(1) = normal fluid correction term (2) = polar fluid correction term literature Cited
Bartlett, E. P., J . Am. Chem. SOC.49, 687 (1927). Bartlett, E. p,, Cupples, H, L., Tremearne,T. H., J. Am. Chem. SOC.60, 1275 (1928). Bhattacharyya, D., Thodos, George, Can. J . Chem. Eng. 43, 150 (1965). Deming, W.E., Shupe, L. E., Phys. Rev. 37, 638 (1931). Din, F., “Thermodynamic Functions of Gases,” T’ol. 3, Butterworths, London, 1962. Halm, R. L., Stiel, L. I., A.I.Ch.E.J. 13,315 (1967a). Halm, R.L., Stiel, L. I., *4.1.Ch.E. ’leeting, New November 196713. Holser, W.T., Kennedy, G. C., Anz. J . Sei. 267, 71 (1959). Kang, T. L., Wirth, L. J., Kobe, K. A., McKetta, J. J., J . Chem. Eng. Data 6, 220 (1961). Kennedy, G. C., Am. J . Sei. 262, 225 (1984). Lewis, G. W., Randall, M,, “Thermodynamics,” 2nd ed., rev. by K. S. Pitzer and L. Brewer, Appendix 1, RIcGraw-Hill, Sew York, 1961.
Lichtblau, I. M., Bretton, R. H., Dodge, B. F., A.Z.Ch.E.J. 10, 486 (1964). Love, A. E., “Treatise on the Mathematical Theory of Elasticity,” 4th ed., p. 145, Dover, Yew York, 1944. Ramsay, W., Young, S., Phil. Trans. Roy. SOC.(London) 177A, 123 (1886). Reid, R. c.,Smith, J. Chem. Eng. progr. 47,418 (1951). Suh, K. W.,Storvick, T. S., A.Z.Ch.E.J., 13, 231 (1967). ~
~
~
~
~
v
, ! ~~ e ~ ~ ~g ~ ~ ~ S, S R j
9, 889 (1937). Uhl, T’. W.,T’oamick, H. P., Chem. Eng. Progr. 69, 33 (1963). S’ogel, -4.I., “Text Book of Practical ,Organic Chemistry, Including Quantitative Organic Analysis,” Longmans, London, 1948 Zubar&, v. ?J,, Bagdonas, -4,v., Teploenergetzka14, 79 (1967). RECEIVED for review July 29, 1968 ACCEPTED April 25, 1969 For supplementary material, order KAPS Document 00508 from ASIS National Auxiliary Publications Service, c/o CCLI Information Sciences, Inc., 22 West 34th St. New York, N. Y., 10001, remitting $1.00 for microfiche or $3.00 for photocopies.
MOLECULAR THERMODY N A M l C S OF SOLUTIONS AT MODERATELY HIGH PRESSURES AAGE FREDENSLUND’ AND G. A. SATHER Department of Chemical Engineering, University of Wisconsin, Madison, Wis. 53706 Working equations are derived for the prediction of excess Gibbs free energies and activity coefficients of mixtures using a molecular theory of solutions. The theory is based on the statistical mechanical formulation of the principle of corresponding states, which is used in conjunction with an equation of state for methane for pressures up to 6000 p.s.i.0. and van der Waals’ approximation of the corresponding states characteristic parameters for the mixture. The molecular theory is shown to be successful in prediction of the excess properties of mixtures of simple molecules a t pressures substantially higher than atmospheric. The excess Gibbs free energy for the C2H6-C2H4system at 40” F. and 42 atm. is, for example, predicted within 15yoof the experimental value. Predictions for the N2-CH4 and 02-COz systems are also reasonable.
URING the last iew decades, the molecular thermody-
D namics of solutions based on the methods of statistical
mechanics has been of increasing interest to chemical engineers. Methods have been developed by Guggenheini (1952), Brown (1957), Prigogine (1957a), Rowlinson (1959), Barker (1963), Hermsen and Prausnitz (1966), Eckert et al. (1967), and Leland et al. (1968) for quantitatively predicting with equilibrium properties of mixtures of simple molecules a t low varying degrees of success the macroscopic configurational pressures from fundamental molecular properties of the pure components. The extension of the molecular theories to systems under hhiger pressures has received little or no attention. The purpose of our treatment is to apply a molecular theory to liquid solutions of simple molecules at pressures substantially higher than atmospheric. The theory used here is within the framework of the cell model. An equation of state for methane recently developed by Veinnx and Kobayashi (1967) is used in conjunction with a statistical mechanical formulation of the principle of corresponding states to predict excess Gibbs free energies of mixing and activity coefficients. Both the familiar two1 Present address, Instituttet for Kemiteknik, Danmarks Tekniske HZjskole, Lyngby, Denmark.
