A thick plate of pure iron at \(1000^{\circ} \mathrm{C}\) is subjected to a carburization process in which the surface of the plate is suddenly exposed to a gas that induces a carbon concentration \(C_{\mathrm{C}, s}\) at one surface. The average diffusion coefficient for carbon and iron at this temperature is \(D_{\mathrm{C}-\mathrm{Fe}}=3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\). Use the correspondence between heat and mass transfer variables in addressing the following questions.

(a) Consider the heat transfer analog to the carburization problem. Sketch the mass and heat transfer systems. Show and explain the correspondence between variables. Provide the analytical solutions to the heat and mass transfer problems.

(b) Determine the carbon concentration ratio, \(C_{\mathrm{C}}(x, t) /\) \(C_{\mathrm{C}, s}\), at a depth of \(1 \mathrm{~mm}\) after \(1 \mathrm{~h}\) of carburization.

(c) From the analogy, show that the time dependence of the mass flux of carbon into the plate may be expressed as \(n_{\mathrm{C}}^{\prime \prime}=ho_{\mathrm{C}, s}\left(D_{\mathrm{C}-\mathrm{Fe}} / \pi t\right)^{1 / 2}\). Also, obtain an expression for the mass of carbon per unit area entering the iron plate over the time period \(t\).