Question

If $C_{s}$ be the velocity of sound in air and $C$ be the rms velocity, then

& A.{{c}_{s}}>c \\\ & B.{{c}_{s}}=c \\\ & C.{{c}_{s}}=c{{\left\\{ \dfrac{\gamma }{3} \right\\}}^{\dfrac{1}{2}}} \\\ & D.\text{no relation} \\\ \end{aligned}$$

Answer 1

Root-mean-square velocity of gases is the root of the mean of the squares of velocity of all the gas particles in the system, this is taken into calculation , because of the random motion and velocities of the gas particles. And we also know from the ideal gases that the speed of the sound in air is given as $v=\sqrt{\dfrac{\gamma RT}{M}}$, to find the necessary equation, we need to compare the two equations.

Formula used:
$v_{rms}=\sqrt{\dfrac{3RT}{M_{m}}}$ and $v=\sqrt{\dfrac{\gamma RT}{M}}$

Complete step by step solution:
The mean speed, most probable speed and root-mean-square speed are properties of the Maxwell- Boltzmann distribution, which studies the molecular collision of the gas molecules, on the basis of statistical thermodynamics. Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate.It is assumed that the particles don’t interact, and exist as independent particles.
The rms is given as $v_{rms}=\sqrt{\dfrac{3RT}{M_{m}}}$, where $R$ is the gas constant, $T$ is the absolute temperature and $M_{m}$ is the molar mass of the gas particles.
Here, we have $C=\sqrt{\dfrac{3RT}{M_{m}}}$
Similarly, the velocity of the sound in air, assuming air as ideal gas, is given as $v=\sqrt{\dfrac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic index, more commonly known as the degree of freedom, $R$ is the universal gas constant, $T$ is the absolute temperature of the gas and $M$ is the molar mass of the gas.
Here, we have $C_{s}=\sqrt{\dfrac{\gamma RT}{M}}$
Taking the ratio between the speeds, we get $\dfrac{C_{S}}{C}=\dfrac{\sqrt{\dfrac{\gamma RT}{M}}}{\sqrt{\dfrac{3RT}{M_{m}}}}$
$\implies \dfrac{C_{S}}{C}=\sqrt{\dfrac{\gamma}{3}}$
$\therefore C_{S}=C\left(\dfrac{\gamma}{3}\right)^{\dfrac{1}{2}}$
Hence the correct answer is option C.{{c}_{s}}=c{{\left\\{ \dfrac{\gamma }{3} \right\\}}^{\dfrac{1}{2}}}

Note:
Rms velocity is taken instead of normal velocity because of the random motion and velocities of the gas particles. From the equation it is clear that $v_{rms}\propto\sqrt T$, $v_{rms}\propto\dfrac{1}{\sqrt M}$. Here , it is assumed that the particles don’t interact, and exist as independent particles.