Question

In a mixture of gases. the average number of degrees of freedom per molecule is 6. The rms speed of the molecules of the gas is c. The velocity of sound in the gas is xc/3. Find x.

Answer 1

Root mean square velocity $c=\sqrt{\frac{3RT}{M}}$ .......(1)

where $M$ is the molecular mass of the air molecule.

Degree of freedom $f=6$

$\gamma =\frac{2+f}{f}=\frac{2+6}{6}=\frac{4}{3}$

Now velocity of sound in air $v=\sqrt{\frac{\gamma P}{\rho }}$

Using $\frac{P}{\rho }=\frac{RT}{M}$ $⟹v=\sqrt{\frac{\gamma RT}{M}}$

Using (1) $v=\sqrt{\gamma \frac{c^2}{3}}$

But $\frac{xc}{3}=\sqrt{\frac{4}{3}\times \frac{c^2}{3}}$

$⟹x=2$