Helium gas is stored at \(293 \mathrm{~K}\) in a 3 -m-outer-diameter spherical container made of 5 -cm-thick Pyrex. The molar concentration of helium in the Pyrex is \(0.00073 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner surface and negligible at the outer surface. Determine the mass flow rate of helium by diffusion through the Pyrex container.

To find the mass flow rate through the Pyrex container, we need to determine the area through which the helium diffuses. We are given that the outer diameter of the container is 3 meters, and the thickness of the Pyrex is 5 centimeters, or 0.05 meters. The inner diameter can be found by subtracting twice the thickness of the Pyrex (0.1 m) from the outer diameter (3 m). Calculate the inner surface area by using the formula for the surface area of a sphere: $$ A = 4 \pi r^2 $$ where \(r\) is the radius of the sphere. Calculate the inner surface area using the inner diameter. .inner_radius:$$ r_{inner} = \frac{3 - 0.1}{2} = 1.45 \mathrm{m} $$ .inner_surface_area:$$ A_{inner} = 4 \pi (1.45)^{2} = 26.49 \mathrm{m}^{2} $$ Now, calculate the outer surface area using the outer diameter. .outer_radius:$$ r_{outer} = \frac{3}{2} = 1.5 \mathrm{m} $$ .outer_surface_area:$$ A_{outer} = 4 \pi (1.5)^{2} = 28.27 \mathrm{m}^{2} $$

Since the concentration of helium varies from the inner surface to the outer surface of the Pyrex, we need to find the average concentration across the 5 cm thickness. Given that the concentration is 0.00073 kmol/m³ on the inner surface and negligible on the outer surface, we can assume a linear distribution and compute the average concentration as: $$ C_{avg} = \frac{C_{inner} + C_{outer}}{2} = \frac{0.00073 + 0}{2} = 0.000365 \mathrm{kmol/ m^{3}} $$

Fick's first law relates the mass flow rate, diffusion area, and diffusion coefficient of a gas. We will need to find the diffusion coefficient for helium in Pyrex. The diffusion coefficient can be found using the following empirical formula, known as the Chapman-Enskog equation: $$ D_{AB} = \frac{1.86 \times 10^{-3} T^{1.5}}{P \sigma^{2} \Omega^{*}_{AB}} $$ where \(D_{AB}\) is the diffusion coefficient (m²/s), \(\sigma\) is the collision diameter of the molecule (m), \(T\) is the temperature (K), \(P\) is the pressure (Pa), and \(\Omega_{AB}^{*}\) is the collision integral, which depends on the reduced temperature. For helium in Pyrex, the typical values for these parameters are: - Collision diameter: \(\sigma = 2.629 \times 10^{-10} m\) - Collision integral: \(\Omega^{*}_{AB} = 0.742\) Note that the pressure is not given, and the Chapman-Enskog equation applies for when diffusion is independent of pressure (dilute gas approximation). The calculation of the diffusion coefficient for helium in Pyrex at 293 K will result in an approximate value. Using these values, we can calculate the diffusion coefficient: $$ D_{AB} = \frac{1.86 \times 10^{-3} (293)^{1.5}}{(2.629 \times 10^{-10})^{2} \times 0.742} = 3.34 \times 10^{-6} \mathrm{m^2/s} $$

Now, we can use Fick's first law to find the mass flow rate of helium through the Pyrex. Fick's first law is given by: $$ J_{A} = -D_{AB} \frac{dC_{A}}{dx} $$ Where \(J_{A}\) is the mass flow rate of component A (helium) (kmol m²/s), \(D_{AB}\) is the diffusion coefficient (m²/s), and \(\frac{dC_{A}}{dx}\) is the concentration gradient of component A (kmol/m³). We can rewrite Fick's first law as: $$ J_{A} = -D_{AB} \frac{C_{inner} - C_{outer}}{\delta} $$ Where \(\delta\) is the thickness of the Pyrex (m). Now we can insert the known values and calculate the mass flow rate of helium: .mass_flow_rate_kmol:$$ J_{A} = -(3.34 \times 10^{-6} \mathrm{m^2/s}) \frac{0.00073 - 0 \mathrm{kmol/m^3}}{0.05 \mathrm{m}} = -4.91 \times 10^{-8} \mathrm{kmol \ m^{-2} \ s^{-1}} $$ To determine the mass flow rate, we can multiply the molar flow rate by the molar mass of helium (4 g/mol or 0.004 kg/mol). .mass_flow_rate_kg:$$ m_{He} = (4.91 \times 10^{-8} \mathrm{kmol \ m^{-2} \ s^{-1}}) \times (0.004 \mathrm{kg/mol}) = 1.96 \times 10^{-10} \mathrm{kg \ m^{-2} \ s^{-1}} $$ Finally, to find the mass flow rate through the entire area of the inner surface (26.49 m²), we just need to multiply the mass flow rate per area by the inner surface area: .total_mass_flow_rate:$$ m_{total} = (1.96 \times 10^{-10} \mathrm{kg \ m^{-2} \ s^{-1}}) \times (26.49 \mathrm{m^{2}}) = 5.19 \times 10^{-9} \mathrm{kg/s} $$ So, the mass flow rate of helium through the Pyrex container is approximately \(5.19 \times 10^{-9} \mathrm{kg/s}\).