Consider a binary system A$-$B forming a regular solid solution. The interaction parameter is $=$

$25k\frac{J}{m}ol.$ The free energy of the regular solution is given by

$G=G_A^0(1-x_B)+G_B^0{x}_B+RT[x_{B}ln{x}_B+(1-x_B)ln(1-x_B)]+x_B(1-x_B)$

$(1)$ Schematically draw the temperature$-$composition phase diagram of this binary system,

show the phase boundary in solid line and the spinodal line in dashed line, also calculate

the critical temperature below which the spinodal decomposition can take place.

$(2)$ At $T=1000K,$ calculate the equilibrium compositions and the boundary compositions for

spinodal decomposition.

$(3)$ For an alloy composition of $x_B=0\mathrm{.}25,$ calculate the thermodynamic factor at $T=1000K.$