The Gibbs free energy diagram at 1358 K for the system, using liquid as the standard state for Mg and solid as the standard state for Si, is shown in Figure 13.15. \Delta G^{\circ }_{(i)}=-47,030 J. at the melting temperature of Mg_{2}Si (1358 K) , and thus, the Gibbs free energy of formation of Mg_{2/3}Si_{1/3}=-47,030/3=-15,676 J, and this is the length of the line de in Figure 13.15. Point b in Figure 13.15 represents the free energy of liquid Si relative to solid Si and lies 9890 J above the point a . Consequently, the length of the line cd is 9890/3 = 3297 J, and the length of the line ce is 3,297 + 15,676 = 18,973 J. Thus, the Gibbs free energy of formation of solid Mg_{2/3}Si_{1/3} from liquid Mg and liquid Si is – 18,919 J. However, at the melting temperature of 1358 K, G^{\circ }_{Mg_2Si,(s)}=G^{\circ }_{Mg_2Si,(l)} , and thus, the Gibbs free energy of formation of liquid Mg_{2/3}Si_{1/3} from liquid Mg and liquid Si at 1358 K is also 973 J. Thus, the line representing the molar Gibbs free energy of formation of melts in the system at 1358 K passes through the point e and from the general expression for the formation of a regular solution
\Delta G^M=RT(X_{Mg}\ln X_{Mg}+X_{Si}\ln X_{Si})+\alpha X_{Mg}X_{Si}
at X_{Si}=1/3 ,
-18,973=8.3144\times 1358\left\lgroup\frac{2}{3}\ln \frac{2}{3}+\frac{1}{3}\ln \frac{1}{3} \right\rgroup +\alpha \frac{1}{3}.\frac{2}{3}
which gives α = – 53,040 J.
Combination of \Delta G^{\circ }_{(i)} and the Gibbs free energy change for the melting of Si gives
\Delta G^{\circ }_{(ii)}=-151,030+69.3T J
for the reaction
2Mg_{(l)}+Si_{(l)}=Mg_2Si_s (ii)
Thus,
-151,030 + 69.3T = -RT \ln K
=-RT\ln \frac{a_{Mg_2Si,(s)}}{a^{2}_{Mg}a_{Si}}
Since liquids on the Mg_2Si liquidus line are saturated with Mg_2Si,a_{Mg_2Si,(s)}=1 , the variations of the activities of Mg and Si with temperature along the Mg_2Si liquidus line are given by
-15,030+69.3T=2RT\ln a_{Mg}+RT\ln a_{Si} (iii)
In a regular solution,
RT\ln a_{i}=RT\ln X_{i}+\Omega (1-X_i)^2
and thus, Equation (iii) becomes
-15,030+69.3T=2RT\ln (1-X_{Si})-2\times 53,040(1-X_{Si})^2 +RT\ln X_{Si}-53,040(1-X_{Si})^2 (iv)
Equation (iv), which is the equation of the Mg_2Si liquidus, is quadratic and gives two values of X_{Si} at each temperature, with one being the liquidus composition in the Mg-Mg_2Si sub-binary and the other being the liquidus composition in the Mg_2Si-Si sub-binary. Equation (iv) is drawn as the broken line abc in Figure 13.16. The Si liquidus line is obtained from Equation 10.22 as
\Delta \overline{G}^{M}_{Si(l)}=-\Delta G^{\circ} _{m,Si}
That is,
RT\ln X_{Si}+\alpha (1-X_{Si})^2=-50,630+30.0T
which gives
T=\frac{50,630-53,040(1-X_{Si})^2}{30.0-8.3144\ln X_{Si}} (v)
Equation (v) is drawn as the broken line cd in Figure 13.16.
Similarly, the Mg liquidus line is given by
RT\ln (1-X_{Si})+\alpha X_{Si^2}=-8790+9.52T
which gives
T=\frac{8790-53,040X^{2}_{Si}}{9.52-8.3144\ln (1-X_{Si})^2} (vi)
The calculated diagram shows good agreement with the actual diagram; the calculated eutectic temperature and eutectic composition in the Mg_2Si-Si sub‑binary are, respectively, 1200 K and X_{Si}= 0.58, which are close to the actual values of 1218 K and X_{Si}=0.53, and the eutectic composition and temperature in the Mg-Mg_2Si sub-binary coincide with the actual values.
