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The molar heat capacities of (benzene+tetrachloromethane), (benzene+trichloromethane), (benzene+acetone), and (acetone+trichloromethane)
J. Chem. Thermodynamics 1997, 29, 1473]1480
The molar heat capacities of ( benzene H tetrachloromethane) , ( benzene H trichloromethane) , ( benzene H acetone) , and ( acetone H trichloromethane) Yiming Ding, a Qingsen Yu, Ruisen Lin, and Hanxing Zong Department of Chemistry, Zhejiang Uni¨ ersity, Hangzhou 310027, P.R. China Using a high-temperature flow calorimeter, we determined the molar heat capacities at constant pressure C p, m of four binary mixtures at three different temperatures Ž410.3, 514.7, and 630.6. K and various compositions. The binaries investigated are: Žbenzene q tetrachloromethane ., Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane .. At T s 630.6 K, all these mixtures showed ideal gas behavior, i.e. heat capacity additivity. However, excess heat capacities were found at lower temperatures, particularly in ŽLewis base q acidic proton. and Žnon-polar q polar. binary systems, such as Žacetone q trichloromethane ., Žbenzene q acetone., and Žbenzene q trichloromethane ., but no significant excess heat capacities were detected in Žnon-polar q non-polar. systems. The excess heat capacity can be attributed to intermolecular interaction between unlike molecules. The heat capacities of the binary mixtures considered in the experiment were also calculated using both cubic Žvan der Waals.-type and virial-type equations of state: PR EOS and truncated virial EOS, respectively. The two EOSs gave comparable results. However, there are some statistically significant biases between the calculated results and experimental values in systems containing a polar component. This may be due to the limitations of the mixing rules used in the calculation. Q 1997 Academic Press Limited KEYWORDS: molar heat capacity; high-temperature flow calorimeter; intermolecular interaction; mixing rules; binary mixture
1. Introduction As a primary thermodynamic property, the molar heat capacity at constant pressure C p, m plays an important role in determining derived property changes associated with the Gibbs free energy, and in chemical engineering design. Most heat capacity values of gases available in the literature are not experimentally determined but have been obtained from theory and spectroscopic data.Ž1. Unfortunately, under real conditions, C p, m is a function of temperature and pressure and it is inconvenient to account for the changes in C p, m that occur along the path of the real process from its initial to final state. While C p, m s of liquids have been extensively measured in relatively simple experiments, the heat capacities of multi-component vapors, even binaries, have rarely been reported.Ž1 ] 3. a To whom correspondence should be addressed. Current address: Department of Chemistry, Brigham Young University, Provo, UT 84602, U.S.A.
0021]9614r97r121473 q 08 $25.00r0rct970264
Q 1997 Academic Press Limited
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A previously described flow calorimeter Ž3,4. used in our research group can measure the heat capacities of vapor mixtures up to T s 800 K and p s 1.0 MPa. Using this calorimeter, we have measured a series of gaseous binary mixtures over a wide temperature range and at various compositions. The difference between heat capacities in the real and the ideal-gas states can be determined from an equation of state ŽEOS. by equations Ž1. or Ž2., depending upon whether the EOS is volume-explicit or pressure-explicit: p
C p , m y C p0, m s yT
H0 Ž
C p , m y C p0, m s T
Vm
H`
2
Vr T 2 . p dp,
Ž 1.
Ž 2 pr T 2 . V dV y T Ž pr T . V2 r Ž pr V . T y R. Ž 2.
Essentially all pure-component equations of state can be applied to mixtures; the difficulty of application, though, lies in obtaining EOS mixture parameters which depend on the composition. In general, two main approaches have been used for estimating multi-component mixture properties from pure-component parameters and mole fractions. One approach is to build thermodynamic models based on rigorous theoretical principles that account for interactions between molecules. An important EOS that has rigorous basis in molecular theory is the virial equation. The truncated virial equation with only the second virial coefficient B can be used to describe any gas mixture up to about one-half of the critical density: Ž5. p s RTr Ž Vm y B . .
Ž 3.
The second virial coefficient for a mixture can be calculated from its pure components using the Lee]Kesler method.Ž6. The second approach is to formulate appropriate mixing rules to yield the least error when the predictions of the equation of state are compared with experimental values. Cubic equations of state based on the characteristics of the van der Waals formulation have been successfully used to describe pure fluids in both gaseous and liquid states. Much effort has been devoted in recent years to developing robust mixing and combining rules to obtain the EOS mixture parameters from the pure fluid parameters.Ž7 ] 9. While the more complex mixing rules have shown significant benefits for systems exhibiting large asymmetry or polarity effects, simple and straightforward mixing rules for mixtures are advantageous. Previous studies Ž1,9 ] 11. have indicated that twoparameter EOSs with classical mixing rules are adequate for most binary mixtures, including non-polar and weakly polar components. The Peng]Robinson cubic equation of state is chosen in this work because it is one of the most commonly used two-parameter EOSs in describing Žliquid q vapor. equilibrium behavior of pure compounds and mixtures.Ž6,11,12. Combined with the classical quadratic mixing rules, the Peng]Robinson EOS can be expressed as: p s RTr Ž Vm y bm . y am Ž T . r Vm Ž Vm q bm . q bm Ž Vm y bm . 4 .
Ž 4.
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C p, m of binary mixtures
For binary mixtures of components i and j, the co-volume am and attractive bm parameters can be calculated from: am s x i2 a i q 2 x i x j a i j q x 2j a j , a i jŽ i/ j. s Ž 1 y k i j . Ž a i a j .
