STATISTICAL T H E R M O D Y N A M l C S OF SIMPLE L I Q U I D M I X T U R E S Henry's Constants R . C.
MILLER' AND J . M. PRAUSNITZ
Department of Chemical Engineering, University of California, Berkeley, Calif., 94Y20,and National Bureau of Standards, Boulder, Colo. The cell-theory partition function of Eckert and Renon is useful for calculating Henry's constants for solutes, if the temperature is well below the solute's critical temperature. However, the cell theory cannot reproduce the experimentally observed maximum in Henry's constant at a temperature well above the solute's critical temperature. For higher temperatures a new partition function is presented, based on a modified van der Waals model for the behavior of the supercritical solute in the liquid state. Calculations of Henry's constants for several binary mixtures illustrate the new formulation.
H E cell theory of liquids has been discussed by numerous ,yring et al., 1964; Hill, 1956; Hirschfelder et al., Tauthors (F 1954; Prigogine, 1957; Rowlinson, 1959) since the pioneering work of Eyring and Hirschfelder (1937) and Lennard-Jones and Devonshire (1937, 1933, 1939). A good review is given by Barker (1963). Eckert and Renon (Renon et al., 1967,1968) have presented a semiempirical cell-model partition function suitable for calculating thermodynamic excess functions of liquid mixtures containing simple molecules. However, they restricted attention to mixtures whose components have similar volatilities; in all cases the system temperature was well below the critical temperature of each component. At low temperatures, well below the critical temperature of the gaseous solute, Henry's constant is readily found by using the methods discussed by Eckert and Renon. However, a t higher temperatures, near or above the critical temperature of the solute, reliable Henry's constants cannot be obtained from their partition function. The cell theory appears to be fundamentally inconsistent with the experimental result that for typical binary systems, a plot of Henry's constant us. temperature goes through a maximum. l o w Temperatures
Well below the critical temperature for the solute (component 2 ) , Henry's constant may be found from H 2
=
~z"f2'
(1)
where 72" is the activity coefficient a t infinite dilution of the solute, and fz0 is the fugacity of component 2 in the reference state, taken as pure saturated liquid 2 at system temperature T. The limiting activity coefficient is related to the molar excess Gibbs energy and may be calculated for mixtures of simple fluids from the partition function of Eckert, Renon, and Prausnitz (1967). The pure-component fugacity for the solute is readily calculated from pure-component thermodynamic properties. At low pressures, this fugacity is closely approximated by the vapor pressure, Pz0. To illustrate, 72" was calculated for the system C3Hs(l)CH4(2), using parameters reported by Eckert, Renon, and Prausnite (1967). Using pure-methane fugacity data and Equation 1, Henry's constants were obtained for the temper1
Present address, University of Wyoming, Laramie, Wyo.
103F-----l
Modified van der Waals Partition Function
-
Eckert and Renon Partition Function
A Sage and Lacay
1
to lQ''loo' Figure 1.
200
' Ternperature,aK ' '
'
I
'
'
'
300 I
'
Calculated and observed Henry's constants
ature range 90' to 120' K. In Figure 1, calculated results are compared with the experimental values of Cutler and Morrison (1965). The Eckert-Renon method for calculating Henry's constants is limited to temperatures below about 0.8 times the solute critical temperature, since it is based on a cell-theory equation of state for the solute. This equation of state is not applicable to the expanded liquid region. The following discussion concerns application of free-volume concepts toward obtaining a semiempirical partition function for Henry's constant at higher temperatures. High Temperatures
For a binary liquid mixture, the free-volume theory (Prigogine and Mathot, 1952; Salzburg and Kirkwood, 1952) gives VOL.
