Phase diagrams

A phase diagram is a graphical representation of phase transitions under different pressures and temperatures. By analyzing the phase diagram of a 

As you can see in the phase diagram of water shown above, there are three regions in the graph representing solid, liquid, and gas phases. The segment BC divides the solid and the liquid phases of water. This segment represents the equilibrium region between the solid and the liquid phases. Because water is 

Substances in their gas phase can usually be liquefied by increasing the pressure. But it reaches a point where this conversion is not possible. The temperature above which liquid phase caimot be achieved regardless of the applied pressure is called the critical temperature. The vapor pressure at the critical temperature is called the critical pressure. 

The colligative properties that you have to be familiar with for the MCAT are  

A phase diagram is a convenient way of representing the phases of a substance as a function of temperature and pressure. For example, the phase diagram for water (Fig. 16.55) shows which state exists at a given temperature and pressure. It is important to recognize that a phase diagram describes conditions and events for a pure substance in a closed system of the type represented in Fig. 16.53, where no material can escape into the surroundings and no air is present. 

To show how to interpret the phase diagram for water, we will consider heating experiments at several pressures, shown by the dashed lines in Fig. 16.56. 

EXPERIMENT 1 Pressure is 1 atm. This experiment begins with the cylinder shown in Fig. 16.53 completely filled with ice at a temperature of -20°C and with the piston exerting a pressure of 1 atm directly on the ice (there is no air 

The phase diagram for water. represents the normal melting point Fj and Pi denote the triple point Ft represents the normal boiling point Fc represents the critical temperature Pc represents the critical pressure. The negative slope of the solid/liquid line reflects the fact that the density of ice is less than that of liquid water. Note that the solid/liquid phase line continues indefinitely. 

Below the critical temperature and pressure, the gaseous state of a substance is often referred to as a vapor. 

1 Phase a state of matter that is uniform throughout, not only in chemical composition but also in 

Component a chemically independent constituent of the system. It is best understood in relation to the phrase number of components which is the minimum number of independent species necessary to define the composition of all the phases present in the system. 

Degree of freedom (or variance) the number of intensive variables that can be changed without disturbing the number of phases in equilibrium. 

For phases that are in equilibrium with one another, the Gibbs phase law holds  

F is the number of degrees of freedom, i.e. the number of variables of state such as temperature and pressure that can be varied independently, P is the number of phases and C is the number of components. Components are to be understood as independent, pure substances (elements or compounds), from which the other compounds that eventually occur in the system can be formed. For example  

For pure water (one component, C = 1) F + P = 3 holds. When three phases are simultaneously in equilibrium with each other, e.g. vapor, liquid and ice, or vapor and two different modifications of ice, then F = 0 there is no degree of freedom, the three phases can coexist only at one fixed pressure and one fixed temperature ( triplepoint ). 

In the system iron/oxygen (C = 2), when two phases are present, e.g. Fe304 and oxygen, pressure and temperature can be varied (F = 2). When three phases are in equilibrium, e.g. Fe, Fe203 and Fe304, only one degree of freedom exists, and only the pressure or the temperature can be chosen freely. 

In phase diagrams for two-component systems the composition is plotted vs. one of the variables of state (pressure or temperature), the other one having a constant value. Most common are plots of the composition vs. temperature at ambient pressure. Such phase diagrams differ depending on whether the components form solid solutions with each other or not or whether they combine to form compounds. 

Experiment 2 pressure is 2.0 torn Again we start with ice as the only component in the cylinder at — 20°C. The pressure exerted by the piston in this case is only 2.0 torr. As heating proceeds the temperature rises to — 10°C, where the ice changes directly to vapor through the process of sublimation. Sublimation occurs when the vapor pressure of ice is equal to the external 

Diagrams of various heating experiments on samples of water in closed systems. 

The phase diagram for water. represents the normal melting point  

Tj and P3 denote the triple point Tb represents the normal tx iling point  

Pressure is 1 atm. This experiment begins with the cylinder shown in Fig. 10.47 completely filled with Ice at a temperature of -20°C and the piston exerting a pressure of 1 atm directly on the Ice (there Is no air space). Since at temperatures below 0°C the vapor pressure of Ice Is less than 1 atm—which Is the constant external pressure on the piston—no vapor is present in the cylinder. As the cylinder is heated, ice is the only component until the temperature reaches 0°C, where the ice changes to liquid water as energy is added. This is the normal melting point of water. Note that under these conditions no vapor exists in the system. 

3 The phase diagram for the chemical A is shown below. Design a process to transform the gas at 300°C and 2 atm to a solid at 0°C and 1 atm without forming any liquid at any point in the process. Label the function of each unit and specify the temperature and pressure of each stream. Begin your process with a stream of chemical A at 300°C and 2 atm. Your process need not include peripheral equipment for energy efficiency. 

4 We wish to reduce the partial pressure of benzene in air from 10 torr to 1 torr. The phase map below shows the path for cooling the contaminated air at constant total pressure, as discussed in Section 4.4. 

Use the phase map below to plot the same process as plotted above - cooling the contaminated air at constant total pressure to reduce the benzene in the vapor phase by 90 mol%. 

