At what temperature $K$ will the $r.m.s.$ speed of hydrogen gas be equal to the $r.m.s.$ speed of oxygen molecules at $47^{\circ}C$?

The $rms$ speed of helium gas at $27\,^{\circ}C$ and $1\,atm$ pressure is $900\,ms^{-1}$. Then the $rms$ speed of helium molecules at temperature $27\,^{\circ}C$ and $2\,atm$ pressure is ...... $m/s$.

Uranium has two isotopes of masses $235$ and $238$ units. If both of them are present in uranium hexafluoride gas, find the percentage ratio of the difference in rms velocities of the two isotopes to the rms velocity of the heavier isotope.

The $r.m.s.$ speed of the molecules of a gas at a pressure $10^5 \ Pa$ and temperature $0^\circ C$ is $0.5 \ km \ sec^{-1}.$ If the pressure is kept constant but temperature is raised to $819^\circ C,$ the velocity will become ........ $km \ sec^{-1}.$

The given diagram shows isotherms for a fixed mass of an ideal gas at temperatures $T_1$ and $T_2$. What is the value of the ratio $\frac{\text{r.m.s. speed of the molecules at temperature } T_2}{\text{r.m.s. speed of the molecules at temperature } T_1}$?

The root mean square velocity of a gas molecule of mass $m$ at a given temperature is proportional to