Question

Find the relation between the velocity of sound in gas $\left( v \right)$ and the rms velocity of molecules of a gas $\left( {{v_{rms}}} \right)$.
A. $v = {v_{rms}}{\left( {\dfrac{\gamma }{3}} \right)^{1/2}}$
B. $v = {v_{rms}}{\left( {\dfrac{2}{3}} \right)^{1/2}}$
C. $v = {v_{rms}}$
D. $v = {v_{rms}}{\left( {\dfrac{3}{\gamma }} \right)^{1/2}}$

Answer 1

The velocity of sound in gas depends on the adiabatic constant, the universal gas constant, the temperature of the gas and the molecular mass of the gas. Now according to the kinetic theory of an ideal gas, the rms velocity of gas molecules depends on the universal gas constant, the temperature of the gas and the molecular mass of the gas. A relation between the two velocities can be obtained by taking the ratio of the two.

Formulas used:
The velocity of sound in gas is given by, $v = \sqrt {\dfrac{{\gamma RT}}{M}}$ where, $\gamma$ is the adiabatic constant, $R$ is the universal gas constant, $T$ is the temperature of the gas and $M$ is the molecular mass of the gas.
The rms velocity of gas molecules is given by, ${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}}$ where, $R$ is the universal gas constant, $T$ is the temperature of the gas and $M$ is the molecular mass of the gas.

Complete step by step solution:
Step 1: Express the relation for the velocity of sound in gas and the rms velocity of gas molecules to obtain a relation between the two.
The velocity of sound in gas is given by, $v = \sqrt {\dfrac{{\gamma RT}}{M}}$ ---------- (1) where $\gamma$ is the adiabatic constant, $R$ is the universal gas constant, $T$ is the temperature of the gas and $M$ is the molecular mass of the gas.
The rms velocity of gas molecules is given by, ${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}}$ ---------- (2) where $R$ is the universal gas constant, $T$ is the temperature of the gas and $M$ is the molecular mass of the gas.
Dividing equation (1) by (2) we get, $\dfrac{v}{{{v_{rms}}}} = \dfrac{{\left( {\sqrt {\dfrac{{\gamma RT}}{M}} } \right)}}{{\left( {\sqrt {\dfrac{{3RT}}{M}} } \right)}}$
Cancelling out the similar terms in the numerator and denominator we get, $\dfrac{v}{{{v_{rms}}}} = \sqrt {\dfrac{\gamma }{3}}$
$\Rightarrow v = {v_{rms}}\sqrt {\dfrac{\gamma }{3}}$

Thus the required relation is $v = {v_{rms}}{\left( {\dfrac{\gamma }{3}} \right)^{1/2}}$ and so the correct option is A.

Note:
The kinetic theory of an ideal gas is based on the concept that the molecules of a gas enclosed in a container randomly collide with each other and with the walls of the container. These collisions are considered to be elastic. So the velocity of the molecules in a gas is expressed in terms of the root mean square speed i.e., ${v_{rms}}$ . And so the speed of sound in gas and the rms speed of gas molecules are related to each other.