The x-axis, molar volume, is a logarithmic scale. For reference, at 1 atm the vapor/liquid region spans from 0.09 liter/mol (liquid molar volume) to 22.4 liter/mol (vapor molar volume). A point within the two-phase region corresponds to the average molar volume, calculated as 

A point midway between the two borders is a two-phase mixture with about 5% vapor, 95% liquid. 

Three phases for the H2O system are shown in this photograph ice (the iceberg), water (the ocean or sea), and vapor (the clouds). These three phases are not in equilibrium with one another. 

Phase diagrams describe how composition varies with temperature and at which temperatures major transitions occur. The oils and fats literature is brimming with phase diagrams of various kinds, and particularly simple phase diagrams have already been described (see Section 1.5.8). 

There are certain limitations in such phase diagrams. Firstly, this type of equilibrium phase diagram gives no indication as to how quickly the phases will crystallise that is, it 

2 Phase Diagrams and Coexisting Phases 5.4.2.1 Phase diagrams 

The phase diagram of the nCB on water is globally compatible with the previous one but with some specific features and a much larger scattering of the points, due to the hydrodynamic mobility of the substrate, and the simultaneous presence of thicknesses f UB and f LB. Actually, the UB is defined as the thinnest striped film. Even if many experiments with decreasing average thickness are performed, we cannot prove that the thinnest possible film, that is, the UB, has actually been observed. 

Considering the LB, the smooth variation obtained on silica (see Figs. 5.6, 5.7, and 5.10) is not recovered. The most reliable system is 6CB on water, where T is the lowest (29.5 0.1°C in pure samples). The trilayer, around 3 nm thick, is observed till 1.5°C below Tni then the LB jumps suddenly to mesoscopic values, around 40 nm at 28°C and 60 nm at 29°C. Closer to the transition, thermal fluctuations cause the system to oscillate continuously between striped films and the LB because the temperature width of the forbidden range at given thickness tends to zero. 

On glycerol, the diffusion of glycerol molecules inside the liquid crystal does not allow concluding unambiguously on the LB side therefore we did not try to explore the vicinity of the NI transition. 

FIGURE 10.21 Phase diagram for water (the pressures and temperatures are not drawn to scale). 

FIGURE 10.22 When a gas and liquid coexist, the interface between them is clearly visible as a meniscus. The meniscus is useful for reading the volume of a liquid in a buret. It disappears as the critical point is reached. 

Sublimation of solid carbon dioxide (dry ice). The white clouds are drops of water vapor (moisture in the air) that condense at the low temperatures near the solid surface. Gaseous carbon dioxide itself is transparent. 

For most substances, including water (see Fig. 10.23c), atmospheric pressure occurs somewhere between the triple-point pressure and the critical pressure, so in our ordinary experience, all three phases—gas, liquid, and solid—are observed. For a few substances, the triple-point pressure lies above P = 1 atm, and under atmospheric conditions, there is a direct transition called sublimation from solid to gas, without an intermediate liquid state. Carbon dioxide is such a substance (see Fig. 10.23b) its triple-point pressure is 5.117 atm (the triple-point temperature is —56.57°C). Solid CO2 (dry ice) sublimes directly to gaseous CO2 at atmospheric pressure. In this respect, it differs from ordinary ice, which melts before it evaporates and sublimes only at pressures below its triple-point pressure, 0.0060 atm. This fact is used in freeze-drying, a process in which foods are frozen and then put in a vacuum chamber at a pressure of less than 0.0060 atm. The ice crystals that formed on freezing then sublime, leaving a dried food that can be reconstituted by adding water. 

Bismuth is a rather rare element in the earth s crust, but its oxides and sulfides appear at sufficient concentrations as impurities in lead and copper ores to make its recovery from these sources practical. Annual production of bismuth amounts to several million kilograms worldwide. Although elemental bismuth is a metal, its electrical conductivity is quite poor and it is relatively brittle. The major uses of bismuth arise from its low melting point (271.3°C) and the even lower melting points of its alloys, which range down to 47°C. These alloys are used as temperature sensors in fire detectors and automatic sprinkler systems because, in case of 

For each of the phase transitions, there is an associated enthalpy change or heat of transition. For example, there are heats of vaporization, fusion, sublimation, and so on. 

Without phase diagrams, condensed phosphate chemistry would be almost impossible to understand. Properly prepared, a phase diagram is an equilibrium diagram, but in industrial processes equilibrium is seldom, if ever, achieved. Often, systems are very slow to reach demonstrable equilibrium. Industrial processes frequently require the production of hundreds of tons of materials in a short period and equilibrium must be compromised. However, if quality crystals are to be grown, even from industrial melts, equilibrium must be approached. This means that melts must be of uniform homogeneity and cooled at a rate that will allow diffusion to crystal surfaces to be orderly rather than forced. For most phosphate systems much reorganization is required for molecules to acquire dimensions and orientations to fit within a crystal lattice. 