718
l&EC FUNDAMENTALS
parameter principle of corresponding states and an extension to a three-parameter corresponding states principle first proposed by Prigogine (1957b) and later used by Eckert (1964) are employed. The fundamental molecular parameters representing the mixtures are calculated using van der Waals’ approximation as described by Leland, Rowlinson, and Sather (1968) and the random mixing and average potential models (Prigogine, 1957a). Statistical Mechanics
The configurational partition function plays a central role in the statistical treatment of mixtures; once the configurational partition function for the system is determined, all configurational properties of the system may be derived from it (Hirschfelder et al., 1954). In this treatment, a twoparameter and a three-parameter principle of corresponding states are derived, each from a different partition function. Two-Parameter Principle of Corresponding States. The configurational partition function for a pure fluid, Qii, may be written as (Prigogine, 1957c) :
where
U is the potential energy of the system, fl
= (kT)-I,
and arcN) symbolizes the integration over the 3N position coordinates. The potential energy, U , is assumed to be the sum of all pair interactions, u ( T ) . For the principle of corresponding states used here, these pair interactions are assumed to be evaluated using a potential of the form
u (T 1 = fij€mF ( T / g i juw )
(2 1
The parameters €00 and um are an energy and a length which, together with the universal function, F , define a reference potential with which all the pair potentials in the mixture are conformal. The parameters f;j and gij are the numbers by which €00 and um are multiplied to generate the i - j pair potential. For a pure fluid, i = j . Inserting Equation 2 into Equation 1 and making use of well known statistical mechanical relationships (Rowlinson, 1959), it may be shown that the configurational Gibbs free energy of any pure fluid a t temperature T and pressure P , Gii’ ( P , T),is given by
Gii’(P, T )= fiiGo’ (Phii/fi;, T/fii) - N k T In hii (3) Here hii = gi?. Go’ is the configurational Gibbs free energy of a reference substance a t temperature T/fci and pressure Phii/fii, referred to here as the pseudo-reduced temperature and pressure, respectively. The reference substance consists of molecules of characteristic energy €00 and characteristic length uoo. Equation 3 is a statement of the two-parameter principle of corresponding states for pure fluids. The essential problem in the extension of Equation 3 to mixtures is the proper averaging of the parameters fii, fjj, f i j and g,i, gjj, gij (i # j ) for the i - i, j - j , and i - j interactions. The aim is to determine a pair of these parameters, f and g, which accurately represent the mixture in the application of the principle of corresponding states. I n the one-fluid model by Scott (1956), also called the random mixture by Brown (1957) and the crude approximation by Prigogine (1957a), the mixture is regarded as a fluid consisting of average molecules of characteristic energy f,em and characteristic length gzum. I n Scott’s two-fluid model (1956), closely related to the semirandom approximation (Brown, 1957) and the average potential model Prigogine (1957a), the two fluids are each of the two species in the binary mixture wit,h properties which reflect the average surroundings in the mixture. The characteristic molecular parameters for each of the species in the mixture are f i ~ m , gium and f j ~ m , gjum, respectively. The detailed expressions for fz, g, and (fi, si) using various models are given below. The parentheses denote that for the two-fluid model there is a set of parameters, two for each “fluid.” For the one-fluid model, Equation 3 becomes G,‘(P,
T , x) = f,Gd
(ph, , - NkT In h, \ fzl
(4)
JZ
where x denotes the dependency of the configurational Gibbs free energy of the mixture on the composition. The principle of coxresponding states applies to “the average fluid” in exactly the same manner as it was applied to pure substances. For Scott’s two-fluid model, the principle of corresponding states is applied separately to each fluid:
The single subscript i denotes component i in the mixture. Equation 5 reduces to the one-fluid model if f, and h, are substituted for f i and hi for all components.
Three-Parameter Principle of Corresponding States. The introduction of a third parameter in a corresponding states treatment offers a large improvement in the correlation of thermodynamic properties of real systems. The third parameter used here, c,, takes into account the hindrance of molecular rotation in solution and the effect of neighboring molecules on translation. This concept was first developed by Prigogine (1957b) for long-chain hydrocarbons, and Eckert (1964) and Eckert et al. (1967) applied it to mixtures of simple molecules. This development is not compatible with the over-all averaging performed in the one-fluid model, so the extension to the three-parameter principle of corresponding states is done only for the two-fluid model. The configurational partition function for pure species i is written in the form (Prigogine, 195713) Qai = [Qi ( T , v ) ] 3 x c ; exp
S i i fiiE0 - N___
kT
where 9i is the partition function per degree of freedom lost by condensation, 3ci is the number of degrees of freedom lost by condensation per-molecule, si; is the normalized coordination number, and Eo is the reduced lattice energy corresponding t’o all elements a t t’heir equilibrium positions, a function of the reduced volume only. The normalization of s,i is such that, sii is one for spherical molecules. The partition function expressed by Equation 6 is more restrictive than Equation 1, since it depends on the existence of a quasilattice. It should, therefore, be applied only to condensed states. The configurational Gibbs free energy may be written in t’erms of a reference configurational Gibbs free energy using the same arguments as before:
where the pseudo-reduced temperature and pressure now are z, For mixgiven by Tca/szz fit and P h a a / s a z f arespectively. tures, the principle of corresponding states is applied to each “fluid” separately as before, and the configurational Gibbs free energy of component i in solution is given by Gi’(P, T , 5 ) =
[flai
ci
(Phi Tci) - , - - A‘kT In hi] sifi STfi
-GO’
(8)
Determination of Parameters
The parameters for the pure components, fit and gzi given by ( ~ ~ ~ / k ) / ( cand ~ / kU )~ , / C T ~may , be determined using the Lennard-Jones potential u(T)
= 4Eza[(uzz/T)n
- (‘Jaa/r)m]
(9 1
which satisfies Equation 2. The analogous parameters for the mixture, f,, g, or (fa, S a ) , are calculated using four different models: random mixing (RM), a one-fluid model; van der Waals’ one-fluid model (vdW-1); the average potential model (APM), a two-fluid model; and van der Waals’ two-fluid model (vdW-2). The vdW-2 model is the only one considered when the threeparameter principle of corresponding states is used. The Rb4 model is based on the assumption of completely random mixing of the various species in the mixture (Brown, 1957). The vdW-1 model may be derived from an expansion of the “soft-sphere” radial distribution function about that of a system of hard spheres, neglecting powers of T-’ beyond the first (Leland et al., 1962). I n the APM and vdW-2 models, VOL.
