Virial equation OF state ( virial equation) || Boyle's temperature & inversion temperature,collision theory & mean free path

The Critical temperature , Boyle's temperature and inversion temperature respectively are given as

Variation OF Z with P& T || Virial Equation OF State

Collision theory|| Arrenius Equation

Write expressions for Boyle temperature and inversion temperature of a gas in terms of van der Waals constants. How are the two related to each other ?

The ratio of the inversion temperature of a gas to its Boyle temperature is

Maxwell’s Speed Distribution Curve||Collision Frequency || Mean free Path

" The ratio of inversion temperature and Boyle's temperature for a real gas is "