Let (C) represent the root mean square speed of molecule in a gas and

1.	Let (C) represent the root mean square speed of molecule in a gas and let (V) the speed of sound waves in that gas. Show that the ratio (C/V)is constant for a given gas.
Answer» Given: (C) represent the root mean square speed of molecule in a gas and let (V) the speed of SOUND waves in that gas. To FIND: C/V ratio is a constant for that gas. Calculation: $C = V_{RMS}= \sqrt{ \dfrac{3RT}{m}}$ Again, velocity of sound in gas (as per LaPlace CORRECTION): $V= \sqrt{ \dfrac{ \gamma RT}{m} }$ Now, REQUIRED ratio is : $\implies \dfrac{C}{V} = \dfrac{ \sqrt{ \frac{3RT}{m} } }{ \sqrt{ \frac{ \gamma RT}{m} } }$ $\implies \dfrac{C}{V} =\sqrt{\dfrac{3}{ \gamma }}$ Since $\gamma$ is constant for a particular gas: $\implies \dfrac{C}{V} = constant$ [Hence Proved]