Chapter 8. Phase Behavior of Mixtures

The presence of multiple components adds a new dimension to the phase behavior of mixtures. In the pure component, molecules are always surrounded by similar species; in a mixture, they are surrounded by both like and unlike species. This gives rise to self-interactions between like molecules, and cross-interaction between unlike molecules. These interactions are much more pronounced in the liquid phase, where molecules are closely packed together. The balance of self- and cross-interactions creates phase behavior that is not seen in pure fluids. If cross-interactions are favorable, components form strong mixtures that are more difficult to separate. If cross interactions are unfavorable, the mixture is weaker and separation is easier. If they are strongly unfavorable, then components may exhibit partial miscibility. Additional variety of phase behaviors comes from the number of phases that can coexist simultaneously. With mixtures we encounter problems of vapor-liquid equilibrium (VLE), but also liquid-liquid (LLE) and liquid-liquid-vapor (LLVE) equilibrium. If a solid component is added, other combinations of equilibria are observed for example, solid-liquid, solid-liquid-vapor, etc. This enormous variety is made possible by the presence of additional components.

The phase behavior of mixtures forms the basis of industrial separations. What makes such separation possible is the fact that when a mixture is brought into a region of multiple coexisting phases, each phase has its own composition. Understanding the phase behavior of multicomponent systems is very important in the calculation of separation processes. In this chapter we review graphical representations of the phase behavior of binary and ternary systems. Since we are dealing with several independent variables, pressure, temperature, and composition, special conventions are used in order to represent information in two-dimensional graphs.

The learning objectives for this chapters are to:

1. Interpret the Txy and Pxy graph of binary mixtures.

2. Use phase diagrams to calculate material balances.

3. Work with ternary-phase diagrams.

8.1 The Txy Graph

A pure substance under constant pressure boils at constant temperature; when heat is added to the boiling liquid, it induces more evaporation, but no temperature change is observed until after all the liquid has evaporated. By contrast, a liquid mixture boils over a range of temperatures. As an example we consider a mixture of n-heptane and n-decane. The normal boiling points of the pure components are 98.1 °C for heptane, and 174.2 °C for decane. We form a solution that contains 50% by mole n-heptane (component 1) and observe the phase changes as the solution is heated, starting at room temperature, under constant pressure at 1 atm. To visualize such an experiment, imagine that the solution is placed in a cylinder fitted with a movable piston so that the pressure remains constant. The cylinder is sealed so no air is present. This is a closed system whose overall composition remains constant at 50% n-heptane by mol, at all times. The initial state, T = 25 °C, P = 1 atm, x1 = 0.5 is shown by point A in Figure 8-1. At A the temperature is well below the boiling point of either compound, and the system is in the liquid phase. Heating the system at constant pressure is equivalent to moving up along the line AE, corresponding to a line of constant overall composition. When we reach 98.1 °C, the boiling point of pure heptane, the solution remains liquid and does not boil. Indeed, it does not begin to boil until about 120 °C. The point where the liquid begins to boil is called the bubble temperature, or bubble point of the solution (point B on the graph). As the liquid continues to boil, the temperature increases until a point is reached where all of the liquid has evaporated (point D). Further heating produces superheated vapor and the state moves higher up towards point E. These steps can be repeated in reverse by cooling the vapor. Starting with point E, cooling causes the temperature of the vapor to drop until condensation occurs at point D. This is the dew point and is identical to the point of complete evaporation of the liquid in the heating experiment. As more liquid condenses the temperature drops and this continues until the bubble point is reached (point B) where all the vapor has condensed. Further cooling produces a compressed liquid and moves the state towards point A.


Figure 8-1: Txy graph of heptane (1) decane (2) at 1 atm.

If this procedure is repeated with different compositions in the range x1 = 0 to 1 we obtain a series of bubble and dew points which define the bubble line and the dew line, respectively. These are shown as lines in Figure 8-1. Above the dew line the system is vapor; below the bubble line the system is liquid. Between the two lines the system consists of a mixture of two phases, each with its own composition. Consider a point C that lies between the bubble and the dew lines. This point represents a system with overall composition z1 = 0.5, which we read off the horizontal axis. In reality, point C consists of a mixture of two phases, a liquid with composition x1 ≈ 0.3 (point L) and a vapor with y1 ≈ 0.8 (point V). Notice that the vapor and liquid compositions are different. In fact, the liquid contains less heptane (30% by mol) than the original liquid (50%), while the vapor contains more (80%). Since the liquid corresponding to C contains more decane that the liquid at B, it boils at a higher temperature, closer to the boiling point of decane. Thus, the reason that the temperature increases during boiling is that liquid progressively becomes more concentrated in the less volatile (heavier) component. The difference between the composition of the liquid and the composition of the vapor is a very important characteristic of multicomponent equilibrium and constitutes the basis for separating the components of a mixture.

