If the potential energy of a gas molecule is $U = \frac{M}{r^6} - \frac{N}{r^{12}}$,where $M$ and $N$ are positive constants,then the potential energy at equilibrium must be

The total internal energy of $2$ moles of a monatomic gas at a temperature $27^{\circ} C$ is $U$. The total internal energy of $3$ moles of a diatomic gas at a temperature $127^{\circ} C$ is:

$A$ cylindrical tube of uniform cross-sectional area $A$ is fitted with two airtight frictionless pistons. The pistons are connected to each other by a metallic wire. Initially, the pressure of the gas is $P_0$ and the temperature is $T_0$; the atmospheric pressure is also $P_0$. Now, the temperature of the gas is increased to $2T_0$. The tension in the wire will be:

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?

State whether the following statements are True or False:
$(i)$ Due to collision,the density of gas molecules changes.
$(ii)$ The average kinetic energy of $1 \ g$ mass of each gas is equal at the same temperature.
$(iii)$ At the same temperature,the $v_{rms}$ of two different gases is the same.
$(iv)$ At constant temperature,if the pressure of a gas is increased,its mean free path decreases.

The plot that depicts the behavior of the mean free time $t$ (time between two successive collisions) for the molecules of an ideal gas,as a function of temperature $(T)$,qualitatively,is (Graphs are schematic and not drawn to scale)