To increase corrosion resistance, chromium is diffused into steel at 980∘ C. If the surface concentration of Cr is maintained constant at 100%, how long will it take (in days) to achieve a Cr concentration of 1.8% at a depth of 0.002 cm below the steel surface? (D0=0.54 cm2/s, EA=286 kJ/mole)
To solve this problem, we can use Fick's Second Law of Diffusion. Fick's Second Law defines the relationship between diffusion, time, and concentration gradient. The equation is given by:
∂C/∂t = D * (∂^2C/∂x^2)
Where:
- ∂C/∂t is the rate of change of concentration with respect to time
- D is the diffusion coefficient
- ∂^2C/∂x^2 is the second derivative of concentration with respect to distance
To find the time it takes to achieve a Cr concentration of 1.8% at a depth of 0.002 cm below the surface, we need to determine the necessary time for the concentration to change from 100% to 1.8%. Since the surface concentration of Cr is maintained constant, we can assume that the concentration gradient (∂^2C/∂x^2) is negative and equal to -1.8%/(0.002 cm).
Now, let's substitute the given values into Fick's Second Law:
∂C/∂t = D * (-1.8% / (0.002 cm)^2)
We also need to convert the diffusion coefficient from cm^2/s to m^2/s, so D = 0.54 cm^2/s * (1 m / 100 cm)^2 = 0.000054 m^2/s.
So, the equation becomes:
∂C/∂t = 0.000054 m^2/s * (-1.8% / (0.002 cm)^2)
Now, we can solve for ∂C/∂t:
∂C/∂t = -4,050 m^2/s * (-1)
Since the time needed to achieve a concentration of 1.8% is unknown, we can integrate both sides of the equation with respect to time and solve for t:
∫(∂C/∂t) dt = ∫(-4,050 m^2/s) dt
∫(∂C/∂t) dt = -4,050 * ∫dt
C = -4,050t + constant
To solve for the constant, we can use the initial condition that at t=0, the concentration C = 100%. So:
100% = -4,050 * 0 + constant
constant = 100%
Therefore, the equation becomes:
C = -4,050t + 100%
Now, we can solve for t when C = 1.8%:
1.8% = -4,050t + 100%
-98.2% = -4,050t
t = (-98.2%)/(-4,050)
t ≈ 0.0242
Since the diffusion coefficient was given in terms of cm^2/s, we can convert it to days by multiplying by (1 day / 86,400 s):
t ≈ 0.0242 * (1 day / 86,400 s)
t ≈ 2.8 * 10^(-7) days
Therefore, it will take approximately 2.8 * 10^(-7) days to achieve a Cr concentration of 1.8% at a depth of 0.002 cm below the steel surface.