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 359.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6695.0 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
See the Related Questions below.
answer, pls
To calculate the time required for carburization, we can use Fick's second law of diffusion, which describes the diffusion of carbon in steel. The equation is as follows:
δC/δt = (D * δ²C/δx²)
Where:
δC/δt is the rate of change of carbon concentration with time,
D is the diffusion coefficient of carbon in steel, and
δ²C/δx² is the second derivative of carbon concentration with respect to 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 be half the value at the surface. Let's denote the surface carbon concentration as C_s and the carbon concentration at a depth of 38.0 x 10^-2 cm as C_d.
Given:
C_s = 6695.0 ppm (carbon concentration at the surface)
C_d = 1/2 * C_s (carbon concentration at a depth of 38.0 x 10^-2 cm)
D = 3.091 x 10^-7 cm²/s (diffusion coefficient of carbon in steel)
To solve this problem, we need to find the diffusion time (t) required for the carbon to diffuse from the surface to the desired depth. We can use the error function (erf) to solve it.
The general formula for the concentration profile in a semi-infinite slab at steady-state diffusion is given by:
C(x, t) = C_s * [1 - erf(x / (2 * sqrt(D * t)))]
Where:
C(x, t) is the carbon concentration at a given depth (x) and time (t).
Since we want C(x, t) at x = 38.0 x 10^-2 cm to be half C_s, we can substitute the values into the equation and solve for t:
C(x, t) = C_s * [1 - erf(x / (2 * sqrt(D * t)))]
C_d = C_s * [1 - erf(38.0 x 10^-2 / (2 * sqrt(D * t)))]
Substituting the given values:
1/2 * C_s = C_s * [1 - erf(38.0 x 10^-2 / (2 * sqrt(D * t)))]
Now, we need to rearrange the equation to solve for t. Divide both sides by C_s:
1/2 = 1 - erf(38.0 x 10^-2 / (2 * sqrt(D * t)))
Rearrange the equation to isolate the erf term:
erf(38.0 x 10^-2 / (2 * sqrt(D * t))) = 1/2
Then, calculate the argument of the error function:
38.0 x 10^-2 / (2 * sqrt(D * t)) = 0.5
Rearrange the equation to solve for t:
sqrt(D * t) = 38.0 x 10^-2 / (2 * 0.5)
sqrt(D * t) = 38.0 x 10^-2 / 1
Square both sides to eliminate the square root:
D * t = (38.0 x 10^-2 / 1)^2
Now, solve for t:
t = (38.0 x 10^-2 / 1)^2 / D
Plug in the values of D:
t = (38.0 x 10^-2 / 1)^2 / (3.091 x 10^-7)
t ≈ 4.722 x 10^7 s
To express the answer in hours, divide by 3600 (seconds in an hour):
t ≈ 4.722 x 10^7 s / 3600 ≈ 13116.67 hours
Therefore, the time required for carburization to achieve a carbon concentration at a depth of 38.0 x 10^-2 cm that is half the value at the surface is approximately 13116.67 hours.