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Prediction of vapor-liquid equilibria of binary-solvent electrolytes
Fluid Phase Equilibria, 53 ( 1989) 199-206 Science Publishers B.V., Amsterdam-Printed
Elsevier
199 in The Netherlands
PREDICTION OF VAPOR-LIQUID EQUILIBRIA OF BINARY-SOLVENT ELECTROLYTES
Kevin L. Gering and Lloyd L. Lee School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
ABmcT An iterative method is presented that is used to predict vapor-liquid equilibria (VLE) of single-salt multisolvent electrolytes of the form solvent-cosolvent-salt. A local composition model f&CM) and an electrolyte model based on the exponential modification of the Mean Spherical Approximation (EXP-MSA) are combined with traditional phase equilibria relations to estimate the pressures and compositions of a vapor phase in equilibrium with a binary-solvent electrolyte. In addition, a pseudo-solvent model is used to obtain a variety of averaged liquid phase electrolyte properties. For our initial model system, methanol-water-LiCl. the above method accurately predicts the salt effects upon vapor composition for dilute to moderate salt concentrations up to 20 mol% LiCl, and yields good approximations of vapor pressure depression for LiCl up to 14 mol%. ‘lluee electrolyte systems are investigated: water-ethylene glycol-LiBr, ammonia-water-LiBr, and methanol-water-LiCl. INTRODUCTION An important class of electrolytes is multisolvent electrolytes, i.e. systems that contain one or more salts, a solvent, and one or more cosolvents. Consider the vapor-liquid equilibria of the ternary system methanol-waterLiCI. The dissociated salt in the liquid phase results in an enrichment of the vapor phase with either the solvent or cosolvent. This phenomenon is often referred to as salring in or salting out of a solvent in a multisolvent electrolyte mixture. The salting in (or out) effect is accompanied by a drop in vapor pressure, commonly referred to as vapor pressure depression (VF’D). which becomes more severe as the salt concentration increases. In this work, a number of concepts are combined to form a versatile, accurate model that is used to predict how the presence of a salt influences the vapor-liquid equilibria and vapor pressure of multisolvent electrolytes. A local composition model (LCM) discussed by Li (1984) and Li et 01 (1986) is used to determine vapor-liquid equilibria and other thermodynamic properties of salt-free binary solvent systems. The exponential modification of the MSA (EXP-MSA), as developed by Anderson and Chandler (1970, 1971a. 1971b) and utilized by Landis (1985) and Gering et al. (1989) is employed to elucidate properties of electrolytes, such as osmotic coefficients and activity coefficients. Certain theoretical aspects are greatly simplified by approximating properties of mixed solvent electrolytes (density, dielectric constants, average molecular weights, solvent activity, etc.) with those of a hypothetical pseudo solvent. Thus, a pseudo-solvent approach is outlined below. In this work we assume that the salt species under consideration completely dissociates in solution for the salt concenaations given, and that the salt exhibits low enough volatility so as to make its vapor phase composition negligible. THEORETICAL DEVELOPMENT As a basis of theoretical approach, consider a liquid phase electrolyte (a+b+2) in equilibrium with a vapor phase (a+b). A small amount of salt 2 is added to the liquid under isothermal conditions such that the mole fraction ratio (xJx& in the liquid phase remains constant at equilibrium. According to the phase rule, we have fixed the thermodynamic state of our system by setting the three parameters: 7-, x2. and x,J+,. Other parameters (e.g. P and ya) can then be evaluated through a scheme utilizing phase equilibria relations. Now, if this isothermal addition of 2 is done in a step-wise fashion, as illustrated in Fig. 1, the vapor composition and pressure will change from state 0 successively to state n. This step-wise procedure is the basis of our calculations.
0378-3812/89/$03.50
0 1989 Elsevier Science Publishers B.V.
200 Stare I
Slate 0 --
I
r
Equilibrium a? T. V’. 1,.
Fig. 1
Srare n
Equilibtium at ‘1, V. 1,.
Y,,)o
Y.. ~211
Lsothemxd succession
of equilibrium states where a nonvolatile salt (2) is incrementally the liquid phase mole fraction ratio x&b is kept constant.
The Gibbs-Duhem x, d tn(% YJ~
1
added while
relation for the above liquid elecuolyte is
+ rb d tn(% Y&
+ r, d WZ Y&
=
+j$!
(1)
This equation can be numerically integrated by the trapezoidal rule, provided there is a small difference between equilibrium states n and n-l, as would be obtained for very small changes in x, , xb , and x2 :
k m(& %fy
+
ib dnkb
Ybfbf
+
i2 dn(x2
%f$
= F
(2)
where i. = (x,_ + x+ )/2 , etc. The reference fugacity of the salt specie in solution is a function of temperature only. Thus, for our isothermal system, f2, and f2,r cancel, giving the final form of the Gibbs-Duhem equation
(3) For Eq.(3). prior knowledge of (~2, . ~2,~ . Y,_~ . yb,, . Ppl) is necessary. The solute activity coefficients, calculations (Gering et al., 1989). hence are q and YZ,, . arc obtained directly from EXP-MSA/pseudo-solvent known a priori of VLE information. In addition, y.,, , yb,, and P,, are determined from previous states, the starting point being the initial salt-free state “0”. where these values can be obtained from a local composition model approach, as is used in this work At equilibrium, the partial molar fugacity of a given component must be equal in all phases @ =fy ), for i=(a,bJ. In terms of activity coefficients, we have at state n
(b)
(4a.b)
Equations (4a.b) have introduced three more unknowns: yay.,“.yb,“, and y.. Note that yb = l-y.. The pure component fugacitiesfiL andf;” (i=a,b) are dependent upon the temperature and pressure of the system, hence are not independent variables; these can be evaluated once the (T,F’) of the system has been determined (or guessed). Since the number of unknowns is greater than the number of applicable equations, then it is necessary to resort to an iterative scheme to solve for the VLE information at a given equilibrium state. The iterative scheme used in this work is given in Fig. 2.
201
At state 0:
Use LCM to defermine (x,, , yJ af T, PO.
(n=O)
Use EOS IOobtain VT, fT,f’.
fi”J for i=(a.b) at State 0.
k Increase
-New
x2
a small increment.
Equilibrium Slate n: n = n + 1
1 Calculate electrolyte properties via EXP-MSA theory and pseudo-solvent model: *IzL) etc. Guess P. Assume ri”. = u,“,,-1 forfist iteration, where i=(a.b).
Use Pure Component Standard States at T. P. to evaluate (f;L,iv). via EOS, where i=(a,b).
I 4 Substitute expressions for u.“. and ~t,L,.fiomEqs.(44) and #b) into Eq.(3 ), then lcTeroot solving technique to solve for y. from Eq.13 j.
Use Eqs.(&) and (46) to solve for v.“. and ybL,
I Calculate EL = (x. ‘1.f.F
and f
= (.z~ybf&
for state n
1
I d Use EOS to obtainz” andz” at (T,P,y,,).
No:
COMPARE: EL =?z” AND zL ir
Maximum or &sired salt concentration reached?
1STkP;’
Fig. 2
Schematic
for Iterative VLE Calculations.
for state n.
NO b
202
MODELING: FORMULATING USEFUL SIMPLIFICATIONS Local Comoosition Model (LCM1 The LCM used in this work is a statistical-mechanical LCM developed by Li (1984) and Li ef al. (1986). where it was applied to describe the composition dependence of an equation of state recently developed by Chung et al. (19X4). This generalized EOS was developed for polar fluids and utilizes reduced density and reduced temperature as independent variables, from which dependent variables (e.g. P) can be calculated. Tbe LCM expression for the Helmholtz free energy was used to derive the composition dependence of the EOS mixture properties. The LCM is used herein to determine the thermodynamic state of a two-component vapor/liquid system (the aforementioned state 0) at a given (T,P). The EOS is used to determine pure component and partial molar fugacities at a given (TJ’,x,y). A full discussion of the LCM and EOS are beyond the scope of this report, but the reader is encouraged to see the above references for further information. Pseudo-solvent Model ADoroach For a mixed-solvent system, we define a hypothetical pseudo solvent (F’S) as one that has physical properties that are averages of the constituent solvent properties. A pseudo-solvent model has been described previously (Gering ef ol., 1989). where weighted averages are used to approximate the electrolyte density, as well as the molecular weight, dielectric constant, and partial molar volume of the pseudo solvent. Another important consideration of multisolvent electrolyte solutions is the degree of salvation of the ions, as increased solvation increases the solvated diameter of an ion. Neutron diffraction and x-ray scattering experiments have clearly demonstrated that cations and anions in solution will be solvated, and that the extent of solvation depends on ion type, ion concentration and the type of solvent(s) present (Marcus, 1985: Licheri el al., 1975; Enderby and Neilson, 1981; Narten and Hahn, 1982; Neilson and Newsome, 1981; Enderby, 1983). Robinson and Stokes (1959, 1973) and Pailthorpe et al. (19X4) are among those who have tried to account for salvation effects in their electrolyte models. Generally, solvated cation and anion diameters decrease as the net ion concentration iIlCfeaseS.
To estimate how the environment of mixed solvents and a varying salt concentration size of a cation in solution, the following expression is proposed:
affects the solvated
where o+0 and CT+~ are average intermolecular cation-solvent distances for salvation shells composed of solvents a and b, respectively (Marcus, 1985), and o:(m) is the regressed cation diameter, determined with respect to solvent a only, as a function of salt molality. The terms ~~ and T* are average lifetimes of solvents a and b in the coordination spheres of a given cation, here Lz*, and are a good measure of the affinity of solvents in a mixture toward a given ion (Marcus, 1985 and Mishastin, 1981). Thus, Eq(5) should yield solvated cation sizes that are representative averages of what would actually be “observed” in a two-solvent electrolyte. However, anions are assumed to remain essentially nonsolvated, regardless of the electrolyte concentration. Anion diameters from EXP-MSA regression analysis were determined by Landis (1985). where they were found to be very similar to crystalline (Pauling) anion diameters. For a single-solvent electrolyte system (a+Z), the solvent activity can be expressed in terms of the osmotic coefficient Inn, = - $, (&zi)& il
for q species of ions
We extend the same relation to a pseudo-solvent system by replacing a. with ups and replacing M. with Mpp The osmotic coefficient +,,, is obtained from EXP-MSA theory. If the vapor phase is assumed to behave as an ideal gas (usually a good assumption for aqueous systems, containing nonvolati!e elecnolytes, with a absolute pressure of less than 10 atm), then the total pressure of a single solvent electrolyte system, P, , canbe approximated by p,
= a. P.
