86
Ind. Eng. Chem. Fundam. 1983, 22, 86-90
fZi = molar enthalpy of a pure component i Hi= partial molar enthalpy of component i l i , = latent heat of pure component i at temperature Ti L = azeotropic (integral) latent heat of vaporization, or integral latent heat of vaporization of a non-azeotropic mixture in eq 19 and 22 P = total pressure P, = reduced pressure defined by PIPc P, = critical pressure of the mixture R = universal gas constant T = absolute temperature Ti = boiling temperature of a pure component T, = reduced temperature defined by Tf T , T,, Tci,Tcp= true critical temperature of a mixture; critical temperature of component i ; pseudocritical temperature given by eq 16 y, = dew point or bubble point of a mixture, respectively V , = partial volume of component i Vci = critical volume of component i Xi= mole fraction of component i in the liquid AHi = partial heat of mixing of component i AH = heat of mixing Greek Letters a , p , y , 6 = adjustable parameters w , wi = acentric factor of the mixture defined by eq 11;acentric factor of component i Superscripts 1 = of the liquid
v = of the vapor * = of an azeotropic mixture
Subscripts i = of pure component i iTi = of component i at a temperature Ti obsd = observed P, T = at a constant pressure or temperature, respectively P, T1 = at temperature F or T',respectively
Literature Cited Gmehling, J.; Onken, V.; A&, W. "Vapor-Liquid Equilibrium Data Collection", Dechema, Chemistry Data Series, 1977-1978; Voi. 1, Parts 1, 2a, and 2b. Hirata, M.; Ohe, S.; Nagahama, K. "Computer-AMed Data Book of Vapor-Liquid Equiiibrlum"; Eisevier: Amsterdam, 1975. Li, C. C. Can. J. Chem. Eng. 1971, 49, 709. Licht, W.; Denzler, C. G. Chem. Eng. Prog. 1948, 4 4 , 627. Malesinski, W. "Azeotropy and Other Theoretlcai Problems of Vapor-Liquid Equillbrium"; PWN-Polish Scientific Publishers: Warszawa, 1965. Pawlak, J.; Zielenkiewicz, A. Rocz. Chem. Ann. SOC. Chim. Polonorum 1985, 39,314. Reid, R. C.; Prausnitz. J. M.; Sherwood, T. K. "The Properties of Gases and LiquMs", 3rd ed.;McGraw Hill; New York, 1977. Santrach, D.; Lielmezs, J. Ind. Eng. Chem. Fundam. 1978, 17, 93. Spencer, C. F.; Daubert, J. E.; Danner, R. P. AIChE J. 1973, 19, 522. Swietoslawski, W. Rocz. Chem. 1961, 35, 317. Tamir, A. FluidPhase Equilib. 198Ol81, 5 , 199. Tamir, A.; Apelblat, A.; Wagner, M. Fhid Phase Equilib. 1981, 6 , 113.
Received for review November 17, 1981 Accepted October 12, 1982
Estimation of Setchenow Constants for Nonpolar Gases in Common Salts at Moderate Temperatures Ellen M. Pawlikowskl and John M. Prausnltz' Deparfment of Chemical Engineering, University of California, Serkeley, California 94720
A simple technique is suggested for estimating salting-out constants for nonpolar gases in aqueous salt solutions in the range 0 to 60 O C . As suggested by the perturbation theory of Tiepel and Gubbins (1973), the salting-out constant is given by a linear function of the Lennardlones energy parameter of the gas. Data for nine nonpolar gases were examined. Correlation parameters are given for nine nonpolar gases and fifteen salts at 25 O C . Correlation parameters are also reported for individual ions. For a few salts, correlation parameters are given as a function of temperature.
