Hydrogen Storage in Alloys - The Future?
Certain alloys such as LaNi5 can store hydrogen at room temperature. A plate of LaNi5 containing no hydrogen is placed in a chamber filled with pure hydrogen and maintained at a constant pressure. At what depth from the surface will the concentration of hydrogen be half the surface concentration after 1 hour? (Express your answer in centimeters). Assume the diffusivity of hydrogen in the alloy is 3.091x10-6 cm2/sec. Use the approximation erf(x) = x for x with values between 0 and 0.6, if appropriate. Use the Erf Table on the Course Info page for values greater than 0.6.
To determine the depth at which the concentration of hydrogen is half the surface concentration after 1 hour, we can use Fick's Law of diffusion. Fick's Law states that the rate of diffusion of a substance is proportional to the concentration gradient.
The equation for Fick's Law in one dimension is:
J = -D * (dc/dx)
Where J is the flux or rate of diffusion, D is the diffusivity of hydrogen in the alloy, dc/dx is the concentration gradient along the depth (x-axis).
Let's assume that the surface concentration of hydrogen is C0, and the concentration at depth x is C(x). We can define the concentration in terms of the surface concentration as follows:
C(x) = C0 * erf(x / (2 * sqrt(D * t)))
Where t is the time of diffusion, and erf is the error function.
We want to find the depth 'x' where C(x) is equal to half the surface concentration C0 after 1 hour.
C(x) = 0.5 * C0
0.5 * C0 = C0 * erf(x / (2 * sqrt(D * t)))
0.5 = erf(x / (2 * sqrt(D * t)))
We can rearrange the equation to solve for x:
x / (2 * sqrt(D * t)) = erf^(-1)(0.5)
Now, using the approximation erf(x) = x for x with values between 0 and 0.6, we can further simplify the equation:
x / (2 * sqrt(D * t)) = 0.5
x = 0.5 * 2 * sqrt(D * t)
Now we can substitute the given values:
D = 3.091x10^(-6) cm^2/sec
t = 1 hour = 3600 seconds
x = 0.5 * 2 * sqrt(3.091x10^(-6) * 3600)
After calculating the value, we will get the depth 'x' in centimeters.