Sound travels in a mixture of two moles of helium and n moles of… | JEE Main 2022 (25 Jul, Shift 2)

Q59JEE Main 2022 · 25 JUL · SHIFT 2Difficult

Molecular Speeds & Maxwell Distribution

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $\sqrt{2}$ times the speed of sound, then the value of n will be:

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Answer: (b)

$\dfrac{v_{rms}}{v_{sound}} = \sqrt{\dfrac{3RT/M}{\gamma RT/M}} = \sqrt{\dfrac{3}{\gamma}} = \sqrt{2} \Rightarrow \gamma_{mix} = \dfrac{3}{2}$
Mixture: $C_V = \dfrac{2\cdot\frac{3}{2}R + n\cdot\frac{5}{2}R}{2+n}$ , $C_P = \dfrac{2\cdot\frac{5}{2}R + n\cdot\frac{7}{2}R}{2+n}$
$\gamma = \dfrac{5 + 3.5n}{3 + 2.5n} = 1.5 \Rightarrow 5 + 3.5n = 4.5 + 3.75n$
$0.25n = 0.5 \Rightarrow n = 2$
So the answer is (b) 2.

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