Origins of the Failure of the Activity Virial Series

We examine the development of the virial equation of state when expressed as a series in the activity with coefficients labeled b_n. Using the one-dimensional hard-rod model as a prototype, we consider steps in its development that introduce inaccuracy that leads it to form a divergent series. We discuss the role of volume dependence of the virial coefficients and present expressions and calculations for volume-dependent coefficients b_n(V) for the hard-rod model up to n = 200. We examine alternative methods for computing properties from the b_n. We recommend that further efforts be made to compute volume-dependent virial coefficients as a means to understand better the virial equation of state and to make it more robust in applications.