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 176.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6580.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 9.5 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

To calculate the time required for carburization, we can use Fick's second law of diffusion. This law relates the change in concentration of a diffusing species with time and is given by:

∂C/∂t = D * (∂²C/∂x²)

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

In this case, we need to find the time required for the carbon concentration at a depth of 9.5 x 10^-2 cm to reach one half the value of the carbon concentration at the surface. Let's denote this desired concentration as C_desired.

Given:
- Carbon concentration at the surface, C_surface = 6580.0 ppm
- Carbon concentration at the initial uniform concentration, C_initial = 176.0 ppm
- Diffusion coefficient of carbon in steel, D = 3.091 x 10^-7 cm^2/s

We need to find the time required, t.

Step 1: Calculate the concentration gradient (∂C/∂x):
The concentration gradient (∂C/∂x) can be calculated using the formula:

(∂C/∂x) = (C_surface - C_initial) / thickness

Given that the thickness is not provided in the question, you need to determine it from the given information or consider a hypothetical thickness to proceed with the calculations.

Step 2: Calculate the desired concentration, C_desired:
C_desired = C_surface / 2

Step 3: Calculate the distance from the surface to the desired depth, x:
x = 9.5 x 10^-2 cm

Step 4: Calculate the second derivative (∂²C/∂x²):
(∂²C/∂x²) = (C_desired - C_initial) / (x^2)

Step 5: Plug all the values into Fick's second law of diffusion and solve for t:
(D * (∂²C/∂x²)) = (∂C/∂t)

(D * (∂²C/∂x²)) = (∂C/∂t)

(D * (C_desired - C_initial) / (x^2)) = (∂C/∂t)

(D * (C_surface/2 - C_initial) / (x^2)) = (∂C/∂t)

(D * (6580.0 ppm/2 - 176.0 ppm) / (x^2)) = (∂C/∂t)

Finally, solve for t by integrating (∂C/∂t) with respect to time, and then calculate the time required.

Note:
- Make sure to convert the units of the diffusion coefficient and depth to be consistent (e.g., both in cm or both in meters).
- The value of C_surface and C_initial should be converted to fractional values (0.6580 and 0.00176, respectively) for accurate calculations.

Unfortunately, without the thickness of the slab provided in the question, we cannot calculate the time required for carburization accurately.