If $c$ is the $rms$ speed of molecules in a gas and $v$ is the speed of sound waves in the gas,show that $\frac{c}{v}$ is constant and independent of temperature for all diatomic gases.

(N/A) The $rms$ speed of molecules of a gas is given by:
$c = \sqrt{\frac{3P}{\rho}} = \sqrt{\frac{3RT}{M_0}}$
(Where $M_0$ is the molar mass of the gas and $R$ is the universal gas constant).
The speed of sound in a gas is given by:
$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M_0}} \quad ... (1)$
(Where $\gamma$ is the adiabatic index or ratio of specific heats).
Taking the ratio of $c$ and $v$:
$\frac{c}{v} = \frac{\sqrt{\frac{3RT}{M_0}}}{\sqrt{\frac{\gamma RT}{M_0}}} = \sqrt{\frac{3}{\gamma}}$
For all diatomic gases,the adiabatic index $\gamma = \frac{C_P}{C_V} = \frac{7}{5} = 1.4$.
Substituting this value:
$\frac{c}{v} = \sqrt{\frac{3}{7/5}} = \sqrt{\frac{15}{7}}$
Since $\sqrt{\frac{15}{7}}$ is a constant value and does not contain the temperature $T$,the ratio $\frac{c}{v}$ is constant and independent of temperature for all diatomic gases.