The topic of active sites on catalysts is of current interest, that is, identifying and counting them, since this knowledge is relevant to improving chemical behavior. Previous work recognizes that catalytic activity is related to active sites via generalized coordination numbers (GCNs). The method we present uses magical formulas, which apply to general nanocatalysis, since it is related to mathematical properties of fcc nanostructures, and is not element specific. This method is demonstrated for platinum nanostructures, of interest for the oxidation-reduction reaction (ORR). We model four polyhedral clusters, nanocages, and trilayer strained heteroepitaxial clusters. These geometric descriptors quantify the variables of morphology, size, and number of wall layers for these shapes. We use magical formulas for the GCNs to count active sites, thus modeling mass activity for platinum clusters and nanocages with (111) facets having tetrahedral, icosahedral, and octahedral morphology, and (100) cubic symmetry. This model predicts that 20 nm octahedral cages with 3 wall layers will have a mass activity ∼ 3 A / m g Pt 30. 3em A/mgₓ. This is a factor of 2 improvement over octahedral cages of 20 nm size with 6 wall layers. We provide a model of monolayer platinum shells with a strained epitaxial interlayer in ternary clusters, including Pd@Ir@Pt, Pd@Rh@Pt, Pd@Cu@Pt, and Ir@Pd@Pt. The maximum predicted mass activity is ∼ 1. 8 A / m g Pt 1. 8A/mgₓ for tetrahedral sub-10 nm core-shell clusters with a copper interlayer. The generality of the approach is illustrated by a suggested study of the nitrogen reduction reaction (NRR), with fcc ruthenium as the catalyst for similar nanoclusters and cages.
Kaatz et al. (Mon,) studied this question.