At a given temperature,the root mean square velocities of oxygen and hydrogen molecules are in the ratio:

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 $47^{\circ}C$?

Consider an ideal gas with the following distribution of speeds:

Speed $(m/s)$	$\%$ of molecules
$200$	$10$
$400$	$20$
$600$	$40$
$800$	$20$
$1000$	$10$

$(a)$ Calculate $v_{rms}$ and hence $T$. (Given mass of one molecule $m = 3.0 \times 10^{-26} \ kg$, Boltzmann constant $k_B = 1.38 \times 10^{-23} \ J/K$)
$(b)$ If all the molecules with speed $1000 \ m/s$ escape from the system, calculate the new $v_{rms}$ and hence the new $T$.

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 temperature of a gas is $-50^{\circ}C$. To what temperature should the gas be heated so that the $rms$ speed is increased by $3$ times?

If ${V_H}$,${V_N}$,and ${V_O}$ denote the root-mean-square velocities of molecules of hydrogen,nitrogen,and oxygen respectively at a given temperature,then: