Ind. Eng. Chem. Res. 1987,26, 1344-1351
1344
ious values of -rCHCIF2 ranging from 0 to 2.0 (g-mol of CHClF,/s)/(L.reactor vol). The results were plotted as shown in Figures 8 and 9. The conversions increase with increasing temperatures (endothermic reaction), the curves being steeper at lower temperatures and flatter at higher temperatures. 4. Conclusion
A combination of elemental mass balances and equilibrium constants has been employed to calculate the thermodynamic equilibrium composition of the reactor effluents for the pyrolysis of monochlorodifluoromethane. The results have been presented in the form of plots for varying diluent ratios in the range 0-8.0. A kinetic study has also been presented for developing a rate equation for the formation of tetrafluoroethylene based on available expressions for the rate constants. To get a good consistency between the thermodynamic and kinetic analyses, it was found necessary to correct one of the reported expressions for the rate constants. Registry No. TFE, 116-14-3;CHClF,, 75-45-6; C3Fs,166-15-4; HC1, 7647-01-0.
Literature Cited
I
li i
j, 700
p
J,
/
1 4 -
800
750
TEMPERATURE
I
(‘C)
’ 85P
W
Figure 9. Temperature-conversion plot for different reaction rates and for diluent (steam) ratio = 7.5.
Equation 41 gives the equilibrium conversion of CHCIFz for given values of diluent ratio and temperature. Equation 40 was solved on a computer for diluent ratios = 1.0 and 7.5, temperature range = 650-850 “C, and var-
Edwards, J. W.; Small, P. A. Ind. Eng. Chem. Fund. 1965, 4 , 396-400. Gozzo, F.; Patrick, C. R. Nature (London) 1964, 202,80. Park, J. D., et al. Ind. Eng. Chem. 1947, 39, 354. Renfrew, M. H.; Lewis, E. E. Ind. Eng. Chem. 1946, 38, 870. Shingu, H.; Hisazumi, M. British Patent 1041738 (cl C O~C),Sept 7, 1966, p 4; Appl. April 9, 1963. Venkateswarlu, Y.; Murti, P. S. Chem. Process. Eng. 1970, Oct, 25.
Received for review January 27, 1986 Revised manuscript received March 2, 1987 Accepted April 10, 1987
Thermodynamics of a Single Electrolyte in a Mixture of Two Solvents Ani1 Rastogi* and Dimitrios Tassios New Jersey Institute of Technology, Newark, New Jersey 07102
It is demonstrated that t h e Debye-Huckel (D-H) term when applied t o single electrolyte-binary solvent mixtures predicts salting in when salting out is observed. An empirical extension of the D-H term is proposed t h a t eliminates this problem. When combined with an NRTL term to account for the long-range forces, the resulting expression provides successful correlation of mean ionic activity coefficients and vapor-phase compositions of several ternary systems. Prediction results for y* are in the right direction, but vapor-phase compositions do not reflect the electrolyte effect. Vapor-liquid equilibrium in electrolytic solutions is of theoretical and industrial importance in various chemical, biological, pollution control, and electrochemical processes. Early in the century Debye and Huckel(l923) proposed the classical excess Gibbs free energy expression for strong electrolytes in a single solvent, but applicable only to dilute solutions. Guggenheim and Turgeon (1935) extended the range of validity of the Debye-Huckel equation to 0.1 m for aqueous solutions. Recently many workers have proposed semiempirical correlations for concentrated aqueous *Present address: Real-Time Simulation, Inc., 27 W. 47th St., New York. NY 10036.
