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If c is rms speed of molecules in a gas ...

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.

Text Solution

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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.
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