= =
e
P
porosity of carbon particles density of solid phase in porous carbon particles, g/cm3
literature Cited
Arnold, D. S., Plank, C. A,, Erickson, E. E Pike, F. P., Ind. Eng. Chem., Chem. Eng. Data Ser. 3, 253 (1958). Bohon, R. L., Clausen, W. F., J . Arner. Chem. SOC.73, 1571 (1951).
Brian, P. L. T., Hales, H. B., A.I.Ch.E. J . 15, 419 (1969). Calderbank, P. H., Trans. Inst. Chem. Eng. 37, 173 (1959). Calderbank. P. H., Moo-Young, - M. B., Chem. Eng. Sci. 16, 39
:1961). as reDorted bv Green. S. J., Sine. K. S. in 1)L ibinin. hI &I.. “Adsirptioil, ‘Surface Area and l%rosity,’J p 526, Academic Press, Xew York, N . Y . , 1967. Higbie, R., Trans. A.I.Ch.E. 31, 365 (193,5). Roughton, G., Ititchie, P. I),, Thompson, J. A,, Chem. Eng. Sci. 7. 111 (ia;7),. Hughmark , G . A,, Ind. Eng. Chem., Process Des. Develop. 6, 218 11967). Kdbel, H.,Siemes, W., Umschau24, 746 (1957)
w.,
Liebermann, L., J . Appl. Phys. 28, 205 (1957). Livingston, H. K., J . Colloid Chem. 4,447 (1949). Manley, D. M. J. P., Brit. J. Appl. Phys. 11,38(1960). Mehta, D. S.,Calvert, S.,Environ. Sci. Techno!. 1, 32.5 (1967). Mehta, D. S.,Calvert, S., Brit. Chem. Eng. 15, 781 (1970). Munemori, M., Sci. Rep. Tohoku Univ. 35, 165 (1951). Nagy, F., Moger, U., Magy. Kem. Foly. 65,406 (1959). Nagy, F., Scha G., Magy. Kem. Foly. 64, 81 (1958). Pozin, M. E., %bp lev, B. H., Gulyaev, S.P., Trans. Leningrad Teknol. Inst. im Eensoveta 43, 52 (1957).
Ray, A. B., Chem. Met. Eng. 28,977 (1923). Satt>erfield,C. N., “Mass Transfer in Heterogeneous Catalysis,” M.I.T. Press, Cambridge, Mass., 1970. Shen, J., Smith, J. hI.,IND.ENG.CHEY.,FUNDAM. 7, 106 (1968). Sherwood, T. K., Farkas, E . J., Chem. Eng. Sci. 21, 573 (1966). Siemes, W., Weiss, W., Dechema Monogr. 32, 451 (19-39). Valentin, IF. H . H., “Absorption in Gas-Liquid lhpersioris,” E. and 1’.N . Spon Ltd., London, 1967. Young, D. M., Crowell, A. D., “Physical Adsorption of Gases,” Butterworths, London, 1962.
V1,.
RECEIVED for review September 18, 1970 ACCEPTEDMarch 17, 1971
A Generalized Correlation for Henry’s Constants in Nonpolar Systems G. T. Preston’ and J. M. PrausniW Deparfment of Chemical Engineering, University of Caliifornia, Berkeley, Calif. 94720
A generalized thermodynamic expression for Henry’s constants is derived. The derivation is based on the statistical mechanics of dilute liquid solutions in conjunction with Scott’s two-fluid theory and a reduced empirical equation of state; it is applicable to solid as well as fluid solutes. When Henry’s constants are plotted against temperature, the correlation correctly predicts the maximum observed for some systems. The correlation is applied successfully to experimentally obtained Henry’s constants covering seven orders of magnitude for 60 nonpolar binary systems.
