A steel containing 0.20 wt% carbon is carburized with its surface carbon concentration held at 1.00 wt%. Which value is closest to the time needed for the carbon concentration to reach 0.60 wt% at a depth of 0.50 mm? Take the diffusion coefficient as 1.6×10⁻¹¹ m²/s, (C − C₀)/(Cₛ − C₀) = 1 − erf(x/2√(Dt)), erf(0.371) = 0.40 and erf(0.477) = 0.50.

Answer: C. About 4.8 hours

(0.60 − 0.20)/(1.00 − 0.20) = 0.50, so 1 − erf(z) = 0.50 and erf(z) = 0.50 give z = x/2√(Dt) = 0.477. Then √(Dt) = 0.50×10⁻³/(2 × 0.477) ≈ 5.24×10⁻⁴ m, Dt ≈ 2.75×10⁻⁷ m², and t = 2.75×10⁻⁷/(1.6×10⁻¹¹) ≈ 1.72×10⁴ s, about 4.8 hours. About 19 hours is the mistake of dropping the 2 in the denominator and setting x/√(Dt) = 0.477, which makes the time four times too long. About 7.9 hours ignores the initial carbon concentration C₀ and uses 0.60/1.00, giving erf(z) = 0.40 and z = 0.371. Because the time is proportional to the square of the depth, how the factors are handled has a large effect.