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The velocity of sound in air is V and the root mean square velocity of the molecules is c. The ratio of v to c is
A. $\dfrac{3}{\gamma }$
B. $\dfrac{\gamma }{3}$
C. $\sqrt {\dfrac{3}{\gamma }} $
D. $\sqrt {\dfrac{\gamma }{3}} $

Last updated date: 23rd Jun 2024
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Hint: Use the formula of speed of sound in gas $\sqrt {\dfrac{{\gamma RT}}{M}} $ and the root mean value $\sqrt {\dfrac{{3RT}}{M}} $ and then find their ratio.

Complete step by step solution:
We know the speed of sound in gas given by:
$\rho = $ $V = \sqrt {\dfrac{{\gamma RT}}{M}} $

And the root mean value is given by:
$C = \sqrt {\dfrac{{3RT}}{M}} $
On dividing v by c, we get,
$\dfrac{V}{C} = \sqrt {\dfrac{\gamma }{3}} $

Hence option d is correct.

Additional Information:
The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior. Sound propagate through a medium following the Newton-Laplace equation: The velocity of sound is $V = \sqrt {\dfrac{K}{\rho }} $ where, K=the modulus of bulk elasticity for gases and $\rho = $density of the medium. So v depends on both of them. Speed of sound decreases with increase in the density of the medium. Speed of sound increases with increase in temperature. Speed of sound increases in proportion to humidity in air. Humidity has a small but significant effect on speed of sound. The speed of sound increases when the sound wave is moving in the direction of wind. The speed of sound decreases when the sound wave is moving in the direction opposite to the direction of wind.

Note: In an ideal gas velocity of sound is only dependent on its temperature.