1r2
,
bm s x i bi q x j bj ,
Ž 5. Ž 6. Ž 7.
where k i j is a binary interaction parameter. The k i j values for many binary mixtures have been previously published.Ž8,11 ] 14. As it is obtained from experimental data with the i]j binary, k i j is usually assumed to be independent of composition, temperature, and pressure.
2. Experimental All the analytical reagents: benzene, tetrachloromethane, trichloromethane, and acetone were bought from the Shanghai Chemical Co., P.R. China. The reagents were purified by fractional distillation using a 1.5 m long column packed with glass rings. All these purified reagents were then stored over a 0.5 nm molecular sieve before use. The densities and refractive indices of the reagents agreed well with those reported in the literature.Ž15. The schematic of the calorimetric system is shown in figure 1. The structure and design of the calorimeter and flash boiler have been previously described.Ž3. The calorimeter was installed in an air bath. The air-bath temperature can be controlled
FIGURE 1. Schematic of the flow calorimeter.
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from room temperature to T s 773 K. Within five layers of heat shields and a vacuum shell, the calorimeter core has a temperature fluctuation of less than 0.02 K over a period of 0.5 h, when a typical measurement can be done. By means of a high-pressure, pulse-free piston pump whose flow rate can be precisely controlled, a constant flow of liquid was injected into the flash boiler to produce a constant flow of mixed vapor. The temperature of the vaporizing chamber in the flash boiler was maintained at 50 K higher than the boiling temperature of the liquid tested. After passage through the tubular coils in the preheated bath and the calorimeter bath, the vapor flowed into the calorimeter, and then through a three-way solenoid value into a cooler. The cooled fluid went to an expansion vessel, from which the liquid returned to the pump. The disconnectable weighing vessel was used to calibrate the flow rate periodically. For every sample, at least five apparent heat capacity values were measured at different flow rates in the range of Ž0.008 to 1.667. mmol . sy1 . The real vapor heat capacity C p, m was obtained by extrapolating the apparent values to infinite flow rates, so as to correct for the heat loss effect at high temperatures.Ž4,16. Given the relatively small pressure drop through the calorimeter Ž-0.6 kPa., the energy errors due to the Joule]Thomson effect were negligible.
3. Results and discussion Measurements were made on gaseous Žbenzene q tetrachloromethane., Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane. at ambient pressure and at temperatures of Ž410.3, 514.7, and 630.6. K. At least six different compositions were prepared for each mixture to determine the vapor heat capacities at constant pressure. The results are listed in table 1. The molar excess heat capacity C p,E m was calculated for each system: C pE, m s C p , m y Ž 1 y x . . C p , m , 1 y x . C p , m , 2 ,
Ž 8.
where C p, m is the molar heat capacity of the gaseous mixture, and C p, m , 1 and C p, m , 2 are the molar heat capacities of the pure components at the same temperature. Given the calorimeter detection uncertainty of "1.0 per cent,Ž3,4. three gaseous mixtures showed measurable excess heat capacities at T s 410.3 K. They are: Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane .. But at T s 514.7 K, only Žacetone q trichloromethane . had a detectable excess heat capacity, and at T s 630.6 K none of the binaries showed any significant C p,E m . As illustrated in table 2, all the excess heat capacities detected in the binaries investigated here have positive values, contrary to the negative C p,E m values we observed before for Žethanol q tetrachloromethane ..Ž3. The difference originiates from the intermolecular forces between unlike molecules. In Žethanol q tetrachloromethane ., the ethanol clusters Žtwo to five ethanol molecules bound together by hydrogen bonding. may be destroyed or weakened by mixing with the non-polar CCl 4 molecules.Ž3. But in Žaromatic q strongly polar. and ŽLewis base q acidic proton. binary gaseous mixtures such as Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane . studied in this experiment, the two kinds of unlike molecules may associate, and the
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TABLE 1. The molar vapor heat capacities C p, m of binary mixtures; k i j is the binary interaction parameter C p, m a C p, m b C p, m c J . Ky1 . moly1 J . Ky1 . moly1 J . Ky1 . moly1
x
0.00 0.20 0.41
118.10 113.19 108.13
0.00 0.20 0.35 0.51
118.10 110.58 104.47 97.85
0.00 0.10 0.20 0.28 0.35
118.10 116.35 114.15 112.34 110.73
0.00 0.10 0.17 0.35
94.06 94.07 93.16 90.15
a
x
C p, m a C p, m b C p, m c J . Ky1 . moly1 J . Ky1 . moly1 J . Ky1 . moly1
Ž1 y x .C 6 H 6 q xCCl 4 , k i j s y0.029 143.24 166.27 0.58 103.87 134.39 153.59 0.80 98.24 124.72 140.20 1.00 93.01 Ž1 y x .C 6 H 6 q xCHCl 3 , k i j s y0.047 143.24 166.27 0.65 91.98 130.58 150.99 0.80 85.59 122.34 139.28 1.00 75.80 112.09 126.74 Ž1 y x .C 6 H 6 q xC 3 H 6 O, k i j s y0.018 143.24 166.27 0.50 107.54 140.46 162.46 0.64 103.77 137.15 158.60 0.80 99.72 135.51 155.49 1.00 94.06 132.47 152.75 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085 110.56 126.44 0.50 87.22 108.28 106.69 0.64 84.25 106.69 120.95 0.80 80.64 102.22 114.07 1.00 75.80
T s 410.3 K.
b
T s 514.7 K.
c
117.24 106.78 97.78
129.30 115.12 102.16
104.06 92.76 81.21
115.72 103.86 87.39
127.90 122.76 117.03 110.54
146.82 141.21 134.70 126.44
97.36 93.09 88.20 81.21
108.11 102.43 95.82 87.39
T s 630.6 K.
E TABLE 2. The excess molar heat capacities C p, m of binary mixture vapors; k i j is the binary interaction parameter
x
E C p, m J . K . moly1
0.00 0.20 0.35
0.00 0.94 1.18
0.00 0.10 0.20 0.00 0.10 0.17 0.00 0.10 0,17
y1
x
E C p, m y1 .
J.K
y1
mol
x
Ž1 y x .C 6 H 6 q xCHCl 3 , k i j s y0.047, T s 410.3 K 0.51 1.32 1.00 0.65 1.37 0.80 1.32 Ž1 y x .C 6 H 6 q xC 3 H 6 O, k i j s y0.018, T s 410.3 K 0.00 0.28 0.97 0.64 0.65 0.35 1.04 0.80 0.85 0.50 1.46 1.00 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085, T s 410.3 K 0.00 0.35 2.61 0.80 1.86 0.50 2.47 1.00 2.26 0.64 2.10 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085, T s 514.7 K 0.00 0.35 1.96 0.80 0.68 0.50 1.50 1.00 1.14 0.64 1.35
E C p, m y1 .
J.K
moly1
0.00
1.05 0.85 0.00 1.47 0.00
1.16 0.00
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freedom of mobility of the gaseous molecules is then decreased. The overall mixture is therefore less sensitive to heat input compared with the corresponding pure component vapors, which results in a positive excess heat capacity. Qualitatively, the excess heat capacity of a binary mixture is related to its binary interaction parameter, which is also related to the intermolecular forces between unlike molecules. The binary interaction parameters can be calculated from Žliquid q vapor. equilibria listed in the DECHEMA Data Series Ž14. using the regression method introduced by Wong et al.Ž8. As shown in tables 1 and 2, all the binaries investigated in this paper have negative k i j values except for Žacetone q ethanol., which has a very small positive value of 0.005; but the k i j value for Žethanol q tetrachloromethane . is 0.228. It seems that the gaseous binary mixtures with negative k i j values have a positive C p,E m , and those with positive k i j values have a negative C p,E m . However, no quantitative relationship has been observed yet between the excess heat capacity and binary interaction parameter. That is because the k i j values are very small and they are directly associated with the Žliquid q vapor. data used in the correlation. In addition, the vapor phase excess heat capacities of the binaries investigated here are also quite small and the uncertainty is relatively large. The heat capacities at constant pressure of all the binary vapors were also calculated at different temperatures and various compositions. When the truncated virial EOS, equation Ž3., was employed in conjunction with equation Ž1., the Lee]Kesler mixing rules were used to obtain the necessary mixture pseudocritical parameters and the second virial coefficient for the mixture. The classical quadratic mixing rules, equations Ž5. to Ž7., were used together with the PR EOS, equation Ž4., to calculate the difference between heat capacities in the real state and the ideal-gas state as the right-hand side of equation Ž2.. Table 3 shows the comparison between the calculated results and experimentally determined values. The absolute average deviation D, the bias b, and the standard deviation s were calculated for all the gaseous mixtures investigated. While the use of two or more binary parameters improved the behavior of EOS in calculating Žvapor q liquid. equilibria,Ž11. it provided few benefits in predicting heat capacity at constant pressure. The PR EOS and the truncated virial EOS gave comparable predictions as illustrated at the bottom of table 3. The bias of the two EOSs results is very small, only Ž0.2 to 0.3. per cent, which is comparable to their standard deviation. This is expected, since the Lee]Kesler mixing rules and the classical quadratic mixing rules, which were used in the truncated virial EOS and the PR EOS for binary mixtures, respectively, have the same theoretical background.Ž1,6. In addition, it was observed during the calculations that the selection of binary interaction parameter values does not much affect the C p, m output, even though it significantly affects the output of Ž p, Vm , T . from EOS.Ž9,11. In general, the calculated results are lower then the experimental values. Except for the non-polar, non-polar Žbenzene q tetrachloromethane ., both the truncated virial EOS and the PR EOS deviated from the experimental results by more than 1.0 per cent. The deviation is particularly large Ž)2.0 per cent. in binary systems
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C p, m of binary mixtures
TABLE 3. Comparison of calculated heat capacities with experimental results, where N is the number of experimental values, D the absolute average deviation, DŽmax. the maximum absolute deviation, b is the bias, and s is the standard deviation N Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3 Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3 Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3
10 2 . D
10 2 . DŽmax.
Truncated Virial EOS 1.0 2.0 ŽT s 514.7 K, x s 0.20. 1.9 3.1 ŽT s 410.3 K, x s 0.35. 1.9 3.2 ŽT s 514.7 K, x s 0.35. 2.1 3.7 ŽT s 410.3 K, x s 0.35. PR EOS 18 0.9 1.7 ŽT s 630.6 K, x s 0.20. 21 2.1 2.9 ŽT s 410.3 K, x s 0.51. 27 2.2 3.3 ŽT s 514.7 K, x s 0.28. 24 2.2 3.7 ŽT s 410.3 K, x s 0.35. Truncated Virial EOS compared with PR EOS: 18 0.3 0.8 ŽT s 410.3 K, x s 0.58. 21 0.3 0.7 ŽT s 410.3 K, x s 0.80. 27 0.3 0.6 ŽT s 410.3 K, x s 0.20. 24 0.3 0.8 ŽT s 410.3 K, x s 0.80. 18 21 27 24
10 2 . b
10 2 . s
y0.8 y1.7 y1.9 y1.8
1.1 1.2 0.9 0.8
y0.9 y1.9 y2.2 y2.1
1.0 1.2 0.8 0.9
0.3 0.3 0.3 0.3
0.3 0.2 0.2 0.3
containing very dissimilar molecules, e. g. the ŽLewis base q acidic proton. mixture. The biases of the two EOSs are comparable, and they are apparently larger than the standard deviation. However, the standard deviations of the two EOSs compared with experimental results are always the same, f1.0 per cent. This corresponds to the uncertainty in the measured C p, m values for the vapors due to the temperature fluctuation in the calorimeter.Ž3,4. Most large deviations of EOSs from experimental values occur at the lowest temperature, 410.3 K. While the intermolecular forces in the vapor phase are much weaker than in the liquid phase, the asymmetry and polarity effects are still significant at temperatures near the boiling temperature. Actually, the intermolecular forces exist not only between unlike molecules, but also between identical molecules, particularly in systems containing benzene. Dimers easily form in such systems through a planar sandwich force field because of the anisotropy of the benzene molecules.Ž17. Therefore, a gaseous binary vapor of A and B is a multi-component mixture containing, in addition to free molecules A and B, also self-associates containing one type of molecule and cross-associates containing both types of molecule. As previously evaluated,Ž9,11,18. the classical mixing rules are applicable only to non-polar and weakly polar systems. If greater accuracy is needed, more complicated mixing rules Ž9,18,19. are required, such as those that couple with excess Gibbs free energy models or with explicit treatment of association, but the disadvantage is that more computer time is needed.
4. Conclusions By using the high-temperature flow calorimeter developed in our laboratory, the
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heat capacities at constant pressure of four binary gaseous mixtures have been determined at three different temperatures ranging from 410.3 K to 630.6 K, and at various compositions. The excess heat capacities were observed at lower temperatures in mixtures containing unlike molecules, such as ŽLewis base q acidic proton. and Žnon-polar q polar. systems, but not in Žnon-polar q non-polar. systems. The heat capacities of the four binaries were also calculated by both virial-type and van der Waals-type EOSs. The truncated virial EOS was combined with the Lee]Kesler method in calculation, and the classical quadratic mixing rules were used instead in the PR EOS prediction. The two calculation methods gave comparable results for all the binaries investigated. The comparison between the calculated and experimental results indicated that the calculation is consistent with experimental values for Žnon-polar q non-polar. mixtures within the experimental uncertainty of 1.0 per cent. However, there are some apparent biases between the calculated results and experimentally determined values in systems containing polar components. More complicated EOS and mixing rules are required if more accuracy in prediction is required. Financial support from the Chinese Petro-Chemical Corporation ŽCPCC. is greatly appreciated. REFERENCES 1. Tester, J. W.; Modell, M. Thermodynamics and Its Applications: 3rd edition. Prentice Hall: New Jersey. 1997. 2. Dong, J.; Lin, R.; Yen, W.-H. Can. J. Chem. 1988, 66, 783]790. 3. Ding, Y.-M.; Yu, Q.; Lin, R.; Zong, H. Thermochim. Acta 1993, 224, 111]117. 4. Ding, Y.-M.; Yu, Q.; Lin, R.; Zong, H. Acta Physica-Chimica Sinica 1993, 9, 542]551. 5. Hou, H.; Holste, J. C.; Hall, K. R.; Marsh, K. N.; Gammon, B. E. J. Chem. Eng. Data 1996, 41, 344]353. 6. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids: 4th edition. McGraw-Hill: New York. 1987. 7. Panagiotopoulos, A. Z.; Reid, R. C. New Mixing Rule for Cubic Equation of State for Highly Polar, Asymmetric Systems. ACS Symp. Ser. 1986, 300, 571]582. 8. Wong, D. S. H.; Orbey, H.; Sander, S. I. Ind. Eng. Chem. Res. 1992, 31, 2033]2039. 9. Salim, P. H.; Trebble, M. A. Ind. Eng. Chem. Res. 1995, 34, 3112]3128. 10. Stryjek, R.; Vera, J. H. Can. J. Chem. Eng. 1986, 323]333. 11. Han, S. J.; Lin, H. M.; Chao, K. C. Chem. Eng. Sci. 1988, 43, 2327]2367. 12. Abbott, M. M.; Smith, J. M.; Van Ness, H. C.: editors. Introduction to Chemical Engineering Thermodynamics: 4th edition. McGraw-Hill: New York. 1992. 13. Trebble, M. A. Can. J. Chem. Eng. 1990, 68, 487]492. 14. Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection. DECHEMA Data Series, DECHEMA: Frankfurt. 1987. 15. Riddick, J. A.; Bunger, W. B. Organic Sol¨ ents: 3rd edition. Wiley-Interscience: New York. 1970. 16. White, D. R.; Downes. C. J. J. Solution Chem. 1988, 17, 733]750. 17. Smith, G. D.; Jaffe, R. L. J. Phys. Chem. 1996, 100, 9624]9630. 18. Malanowski, S.; Anderko, A. Modeling Phase Equilibria: Thermodynamic Background and Practical Tools. Wiley: New York. 1992. 19. Salim, P. H.; Trebble, M. A. Fluid Phase Equilib. 1991, 65, 59]71.
