4194
Ind. Eng. Chem. Res. 1996, 35, 4194-4202
Polyazeotropy in Binary Systems. 2. Association Effects Hugo Segura† and Jaime Wisniak* Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Antonio Aucejo, Juan B. Monto´ n, and Rosa Mun ˜ oz Departamento de Ingenierı´a Quı´mica, Facultad de Quı´mica, Universidad de Valencia, 46100 Burjassot, Valencia, Spain
Polyazeotropy in binary systems is a singular case of vapor-liquid equilibrium where at a given temperature or pressure various azeotropes of alternating deviations from ideality are observed. These deviations are generally caused by liquid-phase nonidealities, but in associated systems and even at low pressures, the vapor-phase nonidealities are remarkable. Associated systems are analyzed using chemical theory, where associated species are considered to be products of polymerization reactions; in those systems, azeotropy can be achieved without the usual requirement of equal composition in the phases. The purpose of this work is to show how the combination of nonidealities in both liquid and vapor phases can induce polyazeotropy and to establish its thermodynamical conditions. Introduction Polyazeotropy in binary systems, i.e., the condition for which more than one stable azeotrope exists at a given temperature or pressure, is a singular case of vapor-liquid equilibrium (VLE). Experimental cases of double azeotropy in organic systems have been reported in the literature for the four systems benzenehexafluorobenzene (Gaw and Swinton, 1968), diethylamine-methanol (Sristava and Smith, 1985), 1,2-butylene oxide-methyl acetate (Leu and Robinson, 1991), and ethanoic acid-2-methylpropyl ethanoate (Christensen and Olson, 1992; Burguet et al., 1996). Lloyd and Wyatt (1955) have reported double azeotropy in the electrolytic system water-dinitrogen pentoxide. No experimental system with more than two azeotropes has been reported in the literature. Several authors have discussed the possibility of polyazeotropy using classical thermodynamics. Guminski (1958) developed the necessary thermodynamical conditions for tangent azeotropy, a limiting case of polyazeotropy where multiple azeotropes yield the same compositional point. This type of azeotropy is characterized by an inflection point in the VLE bubble-point curves. Above (or below) this point, azeotropic behavior disappears, in analogy to the critical temperatures of liquid-liquid equilibrium (LLE), as shown by Wisniak et al. (1996). Van Konynenburg and Scott (1980) have used the van der Waals fluid theory to analyze the possibility of predicting polyazeotropy and show that cubic equations of state are capable of predicting polyazeotropic behavior, particularly in the case of binary VLE. Christensen and Olson (1992) explained the azeotropic behavior observed in the system ethanoic acid-2-methylpropyl ethanoate taking into account the dimerization of ethanoic acid in the vapor phase. In their contribution, they postulated that two kinds of polyazeotropic systems were possible, those induced by the nonideality of the liquid phase and those induced by the nonideality in the vapor phase. More recently, Wisniak et al. (1996) made an in-depth theoretical analysis of polyazeotropy in nonassociating systems and † Postdoctoral Fellow. Permanent address: Departamento de Ingenierı´a Quı´mica, Universidad de Concepcio´n, Concepcio´n, Chile.
S0888-5885(96)00201-1 CCC: $12.00
Table 1. Necessary and Sufficient Conditions for Polyazeotropy (Nonelectrolyte, Nonassociating Binary Systems)a Necessary Conditions (1) G ˜ E inflects in the range 0 < x1 < 1 (2) ln R12 shows stationary points and zeros for integral stable compositions in the range 0 < x1 < 1 Sufficient Conditions (1) Determining the compositions x*1 for which
( ) ∂2G ˜E ∂(x*1)2
)0
T,P
(A) An odd number of azeotropes may exist if
γ∞1 >
Psat 2 > sat
P1
Psat 1 1 2 ∞ or γ < < ∞ 1 γ∞2 Psat γ2 1
(B) An even number of azeotropes may exist when for some x*1
(ln R12)x1)x1* > 0 with γ∞1
sat ∞ γ2 P1
or
(ln R12)x1)x1* < 0 with γ∞1 >
Psat 2 Psat 1
(2) Differential stability condition
( ) ( ) ∂2G ˜L ∂x12
≈
P,T
∂2G ˜E ∂x12
+ P,T
RT g0 x1x2
for some composition at which ln R12 ) 0 a
Wisniak et al. (1996).
