The speed of sound in oxygen (O_2) at a certa | vedclass

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(B) For oxygen $(O_2)$:Molar mass,$M_1 = 32 \,g/mol$.Heat capacity ratio,$\gamma_1 = C_p / C_V = 7/5$ (for diatomic gas).Speed of sound,$v_1 = 460 \,m

Vedclass · Accepted Answer

$1420$

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(B) For oxygen $(O_2)$:Molar mass,$M_1 = 32 \,g/mol$.Heat capacity ratio,$\gamma_1 = C_p / C_V = 7/5$ (for diatomic gas).Speed of sound,$v_1 = 460 \,ms^{-1}$.For helium $(He)$:Molar mass,$M_2 = 4 \,g/mol$.Heat capacity ratio,$\gamma_2 = C_p / C_V = 5/3$ (for monoatomic gas).The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.Taking the ratio of speeds: $\frac{v_1}{v_2} = \sqrt{\frac{\gamma_1 / M_1}{\gamma_2 / M_2}} = \sqrt{\frac{\gamma_1}{\gamma_2} \cdot \frac{M_2}{M_1}}$.Substituting the values: $\frac{460}{v_2} = \sqrt{\frac{7/5}{5/3} \cdot \frac{4}{32}} = \sqrt{\frac{21}{25} \cdot \frac{1}{8}} = \sqrt{\frac{21}{200}} \approx 0.324$.Wait,re-calculating: $\frac{460}{v_2} = \sqrt{\frac{7}{5} \cdot \frac{3}{5} \cdot \frac{4}{32}} = \sqrt{\frac{21}{25} \cdot \frac{1}{8}} = \sqrt{\frac{21}{200}} \approx 0.324$.Actually,$\frac{v_2}{v_1} = \sqrt{\frac{\gamma_2}{\gamma_1} \cdot \frac{M_1}{M_2}} = \sqrt{\frac{5/3}{7/5} \cdot \frac{32}{4}} = \sqrt{\frac{25}{21} \cdot 8} = \sqrt{\frac{200}{21}} \approx \sqrt{9.52} \approx 3.085$.$v_2 = 460 	imes 3.085 \approx 1419.1 \,ms^{-1} \approx 1420 \,ms^{-1}$.

The speed of sound in oxygen $(O_2)$ at a certain temperature is $460 \,ms^{-1}$. The speed of sound in helium $(He)$ at the same temperature will be (assume both gases to be ideal): (in $\,ms^{-1}$)

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