The phase diagram in Figure 8-1 is a Txy graph. In this graph pressure is held constant, the vertical axis represents temperature, and the horizontal axis represents composition. Whether this is the composition of the liquid (x1), of the vapor (y1), or the overall composition (z1) of a two-phase mixture depends on where we are on the phase diagram. From anywhere in the liquid, including the bubble line, the horizontal axis gives x1; from anywhere in the vapor, including the dew line, the horizontal axis gives y1. Between the dew and bubble lines the system consists of two phases. Directly below a point in this region we obtain the overall composition z1.

The Lever Rule

Let us return to point C inside the two-phase region. The line LV that connects the compositions of the vapor and the compositions of the liquid is a tie line. This is a horizontal line and passes through C since all three points, L, C, and V are at the same temperature. Let us assume that the liquid contains a fraction L of the total number of moles and the vapor contains the remaining fraction V = 1 −L. The concentration of heptane in the liquid is x1, in the vapor y1, and let’s say that the overall concentration of heptane is z1 (that would be the concentration of heptane in point C, 50% in our example). The mole balance on heptane and the total mole balance give the following two equations:


Solving for the liquid and vapor fractions we obtain


This has the familiar form of the lever rule encountered in Chapter 2.


Convention for x, y, z

We adopt the following convention for distinguishing the various compositions that appear when a mixture forms two phases: we use xi for the mol fractions of the liquid phase, yi for the mol fractions of the vapor phase, and zi for the overall composition of a two-phase mixture. When a two phase mixture is fully condensed or fully evaporated, the composition of the single phase formed is zi.

8.2 The Pxy Graph

In the Txy graph, we plot the phase behavior of a binary system as a function of temperature at constant pressure. We may also make a plot as a function of pressure at constant temperature. The resulting phase diagram is a Pxy graph. The Pxy graph for the system heptane-decane at 120 °C is shown in Figure 8-3, and it resembles a Txy graph turned upside down. The liquid phase is now at the top of the graph (high pressure) and the vapor is at the bottom (low pressure). As with the Txy graph, the bubble line marks the boundary of the liquid, the dew line marks the boundary of the vapor, and the two lines meet at the saturation pressure of the pure components. In this case, heptane (component 1) is more volatile than decane; its saturation pressure is higher. As a result, the VLE envelope ascends in the direction of increasing x1. The line ABCDE corresponds to the same transitions as those discussed in Figure 8-1 but these are now caused by changing the pressure (expansion or compression) at constant temperature. At A the system is liquid; by decreasing pressure it reaches the bubble point at B; at C it consists of a mixture of liquid and vapor; at D it reaches the dew point; and by further decreasing pressure it evaporates completely. The transformations are reversible and by isothermally compressing the vapor in state E we eventually return to the initial state A.


Figure 8-3: Pxy graph of heptane (1) decane (2) at 120 °C.

To read the Pxy graph we follow the same principles as with Txy graphs. The horizontal axis reads composition (mol fraction of component 1). Points inside the VLE envelope represent a two-phase system, and the horizontal axis gives the overall composition. The tie line is horizontal and points to the composition of the vapor (on the dew line) and the composition of the liquid (on the bubble line). The lever rule applies and can be used to find the liquid and vapor fractions.



Example 8.5 demonstrates the basic idea behind distillation. By partially boiling the solution, we enrich the vapor in the more volatile component and the liquid in the heavier component. By repeating this process as many times as necessary, we can achieve as high of a purity as desired. This is equivalent to a series of flash separators, as shown in Figure 8-6. The liquid from each separator is partially boiled at higher temperature and the vapor from each stage is partially condensed at lower temperature. In this manner, the liquid is continuously enriched in the less-volatile component, and the vapor in the more volatile one. In reality, distillation columns accomplish this task through a series of perforated trays, stacked vertically: the liquid drips down to hotter stages and the vapor bubbles up to cooler stages. The entire column is heated at the bottom. In packed columns, instead of discrete trays the column is filled with a packing material such as beads, rings, or other small objects. As the liquid trickles down and the vapor rises, the packing material forces the two phases into contact and helps them reach equilibrium with each other.


Figure 8-6: Staged separation (distillation). The column is heated at the bottom and cooled at the top (Ti>Ti−1).

8.3 Azeotropes

An anomalous but not uncommon behavior in some binary systems is the occurrence of an azeotrope. An azeotrope is indicated by the presence of a maximum or minimum in the bubble and dew lines. One such example is the system methanol(1)/ CCl4(2), shown in Figure 8-7. The Txy graph shows a minimum at which point the bubble and dew lines meet. There is a range of compositions for which the bubble temperature is lower than the boiling point of either pure compound, indicating that these solutions boil more easily than the pure components. In the Pxy graph the minimum boiling azeotrope is characterized by a maximum in the total pressure, that is, the solution is more volatile than the pure components. Maximum boiling azeotropic behavior is also encountered, though not as often. The Txy graph exhibits a maximum, which lies above the boiling point of either component. This solution is more difficult to boil compared to the pure components. The corresponding Pxy graph exhibits a minimum in the dew and bubble pressure.


Figure 8-7: Minimum boiling azeotrope (top), and maximum boiling azeotrope (bottom). Data from Vapor-Liquid Equilibrium Data Collection, J. Gmehling, U. Onken, and W. Arlt, DECHEMA.

The azeotropic point corresponds to the maximum or minimum on the phase diagram. At this point the bubble and dew lines meet and the composition of the vapor is identical to that of the liquid. A liquid solution that has exactly the azeotropic composition boils as a pure component: the composition of the vapor and the liquid phases are identical, while temperature remains constant throughout boiling. Similarly, a vapor with this concentration condenses as a pure component. The name azeotrope derives from the Greek and translates “not altered by boiling.” Azeotropes are problematic in separations. Suppose we want to separate a solution of methanol in CCl4 containing 10% methanol by mole. This composition lies to the left of the azeotrope. Subjecting this solution to distillation at 1 atm will produce a liquid that approaches pure carbon tetrachloride, and a vapor whose composition approaches the azeotropic point. This means that the vapor fraction cannot exceed the azeotropic concentration, preventing us from obtaining a stream of purified methanol. It is possible to alter the composition of the azeotrope and even remove the azeotrope altogether by changing the pressure and temperature. Alternatively, the addition of a suitable third component could have the same effect. In some cases, the formation of azeotropes is advantageous. Water and hydrochloric acid form an azeotrope; through boiling it is possible to collect a solution that has the azeotropic composition. Because its composition is precisely known (it depends only on the pressure during distillation), this solution may be used as a standard for acid/base titrations.

8.4 The xy Graph

Another graphical representation of binary vapor, liquid data, useful in material balances, is the xy graph, as shown in Figure 8-8. This graph gives the composition of the tie line by plotting the vapor mole fraction against the liquid mol fraction. In this graph, both axes run from 0 to 1 and the diagonal represents points where the vapor composition is equal to that of the liquid. By convention, the more volatile component is chosen as component 1; by this convention the xy graph generally lies above the diagonal. Azeotropes are easily identified on this graph because the xy line crosses the diagonal.


Figure 8-8: xy graphs for two binary systems.

The distance of the xy line from the diagonal is a measure of the difference in the composition of the two phases, and thus a measure of the ease of separation. One way to quantify the difference between the compositions of the two phases is through the known as distribution coefficient, Ki, or K-factor for simplicity, which is defined as


Good separation requires values substantially different from 1. A value larger than unity indicates that component i is preferentially found in the gas phase; a value less than 1 indicates preferential enrichment of the liquid phase. A related property is the relative volatility, αij, which is defined as


In a binary mixture, this takes the form,


The K1 factor of binary mixture can be obtained graphically from the xy graph: if we connect a point of the xy graph to the origin, the slope of this line is K1. At xi = 1 the value of Ki is unity for any component i; at xi = 0 it reaches a limiting value that corresponds to the infinite dilution limit, namely, to the limit that the concentration of the component is reduced to zero by dilution in the other component.1 That is, even though both yi and xi go to zero in this limit, the Ki factor is finite and equal to the slope of the xy line at xi = 0.

1. The limit of infinite dilution should be visualized as a single molecule of component i completely surrounded by molecules of component j.

If the K factors of a binary system are known as a function of pressure and temperature, the entire phase diagram can be constructed. As we will see in Chapter 10, thermodynamics provides methodologies for the estimation of these factors.

