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Ind. Eng. Chem. Res. 2001, 40, 902-907
GENERAL RESEARCH Solid-Liquid Equilibrium Thermodynamics: Checking Stability in Multiphase Systems Using the Gibbs Energy Function J. A. Reyes, J. A. Conesa,* A. Marcilla, and M. M. Olaya Department of Chemical Engineering, University of Alicante, Apartado 99, Alicante, Spain
A new procedure has been suggested and applied to the simultaneous correlation of the experimental data corresponding to all equilibrium regions in ternary systems involving a solid compound. The analysis of the form of the Gibbs energy function permits the validity of the parameters calculated with any particular model to be verified and can be used as a consistence criterion. 1. Introduction Many papers have been found in the literature in which the salting-out effect has been studied, some including some mistakes, as has been pointed out in other papers in which a systematic procedure for studying these types of systems experimentally has been suggested (i.e., Ruiz and Marcilla,1 Marcilla et al.,2-4 Olaya et al.5). Nevertheless, very few articles have been found in which some type of correlation has been carried out involving the solid phase. Generally, the papers dealing with any type of correlation involving solid compounds (either ionic or nonionic) consider only liquid-liquid equilibrium (i.e., Macedo et al.,6 Li et al.,7 Tang et al.,8 Govindarajan and Sabarathinam,9 and Escudero et al.10), using different equations to formulate the activities of the ions in the liquid phase. Very few papers (i.e., Annesini et al.11 and Marcilla et al.12,13) and a book (Prausnitz et al.14) have been found in which the solid phase (normally one pure component) is considered in the equilibrium and the solubilities are either correlated or predicted. In one of these papers,11 different sets of parameters for each liquid-liquid (LL), solidliquid (SL), and liquid-liquid-solid (LLS) equilibrium are reported for a ternary system, and the authors comment that it is not possible to correlate all of these equilibrium data with a single set of parameters. To the contrary, we believe that the parameters of a molecular thermodynamics model must be capable of representing all of the equilibrium data of a system, regardless of the aggregation state and the number of phases present. Thus, the goal of this paper is to show that it is possible to correlate simultaneously LL, SL, and LLS equilibria for ternary systems including an inorganic salt. To do that, all of the equilibrium regions for each system of five ternary systems selected have been correlated together. The NRTL model has been used to formulate the liquid-phase activity. It is obvious that the selection of the molecular thermodynamics model is not relevant for the objective of this paper. * Author to whom all correspondence should be addressed. Tel.: (34) 965 903 789. Fax: (34) 965 903 826. E-mail:
[email protected].
On the other hand, the application of topological concepts such as that of Wasylkiewicz et al.15 is necessary to avoid inconsistencies in the parameters, as has been discussed in a previous paper.12 These concepts have been considered to verfiry the validity of the parameters obtained. The analysis of the shape of the Gibbs energy surface in all of the equilibrium regions of the diagram permits the correct set of parameters to be selected. The GM surface is analyzed in the threedimensional space of compositions. 2. Thermodynamics of Solid-Liquid Equilibrium Gibbs showed that a necessary and sufficient condition for absolute stability in a mixture at fixed temperature, pressure, and overall composition is that the Gibbs energy (GM) surface be at no point below the plane tangent to the surface at a given overall composition. This stability condition has been widely used and explained in many references, more frequently applied to liquid-liquid equilibrium (i.e., Wasylkiewicz et al.15). In the case of liquid-liquid equilibrium, the analytical expression of the Gibbs energy surface is the same for both phases, as the two phases are liquid and exactly the same equation must be applied (let us call this M ). Obviously, this is not the case for equilibria Gliquid involving different state phases such as liquid-vapor M M and a Gliquid will be present) equilibrium (where a Gvapor M M and a Gliquid or solid-liquid equilibrium (where a Gsolid will be present). In these cases, different expressions of the Gibbs excess function can be applied for each phase, yielding two possible Gibbs energy surfaces, depending on the aggregation state of each phase. Obviously, a common reference state must be used for both phases involved. The application of the Gibbs principle leads in this case to the prediction of instability or phase splitting wherever both Gibbs energy surfaces (i.e., one for each phase) present Gibbs energies higher than those provided by the common tangent plane to the two surfaces. The GM curves for binary systems involving solidliquid equilibrium, the solid phase being one pure
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Ind. Eng. Chem. Res., Vol. 40, No. 3, 2001 903
