In the Classroom
Description of Regions in Two-Component Phase Diagrams Robert M. Rosenberg Department of Chemistry, Northwestern University, Evanston, IL 60208-3113
Physical chemistry textbooks frequently present the student with blank two-component phase diagrams and request that the student define the various areas and curves in the diagram and identify the phases they represent (1, 2). The terms, “one-phase area” and “two-phase area” (2, 3) are confusing because an area implies 2 degrees of freedom, and only the one-phase system has two degrees of freedom, as shown below. I propose a way for students to interpret an unlabeled phase diagram through the relationship between the number of degrees of freedom of the system and the geometric construct that represents the system under a specified set of conditions. The general statement of the phase rule for a two-component system is F=C–P+2=4–P where F is the number of degrees of freedom, C is the number of components, and P is the number of phases. Three independent variables, usually temperature, pressure, and one composition variable, are required to specify the state of such a system, so that the phase diagram must be plotted in 3space. A two-dimensional representation is possible if we take a section through the three-dimensional phase diagram at a constant temperature or a constant pressure. The discussion in this paper emphasizes the direct relationship between the dimensions of the geometric construct that represents a system, an area, a curve, or a point, and the number of degrees of freedom of the system. When the pressure or temperature is held constant, the phase rule for a twocomponent system is expressed as F=C–P+1=3–P We consider a system for which the pressure is held constant. •
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The number of degrees of freedom, F, is equal to 2 for a system that consists of one phase, and the system is represented by an area, with both temperature and composition free to change within the limits of the area. The number of degrees of freedom, F, is equal to 1 for a system that consists of two phases, and one variable is free to change while the other is determined by the two curves that represent the state of the system. There are two curves because the two phases in equilibrium have different compositions at any given temperature. The number of degrees of freedom, F, is equal to 0 for a system that consists of three phases, and the system is represented by three points. There are three points because the three phases have different compositions at any given temperature.
Thus, “one-phase area” is a reasonable expression, since F = 2 when the system has one phase. It is confusing, however, to say that a two-phase system is represented by a “twophase area”, since F = 1. It is preferable to describe the system by the two curves that describe the compositions of the two
phases in equilibrium as a function of temperature; one variable can be varied freely within limits, and the other is a function determined by the equilibrium curves. It is similarly confusing to say that a three-phase system is represented by a “three-phase line”, since F = 0. It is preferable to describe the system by the three points that represent the compositions of the three phases in equilibrium, since there are zero degrees of freedom. The points can be joined by a line, but the other points on the line have no relationship to the compositions of the phases. A simple eutectic phase diagram such as that in Figure 1 can be interpreted easily without the necessity of using the geometric criterion suggested here. However, when single solid phases of variable composition are present in the system, as in Figure 2, our geometric criterion is very useful. It is easy to see that the upper part of the diagram represents a single liquid phase, F = 2, and the term one-phase area is appropriate. The difficult question to answer is: which of the regions that represent solid phases represent single solid phases, either solid solutions or nonstoichiometric compounds (berthollides), and which regions represent mixtures of two solid phases? Region III, for example, must represent a single phase, which has been found to be a solid solution, because the bounding line GAH has a discontinuity of slope at A, within the temperature range in which the phase is stable; therefore the region does not represent a system that has F = 1, which would require well-behaved curves with continuous first derivatives throughout the range. Therefore region III represents a one-phase area, with F = 2, a solid solution of P in As. Similarly, region V is a one-phase area, F = 2, because the bounding line BDJ has a discontinuity of slope at D. There is uncertainty whether the region represents a solid solution with roughly equal mole fractions of P and As or a nonstoichiometric compound. Again, region VIII also represents a single phase, F = 2, because the bounding line ELM
Figure 1. Example of unlabeled two-component phase diagram.
JChemEd.chem.wisc.edu • Vol. 76 No. 2 February 1999 • Journal of Chemical Education
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In the Classroom
Figure 2. Phase diagram of the P–As system. Adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, p 305. By permission of ASM International, The Materials Information Society, Materials Park, OH.
has a discontinuity of slope at L. Region VIII is thought to represent a solid solution of As in P. The other regions of the diagram represent the equilibrium between two phases, and F = 1. In region II, liquid I is in equilibrium with solid solution III, with the compositions of the phases in equilibrium given by the curves GC and GA, respectively. Region IV represents the equilibrium between solid solution III and single phase V, with the compositions of the phases in equilibrium given by the curves AH and BI
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respectively. Region VI represents the equilibrium between single phase V and liquid I, with the compositions of the phases in equilibrium given by curves BD and CF. Region VII represents the equilibrium between single phase V and solid solution VIII, with compositions of the phases in equilibrium given by curves DJ and EK. Finally, region IX represents the equilibrium between liquid I and solid solution VIII, with compositions of the phases in equilibrium given by curves FL and EL. The points A, B, and C represent three phases in equilibrium, F = 0, and the three phases are solid solution III of composition A, single phase V, of composition B, and liquid I, of composition C. The points D, E, and F represent three phases in equilibrium, F = 0, and the three phases are single phase V, of composition D, solid solution VIII, of composition E, and liquid I, of composition F. In conclusion, the geometric relationship between the number of degrees of freedom and the geometric constructs in the phase diagram provide a way to distinguish between one-phase regions and two-phase regions in two-component phase diagrams involving single solid phases of variable composition as well as pure phases. Literature Cited 1. Adamson, A. W. Understanding Physical Chemistry, 3rd ed.; Benjamin/Cummings: Menlo Park, CA, 1980; Chapter 12. 2. Sillen, G. L.; Lange, P. W.; Gabrielson, C. O. Problems in Physical Chemistry; Prentice-Hall: New York, 1952; Chapter 7. 3. Ricci, J. E. The Phase Rule and Heterogeneous Equilibrium; Reinhold: New York, 1951; p 23.
Journal of Chemical Education • Vol. 76 No. 2 February 1999 • JChemEd.chem.wisc.edu
