The atmosphere on a planet is possible only if [where $v_{rms}$ is the root mean square speed of gas molecules on the planet and $v_e$ is the escape speed on its surface]

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

The relation between root mean square speed $(v_{rms})$ and most probable speed $(v_{p})$ for the molar mass $M$ of oxygen gas molecule at the temperature of $300\, K$ is:

If the r.m.s. velocity of hydrogen molecules is $4$ times that of an oxygen molecule at $47^{\circ} C$,the temperature of the hydrogen molecules is (Molecular weight of Hydrogen and Oxygen are $2$ and $32$ respectively). (in $^{\circ} C$)

$Assertion :$ The root mean square and most probable speeds of the molecules in a gas are the same.
$Reason :$ The Maxwell distribution for the speed of molecules in a gas is symmetrical.

For a given gas at $1\,atm$ pressure,the $rms$ speed of the molecules is $200\,m/s$ at $127\,^oC$. At $2\,atm$ pressure and at $227\,^oC$,the $rms$ speed of the molecules will be: