175
Ind. Eng. Chem. Fundam. 1981, 20, 175-177
Solubilities of Gases in Water at High Temperatures
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Gas-solubility data for aqueous systems are correlated within the framework of scaled-particle theory over a wide range of temperatures. Henry’s constants are sensitive to the characteristic size parameter which must be temperature dependent to achieve agreement with experimental data. Reliable solubility data for gases in water are scarce, especially at temperatures above 50 “C. We present here a semiempirical correlation of such solubilities based upon scaled-particle theory. The correlation applies to nonreactive gases at low pressures (where Henry’s law holds) for the region 0-300 “C. With this correlation,it is possible to make reliable estimates of solubilities at high temperature, provided that at least one experimental solubility is known at lower temperature, typically 25 “C. As discussed by several authors, notably Pierotti (1976), scaled-particle theory can be used to obtain an expression for Henry’s constant for a solute (2) in solvent (1)
where g , is the work (Gibbs energy) required to make one mole of cavities in the solvent, where each cavity is sufficiently large to hold one molecule of solute; gi is the molar Gibbs energy of interaction between the solute in the cavity and the surrounding solvent molecules; R is the gas constant; T is the system temperature and u1 is the molar volume of the solvent. Henry’s constant is defined by
where f is the fugacity (essentially, the partial pressure) and x is the mole fraction. Scaled-particle theory gives
g, = KO
+ K1~12+ K 2 ~ 1 2+~K3012
(3)
where -In (1 - [)
+ (9/2) (4)
K2 =
(?)[(s) + &)] 18(
K3 = Y3NrP
- 2NrPul (6)
(7)
where reduced density 5 = ru13p/6, p is the density per molecule, u is the molecular diameter, and N is Avogadro’s number. The total pressure is designated by P. Following the procedure given by Wilhelm and Battino (1971), for a polar solvent (1) and a nonpolar solute (2)
Table I. Saturated Densities and Vapor Pressures for Watera v, cm3/g Psat, bar 1.0002 0.00611 1.0003 0.01227 1.0018 0.02337 1.0030 0.03166 1.0121 0.12335 1.0258 0.38547 1.0435 1.01330 1.0649 2.32090 1.0906 4.75974 1.1208 8.92471 1.1565 15.5514 1.1992 25.5045 1.2512 39.7762 1.3168 59.4869 1.4036 85.9196 From “Steam Tables”, 1964. Department of Scientific and Industrial Research, National Engineering Laboratory, Edinburgh, Her Majesty’s Stationery Office. f, OC
0 10 20 25 50 75 100 125 150 175 200 225 250 275 300
When eq 1 , 3 , and 8 are used to calculate Henry’s constants, results are sensitive to diameters ul and cr12 but not to parameter c12. Pierotti (1975) studied the solubilities of gases in water at 25 “C and found that good results were obtained using 612
=
(10) = W l + a21 with u1 = 0.275 nm and tl/k = 85.3 K for water. We investigated solubility data for gases at higher temperatures. To do so, we required, first, precise volumetric and vapor-pressure data for water; these are shown in Table I. Second, we required reliable solubility data over a wide temperature range. These are scatce; we used data for argon (Benson and Krause, 1976; Potter and Clynne, 1978), carbon dioxide (Edwards et al., 1978), helium (Wilhelm et al., 1977; Pray et al., 1952), hydrogen (Wilhelm et al., 1977; Pray et al., 1952), methane (Wilhelm et al., 1977), neon (Benson and Krause, 1976; Potter and Clynne, 1978), nitrogen (Wilhelm et al., 1977; Benson and Krause, 1976; Pray et al., 1952), oxygen (Cramer, 1980; Pray et al., 1952; Benson and Krause, 1976), and xenon (Benson and Krause, 1976; Potter and Clynne, 1978). We found that the experimental data could be fit well only if we allowed the key parameter u12 to be slightly dependent on temperature. For all gases studied here, that temperature dependence is given by 012
where
F(T) = 16280 (l/T)’ where 1.1 is the dipole moment, CY is the polarizability, k is Boltzmann’s constant, and 6 is a parameter characterizing the strength of dispersion forces. 0 196-4313/81/1020-0175$0 1.25/0
(9)
(€lt2)”2
- 141.75(1/T)
+ 1.2978
(12)
with T i n kelvins. At 0 “C, F = 0.997; at 300 “C, F = 1.10. Molecular parameters for 9 gases are given in Table 11. When eq 11 and 12 are used, calculated Henry’s constants are in good agreement with experiment as shown in Figures 0 1981 American Chemical Societv
176
Ind. Eng.
