JEE MainPhysicsModerateKinetic Theory of Gases

Two moles of hydrogen are mixed with n moles of helium. The root
mean square speed of gas molecules in the mixture is $\sqrt{2}$ times
the speed of sound in the mixture. Then n is.

Show answer and solution

V_{rms} = \sqrt{\frac{3RT}{M_{mix}}}

V_{sound\ } = \sqrt{\frac{\gamma RT}{M_{mix}}}

V_{rms} = \sqrt{2}V_{sound\ }

\sqrt{\frac{3RT}{M_{mix}}} = \sqrt{2}\sqrt{\frac{\gamma RT}{M_{mix}}}

\gamma_{mix} = \frac{3}{2}

\gamma_{mix} = \frac{n_{1}C_{P_{1}} + n_{2}C_{P_{2}}}{n_{1}C_{v_{1}} + n_{2}C_{v_{2}}}

\frac{3}{2} = \frac{2 \times \frac{7R}{2} + n \times \frac{5R}{2}}{2 \times \frac{5R}{2} + n \times \frac{3R}{2}}

\Rightarrow \frac{3}{2} = \frac{14 + 5n}{10 + 3n}

\Rightarrow 30 + 9n = 28 + 10n

\Rightarrow n = 2

Source: Grass-G / P-08 Physics paper 23 October.docx

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