(7)
where Pi is tbe vapor pressure of pure (salt-free) solvent a at the system temperature. Vapor pressure depression (VPD) is obtained by subtracting fY from Pi. A more rigorous approach, such as the iterative method described above, must be used to determine the system pressure for systems containing highly volatile mixed solvents and/or
203 volatile electrolytes, as well as systems that exhibit highly nonideal behavior. It is useful to note that Eq.(7) may have meaningful applications to certain mixed solvent electrolyte systems where one component greatly dominates the vapor phase. composition 0: > 0.9). This is done by replacing (I, in Eq.(7) with ops and Pi with Pir, the vapor pressure of the salt-free mixed solvent system at the system temperamre. D P r)rs = aps PPS
(8)
Thus, both Eq.(8) and the iterative method described earlier are used here to predict vapor pressures for multisolvent electrolyte systems.
RESULTS AND DISCUSSION To start, we consider multisolvent electrolytes that possess a volatile component that greatly dominates the vapor phase. Since the vapor composition of such a system is almost pure in one component, it seems likely that the pseudo-solvent model can be useful in predicting the vapor pressure (see Eq.(8)). Electrolyte systems that are well suited for this treatment, under carefully chosen (T,P) ranges, include water-ethylene glycol-LiBr, and ammonia-water-LiBr. P-T-x results for the multisolvent system water-LiBrethylene glycol are given in Fig. 3a, which shows the effect of LiBr composition and system temperature on system pressure. This system was chosen for study because the concentration of ethylene glycol in the vapor rarely exceeds 5 mol% for the (TP) ranges indicated. Values obtained by the EXP-MSA theory and pseudo-solvent model are compared to experimental dam by Uemura and Iyoki (1981). where average absolute deviations in pressure of 5-996 are observed, depending on which isotherm is chosen. It should be noted that a relatively high degree of accuracy is retained by the above model even as the salt concentration increases toward its solubility limit, which ranges from 56% LiBr (by weight) at 2O’C to 64% at 75°C. Such favorable agreement between experimental and calculated values demonstrates the ability of the pseudo-solvent approach to represent such bulk fluid properties as average molecular weight of the solvent, system density, average dielectric constant, and cation diameter. Under certain (T’S) conditions for the ammonia-water-LiBr system, ammonia exhibits a volatility much greater than tbat of water, and hence dominates the vapor phase P-T-x results from pseudo-solvent analysis are shown for this system in Fig. 3b. The calculated results show a higher deviation from experimental results, when compared to the water-ethylene glycol-LiBr system. This may be because the assumption of an ideal gas, inherent in Eq.(8), is no longer valid for the (P.T,x) conditions encountered. However, the general trends of vapor pressure are well predicted. Since the results are plotted on a semilog plot, some differences between experimental and calculated pressures will appear to be somewhat exaggerated. However, there are a multitude of electrolyte systems that would be unfit for the pseudo-solvent model, due to their high degree of nonideality. Thus, the aforementioned iterative method was recently developed to predict phase equilibria for such nonideal systems. The iterative method was tested on the methanol-water-LiCl system, where data was available for the 60°C isotherm (Broul et a1.,1969). Fig. 4a shows experimental and predicted vapor phase compositions for this system. Very good agreement is seen for the entire salt concentration range, where the average absolute deviation is about 2%. For a salt concentration of zero, the calculated VLE values shown were obtained directly from the LCM. Fig. 4b shows experimental and iterated vapor pressures for the above system; we note a decrease in predictive capabilities as the salt concentration increases. Average absolute deviations between iterated and experimental pressures in Fig. 4b are as follows: less than 1% for x2 = 0.0, 4% for x2 = 0.07. 16% for x2 = 0.14, and 35% for x2 = 0.20. InitiaBy, these worsening deviations in pressure appeared to be due to propagated error occurring as the Gibbs-Duhem equation (Eqs.(1,2,3)) was numerically integrated from state to state. However, a variety of integration techniques yielded similar results, ruling out the possibility of a numerical propagation error. Second, some deviations could be due to decreased accuracy of the EXP-MSA theory. However, this is unlikely because much successful work has been performed using the EXP-MSA theory for very concentrated salt solutions (Landis, 1985; Gering et a1.,1989). Finally, some small errors could be introduced via the LCM and EOS used herein. Binary interaction parameters required by the LCM were regressed so that the LCM gave deviations in compositions and pressures (state 0) that were within O-6% and O-2%, respectively. However, it is unlikely that the small deviations asscciated with the LCM and EOS would generate the increasing deviations in pressure seen in Fig. 4b. The iterative technique needs to be refined before it can be successfully applied to electrolyte systems that contain very concentrated salts. We arc currendy investigating the cause of the above pressure deviations.
204 water(a) - ethylene glycol(b) - LiBr(2) LECEND:
l,*:T=ZO’C
.,O +,O .,O .,i,
: T = 4Ov : T = 6OV : T = 80.C :T= 100’2
Q.. . A : Experimental Values t.. . . : Calculated Values from EXP-MSA Theory and Pseudo-solvent Mddel
1 0
20
30 Wt.
ammonia(a)
6.3
0.”
*
Fig. 3
NH, 9
“0
50
50
70
% LiBr
- water(b) - LiBr(2)
0.5
(salt-free
0.6
0.7
basis)
P-T-x diagrams for the (a) water-ethylene glycol-LiBr and (b) ammonia-water-LiBr systems. In (a), the mass ratio of ethylene glycol to water is 0.3445:l; open symbols are experimental data (Uemura and Iyoki, 1981) and filled symbols are calculated values from Eq.(8). In (b) the mass ratio of LiBr to water is 1S:l; open symbols are experimental data (Radermacher, Alefeld. 1982) and filled symbols are values calculated from Q.(8).
1981; Radermacher
and
205
m&ad(a)
- water(b) - LEr(z)
Calculated Values from Iterative Scheme
0.1
0.0 0.0
0.1
0.2
0.1
0.”
0.1
0.6
X M~OH3
(salt-free
X
(salt-free
0:s
04
Fig. 4
M&H
9
0.7
o.*
0.9
I
basis)
0:7
0.9
0:s
basis)
Effect of salt on (a) vapor phase composition and (b) vapor pressure of the methanol-water-LiCl system at 60°C as a function of LiCl and methanol concentrations in the liquid phase. Open symbols: experimental data (Brool, et al. 1%9); filled symbols: values calculated Iinm iterative technique.
206
CONCLUSIONS An iterative method, based on the numerical integration of the Gibbs-Duhem equation, was used to predict vapor-liquid equilibria (VLE) of the model system methanol-water-LiCl. A local composition model (LCM) and an electrolyte model based on the exponential modification of the Mean Spherical Approximation (EXP-MSA) were combined with traditional phase equilibria relations to facilitate phase equilibria calculations. In addition, a pseudo-solvent model was used to estimate vapor pressures for multisolvent electrolyte systems where the vapor phase is assumed to behave as an ideal gas, and is dominated by one component; such systems include waterethylene glycol-LiBr and ammonia-water-LiBr under certain (T,F’) conditions. For our initial model system, methanol-water-LiCl, the above method accurately predicts the salt effects upon vapor composition for dilute to moderate salt concentrations up to 20 mol% LiCI, and yields good approximations of vapor pressure depression for LiCl up to 14 mol%. Certain improvements are foreseen for the iterative method presented herein. These improvements center around decreasing the deviations between experimental and calculated properties (e.g. system pressure) that, up to now, have been seen to increase as the salt concentration increases.
ACKNOWLEDGEMENTS We wish to thank the Gas Research Institute for their generous financial support of this research. LIST OF SYMf3OZ.S GREEK LEITERS:
activity. referen= fugacity. partial molar fugacity. solute molality. molecular weight. pressure.
r” f m M P R T
activity coefficient. solvated cation diameter. average lifetime of solvent in a cation mordination shell. molal osmotic coefficient.
Y 0, ‘T Q” SUPERSCRIPTS:
!!
L V 0
molar volume of solution. liquid phase mole fraction. salt-free mole fraction (liquid). vapor phase mole fraction.
x i Y
liquid phase. vapor phase. salt-free state, or reference value.
SUBSCRIPTS: 2
salt specie. solvent species. nth equilibrium state. pseudo solvent.
a.6 n PS REFERENCES Anderson.
H. C. and D. Chandler. Phys., 53: 547.
Anderson. H. C. and D. Chandler, Phys., 55: 1497.
1970.
J. Chem.
Lichai.
1971a
J. Chem.
Marcus. Y.. 1985. Ion Salvation. Chichester).
Broul, M.. K. Hlavaty, and J. Linek. Czech. Chem. Comm.. 34: 3428. Chandler, D. wd H.C. Anderson, Phys.. 54: 26.
1969.
1971b.
Coil.
1. Chem.
Chtmg, T. H.. M. M. Khan. L. L. Lee, and K. E. Starling. 1984. Fluid Phase Equilibria. 17: 351. Enderby. J. E.. 1983. Ann Rev. Phys. Chem., 34: 155.
G.. G. Piccaluga, and G. Pinna. 1975. Chem. Phys. Len., 35: 119.
Mishastti.
A. I.. 1981. Zh. Fiz. Khim.. 55: 1507.
Narten. A. H. and R. L. Hahn, 1982. 1249.
Pailthorpe, B. A., D. J. Mitchell, wd B. W. Niiam. 1984. 1. Chem. Sot., Faraday Trans. 2. 80: 115. Radetmacher.