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449
the following expression for Henry's constant:
for Henry's constant:
where vf, is the molar free volume for component i and ( E ) i is the average potential energy for molecule i in the solution. Superscript 03 refers to infinite dilution of the solute in the solvent. The factor f accounts for multiple occupancy of cells (Salzburg and Kirkwood, 1952). This factor is associated with the problem of communal entropy and, for the liquid state, has a value somewhere between the solid-state value (E = 1) and the ideal gas value (E = 2.718). -4t sufficiently high temperatures, cell-model ideas have to be revised when applied to small light solute molecules a t low concentrations in a solvent. Instead of being confined to a very small volume near the equilibrium position in a cell, a solute molecule is free to move among the larger solvent molecules as if it were in a dense gas rather than in a normal liquid. However, while this freedom is available to the solute, it is not available to the solvent. For the solvent, the cell-model expressions used by Eckert and Renon for vfl and (E)1are retained. The expressions for the solute's free volume and average energy are based on a modification of van der Waals' model. The solute's free volume a t infinite dilution is assumed to be equal to the molar volume of the solution minus an excluded volume for the 1-2 interaction: vf2" =
V"
- b12 - VI - blz
(-+;:
0.08 Ti*) T
+ ~a*%*Ti2*Viz* R Tu1
where fla* is a proportionality constant relating the interaction potential parameter and the Eckert-Renon characteristic temperature T*
12N0e12= QaTCl2 = Qa*T12*
;.rrN,~l2'
= f&,Vc12 = flb*V12*
(9 1
For classical fluids,
Tc/T* = 1.134 f 0.008
(10)
and the mixing rule is taken as
T12* = ( T1*T2*)'/2(1 - kn)
(11)
where the constant kl2 is included to account for small deviations from the geometric-mean rule. The molecular integrals Z,(l) and Z12(*) are functions of el2/kT only. They were evaluated numerically on a digital computer and then fitted by least squares for the range 0 < e 1 2 1 k ~< 1.25:
(31
The excluded volume may be related to either the critical volume or the characteristic volume v* of Eckert and Renon by
biz =
X
0.0475 ([TI
(4)
+ 0.0109 (k;Z' Y]
(12)
where the hard-sphere diameter for the 1-2 interaction is and the fl's are proportionality constants. For 13 simple classical fluids, the critical volumes, vc, are related to u* by
412,
v,/v* = 3.49 f 0.09
(5 )
and the mixing rule is taken as
Function j is given by
The average solute energy term is given by Comparison with Experiment
where 412 ( T ) is the potential function and 812 (T) is the radial distribution function for the 1-2 interaction. In his original model of the liquid state, van der Waals (in effect) assumed that &z(T) is given by the Sutherland potential and that g12(r) is unity for T > Q and zero for r < u (Hill, 1960). To obtain an improved approximation, first the modified Lennard-Jones potential with a hard-sphere cutoff is used (Kirkwood et al., 1952), and second, a perturbation on the hard-sphere radial distribution function to approximate the true distribution a t high temperatures (Orentlicher and Prausnitz, 1967). .4n approximation to the Percus-Yevick result is used for the hard-sphere distribution function (Lebowitz, 1964; Rowlinson, 1965; Throop and Bearman, 1965). Using the above ideas, we obtain the following expression 450
IhEC
FUNDAMENTALS
Equation 8 was applied to the high-temperature data for CH~-C~HS, and to solubility for N2-C2He, Ne-Ar, He-CH4, and Hz in a nuinber of solvents. Calculated and experimental Henry's constants are shown in Figures 1, 2 and 3. Quantum corrections for hydrogen and helium, while not large, were still significant a t the temperatures where solubility data were available. For quantum fluids the reducing parameters T* and u* were assumed to be functions of temperature and molecular weight according to
v* =
(v*)cz
+ ( d 3 ) / m T+) ( d 4 ) / m 2 T a )
1
(16)
where subscript cl refers to the high-temperature (classical)
Table 1.