Depending upon the experimenter several techniques have been employed to develop phase diagrams. In phosphate science the late G. W. Morey s technique was probably the most reliable, but very time-consuming. It will be seen that even he could miss a compound in a diagram. A compound missed in the original published phase diagram ultimately became the phosphate fiber, [NaCa(P03)3]w. This is no reflection on his work. There is no way to test whether or not all compounds capable of crystallizing in a system have indeed crystallized. 

A region in which a one-phase blend is always stable for any composition, and a region in which two mixed phases are more stable than a homogeneous system, are divided by the critical point (CP). This is defined mathematically as the point at which the second and third derivatives of the Gibbs free energy with respect to the polymer volume fraction are zero. Applying these conditions to the Flory-Huggins equation (Eq. (3.14)), it is possible to define the 

The occurrence of two different phase-separation mechanisms - namely, spinodal decomposition and nudeation and growth - becomes evident from a consideration of the shape of the AG, curve as a function of rp. The initial stage of phase separation is connected with concentration fluctuations. It can be shown that the 

We can now calculate the phase behaviour of a system of hard spheres and depletants by solving the coexistence equations for a phase I in equilibrium with a phase II 

For numerical computations of phase coexistence, it is convenient to work with dimensionless quantities. The dimensionless version of the free volume expression (3.24) for the grand potential is 

In Fig. 3.10 the semi-grand potential is presented as a function of the colloid volume fraction for given depletant reservoir concentration and size ratio q. Four possible scenarios are considered. Indicated are the common tangent constructions that allow to determine conditions where two (or three) phases coexist. A first criterion for two coexisting binodal composition is equality of the slope because it 

3 Phase Transitions of Hard Spheres Plus Depletants Basics 

When two compositions can be connected through the common tangent (the thin straight lines in the figures connecting these compositions), binodal points are found the intercepts of the extrapolated lines correspond to the total pressure —P. Scenario (i) in Fig. 3.10 corresponds to gas-liquid coexistence. In situation (ii) Q(( ) are given for both the fluid state and for the solid state and the common tangent shows the compositions where fluid and solid coexist. A combination of (i) 

One of the interesting things about the phase diagram of CO2 is that its triple point occttrs above atmospheric pressure -5TC, 5.2 atm. This means that there is no liqttid phase at atmospheric pressure, the condition that makes dry ice dry. At elevated pressme, sohd CO2 does melt. In fact, liquid CO2 is used as the solvent in many dry-cleaning operations. 

Sample Problem 12.9 lets you practice interpreting the information in a phase diagram. 

Using the following phase diagram, (a) determine the normal boiling point and the normal melting point of the substance, (b) determine the physical state of the substance at 2 atm and 110°C, and (c) determine the pressure and temperature that correspond to the triple point of the substance. 

Strategy Each point on the phase diagram corresponds to a pressure-temperature combination. 

The normal boiling and melting points are the temperatures at which the substance undergoes phase changes. These points fall on the phase boundary lines. The triple point is where the three phase 

Structure, and the a-polymorph adopts a bcc lattice. Phases that form at high temperatures may be quenched to lower temperatures (i.e. rapidly cooled with retention of structure), allowing the structure to be determined at ambient temperatures. Thermochemical data show that there is usually very little difference in energy between different polymorphs of an element. 

Use Fig. 6.7 to describe what happens to the structure of iron if the pressure is raised from 1 bar while maintaining the temperature at 900 K. 

In general, the bcc structure is the high-temperature form of a metal which is close-packed at a lower temperature. What happens to the density of the metal during this phase change  

Give two examples of metals that have a bcc stmcture 

Line 12 represents the coexistence of vapor and liquid phases and it is referred to as the vapor pressure curve. Point 5 indicates, thus, that the vapor pressure of the liquid at the temperature 5 is equal to P5 which we depict, as we have seen, as P Notice that increasing temperatures lead, as expected, to increasing vapor pressure values. Beyond the temperature of point 2, however, the kinetic energy of the molecules is too high to allow them to form a liquid phase. 

We refer to point 2 as the critical point, and the corresponding values of temperature, pressure and volume, as the critical ones depicted by T, Pg and Vg respectively. Notice that according to our classification, a vapor at a temperature higher than Tg, is a gas. 

Line 13 represents the coexistence of solid and liquid phases and is called the fusion curve. Point 6, thus, depicts the melting point pressure at the temperature 

Line 14 represents the vapor-solid coexistence - or sublimation - curve. Point 7, thus, depicts the vapor pressure of the solid at the temperature T-j. Finally, point 1 represents the coexistence of all three phases vapor, liquid and solid. It is called, appropriately, the triple point T,. 

Champion, P. Guillet, L Poupeau, P. (1981) Diagrammes de Phases des Materiaux Cristallins. Masson Editeur, Paris. 

FERGUSSON, F.D. Jones, T.K. (1966) The Phase Rule. Butterworths Co. Ltd, London. 

Hansen, M. Anderko, K. (1958) Constitution of Binary Alloys, 2nd. ed. McGraw-Hill, New York. 

Haughton, L.J. Prince, A. (1956) The Constitutional Diagrams of Alloys a Bibliography, 2nd. ed. Monograph and Reports Series No. 2, Institute of Metals (loM). 