8.5 VLE at Elevated Pressures and Temperatures

The phase diagrams discussed so far extend over the entire range of the compositional axis, from x1 = 0 (pure saturated component 2) to x1 = 1 (pure saturated component 1). This behavior is observed when both components are below their respective critical points. If one or both components are above their critical point, the VLE region shrinks and does not cover the entire range of compositions. This situation is illustrated in Figure 8-9, which shows the Pxy graph of the system ethanol/water at several temperatures. At 200 °C the phase diagram has the usual form and extends over the entire compositional range. This system also exhibits a maximum boiling azeotrope. Upon increasing temperature, the phase diagram moves upwards but remains qualitatively the same, as long as the temperature remains below the critical of both components. At 275 °C ethanol above its critical temperature (240.77 °C) but water is below its own (374 °C). The phase diagram now does not reach all the way to the ethanol axis. Instead, the dew and bubble lines meet at some intermediate point. The phase diagram is read in the usual way: We draw a tie line, identify the liquid phase on the left (above the bubble line) and the vapor phase on the right (below the dew line). Where the bubble and dew lines meet we have a critical point, which is characterized by its own temperature, pressure, and composition. It is identified as the point where the tie line is tangent to the VLE curve. If temperature is increased, the phase diagram moves up and shrinks further. This behavior continues until the critical temperature of water is reached (not shown on this graph). The dashed line in pane (a) of Figure 8-9 tracks the critical point of the mixture as temperature increases and connects the critical pressures of the pure components (the critical temperature of water is outside the range of this graph). This behavior can also be studied on the xy graph, shown in Figure 8-9(b). When both components are below the critical points, the xy graph extends from x1 = 0 to x1 = 1.2 Once temperature exceeds the critical temperature of compartment 1, the line intersects the diagonal and stops there. This marks the critical point of the mixture, where the composition of the two phases is identical.

2. This line intersects the diagonal at the azeotropic point, but this detail cannot be seen clearly at the magnification of this graph.


Figure 8-9: Pxy and xy graph of ethanol/water at elevated temperatures (data from Frank Barr-David and B. F. Dodge, Journal of Chemical & Engineering Data, 4 (2):107-121, 1959).

Such incomplete phase diagrams occur commonly in systems involving typical gases (e.g., oxygen, nitrogen) in equilibrium with liquids (e.g., water) at room temperature. In this case the liquid is below its critical temperature but the gas above its own. Although such problems fall squarely in the scope of phase equilibrium they are often categorized as solubility problems. The thermodynamics of these systems are examined in Chapter 13.

8.6 Partially Miscible Liquids

Some liquids are only partially miscible in each other. This situation arises when the constituent molecules contain groups that have low affinity for each other. As an example we consider the system n-hexane/ethanol. Both molecules contain CH2 and CH3 groups which are very similar. Ethanol also contains a polar hydroxyl group, –OH, which has little affinity for the alkyl groups. A limited amount of hexane can be accommodated in ethanol and, similarly, small amounts of ethanol can be dissolved in hexane. At certain compositions, however, the system splits into two separate liquid phases due to the lack of affinity between hydroxyls and hydrocarbons. If such a system is brought to boiling it becomes a three-phase system: two liquids and a vapor.

Figure 8-10 shows the Txy of the system hexane(1)/ethanol(2). There are three distinct phases: the vapor, one hexane-rich liquid phase to the right (marked as L1), and one ethanol-rich liquid phase to the left (L2). The rest of the graph represents coexistence between these phases. To find the phases present inside the two-phase regions, we draw a tie line until the phase boundaries are reached. To illustrate the behavior of the system in the two-phase region, consider the following experiment: Starting with pure ethanol at 60 °C, 1.96 bar, we add hexane in small amounts. Initially, all the hexane becomes incorporated into ethanol to form a homogeneous solution. Once the solubility limit of hexane in ethanol is reached, any additional of hexane forms a separate phase. Both phases contain both components, but at different concentrations. The ethanol-rich phase (the phase which originated as pure ethanol) has the composition of point C2, and the hexane-rich phase the composition of point C1. The two-phase system is represented by a single point, C, which lies inside the two-phase region at the overall composition of the two-phase system. This composition is calculated by the lever rule:


where L1 is the fraction of mass (or moles) in the hexane-rich phase, L2 is the fraction of the ethanol-rich phase, and Image, Image, are the compositions (mass or mol fractions) of component i in the two phases. These compositions correspond to points C1 and C2 and are read off the horizontal axis of the graph. If we continue to add hexane, more and more of the ethanol in the ethanol-rich phase is incorporated into the hexane-rich phase. During this part of the process the system consists of two liquid phase. The compositions of these phases remain fixed at those of points C1 and C2, but the relative amounts of each liquid change, with the hexane-rich phase increasing at the expense of the ethanol-rich phase. When the overall composition reaches point C1, enough hexane exists to solubilize all of the available ethanol so that the ethanol-rich phase disappears. Adding more hexane at this point produces a homogeneous solution whose concentration approaches that of pure hexane. The liquid branches of the phase boundary are determined experimentally by measuring the composition of the coexisting liquids at various temperatures. Mutual solubility generally increases with temperature and this is reflected in the (slight) convergence of the liquid branches as temperature increases. Some partially miscible systems become fully miscible at higher temperatures. In the case of hexane/ethanol, however, partial miscibility persists until the liquid reaches the boiling point.


