A liquid metal $($a phase$)$ is contained in a container $($ phase$).$ Assume the melting point of the metal is $T_m=1350K,$ the molar entropy change of melting is $S_M=S_L^0-S_S^0=5\mathrm{.}5Jmo{l}^{-1}{K}^{-1},$

and the molar volume $($for both the liquid and solid phase$)$ is$V_M=9\mathrm{.}6{10}^{-6}\frac{m^3}{m}ol.$ Now, we

super$-$cool the liquid metal to $T=1000K,$

$.$ What is the critical radius of homogeneous nucleation for solidification, if the interface energy between liquid $()$ and solid $()$ phase, ${}_{}=0\mathrm{.}15\frac{J}{m^2}?$

$.$ If the interfacial energy between the liquid and the container is ${}_{}=0\mathrm{.}10\frac{J}{m^2}$ and that between the metal solid and the container is ${}_{}=0\mathrm{.}175\frac{J}{m^2},$ what is the contact angle of the heterogeneous nucleus on the container wall?

$.$ What is the volume of the critical heterogeneous nucleus on the container wall? Draw a schematic diagram of the nucleus on the container wall.