Hulgren, R. Desai, P.D. Hawkins, D.T. Gleiser, M. Kelley, K.K. (1973) 4.1.Selected Values of the Thermodynamics Properties of the Elements University of California, Berkeley/ASM. 

IBLG See questions from Phase Changes and Phase Diagrams  

Unless otherwise noted, all art on this page Is Cengage Learning 2014. 

Pressure is 2.0 torn Again, we start with ice as the only component in the cylinder at -20°C. The pressure exerted by the piston in this case is only 2.0 torr. As heating proceeds, the temperature rises to - 10°C, where the ice changes directly to vapor, a process known as sublimation. Sublimation occurs when the vapor pressure of ice is equal to the external pressure, which in this case is only 2.0 torr. No Uquid water appears under these conditions because the vapor pressure of Uquid water is always greater than 2.0 torr, and thus it cannot exist at this pressure. If Uquid water were placed in a cylinder under such a low pressure, it would vaporize immediately at temperatures above - 10°C or freeze at temperatures below - 10°C. 

Pressure and temperature are two important intensive properties that help determine the phase of a subsfan.ee. A phase diagram indicates the phases of a substance at different pressures and temperatures. Each section of a phase diagram represents a different phase. The lines marking the boundaries of each section represent temperatures and pressures where the corresponding phases are in equilibrium with each other. Like other equilibriums in chemistry, this equilibrium is a dynamic equilibrium, For instance, when water and steam are in equilibrium, water molecules are escaping from the liquid phase at the same rate that they are returning. Notice that there is only one point where a substance can exist in equilibrium as a solid, liquid, and gas. This point is called the triple point. 

There is also a temperature above which a substance cannot be liquefied regardless of the pressure applied. This temperature is called the critical temperature. The pressure required to produce liquefaction while the substance is at the critical temperature is called the critical pressure. Together, the critical temperature and critical pressure define the critical point. Fluid beyond the critical point has characteristics of both gas and liquid, and is called supercritical fluid. 

Comparing the phase diagrams for water and carbon dioxide, we notice some interesting things. Even if it were not labeled, we could approximate the location of the 1 atm mark for either diagram. We know that at atmospheric pressure, water exists in all three phases at different temperatures. Thus, we know that the 1 atmosphere mark must be above the triple point. Since carbon dioxide (dry ice) sublimes (changes from solid to gas) at one atmosphere, we know that the triple point must be above the 1 atm mark. 

Think about this for a single sample of a substance. Pr V, w. and T are interrelated in such a way that if you know three of them, you can derive the other. This means that a phase diagram can also be given as a comparison between volume and pressure, or volume and temperature- What would that look like See the problems on the next page for the answer. 

What is the total heat needed to change 1 gram of water from -10°C to 110°C at 1 atm (AHfllsi(m = 80 cal/g, 

It will be important in discussions in other parts of this text to be able to describe easily which phases will be present under a variety of conditions and if a combination of phases is in equilibrium or not. For example, at what temperature will an element, alloy or mixture of phases melt What is the composition of the material that melts at 

It can be shown that the number of degrees of freedom F which a system has 

When two components are present, an additional degree of freedom exists. The three degrees of freedom are now temperature, pressure and composition. This means that two phases can be present in a two-dimensional region of phase space. One can, for example, change both the composition of the mixture and the temperature and retain the two phases. It is even possible to have three phases present together. For example, two solid phases and a liquid phase can coexist, but only on a line. 

There is a triple poinf at which a liquid phase (in the case of Pb-Sn, a liquid solution of Pb and Sn), and two solid phases coexist. This is known as the eutectic point. It occurs at a specific composition, the eutectic composition, and melts/freezes completely at a specific temperature, the eutectic temperature. In the case of Pb-Sn, the eutectic composition is 61.9 weight % Sn, and the eutectic temperature is 181 °C. Notice that the eutectic solid is a two-phase mixture. In spite of this, both phases melt at the same temperature for the eutectic alloy. The maxima of solid solubility for Sn in solid Pb and Pb in solid Sn occur at the eutectic temperature and are, 19.2 weight % and 97.5 weight % Sn, respectively. A solid mixture having the eutectic composition on average is the only mixture that melts entirely at a single temperature. All other compositions (except pure elements) in a eutectic system partially melt leaving a solid with altered composition (if kinetics allow the solids to maintain equilibrium) in contact with a liquid of different composition. 

Ternary phase diagrams have lines drawn connecting any two phases, which, when mixed, result in either case 1 or 2, above. These are referred to as tie lines . Rephrasing, lines on the phase diagram indicate phases which, when mixed, will be unaffected or will alloy. The two-phase result indicates that the two materials are 

In solid solutions, the separation has to be done step by step, as in distillation processes. The number of steps is determined by the phase diagram and the required purity of the product. As a result, complete product recovery is possible, although the number of stages required increases rapidly as recovery approaches unity. 

Known solid solution forming systems include naphthalene-theonaphthalene thiophene-benzene hexadecane-octadecane and m-chloronitrobenzene-w-fluoronitrobenzene. Further advantages of melt crystallization are the smaller volume of the liquid phase compared to the vapor phase of a substance. A smaller volume leads towards less space or less construction work, which means less capital costs. These advantages are sometimes lost if the process of crystallization and remelting is very slow therefore, the retention time in the apparatus is high. 

