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 244.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 7215.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 20.0 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.

Question

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 244.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 7215.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 20.0 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

Answer 1

Can someone please answer this question?

Answer 2

Please someone show the steps ! I am still not able to get this answer

Answer 3

Step 1:

ξ= (0.5 * carbon concentration at the surface of the steel - carbon concentration at the surface of the steel)/ (carbon concentration in the steel before carburization - carbon concentration at the surface of the steel)

(0.5* 7215.0 - 7215.0 )/ ( 244.5 - 7215.0 )= erf(ξ)

Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ

Step 3:

t= d^2/4*D*ξ^2

d=20.0 x 10-2 cm
D=3.091 x 10-7 cm2/s

Step 4:
t in sec -> hours
Conversion seconds to hours is total/(60x60)

Answer 4

How do you go from step 3 to step 4?

Answer 5

What is the value of erf?

Answer 6

To calculate the time required to carburize the steel, we can use Fick's second law of diffusion. The equation is as follows:

∂C/∂t = D(d²C/dx²)

Where:
- ∂C/∂t is the rate of change of carbon concentration with time.
- D is the diffusion coefficient of carbon in steel.
- ∂²C/∂x² is the second derivative of carbon concentration with respect to depth.

In this case, we want to determine the time it takes for the carbon concentration at a depth of 20.0 x 10^-2 cm to reach one half the value of the carbon concentration at the surface. Let's call this concentration C_half and the surface concentration C_surface.

∂C/∂t = D(d²C/dx²)
(dC/dt) / (D) = (1/C) (dC/dx)

Separating variables and integrating:

∫(1/C)dC = ∫(dX/D)
ln(C) = X/D + Constant
C = e^(X/D + Constant)
C = Ae^(X/D)

Since the initial carbon concentration is uniform through the thickness of the steel, we can use this information to solve for the constant A:

244.5 ppm = Ae^(0 + Constant)
A = 244.5 ppm

Now we can find the value of Constant:

7215.0 ppm = 244.5 ppm * e^(20.0 x 10^-2cm / D + Constant)
Constant = ln(7215.0 ppm / 244.5 ppm)

Finally, we can use this equation to find the time required for the carbon concentration at a depth of 20.0 x 10^-2 cm to be one half the value of the carbon concentration at the surface:

C_half = C_surface / 2
C_half = 7215.0 ppm / 2

7215.0 ppm / 2 = 244.5 ppm * e^(20.0 x 10^-2 cm / D + ln(7215.0 ppm / 244.5 ppm))

To solve for the time t, we need to rearrange the equation:

t = (20.0 x 10^-2 cm^2) / (D) * ln((7215.0 ppm / 244.5 ppm) / (7215.0 ppm / 2 * 244.5 ppm))

Substituting the value of the diffusion coefficient D = 3.091 x 10^-7 cm^2/s, we can calculate the time t.