The outer surface of a steel gear is to be hardened by increasing its carbon content. The carbon is to be supplied from an external carbon-rich atmosphere, which is maintained at an elevated temperature. A diffusion heat treatment at \(850^{\circ} \mathrm{C}(1123 \mathrm{~K})\) for \(10 \mathrm{~min}\) increases the carbon concentration to \(0.90 \mathrm{wt} \%\) at a position \(1.0 \mathrm{~mm}\) below the surface. Estimate the diffusion time required at \(650^{\circ} \mathrm{C}(923 \mathrm{~K})\) to achieve this same concentration also at a) 1.0-mm position. Assume that the surface carbon content is the same for both heat treatments, which is maintained constant. Use the diffusion data in Table \(5.2\) for C diffusion in \(\alpha\)-Fe.

Based on the step by step solution provided, here's a short answer to the problem: To find the required diffusion time at a higher temperature for the same carbon concentration at a specific position below the surface of a steel gear, follow these steps: 1. Calculate the diffusion coefficient (D) at the given temperature 600°C (873K) for carbon diffusion in alpha-iron using the Arrhenius equation. 2. Estimate the semi-infinite solid solution for carbon concentration at a position of 0.5mm below the surface using Fick's second law and the calculated value of D(873K). 3. Calculate the diffusion coefficient (D) at the higher temperature 900°C (1173K) for carbon diffusion in alpha-iron. 4. Estimate the required diffusion time at the higher temperature to achieve the same carbon concentration at a 0.5mm position using Fick's second law and the calculated value of D(1173K).