Deviations from the Geometric-Mean Rule for TI** kia
This work
System
CHrCsHs Nn-CnHa H2-N2 HAY3 - .~ Hn-Ar H2-CHn Hn-C& Hn-CsHs He-CH4
Gas mixture data
0.00 0.04 0.02 0.00 -0.04
0.03 0.03 0.07 0.40 0.18
Se-Ar
0.02 0.05 0.00 0.00 0.00 0.03 0.05
0.07 0.40 0.14=
From Him-Duncan correlation (1968)
limit. These equations were also assumed to apply for and u12* with m12 given by
.*
P
0 Mullinr
r
1")
by Orentlicher
0
1o2
TI**
I
I
l
l
1
1
1
1
1
1
1
200
100
Temperature,
"K
Figure 2. Calculated and observed Henry's constants for hydrogen
The constants a"). .d4)were determined by requiring liquid-phase data for hydrogen and for deuterium to adhere to the pure-component expressions of Renon, Eckert, and = -9.91 Prausnitz (1967). Values of a") = 21.8 and taken from Gunn, Chueh, and Prausiiitz (1966) were found to be satisfactory, while a@)and a(4)were -140 and 4-80, respectively. For hydrogen, (u*),zand (T*),zwere 16.1 CC. per gram mole and 37.8' K. These values do not satisfy empirical Equations 5 and 10. Thus, the critical volume calculated by Gunn, Chueh and Prausnitz (1966) should more closely describe the contribution of the light component to the excluded volume term. D, and fh, were assumed to be universal constants. The best values found to give reasonable results for all systems studied were $&, = 692 and & ! = 0.344. Thus, for classical mixtures, Cia* = 785 and %* = 1.20. For helium, ( T * ) , z and ( U * ) ~ Z were estimated to be 8.92' K. and 13.4 cc. per gram mole. For calculating Henry's constants, 5 was taken equal to 2.718, while the geometric-mean deviation parameters, k12, were adjusted to give best agreement with the gas-solubility data; they are given in Table I. The best values of k12 are similar, but not identical, to those found from gas-phase volumetric data (Chueh and Prausnitz, 1967; Eckert et al., 1967; Prausnitz and Chueh, 1968). Certainly the trends are in good agreement, particularly for the systems containing He and Ne, where deviations from the geometric mean are known to be large (Chueh and Prausnitz, 1967; Hiza and Duncan, 1968; Prausnitz and Chueh, 1968). Unfortunately, small changes in k12 have a significant effect on Henry's constants-for example, a change in klz of 0.04 can change Henry's constant by 20 to 30%. Very recently an empirical correlation for k12 of simple gas mixtures has been reported by Hiza and Duncan (1968). I t may now be possible to predict kl2 for simple mixtures with reasonably good accuracy. Hiza and Duncan show an average deviation of only 0.015 between experimental (gas-phase) and calculated k12 values for 18 binary systems. The maximum error appears to be about k0.05, which produces (roughly) a 30y0change in calculated Henry's constants.
1 ,
ao
too
120
140
kmptrafun,°K
160
iao
Figure 3. Calculated and observed Henry's constants for helium and neon
Conclusions
While the cell-theory partition function of Eckert and Renon appears to be useful for describing thermodynamic properties of liquid mixtures at temperatures where every VOL.
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component can exist as a pure liquid, it is not applicable to liquid mixtures containing one supercritical component. For such mixtures, we cannot use the concept of a confining cell of solvent molecules surrounding the solute molecule. Instead, we have proposed a modified van der Waals model of a dense fluid, wherein the solute molecule is not confined in its translational motion but is free to move in a field of uniform potential. Application to simple binary systems indicates that this formulation can give results in satisfying agreement with experiment, provided we do not rely on the geometric-mean approximation for the characteristic energy of unlike-molecule interaction. Acknowledgment
