QuestionJuly 31, 2025

What approximate carburization time is needed to achieve 5% atomic carbon at a distance from the surface of 0.4 microns? 1 hour 1.5 hours 2 hours 2.5 hours

What approximate carburization time is needed to achieve 5% atomic carbon at a distance from the surface of 0.4 microns? 1 hour 1.5 hours 2 hours 2.5 hours
What approximate carburization time is needed to achieve 5% atomic carbon at a distance from the surface of 0.4 microns? 1 hour 1.5 hours 2 hours 2.5 hours

Solution
4.6(250 votes)

Answer

2 hours Explanation 1. Identify the Diffusion Equation Use Fick's second law for diffusion in a semi-infinite solid: C(x,t) = C_s \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right), where C(x,t) is the concentration at depth x and time t, C_s is the surface concentration, D is the diffusion coefficient, and \text{erfc} is the complementary error function. 2. Set Up the Problem Given C(x,t) = 0.05C_s at x = 0.4 \mu m. Assume C_s = 1 for simplicity, so C(x,t) = 0.05. 3. Solve for Time Rearrange to find t: 0.05 = \text{erfc}\left(\frac{0.4 \times 10^{-6}}{2\sqrt{Dt}}\right). Use tables or numerical methods to solve for t given typical values of D for carbon in steel. 4. Approximate Time from Options Compare calculated t with given options (1 hour, 1.5 hours, 2 hours, 2.5 hours).

Explanation

1. Identify the Diffusion Equation<br /> Use Fick's second law for diffusion in a semi-infinite solid: $C(x,t) = C_s \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$, where $C(x,t)$ is the concentration at depth $x$ and time $t$, $C_s$ is the surface concentration, $D$ is the diffusion coefficient, and $\text{erfc}$ is the complementary error function.<br /><br />2. Set Up the Problem<br /> Given $C(x,t) = 0.05C_s$ at $x = 0.4 \mu m$. Assume $C_s = 1$ for simplicity, so $C(x,t) = 0.05$.<br /><br />3. Solve for Time<br /> Rearrange to find $t$: $0.05 = \text{erfc}\left(\frac{0.4 \times 10^{-6}}{2\sqrt{Dt}}\right)$. Use tables or numerical methods to solve for $t$ given typical values of $D$ for carbon in steel.<br /><br />4. Approximate Time from Options<br /> Compare calculated $t$ with given options (1 hour, 1.5 hours, 2 hours, 2.5 hours).
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