Physics - Sound Wave Question with Solution | TestHub

Speed of sound in a gas is given by, $v_s=\sqrt{\frac{\gamma RT}{M}}$

The formula for RMS speed is, $v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$

It is given that $v_{\mathrm{rms}}=\sqrt{2}{v}_s$

$\Rightarrow \frac{v_s}{v_{\mathrm{rms}}}=\sqrt{\frac{\gamma }{3}}=\frac{1}{\sqrt{2}}\Rightarrow \frac{\gamma }{3}=\frac{1}{2}\Rightarrow \gamma =\frac{3}{2}$

For $\gamma$ of mixture,

${\gamma }_{\mathrm{mix}}=\frac{C_{P,mix}}{C_{V,mix}}=\frac{{\displaystyle \frac{2C_{P,H}+n{C}_{P,He}}{2+n}}}{\frac{2C_{V,H}+n{C}_{V,He}}{2+n}}$

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

$$\begin{gathered} {\gamma }_{\mathrm{mix}}=\frac{{\displaystyle \frac{2C_{P,H}+n{C}_{P,He}}{2+n}}}{\frac{2C_{V,H}+n{C}_{V,He}}{2+n}} \\ \Rightarrow \frac{3}{2}=\frac{2\times {\displaystyle \frac{7}{2}}R+n\times {\displaystyle \frac{5}{2}}R}{2\times {\displaystyle \frac{5}{2}}R+n\times {\displaystyle \frac{3}{2}}R} \end{gathered}$$

$\Rightarrow n=2$