A membrane is to be manufactured to the following specifications. The leak rate of hydrogen is not to exceed 10-3 mol-cm-2h-1 at 700 C when the concentration of hydrogen is maintained at 1.5 x 1019 atoms/cm3 on one side of the membrane and effectively zero on the other side. What is the minimum thickness of iron foil that will meet these requirements (express your results in cm)? The diffusion coefficient of atomic hydrogen in iron at 700 C is 3.091 x 10-4 cm2s-1. Assume steady state.

Question

A membrane is to be manufactured to the following specifications. The leak rate of hydrogen is not to exceed 10-3 mol-cm-2h-1 at 700 C when the concentration of hydrogen is maintained at 1.5 x 1019 atoms/cm3 on one side of the membrane and effectively zero on the other side. What is the minimum thickness of iron foil that will meet these requirements (express your results in cm)? The diffusion coefficient of atomic hydrogen in iron at 700 C is 3.091 x 10-4 cm2s-1. Assume steady state.

Answer 1

To determine the minimum thickness of iron foil that will meet the given specifications, we can use Fick's law of diffusion. Fick's law states that the diffusion rate is proportional to the concentration gradient and the diffusion coefficient.

We can start by calculating the diffusion rate using the formula:

Diffusion Rate = -D * A * ΔC / Δx

Where:
- D is the diffusion coefficient of atomic hydrogen in iron
- A is the area of the membrane (1 cm2)
- ΔC is the difference in concentration of hydrogen on both sides of the membrane
- Δx is the thickness of the membrane (what we are trying to find)

From the given specifications, we have:
- D = 3.091 x 10-4 cm2s-1
- A = 1 cm2
- ΔC = 1.5 x 1019 atoms/cm3

We need to convert the concentration to mol/cm3 for consistency. Since 1 mol of hydrogen has 6.022 x 1023 atoms, we can convert the concentration as follows:

ΔC = 1.5 x 1019 atoms/cm3 * (1 mol / 6.022 x 1023 atoms) = 2.49 x 10-5 mol/cm3

Now, we can rearrange the formula to solve for the thickness Δx:

Δx = -D * A * ΔC / Diffusion Rate

As the leak rate should not exceed 10-3 mol-cm-2h-1, we first need to convert it to mol/cm3s, considering that there are 3600 seconds in an hour:

Diffusion Rate = 10-3 mol-cm-2h-1 * (1 cm3 / 1 cm2) * (1 h / 3600 s) = 2.78 x 10-8 mol/cm3s

Now we can substitute the values into the formula:

Δx = -D * A * ΔC / Diffusion Rate
= -(3.091 x 10-4 cm2s-1) * (1 cm2) * (2.49 x 10-5 mol/cm3) / (2.78 x 10-8 mol/cm3s)

Evaluating this expression will give us the minimum thickness of the iron foil required to meet the given specifications.