When judging if relativity is important in a given phenomenon, we might examine the number $v/c$, with $v$ a typical velocity of the object. If this number is near one, relativity is important. In optics, we examine $\lambda/d$, with $\lambda$ the wavelength of light involved, and $d$ a typical size-scale for the object interacting with light. If this number is near one, physical optics becomes important. In fluid mechanics we have the Reynolds number, etc.
Can we compute a number to tell roughly when a gas will deviate significantly from the ideal gas law?
For simplicity, you may want to deal only with a monatomic gas, but a more general approach would be welcome.