component, were analyzed previously12 using a diagram M vs the mole fraction of solid component in the of Gliquid liquid. In that paper, the standard state was selected as the pure component in the liquid state at the temperature and pressure of the system. Consequently, the Gibbs energy of the pure solid was different from zero. In the case of a mixture of three components, the analysis of the GM curves must be done in threedimensional space. However, what should be the shape of the GM surface? Let us discuss this point using the triangular representation widely used in this field. Consider, in the first place, a ternary system formed by the liquid components 1 and 2 and the solid component 3, which presents two different regions, a one-liquid (1L) region and a one liquid in equilibrium with one solid (1L + 1S) region. Figure 1a shows the conventional equilibrium triangle in the upper part of the prism. The curve delimiting the two regions is that curve going from point A to point B, which contains all of the tangent points of the planes that are tangent to M surface and pass through the point S. The the Gliquid curved surface that starts in the three vertices of the base of the prism of Figure 1a presents a hypothetical M Gliquid surface that can be representative of such sysM tems (note that we have actually represented -Gliquid to obtain the best view). From the morphological point M M curve and the Gsolid curve must fulfill of view, the Gliquid the following conditions: (1) The point representing the M ) 0, because the pure solid (point S) is not at -Gliquid reference state is taken as the liquid component at the temperature and pressure of the system. (2) If, because M binaries 1-3 and 2-3, the of the shape of the Gliquid point S can be connected to points such as A or B by tangent lines, then the pure solid S is separated. In this case, a ruled surface must exist defining the 1L + 1S region in the ternary system, as shown in Figure 1a. The upper part of Figure 1a represents the projection on the conventional triangular diagram of such equiM librium regions. (3) The Gliquid surface in the region 1L should not present minima or curvature changes in such a way that the possibility of drawing a tangent plane joining any two points in the diagram giving a value of M surface energy lower than that provided by the Gliquid does not exist. Let us now consider a more complicated system: a ternary system with 2L, 2L + 1S, and 1L + 1S equilibrium regions. The diagram for such systems must have at least (and as an example) the following special features (see Figure 1b, representing the whole surface, and also panels c-e of Figure 1, representing each of M surface in the the possible regions): (1) The Gliquid region 2L (Figure 1c) presents different pairs of points with a common tangent plane. The connections of these pairs of points give the different tie lines. These pairs of points (i.e., points C and D in Figure 1b for the binary system 1-2) can be joined by tangent planes that present a lower value (more negative, i.e., more favored) of GM than the GM surface. This occurs at different overall compositions along the corresponding tie line between the tangent points (i.e., between the points C and D in the example considered). This is true from points C and D until points E and F (in Figure 1b) are reached. Figure 1c shows two tangent planes, with two tangent points each (i.e., R-β and γ-δ on the GM surface in the 2L region; the projection to the base of the tangent
lines would be the tie lines. (2) Any point on the GM surface pertaining to the region 1L has a tangent plane that is not tangent to any other point in the GM diagram. Figure 1d shows, as an example, two points of this region, R and β. Obviously, in this region, there is only one possibility for GM, i.e., that provided by the GM surface, and no phase splitting is possible. (3) In the 1L + 1S region (Figure 1e), the point S can be joined with different tangent planes to the surface that represents the liquid phase (such as points A and B in the binaries 3-1 and 3-2, respectively, in Figure 1b) in equilibrium with the solid component. For a point of overall composition such as that of point M in Figure 1e, the tangent plane connecting point S with point L has a lower GM value than the GM surface, so phase splitting (in 1L + 1S) is expected. (4) If a plane is found that passes through the point S and is tangent to two M surface, a splitting 2L + 1S will be points of the Gliquid M found, as the G value corresponding to any composition within the projection of such triangle (connecting points E, F, and S in Figure 1b) is lower in the plane defined by S and the two tangent points than that provided by the GM surface. (5) Any equilibrium point in any region of two phases (2L or 1L + 1S) is always defined by the two points of contact of a common tangent plane with the GM surface or the tangent point and S. The line joining these points has a lower energy than that of the M surface and corresponds to a tie line. Gliquid 3. How to Use This Information to Calculate Solid-liquid Equilibrium In this section, the thermodynamic background that will allow for the equilibrium to be calculated correctly is presented (for a more complete description, see Marcilla et al.12). Let us consider the problem of determining the constants for the NRTL model to formulate the liquidphase activity (any other model, UNIQUAC, Margules, Van Laar, any modification of the Wilson equation, etc., could be used for this purpose, and the election of the model is not relevant for the objective of the present paper) in a case involving liquid-solid equilibrium for a ternary system in which the solid phase is one pure component (i.e., component 3). In other words, let us obtain the constants of any of these models from equilibrium data including the solubility of the solid. It must be noted that the ionic character of the solid is irrelevant as far as our example is concerned, and we have considered, to simplify the calculations, that the solid can be treated as a nonionic molecular species. In our case, the Gibbs energy for the mixing for 1 mol of a global mixture is
GM RT
)
M Gsolid
µS
c
∑
xLi
µLi
)s + (1 - s) ) RT RT i)1 RT ∆hf ∆Cp Tf Tf ∆Cp + -1 ln s RTf R T R T RT
[(
+
M Gliquid
)( )
(1 - s)
c
( )]
xLi ln(γLi xLi ) ∑ i)1
(1)
where s is the ratio (moles of solid phase)/(moles of mixture), (1 - s) is the ratio (moles of liquid phase)/ (moles of mixture), µS and µLi are the chemical poten-
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Figure 1. Gibbs energy surfaces for two different systems.
