If c is r.m.s speed of molecules in a gas and `upsilon` is the speed of sound waves in the gas, show that `c//upsilon` is constant and independent of temperature for all diatomic gases.

Explanation

We know that rms speed of moelcules of a gas
`" " c=sqrt((3p)/(rho))=sqrt((3RT)/(M))" " ....(i)`
where M=molar mass of the gas.
Speed of sound wave is gas `v=sqrt((gammarho)/(rho))=sqrt((gammaRT)/(M))" "....(ii)`
On dividing Eq. (i) by Eq.(ii) we ge,
`" " (c)(v)=sqrt((3RT)/(M)xx(M)/(gammaRT))rArr(c)/(v)=sqrt((3)/(gamma))`
where `gamma` =addiabatic constant for diatomic gas
`" " gamma=(7)/(5)" " ` [since `gamma=(C_(rho))/(C_(v))]`
Hence, `" " (c)/(v)=consant `