Question

At what temperature is the velocity of sound in nitrogen is equal to half its velocity in oxygen at 25${}^{\circ }$C? (The ratio of molecular weights of nitrogen and oxygen is 7 : 8)

• A. 45.18 K
• B. 55.18 K
• C. 65.18 K
• D. 75.18 K

Answer 1

The correct option is C 65.18 K

$V=\sqrt{\frac{\gamma p}{d}}$ and $\frac{p}{d}=\frac{RT}{M}$
$V=\sqrt{\frac{\gamma RT}{M}}$
For nitrogen, $V_{N_2}=\sqrt{\frac{\gamma R{T}_{N_2}}{M_{N_2}}}$
For Oxygen, $V_{O_2}=\sqrt{\frac{\gamma R{T}_{O_2}}{M_{O_2}}}$
Given $V_{N_2}=\frac{1}{2}{V}_{O_2}$
$\therefore \frac{V_{O_2}}{V_{N_2}}=\sqrt{\frac{T_{O_2}}{M_{O_2}}\times \frac{M_{N_2}}{T_{N_2}}}$
$2=\sqrt{\frac{298}{16}\times \frac{14}{T_{N_2}}}$
$\left(T_{N_2}\right)4=\frac{298\left(14\right)}{\left(16\right)}$
$T_{N_2}=\frac{1043}{16}=65.18K$