developed the necessary and sufficient conditions of polyazeotropy, summarized in Table 1. They also analyzed the temperature and pressure evolution of polyazeotropic behavior in nonelectrolyte and nonassociating binary systems. A clear distinction was made between polyazeotropy and heteroazeotropy, with polyazeotropy being interpreted as an equilibrium phenomenon involving two stable phases. According to Wisniak et al., polyazeotropic behavior, in general, cannot be predicted a priori from infinite dilution data, as is the case for single azeotropy; some additional knowledge is needed about the VLE of the system, in particular, the geometric behavior of the excess Gibbs energy function with composition. Wisniak et al. also discussed the possibility of representation of various © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4195
azeotropes using current activity coefficient models. The theoretical equations developed predicted well the behavior of the two azeotropes present in the hexafluorobenzene-benzene system and the conditions for their disappearance, as observed experimentally by Ewing et al. (1984). The experimental evidence thus far accumulated indicates that polyazeotropy only occurs when the vapor pressures of the constituents are similar. Wisniak et al. (1996) concluded that if the necessary conditions for polyazeotropy are met, then high vapor pressure differences will tend to give heteroazeotropy instead of polyazeotropy. It is the purpose of this contribution to extend the work of Wisniak et al. to the case of associated solutions, using chemical theory (Prausnitz et al., 1986). Theory According to Prigogine and Defay (1954), an azeotropic transformation occurs when in a multiphase closed system the mass of one of the phases is increased by the contribution of the other phases and no changes in the composition of any of them are observed. This definition is general and particularly interesting because no reference is being made, to the fact usually associated with azeotropy, that the phases have the same composition. Although the last condition is true in systems where nonideality stems only from physical interactions, it can be shown that if chemical interactions are present, azeotropy can be achieved with phases of different composition. In order to illustrate this important point, let us assume the vapor-liquid equilibrium (VLE) of a binary mixture, A(1) + B(2), where component B may dimerize in the vapor and in the liquid phase according to KV(KL)
2B S B2
(1)
In eq 1, KV and KL are dimerization contants in the vapor and the liquid phase, respectively. This example of association, where chemical equilibrium is assumed, is frequently found in treatment of the VLE of mixtures that contain carboxylic acids with nonassociating compounds (Prausnitz et al., 1986). It should be noted that the chemical reaction in eq 1 increases the number of chemical species present in the system, instead of a binary system; the assumed association behavior originates a ternary system of monomer, dimer, and inert (also called the true species). In chemical theory, the material balance of all species is described in terms of apparent compositions, which group into single monomeric species the polymers and/or complexes formed by an associating molecule; in the specific case that we treat here, the introduction of apparent compositions permits us to give a pseudobinary approach to the associated system. The use of apparent compositions is convenient because these are the compositions that, in practice, are measured for associated species. Figure 1 illustrates the results for a single azeotropic system under ideal dimerizing conditions using the equations developed in Appendix A. Table 2 lists the true and apparent compositions of the azeotropic point, the saturation pressure of pure species, and the equilibrium dimerization constants which have been used for drawing Figure 1. It is clear from Table 2 that the true compositions are not the same in the azeotropic point. From an operational viewpoint, the chemical and
Figure 1. Azeotropic condition in a dimerizing system. Table 2. Azeotropic Compositions and Properties for the System Shown in Figure 1 true compositions
app compositions
compd
xi
yi
x0i
y0i
A B B2
0.9233 0.0553 0.0214
0.9154 0.0720 0.0123
0.9040 0.0960
0.9040 0.0960
PA, kPa
P0B, kPa
KV, kPa-1
KL
120.3
98.5
2.0 × 10-2
7.0