The nonexistence of a vapor phase, however, also leads to better control of leakages. Totally closed equipment leads to high environmental safety. 

Melt crystallization does not need any additional substances, therefore, no waste water will be produced and no other chemicals (solvents) have to be reprocessed. Solvent recovery capital and energy cost can represent a major portion of a product isolation process utilizing solution crystallization. 

As we mentioned in the chapter opening, the solid, liquid, and gaseous states of carbon dioxide exist under different temperature and pressure conditions. A phase diagram is a graphical way to summarize the conditions under which the dijferent states of a substance are stable. 

The following is a dramatic demonstration of the effect of pressure on the melting of ice Suspend a block of ice between two chairs.Then loop a length of wire over the block and place weights on the ends of the wire.The pressure of the wire will melt the ice under it. Liquid water will flow over the top of the wire and freeze again, because it is no longer under the pressure of the wire. As a result, the wire will cut through the ice and fall to the floor, but the block of ice will remain as one piece. 

This curve gives the melting points of the solid at various pressures. 

Because the triple point for water occurs at a definite temperature, it is used to define the Kelvin temperature scale.The temperature of water at its triple point is defined to be 273.16 K (0.01 O. 

Phase diagrams for carbon dioxide and sulfur (not to scale) 

In the preceding section, we discussed several features of the equilibrium between a liquid and its vapor. For a pure substance, at least two other types of phase equilibria need to be considered. One is the equilibrium between a sohd and its vapor, the other between solid and liquid at the melting (freezing) point Many of the important relations 

To understand what a phase dicigrcun impKes, consider first curves b (in green) cuid c (in red) and line a (in blue) in Figme 9.6. Each of these shows the pressures and temperatures at which two adjacent pheises me in equfiibrimn. 

Curve b is a portion of the pressure-temperatme cmve of liquid water. At cuiy temperature and pressure along this curve, liquid water is in equihbrium with water vapor. At point X on the curve, these two phases are in equihbrium at 0°C and about 5 mm Hg (more exactly, 0.01°C and 4.56 mm Hg). At point Y corresponding to 100°C, the pressure exerted by the vapor in equilibrium with Uquid water is 1 atm this is the normal boiling point of water. The extension of the curve b beyond point Y gives the equilibrium vapor pressure of water above the normal boiling point. The extension ends at 374°C, the critical temperature of water, where the pressure is 218 atm. 

Curve c represents the vapor pressure curve of ice. At any point along this curve, such as point X (0°C, 5 mm Hg) or point Z, which might represent -3°C and 3 mm Hg, ice and water vapor are in equilibrium with each other. 

Line a gives the temperatures and applied pressures at which liquid water is in equilibrium with ice. 

Point A on a phase diagram is the only one at which all three phases, liquid, solid, and vapor, are in equilibrium with each other. It is called the triple point. For water, the triplepoint temperature is 0.01°C. At this temperature, liquid water and ice have the same vapor pressure, 4.56 mm Hg. 

In the three areas of the phase diagram labeled solid, liquid, and vapor, only one phase is present. To understand this, consider what happens to an equilibrium mixture of two phases when the pressure or temperature is changed. Suppose we start at the point on AB 

The equilibrium between a bquid and its vapor is not the only dynamic equilibrium that can exist between states of matter. Under appropriate conditions, a solid can be in equilibrium with its liquid or even with its vapor. The temperature at which solid and liquid phases coexist at equilibrium is the melting point of the sohd or the freezing point of the liquid. Solids can also undergo evaporation and therefore possess a vapor pressure. 

A phase diagram is a graphic way to summarize the conditions under which equilibria exist between the different states of matter. Such a diagram also allows us to predict which phase of a substance is present at any given temperature and pressure. 

The red curve is the vapor-pressure curve of the liquid, representing equilibrium between the liquid and gas phases. The point on this curve where the vapor pressure is 1 atm is the normal boiling point of the substance. The vapor-pressure curve ends at the critical point (C), which corresponds to the critical temperature and critical pressure of the substance. At temperatures and pressures beyond the critical point, the liquid and gas phases are indistinguishable from each other, and the substance is a supercritical fluid. 

The green curve, the sublimation curve, separates the solid phase from the gas phase and represents the change in the vapor pressure of the sohd as it subhmes at different temperatures. Each point on this curve is a condition of equihbrium between the solid and the gas. 

Imagine that the pressure on the solid phase in the figure is decreased at constant temperature. If the solid eventually sublimes, what must be true about the temperature  

The overall relationships among the solid, liquid, and vapor phases are best represented in a single graph known as a phase diagram. A phase diagram summarizes the conditions at which a substance exists as a solid, liquid, or gas. In this section we will briefly discuss the phase diagrams of water and carbon dioxide. 