Gering,
K. L.. L. L. Lee, and J. L. Savidge. Fluid Phase Equilibria, (preprint).
Radermacher. R. and G Alefeld Warm+Kr#t, 34: 31.
Landis,
L. H.. 1985. Oklahoma.
Li.
H.. 1984. Oklahoma.
M.
Ph.D. Thesis, Ph.D.
Thesis,
Science, 217:
Neilson, G. W. and J. R. Newsome. 1981. 1. Chem. Sot., Faraday Trwu., 77: 1245.
Enderby. J. E. and G. W. Neilson. 1981. Rep. Prog. Phys.. 44: 593. 1989.
(Wiley-Interscience.
R.,
1981.
Ph.D.
Thesis.
Technical
University of Munich. 1982.
&e-t.-
University
of
Robinson. R. A. and R. H. Stokes, 1959. Electrolyte Solutions, (Buoerworths. London).
University
of
Robinson. R. A. and R. H. Stokes, Chem., 2. No. 2/3: 173.
Li. M. H., T. H. Chung. L. L. Lee., and K. E. Starling, 1986. ACS Symposium Series No. 300. Edited by KC. Chao and R.L. Robinson Jr. (American Chemical Society, Washington D.C.) 250.280.
Uemura. T. and S. Iyoki. 279.288.
1981.
1973.
I. Sd.
Refiigerafiorz, 56:
Elsevier
199 in The Netherlands
PREDICTION OF VAPOR-LIQUID EQUILIBRIA OF BINARY-SOLVENT ELECTROLYTES
Kevin L. Gering and Lloyd L. Lee School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
ABmcT An iterative method is presented that is used to predict vapor-liquid equilibria (VLE) of single-salt multisolvent electrolytes of the form solvent-cosolvent-salt. A local composition model f&CM) and an electrolyte model based on the exponential modification of the Mean Spherical Approximation (EXP-MSA) are combined with traditional phase equilibria relations to estimate the pressures and compositions of a vapor phase in equilibrium with a binary-solvent electrolyte. In addition, a pseudo-solvent model is used to obtain a variety of averaged liquid phase electrolyte properties. For our initial model system, methanol-water-LiCl. the above method accurately predicts the salt effects upon vapor composition for dilute to moderate salt concentrations up to 20 mol% LiCl, and yields good approximations of vapor pressure depression for LiCl up to 14 mol%. ‘lluee electrolyte systems are investigated: water-ethylene glycol-LiBr, ammonia-water-LiBr, and methanol-water-LiCl. INTRODUCTION An important class of electrolytes is multisolvent electrolytes, i.e. systems that contain one or more salts, a solvent, and one or more cosolvents. Consider the vapor-liquid equilibria of the ternary system methanol-waterLiCI. The dissociated salt in the liquid phase results in an enrichment of the vapor phase with either the solvent or cosolvent. This phenomenon is often referred to as salring in or salting out of a solvent in a multisolvent electrolyte mixture. The salting in (or out) effect is accompanied by a drop in vapor pressure, commonly referred to as vapor pressure depression (VF’D). which becomes more severe as the salt concentration increases. In this work, a number of concepts are combined to form a versatile, accurate model that is used to predict how the presence of a salt influences the vapor-liquid equilibria and vapor pressure of multisolvent electrolytes. A local composition model (LCM) discussed by Li (1984) and Li et 01 (1986) is used to determine vapor-liquid equilibria and other thermodynamic properties of salt-free binary solvent systems. The exponential modification of the MSA (EXP-MSA), as developed by Anderson and Chandler (1970, 1971a. 1971b) and utilized by Landis (1985) and Gering et al. (1989) is employed to elucidate properties of electrolytes, such as osmotic coefficients and activity coefficients. Certain theoretical aspects are greatly simplified by approximating properties of mixed solvent electrolytes (density, dielectric constants, average molecular weights, solvent activity, etc.) with those of a hypothetical pseudo solvent. Thus, a pseudo-solvent approach is outlined below. In this work we assume that the salt species under consideration completely dissociates in solution for the salt concenaations given, and that the salt exhibits low enough volatility so as to make its vapor phase composition negligible. THEORETICAL DEVELOPMENT As a basis of theoretical approach, consider a liquid phase electrolyte (a+b+2) in equilibrium with a vapor phase (a+b). A small amount of salt 2 is added to the liquid under isothermal conditions such that the mole fraction ratio (xJx& in the liquid phase remains constant at equilibrium. According to the phase rule, we have fixed the thermodynamic state of our system by setting the three parameters: 7-, x2. and x,J+,. Other parameters (e.g. P and ya) can then be evaluated through a scheme utilizing phase equilibria relations. Now, if this isothermal addition of 2 is done in a step-wise fashion, as illustrated in Fig. 1, the vapor composition and pressure will change from state 0 successively to state n. This step-wise procedure is the basis of our calculations.
0378-3812/89/$03.50
0 1989 Elsevier Science Publishers B.V.
200 Stare I
Slate 0 --
I
r
Equilibrium a? T. V’. 1,.
