Sound travels in a mixture of 2 moles of heli | vedclass

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(B) The speed of sound in a gas is given by $v_s = \sqrt{\frac{\gamma RT}{M}}$.The root-mean-square speed of gas molecules is $v_{rms} = \sqrt{\frac{3

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$2$

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(B) The speed of sound in a gas is given by $v_s = \sqrt{\frac{\gamma RT}{M}}$.The root-mean-square speed of gas molecules is $v_{rms} = \sqrt{\frac{3RT}{M}}$.Given that $v_{rms} = \sqrt{2} v_s$,we have $\frac{v_s}{v_{rms}} = \frac{1}{\sqrt{2}}$.Substituting the formulas: $\sqrt{\frac{\gamma}{3}} = \frac{1}{\sqrt{2}} \Rightarrow \frac{\gamma}{3} = \frac{1}{2} \Rightarrow \gamma = \frac{3}{2} = 1.5$.For a mixture,the adiabatic index is $\gamma = 1 + \frac{2}{f_{mix}}$,where $f_{mix}$ is the effective degree of freedom.For Helium (monoatomic),$f_1 = 3$. For Hydrogen (diatomic),$f_2 = 5$.$f_{mix} = \frac{n_1 f_1 + n_2 f_2}{n_1 + n_2} = \frac{2 	imes 3 + n 	imes 5}{n + 2} = \frac{6 + 5n}{n + 2}$.Substituting $f_{mix}$ into the $\gamma$ formula: $\gamma = 1 + \frac{2(n + 2)}{6 + 5n} = \frac{6 + 5n + 2n + 4}{6 + 5n} = \frac{7n + 10}{5n + 6}$.Equating to $1.5$: $\frac{7n + 10}{5n + 6} = \frac{3}{2} \Rightarrow 14n + 20 = 15n + 18$.Solving for $n$: $n = 2$.

Sound travels in a mixture of $2$ moles of helium and $n$ moles of hydrogen. If the root-mean-square (rms) speed of gas molecules in the mixture is $\sqrt{2}$ times the speed of sound,then the value of $n$ will be:

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