Question

The speed of sound in hydrogen at $0^{\circ} C$ is $1200\, ms ^{-1}$. When some amount of oxygen is mixed with hydrogen, the speed decreases to $500\, ms ^{-1}$. If density of oxygen is $16$ times that of hydrogen, the ratio of hydrogen to oxygen by volume in this mixture is $\frac{11}{10} \alpha$. The value of $\alpha$ is_____

Answer 1

$1200=\sqrt{\frac{\gamma p}{\rho_{H_{2}}}}$...(i)
Let $x$ is volume of $H _{2}$ and $y$ volume of $O _{2}$. Then,
$(x+y) \rho_{\text {mix }}=x \rho_{ H _{2}}+y \rho_{ O _{2}}=x \rho_{ H _{2}}+16 y \rho_{ H _{2}}$
$\therefore \rho_{\text {mix }}=\frac{x+16 y}{x+y} \cdot \rho H _{2}$
$\because 500=\sqrt{\frac{\gamma p(x+y)}{(x+16 y) \rho_{H_{2}}}}$...(ii)
Dividing Eq. (i) by Eq. (ii), we get
$\frac{12}{5}=\sqrt{\frac{x+16 y}{x+y}} $
Or $ \frac{x}{y}=\frac{2.2}{1}=\frac{11}{10} \times 2 $
$\Rightarrow \alpha=2$