An undesirable consequence of welding steel by acetylene torch is the rise in hydrogen concentration in the metal. That hydrogen contamination embrittles the steel. The metal is typically subjected to a vacuum heat treatment to restore ductility. A steel part is placed in a vacuum furnace at 600∘C for one hour. The H concentration (initially uniform in the weld) falls to 1/2 its initial value at a depth of 500μm from the surface: It is known that the diffusivity of H in the steel at 600∘ C is 3.091×10−4 cm2/sec. An identical part is heat treated at 700∘ C. Under these conditions the H concentration falls to 1/2 its initial value at a depth of 500μm in 50 minutes. What is the diffusivity of H in steel at 700∘ C?

Question

An undesirable consequence of welding steel by acetylene torch is the rise in hydrogen concentration in the metal. That hydrogen contamination embrittles the steel. The metal is typically subjected to a vacuum heat treatment to restore ductility. A steel part is placed in a vacuum furnace at 600∘C for one hour. The H concentration (initially uniform in the weld) falls to 1/2 its initial value at a depth of 500μm from the surface: It is known that the diffusivity of H in the steel at 600∘ C is 3.091×10−4 cm2/sec. An identical part is heat treated at 700∘ C. Under these conditions the H concentration falls to 1/2 its initial value at a depth of 500μm in 50 minutes. What is the diffusivity of H in steel at 700∘ C?

Give your answer in units of cm2/s, in scientific notation up to two decimal places

Answer 1

To calculate the diffusivity of hydrogen in steel at 700∘ C, we can use Fick's Second Law of Diffusion. Fick's Second Law relates diffusivity, concentration gradient, and time.

The formula for Fick's Second Law is:

∂C/∂t = D * ∂²C/∂x²

Where:
∂C/∂t is the change in concentration with respect to time,
D is the diffusivity,
∂²C/∂x² is the second derivative of concentration with respect to depth (x).

From the problem statement, we know that at a depth of 500μm, the hydrogen concentration is reduced to half its initial value. This means that the concentration gradient (∂C/∂x) at that depth is given by:

(1/2 - 1) / 500μm = -1/1000μm = -10^(-7) / cm

Now we can solve for the diffusivity D at 700∘ C. We can rearrange Fick's Second Law and solve for D:

D = (∂C/∂t) / (∂²C/∂x²)

At 700∘ C, the concentration falls to half its initial value in 50 minutes. The concentration gradient (∂C/∂x) is the same as before, -10^(-7) / cm.

So, the diffusivity D at 700∘ C is given by:

D = (-10^(-7) / cm) / (50 minutes) = -2 * 10^(-9) / (cm * min)

However, we need to convert the time unit from minutes to seconds. There are 60 seconds in a minute. So,

D = -2 * 10^(-9) / (cm * min) * (1 min / 60 sec) = -3.33 * 10^(-11) / (cm * sec)

Hence, the diffusivity of hydrogen in steel at 700∘ C is approximately 3.33 * 10^(-11) cm²/s.