8
NO. 4 NOVEMBER
1969
719
the same kind of averages are performed as in the RM and vdW-1 models, respectively, except, that in the two-fluid models the averages are calculated on each “fluid” separately. The expressions forf,, gz and ( fi, gi) for the various models are given below (Leland et al., 1968, 1969)
RM. fz
= (
CziZjcjfijgijm)n’(n-) ( i
i
CZ..f . ,m)-lKn-) ( C C Z i Z j f i j g i j n ) l / ( n - )
-
z-
CZiZjjijgjn)-/(nw)
i
j
13923
1 5
i
i
a
(10) (11)
= ( C C Z i r j f i j h i j ) ( C Cxixjhij)-’ i
i
i
h, =
i
i
CxiZjhij i
(12) (13)
APM. fi
= (
CZ.f... m)n/(n-d 3
( CZjfijgijn)-/(n-m)
LJgU
(14)
i
i
+ + A3p2+ A4p3+ Asp41p2 + p T [ R + Bip + B2p2 + B3p3 + B4p4+ &p6] + p 2 ( p + PO)’C(P + pol3 - (a+ POYIX CCB + pol3 - ( P + ~ o ) ~ l * eIG x p- C& + E Z ( P+ POYIX
P = [Ai
A2p
+ DZP4- D3p24- D4p3-k D6p41.
( T - To)}4-p2CDi
exp {k - (F1+ F2p)/TJ
i
vdW-1. fi
ties of all other simple substances through the principle of corresponding states, are derived is the one given for methane by Vennix and Kobayashi (1967):
gi = ( C Z j f i j g i j m ) - l / ( n - m ) ( Cxjfijgijn)li(n-“‘) i
(15)
i
The per cent deviation from experimental values of density up to 0.36 gram per cc. for pressures up to 6000 p.s.i.a. is claimed to be less than 0.1%. Working Equations
The working equations for the prediction of the excess Gibbs free energy and activity coefficients of mixtures may now be derived. From well known thermodynamic equalities, it may be shown that the configurational Gibbs free energy is related to the P - p - T behavior by:
vdm-2. fi
= ( Czifijhij)( Czjhii)-’ i
G’(p, T ) =
(16)
i
hi = &hij
(17)
(23)
[a[( a p)
- RT]dp+
aP
RTInNp
(24)
T
Integrating the first integrand by parts,
1
The quantides g i j = uij/um and f i j = ~ i j / e ~i ,# j, entering into the above eight equations are calculated using the mixing rules u” = 1 (18) a at + a3 and e . . --
( € .2, 2 ’ € j j ) ’ / 2 ( 1
- Kij)
(19)
The deviation from the geometric mean, ~ f j may , be obtained from independent sources such as the second cross-virial coefficient (Sprow and Prausnitz, 1966). Values of c i for various simple fluids are given by Eckert (1964), and ci is assumed to be the same in solution as in the pure fluid. The values of c i used in the calculations here are normalized such that ci for methane is 1. For nearly spherical molecules, s2i, the normalized coordination number for the pure fluids, may be assumed equal to 1. Hermsen and Prausnitz (1966a) list values of sii for various hydrocarbons. The calculation of si, the coordination number of species i in the mixture, is discussed by Eckert (1964). Under certain simplifying assumptions, si is given in terms of the probability of finding a moleculej in the cell surrounding a molecule i , p i , the cell collision diameter, ui, and sii: si
= (u?/ Cpjuj)sii
G’(p, T ) - RT In Np =
Equation 25 permits the evaluation of the configurational Gibbs free energy of methane, Go’(p, T ) , directly from the equation of state, Equation 23. The results of the integration are shown in Appendix A. Making use of Equations 3 and 5 for the two-parameter principle of corresponding states and Equations 7 and 8 for the three-parameter principle of corresponding states, the configurational Gibbs free energies of the pure components and the mixture are calculated. The excess Gibbs free energy is given by:
GE = C Z ~ [ G ~ ’ (TP, ,Z) - Gil’(P, T ) ] Let
r be defined by
r = G/RT
i.e.,
(20 )
i
~i
= Cpjuij
roil= Go’ (Poi,
(21)
The above three equations must be solved by trial and error.