physical azeotropes are not distinguishable; both of them limit the distillation process in the same manner because the dimer cannot be separated from the monomer by physical operation. The relative effect of association in both phases may be obtained using the simple mechanism of association discussed above in which the mixture of monomer, dimer, and inert is assumed to form ideal ternary solutions in the vapor and in the liquid phase. In Figure 2, the total pressure exerted by an associating system at constant temperature has been plotted for different values of the ratio KV/KL. The curves have been drawn assuming that the pure components have the same vapor pressure (100 kPa). When the ratio among the equilibrium constants is zero, that is, when the dimerization of B occurs only in the liquid phase, the expected positive deviation from ideality (Prausnitz et al., 1986) is obtained with a positive azeotrope. A larger value of the ratio KV/KL produces a negative azeotrope, and dimerization in the vapor phase is dominant. For an intermediate value of the ratio KV/KL, the notable case of an infinite number of azeotropes, not reported before, is obtained. The study of this rare case of polyazeotropy will be the matter of another publication; the important point to be learned here is that polyazeotropy can be achieved by considering only chemical contributions to the deviation of solutions from ideality, in spite of the fact that the association mechanism assumed here is a very simplified picture of the physical situation, fundamentally in the liquid phase, for reasons that will be explained latter. It should also be noted that depending on which phase the association regime is dominant, different deviations from ideality can be obtained. It
4196 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
Figure 2. Evolution of azeotrope for different degrees of dimerization in the vapor phase in an isothermal VLE system (KL ) 7; units of KV, bar-1).
is well-known that equilibrium constants depend only on temperature; hence, for a particular temperature, a combination that simultaneously gives positive and negative deviations from ideality, as observed in polyazeotropy, is certainly possible. As is expected, the combination of physical and chemical effects of real solutions can give intermediate cases of polyazeotropy; one of these cases seems to be the two azeotropes observed for the 2-methylpropyl ethanoate-ethanoic acid system. Evidence of Association in the 2-Methylpropyl Ethanoate-Ethanoic Acid System. The mechanism of association of ethanoic acid is well-known (Seaton, 1993). In the vapor phase, ethanoic acid exists as a monomer, dimer, and trimer; at pressures not significantly above atmospheric, a negligible amount of trimer is formed (Tsonopoulos and Prausnitz, 1970). The dimerization constant is reported in a recent work of Fu et al. (1995)
Figure 3. Effective fugacity coefficients for the 2-methylpropyl ethanoate-ethanoic acid system at 390.15 K. Calculated from the experimental data of Burguet et al. (1996).
chemical theory and the ideal gas approximation in our calculations due to the availability of eq 2. Generally, the mechanism of association in the liquid phase is complex due to the possibility of simultaneous interaction among molecules. As pointed by Prausnitz et al. (1986), associated solutions can be explained in term of self-association, formation of complexes, and chemical interactions between polymers and complexes. Chemical theory is usually a very simplified picture of the behavior of real solutions due to the lack of experimental information for the equilibrium constants and, even more, due to the lack of evidence on the nature of the chemical species present. Seaton (1993) quotes experimental results that suggest the following mechanism for the equilibrium dimerization of ethanoic acid in the liquid phase: O CH3C
O OH
O H O OH
CH3C OH
7290 ln K (atm ) ) -17.362 + T V
-1
(2)
According to chemical theory (Prausnitz et al., 1986), the effective fugacity coefficient, φˆ 0i , of the apparent species can be directly calculated, when ideal gas behavior is assumed, from the following relation:
φˆ 0i )
yi y0i
(3)
In Figure 3, the effective fugacity coefficients of the 2-methylpropyl ethanoate-ethanoic acid system obtained from chemical theory, assuming ideal gas mixture and validity of eq 2, are compared with those obtained by the method of Hayden and O’Connell (1975). Excellent agreement is observed, and it can be concluded that the physical effects in the vapor phase can be neglected. The chemical effects in the vapor phase are considerable and cannot be neglected in the computation of activity coefficients. We prefer to retain the
CCH3
CH3C O H O
O
CH3C
This mechanism shows some differences if compared to the mechanism proposed by Prausnitz et al. (1986) for dimerization in the vapor phase; in the latter, the formation of the intermediate open-chain dimer, which can give higher polymers, is ignored. This assumption seems to be confirmed by the fact that at low pressures, dimers are the more abundant chemical species. On the other hand, the evidence of an open-chain dimer in the liquid phase suggests the possibility that higher degrees of polymerization may be present. Seaton (1993) states that the degree of association is 1.88 at 273 K and 1.45 at 373 K. As a first approximation, it will be supposed that association proceeds the formation of dimer only, in an ideal solution with the monomer. From the degree of association the following liquid-phase dimerization constant is estimated:
ln KL ) -7.400 +
3177 T
(4)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4197
Figure 5. Relative volatility for single and multiple azeotropy. Circles indicate azeotropic compositions.