Thermodynamic properties of a dilute liquid solution call be determiiied from Henry’s law which says t h a t the fugacity of the solute is proportioiial to its mole fraction in the liquid lilirise. (At lo^ pressures the fugacity of the solute is equal t o its pai,tinl pressure.) The coiistaitt of proportionality, called Heiiry’s coiistant, is a funct’ion of ternperature and, to a 1es;her esteiit, also of pressure. Sumerous workers have studied the therinodyiiamics of dilute solutioiis; we do not give aii exhaustive review here. For eiigiiieeiiiig work, iiotable contributions were made by 1)udge aiid Newtoii (1!337), Eley (1939): Uhlig (1937) and Krishevsky aiid coworkers (1935, 1945). More recent work iiicludes t h a t of Prausnitz and Shair (1961), Hildebrand, et al. (1970), Kobatake arid -4lder (1962), Pierotti (1963, 1965), aiid Miller and I’rausnitz (1969). Many of these and other Ytudier are critically reviewed by I h t t i n o atid Clever (1966). See also the receiit tests of Prausnitz (1969) and King (1 969). For niasimuni utilitj-, a method for correlating Henry’s c-oiistaiits should be applicable to a variety of binary systems Present address, Garrett Research arid Development Co., L a Verile, Calif. 91750. * T(Jwhom correspotideiice should be sent.
over a wide range of temperature. Previous treatments are restricted in their application, often because they rely on some particular description of the liquid state. A method of correlatioii is presented here which, in principle, is not so restricted; for implemeiitation, however, it depends oti a reliable generalized equation of state. 111a binary mixture, the fugacity of cornponelit 2 , the solute, can be written = y2*(P)Z2H2,1(P expJpr ‘)
aY
(1)
where subscript 1 stands for the solvent; y2*(P)is the act’ivity coefficient of the solute in the solution a t pressure P, defined such that yr* + 1 as x 2 + 0; P‘ is the (arbit,rary) reference pressure for Henry’s constant N 2 , 1 ( P ‘ ) , aiid is the part’ial molar volume of the solute in the mixture a t infinite dilutioii, all a t system temperature T . Statistical Mechanics
T o obtain a useful expression for Heiiry’s constant, we ube Hill’s work on the statistical mechanics of dilute solutions (1957) 1960), from which it can be shoivii that Ind. Eng. Chem. Pundam., Vol. 10, No. 3, 1971
389
where f l ( P ) is the fugacity of component 1 a t pressure P, N is the total number of molecules in the liquid mixture, and A t = h/(27rn~,kT)"'. A derivation of eq 2 is given in the Appendix. The partition function for a mixture containing N2 solute molecules, designated by AS,, is given by
Substituting eq 12 into eq 9, we obtain
A.vz(N, P, T) =
&(S- N ? , X2, V , T) exp(-PV/kT)
v
(3)
From classical thermodynamics, the fugacity of the pure solvent is given by (Prausnitz, 1969)
where Q is the caiioiiical partition function, k is Boltzmanii's constant, and V is the total volume. K e can replace the summation of eq 3 by its maximum term, since fluctuations of V about the macroscopic volume of the mixture are insignificant. Equation 3 then becomes A,,T~(~Y, P , T)
=
&(K - N ? , N?, V , T) exp(-PVIkT)
For the pure saturated solvent (Ar, A.
=
=
0; P
=
(4)
P1Sv1S
__
RT
Substituting eq 14 into eq 13, we obtain finally
Pls)we have
Q(A-, vis, T) exp(-NPlSvlS/RT)
(5)
where R, the gas constant, replaces the product of k and Avogadro's number. For the pure solvent &, we write
where pl e o , , f , s is a "one-molecule" configurational partition function for the solvent. Then q1 O o n f ; gx exp(-NPISvlS/RT) N!A13h'
A0 =
=
QI
P2"VlS
___
RT
1 (15)
The complete analogy between eq 14 and 15 is not surprising if we recall the physical significance of H2,1: it is the fugacity of a hypothetical "pure" fluid whose properties are the same as those of solute 2 in solvent 1 at infinite dilution.