I L E C
FUNDAMENTALS
,
etc.,
PO,, and TO,,are the pseudo-reduced pressure and temperature for pure component i, and PO,and TO,are the pseudoreduced variables for component i in solution. The activity coefficient of component k in a solution is given by lnyk = a ( n T r E ) / h k
Reference Equation of State
The equation of state from which all configurational thermodynamic propert,ies of methane, and thus the proper-
Toi1
R Toi
i
720
(26)
i
where
nTrE =
CnI[rl’- rizil i
(281 (29 )
The differentiation indicated in Equation 28 is now performed:
Pure-Cwnponmt Charocterislk
Mixinq Ruler
Mirlura Model
MoiecuIor
vdw-l
VdWZ
Poromoters fw Unlike
From Equations 4 and 5 it follows that
rz’= ro,’- In h,
(31)
Thus
ar; an,
-
are,' ank
Equation of
1 ahi ha ank
(32 1
and Activity of Reference FiuM
Since roc’ = ro,’(P0,, To,),
are,' c--._. + -are,' . - ape, ank aPo, aTo,
-
ank
aTO,
ank
(33)
The derivative dro,’/aPo, is replaced by
(aro,’/apo, I/(apo,/apo, since the equation of state is explicit in pressure. pseudo-reduced density, po,, is given by pot = h,(Mo’JI,)
Summary of Method
The (34)
where JI is the molecular weight. The following observations are made:
(35)
Inserting Equations 31 through 36 into Equation 30, the working equation for the activity coefficients for the twoparameter principle of corresponding states becomes
PO,
Figure 1. Schematic diagram of calculational procedure used to obtain theoretical values of Gibbs free energies and activity coefficients for mixtures
(37 )
For the three-parametei principle of corresponding states, aPo,/ank and dro,/dnx depend not only on f a and h,, but also on sz. The working equation analogous to Equation 37 becomes:
The determination of the excess Gibbs free energies and activity coefficients is shown schematically in Figure 1. The starting points of the method are: 1. The characteristic molecular parameters for the pure components, which together with ~ %and j the desired temperature, pressure, and composition form the numeric input to the calculation. 2. The concepts of statistical mechanics resulting in a formulation of the principle of corresponding states. Statistical mechanics may also be used in the formulation of the mixture models. 3. Mixing rules for the characteristic molecular parameters for unlike interactions. 4 A model for the mixture, here the RM, vdW-1, APRI, and vdW-2 models. After receiving input from items 1 and 3, the model calculates the over-all characteristic parameters for the mixture, fa, h,, and perhaps sa. The pure-component and mixture Characteristic parameters are fed into the principle of corresponding states, which permits the properties of the refeience to be calculated from the reference equation of state through the use of pseudoreduced properties-e.g., Equation 39. The reference properties are fed back to the principle of corresponding states, resulting in the corresponding properties of the pure components and the mixture. These properties are then used in the calculation of the excess Gibbs free energy and activity coefficients. The calculations described above were done with the aid of a CDC 1604 computer. Results
The pseudo-reduced density is the same as given in Equation 34. Equations 37 and 38 contain two kinds of derivatives, those which are model-dependent (aftlank, ahi/ank and as,/an,) and those which are derived from the methane equation of state (aro,’/aTo,, aro,’/apo,; and aPo,/apo,). The model-dependent derivatives are shown for the vdW-1 and vdW-2 models in Appendix B, and expressions for the model-independent derivatives are given in Appendix A.
The application of the theory to the S Z - C H ~system a t 122.04’ K. and pressures of 6.81, 10.21, and 13.61 atm. is considered first. The experimental values for this system are presented in Table I. The characteristic molecular parameters for the pure components were obtained from Eckert et al. (1967), who Table I. Experimental Results for Ns-CH4 System
Data from Prausnitz et al. (1967b) XN~
0.144 0.272 0.432 VOL.
P,Atm.
Y N ~
YCH~
6.81 10.21 13.61
1.719 1.453 1.242
1.019 1.065 1.159
8
G~/RT
0.094 0.147 0.177
NO. 4 NOVEMBER 1 9 6 9
721
Table II.
Predictions for N2-CH4 System T = 122.04" K . GE/R
--P,Atm.
h'CP
6.81 10.21 13.61 10.21 13.61 6.81 13.61 6.81 10.21 13.61 6.81 10.21 13.61
ki j
RM
O.Oo0
0.074 0.066 0.010 0.182 0.193
0.024 0.000
T YN21
YCHh
vdW-1
APM
vdW-2
vdW-2
vdW-2
0.035 0.004 -0.065 0.120 0.118
0.061 0.044 -0,019 0.159 0.162
0.042 0.013 -0.057 0.128 0.125 0.061 0.090 0.122 0.183 0.145 0.134 0.182 0.262
1.637 1.119 0.823 1.689 1.219 1.781 1.135 2.509 1.694 1.219 2.723 1.872 1.341
0.967 0.976 1.049 0.980 1.072 0.975 1.064 0.988 1.055 1.111 0.988 1.016 1.268
0.016 0.024
Table 111.
