Question

The speed of sound through oxygen gas at $T \, K$ is $v \,m \,s^{-1}$. If the temperature becomes $2T$ and oxygen gas dissociated into atomic oxygen, the speed of sound

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

Answer 1

The Correct Option is D

The speed of sound in a gas at $T\, K$ is given by $\upsilon = \sqrt{\frac{\gamma RT}{M}}$ where $\gamma$ is the ratio of specific heat, $R$ is the universal gas constant and $M$ is the molecular mass of a gas. For oxygen gas $\upsilon = \sqrt{\frac{\frac{7}{5}RT}{M_{O_2}}}$ Since $O_2$ is a diatomic gas, so For diatomic gas $\gamma = \frac{7}{5}$ [For no vibrational mode] For atomic oxygen $\upsilon' = \sqrt{\frac{\frac{5}{3}R\left(2T\right)}{M_{O}}} $ $ \therefore \:\:\: \frac{\upsilon'}{\upsilon} = \sqrt{\frac{5}{3} \frac{\left(2RT\right)}{M_{O}}} \times \sqrt{\frac{5M_{O_2}}{7RT}}\qquad$ $ \frac{\upsilon'}{\upsilon} = \sqrt{\frac{50}{21} \frac{M_{O_2}}{M_{O}}}$ $= \sqrt{\frac{50(2)}{21}} \left[ \because \:\:M_O = \frac{M_{O_2}}{2} \therefore \frac{M_{O_2}}{M_O} = 2 \right]$ $ \sqrt{\frac{100}{21}}$ $ \upsilon' = \frac{10}{\sqrt{21}} \upsilon$