ChE curriculum
VAPOR-LIQUID EQUILIBRIA Using the Gibbs Energy and the Common Tangent Plane Criterion
María del Mar Olaya, Juan A. Reyes-Labarta, María Dolores Serrano, Antonio Marcilla University of Alicante • Apdo. 99, Alicante 03080, Spain hase thermodynamics is often perceived as a difficult overall composition. This is the case with the binary system subject with which many students never become fully in Figure 1(a); it is homogeneous for all compositions. The gM comfortable. It is our opinion that the Gibbsian geo- vs. composition curve is concave down, meaning that no split Pmetrical framework, which can be easily represented in Excel occurs in the global mixture composition to give two liquid spreadsheets, can help students to gain a better understanding phases. Geometrically, this implies that it is impossible to find of phase equilibria using only elementary concepts of high two different points on the gM curve sharing a common tangent school geometry. line. In contrast, the change of curvature in the gM function Phase equilibrium calculations are essential to the simula- as shown in Figure 1(b) permits the existence of two conju- tion and optimization of chemical processes. The task with gated points (I and II) that do share a common tangent line these calculations is to accurately predict the correct number and which, in turn, lead to the formation of two equilibrium of phases at equilibrium present in the system and their com- liquid phases (LL). Any initial mixture, as for example zi in positions. Methods for these calculations can be divided into Figure 1(b), located between the inflection points s on the M 2 M dx2 two main categories: the equation-solving approach (K-value g curve, is intrinsically unstable (d g / i <0) and splits method) and minimization of the Gibbs free energy. Isofugac- Antonio Marcilla is a professor of chemical engineering at Alicante Uni- ity conditions and mass balances form the set of equations in versity. He has presented courses in Unit Operations, Phase Equilibria, the equation-solving approach. Although the equation-solving and Chemical Reactor Laboratories. His research interests are pyrolisis, approach appears to be the most popular method of solution, it liquid-liquid extraction, polymers, and rheology. He is also currently involved with the study of polymer recycling via catalytic cracking and does not guarantee minimization of the global Gibbs energy, the problem of simultaneous correlation of fluid and condensed phase which is the thermodynamic requirement for equilibrium. equilibria. This is because isofugacity criterium is only a necessary but María del Mar Olaya completed her B.S. in chemistry in 1992 and Ph.D. in chemical engineering in 1996. She teaches a wide range of courses not a sufficient equilibrium condition. Minimization of the from freshman to senior level at the University of Alicante, Spain. Her global Gibbs free energy can be equivalently formulated as research interests include phase equilibria calculations and polymer the stability test or the common tangent test. structure, properties, and processing. Juan A. Reyes-Labarta received both his B.S. and Ph.D. in chemical The Gibbs stability condition is described and has been engineering in 1993 and 1998, respectively, at the University of Alicante used extensively in many references.[1, 2] It has been more (Spain). After post-doctoral stays at Carnegie Mellon University (USA) and the Institute of Polymer Science and Technology-CSIC (Spain), he is now frequently applied in liquid-liquid equilibrium rather than a full-time lecturer in Separation Processes and Molding Design. vapor-liquid equilibrium calculations. Gibbs showed that a María Dolores Serrano is a recent chemical engineering graduate of the necessary and sufficient condition for the absolute stability of University of Alicante. She is currently a post-graduate student, working a binary mixture at a fixed temperature, pressure, and overall on different aspects of phase equilibria. composition is that the Gibbs energy of mixing (gM) curve at no point be below the tangent line to the curve at the given © Copyright ChE Division of ASEE 2010 236 Chemical Engineering Education (a) (b)
Figure 1. Dimensionless Gibbs energy of mixing (gM) for a binary liquid mixture as a function of the molar fraction of component
i (xi): (a) for a homogeneous system, and (b) for a heteroge- neous system (LL).