Predictions for CZHB-CZH~ System
RM
VdW-1
APM
vdW-2
GE/RT Pred. NCP = 3 vd W-2
0.052 0.046 0.061 0.062 0.045 0.067
-0.001 0.000 0.000 0.OOO 0.000 -0.003
0.030 0.027 0.036 0.036 0.026 0.038
0.004 0.004 0.005 0.005 0.003 0.003
0.019 0.017 0.022 0.021 0.016 0.022
G E / R T (Pred.) N C P = 2 GEIRT,
T,
O
C.
4.62 4.62 4.62 4.62 4.62 -17.60 a
P , Atm.
31.6 31.2 35.0 38.4 41.8 20.8
X c 2 ~ ( van Laar
0.700 0.750 0.581 0.401 0.216 0.500
0.023 0.020 0.027 0.027 0,020 0.014"
Experimental value from Hanson et al. (19.53).
also found the deviation from the geometric mean, ~ i j to , be 0.024 from best fit to the excess functions. The true value of ~ i is j not known. The pure component normalized coordination numbers are taken to be one. The predicted values are given in Table 11. Both the two-parameter (NCP = 2 ) and the three-parameter (KCP = 3) principle of corresponding states are used. The predictions of GE/RT are very poor for all four models when NCP = 2 and ~~j = 0.000, On the other hand, the predictions are about equally good for all four models when KCP = 2 and ~~j = 0.024. The individual activity coefficients, however, are in much better agreement with experiment when the vdW-2 model is used than when the vdW-1 model is used. For the vdW-1 model a t 6.81 atm. and K i j 3 0.024, YN* = 2.112 and Y C H ~= 0.891. When KCP = 3 and ~ i =j 0.000, the predictions of GE/RT are still low, but much better than when NCP = 2 and K~~ = 0.000. A small, positive value of K L j , ~~j = 0.016, brings the predictions into good agreement with the experimental values. Some deviation from the experimental values may be expected here, since the N2-CH4 system is rather dilated a t 122.04' K. Mixtures of slightly more complicated molecules, the CzH&2H4 and CzH4-CzH2 systems, a t pressures up to 42 atm., are considered next. The experimental values for these systems mere calculated from the van Laar equations (Chueh and Prausnitz, 1968) with c0efficient.sobtained from Prausnitz (1968). For lack of better information, ~~j is taken to be 0.000. The molecular parameters of the pure components, ei,/k, uir, and ci, are given by Eckert et al. (1967), and the normalized coordination numbers, sii, are taken from Hermsen and Prausnitz (1966a). The experimental and predicted values of GE/RT for the C2H&2H4 system are shown in Table 111. 722
-___
l&EC FUNDAMENTALS
When NCP = 2, the values of GE/RT predicted from the RM and APM models are apparently in much better agreement with the experimental values than those predicted from the vdW-1 and vdW-2 models. This is probably fortuitous, since the two-parameter principle cannot be expected to hold too well for CzH8 and CzH4. Furthermore, the agreement between experiment and theory is by far the best, when the vdW-2 model is used in conjunction with the three-parameter principle of corresponding states. The coordination number is not available for CzHz, and therefore only the two-parameter principle of corresponding states could be used for the C2H4-C2H2 system. The predicted values of GE/RT are only about half the values calculated from the van Laar equations, RS may be seen from Table IV. The three-parameter principle of corresponding states used here is dependent on the existence of a quasi-lattice due to the particular form of the partition function employed. This method should therefore not be applied to systems containing supercritical components, since the supercritical molecules have much more mobility than the quasi-lattice allows. Predictions for the 02-COz system a t - 5 O O C . were nevertheless attempted. The molecular parameters foi 02 and COz were obtained from Eckert et al. (1967), and the values of sii were assumed equal t o 1. The experimental values for GE/RT were obtained by Fredenslund (1968). ~~
~
Table IV.
Predictions for CzH4-CzHz System at 4 . 6 2 O C. G E : R T (Pred.) N C P
P,
=
2
GE,RT
Atm.
zca,
van Laar
RM
vdW-1
APM
vdW-2
38.4 41.7
0.287 0.475
0.083 0.091
0.062 0.046
0.060 0.044
0.052 0.039
0.051 0.038
Table V.
Predictions for 02-CO2 System at -50'
C. GE,RT
GE
p, Atm.
10 20 30 40 50 60
70 80 90 100 110 120 130
RT
Exptl.
0.006 0.016 0.0265 0.052 0.0485 0.102 0.071 0.166 0.0955 0.207 0.122 0.245 0.1515 0.284 0.1835 0.320 0.2183 0.354 0.255 0.390 0.294 0.433 0.338 0.475 0.393 0.512
Table VI.
GE R T (Pred.) X C P = 2
RRI
vdW-1
0.029 0.117 0.202 0.284 0.371 0.461 0.558
0.021 0.080 0.136 0.189 0.245 0.304 0.368 0.656 0.435 0.755 0.505 0.850 0.572 0.937 0.637 1.018 0.700 1.093 0.762
Pred.
ucp
= 3
APM
vdW-2
vdW-2
0.017 0.105 0.192 0.250 0.309 0.418 0.493 0.579 0.659 0.733 0.802 0.867 0.928
0.012 0.087 0.120 0.203 0.248 0.344 0.409 0.467 0.540 0.601 0.658 0.713 0.766
0.012 0.057 0.109 0.162 0.218 0.276 0.335 0.395 0.452 0.503 0.549 0.590 0.627
Experimental and Predicted Activity Coefflcientr for 02-CO2 System at -50' C. 7%(Exptl.1
P , Atm.
