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 √2 times the speed of…

Speed of sound in a gas is given by, ${v}_{s}=\sqrt{\frac{\gamma RT}{M}}$vs​=MγRT​​

The formula for RMS speed is, ${v}_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$vrms​=M3RT​​

It is given that ${v}_{\mathrm{rms}}=\sqrt{2}{v}_{s}$vrms​=2​vs​

$$\Rightarrow \frac{{v}_{s}}{{v}_{\mathrm{rms}}}=\sqrt{\frac{\gamma }{3}}=\frac{1}{\sqrt{2}}\Rightarrow \frac{\gamma }{3}=\frac{1}{2}\Rightarrow \gamma =\frac{3}{2}$$⇒vrms​vs​​=3γ​​=2​1​⇒3γ​=21​⇒γ=23​

For $\gamma$γ of mixture,

$${\gamma }_{\mathrm{mix}}=\frac{{C}_{P,mix}}{{C}_{V,mix}}=\frac{\frac{2{C}_{P,H}+n{C}_{P,He}}{2+n}}{\frac{2{C}_{V,H}+n{C}_{V,He}}{2+n}}$$γmix​=CV,mix​CP,mix​​=2+n2CV,H​+nCV,He​​2+n2CP,H​+nCP,He​​​

Since Hydrogen is a diatomic gas, ${C}_{V,H}=\frac{5}{2}R,{C}_{P,H}=\frac{7}{2}R$CV,H​=25​R,CP,H​=27​R and Helium is monatomic gas, ${C}_{V,He}=\frac{3}{2}R,{C}_{P,He}=\frac{5}{2}R$CV,He​=23​R,CP,He​=25​R. Therefore,

$${\gamma }_{\mathrm{mix}}=\frac{\frac{2{C}_{P,H}+n{C}_{P,He}}{2+n}}{\frac{2{C}_{V,H}+n{C}_{V,He}}{2+n}} \Rightarrow \frac{3}{2}=\frac{2\times \frac{7}{2}R+n\times \frac{5}{2}R}{2\times \frac{5}{2}R+n\times \frac{3}{2}R}$$γmix​=2+n2CV,H​+nCV,He​​2+n2CP,H​+nCP,He​​​⇒23​=2×25​R+n×23​R2×27​R+n×25​R​

$$\Rightarrow n=2$$⇒n=2