(Recei¨ ed 17 April 1997; in final form 4 June 1997)
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The molar heat capacities of ( benzene H tetrachloromethane) , ( benzene H trichloromethane) , ( benzene H acetone) , and ( acetone H trichloromethane) Yiming Ding, a Qingsen Yu, Ruisen Lin, and Hanxing Zong Department of Chemistry, Zhejiang Uni¨ ersity, Hangzhou 310027, P.R. China Using a high-temperature flow calorimeter, we determined the molar heat capacities at constant pressure C p, m of four binary mixtures at three different temperatures Ž410.3, 514.7, and 630.6. K and various compositions. The binaries investigated are: Žbenzene q tetrachloromethane ., Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane .. At T s 630.6 K, all these mixtures showed ideal gas behavior, i.e. heat capacity additivity. However, excess heat capacities were found at lower temperatures, particularly in ŽLewis base q acidic proton. and Žnon-polar q polar. binary systems, such as Žacetone q trichloromethane ., Žbenzene q acetone., and Žbenzene q trichloromethane ., but no significant excess heat capacities were detected in Žnon-polar q non-polar. systems. The excess heat capacity can be attributed to intermolecular interaction between unlike molecules. The heat capacities of the binary mixtures considered in the experiment were also calculated using both cubic Žvan der Waals.-type and virial-type equations of state: PR EOS and truncated virial EOS, respectively. The two EOSs gave comparable results. However, there are some statistically significant biases between the calculated results and experimental values in systems containing a polar component. This may be due to the limitations of the mixing rules used in the calculation. Q 1997 Academic Press Limited KEYWORDS: molar heat capacity; high-temperature flow calorimeter; intermolecular interaction; mixing rules; binary mixture
1. Introduction As a primary thermodynamic property, the molar heat capacity at constant pressure C p, m plays an important role in determining derived property changes associated with the Gibbs free energy, and in chemical engineering design. Most heat capacity values of gases available in the literature are not experimentally determined but have been obtained from theory and spectroscopic data.Ž1. Unfortunately, under real conditions, C p, m is a function of temperature and pressure and it is inconvenient to account for the changes in C p, m that occur along the path of the real process from its initial to final state. While C p, m s of liquids have been extensively measured in relatively simple experiments, the heat capacities of multi-component vapors, even binaries, have rarely been reported.Ž1 ] 3. a To whom correspondence should be addressed. Current address: Department of Chemistry, Brigham Young University, Provo, UT 84602, U.S.A.
0021]9614r97r121473 q 08 $25.00r0rct970264
Q 1997 Academic Press Limited
1474
Y. Ding et al.
A previously described flow calorimeter Ž3,4. used in our research group can measure the heat capacities of vapor mixtures up to T s 800 K and p s 1.0 MPa. Using this calorimeter, we have measured a series of gaseous binary mixtures over a wide temperature range and at various compositions. The difference between heat capacities in the real and the ideal-gas states can be determined from an equation of state ŽEOS. by equations Ž1. or Ž2., depending upon whether the EOS is volume-explicit or pressure-explicit: p
C p , m y C p0, m s yT
H0 Ž
C p , m y C p0, m s T
Vm
H`
2
Vr T 2 . p dp,
Ž 1.
Ž 2 pr T 2 . V dV y T Ž pr T . V2 r Ž pr V . T y R. Ž 2.
Essentially all pure-component equations of state can be applied to mixtures; the difficulty of application, though, lies in obtaining EOS mixture parameters which depend on the composition. In general, two main approaches have been used for estimating multi-component mixture properties from pure-component parameters and mole fractions. One approach is to build thermodynamic models based on rigorous theoretical principles that account for interactions between molecules. An important EOS that has rigorous basis in molecular theory is the virial equation. The truncated virial equation with only the second virial coefficient B can be used to describe any gas mixture up to about one-half of the critical density: Ž5. p s RTr Ž Vm y B . .
Ž 3.
The second virial coefficient for a mixture can be calculated from its pure components using the Lee]Kesler method.Ž6. The second approach is to formulate appropriate mixing rules to yield the least error when the predictions of the equation of state are compared with experimental values. Cubic equations of state based on the characteristics of the van der Waals formulation have been successfully used to describe pure fluids in both gaseous and liquid states. Much effort has been devoted in recent years to developing robust mixing and combining rules to obtain the EOS mixture parameters from the pure fluid parameters.Ž7 ] 9. While the more complex mixing rules have shown significant benefits for systems exhibiting large asymmetry or polarity effects, simple and straightforward mixing rules for mixtures are advantageous. Previous studies Ž1,9 ] 11. have indicated that twoparameter EOSs with classical mixing rules are adequate for most binary mixtures, including non-polar and weakly polar components. The Peng]Robinson cubic equation of state is chosen in this work because it is one of the most commonly used two-parameter EOSs in describing Žliquid q vapor. equilibrium behavior of pure compounds and mixtures.Ž6,11,12. Combined with the classical quadratic mixing rules, the Peng]Robinson EOS can be expressed as: p s RTr Ž Vm y bm . y am Ž T . r Vm Ž Vm q bm . q bm Ž Vm y bm . 4 .
Ž 4.
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C p, m of binary mixtures
For binary mixtures of components i and j, the co-volume am and attractive bm parameters can be calculated from: am s x i2 a i q 2 x i x j a i j q x 2j a j , a i jŽ i/ j. s Ž 1 y k i j . Ž a i a j .
1r2
,
bm s x i bi q x j bj ,
Ž 5. Ž 6. Ž 7.
where k i j is a binary interaction parameter. The k i j values for many binary mixtures have been previously published.Ž8,11 ] 14. As it is obtained from experimental data with the i]j binary, k i j is usually assumed to be independent of composition, temperature, and pressure.
