An Exponential Growth Learning Trajectory: Students’ Emerging Understanding of Exponential Growth Through Covariation

This article presents an Exponential Growth Learning Trajectory (EGLT), a trajectory identifying and characterizing middle grade students’ initial and developing understanding of exponential growth as a result of an instructional emphasis on covariation. The EGLT explicates students’ thinking and learning over time in relation to a set of tasks and activities developed to engender a view of exponential growth as a relation between two continuously covarying quantities. Developed out of two teaching experiments with early adolescents, the EGLT identifies three major stages of students’ conceptual development: prefunctional reasoning, the covariation view, and the correspondence view. The learning trajectory is presented along with three individual students’ progressions through the trajectory as a way to illustrate the variation present in how the participants made sense of ideas about exponential growth.     

The chemistry of metal carbonyl anions : III. Sodium-potassium alloy: An efficient reagent for the production of metal carbonyl anions

Reduction of several metal carbonyl dimers including Mn₂](CO)₁₀, [C₅H₅Fe(C0)₂]₂, Co₂(CO)₈, and [C₅H₅M(CO)₃]₂ (M = Cr, Mo and W) by sodium—potassium alloy (NaK) in tetrahydrofuran at room temperature provides a rapid and clean method for the production of the corresponding metal carbonyl anions in high yield. Isolation and characterization of [n-Bu₄N] [Fe(CO)₂C₅H₅] from the iron dimer reduction is described. Reductions of other carbonyls including M(CO)₆ (M = Cr, Mo and W) and Re₂(CO)₁₀ proceed more slowly than previously established methods and provide principally M₂(CO)₁₀^2? and Re(CO)₅₅^?. Methods for the preparation of Re(CO)₅^? are critically considered. The reaction of NaK with [C₅H₅NiCO]₂ is discussed in relation to previously reported results. Infrared solution spectra of a number of carbonyl anions in THF, obtained in a special infrared solution cell, are reported.     

Structural analysis of porphyrin molecular squares using molecular mechanics and density-functional methods

"Molecular squares" formed from Re(CO)(3)Cl corners and porphyrin sides have potential applications as hosts for catalytic sites and as building blocks for membranes. In these materials, knowledge of the conformations of the squares is important. Molecular-mechanics (MM) and density-functional (DF) calculations have been used iteratively in this work to find the minimum-energy configurations of several porphyrin molecular squares. MM predicts that the steric and torsional interactions at connecting junctures of the square framework determine the overall geometry. Torsional degrees of freedom around these junctures were therefore analyzed using DF methods, giving further insight and helping choose among MM force-field options. Single-point DF calculations on the entire squares showed that the energy and conformation of the entire square could be reliably obtained by performing DF calculations on the critical elements of the square and then piecing them together. This "piecewise" strategy allows for both the major torsional motions and the most important local relaxations of large supramolecular species such as molecular squares.     

Computational insights into the acceptor chemistry of phosphenium cations

Phosphines are traditionally considered as Lewis bases or ligands in transition metal and main group complexes. Despite their electron-rich (lone pair-bearing) nature, an extensive coordination chemistry for Lewis acidic phosphorus centers is being developed; such chemistry provides a new synthetic approach for phosphorus-element bond formation, leading to new types of structures and modes of bonding. Complexes of Ph2P+ with a variety of donor elements (P, N, C) give experimentally short donor-acceptor bond lengths, when compared to other cationic phosphorus Lewis acid complexes. We have calculated that the energy of the lowest unoccupied molecular orbital (LUMO) in Ph2P+ is lower than that of (Me2N)2P+, which partially explains the greater exothermicity of reactions of donors with the diaryl acceptor. Furthermore, the energies required to distort the diphenylphosphenium cation from its ground-state geometry are significantly smaller than those of the diamido cations and, thus, enhance the exothermicity of donor coordination. These computational data, in conjunction with evidence from experimental solid-state structures, indicate that Ph2P+ is a significantly better Lewis acid relative to the more common diaminophosphenium analogues (R2N)2P+ and are used to elucidate the nature of the bonding in donor-phosphenium complexes.     

Binary phase diagrams of some bivalent metal carboxylate systems

Binary phase diagrams are presented for mixtures of lead(II), zinc(II), cadmium (II), manganese(II) and mercury(II) octadecanoates. Evidence is presented to show that addition of one metal carboxylate to another, of the same chain length, does not alter the degree of order of the hydrocarbon chains either in the solid or liquid phase.     

Mass spectra of some aminonaphthyridines and aminoquinolines

The mass spectra of all the aminoquinolines, the 2–, 3– and 4-amino-1,5-naphthyridines, some amino-1,6-naphthyridines, and two amino-1,8-naphthyridines with methyl substituents are reported. The major fragment in the aminoquinolines is formed by the loss of HCN from the molecular ion. The most abundant fragment in the aminonaphthyridines is formed by the loss of HCN from the molecular ion except in the 2-amino-1,5-naphthyridine isomer. In both 1,8-naphthyridine isomers investigated, the loss of C₂H₂ is an alternate fragmentation pathway of significance. In all of the compounds investigated, the loss of the primary amino group from the molecular ion was found to be an insignificant fragmentation.     

The infrared spectra of some dioxygen and carbon monoxide metallopophyrin complexes

The infrared stretching values for O₂ bound to Co, Fe, and Mn porphyrins are reported. It is noted for similar complexes, ν(O₂) is insensitive to variations in the metal center or axial base. The value of ν(O₂) for O₂ adducts of tetraphenylporphyrinatomanganese (II) confirms earlier work that the O₂ moiety is bound in a peroxo-like manner. Also ν(CO) values are reported for the iron (II) porphyrin complexes.     

67Zn solid-state and single-crystal NMR spectroscopy and X-ray crystal structure of zinc formate dihydrate.

The crystal structure, quadrupole coupling parameters, and the orientation of the electric field gradient tensors for each site of zinc formate dihydrate have been determined. There are two distinct sites in the asymmetric unit: one containing four in-plane waters with two bridging formats, the other containing six bridging formates. The solid-state NMR lineshapes have been assigned to their respective sites by using isotopic labeling and cross-polarization methods. The hydrated site corresponds to the lineshape having a quadrupole coupling constant (Cq) of 9.6 MHz and the anhydrous site has a Cq of 6.2 MHz. The absence of chemical shielding contributions to the observed lineshapes has been verified with a high-field solid-state NMR experiment performed at 18.8 T.