- 1. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 667 F rom the point of view of separation processes involving vapour-liquid systems (e.g. fractional distillation), it is important to know if an azeotrope exists in a particular system. If complete experimental vapour-liquid equilibrium data are available, this information is known explicitly. If, however, the data are embedded in a correlation for liquid-phase activity coefﬁcients (γi) (or, equivalently, for the excess molar Gibbs function (gE)), or if the data come from a model for γi or gE, this information is only known implicitly. Models may include a solution model such as a regular solution (Hildebrand et al., 1970), or a group-contribution method based on molecular structure, such as UNIFAC (Fredenslund et al., 1975), for use when no experimental or model parameters are available. The possible existence of an azeotrope may be considered for a system either at constant temperature (T) or at constant pressure (P). The former may be more important from a theoretical On Criteria for Occurrence of Azeotropes in Isothermal and Isobaric Binary Systems Ronald W. Missen Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College Street, Toronto, ON, Canada M5S 3E5 point of view, but since separation processes involving vapour- liquid systems operate essentially at constant pressure, this situation is more important from this practical point of view. Attention has been paid in the literature to the isothermal case, but we address both situations here. These considerations lead to the question of obtaining criteria for the occurrence of azeotropes in general. Some guidance in the literature for isothermal systems comes from the “Bancroft rule” (King, 1969) and from criteria developed by Prigogine and Defay (1954) for regular solutions. The former, which states, in effect, that if two species have the same saturation vapour pressure p* (at T′, say), then there is an azeotrope at T′, is self-evident, since any departure from ideality in such a case leads to an azeotrope. Comprehensive criteria for isothermal systems are given by Brandini (1974) in terms of activity coefﬁ- cients at inﬁnite dilution (γ∞). We include here a different Criteria are developed for the occurrence of azeotropes in binary nonelectrolyte systems for both isothermal and isobaric situations in terms of liquid-phase activity coefﬁcients at inﬁnite dilution (γ∞). In the case of isothermal systems at temperature T, for a positive azeotrope, γ∞ lvc ≥ p*mvc/ p*lvc, where lvc refers to the less volatile component, mvc to the more volatile component, and p* to saturation vapour pressure at T; for a negative azeotrope, γ∞ mvc ≤ p*lvc/p*mvc. In the case of isobaric systems at pressure P, for a minimum-boiling azeotrope, γ∞ lvc(TBmvc) ≥ P/p*lvc(TBmvc), where TB refers to the boiling point at P; for a maximum-boiling azeotrope, γ∞ mvc(TBlvc) ≤ P/p*mvc(TBlvc). The criteria are also given in terms of the parameters of selected correlations for the excess molar Gibbs function (gE). Examples of the use of the criteria are provided. Various methods that generate values of γ∞ can be used in conjunction with the criteria, for example, in screening procedures. Des critères basés sur les coefﬁcients d’activité en phase liquide pour une dilution inﬁnie (γ∞) ont été mis au point pour prédire l’apparition des azéotropes dans les systèmes non électrolytiques binaires dans des conditions isothermes et isobares. Dans le cas des systèmes isothermes à la température T, pour une azéotrope positive, γ∞ lvc ≥ p*mvc/p*lvc , où lvc réfère au composant moins volatil, mvc au composant plus volatil et p* à la pression de vapeur saturante à T; pour une azéotrope négative, γ∞ mvc ≤ p*lvc/p*mvc . Dans le cas de systèmes isobares à la pression P, pour une azéotrope d’ébullition minimum, γ∞ lvc(TBmvc) ≥ P/p*lvc(TBmvc) , où TB réfère au point d’ébullition à P; pour une azéotrope d’ébullition maximum, γ∞ mvc(TBlvc) ≤ P/p*mvc(TBlvc). On donne également les critères pour les paramètres de corrélations sélectionnées pour la fonction molaire d’excès de Gibbs (gE). Des exemples d’utilisation des critères sont donnés. Diverses méthodes donnant des valeurs de γ∞ peuvent être utilisées avec ces critères, par exemple dans les méthodes de tamisage. Keywords: isothermal azeotrope criteria; isobaric azeotrope criteria; binary systems
- 2. 668 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 development of these together with a graphical interpretation. We also give a corresponding new treatment for isobaric systems. Examples to illustrate application of the criteria developed are provided. ISOTHERMAL CRITERIA FOR OCCURRENCE OF AZEOTROPES Qualitative Considerations Figure 1 illustrates schematically P(x1) relationships for a ﬁxed- T plane intersecting a P-T-x1 saturated-liquid surface, where x1is mole fraction of component 1. It shows P(x1) for various degrees of departure from Raoult’s law (A, for an ideal system) in both positive and negative directions. Curves B, C, and D show increasing degrees of positive deviation: B for no azeotrope and D for a positive azeotrope (at the maximum, where x1 = y1 (not shown, the mole fraction in the coexisting vapour phase, whether actual or virtual)); C represents the boundary between these two types of behaviour, as the indication of incipient occurrence of a positive azeotrope with respect to increasing extent of nonideality. (For still greater positive deviation than in D, single liquid-phase stability eventually breaks down, and increasing degrees of partial miscibility set in, leading ultimately to complete immiscibility, the greatest degree of positive deviation from Raoult’s law.). Correspondingly, curves E and G (negative azeotrope at the minimum) illustrate increasing degrees of negative deviation, with F representing incipient negative azeotrope formation with respect to increasing nonide- ality. (There is no corresponding single-phase stability breakdown in this direction.) Assumptions We assume the following: (A1) The system is a binary nonelectrolyte system of components 1 and 2 that are completely miscible in the liquid phase. (A2) Information for γi or gE is for a saturated liquid phase, actually or potentially in equilibrium with a vapour phase. (A3) The γi or gE data refer to Raoult’s law ideality; that is, γi → 1 as xi → 1. (A4) The γi or gE data refer either to isothermal or to isobaric (case to follow) conditions. (A5) The dependence of γi or gE on P is negligible, but the dependence on T must be allowed for. (A6) The pressure is sufﬁciently low that the Poynting correc- tion (Sandler, 1999) is equal to one. Criteria in Terms of Activity Coefﬁcients at Inﬁnite Dilution The activity coefﬁcients at inﬁnite dilution are designated γ1 ∞ (at x1 = 0) and γ2 ∞ (at x2 = 0) for components 1 and 2, respectively. Our goal is to obtain criteria in terms of these quantities. In the absence of any fugacity corrections at low pressure (assumption (A6)), the total pressure at T is given by P = γ1(T,x1)x1p1* (T) + γ2(T,x1)x2p2* (T) (1) From Equation (1), on differentiation at constant T, we obtain ∂ ∂ = + ∂ ∂ − + ∂ ∂ P x p x p x p x p x1 1 1 1 1 1 1 2 2 2 2 2 1 * * * *γ γ γ γ (2) = 0 (at an azeotrope) (3) Consider incipient positive azeotrope formation with component 1 as the more volatile component (p1* > p2* at T). This occurs as x1 → 1 at C in Figure 1. In such a case, x2 → 0, γ1 → 1, γ2 → γ2 ∞, x1p1* ∂γ1/∂x1 = x1p1*γ1 ∂lnγ1/∂x1 → p1* ∂lnγ1/∂x1 → 0, and x2p2* ∂γ2/∂x1 → 0. From Equations (2) and (3), we conclude that a criterion for incipient formation of a positive azeotrope is γ2 ∞ = p1*/p2* ( > 1). Instead of component 1 being the more volatile component, an arbitrary designation may have component 2 in this role. The situation depicted in Figure 1 is then reversed (although not shown). Incipient azeotrope formation occurs as x1 → 0 or x2 → 1, in which case, γ1 → γ1 ∞ and γ2 → 1. From Equations (2) and (3), the criterion for an incipient azeotrope is γ1 ∞ = p2*/p1* ( > 1). In either