Sound travels in a mixture of two moles of helium and n moles of...

Speed of sound in a gas is given by, ${v}_{s} = \sqrt{\dfrac{\gammaRT}{M}}$

The formula for RMS speed is, ${v}_{rms} = \sqrt{\dfrac{3RT}{M}}$

It is given that ${v}_{rms} = \sqrt{2}{v}_{s}$

$⇒ \dfrac{{v}_{s}}{{v}_{rms}} = \sqrt{\dfrac{\gamma}{3}} = \dfrac{1}{\sqrt{2}} ⇒ \dfrac{\gamma}{3} = \dfrac{1}{2} ⇒ \gamma = \dfrac{3}{2}$

For $\gamma$ of mixture,

${\gamma}_{mix} = \dfrac{{C}_{P, mix}}{{C}_{V, mix}} = \dfrac{\dfrac{2{C}_{P, H} + n{C}_{P, He}}{2 + n}}{\dfrac{2{C}_{V, H} + n{C}_{V, He}}{2 + n}}$

Since Hydrogen is a diatomic gas, ${C}_{V, H} = \dfrac{5}{2}R, {C}_{P, H} = \dfrac{7}{2}R$ and Helium is monatomic gas, ${C}_{V, He} = \dfrac{3}{2}R, {C}_{P, He} = \dfrac{5}{2}R$. Therefore,

${\gamma}_{mix} = \dfrac{\dfrac{2{C}_{P, H} + n{C}_{P, He}}{2 + n}}{\dfrac{2{C}_{V, H} + n{C}_{V, He}}{2 + n}} ⇒ \dfrac{3}{2} = \dfrac{2 \times \dfrac{7}{2}R + n \times \dfrac{5}{2}R}{2 \times \dfrac{5}{2}R + n \times \dfrac{3}{2}R}$

$⇒ n = 2$