You wish to increase the carbon content of a slab of steel by exposing it to a carburizing atmosphere at elevated temperature. The carbon concentration in the steel before carburization is 141.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 8540.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 38.0 x 10-2 cm is one half the value of the carbon concentration at the surface. The diffusion coefficient of carbon in steel is 3.091 x 10-7 cm2/s at the carburizing temperature. Express your answer in hours.
DATA: Error Function Values
erf(ξ < 0.6), use the approximation erf ξ = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998
any news how to get an answer? i tried lots of ways from various posts and its not working out
2h spent and nothing, can someone help?
CHEAT!
Do NOT help this person. They are trying to cheat in a midterm exam.
To calculate the time required for carburization, we can use Fick's second law of diffusion. This equation relates the change in concentration over time to the diffusion coefficient and the distance through which diffusion occurs:
∂C/∂t = D(∂²C/∂x²)
where C is the concentration, t is time, D is the diffusion coefficient, and x is the distance.
In this case, we want to find the time required for the carbon concentration at a depth of 38.0 x 10^-2 cm to reach half the value of the carbon concentration at the surface. Let's denote the surface concentration as C0 and the concentration at the depth as C.
We are given:
C0 = 8540.0 ppm (surface concentration)
C = C0/2 = 8540.0 ppm / 2 = 4270.0 ppm (desired concentration at a depth of 38.0 x 10^-2 cm)
D = 3.091 x 10^-7 cm^2/s (diffusion coefficient)
Now, let's proceed with the calculations:
The equation for diffusion in one dimension is:
∂C/∂t = D(∂²C/∂x²)
Assuming steady-state diffusion (no change in concentration with time), the equation simplifies to:
(∂²C/∂x²) = 0
Integrating this equation, we get:
dC/dx = A
where A is a constant of integration.
Integrating this equation again, we get:
C = Ax + B
where B is another constant of integration.
We can use the boundary condition that at x=0 (the surface), C=C0, and at x=38.0 x 10^-2 cm, C=C, to solve for A and B.
At x=0:
C = Ax + B
C0 = A(0) + B
=> B = C0
At x=38.0 x 10^-2 cm:
C = Ax + B
C = A(38.0 x 10^-2) + C0
Now, we can substitute the given values to solve for A:
4270.0 ppm = A(38.0 x 10^-2) + 8540.0 ppm
Simplifying the equation:
4270.0 ppm - 8540.0 ppm = A(38.0 x 10^-2)
-4270.0 ppm = A(38.0 x 10^-2)
A = -4270.0 ppm / (38.0 x 10^-2)
A = -11237.0 ppm/cm
Now, we can rewrite the concentration equation as:
C = -11237.0 ppm/cm * x + 8540.0 ppm
To find the time required, we need to use the relationship between distance (x) and time (t):
ξ = x / (2√(Dt))
Using this relationship, we can solve for t:
t = x² / (4D)
Now, we can substitute the given values to find the time required:
t = (38.0 x 10^-2 cm)² / (4 * 3.091 x 10^-7 cm^2/s)
t = (1444 x 10^-4 cm²) / (1.2364 x 10^-6 cm²/s)
t ≈ 1169400 s
To express the answer in hours, we divide the time by the number of seconds in an hour:
t = 1169400 s / (3600 s/h)
t ≈ 324.28 hours
Therefore, the time required to carburize the steel so that the carbon concentration at a depth of 38.0 x 10^-2 cm is one half the value of the carbon concentration at the surface is approximately 324.28 hours.