case, if the component designa- tions are changed to more volatile component (mvc) and less volatile component (lvc), instead of components 1 and 2, the criterion for an incipient positive azeotrope can be written as one equation: p*2 P X1 F T CONSTANTT CONSTANTT CONST p*1 0 1 G E A B D C Figure 1. Schematic representation of P(x1) at ﬁxed T showing positive and negative deviations from Raoult’s law. A Raoult’s law B Positive deviation, no azeotrope C Positive deviation, incipient positive azeotrope D Positive deviation, positive azeotrope at maximum E Negative deviation, no azeotrope F Negative deviation, incipient negative azeotrope G Negative deviation, negative azeotrope at minimum
- 3. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 669 γlvc mvc lvc p p ∞ = * * criterion for incipient positive azeotrope formaation (4) As the degree of positive departure from ideality increases, in the sense of curve D in Figure 1, azeotrope formation occurs. In every case, γ∞ lvc increases relative to the value in Equation (4). Thus, a criterion for the existence of a positive azeotrope is γlvc mvc lvc p p ∞ ≥ * * criterion for positive azeotrope (5) If γ∞ lvc < p*mvc/p*lvc, no azeotrope exists. Similarly, consider incipient negative azeotrope formation as x1 → 0 at F in Figure 1, with component 1 as the mvc (or the reverse as x1 → 1, if component 2 is the mvc). Considerations based on Equations (2) and (3) corresponding to those above lead to the following criterion for an incipient negative azeotrope: γmvc lvc mvc p p ∞ = * * criterion for incipient negative azeotrope formaation (6) As the degree of negative departure from ideality increases in the sense of curve G in Figure 1, azeotrope formation occurs. In every case, γ∞ mvc decreases relative to the value in Equation (6). Table 1. Expressions for γ1 ∞ and γ2 ∞ from selected gE equations gE equation γ1 ∞ γ2 ∞ Redlich-Kister, Equation (8) Van Laar Wilson NRTL exp ( ) exp exp exp 1 1 1 1 0 0 1 RT g RT g A B k K k k k K k = = ∑ ∑− Λ 22 21 21 12 21 12 12 12 1 1 1 1exp( ) exp( ) exp exp - exp − − + ( ) Λ Λ Λ τ τ α τ τ 2 2 21 12 2 1+ ( ) τ α τexp - Table 2. Examples of use of criteria (5) and (7) for positive and negative azeotropes System T/ºC mvc dev. gE criterion (5) criterion (7) ref. corr.a γ∞ lvc P*mvc pos. γ∞ lvc P*lvc neg. P*lvc azeo. P*mvc azeo. n-C5H12 (1) 40 (1) + VL 4.07 1.52 Y ← N/A → (a) + 40 (1) + W 4.02 1.52 Y ← N/A → (a) propionaldehyde (2) 40 (1) + NRTL 3.85 1.52 Y ← N/A → (a) n-butyraldehyde (1) 45 (1) + VL 3.05 2.30 Y ← N/A → (a) + 45 (1) + W 3.12 2.30 Y ← N/A → (a) n-C7H16 (2) 45 (1) + NRTL 3.04 2.30 Y ← N/A → (a) isobutyraldehyde (1) 45 (1) + VL 2.72 3.41 N ← N/A → (a) + 45 (1) + W 2.76 3.41 N ← N/A → (a) n-C7H16 (2) 45 (1) + NRTL 2.73 3.41 N ← N/A → (a) DMA (1) + CH3OH (2) 40 (2) – RK ← N/A → 0.406 0.019 N (b) CH3F (1) + HCl (2) –90.8 (2) – RK ← N/A → 0.267 0.668 Y (c) C6F6(1) + C6H6(2) 60 (2) ± RK 1.20 1.04 Y 0.898 0.979 Y (d) (a) Eng and Sandler (1984); (b) Zielkiewicz (2003); (c) Senra et al. (2002); (d) Gaw and Swinton (1968) aRK (Redlich-Kister); VL (van Laar); W (Wilson) Figure 2. Regions of occurrence of positive and negative azeotropes in binary systems at ﬁxed T according to criteria (5) and (7). (note change of variables in lower left and upper right ﬁelds)
- 4. 670 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 a gE expression is available, but is outside the scope of this paper. As noted above, Brandini (1974) presents essentially the same criteria as (5) and (7) for isothermal systems, but expresses them in terms of components 1 and 2, and provides no graphical interpretation corresponding to Figure 2. The designation as components 1 and 2 requires four statements rather than the two in criteria (5) and (7). Criteria in Terms of Parameters of gE Correlations As noted above, the information required to apply the criteria for existence of azeotropes may be embedded in equations for the excess molar Gibbs function (gE). In Table 1, expressions are provided for γ1 ∞ and γ2 ∞ from