Figure 8-10: Txy graph of partially miscible liquids: hexane(1)/ethanol(2) at 1.96 bar.

To demonstrate the phase behavior of a two-liquid system in the boiling region, suppose we start with state C in the two-phase region (see Figure 8-10). The overall composition is z1 = 0.2 at 60 °C, 1.96 bar and the system consists of two liquids whose compositions are given by points C1 and C2. The onset of boiling (bubble point) is shown by point D. At this point, the liquid boils and produces a vapor whose composition corresponds to point V, while the two liquids are represented by points A and B. Thus at the bubble point there are three phases present, two liquids and the vapor. As long as two liquids are present in boiling, the state of the system remains pegged at point D: the boiling temperature is constant, and the composition of all phases is also constant, and are given by points A, V, and B. In other words, the system forms an azeotrope. The only change observed during this stage of the process is the continuous decrease of the amounts of the liquids, and the increase in the amount of the vapor. Once any of the liquid phases is completely depleted, the state moves up. Which phase is depleted first depends on the overall composition of the starting mixture. For example, state E consists of vapor in equilibrium with the ethanol-rich phase, as indicated by the phases at the end points of the tie line that passes through E; the hexane-rich phase has completely evaporated. At point F the system reaches its dew point. Further heating moves the state into the region of superheated vapor. The line CG on the Txy graph represents a path of heating under constant pressure and constant overall composition.

The corresponding Pxy graph is shown in Figure 8-11. Qualitatively, it resembles a Txy graph turned upside down. Since pressure has little effect on the mutual solubility of liquids, the boundaries between the two liquid phases, lines AAʹ and BBʹ, are essentially vertical. To read this graph, we follow the same principles as with the Txy graph. First we label the single phases (hexane-rich liquid, ethanol-rich liquid, vapor). All other regions are areas where two or three phases are present. These phases are identified by drawing tie lines until they intersect a phase boundary.


Figure 8-11: Pxy graph of partially miscible liquids – hexane(1)/ethanol(2).

8.7 Ternary Systems

When a soluble third component is added to two partially miscible liquids, we obtain a ternary system in which the third component (solute) is partitioned between the two liquid phases (solvents). The phase behavior of such systems is important in liquid-liquid extraction, a process that takes advantage of the differences in solubility to transfer a solute from one solvent into another. Since two mole fractions are required to represent composition in ternary systems, it is not possible to present temperature-composition or pressure-composition graphs in a two-dimensional plot. Instead, we map out the composition of the phases at constant pressure and temperature. This is done in triangular diagrams such as the one shown in Figure 8-13. In this graph the three components are placed at the vertices of an equilateral triangle. Points inside the triangle represent possible concentrations of the three components. These concentrations are expressed on a percent basis, either by mass or by mole (the compositions in Figure 8-13 are given on a mass basis). The concentration of a component at any point inside the triangle is measured by the fractional distance of that point from the side opposite to the vertex of the component. The vertex corresponds to concentration 100% of that component, while points on the side opposite to the vertex correspond to 0% of that component. For example, point A lies at 44% of the maximum distance above the horizontal axis, which makes the concentration of acetic acid 44% by weight. Because of geometric similarity, fractional distances from a side do not have to be measured along vertical lines, thus for convenience we use grid lines parallel to one of the other sides. For example, the concentration of acetic acid at point A can be read by following the grid line AB to the right side of the triangle which is marked 0 at the bottom and 100 at the top. The concentration of water is 20%; this is found by following the grid line AC. The concentration of the third component can be determined the same way, or from the balance of the other two.


Figure 8-13: Phase diagram of acetic acid/water/methylisobutyl ketone at 25 °C. Concentrations are in weight percent.

As a phase diagram, Figure 8-13 provides information similar to that obtained from Pxy and Txy graphs.3 The shaded area represents a single liquid phase, and the white area represents separation into two liquid phases. Tie lines are drawn to indicate the composition of the two phases. For example, point E lies inside the two-liquid range, therefore, it phase separates into two liquid phases, L1, which contains mostly water, and L2, which contains mostly methylisobutyl ketone. The lever rule applies and can be used to obtain the relative amounts of the two liquid phases. Notice that the tie lines are not generally horizontal, and unless they are shown on the graph, we cannot tell the compositions at equilibrium.