The phase diagram of helium is shown here. Helium is the only known substance that has two different liquid phases called helium-I and helium-II. (a) What is the maximum temperature at which helium-II can exist (b) What is the minimum pressure at which solid helium can exist (c) What is the normal boiling point of helium-I (d) Can solid helium sublime (e) How many triple points are there  

Hard-Boiling an Egg on a Mountaintop, Pressure Cookers, and Ice Skating 

Phase equilibria are affected by external pressure. Depending on atmospheric conditions, the boihng point and fieezing point of water may deviate appreciably from 100°C and 0°C, respectively, as we see below. 

The effect of pressure on boiling point also explains why pressure cookers save time in the kitchen. A pressure cooker is a sealed container that allows steam to escape only when it exceeds a certain pressure. The pressure above the water in the cooker is the sum of the atmospheric pressure and the pressure of the steam. Consequently, the water in the pressure cooker will boil at a higher temperature than 100°C and the food in it will be hotter and cook faster. 

We can also start to understand why some substances, such as soHd carbon dioxide, sublime to a vapor without first forming a liquid. There is no fundamental requirement for the three lines to lie exactly in the positions we have drawn them in Fig. 3.3 the Hquid line, for instance, could lie where we have drawn it in Fig. 3.4. Now we see that at no temperature (at the given pressure) does the liquid phase have the lowest molar Gibbs energy. Such a substance converts spontaneously directly from the solid to the vapor. That is, the substance sublimes. 

The transition temperature between two phases, such as between liquid and solid or between conformations of a protein, is the temperature, at a given pressure, at which the two phases are in equilibrium and therefore their molar Gibbs energies are equal. At 1 atm, for instance, ice and liquid water are in equilibrium at 0°C and G (H20,1) = G ,(H20,s). 

To prepare for being able to describe phase transitions in biological macromolecules, 

The phase diagram of a substance is a map showing the conditions of temperature and pressure at which its various phases are thermodynamically most stable (Fig. 3.5). For example, at point A in the illustration, the vapor phase of the substance is thermodynamically the most stable, but at C the liquid phase 

The boundaries between regions in a phase diagram, which are called phase boundaries, show the values of p and T at which the two neighboring phases are in equilibrium. For example, if the system is arranged to have a pressure and 

Two other special features on the diagram are designated by black dots. The dot at point D, known as the critical point, represents the critical temperature and the critical pressure (the point at which the liquid state no longer exists, regardless of the amount of pressure). The other dot represents the intersection of the three lines, known as the triple point. The triple point represents the temperature and pressure at which all three phases coexist simultaneously. 

All of the phase changes listed earlier in the chapter are also detectable on the phase diagrams. The equilibrium lines, mentioned earlier in this section, represent those places where phase changes occur. For example, at any of the locations along line A-C, a transition from left to right represents melting. 

Which point represents the equilibrium between the liquid and gas phases during vaporization  

Answer The correct answer is D. The entire line from C to the critical point E represents equilibrium between the liquid and gas phases. 

we can fix, for example, temperature and the vapor composition, and expect the system to set its own values of liquid composition as well as total pressure. If, on the other hand, we choose to prescribe pressure and vapor composition, the system will respond with a particular temperature (i.e., its boiling point), as well as a particular liquid composition. A third possibility is to fix both temperature and total pressure, in which case the system will set its own values of both liquid and vapor composition. All three cases are encoxmtered in practice and are expressed in terms of appropriate phase diagrams, which are taken up below. 

In the phase equilibria considered so far, the principal focus rested on the distribution of a single key component, usually referred to as the solute, between the constituent phases. Thus, in the gas-liquid and liquid-solid equilibria discussed in Sections 6.2.1 and 6.2.2, our concern was with only one of the components present, while the remaining bulk components, such as liquid solvent or solid adsorbent, were left out of consideration. 

Binary vapor-liquid equilibria of ideal systems (a) boiling-point diagram (b) vapor-pressure diagram and (c) x-y diagram. 

Mass Transfer and Separation Processes Principles and Applications 

2 Ideal Solutions and Raoult s Law Deviation from Ideality 

Another experimental quantity of interest is the entropy of transition from smectic-A to nematic (cholesteric) structure. Just as in the 

One of the most unambiguous and simple methods of investigating the association of molecules, as in charge-transfer complexes, solvates, etc. is by means of the freezing point or the vapour pressure of the mixture as a function of its composition. These classical methods have been displaced to some extent by spectroscopic methods, which, however, fail if one is dealing 

A plant is to make 10,000 lb/hr of urea crystals from a solution that contains 75% dissolved salt. The material balance and operating conditions are shown on the sketch. Key crystallization data are given by Bennett (1981, p. 452) as 

Other information deduced from pilot plant work is  

The feed contains 75% solids, but 1200 lb/hr of wash water from the centrifuge also is returned to the crystallizer. 

The liquor contains 66.8% dissolved urea and has a specific gravity of 1.17 at the operating temperature of 130°F. 

By considering the temperature dependence of the free energy of mixing. 

The phase boundary is determined by the common tangent of the free energy at the compositions (j) and j) corresponding to the two equilibrium phases 

For the simple example of a symmetric polymer blend with Aa — Ab = A . 