tials of the pure solid (deduced using a thermodynamic cycle, i.e., Prausnitz et al.14) and of component i in the liquid phase, respectively, using the same reference state, namely, the pure liquid at the same temperature and pressure of the system. xi represents the mole fraction, and the subindex i represents the component.
The upper index S refers to the solid phase and L to the liquid one. Equation 1 solves the problem, and for any global mixture that separates into a solid and a liquid phases, a minimum in GM with respect to s must exist. The composition of such a minimum, if it exists, does not
Ind. Eng. Chem. Res., Vol. 40, No. 3, 2001 905 Table 1. Aij and OFa,b for the System Water (1) + Butanol (2) + NaCl (3) at 25 °C j
i a
1 2 3
1
2
3
-137.52 -1238.6
1327.5 4182.0
905.65 243.56 -
OF(2L) ) 4.977 × 10-3. b OF(1L + 1S) ) 2.228 × 10-5.
Table 2. Aij and OFa,b for the System Water (1) + Acetone (2) + NaCl (3) at 25 °C j
i a
1 2 3
1
2
3
-255.75 -1026.36
1037.0 5838.5
305.32 -43.22 -
OF(2L)) 1.508 × 10-5. b OF(1L + 1S) ) 2.665 × 10-4.
Table 3. Aij and for the System Water (1) + Ethanol (2) + NaCl (3) at 25 °C OFa,b
Figure 3. Equilibrium data for the system water (1) + acetone (2) + NaCl (3) at 25 °C. Table 5. Aij and OFa,b for the System Water (1) + 1-Butanol (2) + Potassium Benzylpenicillin (3) at 20 °C j
j
i a
1 2 3
1
2
3
-389.32 -849.12
10.386 713.21
-125.51 -450.79 -
i a
1 2 3
1
2
3
304.94 -222.21
596.74 -106.60
-2748.87 663.09 -
OF(2L)) 3.519 × 10-3. b OF(1L + 1S) ) 1.018 × 10-2.
10-5.
OF(2L)) -. OF(1L + 1S) ) 5.153 × b
Table 4. Aij and OFa,b for the System Ethanol (1) + 1-Butanol (2) + NaCl (3) at 25 °C j
i a
1 2 3
1
2
3
24.02 -424.48
80.98 -289.2
941.5 902.13 -
OF(2L)) -. b OF(1L+1S) ) 1.784 × 10-9.
Figure 4. Equilibrium data for the system water (1) + ethanol (2) + NaCl (3) at 25 °C.
Figure 2. Equilibrium data for the system water (1) + 1-butanol (2) + NaCl (3) at 25 °C.
depend on the composition of the initial mixture, although the actual value of GM, logically, varies with the global composition. The application of this criterion has been explained elsewhere.12,13 4. Correlation of Five Experimental Systems Apart from illustrating some topological properties of the LLS equilibrium, the main goal of the present work
is to demonstrate that it is possible to simultaneously correlate the LL, SL, and LLS equilibria with any phase-equilibrium thermodynamics model. Experimental data for the following five ternary systems involving a solid phase have been correlated using the suggested procedure (experimental data from Marcilla et al.2 for systems 1, 3, and 4; from Olaya et al.5 for system 2; and from Annesini et al.11 for system 5): System 1swater + 1-butanol + NaCl (2L, 2L + 1S, and 1L + 1S) at 25 °C. System 2swater + acetone + NaCl (2L, 2L + 1S, and 1L + 1S) at 25 °C. System 3swater + ethanol + NaCl (1L + 1S) at 25 °C. System 4sethanol + 1-butanol + NaCl (1L + 1S) at 25 °C. System 5swater + 1-butanol + potassium benzylpenicillin (2L and 1L + 1S) at 20 °C. The possible ionic dissociation of any of the molecules involved has not been considered, as it is irrelevant for
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Figure 5. Equilibrium data for the system ethanol (1) + 1-butanol (2) + NaCl (3) at 25 °C.
