Two new methods of partitioning the second virial coefficient B into free, bound, and metastable parts, which differ from the well known partitioning of Stogryn and Hirschfelder, are presented. It is shown that the proper partitioning to use depends on the specific physical problem of interest. In particular, in the kinetic theory of moderately dense gases due to Curtiss, Snider, and co-workers, certain collision integrals reduce unambiguously to linear sums of B and its temperature derivatives for repulsive potentials, but it has not been clear to what such integrals reduce for realistic potentials. It is shown that such integrals reduce to the previously derived expressions with B replaced by one of our two new definitions of its free part. This contrasts with previous applications to real gases in which Curtiss and co-workers have used the full B and Kuznetsov has used the free part of B as defined by Stogryn and Hirschfelder. Also, original numerical calculations for the collision integrals are presented and the numerical consistency of the theory is verified.