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 390.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 8180.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 96.5 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

I got the right answer :)

how did you do it? please help to find value of erf and how to use approximation for values of erf(1) and erf(2)??

Step 1:

(0.5*8180.0 - 8180.0)/ (390.5 - 8180.0)= erf(ξ)

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

Step 3:

t= d^2/4*D*ξ^2

d=96.5 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)

To calculate the time required to carburize the steel, we can use Fick's second law of diffusion:

∂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.
∂²C/∂x² is the second derivative of carbon concentration with respect to depth.

Based on the given information, the carbon concentration at the surface (x = 0) is 8180.0 ppm, and at a depth of 96.5 x 10^(-2) cm (x = 96.5 x 10^(-2) cm), it should be half of the surface concentration, i.e., 8180.0 ppm / 2 = 4090.0 ppm.

To solve Fick's second law, we need to assume that the carbon concentration in the steel is initially uniform and use the boundary conditions:

At t = 0, C = 390.5 ppm, at all x values.
At x = 0, C = 8180.0 ppm.
At x = 96.5 x 10^(-2) cm, C = 4090.0 ppm.

We can rewrite the equation as:

∂C/∂t = D * ∂²C/∂x²

∂C/∂t = D * (1/x) * ∂(x * ∂C/∂x)/∂x

Assuming C(x, t) = f(x) * g(t), we can separate variables:

(∂f/∂x) * (1/f) = (1/D) * (∂g/∂t) * (∂t/x)

Integrating both sides gives:

ln(f) = (1/D) * g * ln(x) + constant

Taking exponential of both sides gives:

f = A * x^(g/D)

Since the carbon concentration should remain finite as x approaches 0, we can conclude that g/D should be positive or zero.

Now, to find the time required, we need to determine the values of A, g, and D.

We need to find A from the initial condition C = 390.5 ppm at t = 0.

C(0, 0) = A * 0^(g/D) = 390.5 ppm

This implies A = 390.5 ppm.

Next, we can find g/D from the boundary condition C = 4090.0 ppm at x = 96.5 x 10^(-2) cm.

C(96.5 x 10^(-2) cm, t) = 390.5 ppm * (96.5 x 10^(-2) cm)^(g/D) = 4090.0 ppm

Simplifying:

(96.5 x 10^(-2) cm)^(g/D) = 4090.0 ppm / 390.5 ppm = 10.476

Taking the natural logarithm of both sides:

(g/D) * ln(96.5 x 10^(-2)) = ln(10.476)

(g/D) = ln(10.476) / ln(96.5 x 10^(-2))

Now, let's calculate g/D:

(g/D) ≈ 0.690 / (-2.588)

(g/D) ≈ -0.267

Since g/D should be positive or zero, we can conclude that g/D = 0.

Therefore, f = A * x^(g/D) simplifies to f = A.

It means that the carbon concentration is independent of depth, and the carburization process happens uniformly throughout the steel.

Finally, we can find the time required using the inverse of the diffusion coefficient at the given depth:

t = (x^2) / (2 * D)

t = (96.5 x 10^(-2) cm)^2 / (2 * (3.091 x 10^(-7) cm^2/s))

t ≈ 1574.78 cm^2 / (6.182 x 10^(-7) cm^2/s)

t ≈ 2.548 x 10^9 s

To convert seconds to hours, divide by 3600 (since there are 3600 seconds in an hour):

t ≈ 7.078 x 10^5 hours

Therefore, the time required to carburize the steel is approximately 7.078 x 10^5 hours.