...the ideal gas law predicted that has the temperature approached zero, the volume would approach zero. This, and the experimental fact that the law was useful only as a first approximation lead to a famous refinement, the van der Waals equation:
This is, in fact, a cubic equation. It includes a term treating molecules as perfect spheres with fixed volume, as well as a term for the attractive force between molecules which are known, to this day, as "van der Waal's forces" I once wrote an HP41C program to solve it in terms of moles of gas as a function of pressure when I was doing a hydrogenation back when I was a student. The simplest way to have solved the equation would have been to use Newton's method for solving equations, but I knew there was an exact root equation, since Galois proved that cubic equations were the highest order of equations that could be solved by roots. But since I'm hardly Galois, I had no idea what that root equation was, but I tracked it down in the 1943 edition of the Handbook of Chemistry and Physics and spent a day or two getting the HP41C program to solve it debugged.
It worked OK for the hydrogenations, close enough for organic chemistry anyway. It was a waste of time to do this, by the way, but it I did it for fun and to satisfy a certain neuroses
The Peng Robinson equation was a huge refinement of previous refinements of the van der Waals equation, notably the Redlich-Kwong (RK) equation introduced in the 1940's and the Souave Redlich Kwong equation, a further refinement introduced in the early 1970's. The Peng Robinson paper was published in 1976. It is one of the most cited papers in chemistry; Google Scholar lists close to 8,000 citations.
Here is the equation:
There are Peng Robinson solvers all over the internet, and of course, commercial software to do it as well. But they're no fun. Also it's not clear that one can easily program in alternate functions, like those listed in the paper cited in the OP.
The Peng Robinson equation usually uses "the reduced" temperatures and pressures, which is the ratio, with the thermodynamic temperature scale used, between the lab temperature and the critical temperature, in the case of temperature, and ratio of the system pressure and the critical pressure.
Mathematica, by the way, spits out the root solution formula for cubic equation in algebraic format in less than a second, significantly less than a second. It is easy to assign values to the equation's terms and get a numerical solution for both the real and the two imaginary roots. (I programmed to HP41C to give the imaginary roots as well, something a Newtonian solution doesn't give.)
It's great fun.
I'm sure this is more than you wanted to know, but I hope you enjoyed it anyway.
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