TO?
0 2
10 20 30 40
0.006
4.165 3.761 3.502 3.311 3.124 2.937 2.737 2,554 2.380 2,223 2,074 1.917 1.738
50 60 70
80 90 100 110 120 130
0.0265 0.0485 0.071 0.0955 0.122 0.1515 0.1835 0.2183 0.255 0.294 0.338 0.393
coz 1.008 1.018 1.044 1,091 1.115 1.138 1.168 1.198 1,234 1.284 1.362 1.469 1.624
yZ (Pred.1 0 2
3.107
2.851 2.557 2.269 1.999 1.752 1.521 1.290
coz 1.005 1.024 1.048 1.079 1.118 1.167 1.231 1.312 1.418 1.551 1.723 1.969 2.383
The activity coefficients for 0 2 given by Fredenslund (1968) are normalized with respect to the infinitely dilute solution, the so-called unsymmetric convention (Prausnitz et al., 1967a). They were converted to the symmetric convention (the normalization with respect to the pure liquid a t the temperature of the system) which is a hypothetical state for 02 a t -50' C. The extrapolated, hypothetical standard state fugacity was found t o be 120 =t 10 atm. The value of K ? used ~ for the OZ-CO~system is 0.03, which was obtained from data on the second cross-virial coefficient. The experimental and predicted values for GE/RT are shown in Table V. The calculated values of GE/RT are in good agreement with the experimental values when S C P = 2 and the vdW-l and vdW-2 models are used, the agreement being somewhat less satisfactory for the R n l and APM models. The best agreement between theory and experiment is obtained, in spite of the theoretical limitations, when the three-parameter principle of corresponding states is used. This is also the case for the activity coefficients. When XCP = 2, the calculated activity coefficient for 0 2 a t 30 atm., for example, is 14.00 for the vdW-1 model and 9.10 for the vdW-2 model, compared with an experimental value of 3.502. As may be seen from Table VI, the predicted value for XCP = 3 is 3.685. The predicted and experimental activity coefficients are in reasonably good agreement up to about 80 atm.
method to predict solution properties at high pressures quantitatively from the pure-component characteristic molecular parameters. No definite conclusion may be reached as to the relative validity of the van der Waals models and the random mixing and average potential models on the basis of the work shown here. However, extensive calculations by Fredenslund (1968) and Leland et al. (1968, 1969) on systems of simple molecules of different sizes a t low pressures clearly indicate that predictions based on mixture paramete1 s calculated with the van der Waals approximation are in better agreement with experiment than predictions using the random mixing and average potential models. When the theory is applied to slightly complex systems, such as the CzHfi-CzH4 system, it is necessary to use a threeparameter principle of corresponding states rather than the two-parameter principle. The method may be applied to liquids containing supercritical solutes-for instance, the 02-COz system. The increased mobility of the solute should be taken into account, however. Appendix A
The detailed expressions for the quantities GO,I , are, '/aTo,, dro,'/dpo,, and aPo,/dpo, entering into the calculation of the reference properties are shown here. Throughout this appendix, subscripts 0 and i and superscript I , designating a configurational property, are omitted. From Equation 25,
where G* is the perfect gas configurational Gibbs free energy a t the temperature and pressure of the system. Using the equation of state, Equation 23, it is easy t o show that
P
RT
P2
P
---=
Cl
+ CPP + + + + C3P2
c 4 p 3
c5p4
Conclusions
A molecular theory of solutions of simple molecules, based on the statistical mechanical formulation of the principle of corresponding states together with an equation of state for methane, has been developed. It is possible to use this VOL.
8
NO.
4 NOVEMBER 1 9 6 9
723
Thus
+ 9 b ( T ) }exp ( - E z o ) { L ( T )+ zo2La(T)}]
& ( P , T ) = SCexp ( - E x ) { L ( T )4-z L 2 ( T )
The configurational Gibbs free energy may now be calculated:
G - G* = P / P - R T
+ 11( P , T )+ 12 ( P , 2’) +
13(p,
T)
The derivative dP/dp is determined directly from the equation of state:
aP/dp = 2C1p
+ 4C3p34- 5C4p4+ 6C5p6+ RT -
3c2p2
+
-
~ E z ( P P O ) ~ ( T To)p2(z- a’)(P’ exp { G - LEI
+ E2zl(T - TO)]+
+ + 3ppo) - a’)(p’ - -k + P O ) ~ ( ~+( P PO)^^' - 6 b + + 3a’(p + PO)^)^ X exp { G - [E1 + E2zI(T - TO)} + [2P (2P2
p2(p
- z) X
P?
