III.C$.($a$-10$ pts$)$ For the solidification of iron, calculate the critical radius $\backslash (\backslash$mathrm$\{$r$\}^\{*\}\backslash )$ and the activation free energy $\backslash (\backslash$Delta $\backslash$mathrm$\{$G$\}^\{*\}\backslash )$ if nucleation is homogeneous for an undercooling $($supercooling$)$ of $300$ K $.$ Values for the enthalpy of freezing and specific surface free energy are $\backslash (-1\mathrm{.}85\backslash$times $10^\{9\}\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ and $0\mathrm{.}204\backslash (\backslash$mathrm$\{$J$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ),$ respectively.

$($b$-10$ pts$)$ What is the number of BCC unit cells found in a nucleus of critical size?

$($c$-10$ pts$.)$ What us heterogeneous nucleation? Why is it important in engineering applications?

$($d$-10$ pts$.)$ Describe one engineering application where one would want to suppress nucleation $($homogenous or heterogeneous$)$ completely.

$($e$-10$ pts$.)$ Plot the variation of $\backslash (\backslash$Delta $\backslash$mathrm$\{$G$\}\backslash )$ as a function of r $.$

Note: Refer to the Fe$-$C phase diagram to obtain the melting temperature $\backslash (\backslash$mathrm$\{$T$\}\_\{\backslash$mathrm$\{$m$\}\}\backslash )$ of iron. Data: Lattice parameter of BCCFe is $0\mathrm{.}3571$ nm $.$