Question

The velocity of sound in air is 330 m/s. What will be the root mean square velocity of air molecules $ (\gamma = 1.4) $ approximately equal to?
A. 400 m/s
B. 471.4 m/s
C. 231 m/s
D. 462 m/s

Answer 1

Hint
The root mean square velocity is a measure of the average speed of the molecules in a gas. It is inversely proportional to the adiabatic index and directly proportional to the ideal velocity.
Formula used: $ {v_{rms}} = v\sqrt {\dfrac{3}{\gamma }} $ , where $ {v_{rms}} $ is the root mean square velocity, v is the ideal velocity or the speed in air of sound and $ \gamma $ is the adiabatic index.

Complete step by step answer
As sound navigates through the air, the particles in the air will start to move with random velocities. Root mean square is the root of the average of the square of velocities of all the molecules. And in the question, we are required to find this value.
We are provided with the following data in this question:
Velocity of sound in air $ v = 330 $ m/s
Adiabatic index $ \gamma = 1.4 $ [Its value is determined experimentally].
Now, we are aware that the relation between the ideal velocity and the root mean square velocity is given as:
 $\Rightarrow {v_{rms}} = v\sqrt {\dfrac{3}{\gamma }} $
Putting the known values in this equation gives us:
 $\Rightarrow {v_{rms}} = 330\sqrt {\dfrac{3}{{1.4}}} $
Upon solving further, we get:
 $\Rightarrow {v_{rms}} = 330 \times 1.46 $
 $\Rightarrow {v_{rms}} = 483 $ m/s
This value is closest to $ 471.4 $ m/s in the options given. Hence, the final answer is option (B).

Note
We prefer to use the root mean squared value of velocity of molecules of gases for calculations because of the random movements possessed by the molecules. As they move in all the directions, all the velocities add up to a zero. This would not make sense to use in calculations.