Question

The speed of sound through oxygen at $T K$ is $v\, ms ^{-1}$. As the temperature becomes $2 T$ and oxygen gas dissociates into atomic oxygen, the speed of sound :

• A. remains the same
• B. becomes $ 2v $
• C. becomes $ \sqrt{2v } $
• D. none of these

Answer 1

Answer: (B)

The rms velocity of sound in gas is
$v_{ rms (\text { molecules })}=\sqrt{\frac{\gamma R T}{M}}$
$=\sqrt{\frac{1.4 \times R T}{M}}$ ...(1)
when the oxygen dissociates, its molecular mass becomes atomic mass, so,
$M=\frac{M}{2}$ and $T=2 T$ given, and $\gamma=1.66$
$v_{ rms }(\text { atomic })=\sqrt{\frac{1.66 \times R \times 2 T}{M / 2}}$
$=\sqrt{\frac{1.66 \times R \times 2 T \times 2}{M}}$ ...(2)
From equations (1) and (2)
$\frac{v_{ rms }(\text { atomic })}{v_{ rms (\text { molecular })}}=\sqrt{\frac{\sqrt{1.66 \times R \times 2 T \times 2}}{M}}$
$\times \sqrt{\frac{M}{1.4 \times R T}}$
$=\sqrt{\frac{1.66 \times 4}{1.4}}=2.18$
Hence, $v_{ rms }$ (atomic) $=2.18\, v_{ rms }$ (molecular)
$\approx 2 v_{ rms }$ (molecular)