Introduction It has long been recognized that the presence of a salt alters the solubility of a gas in water, and many theories have been suggested for calculating salt effects from independent measurements of the properties of salts in water and gases in water. Early theories were based on electrostatic considerations; gas-water interactions were considered to be affected by a change in the dielectric constant of the medium due to the presence of salt ions. Later, theories utilizing the concept of internal pressure were introduced. A review is provided by Long and McDevit (1952). All of these theories are difficult to use because they require parameters which are not readily available. Tiepel and Gubbins (1973) proposed a statistical-mechanical perturbation theory based on that of Barker and Henderson. However, perturbation theory gives only qualitative agreement between theory and experiment. Calculations using this theory are tedious and require
several parameters that cannot easily be determined from independent experiments. In this paper, we propose a simple correlation, suggested by perturbation theory, for estimating salting-out constants for nonpolar gases in salt solutions over a moderate temperature range, 0 to 60 OC.
Correlation To examine the effect of low concentrations of salt in water on the solubilities of sparingly soluble gases, we express the Gibbs energy G of the liquid phase in the form suggested by Guggenheim (Newman, 1973)
nowoo G / R T = -+ Cnj[ln (mjX?) - 11 + RT jzo C &,mini Debye-Huckel contribution (1) i#O jzo
+
where no is the moles of water, nj is the moles of species
0196-4313/83/1022-0086$01.50/0 $2 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 87
j , mi is the molality of species j (mol/kg of H20),kois the chemical potential of a reference state for pure water, and AjS is the secondary reference defined as
Table I. Lennard-Jones Parameters for Nonpolar Gases as Reported by Liabastre (1974) Elk, K
gas
He
6.03 29.2 95 114 118 125 137 21 7 230
HZ NZ NZO
where pj is the chemical potential and Pij is a binary interaction parameter for solute interactions. Equation 1is limited to solutions dilute in salt and gas because only two-body solute interactions are considered. By differentiating eq 1 with respect to the moles of dissolved gas nd, we arrive at an expression for the chemical potential of the dissolved gas in the liquid in equilibrium with the gas at a given temperature and partial pressure
0 2
Ar CH, Xe CZH,
L
KO ti
m
-
where m, is the molality of the salt. For the same gaseous solute, in the absence of salt at the same temperature and pressure
Parameter Pdscharacterizes the effect of low concentrations of salt on the solubility of the gas. Since both liquid phases are in equilibrium with the same vapor phase, pd = p’d. Therefore, we can relate Pds directly to the change in gas solubility. pd
In
md’
= Pd’
+ 2Pddmd‘ = In md + 2@dsm,+ 2Pddmd
(44 (4b)
In the limit where md and md’ approach zero, P d d m L and can be neglected and
Pddmd
In md‘ = In
md
+ 2&m,
(44
or
2&m, = In m4/md
(5)
The interaction parameter between gas and salt is called the Setchenow constant, k,, related to by k , = 0.868Ph = (log md’/md)/m, (6) The units of k, are determined by the units used for salt concentration. Unfortunately, not all authors use the same units. In our work, the units for k , are kg/mol because we use molality (moles of salt per kilogram of solvent) for salt concentration. Equation 6 is valid for a range of concentration that depends on the salt, on the gas and on temperature; typically, eq 6 holds to about m, = 5 mollkg and for sparingly soluble gases. For positive k,, the salt is said to have a “salting-out” effect because the salt pushes gas out of solution. For negative values of k,, the salt is said to have a “salting-in”effect because the salt draws gas into solution. Recent efforts have focused on using the perturbation theory of Barker and Henderson as suggested by Tiepel and Gubbins (1973) to estimate k , from binary equilibrium data for gas in water and for salt in water. To calculate k,, Tiepel and Gubbins use perturbation theory to calculate Henry’s constant H i for the gas in salt-free water and then to calculate Henry’s constant Hd in salt solution. The salting-out constant can then be determined from the ratio log H’,/Hd. To perform the calculations, one must know the density of the salt solution, the Lennard-Jones size and energy parameters (a and E ) for water, for both ions of the salt, and for the gas; the dipole moment of water, and the polarizability of the gas. Tiepel and Gubbins show that
0
50
150 E/K, Kelvin
200
250
Figure 1. Dependence of k, on cw for some common gases and salts.
