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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?

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    $\begingroup$ Even if you restrict yourself to a cubic equation of state, you can certainly do better than van der Waals. Some tests I did a few years back point to the Redlich-Kwong equation being one of the best two-parameter equations of state for a good range of pressures and temperatures. Of course, if you need to do better than that, you use virial... $\endgroup$
    – user95
    Commented May 13, 2012 at 14:48
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    $\begingroup$ While handling volumetric and thermal departure functions of most hydrocarbons ok, Redlich-Kwong is a poor choice for vapor-liquid-equilibrium calculations. Much better without sacrificing the basic cubic equation structure is the Soave-Redlich-Kwong or Peng-Robinson EOSs. $\endgroup$
    – Jason Waldrop
    Commented Aug 10, 2012 at 21:31
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    $\begingroup$ Mathematically we can't even solve the gravitational 3-body problem in general with a closed form solution. And that has simpler forces (well, gravity, with a neat inverse-square law) than those that act between molecules in a gas. So that's not very hopeful. $\endgroup$
    – matt_black
    Commented Nov 12, 2012 at 23:56
  • $\begingroup$ Simple mathematical expressions reflect the underlying mathematical forms of the physical equations are simple. However, the underlying physical theory, namely quantum theory, is not simple :( Hence, we would naturally expect the exact solution (presumably it exists), even its accurate approximation, not to be simple... $\endgroup$
    – user26143
    Commented Oct 20, 2016 at 22:49

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It's been known since 1941 that the answer to your question is in the negative, i.e. that there will never be a closed form equation of state for a nonideal gas.

In 1941 Mayer and Montroll developed what is now known as the cluster expansion for the partition function of a nonideal gas whose particles have pairwise interactions. This cluster expansion provides a 1-1 correspondence between various integrals over the interaction potential and virial coefficients in the Kamerlingh Ohnes equation of state which has in principle an infinite number of terms in it. Therefore, it would not be considered a closed form equation of state. In practice the virial coefficients are known to decay so it is usually safe to truncate the expansion for practical calculations.

One could extend the cluster expansion to three-body and higher-order interaction potentials but this will not change the fundamental argument above.

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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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    $\begingroup$ Of course, on scale of the individual molecules, thermodynamics itself becomes meaningless. $\endgroup$
    – Jiahao Chen
    Commented May 12, 2012 at 4:23
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I'll add to Aesin’s answer that in this case, the burden of proof rests on the side of an analytical (or closed-form) equation of state. Statistical mechanics explicitly guarantees that there is a relationship between $p$, $V$ and $T$, i.e. that they are not independent state variables. However, no further generic statement can be made about it, and only by making appropriate approximations can one actually write an actual equation of state (EOS). One such EOS is the ideal gas law, others are as you have cited, but if one were to prove the existence of a closed-form generic EOS for fluids, that would be a major upheaval of the current understanding of thermodynamics of fluids.

For an overview of the thermodynamics of fluids, you can refer to your favorite statistical mechanics or statistic thermodynamics textbook (my own preference goes to McQuarrie). For a introduction to issues specific to the liquid state (but also valid in general for fluids far from the ideal gas behaviour), I would recommend Theory of Simple Liquids, by Hansen and McDonald.

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Using the corresponding states approach (normalizing to the critical pressure and critical temperature) usually gives pretty accurate results for design purposes.

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There are different factors contributing to the pVT state, and some of them have (different!) temperature dependencies and are interacting.

  • volume of particles
  • electrostatic repulsion of electron shells
  • London dispersion
  • dipolar interaction
  • energy distribution of particles
    • kinetic energy
    • rotational excitation (quantised, starts at moderate temperatures)
    • vibrational excitation (quantised, higher temperatures)
    • electronic excitation (even higher temp.)
  • and a few more things, like spin states et cetera

To model this in a single equation, you surely need a lot of parameters. Benedict, Webb and Rubin use eight, and thats not the end of it.

If it is already impossible to find a closed equation for one gas, think about gas mixtures, or even reaction mixtures.

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Some general comments on the current use of the VDW equation in both teaching and research (added as an answer at the OP's suggestion):

The OP writes: "[VDW] is only still around for historical purposes, as it is largely inaccurate." That's not quite right—it's like concluding that, since the ideal gas law is inaccurate, it must only persist for historical reasons. Rather, the ideal gas law persists because it provides the simplest model for gas behavior, and VDW persists because it provides the next-most-simple model. Both serve as powerful pedagogical tools for progressively developing student understanding; VDW, in particular, provides a simple and intuitive way to introduce intermolecular interactions, and their effects.

I'd also argue that VDW serves a useful purpose in research, as an ansatz (a simple paradigm) for understanding gas behavior. Specifically, the fact that certain properties can be predicted by this simple two-parameter model, and others can't, provides insight into the nature of those respective properties (into how they arise).

I.e., it's very useful to know whether a certain behavior can be predicted by a minimal model, vs. requiring a more complex one. Knowing this tells you something about that behavior. It's often desirable to know the simplest model that can predict a certain property.

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