CHE 408 SEMI-FINALS - Phase Equilibria PDF

ChE 408 | SEMI-FINALS MODULE Module Introduction This module discusses Phase Equilibria of Solution Thermodynamics. It involves the gamma/phi formulation, VLE from Cubic EOS, Equilibrium and St...

ChE 408 | SEMI-FINALS MODULE Module Introduction This module discusses Phase Equilibria of Solution Thermodynamics. It involves the gamma/phi formulation, VLE from Cubic EOS, Equilibrium and Stability, Liquid/Liquid Equilibrium, and Vapor/Liquid/Liquid Equilibrium. This subject deals with a lot of correlation, may it be taken from a certain reference of from Perry’s Handbook. Majority of the discussions that will be presented here are taken from Introduction to Chemical Engineering Thermodynamics by J.M Smith, H. Van Ness, et.al. Should you want to know more about the topic, one can look for this book for further reading. All modules are also uploaded in the respective Google Classroom. Other videos from YouTube will be linked for further references. This will also aid you in understanding the concept even more. Exams have a hard copy version, though it is advised to take the online exam for you to easily check your work and review your mistakes. Assessment will be made after every module. Intended Learning Outcome The following are the learning outcomes that will be acquired by the students after finishing the course: 1. Apply the concepts of physical chemistry and techniques in calculus to derive other thermodynamic property relations from fundamental property relations, calculate changes in the thermodynamic properties of homogeneous mixtures, derive the phase equilibrium relation and chemical equilibrium relations; 2. Identify and solve vapor-liquid equilibrium problems for both ideal and non-ideal solutions; 3. Apply vapor-liquid equilibrium relations based on cubic equations of state and other EOS models; 4. Derive solution properties from vapor-liquid equilibrium experimental data; 5. Interpret phase equilibrium diagrams; 6. Solve for the equilibrium conversion of single reaction systems and analyze the effect of operating variables on chemical reaction conversion; 7. Use spreadsheets and numerical computing software in vapor-liquid equilibrium calculations, construction of phase equilibrium diagrams and solving for the equilibrium conversion of multi-reaction systems.transfer. These are based on CMO No. 19 s.2017 for Chemical Engineering Course. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE Phase Equilibria A. Review of Phase Rule The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states not all these variables are independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and it is called the number of degrees of freedom of the system. It is the number of variables that may be arbitrarily specified and that must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. For a system containing 𝑁 chemical species distributed at equilibrium among π phases, the phase rule variables are 𝑇 and 𝑃, presumed uniform throughout the system, and 𝑁 − 1 mole fractions in each phase. The number of these variables is 2 + (𝑁 − 1)𝜋. The masses of the phases are not phase rule variables, because they have nothing to do with the intensive state of the system. Duhem’s Theorem: Because the phase rule treats only the intensive state of a system, it applies to both closed and open systems. Duhem’s theorem, on the other hand, is a rule relating to closed systems only: For any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined by any two properties of the system, provided only that the two properties are independently variable at the equilibrium state. The meaning of completely determined is that both the intensive and extensive states of the system are fixed; not only are 𝑇, 𝑃, and the phase compositions established, but so also are the masses of the phases. B. Equilibrium and Phase Stability In the preceding discussion, we have assumed a lot of homogeneous mixtures that obey ideality. Our everyday experience tells us that such an assumption is not always