The thermodynamics of transition
3.4 Phase diagrams
To prepare for being able to describe phase transitions in biological macromolecules, first we need to explore the conditions for equilibrium between phases of simpler substances.
Molar Gibbs energy, Gm
Temperature, T g l
s
Gas
Liquid Solid
Tf Tb
Fig. 3.3 The variation of molar Gibbs energy with temperature. All molar Gibbs energies decrease with increasing temperature. The regions of temperature over which the solid, liquid, and gaseous forms of a substance have the lowest molar Gibbs energy are indicated in the band at the top of the illustration.
Molar Gibbs energy, Gm
Temperature, T
s g
Gas
Liquid
Solid
Fig. 3.4 If the line for the Gibbs energy of the liquid phase does not cut through the line for the solid phase (at a given pressure) before the line for the gas phase cuts through the line for the solid, the liquid is not stable at any temperature at that pressure.
Such a substance sublimes.
Thephase diagramof 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 ther- modynamically the most stable, but at C the liquid phase is the most stable.
The boundaries between regions in a phase diagram, which are called phase boundaries, show the values of pand Tat which the two neighboring phases are in equilibrium. For example, if the system is arranged to have a pressure and tem- perature represented by point B, then the liquid and its vapor are in equilibrium (like liquid water and water vapor at 1 atm and 100°C). If the temperature is re- duced at constant pressure, the system moves to point C, where the liquid is sta- ble (like water at 1 atm and at temperatures between 0°C and 100°C). If the tem- perature is reduced still further to D, then the solid and the liquid phases are in equilibrium (like ice and water at 1 atm and 0°C). A further reduction in temper- ature takes the system into the region where the solid is the stable phase.
(a) Phase boundaries
The pressure of the vapor in equilibrium with its condensed phase is called the vapor pressureof the substance. Vapor pressure increases with temperature because, as the temperature is raised, more molecules have sufficient energy to leave their neighbors in the liquid.
The liquid-vapor boundary in a phase diagram is a plot of the vapor pressure against temperature. To determine the boundary, we can introduce a liquid into the near-vacuum at the top of a mercury barometer and measure by how much the column is depressed (Fig. 3.6). To ensure that the pressure exerted by the vapor is truly the vapor pressure, we have to add enough liquid for some to remain after the vapor forms, for only then are the liquid and vapor phases in equilibrium. We can change the temperature and determine another point on the curve and so on (Fig. 3.7).
Now suppose we have a liquid in a cylinder fitted with a piston. If we apply a pressure greater than the vapor pressure of the liquid, the vapor is eliminated, the piston rests on the surface of the liquid, and the system moves to one of the points in the “liquid” region of the phase diagram. Only a single phase is present. If in- stead we reduce the pressure on the system to a value below the vapor pressure, the system moves to one of the points in the “vapor” region of the diagram. Reducing the pressure will involve pulling out the piston a long way so that all the liquid evaporates; while any liquid is present, the pressure in the system remains constant at the vapor pressure of the liquid.
Pressure
Solid
E C
B A
D
Liqiud
Vapor Phase
boundary
Temperature, T Fig. 3.5 A typical phase diagram, showing the
regions of pressure and temperature at which each phase is the most stable. The phase boundaries (three are shown here) show the values of pressure and temperature at which the two phases separated by the line are in equilibrium. The significance of the letters A, B, C, D, and E (also referred to in Fig. 3.8) is explained in the text.
Vapor pressure
(a) (b) (c)
Fig. 3.6 When a small volume of water is introduced into the vacuum above the mercury in a barometer (a), the mercury is depressed (b) by an amount that is proportional to the vapor pressure of the liquid. (c) The same pressure is observed however much liquid is present (provided some is present).
COMMENT 3.1 The text’s web site contains links to online databases of data on phase transitions. ■
SELF-TEST 3.1 What would be observed when a pressure of 50 Torr is applied to a sample of water in equilibrium with its vapor at 25°C, when its vapor pres- sure is 23.8 Torr?
Answer:The sample condenses entirely to liquid.
The same approach can be used to plot the solid-vapor boundary, which is a graph of the vapor pressure of the solid against temperature. The sublimation va- por pressureof a solid, the pressure of the vapor in equilibrium with a solid at a particular temperature, is usually much lower than that of a liquid because the mol- ecules are more strongly bound together in the solid than in the liquid.
A more sophisticated procedure is needed to determine the locations of solid- solid phase boundaries like that between the different forms of ice, for instance, be- cause the transition between two solid phases is more difficult to detect. One ap- proach is to use thermal analysis, which takes advantage of the heat released during a transition. In a typical thermal analysis experiment, a sample is allowed to cool and its temperature is monitored. When the transition occurs, energy is released as heat and the cooling stops until the transition is complete (Fig. 3.8). The transi- tion temperature is obvious from the shape of the graph and is used to mark a point on the phase diagram. The pressure can then be changed and the corresponding transition temperature determined.
