Question

The speed of sound through oxygen at $TK$ is $vm{s}^{-1}$. As the temperature becomes $2T$ 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
$v_{rms(\text{ molecules })}=\sqrt{\frac{\gamma RT}{M}}$
$=\sqrt{\frac{1.4\times RT}{M}}$ ...(1)
when the oxygen dissociates, its molecular mass becomes atomic mass, so,
$M=\frac{M}{2}$ and $T=2T$ given, and $\gamma =1.66$
$v_{rms}(\text{ atomic })=\sqrt{\frac{1.66\times R\times 2T}{M/2}}$
$=\sqrt{\frac{1.66\times R\times 2T\times 2}{M}}$ ...(2)
From equations (1) and (2)
$\frac{v_{rms}(\text{ atomic })}{v_{rms(\text{ molecular })}}=\sqrt{\frac{\sqrt{1.66\times R\times 2T\times 2}}{M}}$
$\times \sqrt{\frac{M}{1.4\times RT}}$
$=\sqrt{\frac{1.66\times 4}{1.4}}=2.18$
Hence, $v_{rms}$ (atomic) $=2.18v_{rms}$ (molecular)
$\approx 2v_{rms}$ (molecular)