Question

In a mixture of gasses, the average number of degree of freedoms per molecules is $6$. The rms speed of the molecules of the gas is $c$. The velocity of sound in the gas is:

• A. $\frac{c}{\sqrt{3}}$
• B. $\frac{c}{\sqrt{2}}$
• C. $\frac{2c}{3}$
• D. $\frac{3c}{4}$

Answer 1

The correct option is C $\frac{2c}{3}$

Given: Degree of freedom per molecules is 6, rms speed of the molecules is c.

Solution:

We know that $\gamma =1+\frac{2}{f}$

$\Rightarrow \gamma =1+\frac{1}{3}$

So, $\gamma =\frac{4}{3}$

We know that velocity of sound in air is $v=(\frac{\gamma P}{\rho }{)}^{0.5}$

From ideal gas equation, $\frac{P}{\rho }=\frac{RT}{M}$

$\therefore v=(\frac{\gamma RT}{M}{)}^{0.5}$ -(i)

Also rms velocity is given by $c=(\frac{3RT}{M}{)}^{0.5}$ -(ii)

Solving (i) and(ii) , we get $v=(\frac{\gamma }{3}{)}^{0.5}\times c$

Putting $\gamma =\frac{4}{3}$, $v=\frac{2c}{3}$

Hence option C is correct.