Ind. Eng. Chem. Res. 2006, 45, 8217-8222
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Possible Existence of a Negative (Positive) Homogeneous Azeotrope When the Binary Mixture Exhibits Positive (Negative) Deviations from Ideal Solution Behavior (That is, When gE is Positive (Negative)) Jean-Noe1 l Jaubert* and Romain Privat Institut National Polytechnique de Lorraine, Ecole Nationale Supe´ rieure des Industries Chimiques, Laboratoire de Thermodynamique des Milieux Polyphase´ s, 1 rue GrandVille, 54000 Nancy, France
The objective of this note is to give proof that the existence of a positive (or, respectively, negative) homogeneous azeotrope is not linked to the sign (positive or negative) of the molar excess Gibbs energy function (gE) but rather to the curvature of gE. We demonstrate that when gE is concave (or, respectively, convex), a binary system may exhibit a positive (or, respectively, negative) azeotrope. Because a concave or convex function may indifferently be positive or negative, when gE is positive (or, respectively, negative), a binary system may exhibit a negative (or, respectively, positive) azeotrope. Introduction In some excellent textbooks of thermodynamics,1-9 it is written that, when a binary system exhibits positive (negative) deviations from ideal solution behavior (i.e., when gE is positive (negative)), the system may not exhibit a negative (positive) azeotrope. In other words, some authors make a direct link between the adjective positive (negative) used to describe an azeotrope and the sign of the associated gE function. Here, we want to give proof that such a direct link may be incorrect. Indeed, even when gE is positive (negative), a binary system may exhibit a negative (positive) azeotrope. However, a direct link exists between the curvature of gE (concave or convex) and the possible existence of a homogeneous azeotrope (positive or negative). In this note, it is demonstrated that when a binary system exhibits a positive (negative) azeotrope, the curve gE vs x1 is necessarily concave (convex) at the azeotropic composition, regardless of the sign of gE. As a conclusion, the following theorem is demonstrated: “for a binary system at the azeotropic composition, the bubble-P curVe on an isothermal Pxy diagram has always the same curVature (concaVe for a positiVe azeotrope and conVex for a negatiVe azeotrope) as the gE Vs x1 function”. Lecat’s Definition of a Homogeneous Positive (or Respectively Negative) Azeotrope The term “azeotrope” was originally derived from the Greek, as reported by Wade and Merriman10 in 1911. The adjective positive or negative that is used to describe a homogeneous azeotrope was introduced by Maurice Lecat11 in 1926. More information may be found in the encyclopedia by Franc¸ ois Auguste Victor Grignard (winner of the 1912 Nobel prize in Chemistry), in which Lecat wrote a chapter of 147 pages that was devoted to the distillation.12 According to Lecat, an azeotrope is called positive when a maximum occurs on an isothermal Pxy diagram. Conversely, an azeotrope is called negative when a minimum occurs on an isothermal Pxy diagram. In other words, a positive azeotrope is simply a minimum boiling-point azeotrope that is characterized by a minimum on * To whom the correspondence should be addressed. Fax number: +33 3 83 17 51 52. E-mail:
[email protected].
