A steel part is carburized at temperature T₁ and then again at a higher temperature T₂. The diffusivity D follows D = D₀ exp(−Qd/RT). Which outcome is expected at T₂?
AD decreases because higher temperature destabilizes the lattice
BD increases because the exponential term becomes less negative, raising D dramatically
CD stays the same because D₀ is fixed
DD increases linearly with temperature
In the Arrhenius expression D = D₀ exp(−Qd/RT), raising T makes the exponent −Qd/RT less negative, so exp(−Qd/RT) grows. Because the relationship is exponential, even a modest temperature increase produces a large jump in D — often an order of magnitude for a 50°C rise.
Question 2 True / False
Carbon diffusing into iron (interstitial diffusion) is faster than iron atoms exchanging sites (substitutional diffusion) because interstitial atoms do not require a neighboring vacancy to move.
TTrue
FFalse
Answer: True
Substitutional diffusion depends on a vacancy being adjacent to the diffusing atom — a relatively rare event. Interstitial atoms (like C in the iron lattice) are small enough to hop between existing gaps without waiting for vacancies, giving interstitial diffusion a lower activation energy Qd and therefore higher D at any given temperature.
Question 3 Short Answer
During carburization of steel, how does the carbon concentration profile change with increasing exposure time at constant temperature?
Think about your answer, then reveal below.
Model answer: The profile broadens and carbon penetrates deeper into the steel. The complementary error function solution C(x,t) = Cs − (Cs − C₀)·erf(x / 2√(Dt)) shows that the characteristic diffusion depth scales as √(Dt), so longer time t pushes the concentration profile further from the surface while the surface concentration remains fixed at Cs.
The erfc solution comes directly from Fick's second law for a semi-infinite solid with constant surface concentration. The √(Dt) scaling is fundamental: doubling the time does not double the penetration depth — it increases it by only √2. This sublinear growth explains why extremely deep case hardening requires very long times or much higher temperatures.