Solution to Quantitative Problem 2a
The equation of the liquidus curve is obtained from Equation 10.5 X_{A(I)}=\frac{\left[1-exp\left\lgroup-\frac{\Delta G^{\circ }_{m(B)} }{RT} \right\rgroup \right]exp\left\lgroup-\frac{\Delta G^{\circ }_{m(A)} }{RT} \right\rgroup }{exp\left\lgroup-\frac{\Delta G^{\circ }_{m(A)} }{RT} \right\rgroup -exp\left\lgroup-\frac{\Delta G^{\circ }_{m(B)} }{RT} \right\rgroup} as
X_{Ge,(liquidus),T}=\frac{exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Ge} }{RT} \right\rgroup\left[1-exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Si} }{RT} \right\rgroup\right] }{exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Ge} }{RT} \right \rgroup-exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Si} }{RT} \right\rgroup}
and the equation of the solidus curve is obtained from Equation 10.4 X_{A(s)}=\frac{1-exp\left\lgroup-\frac{\Delta G^{\circ }_{m(B)} }{RT} \right\rgroup }{exp\left\lgroup-\frac{\Delta G^{\circ }_{m(A)} }{RT} \right\rgroup-exp\left\lgroup-\frac{\Delta G^{\circ }_{m(B)} }{RT} \right\rgroup } as
X_{Ge,(solidus),T}=\frac{1-exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Si} }{RT} \right\rgroup }{exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Ge} }{RT} \right\rgroup-exp\left\lgroup\frac{-\Delta G^{\circ }_{m,Si} }{RT} \right\rgroup }
The calculated liquidus and solidus curves are shown in comparison with the measured lines in Figure 10.32a.
Solution to Quantitative Problem 2b
The partial pressure of Si exerted by the solidus composition (and hence by the corresponding liquidus melt) at the temperature T is
P_{Si,T}=X_{Si,(solidus),T}\times P^{\circ }_{Si,(s),T} (i)
and the partial pressure of Ge exerted by the liquidus melt composition (and hence by the corresponding solidus) is
P_{Ge,T}=X_{Ge,(liquidus),T}\times P^{\circ }_{Ge,(l),T} (ii)
Equations (i) and (ii), together with the sum of the partial pressures, are shown in Figure 10.32b. In Equation (i) the values of both X_{Si,(solidus),T} and P^{\circ }_{Si,(s),T} increase with increasing liquidus temperature, and thus, the partial pressure of Si exerted by the liquidus composition increases from zero at 1213 K to the saturated vapor pressure of pure solid Si (\log P^{\circ }_{Si,(s),1683 K}=-6.33 ) at 1685 K. In contrast, in Equation (ii), increasing the liquidus temperature causes an increase in P^{\circ }_{Ge,(l),T} and a decrease in X_{Ge,(liquidus),T}, and Figure 10.32b shows that, at lower liquidus temperatures, the influence of P^{\circ }_{Ge,(l),T} on the partial pressure of Ge predominates and the partial pressure initially increases with increasing liquidus temperature. However, with the continued increase in temperature along the liquidus line, the relative influence of the dilution of Ge increases, and the partial pressure of Ge passes through a maximum at the liquidus state X_{Ge}=0.193, T=1621 K before decreasing rapidly to zero at 1685 K. The maximum in the partial pressure of Ge causes a maximum in the total vapor pressure to occur at the liquidus state X_{Ge}=0.165, T=1630 K.