and gL. Obviously, a common reference state must be used for each of the components in the calculations, and for both of the phases involved (for example the pure component as liquid at the same P and T). VLE exists at a given T and P, whenever
any initial mixture, such as zi in Figure 2, is thermodynami- cally unstable and splits into a vapor and liquid phases having
compositions yi and xi, respectively, with a common tangent line to both gV and gL curves and a lower value for the overall Gibbs energy of mixing (M’’
Vol. 44, No. 3, Summer 2010 237 This activity is framed in the subject of Principals of Gibbs Energy for the Liquid Phase Separation Processes in Chemical Engineering in a fourth- For liquid mixtures the ideal Gibbs energy of mixing (di- year course of a five-year program of chemical engineering. mensionless) is This task takes place during the first part of the course and consists of: gid,L x ln x ()5 =∑ i ⋅ i a) A lecture of the theoretical fundamentals of VLE, presen- i tation and analysis of T-x,y diagrams and sketch of Gibbs stability criterion. where the reference state for each component i is the pure b) A guided classroom solution of different VLE problems liquid at the temperature and pressure of the system. with increasing difficulty. Previous presentation of a simple Many equations have been proposed to model the excess non-azeotropic VLE binary system, and consequent solu- Gibbs energy and can be found in the literature. For example, tion of a binary homogenous azeotropic mixture, where the van Laar equation for binary systems: students work on their own computers using Excel and AA⋅ ⋅x⋅ x Matlab. gEL, = 12 21 1 2 ()6 c) Development of a project in groups of students where an A12 ⋅ x1 + A 21⋅ x 2 analogous analysis of VLE, but also of LLE, must be done for a heterogeneous azeotropic binary mixture. where A12 and A21 are the binary interaction parameters of d) As optional projects, VLE for ternary or more compli- the model that must be obtained by equilibrium data cor- cated mixtures could also be considered, but the visual- relation. ization becomes more complicated. THEORY Guess parameters The equilibrium condition for component i being simultane- ously present as a liquid and vapor equilibrium phase can be written as the isofugacity condition Guess Tb i v L fi = fi ()1
c P v 0 V oL oL i p , J i ϕi ⋅P ⋅ yi=p i ⋅ϕi ⋅exp dP⋅γi ⋅x i (2) i i ∫Po i RT
v where ϕi is the fugacity coefficient for the vapor phase, P is y p 0 J x P i the total pressure, yi and xi are the molar fractions of compo- i i i oL i nent i in the vapor and liquid phases, respectively, pi is the oL vapor pressure of component i, ϕi is the fugacity coefficient po for the liquid phase at i , γi is the activity coefficient, defined vc by the selected model, and i is the molar volume of the no n ¦ yi 1 condensed phase as a function of pressure. The exponential i 1 correction is the Poynting factor which takes into account that the liquid is at a pressure P different from the liquid satura- po [7] tion pressure i . yes v oL At moderate pressures, ϕi , ϕi , and the Poynting factor are near unity, and Eq. (2) can be rewritten as NP n exp 2 o Min( ¦ ¦ (yi yi ) P⋅ yi = p i ⋅γi ⋅ x i ()3 no k 1i 1 NP exp 2 ¦ (Ti Ti ) ) The constant pressure T-x,y diagram can be obtained by k 1 calculating the bubble temperatures (Tb) of the various liquid mixture compositions (xi), and also by calculating the compo- yes sition of the equilibrium vapor phase (yi). The calculation is to be carried out according to the flow chart inFigure 3. END On the other hand, the Gibbs energy of mixing (dimension- less) is the sum of two contributions, the ideal and excess Figure 3. Flow chart for the calculation of the activity Gibbs energies: coefficient model parameters by equilibrium data regres- sion. The discontinuous line represents the algorithm for M id E g= g + g ()4 the bubble point equilibrium calculations.