the aim of this paper. The NRTL model has been used to formulate the activity coefficients of all of the compounds involved, and all of the equilibrium regions for each system have been simultaneously correlated with a single set of parameters. The parameter Rij of the NRTL model is, initially, a fitting parameter. Nevertheless, a good correlation is obtained if the values recommended by Renon and Praustniz16 are used. In the present work, the value Rij ) 0.2, frequently used in the literature, has been selected. The objective function used in the optimization (eq 4) is the result of two contributions: one corresponding to the two-liquid zone [OF(2L)] and the other one corresponding to the zone in which a solid is also present in the equilibrium [OF(1L + 1S)]. The corresponding equations are n
OF(2L) )
3
{[((Xi)k,or)exp - ((Xi)k,or)cal]2 + ∑ ∑ k)1i)1 [((Xi)k,ac)exp - ((Xi)k,ac)cal]2} (2) n
OF(1L + 1S) )
3
{[((Xi)k,L)exp - ((Xi)k,L)cal]2} ∑ ∑ k)1i)1
OF ) OF(2L) + OF(1L + 1S)
(3) (4)
The zone corresponding to the equilibrium between two liquid phases has been solved following the procedure described by Gomis,17 which involves the resolution of the system of equations that results when the activities of the three components in the two phases are equalized. For this purpose, the concentrations that satisfy the following equation are calculated and sub-
stituted into eq 2 in order to calculate the OF due to this equilibrium. n
OF(a) )
3
[(ai)k,or - (ai)k,ac]2 < 10-6 ∑ ∑ k)1i)1
(5)
The calculation corresponding to the solid-liquid equilibrium implies some modifications. In this case, in addition to the isoactivity criterion, the stability criterion has been included, i.e., the minimization of eq 1 with respect to s. The optimization of the parameters of the model (Aij ) binary interaction parameters) has been done using the Simplex method with the objective function defined in eq 4. Tables 1-4 show the parameters Aij and the OF obtained, and Figures 2-6 present the corresponding graphs for the systems considered, where both experimental and calculated points are plotted. It can be observed that the fitting of all of the regions using the same set of Aij is possible and yields very good results. In the case of the ternary system water + acetone + NaCl (Figure 3), the fit of the data corresponding to the region 1L + 1S (both aqueous and organic) is very good. In this case, an additional difficulty arises, as it was difficult to fit the data when passing from the region 1L + 1S to the regions 2L + 1S M surface and 2L. This is due to the fact that the Gliquid presents a very low curvature in that region, and so, the location of the minimum in the Gibbs energy function corresponding to the aqueous phase in the tie triangle is somewhat imprecise. Table 5 shows the parameters and the corresponding objective functions obtained when the different equilib-
Ind. Eng. Chem. Res., Vol. 40, No. 3, 2001 907
Literature Cited
Figure 6. Equilibrium data for the system water (1) + 1-butanol (2) + potassium benzylpenicillin (3) at 20 °C.
