a system, the constitution or phase diagram is the most popular and convenient.
The complexity of a phase diagram is determined primarily by the number of components which occur in the system, where components are chemical species of fixed composition. The simplest components are chemical elements and stoichiometric com-pounds. Systems are primarily categorized by the number of components which they contain, e.g., one-component (unary) systems, two-component (binary) systems, three-component (ternary) systems, four-component (quaternary) systems, etc.
The phase diagram of a one-component system (i.e., a system of fixed composition) is a two-dimensional representation of the dependence of the equilibrium state of existence of the system on the two independent variables. Temperature and pressure are normally chosen as the two independent variables; Fig. 1.4 shows a schematic representation of part of the phase diagram for H2O. The full lines in Figure 1.4 divide the diagram
Figure 1.4 Schematic representation of part of the phase diagram for H2O.
into three areas designated solid, liquid, and vapor. If a quantity of pure H2O is at some temperature and pressure which is represented by a point within the area AOB, the equilibrium state of the H2O is a liquid. Similarly, within the areas COA and COB the equilibrium states are, respectively, solid and vapor. If the state of existence lies on a line, e.g., on the line AO, then liquid and solid H2O coexist in equilibrium with one another, and the equilibrium is said to be twophase, in contrast to the existence within any of the three areas, which is a one-phase equilibrium. A phase is defined as being a finite volume in the physical system with-in which the properties are uniformly constant, i.e., do not experience any abrupt change in passing from one point in the volume to another. Within any of the onephase areas in the phase diagram, the system is said to be homogeneous. The system is hetero-geneous when it contains two or more phases, e.g., coexisting ice and liquid water (on the line AO) is a heterogeneous system comprising two phases, and the phase boundary between the ice and the liquid water is that very thin region across which the density changes abruptly from the value for homogeneous ice to the higher value for liquid water.
The line AO represents the simultaneous variation of P and T required for maintenance of the equilibrium between solid and liquid H2O, and thus represents the influence of pressure on the melting temperature of ice. Similarly the lines CO and OB represent the simultaneous variations of P and T required, respectively, for the maintenance of the equilibrium between solid and vapor H2O and between liquid and vapor H2O. The line CO is thus the variation, with temperature, of the saturated vapor pressure of solid ice or, alternatively, the variation, with pressure, of the sublimation temperature of water vapor.
The line OB is the variation, with temperature, of the saturated vapor pressure of liquid water, or, alternatively, the variation, with pressure, of the dew point of water vapor. The
12 Introduction to the Thermodynamics of Materials
three two-phase equilibrium lines meet at the point O (the triple point) which thus represents the unique values of P and T required for the establishment of the three-phase (solid+liquid+vapor) equilibrium. The path amb indicates that if a quantity of ice is heated at a constant pressure of 1 atm, melting occurs at the state m (which, by definition, is the normal melting temperature of ice), and boiling occurs at the state b (the normal boiling temperature of water).
If the system contains two components, a composition axis must be included and, consequently, the complete diagram is three-dimensional with the coordinates composition, temperature, and pressure. Three-dimensional phase diagrams are discussed in Chapter 14. In most cases, however, it is sufficient to present a binary phase diagram as a constant pressure section of the three-dimensional diagram. The constant pressure chosen is normally 1 atm, and the coordinates are composition and temperature.
Figure 1.5, which is a typical simple binary phase diagram, shows the phase relation-ships occurring in the system Al2O3–Cr2O3 at 1 atm pressure. This phase diagram shows that, at temperatures below the melting temperature of Al2O3 (2050°C), solid Al2O3 and solid Cr2O3 are completely miscible in all proportions. This occurs because Al2O3 and Cr2O3 have the same crystal structure and the Al3+ and Cr3+ ions are of similar size. At temperatures above the melting temperature of Cr2O3 (2265°C) liquid Al2O3 and liquid Cr2O3 are completely miscible in all proportions. The diagram thus contains areas of
Figure 1.5 The phase diagram for the system Al2O3–Cr2O3.
complete solid solubility and complete liquid solubility, which are separated from one another by a two-phase area in which solid and liquid solutions coexist in equilibrium with one another. For example, at the temperature T1 a Cr2O3–Al2O3 system of composition between X and Y exists as a two-phase system comprising a liquid solution of composition l in equilibrium with a solid solution of composition s. The relative proportions of the two phases present depend only on the overall composition of the system in the range X–Y and are determined by the lever rule as follows. For the overall composition C at the temperature T1 the lever rule states that if a fulcrum is placed at f on the lever ls, then the relative proportions of liquid and solid phases present are such that, placed, respectively, on the ends of the lever at s and l, the lever balances about the fulcrum, i.e., the ratio of liquid to solid present at T1 is the ratio fs/lf.
Because the only requirement of a component is that it have a fixed composition, the designation of the components of a system is purely arbitrary. In the system Al2O3–Cr2O3 the obvious choice of the components is Al2O3 and Cr2O3. However, the most convenient choice is not always as obvious, and the general arbitrariness in selecting the components can be demonstrated by considering the iron-oxygen system, the phase diagram of which is shown in Fig. 1.6. This phase diagram shows the Fe and O form two stoichiometric compounds, Fe3O4 (magnetite) and Fe2O3 (hematite), and a limited range of solid solution (wustite). Of particular significance is the observation that neither a stoichiometric compound of the formula FeO nor a wustite solid solution in which the Fe/O atomic ratio is unity occurs. In spite of this it is often found convenient to consider the stoichiometric FeO composition as a thermodynamic component of the system. The available choice of the two components of the binary system can be demonstrated by considering the composition X in Fig. 1.6. This composition can equivalently be considered as being in any one of the following systems:
14 Introduction to the Thermodynamics of Materials
Figure 1.6 The phase diagram for the binary system Fe–O.
1. The system Fe–O (24 weight % O, 76 weight % Fe)
2. The system FeO–Fe2O3 (77.81 weight % FeO, 22.19 weight % Fe2O3) 3. The system FeO–Fe3O4 (67.83 weight % FeO, 32.17 weight % Fe3O4) 4. The system Fe–Fe3O4 (13.18 weight % Fe, 86.82 weight % Fe3O4) 5. The system Fe–Fe2O3 (20.16 weight % Fe, 79.84 weight % Fe2O3) 6. The system FeO–O (97.78 weight % FeO, 2.22 weight % O)
The actual choice of the two components for use in a thermodynamic analysis is thus purely a matter of convenience. The ability of the thermodynamic method to deal with descriptions of the compositions of systems in terms of arbitrarily chosen components, which need not correspond to physical reality, is a distinct advantage. The thermodynamic behavior of highly complex systems, such as metallurgical slags and molten glass, can be completely described in spite of the fact that the ionic constitutions of these systems are not known completely.