P. M. CUKORAND J. M. PRAUSNITZ
598
Solubilities of Gases in Liquids at Elevated Temperatures. Henry's Constants for Hydrogen, Methane, and Ethane in Hexadecane,
Bicyclohexyl, and Diphenylmethane by P. M. Cukor and J. M. Prausnitz* Department of Chemical Engineering, Universitu of California, Berkeley, California (Received March 6 , 1.971) Publication costs assisted by the Petroleum Research Fund
A new gas solubility apparatus has been constructed for studying gas-liquid equilibria at elevated temperatures. Henry's constants for methane, ethane, and hydrogen in n-hexadecane, bicyclohexyl, and diphenylmethane are reported for the temperature range 25-200". I n these three solvents, the solubility of hydrogen rises and that of ethane falls with increasing temperature. The solubility of methane goes through a flat minimum in the region near 150".
Although accurate knowledge of gas solubilities in liquids is of primary importance in many chemical processes, no widely applicable theory exists for their prediction. Further, experimental solubilities a t temperatures well removed from 25" are scarce and predictions at elevated temperatures are generally unreliable. In order to build a theory of gas solubility and to understand more fully the effects of temperature on gas-liquid equilibria, an apparatus has been constructed for the determination of gas-liquid equilibria at elevated temperatures. The experimental technique is based on the method of Dymond and Hildebrand. Solvent is continuously circulated through a vapor space containing a measured quantity of solute. A gas burette is used to determine precisely the number of moles of solute added. The equilibrium total pressure is measured using a fused-quartz Bourdon tube as a null instrument. This tube is mounted in a furnace whose temperature is maintained a t 235". Nitrogen is used to balance exactly the pressure which the system gases exert on the Bourdon tube; the nitrogen pressure is read a t 44" on a precision pressure gage. This procedure prevents solvent vapors from distilling into the pressure gage. The pressure-measuring equipment, manufactured by Texas Instruments, Inc., is accurate to within *0.04 mm Hg over a pressure range of 0-1000 mm Hg. Details of the design and operation of the gas solubility apparatus are given elsewhere.2ta
Data Reduction The solubility is calculated by determining n', the number of moles of solute in the vapor phase at equilibrium. This value is subtracted from no, the number of moles of gas added as calculated from the gas burette readings. The difference is An, the number of moles of gas which has dissolved. The number of moles of The Journal of Physical Chemistry, Vol. 76, N o . 4, 1979
solvent, n,, is calculated from the solvent density and the calibrated volume of the liquid chamber less the quantity of solvent removed due to expansion of the liquid with temperature. The (mole fraction) solubility a t the partial pressure of the solute is given by xz
An
= ____
n,
4-An
Since x2 is small, Henry's constant for solute 2 dissolved in solvent 1is calculated from
To determine n', the vapor-phase mole fraction yi was calculated from the equations
fiL flV
flL
= flV
(3)
fZL
= f2V
(4)
= (1 - Zz)P1#
(5)
- y2)P
(6)
= Pl(1
fzL = Hz,ix2
(7)
= CpzzlzP
(8)
fiV
where f is the fugacity, L and V refer to the liquid and vapor phase respectively, (o is the vapor-phase fugacity coefficient, PIs is the solvent vapor pressure at the system temperature, and P is the total pressure. The virial equation, truncated after the second virial coefficient, was used to relate cp1 and cp2 to temperature, (1) J. Dymond and J. H. Hildebrand, Ind. Eng. Chem., Fundam., 6 , 130 (1967). (2) P. M.Cukor and J. M. Prausnitz, Ind. Eng. Chem., Fundam., 10, 638 (1971). (3) P. M. Cukor, Dissertation, University of California, Berkeley, 1971.
SOLUBILITIES OF GASESIN
599
LIQUIDS
pressure, and composition. Virial coefficients were calculated using the correlation of Pitzer : and CurL4 0 Mixing rules suggested by Prausnitz and Chueh6 were used to determine the cross coefficient Biz. The solution of equations 1-8 for the compositions x 2 and y2 is a trial-and-error procedure. Only the temperature and pressure are known. A reasonable first approximation is to assume that (1 - y2) = Pls/P. Since P, T , and V , the volume of the vapor space, are known, n’ may be calculated from
where B M ,the mixture virial coefficient, is given by
+
+
BM = (1 - ~ 2 ) ~ B i i2 ~ 2 ( 1- ~2)B12 ~2’B22 (10) The liquid phase composition is calculated from equation l and Henry’s constant is determined from equations 2 and 8. Fugacity coefficients are calculated using the initial values of y . A new approximation for (1 - y2) is calculated by combining equations 5-8 to obtain
Substitution of the new value of y2 in eq 9 permits a second approximation to be made for 2 2 and H2,1. New values of the vapor composition are calculated until the values of the vapor and liquid compositions remain constant. I n reviewing the literature on gas solubilities we found that when calculating solubility many authors assume that the vapor-phase mole fraction of solvent is given by (1 - y 2 ) = Pls/P. This assumption can lead to serious error in reported values of x2 either if the solubility is so low that no = n’ or if the vapor pressure of the solvent is more than about one-tenth of the total pressure at the system temperature. For example, our calculations for the system hydrogen-bi861 mm, Pi8 320 mm) cyclohexyl a t 475°K ( P show that H2,1 = 1197 atm if we assume that (1 - y z ) = PiS/P = 0.37079. However, the correct value of (1 - y2) is 0.38956 which results in a value of H2.1 of 964 atm. This represents an error of nearly 24% in the solubility whereas the error in (1 - y2) is only 4.8%. We expect, therefore, that many solubility data which have been reported previously may be too low.
