Question

The velocity of sound in a gas is 300 m/s. The root mean square velocity of the molecules is of the order of
A. 4 m/s
B. 40 m/s
C. 400 m/s
D. 4000 m/s

Answer 1

Hint: \[The{\text{ }}velocity{\text{ }}of{\text{ }}sound{\text{ }}in{\text{ }}any{\text{ }}gas{\text{ }}is{\text{ }} = \;\sqrt {\dfrac{{\gamma RT}}{M}} \],
Where $\gamma $ is the ratio of specific heats at constant pressure and constant volume.
i.e., $\gamma = \dfrac{{{C_p}}}{{{C_v}}}$,
R is the Universal gas constant,
T is the Temperature,
And M is the molecular mass.
\[Root{\text{ }}mean{\text{ }}square{\text{ }}speed{\text{ }} = \;\;\sqrt {\dfrac{{3RT}}{M}} \].

Complete step by step solution:
Root mean square speed means square root of the average of the square of the velocity of the molecules of gas.
${\upsilon _{rms}} = \sqrt {\dfrac{{{v_1}^2 + {v_2}^2 + {v_3}^2 +. ......{v_n}^2}}{n}} $,
n is the number of particles.
Or in other words,
As we know that Kinetic Energy for the gas molecule at temperature T is,
${E_K} = \dfrac{3}{2}RT$,
Where R is Universal gas constant,
T is temperature in Kelvin,
${E_K} = \dfrac{1}{2}m{\upsilon ^2}$,
Where, m is the mass of particle,
$\upsilon $ is the root mean square velocity of particle
Now, by combining these two,
We will get,
${\upsilon ^2} = \dfrac{{3RT}}{M}$,
${\upsilon _{rms}} = \sqrt {\dfrac{{3RT}}{M}} $.

As we know that the velocity of sound of different gases is different.
Let v be the velocity of sound in air.
Let c be the Root mean square speed of air.
$v = \sqrt {\dfrac{{\gamma RT}}{M}} $ $\left( {equation \to 1} \right)$
$c = \sqrt {\dfrac{{3RT}}{M}} $ $\left( {equation \to 2} \right)$
Now, Dividing equation 1 and equation 2,
$\dfrac{v}{c} = \sqrt {\dfrac{\gamma }{3}} $ $\left( {equation \to 3} \right)$,
Now, we have to find $\gamma $.
Here, we have to find the $\gamma $ for diatomic molecules,
For diatomic molecule,
${C_p} = \dfrac{5}{2}R$,
${C_P}$ is specific heat at constant pressure,
${C_v} = \dfrac{3}{2}R$,
${C_v}$ is specific heat at constant volume,

Specific heat is the amount of heat per unit mass required to raise the temperature by $1$ degree Celsius.
So, $\gamma = \dfrac{{\dfrac{{5R}}{2}}}{{\dfrac{{3R}}{2}}}$,
$\gamma = 1.4$,
Now put the value of $\gamma $ in equation 3,
$\dfrac{v}{c} = \sqrt {\dfrac{{1.4}}{3}} $,
\[Velocity{\text{ }}of{\text{ }}sound{\text{ }}in{\text{ }}gas{\text{ }} = {\text{ }}300m/s\left( {given} \right)\]
$c = 300 \times \sqrt {\dfrac{3}{{1.4}}} $,
$c = 400m/s(approx)$

$\therefore $ Option(C) is correct.

Note: The value of $\gamma $ depends on whether the molecule is mono-atomic or diatomic. Root means square speed of a molecule depends on the temperature and molecular mass.