The velocities of sound at the same pressure | vedclass

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Question

(A) The velocity of sound in a gas is given by the formula $v = \sqrt{\frac{\gamma P}{\rho}}$,where $\gamma$ is the adiabatic index,$P$ is the pressur

Vedclass · Accepted Answer

$\frac{1}{{\sqrt 2 }}$

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(A) The velocity of sound in a 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 both gases are monatomic,they have the same adiabatic index $\gamma = 5/3$.Given that the pressure $P$ is the same for both gases,the velocity $v$ is inversely proportional to the square root of the density: $v \propto \frac{1}{\sqrt{\rho}}$.Therefore,the ratio of the velocities is $\frac{v_1}{v_2} = \sqrt{\frac{\rho_2}{\rho_1}}$.Given $\frac{\rho_1}{\rho_2} = 2$,we have $\frac{\rho_2}{\rho_1} = \frac{1}{2}$.Substituting this into the ratio formula: $\frac{v_1}{v_2} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$.

The velocities of sound at the same pressure in two monatomic gases of densities ${\rho _1}$ and ${\rho _2}$ are $v_1$ and $v_2$ respectively. If ${\rho _1}/{\rho _2} = 2$,then the value of $\frac{{{v_1}}}{{{v_2}}}$ is:

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