four such equations: Redlich-Kister (1948), van Laar (Carlson and Colburn, 1942), Wilson (1964), and NRTL (Renon and Prausnitz, 1968). The van Laar results are readily apparent from the Carlson-Colburn forms of the activity coefﬁcients. The Redlich-Kister equation is an expansion in terms of composition that can be represented by: g x x g x xE k K k k = − = ∑1 2 0 1 2( ) (8) The results in Table 1 can be derived from Equation (8). Examples of Use of Criteria Table 2 shows randomly selected examples of the use of criteria (5) and (7) to indicate presence or absence of an azeotrope. The ﬁrst nine entries come from the work of Eng and Sandler (1984) on three aldehyde systems (column 1) at the T given in column 2. In each of these cases, component (1) is the more volatile component (mvc), and the deviation from ideality is positive. For each system, they compared the ﬁtting of several gE correla- tions (corr.), including van Laar (VL), Wilson (W) and NRTL equations. The use of criterion (5) is applicable here, and columns 6 to 8 give the results – azeotropes in the ﬁrst two systems, but no azeotrope in the third, as consistently shown by each gE correlation. The next two entries (Zielkiewicz, 2003; Senra et al., 2002) refer to negative deviation and the use of criterion (7) in conjunction with the RK equation. The last entry in Table 2 is for the rare case of a double azeotrope, in the hexaﬂuorobenzene + benzene system at 60°C (Gaw and Swinton, 1968). This is the situation in which both a positive and a negative azeotrope occur at a particular T. In this case, both criteria (5) and (7) are required, and they indicate the existence of the two azeotropes observed. In all 12 cases in Table 2, the results of application of criterion (5) or (7) agree with experimental results. ISOBARIC CRITERIA FOR OCCURRENCE OF AZEOTROPES Qualitative Considerations Figure 3 illustrates schematically T(x1) relationships for ﬁxed-P planes intersecting a P-T-x1 saturated-liquid surface. The behaviour labelled by C′, D′, F′, and G′ corresponds to that labelled by C, D, F, and G in Figure 1. Thus, D′ represents a minimum-boiling azeotrope at ﬁxed P, which corresponds to a positive azeotrope at ﬁxed T. Similarly, G′ represents a maximum- boiling azeotrope, which corresponds to a negative azeotrope. The behaviour indicated at C′ and F′ represents incipient formation (with respect to increasing extent of nonideality) of Thus, a criterion for the existence of a negative azeotrope is γmvc lvc mvc p p ∞ ≤ * * criterion for negative azeotrope (7) If γ∞ mvc > p*lvc/p*mvc, no azeotrope exists. Figure 2 shows the results expressed by the criteria of (5) and (7) in graphical form. In the lower left (closed) ﬁeld, the diagonal line representing incipient azeotrope formation divides the regions of negative azeotrope occurrence and no azeotrope according to (7). Similarly, the continuation of the diagonal line through the upper right (open-ended) ﬁeld divides the regions of positive azeotrope occurrence and no azeotrope according to (5). The horizontal axis at γ = 1 represents ideal (Raoult’s law) behaviour, and forms one border of each “no azeotrope” ﬁeld. The vertical axis at a pressure ratio = 1 represents the Bancroft rule, and forms one border of each azeotrope ﬁeld. In a model sense, Figure 2 indicates the extent of nonideality, in terms of γ∞ lvc or γ∞ mvc that can be tolerated to avoid the occurrence of an azeotrope for a given ratio of vapour pressures. In Figure 2, the indeﬁnite extension of criterion (5) for a positive azeotrope can be deceiving. Assumption (A1) notwith- standing, increasing nonideality, represented by increasing γ∞ lvc, leads eventually to a miscibility gap in the liquid phase. This cannot be predicted by criterion (5), and, if suspected, must be tested independently by a stability analysis. This is facilitated if TB1 C' D' X1 G' F' T P CONSTANTP CONSTANTP CONST TB2 0 1 Figure 3. Schematic representation of T(x1) at ﬁxed P for various situations: D′ minimum-boiling azeotrope C′ incipient formation of minimum-boiling azeotrope G′ maximum-boiling azeotrope F′ incipient formation of maximum-boiling azeotrope