2. Experimental All the analytical reagents: benzene, tetrachloromethane, trichloromethane, and acetone were bought from the Shanghai Chemical Co., P.R. China. The reagents were purified by fractional distillation using a 1.5 m long column packed with glass rings. All these purified reagents were then stored over a 0.5 nm molecular sieve before use. The densities and refractive indices of the reagents agreed well with those reported in the literature.Ž15. The schematic of the calorimetric system is shown in figure 1. The structure and design of the calorimeter and flash boiler have been previously described.Ž3. The calorimeter was installed in an air bath. The air-bath temperature can be controlled
FIGURE 1. Schematic of the flow calorimeter.
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from room temperature to T s 773 K. Within five layers of heat shields and a vacuum shell, the calorimeter core has a temperature fluctuation of less than 0.02 K over a period of 0.5 h, when a typical measurement can be done. By means of a high-pressure, pulse-free piston pump whose flow rate can be precisely controlled, a constant flow of liquid was injected into the flash boiler to produce a constant flow of mixed vapor. The temperature of the vaporizing chamber in the flash boiler was maintained at 50 K higher than the boiling temperature of the liquid tested. After passage through the tubular coils in the preheated bath and the calorimeter bath, the vapor flowed into the calorimeter, and then through a three-way solenoid value into a cooler. The cooled fluid went to an expansion vessel, from which the liquid returned to the pump. The disconnectable weighing vessel was used to calibrate the flow rate periodically. For every sample, at least five apparent heat capacity values were measured at different flow rates in the range of Ž0.008 to 1.667. mmol . sy1 . The real vapor heat capacity C p, m was obtained by extrapolating the apparent values to infinite flow rates, so as to correct for the heat loss effect at high temperatures.Ž4,16. Given the relatively small pressure drop through the calorimeter Ž-0.6 kPa., the energy errors due to the Joule]Thomson effect were negligible.
3. Results and discussion Measurements were made on gaseous Žbenzene q tetrachloromethane., Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane. at ambient pressure and at temperatures of Ž410.3, 514.7, and 630.6. K. At least six different compositions were prepared for each mixture to determine the vapor heat capacities at constant pressure. The results are listed in table 1. The molar excess heat capacity C p,E m was calculated for each system: C pE, m s C p , m y Ž 1 y x . . C p , m , 1 y x . C p , m , 2 ,
Ž 8.
where C p, m is the molar heat capacity of the gaseous mixture, and C p, m , 1 and C p, m , 2 are the molar heat capacities of the pure components at the same temperature. Given the calorimeter detection uncertainty of "1.0 per cent,Ž3,4. three gaseous mixtures showed measurable excess heat capacities at T s 410.3 K. They are: Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane .. But at T s 514.7 K, only Žacetone q trichloromethane . had a detectable excess heat capacity, and at T s 630.6 K none of the binaries showed any significant C p,E m . As illustrated in table 2, all the excess heat capacities detected in the binaries investigated here have positive values, contrary to the negative C p,E m values we observed before for Žethanol q tetrachloromethane ..Ž3. The difference originiates from the intermolecular forces between unlike molecules. In Žethanol q tetrachloromethane ., the ethanol clusters Žtwo to five ethanol molecules bound together by hydrogen bonding. may be destroyed or weakened by mixing with the non-polar CCl 4 molecules.Ž3. But in Žaromatic q strongly polar. and ŽLewis base q acidic proton. binary gaseous mixtures such as Žbenzene q trichloromethane ., Žbenzene q acetone., and Žacetone q trichloromethane . studied in this experiment, the two kinds of unlike molecules may associate, and the
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TABLE 1. The molar vapor heat capacities C p, m of binary mixtures; k i j is the binary interaction parameter C p, m a C p, m b C p, m c J . Ky1 . moly1 J . Ky1 . moly1 J . Ky1 . moly1
x
0.00 0.20 0.41
118.10 113.19 108.13
0.00 0.20 0.35 0.51
118.10 110.58 104.47 97.85
0.00 0.10 0.20 0.28 0.35
118.10 116.35 114.15 112.34 110.73
0.00 0.10 0.17 0.35
94.06 94.07 93.16 90.15
a
x
C p, m a C p, m b C p, m c J . Ky1 . moly1 J . Ky1 . moly1 J . Ky1 . moly1
Ž1 y x .C 6 H 6 q xCCl 4 , k i j s y0.029 143.24 166.27 0.58 103.87 134.39 153.59 0.80 98.24 124.72 140.20 1.00 93.01 Ž1 y x .C 6 H 6 q xCHCl 3 , k i j s y0.047 143.24 166.27 0.65 91.98 130.58 150.99 0.80 85.59 122.34 139.28 1.00 75.80 112.09 126.74 Ž1 y x .C 6 H 6 q xC 3 H 6 O, k i j s y0.018 143.24 166.27 0.50 107.54 140.46 162.46 0.64 103.77 137.15 158.60 0.80 99.72 135.51 155.49 1.00 94.06 132.47 152.75 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085 110.56 126.44 0.50 87.22 108.28 106.69 0.64 84.25 106.69 120.95 0.80 80.64 102.22 114.07 1.00 75.80
T s 410.3 K.
b
T s 514.7 K.
c
117.24 106.78 97.78
129.30 115.12 102.16
104.06 92.76 81.21
115.72 103.86 87.39
127.90 122.76 117.03 110.54
146.82 141.21 134.70 126.44
97.36 93.09 88.20 81.21
108.11 102.43 95.82 87.39
T s 630.6 K.