Substituting the compressibility factor zZrn eq 15 becomes
exp[- ( N - NJPlmt)rn/RT] X (N - iy,)!A,a(.V--A',)
,.v-sz
-
Henry's Constant Correlation
(7)
Applying the two-fluid theory of Scott (1956) t o describe the mixture, we have A,,
- 1 (14)
PzmvlS/RT,
For numerical results it is necessary to make some assumptions for expressing zZm as a function of temperature and volq? c o n f i g . m i v 2 ~ X ~ [ - N N ~ P ~ ~ V ~ / R T I ume. For simplicity, we assume validity of two-parameter (8) N2!A23h'2 corresponding states theory; that is, we assume that 22" is a universal function of reduced temperature and reduced where subscript m refers to properties of the mixture; in the volume. Equation 16 can then be written in the general form general case such properties are composition dependent. The factorization of the mixture partition function in eq 8 is useful because it is convenient to assume t h a t the factor for each component follows corresponding-states theory (Hill, 1960, Chapter 16). where F is ail unspecified function. For a n infinitely dilute mixture AT? = 1, q 1 o o n f , e . = T o evaluate the right-hand side of eq 17 we use, in reduced form, an empirical (Strobridge) equation of state presented q1 c o n i i g , v,,, = VI', and P1, = PI'; we denote Pzmby P2mand by Gosman and coworkers (1969) for representing the ex92 e o n l ; a , 712 by q2 oolrfigm. Combining eq 2, 7, and 8 we obtain tensive P V T data of gaseous and liquefied argon. Details of this equation are given in the Appendix. The resulting generalillustrated in Figure 1, is ized equation for H2,1(P1S)! T o evaluate we write
-RT
p - -R = T k T ( b~In)Q V
T,.V
2'
Substituting eq 6 into eq 10, we obtain
where Integrating eq 11 and noting that qcorliir-+ Nv/NA as v + 390
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
m
I
I
I
I
I
I
1
T, 127
I
I'1
= I
TIT,,"; I
U, = v:/v,;O
I
I
I
I
I
I
[
I
:4 I
100
150
200
250
vc", , Figure 2.
2
I 300 cc/rnol
I 350
I 400
450
Typical d a t a reduction for best TcZm and vc2"
0
T h e saturated liquid molar volume of the solvent, required in eq 18,was calculated using a computer subroutine presented b y Prausnitz and Chueh (1968), based on the correlation of Lyckmaii, et al. (1965).
-2
-8
-10
-12 -I 4
I, I/ M LA N E TR E I !I/ /I+-
Results
EDATA X T R A P O L A T E D USING
I I/
-I 6 I I I 0.2 0.4 0.6 0.8 1.0
Figure 1
.
SOLUBILITY
I 1.2
I
1.4 T / T ~
I 1.6
I I 1.8 2.0 2.2
Generalized correlation for Henry's constants
The 16 dimensioiiless constants are giveii iii Table I. Equation Table I. Constants for Eq 18 Ci I
I
0.42457138 -0.97214424 -0,48309824 -0.12611504 0.02042006 0.15446554 -0,06794337 0.06654287
9 10 11 12 13 14 15 16
Figure 3, which presents typical results for several systems, compares observed Henry's constants with those calculated using eq 18. .Is indicated in the next-to-last columii of Table 11, better agreement than that seen in Figure 3 wa5 obtained for some systems; in a few cases, however, the agreement with observed results was poor. As discussed by Gunn, et al. (1966), and by Chueh aiid Prausnitz (1967), when the solute is a quaiitum gas (iieon, hydrogen, helium), t,he effective critical properties are slightly temperature dependent. For quaiitum gas solutes, therefore, we write eq 19 and 20.
Ci
2000
0.30730388 0.27319810 -0,32214081 -0.07442808 0.35402549 0.05514906 0.02432847 0.86320302
50 'CARBON DIOXIDE ( 2 1 -PROPRANE I 1 1 t-
18 holds for reduced temperatures above 0.45 and reduced volumes above 0.25. Data Reduction
For each biliary system in Table 11, experimental data a t several temperatures were used to determine best values of T,." and by a graphical procedure. At each temperature the observed Henry's constant and a set of arbitrary values of elc2" were substituted into eq 18 to obtain a corresponding set of TCzm's. plot of T," US. v , ? ~ was prepared for each temperature, as shown in Figure 2. The intersection of all (or most) of the curves dictated the choice of the "best" T,," and zlc2-
21Clrn.
