GASSOLUBILITIES
601
constant may go through a maximum becausc the liquid expands with rising temperaturc, favoring the dissolution of gas molecules.
Acknowledgment. The authors are grateful to the
National Science Foundation, to the donors of the Petroleum Research Fund, and to Gulf Oil Research and Development Company for financial support, and to Cecil Chappelow for assistance in calibrating the apparatus.
Gas Solubilities from a Perturbed Hard-Sphere Equation of State by P. M. Cukor and J. M. Prausnitz* Department of Chemical Engineering, University of California, Berkeley, California
(Received March 6, 1971)
Publication costs assisted by the Petroleum Research Fund
Solubilities of gases in liquids are correlated using a binary hard-sphere equation of state coupled with a van der Waals attraction term. The correlation is useful for rough estimates of solubilities in nonpolar systems over a wide range of temperature.
Following the work of Longuet-Higgins and Widom,’ Lebowitz2 has proposed that the equation of state of a binary mixture is given by
where P is the pressure, T is the absolute temperature, p , is the number density, k is Boltzmann’s constant and
where xi is the mole fraction and di is the hard-sphere diameter of component i. The constant a,, characterizing the attractive forces in the mixture, is given by
a,
=
al1xl2
+
2alzx1x2
+
~ 2 2 x 2 ~
(3)
where aij characterizes attractive forces between molecule i and molecule j. The function xm is given by Lebowits.2 Following the work of Pierot@ and Snider and H e r r i n g t ~ n ,we ~ have used equation 1 to obtain an expression for Henry’s constant. Upon equating the chemical potential of component 2 in the (ideal) gas phase to that in the liquid phase as z24 0, we obtain (4)
where H2,1is Henry’s constant of solute 2 in solvent 1,
R is the gas constant, v1 is the molar volume of the solvent, and A12 is a characteristic binary constant which, to a fair approximation, is independent of temperature. The function f is given by
where t1is obtained from equation 2 with xz = 0. Experimental Henry’s constants5-’ for thirteen gases in sixteen solvents were reduced to obtain values of A12. Hard-sphere diameters for the solvents were obtained from the group contribution method of Bondis for van der Waals volumes and the equation di = 1.47Vw,‘/*
(6) where Vwi is the van der Waals volume of moleculeoi in cm3/mol and di is the hard-sphere diameter in A. van der Waals volumes for solute molecules are given in Table I. From the van der Waals theory of simple, binary mixtures, we expect A12 to be related to (TclTcz)/ (Pcl/Pcz)’~z where To is the critical temperature and Pois the critical pressure. (1) H. C. Longuet-Higgins and B. Widom, Mol. Phys., 8, 549 (1964). (2) J. L. Lebowitz, Phys. Rev. A, 133, 895 (1964). (3) R. L. Pierotti, J . Phys. Chem., 67, 1840 (1963). (4) N. S. Snider and T. H. Herrington, J . Chem. Phys., 47, 2248 (1967); see also L. A. K. Staveley, ibid., 53,3136 (1970), and R. C. Miller, ibid., 55, 1613 (1971). (5) J. M. Prausnitz and P. L. Chueh, “Computer Calculations for High-pressure Vapor-Liquid Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1968. (6) J. H. Hildebrand and R. L. Scott, “Regular Solutions,” PrenticeHall, Englewood Cliffs, N. J., 1982. (7) P. M . Cukor, Dissertation, University of California, Berkeley, 1971. (8) A. Bondi, “Physical Properties of Molecular Crystals, Liquids, and Glasses,” Wiley, New York, N. Y., 1968.
The Journal of Physical Chemistry, Vol. 76,No. 4p1972
602
P. M. CUKORAND J. M. PRAUSNITZ
1
0
N2
Ji
Y N
0
E
\
c E
IO
N
a
Only pure-component critical propcrties and the solvent density arc required. In data reduction we uscd the corrclation of Bondi and Simkins to calculate the densities of the pure solvents as a function of tcmpcm ture. A statistical comparison of predicted and measured Henry’s constants for eight systems is given in Table 11. A total of 27 data points over a temperature range of 298-505”K were used in thc comparison. Expcrimental data for the systems included in Tablc I1 were not used in establishing the constants a! and 0 of equation 7. Table I1 also reports results calculated by Table 11: Comparison of Predicted and Measured Henry’s Constants Temp range,
I
OK
4
This work Shair’s correlation Figure 1. van der Waals interaction constants for liquid (1)-gas (2) systems.
Mean deviation
298-505 298-505 =
l
n
Mean dev? %
Max dev, %
23.1 24.4
46 89
;,E lHi(exp) -
Hi(calcd)/
Z=1
(nis the number of data points.)
Table I: van der Waals Volumes for Solute Molecules” Solute
V,”,cma/mol
6.5 4.8 7.5 12.5 15.0 13.0 24.0 14.1 17.1 32.0 29.0 45.0 24.0
V , values were adjusted so as to minimize scatter in the correlation shown in Figure 1.
The results of data reduction are shown in Figure 1; the data show that the binary constant A I 2 is given by
with a = 1.0787 and p = -5.9081; T , is in OK, P, in atmospheres, and A12in 1.2 atm/mo12. Equation 7 may be used to estimate Henry’s constants in nonpolar systems over an appreciable range of temperature.
The Journal of Physical Chemistry, Vol. 76, N o . 4>1978
Shair’s method based on solubility parameters.l0 For the systems considered the two correlations give nearly equal average deviations between calculated and experimental results. Unfortunately, neither the hardsphere equation nor Shair’s correlation gives uniformly accurate estimates of Henry’s constants. In some cases the former method is more reliable; in others the latter is better. However, the individual deviations between calculated and experimental solubilities suggest that: 1. The hard-sphere model tends to give poor results when the solvent molecule is a straight chain of more than seven atoms. 2. For slightly polar solvents (e.g., ethyl ether) the hard-sphere model tends to be superior to Shair’s correlation. 3. Shair’s correlation gives a temperature dependence on solubility which is stronger than that of the hard-sphere model. Both correlations, however, can predict maxima in plots of Henry’s constant us. temperature, consistent with experiment.
Acknowledgment. For financial support the authors are grateful to Gulf Research and Development Company, to the National Science Foundation, and to the donors of the Petroleum Research Fund. (9) A. Bondi and D. J. Simkin,A I C h E J.,6 , 191 (1960). (10) J. M. Prausnita and F. H. Shair, ibid., 7, 682 (1961).
