Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problem 1.17.

The ideal gas law isn’t entirely accurate for any real gas. For low density gases, one way of accounting for deviations from the ideal gas law is to use a *virial expansion*:

where and are the virial coefficients, and depend on the particular gas we’re modelling. For nitrogen molecules the measured values of are

100 | 0.0082 | ||

200 | 0.0164 | ||

300 | 0.0246 | ||

400 | 0.0328 | ||

500 | 0.0410 | ||

600 | 0.0492 | ||

Using the ideal gas law and the gas constant to get values for at each temperature gives the third column in the table, and then we can use these values to calculate the terms in the fourth column. [Note that I’ve converted Schroeder’s values to SI units.] The corrections are very small so the ideal gas law should work well under these conditions.

As to why is negative for low temperatures and positive for high temperatures, it is known that gas molecules feel a weak attraction when fairly close to each other. At low temperatures, the molecular speed is lower, so this attraction would have a chance to be more strongly felt. Thus the molecules would tend to be closer to each other than if they didn’t interact, resulting in a slightly smaller volume. A negative value of (at a given and ) means a smaller volume.

For higher temperatures, the molecules are moving too fast for this attraction to have any effect, so molecules simply bounce off each other. Because the molecules have a non-zero volume (as opposed to the point molecules assumed by the ideal gas law), a slightly larger volume is needed at a given (high) temperature and pressure.

Another equation of state (that is, a relation between , and ) is the van der Waals equation:

where the parameters and are constant for a given gas. To compare this to the virial expansion above, we can write this as

By Taylor-expanding the first term in brackets, assuming , we get

Comparing with 1 we see that the van der Waals model predicts

By fitting the curve 8 to the data in the table above, we can get estimates for and . I used Maple’s *Fit* function (which does a least squares fit), with the result:

Comparing the value of with the values of in the above table, we see that our assumption of is consistent, so we’re safe.

The following plot illustrates how good the fit is:

The green curve is the van der Waals fit 8 and the red crosses are the data from the table above. [Note that is plotted using Schroeder’s units, which are just SI units multiplied by .] The fit is actually fairly good, so the van der Waals equation is a decent model for these data.