Figure 3. Calculated a n d observed Henry's constant: points, observed; , calculated; T,,", vc," from Table II; - -, calculated; T,,", vC2" from approximate correlations
-
-
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
391
Table II. Critical Constants for Infinitely Dilute Solutes Solute
r,
Solvent
Argon Nitrogen
Ethane Oxygen
Nitrogen Nitrogen
Methane Carbon tetrafluoride Ethane Butane Hexane Heptane Nitrogen
Nitrogen Nitrogen Nitrogen Nitrogen Oxygen Methane Methane Methane Methane Methane Methane Methane Methane Methane Ethane Ethane Ethane Ethane Ethane Ethane Ethane Ethane Ethylene Acetylene5 Acetyleneb Propane Propane Propane Butane Carbon dioxide*
Carbon tetrafluoride Ethane Propane Butane Pentane Hexane Heptane Carbon dioxide Hydrogen sulfide Ethylene Propane Butane Pentane Heptane Benzene Cyclohexane Hydrogen sulfide Ethane llethane Ethane Butane Pentane Benzene Heptane Methane
OK
rc2-,
% error OK
ve2-, cm3/mole
in
Data source
H2.1a
82-110 65-83
110 6 167
89.8 81.5
1 12
Eckert (1964) Armstrong, et al. (1955) Pool, et al. (1962)
122-172 70-110
141.7 193.5
91.7 147.5
3 59
C
122-277 310-400 344-444 305-455 65-83
199 214 192 220 144
152 248 297 361 91.1
49 4 3 7 6
c c c c
105-110
249
150
200-283 194-344 310-394 310-410 310-444 200-51 1 209-272
226 2 252 245 262 272 300 214.3
138 190 219 270 316 386 87.7
3 6 2 2 4 23 1
210-344
242
96
5
200-255 310-366 310-394 310-444 310-477 323-473 333-505 288-344
305 321 324.5 341 347 37 1 3 57 327
123 184 222 275 366 235 269 100
200-255 95-145 150-175 344410 344-444 377-477 422-505 110-170
290 229.7 323 372 7 379 8 423 5 446 5 235
141.5 96.7 157 220 263.5 240.5 370 94
1 4 11 1 1 3 2 32
6
2 m
Eckert (1964)
Armstrong, et al. (1955) Pool, et al. (1962) Thorp and Scott (1956)
C
Hanson, et al. (1953)
i
C
4 7 6 7 9 5
c
Carbon diovideb
Ethane
120-190
304
152
12
Carbon dioxideb Carbon dioxide
Ethylene
135-170
288
118
9
Propane
130-355
278
187
24
Carbon dioxide Carbon dioxide5 Carbon dioxideb
Propane
294-355
274
181
2
Propane
130-190
343
206
24
Propylene
120-170
29 1
166
11
Carbon dioxide
Butane
150-410
290
232
30
Carbon dioxide
Butane
3 10-41 0
285
225
1
C
c c c c
Hanson, et al. (1953) x e u m a n n and Mann (1969) Clark and Din (1950) c C C
C
Cheung and Zander (1968) Davis, et al. (1962) Sterner (1960) Brewer and Kurata (1958) Donnelly and Katz (1954) Jensen and Kurata (1969) Cheung and Zander (1968) Clark and Din (1953) Clark and Din (1953) Jensen and Kurata (1969) Cheung and Zander (1968) C
c
Jensen and Kurata (1969) Cheung and Zander (1968) Cheung and Zander (1968) Haselden and Snowden (1962) Jeiisen and Kurata (1969) Cheung and Zander (1968) C
~~~
392
_____
~~
_ _ _ _ _ _ ~ ~
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
~-
C
Table II (Continued)
% error Solute
Carbon dioxide* Carbon dioxide Carbon dioxide Hydrogen sulfide* Hydrogen sulfide6 Hydrogen sulfide Hydrogen sulfide Hydrogen sulfide* Hydrogen sulfide Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Helium Helium Helium Helium Helium Helium Helium Helium
Solvent
To*-,
O K
O K
vcpm, cm3/mole
in H2;la
Data source
Jensen and Kurata (1969) Cheung and Zander (1968)
Butane
150-190
386
262
25
Pentane
310-366
366
294
5
C
Hydrogen sulfide Methane
293-353
307
95
3
C
120-170
247
92
8
Cheung and Zander (1968)
Ethane
120-160
334
151
41
Cheung and Zander (1968)
Butane
140-394
334
240
6
Cheung and Zander (1968)
Butane
352-394
334
240
1
C
Butane
140-160
300
225
8
Cheung and Zander (1968)
Pentane
377-444
320
200
73
C
Nitrogen Methane Ethane Ethylene Propane Propylene Butane Hexane Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane Carbon dioxide
79-109 103-172 172-283 158-255 200-344 200-297 200-297 278-478 23-32 86-108 69-112 90-1 44 125-170 170-290 198-323 245-275
72 133 138 133 156 167 196 192 17 75.5 50.0 75 78 114 146 128
84 105 138 119 184 172 237 31 1 55 81.7 89.6 80 102.7 150 200 100.6
6 19 14 9 10 11 21 6 5 8 11 10 1 11 12 1
C
C
J-
,.(Hca~c;-
76 error in H Q ,E~ D.