electrolytic solutions (Bromley, 1972, 1973; Meissner and Kusik, 1972; Pitzer, 1973, 1977; Pitzer and Mayorga, 1973, 1974; Pitzer and Kim, 1974; Cruz and Renon, 1978; Meissner et al., 1972). Correlation and prediction in multicomponent systems-the typical industrial case-has been studied by several authors. For weak volatile electrolytic systems, such as NH3-CO2-H,S-H20, considerable work has been carried out by Prausnitz and co-workers (Prausnitz et al., 1975, 1978). For strong electrolytic solutions, correlation and prediction in ternary systems-one electrolyte in two solvents-has been discussed among others by Hala (1969),
0888-5885/87/2626-1344$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7,1987 1345 Table I. Ternary Data Sources no. system 1 HCl-Hz0-EtOH 2 HCl-H,O-MeOH 3 HCl-H,O-MeOH 4 LiCl-H20-EtOH 5 LiCl-H,O-MeOH 6 LiCl-H,O-MeOH 7 LiCl-H,O-MeOH 8 NaBr-H,O-MeOH 9 NaBr-H20-MeOH 10 NaC1-H20-MeOH 11 KC1-H20-MeOH 12 LiCl-H20-MeOH 13 NaBr-H20-MeOH 14 NaF-H,O-MeOH
T or P 25 "C 25 "C 25 "C 25 OC 25 "C 25 "C 60 "C 25 "C 40 OC 25 "C 1 atm 1 atm 1atm 1 atm
Table 11. Binary Data Sources no. system T or P 1 HCl-HZO 25 OC 2 LiCI-H,O 25 "C LiC1-H20 3 60 OC 4 25 "C NaBr-H20 5 40 OC NaBr-H20 6 NaCl-H20 25, 60, 70 "C 80,90,100 "C 7 CaC12-MeOH 25 "C HCl-MeOH 8 25 "C HCl-EtOH 25 "C 9 10 LiBr-MeOH 15, 25 OC LiC1-MeOH 11 25 "C 12 60 "C LiC1-MeOH 13 25 OC NaBr-MeOH 14 25 "C NaC1-MeOH 15 25 "C H20-MeOH 40 "C 16 H,O-MeOH 17 HZO-EtOH 25 "C 18 H,O-MeOH 1 atm
m range 0.006-2.5 0.02-0.5 0.001-2.0 0.5-4.0 0.02-1.0 1.0 0.58-14.1 1.0-7.1 1.0-6.2 0.02-0.5 0.012-2.0 0.085-3.8 0.076-3.8 0.012-0.95
m range
type of data m-X-y, m-X-y, m-X-y, m-X-y-P-T m-x-f, m-X-y-P-T m-X-y-P-T m-X-y-P-T m-X-y-P-T m-X-y, m-X-y-P-T m-X-y-P-T m-X-y-P-T m-X-y-P-T
0.1-2.0 0.1-6.0 0.88-9.0 0.1-4.0 4.0 0.1-4.0
type of data m vs. y h m vs. y, and P m vs. P m vs. y, and P m vs. P m vs. y+ and P
0.3-2.6 0.002-0.56 0.005-0.1 0.3-6.6 0.33-3.67 0.33-7.4 0.56-1.56 0.0001-0.1 0.0 0.0 0.0 0.0
m vs. P m vs. y+ m vs. y+ m vs. P m vs. P m vs. P m vs. P m vs. y* X-y-P-T X-y-F-T X-y-P-T X-y-P-T
Beckerman and Tassios (1976), and Chen et al. (1980). Hala uses a combination of Debye-Huckel (D-H) and Margules expressions. Beckerman and Tassios dealt with solvent activity coefficients by using only the NRTL model. Chen et al. used a Debye-Huckel and a local composition model combination. Reasonable correlations of vapor-phase compositions are obtained, but results for predictions are limited and uncertain. In all these works, only correlation of VLE data is achieved; no attempt is made to describe mean ionic activity coefficient (y*). In this study we present an expression that combines an extended D-H equation to account for Coulombic forces with the NRTL expression to represent long-range forces that can be used for both VLE and mean ionic activity coefficients of mixtures of a single electrolyte in mixed solvents. The expression is applied to a large databank containing ternary systems (Table I) and the corresponding binary systems (Table 11). Extension to multicomponent solvents is straightforward. The correlation and prediction of both vapor-phase compositions and Y* values is examined.
