a$)$ Estimate the size of critical nucleus of tin when it is super cooled by $20C.$ Assume nucleation to be homogeneous. The enthalpy change for solidification of tin is $420M\frac{J}{m^3}.$ The liquid$/$solid interfacial energy for tin $(Y_{\text{l}\text{i}\text{q}-s\frac{n}{s}ol-sn})$ is $0\mathrm{.}055\frac{J}{m^2}.$ The melting point of tin is $232C.$

b$)$ By which factor will the energy barrier for nucleation $(G^{**})$ be reduced if tin nucleates heterogeneously on a mold surface when the energies of the mold$-$liquid tin $(Y_{\text{m}\text{o}\text{l}\frac{d}{l}iq-sn})$ and the mold$-$solid tin $(Y_{\text{m}\text{o}\text{l}\frac{d}{s}ol-sn})$ interfaces are $0\mathrm{.}082$ and $0\mathrm{.}038\frac{J}{m^2},$ respectively?