T,
Hobsd)' obsd
x
C C C C
C C
C
d d d d d d d d
100. * Solid solute. Data quoted by Prausnitz and Chueh (1968), Appendices B and
no. of points Data quoted by Solen, et al. (1970).
'
450
(19)
400
I
I
I
I
I
t
TcZm,''and v ~ are~ the~ classical, ~ ~ high-temperature ' limits of T,," and vc2"; the quantity A i l 2 is given b y
where MI and A i 2 are, respectively, the solvent and solute molecular weights. Chueh and Prausnitz report c1 = 21.8'K and c3 = -9.91'K. For systems containing hydrogen or heand vc2",c2. lium, the values in Table I1 are Tc2m,cz To increase the utility of our correlation, we have also considered systems for which experimental d a t a are available at only one or two temperatures. For such systems it is not
Critical temperature for infinitely dilute solutes: H, CzHs, C~HI; X I CO?,GHzi 0, CHI; 4 N?; 0 , HP; 0, He
Figure 4.
e, CIHlo; A, CSH,; Ar,
0 2 ,
Ind Eng. Chem. Fundom., Vol. 10, No. 3, 1971
393
Table 111. Critical Temperature for Infinitely Dilute Solutes from limited Data (vcZm = 21.888 0.79827vc,, v in cm3/mole)
+
Solute
r,
OK
To,",
Data source
OK
Argon
Kitrogen
83.8
144.0
Argon Argon
Oxygen Methane
83-90 90.7
177 162.0
Argon Argon Argon Argon Argon Argon
Heptane Isooctane Cyclohexane Benzene Toluene Carbon tetrachloride Carbon disulfide Methane
298.2 298.2 298.2 298.2 298.2 298.2
252.1 244.4 234.5 230.0 232.0 235.0
a a
298.2
242.0
a
115.5
203.4
Carbon tetrafluoride Heptane Isooctane Cyclohexane Benzene Toluene Isooctane Cyclohexane Benzene Toluene Argon
117.1
198.8
298.2 298.2 298.2 298.2 298.2 298.2 298.2 298.2 298.2 83.8
296.2 286.7 283.3 279.3 281 .O 336.3 342.3 331.5 341 .O 160.2
Thorp and Scott (1956) Thorp and Scott (1956) a
83.8
137.4
70.0 110.0 298.2 298.2 298.2
132.9 148.0 202.8 197.4 200.9
298.2
204.8
a
83.8
138.9
90.7
153
Sprow and Prausnitz (1966) Pool, et al. (1962) Sprow and Prausnitz (1966) Mathot, et a2. (1956) Pool, et al. (1962)
Argon Krypton Krypton Krypton Krypton Krypton Krypton Krypton Xenon Xenon Xenon Xenon Kitrogen Yitrogen
Carbon monoxide
Nitrogen
Carbon tetrafluoride Cyclohexane Benzene Carbon tetrachloride Carbon disulfide Nitrogen
Nitrogen Nitrogen Yitrogen Nitrogen Carbon monoxide
Sprow and Prausnitz (1966) Pool, e t al. (1962) Pool, et al. (1962) Sprow and Prausnitz (1966) a
a a
a
a
a a
a a
a a a Sprow and Prausnitz (1966) Pool, e t al. (1962) Sprow and Prausnitz (1966) Pool, e t al. (1962) Eckert (1964) a a
a
Carbon monoxide
Methane
Oxygen Oxygen Oxygen Oxygen
83-90 298.2 298.2 298.2
173.6 242.9 226,s 230,O
298.2
237.6
a
Methane
Argon Isooctane Benzene Carbon tetrachloride Carbon disulfide -4rgon
90.7
194.9
Methane
Krypton
115.5
206.6
Xethane
Nitrogen
90.7
162.5
Methane
Carbon monoxide
90.7
167
Sprow and Prausnitz (1966) Thorp and Scott (1956) Sprow and Prausnitz (1966) Sprow and Prausnitz (1966) X a t h o t (1956)
Xethane Methane Methane
Isooctane Cyclohexane Benzene
298.2 298,2 298.2
Oxygen
394
Solvent
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