238 Chemical Engineering Education Substituting Eq. (5) and Eq. (6) into Eq. (4), and consider- For non-isothermal VLE, however, the Gibbs energy for ing the selected reference state, the expression obtained for both p phases (V and L) must be calculated as a function of the Gibbs energy of the liquid phase at the temperature of temperature T using a reference state (i.e., the pure liquid) p the system T is: at a unique reference temperature T0 g (T0). In this case, the AA⋅ ⋅x⋅ x Gibbs energy for both vapor and liquid phases includes an gL T =gM,L = x ln x+ x ln x + 12 21 1 2 ()7 () 1 1 2 2 entropic term from T0 to T. A12 ⋅ x1 +A 21 ⋅ x 2 For isobaric conditions, Eq. (11) becomes: Gibbs Energy for the Vapor Phase 1 T T Cp gp T = gp T − L ⋅dTdT ()12 ()0 () ∫ ∫ The vapor phase is considered ideal and, therefore, the RT T0T 0 T following equation is used for the Gibbs energy of mixing of the vapor phase: where CpL is the heat capacity of the liquid. In the above MV,,id V expression gp(T) is calculated with Eq. (7) for p=L or with g= g = y1ln y 1 + y2ln y 2 ()8 Eq. (10) for p=V. To compare Gibbs energy curves for liquid and vapor phases both must be obtained from a common reference state. For PROBLEM STATEMENT VLE calculations at constant T and P a convenient reference For the binary homogeneous azeotropic system ethanol (1) + state is the pure component as liquid at the same T and P of benzene (2) at P=1 atm: the system. In this case, the Gibbs energy of pure component a) Calculate the parameters for the selected thermodynamic go, V ≠ 0 go, L = 0 i in the vapor and liquid phases are i and i , model regressing T-x,y experimental data. respectively. The difference in the Gibbs energy between a pure vapor and a pure liquid can be approximated using the b) Build the temperature vs. composition (T-x,y) diagram with an Excel spreadsheet. following equation: P c) Represent graphically in a 3-D diagram (for example, us- go,, V go L ln ()9 ing Matlab) the g (vapor and liquid) vs. composition and i −i = o pi temperature surfaces for this system.
Consequently the following equation is used for the Gibbs d) For a more precise analysis of the above 3-D figure, energy of the vapor phase, referred to the liquid aggregation plot g curves in Excel for the vapor and liquid mix- state at the temperature of the system T: tures of the following isotherms: 90.0 ˚C, 79.0˚ C, 72.0˚ C, temperature of the calculated azeotrope, and gV T =y ⋅ go,V +y ⋅ go,V +yln y + yln y ()10 () 1 1 2 2 1 1 2 2 60.0 ˚C. The number of phases present and their o, V compositions must be deduced using the Gibbs According to Eq. (9), the sign of g may be positive or i common tangent test. negative; depending on the ratio between the total pressure of the system and the vapor pressure of component i, which, e) Show that the results obtained using the Gibbs in turn, depends on the temperature: stability criteria are consistent with those obtained o o,, V o L o, V using the T-x,y diagram. −If P > pi →gi − gi >0 → gi is positive −If P < po → go,, V − go L < 0→ go, V is negative The Van Laar equation can be used to represent the excess i i i i E Gibbs energy (g ) and the activity coefficient (γi) of the liquid After both the gV and gL isotherm curves have been calcu- mixtures. The vapor phase can be considered as ideal. lated and represented, the Gibbs stability or common tangent plane test can be easily applied to them to ascertain which SOLUTION of the phases are most stable and what the equilibrium com- The Aij parameters for the Van Laar model [Eq. (6)] calculated positions are. for the ethanol (1) + benzene (2) binary system have been cal- The reference state is defined to be arbitrary and for a system culated by fitting VLE data[8-10] according to the diagram flow at constant T and P can be selected to be the liquid state at shown in Figure 3. Figure 4 (next page) shows an example the temperature of the system for both components in both of the possible spreadsheet distribution to obtain the model phases. If Gibbs energy surfaces for both p phases are gener- parameter values and the T-x,y diagram by successive bubble ated to analyze the equilibrium as a function of temperature temperature calculations with Solver function in Excel using T, however, a reference state must be used in the calculations various liquid mixture compositions (xi) and calculating the equilibrium vapor phase composition. The parameter values at a unique temperature T0. T T obtained are A12=1.965 and A21=1.335 (dimensionless). As can p p 1 1 g() T0 = g() T −S ⋅ dT +V ⋅ dP ()11 be seen in Figure 4, a homogeneous azeotropic point occurs for ∫T ∫T RT 0 RT 0 this system at a minimum boiling temperature.
