Two flasks of equal volume are filled with $N_2$ gas and connected by a thin tube of negligible volume. When both flasks are immersed in boiling water,the pressure of the gas inside the system is $0.5 \, atm$. Calculate the pressure of the system when one flask is immersed in ice water and the other is kept in boiling water. (in $, atm$)

Fill in the blanks:
$(1)$ Each line of the graph of pressure versus temperature (in Kelvin) at a specific molar volume is called ........... .
$(2)$ $A$ gas that strictly follows Boyle's,Charles's,and Avogadro's laws is called ........... .
$(3)$ The ratio of the most probable speed to the average speed is ........... .
$(4)$ The ratio of $U_{rms}$ to $U_{mp}$ is ........... .

If the density of a mixture of nitrogen and oxygen gases at $400 \ K$ and $1 \ atm$ pressure is $0.920 \ g \ L^{-1}$, what is the mole fraction of nitrogen in the mixture? ($R=0.082 \ L \ atm \ mol^{-1} \ K^{-1}$; assume ideal gas behavior for oxygen and nitrogen)

The total pressure in a container containing a gas and a few drops of water is $830 \, mm \, Hg$. If the temperature of the container is decreased by $1 \%$,what will be the new pressure of the container in $mm \, Hg$? The vapor pressure of water at the two temperatures is $30 \, mm \, Hg$ and $25 \, mm \, Hg$ respectively.

Consider the following statements:
$Statement-I$ : If the intermolecular forces are stronger than thermal energy,the substance prefers to be in a gaseous state.
$Statement-II$ : Among all elements,the total number of elements available as gases at room temperature is $10$.

What will be the pressure of the gaseous mixture when $0.5 \, L$ of $H_{2}$ at $0.8 \, bar$ and $2.0 \, L$ of dioxygen at $0.7 \, bar$ are introduced in a $1 \, L$ vessel at $27^{\circ} C$ (in $, bar$)?