270.6 267.9 268.1
a
a a
a a a
Table 111 (Continued)
r,
Solvent
Solute
Carbon tetrachloride Carbon disulfide Krypton
Methane Methane Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Carbon tetrafluoride Ethane Ethane Ethane Ethane Ethane Propane Cyclopropane Cyclopropane Cyclopropane Carbon dioxide Carbon dioxide Carbon dioxide Carbon dioxide
Data source
OK
298.2
268.1
a
298.2
284.9
a
117.1
229.7
Methane
105-110
Heptane
298.2
245.2
Isooctane
298.2
244.5
a
Cyclohexane
298.2
216.8
a
Benzene
298.2
210.0
a
Carbon tetrachloride Carbon disulfide Isooctane Cyclohexane Benzene Carbon tetrachloride Carbon disulfide Cyclohexane Hexane Cyclohexane Benzene Heptane Isooctane Cyclohexane Carbon disulfide Heptane
298.2
229.7
a
298,2
201.1
a
298,2 298.2 298.2 298,2
338.9 346.8 346.1 347.9
a
298.2
366.7
a
298.2 298.2 298.2 298.2 298,2 298.2 298,2 298.2
396.6 399.3 412.9 422.1 317.1 309.6 303.2 321.8
298.2
309.8
a
298.2
313,6
a
298.2
288.7
a
298,2
277.9
a
298.2
281,3
a
298.2
302.2
a
298.2
270,2
a
possible to find two meaningful adjustable parameters; it is necessary to fix one of them d priori. The uc:”’s in Table 11 are approximately related to solvent critical volumes b y 21.888
To-,
Thorp and Scott (1956) Thorp and Scott (1956) a
Sulfur hexafluoride Isooctane Sulfur hexafluoride Cyclohexane Sulfur hexafluoride Benzene Sulfur hexafluoride Toluene Sulfur hexafluoride Carbon tetraSulfur chloride hesafluoride Carbon Sulfur herafluoride disulfide Data quoted by Hildebrand, et al. (1970).
t’c2m =
OK
+ 0.79827~~~
(22)
Equations 18 and 22 were used with the limited experimental data to obtain the ralues of T,?” given in Table 111. For some binary systems not’ included in Table I1 or Table 111, Henry’s constants can be estimated by using eq 22 for uc,a and obtaining T,,“ from Figure 4, an approximate correlation which was prepared by applying eq 18 and 22 to literature data for several of the systems of Tables I1 and 111. The solutes used for this correlation are listed in Figure 4;
215
a a a
the solvents were nitrogen, carbon monoxide, argon, oxygen. ethylene, hydrogen sulfide, benzene, cyclohexane! aiid normal alkanes C1-C,. The dashed lines of Figure 3 indicate the reliability of Henry’s constants estimated using the approximate correlations, Figure 4 and eq 22. Equation 18 is valid for T/T,,” >, 0.45. We have exteiided the Henry’s constant correlation of Figure 1 for use when 0.3 5 T/Tc,m< 0.48; this can occur when the solute of a system is far below its triple poiiit. Experimental values of H2,\ were determined from solid-liquid solubility data for ethylene in oxygen (AlcKinley and TTaiig, 1958) aiid for butadiene in methane and butadiene in argon (Preston, 1970). These observed Henry’s coilstants were theii plotted Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
395
in Figure 1 using reducing parameters found from eq 22 and Figure 4. The solid-solute systems included in Table I1 are all within the range of validity of eq 18 as written.
