Question

At 873 K, hydrogen diffuses under steady state condition through a 5 mm thick palladium sheet with a cross-sectional area of 0.3 m\(^2\). The concentrations of hydrogen at high and low pressure ends of the sheet are 3 kg/m\(^3\) and 0.5 kg/m\(^3\), respectively. The amount of hydrogen (in kg per day) passing through the sheet is (rounded off to two decimal places) ............
Given: At 873 K, diffusivity of hydrogen $= 1.8 \times 10^{-8}\ {m}^2 \cdot {s}^{-1}$

Hint:

When using Fick's law for diffusion, ensure that the units are consistent, and always multiply by the time period (seconds in a day) when calculating the total amount of substance passing through.

Answer 1

We can calculate the amount of hydrogen passing through the palladium sheet using Fick's law of diffusion, which is given by:
\[ J = \frac{D}{L} (C_1 - C_2) \] where:
- \( J \) is the mass flux of hydrogen (kg/s),
- \( D = 1.8 \times 10^{-8} \, {m}^2/{s} \) is the diffusivity of hydrogen,
- \( L = 5 \times 10^{-3} \, {m} \) is the thickness of the sheet,
- \( C_1 = 3 \, {kg/m}^3 \) and \( C_2 = 0.5 \, {kg/m}^3 \) are the concentrations of hydrogen at the high and low pressure ends, respectively.
Substituting the values into Fick's law:
\[ J = \frac{1.8 \times 10^{-8}}{5 \times 10^{-3}} \times (3 - 0.5) = 7.2 \times 10^{-6} \times 2.5 = 1.8 \times 10^{-5} \, {kg/s} \] Now, to find the amount of hydrogen passing through the sheet per day, we multiply by the number of seconds in a day:
\[ {Amount of hydrogen per day} = 1.8 \times 10^{-5} \times 86400 = 0.19 \, {kg/day}. \] Thus, the amount of hydrogen passing through the sheet is \( 0.19 \, {kg/day} \).