Phase equilibrium calculations are essential to the

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1 ChE classroo Application of the double-tangent construction of coexisting phases to Any Type of Phase Equilibriu For Binary Systes Modeled With the Gaa-Phi Approach Jean-Noël Jaubert and Roain Privat Université de Lorraine, École Nationale Supérieure des Industries Chiiques BP 0451, Nancy cedex 9, France Phase equilibriu calculations are essential to the siulation and optiization of cheical processes. The success of such calculations depends on correct predictions of the nuber and copositions of phases present at a given teperature, pressure, and overall fluid coposition. Equality of cheical potentials for each coponent in all phases is, however, only a necessary but not a sufficient condition to reach a therodynaically stable equilibriu. As a consequence, if a stability analysis is not perfored, it is not possible to ascertain that the calculated equilibriu state is the one having the lowest Gibbs energy and consequently, an incorrect nuber of phases or incorrect phase copositions ay be predicted. The double-tangent construction of coexisting phases is an elegant approach to visualizing all the ultiphase systes that isfy the equality of cheical potentials and to select the stable state. This paper shows how the olar Gibbs energy change on ixing (g ) can be used to easily perfor the double-tangent construction of coexisting phases for binary systes odeled with the gaa-phi approach. Many exaples of instances where incorrect predictions (false two-phase states) ay be ade will be described. The atheatical equations to handle are siple and can easily be used by the students during class exercises. Two binary systes involving vapor-liquid (VL), liquid-liquid (LL), vapor-liquid-liquid (VLL), solid-liquid (SL), solid-solid (SS), and solid-solid-liquid (SSL) equilibria are discussed in detail. The Gibbs double-tangent construction of coexisting phases This section is devoted to recalling how to properly perfor the Gibbs double-tangent construction of coexisting phases that akes it possible to graphically deterine the coposition of the phases in equilibriu in a syste of known teperature, pressure, and overall coposition. Jean-Noël Jaubert is a professor of cheical engineering therodynaics at ENSIC (Ecole Nationale Supérieure des Industries Chiiques), a state-run institution of higher education characterized by a highly selective adission procedure. He received his doctorate in 1993 and has published ore than 100 research papers. His research interests include the developent of predictive therodynaic odels based on the group contribution concept, the use of exergetic life cycle assessent in order to reduce CO eissions, the easureent and correlation of liquid-vapor equilibriu under high pressure, and enhanced oil recovery. Roain Privat currently works as an associate professor of therodynaics and cheical engineering at the Cheical Engineering University of Nancy (Ecole Nationale Supérieure des Industries Chiiques), Nancy, France. In 008, he received his Ph.D. He currently perfors his research work at the Laboratory for Reactions and Cheical Processes (L.R.G.P.) in the fields of therodynaic odeling and coputational therodynaics (developent of the predictive odels, and search for new algoriths for phase-equilibriu calculation, etc.). Copyright ChE Division of ASEE Cheical Engineering Education

