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 263.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6520.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 29.0 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 263.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6520.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 29.0 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

This is a question from the second midterm of 3.091x by edx. Stop cheating, it's silly and pointless.

Answer 2

To calculate the time required to carburize the steel, we need to use Fick's second law of diffusion. This law relates the diffusion flux, the diffusion coefficient, and the change in concentration with respect to time.

The general form of 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.

In this case, we are given the initial and surface carbon concentrations, and we want to find the time required for the concentration at a specific depth to reach half the surface concentration.

Let's define some variables:
- C0: Initial carbon concentration (263.5 ppm)
- Cs: Surface carbon concentration (6520.0 ppm)
- C: Carbon concentration at depth (to be determined)
- x: Distance from the surface to the depth (29.0 x 10^-2 cm)
- t: Time required for carburization (to be determined)
- D: Diffusion coefficient of carbon in steel (3.091 x 10^-7 cm²/s)

We can use the given error function values to simplify the solution:

erf(ξ < 0.6), use the approximation erf(ξ) = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998

Now, we can solve for the time t.

First, let's calculate the difference in concentration (ΔC) between the surface and the desired depth:

ΔC = Cs - C

Next, calculate the second derivative of concentration (∂²C/∂x²):

∂²C/∂x² = (ΔC / (2 * D * t)) * erf(x / (2 * √(D * t)))

Plugging in the values, we have:

(ΔC / (2 * D * t)) * erf(x / (2 * √(D * t))) = 1/2

Substituting the given error function approximation (erf(ξ) = ξ) for the small value of ξ:

(ΔC / (2 * D * t)) * (x / (2 * √(D * t))) ≈ 1/2

Rearranging the equation:

(ΔC * x) / (4 * D * t^(3/2)) ≈ 1/2

Now, solving for t:

t = ((ΔC * x) / (8 * D))^(2/3)

Substituting the given values:

t = ((ΔC * 29.0 x 10^-2 cm) / (8 * 3.091 x 10^-7 cm²/s))^(2/3)

Calculating ΔC:

ΔC = Cs - (1/2) * Cs = (1/2) * Cs

Substituting ΔC and the other given values, we can calculate t:

t = (((1/2) * 6520.0 ppm * 29.0 x 10^-2 cm) / (8 * 3.091 x 10^-7 cm²/s))^(2/3)

Calculating this expression will give us the time required for carburization in seconds. To convert it to hours, we can divide it by 3600 (the number of seconds in an hour):

t_hours = (1 / 3600) * (((1/2) * 6520.0 ppm * 29.0 x 10^-2 cm) / (8 * 3.091 x 10^-7 cm²/s))^(2/3)

Finally, calculate t_hours to get the answer in hours.