Z)
(1:
PO)‘
[ @Dip
3&P2 f 4D3p3
5D4p4 4-6Dsp6 -
+ + + D4p3+ Dap4)1X
(F2/T)p2(D1 DBP D3p2
exp {k -
( R4-F2)/Tl
I n the determination of the other two derivatives, let
n=-G - G* RT
Dividing the previous expression for (G follows that
- G*)
by R T , i t
Differentiating this equation with respect to p and T , the results after slight rearrangements are:
Constants and Functions Used in Appendix A Ci
= A i + Bi = exp { k F1/T}
Q
= =
r
T/F2 DiQ = D2Q = DaQ =DA
H I (T) Hz ( T ) H3 ( T ) Hd ( T ) Ha(T) = D5Q
724
l&EC
FUNDAMENTALS
-
++ D2Q2 + 2DaQ3 6 D A 4+ 24D6Q6 2D3Q24-6D4Q3 24D6Q4 + 4DbQ2 3D4Q2 12DbQ3
The detailed expression for this derivative is given by Fredenslund (1968).
+
+ +
+ ++
2DzT/Fz 6D3T2/Fz3 24DaT3/Fz4-k 120Dr.T 4 iF 2 Hz' ( T ) = Dz/Fz 4D3F/Fz2 18D4T2/Fz3 96D6T3/Fz4 6D4T/Fz2 36DiiT2/Fz3 H3'(T) D3/Fz 8DsT/F*' H n ' ( T ) = Da/Fz H s ' ( T ) = DdFz L1' ( T ) = - 6/EzTtZ 2y/EZ2TI3 - 6/EzaTt4 L; ( T ) = y/EzTI2 - 4/Ez2T'3 L3/(T) = - 1/E2Tt2
H?(T) = Di/Fz
4 +
+
+
X CY'
0' T'
?E Y
6
+ + -
= (P pol3 = ( a + pol3 = (P POY = T To = po3
= 5 exp (G
- E1T')
= EzT' = a'
+ P'
=
It is easy to show that the model-dependent derivatives appearing in Equations 37 and 38 are related to the derivatives calculated here by
aro, -
- aC(Go,
- Gt*)/RTo,I
dT0,
aT0,
and for the two-parameter principle of corresponding states
-- are,
GO, - Gt*)/RTo,l+
aPo,
aPo,
f. PO'
whereas for the three-parameter principle of corresponding states
are, - - al:(Go, - G,*)/RTo,l I aPo,
Appendix
aPZ
fast CZPOi
B
Nomenclature
A B C
freedom constant in equation of state for methane constant in equation of state for methane lattice energy f characteristic molecular energy parameter F1, FZ = constants in equation of state for methane F = universal function = characteristic molecular length parameter Q G = Gibbs free energy G = constant in equation of state for methane h = characteristic molecular volume parameter (= g3) k = constant in equation of state k = Planck's constant m = exponent in the Lennard-Jones potential iM = molecular weight n = exponent in the Lennard-Jones potential n = number of moles N = Avogadro's number P = probability P = pressure Q = partition function r = intermolecular distance R = gas constant S = normalized coordination number T = temperature U = intermolecular potential U = potential energy of a system 21 = liquid molar volume 2 = mole fraction in liquid
D E E
p Y
I' E
VdW-1
= = = =
GREEKLETTERS
a
The derivatives afi/ank and ahi/ank, and in the case of the three-parameter principle of corresponding states, &,/ank, are shown for the vdW-1 and vdW-2 models.
= constant in equation of state for methane = constant in equation of state for methane = parameter for number of external degrees of
K
P U
U
\k
= = = =
constant in equation of state for methane constant in equation of state for methane
(KT)-1
activity coefficient = reduced Gibbs free energy = characteristic molecular energy = deviation from geometric mean = density = cell collision diameter = characteristic molecular length = partition function per degree of freedom
SUBSCRIPTS
vdW-2
00 .. aa
i
lj ank
nT
From Equations 20, 21, and 22 it may be seen that for a binary mixture Si = pj) Thus as,
as,
api
as. a p .
ank
api ank
a p j ank
-=-.-+-'.A
ij
= pure component j = component j in mixture = unlike interaction
X
= mixture property for one-fluid model
0 k
= property of reference fluid = component k
T
= total
3
ahi- hrk - hi --
SUPERSCRIPTS I
(k = i o r j )
= pure reference component = pure component i = component i in mixture
E
*
= configurational property = excess property
= perfect gas quantity VOL.
8
NO.
4
NOVEMBER
1969
725
Literature Cited
Barker, J. A., “Lattice Theories of the Liquid State,” Pergamon, Oxford, 1963. Brown, W. B., Phil. Trans. Roy. SOC.A260, 221 (1957). Chueh. P. L.. Prausnitz. J. M.. Ind. Ena. Chem. 60. No. 3. 34 (1968). ’ Eckert, C. A., Ph.D. thesis, University of California, Berkeley, 1964.
Eckeit, C. A., Renon, H., Prausnitz, J. M., IXD.ENG.CHEM. FUNDAMENTALS 6, 58 (1967). Fredenslund, A., Ph.D. thesis, University of Wisconsin, Madison, 1462 *”..”. Guggenheim, E. A,, “Mixtures,” Oxford University Press, London, 1952. Hanson, G. H., Hogan, R. J., Ruehlen, F. N., Cines, 1cI. R., Chem. Eng. Progr. Symp. Ser. 49, No. 6, 37 (1953). Hermsen, R. W., Prausnitz, J. M., Chem. Eng. Sci. 21, 791 (1966a). Hermsen, R. W., Prausnitz, J. &I., Chem. Enq. Sci. 21, 803 (1966b). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp. 110, 134, Wiley, New York -19.54 --~ I
Leiand, T. W., Chappelear, P. S., Gamson, B. W., A.I.Ch.E.J. 8, 482 (1962). Leland, T. W., Rowlinson, J. S., Sather, G. A,, Trans. Faraday SOC. 64, 1447 (1968). Leland, T. W., Rowlinson, J. S.,Sather, G. A., Watson, I. D., Trans. Faraday SOC.,66, 2034 (1969).
