1.5 INDEPENDENT AND DEPENDENT THERMODYNAMIC PROPERTIES
1.5 Independent and Dependent Thermodynamic Properties ◄ 19
We next wish to examine how to constrain the state of systems with more than one phase present. If we have a pure substance with two phases present, the phase rule says we need just one property in each phase to constrain the values of all the other proper- ties for that phase. However, the properties temperature and pressure present a special case, since they are equal in both phases. Most other properties are different between phases.12 Thus, if we know either T or P of the system, we constrain the properties in each of the phases.
To illustrate this concept, consider a pure system of boiling water where we have both a liquid and a vapor phase. In this text, we use water to indicate the chemical spe- cies H2O in any phase: solid, liquid, or gas.13 The phase rule tells us that for the liquid phase of water, we need only one property to constrain the state of the phase. If we know the system pressure, P, all the other properties 1T, vl , ul, c2 of the liquid are constrained. The subscript “l” refers to the liquid phase. It is omitted on T since the temperatures of both the liquid and vapor phases are equal. For example, for a pressure of 1 atm, the temperature is 100 [°C]. We can also determine that the volume of the liquid is 1.0431023 3m3/kg4, the internal energy is 418.94 [kJ/kg], and so on. The sys- tem pressure of 1 atm also constrains the properties of the vapor phase. The temperature remains the same as for the liquid, 100 [°C]; however, the values for the volume of the vapor 11.633m3/kg4 2, the internal energy (2,506.5 [kJ/kg]), and so on are different from those of the liquid.
The pressure (and temperature) in each phase of a two-phase system is equal; hence, if we know P (or T), we know the values of all the intensive properties in both phases.
However, we have not yet constrained the state of the system. To do so, we need to know the proportion of matter in each phase. Thus, a second independent intensive property that is related to the mass fraction in each phase is required. Specifying that a system of boiling water is at 1 atm does not tell us how much liquid and how much vapor are present. We could have all liquid with just one bubble of vapor, all vapor with just one drop of liquid, or anything in between.
To constrain the state of the system, we can specify, for example, the fraction of water that is vapor. This quantity is termed the quality, x:
x5 nv
nl1nv
(1.14) where nv and nl are the number of moles in the liquid and vapor phases, respectively.
Any intensive property can then be found by proportioning its value in each phase by the fraction of the system that the phase occupies. For example, if we know x, the molar volume of a liquid–vapor system can be calculated as follows:
v5 112x2vl1xvv (1.15)
Note that the molar volume we calculate from Equation (1.15) is not representative of that from either phase but rather is a weighted average that we report as the value of
12 We will learn in Chapter 6 that Gibbs energy, g, is another property that takes the same value for different phases in a system that contains a pure substance.
13 There are different usages of the word water; some texts reserve water only for the liquid phase.
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20 ► Chapter 1. Measured Thermodynamic Properties and Other Basic Concepts
the system. Other intensive properties (e.g., u, s, h . . . ) can similarly be found once the quality is known.
Conversely, if we specify the molar volume, v, of a two-phase mixture in addition to the pressure, we have two independent properties and have completely constrained the system. The molar volume, at a given pressure, is characteristic of the mole propor- tion in each phase. In fact, knowing the volume allows us to back-calculate the quality through Equation (1.15), since the volumes in each phase, vv and vl, are constrained by the pressure. However, we cannot choose both T and P as the two properties to constrain a two-phase system, since these properties are not independent. Once we know T, P is constrained; it is the saturation pressure. Since they have equal values in each phase, neither property tells us the proportion of matter belonging to each phase.
A pure substance can also have three phases present. According to the Gibbs phase rule, each phase in such a system has zero degrees of freedom. They do not have any independent properties; therefore, all intensive properties in each of the three phases are specifi ed. Consider a system in which a pure substance exists in the solid, liquid, and vapor phases. The properties of each phase can have only one value. Since the tempera- ture and pressure are equal in all the phases, they are fi xed for the entire system. For example, the values for P and T for water in a system with solid, liquid, and vapor are fi xed at 611.3 [Pa] and 0.01[°C], respectively. This state is known as the triple point.14 In this case, we can specify neither T nor P, since neither property is independent. In other words, both properties we specify to constrain the state must be related to the fraction of matter in each of the three phases present. A pure substance cannot have more than three phases, as such a state would violate the Gibbs phase rule.