an isobaric Txy diagram and by a maximum on the corresponding isothermal Pxy diagram. Conversely, a negative azeotrope is simply a maximum boiling-point azeotrope that is characterized by a maximum on an isobaric Txy diagram and by a minimum on the corresponding isothermal Pxy diagram. We see no reason to change these definitions, and they will be used throughout this article. At this step, it is interesting to wonder why Lecat used the adjectives “positive” and “negative” to characterize a homogeneous azeotrope. Indeed, which property or which parameter is positive (negative) when a binary system exhibits an azeotrope? To answer this question, one must carefully read the paper by Lecat.12 Doing so, we quickly realized that, in Lecat’s mind, a positive function is, in fact, a concave function. Conversely, in Lecat’s mind, a negative mathematical function f(x) is not as classically defined by mathematicians, a function for which f(x) < 0 for all x but a convex function (i.e., a function for which the second derivative f ′′(x) > 0). Because, in an isothermal phase diagram, in the vicinity of a minimum (maximum) boiling-point azeotrope, the bubble and the dew curves are necessarily concave (convex), Lecat decided to call such an azeotrope a positive (negative) azeotrope. However, the correct terminology should be a concave (convex) azeotrope. In his paper, Lecat never defines a positive azeotrope as an azeotrope for which gE > 0. As a conclusion, Lecat never defines any property or any parameter effectively greater than zero (>0) or less than zero ( 0 (i.e. the bubble-P curve is convex)
T x1 ) xaz 1
(2)
Step 1: Calculation of (∂PB/∂x1)T. In the so-called gammaphi (γ-φ) approach, and assuming, for sake of simplicity, that the vapor phase behaves as an ideal gas mixture, one has
PB(T,x1) ) Ps1(T)x1γ1(T,x1) + Ps2(T)x2γ2(T,x1)
(3)
In the γ-φ approach, the activity coefficients (γ) are calculated from a gE model. Because such gE models (Porter, Margules, Van Laar, Redlich-Kister, Wohl, Wilson, NRTL, UNIQUAC, UNIFAC, etc.) are not dependent on pressure, the resulting activity coefficients are pressure-independent. Differentiation of eq 3 leads to
( ) [( ) ] [( ) ] ∂PB ∂x1
) Ps1 x1
T
Moreover,
∂γ1 ∂x1
+ γ1 + Ps2 x2
T
∂γ2 ∂x1
) γ1
T
) γ2
T
Thus, one has
( ) ∂PB ∂x1
) Ps1γ1 - Ps2γ2 + Ps1x1γ1
T
∂ ln γ1 ∂x1
T
(4)
∂ ln γ2 ∂x1
T
(5a) (5b)
( ) ∂ ln γ1 ∂x1
+ T
Ps2x2γ2
( ) ∂ ln γ2 ∂x1
(6)
T
Moreover, the well-known activity coefficient form of the Gibbs-Duhem equation is
x2 Demonstration In this section, the following theorem is demonstrated: “for a binary system, in the immediate vicinity of the azeotropic composition, the bubble-P curve on an isothermal Pxy diagram has always the same curvature (concave for a positive azeotrope and convex for a negative azeotrope) as the gE vs x1 function”. It is obvious (see Figure 1) that when a binary system exhibits a positive (negative) homogeneous azeotrope, at the azeotropic composition (xaz 1 ) the bubble-P curve on an isothermal Pxy diagram reaches a maximum (minimum). As a consequence, the bubble-P curve is concave (convex). By noting PB, the bubble pressure, it is thus possible to write
- γ2
T
( ) ( ) ( ) ( ) ∂γ1 ∂x1
Figure 1. Isothermal Pxy diagrams for two binary systems exhibiting a homogeneous azeotrope. Panel (a) shows a positive homogeneous azeotrope in a mixture of ethanol (1) and benzene (2) at 370 K. The azeotrope is merged with the maximum on the isothermal Pxy diagram. In the vicinity of the azeotrope, the bubble P-curve is concave (on this particular example, it is also concave, for all x1 ∈ [0,1]). Panel (b) shows a negative homogeneous azeotrope in a mixture of acetone (1) and chloroform (2) at 320 K. The azeotrope is merged with the minimum on the isothermal Pxy diagram. In the vicinity of the azeotrope, the bubble-P curve is convex (in this particular example, it is also convex, for all x1 ∈ [0,1]).