E TABLE 2. The excess molar heat capacities C p, m of binary mixture vapors; k i j is the binary interaction parameter
x
E C p, m J . K . moly1
0.00 0.20 0.35
0.00 0.94 1.18
0.00 0.10 0.20 0.00 0.10 0.17 0.00 0.10 0,17
y1
x
E C p, m y1 .
J.K
y1
mol
x
Ž1 y x .C 6 H 6 q xCHCl 3 , k i j s y0.047, T s 410.3 K 0.51 1.32 1.00 0.65 1.37 0.80 1.32 Ž1 y x .C 6 H 6 q xC 3 H 6 O, k i j s y0.018, T s 410.3 K 0.00 0.28 0.97 0.64 0.65 0.35 1.04 0.80 0.85 0.50 1.46 1.00 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085, T s 410.3 K 0.00 0.35 2.61 0.80 1.86 0.50 2.47 1.00 2.26 0.64 2.10 Ž1 y x .C 3 H 6 O q xCHCl 3 , k i j s y0.085, T s 514.7 K 0.00 0.35 1.96 0.80 0.68 0.50 1.50 1.00 1.14 0.64 1.35
E C p, m y1 .
J.K
moly1
0.00
1.05 0.85 0.00 1.47 0.00
1.16 0.00
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freedom of mobility of the gaseous molecules is then decreased. The overall mixture is therefore less sensitive to heat input compared with the corresponding pure component vapors, which results in a positive excess heat capacity. Qualitatively, the excess heat capacity of a binary mixture is related to its binary interaction parameter, which is also related to the intermolecular forces between unlike molecules. The binary interaction parameters can be calculated from Žliquid q vapor. equilibria listed in the DECHEMA Data Series Ž14. using the regression method introduced by Wong et al.Ž8. As shown in tables 1 and 2, all the binaries investigated in this paper have negative k i j values except for Žacetone q ethanol., which has a very small positive value of 0.005; but the k i j value for Žethanol q tetrachloromethane . is 0.228. It seems that the gaseous binary mixtures with negative k i j values have a positive C p,E m , and those with positive k i j values have a negative C p,E m . However, no quantitative relationship has been observed yet between the excess heat capacity and binary interaction parameter. That is because the k i j values are very small and they are directly associated with the Žliquid q vapor. data used in the correlation. In addition, the vapor phase excess heat capacities of the binaries investigated here are also quite small and the uncertainty is relatively large. The heat capacities at constant pressure of all the binary vapors were also calculated at different temperatures and various compositions. When the truncated virial EOS, equation Ž3., was employed in conjunction with equation Ž1., the Lee]Kesler mixing rules were used to obtain the necessary mixture pseudocritical parameters and the second virial coefficient for the mixture. The classical quadratic mixing rules, equations Ž5. to Ž7., were used together with the PR EOS, equation Ž4., to calculate the difference between heat capacities in the real state and the ideal-gas state as the right-hand side of equation Ž2.. Table 3 shows the comparison between the calculated results and experimentally determined values. The absolute average deviation D, the bias b, and the standard deviation s were calculated for all the gaseous mixtures investigated. While the use of two or more binary parameters improved the behavior of EOS in calculating Žvapor q liquid. equilibria,Ž11. it provided few benefits in predicting heat capacity at constant pressure. The PR EOS and the truncated virial EOS gave comparable predictions as illustrated at the bottom of table 3. The bias of the two EOSs results is very small, only Ž0.2 to 0.3. per cent, which is comparable to their standard deviation. This is expected, since the Lee]Kesler mixing rules and the classical quadratic mixing rules, which were used in the truncated virial EOS and the PR EOS for binary mixtures, respectively, have the same theoretical background.Ž1,6. In addition, it was observed during the calculations that the selection of binary interaction parameter values does not much affect the C p, m output, even though it significantly affects the output of Ž p, Vm , T . from EOS.Ž9,11. In general, the calculated results are lower then the experimental values. Except for the non-polar, non-polar Žbenzene q tetrachloromethane ., both the truncated virial EOS and the PR EOS deviated from the experimental results by more than 1.0 per cent. The deviation is particularly large Ž)2.0 per cent. in binary systems
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TABLE 3. Comparison of calculated heat capacities with experimental results, where N is the number of experimental values, D the absolute average deviation, DŽmax. the maximum absolute deviation, b is the bias, and s is the standard deviation N Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3 Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3 Ž1 y x .C 6 H 6 q xCCl 4 Ž1 y x .C 6 H 6 q xCHCl 3 Ž1 y x .C 6 H 6 q xC 3 H 6 O Ž1 y x .C 3 H 6 O q xCHCl 3
10 2 . D
10 2 . DŽmax.
