DG 5 2 278 J Would this be considered a spontaneous process? Because the pressure is not kept
6.6 Phase Diagrams and the Phase Rule
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6.6 Phase Diagrams and the Phase Rule
Although phase transitions can seem complicated, there is a simplification: the phase diagram. Phase diagrams are graphical representations of what phases are sta- ble under various conditions of temperature, pressure, and volume. Most simple phase diagrams are two-dimensional, with pressure on one axis and temperature on the other.
The phase diagram itself is composed of lines that indicate the temperature and pressure values at which phase equilibrium occur. For example, Figure 6.3 is a par- tial phase diagram of H2O. The diagram shows the stable phase in each region of the diagram. The lines on the phase diagram represent the phase transitions. Any point on a line represents a particular pressure and temperature at which multiple phases can exist in equilibrium. Any point not on a line indicates a phase that is the predominant stable phase of the compound H2O under those conditions.
Consider the points labeled in Figure 6.3. Point A represents a value for pressure pA and temperature TA in which the solid form of H2O is stable. Point B represents a set of pressure and temperature conditions pB and TB where melting occurs: Solid can exist in equilibrium with liquid. Point C represents pressure and temperature conditions in which liquid is the stable phase. Point D represents pressure and tem- perature conditions in which liquid can exist in equilibrium with the gas: Boiling occurs. Finally, point E represents a set of pressure and temperature conditions in which the stable phase of H2O is gaseous.
The phase diagram implies that under many conditions solid and liquid can exist in equilibrium, and under many conditions liquid and gas can exist in equilibrium.
This is certainly the case. But what are these lines giving us? Because they are a plot of how the pressure changes with change in temperature for the phase equilib- ria, the lines represent dp/dT. This quantity can be calculated using the Clapeyron or the Clausius-Clapeyron equation. Single-component phase diagrams are nothing
Figure 6.3 A qualitative, partial phase diagram (pressure versus temperature) of H2O. Specific points in a phase diagram (like points A, B, C, D, and E here) indi- cate conditions of pressure and tempera- ture and what phase(s) of the component are stable under those conditions.
Temperature
Pressure
Solid
Liquid
A B C D E
Gas
Ethanol has a vapor pressure of 43.7 mmHg at 20.0°C. It has a molar volume of 58.40 mL at that temperature. A system containing C2H5OH(,) and C2H5OH(g) at equilibrium is pressurized with 135.0 atm of Ar. What is the new vapor pressure of ethanol?
Solution
We can use equation 6.18 for this, but we need to watch our units: The molar vol- ume should be expressed in L, and the proper value of R has L and atm units in it.
Thus, we have
p*5(43.7 mmHg)e(0.05840 L/mol)(135 atm)/(0.08205 L#atm mol#K)(293.2 K)
p* 5 (43.7 mmHg)(1.38380…) p* 5 60.6 mmHg
This is a 39% increase in the vapor pressure of the ethanol.
ExamplE 6.9
Not only have we converted volume to L, but we have converted temperature to kelvins.
Another, perhaps more useful form of this equation is expressed in terms of the new vapor pressure of the vapor, p*:
p*5peV(,) DP/RT (6.18)
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pressure and temperature vary, a three-dimensional plot would be necessary and the equation of state for all phases would be needed.
One other thing to notice from the example is that the slope is negative.
Almost all compounds have a positive slope for the solid-liquid equilibrium line, because solids usually have less volume than the same amount of liquid. The negative slope is a consequence of the increase in volume experienced by H2O when it solidifies.
The solid-gas equilibrium line represents those conditions of pressure and tem- perature where sublimation occurs. For H2O, obvious sublimation occurs at pres- sures lower than those that are normally experienced. (Sublimation of ice does occur slowly at normal pressures, which is why ice cubes get smaller over time in your freezer. The so-called freezer burn of frozen foods is caused by sublimation of ice from the food. This is why it’s important to wrap frozen food tightly.) However, for carbon dioxide, normal pressures are low enough for sublimation. Figure 6.4 shows a phase diagram for CO2, with the 1-atm position marked. Liquid CO2 is stable only under pressure. Some gas cylinders of carbon dioxide are high enough in pressure that they actually contain liquid CO2.
