Let's solve the carburizing time step-by-step. --- # **Problem Restatement** **Given:** - Initial carbon content, \( C_ = .20\% \) - Desired carbon content at depth \( x = 2\, \text{mm} \), \( C_x = .45\% \) - Surface carbon content, \( C_s = 1.30\% \) - Temperature = \( 100^\circ C \) - Diffusion coefficient for C in Fe at \( 100^\circ C \): \( D = 2.9 \times 10^{-11} \, \text{m}^2/\text{s} \) - Depth, \( x = 2\, \text{mm} = .002\, \text{m} \) --- # **Step 1: Use the Diffusion Equation** The solution for carburizing (semi-infinite solid, constant surface concentration) is: \[ \frac{C_x - C_}{C_s - C_} = 1 - \text{erf}\left(\frac{x}{2 \sqrt{Dt}}\right) \] --- # **Step 2: Plug in Values** \[ \frac{.45 - .20}{1.30 - .20} = \frac{.25}{1.10} = .227 \] So, \[ .227 = 1 - \text{erf}\left(\frac{x}{2 \sqrt{Dt}}\right) \] \[ \text{erf}\left(\frac{x}{2 \sqrt{Dt}}\right) = 1 - .227 = .773 \] --- # **Step 3: Find the z-value Corresponding to erf(z) = .773** From the table in the image: - For \( z = .85 \), \( \text{erf}(z) = .7707 \) - For \( z = .90 \), \( \text{erf}(z) = .797 \) Interpolate for \( \text{erf}(z) = .773 \): \[ z = .85 + \frac{.773 - .7707}{.797 - .7707} \times (.90 - .85) \] \[ = .85 + \frac{.0023}{.0263} \times .05 \] \[ = .85 + .0044 \] \[ \approx .854 \] --- # **Step 4: Relate z to t** \[ z = \frac{x}{2 \sqrt{Dt}} \] \[ .854 = \frac{.002}{2\sqrt{2.9 \times 10^{-11} t}} \] Solve for \( t \): \[ 2 \sqrt{2.9 \times 10^{-11} t} = \frac{.002}{.854} \] \[ 2 \sqrt{2.9 \times 10^{-11} t} = .00234 \] \[ \sqrt{2.9 \times 10^{-11} t} = .00117 \] \[ 2.9 \times 10^{-11} t = (.00117)^2 \] \[ 2.9 \times 10^{-11} t = 1.3689 \times 10^{-6} \] \[ t = \frac{1.3689 \times 10^{-6}}{2.9 \times 10^{-11}} \] \[ t \approx 4.72 \times 10^4 \, \text{seconds} \] --- # **Step 5: Convert to Hours** \[ t = \frac{4.72 \times 10^4}{360} \approx 13.1\, \text{hours} \] --- # **Final Answer** ### **The carburizing time necessary is:** \[ \boxed{13.1\, \text{hours}} \] --- **Summary:** - Used the error function solution for diffusion. - Plugged in values and solved for \( t \) using the given data and interpolation for the error function. - Converted time to hours for a practical answer.