►1.6 THE PvT SURFACE AND ITS PROJECTIONS FOR PURE SUBSTANCES
In this section, we explore graphical depictions of the relation between the measured variables P, v, and T. Figure 1.6 shows a PvT surface for a typical pure substance. This three-dimensional graph is constructed by plotting molar volume on the x-axis, tempera- ture on the y-axis, and pressure on the z-axis. The state postulate tells us that these three intensive properties are not all independent. The “surface” that is plotted identifi es the values that all three measured properties of a given pure substance can simultaneously have. While each species has its own characteristic PvT surface, the general qualitative features shown in Figure 1.6 are common to all species.15
Below the PvT surface in Figure 1.6, two-dimensional projections in the Pv plane and PT plane are shown. These projections are often referred to as Pv diagrams and PT diagrams, respectively. We also project the PvT surface onto the Tv plane; however, it is not shown. It is often more convenient to describe thermodynamic states and processes using two-dimensional projections. It should be noted that the PvT surface and its pro- jections are not drawn to scale in Figure 1.6, but rather exaggerated to illustrate the salient features.
Each of the depictions in Figure 1.6 shows three single-phase regions labeled “vapor,”
“liquid,” and “solid.”16 In these regions, P and T are independent, so we can specify each of these properties independently to constrain the state of the system. Once P and T
14 It is actually the triple point value of 0.01°C that, together with the same scale per degree as the Kelvin temperature scale, has been chosen to specify the Celsius temperature scale.
15 Figure 1.6 shows the behavior for a species that contracts upon freezing. A few substances such as water, silicon, and some metals expand upon freezing and will have a freezing line with a negative slope on the PT projection.
16 In fact, many species have several different solid phases.
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1.6 The PvT Surface and Its Projections for Pure Substances ◄ 21
are identifi ed, the state is fi xed and, consequently, the other properties are constrained.
Thus, the molar volume can have only one value.
Joining the single phases are two-phase regions where two phases can coexist at equilibrium. Liquid–vapor, solid–vapor, and solid–liquid two-phase regions are identi- fi ed. Knowing P and T allows us to identify the phase(s) the substance is in; thus, we often call these projections phase diagrams. As we discussed in association with the Gibbs phase rule, in the two-phase regions the properties T and P are no longer inde- pendent, since pressure and temperature have equal values in the different phases and picking either one of these properties constrains the other. Therefore, these regions are represented by lines on the PT diagram. On the other hand, we can constrain the system by specifying P and v, since v is characteristic of the fraction of matter present in each phase. Thus, a given value of v in the shaded regions in the Pv diagram represents the differing proportions of each phase present. The line in the Pv diagram that sepa- rates the two-phase region from the single-phase liquid on one side and the single-phase vapor on the other is known as the liquid–vapor dome.
The triple point is labeled on the PT diagram in Figure 1.6. In this state, a pure substance can have vapor, liquid, and solid phases all coexisting together. The phase rule tells us that each phase has zero degrees of freedom. Consequently, both the system temperature and pressure are fi xed as a point on the PT diagram. The Pv projection shows the three-phase region as a line, the triple line, since the molar volume changes as the proportion of each phase changes.
Figure 1.6 The PvT surface of a pure substance and two-dimensional projections in the Pv and PT plane.
Solid–
liquid
Solid–
vapor
Solid–vapor
Super critical
Critical point
Critical point Critical
point
Liquid
Liquid
Liquid
Liquid–
vapor Solid
Solid P
v T
v
Solid
Vapor
Vapor
Vapor Triple line
Liquid–
vapor Triple line
Triple line P
P
T T = Tc
Solid–liquid
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22 ► Chapter 1. Measured Thermodynamic Properties and Other Basic Concepts
The projections of PvT surfaces are useful for identifying the thermodynamic state of a system. To illustrate this point, we show fi ve different states, all at identical pressures, in Figure 1.7. On the left of the fi gure, each state is identifi ed in the context of a piston–
cylinder assembly. If the system represented by state 1 undergoes a set of isobaric pro- cesses whereby energy is input to the system, it will go from state 1 to 2 to 3 to 4 to 5.
These states are also identifi ed by number on the Pvdiagram and the PT diagram on the right of the fi gure. Note that the lower half of the Pvdiagram is omitted for clarity.
State 1 represents subcooled liquid, where pressure and temperature are inde- pendent properties. As energy is put into the system, the temperature will rise until the liquid becomes saturated, as illustrated on the PT diagram. The volume also increases;
however, the magnitude of the change is small, since the volume of a liquid is relatively insensitive to temperature.
