A student plotted the following four graphs r | vedclass

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(D) The speed of sound $v$ in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma P}{\rho}}$,where $\gamma$ is the adiabatic index,$P$ is the

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(D) The speed of sound $v$ in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma P}{\rho}}$,where $\gamma$ is the adiabatic index,$P$ is the pressure,and $\rho$ is the density of the gas.Since the density $\rho = \frac{m}{V}$ and for an ideal gas $PV = nRT$,we have $\rho = \frac{PM}{RT}$,where $M$ is the molar mass and $T$ is the temperature.Substituting this into the velocity formula: $v = \sqrt{\frac{\gamma P}{PM/RT}} = \sqrt{\frac{\gamma RT}{M}}$.This shows that the speed of sound $v$ depends only on the temperature $T$ and the nature of the gas (molar mass $M$ and $\gamma$),and is independent of the pressure $P$ as long as the temperature remains constant.Therefore,the graph showing a constant velocity $v$ with respect to pressure $P$ is correct,which corresponds to the fourth graph.

$A$ student plotted the following four graphs representing the variation of velocity of sound in a gas with the pressure $P$. Which one is correct?

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