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 325.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6990.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 87.0 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 325.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6990.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 87.0 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

To calculate the time required to carburize the steel, we can use Fick's second law of diffusion. This law relates the change in concentration of a substance over time to the diffusion coefficient and the distance of diffusion.

The equation for Fick's second law is:

∂C/∂t = D * (∂²C/∂x²)

where:
- ∂C/∂t is the change in concentration with respect to time
- D is the diffusion coefficient
- ∂²C/∂x² is the second derivative of the concentration with respect to distance

To solve this equation, we need to consider a few assumptions:
1. The carbon concentration is initially uniform through the thickness of the steel.
2. The concentration at the surface of the steel is 6990.0 ppm.
3. We want to find the time required for the carbon concentration at a depth of 87.0 x 10⁻² cm to be one half the value of the carbon concentration at the surface.

Let's set up the equation and solve it step by step.

Step 1: Calculate the concentration difference between the surface and the desired depth.
ΔC = (6990.0 ppm) - (6990.0 ppm / 2)
= 6990.0 ppm / 2
= 3495.0 ppm

Step 2: Convert the distance from cm to meters.
x = 87.0 x 10⁻² cm
= 0.87 m

Step 3: Substitute the given values into Fick's second law.
∂C/∂t = D * (∂²C/∂x²)
ΔC / t = D * (∂²C/∂x²)

Step 4: Rearrange the equation to solve for t.
t = ΔC / (D * (∂²C/∂x²))

Step 5: Calculate (∂²C/∂x²), the second derivative of the carbon concentration with respect to distance.
∂²C/∂x² = (∂/∂x) * (∂C/∂x)

As the carbon concentration initially is uniform, (∂C/∂x) will be zero. Therefore, (∂²C/∂x²) will also be zero.

Step 6: Substitute the values into the equation.
t = ΔC / (D * (∂²C/∂x²))
t = ΔC / (D * 0)

As (∂²C/∂x²) is zero, the time required to carburize the steel is infinite.

Since the carbon concentration does not change with time at a depth of 87.0 x 10⁻² cm, it will never reach one-half the value of the carbon concentration at the surface under the given conditions.