Chem. Fundam., Vol. 20, No. 2, 1981
I
I
-0 . 5
I
I
I
i
I
I
Oxygen
- Calculated Benson, Kraurs 0
Poftsr, Clynna 0 Pray, Schwsicksrf, Minnich Bsnron, Krause 6 Cramsr Bsnron, Krause A P o f t e r , Clynne
4.0
3,8
APotttr,.Clynne
fi
3.8 1
0
50
I
I
100
I
150 200
I
0
I
I
250
300
50
I
I
I
I
100
150
200
250
350
I
300 350
I, o c
I, oc
Figure 2. Calculated and observed Henry’s constants (bar).
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Figure 1. Calculated and observed Henry’s constants (bar). I
I
I
I
I
I
Table 11. Scaled-Particle-Theory Parameters for Calculation of Henry’s Constantsa fluid
a
u,
nm
elk, K
10 2 4 & , em3/
molecule
water 0.27 5 85.3 1.590 0.3 56 122.0 argon 1.630 0.332 300.0 carbon di oxide 2.590 2.0 0.264 helium 0.204 0.267 hydrogen 0.802 10.0 2.700 0.413 me thane 160.0 neon 0.393 34.9 0.309 nitrogen 0.373 1.730 95.0 oxygen 0.362 118.0 1.560 xenon 0.432 4.100 221.0 f I 2 = (E, c ~ ) ” ~dipole ; moment of water = 1.84 D.
Corbondioxide
I
0
a
0
-
3 .O
-
2.6,r
-Calculated 0 Pray, Schweickert, Minnich W i l h e l m , B a t t i n o , Wilcock 0 Wtlhelm, Battino, Wilcock A E d w a r d s , Mourer, N e w m a n , Prousnitz I I 1 I I
,
1,2, and 3. Here “good” means that the deviation between calculation and experiment is usually within *15%. (Deviations for helium are larger but there is good reason to question the reliability of the data.) This is considered good because experimental uncertainties are large, especially for sparingly soluble gases at high temperatures. Table I11 gives calculated Henry’s constants for 9 gases. To estimate Henry’s constants at high temperatures for a nonreactive gas not listed in Table 11, it is necessary to estimate first values of u2 and t2. This can be done by utilizing one or more experimental Henry’s constants at ambient temperatures. Since results are not sensitive to c2, a rough estimate of t2 (see for example, Hirschfelder et al., 1954) is sufficient. Using the scaled-particle-theory equations given above, the experimental result(s) a t ambient temperature(s) is (are) used to find u2. This value, Table 111. Calculated Henry’s Constants, t , OC Ar CO, He 0 10 20 25 50 75 100 125 150 175 20 0 225 250 27 5 300
25.9 31.0 35.8 38.1 48.1 54.7 57.6 57.1 53.9 48.8 42.7 36.1 29.5 23.3 17.5
1.19 1.52 1.88 2.23 3.08 4.09 4.99 5.70 6.17 6.40 6.38 6.15 5.74 5.17 4.48
120.0 127.0 131.0 133.0 135.0 130.0 120.0 108.0 94.7 81.0 67.8 55.5 44.3 34.3 25.5
bar H2
58.0 62.7 66.6 68.2 73.6 74.8 72.7 68.1 61.9 54.7 47.3 39.8 32.7 26.0 19.8
CH,
Ne
N,
0,
Xe
23.9 30.4 37.1 40.4 56.0 67.7 73.9 74.9 71.1 64.3 55.6 46.2 36.9 28.1 20.4
99.6 109.0 117.0 120.0 130.0 131.0 125.0 115.0 101.0 86.9 72.7 59.2 46.8 35.8 26.2
59.7 70.9 81.4 86.2 106.0 118.0 120.0 115.0 105.0 91.6 77.2 62.8 49.3 37.2 26.6
32.1 38.3 44.2 47.0 59.0 66.7 69.5 68.4 63.9 57.3 49.6 41.5 33.6 26.2 19.4
6.43 8.69 11.20 12.50 19.50 26.00 30.80 33.40 33.80 3 2.20 29.30 25.40 21.10 16.80 12.70
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Xnd. Eng. Chem. Fundam. 1981, 20, 177-180
react with water, calculated Henry's constants are too high. For example, aqueous solubility data for carbon monoxide (Wilhelm et al., 1977) are very well represented over the range 0-85 "C using up = 0.395 nm and t 2 / k = 134 K. When these parameters are used to calculate Henry's constants for carbon monoxide in the region 250-300 "C, calculated results are about twice those measured by Jung et al. (1971). However, these authors report appreciable chemical reaction between water and carbon monoxide at temperatures above 250 "C, explaining the higher observed solubility. While scaled-particle theory may provide a useful basis for correlating phase-equilibrium data, it is clear that there is a large gulf between the assumptions of the theory and the properties of real systems, especially aqueous systems. This gulf is partly absorbed in the adjustable parameters but at the cost of physical plausibility. It is strange, for example, that the best up for carbon dioxide (0.332 nm) should be smaller than the best up for carbon monoxide (0.395 nm). Scaled-particle theory is strongly sensitive to the value of up;therefore, it is important to evaluate this parameter carefully from reliable experimental data, while recognizing that its physical significance is only approximate.
177
Acknowledgment
The authors are grateful to the US. Department of Energy for financial support and to the Deutscher Akademischer Austauschdienst for a fellowship to G.S. L i t e r a t u r e Cited Benson, B. B.; Krause, D. J. Chem. Phys. 1878, 64, 691. Cramer, S. D. Ind. Eng. Chem. Process Des. Dev. 1980, 79, 300. Edwards, T. J.; Mawer, G.; Newman, J.; Prausnitz, J. M. AIChE J . 1978, 24, 1966. Jung, J.; Knacke, 0.; Neuschiitz, D. Chem. Ing. Tech. 1971, 43, 112. Hayduk, W.; Laudie, H. AIChE J. 1873, 19, 1933. Hayduk, W.; Buckley, W. D. Can. J. Chem. Eng. 1871, 49, 667. Curtiss, C. F.;Bird, R. 8. "Molecular Theory of Gases and Hlrschfeider, J. 0.; Liquids", Wlley: New York, 1954. Pierotti, R. A. Chem. Rev. 1978, 76, 717. Potter, R. W.; Clynne, M. A. J. So/ut&n Chem. 1878, 7 , 837. Pray, H. A.; Schweickert, C. E.; Minnich, B. H. Ind. Eng. Chem. 1852, 4 4 , 1150. Wiihelm. E.; Battlno, R. J . Chem. 73ermcdyn. 1971, 3 , 379. Wilhelm, E.; Battlno. R.; Wilcock, R. J. Chem. Rev. 1977, 77, 226.
Chemical Engineering Department and Lawrence Berkeley Laboratory University California, Berkeley Berkeley, California 94720
Gunther Schulze J. M. Prausnitz*
Received for review June 17,1980 Accepted February 17,1981
Azeotropic Composition from the Activity Coefficients of Components
+
An equation for calculating the composition of a binary azeotrope, x p = (1 a)-',has been tested on 45 systems exhibiting small, medium, and large deviations from regularity. When corrections for the real behavior of the vapor phase were taken into account for the case in which one component is associated in the vapor phase, e.g., acetic acid, the results obtained for the azeotropic composition improved considerably. The effect of the differences in the vaporization entropies of the components on the azeotropic composition, as exemplified by the systems containing acetic acid and n-paraffins, was found to be small compared to that of the dimerization of the acM in the vapor phase.
and observed (experimental)azeotropic compositions Ax2 = xp(calcd) - ~2(0bsd). It is expected that as a result of this study, a clearer picture will emerge regarding the suitability of Kireev's equation for computation of the azeotropic composition. To calculate the azeotropic composition, eq 1requires knowledge of the values of pFs, which in turn can be found from equations correlating the vapor pressure and temperature. However, the p"-T data are not available for a large number of organic compounds. This in itself may be considered as a disadvantage. In view of this fact, it also seemed worthwile to compare in a general way the results obtained by eq 1with those computed by using the equations developed by Prigogine (1954) and by Malesinski (1965). The results obtained by these equations for the azeotropic composition of some binary systems were reported by Kurtyka and Kurtyka (1980). But in both cases the azeotropic composition can be computed from the easily accessible data, namely the boiling temperatures of the pure components and of the azeotropes.
Introduction
The composition of an azeotrope in a binary regular solution is given by the equation xp = (1 + a)-1 (1) In eq 1, x 2 is the mole fraction of component 2 in the azeotrope and a is the square of the ratio of logarithms of the activity coefficients y 2 and yl,Le., (In y2/ln y1)1/2. At the azeotropic point the composition of the liquid and the vapor phase are equal, x 2 = y2, and in the case where the vapor phase is ideal, the expressions for y1 and y2take a simple form, namely 71 = P/P1° (2) 72
= P/PZ0
(3)
where p l 0 and p p oare the vapor pressures of component 1 and 2, respectively at the boiling temperature of the azeotrope, and p is the total pressure of the mixture. Equation 1, although in somewhat different form, due to Kireev (1941), has not been tested for computing the azeotropic composition of the systems exhibiting small, medium, and large deviations from regularity. Moreover, it also became interesting and useful to modify eq 1in a way that would take into account vapor phase nonideality and thus show to what extent that correction affects the deviations between the calculated 0196-4313/81/1020-0177$01.25/0
Results and Discussion
The azeotropic data, except those for the systems containing water (Andon et al., 1957), were taken from Horsley's book (1973). When doubts arose, original papers were consulted. 0
1981 American Chemical Society
