The permeability coefficient of a type of small gas molecule in a polymer is dependent on absolute temperature according to the following equation: $$ P_{M}=P_{M_{0}} \exp \left(-\frac{Q_{p}}{R T}\right) $$ where \(P_{M_{0}}\) and \(Q_{p}\) are constants for a given gas-polymer pair. Consider the diffusion of hydrogen through a poly(dimethyl siloxane) (PDMSO) sheet \(20 \mathrm{~mm}\) thick. The hydrogen pressures at the two faces are \(10 \mathrm{kPa}\) and \(1 \mathrm{kPa}\), which are maintained constant. Compute the diffusion flux [in \(\left.\left(\mathrm{cm}^{3} \mathrm{STP}\right) / \mathrm{cm}^{2} \cdot \mathrm{s}\right]\) at \(350 \mathrm{~K}\). For this diffusion system $$ \begin{aligned} P_{M_{0}} &=1.45 \times 10^{-8}\left(\mathrm{~cm}^{3} \mathrm{STP}\right)(\mathrm{cm}) / \mathrm{cm}^{2} \cdot \mathrm{s} \cdot \mathrm{Pa} \\ Q_{p} &=13.7 \mathrm{~kJ} / \mathrm{mol} \end{aligned} $$ Also, assume a condition of steady state diffusion.

Step by step solution

01

Compute permeability coefficient

Using the given equation for permeability coefficient: $$ P_M = P_{M_0} \exp \left(-\frac{Q_p}{RT}\right) $$ Where, \(P_{M_0} = 9.0 \times 10^{-5} (\text{cm}^3\text{STP})(\text{cm}) / \text{cm}^2 \cdot \text{s} \cdot \text{Pa}\), \(Q_p = 42,300 \text{ J/mol}\), \(R = 8.314 \text{ J/mol}·\text{K}\) (universal gas constant), and \(T = 350 \text{ K}\) (temperature). Now, calculate \(P_M\): $$ P_M = 9.0 \times 10^{-5} \exp \left(-\frac{42,300}{(8.314)(350)}\right) $$

02

Calculate diffusion flux

We are given that the water vapor pressures at the two faces are 20 kPa and 1 kPa, and the polystyrene sheet is 30 mm thick. Assuming steady-state diffusion, we can calculate the diffusion flux (\(J\)) using Fick's first law: $$ J = - P_M \frac{ΔP}{l} $$ Where \(ΔP = P_1 - P_2 = (20 - 1) \times 10^3 \, \text{Pa}\) (pressure difference between the two faces), \(l = 30 \times 10^{-1} \, \text{cm}\) (thickness of the sheet). Now, substitute the values and calculate \(J\): $$ J = - P_M \frac{(20 - 1) \times 10^3}{30 \times 10^{-1}} $$ Use the value of \(P_M\) obtained in step 1 to find the diffusion flux \(J\).

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