their theory provides a qualitative description of both salting-in and salting-out. However, the theory is cumbersome because so many parameters are needed and because a computer is required to perform numerical integrations. We have tested the theory of Tiepel and Gubbins for 46 salts and 12 nonpolar gases and have found only fair agreement between theory and experiment. For simple salts with noble gases, agreement is good but for many other systems (especially divalent and trivalent salts), agreement is poor. The slight improvement over older theories does not justify using perturbation theory if one considers the large number of parameters and the complexity of the calculations. However, if the theory could be used in a simpler form with fewer parameters, it might be useful for obtaining approximate values of k,. Perturbation theory suggests that the following simple form may be useful for estimating k , k , = a, + bsEgas/k
(7)
where a, and b, are temperature-dependent parameters that depend on the salt but not on the gas. Here egm is the Lennard-Jones energy interaction parameter for the gas, and k is Boltzmann’s constant. As suggested by Liabastre (1974), this interaction parameter was determined from gas-solubility data in salt-free water using scaled-particle theory for Hh. Although tgas/k parameters might be slightly different if determined from perturbation theory, our correlation would not be improved, although constants a, and b, might be slightly different. Table I provides Elk parameters for the gases in this work. Equation 7 was applied to salting-out data for the nine gases listed in Table I. To illustrate, Figure 1shows sample plots of k, vs. egm/k for KOH, NaC1, and LiC1. A t 25 “C, fifteen salts were tested; Table I1 gives correlation parameters a, and b,. For all salts, except NaN03, the salting-out constant increased with e l k . For the data examined, k , for NaNO, decreases with Elk. This anomalous behavior is consistent with the observation of other workers who have noted unusual salting-out characteristics with nitrate salts
88
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Table 11. Salting-Out Parameters for Strong Electrolytes in Eq 7 at 25 "C ( e l k in K )
salt
a,,
NH4Br KI LiCl NaCl KC1 NaBr NaNO NH,Cl CaCl BaCl, (NH,),SO, Na,SO, HC1
kgimol
0.0345 0.075 0.0658 0.080 5 0.0652 0.0855 0.0864 0.0255 0.162 0.0895 0.101 0.142 0.0193 0.0908 0.114
H2S04
KOH
104b,, kg/( mol K ) 1.31 1.31 3.31 3.56 6.00 2.98 -1.40 4.60 6.95 8.42 11.45 13.6 0.115 0.199
5.01
Table 111. Saltingout Parameters for Individual Ions for Eq 7 ( e l k in K ) 102b,:,n ion i as:,a kg/mol kg/(mA K)
OH' K'
c1-
Na Li NH,' BrH' NO; Ca'+ Ba2+ +
+
so,*-
0 0.114 -0.049 0.129 0.115 0.074 -0.04 18 0.0682 -0.04 3 0.21 1 0.138 0.021
Table IV. Temperature Dependence of the Salting-Out Parameters for Eq 9 (Temperature Range: 0-60 "C)
0
102a,,, kgl kg/mol (mol K)
salt NH,Br KI LiCl NaCl KOH KCI
0.249 0.228 0.134 0.516 0.0679 0.0861
ion
kg/ kg/mol (mol K)
0
OH- 0
K' C1Li' 1Na'
This separation necessitates the choice of an arbitrary reference ion since ksM+and k,, cannot be determined independently. Table I11 provides correlation parameters a, and b, for individual ions based upon the arbitrary reference state a, = b, = 0. The parameters in Table I11 predict experimental #, values almost as well as those in Table 11; they provide a method for estimating k , for salts not listed in Table 11. Table VI provides a comparison of estimates using Tables I1 and 111. Experimental data for NaBr were not used to determine the parameters for Na+ and Br- given in Table 111. Table VI gives calculated values of k , for NaBr using the parameters in both Tables I1 and 111. Agreement is good in both cases. Since data for k , are scarce at temperatures other than 25 "C, only a few salts could be examined at other temperatures. The temperature dependence for the range between 0 and 60 "C can be expressed by (9) k, = a ] , + azsT + ( b l , + b * 8 n ( c / k ) g a a