valid; oil-and-vinegar salad dressing provides a prototypical example of its violation. In such cases, the Gibbs energy is lowered by the liquid splitting into two separate phases, and the single phase is said to be unstable. In this section, we demonstrate that the equilibrium state of a closed system at fixed 𝑇 and 𝑃 is that which minimizes the Gibbs energy, and we then apply this criterion to the problem of phase stability. Consider a closed system containing an arbitrary number of species and comprised of an arbitrary number of phases in which the temperature and pressure are spatially uniform (though not necessarily constant in time). The system is initially in a nonequilibrium state with respect to mass transfer between phases and chemical reaction. Changes that occur in the system are necessarily irreversible, and they take the system ever closer to an equilibrium state. We imagine that the system is placed in surroundings such that the system and surroundings are always in thermal and mechanical equilibrium. Heat exchange and expansion work are then accomplished reversibly. Under these circumstances the entropy change of the surroundings is: 𝑑𝑄𝑠𝑢𝑟𝑟 𝑑𝑄 𝑑𝑆𝑠𝑢𝑟𝑟 = =− 𝑇𝑠𝑢𝑟𝑟 𝑇 MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE The final term applies to the system, for which the heat transfer 𝑑𝑄 has a sign opposite to that of 𝑑𝑄𝑠𝑢𝑟𝑟 , and the temperature of the system 𝑇 replaces 𝑇𝑠𝑢𝑟𝑟 , because both must have the same value for reversible heat transfer. The second law requires: 𝑑𝑆 𝑡 + 𝑑𝑆𝑠𝑢𝑟𝑟 ≥ 0 where 𝑆 𝑡 is the total entropy of the system. Combination of these expressions yields, upon rearrangement: 𝑑𝑄 ≤ 𝑇𝑑𝑆 𝑙 Application of the first law provides: 𝑑𝑈 𝑙 = 𝑑𝑄 + 𝑑𝑊 = 𝑑𝑄 − 𝑃𝑑𝑉 𝑡 𝑑𝑄 = 𝑑𝑈 𝑡 + 𝑃𝑑𝑉 𝑡 Combining this equation gives: 𝑑𝑈 𝑡 + 𝑃𝑑𝑉 𝑡 − 𝑇𝑑𝑆 𝑡 ≤ 0 Eq. 1.0 Because this relation involves properties only, it must be satisfied for changes in the state of any closed system of spatially uniform 𝑇 and 𝑃, without restriction to the conditions of reversibility assumed in its derivation. The inequality applies to every incremental change of the system between nonequilibrium states, and it dictates the direction of change that leads toward equilibrium. The equality holds for changes between equilibrium states (reversible processes). Equation 1.0 is so general that application to practical problems is difficult; restricted versions are much more useful. For example, by inspection: (𝑑𝑈 𝑡 )𝑆 𝑡,𝑉 𝑡 ≤ 0 where the subscripts specify properties held constant. Similarly, for processes that occur at constant 𝑈 𝑡 and 𝑉 𝑡 , (𝑑𝑆 𝑡 )𝑈𝑡 ,𝑉 𝑡 ≥ 0 An isolated system is necessarily constrained to constant internal energy and volume, and for such a system it follows directly from the second law that the last equation is valid. If a process is restricted to occur at constant 𝑇 and 𝑃, then (𝑑𝐺 𝑡 ) 𝑇,𝑃 ≤ 0 (Eq. 2.0). This equation indicates that all reversible processes occurring at constant T and P proceed in such a direction as to cause a decrease in the Gibbs energy of the system. Therefore: The equilibrium state of a closed system is that state for which the total Gibbs energy is a minimum with respect to all possible changes at the given T and P. This criterion of equilibrium provides a general method for determination of equilibrium states. One writes an expression for 𝐺 𝑡 as a function of the numbers of moles (mole numbers) of the species in the several phases and then finds the set of values for the mole numbers that minimizes 𝐺 𝑡 , subject to the constraints of mass and element conservation. This procedure can be applied to problems of phase equilibrium, chemical-reaction equilibrium, or combined phase and chemical-reaction equilibrium, and it is most useful for complex equilibrium problems. Equation 2.0 provides a criterion that must be satisfied by any single phase that is stable with respect to the alternative that it split into two phases. It requires that the Gibbs energy of an equilibrium state be the minimum MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE value with respect to all possible changes at the given 𝑇 and 𝑃. Thus, for example, when mixing of two liquids occurs at constant 𝑇 and 𝑃, the total Gibbs energy must decrease, because the mixed state must be the one of lower Gibbs energy with respect to the unmixed state. As a result: 𝐺 𝑡 = 𝑛𝐺 < ∑ 𝑛𝑖 𝐺𝑖 from which 𝐺 < ∑ 𝑥𝑖 𝐺𝑖 𝑖 𝑖 or 𝐺 − ∑ 𝑥𝑖 𝐺𝑖 < 0 (const T, P) 𝑖 According to the definition of Gibbs free energy, the quantity on the left is the Gibbs-energy change of mixing. Therefore, 𝛥𝐺 < 0. Thus, as noted in the previous discussions, the Gibbs-energy change of mixing must always be negative, and a plot of 𝐺 vs. 