An adiabatic vessel contains $$n1$$ moles of diatomic gas. Moment of inertia of each molecule is $$I=2.76$$×$$10$$−$$46kgm2$$ and root mean square angular velocity is ω$$0=5$$×$$1012rads$$−$$1$$ . Another adiabatic vessel contains $$n2=$$ $$5$$ moles of a monoatomic gas at a temperature $$470K$$. Assume gases to be ideal, calculate $$root-mean$$ square angular velocity of diatomic molecules (in$$ multiple of $$1012rads$$−$$1) when the two vessels are connected by a thin tube of negligible volume.

Question

An adiabatic vessel contains  $$n1$$ moles of diatomic gas. Moment of inertia of each  molecule is $$I=2.76$$×$$10$$−$$46kgm2$$ and root mean square angular velocity is ω$$0=5$$×$$1012rads$$−$$1$$ .  Another adiabatic vessel contains   $$n2=$$ $$5$$ moles of a monoatomic gas at a  temperature $$470K$$. Assume gases to be ideal, calculate $$root-mean$$ square  angular velocity of diatomic molecules (in$$ multiple of $$1012rads$$−$$1) when the two  vessels are connected by a thin tube of negligible volume.

Infinity Learn · Accepted Answer

By energy conservation,$$f1n1+f2n2Tf=f1n1T1+f2n2T2$$      (1)Here$$, $$f1=5$$,  $$n1=3$$, $$T1=250K$$  and   $$f2=3$$,  $$n2=5$$, $$T2=470K$$ Substituting in $$(1),$$Tf=5$$×$$3$$×$$250+3$$×$$5$$×$$4705$$×$$3+3$$×$$5=360K$$ If RMS angular velocity of diatomic gas molecules is  ωRMSf, then according to law  of equipartition of energy,$$12I$$ω$$RMSf2=kT$$ Or,⇒ω$$RMSf=2kTI=2$$×$$1.38$$×$$10$$−$$23$$×$$3602.76$$×$$10$$−$$46=6$$×$$1012rad/s$$.

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