Question

If the speed of sound in a mixture of $2$ moles of Helium and $2$ moles of Hydrogen at temperature $\frac{972}{5}\, K$ is $n \times 100 \,ms ^{-1}$, then the value of $n$ is (Take, $R=\frac{25}{3} \,J\, mol ^{-1} K ^{-1}$ )

• A. 9
• B. 10
• C. 100
• D. 90

Answer 1

Answer: (A)

Given, $T=\frac{972}{5} K , R=\frac{25}{3} \,J\, mol ^{-1} K ^{-1}$
Molecular mass of the mixture
$=\frac{n_{H e} M_{H e}+n_{H} M_{H}}{n_{H e}+n_{H}}$
$\because n_{H e}=2$ moles, $n_{H}=2$ moles
$M_{H e}=4, M_{H}=2$
$\therefore $ Molecular mass of the mixtures,
$M =\frac{2 \times 4+2 \times 2}{2+2}$
$=\frac{8+4}{4}=3 \,gm / mole$
$\because$ Degree of freedom of mixture
$f =\frac{2 \times 3+2 \times 5}{2+2}=4$
$\because f_{\text {mix }} =1+\frac{2}{f}=1.5$
$ \because \text { Speed of sound } =\sqrt{\frac{f_{text{mix}} R T}{M}} $
$=\sqrt{\frac{1.5 \times \frac{25}{3} \times \frac{972}{5}}{3 \times 10^{-3}}} $
$=900=n \times 100\, m / s$
Hence, $ n =9$