy of the experimental data and the fit desired. It offers an increased flexibility using the desired
Table 5 Coefficients (ak ) from Eq. (6) for binary liquid mixtures at 298.15 K. Binary system
Propyl acetate (1) + o-xylene (2) Propyl acetate (1) + m-xylene (2) Propyl acetate (1) + p-xylene (2) Propyl acetate (1) + ethyl benzene (2)
Binary coefficients a0
a1
a2
−0.1868 −2.1288 −0.0976 0.8150
−0.0137 0.2000 −0.0072 −0.0768
−0.0197 −5.3475 −0.0289 −0.0183
154
A. Awasthi et al. / Fluid Phase Equilibria 287 (2010) 151–154
From Fig. 1(a–c), it can be seen that the curves representing these points are asymmetrical in nature. It is interesting to note that a maximum or minimum in excess surface tension values is obtained at almost equimolar concentration of the binary liquid mixtures. A positive deviation is obtained for propyl acetate + ethyl benzene while negative deviations are observed for propyl acetate + o-xylene and propyl acetate + p-xylene over the entire composition range. Positive values of E are due to weak intermolecular interactions while negative values of E are attributed to strong intermolecular interactions between the molecules [24]. Thus, propyl acetate with o-xylene/p-xylene show strong intermolecular associations while propyl acetate + ethyl benzene demonstrates a weak intermolecular interaction between the molecules.
Greek letters surface tension of mixture mix i surface tension of ith component 20, i surface tension at 293.15 K (10−3 N m−1 ) of ith component E excess value of surface tension ˛ci Riedel coefficient (Ref. [25]) of ith component
4. Conclusions
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The present study proves that the corresponding-states groupcontribution methods can be used to predict surface tension of mixture with input data of pure components only. The CSGCST1 method requires the boiling point of the pure components while CSGC-ST2 method needs the value of surface tension at 20 ◦ C (293.15 K) of the pure compounds only. The present work shows the reliability and wide applicability of corresponding-states groupcontribution methods to predict the surface tension not only for pure liquids but also for multicomponent liquid mixtures with high prediction accuracies. List of symbols xi mole fraction of ith component a–e constants of CSGC-ST1 (Eq. (1)) g constants of CSGC-ST2 (Eq. (2)) ∗ Pci assumed-critical pressure of ith component Tci∗ assumed-critical temperature of ith component Tri∗ reduced temperature of ith component ∗ reduced temperature at 293.15 K of ith component Tri,20 Tbi normal boiling point of ith component T absolute temperature Ti contribution values of ith group to Tc∗ ak Redlich–Kister coefficients j group (shown as subscript in Eq. (4)) AT –DT parameters of Eq. (4): Ref. [14]
Acknowledgement The authors are thankful to the University Grants Commission, New Delhi for providing financial assistance in a Major Research Project: File No. 33-34/2007 (SR). References