The mean free path for a gas,with molecular diameter $d$ and number density $n$,can be expressed as:

In the absence of intermolecular forces of attraction,the observed pressure $P$ will be

Write the condition for a real gas to behave as an ideal gas.

$A$ $25 \times 10^{-3} \, m^3$ volume cylinder is filled with $1 \, mol$ of $O_2$ gas at room temperature $(300 \, K)$. The molecular diameter of $O_2$ is $0.3 \, nm$ and its root mean square speed is $200 \, m/s$. What is the average collision rate (per second) for an $O_2$ molecule?

Given below are two statements:
Statement $(I)$: The mean free path of gas molecules is inversely proportional to the square of the molecular diameter.
Statement $(II)$: The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas.
In the light of the above statements,choose the correct answer from the options given below:

There are two identical chambers,completely thermally insulated from surroundings. Both chambers have a partition wall dividing the chambers into two compartments. Compartment $1$ is filled with an ideal gas and Compartment $3$ is filled with a real gas. Compartments $2$ and $4$ are vacuum. $A$ small hole (orifice) is made in the partition walls and the gases are allowed to expand into the vacuum.
Statement $-1$: No change in the temperature of the gas takes place when an ideal gas expands in a vacuum. However,the temperature of a real gas goes down (cooling) when it expands in a vacuum.
Statement $-2$: The internal energy of an ideal gas is only kinetic. The internal energy of a real gas is kinetic as well as potential.