If the temperature of an ideal gas is increased from $27^{\circ}C$ to $927^{\circ}C$,what happens to its $r.m.s.$ velocity?

$A$ gas $A$ has $N$ molecules at temperature $T$ with mass $m$. Another gas $B$ has $2N$ molecules at the same temperature $T$ with mass $2m$. Both are contained in a vessel. If the $rms$ velocity of molecules of gas $B$ is $v$,and the $x$-component of the mean square velocity of molecules of gas $A$ is $w^2$,find the ratio $w^2/v^2$.

The temperature at which the root mean square speed of a gas will be half of its value at $0\,^{\circ}C$ is ....... $^{\circ}C$.

$A$ sample of gas is at $0^{\circ}C$. To what temperature must it be raised in order to double the $r.m.s.$ speed of the molecules? (in $^{\circ}C$)

The temperature of an ideal gas is reduced from $927^\circ C$ to $27^\circ C$. The $r.m.s.$ velocity of the molecules becomes

The density of hydrogen gas at standard atmospheric pressure $(P = 1.01 \times 10^{5} \ Pa)$ is $0.09 \ kg/m^{3}$. Find the root mean square velocity $(v_{rms})$ and the average kinetic energy of $1 \ mole$ of the gas.