$A$ gas bulb of $1 \ L$ capacity contains $2.0 \times 10^{21}$ molecules of nitrogen gas at a pressure of $7.57 \times 10^3 \ N \ m^{-2}$. Calculate the root mean square speed in $cm \ sec^{-1}$. (in $.5$)

The following graph is obtained for a gas at different temperatures $T_1$,$T_2$,$T_3$. What is the correct order of temperature? ($x$-axis $=$ velocity; $y$-axis $=$ number of molecules)

The r.m.s. velocity of $CO_2$ at a temperature $T$ (in kelvin) is $x \ cm \ s^{-1}$. At what temperature (in kelvin), the r.m.s. velocity of nitrous oxide would be $4 x \ cm \ s^{-1}$ (in $T$)? (Atomic weights of $C, N$ and $O$ are respectively $12, 14$ and $16$)

Helium diffuses four times faster than which of the following?

Root mean square $(rms)$ speed of $O_2$ is $500 \ m/s$ at a constant temperature. Calculate the $rms$ speed and the average kinetic energy of $H_2$ at the same temperature. (Consider, $R=8.33 \ J \ K^{-1} \ mol^{-1}$)

Case $-1:$ An ideal gas of molecular weight $M$ at temperature $T$.
Case $-2:$ Another ideal gas of molecular weight $2M$ at temperature $T/2$.
Identify the correct statement in the context of the above two cases.