Truncated Virial EOS 1.0 2.0 ŽT s 514.7 K, x s 0.20. 1.9 3.1 ŽT s 410.3 K, x s 0.35. 1.9 3.2 ŽT s 514.7 K, x s 0.35. 2.1 3.7 ŽT s 410.3 K, x s 0.35. PR EOS 18 0.9 1.7 ŽT s 630.6 K, x s 0.20. 21 2.1 2.9 ŽT s 410.3 K, x s 0.51. 27 2.2 3.3 ŽT s 514.7 K, x s 0.28. 24 2.2 3.7 ŽT s 410.3 K, x s 0.35. Truncated Virial EOS compared with PR EOS: 18 0.3 0.8 ŽT s 410.3 K, x s 0.58. 21 0.3 0.7 ŽT s 410.3 K, x s 0.80. 27 0.3 0.6 ŽT s 410.3 K, x s 0.20. 24 0.3 0.8 ŽT s 410.3 K, x s 0.80. 18 21 27 24
10 2 . b
10 2 . s
y0.8 y1.7 y1.9 y1.8
1.1 1.2 0.9 0.8
y0.9 y1.9 y2.2 y2.1
1.0 1.2 0.8 0.9
0.3 0.3 0.3 0.3
0.3 0.2 0.2 0.3
containing very dissimilar molecules, e. g. the ŽLewis base q acidic proton. mixture. The biases of the two EOSs are comparable, and they are apparently larger than the standard deviation. However, the standard deviations of the two EOSs compared with experimental results are always the same, f1.0 per cent. This corresponds to the uncertainty in the measured C p, m values for the vapors due to the temperature fluctuation in the calorimeter.Ž3,4. Most large deviations of EOSs from experimental values occur at the lowest temperature, 410.3 K. While the intermolecular forces in the vapor phase are much weaker than in the liquid phase, the asymmetry and polarity effects are still significant at temperatures near the boiling temperature. Actually, the intermolecular forces exist not only between unlike molecules, but also between identical molecules, particularly in systems containing benzene. Dimers easily form in such systems through a planar sandwich force field because of the anisotropy of the benzene molecules.Ž17. Therefore, a gaseous binary vapor of A and B is a multi-component mixture containing, in addition to free molecules A and B, also self-associates containing one type of molecule and cross-associates containing both types of molecule. As previously evaluated,Ž9,11,18. the classical mixing rules are applicable only to non-polar and weakly polar systems. If greater accuracy is needed, more complicated mixing rules Ž9,18,19. are required, such as those that couple with excess Gibbs free energy models or with explicit treatment of association, but the disadvantage is that more computer time is needed.
4. Conclusions By using the high-temperature flow calorimeter developed in our laboratory, the
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heat capacities at constant pressure of four binary gaseous mixtures have been determined at three different temperatures ranging from 410.3 K to 630.6 K, and at various compositions. The excess heat capacities were observed at lower temperatures in mixtures containing unlike molecules, such as ŽLewis base q acidic proton. and Žnon-polar q polar. systems, but not in Žnon-polar q non-polar. systems. The heat capacities of the four binaries were also calculated by both virial-type and van der Waals-type EOSs. The truncated virial EOS was combined with the Lee]Kesler method in calculation, and the classical quadratic mixing rules were used instead in the PR EOS prediction. The two calculation methods gave comparable results for all the binaries investigated. The comparison between the calculated and experimental results indicated that the calculation is consistent with experimental values for Žnon-polar q non-polar. mixtures within the experimental uncertainty of 1.0 per cent. However, there are some apparent biases between the calculated results and experimentally determined values in systems containing polar components. More complicated EOS and mixing rules are required if more accuracy in prediction is required. Financial support from the Chinese Petro-Chemical Corporation ŽCPCC. is greatly appreciated. REFERENCES 1. Tester, J. W.; Modell, M. Thermodynamics and Its Applications: 3rd edition. Prentice Hall: New Jersey. 1997. 2. Dong, J.; Lin, R.; Yen, W.-H. Can. J. Chem. 1988, 66, 783]790. 3. Ding, Y.-M.; Yu, Q.; Lin, R.; Zong, H. Thermochim. Acta 1993, 224, 111]117. 4. Ding, Y.-M.; Yu, Q.; Lin, R.; Zong, H. Acta Physica-Chimica Sinica 1993, 9, 542]551. 5. Hou, H.; Holste, J. C.; Hall, K. R.; Marsh, K. N.; Gammon, B. E. J. Chem. Eng. Data 1996, 41, 344]353. 6. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids: 4th edition. McGraw-Hill: New York. 1987. 7. Panagiotopoulos, A. Z.; Reid, R. C. New Mixing Rule for Cubic Equation of State for Highly Polar, Asymmetric Systems. ACS Symp. Ser. 1986, 300, 571]582. 8. Wong, D. S. H.; Orbey, H.; Sander, S. I. Ind. Eng. Chem. Res. 1992, 31, 2033]2039. 9. Salim, P. H.; Trebble, M. A. Ind. Eng. Chem. Res. 1995, 34, 3112]3128. 10. Stryjek, R.; Vera, J. H. Can. J. Chem. Eng. 1986, 323]333. 11. Han, S. J.; Lin, H. M.; Chao, K. C. Chem. Eng. Sci. 1988, 43, 2327]2367. 12. Abbott, M. M.; Smith, J. M.; Van Ness, H. C.: editors. Introduction to Chemical Engineering Thermodynamics: 4th edition. McGraw-Hill: New York. 1992. 13. Trebble, M. A. Can. J. Chem. Eng. 1990, 68, 487]492. 14. Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection. DECHEMA Data Series, DECHEMA: Frankfurt. 1987. 15. Riddick, J. A.; Bunger, W. B. Organic Sol¨ ents: 3rd edition. Wiley-Interscience: New York. 1970. 16. White, D. R.; Downes. C. J. J. Solution Chem. 1988, 17, 733]750. 17. Smith, G. D.; Jaffe, R. L. J. Phys. Chem. 1996, 100, 9624]9630. 18. Malanowski, S.; Anderko, A. Modeling Phase Equilibria: Thermodynamic Background and Practical Tools. Wiley: New York. 1992. 19. Salim, P. H.; Trebble, M. A. Fluid Phase Equilib. 1991, 65, 59]71.
(Recei¨ ed 17 April 1997; in final form 4 June 1997)
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