Vol. 44, No. 3, Summer 2010 239 Figure 4 (right). Spreadsheet ex- ample of the Van Laar parameters calculation re- gressing T-x,y ex- perimental data for the ethanol (1) + benzene (2) binary system at P=760 mmHg.
Figure 5 (below). Gibbs energy surfaces for vapor
Vg (ideal) and
liquid Lg (Van Laar) mixtures as a function of the temperature and composition for the ethanol (1) + benzene (2) binary system at P=1 atm.
(gV lower than gL) and the liquid phase the stable aggregation state at lower temperatures (gL lower than gV). For a more detailed analysis of this figure some sectional planes have been selected, corresponding to the following isotherms: 90 ˚C, 79 ˚C, 72 ˚C, 68.01 ˚C (calculated azeotrope), and 60 ˚C. Figure 6 shows the Gibbs energy curves plotted for the vapor and the homogeneous liquid mixtures at each one of these temperatures. After students have completed these representations, an analysis of Figure 6 is done, taking into account the Gibbs stability criteria: o o • For T=90 ˚C, P=1atm < p2 < p1 ; therefore, taking into o, V o, L V account Eq. (9), gi < gi for i=1, 2, and the entire g vs. composition curve is lower than gL [Figure 6(a)], showing that the vapor phase is the stable aggregation state over the entire composition space at this temperature. po o • For T=79 ˚C, 2 < P=1atm < p1; therefore, for the ethanol Figure 5 shows the 3D graph used to represent the Gibbs o, V o, L o, V o, L L V g g g g V energy surfaces of the liquid (g ) and vapor phases (g ) as a component, 1 < 1 , but for benzene, 2 > 2 . The g L function of temperature and composition. The selected refer- and g curves have two points sharing a common tangent ence state for each one of the components is the liquid state at line, corresponding to the VL equilibrium y1= 0.0355 and x =0.00497. At molar fractions below z =0.00497 the azeotrope boiling point (T0). The Gibbs energy surfaces 1 1 for L and V have been calculated using Eq. (12) where gP(T) the liquid is the stable aggregation state, and at values when p=liquid is calculated with Eq. (7) and when p=vapor higher than z1=0.0355 the vapor is the stable phase o, V [Figure 6(b)]. Any global mixture between those values is calculated with Eq. (10). The values for gi are calculated with Eq. (9), where the vapor pressures for ethanol and ben- will split in the VLE. zene have been obtained using the Antoine equation, with the o o o, V o, L • For T=72 ˚C, p2 < p1 < P=1atm; therefore, g > g [11] 2 2 constants given in Table 1. The entropy changes of Eq. (12) for both ethanol and benzene components. Two regions [12] are calculated with Cp(T) given in Table 2. of the gV and gL curves each contain one point of VL
As can be seen in Figure 5, the g surfaces cross each other equilibrium: [y1= 0.269, x1= 0.0708] and [y1’= 0.681, so that the vapor phase is the stable phase at high temperatures x1’= 0.861] having common tangent lines that connect 240 Chemical Engineering Education the conjugated y-x equilibrium (a) T=90ºC (b) T=79ºC compositions, as can be seen in 0.2 Figure 6(c). The vapor is the 0.2 0 stable aggregation state at in- 0.0 0.0 0.2 0.4 0.6 0.8 1.0 termediate concentrations and -0.2 0 0.2 0.4 0.6 0.8 1 the liquid is the stable phase -0.4 -0.2 g -0.6 near each pure component. y1 g -0.4 x1 V -0.8 • The g curve rises as the tem- -0.02 0 0.02 0.04 -1 -0.6 perature decreases until the -0.06