2 Fro basic therodynaics, [1] it is well known that by plotting, at constant teperature and pressure, any olar property of a binary syste vs. z 1 (the overall coposition), and by adding the tangent line for a coposition z 1*, the tangent intercepts (at z 1 = 1 and z 1 = 0 ) directly give the values of the two olar partial properties: 1 (T,P, z 1* ) and (T,P,z z 1* ). This graphical technique can be used to deterine the coposition of the coexisting phases by plotting g (the olar Gibbs energy) as a function of z 1 at constant T and P. Indeed, for a binary syste, the equilibriu condition between two phases α and β is: g 1 g α β ( T,P,z 1 ) = g 1 T,P,z 1 α T,P,z 1 β = g ( T,P,z 1 ) ( 1) where g i denotes the cheical potential of coponent i. As a consequence, the presence of a double tangent in a g vs. z 1 plot allows defining two phases, the coposition of which are z 1α and z 1β, which isfy Eq. (1). Let us recall, however, that Eq. (1) is a necessary but not a sufficient condition to reach a stable equilibriu. This eans that the graphically deterined copositions do not necessarily correspond to a stable equilibriu. It is nevertheless ipossible to plot g vs. z 1 since the olar Gibbs energy cannot be defined absolutely by the classical therodynaics, and an elegant way to bypass this liitation is to plot instead: g( T,P,z) = g( T,P,z) z i g i=1 ref. state ( T,P) The corresponding partial olar properties are given by: ref. state g i ( T,P,z) = g i ( T,P,z) g T,P Eq. (1) thus writes: α g 1 T,P,z 1 α g T,P,z 1 β = g 1 ( T,P,z 1 ) β = g ( T,P,z 1 ) ( 3) ( 4) We can thus assert that the presence of a double tangent on an isotheral and isobaric g(t, P, z) vs. z 1 plot allows defining two phases that isfy Eq. (1). In Eq. (), the pure coponent reference state can be chosen freely and can, for instance, be the pure liquid, the pure gas, or the stable (actual) state. In this study, the selected reference state is the stable (actual) state of the pure coponent at the pressure and teperature of the studied syste so that g(t, P, z) is siply the Gibbs energy change on ixing g. Let us indeed recall that the property change on ixing M, of any extensive property M, is by definition the difference between the property M of the actual ixture and the su of the properties of the pure coponents that ake it up all at the sae teperature and pressure as the ixture. [,3] We here want to ephasize that such a quantity is called change on ixing because it is the change that would be easured if the ixing took place in a ixing device of a laboratory. As an exaple, the enthalpy change on ixing can be easured using a calorieter; two streas one of pure fluid 1 and the second of pure fluid, both at a teperature T and at pressure P enter a ixing device, and a single ixed strea, also at T and P, leaves. Heat is added or reoved to aintain the teperature of the outlet strea. This quantity of heat is by definition the enthalpy change on ixing. This well-known exaple was selected to ephasize that all the states are physically realizable: The obtained ixture (liquid, gaseous, solid, or in the two-phase area) and the two pure fluids (either liquid, gaseous, or solid) are in their stable state at T and P. We can thus define the olar Gibbs energy change on ixing g by the following equation: g ( T, P, z) = g ( T, P, z) z i g stable (T, P ) 5 olar Gibbs energy of the stable (actual) ixture We believe it was necessary to recall the proper definition of the olar Gibbs energy change on ixing because in any textbooks in which the double-tangent construction of coexisting phases is explained, g(t, P, z) defined in Eq. () is istakenly called Gibbs energy change on ixing when the pure-coponent reference state is the pure liquid or the pure gas. When the different phases in equilibriu are in the sae aggregation state, e.g., liquid-liquid equilibriu (LLE), a single expression of g ust be considered ( g liquid for an LLE). On the other hand when the phases in equilibriu are in different aggregation states, e.g., vapor-liquid equilibriu (VLE), solid-liquid equilibriu (SLE), etc., it is necessary to plot as any g curves as aggregation states ( g liquid VLE, g liquid g liquid and g solid i=1 and g gas for a for an SVE). for an SLE, g gas and g solid are, respectively, the g expressions of, g gas, and g solid the one-phase liquid or gas or solid postulated syste. For a given value of the overall coposition z 1, only the sallest value between g liquid, g gas, and g solid that identifies the ost stable hoogeneous syste ust by reported in the g vs. z 1 plot (this requireent is a stateent of the second law of therodynaics: when several hoogeneous states are possible at a given T, P, and overall coposition only the transforation fro high to low Gibbs energy levels is authorized thus justifying why the stable state is associated to the lowest Gibbs energy change on ixing value). The olar Gibbs energy change on ixing of the one-phase syste is thus defined by: g ( T,P,z)=in g liquid ( T,P,z),g gas ( T,P,z),g solid ( T,P,z) { } 6 The next section explains how to calculate such a property change on ixing regardless of the aggregation state. Vol. 48, No. 1, Winter

3 Matheatical expressions of g For a binary syste, g is defined by: g given agg. state ( T,P,z) = z i g i,given agg. state T,P,z i=1 given agg. state = liquid or gas or solid ( 7) In order to calculate g, it is thus necessary to know the atheatical expression of the cheical potential change on ixing, g i, whatever the considered aggregation state (liquid, gas or solid): g i,given agg. state ( T,P,z) = ln ˆf i,given agg. state stable f ( T,P,z) T,P ( 8) Special attention to g solid The type of solid-liquid equilibriu (SLE) ainly depends on the utual solubility of the coponents in the solid phase. In this paper, we will assue total iiscibility in the solid phase (a solid phase is assued to be one pure coponent) since it is the widest spread SLE behavior in the case of organic copounds. More coplicated cases could obviously be iagined and analyzed. By forulating this hypothesis, the g solid (T,P,z) quantity can only be calculated for z i =1 and becoes coposition-independent. Consequently, the curve reduces to a single point. Basic therodynaics relations In order to use Eq. (8), the basic therodynaic relations between the fugacity, the fugacity coefficients, and the activity coefficients ust be known. For clarity, let us thus recall that: the fugacity ˆf i and the fugacity coefficient ˆϕ i of a coponent i in a ulti-coponent phase are related through: ˆf i ( T,P,z) = P z i ˆϕ i T,P,z For a pure coponent, Eq. (9) is written: ( T,P) = P ϕ T,P f ( 9) ( 10) The fugacity of a pure coponent changes with pressure according to: f ( T,P ) = f T,P 1 exp 1 P v P 1 ( T,P) dp ( 11) The activity coefficient γ i of a coponent i in a liquid solution isfies: ˆf i, liquid f liquid T, P = x i γ i ( 1) The Gibbs energy of fusion (elting), defined as liquid g,i = g i,pure T,P g i,pure solid (T,P), can be considered as pressure-independent. It can be calculated knowing the enthalpy (heat) of fusion h,i at the noral elting teperature T,i by [4] : g,i = h,i T,i 1 T T,i 13 General expressions of g i In Eq. (8), the aggregation state of the stable pure coponent depends on teperature and pressure and three different cases ust be considered. CASE 1: pure coponent i is gaseous, i.e., T > T b,i (P) where T b,i (P) is the boiling teperature of pure i at the selected pressure. Fro Eqs. (8) to (13), we obtain: g i, liquid g i,gas g i,solid = ln = ln ( T,P) = ln ˆf i, liquid f gas ˆf i,gas f gas ˆf i,solid f gas = ln P i x i γ i C i P = ln y i ϕ i,gas gas ϕ ( T,P) lny i (under oderate P) ( T,P,z) = ln f solid gas f ( T,P) g gas = g subliation, i T,P = gliquid T, P = ln P i C i P h ( T ), i,i,i g,i T ( 14) In Eq. (14): 44 C i = ϕ T,P i( T ) ϕ gas T, P exp 1 P v liquid P 1 ( T,P) dp. ( 14a) Cheical Engineering Education

