Question

[Key Concept: For a gas mixture, γ is the ratio of mole-weighted C_P and C_V, and the molar mass is the mole-weighted mean.]

$$V = \sqrt{\frac{\gamma RT}{M}}$$

$$\gamma_{mix} = \frac{C_{P_{mix}}}{C_{V_{mix}}}$$

$$\begin{matrix} & \ = \frac{n_{1}C_{P_{1}} + n_{2}C_{P_{2}}}{n_{1}C_{V_{1}} + n_{2}C_{V_{2}}} \\ & \ = \frac{1\left( \frac{5R}{2} \right) + 2\left( \frac{7R}{2} \right)}{1\left( \frac{3R}{2} \right) + 2\left( \frac{5R}{2} \right)} \\ & \ = \frac{19}{13} \approx 1.46 \\ & M_{mix} = \frac{n_{1}M_{1} + n_{2}M_{2}}{n_{1} + n_{2}} \\ & \ = \frac{1(4) + 2(32)}{1 + 2} = \frac{68}{3}gm/mol \\ & \ = \frac{68 \times 10^{- 3}}{3}\text{ }kg/mol \\ & \ \Rightarrow \ \text{ }V = \sqrt{\frac{\gamma_{mix} \times RT}{M_{mix}}} = 401\text{ }m/s \end{matrix}$$

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