Chromium is diffused into steel at 980o C to increase its corrosion resistance. If during the diffusion the surface concentration of chromium is maintained constant, how long will it take to achieve 1.8% of the surface concentration at a depth of 0.002 cm below the steel surface? Express your answer in seconds. The diffusivity has a pre-exponential term equal to 0.54 cm2/s and an activation energy of 286 kJ/mole.
To calculate the time it takes to achieve a certain concentration at a specific depth during diffusion, we can use Fick's second law of diffusion. The equation is as follows:
C = Co * (1 - erf((x)/(2 * sqrt(D * t))))
Where:
C = concentration at the depth x
Co = surface concentration
erf = the error function
x = depth at which concentration is measured
D = diffusion coefficient
t = time
In this case, we are given:
C = 0.018 (1.8% of the surface concentration)
Co = Surface concentration (constant)
x = 0.002 cm (depth)
We need to find 't', the time it takes to achieve the 1.8% concentration at a depth of 0.002 cm.
First, we need to calculate the diffusion coefficient, 'D', using the provided information:
D = D0 * exp((-Q)/(R * T))
Where:
D0 = pre-exponential term
Q = activation energy
R = gas constant (8.314 J/(mol*K))
T = temperature in Kelvin (980°C = 1253.15 K)
Given:
D0 = 0.54 cm²/s
Q = 286 kJ/mol
Converting Q to J/mol:
Q = 286 kJ/mol * 1000 J/kJ = 286,000 J/mol
Now we can calculate D:
D = 0.54 cm²/s * exp((-286,000 J/mol) / (8.314 J/(mol*K) * 1253.15 K))
Let's calculate D:
D = 0.54 cm²/s * exp((-286,000 J/mol) / (10,358.37 J/(mol*K)))
D = 0.54 cm²/s * exp(-27.63)
D ≈ 0.076 cm²/s
Now that we have 'D', we can rearrange the Fick's second law equation to solve for 't':
t = ((x * x) / (4 * D)) * (-ln((C / Co) - 1))
t = ((0.002 cm)²) / (4 * (0.076 cm²/s))) * (-ln((0.018 / Co) - 1))
However, we lack information about the surface concentration 'Co', so we cannot provide a numerical answer.