Activity Coefficients in Electrolytic Solutions The main thermodynamic quantities involved in electrolytic solutions are the solvent activity and osmotic coefficients and the electrolyte mean ionic activity coefficients. These are developed starting with the equality of component fugacities between two phases in equilibrium:
no. of data pts
ref Harned and Owen, 1958 Akerlof, 1930 Harned and Owen, 1958 Ciparis, 1966 Akerlof, 1930 Ciparis, 1966 Hala, 1969 Ciparis, 1966 Ciparis, 1966 Akerlof, 1930 Rousseau et al., 1976 Rousseau et al., 1976 Rousseau et al., 1976 Rousseau et al., 1976
44 24 24 31 45 5 25 16 10 35 33 24 23 24
no. of data pts
ref Stokes, 1955 Stokes, 1955
15 27 17 19 4 11
Robinson and Robinson and Hala, 1969 Robinson and Ciparis, 1966 Robinson and
7 22 8 8 6 12 9 7 8 6 10 34
Bixon et al., 1979 Harned and Owen, 1958 Taniguichi and Janz, 1957 Skabichevskii, 1969 Skabichevskii, 1969 Hala, 1969 Bixon et al., 1979 Covington and Dickinson, 1973 Ciparis, 1966 Ciparis, 1966 Ciparis, 1966 Rebolleda and Ocon, 1958
Stokes, 1955 Stokes, 1955
Assuming ideal vapor behavior, the fugacity of solvent i can be represented by XiYlPp = yip
(2)
In the case where solvent i is mixed with a nonvolatile electrolyte, y i = 1 and (3)
and the mean ionic activity coefficient (r*)of an electrolyte is given by In -y* = 4 - 1 +
m'
(4 - 1) d In m
(4)
where $I is the osmotic coefficient 4 = --1000 In ai mMw
Proposed Expression In a mixture of one electrolyte and two solvents, the excess Gibbs free energy function is derived by considering the various interactions in the liquid solution, namely, ion-ion, ion-solvent, and solvent-solvent molecules. As a first approximation, we consider a combination of the Debye-Huckel and NRTL equations to represent these interactions:
where
1346 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
+ pZ112)2 - 2(1 + pl’”) +
11
4- NT[(O.OO1umMw+ 1) X ln (1 4- P Z ’ / ~ )+ 2 In (O.OO1umMw+ 1) - O.OO1umMw] (7)
and p = 1.0; A, = Debye-Huckel constant, defined in Appendix B; M , = molecular weight of the solvent mixture (electrolyte free); and NT = total number of solvent molecules. For the NRTL equation, the ternary expression is obtained by extending the binary expression of Cruz and Renon (1978)
where N s is the number of components in the system, with component 1 being the electrolyte YE z*i = ZAi + ZBL
where A, = Debye-Huckel constant and a = 1.5/12+2-(as suggested by Bromley (1972). In arriving at eq 13, different forms of the physical term similar to the extended form of Gronwall et al. (1928,1931) were considered. Extensive binary and ternary data correlation work suggested that eq 13 gives the best results. It was also observed that the optimum value of constant a for an individual binary mixture did not have a significant effect on binary or ternary data correlation, and consequently, a fixed value of the constant a, as suggested by Bromley (1973),is used in this work. The details of the development of eq 13 are given in Appendix A. The inclusion of this “physical” term provides for the experimentally observed salting out effect as illustrated in Figure 1. The combination of the D-H and the physical terms, eq 7 and 13, will be referred to hereafter as the extended D-H term. The activity coefficients expressions can be obtained by differentiating the excess Gibbs free energy function:
(9)
UA
UB
G+i = GAi + -GBi uA
(10)
A and B refer to the cation and anion, respectively, i 1 2 refers to the solvents, z k i = AgkiGki
aik
=
aki
&ik
