A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural "Clifford" Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the K3 surfaces associated to infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.