# Is there a number that describes a gas's departure from the ideal gas law?

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.

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The simplest extension of the ideal gas law is the van der Waals equation of state:

$$(p + \frac{a'}{v^2})(v - b') = kT$$

where $a'$ and $b'$ account for the interatomic attraction and the finite particle size respectively. These two parameters are thus candidates for "number(s) that describe deviations from the ideal gas law".

The proper measure of such deviations are referred to as virial coefficients which are parameters in the virial expansion - a power series expansion of the pressure in terms of the temperature. You can find details at the referenced links or in a stat mech textbook such as Pathria or Huang.

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@space_cadet Thanks for that summary and link. In their light, my question becomes, "Is it possible to make an order-of-magnitude estimate for the second and third virial coefficients for a given gas at a given temperature?" –  Mark Eichenlaub Dec 27 '10 at 9:33
Historical side-note: this was originally developed by Mayer and called Mayer cluster expansion (because of the graph theory underlying the expansion as seen at the bottom of the article on Virial coefficients). Later it was realized that same approach can be generalized to polymer models. –  Marek Dec 27 '10 at 9:59
@Mark: these coefficients can be computed quite precisely to much higher orders (I am sure they can be found somewhere). To do it yourself would be tedious, but the procedure is fairly straightforward: take some nice gas potential (such as Lennard-Jones) and compute $n$-particle partition function. This is just integration and can be carried out numerically, if nothing else helps. –  Marek Dec 27 '10 at 10:01
Lennard-Jones is alright for nonpolar gases, but the computation of the corresponding virial coefficients is nontrivial. The second virial coefficient is pretty difficult (the closed form of the second virial coefficient involves modified Bessel functions), and the third virial coefficient requires the computation of a difficult triple integral. –  user172 Dec 30 '10 at 17:21
For polar gases, one traditionally uses the modification of the Lennard-Jones potential, the Stockmayer potential. The computations, as can be expected with an elaborate modification of Lennard-Jones, are quite intricate. –  user172 Dec 30 '10 at 17:22

I seem to recall it was primarily about heat capacity. A perfect gas has heat capacities of 5/3 or 1 (constant pressure versus volume), because the only energy sinks/sources are translational kinetic energy of the particles (atoms/molecules), and pressure times dV. If these molecules have other degrees of freedom, vibrational or rotational usually, then there are other places for energy (heat) to be stored and the heat capacity increases. This can be made intuitive by thinking where extra heat can go. In an imperfect gas, some of the additional energy doesn't go to translational kinetic energy of the particles (where it contributes to pressure), but instead it goes to other internal (to the particles) degrees of freedom. Even monoatomic gases have some other degrees of freedom (quantum states(orbitals) for the electron(s)), but these are usually of high ennough energy as to not be activated at room temperature. For a given material the heat capacity is a function of temperature and pressure, any departure from the values for a perfcet gas, means that an adiabatic change in pressure/volume will have a different curve than for a perfect gas.

Also the comment about van der Waals, is valid, the particles (atoms/molecules) do take up volume, so not all of the volume is not available for the other particles to bounce around in. This means, the pressure/volume/temperature relationship is modified by this effect as well. But this effect is proportional to density, so at low pressure the heat capacity will be the dominate cause of departure from a perfect gas.

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There are quantities called the compressibility factor

$$Z=\frac{P\bar{V}}{RT}$$

and the fugacity

$$f=P\int_{P^*}^P \frac{Z-1}{P}\mathrm dP$$

that can be considered as measures of departure from ideality. For an ideal gas, $Z=1$ and $f=P$. The fugacity is intimately related to the "chemical potential" (molar Gibbs free energy) of a nonideal gas; books such as those by McQuarrie and Hirschfelder-Curtiss-Bird should have a more complete discussion on these.

I'll just make the additional note that though the van der Waals equation of state (EOS) is standard, there is a more modern EOS that is not more complicated than van der Waals's and usually gives more accurate property predictions (considering its relatively simple form), the Redlich-Kwong equation:

$$P=\frac{RT}{\bar{V}-b}-\frac{a}{\sqrt{T}\bar{V}(\bar{V}+b)}$$

The expression behaves surprisingly well in phase equilibria studies, but certainly not as good as more elaborate multiparameter EOSs. Its advantage is that the EOS is easily manipulable. As can ascertained from the original article, the proposers of this EOS were associated with Shell Petroleum, and they have used it for the phase equilibria of petroleum components.

On the other hand, there has been much research on "cubic equations of state" since then; see this survey article for instance.

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Disclaimer: my undergraduate chemistry thesis a number of years ago tackled the comparison of van der Waals and related EOSs (in particular, comparing the predicted and actual compression factors of various gases), so the concepts are more or less fresh to me. –  user172 Dec 30 '10 at 6:10