# &ki
and the subscript k refers to the two ions. The procedure for the development of these expressions is given in ApDendix A. When this combination of the Debye-Huckel and the NRTL equations is applied in the correlation of ternary -yt and VLE data, the results are poor because the D-H term predicts salting in instead of the observed salting out. This is demonstrated for the system LiC1-water-ethanol a t 25 O C in Figure 1 which presents the conbribution of the D-H term to the ratio, yE/yw,as a function of X E ~ O H at m = 1.0. Since ethanol is salted out, this contribution should be greater than one because the NRTL term alone cannot account for this effect. Sander et al. (1984) encountered the same problem in correlating mixed solvent data with an electrolyte. To avoid salting in effect due to the D-H term, they suggested that the Debye-Huckel constant, A,, should be regarded as a composition-independent parameter. For their work, Sander used A, = 2. Others (Rousseau and Boone, 1978; Rousseau et al., 1972, 1976; Mock et al., 1986) have completely dropped the D-H term in the correlation of VLE data. However, it should be noted that it is important to include the D-H term in order to study ionic equilibrium and complex formation or to determine salt solubility in mixed solvent mixtures. Any expression without a proper D-H term has only limited applications. To alleviate the problem of salting in due to the D-H term in this work, an empirical term, referred to as the “physical” term, is added:
In yi = In y+*- In (O.OO1umMw+ 1)
(16)
where ?a = mean molal activity coefficient and y+* = rational activity coefficient. Substitution of eq 7-13 in eq 14 and 15 results in the following expressions for a ternary mixture:
In Y+* = In
Y+*ext.D-H
+ In
yt*NRTL
(17)
For solvent i ,
In (O.OO1umMwql)- O.OO1umMw (21)
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1347 where
to
u1(pP/')
1
=
0
0
0
t4
0
0
9p
0
z
t2
2
0
$,(an =
-
(1
4 + u1)'/' + UI UI -
(1
2
+ u1)'/2
000 0 0
AD A
1
A
A
- 2(1 + C Z ~ ) ~ / ' (23)
b
A '
A
A
I 1 0
0 5
- 2 0
+ p 1 q 2 - 2(1 + PI'/') + In (1 + PPI') +
i]
(24) 1.0
^4
0.6
DP
(mmne ) a
2xAxixj{
Z..G .G.. ZjiG*i 11 *I Y ( x j G j i+ Xi)3 + (XiGij+
]
(26) 0
a
4 m
Figure 2. Correlation and prediction of mean molal activity coefficient and vapor pressure depression data for the system NaCl-H20 corat 25 OC: (-) experimental (Robinson and Stokes, 1955); (0) relation; ( 0 )prediction using binary parameters of 60 "C.
Binary Systems A
60
A list of binary systems used in this study is presented in Table 11, and detailed correlation results, using eq 17-27, are given by Rastogi (1981). The binary y+ and/or vapor pressure depression [AP= (P- PO)] data are correlated for the two parameters, Gii and Z+i, or for the temperature-independent parameters, Agh and &si. For simplicity, we set f f A 2 = f f A 3 = 0.2 and C Y B = ~ ffB3 = 0.0. The obtained results are reasonably good. Absolute errors (in y+ for aqueous and AP for nonaqueous).for m up to 6 are below 5% except for the LiCl where a larger error (up to 11.5) was found. Typical results are presented in Figures 2 and 3 for the system NaC1-H20 at 25 "C and LiBr-MeOH at 15 OC. The binary parameters (&h and &E) show reasonable temperature independency within a 30-40 OC range for aqueous electrolyte mixtures and within a 15-20 "C range for nonaqueous electrolyte mixtures as shown in Figures 2 and 3. Aqueous-electrolyte binary data reduction also indicates a multiplicity of roots for.binary parameters (Agh and AgBi). No multiplicity of roots is observed for the nonaqueous electrolyte binaries.
O
a
6
m
Figure 3. Correlation and prediction of vapor pressure depression data for the system LiBr-MeOH at 15 OC: (-1 experimental (Skabichevskii, 1969); (0)correlation; (A)prediction using binary parameters of 30 OC.