3. However, on triangular plots both temperature and pressure are fixed.

The phase diagram of the system acetic acid/water/MIBK is typical of many ternary systems. The left side of the triangle represents mixtures of water and acetic acid (the concentration of methylisobutyl ketone along this side is 0). This side is entirely in the single-phase region, thus we conclude that acetic acid and water are fully miscible. Similarly, from the right side of the triangle we conclude that methylisobutyl ketone is fully miscible with acetic acid. By contrast, water and methylisobutyl ketone are only partially miscible because the bottom side, which represents mixtures of water and ketone (since the concentration of acetic acid along this side is 0), is in contact with the two phase region. This type of phase diagram in which a solute (here acetic acid) is fully miscible into two solvents (water and methylisobutyl ketone) but the two solvents are only partially miscible with each other is the most common type of ternary-phase diagram and is classified as type I. More complex phase diagrams are also observed. In some, the two-phase region touches two axes, in some it touches no axis at all, and some exhibit two or more regions where phase separation occurs.

Because plotting data on equilateral triangles has certain inconveniences, sometimes ternary data are presented on a right triangle, as shown by the smaller inset in Figure 8-13. In this graph, the vertical axis represents one component, the horizontal axis the second component and the third component is obtained by mass balance from the two known fractions.

8.8 Summary

The most important feature of phase equilibrium in multicomponent systems is that the composition of phases is not the same.4 If a mixture is brought into the two-phase region, we obtain phases that are preferentially enriched in one or more components. Separation methods such as distillation, absorption, and extraction operate based on this principle.

4. There are some exceptions, azeotropes, for example, and critical points.

The phase diagram is a graphical representation of the phase of the system at a given pressure, temperature, and composition. Phase diagrams of binary systems are presented as Pxy or Txy graphs. A phase diagram consists of areas where a single phase exists, and areas where multiple phases (two or more) exist at equilibrium. The bubble line marks the boundary of the liquid phase and the dew line that of the vapor phase. Between the bubble and dew lines, the system consists of two phases at equilibrium. Tie lines connect the composition of the phases. The lever rule is a simple relationship between the composition of phases at equilibrium, the overall composition, and the relative amounts of each phase. It is nothing but a statement of mass conservation. The phase diagram of ternary systems is usually presented on an equilateral triangle. This graph shows the phase boundaries as a function of composition, at a given temperature and pressure. A series of such graphs is needed to study the phase behavior of a ternary system at various temperatures or pressures.

The solution of problems in phase equilibrium is facilitated enormously if the phase diagram is known. Several systems have been studied experimentally and the results have been collected in extensive databases. However, given the inexhaustible variety of components, compositions, temperatures, and pressures, the chemical engineer will invariably be faced with components, or conditions, for which such data are not available. A main goal in much of the rest of this book is the development of methods to predict phase behavior where data are not available.

8.9 Problems

Problem 8.1: The data below give the dew and bubble temperature of methanol/ ethanol mixtures at 1 bar.


a) A solution containing 30% methanol (by mol) is flashed to 1 bar, 70.82 °C. Determine the phase of the system; if two phases, report the compositions and relative amounts of each phase.

b) What is the maximum mol fraction of methanol that can be achieved when a solution with 30% methanol is flashed to 1 bar? The maximum mol fraction of ethanol?

c) Make a Txy plot for this system and annotate it properly.

Note: Use linear interpolations and report mol fractions to the third decimal point.

Problem 8.2: Use the data for the system methanol-ethanol (from the previous problem) to design a separation process that takes a mixture with 20% methanol and produces a stream that contains 90% methanol by continuously flashing the vapor stream until the required purity is reached (the liquid streams are not recycled). The process is to be operated at 1 bar with a ratio V/L = 0.5 in all flash separators.

a) How many flash separators are needed?

b) Calculate the percentage of methanol recovered in the 90% stream relative to the methanol in the feed.

c) Change the vapor liquid ratio to V/L = 0.75 and repeat parts a and b.

d) Your boss has asked you to recommend a value for the vapor, liquid ratio V/L. What is your recommendation?

Problem 8.3: The data below are for the system 1,4 dioxane (1)/methanol(2) at 308.5 K.

a) What is the saturation pressure of methanol at 308.5 K?

b) What is the phase of a mixture that contains 60% dioxane at 308.5 K, 0.1 bar?

c) Determine the bubble and dew pressures of a mixture that contains 60% dioxane at 308.5 K.

