# What meaning do multiple real solutions of a truncated virial equation have, if any?

Let's say I truncate the virial equation after the third term for use as my equation of state:

$${P\over\rho RT}=1+B\rho+C\rho^2$$

I have tabulated values for $B(T)$ and $C(T)$. I know $P$ and $T$ and want to solve for $\rho$. The equation is cubic in $\rho$ meaning I could potentially have three real solutions. I suspect that since the virial equation is intended to calculate the densities of vapors or supercritical gasses at relatively low pressures, that the lowest real root is the "real world" value. Is this correct? Do the other real values, if they exist, have any meaning? Can this be generalized to higher order virial expansions?

I don’t have much practical experience with virial equations, though I teach it :)

My recommendations would be to identify the correct root based on physical knowledge of the system: first, the root has to be real and its value of $\rho$ has to be positive. This part is obvious. The second thing we know is that the virial equation represents a deviation from the ideal gas law. So, I would take the root that is closest to ideal behavior.

I've tried to confirm it with an example, so I investigate water at 298 K (based on virial coefficients given in K.M. Benjamin et al., J. Phys. Chem. C 2007, 111, 16021-16027). Here's a plot hastily made:

At least in these conditions, this rule works.

• OK, if I'm reading your plot correctly, the largest solution decreases with increasing pressure, not a physically realistic solution behavior. I suppose this is enough to discredit it as a meaningful root. You have answered my question regarding truncation after the 3rd term although I still wonder about higher order terms. Since as @Paul J. Gans says below, practically speaking 4th and higher virial coefficients aren't available it is kind of an academic point. – Jason Waldrop Sep 18 '12 at 16:17

The virial equation is set up for the gas phase only. The virial coefficients $B$ and $C$ are found from gas phase measurements.

If you get three solutions, I suspect that it is the largest one that you want. I'd do a trial solution with something like water vapor to see.

Higher order virial equations are problematic since it is exceptionally difficult to determine higher virial coefficients from experimental data. Higher order virial equations do have theoretical interest, but that's another story.

• I think you meant the smallest solution is the one I want since I wrote the virial equation in terms of molar density rather than molar volume. As an aside, you obtain a quadratic equation in $\rho$ if you truncate after the 2nd term and thus two solutions. The behavior of this quadratic system is that one solution is near zero (the physically correct one) and increasing with pressure whereas the other is positive and decreasing (the extraneous solution). So, the quadratic case can be understood rather easily. – Jason Waldrop Sep 14 '12 at 15:46
• You are quite right. It is the smallest value of the density. My bad. Apologies to all. – Paul J. Gans Sep 15 '12 at 1:37