Question

The speed of sound in hydrogen at $0^{\circ }C$ is $1200m{s}^{-1}$. When some amount of oxygen is mixed with hydrogen, the speed decreases to $500m{s}^{-1}$. 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

Answer: 2

$1200=\sqrt{\frac{\gamma p}{{\rho }_{H_2}}}$...(i)
Let $x$ is volume of $H_2$ and $y$ volume of $O_2$. Then,
$(x+y){\rho }_{\text{mix }}=x{\rho }_{H_2}+y{\rho }_{O_2}=x{\rho }_{H_2}+16y{\rho }_{H_2}$
$\therefore {\rho }_{\text{mix }}=\frac{x+16y}{x+y}\cdot \rho H_2$
$\because 500=\sqrt{\frac{\gamma p(x+y)}{(x+16y){\rho }_{H_2}}}$...(ii)
Dividing Eq. (i) by Eq. (ii), we get
$\frac{12}{5}=\sqrt{\frac{x+16y}{x+y}}$
Or $\frac{x}{y}=\frac{2.2}{1}=\frac{11}{10}\times 2$
$\Rightarrow \alpha =2$