-
0
7 650
i
5001
“TY,; 1 I en; ; ; ;I ; : ; ; ; ; :
I50
n - Hexodecane
IO0 295 315
335
355 375 395 TEMPERATURE
415
435
455 475
355 375 395 415 435 TEMPERATURE, ‘K
455 475
,
O K
Figure 1. Henry’s constants for methane.
d
1601
-
Results The experimental values of Henry’s constants are shown in Figures 1, 2, and 3. The accuracy of these results is estimated to beat least 2% and probably closer to 1%.6 For each gas, the value of H increases (solubility decreases) as the solvent structure changes from (paraffinic) hexadecane to (naphthenic) bicyclohexyl to (aromatic) diphenylmethane. This behavior is in agreement with the trends predicted by Scatchard-
295
315
335
Figure 2. Henry’s constants for ethane.
Hildebrand theory: the difference between the solubility parameters (shown in Table IV) of each of the (4) K. 8. Pitzer and R. F. Curl, Jr., J . Amer. Chem. Soc., 79, 2369
(1957). (5) J. M. Prausnitz and P. L. Chueh, “Computer Calculations for High-pressure Vapor-Liquid Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1968. (6) .Tabulated Henry’s constants (Tables 1-111) will appear following these pages in the microfilm edition of this volume of the journal. Single copies may be obtained from the Business Operations Office, Books and Journals Division, American Chemical Society, 1155 Sixteenth St., N. W., Washington, D. C. 20036,by referring to author, title of article, volume, and page number. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche. The Journal of Physical Chemistry, Vol. 76,No. 4 , 1973
600
P. M. CUKORAND J. M. PRAUSNITZ Table V: Heats of Solution, Ahzm (cal/mol) o
n- Hexodecone
Methane
T,OK
Ethane
Hydrogen
in n-Hexadecane 325 400 450
- 705 -420
325 400 450
- 1002 - 284
325 400 450
- 304
- 2389 - 2166 - 1514
14.1
840 1213 1484
in Bicyclohexyl I
- 2579 - 2140 - 1218
792 1608 2380
in Diphenylmethane - 1934 -0.92 - 1691 387 - 1094
1135 1575 1873
661
300m 0295 315 335
355 375 395 415 435 455 475
TEMPERATURE ,
Figure 3.
OK
Henry’s constants for hydrogen.
Table IV : Solubility Parameters a t 25”
3.25O 5.686 6.600 7.46 8.16 9.08 Taken from J. M. Prausnitz and F. H. Shair, AZChE 682 (1961).
J., 7,
three gases and a given solvent is smallest in the case of n-hexadecane and greatest in the case of diphenylmethane. The temperature coefficient of Henry’s constant is given by
temperature a t a decreasing rate as the temperature rises. It appears that a t temperatures somewhat higher than 475°K the value of H may go through a maximum. For methane the enthalpy of solution changes sign a t about 425°K and Henry’s constant goes through a maximum. It is likely that this behavior may be observed for any gas in any nonpolar solvent provided the range of temperature is chosen properly. For example, Prausnitz and Chuehs report values of Henry’s constants as a function of temperature for a number of systems, Table VI summarizes the values of T m ,the temperature for which a maximum in the Henry’s constant is observed, for several gases dissolved in nheptane. As the critical temperature of the solute falls SO does Tm.
Table VI: Temperature T , Where Henry’s Constant Is a Maximum for Solutes in n-Heptane Solute
Nz CH4 CzHe
Tmr OK
Critical temp, OK
353 41 1 450
126 191 305
where Ah2” = i 2 L m-
h2G
(13)
where iZLm is the partial molar enthalpy of solute a t infinite dilution in solvent 1 and hzG is the enthalpy of ideal gas 2 at the same temperature. The natural logarithms of the experimental Henry’s constants were fitted to a second-degree polynomial in temperature and differentiated. Table V summarizes the values of A&” for the nine systems a t several temperatures. The temperature dependence of Henry’s constant is similar for a given solute in all t h e e solvents. For hydrogen, Henry’s constant falls as the temperature increases; for ethane Henry’s constant increases with The Journal of Physieal Chemistry, Vol. 76, N o . 4 , 1979
The critical temperature of hydrogen is so low that T , for hydrogen is far below the temperatures considered here. For hydrogen, therefore, Henry’s constant declines as the temperature increases. Recently Preston and Funk7 have shown that for simple nonpolar systems maxima in Henry’s constant do not occur if the data are plotted at constant liquid volume rather than at constant pressure. For (essentially) constant-pressure measurements, Henry’s (7) G. T. Preston, E. W. Funk, and J. M. Prausnitz, Phgs. Chem. Liquids, 2, 193 (1971).
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
