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 time (approximately)

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

Answer 1

The correct option is C $3t$

Velocity of sound, $v=\sqrt{\frac{\gamma RT}{M}}$
At same temperature,
$\Rightarrow \frac{v_{O_2}}{v_{H_e}}=\sqrt{\frac{{\gamma }_{O_2}}{{\gamma }_{He}}\times \frac{M_{He}}{M_{O_2}}}$
$\Rightarrow \frac{v_{O_2}}{v_{He}}=\sqrt{\frac{\left(\frac{7}{5}\right)}{\left(\frac{5}{3}\right)}\times \frac{4}{32}}$
$\Rightarrow \frac{v_{O_2}}{v_{He}}=\sqrt{\frac{21}{25}\times \frac{4}{32}}$
$\Rightarrow v_{O_2}=0.32v_{He}$ ..............(1)
Time taken by sound in Helium gas to cover distance $l$,
$t=\frac{l}{v_{He}}$ .............(2)
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 (1)]
$\Rightarrow t^{{}^{\prime }}=\frac{t}{0.32}$ [from (2)]
$\Rightarrow t^{{}^{\prime }}\approx 3t$