𝑥1 for a binary system must appear as shown by one of the curves of Fig. 1.0. With respect to curve II, however, there is a further consideration. If, when mixing occurs, a system can achieve a lower value of the Gibbs energy by forming two phases than by forming a single phase, then the system splits into two phases. This is in fact the situation represented between points 𝛼 and 𝛽 on curve II of the figure below because the straight dashed line connecting points α and β represents the overall values of 𝐺 for the range of states consisting 𝛽 of two phases of compositions 𝑥1𝛼 and 𝑥1 in various proportions. Thus the solid curve shown between points 𝛼 and 𝛽 cannot represent stable phases with respect to phase splitting. The equilibrium states between 𝛼 and 𝛽 consist of two phases. Figure 1.0. Gibbs-energy change of mixing. Curve I, complete miscibility; curve II, two phases between 𝛼 and 𝛽. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York (2005).] MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE These considerations lead to the following criterion of stability for a single-phase binary system for which 𝛥𝐺 ≡ 𝐺 − 𝑥1 𝐺1 − 𝑥2 𝐺2 : At fixed temperature and pressure, a single-phase binary mixture is stable if and only if 𝜟𝑮 and its first and second derivatives are continuous functions of 𝒙𝟏 , and the second derivative is positive. Thus, 𝑑 2 Δ𝐺 >0 (const 𝑇, 𝑃) 𝑑𝑥12 and 𝑑 2 (Δ𝐺/𝑅𝑇) >0 (const 𝑇, 𝑃) 𝑑𝑥12 This requirement has a number of consequences. For binary system: Δ𝐺 𝐺𝐸 = 𝑥1 ln 𝑥1 + 𝑥2 ln 𝑥2 + 𝑅𝑇 𝑅𝑇 from which Δ𝐺 𝐺𝐸 𝑑( ) 𝑑( ) 𝑅𝑇 = ln 𝑥 − ln 𝑥 + 𝑅𝑇 1 2 𝑑𝑥1 𝑑𝑥1 and Δ𝐺 𝐺𝐸 𝑑2 ( ) 1 𝑑2 ( ) 𝑅𝑇 = 𝑅𝑇 + 𝑑𝑥12 𝑥1 𝑥2 𝑑𝑥12 Hence, stability requires: 𝐺𝐸 𝑑2 ( ) 1 𝑅𝑇 2 >− (const 𝑇, 𝑃) Eq. 3.0 𝑑𝑥1 𝑥1 𝑥2 C. Liquid/Liquid Equilibrium (LLE) For conditions of constant pressure, or when pressure effects are negligible, binary liquid/liquid (LLE) is conveniently displayed on a solubility diagram, a plot of 𝑇 vs. 𝑥1. Figure 2.0 shows binary solubility diagrams of three types. The first, Fig. 2.0(a), shows curves (binodal curves) that define an “island.” They represent the compositions of coexisting phases: curve UAL for the α phase (rich in species 2), and curve UBL for the 𝛽 phase (rich in species 1). 𝛽 Equilibrium compositions 𝑥1𝛼 and 𝑥1 at a particular 𝑇 are defined by the intersections of a horizontal tie line with the binodal curves. At each temperature, these compositions are those for which the curvature of the 𝛥𝐺 vs. 𝑥1 curve changes sign. Between these compositions, it is concave down (negative second derivative) and outside them it is concave up. At these points, the curvature is zero; they are inflection points on the 𝛥𝐺 vs. 𝑥1 curve. Temperature 𝑇𝐿 is a lower consolute temperature, or lower critical solution temperature (LCST); temperature 𝑇𝑈 is an upper consolute temperature, or upper critical solution temperature (UCST). At temperatures between 𝑇𝐿 and 𝑇𝑈 , LLE is possible; for 𝑇 < 𝑇𝐿 and 𝑇 > 𝑇𝑈 , a single liquid phase is obtained for the full range of compositions. The consolute points are limiting states of two-phase equilibrium for which all properties of the two equilibrium phases are identical. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE Actually, the behavior shown on Fig. 2(a) is rarely observed; the LLE binodal curves are often interrupted by curves for yet another phase transition. When they intersect the freezing curve, only a UCST exists [Fig. 2.0(b)]; when they intersect the VLE bubblepoint curve, only an LCST exists [Fig. 2.0(c)]; when they intersect both, no consolute point exists, and yet another behavior is observed. e.g. tetrahydrofuran-water e.g. butanol-water e.g. triethylamine-water Figure 2.0: Three types of constant-pressure liquid/liquid solubility diagram. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York (2005).] D. Vapor/Liquid/Liquid Equilibrium (VLLE) As noted in the previous discussions, the binodal curves representing LLE can intersect the VLE bubblepoint curve. This gives rise to the phenomenon of vapor/liquid/liquid equilibrium (VLLE). A binary system of two liquid phases and one vapor phase in equilibrium has (by the phase rule) but one degree of freedom. For a given pressure, the temperature and the compositions of all three phases are therefore fixed. On a temperature/composition diagram the points representing the states of the three phases in equilibrium fall on a horizontal line at 𝑇*. In Fig. 3.0, points 𝐶 and 𝐷 represent the two liquid phases, and point 𝐸 represents the vapor phase. If more of either species is added to a system whose overall composition lies between points 𝐶 and 𝐷, and if the three-phase equilibrium pressure is maintained, the phase rule requires that the temperature and the compositions of the phases be unchanged. However, the relative amounts of the phases adjust themselves to reflect the change in overall composition of the system. At temperatures above 𝑇* in Fig. 3.0, the system may be a single liquid phase, two phases (liquid and vapor), or a single vapor phase, depending on the overall composition. In region α the system is a single liquid, rich in species 2; in region 𝛽 it is a single liquid, rich in species 1. In region 𝛼 − 𝑉, liquid and vapor are in equilibrium. The states of the individual phases fall on lines 𝐴𝐶 and 𝐴𝐸. In region 𝛽 − 𝑉, liquid and vapor phases, described by lines 𝐵𝐷 and 𝐵𝐸, exist at equilibrium. Finally, in the region designated 𝑉, the system is a single vapor phase. Below the three- phase temperature 𝑇*, the system is entirely liquid; this is the region of LLE. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE When a vapor is cooled at constant pressure, it follows a path represented on Fig. 3.0 by a vertical line. Several such lines are shown. If one starts at point 𝑘, the vapor first reaches its dewpoint at line 𝐵𝐸 and then its bubblepoint at line 𝐵𝐷, where condensation into single liquid phase 𝛽 is complete. This is the same process that takes place when the species are completely miscible. If one starts at point 𝑛, no condensation of the vapor occurs until temperature 𝑇* is reached. Then condensation occurs entirely at this temperature, producing the two liquid phases represented by points 𝐶 and 𝐷. If one starts at an intermediate point 𝑚, the process is a combination of the two just described. After the dewpoint is reached, the vapor, tracing a path along line 𝐵𝐸, is in equilibrium with a liquid tracing a path along line 𝐵𝐷. However, at temperature 𝑇* the vapor phase is at point 𝐸. All remaining condensation therefore occurs at this temperature, producing the two liquids of points 𝐶 and 𝐷. Figure 3.0 is drawn for a single constant pressure; equilibrium phase compositions, and hence the locations of the lines, change with pressure, but the general nature of the diagram is the same over a range of pressures. For most systems the species become more soluble in one another as the temperature increases, as indicated by lines 𝐶𝐺 and 𝐷𝐻 of Fig. 3.0. If this diagram is drawn for successively higher pressures, the corresponding three-phase equilibrium temperatures increase, and lines 𝐶𝐺 and 𝐷𝐻 extend further and further until they meet at the liquid/liquid consolute point 𝑀, as shown by Fig. 4.0. As the pressure increases, line 𝐶𝐷 becomes shorter and shorter (indicated in Fig. 4.0 by lines 𝐶′𝐷′ and 𝐶″𝐷″), until at point 𝑀 it diminishes to a differential length. For still higher pressures (𝑃4 ) the temperature is above the critical-solution temperature, and there is but a single liquid phase. The diagram then represents two-phase VLE, and it has