1348 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Table 111. Correlation of Isothermal Ternary ya Data av abs. max m error in system available used T, "C Ya 2.5 2.5 25 2.0 HCl-HZO-EtOH 25 2.2 HCl-H20-MeOH 2.0 2.0 1.0 1.0 25 7.7 LiCl-H,O-MeOH 0.5 0.5 25 6.1 NaCl-H20-MeOH 4.5 overall Table IV. Correlation of Ternary VLE Data max m av abs. availerror in svstem able used T o r P Y X 1000 1.0 25 "C 4 4.0 LiCl-H20-EtOH 25 "C 1 1.0 1.0 LiC1-H20-MeOH 6.2 40 OC 7 6.2 NaBr-H,O-MeOH 25 OC 5 7.1 7.1 NaBr-H,O-MeOH 60 "C 3 14.1 6.0 LiC1-H20-MeOH 1 atm 4 3.8 3.8 LiC1-H,O-MeOH 1 atm 20 3.8 3.8 NaBr-HzO-MeOH 1 atm 10 2.0 2.0 KCl-H,O-MeOH 0.95 0.95 1 atm 5 NaF-H20-MeOH 7 overall
-NP
[
Tic-
TIE
TIE
1:
(28)
where the subscripts E and C refer to experimental and calculated, respectively. In eq 28, a weighting factor of 10 is used for the deviation in the vapor-phase composition, in order to make the magnitude of this term equal to that of the relative fractional deviation in P and Ti. The activity coefficient expressions 18-26 have six adjustable parameters for a ternary mixture. To obtain meaningful parameter values, the data are regressed only for four parameters with preset values of &23 and A g 3 2 obtained from the solvent-solvent binary data correlation ( a 2 3 = -1.0 (Marina and Tassios, 1973)). If the experimental data are isothermal, the parameters evaluated are Gi2, Z12, Gi3, and Z13. For isobaric data, the temperature-independent parameters &A2, &A39 and &3 are evaluated. A nonlinear subroutine LSQ2 was used in biTable V. Prediction of system HC1-H20-MeOH LiCl-H20-MeOH HCl-HZO-EtOH
A A
%
salting out 83 96 79 89 91 84 50 71 62 78 overall
Ternary Systems In correlating ternary data, the best results were obtained with the objective function
s=1
A
0.6
0
2
1
m
Figure 4. Correlation and prediction of mean molal activity coefficient data for the system HCl-H20-EtOH at 25 "C and constant X',,, = 0.0417: (-) experimental (Harned and Owen, 1958); (0) correlation; (A)prediction using binary parameters.
nary or ternary data regression using the systems presented in Tables I and 11. yI vs. m data for four isothermal ternary mixtures are correlated successfully with four parameters (Giz, Zi2, GA3, and ZIJ as demonstrated in Table 111. The typical results of ternary -yI data correlation, shown in Figure 4 for the HC1-H,O-EtOH system at 25 OC,suggest however that the deviation in Ti tends to increase as the nonaqueous solvent (MeOH or EtOH) concentration increases. Results for the correlation at vapor-phase composition of the five isothermal and four isobaric ternary VLE data of Table I are presented in Table IV in terms of the average absolute error in y:
AY =
1
NP
- YEli
&YC
The errors are very low, probably comparable to correlation on nonelectrolytic systems. A more important test of model performance in this case is really the degree of salting out obtained. Table IV presents, therefore, the % salting out % salting out =
- AY3E)
(&3C
AY3E
where A Y ~ C= y3c(salt) - y3c(no salt) A Y ~ E= Y,~(salt)- Y , E ( ~ O salt)
Both the errors in y and the % salting out value indicate
in Ternary Mixtures Using Binary Parameters no. max of vts m T, " C (YZR &PR &W G+P -150.9 336.47 0.1061 48 2.0 25 -1.0 -150.9 336.47 0.0557 45 1.0 25 -1.0 105.7 23 0.1 25 -1.0 383.87 0.094 y,
z,
G+R
Z,,
-33.295 -81.532 -34.79
10.9 6.17 117.0
39.70 56.38 12.35
Table VI. Prediction of Vapor-Phase Composition and Total Pressure Using Binary Parameters no. max of m &A2 Or &e2 or &A3 Or &B3 Or system pts used T o r P a 2 3 &23 &32 G ~ z Z*Z Ga3 Z+3 LiCl-H20-EtOH 10 0.5 25 " C -1.0 105.8 383.8 0.0557 -81.532 4.1 144.2 LiCl-H20-MeOH 5 1.0 25 OC -1.0 -150.9 336.47 0.0557 -81.532 6.1745 56.382 NaBr-H20-MeOH 6 1.9 25 "C -1.0 -150.9 336.47 36.949 1.261 6.876 54.598 LiCl-H20-MeOH 10 2.0 60 "C -1.0 140.3 235.92 0.0353 -287.85 5.09 91.21 LiCl-H,O-MeOH" 19 2.0 1 atm -1.0 97.08 312.47 81.74 -11.59 19.5 170.5 97.08 312.50 163.3 1 atm -1.0 12.55 -21.56 NaBr-H20-MeOHb 19 2.0 177.3 Binary parameters evaluated at 60 "C. *Binary parameters evaluated at 25 "C.