-1.2 M g -0.10 azeotropic temperature (68.01 -0.8 V x1, y1 ˚C) is reached. Here, both g -0.14 L and g curves are tangent in -1.0 -0.18 x , y one point [Figure 6(d)]. This x1, y1 1 1 point corresponds to the homo- (c) T=72ºC (d) T=68.01ºC (calculated azeotrope) geneous azeotrope, for which 0.30 0.4
the vapor and liquid phases 0.3 0.10 x1 y1 y1' x1' in equilibrium have identical 0.2
compositions (y1= x1=0.441). 0.1 -0.10 0 0.2 0.4 0.6 0.8 1 x1=y1 L g • For T=60 ˚C, the g curve g 0.0 V 0 0.2 0.4 0.6 0.8 1 lies below the g curve over -0.30 -0.1 the entire composition space, -0.2 -0.50 demonstrating that a homoge- -0.3
neous liquid phase is the most -0.70 -0.4
x1, y1 stable aggregation state for any x1, y1 global mixture composition [Figure 6(e)]. (e) T=60ºC It must be highlighted that all of 0.8 the above, deduced from the Gibbs 0.6 energy curves, is obviously consis- Tangent line 0.4 tent with the T-x,y diagram shown g 0.2 Equilibrium compositions in Figure 4. Treatment of the VLE L 0.0 g calculation using the Gibbs common -0.2 0 0.2 0.4 0.6 0.8 1 V tangent plane criteria provides stu- g -0.4 dents with a deeper understanding x1, y1 of the problem, however, because the insight into the reasons for the Figure 6. Analysis of the Gibbs energy curves for vapor ( Vg ) and liquid ( Lg ) mix- V or L phase stability or the VLE tures at different temperatures for the ethanol (1) + benzene (2) binary system at P=1 atm showing the common tangent equilibrium condition. splitting is much more evident than with using the isofugacity condition. Although solving the isofugacity condition together with the mass balance equa- Table 1 Antoine Equation Constants for Ethanol and Benzene[10] tions (K-value method) constitutes the most popular method o o of calculation, our experience has shown that the Gibbsian log(p )=A-B/(T+C) (p in bar, T in ˚C) geometrical framework is a very useful tool for educational A B C purposes. Students state that the geometric analysis of chemi- Ethanol 5.33675 1648.220 230.918 cal equilibrium, with an available and easy to use program Benzene 3.98523 1184.240 217.572 such as Excel, permits a clear understanding of the VLE split- ting in terms of Gibbs energy minimization. Table 2 Heat Capacity Constants of Liquid for Ethanol and Benzene[12] EXTENSION TO HETEROGENEOUS CpL=A+BT+CT2+DT3 (J/mol/K, T in K) AZEOTROPIC MIXTURES A B C D An extension of this exercise is proposed where Ethanol 59.342 0.36358 -0.0012164 1.803·10-6 the VLE is studied for a heterogeneous instead of a homogeneous azeotropic system. The binary system Benzene -31.662 1.3043 -0.0036078 3.8243·10-6 Vol. 44, No. 3, Summer 2010 241 water-n-butanol at P = 1 atm is an example of a heterogeneous must be specified to calculate the LL equilibrium compo- azeotrope. The vapor phase can be considered ideal as in the sitions. Figure 9(a) shows the isotherm corresponding to L L1 previous example. The NRTL equation can be used to repre- T=91 ˚C where two points of the g curve (x1 = 0.623, E L2 sent the excess Gibbs energy (g ) and the activity coefficient x1 = 0.978) have a common tangent line, demonstrating
