Waves

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Question 136 of 142

Sound travels in a mixture of two moles of helium and nn moles of hydrogen. If rms speed of gas molecules in the mixture is 2\sqrt{2} times the speed of sound, then the value of nn will be

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Answer: (B) 22

Speed of sound in a gas is given by, vs=γRTM{v}_{s}=\sqrt{\dfrac{\gamma RT}{M}}
The formula for RMS speed is, vrms=3RTM{v}_{rms}=\sqrt{\dfrac{3RT}{M}}
It is given that vrms=2vs{v}_{rms}=\sqrt{2}{v}_{s}
⇒vsvrms=γ3=12⇒γ3=12⇒γ=32\Rightarrow \dfrac{{v}_{s}}{{v}_{rms}}=\sqrt{\dfrac{\gamma }{3}}=\dfrac{1}{\sqrt{2}}\Rightarrow \dfrac{\gamma }{3}=\dfrac{1}{2}\Rightarrow \gamma =\dfrac{3}{2}
For γ\gamma of mixture,
γmix=CP,mixCV,mix=2CP,H+nCP,He2+n2CV,H+nCV,He2+n{\gamma }_{mix}=\dfrac{{C}_{P,mix}}{{C}_{V,mix}}=\dfrac{\dfrac{2{C}_{P,H}+n{C}_{P,He}}{2+n}}{\dfrac{2{C}_{V,H}+n{C}_{V,He}}{2+n}}
Since Hydrogen is a diatomic gas, CV,H=52R,CP,H=72R{C}_{V,H}=\dfrac{5}{2}R,{C}_{P,H}=\dfrac{7}{2}R and Helium is monatomic gas, CV,He=32R,CP,He=52R{C}_{V,He}=\dfrac{3}{2}R,{C}_{P,He}=\dfrac{5}{2}R. Therefore,
γmix=2CP,H+nCP,He2+n2CV,H+nCV,He2+n⇒32=2×72R+n×52R2×52R+n×32R{\gamma }_{mix}=\dfrac{\dfrac{2{C}_{P,H}+n{C}_{P,He}}{2+n}}{\dfrac{2{C}_{V,H}+n{C}_{V,He}}{2+n}} \Rightarrow \dfrac{3}{2}=\dfrac{2\times \dfrac{7}{2}R+n\times \dfrac{5}{2}R}{2\times \dfrac{5}{2}R+n\times \dfrac{3}{2}R}
⇒n=2\Rightarrow n=2

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