The oxygen molecule has a mass of `5.30xx10^(-26)kg` and a moment of inertia of `1.94xx10^(-46 )kgm^(2)` about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is `(2)/(3)` of its kinetic energy of translation. Find the average angular velocity of the molecules.

Explanation

Rotational kinetic energy = `(2)/(3)` translational kinetic energy
`(1)/(2)Iomega^(2)=(2)/(3)xx(1)/(2)mv^(2)`
`omega^(2)=(2)/(3)xx(mv^(2))/(I)`
`therefore omega=sqrt((2)/(3)(mv^(2))/(I))`
`=sqrt((2)/(3)xx(5.3xx10^(-26)xx(500)^(2))/(1.94xx10^(-46)))`
`=sqrt((265xx10^(-22))/(3xx1.94xx10^(-46)))`
`therefore omega=sqrt(45.53xx10^(24))`
`therefore omega=6.7477xx10^(12)" rad s"^(-1)~~6.75" rad s"^(-1)`