A sheet of \(\mathrm{BCC}\) iron \(2 \mathrm{mm}\) thick was exposed to a carburizing gas atmosphere on one side and a decarburizing atmosphere on the other side at \(675^{\circ} \mathrm{C}\). After having reached steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were determined to be 0.015 and 0.0068 wt \(\% .\) Compute the diffusion coefficient if the diffusion flux is \(7.36 \times 10^{-9}\) \(\mathrm{kg} / \mathrm{m}^{2}\) -s. Hint: Use Equation 4.9 to convert the concentrations from weight percent to kilograms of carbon per cubic meter of iron.

First, we need to convert the given concentrations from weight percent to kilograms of carbon per cubic meter of iron using Equation 4.9. The given concentrations are 0.015 wt% and 0.0068 wt%. Equation 4.9 is written as: C(x) = [ρ₁ * (C_wt / 100)] / (M₁ + (C_wt / 100) * (M₂ - M₁)) Where C(x) is the concentration in kg/m³, ρ₁ is the density of BCC iron (7870 kg/m³), C_wt is the weight percent of carbon, M₁ is the atomic weight of iron (55.85), and M₂ is the atomic weight of carbon (12.01). Let's calculate the concentrations in kg/m³ for both surfaces. For the carburizing side (0.015 wt%): C₁(x) = [(7870) * (0.015 / 100)] / [(55.85) + ((0.015 / 100)(12.01 - 55.85))] = 133.18 kg/m³ For the decarburizing side (0.0068 wt%): C₂(x) = [(7870) * (0.0068 / 100)] / [(55.85) + ((0.0068 / 100)(12.01 - 55.85))] = 58.65 kg/m³

Now that we have the concentrations in kg/m³, we can calculate the concentration gradient. ΔC = C₁(x) - C₂(x) ΔC = 133.18 kg/m³ - 58.65 kg/m³ = 74.53 kg/m³ Since the sheet is 2mm thick, we will convert it into meters: Thickness = 2 mm = 0.002 m Now, we can find the concentration gradient (dC/dx) in kg/m⁴: dC/dx = ΔC / Thickness dC/dx = 74.53 kg/m³ / 0.002 m = 37265 kg/m⁴

Finally, using Fick's first law, we can calculate the diffusion coefficient. Fick's first law is: J = -D * (dC/dx) Where J is the diffusion flux, D is the diffusion coefficient, and dC/dx is the concentration gradient. We will solve for D: D = -J / (dC/dx) We have been given the diffusion flux, J = 7.36 * 10⁻⁹ kg/m²s. Now, we can calculate D: D = -(7.36 * 10⁻⁹ kg/m²s) / (-37265 kg/m⁴) D = 1.98 * 10⁻¹³ m²/s The diffusion coefficient of carbon in the BCC iron sheet is 1.98 * 10⁻¹³ m²/s.