You are hardening a tool made of BCC iron by increasing the carbon concentration

Question

You are hardening a tool made of BCC iron by increasing the carbon concentration at its surface by heating the tool in the presence of a carbon-rich gas (for example, methane). A heat treatment at 600 degree C for 50 minutes results in a carbon concentration of 0.75 wt% at a position 0.5 mm below the surface. How long would it take to obtain a carbon concentration of 0.75 wt% at a position 0.5 mm below the surface if the heat treatment were conducted at 900 degree C? Express your answer using three significant figures.

Explanation / Answer

The following relation holds for the diffusion processes:

x2 / (D*t) = constant

Here, x = depth

D is diffusion co-efficient

In our case, x does't change so x is also a constant. The relation can thus be expressed for this case as:

D*t = constant

Thus,

D1 * t1 = D2 * t2 ------ (I)

D is related to temperature T by the relation:

D = D0 * exp(-Ea / RT)

Here Ea is the activation energy of the diffusion

D1 = D0 * exp(-Ea / RT1)

D2 = D0 * exp(-Ea / RT2)

D2/D1 = exp( [Ea / RT1 ] - [Ea / RT2] ) = t1/t2

Put the values:

Ea = 80,000 J/mol

R = 8.31 J/(mol.K)

T1 = 873 K

T2 = 1173 K

t1 = 50 minutes

Solving we get:

t2 = 2.98 minutes

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