Problem $3$

A mild steel rod with an initial carbon concentration of $\backslash (0\mathrm{.}2\backslash \%\backslash )$ by weight is to be case hardened. It is packed in a carbonaceous material and maintained at a high temperature. Thermodynamic data for the chemical system indicates an equilibrium concentration of carbon in iron at the phase interface of $\backslash (1\mathrm{.}5\backslash \%\backslash ).$ Calculate the time required for the location $1$ mm below the surface to have a carbon mass concentration of $\backslash (0\mathrm{.}8\backslash \%\backslash ).$ The diffusion coefficient of carbon in steel at the process temperature is $\backslash (5\mathrm{.}6\backslash$times $10^\{-10\}\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\}\backslash ).$