The liquid-gas equilibrium line represents conditions of pressure and temper- ature where those phases can exist at equilibrium. Notice that it has the form of an exponential equation; that is, p~e21/T. This is consistent with equation 6.16.
The vaporization line in the phase diagram is a plot of the Clapeyron equation or the Clausius-Clapeyron equation. Notice, however, that this line ends at a particular pressure and temperature, as shown in Figure 6.5. It is the only line that doesn’t have an arrow on its end to indicate that it continues. That’s be- cause beyond a certain point, the liquid phase and the gas phase become indis- tinguishable. This point is called the critical point of the substance. The pressure and temperature at that point are called the critical pressure pC and critical tem- perature TC. For H2O, pC and TC are 218 atm and 374°C. Above that tempera- ture, no pressure can force the H2O molecules into a definite liquid state. If the H2O in the system exerts a pressure higher than pC, then it cannot exist as a definite liquid or gas. (It can exist as a solid if the temperature is low enough.) The state of the H2O is called supercritical. Supercritical phases are important in some industrial and scientific processes. In particular, there is a technique called The line between the solid and liquid phases for the H2O phase diagram in Figure 6.3 is a fairly straight line, indicating a constant slope. Use the answers to Example 6.4, the melting of ice, to calculate the value for the slope of that line.
Solution
Recall that one definition of the slope of a line is Dy/Dx. The y-axis represents pres- sure and the x-axis represents temperature, so for Dp/DT we expect a slope where the units are bar/K or atm/K. Example 6.4 showed that it takes 1.353103 bar to change the melting point of water by 210.0°C, which is 210.0 K. Therefore, Dp/DT is equal to (1.353103 bar)/(210.0 K) or 21.353102 bar/K. This is a fairly large slope.
ExamplE 6.10
F i g u r e 6.4 A phase diagram for carbon dioxide, CO2. Notice that as the temperature of solid CO2 is increased at standard pressure, the solid goes directly into the gas phase. Liquid CO2 is stable only at increased pressure.
1
Temperature (°C) –78.5
5.11 73
–56.4 31.1
Liquid
Solid
Pressure (atm) Gas
Figure 6.5 The triple point and the critical point for H2O. The liquid-gas equi- librium line is the only one that ends at a certain set of conditions for all substances.
For H2O, the line ends at 374°C and 215 bar. At higher temperatures or vapor pressures, there is no distinction between a “liquid” and a “gas” phase.
215
Temperature (°C) 374
Pressure (bar)
0.00611
0.01 Solid
Liquid
Gas Triple point
Critical point
more than plots of the Clapeyron equation or the Clausius-Clapeyron equation for a substance. This is true for pressure-temperature phase diagrams, which is what we will consider almost exclusively here. For a phase diagram where volume as well as
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supercritical fluid chromatography in which compounds are separated using su- percritical CO2 or other compounds as a “solvent.” (TC and pC for CO2 are about 304 K and 73 bar.)*
One other point in the phase diagram is worthy of mention. Figure 6.5 indicates a set of conditions where solid, liquid, and gas are in equilibrium with each other.
This is called the triple point. For H2O, the triple point is 0.01°C, or 273.16 K, and 6.11 mbar, or about 4.6 torr. Because H2O is so common, the triple point for H2O is recognized internationally as a verifiable temperature standard. All materials have triple points, a unique set of pressure and temperature conditions in which all three phases can exist in equilibrium with each other. Table 6.3 lists conditions of critical points for some substances.
The phase diagram for H2O is commonly used as an example for several reasons:
It is a common material, and the phase diagram shows some unusual characteris- tics. Figure 6.6 shows a more expansive phase diagram for the compound H2O. One of the noteworthy points is that there are actually several types of solid H2O, that is, ice. Note the pressure and temperature scales, however. We are not likely to experi- ence these forms of ice outside the laboratory.
Table 6.3 Critical temperatures and pressures for various substances Substance TC (K) pC (bar) Ammonia 405.7 111 Hydrogen 32.98 12.93 Methane 191.1 45.2
Nitrogen 126 33.1
Oxygen 154.6 50.43
Sulfur 1314 207
Water 647.3 215.15
Figure 6.6 This phase diagram of water extends to higher temperatures and pressures than Figure 6.3. Notice that there are several possible crystal structures of solid H2O, most of which exist only at high pressures. At least 15 forms of solid H2O are known.
