$($a$)$ For the solidification of iron, calculate the critical radius $r^{**}$ and the activation free energy $G^{**}$ if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are $-1\mathrm{.}85{10}^9\frac{J}{m^3}$ and $0\mathrm{.}204\frac{J}{m^2},$ respectively. Use the supercooling value $T=277K,$ and the melting point of iron is $1538C.$

Critical Radius, $r^{**}=$ nm

Activation Free Energy $G^{**}=$ J

$($b$)$ Now calculate the number of atoms found in a nucleus of critical size. Assume a lattice parameter of $0\mathrm{.}292nm$ for solid iron at its melting temperature.

Number of Atoms for Critical Size $=$ atoms$/$critical nucleus