## Meltingsolidification of an ideal twocomponent system

The melting characteristics of a binary system that is ideal in both liquid and solid states were derived analytically in Sect. 8.4.1. Here, the same analysis is performed graphically.

All graphical determinations of binary phase diagrams begin with the free energy Vs composition curves for all possible phases in the system. For ideal systems, these curves are given by Eqs (8.22) - (8.24) with hex = 0 in both solid and liquid phases:

g(L) = xa§a(l) + xb§b(l) + rt(xa 1" xa + xB In xB) (8.27a)

g(S) = XAgA(S) + XBgB(S) + RT(xa ln Xa + Xb lnxB) (8.27b)

In order to construct the free energy - composition curves using these equations, the four pure-component molar free energies must be specified. Equation (8.13) relates the molar free energies of the pure liquids and pure solids:

gA(L) = gA(S) + |l - tT- Wma; gB(L) = gB(S) + |l - |AhMB (8.28)

As discussed in Sect. 1.6, the free energy (in common with u, h, and h) has no absolute value. Therefore, two molar free energies, say gA(S) and gB(S), can be specified arbitrarily. Of course, this choice affects the shapes and positions of the free energy curves, but the compositions at the common tangency points are unaffected4. For convenience, gA(S) = 0 and gB(S) = 0 are choosen.

The melting properties of the two components are those of uranium (component A) and zirconium (component B) and are given in Table 8.2.

Table 8.2 Melting properties of Uranium and Zirconium

Component Element_Tm, K AhM, kJ/mole

A uranium 1406 15.5

### B zirconium 2130 23.0

To illustrate the common tangent construction, a temperature is 1500 K is chosen. Equation (8.27) gives gA(L) = -1.04 kJ/mole. The minus sign indicates that liquid U is more stable than the solid at a temperature above the melting point. For Zr, Eq (8.27) yields gB(L) = 6.80 kJ/mole.

Plots of Eqs (8.26a) and (8.26b) are shown in Fig. 8.8. The common-tangency points are xBL = 0.10 on the liquid curve and xBS = 0.17 on the solid curve. For xB < xBL The free energy of the liquid is lower than that of the solid, so the system is a single phase liquid. Similarly for xB > xBS, g(L) > g(S) and the system is a solid solution. At this temperature, the two-phase region occupies the overall composition interval xbl < xb < xBS. The same results are obtained from the analytical method of Sect. 8.4.1.

4 This can be proved by substituting Eqs (8.27) into (8.26) and then into Eq (8.25). The resulting common-tangent equation does not contain gA(S) and gB(S).

Problems 8.11 and 8.17 use free energy plots and the common-tangent rule to deduce a more complex type of phase diagram than the ideal systems treated above. The following section shows how this method generates a particularly common phase diagram, called a eutectic phase diagram.

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