- 5. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 671 minimum-boiling and maximum-boiling azeotropes, respec- tively. TB1 and TB2 are the boiling points of components 1 and 2, respectively, at P. The more volatile component (mvc) has the lower boiling point: TB1 < TB2. Criteria in Terms of Activity Coefﬁcients at Inﬁnite Dilution The activity coefﬁcient intercepts at x1 = 0 and x1 = 1, γ1 ∞ and γ2 ∞, respectively, again represent the activity coefﬁcients at inﬁnite dilution. Our goal is to obtain criteria for the occurrence of azeotropes in terms of these quantities. From Equation (1), which represents P(T,x1), with dP = 0, we form the derivative ∂ ∂ = − + ∂ ∂ − + ∂ ∂ ∂ T x p x p x p x p x x p1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 γ γ γ γ γ γ γ * * ln * * ln * lln * * ln *γ γ γ γ γ1 1 1 1 2 2 2 2 2 2 2 ∂ + + ∂ ∂ + T x dp dT x p T x dp dT (9) = 0 (at an azeotrope) (10) where (Smith and Missen, 1991) ∂ ∂ = − l T h RT i i E 2 nγ (11) where – hi E is the excess partial molar enthalpy of species i, and R is the gas constant. Corresponding to the isothermal case, consider ﬁrst incipient formation of a minimum-boiling azeotrope as x1 → 1 at C′ in Figure 3. Then, x2 → 0, T → TB1, γ1 → 1, γ2 → γ2 ∞ (TB1), – hi E → 0, and Equations (9) and (10) reduce to − − = ∞ p T T p T dp dT B1 B1 2 B1 2 1 1 *( ) ( ) *( ) * γ 0 (12) From Equation (12), since dp1*/dT ≠ 0, we obtain the following criterion for incipient formation of a minimum-boiling azeotrope with component 1 as the mvc: γ2 1 1 1 2 1 2 1 ∞ = =( ) *( ) *( ) *( ) T p T p T P p T B B B B (13) If component 2 is arbitrarily designated as the mvc, then the criterion for incipient formation becomes γ1 2 2 B2 1 B2 1 B2 ∞ = =( ) *( ) *( ) *( ) T p T p T P p T B (14) Equations (13) and (14) can be combined into one criterion by using designations mvc and lvc instead of 1 and 2: Table 4. Examples of use of criterion (18) for maximum-boiling azeotropes System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞ mvc P kPa ºC ºC corr.a (TBlvc) P*mvc(TBlvc) azeo ref. 1-butanol (1) 101.3 117.7 77.1 (2) VL 0.44 0.31 N (a) + 101.3 117.7 77.1 (2) W 0.40 0.31 N (a) 1-butylamine (2) 101.3 117.7 77.1 (2) NRTL 0.43 0.31 N (a) 2-butanol (1) 101.3 99.6 77.1 (2) VL 0.41 0.51 Y (b) + 101.3 99.6 77.1 (2) W 0.35 0.51 Y (b) 1-butylamine (2) 101.3 99.6 77.1 (2) NRTL 0.42 0.51 Y (b) CH3OH (1) + diethylamine (2) 97.3 63.5 53.8 (2) M3 0.37 0.72 Y (c) CH3OH (1) + 1-butylamine (2) 97.3 63.5 76.6 (1) M2 0.79 0.61 N (c) (a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967) aVL (van Laar); W (Wilson); M2, M3 (two-constant, three-constant Margules) Table 3. Examples of use of criterion (16) for minimum-boiling azeotropes System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞ lvc P azeo ref. kPa ºC ºC corr.a (TBmvc) P*lvc(TBmvc) 1-butanol (1) 101.3 117.7 68.8 (2) VL 8.6 7.9 Y (a) + 101.3 117.7 68.8 (2) W 12.4 7.9 Y (a) n-hexane (2) 101.3 117.7 68.8 (2) NRTL 6.7 7.9 Nb (a) 2-butanol (1) 101.3 99.6 68.8 (2) VL 6.8 3.6 Y (b) + 101.3 99.6 68.8 (2) W 8.3 3.6 Y (b) n-hexane (2) 101.3 99.6 68.8 (2) NRTL 5.8 3.6 Y (b) n-hexane (1) 101.3 68.8 77.1 (1) VL 1.83 1.36 Y (a) + 101.3 68.8 77.1 (1) W 1.83 1.36 Y (a) 1-butylamine (2) 101.3 68.8 77.1 (1) NRTL 1.82 1.36 Y (a) CH3OH (1) + triethylamine (2) 97.3 63.5 88.3 (1) M2 2.09 2.37 N (c) C3H7OH (1) + H2O (2) 101.3 87.8 100.0 (1) VL 3.10 1.57 Y (d) (a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967); (d) Carlson and Colburn (1942) aVL (van Laar); W (Wilson); M2 (two-constant Margules) bsee text
