Question:
Two moles of helium are mixed with \(n\) moles of hydrogen. The root mean square (rms) speed of gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound in the mixture. Then, the value of \(n\) is
Two moles of helium are mixed with \(n\) moles of hydrogen. The root mean square (rms) speed of gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound in the mixture. Then, the value of \(n\) is
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For monatomic gas \(\gamma = 5/3\), for diatomic \(\gamma = 7/5\).
Updated On: Apr 20, 2026
- 1
- 3
- 2
- 3/2
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
RMS speed: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), speed of sound: \(v_{sound} = \sqrt{\frac{\gamma RT}{M}}\). Given: \(v_{rms} = \sqrt{2}\, v_{sound}\).
Step 2: Detailed Explanation:
Equate: \[ \sqrt{\frac{3RT}{M}} = \sqrt{2}\,\sqrt{\frac{\gamma RT}{M}} \Rightarrow 3 = 2\gamma \Rightarrow \gamma = 1.5 \] For mixture: \[ \gamma = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 C_{v1} + n_2 C_{v2}} \] Heat capacities: He (monatomic): \(C_v = \frac{3}{2}R,\; C_p = \frac{5}{2}R\); H\(_2\) (diatomic): \(C_v = \frac{5}{2}R,\; C_p = \frac{7}{2}R\). Substitute: \[ \gamma = \frac{2\times\frac{5}{2}R + n\times\frac{7}{2}R}{2\times\frac{3}{2}R + n\times\frac{5}{2}R} = \frac{10 + 7n}{6 + 5n} \] Solve: \[ \frac{10 + 7n}{6 + 5n} = 1.5 \Rightarrow 10 + 7n = 9 + 7.5n \Rightarrow 1 = 0.5n \Rightarrow n = 2 \]
Step 3: Final Answer:
\[ \boxed{2} \]
RMS speed: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), speed of sound: \(v_{sound} = \sqrt{\frac{\gamma RT}{M}}\). Given: \(v_{rms} = \sqrt{2}\, v_{sound}\).
Step 2: Detailed Explanation:
Equate: \[ \sqrt{\frac{3RT}{M}} = \sqrt{2}\,\sqrt{\frac{\gamma RT}{M}} \Rightarrow 3 = 2\gamma \Rightarrow \gamma = 1.5 \] For mixture: \[ \gamma = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 C_{v1} + n_2 C_{v2}} \] Heat capacities: He (monatomic): \(C_v = \frac{3}{2}R,\; C_p = \frac{5}{2}R\); H\(_2\) (diatomic): \(C_v = \frac{5}{2}R,\; C_p = \frac{7}{2}R\). Substitute: \[ \gamma = \frac{2\times\frac{5}{2}R + n\times\frac{7}{2}R}{2\times\frac{3}{2}R + n\times\frac{5}{2}R} = \frac{10 + 7n}{6 + 5n} \] Solve: \[ \frac{10 + 7n}{6 + 5n} = 1.5 \Rightarrow 10 + 7n = 9 + 7.5n \Rightarrow 1 = 0.5n \Rightarrow n = 2 \]
Step 3: Final Answer:
\[ \boxed{2} \]
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