Figure 4. Comparison between chemical theory and experimental data at (a) 353.15 K and (b) 390.15 K. 2-Methylpropyl ethanoate (1)-ethanoic acid (2) system. Data of Burguet et al. (1996).
Using the data for chemical equilibrium constants in eqs 2 and 4, the VLE diagram of the 2-methylpropyl ethanoate (1)-ethanoic acid (2) system has been calculated and compared with the experimental data at 353.15 and 390.15 K, as reported by Burguet et al. (1996). As shown in Figure 4, in none of these cases is the agreement is satisfactory, but the inflectant behavior for the bubble pressure curve in Figure 4a (353.15 K) is remarkable, because it shows some evidence of simultaneous positive and negative deviations from ideality, a behavior observed in polyazeotropy. Seaton (1993) has estimated the physical properties of the pure monomer and the pure dimer of ethanoic acid. Large differences are observed in the critical properties; for example, the critical volume of the dimer is twice that of the monomer. Furthermore, 2-methylpropyl ethanoate and ethanoic acid are polar compounds. Thus, physical interactions of weak physical forces, size, and form may be expected in the liquid
phase. The need for physical corrections is shown in Figure 5, where the excess Gibbs energy has been calculated from experimental data (Burguet et al., 1996) considering dimerization in the vapor phase. In addition, Burguet et al. have obtained satisfactory correlation of their data using excess models that include physical and chemical contributions for the liquid phase, such as the model proposed by Fu et al. (1995) and Flory’s treatment for associated solutions discussed in detail by Prausnitz et al. (1996). The solid line in the figure in question shows another important fact. As demonstrated by Wisniak et al. (1996), a necessary condition for polyazeotropy in nonassociated systems is the inflection of the liquid-phase excess Gibbs energy function between neighboring azeotropes. This inflection does not appear in the data of the 2-methylpropyl ethanoate-ethanoic acid system; hence, it is clear that the conditions proposed by Wisniak et al. (1996) must be generalized to include associated VLE. Necessary and Sufficient Conditions of Polyazeotropy in Associated Systems. In order to consider the influence of physical effects in the liquid phase, the phase equilibrium relations are written in a more rigorous form (Smith and Van Ness, 1987):
yiP ) xiγiPsat i
(i ) A, B, B2)
(5)
where γi is the activity coefficient of the true species and the vapor phase is considered to be ideal for a mixture of monomers, dimer, and associated species. Equation 5 is of limited usefulness because activity coefficients of true species are usually not known. Following the reasoning of Marek and Standart (1954), eq 5 is more conveniently written in terms of apparent compositions by making transformations that include material balances and a Lewis-Randall reference state
y0i φˆ 0i P ) x0i γ0i f 0i
(i ) A, B)
(6)
where φˆ 0i is the effective fugacity coefficient of apparent species and f 0i is the apparent pure compound saturation fugacity, defined as follows:
4198 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
φˆ 0i )
yi y0i
f 0i ) lim[φˆ 0i P] ) P0,sat φ0i i
(7)
yif1