Prausnitz, J. M., University of California, Berkeley, Calif., private communication, 1968. Prausnitz, J. U., Eckert, C. .A,, Orye, R. V., O’Connell, J. P.. “ Computer Calculations for Multicomponent Vapor-Liquid Equilibria,” pp. 7-11, Prentice-Hall, Englewood Cliffs, K.J., 1967a. Prtusnitz, J. M., Eckert, C. A., Orye, R. T., O’Connell, J. P., Computer Calculations for Multicomponent Trapor-Liquid Equilibria,” p. 180, Prentice-Hall, Englewood Cliffs, Y. J., 1967b. Prigogine, l., “Molecular Theory of Solutions,” pp. 166-232, North-Holland, Amsterdam, 1957a. Prigogine, I., “Molecular Theory of Solutions,” pp. 323-47, North-Hollan?, Amsterdam, 1957b. Prigogine, I., Molecular Theory of Solutions,” Chap. 11, North-Holland, Amsterdam, 1957~. Rowlinson, J. S.,“Liquids and Liquid Mixtures,” pp. 271-323, Butte1 worths London. 1959 .....~ ~~.~~~ ,~ ~ . ~ ~~ - ...~ -. . , Scott, R. L., J . Chem. Phys. 26, 193 (1956). Sprow, F. B., Prausnitz, J. M.,A.I.Ch.E.J. 12, 780 (1966). Fennix, A. J., Kobayashi, R., “Arf,Equationof State for Methane in the Gas and Liquid Phases, 60th annual A.1.Ch.E. Meeting, New York, Nov. 26-30, 1967. RECEIVED for review September 23, 1968 ACCEPTED June 18, 1969 Work supported by fellowships and grants from the Amoco Chemicals Corp., the National Science Foundation through the University of Wisconsin Research Committee, and the University of Wisconsin Engineering Experiment Station. ~
PREDICTION OF G A S SOLUBILITY I N MOLTEN SALTS A. K . K. L E E ’ A N D E.
F. J O H N S O N
Plasma Physics Laboratory and Department of Chemical Engineering, Princeton University, Princeton, N . J .
A method, proposed for predicting the solubility of gases in molten salts, is based on an application of the scaled-particle theory and a theory of corresponding states to molten salts. Calculated values from the proposed solubility equation are in good agreement with observed values.
HE scaled-particle theory is an equilibrium theory of rigid Tsphere fluids, which derives its name from a scale-up process applied to the size of a molecule in the fluid until the molecule reaches the scale of its neighbors. By extending the ideas of the scaled-particle theory Reiss et al. (1959, 1960) have shown that the reversible work required to introduce a hard-sphere solute molecule into a solvent can be used to calculate solubilities. The solubilities of some nonpolar gases in nonpolar solvents have been computed by Pierotti (1963) in good agreement with experiment. I n view of the growing importance and versatility of molten salts as solvents (Bloom and Hastie, 1965) and heat transfer media and in view of the lack of solubility data in general, it is desirable to develop methods for predicting solubilities in salts. The prediction of gas solubility in molten salts is of special interest in the fields of nuclear reactor technology and electrochemistry. A simple equation used by Blander et al. (1959) and Bratland et al. (1966) to calculate the gas solubility in molten salts is of the form
In K , = -NA-y/RT
(11
where N is the Avogadro number, A the surface area of the gas molecule, y the macroscopic surface tension of the solvent, 1
726
Present address, Laurentian University, Sudbury, Canada. I&EC
FUNDAMENTALS
R the gas constant, and T the temperature. K , is a form of the Henry’s law constant,
Kc = cd/ c g
(2 )
where C, is the concentration of gas in the gas phase in equilibrium with the liquid a t a concentration Cd. Generally, this approximate equation does not give good agreement with experiment. The method proposed below for predicting gas solubility in molten salts is based on the scaled-particle theory (Reiss et al., 1959) and the theory of corresponding states for molten salts (Reiss et al., 1961). Values computed by this method are compared with experimental values. Unfortunately, experimental data on gas solubility in molten salts are very limited (Blander et al., 1959; Bratland et al., 1966, 1967; Burkhard and Corbett, 1957; Camp and Johnson, 1962; Copeland and Seibles, 1966, 1968; Copeland and Zybko, 1965, 1966; Grimes et al., 1958; Ryabukhin, 1962; Schenke et al., 1966; Shaffer et al., 1959; Watson et al., 1962; Woelk, 1960) and only those for single-component salts can be used for comparison. Proposed Method
Since most solutions of gases in molten salts are very dilute, the theory of dilute solutions is applicable. For a
