Let's solve the problem step by step: --- ## **Given Data** - **Temperature, \(T\):** 293 K - **Pressure inside, \(P_1\):** 500 kPa = 500,000 Pa - **Inner diameter, \(d_1\):** 2 m → Inner radius, \(r_1 = 1\) m - **Wall thickness:** 1 cm = .01 m - **Outer radius, \(r_2 = r_1 + .01 = 1.01\) m** - **Material:** Fused silica (\(\text{SiO}_2\)) - **Well-ventilated outside:** Partial pressure of He outside, \(P_2 \approx \) Assume: - **Diffusion coefficient for He in SiO₂, \(D\):** Typical value: \(D = 3 \times 10^{-12}\) m²/s - **Solubility of He in SiO₂, \(S\):** \(S = 7.8 \times 10^{-4}\) mol/(m³·Pa) (These are typical values; use your specific values if given.) --- ## **Step 1: Molar Flow Rate by Diffusion** For a **spherical shell**, the steady-state molar flow rate is: \[ N = \frac{4 \pi D S (r_1 r_2)}{r_2 - r_1} (P_1 - P_2) \] Where: - \(N\) = molar flow rate (mol/s) - \(D\) = diffusion coefficient (m²/s) - \(S\) = solubility (mol/(m³·Pa)) - \(r_1, r_2\) = inner and outer radii (m) - \(P_1, P_2\) = partial pressures inside and outside (Pa) --- ### **Plug in the Values** - \(D = 3 \times 10^{-12}\) m²/s - \(S = 7.8 \times 10^{-4}\) mol/(m³·Pa) - \(r_1 = 1\) m - \(r_2 = 1.01\) m - \(P_1 = 500,000\) Pa - \(P_2 = \) Pa \[ N = \frac{4 \pi \times (3 \times 10^{-12}) \times (7.8 \times 10^{-4}) \times (1 \times 1.01)}{1.01 - 1} \times (500,000 - ) \] --- Calculate numerator: \[ 4\pi \approx 12.566 \] \[ 3 \times 10^{-12} \times 7.8 \times 10^{-4} = 23.4 \times 10^{-16} = 2.34 \times 10^{-15} \] \[ 2.34 \times 10^{-15} \times 1.01 = 2.3634 \times 10^{-15} \] \[ 12.566 \times 2.3634 \times 10^{-15} \approx 29.71 \times 10^{-15} \] \[ \text{Numerator: } 29.71 \times 10^{-15} \times 500,000 = (29.71 \times 5) \times 10^{-15+5} = 148.55 \times 10^{-10} = 1.4855 \times 10^{-8} \] Denominator: \[ 1.01 - 1 = .01 \] \[ N = \frac{1.4855 \times 10^{-8}}{.01} = 1.4855 \times 10^{-6} \text{ mol/s} \] --- ## **Final Answers** ### **(a) Molar Flow Rate, \(N\):** \[ \boxed{1.49 \times 10^{-6} \ \text{mol/s}} \] --- ### **(b) Mass Flow Rate, \(\dot{m}\):** Molar mass of He, \(M = 4.00\) g/mol = .004 kg/mol \[ \dot{m} = N \cdot M \] \[ \dot{m} = (1.49 \times 10^{-6}) \times .004 = 5.96 \times 10^{-9} \text{ kg/s} \] --- ## **Summary Table** | Quantity | Value | |--------------------|-----------------------------| | Molar flow rate | \(1.49 \times 10^{-6}\) mol/s | | Mass flow rate | \(5.96 \times 10^{-9}\) kg/s | --- ### **Explanation** - Used steady-state spherical diffusion equation. - Used typical values for \(D\) and \(S\) for He in SiO₂. - Assumed outside He pressure is negligible (well-ventilated). - Calculated step by step for clarity. If you have different values for \(D\) or \(S\), substitute them into the formula above!