When the temperature of an ideal gas is increased from $27^\circ C$ to $227^\circ C$,its $r.m.s.$ speed changes from $400 \ m/s$ to $V_s$. The value of $V_s$ is ........ $m/s$.

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

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

Column-$I$ represents the formula for ${v_{rms}}$ and Column-$II$ represents the corresponding condition (phenomena). Match them correctly:

Column-$I$	Column-$II$
$(a)$ ${v_{rms}} = \sqrt {\frac{3P}{\rho}}$	$(i)$ For $1 \text{ mole ideal gas}$
$(b)$ ${v_{rms}} = \sqrt {\frac{3RT}{M_0}}$	$(ii)$ For one molecule of gas
$(c)$ ${v_{rms}} = \sqrt {\frac{3{k_B}T}{m}}$	$(iii)$ On the basis of kinetic theory

The $rms$ speed of hydrogen molecules at room temperature $(300 \ K)$ will be:

Three vessels $A$,$B$,and $C$ contain gases at the same temperature. Vessel $A$ contains $O_2$ gas,vessel $B$ contains $N_2$ gas,and vessel $C$ contains a mixture of $O_2$ and $N_2$ gases. If the average velocity of $O_2$ in vessel $A$ is $V_1$ and the average velocity of $N_2$ in vessel $B$ is $V_2$,what is the average velocity of $O_2$ in vessel $C$?