Fig. 1
Srare n
Equilibtium at ‘1, V. 1,.
Y,,)o
Y.. ~211
Lsothemxd succession
of equilibrium states where a nonvolatile salt (2) is incrementally the liquid phase mole fraction ratio x&b is kept constant.
The Gibbs-Duhem x, d tn(% YJ~
1
added while
relation for the above liquid elecuolyte is
+ rb d tn(% Y&
+ r, d WZ Y&
=
+j$!
(1)
This equation can be numerically integrated by the trapezoidal rule, provided there is a small difference between equilibrium states n and n-l, as would be obtained for very small changes in x, , xb , and x2 :
k m(& %fy
+
ib dnkb
Ybfbf
+
i2 dn(x2
%f$
= F
(2)
where i. = (x,_ + x+ )/2 , etc. The reference fugacity of the salt specie in solution is a function of temperature only. Thus, for our isothermal system, f2, and f2,r cancel, giving the final form of the Gibbs-Duhem equation
(3) For Eq.(3). prior knowledge of (~2, . ~2,~ . Y,_~ . yb,, . Ppl) is necessary. The solute activity coefficients, calculations (Gering et al., 1989). hence are q and YZ,, . arc obtained directly from EXP-MSA/pseudo-solvent known a priori of VLE information. In addition, y.,, , yb,, and P,, are determined from previous states, the starting point being the initial salt-free state “0”. where these values can be obtained from a local composition model approach, as is used in this work At equilibrium, the partial molar fugacity of a given component must be equal in all phases @ =fy ), for i=(a,bJ. In terms of activity coefficients, we have at state n
(b)
(4a.b)
Equations (4a.b) have introduced three more unknowns: yay.,“.yb,“, and y.. Note that yb = l-y.. The pure component fugacitiesfiL andf;” (i=a,b) are dependent upon the temperature and pressure of the system, hence are not independent variables; these can be evaluated once the (T,F’) of the system has been determined (or guessed). Since the number of unknowns is greater than the number of applicable equations, then it is necessary to resort to an iterative scheme to solve for the VLE information at a given equilibrium state. The iterative scheme used in this work is given in Fig. 2.
201
At state 0:
Use LCM to defermine (x,, , yJ af T, PO.
(n=O)
Use EOS IOobtain VT, fT,f’.
fi”J for i=(a.b) at State 0.
k Increase
-New
x2
a small increment.
Equilibrium Slate n: n = n + 1
1 Calculate electrolyte properties via EXP-MSA theory and pseudo-solvent model: *IzL) etc. Guess P. Assume ri”. = u,“,,-1 forfist iteration, where i=(a.b).
Use Pure Component Standard States at T. P. to evaluate (f;L,iv). via EOS, where i=(a,b).
I 4 Substitute expressions for u.“. and ~t,L,.fiomEqs.(44) and #b) into Eq.(3 ), then lcTeroot solving technique to solve for y. from Eq.13 j.
Use Eqs.(&) and (46) to solve for v.“. and ybL,
I Calculate EL = (x. ‘1.f.F
and f
= (.z~ybf&
for state n
1
I d Use EOS to obtainz” andz” at (T,P,y,,).
No:
COMPARE: EL =?z” AND zL ir
Maximum or &sired salt concentration reached?
1STkP;’
Fig. 2
Schematic
for Iterative VLE Calculations.
for state n.
NO b
202
MODELING: FORMULATING USEFUL SIMPLIFICATIONS Local Comoosition Model (LCM1 The LCM used in this work is a statistical-mechanical LCM developed by Li (1984) and Li ef al. (1986). where it was applied to describe the composition dependence of an equation of state recently developed by Chung et al. (19X4). This generalized EOS was developed for polar fluids and utilizes reduced density and reduced temperature as independent variables, from which dependent variables (e.g. P) can be calculated. Tbe LCM expression for the Helmholtz free energy was used to derive the composition dependence of the EOS mixture properties. The LCM is used herein to determine the thermodynamic state of a two-component vapor/liquid system (the aforementioned state 0) at a given (T,P). The EOS is used to determine pure component and partial molar fugacities at a given (TJ’,x,y). A full discussion of the LCM and EOS are beyond the scope of this report, but the reader is encouraged to see the above references for further information. Pseudo-solvent Model ADoroach For a mixed-solvent system, we define a hypothetical pseudo solvent (F’S) as one that has physical properties that are averages of the constituent solvent properties. A pseudo-solvent model has been described previously (Gering ef ol., 1989). where weighted averages are used to approximate the electrolyte density, as well as the molecular weight, dielectric constant, and partial molar volume of the pseudo solvent. Another important consideration of multisolvent electrolyte solutions is the degree of salvation of the ions, as increased solvation increases the solvated diameter of an ion. Neutron diffraction and x-ray scattering experiments have clearly demonstrated that cations and anions in solution will be solvated, and that the extent of solvation depends on ion type, ion concentration and the type of solvent(s) present (Marcus, 1985: Licheri el al., 1975; Enderby and Neilson, 1981; Narten and Hahn, 1982; Neilson and Newsome, 1981; Enderby, 1983). Robinson and Stokes (1959, 1973) and Pailthorpe et al. (19X4) are among those who have tried to account for salvation effects in their electrolyte models. Generally, solvated cation and anion diameters decrease as the net ion concentration iIlCfeaseS.