The substance is known to be in a saturated condition when it is in the two-phase region at vapor–liquid equilibrium. A saturated liquid is “ready” to boil; that is, any more energy input will lead to a bubble of vapor. It is labeled as state 2 on the left of the liquid–
vapor dome in the Pv diagram. Since we are now in a two-phase region, the tempera- ture is no longer independent. At a given pressure, the temperature at which a pure substance boils is known as the saturation temperature. The saturation temperature at any pressure is given by the line in the PT diagram on which state 2 is labeled. The Changes of State During a Process
State 1 Subcooled liquid
State 3
Saturated liquid
Saturated liquid Saturated vapor
Superheated vapor Saturated vapor
State 2
State 4 State 5
Liquid
Liquid Liquid – vapor
Vapor
Vapor Solid
Critical point
Critical point P
m m
m m
mI = mv
m m m
m
m m
P
T v
Triple point 3
1 2 4 5
1 2,3,4 5
Figure 1.7 Five states of a pure substance and their corresponding locations on Pv and PT projections. All states are at the same pressure. State 1 is a subcooled liquid; state 2, a satu- rated liquid; state 3, a saturated liquid–vapor mixture; state 4, a saturated vapor; and state 5, a superheated vapor. The volume of liquid is exaggerated for clarity.
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1.6 The PvT Surface and Its Projections for Pure Substances ◄ 23 temperature at which a species boils at 1 atm is termed the normal boiling point. State 3 represents the state where half the mass in the system has vaporized. Therefore, it is identifi ed as halfway across the liquid–vapor dome in the Pv projection. The molar volume represented by state 3 is not realized by either phase of the system; rather, it is an aver- age of the liquid and vapor that we use to characterize the molar volume of the system. In fact, the fraction of mass in the liquid phase in state 3 has the same molar volume as that in state 2. Similarly, state 4 is saturated vapor, the point at which any energy that is removed would cause a drop of liquid to condense. Note that states 2, 3, and 4 are represented by an identical point on the PT diagram, since P and T are not independent in the two-phase region—illustrating the fact that we cannot use P and T to constrain the state of the system.
Finally, state 5 is superheated vapor, which exists at a higher temperature than the saturated vapor. The increase in volume with temperature in the vapor phase is much more pronounced than it was for the liquid. How would you draw this process on a Tvdiagram?
Saturation Pressure vs. Vapor Pressure
The saturation pressure is the pressure at which a pure substance boils at a given tem- perature. A related quantity, the vapor pressure of a substance, is its contribution to the total pressure in a mixture at a given temperature. This contribution is equal to the partial pressure of the substance in an ideal gas mixture.
Figure 1.8 provides a schematic representation of each of these quantities. The two piston–cylinder assemblies depicted on the left represent cases for which the saturation pressure is defi ned. In these systems, pure species a is in vapor–liquid equilibrium at temperatures T1 and T2, respectively, where T2 is greater than T1. In each case, there is a unique pressure at which the two phases can be in equilibrium—defi ned as the satura- tion pressure, Psata . For example, pure water at 293 K (20°C) has a saturation pressure of 2.34 kPa. Said another way, for pure water to boil at 293 K, the pressure of the system must be 2.34 kPa. If the pressure is higher, water will exist only as a liquid. Conversely,
Figure 1.8 Graphical representation of the saturation pressure of pure a and the vapor pres- sure of a in a mixture of a and b. Two temperatures, T1 and T2 are shown.
Saturated vapor
Saturated liquid
Saturated vapor Saturated vapor Saturated vapor
Saturated liquid Liquid a Liquid a
Saturation Pressure: Pasat is defined for a system of pure a
Vapor Pressure: 1 pa is defined for a vapor mixture of a and b
1. The case depicted assumes ideal gas behavior and that species b does not condense into the liquid phase:
P = pa + pb = Pasat + pb
T =T1
T =T1 T =T2 >T1 T =T2 >T1
P = Pasat
m m
m m
m m
m m
a a
a a a
a
a a a
a a a a a
a a
a a
a a
a
a a
a a
a a
a a
a a
a a
a a
a a
b a b
b b
b b b
b b
b b b
b b
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24 ► Chapter 1. Measured Thermodynamic Properties and Other Basic Concepts
if the pressure is lower, it will be a single-phase vapor. At a higher temperature, the saturation pressure will be higher, as depicted for T2 in Figure 1.8. For example, at 303 K (30°C), water has a saturation pressure of 4.25 kPa. This incremental increase in tem- perature nearly doubles the saturation pressure.