In eq 7, P0,sat is the observed vapor pressure of the i “pure” associating compound. Equations 6 and 7 are usually used in the analysis of associating systems; Tamir and Wisniak (1976) applied them for the reduction and analysis of VLE data of carboxylic acids. Depending on the mechanism of association in the vapor phase, the apparent fugacity coefficients can have different and very complex functionalities, but, in general, they will depend on the apparent vapor-phase composition and on the temperature. The advantage of eq 6 is that it has a full γ-φ format and does not depend on true compositions. If necessary, physical corrections to the vapor-phase fugacity coefficients can be introduced using virial equations of state, such as the correlation of Hayden and O’Connell (1975), which handle association effects in the vapor phase. Activity coefficients of apparent species in the liquid phase can be modeled combining chemical theory with physical interactions (Prausnitz et al., 1986) or using standard excess energy models or some refinements of them, as shown in the paper of Fu et al. (1995), where the UNIQUAC model of Abrams and Prausnitz (1975) has been modified to include phase association. Equation 6 expresses the equilibrium condition of isofugacity between phases. From them, the following relations for the apparent Gibbs energy are obtained:
+ ∑x0i Ci(P,T) ) ∑i x0i ln ˆf 0,L i i RT∑x0i ln(x0i γ0i f 0i ) + ∑x0i Ci(P,T) i i
G ˜ 0,L ) RT
+ ∑y0i Ci(P,T) ) ∑i y0i ln ˆf 0,V i i RT∑y0i ln(y0i φˆ 0i P) + ∑y0i Ci(P,T) i i
(8)
temperature and pressure, when noncritical azeotropy is being analyzed; both second derivatives must be positive as a requirement of stability. Equations 10 and 11 express the local concavity of the bubble pressure curve and of the bubble temperature curve, respectively. Comparison of eqs 10 and 11 indicates that when partial volumes and enthalpies are positive, the azeotrope concavity in a T-x diagram has a sign opposite to the sign of the concavity on the same azeotrope in a P-x diagram, as expected. In addition, it should be noted that the sign of the concavity in the azeotropic composition is given by the difference
∆)
|
[( ) ( )] [( ) ( )]
1 ∂2G ˜L ∂2P ) 2 ∂x12 T x1∆V h 1 + x2∆V h 2 ∂x2 2
∂G ˜ ∂y22
2
|
2
P,T
(∂2G ˜ V/∂y22)P,T
(10)
( ) ( )
∂G ˜ ∂y22
( ) ∂2G ˜ 0,V ∂(y01)2
P,T
2
L
(∂2G ˜ V/∂y22)P,T
(
) RT
T,P
T,P
)
∂2 ln[φˆ 01/φˆ 02] ∂(y01)2
+
both of them are positive quantities in low-pressure VLE. Furthermore, eqs 10 and 11 contain the second compositional derivative of the Gibbs energy at constant
RT x01x02
(14)
RT ) y01y02
( )
T,P
∂2G ˜ 0,E V
[( ) ( )] [( ) ( )] ∂2G ˜ 0,E V ∂(y02)2
(11)
where ∆V hi ) V h Vi - V h Li is the partial change of volume V L and ∆H hi ) H hi - H h i is the partial change of enthalpy;
+
+
RT (15) y01y02
where G ˜ 0,E is the excess Gibbs energy of the vapor V phase. Subtracting eq 14 from eq 15 and applying the difference to an associating azeotrope point transforms eqs 10 and 11 into
2
(∂ G ˜ /∂x2 )P,T
T,P
∂(x01)2
∂(y01)2 T,P
P,T
V
∂2G ˜ 0,E L
)
On the other hand, the second compositional derivative of eq 9 is given by
|
-
(13)
where G ˜ 0,E L is the excess Gibbs energy expressed as a function of apparent liquid-phase compositions and Ci is a integration constant (usually known as the reference state). From eq 7, it is clear that the pure saturation fugacity, f 0i , does not depend on composition; hence,
∂2G ˜ 0,E 1 L ∂2P ) 0 2 0 0 0 0 0 2 ∂(x1) T x1∆V h 1 + x2∆V h 2 ∂(x2)
L
1 ∂G ˜ ∂T )2 ∂x12 P ∂x x1∆H h 1 + x2∆H h2 2 2
(∂2G ˜ L/∂x22)P,T
(12)
P,T
+ ∑x0i Ci(P,T) ) ∑i x0i ln ˆf 0,L i i 0 + RT x G ˜ 0,E ∑i i ln(x0i f 0i ) + ∑i x0i Ci(P,T) L
(9)
P,T
V
P,T
G ˜ 0,L ) RT
∂2 G ˜ 0,L ∂(x01)2
-
∂2G ˜V ∂y22
-
The analysis of polyazeotropy in associated systems starts from eq 12, expressed in terms of apparent Gibbs energy because the condition of equal compositions is observed for apparent compositions and not necessarily for true compositions, as has been demonstrated above. Equation 8 can be written as
G ˜ 0,V ) RT