4 This quantity is close to unity under low or oderate pressure and is usually neglected. CASE : pure coponent i is liquid, i.e., T,i (P) < T < T b,i (P). We thus obtain: g i, liquid g i,gas g i,solid = ln = ln ( T,P) = ln In Eq. (15) ˆf i, liquid f liquid ˆf i,gas f liquid ˆf i,solid f liquid = g, i C i = ϕ i,gas T,P,y ϕ T,P i( T ) = ln x γ i i = ln P y C i i P i ( T,P,z) = ln f exp 1 = h T,i,i solid liquid f P i v liquid P ( T,P),i ( T,P) dp. ( 15) ( 15a) This quantity is close to unity under low or oderate pressure and is usually neglected. CASE 3: pure coponent i is solid, i.e., T < T,i (P). We thus obtain: g i, liquid g i,gas g i,solid = ln = ln ( T,P) = ln ˆf i, liquid f solid = ln( x γ )+ g T,i i i = ln( x i γ i )+ h,i T,i ˆf i,gas f solid,i = ln P y C i i P i + g,i T = ln P y C i i P i + h,i T,i ˆf i,solid f solid Our personal experience ( T,P,z) = ln f solid solid f,i ( T,P) = 0 16 We believe that relations given in Eqs. (14) to (16) are of the highest iportance because they are unknown by ost of the students, including the sartest ones. We indeed asked the following question to 50 of our best graduate students: What is the atheatical expression of the Gibbs energy change on ixing of a vapor phase that can be considered as perfect? Ninety percent of the answered: g perfect gas ( T,y) = y 1 ln y 1 + y ln y ( 17) Unfortunately, in spite of what is written in too any cheical engineering therodynaics textbooks, this is not always correct. Eq. (17) is only true when the obtained perfect-gas ixture (after ixing of stable pure coponents) and the pure coponents are all in the gaseous state. However, a perfect gas ixture can be obtained by ixing a pure gas and a pure liquid or even two pure liquids. As an exaple by ixing at t = 0 C and P = kpa, 9 ol of gaseous toluene(1) and 1 ol of liquid ethylbenzene(), we obtain a gas ixture that can be considered as a perfect-gas ixture since the pressure is low. For such a case, according to Eqs. (14) and (15): g perfect gas ( T,y) P = y 1 ln y 1 + y ln y + y ln ( 18) P Moreover, in such a case, the olar enthalpy change on ixing is far fro zero. [3] One has: h y h perfect gas vaporization, ( 0 C) +4 kj ol 1 ( 19) In return, the olar excess enthalpy h E, is obviously null since a perfect-gas ixture is a particular ideal solution. Be sure that our students open their eyes wide when they are told that the ixing enthalpy of a perfect-gas ixture can be different fro zero. A siilar aazeent arises when we add that for a perfect-gas ixture, the excess enthalpy is necessarily zero, so that h E ay be different fro h. Applications The goal of this section is to highlight that the Gibbs double-tangent construction of coexisting phases is not only a very useful tool to visualize all the solutions of Eq. (1) but also to study the stability of a binary syste of known teperature, pressure, and overall coposition. In advanced textbooks of therodynaics, [5] such a stability analysis is generally perfored with the help of an equation of state capable of representing the liquid and the gas phases. In this case, the obtained atheatical equations are, however, coplex and not easy to handle. [6,7] We here want to show that the sae kind of analysis can be perfored with less effort using the gaa-phi approach. [8,9] Application 1 In this section, it was decided to study the phase behavior of the binary syste butan-- one(1) + water() under atospheric pressure. For this syste, experiental data show the siultaneous existence of liquid-liquid equilibriu and positive azeotropy. Such a coplexity is thus propitious to underline the usefulness of the double-tangent construction. Since the pressure is low, the gas phase was assued to be a perfect-gas ixture. The non- ideality of the liquid phase was accounted through the NL equation [10] with the following teperature-independent paraeters: Vol. 48, No. 1, Winter