∂γ2 ∂x1
( ) ∂ ln γ2 ∂x1
) -x1
T
Equation 6 becomes
( ) ∂PB ∂x1
T
) Ps1γ1 - Ps2γ2 + x1
( ) ∂ ln γ1 ∂x1
(7)
T
( )
∂ ln γ1 (Ps γ - Ps2γ2) (8) ∂x1 T 1 1
i.e.,
The criterion of stability for a single-phase binary system can be written as14
Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8219
( ) ∂ ln γ1 ∂x1
1 >x T 1
(10)
Thus, the Γ(T,x1) function that appears in eq 9 clearly must be positive for the liquid phase to remain stable. As a conclusion, when a binary system exhibits a homogeneous azeotrope, by combining eqs 1 and 9, one has the following well-known equation: s az Ps1γaz 1 - P2γ2 ) 0
{
} ( ) ( ) ( ) ( )
∂2[gE/(RT)] ∂x12
)
T
∂ ln γ1 ∂x1
-
T
1 ∂ ln γ1 ) x2 ∂x1
∂ ln γ2 ∂x1
)-
T
T
1 ∂ ln γ2 x1 ∂x1
(18)
T
This equation, in combination with eq 14, yields
(11)
Step 2: Calculation of (∂2PB/∂x12)T. Starting with eq 9, a simple differentiation gives
( ) [ ( ) ( )] ∂2PB ∂x12
)Γ
∂γ1 ∂x1
Ps1
T
-
∂γ2 ∂x1
Ps2
T
+
(Ps1γ1
-
Ps2γ2)Γ′
(12)
T
( )|
with
Γ′ )
∂2PB
( ) ∂Γ ∂x1
∂x12
T
This equation, in combination with eqs 11 and 5, yields
( )| ∂2PB 2
∂x1
[
( )| ( )|
s az ) Γ(T,xaz 1 ) P 1 γ1
T x1)x1az
∂ ln γ1 ∂x1 Ps2γaz 2
-
T x1)x1az
∂ ln γ2 ∂x1
]
(13)
T x1)x1az
A new application of the Gibbs-Duhem equation (eq 7) gives
( )|
)
T x1)x1az
Γ(T,xaz 1)
( )|
1 ∂ ln γ1 ∂x1 xaz 2
az s az az [Ps1γaz 1 x2 + P2γ2 x1 ] (14)
T x1)x1
az
Moreover, by definition,
gE ) x1 ln γ1 + x2 ln γ2 RT
(15)
Differentiation of eq 15 gives
{
and
T x1)x1az
{
}|
∂2[gE/(RT)] ∂x12
T x1)x1az
always have the same sign. Thus, eq 2 clearly indicates that, in the immediate vicinity of the azeotropic composition (x1 ) x1az), the curve gE vs x1 is necessarily concave for a positive azeotrope and necessarily convex for a negative azeotrope. However, regardless of the type of azeotrope, gE may be positive or negative. It is obvious that when the curve gE vs x1 is concave (convex), the binary system does not necessarily exhibit a positive (negative) homogeneous azeotrope. A concave (convex) gE is a necessary but not sufficient condition for a binary system to exhibit a homogeneous positive (negative) azeotrope. Discussion and Example
∂2PB ∂x12
The vapor pressures, the mole fractions and the activity coefficients are all positive quantities. Moreover, we know, from the stability condition, that Γ(T,x1) > 0. As a conclusion, the terms
}
∂[gE/(RT)] ∂x1
) ln γ1 - ln γ2 + x1
T
( ) ( ) ∂ ln γ1 ∂x1
+ x2
T
∂ ln γ2 ∂x1 T (16)
According to eq 7, the last two terms sum to zero; hence,
{
}
∂[gE/(RT)] ∂x1
) ln γ1 - ln γ2
(17)
T
A second differentiation and a second application of the GibbsDuhem equation gives
We believe that confusion has sometimes appeared in textbooks of thermodynamics, because when an azeotropic system exhibits positive (negative) deviations from ideal solution behavior (i.e., when gE is positive (negative)), the curve gE vs x1 generally is concave (convex). Under these conditions, it becomes identical to say gE positive (negative) or gE concave (convex). The fact that, in Lecat’s mind, positive (negative) means concave (convex) also added to the confusion. Example of the Existence of a Negative Homogeneous Azeotrope for a Binary Mixture Exhibiting Positive Deviations from Ideal Solution Behavior. We know azeotropic binary systems for which the curvature of the function gE vs x1 changes with composition but without changing sign. A wellknown example15-17 is the pentafluoroethane (1)/ammonia (2) system. For this system, at T ) 49.9 °C, for all x1 values (0 < x1 < 1), gE is positive (see Figure 2), which means that this system exhibits positive deviations from ideal solution behavior. However, for small values of x1, the curve gE vs x1 is concave. When x1 increases, an inflection point appears and the curve gE vs x1 becomes convex. In the concave part, the system exhibits a first azeotrope, which is necessarily a positive azeotrope and in the convex part, the system exhibits a second azeotrope which is necessarily a negative azeotrope. Thus, for this system, a negative azeotrope appears, although gE is positive. Application of the Proposed Theorem to a Saddle-Point Azeotrope. Until now, we only mentioned the possible existence
