In solidification, for homogenous nucleation, the change of free energy is represented in equation and graph as below

$G_{\text{h}\text{o}\text{m}}=-\frac{4}{3}{r}^3{G}_v+4r^2{}_{SL}$

a$.$ Proof that the critical radius and the nucleation barrier can

be written as

$r^{**}=\frac{2{}_{SL}}{G_v}$

$G^{**}=\frac{16{}_{SL}^3}{3(G_v{)}^2}$

b$.$ Using the Turnbull's approximation to show that the critical radius and the nucleation barrier can be written as

$r^{**}=\frac{2{}_{SL}{T}_m}{L_{v}T},$ and $,G^{**}=\frac{16{}_{SL}^3{T}_m^2}{3L_v^2{T}^2}$