To estimate how the environment of mixed solvents and a varying salt concentration size of a cation in solution, the following expression is proposed:
affects the solvated
where o+0 and CT+~ are average intermolecular cation-solvent distances for salvation shells composed of solvents a and b, respectively (Marcus, 1985), and o:(m) is the regressed cation diameter, determined with respect to solvent a only, as a function of salt molality. The terms ~~ and T* are average lifetimes of solvents a and b in the coordination spheres of a given cation, here Lz*, and are a good measure of the affinity of solvents in a mixture toward a given ion (Marcus, 1985 and Mishastin, 1981). Thus, Eq(5) should yield solvated cation sizes that are representative averages of what would actually be “observed” in a two-solvent electrolyte. However, anions are assumed to remain essentially nonsolvated, regardless of the electrolyte concentration. Anion diameters from EXP-MSA regression analysis were determined by Landis (1985). where they were found to be very similar to crystalline (Pauling) anion diameters. For a single-solvent electrolyte system (a+Z), the solvent activity can be expressed in terms of the osmotic coefficient Inn, = - $, (&zi)& il
for q species of ions
We extend the same relation to a pseudo-solvent system by replacing a. with ups and replacing M. with Mpp The osmotic coefficient +,,, is obtained from EXP-MSA theory. If the vapor phase is assumed to behave as an ideal gas (usually a good assumption for aqueous systems, containing nonvolati!e elecnolytes, with a absolute pressure of less than 10 atm), then the total pressure of a single solvent electrolyte system, P, , canbe approximated by p,
= a. P.
(7)
where Pi is tbe vapor pressure of pure (salt-free) solvent a at the system temperature. Vapor pressure depression (VPD) is obtained by subtracting fY from Pi. A more rigorous approach, such as the iterative method described above, must be used to determine the system pressure for systems containing highly volatile mixed solvents and/or
203 volatile electrolytes, as well as systems that exhibit highly nonideal behavior. It is useful to note that Eq.(7) may have meaningful applications to certain mixed solvent electrolyte systems where one component greatly dominates the vapor phase. composition 0: > 0.9). This is done by replacing (I, in Eq.(7) with ops and Pi with Pir, the vapor pressure of the salt-free mixed solvent system at the system temperamre. D P r)rs = aps PPS
(8)
Thus, both Eq.(8) and the iterative method described earlier are used here to predict vapor pressures for multisolvent electrolyte systems.
RESULTS AND DISCUSSION To start, we consider multisolvent electrolytes that possess a volatile component that greatly dominates the vapor phase. Since the vapor composition of such a system is almost pure in one component, it seems likely that the pseudo-solvent model can be useful in predicting the vapor pressure (see Eq.(8)). Electrolyte systems that are well suited for this treatment, under carefully chosen (T,P) ranges, include water-ethylene glycol-LiBr, and ammonia-water-LiBr. P-T-x results for the multisolvent system water-LiBrethylene glycol are given in Fig. 3a, which shows the effect of LiBr composition and system temperature on system pressure. This system was chosen for study because the concentration of ethylene glycol in the vapor rarely exceeds 5 mol% for the (TP) ranges indicated. Values obtained by the EXP-MSA theory and pseudo-solvent model are compared to experimental dam by Uemura and Iyoki (1981). where average absolute deviations in pressure of 5-996 are observed, depending on which isotherm is chosen. It should be noted that a relatively high degree of accuracy is retained by the above model even as the salt concentration increases toward its solubility limit, which ranges from 56% LiBr (by weight) at 2O’C to 64% at 75°C. Such favorable agreement between experimental and calculated values demonstrates the ability of the pseudo-solvent approach to represent such bulk fluid properties as average molecular weight of the solvent, system density, average dielectric constant, and cation diameter. Under certain (T’S) conditions for the ammonia-water-LiBr system, ammonia exhibits a volatility much greater than tbat of water, and hence dominates the vapor phase P-T-x results from pseudo-solvent analysis are shown for this system in Fig. 3b. The calculated results show a higher deviation from experimental results, when compared to the water-ethylene glycol-LiBr system. This may be because the assumption of an ideal gas, inherent in Eq.(8), is no longer valid for the (P.T,x) conditions encountered. However, the general trends of vapor pressure are well predicted. Since the results are plotted on a semilog plot, some differences between experimental and calculated pressures will appear to be somewhat exaggerated. However, there are a multitude of electrolyte systems that would be unfit for the pseudo-solvent model, due to their high degree of nonideality. Thus, the aforementioned iterative method was recently developed to predict phase equilibria for such nonideal systems. The iterative method was tested on the methanol-water-LiCl system, where data was available for the 60°C isotherm (Broul et a1.,1969). Fig. 4a shows experimental and predicted vapor phase compositions for this system. Very good agreement is seen for the entire salt concentration range, where the average absolute deviation is about 2%. For a salt concentration of zero, the calculated VLE values shown were obtained directly from the LCM. Fig. 4b shows experimental and iterated vapor pressures for the above system; we note a decrease in predictive capabilities as the salt concentration increases. Average absolute deviations between iterated and experimental pressures in Fig. 4b are as follows: less than 1% for x2 = 0.0, 4% for x2 = 0.07. 16% for x2 = 0.14, and 35% for x2 = 0.20. InitiaBy, these worsening deviations in pressure appeared to be due to propagated error occurring as the Gibbs-Duhem equation (Eqs.(1,2,3)) was numerically integrated from state to state. However, a variety of integration techniques yielded similar results, ruling out the possibility of a numerical propagation error. Second, some deviations could be due to decreased accuracy of the EXP-MSA theory. However, this is unlikely because much successful work has been performed using the EXP-MSA theory for very concentrated salt solutions (Landis, 1985; Gering et a1.,1989). Finally, some small errors could be introduced via the LCM and EOS used herein. Binary interaction parameters required by the LCM were regressed so that the LCM gave deviations in compositions and pressures (state 0) that were within O-6% and O-2%, respectively. However, it is unlikely that the small deviations asscciated with the LCM and EOS would generate the increasing deviations in pressure seen in Fig. 4b. The iterative technique needs to be refined before it can be successfully applied to electrolyte systems that contain very concentrated salts. We arc currendy investigating the cause of the above pressure deviations.