These questions can be answered more easily if you plot the Pxy graph based on these data.


Problem 8.4: Use the data below for the system ethyl propyl ether (1)-chloroform (2) to answer the following questions:

a) What is the boiling point of chloroform at 0.5 bar?

b) Is this a maximum boiling or minimum boiling azeotrope?

c) What is the composition at the azeotropic point?

d) A mixture of the two components contains 80% by mol ethyl propyl ether. What is the phase of this mixture at 48.3 °C, 0.5 bar? If a two-phase system, report the composition of the two phases and their relative amounts.

e) One mol of a solution of these two components, whose bubble point at 0.5 bar is 48.3 °C, is mixed with chloroform until the final mixture contains 50% chloroform (by mol). How much chloroform is needed?


Problem 8.5: With reference to Figure 8-7, consider the following experiment: We begin with 1 mol of carbon tetrachloride at 35 °C, 0.35 bar and add methanol dropwise at constant temperature and pressure until the the mixture contains 99% by mol methanol. Describe all phase changes observed along this path and report the amount of methanol (in moles) that has been added up to the point that a phase change is observed.

Problem 8.6: A mixture that contains 40% by mole n-heptane in n-decane is to be separated in a series of flash separators until a stream is obtained that contains at least 95% n-heptane. Determine the number of separators needed, their temperature, and the recovery of n-heptane if all separators are at 1.013 bar and V/L = 3 in all separators. Txy data are given below:


Problem 8.7: A mixture of normal heptane (40% by mol) in normal decane is to be separated in two flash separators. The feed stream is led to separator 1; the vapor stream of separator 1 is fed to separator 2 while the liquid stream from separator 2 is recycled into separator 1. Both separators operate at 1 atm. Heat exchangers are used to ensure that all streams that enter a separator are at the same temperature as the separator. Determine if the problem is fully specified and if not, make any additional specifications and solve the material balances. Report the purity of heptane and decane in the product streams and the % recovery of each component. Additional data: Use the fitted equations for x and y as functions of T that are given in Example 8.5.

Problem 8.8: Two flash separators in series operate at 1 atmosphere total pressure. The feed (F1) into the first drum is a binary mixture of methanol/water that is 55 mol % methanol with flow rate 10,000 kg moles/h. The first flash drum operates at 75 °C and its liquid stream is fed into the second drum. The second flash drum operates at temperature, T2 and the liquid product composition (x2) is 15 mol % methanol.

a) What is the fraction vaporized in the first flash drum and the total fraction of the initial feed vaporized?

b) What are y1, y2, x1, and T2?

In this problem, the subscripts 1 and 2 refer to separator 1 and 2.

Additional information: The bubble and dew temperature (in °C) as a function of the mol fraction of methanol (x) are given by the following equations:


Problem 8.9: Carbon tetrachloride (1) and water (2) are essentially immiscible liquids. A stream contains 80% CCl4 by mole and the remaining is water. The temperature is 100 °C.

a) If the pressure is 4 bar, what is the phase of the stream, liquid, vapor, or vapor, liquid mixture?

b) If the pressure is 2 bar, what is the phase?

c) A liquid with overall composition x1 = 0.8 is brought to its bubble pressure at 100 °C. Which liquid boils off first?

d) What is the vapor fraction when the first liquid boils off? Explain your answers.

Additional data: The saturation pressure of CCl4 at 100 °C is 1.95 bar.

Problem 8.10: The system 2-butanol(1)/water(2) at 1.013 bar exhibits partial miscibility. Data for this system are given below. Answer the following question neglecting the variability of mutual solubility with temperature.

a) Make a Txy graph and annotate single- and two-phase regions.

b) You are given 1 mol of a mixture that contains 2-butanol and water at 1 atm, 75 °C. The system consists of a single phase but its composition is not known. You add water dropwise at 1 atm, 75 °C, and you notice that a tiny amount of a second liquid phase appears when you have added 0.46 mol of water. What was the composition of the solution you started with?

c) Twenty mol of 2-butanol is mixed with 80 mol of water and the system is brought to boil until 75 mol is in the vapor phase. How many phases are present at this point and what is their composition?

d) A mixture that contains 82% 2-butanol by mol is flashed to 1 atm. If the desired mol fraction of 2-butanol at the exit of the flash separator is 90%, determine the temperature of the separator and the amount of 2-butanol that is recovered in the 2-butanol rich stream.


