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

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(A) The speed of sound in an ideal gas is given by the formula: $v = \sqrt{\frac{\gamma RT}{M}}$.For oxygen $(O_2)$,which is a diatomic gas,the adiaba

Vedclass · Accepted Answer

$1420$

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(A) The speed of sound in an ideal gas is given by the formula: $v = \sqrt{\frac{\gamma RT}{M}}$.For oxygen $(O_2)$,which is a diatomic gas,the adiabatic index $\gamma_{O_2} = 1.4 = \frac{7}{5}$ and the molar mass $M_{O_2} = 32 \; g/mol$.Thus,$v_{O_2} = \sqrt{\frac{7RT}{5 	imes 32}}$.For helium $(He)$,which is a monatomic gas,the adiabatic index $\gamma_{He} = 1.67 = \frac{5}{3}$ and the molar mass $M_{He} = 4 \; g/mol$.Thus,$v_{He} = \sqrt{\frac{5RT}{3 	imes 4}}$.Taking the ratio of the speeds:$\frac{v_{He}}{v_{O_2}} = \sqrt{\frac{\gamma_{He} / M_{He}}{\gamma_{O_2} / M_{O_2}}} = \sqrt{\frac{5/3}{4} 	imes \frac{32}{7/5}} = \sqrt{\frac{5}{12} 	imes \frac{160}{7}} = \sqrt{\frac{800}{84}} = \sqrt{\frac{200}{21}} \approx 3.086$.Therefore,$v_{He} = 460 	imes 3.086 \approx 1420 \; m/s$.

The speed of sound in oxygen $(O_2)$ at a certain temperature is $460 \; m/s$. The speed of sound in helium $(He)$ at the same temperature will be (assume both gases to be ideal): (in $; m/s$)

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