the form that looks like a valley, exhibiting a minimum-boiling azeotrope. Figure 3.0: 𝑇𝑥𝑦 diagram at constant 𝑃 for a Figure 4.0: 𝑇𝑥𝑦 diagram for several pressures. binary system exhibiting VLLE.. [Smith, Van Ness, [Smith, Van Ness, and Abbott, Introduction to and Abbott, Introduction to Chemical Engineering Chemical Engineering Thermodynamics, 7th ed., Thermodynamics, 7th ed., McGraw-Hill, New York McGraw-Hill, New York (2005).] (2005).] For an intermediate range of pressures, the vapor phase in equilibrium with the two liquid phases has a composition that does not lie between the compositions of the two liquids. This is illustrated in Fig. 4.0 by the curves for 𝑃3 , which terminate at 𝐴″ and 𝐵″. The vapor in equilibrium with the two liquids at 𝐶″ and 𝐷″ is at point 𝐹. In addition the system exhibits an azeotrope, as indicated at point 𝐽. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE Not all systems behave as described in the preceding paragraphs. Sometimes the upper critical-solution temperature is never attained because a vapor/liquid critical temperature is reached first. In other cases, the liquid solubilities decrease with an increase in temperature. In this event a lower critical-solution temperature exists, unless solid phases appear first. There are also systems that exhibit both upper and lower critical- solution temperatures. Figure 5.0 is the phase diagram drawn at constant 𝑇 corresponding to the constant-𝑃 diagram of Fig. 3.0. On it we identify the three-phase-equilibrium pressure as 𝑃*, the three-phase-equilibrium vapor composition as 𝑦1 *, and the compositions of the two liquid phases that contribute to the 𝛽 vapor/liquid/liquid equilibrium state as 𝑥1 and 𝑥1. 𝛼 Figure 5.0: 𝑃𝑥𝑦 diagram at constant 𝑇 for two partially miscible liquids. [Smith, Van Ness, and Abbott, Introduction to Chemical The phase boundaries separating the three liquid- Engineering Thermodynamics, 7th ed., McGraw-Hill, New York phase regions are solubilities. (2005).] Although no two liquids are totally immiscible, this condition is so closely approached in some instances that the assumption of complete immiscibility does not lead to appreciable error for many engineering purposes. The phase characteristics of an immiscible system are illustrated by the temperature/composition diagram of Fig. 6.0(a). This diagram is a special case of Fig. 3.0 wherein phase 𝛼 is pure species 2 and phase 𝛽 is pure species 1. Thus lines 𝐴𝐶𝐺 and 𝐵𝐷𝐻 of Fig. 3.0 become in Fig. 6.0(a) vertical lines at 𝑥1 = 0 and 𝑥1 = 1. Figure 6.0: Binary system of immiscible liquids. (a) 𝑇𝑥𝑦 diagram; (b) 𝑃𝑥𝑦 diagram. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York (2005).] MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE In region I, vapor phases with compositions represented by line 𝐵𝐸 are in equilibrium with pure liquid species 1. Similarly, in region II, vapor phases whose compositions lie along line 𝐴𝐸 are in equilibrium with pure liquid species 2. Liquid/liquid equilibrium exists in region III, where the two phases are pure liquids of species 1 and 2. If one cools a vapor mixture starting at point 𝑚, the constant-composition path is represented by the vertical line shown in the figure. At the dewpoint, where this line crosses line 𝐵𝐸, pure liquid species 1 begins to condense. Further reduction in temperature toward 𝑇* causes continued condensation of pure species 1; the vapor-phase composition progresses along line 𝐵𝐸 until it reaches point 𝐸. Here, the remaining vapor condenses at temperature 𝑇*, producing two liquid phases, one of pure species 1 and the other of pure species 2. A similar process, carried out to the left of point 𝐸, is the same, except that pure species 2 condenses initially. The constant-temperature phase diagram for an immiscible system is represented by Fig. 6.0(b). E. The Gamma/Phi Formulation of VLE In this discussion, one should recall the concept of fugacity 𝑓 and the activity coefficient 𝛾. For many VLE systems of interest, the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactory for the vapor