5 g 1 = J ol 1 g 1 = J ol 1 α 1 = ( 0) The vapor pressures of the two pure coponents were correlated with the Antoine equation and the resulting expressions are: log( P 1 / bar) = T / C ( 1) log( P / bar) = T / C Our first task was thus to build the liquid-liquid and the liquid-vapor phase diagras. The first one was classically calculated by varying the teperature T. For each teperature, the following two-equation syste was solved in β order to deterine the coposition ( x 1α and x 1 ) of the two liquid phases (L α and L β ) in equilibriu: α x 1α γ 1 T,x 1 α x α γ T,x 1 β β γ 1 ( T,x 1 ) β = x β γ ( T,x 1 ) ( ) The vapor-liquid phase diagra was calculated with a so-called bubble-t algorith: the pressure is fixed at P = 1 at and the coposition of the liquid phase x 1 is varied between zero and one. Knowing P and x 1, the teperature T is the solution of the equation: P P 1 x 1 γ 1 T,x 1 x γ T,x 1 P = 0 3 and the gas phase coposition y 1 is given by: y 1 = P 1 x 1 γ 1 ( T,x 1 ) ( 4) P The corresponding phase diagras can be seen in Figure 1. The liquid-liquid phase diagra in Figure 1a shows that the studied binary syste exhibits an upper critical point (UCP), the coordinates of which are: T UCP = 49.8 K and x 1,UCP = As expected fro the experiental data, Figure 1b highlights that the studied syste also exhibits a positive hoogeneous azeotrope, [11] the coordinates of which are: T az = K and az x 1 = The liquid-vapor phase diagra shown in Figure 1b was built by solving Eqs. (3) and (4) that is, by assuing that VLE could exist in the whole coposition range. As a result, we notice the existence of a closed loop on the dew-point curve that characterizes a etastable gaseous phase. Moreover, it is observable that in a certain coposition range, the bubble-point curve is an increasing function of x 1, eanwhile the Figure 1. Isobaric phase diagras (P = 1 at) of the binary syste butan- -one(1) + water(). a) Liquid-liquid phase diagra. ( ) upper critical point. b) Liquid-vapor phase diagra as directly obtained using a bubble-t algorith (without testing the stability). (+) positive hoogeneous azeotrope. c) Superiposition of the two previous phase diagras in order to highlight the three-phase LLV line. Dashed lines: nonstable parts (etastable and unstable parts) of the dew and bubble curves. d) Correct (stable) phase diagra obtained after reoving all the non-stable parts. (s) phases involved in a 3-phase equilibriu. 46 Cheical Engineering Education

6 Figure. Highlight of the eight selected teperatures (horizontal lines) at which it was decided to perfor a stability analysis under P=1 at. T 1 = K, T =T az = K, T 3 = K, T 4 = K is the three-phase teperature, T 5 = K, T 6 = K is the noral boiling teperature (T b,1 ) of pure butan--one(1), T 7 = K is the noral boiling teperature (T b, ) of pure water(), T 8 = K. dew-point curve is a decreasing function of y 1. Such a behavior breaks the Gibbs-Konovalow theore and indicates that this increasing part of the bubble-point curve is an unstable liquid phase. An easy way to locate the three-phase line that is the consequence of the overlapping of both the LL and the LV phase diagras is to superipose the. This is what has been done in Figure 1c. By the end, after reoving all the non-stable parts of the resulting diagra, the correct (stable) isobaric phase diagra obtained is shown in Figure 1d. In order to perfor the stability analysis of the considered syste with the help of the double-tangent construction of coexisting phases, it was decided to plot the Gibbs energy change on ixing of the previous syste at P = 1 at and at eight teperatures identifiable by horizontal lines in Figure. Each tie, g for both the liquid and the gas phase was plotted on the whole coposition range (the positive values of g were, however, in general not drawn). Under T 1 = K and P = 1 at, the two pure coponents are in a liquid state so that according to Eq. (15): g liquid g gas ln( x 1 γ 1 )+ x ln( x γ ) T,P,y = y 1 ln P y 1 P 1 + y ln P y P Figure 3. Gibbs double-tangent construction of coexisting phases. Exaple of the butan--one(1) + water() binary syste under P = 1 at and at T 1 = K. The double tangent identifies a stable liquid-liquid equilibriu. The crosses (+) identify the starting and the ending point of the double tangent. ( 5) In Figure 3, it is observed that regardless of the overall coposition z 1, g liquid < g gas so that according to Eq. (6), g syste (T 1,P =1 at,z) = g liquid (T 1,P =1 at,x). At this teperature, the binary syste is thus necessarily in the liquid state, i.e., ade up of one or two liquid phases. Vol. 48, No. 1, Winter