f2
(A.1)
= Hz,la2
where a2 = ~ 2 ~ x the 2 , activity of the solute, and a2 + x2 as 22 + 0. Hill (1957) shows that
Discussion
Equation 16 is general; it is valid for a n y solute (gas, liquid, or solid) in a liquid solution. However, numerical results depend on the particular choice of the equation of state; therefore, the values of T,," and v,," in Tables I1 and 111 pertain only to eq 18. It is a remarkable feature of eq 18 t h a t it provides a simple treatment for correlating Henry's const'ants over about seven orders of magnitude. The correlat'iori correctly predicts that Henry's constants increase with temperature for most systems but decrease with t,emperature for systems containing hydrogen or helium. For the system methane-heptane, among others, the correlation predicts that a plot of H ~ , 1 ( ~ 1 ' ) us. T goes through a maximum, in agreement with experiment. For dat'a reduction we have chosen the empirical GosmanStrobridge equation of state reduced by a two-parameter theory of corresponding states. -1s a result, while the general basis of our correlation, eq 16, is not restricted, our particular application, eq 17 and 18, is limited to simple nonpolar systems. For some systems, the average errors in Table I1 suggest that it is impossible to achieve good results over a wide range of temperature using a reduced equation of state with only t,wo parameters; therefore, we recommend caution when estimating Henry's const,ants a t temperatures lower than those listed in Table I1 for a given binary system. Also, for systems where experimental data are available over two separate temperature ranges, Table I1 gives the best T,," and vc2" for each range, as well as the best T,," and vc2" for all temperatures. The limitations of two-parameter corresponding states theory are well known (Leland and Chappelear, 1968). Much better quantit'ative result's can probably be obt'ained if a three-parameter corresponding-states treatment is applied to eq 16, or if shape fact,ors are introduced into t'he twoparameter corresponding-states treatment (Leland and Chappelear, 1968).
where X i = e"'/RT, the absolute activity, and A.v,(N, P, T ) is defined by eq 3. By definition
=X,/Xiid
(-4.3)
where i t i dis the absolute activity a t the standard state (ideal gas at the same temperature and at unit pressure, L e . ,f z i d 3 1 ) .
where A i
=
h / ( 2 ~ r n ~ k T )Combining "~. eq A.1LA4.4,we obtain
B. Equation of State. Gosman, et al. (1969), determined the empirical constants required to reproduce P V T d a t a of argon using Strobridge's equation of state, which is
+
If we substitute
Conclusions
Henry's constants for gas, liquid, or solid solutes in liquid solverits can be correlated through a generalized function of reduced temperature and reduced volume. The correlation depends on an equation of state but does not assume a particular physical structure for the liquid. The characteristic temperature and volume for the infinitely dilute solute can be determined from limited experimental data, thereby facilitating interpolation and judicious extrapolation for predicting Henry's constants a t temperatures other than t'hose where experimental results are available. For some nonpolar systems, where the characteristic temperature and volume can be obtained from approximate correlations, Henry's constants can be estimated even in the absence of a n y mixture data. Acknowledgment
where T,, and v,, are the critical properties of argon, eq 13.1 becomes
where C ~ TC, ~ Tetc., , are defined in the text following eq 18. The integral in eq 16 is
The authors are grateful to the National Science Foundation and to Gulf Research and Development Co. for financial support, and to the Computer Center of the University of California for the use of its facilities. Appendix
A. Derivation of Eq 2. TTe write eq 1 as 396
Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
Combining eq B.3 and B.4, we obtain eq 18.
Nomenclature = solute activity cl, cg = constants for quantum-corrected critical properties, a2
O K
constant for reduced equation of state temperature-dependent coefficient for reduced equaCir tion of state = fugacity of component i, a t m fi = Planck constant, erg sec h H z , , = Henry’s constant for component 2 in binary mixture, atm k = Boltzmann constant, ergs/’K m = molecular mass, g .If = molecular weight = constant in Gosman’s equation of state for argon nt = total number of molecules n; N2 = number of solute molecules Na = Avogadro’s number, molecules/mole P = pressure, a t m Pp = reference pressure, aim qconfin= “one-molecule” configurational partition function, cm3 Q = canonical partition function = gas constant, a t m cma/mole ‘K or cal/’mole ‘K R T = temperature, O K = molar volume, = S A V I N ,cm3/niole 21 = partial molar volume of solute a t infinit’e dilution, 2?2m cm3jmole V = total volume, cm3 = mole fraction solute in solution z? 2 = compressibility factor
Ci
= =
GREI:KLETTERS yE*
=
A
= = = =
x
,I @
liquid-phase solute activity coefficient (unsymmetric convention) a partition function absolut’e activity, e P / R T thermal de Broglie wavelength, = h/(2?rmkT)’:2,cni chemical potent,ial, cal/mole
SUESCRIPTS solvent solute 0 or 1 number of solute molecules X2, in c = critical property m = mixture property r = reduced property 1 2
= = =
SUPERSCRIPT^ cl = classical, high-temperature h i i t id = ideal gas state S = saturated liquid property ~0 = property of the infinitely dilute solute literature Cited
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