Any point lying on a phase boundary represents a pressure and temperature at which there is a “dynamic equilibrium” between the two adjacent phases. A state ofdynamic equilibrium is one in which a reverse process is taking place at the same rate as the forward process. Although there may be a great deal of activity at a molecular level, there is no net change in the bulk properties or appearance of the sample. For example, any point on the liquid-vapor boundary represents a state of dynamic equilibrium in which vaporization and condensation continue at match- ing rates. Molecules are leaving the surface of the liquid at a certain rate, and mol- ecules already in the gas phase are returning to the liquid at the same rate; as a re- sult, there in no net change in the number of molecules in the vapor and hence no net change in its pressure. Similarly, a point on the solid-liquid curve represents conditions of pressure and temperature at which molecules are ceaselessly breaking away from the surface of the solid and contributing to the liquid. However, they are doing so at a rate that exactly matches that at which molecules already in the liquid are settling onto the surface of the solid and contributing to the solid phase.
0 20 40 60 80 100
Vapor pressure/kPa
101.325 kPa at 100°C
3.167 kPa at 25°C
0 20 40 60 80 100 Temperature/°C
Fig. 3.7 The experimental variation of the vapor pressure of water with temperature.
Temperature
Time B
D E Tf
Fig. 3.8 The cooling curve for the B–E section of the horizontal line in Fig. 3.5. The halt at D corresponds to the pause in cooling while the liquid freezes and releases its enthalpy of transition. The halt lets us locate Tfeven if the transition cannot be observed visually.
(b) Characteristic points
We have seen that as the temperature of a liquid is raised, its vapor pressure in- creases. First, consider what we would observe when we heat a liquid in an open vessel. At a certain temperature, the vapor pressure becomes equal to the external pressure. At this temperature, the vapor can drive back the surrounding atmosphere and expand indefinitely. Moreover, because there is no constraint on expansion, bubbles of vapor can form throughout the body of the liquid, a condition known asboiling. The temperature at which the vapor pressure of a liquid is equal to the external pressure is called the boiling temperature. When the external pressure is 1 atm, the boiling temperature is called the normal boiling point,Tb. It follows that we can predict the normal boiling point of a liquid by noting the temperature on the phase diagram at which its vapor pressure is 1 atm.
Now consider what happens when we heat the liquid in a closed vessel. Be- cause the vapor cannot escape, its density increases as the vapor pressure rises and in due course the density of the vapor becomes equal to that of the remaining liq- uid. At this stage the surface between the two phases disappears (Fig. 3.9). The temperature at which the surface disappears is the critical temperature,Tc. The vapor pressure at the critical temperature is called the critical pressure,pc, and the critical temperature and critical pressure together identify the critical pointof the substance (see Table 3.1). If we exert pressure on a sample that is above its criti- cal temperature, we produce a denser fluid. However, no surface appears to sepa- rate the two parts of the sample and a single uniform phase, a supercritical fluid, continues to fill the container. That is, we have to conclude that a liquid cannot be produced by the application of pressure to a substance if it is at or above its critical tem- perature. That is why the liquid-vapor boundary in a phase diagram terminates at the critical point (Fig. 3.10).
A supercritical fluid is not a true liquid, but it behaves like a liquid in many respects—it has a density similar to that of a liquid and can act as a solvent. For example, supercritical carbon dioxide is used to extract caffeine in the manufacture of decaffeinated coffee, where, unlike organic solvents, it does not result in the for- mation of an unpleasant and possibly toxic residue.
The temperature at which the liquid and solid phases of a substance coexist in equilibrium at a specified pressure is called the melting temperature of the sub- stance. Because a substance melts at the same temperature as it freezes, the melt- ing temperature is the same as the freezing temperature. The solid-liquid bound- ary therefore shows how the melting temperature of a solid varies with pressure.
Table 3.1
Critical constants*pc/atm Vc/(cm3mol1) Tc/K
Ammonia, NH3 111 73 406
Argon, Ar 48 75 151
Benzene, C6H6 49 260 563
Carbon dioxide, CO2 73 94 304
Hydrogen, H2 13 65 33
Methane, CH4 46 99 191
Oxygen, O2 50 78 155
Water, H2O 218 55 647
*The critical volume, Vc, is the molar volume at the critical pressure and critical volume.
Increasing temperature Fig. 3.9 When a liquid is heated in a sealed container, the density of the vapor phase increases and that of the liquid phase decreases, as depicted here by the changing density of shading. There comes a stage at which the two densities are equal and the interface between the two fluids disappears. This disappearance occurs at the critical temperature. The container needs to be strong:
the critical temperature of water is at 373°C and the vapor pressure is then 218 atm.
The melting temperature when the pressure on the sample is 1 atm is called the normal melting pointor the normal freezing point,Tf. A liquid freezes when the energy of the molecules in the liquid is so low that they cannot escape from the attractive forces of their neighbors and lose their mobility.