- 6. 672 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 γlvc B mvc lvc B mvc T P p T ∞ =( ) ( )* criterion for incipient formation of mminimum boiling azeotrope− (15) Since the actual occurrence of a minimum-boiling azeotrope, as in curve D′ in Figure 3, results from a greater positive departure from ideality (that is, a greater value of γ∞ lvc) than represented in Equation (15), a criterion for this is γlvc B mvc lvc B mvc T P p T ∞ ≥( ) ( )* criterion for minimum - boiling azeotrrope (16) If γ∞ lvc(TBmvc) < P/p*lvc(TBmvc), there is no azeotrope. Next, consider incipient formation of a maximum-boiling azeotrope as x1 → 0 at F′ in Figure 3. Or, conversely, if component 2 is arbitrarily designated as the mvc, incipient formation occurs as x2 → 0 (not shown in Figure 3). In either case, it occurs as xmvc → 0. Arguments similar to those above for a minimum- boiling azeotrope lead to the following criterion for incipient formation: γ∞ mvc B lvc mvc B lvc T P p T ( ) ( )* = criterion for incipient formation off maximum - boiling azeotrope (17) Since the actual occurrence of a maximum-boiling azeotrope, as in curve G′ in Figure 3, results from a greater negative departure from ideality (that is, a smaller value of γ∞ mvc)than represented in Equation (17), a criterion for this is γmvc B lvc mvc B lvc T P p T ∞ ≤( ) ( )* criterion for maximum - boiling azeotrrope (18) If γ∞ mvc(TBlvc) > P/p*mvc(TBlvc), there is no azeotrope. Figure 4 shows the results expressed by criteria (16) and (18) in graphical form. In the lower left (closed) ﬁeld, the diagonal line representing incipient azeotrope formation divides the regions of maximum-boiling azeotrope occurrence and no azeotrope according to (18). Similarly, the continuation of the diagonal line through the upper right (open-ended) ﬁeld divides the regions of minimum-boiling azeotrope occurrence and no azeotrope according to (16). Figure 4 is similar to Figure 2, but the variables and azeotrope ﬁelds have different signiﬁcance. In the use of Figure 4 or criterion (16) for a minimum-boiling azeotrope, the possible occurrence of a liquid-phase miscibility gap for sufﬁciently large values of γ∞ lvc(TBmvc) must be consid- ered, as discussed above for a positive azeotrope. Criterion (16) cannot predict this. In the use of criteria (16) and (18), values of γ∞ lvc(TBmvc) and γ∞ lvc(TBlvc) are the ones naturally obtained from experimental vapour-liquid equilibrium data at P. If, however, γ∞ lvc and γ∞ mvc are obtained otherwise, for example, from gE(T′), their values, γ∞ lvc(T′) and γ∞ lvc(T′) must be adjusted to give the values required for criteria (16) and (18). This temperature adjustment is done by means of Equation (11). Thus, for γ∞ lvc ln γ γ lvc B mvc lvc T T lvc ET T h T T dT B mvc ∞ ∞ ′ ∞ ′ = − ∫ ( ) ( ) ( )1 R 2 (19) and for γ∞ mvc ln γmvc B lvc mvc T T mvc ET T h T T dT B lvc ∞ ∞ ′ ∞ ′ = − ∫ ( ) ( ) ( ) ³ 1 R 2 (20) where – hlvc E∞ and – hE∞ mvc are the excess partial molar enthalpies at inﬁnite dilution of lvc and mvc, respectively. The excess enthalpy (heat of mixing) compilations of Christensen et al. (1982, 1988) and of Christensen et al. (1984) and Gmehling and Holderbaum (1989,1991) are useful for this purpose. Criteria in Terms of Parameters of gE Correlations As noted for isothermal systems, since γi ∞ can be obtained from gE correlations, the criteria for the occurrence of azeotropes in isobaric systems can be applied using the parameters of the gE correlations, as given in Table 1. The last three of these are used in the examples of applications of the criteria in the following section. Examples of Use of Criteria Tables 3 and 4 show randomly selected examples of the use of criteria (16) and (18), respectively, to indicate presence or absence of azeotropes. In these tables, vapour pressures required were calculated from Antoine constants provided by Boublík et al. (1984) for the butanols, and by Nakanishi et al. (1967) for methanol and the amines. In Table 3, the ﬁrst nine entries for three systems come from the work of Dominguez et al. (1997, 2002) for positive deviations from ideality for systems involving butanols, n-hexane, and 1-butylamine (column 1) at 101.3 kPa (column 2). The boiling points (TB) of the components are listed in columns 3 and 4. The MINIMUM - BOILING AZEOTROPE NO AZEOTROPE P CONSTANT γlvcγlvcγ∞(TBmvc) P/plvc*(TBmvc) MAXIMUM - BOILING AZEOTROPE NO AZEOTROPE γmvγmvγc∞(TBlvc) P/pmvc (TBlvc)0 1P/p0 1P/p *0 1*(T0 1(TBlvc 0 1 Blvc)0 1) 1 Figure 4. Regions of occurrence of minimum-boiling and maximum- boiling azeotropes in binary systems at ﬁxed P, according to criteria (16) and (18). (note change of variables in lower left and upper right ﬁelds)