Ice VI
Ice V
Ice I Ice III
Ice I 6000
0200
Temperature (K)
700 Ice
II
Pressure (bar)
4000
2000
215
1 2
0.006
600 647.30 500
400 273.16 373.15 273.15 300
Critical point
Triple point
Liquid (water)
Gas (steam) Liquid (water)
*One method of decaffeinating coffee beans is by using supercritical CO2.
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At the molecular level, what are the differences between the various forms of ice? Figure 6.7 shows the experimental crystal structures of “normal” ice (ice I) and ice as it exists at about 10,000 atm pressure and a temperature of 2150°C (ice XV). It should be obvious from the two crystal structures that they are different: The atoms and molecules are in different orientations with respect to each other, which is what defines a different solid phase. These solid phases have different physical properties, some of which—like melting point—are apparent from the phase diagram. Others are not so apparent. For example, the density of ice IX, which exists at around a pressure of 120 atm and a temperature of 150 K, is about 1.16 g/mL, about 26% higher than normal ice (and hence would sink, not float, in liquid water!).
Figure 6.8 shows a phase diagram of helium. Because helium is a gas at tem- peratures down to 4.2 K, the temperature axis on this diagram does not have a large temperature range. At the other extreme, Figure 6.9 shows a phase diagram of carbon. Notice the regions where diamond is the stable phase.
Although pressure and temperature are the common variables for phase dia- grams in chemistry, volume can also be plotted on an axis in a phase diagram, as shown in Figure 6.10. There are also three-dimensional phase diagrams that plot pressure, volume, and temperature; Figure 6.11 shows an example of that.
Figure 6.7 Different structures of ice at the molecular level. (a) Crystal structure of ice I, the “normal” form of ice. (b) The crystal structure of ice XV, a recently discovered phase of solid H2O that exists around 10,000 atm and 2150°C. Note how different solid phases have different arrangements of atoms and molecules.
(a)
(b)
(a)
(b)
Figure 6.8 The phase diagram of helium, He, does not need a large tempera- ture range. Notice that solid He does not exist unless pressures are large.
40
0 0.0001
Temperature (K)
100
Pressure (bar)
Superfluid B phase
0.001 0.01 0.1 1 10
30
20
10
Solid
Gas Superfluid
A phase
Normal liquid
Figure 6.9 A phase diagram of carbon, showing where the graphite allotrope is stable and where the diamond allotrope is stable.
103
10–2
0 Temperature (3103 K) 6
Pressure (kbar)
5 4 3 2 1 102
10
1 10–1
Gas Diamond
Graphite
Liquid
T
Volume V1
Temperature
V2 Gas
P
Liquid
Figure 6.10 An example of a temperature- volume phase diagram. At a certain pressure P, the phase diagram specifies what phase must be present except between V1 and V2
(for the given pressure). Under these con- ditions, a varying amount of liquid phase (shaded area) may be present and still satisfy the given conditions of T and P. In part because of this ambiguity, temperature- volume phase diagrams aren’t as common as pressure-temperature phase diagrams.
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Use the phase diagram of CO2, Figure 6.4, to describe the changes in phase as one makes the following changes in conditions.
a. 50 K to 350 K at a pressure of 1.00 bar b. 50 K to 350 K at a pressure of 10 bar c. 1 bar to 100 bar at a temperature of 220 K Solution
a. Figure 6.12 shows the change in conditions for this isobaric process. Starting at point A, the temperature is increased as we move from left to right, indicating that we are warming the solid CO2, until we reach the line at point B indicating the equilibrium between solid and gas phases. At this point, the solid CO2 sublimes directly into the gas phase. (This occurs at about 196 K, or 277°C.) As the temperature increases to 350 K, we are warming gaseous CO2 until we reach point C, the final conditions.
b. Figure 6.13 shows the change in conditions for the isobaric warming of CO2 at 10 bar. In this case, we start with a solid at point A, but because we are above the triple point for CO2, at point B we are in an equilibrium with solid and liquid CO2 present. As we add heat, solid melts until all solid becomes liquid, and then the liquid CO2 warms. We continue warming until point C is reached, which represents the conditions where CO2 liquid is in equilibrium with CO2 gas. When all of the liquid is converted to gas, the gas warms until the final conditions at point D are reached.
c. Figure 6.14 illustrates the isothermal process. The starting point A is at low enough pressure that the CO2 is in the gas phase. However, as the pressure is increased, the CO2 passes into the liquid phase (briefly) and then into the solid phase. Note that if the temperature were only a few degrees lower, this change would have occurred on the other side of the triple point and the phase transition would have been a direct gas-to-solid condensation.
ExamplE 6.11
Volume
Pressure
1-phase region 2-phase region
Temperature Figure 6.11 A three-dimensional phase diagram can plot the phases present in a system for given sets of pressures, tem- peratures, and volumes.
1
Temperature (°C) –78.5
5.11 73
–56.4 31.1
CO2 (s)
CO2 (g) CO2 ( )
350 KTo C
Pressure (atm) A B
Figure 6.12 An illustration of the isobaric change for CO2 specified in Example 6.11a. Compare this to Figure 6.13.
1
Temperature (°C) –78.5
5.11 73
–56.4 31.1
CO2 (s)
CO2 (g) CO2 ( )
350 KTo
Pressure (atm)
A B C D
Figure 6.13 An illustration of the isobaric change for CO2 specified in Example 6.11b.
Compare this to Figure 6.12.
1
Temperature (°C) –78.5
5.11 73
–56.4 31.1
CO2 (s)
CO2 (g) CO2 ( ) To 100 bar
Pressure (atm)
A B C D
Figure 6.14 An illustration of the change specified in Example 6.11c.
Phase diagrams are very useful in helping to understand how single-component systems act under a change in condition: Simply plot the change on the phase dia- gram and observe which phase transitions occur for that change. Single-component phase diagrams are especially easy to interpret.
Phase diagrams of single-component systems are useful in illustrating a simple idea that answers a common question: How many variables must be specified in order to determine the phase(s) of the system when it’s at equilibrium? These vari- ables are called degrees of freedom. What we need to know is how many degrees of freedom we need to specify in order to characterize the state of the system. This information is more useful than one might think. Because the position of phase transitions ( especially transitions that involve the gas phase) can change quickly
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with pressure or temperature, knowing how many state variables must be defined is important.
Consider the two-dimensional phase diagram for H2O. If you knew that only H2O was in the system at equilibrium and that it was in the solid phase, then any point in the shaded region of Figure 6.15 would be possible. You would have to specify the temperature, the pressure, and the phases of the system. However, suppose you knew that you had solid and liquid H2O in the system at equilibrium.
Then you know that the condition of the system must be indicated by the line in the phase diagram that separates the solid and liquid phase. For a given set of phases, you need only specify temperature or pressure, because knowing one gives you the other (because the system—with the phases in equilibrium—must have condi- tions corresponding to that line). The number of degrees of freedom has dropped because the number of phases in your system has increased.
Suppose you know the phases of H2O in your system at equilibrium. You don’t have to specify any degrees of freedom because there is only one set of conditions in which that will occur: For H2O in the solid, liquid, and gas phases, those condi- tions are 273.16 K and 6.11 mbar. (See Figure 6.5: There is only one point on that phase diagram where solid, liquid, and gas exist in equilibrium, and that is the triple point.) There is a relationship between the number of phases occurring at equilib- rium and the number of degrees of freedom necessary to specify the point in the phase diagram that describes the state of the system.
In the 1870s, J. Willard Gibbs (for whom Gibbs energy is named) deduced the simple relationship between the number of degrees of freedom and the number of phases. For a single-component system,
degrees of freedom532P (6.19)
where P represents the number of phases present at equilibrium. Equation 6.19 is a simplified version of what is known as the Gibbs phase rule. In this rendition, it assumes that one of the state variables of the system, usually the volume, can be determined from the others (via an equation of state). You should verify that this simple equation provides the correct number of degrees of freedom for each situa- tion described above.