204 water(a) - ethylene glycol(b) - LiBr(2) LECEND:
l,*:T=ZO’C
.,O +,O .,O .,i,
: T = 4Ov : T = 6OV : T = 80.C :T= 100’2
Q.. . A : Experimental Values t.. . . : Calculated Values from EXP-MSA Theory and Pseudo-solvent Mddel
1 0
20
30 Wt.
ammonia(a)
6.3
0.”
*
Fig. 3
NH, 9
“0
50
50
70
% LiBr
- water(b) - LiBr(2)
0.5
(salt-free
0.6
0.7
basis)
P-T-x diagrams for the (a) water-ethylene glycol-LiBr and (b) ammonia-water-LiBr systems. In (a), the mass ratio of ethylene glycol to water is 0.3445:l; open symbols are experimental data (Uemura and Iyoki, 1981) and filled symbols are calculated values from Eq.(8). In (b) the mass ratio of LiBr to water is 1S:l; open symbols are experimental data (Radermacher, Alefeld. 1982) and filled symbols are values calculated from Q.(8).
1981; Radermacher
and
205
m&ad(a)
- water(b) - LEr(z)
Calculated Values from Iterative Scheme
0.1
0.0 0.0
0.1
0.2
0.1
0.”
0.1
0.6
X M~OH3
(salt-free
X
(salt-free
0:s
04
Fig. 4
M&H
9
0.7
o.*
0.9
I
basis)
0:7
0.9
0:s
basis)
Effect of salt on (a) vapor phase composition and (b) vapor pressure of the methanol-water-LiCl system at 60°C as a function of LiCl and methanol concentrations in the liquid phase. Open symbols: experimental data (Brool, et al. 1%9); filled symbols: values calculated Iinm iterative technique.
206
CONCLUSIONS An iterative method, based on the numerical integration of the Gibbs-Duhem equation, was used to predict vapor-liquid equilibria (VLE) of the model system methanol-water-LiCl. A local composition model (LCM) and an electrolyte model based on the exponential modification of the Mean Spherical Approximation (EXP-MSA) were combined with traditional phase equilibria relations to facilitate phase equilibria calculations. In addition, a pseudo-solvent model was used to estimate vapor pressures for multisolvent electrolyte systems where the vapor phase is assumed to behave as an ideal gas, and is dominated by one component; such systems include waterethylene glycol-LiBr and ammonia-water-LiBr under certain (T,F’) conditions. For our initial model system, methanol-water-LiCl, the above method accurately predicts the salt effects upon vapor composition for dilute to moderate salt concentrations up to 20 mol% LiCI, and yields good approximations of vapor pressure depression for LiCl up to 14 mol%. Certain improvements are foreseen for the iterative method presented herein. These improvements center around decreasing the deviations between experimental and calculated properties (e.g. system pressure) that, up to now, have been seen to increase as the salt concentration increases.
ACKNOWLEDGEMENTS We wish to thank the Gas Research Institute for their generous financial support of this research. LIST OF SYMf3OZ.S GREEK LEITERS:
activity. referen= fugacity. partial molar fugacity. solute molality. molecular weight. pressure.
r” f m M P R T
activity coefficient. solvated cation diameter. average lifetime of solvent in a cation mordination shell. molal osmotic coefficient.
Y 0, ‘T Q” SUPERSCRIPTS:
!!
L V 0
molar volume of solution. liquid phase mole fraction. salt-free mole fraction (liquid). vapor phase mole fraction.
x i Y
liquid phase. vapor phase. salt-free state, or reference value.
SUBSCRIPTS: 2
salt specie. solvent species. nth equilibrium state. pseudo solvent.
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