Question

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 1

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]