- 7. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 673 mvc is indicated next, followed, in order, by the gE (or γ) correla- tion (corr.) used to ﬁt experimental equilibrium data, the value of γ∞ lvc at TBmvc, the value of the ratio P/p*lvc(TBmvc), and the conclusion as to whether an azeotrope occurs. For each of these three systems (nine entries), the authors ﬁtted the van Laar (VL), Wilson (W), and NRTL equations, among others, to the experimental data, and provided resulting values of γ∞ lvc(TBmvc) In each case except one, the existence of a minimum-boiling azeotrope is indicated, in agreement with experimental results. The one apparent exception is for the third entry. In this case, however, the NRTL estimate of γ∞ lvc(TBmvc) (together with two others not listed here) is low in comparison with the estimates from the van Laar and Wilson equations (graphical extrapolation indicates a value ≥ 9, in line with these last two). (This example shows that the experimental data from VLE measurements across the composition range do not always provide for a good statistical ﬁt at the extremes for γ1 ∞ and γ2 ∞ for a given correla- tion.) The last two entries (Nakanishi et al., 1967; Carlson and Colburn, 1942) similarly illustrate situations in which no azeotrope and an azeotrope, respectively, occur. In Table 4, the ﬁrst six entries for two systems come from the work of Dominguez et al. (1997, 2002) for negative deviations from ideality. For the system 1-butanol (1) + 1-butylamine (2), the conclusion is that there is no azeotrope. For the system 2- butanol (1) + 1-butylamine (2), the conclusion is the opposite – there is a maximum-boiling azeotrope. Both of these conclu- sions agree with the experimental result. The last two entries, from the work of Nakanishi et al. (1967), also provide one example in which an azeotrope occurs and one in which there is no azeotrope. CONCLUSION Any method that generates values of activity coefﬁcients at inﬁnite dilution, γ1 ∞ and γ2 ∞, in a binary system can be used to determine whether an azeotrope exists in the system at a speciﬁed (constant) T, in accordance with Figure 2 (based on criteria (5) and (7)); or in the system at a speciﬁed (constant) P in accordance with Figure 4 (based on criteria (16) and (18)). (For isothermal systems, the criteria are essentially the same as those given by Brandini (1974).) In a model sense, Figure 2 or Figure 4 shows the extent of nonideality that can be tolerated to avoid an azeotrope for a given pressure ratio; the greater the ratio, the greater the extent of nonideality allowable, consistent with qualitative considerations. Gmehling et al. (1994) have described various experimental methods for determining γ∞, including the use of gas-liquid chromatography and ebulliometry (as well as from VLE measure- ments). Malanowski and Anderko (1992) note these and describe two estimation methods. Applications should prove useful for screening for various purposes. ACKNOWLEDGEMENT Financial assistance has been received from the Natural Sciences and Engineering Research Council of Canada. NOMENCATURE A,B parameters in Van Laar equation gE excess Gibbs function (J mol-1) gk parameter in Redlich-Kister equation (8) – hi E excess partial molar enthalpy of component i (J mol-1) p* saturation vapour pressure (kPa) P pressure (kPa) R gas constant, 8.3145 (J mol-1 K-1) T temperature (K or °C) y mole fraction in vapour phase x mole fraction in liquid phase Greek Symbols γ activity coefﬁcient Λ12, Λ21 parameters in Wilson equation τ12, τ21 parameters in NRTL equation Superscripts ∞ at inﬁnite dilution Subscripts B boiling point i component i lvc less volatile component mvc more volatile