Question

A thin plastic membrane separates hydrogen from air. The molar concentrations of hydrogen in the membrane at the inner and outer surfaces are determined to be \(0.045\) and \(0.002 \mathrm{kmol} / \mathrm{m}^{3}\), respectively. The binary diffusion coefficient of hydrogen in plastic at the operation temperature is \(5.3 \times\) \(10^{-10} \mathrm{~m}^{2} / \mathrm{s}\). Determine the mass flow rate of hydrogen by diffusion through the membrane under steady conditions if the thickness of the membrane is (a) \(2 \mathrm{~mm}\) and $(b) 0.5 \mathrm{~mm}$.

Short answer

Question: Calculate the mass flow rate of hydrogen through the membrane under steady conditions for both (a) a 2 mm thick membrane and (b) a 0.5 mm thick membrane. Answer: For (a) case with a 2 mm thick membrane, the mass flow rate of hydrogen is: \((\dot{m}_1)_{H} = J_{1}\times M_{H}\) For (b) case with a 0.5 mm thick membrane, the mass flow rate of hydrogen is: \((\dot{m}_2)_{H} = J_{2}\times M_{H}\) You will need to perform the calculations with the given values provided in the solution to find the mass flow rates for both cases.

Step-by-step solution

Write down the given values.

We are given the following values: - Inner surface molar concentration, \(C_{1} = 0.045\ \mathrm{kmol/m^3}\) - Outer surface molar concentration, \(C_{2} = 0.002\ \mathrm{kmol/m^3}\) - Binary diffusion coefficient of hydrogen in plastic, \(D_{AB} = 5.3\times10^{-10}\ \mathrm{m^2/s}\) - Molar mass of hydrogen, \(M_{H} = 2.016\ \mathrm{g/mol}\) For cases (a) and (b), we have different membrane thicknesses: - \((a): \delta x_{1} = 2\ \mathrm{mm}\) - \((b): \delta x_{2} = 0.5\ \mathrm{mm}\)

Calculate the molar flux for both cases.

Using Fick's first law of diffusion, we can calculate the molar flux for both cases: \(J_{1} = -D_{AB}\left(\frac{C_{1} - C_{2}}{\delta x_{1}}\right)\) for (a) \(J_{2} = -D_{AB}\left(\frac{C_{1} - C_{2}}{\delta x_{2}}\right)\) for (b) Note that we'll need to convert the membrane thicknesses into meters. - \((a): \delta x_{1} = 2\ \mathrm{mm} = 0.002\ \mathrm{m}\) - \((b): \delta x_{2} = 0.5\ \mathrm{mm} = 0.0005\ \mathrm{m}\)

Calculate the mass flow rate for both cases.

After we have calculated the molar flux for both cases, we can use the molar mass of hydrogen to convert it into mass flow rate: - Mass flow rate for case (a): \((\dot{m}_1)_{H} = J_{1}\times M_{H}\) - Mass flow rate for case (b): \((\dot{m}_2)_{H} = J_{2}\times M_{H}\) Performing the calculations with the given values will give us the mass flow rate of hydrogen through the membrane under steady conditions for both thicknesses.