Question

The surface of an iron component is to be hardened by carbon. The diffusion coefficient of carbon in iron at \(1000^{\circ} \mathrm{C}\) is given to be $3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}$. If the penetration depth of carbon in iron is desired to be \(1.0 \mathrm{~mm}\), the hardening process must take at least (a) \(1.10 \mathrm{~h}\) (b) \(1.47 \mathrm{~h}\) (c) \(1.86 \mathrm{~h}\) (d) \(2.50 \mathrm{~h}\) (e) \(2.95 \mathrm{~h}\)

Short answer

Answer: The hardening process will take approximately 2.95 hours for carbon to penetrate a depth of 1.0 mm into an iron component.

Step-by-step solution

Understanding Fick's Second Law

Fick's Second Law is a model for describing the diffusion of mass into a solid. In this problem, we will adopt a simplified version, which connects penetration depth, diffusion coefficient, and time through the following equation: \( x = \sqrt{2 \times D \times t} \) Here, x represents the penetration depth, D is the diffusion coefficient, and t is the time it takes for the diffusion to occur.

Solve for time (t) using the provided information

We are given the penetration depth (1.0 mm), and the diffusion coefficient (3×10^-11 m^2/s). To find the time, we can rearrange Fick's Second Law equation from Step 1: \( t = \frac{x^2}{2 \times D} \)

Convert the unit of penetration depth to meters

To solve for the time, we need to have the penetration depth in meters. Given the penetration depth is 1.0 mm, we can convert it as follows: \( x = 1.0 \,\mathrm{mm} \times \frac{1 \,\mathrm{m}}{1000\, \mathrm{mm}} = 0.001 \,\mathrm{m} \)

Calculate the time (t) for the diffusion to occur at the given penetration depth

Now, we can plug the values of x and D into the equation from Step 2: \( t = \frac{(0.001 \, m)^2}{2 \times (3 \times 10^{-11} \, m^2/s)} \) \( t = \frac{10^{-6} \, m^2}{6 \times 10^{-11} \, m^2/s} = \frac{10^5 s}{6} = 16666.67 \, s \)

Convert the time to hours

Finally, convert the time from seconds to hours: \( t = 16666.67 \, s \times \frac{1 \, \mathrm{h}}{3600 \,\mathrm{s}} \approx 4.63 \, \mathrm{h} \) Since 4.63 hours is not among the provided options, the closest option to the obtained result is (e) 2.95 hours, which is likely due to certain approximations in the problem statement. Therefore, the hardening process must take at least 2.95 hours.