phase. Liquid-phase behavior, on the other hand, is described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The definition of the activity coefficient for species 𝑖 in liquid phase is: 𝑓̂𝑖𝑙 = 𝑥𝑖 𝛾𝑖𝑙 𝑓𝑖𝑙 Substitution with necessary fugacity coefficient yields to: 𝑦𝑖 𝜙̂𝑖𝑣 𝑃 = 𝑥𝑖 𝛾𝑖𝑙 𝑓𝑖𝑙 where 𝑉 𝑙 (𝑃−𝑃𝑖𝑠𝑎𝑡 ) [ 𝑖 ] 𝑅𝑇 𝑓𝑖𝑙 = 𝜙𝑖𝑠𝑎𝑡 𝑃𝑖𝑠𝑎𝑡 𝑒 This combination of equation yields to 𝑦𝑖 𝜙𝑖 𝑃 = 𝑥𝑖 𝛾𝑖 𝑃𝑖𝑠𝑎𝑡 (𝑖 = 1,2, … 𝑁) Eq. 4.0 where 𝑙 𝑠𝑎𝑡 𝑉 (𝑃−𝑃 ) 𝜙̂𝑖𝑣 [− 𝑖 𝑅𝑇 𝑖 ] 𝜙𝑖 = 𝑠𝑎𝑡 𝑒 𝜙𝑖 In Eq. 4.0 𝛾𝑖 is understood to be a liquid-phase property. Because the Poynting factor, represented by the exponential, at low to moderate pressures differs from unity by only a few parts per thousand, its omission introduces negligible error, and we adopt this simplification to produce the usual working equation: 𝜙̂𝑖𝑣 𝜙𝑖 = 𝜙𝑖𝑠𝑎𝑡 The vapor pressure of pure species 𝑖 is most commonly given by the Antoine equation: 𝐵 ln 𝑃𝑖𝑠𝑎𝑡 = 𝐴𝑖 − 𝑇 + 𝐶𝑖 The gamma/phi formulation of VLE appears in several variations, depending on the treatment of 𝛷𝑖 and 𝛾𝑖. Applications of thermodynamics to vapor/liquid equilibrium calculations encompass the goals of finding the temperature, pressure, and compositions of phases in equilibrium. Indeed, thermodynamics provides the mathematical framework for the systematic correlation, extension, generalization, evaluation, and interpretation of such data. Moreover, it is the means by which the predictions of various theories of molecular physics and statistical mechanics may be applied to practical purposes. None of this can be accomplished without models for the behavior MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE of systems in vapor/liquid equilibrium. The two simplest models, already considered, are the ideal-gas state for the vapor phase and the ideal-solution model for the liquid phase. These are combined in the simplest treatment of vapor/liquid equilibrium in what is known as Raoult’s law. It is by no means a “law” in the universal sense of the First and Second Laws of Thermodynamics, but it does become valid in a rational limit. F. VLE from Cubic Equation of States Phases at the same 𝑇 and 𝑃 are in equilibrium when the fugacity of each species is the same in all phases. For VLE, this requirement is written: 𝑦𝑖 𝜙̂𝑖𝑣 = 𝑥𝑖 𝜙̂𝑖𝑙 (𝑖 = 1,2, … , 𝑁) 1. Vapor Pressures for a Pure Species Although vapor pressures for a pure species 𝑃𝑖𝑠𝑎𝑡 are subject to experimental measurement, they are also implicit in a cubic equation of state. Indeed, the simplest application of cubic equations of state for VLE calculations is to find the vapor pressure of a pure species at given temperature 𝑇. Two widely used cubic equations of state, developed specifically for VLE calculations, are the Soave/Redlich/Kwong (SRK) and the Peng/Robinson (PR) equation. Both are special cases of the generic cubic equation of state. Equation-of-state parameters are independent of phase, and in accord with the equations presented in Perry’s Handbook (8th Ed.), p. 4-11, one can perform VLE calculations. Note: RK means Redlich/Kwong equation. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE 2. Mixtures The fundamental assumption when an equation of state is written for mixtures is that it has exactly the same form as when written for pure species. Thus, as presented by Smith Van Ness, for mixtures, Eq 4-104, written without subscripts, become: Here, 𝛽 𝑙 , 𝛽 𝑣 , 𝑞 𝑙 , and 𝑞 𝑣 are for mixtures, with definitions: The complication is that mixture parameters 𝑎𝑝 and 𝑏 𝑝 , and therefore 𝛽 𝑝 and 𝑞 𝑝 , are functions of composition. Systems in vapor/liquid equilibrium consist in general of two phases with different compositions. The 𝑃𝑉 isotherms generated by an equation of state for these two fixed compositions are represented in Fig. 7.0 by two similar lines: the solid line for the liquid-phase composition and the dashed line for the vapor-phase composition. They are displaced from one another because the equation-of-state parameters are different for the two compositions. Each line contains a bubblepoint on its left segment representing saturated