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## Boris S. Bokstein, Mikhail I. Mendelev, and David J. Srolovitz

Print publication date: 2005

Print ISBN-13: 9780198528036

Published to Oxford Scholarship Online:

DOI: 10.1093/oso/9780198528036.001.0001

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# Phase equilibria II

Chapter:
(p.74) 4 Phase equilibria II
Source:
Thermodynamics and Kinetics in Materials Science
Publisher:
Oxford University Press
DOI:10.1093/oso/9780198528036.003.0006

In Chapter 2, we considered phase equilibria in a single component system. In the previous chapter, we considered the thermodynamics of multicomponent solutions. We are now prepared to discuss phase equilibria in multi-component systems. We consider the case of phase equilibria in a two-component systems in which all phases are condensed (i.e. solids or liquids). Under normal laboratory conditions experience suggests that the free energy of condensed phases is only very weakly dependent on the pressure and, hence, pressure is usually assumed to be constant. At constant pressure, the Gibbs phase rule (Eq. 2.3) for a two-component system takes the following form: . . . F = 3 – p, (4.1) . . . where F is the number of degrees of freedom (i.e. the number of independent parameters that must be set in order to fully determine the state of a system) and P is the number of phases present in equilibrium. This equation implies that the maximum number of independent parameters is 2; corresponding to the single-phase case. The most commonly used parameters are the temperature and the composition of the alloy. F = 2 implies that these two parameters can be varied over a finite range while the system remains the same, single phase. If two phases are in equilibrium with each other (P = 2), F = 1. This implies that the same two phases will be in equilibrium as we vary one parameter (the other parameter is not free). More specifically, we can retain the same two phase equilibrium while varying the alloy concentration over some range, if the temperature is described by some function, T(x2). T(x2) describes the curve (locus of points) along which this two-phase equilibrium occurs. Therefore, it is convenient to represent the phase equilibria in two-component condensed systems in T-x2 coordinates (as done below). Equation (4.1) implies that the maximum number of phases which can be in equilibrium with each other is three. This corresponds to F = 0—any change of temperature and/or alloy composition will lead to the disappearance of at least one of the phases.

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