Question

How many degrees of freedom have the gas molecules, if under standard conditions the gas density is $\rho =1.3kg/m^3$ and velocity of sound propagation is v = 330 m/s ?

Answer 1

In standard condition, pressure of gas is $P={10}^{5}N/m^2$.

$v_{rms}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3\left({10}^5\right)}{1.3}}=480m/s$

Using $\frac{v_{rms}}{v_{sound}}=\sqrt{\frac{3}{\gamma }}$

or $\frac{480}{330}=\sqrt{\frac{3}{\gamma }}\Rightarrow \gamma =1.4$

also $\frac{v_{rms}}{v_{sound}}=\sqrt{\frac{3}{\gamma }}=\sqrt{\frac{3f}{f+2}}$

or $\frac{3}{\gamma }=\frac{3f}{f+2}$

or $f+2=\gamma f=1.4f$

$\therefore f=\frac{2}{0.4}=5$