Question

Two moles of Helium are mixed with \(n\) moles of Hydrogen. The rms speed of the gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound in the mixture. The value of \(n\) is

• A. \(1\)
• B. \(3\)
• C. \(2\)
• D. \(\dfrac{3}{2}\) \bigskip

Hint:

For any ideal gas, \[ \frac{v_{\text{rms}}}{v_s} = \sqrt{\frac{3}{\gamma}}. \] If \(v_{\text{rms}}=\sqrt{2}\,v_s\), then \[ \gamma=\frac{3}{2}. \] For gas mixtures, first calculate the total \(C_P\) and \(C_V\), then use \[ \gamma=\frac{C_P}{C_V}. \]

Answer 1

The Correct Option is C

Concept: For an ideal gas, \(v_{\text{rms}} = \sqrt{3RT/M}\) and \(v_s = \sqrt{\gamma RT/M}\). Their ratio gives \(\gamma\). For a mixture, \(\gamma = C_{p,\text{mix}}/C_{v,\text{mix}}\). Equate and solve for \(n\).

Step 1: \(v_{\text{rms}}/v_s = \sqrt{3/\gamma} = \sqrt{2} \Rightarrow 3/\gamma = 2 \Rightarrow \gamma = 3/2\).

Step 2: He (monatomic): \(C_V = 3R/2, C_P = 5R/2\). H₂ (diatomic): \(C_V = 5R/2, C_P = 7R/2\). For mixture: \(C_P = (2\cdot5R/2 + n\cdot7R/2) = (10+7n)R/2\), \(C_V = (2\cdot3R/2 + n\cdot5R/2) = (6+5n)R/2\). \(\gamma = (10+7n)/(6+5n) = 3/2 \Rightarrow 20+14n = 18+15n \Rightarrow n=2\).

Step 3: Write the final answer. \(\boxed{n=2}\)