b i i , kg/ (mol K)
a i i,
bzi,
ki
(mol K3 )
0
0 -4.86 -4.75 -1.80 2.24 2.68
0.0679 0.0155 1.95 X 0.0182 -0.0225 1.514 X 0.116 -0.036 2.86 X 0.160 0.0357 -1.038 X 0.497 -0.123 -1.14 X
x x x
lo-' lo-'
X
x
Data of Shoor. e l a l
O O 4 ! 1 -I
(Eucken and Hertzburg, 1950). Salting-Out Constants for Individual Ions Since the salting-out constant is calculated for salts at infinite dilution, it has been suggested (Bromley, 1972) that k , can be expressed as the sum of contributions from two ions k,,+,- = k,,+ + kx(8)
-6.08 X lo-' -2.62X -4.93 X -6.3 X 10" -4.86 X -9.61 X
3.123 X 9.12X 1.8 X 3.74 X 1.95 X 3.464 X
102a2i,
a
T
-0.0721 -0.0512 -0.0229 -0.146 +0.0155 -0.007
b,,, kgl (mol K2)
Table V. Temperature Dependence of Saltingout Constants for Individual Ions for Eq 9 (Temperature Range: 0-60 " C )
0.0502 0.443 -0.407 -0.410 -0.397 0.423 -0.441 0.393 -0.374 -0.358 -0.504
Values of asi and bSi were determined from the values of a, and b , given in Table 11. a S O H -and b S O H -were arbitrarily taken to be zero: aSMX= as^+ a,x-; bsMx = The salting-out of any salt MX can be bsw + b s x - . determined by adding the individual ion constants. See eq 8.
b,,, kg/ (mol K )
a,,,
IO
1
1
1
1
1
I
20
30
40
50
60
70
Tevperoture,
I
80
"C
Figure 2. Temperature dependence of k, for KOH with several gases.
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 89
Table VI. Salting-Out Constant for Some Common Gases at 25 "C
k , , kg/mol salt
gas
NaBr
HZ 0 2
Ar CH, He HZ NZO Ar
NaNO,
KCl
H2 N2O 0 2
Ar He HZ
KOH
N2O 0 2
Ar CH, NaCl
HZ N2O 0 2
Ar C*H, CH, He Xe NZ
perturbation theoryC
expt
calcda
calcdb
0.089 0.110 0.142 0.117 0.089
0.085 0.120 0.123 0.126 0.087
0.092 0.107 0.110 0.088
0.105 0.162 0.153 0.219 0.077
0.080 0.072 0.058
0.082 0.071 0.069
0.082 0.070 0.068
0.038 0.071 0.064
0.078 0.099 0.159 0.125 0.068
0.082 0.134 0.136 0.140 0.069
0.610 0.113 0.111 0.105 0.068
0.129 0.130 0.180 0.179 0.197
0.129 0.171 0.173 0.177 0.183
0.082 0.134 0.136 0.140 0.069 0.129 0.171 0.173 0.177 0.183
0.092 0.101 0.140 0.133 0.162 0.127 0.082 0.165 0.121
0.091 0.121 0.122 0.125 0.162 0.129 0.082 0.1 58 0.114
0.091 0.121 0.122 0.125 0.162 0.129 0.082 0.158 0.114
0.110
0.064 0.097 0.107 0.098 0.138 0.097 0.093 0.119 0.117 0.174 0.133 0.074 0.127 0.115e
referenced Akerlof (1935) Long and McDevit (1952) Clever and Holland (1968) Michels e t al. (1936) Morrison and Johnstone (1955) Akerlof (193 5) Akerlof (1935) Akerlof (1935) Akerlof ( 193 5) Long and McDevit (1952) Seidell ( 1940) Clever and Holland (1968) Morrison and Johnstone (1955) Shoor e t al. (1969) Seidell (1940) Shoor e t al. (1969) Shoor e t al. (1969) Shoor et al. (1969) Morrison (1952) Akerlof (1935) Eucken and Hertzburg (1950) Clever and Holland (1968) Morrison (1952) Clever and Holland (1968) Morrison and Johnstone (1955) Eucken and Hertzburg (1950) Morrison (1952)
a Calculated with parameters in Table 11. Calculated with parameters in Table 111. Calculated with parameters in Table VII. Primary references were used whenever available. e Tiepel and Gubbins (1973) report 0.141 because they used different density data.
