The value closest to the thermal velocity of a Helium atom at room temperature $(300\,K)$ in $m/s$ is $[k_B = 1.4 \times 10^{-23}\,J/K; m_{He} = 7 \times 10^{-27}\,kg]$.

The molecules of a given mass of a gas have a $r.m.s.$ velocity of $200 \, m/s$ at $27^{\circ}C$ and $1.0 \times 10^5 \, N/m^2$ pressure. When the temperature is $127^{\circ}C$ and pressure is $0.5 \times 10^5 \, N/m^2$,the $r.m.s.$ velocity in $m/s$ will be

At what temperature $(K)$ will the $r.m.s.$ velocity of hydrogen gas be equal to the escape velocity from the Earth?

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 $^{\circ} C$)?

$A$ cubical box with porous walls containing an equal number of ${O_2}$ and ${H_2}$ molecules is placed in a large evacuated chamber. The entire system is maintained at constant temperature $T$. The ratio of the ${v_{rms}}$ of ${O_2}$ molecules to that of the ${v_{rms}}$ of ${H_2}$ molecules,found in the chamber outside the box after a short interval,is:

Derive the equation of $v_{rms}$ in terms of the mass of a molecule.