$.$From the following phase diagram $($Figure $1),$ in the composition of $70wt\%Pb-30wt\%Mg($a$),$ calculate the weight fraction of the $Mg2Pb$ phase $($eutectic $Mg2Pb)$ in the process tissue when the liquid phase is equilibrated to room temperature. $($The eutectic point is $65wt\%Pb-35wt\%Mg,465C,$ and the $Mg2Pb$ phase is assumed to have a composition of $80wt\%Pb-20wt\%Mg.)$

$($b$)$ When the cooling rate is increased, answer how the compositions of the a phase and the Mg$2$Pb phase change.

$($c$)$ From the phase diagram, answer whether you can tell the difference between the diffusion rate of $Pb$ in $Mg$ and the diffusion rate of $Mg$ in $Pb,$ and briefly explain why.