7 Figure 4. Coupling of stability analysis and Gibbs double-tangent construction of coexisting phases. Exaple of the butan--one(1) + water() binary syste at P = 1 at and T = T az = K. Full-line double tangent: stable LLE. Dashed double tangents: non-stable VLE. (+) starting and ending points of a double tangent. ( ) particular double tangent of null length. Figure 5. Butan--one(1) + water() syste at T 3 = K. Full-line double tangents: stable equilibria. Dashed double tangents: non-stable equilibria. (+) starting and ending points of a double tangent. In the present case, such a liquid syste is, however, not hoogeneous in the whole coposition range. The presence of a double tangent indicates that LL deixing occurs. The x-coordinates of the tangency points allow deterining the coposition of the two liquid phases in equilibriu. The deterined LLE is stable since both phases isfy the Gibbs stability condition (liquid phase copositions are thus identical to those obtained fro Figure 1.d). Gibbs indeed showed [1] that a necessary and sufficient condition for stability of a binary ixture of known teperature, pressure, and overall coposition is that the tangent to the g curve (plotted at the ixture T and P) at the given overall coposition does not lie above the curve at any coposition. The atheatical expression of g we are talking about is obviously given by Eq. (6). In return, if we can find z 1 [0;1] so that the tangent to the g curve at the given overall coposition is located above (and generally intersects) the g curve, the considered overall coposition is non-stable. In other words, after perforing a phase equilibriu calculation, the calculated equilibriu will be declared stable if the corresponding double tangent never lies above the g curve. This is clearly the case for the LLE shown in Figure 3. At T =T az =347.51K and P=1at, the two pure coponents are in a liquid state so that Eq. (5) still applies. The corresponding curves can be seen in Figure 4. It is noticeable that by increasing the teperature fro T 1 to T, g gas rapidly grows up and at T, the g gas curve becoes tangent to the g liquid curve at a unique point (see the sybol * in Figure 4). At this point, a double tangent, the length of which is zero, can be drawn indicating that a liquid and a gas phase with the sae coposition are in equilibriu. For such a VLE, the Gibbs stability condition is isfied and we are thus at the azeotropic point. In Figure 4, the stable LLE (already present at T 1 ) and which is characterized by a double tangent (DT 1 ) starting and ending on the g liquid curve, is present again. In addition, it is possible to draw two other double tangents starting on the g liquid curve and ending on the g gas curve (see the dashed double tangents DT and DT 3 in Figure 4). Such VLE are, however, not stable although they isfy Eq. (1). We indeed can find copositions for 48 Cheical Engineering Education

8 which the double tangent lies above the g curve. We thus here verify that the equality of the cheical potentials is a necessary but not a sufficient condition to ensure a stable equilibriu. Figure 4 highlights that the double tangent DT starts in a region where the g liquid curve is concave upward (convex), which refers to a etastable hoogeneous liquid phase. In Figure, this point is located on the decreasing part of the dashed bubble-curve. On the other hand, the double tangent DT 3 starts in a region where the g liquid curve is concave downward, which corresponds to an unstable hoogeneous liquid phase. In Figure, this point is located on the increasing part of the dashed bubble-curve. We can notice that these two double tangents end at the g gas curve, which is always convex. Such gas phases are thus etastable and are located on Figure on the dashed closed loop. At T 3 =348.15K and P=1 at, the two pure coponents are in a liquid state so that Eq. (5) still applies. The corresponding curves can be seen in Figure 5. It is noticeable that by again increasing the teperature, we now reach T 3 >T az and the g gas curve, which continues increasing in size, now twice crosses the g liquid curve (see Figure 5). As expected, two stable VLE, located on each side of the azeotropic coposition, have appeared (see double tangents DT 4 and DT 5 in Figure 5). The two double tangents DT and DT 3, which represent non-stable VLE, are still present. We can, however, notice that the etastable liquid and gas phases that bound DT have copositions that are now extreely close to stable copositions (the starting point of non-stable DT is very close to the starting point of stable DT 1 and the ending point of DT is very close to the ending point of stable DT 4 ). At T 4 =349.03K and P =1at, the two pure coponents are in a liquid state so that Eq. (5) still applies. The corresponding curves can be seen in Figure 6. T 4 is the three-phase teperature under P = 1 at. At this teperature, three out of the five double tangents that existed at teperature T 3 (DT 1, DT, and DT 4 ) erge together and give birth to a stable tripletangent. Each of the three tangency points defines a phase of the VLLE. The etastable phases that bounded DT at teperature T 3 thus becoe stable at T 4. Figure 6. Butan--one(1) + water() syste at T 4 = K. Full-line double (or triple) tangents: stable equilibria. Dashed double tangent: nonstable equilibriu. (+) starting and ending points of a double (or triple) tangent. Figure 7. Butan--one(1) + water() syste at T 5 = K. Full-line double tangents: stable equilibria. Dashed double tangent: non-stable equilibriu. (+) starting and ending points of a double tangent. At T 5 = K and P =1at, the two pure coponents are in a liquid state so that Eq. (5) still applies. The corresponding curves can be seen in Figure 7. As expected, at teperature T 5 higher than the three-phase teperature, the LLE Vol. 48, No. 1, Winter

