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°C (1123 K) for 10 min increases the carbon concentration to 0.90 wt% at a position 1.0 mm below the surface. Estimate the diffusion time required at 650°C (923 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 a-Fe.

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Activation Energy Qa Calculated Values Diffusing Species Host D(m²ls) Do(m²ls) T(C) Metal kJ/mol eV/atom 3.0 × 10-21 1.8 × 10-15 Fe a-Fe 2.8 × 10-4 251 2.60 500 (BCC) 900 1.1 × 10-17 7.8 x 10-16 y-Fe (FCC) 5.0 x 10-5 Fe 284 2.94 900 1100 2.4 x 10-12 1.7 x 10-10 6.2 × 10-7 0.83 500 a-Fe 80 900 5.9 x 10-12 5.3 x 10-11 2.3 x 10-5 1.53 y-Fe 900 148 1100 7.8 x 10-5 4.2 x 10-19 Cu Cu 211 2.19 500 2.4 x 10-5 Zn Cu 189 1.96 500 4.0 x 10-18 2.3 × 10-4 Al Al 144 1.49 500 4.2 x 10-14 6.5 × 10-5 136 Cu Al 1.41 500 4.1 × 10-14 1.2 x 10-4 Mg 131 500 1.9 × 10-13 Al 1.35 2.7 × 10-5 Cu Ni 256 2.65 500 1.3 × 10-22