av % error in y, 13.5 15.3 8.4
A Y ~ E A y s C @ av, av av mmHg 0.017 -0.003 1.4 1.9 0.027 -0.006 4.9 0.022 -0.01 14.2 0.021 -0.019 0.016 -0.001 17.4 16.5 0.019 -0.002
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1349 An extended D-H model for Coulombic forces with the NRTL model for the long-range forces is presented that provides successful correlation of single electrolyte-binary solvent mixtures. The prediction results, however, are in the right direction for y+ but fail to account for the effect of electrolyte in the prediction of y's.
1.
2
u
2 2
Nomenclature .5
0
0
.5
1
'MeOH,EXP
Figure 5. Comparison of experimental and correlated vapor-phase compositions for the system LiC1-H20-MeOH at 60 O C : (A)correlation; (0) Hala (1969). 1.
-1
x 4
s
activity of solvent i A, = Debye-Huckel constant, (kg/g-mol)'/* A1-A6 = constants defined in eq B-8 q - a 6 = constants defined in eq B-3 AD1, AD2 = constants defined in eq B-9 a', b', c ' = pure component liquid molar volume constants in eq B-7 d = density of the solvent mixture (electrolyte free), g/cm3 D = dielectric constant of the solvent mixture (electrolyte free) f , = fugacity of component i in the mixture gE = molar excess Gibbs free energy, cal/g-mol GE = total excess Gibbs free energy, cal GI, = NRTL adjustable parameter for solvent-solvent binary, cal/g-mol Ag,, = NRTL temperature-independent parameter for solvent-solvent binary GAl, G B 1 = binary adjustable temperature-independent parameters defined in eq 12 AgA,,AgBl = binary adjustable temperature-independent parameters in eq 12, kJ/g-mol G+, = adjustable parameters for binary 1-2 or 1-3 in eq 10 I = ionic strength = 1/2xkmkZkz,g-mol/kg of solvent K = Boltzmann constant, 1.38054 X J / ( K molecules) NP = total number of points in a system Ns = total number of components in a system NT = total number of moles of the solvent or solvent mixture (electrolyte free) m = molality of an electrolyte, g-mol/kg of solvent M, = molecular weight of the solvent mixture, g/g-mol P = total pressure of the system, mmHg Pp = vapor pressure of the pure component i , mmHg R = gas constant, 1.987 cal/(g-mol K) or 8.314 X kJ/ (g-mol K) T = temperature, K V = molar volume, cm3/g-mol X, = liquid-phage mole fraction of component m X', = liquid-phase mole fraction of solvent i , electrolyte free y m = vapor-phase mole fraction of component m Z+, Z- = valency of cation and anion, respectively Z , = NRTL solvent-solvent binary constant, cal/g-mol Zfl, ZAI,and ZBI = binary parameters defined in eq 9-12, kJ/g-mol z, = charge of an electron a, =
.5
0 .5
'MeOH, EX P
Figure 6. Comparison of experimental and correlated vapor-phase compositions for the system NaBr-H20-MeOH at 25 O C : (A)correlation; (0) Chen et al. (1980).
that the proposed model provides very satisfactory correlation results. In addition, the plots a t calculated vapor-phase composition vs. the experimental ones, presented in Figures 5 and 6, suggest that the model provides better results than that of Hala (1969) and Chen et al. (1980). The results of Table I11 and IV suggest that the model should be applicable up to I = 6. The data for LiC1-water-EtOH a t I = 4 could not, however, be successfully correlated. In addition, as the temperature increases, the solvent dielectric constant decreases, lowering the limit of complete dissociation. This may explain the better results obtained with isothermal data as compared to isobaric ones (Table IV). The model is also applicable in the prediction of ternary behavior from binary data. The obtained Y~ values, however, are of fair quality as Table V and Figure 4 indicate, while vapor-phase compositions tend to be the same as the electrolyte-free values (Table VI).
