Q139 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Physics

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

  • Correct answer
    A
    9
  • B
    10
  • C
    100
  • D
    90

Solution

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

Hence, the correct answer is option: 'A'.

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