If c is rms speed of molecules in a gas and v is the speed of sound waves in the gas, show that (c)/(v) is constant and independent of temperature for all diatomic gases.

If c is rms speed of molecules in a gas and v is the speed of sound wa

1.	If c is rms speed of molecules in a gas and v is the speed of sound waves in the gas, show that (c)/(v) is constant and independent of temperature for all diatomic gases.
Answer» <P> Solution :rms speed of MOLECULE of a gas, `c = sqrt((3 P )/( rho ))` (By fourmula) `= sqrt (( 3 PV )/( M ))` `= sqrt ((3 mu RT)/( mu M _(0)))` `c = sqrt (( 3 RT)/( M_(0))) ""...(1)` (Where `M _(0)=` molecular WEIGHT of a given gas, R = gas constant) Speed of sound in a gas, `v = sqrt (( gamma P )/( rho ))= sqrt (( gamma P V )/( M )) = sqrt (( gamma mu RT )/( mu M _(0))) = sqrt (( gamma RT)/( M _(0))) ""...(2)` Taking ratio of equation (1) and (2), `c/v = sqrt ((3)/( gamma )) = sqrt ((3)/( 7//5)) = sqrt ((15)/( 7)) = ` constant (Where `gamma = (C _(P))/( C _(V)) = 7/5 ` for diatomic GASES) `implies c/v` does not depend on temperature.