There is a set of conditions under which three different phases (typically solid, liquid, and vapor) all simultaneously coexist in equilibrium. It is represented by the triple point, where the three phase boundaries meet. The triple point of a pure sub- stance is a characteristic, unchangeable physical property of the substance. For wa- ter the triple point lies at 273.16 K and 611 Pa, and ice, liquid water, and water vapor coexist in equilibrium at no other combination of pressure and temperature.1 At the triple point, the rates of each forward and reverse process are equal (but the three individual rates are not necessarily the same).
The triple point and the critical point are important features of a substance because they act as frontier posts for the existence of the liquid phase. As we see from Fig. 3.11a, if the slope of the solid-liquid phase boundary is as shown in the diagram:
The triple point marks the lowest temperature at which the liquid can exist.
The critical point marks the highest temperature at which the liquid can exist.
We shall see in the following section that for water, the solid-liquid phase bound- ary slopes in the opposite direction, and then only the second of these conclusions is relevant (see Fig. 3.11b).
Pressure
Temperature Normal freezing point
Triple
point Normal
boiling point
Critical point 1 atm
Solid
Liquid
Vapor
Fig. 3.10 The significant points of a phase diagram. The liquid-vapor phase boundary terminates at the critical point.
At the triple point, solid, liquid, and vapor are in dynamic equilibrium. The normal freezing pointis the temperature at which the liquid freezes when the pressure is 1 atm; the normal boiling pointis the temperature at which the vapor pressure of the liquid is 1 atm.
Pressure
Temperature Temperature
Anomalous
Normal
Liquid
Liquid
(a)
Pressure
(b)
Fig. 3.11 (a) For substances that have phase diagrams resembling the one shown here (which is common for most substances, with the important exception of water), the triple point and the critical point mark the range of temperatures over which the substance can exist as a liquid. The shaded areas show the regions of temperature in which a liquid cannot exist as a stable phase. (b) A liquid cannot exist as a stable phase if the pressure is below that of the triple point for normal or anomalous liquids.
1The triple point of water is used to define the Kelvin scale of temperatures: the triple point is defined as lying at 273.16 K exactly. The normal freezing point of water is found experimentally to lie approximately 0.01 K below the triple point, at very close to 273.15 K.
(c) The phase diagram of water
Figure 3.12 is the phase diagram for water. The liquid-vapor phase boundary shows how the vapor pressure of liquid water varies with temperature. We can use this curve, which is shown in more detail in Fig. 3.13, to decide how the boiling tem- perature varies with changing external pressure. For example, when the external pressure is 149 Torr (at an altitude of 12 km), water boils at 60°C because that is the temperature at which the vapor pressure is 149 Torr (19.9 kPa).
SELF-TEST 3.2 What is the minimum pressure at which liquid is the thermo- dynamically stable phase of water at 25°C?
Answer: 23.8 Torr, 3.17 kPa (see Fig. 3.13) Temperature, T/K 200
0 400 600 800
Pressure,p/Pa
1012
109
106
103 1
X
I VIII
II III
Liquid
Vapor V
VII XI
VI
Fig. 3.12 The phase diagram for water showing the different solid phases.
273.15 273.16
Temperature, T/K
272 273
Pressure,p/bar
1 0.006 130
Triple point Liquid
Ice
Fig. 3.13 The solid-liquid boundary of water in more detail.
The graph is schematic and not to scale.
The solid-liquid boundary line in Fig. 3.12, which is shown in more detail in Fig. 3.13, shows how the melting temperature of water depends on the pressure. For example, although ice melts at 0°C at 1 atm, it melts at 1°C when the pressure is 130 atm. The very steep slope of the boundary indicates that enormous pressures are needed to bring about significant changes. Notice that the line slopes down from left to right, which—as we anticipated—means that the melting temperature of ice falls as the pressure is raised. We can trace the reason for this unusual be- havior to the decrease in volume that occurs when ice melts: it is favorable for the solid to transform into the denser liquid as the pressure is raised. The decrease in volume is a result of the very open structure of the crystal structure of ice: as shown in Fig 3.14, the water molecules are held apart, as well as together, by the hydro- gen bonds between them, but the structure partially collapses on melting and the liquid is denser than the solid.
Figure 3.12 shows that water has one liquid phase2 but many different solid phases other than ordinary ice (“ice I,” shown in Fig 3.14). These solid phases dif- fer in the arrangement of the water molecules: under the influence of very high pressures, hydrogen bonds buckle and the H2O molecules adopt different arrange- ments. These polymorphs, or different solid phases, of ice may be responsible for the advance of glaciers, for ice at the bottom of glaciers experiences very high pres- sures where it rests on jagged rocks. The sudden apparent explosion of Halley’s comet in 1991 may have been due to the conversion of one form of ice into an- other in its interior. Figure 3.12 also shows that four or more phases of water (such as two solid forms, liquid, and vapor) are never in equilibrium. This observation is justified and generalized to all substances by the phase rule, which is derived in Fur- ther information3.1.