9 becoes non-stable (see Figure 7). It is indeed possible to find copositions for which the dashed double tangent characterizing this equilibriu lies above the g curve. It is recalled that the g curve to be considered is: g ( T,P,z) = in{ g liquid ( T,P,z), g gas ( T,P,z) }. ( 5a) At this teperature and pressure, only two stable VLE located on each side of the azeotrope are observed. stable liquid gas At T 6 =35.73 K=T b,1 and P=1at, pure coponent 1 is in VLE so that g 1 pure (T,P)= g 1 pure (T,P)= g 1 pure (T,P). On the other hand, coponent is still in a liquid state. Eqs. (14) or (15) can thus indifferently be used for coponent 1 whereas Eq. (15) ust be applied to coponent. We thus obtain: g liquid ln P 1 x 1 γ 1 + x ln( x γ ) ln( x 1 γ 1 ) P + x ln ( x γ ) Eq. (15 ) Eg. (14 ) ( 6) g gas = y 1 lny 1 + y ln P y P Eq.(14 ) = y ln P y 1 + y 1 P 1 ln P y P Eq.(15 ) The corresponding curves can be seen in Figure 8. The double tangent representing the VLE located on the right-hand side of the azeotropic coposition (DT 5 in Figure 7) disappears at T 6. Its length becoes null (see the sybol * in Figure 8) so that the g liquid curve and the g gas curve have a coon tangent. This behavior is induced by the VLE of pure coponent 1 and arises therefore at a coposition z 1 =1. stable At T 7 =373.14K=T b, and P=1at, pure coponent is in VLE so that g pure hand, coponent 1 is gaseous. liquid gas (T,P) = g pure (T,P) = g pure (T,P). On the other Eqs. (14) or (15) can thus indifferently be used for coponent whereas Eq. (14) ust be used for coponent 1. We thus obtain: g liquid g gas ln P 1( T ) x 1 γ 1 + x ln P x γ ln P 1 P P Eq. (14 ) = y 1 lny 1 + y lny = y lny + y ln P y 1 1 P Eq.(14 ) The corresponding curves can be seen in Figure 9. At T 7, the double tangent that highlights the VLE located on the left-hand side of the azeotrope (see DT 6 in Figures 7 and 8) disappears and the g liquid curve and the g gas curve becoe tangent at z 1 = 0 (the observed phenoenon is siilar to the one discussed at T 6 ). For all the other copositions, the syste is hoogeneous and gaseous. Eq. (15 ) x 1 γ 1 + x ln( x γ ) P Eq. (15 ) ( 7) Figure 8. Butan--one(1) + water() syste at T 6 = K = T b,1. Full-line double tangent: stable equilibriu. Dashed double tangent: non-stable equilibriu. (+) starting and ending points of a double tangent. ( ) particular double tangent of null length. 50 Cheical Engineering Education

10 Under T 8 = K and P = 1 at, both pure coponents are gaseous so that according to Eq. (14): g liquid g gas ln P 1 x 1 γ 1 + x ln P P = y 1 lny 1 + y lny x γ P The corresponding curves can be seen in Figure 10 (page 5). At this teperature, the syste is solely in a gaseous state. A double tangent characterizing a non-stable LLE is still present. It will disappear when the upper critical solution teperature is reached. Application In this section, it was decided to study the solid-liquid phase diagra of the binary syste water(1) + ethylene glycol(). Condensed phases are seldo sensitive to pressure effects explaining why activity coefficients in the liquid and solid phases are considered as pressure-independent. Consequently, not only the liquidus and the solidus branches of the phase diagra but also the g curves are pressure-independent (fro now on, pressure will thus not be entioned further). For siplicity, it was here liquid liquid assued that the liquid water + ethylene glycol phase was ideal ( γ 1 = γ =1) and that total iiscibility occurred in the solid phase. In Figure 11 (page 53), the SLE behavior of the siple eutectic syste water(1) + ethylene glycol () is shown. Solid phases S 1 and S, respectively, contain pure water and pure ethylene glycol. The phase diagra was built with the following data: T,1 = 73.15K T, = 55.75K h,1 T,1 h, T, = 6009 J ol 1 =11600 J ol 1 The atheatical equation of each liquidus branch is [4] : h T =,i ( T,i ) T,i h,i T,i ( 8) ( 9) ( 30) R T,i lnx i By varying x 1 between zero and one, the phase diagra shown in Figure 11 can be built. The two liquidus branches intersect at an eutectic point (E in Figure 11) where 3 phases ( solid phases and one liquid phase) are siultaneously in equilibriu. The coordinates of this point were found to be: T E = 3K x 1E = 0.55 ( 31) Below T E, the stable state is a solid-solid equilibriu and the non-stable liquidus branches are aterialized by dashed lines in Figure 11. At this step, it was decided to plot the Gibbs energy change on ixing of the considered syste at seven different teperatures: T 1 = 10 K, T = 0 K, T 3 = T E, T 4 = 30 K, T 5 = T,, T 6 = T,1, and T 7 = 80 K. Three different aggregation states are involved in the phase diagra shown in Figure 11: liquid, pure solid S 1, and Figure 9. Butan--one(1) + water() syste at T 7 = K = T b,. Dashed double tangent: non-stable equilibriu. (+) starting and ending points of a double tangent. ( ) particular double tangent of null length. Vol. 48, No. 1, Winter

