Question

The velocity of sound in hydrogen at $0^{o}C$ is $1200\text{m/s}$. When some amount of oxygen is mixed with hydrogen, the velocity decreases to $500\text{m/s}$. Determine the ratio of $H_{2}to{O}_2$ by volume in this mixture, given that the denstiy of oxygen is 16 times that of hydrogen

Answer 1

Given that velocity of sound in hydrogen at $0^{o}C$ is $1200\text{m/s}$
$1200=\sqrt{\frac{\gamma P}{{\rho }_H}}...\left(1\right)$
let there be $x$ volume of $H_2\text{and}y$ volume of $O_2)$ in the mixture. Then conserving mass we get,
$(x+y){\rho }_{mix}=x{\rho }_H+y{\rho }_o=x{\rho }_H+16y{\rho }_H$
$\Rightarrow {\rho }_{mix}=\frac{(x+16y)}{x+y}{\rho }_H$

Given that velocity of mixture is $500\text{m/s}$

$\therefore 500=\sqrt{\frac{\gamma P(x+y)}{(x+16y){\rho }_H}}...\left(2\right)$
Dividing (2) by (1),
$\frac{12}{5}=\sqrt{\frac{x+16y}{x+y}}\frac{144}{25}=\frac{x+16y}{x+y}\frac{x}{y}=\frac{2.15}{1}$

Why this question? Note: When we mix $n$ different gases with $n$ different densities, the mass is conserved,i.e., the mass of the system is the sum of masses of each components.