A schematic illustrating when we use vapor pressure is shown for the two systems depicted on the right in Figure 1.8, where the vapor phase contains a mixture of species a and b. The vapor pressure of species a represents its contribution to the total pressure of the mixture. The two temperatures shown, T1 and T2, are identical to those for pure species a on the left of the fi gure. For convenience, we assume species b is does not noticeably condense in the liquid and that the vapor behaves as an ideal gas.17 Then the vapor pressure of species a is identical to the corresponding saturation pressure at the same temperature.
For example, now consider an open container of water sitting in a room at 293 K and 1 bar. Some of the water will evaporate and go into the air. The partial pressure of water at equilibrium with the air will be equal to the saturation pressure of pure a, 2.34 kPa. Since water is but one of many components in the mixture, we say water has a vapor pressure of 2.34 kPa. In contrast, the total pressure of the system is around 1 atm. The vapor pressure presented in Figure 1.8 depends only on the temperature of the water, not on the total pressure of the system. In other words, the vapor pressure of a is inde- pendent of how much b is present. While we can use saturation pressures to determine the vapor pressure in a given mixture, the term saturation pressure refers to the pure species. You should learn the difference between saturation pressure and vapor pressure because they are often confused.
A magnifi ed view of the upper part of the Pv phase diagram is shown in Figure 1.9.
Four isotherms are shown. Along all four isotherms, the volume increases as the pres- sure decreases. At the lowest two temperatures, the isotherms start in the liquid phase.
In the liquid phase, the volume change is relatively small as the pressure drops. Along a given isotherm, the pressure decreases until it reaches the saturation pressure. This point is marked by the intersection with the left side of the liquid–vapor dome. At this point, any increase in volume leads to a two-phase liquid–vapor mixture, where the value of liquid volume is given by the intersection of the isotherm with the left side of the dome and the vapor volume is given by the intersection of the isotherm with the right side of the dome. The pressure remains constant in the two-phase region, since P and T are no longer independent. After complete vaporization, the pressure again decreases. The corresponding increase in volume of the vapor is noticeably larger than that of the liquid.
As the temperatures of the isotherms increase, the saturated liquid volumes get larger and the saturated vapor volumes get smaller. Finally, at the critical point, located at the top of the liquid–vapor dome, the values of vl and vv become identical. The critical point represents a unique state and is identifi ed with the subscript “c.” Thus, it is con- strained by the critical temperature, Tc and the critical pressure, Pc. Values for these criti- cal properties of many pure substances are reported in Appendix A. The critical point represents the point at which liquid and vapor regions are no longer distinguishable. The critical point is also labeled in the depictions in Figure 1.6.
The Critical Point
17 We will learn how to treat the more general case in Chapters 7 and 8.
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1.6 The PvT Surface and Its Projections for Pure Substances ◄ 25
The critical isotherm goes through an infl ection point at the critical point. Math- ematically, this condition can be written as:
¢'P 'v≤
Tc
50 (1.16)
and, ¢'2P
'v2≤
Tc
50 (1.17)
The partial derivatives in Equations (1.16) and (1.17) specify that we need to keep the temperature constant at its value at the critical point.
The isotherm above the critical point is representative of a supercritical fl uid. This isotherm continuously decreases in pressure as the volume increases. A supercritical fl uid has partly liquidlike characteristics (e.g., high density) and partly vaporlike charac- teristics (compressibility, high-diffusivity). Not surprisingly, there are many interesting engineering applications for substances in this state. There can be confusion between the terms gas and vapor. We refer to a gas as any form of matter that fi lls the container;
it can be either subcritical or supercritical. When we speak of vapor, it is gas that if iso- thermally compressed will condense into a liquid and is, therefore, always subcritical.
Figure 1.9 Magnified view of the Pv diagram. Four isotherms are shown—two below the critical temperature (subcritical), one at the critical temperature, and one above the critical temperature (supercritical).
Liquid-vapor
Vapor Liquid
Critical point
T > Tc P
v T < Tc
T = Tc
Consider a two-phase system at a specifi ed T that contains 20% vapor, by mass, and 80% liquid.
Identify the state on a Tv phase diagram. Explain why graphical determination of the state is termed the lever rule.
SOLUTION The quality of the system, defi ned as the fraction of matter in the vapor, is 0.2. The molar volume can be written in terms of the quality according to Equation (1.15):
v5vl1x1vv2vl2 5vl10.21vv2vl2 (E1.2A) EXAMPLE 1.2
Determination of Location of a Two- Phase System on a Phase Diagram
(Continued)
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