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Crystal defects and diffusion

Carbon diffuses at steady state through a 5.0 mm thick steel plate whose two faces are held at carbon concentrations of 1.2 kg/m³ and 0.8 kg/m³. If the diffusion coefficient is 3.0×10⁻¹¹ m²/s, which value is closest to the diffusion flux of carbon? Assume the concentration varies linearly through the thickness.

  1. AAbout 2.4×10⁻¹² kg/(m²·s)
  2. BAbout 7.2×10⁻⁹ kg/(m²·s)
  3. CAbout 2.4×10⁻⁹ kg/(m²·s)
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Answer: C. About 2.4×10⁻⁹ kg/(m²·s)

At steady state, Fick's first law J = −D(dC/dx) applies. The concentration gradient is (0.8 − 1.2)/(5.0×10⁻³ m) = −80 kg/m⁴, so J = 3.0×10⁻¹¹ × 80 = 2.4×10⁻⁹ kg/(m²·s), flowing from the high-concentration side to the low-concentration side. About 2.4×10⁻¹² is a unit-conversion error from dividing by the thickness left as 5.0 in mm, and about 7.2×10⁻⁹ uses the concentration on one side, 1.2 kg/m³, instead of the concentration difference. It is the concentration difference that drives the flow, so getting the gradient right matters. The same approach is used in hydrogen permeation tests and in evaluating gas separation membranes.

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  1. The diffusion coefficient of carbon in γ-iron follows the Arrhenius form D = D₀exp(−Q/RT) with D₀ = 2.3×10⁻⁵ m²/s and activation energy Q = 148 kJ/mol. Which value is closest to the diffusion coefficient at 927 °C? Take the gas constant as R = 8.314 J/(mol·K).
  2. The diffusion coefficient of an element was 1.0×10⁻¹⁴ m²/s at 800 K and 4.0×10⁻¹³ m²/s at 1000 K. Assuming the Arrhenius form D = D₀exp(−Q/RT), which value is closest to the activation energy Q? Take R = 8.314 J/(mol·K).
  3. 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.
  4. In a diffusion treatment at a fixed temperature, by what factor must the treatment time be increased to double the depth at which a given concentration is reached?
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