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.

To determine the minimum thickness of the iron foil, we need to use Fick's Law of Diffusion, which states that the flux of atomic hydrogen through a material is directly proportional to the concentration gradient and the diffusion coefficient, and inversely proportional to the thickness of the material.

The formula for Fick's Law of Diffusion is:

J = -D * (dC/dx)

Where:
J is the flux of atomic hydrogen (mol/cm2s)
D is the diffusion coefficient of atomic hydrogen in iron at 700°C (3.091 x 10-4 cm2/s)
dC/dx is the concentration gradient of atomic hydrogen (atoms/cm3cm)
x is the thickness of the iron foil (cm)

In this case, we want to find the minimum thickness, so we need to set the flux equal to the maximum acceptable leak rate:

J = 10-3 mol/cm2h-1 (Since the leak rate is given in mol/cm2h-1)

Let's assume the thickness of the iron foil is h cm.

We know that the concentration gradient (dC/dx) is given by the difference in concentration on both sides of the membrane divided by the thickness:

dC/dx = (1.5 x 1019 atoms/cm3 - 0 atoms/cm3) / h

Let's substitute the given values into Fick's Law equation:

10-3 mol/cm2h-1 = - (3.091 x 10-4 cm2/s) * [(1.5 x 1019 atoms/cm3 - 0 atoms/cm3) / h]

Simplifying the equation:

10-3 = - (3.091 x 10-4) * (1.5 x 1019) / h

-10-3 = - (3.091 x 10-4) * (1.5 x 1019) / h

Cross-multiplying:

h = - (3.091 x 10-4) * (1.5 x 1019) / (-10-3)

h = (3.091 x 10-4) * (1.5 x 1019) / (10-3)

Calculating the value:

h ≈ 4.6365 x 10-2 cm

Therefore, the minimum thickness of the iron foil that will meet the given requirements is approximately 4.6365 x 10-2 cm.