Equations 8 and 9 will be used in the analysis of the azeotropic conditions presented below. According to Malesinki (1965), the following equations express the concavity of the equilibrium points in an azeotropic condition; these equations have been used by Wisniak et al. (1996) in their analysis of polyazeotropy in nonassociating systems:
( ) ( ) ∂2 G ˜L ∂x22
|
-
P,T 2 0,L
(∂ G ˜
P,T
(∂2G ˜ 0,V/∂(y02)2)P,T
∂2G ˜ 0,E ∂2T 1 L ) ∂(x01)2 P x01∆H h 01 + x02∆H h 02 ∂(x02)2
P,T 2 0,L
∂2 G ˜ 0,E V ∂(y02)2
(∂ G ˜
P,T
/∂(x02)2)P,T
(16)
/∂(x02)2)P,T
(∂2G ˜ 0,V/∂(y02)2)P,T
(17)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4199
systems is then
τ(x01 ) 1) > 0 w τ(x01 ) 0) < 0 or
τ(x01 ) 1) < 0 w τ(x01 ) 0) > 0
(20)
The above condition can be written in terms of activity coefficients at infinite dilution, γ∞i , as
φ02 P0,sat 2
1 γ02
∞
> P0,sat φˆ 02 1
∞
w
γ0∞ 1
w
∞ γ01
φˆ 01 P0,sat 2
∞
P0,sat φ01 1
(21)
At low pressures, fugacity coefficients can be evaluated from chemical theory using only vapor association equilibrium constants. Equations 21 can be considered to be an extension of the criteria of Brandani (1974) to associated systems. As can be deduced from Figure 6, the possibility of prediction of polyazeotropic behavior using infinite dilution data is complicated by the fact that the logarithm of the relative volatility has the same sign in diluted compositions when the number of azeotropes is even. As suggested by Wisniak et al. (1996), for polyazeotropic cases, some additional VLE data of the particular system are needed in the midrange compositions. From Figure 6, it is also concluded that eq 19 does not change sign among consecutive azeotropes but changes sign if all the compositional range is considered. This observation is important because the composition in which eq 18 becomes zero can be used to determine the sign of eq 19. If changes of sign are observed, then polyazeotropy could be possible. In their analysis of the system 2-methylpropyl ethanoate-ethanoic acid, Christensen and Olson (1992) have concluded that a polyazeotropic condition can be achieved when the bubble pressure (or bubble temperature) curve shows two or more stationary points. They have illustrated this contention assuming an “artificial” 2methylpropyl ethanoate-ethanoic acid system, with a fictitious excess Gibbs energy given by a Margules-type equation 0 0 0 0 G ˜ 0,E L ) -8000(x1 - x2)x1x2
J (mol )
(22)
Christensen and Olson have used eq 22 to calculate the bubble pressure curve of the system at 390.15 K, correcting the vapor phase for nonideal behavior using the method of Hayden and O’Connell (1975). The resulting bubble pressure curve exhibited two maximums and two minimums from which Christensen and Olson concluded that at 390.15 K, the artificial 2-methylpropyl ethanoate-ethanoic acid system has four azeotropes. The conclusion obtained by Christensen and Olson will be true only if all the stationary points, particularly maximum pressures (or minimum temperatures), are stable or when these stationary points appear in experimental VLE data of good quality, because experimental data reflect stable states. It must
4200 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 3. Necessary and Sufficient Conditions for Polyazeotropy (Extension to Associated Binary Systems) Necessary Conditions 0 ˜ 0,E (1) G ˜ 0,E L - G V inflects in the range 0 < x1 < 1 (2) ln R12 shows stationary points and zeros for integral stable compositions in the range 0 < x01 < 1 Sufficient Conditions (1) Determining the compositions x0* 1 for which
( ) ( ) ∂2G ˜ 0,E L
∂2G ˜ 0,E V
∂(x01)2 T,P
-
∂(y01)2
)0 T,P
(A) An odd number of azeotropes may exist if ∞ γ01
P0,sat φˆ 01 2
γ2
P0,sat φˆ 02 1
or
∞
∞
γ01 >
P0,sat φˆ 01 2
∞
P0,sat φ01 1
w
P0,sat φ02 2
1 γ02
< ∞
P0,sat φˆ 02 1
(B) An even number of azeotropes may exist when for some ∞ γ01
(ln R12)x1)x10* > 0 with Figure 7.