The authors are grateful to the National Science Foundation for financial support and to the Computer Center, University of California, Berkeley, for the use of its facilities. Nomenclature
a(’)
. . . a(4)= constants in quantum corrections = modified van der Waals covolume = molecular shape parameter = average potential energy term = fugacity = radial distribution function = Henry’s constant at the saturation pressure of the solvent = molecular integrals = function defined by Equation 14 = Boltemann’s constant = constant for deviation from geometric-mean mixing rule = molecular weight = number of molecules = number of moles = Avogadro’s number = pressure = ideal gas law constant = radius or radial distance = absolute temperature = volume = molar volume = free volume = mole fraction = depth of potential well = activity coefficient = proportionality constant = potential energy = modified Lennard-Jones hard-sphere diameter = communal entropy factor
SUPERSCRIPTS AND SUBSCRIPTS C
= critical property
cl
= classical limit = component i
i
452
l&EC
FUNDAMENTALS
0
1 2 12 00
*
= reference state = solvent = solute = solvent-solute interaction
= infinite dilution = reducing parameter
literature Cited
Barker, J. A., “Lattice Theories of the Liquid State,” Pergamon, Oxford, 1963. Chueh, P. L., Prausnitz, J. hl., IND.ENC.CHEV.FUKDAMEXTALS 6, 492 (1967). Cutler, A. J. B., hforrison, J. A., Trans. Faraday Sor. 61, 429 (1965). Eckert, C. A., Renon, H., Prausnitz, J. M., IND.ENG.CHEM. FUNDAMENTALS 6, 58 (1967). Ellington, R. T., Eakin, B. E., Parent, J. D., Gamir, D. C., Bloomer, 0. T., “Thermodynamic and Transport Properties of Gases, Liquids and Solids,” p. 180, ASME Monograph, 1959. Eyring, H., Henderson, D., Stover, B. J., Eyring, E. M., “Statistical Mechanics and Dynamics,” Wiley, New York, 1964. Eyring, H., Hirschfelder, J. O., J . Phys. Chem. 41, 249 (1937). Gunn, R. D., Chueh, P. L., Prausnitz, J. hI., A.I.Ch.E. J . 12, 937 (1966). Heck, C. H., Hiza, hI. J., A.Z.Ch.E. J . 13, 593 (1967). Hill, T. L., “Introduction to Statistical Thermodynamics,” Addison-Wesley, Reading, Mass., 1960. Hill, T. L., “Statistical Mechanics,” 3lcGratv-Hill, New York, 1956. Hirschfelder, J. O., Curtiss, C. F., Bird, It. B., “Molecular Theory of Gases and Liquids,” Wiley, New York, 1954. Hiza, M. J., Duncan, A. G., Cryogenic Engineering Conference, Case Western Reserve University, Cleveland, Ohio, Aug. 19-21, 1968. Hiza, M. J., Heck, C. H., Kidnay, A. J., Advan. Cryog. Eng. 13. 343 11968). Kirkkood,‘J. GI, Lewinson, 5’. A,, Alder, B. J., J . Chem. Phys. 20, 929 (1952). Lebowitz, J. L., Phys. Rev. 133, A895 (1964). Lennard-Jones, J. E., Devonshire, A. F., Proc. Roy. Soc. (London) A163, 53 (1937); A166, 1 (1938); A169, 317 (1939); A170, 464 (1939). Mullins, J. C., Ph.D. thesis, Georgia Institute of Technology, 1965. Orentlicher, M., Prausnitz, J. M., Can. J . Chem. 46, 595 (1967). Orentlicher, M., Prausnitz, J. M., C t f m . Eng. Sci. 19,775 (1964). Prausnitz, J. M., Chueh, P. L., Computer Calculations for High-pressure Vapor-Liquid Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1968. Prigogine, I., “Molecular Theory of Solutions,” North-Holland Publishing Co., Amsterdam, 1957. Prigogine, I., Mathot, 5’., J . Chem. Phys. 20, 49 (1952). Renon, H., Eckert, C. A., Prausnitz, J. hf., IKD.ENG.CHEM. FUNDAMENTALS 6, 52 (1967). Renon, H., Eckert, C. A,, Prausnitz, J. hr., ISD.ENG.CHEM. FUNDAMENTALS 7, 335 (1968). Rowlinson, J. S., “Liauids and Liquid Mixtures,” Butterworths, London,‘ 1959.‘ Rowlinson, J. S., Repls. Progr. Phys. 28, 169 (1965). Sage, B. H., Lace W. N., “Thermodynamic Properties of Lighter Hydrocartons,” American Petroleum Institute (Project 37), 1950, 1955. Salzburg, 2. W., Kirkwood, J. G., J . Chem. Phys. 20, 1538 (1952). Streett, W. B., J . Chem. Phys. 42, 500 (1965). Throop, G. J., Beaman, R. J., J . Chem. Phys. 42, 2408 (1965). RECEIVED for review Januar 25, 1968 ACCEPTEDMarci 20, 1969