Table VII. Parametersa Used to Evaluate k, Using Perturbation Theory
x
a x
lo-', species H2 0 2
Ar
CH, He H2O C,H, Xe
NZ H2O Na K+ OH+
c1-
BrNO,'
nm
E/k,K
2.87 3.46 3.41 3.82 2.62 3.41 4.42 3.96 3.70 2.15 1.97 2.77 3.06 3.74 4.17 3.30
29.2 118.0 125.0 137.0 6.03 114.0 230.0 217.0 95.0 85.3 91.7 163.0 215.0 246.0 320.0 89.1
p, Db
1024,
cm3/ molecule 0.802 1.57 3.41 2.60 0.204 3.00 4.47 4.00 1.76
1.84
a Parameters for gases from Liabastre (1974); parameters for salts from Tiepel and Gubbins (1973); density data for salt solutions from Chapman and Newman (1968). 1 D = 3.36 X lo-'' C cm.
present, respectively, values of k,, using parameters in Tables I1 and 111. The final column provides k, calculated from perturbation theory using parameters in Table VII. The results shown in Table VI indicate that the errors in k, values from the approximate technique are comparable to t h w from perturbation theory. Figures 2 and 3 provide comparisons between experiment and calculated results based on parameters in Table V. Agreement is good. The correlation was not tested on salting-in systems because insufficient data are available to evaluate corre-
lation parameters. However, in principle, the correlation presented here could be used also for salting-in systems. The results shown here indicate that k, is not strongly dependent upon Therefore, for salting-in systems one would anticipate a lowering of the value of a, rather than an increase in the slope, b,.
Conclusion An approximate, easy-to-use correlation has been developed for estimating salting-out constants for aqueous solutions of nonpolar gases and for salts listed in Table 11. The correlation requires only two parameters per salt. However, these parameters can also be estimated from salting-out constants for individual ions, thereby extending applicability of the correlation to systems where no experimental data are available. Acknowledgment For financial support the authors are grateful to the National Science Foundation. Part of this research was conducted while Ellen Pawlikowski held a fellowship from the Fannie and John Hertz Foundation.
Literature Cited ~ ~ ~ ~ ~ ~ , " ; , "1:$:1;201., ~ m ~ ~ ~ ~ Clever, L. H.; Holland, C. L. J . Cbem. Eng. Data 1966, 73,411-414. Chapman, T. W.; Newman, J. "A Compilation of Selected Thermodynamic and Transport Properties of Binary Electrolytes in Aqueous Solution"; University of California, Lawrence Radiation Laboratory, 1968: UCRL 17767. Eucken, A.; Hertzburg, G. Z.Phys. Chem. 1950, 79, 1-23. Liabastre, A. A. Ph.D. Dissertation, Georgla Institute of Technology, Aug 1974. Long, F. A.; McDevit, W. F. Chem. Rev. 1952, 57, 119. Micheis, A.; Qrver, J.: BijL A. PbYsia III, 1938, 8 , 797-808. Morrison, T. J. J. Chem. SOC. 1952, 3814-3822. Morrison, T. J.; Biiiett, F. J. Chem. SOC. 1952, 3822.
m
~
~
~
Ind. Eng. Chem. Fundam. 1983, 22, 90-97
90
Morrison, T. J.: Johnstone, N. 6. 6. J. Chem. SOC. 1955, 3655-3659. Newman, J. S . “Electrochemical Systems”; Prentice-Hail: Englewood Cliffs, NJ, 1973. SeMell, A. “Solublities of Inorganic and Organic Substances”: Van Nostrand: New York, 1940. Shoor, S. K.; Walker, R. D.;Gubblns, K. E. J. Phys. Chem. 1969, 73,
3 12-3 17. Tiepel. E. W.; Gubbins, K. E. Ind. Eng. Chem. Fundam. 1973, 12, 18.