be pointed that if a system presents an unstable maximum bubble pressure (or minimum bubble temperature) where positive deviation from ideality is observed, it will correspond to a compositional range where heteroazeotropy instead of local azeotropy is observed. Unstable bubble pressures can be easily obtained when VLE data are estimated using a liquidphase model like that given in eq 22. In order to distinguish heteroazeotropy from polyazeotropy, it is necessary to test the stability of the liquid phase. This test can be done by remembering that eq 14 yields a negative value for any unstable composition, in which case liquid-phase immiscibility will be observed. When this test is applied to eq 22, it is found that eq 14 becomes negative in the range 0.1 < x01 < 0.35. In other words, Christensen and Olson have erroneously qualified their artificial mixture as a tetraazeotropic system. From stability analysis, it is concluded that the system is heteroazeotropic and only one stable azeotrope exists, as shown in Figure 7. Hence, a general analysis of polyazeotropy must include a study of the stability of the system, as has been pointed by Wisniak et al. (1996). Considering all the arguments discussed in this section, we may summarize a suitable set of necessary and sufficient polyazeotropic conditions, as given in Table 3. Conclusions In this work, we have presented theoretical and experimental evidence that show that polyazeotropy can be promoted by association effects. This singular behavior appears when the deviations from ideality in the liquid and in the vapor phase allow simultaneous positive and negative deviations, in the ranges of pressure and temperature where the system is in equilibrium, and the vapor pressures of the components do not differ significantly. A suitable set of necessary and sufficient conditions for polyazeotropy, where association has been considered, has been developed. These conditions generalize the previous work of Wisniak et al. (1996). It is shown that the equilibrium phases present at every azeotropic point in an associat-
P0,sat φˆ 01 2
∞
P0,sat φˆ 02 1
∞
(2) Differential stability condition
( ) ( ) ∂2G ˜ 0,L ∂x12
˜
P,T
∂2G ˜ 0,E L ∂x12
+
P,T
RT g0 x01x02
for some composition at which ln R12 ) 0
ing system have equal apparent compositions and different true compositions. Equation 18 plays a key role in associated azeotropy. Mathematically, it expresses the difference of the second compositional derivatives between excess Gibbs energy functions corresponding to the liquid and to the vapor phase, conversely. As is known, all the nonidealities are reflected by the mathematical behavior of excess functions. It has been established that a necessary (but not sufficient) condition of polyazeotropy in associated systems, where nonidealities of the vapor phase must be considered, is that eq 18 becomes zero at some composition between neighboring azeotropes. This condition of polyazeotropy does not necessarily impose a inflectant behavior to any of the excess functions, as can be seen for the excess energy of the 2-methylpropyl ethanoateethanoic acid system in the liquid phase. When the vapor phase can be considered ideal and nonassociated, then the necessary condition of polyazeotropy reduces to an inflectant behavior for the liquid-phase excess Gibbs energy function. Single azeotropy of associated systems can be predicted a priori using infinite dilution data and an association constant in the vapor phase, but as in nonassociated systems, the prediction of polyazeotropy is not possible without midrange compositional information. Nomenclature aˆ ) activity f ) fugacity g ) dimensionless Gibbs energy (G ˜ /RT) G ) Gibbs energy H ) enthalpy K ) chemical equilibrium constant
