Question

The ratio of the speed of sound in nitrogen gas to that in helium gas, at $300K$ is?
$(A)\sqrt {\left( {\dfrac{2}{7}} \right)} $
$(B)\sqrt {\left( {\dfrac{1}{7}} \right)} $
$(C)\dfrac{{\sqrt 3 }}{5}$
$(D)\dfrac{{\sqrt 6 }}{5}$

Answer 1

Hint: Sound can travel through any medium. Sound travels in the form of waves which are travelling in the form of vibration of particles. It can travel through gaseous, liquid or solid material. Propagation of sound waves can be affected in different media and its speed varies in different media according to the material and density of the media through which it is travelling.

Complete answer:
We know that sound waves travel due to the vibrations produced in the atoms and molecules of the media. Sound needs a medium to travel because it cannot travel in vacuum. This happens because there are no atoms or molecules present in the vacuum to vibrate.
So, the speed of the sound waves in a media generally depends upon the density of the material. Hence, the sound waves can travel fastest in the gaseous medium.
The formula we will use to solve this question is given as:
$v = \sqrt {\dfrac{{\gamma RT}}{M}} $
Where, the temperature of both the gases is same, so the formula becomes:
$v\alpha \sqrt {\dfrac{\gamma }{M}} $
Where, $\gamma $ is the heat capacity ratio which is different for monoatomic gas (helium gas) and diatomic gas (nitrogen gas) and M is the molecular mass of these gases.
$\dfrac{{{v_{{N_2}}}}}{{{v_{He}}}} = \sqrt {\dfrac{{{\gamma _{{N_2}}}}}{{{\gamma _{He}}}} \times \dfrac{{{M_{He}}}}{{{M_{{N_2}}}}}} $
Where values of $\gamma $ are:
${\gamma _{mono}} = \dfrac{5}{3}$
${\gamma _{dia}} = \dfrac{7}{5}$
Therefore, put these values in the formula given above:
$\dfrac{{{v_{{N_2}}}}}{{{v_{He}}}} = \sqrt {\dfrac{{\left( {\dfrac{7}{5}} \right)}}{{\left( {\dfrac{5}{3}} \right)}} \times \dfrac{4}{{28}}} $
$\dfrac{{{v_{{N_2}}}}}{{{v_{He}}}} = \dfrac{{\sqrt 3 }}{5}$
Hence, the correct option is $(C)\dfrac{{\sqrt 3 }}{5}$ .

Note:
 In this formula, $\gamma $ is present which is known as heat capacity ratio. It is the ratio of heat capacity at constant pressure to the heat capacity at constant volume. The value of this heat capacity ratio is different for different types of gases such as monatomic, diatomic, triatomic and polyatomic gases.