Q143 / 180 · COMEDK 2013 · Physics
Oxygen is 16 times heavier than hydrogen. Equal volumes of hydrogen and oxygen are mixed. The ratio of speed of sound in the mixture to that in hydrogen is
- A$\sqrt{8}$
- Correct answerB$\sqrt{2 / 17}$
- C$\sqrt{1 / 8}$
- D$\sqrt{32 / 17}$
Solution
Velocity of sound in a gas is given by
$$
v=\sqrt{\frac{\gamma R T}{M}}
$$
where, $\gamma=$ specific heat ratio
and $M=$ molecular mass of gas.
Fror hydrogen, $v_{H}=\sqrt{\frac{\gamma R T}{M_{1}}}$
For mixture of gases, molecular mass,
Here,
$$
M_{\mathrm{mix}}=\frac{n_{1} M_{1}+n_{2} M_{2}}{n_{1}+n_{2}}
$$
$$
n_{1}=n_{2}
$$
(for equal volumes)
and $\quad M_{2}=16 M_{1} \quad$ (given)
$$
M_{\text {mix }}=\frac{M_{1}+16 M_{1}}{2}=\frac{17 M_{1}}{2}
$$
$\therefore$ Velocity of sound, $v_{\operatorname{mix}}=\sqrt{\frac{\gamma R T}{M_{\operatorname{mix}}}}$
Dividing Eq. (ii) by Eq. (i) we get
$$
\begin{aligned}
\frac{v_{\operatorname{mix}}}{v_{H}} &=\sqrt{\frac{\gamma R T}{M_{\operatorname{mix}}}} \times \sqrt{\frac{M_{1}}{\gamma R T}} \\
&=\sqrt{\frac{M_{1}}{M_{\text {mix }}}}=\sqrt{\frac{2}{17}}
\end{aligned}
$$
Hence, the correct answer is option: 'B'.