The ideal gas equation (daresay "law") is a fascinating combination of the work of dozens of scientists over a long period of time.
I encountered Van der Waals interpretation for non-ideal gases early on, and it was always somewhat in a "closed-form" $$\left( p + \frac{n^2a}{V^2} \right)(V - nb) = nRT$$
with $a$ being a measure of the charge interactions between the particles and $b$ being a measure of the volume interactions.
Understandably, this equation is only still around for historical purposes, as it is largely inaccurate.
Fast-forwarding to the 1990s, Wikipedia has a listing of one of the more current manifestations (of Elliott, Suresh, and Donohue):
$$\frac{p V_\mathrm{m}}{RT} = Z = 1 + Z^{\mathrm{rep}} + Z^{\mathrm{att}}$$
where the repulsive and attractive forces between the molecules are proportional to a shape number ($c = 1$ for spherical molecules, a quadratic for others) and reduced number density, which is a function of Boltzmann's constant, etc (point being, a lot of "fudge factors" and approximations are getting thrown into the mix).
Rather than seeking an explanation of all of this, I am wondering whether a more "closed form" solution lies at the end of the tunnel, or whether the approximations brought forth in the more modern models will have to suffice?
Answer
At the end of the tunnel, you're still trying to approximate the statistical average of interactions between individual molecules using macroscopic quantities. The refinements add more parameters because you're trying to parametrise the overall effect of those individual interactions for every property that is involved for each molecule.
You're never going to get a unified "parameter-free" solution for those without going down to the scale of the individual molecules (e.g. ab initio molecular dynamics), as far as I can tell.
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