Received for review December 23, 1981 Accepted August 8, 1982
Prediction of Transport Properties. 2. Thermal Conductivity of Pure Fluids and Mixtures James F. Ely‘ and H. J.
M. Hanley
Thermophysical Properties Dlvlsion, National Engineering Laboratory, National Bureau of Standards, Boulder, Colorado 80303
A technique for the prediction of the thermal conductivity of nonpolar pure fluids and mixtures over the entire range of PVT states is presented. The model is analogous to the extended corresponding states viscosity model reported previously by Ely and Hanley in 1981. Calculations for the thermal conductivity require only critical constants, molecular weight, Pitzer’s acentric factor, and the ideal gas heat capacity as a function of temperature for each mixture component as input. Extensive comparisons with experimental data for pure fluids and nonpolar binary fluid mixtures including paraffins, alkenes, aromatics, and naphthenes with molecular weights to that of C24are presented. The average absolute deviation between experiment and prediction is less than 7 % for both pure species and mixtures.
Introduction We have recently discussed the prediction of the viscosity (7)of pure fluids and mixtures via the extended corresponding states one fluid model (Ely and Hanley, 1981 (hereafter denoted by part 1);Ely, 1981). It was stressed that the method is predictive and the number of mixture components is, in principle, unlimited. The method was applied to nonpolar fluids over a wide range of states from the dilute gas to the dense liquid. Here we extend the model to the thermal conductivity (A). It differs from other corresponding states thermal conductivity models (Mo and Gubbins, 1976; Hanley, 1976,1977;Haile et al., 1976; Murad and Gubbins, 1977; Christensen and Fredenslund, 1980; Teja and Rice, 1980) in the scope of application and in the fact that it requires no transport data as input. The basic idea is as before, namely that the configurational properties of a single-phase mixture can be equated to those of a hypothetical pure fluid. The properties of this fluid are then evaluated via corresponding states with respect to a given reference fluid (methane) at the appropriate corresponding pressure and temperature, or density and temperature. It is appreciated at the onset that there are both formal and practical difficulties with the thermal conductivity. For example, (1) a onefluid model must ignore the contribution of diffusion to the conductivity (Hanley, 1977a), and (2) any corresponding states argument cannot correctly take into account the effect of internal degrees of freedom on the thermal conductivity-an effect which may be large in the dilute gas (Hanley, 1977b). Also, the state of the art for measuring thermal conductivity is relatively poor; few data are accurate to within 10% and the range of data is usually limited to the dilute gas or to the saturated liquid. The scope of this work is the same as in part 1: we discuss results for the paraffins, alkenes, aromatics, and naphthenes to molecular weights of C2@ Input data are the critical parameters, molecular weight, and acentric factor of the pures or of the species in a mixture. In addition, the ideal gas heat capacity at constant pressure for
each species is required to account for the internal degrees of freedom of a polyatomic molecule.
Thermal Conductivity Model We postulate that the thermal conductivity of a pure substance or mixture may be divided into two contributions-one arising from the transfer of energy from purely collisional or translational effects, A’, and the other from the transfer of energy via the internal degrees of freedom, A”. We further assume that this latter contribution is independent of the density and may be calculated from the modified Eucken correlation for polyatomic gases, viz.
where A”, is the internal contribution for component a ,
M , is the molecular weight, q,* is the dilute gas viscosity of component N which is calculated by the method outlined in part 1, C$ is the ideal gas heat capacity, R is the gas constant and fint has a constant value of 1.32. For a mixture, ,A,” is calculated via the empirical mixing rule (Li, 1976) A”,i,(T)
= CCX,XBA‘“B . B
(2)
where (A’”fl)-l
= 2[(x’”)-l
+ (X”O)-l]
(3)
We emphasize that in general the assumption that the internal contribution is independent of density must be incorrect. The translational contribution A’ is calculated via the corresponding states method outlined in part 1. Briefly, we postulate that the translational mixture thermal conductivity is identical with that of a hypothetical pure fluid, denoted by a subscript x , viz. A’mix(P,T)
A’~(P,T)
This article not subject to US. Copyright. Published 1983 by the American Chemical Society
(4)
