BITSAT 2024 Shift 1 Paper

In a mixture of gases where the average number of degrees of freedom per molecule is 6 , the velocity of sound in the gas can be determined using the following relationships:The root mean square (rms) speed $v_{\text{rms}}$ is given by:$v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$The velocity of sound $v_{\text{sound}}$ is given by:$v_{\text{sound}} = \sqrt{\frac{\gamma R T}{M}}$The ratio of the velocity of sound to the rms speed is:$\frac{v_{\text{sound}}}{v_{\text{rms}}} = \sqrt{\frac{\gamma}{3}}$Given that the degrees of freedom $f = 6$, we calculate the adiabatic index $\gamma$ as follows:$\gamma = 1 + \frac{2}{f} = 1 + \frac{2}{6} = \frac{4}{3}$Substituting $\gamma = \frac{4}{3}$ into the equation for the velocity of sound:$v_{\text{sound}} = \sqrt{\frac{\frac{4}{3}}{3}} v_{\text{rms}} = \frac{2}{3} v_{\text{rms}}$Therefore, the velocity of sound in the gas is:$\frac{2c}{3}$