Question

A sound wave travels a distance $l$ in helium gas in time $t$ at a particular temperature. If at the same temperature a sound wave is propagated in oxygen gas, it will cover the same distance $l$ in what time (approximately)?

• A. $2t$
• B. $4t$
• C. $3t$
• D. $5t$

Answer 1

The correct option is C $3t$

Velocity of sound, $v=\sqrt{\frac{\gamma RT}{M}}$

where $\gamma$ is the adiabatic index, $R$ is universal gas constant, $T$ is absolute temperature and $M$ is molecular mass of the gas.

At the same temperature,

$\Rightarrow \frac{v_{O_2}}{v_{H_e}}=\sqrt{\frac{{\gamma }_{O_2}}{{\gamma }_{He}}\times \frac{M_{He}}{M_{O_2}}}$

$\Rightarrow \frac{v_{O_2}}{v_{He}}=\sqrt{\frac{\left(\frac{7}{5}\right)}{\left(\frac{5}{3}\right)}\times \frac{4}{32}}$

$\Rightarrow \frac{v_{O_2}}{v_{He}}=\sqrt{\frac{21}{25}\times \frac{4}{32}}$

$\Rightarrow v_{O_2}=0.32v_{He}....\left(1\right)$

Time taken by sound in Helium gas to cover distance $l$,

$t=\frac{l}{v_{He}}.....\left(2\right)$

Now, the time taken by sound in Oxygen gas to cover distance $l$,

$t^{{}^{\prime }}=\frac{l}{v_{O_2}}$

$\Rightarrow t^{{}^{\prime }}=\frac{l}{0.32\times v_{He}}$ [from $\left(1\right)$]

$\Rightarrow t^{{}^{\prime }}=\frac{t}{0.32}$ [from $\left(2\right)$]

$\Rightarrow t^{{}^{\prime }}\approx 3t$