11 Figure 10. Butan--one(1) + water() syste at T 8 = K. Dashed double tangent: non-stable equilibriu. (+) starting and ending points of a double tangent. pure solid S. Consequently, three g curves need to be plotted at each teperature. Since each solid phase is assued to be pure, the g S1 and g S curves reduce each to a single point. They indeed only have a physical eaning for z 1 = 1 and z 1 = 0, respectively. Fro a atheatical point of view, it is not possible to define the tangent to a curve ade up of a single point and the double-tangent construction needs to be slightly revisited. When the g solid curve reduces to a single point, a solidliquid equilibriu involving a pure solid phase is characterized by a segent that is both tangent to the g liquid curve and that passes through the point representing the solid phase. Matheatically, such a segent is not a double tangent to the g curve but contains the sae inforation (it links two phases in equilibriu). Siilarly, a solid-solid equilibriu between two pure solid phases is characterized by a segent joining the two points characterizing the two pure solid phases. Once again and strictly speaking, such a segent cannot be called a double tangent but it has exactly the sae eaning. For the sake of siplicity, any segent joining two phases in equilibriu on a g plot will be subsequently called a double tangent, even if the equilibriu involves a pure solid phase. For T = T 1 or T or T 3 or T 4, the two pure coponents are solid so that according to Eq. (16): g liquid g S 1 ( T,x) lnx 1 + h,1 T,1,1 + x lnx + h, T, = 0 = 0 g S, The corresponding curves can be seen in Figure 1 (page 54). At teperature T 1, it is possible to draw three double tangents noted DT 1, DT, and DT 3 (see Figure 1a). DT 1 and DT characterize non-stable SLE; they indeed respectively lie above the g curve defined as g ( T,z) = in g liquid ( T,z), g S1, g S T ( 3) { } ( 3a) for z 1 =1 and 0. On the other hand, DT 3 highlights a stable solid-solid equilibriu as expected at this teperature fro Figure 11. At teperature T (see Figure 1b) the situation is siilar except that the g liquid curve increased in size, aking the 3 double tangents get closer to each other. They erge at T 3 (see Figure 1c) and give birth to a unique triple tangent that characterizes the 3-phase (eutectic) teperature. By again increasing the teperature (see Figure 1d), the g liquid curve, which continues growing up, now twice crosses DT 3 which thus becoes non-stable. In return DT 1 and DT becoe stable and characterize the two SLE visible in Figure 11 on each side of the eutectic point. For T 5 = T, pure coponent is in SLE but coponent 1 is still solid. Eq. (15) or (16) can thus indifferently be used for 5 Cheical Engineering Education

12 coponent whereas Eq. (16) ust be applied to coponent 1. We thus obtain: g liquid ( T, x) ln x 1 + h,1 ( T,1 ) 1 T,1 + x ln x + h, ( T, ) 1 T, Eq.(16 ) ln x 1 + h,1 ( T,1 ) 1 T,1 + x ln x Eq.(15 ) g S 1 = 0 g S = 0 = h ( T ),, 1 T Eq. (16 ), Eq. (15 ) ( 33) Between T 4 and T 5, the g liquid curve continues its growth and the abscissa of its iniu oves on the left (towards sall ole fractions). Consequently, the length of the DT 1 double tangent diinishes rapidly. Such a DT, which characterizes the stable S LE, disappears at T 5 = T,. Its length becoes zero (see the sybol * in Figure 13, page 55) and the g liquid and the g S curves are superiposed for z 1 = 0. As a consequence, DT which characterizes the S 1 LE is the unique DT to reain stable at this teperature. For T 6 = T,1 pure coponent 1 is in SLE but coponent is liquid. Eq. (15) or (16) can thus indifferently be used for coponent 1 whereas Eq. (15) ust be applied to coponent. We thus obtain: g liquid g S ( T,x) lnx 1 + h,1 T,1 + x lnx lnx,1 + x lnx 1 Eq. (15 ) Eq. (16 ) = 0 = h T,1,1 Eq. (16 ),1 = h, T, g S Eq. (15 ), 0.35at T,1 ( 34) Figure 11. Solid-liquid and solid-solid equilibria of the water(1) + ethylene glycol() syste. Vol. 48, No. 1, Winter

13 Fro T 5 to T 6, the g liquid curve continues to grow up so that the length of the DT double tangent diinishes: it vanishes at T 6 = T,1 (see the sybol * in Figure 14) and the g liquid and g S1 curves are superiposed for z 1 =1. As a consequence, at this teperature pure coponent 1 is in SLE. The DT 3 double tangent that stands above the g liquid curve still characterizes a non-stable SSE. For T 7 = 80 K, the two pure coponents are liquid so that according to Eq. (15): g liquid ( T,x) lnx 1 + x lnx g S 1 = h,1 ( T,1 ), at T 7 g S = h, ( T, ), 0.47at T 7 The corresponding curves can be seen in Figure 15 (page 56). At this teperature, as expected fro Figure 11, the syste is ( 35) 54 Cheical Engineering Education

