Chemistry - Why are equations of state for a non-ideal gas so elusive?

Solution 1:

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.

Solution 2:

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.

---

Solution 3:

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.

---

Solution 4:

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

---

Solution 5:

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.