For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the $\ln P$ versus $\ln V$ graph is plotted for an adiabatic process, as shown.

Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic),the second contains chlorine (diatomic),and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not,in which case is $v_{rms}$ the largest?

$A$ cylindrical tube $AB$ of length $l$, closed at both ends, contains an ideal gas of $1 \text{ mol}$ having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $AB$ and passing through the edge at end $A$. If $P_{A}$ and $P_{B}$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube):

The average translational kinetic energy of $N$ molecules in a gas is $E_1$. The kinetic energy of the electron $(e)$ accelerated from rest through potential difference $V$ volt is $E_2$. The temperature at which $E_1=E_2$ is possible is ( $R=$ gas constant,$N=$ number of molecules).

Match Column $- I$ and Column $- II$ and choose the correct match from the given choices.

Column $- I$	Column $- II$
$(A)$ Root mean square speed of gas molecules	$(P)$ $\frac{1}{3} n m \bar{v}^{2}$
$(B)$ Pressure exerted by ideal gas	$(Q)$ $\sqrt{\frac{3 RT}{M}}$
$(C)$ Average kinetic energy of a molecule	$(R)$ $\frac{5}{2} RT$
$(D)$ Total internal energy of $1$ mole of a diatomic gas	$(S)$ $\frac{3}{2} k_{B} T$

For the equation $\beta = - \frac{1}{V} \left( \frac{dV}{dP} \right)$, identify the correct graph of $\beta$ versus $P$ for an ideal gas at constant temperature.