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 100.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 9775.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 75.0 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 100.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 9775.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 75.0 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

CHEAT!

Do NOT help this person. They are trying to cheat in a midterm exam.

Answer 2

To calculate the time required for carburization, we can use Fick's second law of diffusion. The equation is given by:

C(x, t) = C0 + (Cs - C0) * erfc[(x / 2 * sqrt(D * t))]

Where:
- C(x, t) is the carbon concentration at a distance x from the surface after time t.
- C0 is the initial carbon concentration (100.0 ppm in this case).
- Cs is the carbon concentration at the surface (9775.0 ppm in this case).
- erfc is the complementary error function, defined as 1 - erf.
- D is the diffusion coefficient of carbon in steel (3.091 x 10^-7 cm^2/s in this case).
- x is the distance from the surface (75.0 x 10^-2 cm in this case).
- t is the time we want to calculate.

We want to find the time at which the carbon concentration at a depth of 75.0 x 10^-2 cm is half the value at the surface.
So, we can set up the equation:

C(75.0 x 10^-2 cm, t) = Cs / 2

Plugging in the values:

Cs / 2 = C0 + (Cs - C0) * erfc[(x / 2 * sqrt(D * t))]
4887.5 ppm = 100.0 ppm + (9775.0 ppm - 100.0 ppm) * erfc[(75.0 x 10^-2 cm / 2 * sqrt(3.091 x 10^-7 cm^2/s * t))]

Now, we can solve for t. Rearranging the equation:

erfc[(75.0 x 10^-2 cm / 2 * sqrt(3.091 x 10^-7 cm^2/s * t))] = (4887.5 ppm - 100.0 ppm) / (9775.0 ppm - 100.0 ppm)

erfc[(75.0 x 10^-2 cm / 2 * sqrt(3.091 x 10^-7 cm^2/s * t))] = 0.49876

Using the given error function values:

75.0 x 10^-2 cm / 2 * sqrt(3.091 x 10^-7 cm^2/s * t) = 0.6

Solving for t:

t = (0.6 / (75.0 x 10^-2 cm / 2 * sqrt(3.091 x 10^-7 cm^2/s)))^2

t = (0.6 / (0.375 cm * sqrt(3.091 x 10^-7 cm^2/s)))^2

t ≈ 314.95 seconds

Finally, to convert seconds to hours:

t = 314.95 seconds * (1 hour / 3600 seconds)

t ≈ 0.0875 hours

Therefore, the time required to carburize the steel so that the carbon concentration at a depth of 75.0 x 10^-2 cm is half the value at the surface is approximately 0.0875 hours.