A$.$ Starting with the equation below, derive expressions for the critical nucleus size and $($:$x^{**}\}$ and activation free energy.

$G_{cryst}=(4xl)+(2x^2)+(x^{2}l)G_v$

B$.$ Derive expressions for the temperature$-$dependence of $l^{**},x^{**},$ and the activation free energy.

C$.$ Consider the scenario in which the sample is quenched to a temperature below $T_m^{}$ and allowed to crystallize. If this process is repeated and the temperature of crystallization is decreased, how will this affect $l^{**},x^{**}$ and the activation free energy?