The temperature at which the r.m.s. velocity of hydrogen molecules is $4.5$ times that of an oxygen molecule at $47^{\circ}C$ is (molecular weights of hydrogen and oxygen are $2$ and $32$ respectively). (in $^{\circ}C$)

What is $rms$ value? What is $v_{rms}$? Derive the equation of $v_{rms}$ in terms of pressure.

The r.m.s. speed of the molecules of a gas at $100^{\circ} C$ is $v$. The temperature at which the r.m.s. speed will be $\sqrt{3} v$ is: (in $^{\circ} C$)

At what temperature will the $rms$ speed of oxygen molecules become just sufficient for escaping from the Earth's atmosphere $?$ (Given: Mass of oxygen molecule $(m) = 2.76 \times 10^{-26} \, kg$,Boltzmann's constant $k_B = 1.38 \times 10^{-23} \, JK^{-1}$)

The temperature of a gas is $-80^{\circ} C$. To what temperature should the gas be heated so that the r.m.s. speed is increased by $2$ times (in $^{\circ} C$)?

The gas having an average speed four times that of $SO_2$ (molecular mass $64$) is