Question

If $c\text{is}r.m.s.$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 1

Find r.m.s. speed of molecules in a gas.

$r.m.s.$ speed of molecules in a gas is given by,

$c=\sqrt{\frac{3P}{\rho }}$

According to ideal gas equation,

$PV=nRT$

For $1mol$ gas , $n=1$
$P=\frac{RT}{V}$

So, $\frac{P}{\rho }=\frac{\frac{RT}{V}}{\frac{M}{V}}=\frac{RT}{M}$

Hence, $r.m.s.$ speed $c=\sqrt{\frac{3RT}{M}}$

Find speed of sound wave in gas.

Speed of sound wave in a gas, $v=\sqrt{\frac{\gamma P}{\rho }}$

$v=\sqrt{\frac{\gamma RT}{M}}$

Here, $\gamma =\text{adibatic constant for diatomic gas}$

$\gamma =\frac{7}{5}$(for diatomic gas)

So, $v=\sqrt{\frac{7RT}{5M}}$


Find ratio of $c/v$.

The ratio of $c\text{and}v$,

$\frac{c}{v}=\frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{7RT}{5M}}}=\sqrt{\frac{15}{7}}$

Which is constant and independent of the temperature.

Final Answer: $\sqrt{\frac{15}{7}}$