An FCC iron-carbon alloy initially containing 0.10 wt \(\% \mathrm{C}\) is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at 1.10 wt\%. If after 48 h the concentration of carbon is \(0.30 \mathrm{wt} \%\) at a position \(3.5 \mathrm{mm}\) below the surface, determine the temperature at which the treatment was carried out.

To solve this problem, we need Fick's second law of diffusion, as well as the diffusion coefficient equation for carbon in iron, which depends on temperature. Here are the equations: 1. Fick's second law of diffusion: \(C(x,t) = C_{\infty} - (C_{\infty} - C_{0}) \cdot \operatorname{erf}\Big(\frac{x}{2\sqrt{Dt}}\Big)\) 2. Diffusion coefficient equation for carbon in FCC iron: \(D = D_{0} \cdot \exp\Big(-\frac{Q_{\text{d}}}{RT}\Big)\) For carbon in FCC iron, the constants are as follows: - \(D_{0} = 2.3 × 10^{-5} \, \mathrm{m^2/s}\) - \(Q_{\text{d}} = 1.4 \times 10^{5} \, \mathrm{J/mol}\)

We can use Fick's second law of diffusion to find the diffusion coefficient \(D\), which is needed to find the temperature. First, rearrange the equation to solve for D: \(D = \frac{x^2}{4t}\Big(\operatorname{erf}^{-1}\Big(\frac{C_{\infty} - C(x,t)}{C_{\infty} - C_{0}}\Big)\Big)^2\) Substitute the given values: - \(C_{\infty} = 1.10 \,\mathrm{wt\%}\) - \(C_{0} = 0.10 \,\mathrm{wt\%}\) - \(C(x,t) = 0.30 \,\mathrm{wt\%}\) - \(x = 3.5 \,\mathrm{mm} = 3.5 \times 10^{-3} \,\mathrm{m}\) - \(t = 48 \,\mathrm{h} = 48 × 3600 \,\mathrm{s}\) - \(\operatorname{erf}^{-1}()\) is the inverse error function \(D = \frac{(3.5 \times 10^{-3}\,\mathrm{m})^2}{4(48 × 3600 \,\mathrm{s})(\operatorname{erf}^{-1}(\frac{1.10 - 0.30}{1.10 - 0.10}))^2} ≈ 5.74 × 10^{-12} \, \mathrm{m^2/s}\)

With the diffusion coefficient calculated, we can now use the diffusion coefficient equation for carbon in FCC iron to find the temperature of the treatment. Rearrange the diffusion coefficient equation to solve for the temperature T: \(T = \frac{Q_{\text{d}}}{R\cdot \ln(D_{0}/D)}\) Use the known values: - \(D = 5.74 × 10^{-12} \, \mathrm{m^2/s}\) - \(D_{0} = 2.3 × 10^{-5} \, \mathrm{m^2/s}\) - \(Q_{\text{d}} = 1.4 \times 10^{5} \, \mathrm{J/mol}\) - \(R = 8.314 \, \mathrm{J/(mol\cdot K)}\) \(T = \frac{1.4 × 10^5\,\mathrm{J/mol}}{8.314\,\mathrm{J/(mol\cdot K)}\cdot \ln\Big(\frac{2.3 × 10^{-5}\,\mathrm{m^2/s}}{5.74 × 10^{-12}\,\mathrm{m^2/s}}\Big)} ≈ 1226 \, \mathrm{K}\) The carburization temperature for the given FCC iron-carbon alloy is approximately 1226 Kelvin.