Conclusions It is demonstrated that in single electrolyte-mixed solvent systems, the D-H term predicts salting in where salting out is observed.
Greek Symbols cyz3 = a constant used in NRTL equation for solvent-solvent binary (= -1.0 or 0.2, 0.3, 0.47) aAi,cyBi = NRTL nonrandomness constants for ion-solvent (=0.2 and 0.0, respectively) y i = activity coefficient of solvent i y+ = mean molal or ionic activity coefficient y+* = rational activity coefficient 4 = osmotic coefficient in a binary (1-2 or 1-3) mixture defined in eq 5 ul(pI1lz) = defined in eq 22 +(al) = defined in eq 23 u (PI'/*) = defined in eq 24 1 G1(aI) = defined in eq 25 vA, vB = number of cations and anions, respectively u = total number of ions ( = Y A + vg) Superscripts o = pure component L = liquid phase V = vapor phase
1350 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
Subscripts 1, 2, 3 = electrolyte, solvent 2, and solvent 3, respectively A, B = cation and anion, respectively C = calculated property E = experimental i, j = solvent 2 or 3 k = cation or anion li = binary 1-2 or 1-3 i j = solvent-solvent binary l,m,n = ions or electrolyte or solvent 2 or 3
Appendix A (I) Development of Ternary Expression for GE.(a) Extended Debye-Huckel Term. The semiempirical extension of the Debye-Huckel equation that is proposed in this work is analogous to that proposed by Gronwall et al. (1928, 1931). This additional term represents salting out effect in mixed solvents: In 7iext.D-H -
where p , a, and n are adjustable parameters. The DebyeHucke1 constant, A,, is obtained by using density and dielectric constant of the solvent mixture, electrolyte free. The expression for the extended Debye-Huckel Gibbs free energy is obtained by combining eq A-1 with eq 16 and integrating, as
[ $1
= v L N ’ In Ti* dN1
(A-2)
ext.D-H
It should be noticed that the final form of the extended Gibbs free energy expression will depend upon the constants p , a and n in eq A-1. The integration of eq A-2 can be accomplished by fixing a value of n which can be an integer or a noninteger. (b) NRTL Term. Renon and Prausnitz (1968) proposed the expression
(A-9)
(11) Development of Ternary Activity Coefficient Expressions. The excess Gibbs free energy functions A-2 and 8 with eq 14 and 15 are used to derive the activity coefficient expressions. The final expressions of the activity coefficients will depend upon constants p , a, and n in the extended Debye-Huckel term. However, the combined expressions of the extended Debye-Huckel and NRTL will have five parameters p, a, n, Gii, and Zii for a binary mixture. The following stepwise procedure was used to (a) reduce five parameters to two NRTL parameters only for a binary system and (b) to obtain activity coefficient eq 18-26. (1)Preset p, a, and n in eq A-1 and obtain the extended Debye-Huckel art of the GE expression. (2) Combine and differentiate the GEd to obtain Gfxt,DH with GNRTL E? In y i (i = solvent) expressions. The final activity coefficient has only binary NRTL parameters (G*i, Zii, &ij, &ji, and aij). (3) Correlate binary and ternary VLE and y+ vs. m data to evalute the NRTL parameters by using activity coefficient expressions. Steps 1-3 were repeated for different combinations of fixed values of constants p , a, and n. It was found that the best results were obtained by fixing the constants at p = 1.0 and a = 1.5/1Z+2-1as suggested by Bromley and n = based upon the results of binary and ternary data correlation. Appendix B The DebyeHuckel constant at the system temperature and pressure is given by A, =
[
[A]
3/2
2 ~ N d ]IJ2 (1000)(2.303)2
(B-1)
or
[
A, = 1.8246 X 106d1J2where Zmland Gnl are defined in eq 11 and 12. For an electrolytic mixture, eq A-3 is converted by utilizing the assymetric convention. GWfW‘)= Gb;RTL - NA lim RT In YA’ - N B lim RT In YB’ (A-4) NB+
NA-0
The resulting equation for a ternary electrolytic mixture is given by eq 8. As proposed by Cruz and Renon, the following order of energy parameters has been used in deriving eq 8: (solvent-ion) < (solvent or electrolyte solvent or electrolyte)