rium regions for the ternary system water + 1-butanol + potassium benzylpenicillin (KBP) at 20 °C (Annesini et al.)11 are fitted simultaneously. In Figure 6, experimental and calculated equilibrium data are plotted. In the original work,11 the authors failed in fitting this ternary system using the same set of parameters, and they commented that it was not possible to describe both liquid-liquid and solid-liquid equilibrium results with the same set of parameters. Therefore, the analysis of the Gibbs energy function and the application of the suggested procedure allows us to describe all of the equilibrium regions with the same set of parameters, being complementary to the isoactivity criterion. 6. Conclusions The analysis of the shape of the Gibbs energy function and the application of the suggested procedure allows us to obtain improved parameters that can describe all of the equilibrium regions, being complementary to the isoactivity criterion. A systematic and complete analysis of the entire region of GM is necessary to ensure a correct interpretation of the phase behavior and to validate the binary parameters in the whole system. 7. Notation Aij ) Binary interaction parameters fi ) Fugacity of a component γ ) Activity coefficient x ) Molar composition f 0 ) Fugacity in the standard state GM ) Gibbs energy of mixing (kJ/mol) µi ) Chemical potential (kJ/mol) µS ) Chemical potential of the pure solid (kJ/mol) s ) (moles of solid phase)/(moles of mixture) ∆hf ) melting heat (kJ/mol) ∆Cp ) difference between heat capacity of the solid and that of the liquid phase at the melting point Tf ) melting point
(1) Ruiz, F.; Marcilla, A. Some Comments on the Study of the Salting Out Effect in Liquid-Liquid Equilibrium. Fluid Phase Equilib. 1993, 89, 387-395. (2) Marcilla, A.; Ruiz, F.; Olaya, M. M. Liquid-Liquid-Solid Equilibria of the Quaternary System Water-Ethanol-1-ButanolSodium Chloride at 25 °C. Fluid Phase Equilib. 1995, 105, 7191. (3) Marcilla, A.; Ruiz, F.; Garcia, A. N. Liquid-Liquid-Solid Equilibrium of the Quaternary System Water-Ethanol-AcetoneSodium Chloride at 25 °C. Fluid Phase Equilib. 1995, 112, 273289. (4) Marcilla, A.; Conesa, J. A.; Olaya, M. M. Salt Effect in the Quaternary System Water + Ethanol + 1-Butanol + Potassium Chloride at 25 °C. J. Chem. Eng. Data 1997, 42 (5), 858-864. (5) Olaya, M. M.; Garcia, A. N.; Marcilla, A. Liquid-LiquidSolid Equilibrium of the Quaternary System Water-AcetoneButanol-Sodium Chloride at 25 °C. J. Chem. Eng. Data 1996, 41, 910-917. (6) Macedo, E. A.; Skovborg, P.; Rasmussen, P. Calculation of Phase Equilibria for Solutions of Strong Electrolytes in SolventWater Mixtures. Chem. Eng. Sci. 1990, 45 (4), 875-882. (7) Li, Z.; Tang, Y.; Li, Y. Salting effect in partially miscible systems of n-butanol-water and butanone-water. 1. Determination and correlation of liquid-liquid equilibrium data. Fluid Phase Equilib. 1995, 103, 143-153. (8) Tang, Y.; Li, Z.; Li, Y. Salting effect in partially miscible systems of n-butanol-water and butanone-water. 2. An extended Setschenow equation and its application. Fluid Phase Equilib. 1995, 105, 241-258. (9) Govindarajan, M.; Sabarathinam, P. L. Salt effect on liquidliquid equilibrium of the methyl isobutyl ketone-acetic acidwater system at 35 °C. Fluid Phase Equilib. 1995, 108, 269-292. (10) Escudero, I.; Cabezas, J. L.; Coca, J. Liquid-Liquid Equilibria for 2,3-Butanediol + Water + 4-(1-Methylpropyl)phenol + Toluene at 25 °C. J. Chem. Eng. Data 1996, 41, 2-5. (11) Annesini, M. C.; Gironi, F.; Marrelli, L. Phase Equilibria of Potassium Benzylpenicillin in Water + Butan-1-ol. J. Chem. Eng. Data 1994, 39, 502-505. (12) Marcilla, A.; Conesa, J. A.; Olaya, M. M. Comments on the Problematic Nature of the Calculation of Solid-Liquid Equilibrium. Fluid Phase Equilib. 1997, 135 (2), 169-175. (13) Marcilla, A.; Conesa, J. A.; Olaya, M. M. Thermodinamical consistent method for the calculation of solid-liquid-liquid equilibrium. Equifase 99, Congreso Iberoamericano de Ingenierı´a Quı´mica, Vigo, Spain, June 20-24, 1999. (14) Prausnitz, J. M.; Lichtentaler, R. N.; Gomes De Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1986. (15) Wasylkiewicz, S. W.; Sridhar, L. S.; Doherty, M. F.; Malone, M. F. Global Stability Analysis and Calculation of LiquidLiquid Equilibrium in Multicomponent Mixture. Ind. Eng. Chem. Res. 1996, 35, 1395-1408. (16) Renon, M.; Prausnitz, J. M. Local compositions in thermodynamic excess function for liquid mixtures. AIChE J. 1968, 12, 678. (17) Gomis, V. Equilibrio liquido-liquido en sistemas cuaternarios con dos pares de compuestos. Ph.D. Thesis, University of Alicante, Alicante, Spain, 1983.
Received for review April 26, 2000 Revised manuscript received October 5, 2000 Accepted October 25, 2000 IE000435V