Problem 8.11: Acrylonitrile and water are partially miscible liquids. At 1 atm, a two-phase mixtures boils at 70.6 °C and the compositions of the three phases are:


The normal boiling point of acrylonitrile is 78 °C.

a) Seven mol of water is mixed with 5 mol of acrylonitrile at 1 bar, 25 °C. Assuming that the mutual solubility at 25 °C is the same as at 70.6 °C, how many phases are formed and what are their compositions and amounts?

b) How much water should be added to the mixture of part (a) to form a single-phase system?

c) Draw a qualitative Txy graph and mark the states involved in parts a and b.

Problem 8.12: Water and 1-butanol are partially miscible in each other. At 1 atm, the two-phase system boils at 93.0 °C and the composition of the three phases are (in mol %):


The boiling point of pure butanol at 1 atm is 117.7 °C.

a) A solution is formed by mixing 5 mol of water with 2 mol of 1-butanol at 50 °C.

Assuming the mutual solubility of the two components at 50 °C is the same as at 93.0 °C, how many phases are formed? If more than one, indicate the composition and amount of each phase.

b) If the mixture of part (a) is brought to boiling, which liquid phase will disappear first, the butanol-rich or the water-rich phase?

c) It is desired to convert the mixture of part (a) into a single-phase solution. This can be done either by adding more water or by adding more butanol. Calculate the minimum amounts of water and butanol that are needed to produce a single-phase system.

Problem 8.13: Butyraldehyde(1) and water (2) are partially miscible liquids. At 1 bar, the two-phase system boils at 68 °C, and the composition of the three phases are: y1 = 90.3%, Image and Image (all in mol percent, with superscripts A and B indicating the butyraldehyde-rich and water-rich phases, respectively). The normal boiling point of butyraldehyde is 75.7 °C.

a) Forty mol of butyraldehyde are mixed with 10 mol water at 1 bar and the system is boiled at constant pressure. Which phase boils off first?

b) With the overall composition of the previous part, how many phases are present and in what amounts (moles) when half of the total moles are in the vapor phase?

c) How many phases are present and in what amounts (moles) when half of the total moles are in the butyraldehyde-rich liquid?

Problem 8.14: Nitroethane (component 1) and octane (component 2) are only partially miscible at 31 °C. At this temperature the triple point is at 6.8 kPa, the mol fraction of nitroethane in the vapor is 0.63, and in the two liquids 0.25 and 0.85, respectively. The saturation pressure of the pure components at 31 °C are, Image and Image All questions below refer to 31 °C.

a) Draw a qualitative Pxy graph of this system and mark all the phases.

b) 4.5 mol of nitroethane is mixed with 1.5 mol of octane at 1 bar. What phases are present? Determine the fraction of the total moles in each phase.

c) The mixture of part (b) is titrated by adding octane until the system forms a single phase. How many moles of octane were added?

d) The mixture of part (b) is brought to a boil by reducing pressure. Which liquid phase disappears first?

e) In part (d), what is the vapor fraction at the point that the first liquid phase completely boils off?

Problem 8.15: Glycerol(1) and acetophenone(2) are partially miscible. The bubble point of the two-liquid system at 140 °C is 0.15 bar. The mole fraction of acetophe-none in the one liquid phase is 10%, in the other liquid phase 85%, and in the vapor 95%.

a) Draw a qualitative Pxy graph of this system. Place on the horizontal axis the mole fraction of the more volatile component. Annotate the graph properly, place all the available information on the graph, and identify the various phases.

b) Twenty mole of acetophenone are mixed with 60 mole of glycerol at 140 °C, 0.2 bar. How many phases are present? Show the state on the Pxy graph.

c) The solution of part b is brought into boiling by reducing the pressure while keeping the temperature at 140 °C. Which phase boils off first? How much vapor is present at the point that the first liquid boils off?

d) How many phases are present when 8% of the original system is in the vapor and what is their composition?

Additional data: Saturation pressures at 140 °C: Image: 0.00313 bar; Image: 0.17 bar.

Problem 8.16: The system 3-methyl-1-butanol (1)/ethanol (2)/water (3) exhibits partial miscibility. The data below give the equilibrium composition of the two phases at 20 °C (Kadir et al., J. Chem. Eng. Data 2008, 53, 910–912):


a) Draw a phase diagram for this system.

b) Of the three binary systems that can be formed with these three components, which are fully miscible and which are not?

c) One mol of 3-methyl-1-butanol (1) is mixed with 0.241 mol of ethanol and 4.15 mol of water at 20 C. Determine the number of phases, their composition, and the number of moles in each phase.

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