(γi) of the liquid mixtures. The equation parameters can be that a liquid phase splitting is the most stable situation obtained by fitting the VLE and VLLE data at 1 atm[10,13,14] for any global mixture composition z comprised between L1 L2 following the same calculation algorithm shown in the pre- them, x1 P= 1 atm 120 115 110 105 V L+V 100 95 92.7ºC V+L T (ºC)T 90 Azeotrope VLL 85 80 L 75 L+L 70 0.0 0.2 0.4 0.6 0.8 1.0 x1, y1 Figure 8. Gibbs energy surfaces for vapor ( Vg ) and liquid ( Lg ) mixtures as a function of the temperature Figure 7. Temperature versus liquid (x) and vapor (y) mo- and composition for the water (1) + n-butanol (2) binary lar fractions for the binary system water (1) + n-butanol system at P=1 atm. The isotherm curves (T=92.7oC) of the (2) at P=1atm, including the azeotropic isotherm line. azeotrope have also been included on the surfaces. 242 Chemical Engineering Education have a common tangent line. The existence of this VLL the T-x,y diagrams. This graphical analysis proves that the equilibrium is consistent with the T-x,y representation vapor is the stable phase at high temperatures, the liquid phase of this system (Figure 7). is the stable aggregation state for lower temperatures, and that With this example, the students demonstrate the reason for the azeotrope (VLE) corresponds to a temperature where both the VLL splitting in terms of stability or the minimum Gibbs liquid and vapor Gibbs energy curves are tangent in one point. energy of the system. Students use this previous spreadsheet to develop an extension to consider the VLLE of a heterogeneous azeotropic system, CONCLUSIONS which is tackled as a project in groups. Their reports of results Dealing with the VLE calculation in terms of the Gibbs show that the reasons for the V or L phase stability or the VLE common tangent criteria provides students with a deeper under- and VLLE splitting is much more evident with the Gibbsian standing of the problem than using the isofugacity condition. framework than using the isofugacity condition. An exercise of application of the Gibbs common tangent criteria to VLE has been proposed for a homogeneous azeo- NOMENCLATURE F tropic binary system at a constant pressure using simple tools fi Fugacity of component i in phase F such as Excel spreadsheets and Matlab graphics. The Gibbs P Pressure energy surface and curves at different temperatures have been p phase analyzed to compare distinct situations that are consistent with f Degrees of freedom (Phase Rule) (a) T=91ºC (b) T=94ºC x1 y1 y1' x1' L1 L2 0.00 x1 x1 0.00 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -0.05 -0.05 -0.10 g -0.10 g -0.15 -0.15 -0.20 -0.20 -0.25 x1, y1 -0.25 x1, y1 (c) T=92.7ºC (calculated azeotrope) T = 92.7ºC L1 L2 x1 y1 x1 0 0 0.2 0.4 0.6 0.8 1 -0.05 Tangent line -0.1 g Equilibrium compositions L -0.15 g gV -0.2 -0.25 x1, y1 Figure 9. Analysis of the Gibbs energy curves for vapor ( Vg ) and liquid ( Lg ) mixtures at different temperatures for the water (1) + n-butanol (2) binary system at P=1 atm, showing the common tangent equilibrium condition. Vol. 44, No. 3, Summer 2010 243 Tb Boiling temperature REFERENCES n Number of components 1. Wasylkiewicz, S.K., S.N. Sridhar, M.F. Doherty, and M.F. 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Chemistry Data The authors gratefully acknowledge financial support from Series, DECHEMA, (2003) the Vicepresidency of Research (University of Alicante) and 14. Iwakabe, K., and H. Kosuge, “A Correlation Method For Isobaric Va- por-liquid and Vapor-liquid-liquid Equilibria Data of Binary Systems,” Generalitat Valenciana (GV/2007/125). Fluid Phase Equilib., 192, 171- 186 (2001) p 244 Chemical Engineering Education