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4201
number of moles of apparent species and must be taken to be zero for nonmonomeric species. On the basis of 1 mol,
n ) number of moles P ) absolute pressure R ) universal gas constant T ) absolute temperature V ) volume x, y ) compositions of the liquid and vapor phases
0 nV,0 A ) yA
Greek Symbols R12 ) relative volatility between components 1 and 2 ξ ) degree of advancement φ ) pure fugacity coefficient φˆ ) effective fugacity coefficient γ ) activity coefficient ∂ ) partial derivative ∆ ) difference operator τ ) function defined in eq 19
0 nV,0 B ) yB
y0A + y0B ) 1
where the compositions y0A and y0B are the apparent compositions. Combining eqs A.3 and A.4 yields the true compositions as a function of ξV and apparent concentrations
Subscripts L ) pertaining to the liquid phase V ) pertaining to the vapor phase
yA )
Superscripts 0 ) pertaining to an apparent quantity ∼ ) molar property ˆ ) pertaining to an effective property - ) partial property sat ) saturation property E ) excess property L ) pertaining to the liquid phase V ) pertaining to the vapor phase
yB )
K )
(aˆ VB)2
(A.1)
V
KV )
yB2P
ξ )
ξV 1 - ξV
2(1 + 4KVP)
(A.5)
xA )
xB )
nBV2 ) nBV,0 + ξV 2
In chemical theory, it is of interest to express the material balance in terms of apparent compositions, i.e., monomeric species. Complexes and polymers can be related stoichiometrically to them. In eq A.3, the yield of a given concentration of monomeric species to the associated one can be calculated by setting nV,0 equal i to zero for nonmonomeric species. Thus, nV,0 is the i
1 - ξL
1 - ξL ξL 1 - ξL
(A.7)
where x0A and x0B are the apparent liquid-phase compositions and ξL is the degree of advancement of the dimerization reaction in the liquid phase, given by a relation similar to eq A.6:
ξL ) (A.3)
x0A
x0B - 2ξL
xB 2 )
V nVB ) nV,0 B - 2ξ
(A.6)
Let us now suppose that component B also dimerizes in the liquid phase, assumed here as an ideal solution of monomer, dimer, and inert. By repeating the arguments used to obtain eqs A.5 and A.6, we get
(A.2)
The material balance of the vapor phase can be solved using the standard calculations of chemical equilibrium (Walas, 1985). Introducing the degree of advancement, ξV, the number of moles of each specie is given by
nVA ) nV,0 A
1 - ξV
1 + 4KVPy0B - x1 + 8KVPy0B - 4KVP(y0B)2
where aˆ Vi is the activity of species i and KV is the equilibrium constant of the dimerization reaction in the vapor phase. Assuming ideal gas behavior, eq A.1 simplifies to
y B2
1 - ξV
Replacing eqs A.5 in eq A.2 and solving the resultant quadratic ξV yields
In this appendix, the pertinent relations for the VLE of the dimerization given in eq 1 are developed. According to chemical theory, the reactions of association are assumed to be in equilibrium; thus,
aˆ BV2
y0A
y0B - 2ξV
yB2 )
Appendix A: Ideal Mechanism of Dimerization in the Vapor and Liquid Phases
V
(A.4)
1 + 4KLx0B - x1 + 8KLx0B - 4KL(x0B)2 2(1 + 4KL)
(A.8)
Phase equilibrium relations can be achieved from the true compositions given by eqs A.5 and A.7 and Raoult’s law
xiP ) xiPsat i
(i ) A, B, B2)
(A.9)
where Psat i is the vapor pressure of the true species. For the dimerizing compound, the saturation pressure can
4202 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
be calculated from the apparent vapor pressure P0,sat B (i.e., the saturation pressure observed for a mixture of monomers and dimers of pure apparent B) as follows:
Psat B
) lim xB0f1
(1 + x1 + 4KL) P0,sat B
yBP ) xB
1+
x1 +
(A.10)
4KVP0,sat B
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Received for review April 9, 1996 Revised manuscript received July 5, 1996 Accepted July 28, 1996X IE9602015
X Abstract published in Advance ACS Abstracts, October 1, 1996.