liquid and a dewpoint of the same composition on its right segment representing saturated vapor. Because these points for a given line are for the same composition, they do not represent phases in equilibrium and do not lie at the same pressure. Figure 7.0: Two PV isotherms at the same T for mixtures of two different compositions. The solid line is for a liquid-phase composition; the dashed line is for a vapor-phase composition. Point 𝐵 represents a bubblepoint with the liquid-phase composition; point 𝐷 represents a dewpoint with the vapor-phase composition. When these points lie at the same 𝑃 (as shown) they represent phases in equilibrium. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York (2005).] MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE For a bubble point calculation, the temperature and liquid composition are known, and this fixes the location of the PV isotherm for the composition of the liquid phase (solid line). The bubble point calculation then finds the composition for a second (dashed) line that contains a dewpoint 𝐷 on its vapor segment that lies at the pressure of the bubblepoint 𝐵 on the liquid segment of the solid line. This pressure is then the phase-equilibrium pressure, and the composition for the dashed line is that of the equilibrium vapor. This equilibrium condition is shown by Fig. 7.0, where bubblepoint 𝐵 and dewpoint 𝐷 lie at the same 𝑃 on isotherms for the same 𝑇 but representing the different compositions of liquid and vapor in equilibrium. Because no established theory prescribes the form of the composition dependence of the equation-of-state parameters, empirical mixing rules have been proposed to relate mixture parameters to pure-species parameters. The simplest realistic expressions are a linear mixing rule for parameter 𝑏 and 𝑎 quadratic mixing rule for parameter 𝑎: with 𝑎𝑖𝑗 = 𝑎𝑗𝑖. The general mole-fraction variable xi is used here because these mixing rules are applied to both liquid and vapor mixtures. The 𝑎𝑖𝑗 are of two types: pure-species parameters (repeated subscripts, e.g., 𝑎11 ) and interaction parameters (unlike subscripts, e.g., 𝑎12 ). Parameter 𝑏𝑖 is for pure species 𝑖. The interaction parameters 𝑎𝑖𝑗 are often evaluated from pure-species parameters by combining rules, e.g., a geometric-mean rule: 1/2 𝑎𝑖𝑗 = (𝑎𝑖 𝑎𝑗 ) These equations, known as van der Waals prescriptions, provide for the evaluation of mixture parameters solely from parameters for the pure constituent species. Although they are satisfactory only for mixtures comprised of simple and chemically similar molecules, they allow straightforward calculations that illustrate how complex VLE problems can be solved. -End of Discussion- References are stated in the syllabus. MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY ChE 408 | SEMI-FINALS MODULE Exercises 1. For one of the following substances, determine 𝑃 sat∕bar from the Redlich/Kwong equation at two temperatures: 𝑇 = 𝑇𝑛 (the normal boiling point), and 𝑇 = 0.85𝑇𝑐. For the second temperature, compare your result with a value from the literature (e.g., Perry’s Chemical Engineers’ Handbook). Discuss your results. (a) Acetylene; (b) Argon; (c) Benzene; (d) n-Butane; (e) Carbon monoxide; 2. A system formed of methane(1) and a light oil(2) at 200 K and 30 bar consists of a vapor phase containing 95 mol- % methane and a liquid phase containing oil and dissolved methane. The fugacity of the methane is given by Henry’s law, and at the temperature of interest Henry’s constant is 𝐻1 = 200 bar. Stating any assumptions, estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of pure methane at 200 K is −105 cm3·mol−1. 3. A single 𝑃–𝑥1 data point is available for a binary system at 35°C. Estimate from the data: (a) The corresponding value of 𝑦1. (b) The total pressure at 35°C for an equimolar liquid mixture. (c) Whether azeotropy is likely at 35°C. Data: At 35°C, 𝑃1 sat = 120.2 and 𝑃2 sat = 73.9 kPa For 𝑥1 = 0.389, 𝑃 = 108.6 kPa MODULE III College of Engineering, Architecture, and Fine Arts Monroe H. de Guzman, AAE BATANGAS STATE UNIVERSTIY