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Figure 2. Behavior of the pentafluoroethane (1)/ammonia (2) system at T ) 49.9 °C, using the NRTL equation fitted to vapor-liquid equilibrium (VLE) data:15 (a) plot of gE/(RT), ln γ1, and ln γ2 vs x1, and (b) plot of the isothermal Pxy diagram (the dashed line is Raoult’s line and the filled circles are the experimental VLE data).
of a positive (negative) azeotrope. However, binary systems may also exhibit what is either called a saddle-point azeotrope or an intermediate boiling-point azeotrope. In such a point, as for any azeotrope, the liquid and the vapor phase have the same composition. However, the boiling point of the azeotropic mixture lies midway between the boiling points of its individual components. A saddle point azeotrope may appear for systems that exhibit a double azeotrope when the two azeotropes merge into a single point (when the compositions of the two azeotropess one positive and one negativesbecome identical, the resulting point is a saddle azeotrope). As a consequence, a saddle-point azeotrope has the properties of both a positive azeotrope and a negative azeotrope. At this point (amalgamation of a positive and of a negative azeotrope), the bubble-P curve must be simultaneously concave (which is a necessary condition for the existence of a positive azeotrope) and convex (necessary condition for the existence of a negative azeotrope). Consequently, the curvature of the bubble-P curve changes at the saddle-point azeotrope composition and an inflection point appears. At the saddle-point azeotrope, as for any azeotrope, the bubble-P curve has an horizontal tangent (another possibility for the bubble-P curve to exhibit an inflection point with an horizontal tangent is discussed in the Appendix at the end of this note). From a mathematical point of view, a saddle point azeotrope is thus characterized by
( )| ∂PB ∂x1
T x1)x1az
)0
and
( )| ∂ 2P B ∂x12
)0
T x1)x1az
According to the theorem established in this note, at the saddlepoint azeotrope composition, one also has
{
}|
∂2[gE/(RT)] ∂x12
)0
T x1)x1az
and an inflection point necessarily exists at this composition in the curve gE vs x1. An example of such behavior is shown in Figure 3 for the pentafluoroethane (1)/ammonia (2) system. Such a diagram was plotted using the NRTL equation. The NRTL parameters were determined using available experimental data at T ) 49.9 °C15 and the temperature at which the saddle-point azeotrope appears (T ) 31.6 °C) was determined using these parameters. Below T ) 31.6 °C, no true azeotropy occurs. Concluding Remarks In this note, we have given proof that, for a binary system in the immediate vicinity of the azeotropic composition, the bubble-P curve on an isothermal Pxy diagram always has the same curvature (concave for a positive azeotrope and convex for a negative azeotrope) as the gE vs x1 function. At this step,
Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8221
Figure 3. Isothermal Pxy diagram and corresponding gE/(RT) vs x1 curve for the pentafluoroethane (1)/ammonia (2) system at T ) 31.6 °C. At this temperature, the system exhibits a saddle-point azeotrope.
we could wonder whether the bubble-P curve on an isothermal Pxy diagram always has the same curvature as the gE vs x1 function for compositions different from the azeotropic composition. Our answer is simple: in most cases, this is true, but exceptions may exist. This means that, in most cases, when gE is concave (convex), the bubble curve on an isothermal Pxy diagram is also concave (convex). When an inflection point exists on the bubble curve, it also generally exists on the gE vs x1 plot. This concluding remark can help chemical engineers who want to model binary systems for which the bubble-P curve on an isothermal phase diagram does not behave classically. For example, if the bubble-P curve has several inflection points, it is necessary to select a gE model that is mathematically able to reproduce such behavior. To be concrete, the diethylamine (1)/methanol (2) system exhibits a negative azeotrope inflection (xaz ≈ 0.7) on the 1 ≈ 0.25) and an inflection point (x1 17 bubble-P curve at T ) 75 °C. For such a binary system, gE is negative and the curve gE vs x1 must be initially convex, to reproduce the negative azeotrope, and then concave, to reproduce the inflection point. Thus, gE is very similar to that observed in Figure 2a, except that it has a negative value.
Figure A1. (a) Liquid-liquid phase diagram TxR1 xβ1 for the methyl acetate (1)/water (2) system, which has an upper critical solution temperature (UCST). (b) Isothermal Pxy diagram for the same system at T ) TUCS ) 381.15 K.
Given the conditions of eq 9 being
and eq 12 being
( ) ∂2PB ∂x12
) Γ(T,x1)∆′(T,x1) + ∆(T,x1)Γ'(T,x1)
T
with
Γ′ )
In a previous section, we have seen that, at the saddle point azeotrope composition, the bubble-P curve exhibited an inflection point with a horizontal tangent. Here, we want to discuss another possibility for the bubble-P curve to exhibit such behavior.
∂Γ ∂x1
T
and
∆′ ) Appendix: Other Possibility for the Bubble-P Curve To Exhibit an Inflection Point with a Horizontal Tangent
( )
( ) ∂∆ ∂x1
T
the following situations are observed. (1) When ∆(T,x1) ) 0 (which is a necessary and sufficient condition for a binary system to exhibit a homogeneous azeotrope) and, simultaneously, ∆′(T,x1) ) 0, it is obvious that (∂PB/∂x1)T ) 0 and simultaneously (∂2PB/∂x12)T ) 0. Thus, the bubble-P curve exhibits an inflection point with a horizontal
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Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006
tangent. This inflection point corresponds to a saddle-point azeotrope (more details have been given in a previous section). (2) When Γ(T,x1) ) 0 (the stability criterion of a single-phase binary system becomes zero) and, simultaneously, Γ′(T,x1) ) 0, it is obvious that (∂PB/∂x1)T ) 0 and, simultaneously, (∂2PB/ ∂x12)T ) 0. Thus, the bubble-P curve exhibits an inflection point with a horizontal tangent. This inflection point corresponds to a critical end point. A proof of this statement is given below. From classical thermodynamics,18 a critical binary liquid mixture is characterized by two equations:
( ) ( ) ∂2gm ∂x12
T,P
∂3gm ∂x13
T,P
)0
(20a)
)0
(20b)
where gm is the molar Gibbs energy on mixing (gm ) RT × ∑ xi ln ai). An equivalent, but more useful, characterization of a binary liquid-liquid critical point is provided by introducing the activity coefficients into the previous equations. We then obtain
{(
( )
∂ ln γ1 1 + )0 ∂x1 T x1 S ∂2 ln γ1 1 )0 ∂x12 T x12
)
{
Γ(T,x1) ) 0 Γ′(T,x1) ) 0
(21)
When Γ(T,x1) ) 0 and, simultaneously, Γ′(T,x1) ) 0, from eqs 9 and 12, the bubble-P curve on an isothermal diagram possesses an inflection point that is also a stationary point (i.e., that has a horizontal tangent). From eq 21, this inflection point is obviously a liquid-liquid critical point. Because two identical liquid phases are in equilibrium with a gas phase, such a critical point is, in fact, a critical endpoint. An illustration of this behavior is shown in Figure A1 for the methyl acetate (1)/water (2) system. We know from experiments19 that such a system has a upper critical solution temperature (UCST) near 381.15 K and a methyl acetate mole fraction of 0.212. The Margules equation was used to reproduce the coordinates of the UCST, and the resulting parameters were used to plot the Pxy diagram at T ) 381.15 K (see Figure A1). As expected, the bubble-P curve exhibits an inflection point with a horizontal tangent. Acknowledgment Yannick Privat, author R.P.’s twin brother and holder of the agre´ gation in Mathematics (France’s top-level competitive examination for the recruitment of teachers), is gratefully
acknowledged for advising the authors so that the derivations included in this note seem sound to the reader. Literature Cited (1) Astarita, G. Thermodynamics: An AdVanced Textbook for Chemical Engineers; Springer: New York, 1989. (2) Sandler, S. I. Chemical and Engineering Thermodynamics, 3rd Edition; Wiley: New York, 1999. (3) Ott, J. B.; Boerio-Goates, J. Chemical Thermodynamics: AdVanced Applications; Academic Press: San Diego, CA, 2000. (4) Infelta, P. Introductory Thermodynamics, BrownWalker Press: Boca Raton, FL, 2004. (5) Khoury, F. M. Multistage Separation Processes, 3rd Edition; CRC Press; Boca Raton, FL, 2004. (6) Silbey, R. J.; Alberty, R. A.; Bawendi, M. G. Physical Chemistry, 4th Edition; Wiley: New York, 2004. (7) O’Connell, J. P.; Haile, J. M. Thermodynamics: Fundamentals for Applications; Cambridge University Press: Oxford, U.K., 2005. (8) Malijevsky, A.; Novak, J. P.; Labik, S.; Malijevska, I. Physical Chemistry in Brief; 2005. (Book only available at the following URL: http:// www.vscht.cz/fch/en/tools/breviary-online.pdf.) (9) http://www.iupac.org/didac/Didac%20Eng/Didac05/Content/ ST07.htm. (10) Wade, J.; Merriman, R. W. The Influence of water on the boiling point of ethyl alcohol at pressures above and below the atmospheric pressure. J. Chem. Soc., Trans. 1911, 99, 997-1011. (11) Lecat, M. Sur l’aze´otropisme, particulie`rement des syste`mes binaires a` constituants chimiquement voisins (Azeotropism of binary systems of chemically similar constituents). C. R. Acad. Sci. Paris 1926, 183, 880882. (12) Lecat, M. Traite´ de chimie organique; Grignard, V., Ed.; Masson et Compagnie: Paris: 1935; Vol. 1, pp 121-267. (13) Scatchard, G. Change of volume on mixing and the equations for nonelectrolyte mixtures. Trans. Faraday Soc. 1937, 33, 160-166. (14) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 7th edition; McGraw-Hill: New York, 2004, p 579. (15) Chai Kao, C.-P.; Paulaitis, M. E.; Yokozeki, A. Double azeotropy in binary mixtures of NH3 and CHF2CF3. Fluid Phase Equilib. 1997, 127, 191-203. (16) Yokozeki, A.; Zhelazny, V. P.; Kornilov, D. V. Phase behaviors of ammonia/R-125 mixtures. Fluid Phase Equilib. 2001, 185, 177-188. (17) Shulgin, I.; Fischer, K.; Noll, O.; Gmehling, J. Classification of homogeneous binary azeotropes. Ind. End. Chem. Res. 2001, 40, 27422747. (18) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd Edition; Prentice Hall PTR: Englewood Cliffs, NJ, 1998; p 273. (19) Hill, A. E. International Critical Tables; Washburn, E. W., Ed.; McGraw-Hill: New York, 1928; Vol. 3, p 387.
ReceiVed for reView July 7, 2006 ReVised manuscript receiVed September 21, 2006 Accepted October 2, 2006 IE060874F
