The speed of sound in an ideal gas at a given | vedclass

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(B) The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$,and the rms speed of gas molecules is given by $v_{	ext{rms}} = \

Vedclass · Accepted Answer

$\frac{5}{\sqrt{21}}$

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(B) The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$,and the rms speed of gas molecules is given by $v_{	ext{rms}} = \sqrt{\frac{3RT}{M}}$.Taking the ratio,we get $\frac{v}{v_{	ext{rms}}} = \sqrt{\frac{\gamma}{3}}$.For Helium (monatomic gas),$\gamma_{	ext{He}} = \frac{5}{3}$. Thus,$X = \sqrt{\frac{5/3}{3}} = \sqrt{\frac{5}{9}}$.For Oxygen (diatomic gas),$\gamma_{	ext{O}_2} = \frac{7}{5}$. Thus,$X^{\prime} = \sqrt{\frac{7/5}{3}} = \sqrt{\frac{7}{15}}$.Now,the ratio $\frac{X}{X^{\prime}} = \frac{\sqrt{5/9}}{\sqrt{7/15}} = \sqrt{\frac{5}{9} 	imes \frac{15}{7}} = \sqrt{\frac{5 	imes 5}{3 	imes 7}} = \sqrt{\frac{25}{21}} = \frac{5}{\sqrt{21}}$.

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