component 1, 2 component 1, 2 REFERENCES Boublík, T., V. Fried and E. Hála, “The Vapour Pressures of Pure Substances,” 2nd ed., Elsevier, Amsterdam, 1984. Brandini, V., “Use of Inﬁnite-Dilution Activity Coefﬁcients for Predicting Azeotrope Formation at Constant Temperature and Partial Miscibility in Binary Liquid Mixtures,” Ind. Eng. Chem. Fundam., 13, 154– 156 (1974). Carlson, H. C. and A. P. Colburn, “Vapor-Liquid Equilibria of Nonideal Solutions,” Ind. Eng. Chem., 34, 581– 589 (1942). Christensen, C., J. Gmehling, P. Rasmussen and U. Weidlich, “Heats of Mixing Data Collection,” Chemistry Data Series, Vol. III, Parts 1 and 2, DECHEMA, Frankfurt am Main, FRG (1984). Christensen, J. J., R. W. Hanks and R. M. Izatt, “Handbook of Heats of Mixing,” Wiley, New York, NY (1982). Christensen, J. J., R. L. Rowley and R. M. Izatt, “Handbook of Heats of Mixing, Supplementary Volume,” Wiley, New York, NY (1988). Dominguez, M., A. M. Mainar, H. Artigas, F. M. Royo and J. S. Vrieta, “Isobaric VLE Data of the Ternary System (1-Butanol + n-Hexane + 1-Butylamine) and the Three Constituent Binary Mixtures at 101.3 kPa,” J. Chem. Eng. Japan, 30, 484– 490 (1997). Dominguez, M., S. Martin, H. Artigas, M. C. López and F. M. Royo, “Isobaric Vapor-Liquid Equilibrium for the Binary Mixtures (2-Butanol + n-Hexane) and 2-Butanol + 1-Butylamine) and for the Ternary System (2-Butanol + n-Hexane + 1-Butylamine) at 101.3 kPa,” J. Chem. Eng. Data, 47, 405– 410 (2002). Eng, R. and S. I. Sandler, “Vapor-Liquid Equilibria for Three Aldehyde/Hydrocarbon Mixtures,” J. Chem. Eng. Data, 29, 156– 161 (1984). Fredenslund, Aa., R. L. Jones and J. M. Prausnitz, “Group- Contribution Estimation of Activity Coefﬁcients in Nonideal Liquid Mixtures,” AIChE J., 21, 1086– 1099 (1975). Gaw, W. J. and F. L. Swinton, “Thermodynamic Properties of Binary Systems Containing Hexaﬂuorobenzene,” Trans. Faraday Soc., 64, 2023– 2034 (1968). Gmehling, J. and T. Holderbaum, “Heats of Mixing Data Collection,” Chemistry Data Series, Vol. III, Part 3 (1989), Part 4 (1991), DECHEMA, Frankfurt am Main, FRG.
- 8. 674 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 Gmehling, J., J. Menke and M. Schiller, “Activity Coefﬁcients at Inﬁnite Dilution,” Part 3, DECHEMA, Frankfurt am Main, FRG (1994), pp. XIV– XXIII. Hildebrand, J. H., J. M. Prausnitz and R. L. Scott, “Regular and Related Solutions,” Prentice-Hall, Englewood Cliffs, NJ (1970). King, M. B., “Phase Equilibrium in Mixtures,” Pergamon, Oxford (1969), p. 373. Malanowski, S. and A. Anderko, “Modelling Phase Equilibria,” Wiley, New York, NY (1992), pp. 133– 136. Nakanishi, K., H. Shirai and T. Minamiyama, “Vapor-Liquid Equilibria of Binary Systems Containing Alcohols,” J. Chem. Eng. Data, 12, 591– 594 (1967). Prigogine, I. and R. Defay, “Chemical Thermodynamics,” transl. D. H. Everett, Longmans, London (1954), pp. 464– 465. Redlich, O. and A. T. Kister, “Algebraic Representation of Thermodynamic Properties and the Classiﬁcation of Solutions,” Ind. Eng. Chem., 40, 345– 348 (1948). Renon, H. and J. M. Prausnitz, “Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures,” AIChE J., 14, 135– 144 (1968). Sandler, S. I., “Chemical and Engineering Thermodynamics,” 3rd ed., Wiley, New York, NY (1999), p. 291. Senra, A. M. P., I. M. A. Fonseca, A. G. M. Ferreira and L. Q. Lobo, “Vapour-liquid equilibria of {xCH3F + (1 – x)HCl} at temperatures of 159.01 K and 182.33 K,” J. Chem. Thermodynamics, 34, 1557– 1566 (2002). Smith, W. R. and R. W. Missen, “Chemical Reaction Equilibrium Analysis,” Wiley-Interscience, New York, NY (1982); Krieger, Malabar, FL (1991), pp. 43, 155– 156. Wilson, G. M., “Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing,” J. Am. Chem. Soc., 86, 127– 130 (1964). Zielkiewicz, J., “(Vapour + liquid) equilibrium in (N,N- dimethylacetamide + methanol + water) at the temperature 313.15 K,” J. Chem.Thermodynamics, 35, 1993– 2001 (2003). Manuscript received February 17, 2005; revised manuscript received June 22, 2005; accepted for publication July 27, 2005.