What is Inversion Temperature in terms of Joule Thomson Effect? Does Critical Temperature depend upon Inversion Temperature?
This is half-part answer to your question:

Joule Thompson Inversion Temperature is the temperature when Joule Thomson coefficient $\mu_\mathrm{JT}$ is null.

$$\mu_\mathrm{JT} =  \left(\frac{\partial T}{\partial p}\right)_h$$

It means that the gas does not chill ($\mu_\mathrm{JT}&gt;0$) or warm ($\mu_\mathrm{JT}&lt;0$) when it experiences expansion. Pressurized Hydrogen has a negative Joule-Thomson coefficient, which must be taken in account if you plan to stock it and expand it, because it will significantly heat when expended.

This factor is influenced by gas molecular interactions (repulsive and attractive). Therefore, for an Ideal Gas implies that $\mu_\mathrm{JT} = 0$.

You will find detailed explanation in Atkins, Physical Chemistry, Chapter 2: First Law - The Joule-Thomson Effect.