14 Figure 1. (facing page) Coupling of stability analysis and Gibbs double-tangent construction of coexisting phases. Exaple of the water(1) + ethylene glycol() syste. Full-line double (or triple) tangents: stable equilibria. Dashed double tangents: non-stable equilibria. (+) starting and ending point of a double (or triple) tangent.(a) T 1 =10 K, (b) T =0 K, (c) T 3 =T E =3 K, (d) T 4 =30K. hoogeneous and totally liquid: the g liquid curve is concave upward and is erged with in { g liquid (T,z), g S1 (T), g S (T) }. A double tangent characterizing a non-stable SSE is still present. Figure 13. Water(1) + ethylene glycol() syste at T 5 = T,. Full-line double tangent: stable solid-liquid equilibriu. Dashed double tangent: non-stable SSE. (+) starting and ending point of a double tangent. ( ) particular stable double tangent of null length. Conclusion In this paper, the double-tangent construction of coexisting phases has been applied to any type of phase equilibriu (VLE, LLE, VLLE, SLE, SSE, SSLE) through two binary systes odeled with the gaa-phi approach. The atheatical expressions of g to be used are siple enough to be easily ipleented by students in tools such as Excel or Matlab. Such a graphical technique is very useful to build any type of isotheral or isobaric phase diagras including the ost coplex ones (with hoogeneous azeotrope, with 3-phase lines...). The coupling of the double-tangent construction and of the stability analysis akes it possible for the students to well understand that the equality of cheical potentials for each coponent in all phases is only a necessary but not a sufficient condition to reach a therodynaically stable equilibriu. Although well-described in the literature, [5,6] such a construction cannot be perfored with an equation of state during class exercises. The equations to handle are indeed coplex and need good atheatical skills. This technique was also applied with the gaa-phi approach by Marcilla s research group in two very interesting papers. [8,9] The pure-coponent reference state they selected in Eq. () was, however, the pure liquid. This choice is obviously adequate but ay, in our opinion, introduce confusion in students who find it hard to understand the difference between Figure 14. Water(1) + ethylene glycol() syste at T 6 = T,1. Dashed double tangent: non-stable SSE. (+) starting and ending point of a double tangent. ( ) particular stable double tangent of null length. Vol. 48, No. 1, Winter

15 g, properly defined in Eq. (5), and: g liq ( T,P,z) = g T,P,z z i liquid g T,P i=1 ( 36) Moreover, at a given T and P, the pure liquid is not always stable and by experience, the students do not like working with hypothetical states. In return, to our knowledge, this paper is the first one in which such a technique is used taking as reference the stable pure coponent at the considered T and P. Such a choice is extreely convenient since: 1) It never creates hypothetical states. ) It does not lead to the definition of a new therodynaic function. The students only need to know the correct definition of the olar Gibbs energy change on ixing. Indeed, in such a case: g stable ( T,P,z) = g T,P,z z i stable g T,P i=1 = g ( T,P,z) ( 37) equations are applied to ordinary cheicals? Part : study of solid-liquid equilibria in binary systes, Fluid Phase Equilib., 318, 61 (01) 8. Marcilla, A., J.A. Conesa, and M.M. Olaya, Coents on the probleatic nature of the calculation of solid-liquid equilibriu, Fluid Phase Equilib., 135, 169 (1997) 9. Del Mar Olaya, M., J.A. Reyes-Labarta, M.D. Serrano, and A. Marcilla, Vapor-liquid equilibria using the Gibbs energy and the coon tangent plane criterion, Che. Eng. Ed., 44(3), 36 (010) 10. Renon, H., and J.M. Prausnitz, Local copositions in therodynaic excess functions for liquid ixtures, AIChE J., 14, 135 (1968) 11. Jaubert, J.N., and R. Privat, Possible existence of a negative (positive) hoogeneous azeotrope when the binary ixture exhibits positive (negative) deviations fro ideal solution behavior (that is when ge is positive (negative)), Ind. Eng. Che. Res., 45, 817 (006) 1. Gibbs, J.W., A Method of Geoetrical Representation of the Therodynaic Properties of Substances by Means of Surfaces, Transaction of the Connecticut Acadey, II, 38 (1873) p References 1. Sith, J.M., H.C. Van Ness, and M.M. Abott, Introduction to Cheical Engineering Therodynaics, 7th Ed., New York: Mc Graw Hill (005). Sandler, S.I., Cheical, Biocheical, and Engineering Therodynaics, 4th Ed., New York: John Wiley & Sons, Inc. (006) 3. Privat, R., and J.N. Jaubert, Discussion around the paradig of ideal ixtures with ephasis on the definition of the property changes on ixing, Che. Eng. Sci., 8, 319 (01) 4. Gehling, J., B. Kolbe, M. Kleiber, and J. Rarey, Cheical Therodynaics for Process Siulation, Weinhei: Wiley-VCH (01) 5. Michelsen M.L., and J. Mollerup, Therodynaic Models: Fundaentals and Coputational Aspects, Holte: Tie-line Publications (007) 6. Baker, L.E., A.C. Pierce, and K.D. Luks, Gibbs energy analysis of phase equilibria, SPE Journal, (5), 731 (198) 7. Privat, R., E. Conte, J.N. Jaubert, and R. Gani, Are safe results obtained when SAFT Figure 15. Water(1) + ethylene glycol() syste at T 7 = 80 K. Dashed double tangent: non-stable SSE. (+) starting and ending point of a double tangent. 56 Cheical Engineering Education