A specimen of LaNi5 containing hydrogen is placed in a vacuum furnace. After 1 hour, at what depth from the surface of the specimen has the concentration of hydrogen reached 1/3 the initial concentration?
The diffusion coefficient of hydrogen in the alloy has a value of 3.091*10-6 cm2/s. Assume that the initial concentration of hydrogen is uniform throughout the specimen and that the concentration of hydrogen is maintained at zero in the vacuum furnace.
DATA: Error Function Values (given without regard as to whether you need these data to solve the problem) for values of z less than 0.6, use the aproximation erf(z) = z; erf(1.0) = 0.843; erf(2.0) = 0.998
Express your answer in units of cm:
0.0703
Dear Cheunology, can you please make me understand on how to solve this problem. in detail????
To solve this problem, we need to use Fick's second law of diffusion, which describes the diffusion of a substance in a solid material over time. The equation is as follows:
c(x, t) = c0 * erfc((x - x0) / (2 * sqrt(D * t)))
Where:
- c(x, t) is the concentration of hydrogen at depth x from the surface after time t
- c0 is the initial concentration of hydrogen
- x0 is the initial depth from the surface (usually 0 in this case)
- D is the diffusion coefficient of hydrogen in the alloy
- t is the time
We need to find the depth (x) at which the concentration of hydrogen is 1/3 (or 1/3*c0) of the initial concentration after 1 hour (3600 seconds).
Let's plug in the given values:
c(x, 3600) = c0 * erfc((x - x0) / (2 * sqrt(D * t)))
1/3 * c0 = c0 * erfc((x - 0) / (2 * sqrt(3.091*10^-6 * 3600)))
Now, let's solve for x:
1/3 = erfc(x / (2 * sqrt(3.091*10^-6 * 3600)))
To solve for x, we will need to use the inverse error function (erfcinv). However, the given error function values are only provided for z less than 0.6. So, we need to express the equation in terms of z. Let's rearrange the equation:
1/3 = erfc(x / (2 * sqrt(3.091*10^-6 * 3600)))
erfcinv(1/3) = x / (2 * sqrt(3.091*10^-6 * 3600))
Now we can substitute the given approximation for erfc(1/3) and solve for x:
erfcinv(1/3) = x / (2 * sqrt(3.091*10^-6 * 3600))
erfcinv(1/3) = x / (6.2481 * 10^-3)
Multiply both sides by 6.2481 * 10^-3 to isolate x:
x = erfcinv(1/3) * 6.2481 * 10^-3
Now, let's calculate x using the given approximation for erfcinv(1/3):
x = 0.231 * 6.2481 * 10^-3
x = 0.0014488 cm
Therefore, at a depth of approximately 0.0014488 cm from the surface, the concentration of hydrogen will be 1/3 of the initial concentration after 1 hour.