# How does one experimentally determine virial coefficients?

The Virial expansion is a systematic correction to the ideal gas law which takes the form of an expansion around the number density, $\rho$, of the gas in question. It looks like this: $$\frac{p}{k_bT}=\rho+B_2(T)\rho^2+B_3(T)\rho^3+\cdots$$ The form of this equation is not surprising. An ideal gas is simply one for which the coefficients $B_2$, $B_3$, etc. are equal to zero. These are called the second and third Virial coefficients.

According to Wikipedia (and I have heard this elsewhere), the Virial coefficients systematically correct the ideal gas law by considering specific interactions. To quote the Wikipedia article,

The second virial coefficient $B_{2}$ depends only on the pair interaction between the particles, the third ($B_{3}$) depends on 2- and non-additive 3-body interactions, and so on.

This quote gives me quite a good idea of how I would determine these coefficients theoretically using $\textit{ab initio}$ calculations, but I can't imagine how I could calculate only the 2-body interactions experimentally...

Indeed, this paper seems to present one way to determine them by the so-called Burnett method, but I don't have a way of accessing this paper and have never heard of this method.

So, if anyone could give me a summary of how to determine these coefficients, I would appreciate that and am quite interested to hear the answers.

• The Burnett method is described in this paper. nvlpubs.nist.gov/nistpubs/jres/75C/jresv75Cn3-4p165_A1b.pdf Essentially a near perfect gas is used to calculate an apparatus constant. The apparatus can then be used to measure a gas which does not have near ideal gas behavior. The apparatus has two chambers and a pressure gauge and kept and some temperature. The second chamber is isolated from the first and evacuated. The first chamber is filled with the gas at some measured pressure. The valve between the two chambers is opened and the pressure measured again.
– MaxW
Commented Sep 20, 2016 at 4:15
• re: ab initio calculations I think this would be a lot harder than you imagine. This would in essence be calculating the critical temperature, pressure and volume for a compound.
– MaxW
Commented Sep 20, 2016 at 4:35
• Well I think it would require quite a lot of computational time and I would almost certainly end up using MD due to the necessity of a large number of particles, but what I really meant is that conceptually, it is quite easy to imagine to what would need to be done. In practice it'd probably be tedious like anything else. And thanks for the link. Commented Sep 20, 2016 at 5:52

## 1 Answer

These coefficients can be calculated based on 2 ways:

1) Based on experimental data

In this method, from pressure-volume-temperature (PVT) data these coefficients will be calculated. Consider the following equation: $$\vartheta\left(\frac{P\vartheta}{RT} - 1\right) = B(T) + \frac{C(T)}{\vartheta} +\ldots$$ $$\vartheta = \frac{1}{\rho} =\mathrm{Molar\ volume}$$

This equation is a re arange form of Virial equation. Now accroding to this equation, if you plot the left hand side of aforementioned equation versus molar density, the intercept of that line represents second virial coefficient and the slope of that line is the third virial coefficient. If the curve was not a straight line, we should consider more virial coefficients for them.

2) From potential functions.

In this method, the virial coefficients are calculated by potential functions such as L-J model, Kihara model, Exp-6 model and etc. The basis of This method is statistical thermodynamics.