The above equation can be solved for the interaction parameter corresponding to the phase boundary—the binodal (solid line in the bottom part of Fig. 4.8) of a symmetric blend  

If a substance exists in more than one crystalline form, it is 

Analyze We are asked to read a graph of vapor pressure versus temperature to determine the boiling point of a substance at a particular pressure. The boiling point is the temperature at which the vapor pressure is equal to the external pressure. 

Plan We need to convert 0.80 atm to torr because that is the pressure scale on the graph. We estimate the location of that pressure on the graph, move horizontally to the vapor pressure curve, and then drop vertically from the curve to estimate the temperature. 

Comment We can make a flask of diethyl ether boil at room temperature by using a vacuum pump to lower the pressure above the liquid to about 0.8 atm. 

At what external pressure will ethanol have a boiling point of 60 °C  

71 mWcm ) on PBHT/PCeiBM blend composition for devices thermally treated at 140°C after spin casting (filled circles), subsequently melt quenched from 290 °C (open triangles), and then after further annealing at 140°C (open circles). In accordance with previous reports,Jsc is optimized after annealing at blend compositions comprising 50-60 wt% of the polymer. Error bars represent estimated percentage error based on comparison of similar devices. (From Ref. [130].) 

4 Kalinowski, J. (2004) Organic Light-Emitting Diodes Principles, Characteristics, and Processes, Optical Fngineering, 

24 Hansen, C.M. (2007) Chapter 1, in Hansen Solubility Parameters - A User s Handbook, 2nd edn, CRC Press, Boca Raton, PL. 

32 van Krevelen, D.W. (1972) Properties of Polymers Correlations with Chemical Structure, Elsevier, Amsterdam. 

34 Fredenslund, A., Gmehling, J., and Rasmussen, P. (1977) Vapor-Liquid Equilibria Using UNIFAC, Elsevier, Amsterdam. 

Solids, too, undergo evaporation and therefore possess a vapor pressure. Consider the following dynamic equilibrium  

The various phase changes that a substance can undergo. 

The phase diagram of carbon dioxide. Note that the solid-liquid boundary line has a positive slope. The liquid phase is not stable below 5.2 atm, so that only the solid and vapor phases can exist under atmospheric conditions. 

Under atmospheric conditions, solid carbon dioxide does not melt it can only sublime. The cold carbon dioxide gas causes nearby water vapor to condense and form a fog. 

1 How much energy (in kJ) is required to convert 25.0 g of liquid water at room temperature (25°C) to steam at 110 C  

CHAPTER 12 Intermolecular Forces and the Physical Properties of Liquids and Solids 

For ideal gases, a total absence of intermolecular forces is assumed. Neither attractive nor repulsive forces are taken into accoxmt and collisions between molecules are taken to be entirely elastic in nature. These assumptions hold well at low pressures because of the large distances between particles and the vanishingly short time they spend in collision with each other. In liquids, the molecules are closely packed and in constant intimate contact witii each 

Several important subsidiary laws flow from this relation. In the first instance, we can extend Raoult s law to the second component of the binary mixture and obtain 

Deviations from ideal behavior occur when there is a marked difference in the molecular structure of tiie participating species. Typical combinations that give rise to nonideal behavior are pairs of polar and nonpolar substances, in which the attractive and repulsive forces vary with compositional changes. When repulsive forces predominate, the vapor pressures rise above the values predicted by ideal solution theory and we speak of positive deviations. Most vapor-liquid equilibria fall in this category. A predominance of 

You just saw that the heat capacity of ice is Cg ice = 2.09 J/g °C and that the heat of fusion of ice is 6.02 kJ/mol. When a small ice cube at -10 °C is put into a cup of water at room temperature, which of the following plays a greater role in cooling the liquid water the warming of the ice from -10 °C to 0 °C, or the melting of the ice  

The conductivity tr of ionic crystals can be assumed to be due to the mobilities of the individual ions and electrons  

In this formula q, Omob and denote respectively, the charge, the concentration and the mobility of charge carriers. In fluorite-like phases these charge carriers are mainly fluoride ions. 

The value of Ag b obfained wifh equation (12.4) can replace the enthalpy of vaporizafion in fhe Clausius-Clapeyron equation (12.2), so that sublimation pressures can be calculated as a function of temperature. 

Recall the discussion of dew and frost formation (see the Focus On feature for Chapter 6, Earth s Atmosphere, atwww.masteringchemistry.com). Do the surroundings absorb or lose heat when water vapor condenses to dew or frost Is the quantity of heat per gram of H20(g) condensed the same whether the condensate is dew or frost Explain. 

Even at temperatures well below its melting point of 114 °C, solid iodine exhibits an appreciable sublimation pressure. Here, purple iodine vapor is produced at about 70 °C. Deposition of the vapor to solid iodine occurs on the colder walls of the flask. 

The outline of a phase diagram is suggested by the distribution of points. 

The extreme range of temperatures and pressures required for the entire phase diagram precludes plotting it to scale. This is why the axes are labeled not to scale.  

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Table 1 gives the measured data, estimates of the true values corresponding to the measurements, and deviations of the measured values from model predictions. Figure 1 shows the phase diagram corresponding to these parameters, together with the measured data. 

