Calculating Processing Time in Unsteady Diffusion $|$In some applications, it is desired that the surfaces of parts made of steel $($iron$-$carbon alloys$)$ be harder than the inner parts. One way to achieve this is to increase the carbon concentration on the surface of the steel workpiece by exposing it to an atmosphere of a hydrocarbon$-$rich gas such as methane $($CH$4)$ at high temperature, a process called carburizing.Let's assume that an alloy with a uniform carbon concentration of $0\mathrm{.}25\%$ by weight is heated to $1100\backslash$deg C before the carburizing process. If the surface of the alloy is suddenly exposed to an atmosphere containing $1\mathrm{.}2\%$ by weight carbon and the carbon content on the surface is maintained at this value, calculate the carburizing time required for the carbon content to reach $0\mathrm{.}8\%$ at a depth of $0\mathrm{.}5$ mm from the surface. The diffusion coefficient of carbon in iron at this temperature is $1\mathrm{.}6$ x $10$ m$^2$ s and assume that the steel part is semi$-$infinite.