The internal energy of real gases is determined by which of the following?

Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at $2.0 \; atm$ and temperature $17\,^{\circ} C$. Take the radius of a nitrogen molecule to be roughly $1.0 \; \mathring{A}$. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of $N_{2} = 28.0 \; u$).

Calculate the value of mean free path $(\lambda)$ for oxygen molecules at temperature $27^{\circ} C$ and pressure $1.01 \times 10^{5} Pa$. Assume the molecular diameter $d = 0.3 nm$ and the gas is ideal. Given Boltzmann constant $k = 1.38 \times 10^{-23} J K^{-1}$. (Result in $nm$)

For a molecule of an ideal gas,the number density is $2 \sqrt{2} \times 10^8 \text{ cm}^{-3}$ and the mean free path is $\frac{10^{-2}}{\pi} \text{ cm}$. The diameter of the gas molecule is

An ideal gas is enclosed in a cylinder at a pressure of $2 \, atm$ and a temperature of $300 \, K$. The mean time between two successive collisions is $6 \times 10^{-8} \, s$. If the pressure is doubled and the temperature is increased to $500 \, K$,the mean time between two successive collisions will be close to:

$A$ molecule is travelling in air at $300 \, K$ and $1 \, atm$, and the radius of the molecule is $0.6 \times 10^{-10} \, m$. Calculate the approximate mean free path of the molecule. (The number density is $2.44 \times 10^{25} \, \text{molecules}/m^3$)