A pure substance melts at a fixed temperature. A binary solution changes from solid to liquid over a range of temperatures. In the melting range, components A and B in the solid phase are in equilibrium with A and B in the coexisting liquid phase. The equilibrium conditions are:
1 The composition of the gas phase in equilibrium with the solid or liquid (as given by Eqs (8.5) and (8.6)) is ignored in this representation. In order to graphically depict the equilibrium partial pressures as functions of temperature and composition would require a third dimension on the plot, which would make the representation unwieldy.
2 The total pressure and the equilibrium partial pressures of the components of the condensed phase are independent of each other. For example, an inert gas can be added to the gas phase without affecting the thermodynamics.
The chemical potentials are related to mole fractions by Eq (8.3). If the A-B solutions are ideal in both the solid and liquid phases, the above conditions become:
Since xAL+ xBL = 1 and xAS+ xBS = 1, the above equations contain two unknowns. The solution gives the composition of the solid and liquid as functions of temperature, the plot of which is a binary phase diagram. Eliminating the mole fractions of component A gives the following solutions of Eqs (8.11):
1 — e a ( 1 — e a xBL ="1-r and xBS = ^l"1-- I (8.12)
a and P are the temperature functions:
The temperature dependence of the free energy of a pure phase was derived in Sect. 3.7.4 To a good approximation, the free energy difference between liquid and solid is given by Eq (3.27b):
where TM is the melting temperature of the pure substance and AhM is its heat of fusion. Combining the above two equations gives a and P as explicit functions of temperature:
Figure 8.3 shows the phase diagram for an ideal binary system calculated from Eqs (8.12) using specific values of the melting properties of metals A and B (chosen as U and Zr, see Table 8.2). The upper line (representing T Vs xBL) is called the liquidus. All points lying above this line are completely liquid. Similarly, all points below the lower curve (the solidus, or T Vs xBS) are completely solid, which in this case is an ideal solid solution. In the region bounded by the solidus and the liquidus, two phases coexist.
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