Students’ conceptual understanding of a function and its derivative in an experimental calculus course

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its derivative as viewed by students of a reformed Calculus 1 course offered in two experimental sections at the Lebanese American University in Beirut, Lebanon. Results have shown that the general approach adopted in the course proved to be unpopular for a great majority of the students, but rewarding for others. Interviews conducted with some students and a study of their performance on very specific exam questions reveal that for most students, the algebraic representation of a function still dominated their thinking; however, these students showed an almost complete understanding of the derivative, particularly the idea of the instantaneous rate of change and/or the slope of a curve at a given point. Furthermore, very few of these students referred to the mechanical methods for finding derivatives.     

Cation transport by a redox-active synthetic ion channel

The synthesis, cation binding and transmembrane conductive properties of a novel group of synthetic ion channels containing a redox-active centre are described. Experiments using a black lipid membrane preparation revealed that these compounds function effectively as ion channels. Subsequent 23Na NMR spectroscopy studies focused on a synthesized ion channel with a ferrocene centre. When incorporated in vesicular bilayers, this channel was demonstrated to support a Na+ flux that was at least six times faster than ion transport by monensin. Since oxidation of the ferrocene moiety completely inhibited the Na+ transport, the redox-active centre provides a potential mechanism for controlling ion flux.     

The site of protonation of bifunctional bases with competing basic centers. I. Aromatic nitriles

The experimental values of the gas-phase proton affinities for a variety of 4-substituted benzonitriles, 4-substitutedN, N-dimethylanilines, and 4-substituted benzaldehydes have been examined by means of correlation analysis techniques and by ab initio quantum mechanical methods (MP2/ 6-31G(d) level). From this study it is concluded that in the gas phase, 4-(dimethylamino)-benzonitrile essentially protonates on the dimethylamino group, while protonated 4-cyanobenzaldehyde is very nearly a 21 mixture of the carbonyl- and cyano-protonated forms.This work is dedicatedin memoriam to Professor Robert W. Taft.     

Kinetics of the Formation of Goethite in the Presence of Sulfates and Chlorides of Monovalent Cations

The synthesis of goethite by oxidation of Fe²⁺in presence of metallic iron was undertaken in an aqueous medium containing indifferent salts such as Na₂SO₄, (NH₄)₂SO₄, NaCl, and NH₄Cl. Temperature and bubbling air rate were maintained, respectively, at 70°C and 1 L/min. The influence of anions and cations on the kinetics of each step of the process has been followed distinctly, the iron dissolution rate has been determined by the variation of the medium acidity, and the precipitation of goethite has been determined by gravimetric measurements. With respect to Cl⁻, the SO₄²⁻anion decreases the rate of the two reactions. NH₄⁺acts as an inhibitor when it is present at low concentrations and as an accelerator for higher concentrations; the limit corresponding to the change of NH₄⁺behavior depends on the nature of the counter ion. The reaction product is composed of pure goethite in the presence of sulfate salts, whereas a mixture of goethite and lepidocrocite, respectively, 60–70 and 40–30%, was observed in the presence of chloride salts.     

Energy funneling of IR photons captured by dendritic antennae and acceptor mode specificity: anti-stokes resonance raman studies on iron(III) porphyrin complexes with a poly(aryl ether) dendrimer framework

A series of poly(aryl ether) dendrimer chloroiron(III) porphyrin complexes (LnTPP)Fe(III)Cl (number of aryl layers [n]=3 to 5) were synthesized, and their Boltzmann temperatures under IR irradiation were evaluated from ratios of Stokes to anti-Stokes intensities of resonance Raman bands. While the Boltzmann temperature of neat solvent was unaltered by IR irradiation (LnTPP)Fe(III)Cl (n=3 to 5), all showed a temperature rise that was larger than that of the solvent and greater as the dendrimer framework was larger. Among vibrational modes of the metalloporphyrin core, the temperature rise of an axial Fe-Cl stretching mode at 355 cm-1 was larger than that for a porphyrin in-plane mode at 390 cm-1. Although most of the IR energy is captured by the phenyl nu8 mode at 1597 cm-1 of the dendrimer framework, an anti-Stokes Raman band of the phenyl nu8 mode was not detected, suggesting the extremely fast vibrational relaxation of the phenyl mode. From these observations, it is proposed that the energy of IR photons captured by the aryl dendrimer framework is transferred to the axial Fe-Cl bond of the iron porphyrin core and then relaxed to the porphyrin macrocycle.