Figure 6-1. Calculated phase diagram for the acetone(I)-methanol(2) system.

So far we have considered only a single component. However, reservoir fluids contain a mixture of hundreds of components, which adds to the complexity of the phase behaviour. Now consider the impact of adding one component to the ethane, say n-heptane (C7H.,g). We are now discussing a binary (two component) mixture, and will concentrate on the pressure-temperature phase diagram. 

Figure 5.20 Pressure-temperature phase diagram mixture of ethane and n-heptane...

The example of a binary mixture is used to demonstrate the increased complexity of the phase diagram through the introduction of a second component in the system. Typical reservoir fluids contain hundreds of components, which makes the laboratory measurement or mathematical prediction of the phase behaviour more complex still. However, the principles established above will be useful in understanding the differences in phase behaviour for the main types of hydrocarbon identified. 

Figure 5.21 helps to explain how the phase diagrams of the main types of reservoir fluid are used to predict fluid behaviour during production and how this influences field development planning. It should be noted that there are no values on the axes, since in fact the scales will vary for each fluid type. Figure 5.21 shows the relative positions of the phase envelopes for each fluid type. 

A volatile oil contains a relatively large fraction of lighter and intermediate oomponents which vaporise easily. With a small drop in pressure below the bubble point, the relative amount of liquid to gas in the two-phase mixture drops rapidly, as shown in the phase diagram by the wide spacing of the iso-vol lines. At reservoir pressures below the bubble point, gas is released In the reservoir, and Is known as solution gas, since above the bubble point this gas was contained in solution. Some of this liberated gas will flow towards the producing wells, while some will remain in the reservoir and migrate towards the crest of the structure to form a secondary gas cap. 

Black oils are a common category of reservoir fluids, and are similar to volatile oils in behaviour, except that they contain a lower fraction of volatile components and therefore require a much larger pressure drop below the bubble point before significant volumes of gas are released from solution. This is reflected by the position of the iso-vol lines in the phase diagram, where the lines of low liquid percentage are grouped around the dew point line. 

When oil and gas are produced simultaneously into a separator a certain amount (mass fraction) of each component (e.g. butane) will be in the vapour phase and the rest in the liquid phase. This can be described using phase diagrams (such as those described in section 4.2) which describe the behaviour of multi-component mixtures at various temperatures and pressures. However to determine how much of each component goes into the gas or liquid phase the equilibrium constants (or equilibrium vapour liquid ratios) K must be known. 

After having proved the principles a dynamic test facility has been constructed. In this facility it is possible to inject 3 tracers in a flownng liquid consisting of air, oil and water. By changing the relative amounts of the different components it is possible to explore the phase diagram and asses the limits for the measurement principle. Experiments have confirmed the accuracy in parameter estimation to be below 10%, which is considered quite satisfactorily for practical applications. The method will be tested on site at an offshore installation this summer. 

Fig. IV-17. A schematic phase diagram illustrating the condensed mesophases found in monolayers of fatty acids and lipids.

Fig. XII-8. A schematic friction phase diagram showing the trends found in the friction forces of surfactant monolayers. (From Ref. 53.)...

Discuss the dependence of the friction phase diagram on temperature, mono-layer density, velocity, load and solvent vapor. Explain why each of these variables will drive one to the right or left in Fig. XII-8. 

Fig. XIV-11. Schematic phase diagram of a microemulsion-fortning system. (From Ref. 77.)...

Fig. XVII-17. Schematic phase diagram for O2 on graphite (see text). (From Ref 95. Reprinted with permission from American Chemical Society, copyright 1996.)...

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

Figure A2.5.1. Schematic phase diagram (pressure p versus temperature 7) for a typical one-component substance. The full lines mark the transitions from one phase to another (g, gas liquid s, solid). The liquid-gas line (the vapour pressure curve) ends at a critical point (c). The dotted line is a constant pressure line. The dashed lines represent metastable extensions of the stable phases.

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Figure A2.5.3. Typical liquid-gas phase diagram (temperature T versus mole fraction v at constant pressure) for a two-component system in which both the liquid and the gas are ideal mixtures. Note the extent of the two-phase liquid-gas region. The dashed vertical line is the direction x = 1/2) along which the fiinctions in figure A2.5.5 are detemiined.

Figure A2.5.5. Phase diagrams for two-eomponent systems with deviations from ideal behaviour (temperature T versus mole fraetion v at eonstant pressure). Liquid-gas phase diagrams with maximum (a) and minimum (b) boiling mixtures (azeotropes), (e) Liquid-liquid phase separation, with a eoexistenee eurve and a eritieal point.

A third kind of phase diagram in a two-eomponent system (as shown in figure A2.5.5(e) is one showing liquid-liquid phase separation below a oritieal-solution point, again at a fixed pressure. (On aT,x diagram, the eritieal point is always an extremum of tire two-phase eoexistenee eurve, but not always a maximum. 

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

The boundaries separating these principal types of phase behaviour are shown on X,C, diagram (for equalsized molecules) in figure A2.5.13. For molecules of different size, but with the approximation of equation (A2.5.10). more global phase diagrams were calculated using a third parameter,... 

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