Open Problems

Periodica Mathematica Hungarica -     

Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

Neutral oxygen atom density in the MESOX air plasma solar furnace facility

The density of neutral oxygen atoms in the MESOX set-up, one device of the PROMES-CNRS solar facilities, was determined by a fiber-optics catalytic probe (FOCP). Plasma was created in a flowing air within a quartz tube with the outer diameter of 5 cm by a 2.45 GHz microwave generator with the output power up to 1000 W. The flow of air was varied between 4 and 20 l/h. The O-atom density was found to increase monotonously with the increasing discharge power, and it decreased with the increasing flow rate. The degree of dissociation of oxygen molecules in the plasma column depended largely on the flow rate. At the air flow of 4 l/h it was about 80% but it decreased to about 20% at the flow of 20 l/h.     

Beiträge zur Chemie des Phosphors. 203 [1].(t- BuSi)4P3Cl3, das erste Silaphosphan mit Norcuban-Struktur

Contributions to the Chemistry of Phosphorus. 203. (t-BuSi)₄P₃Cl₃ – The First Silaphosphane with a Structure Analogous to Norcubane The reaction of t-BuSiCl₃ with LiAl(PH₂)₄ yields the compound (t-BuSi)₄P₃Cl₃ ( 1 ), which has been isolated in pure form. According to NMR spectroscopic investigations and an X-ray single-crystal analysis, 1 is 2,4,6,7-tetra-tert-butyl-2,4,7-trichloro-1,3,5-triphospha-2,4,6,7-tetrasilatricyclo[3.1.1.0^3,6]heptane and thus the first silaphosphane with a structure analogous to norcubane. The compound crystallizes monoclinically in the space group P2₁/n with a = 1102.1 pm, b = 1609.2 pm, c = 1612.8 pm, β = 90.90° and Z = 4. The main structural feature is a cube alternately composed of Si and P, from which one phosphorus atom has been removed. The Si atoms are each substituted by t-Bu groups, the three adjacent to the missing corner additionally by Cl atoms.     

Chain and Ring Phosphorus Compounds—Analogies between Phosphorus and Carbon Chemistry

Up to about 15 years ago compounds with a skeleton of phosphorus chains or rings were regarded as “exotic” in the field of nonmetal chemistry. Aside from a number of examples of molecules with two P atoms directly bonded to each other and a few sporadically discovered monocyclic ring compounds, only solids of undefined composition and structure were known. Since then the state of our knowledge in this sector has made considerable progress: between PH₃ and its derivatives on the one hand, and the high-molecular modifications of elementary phosphorus on the other, an unexpected variety of well defined compounds have been discovered, showing many similarities to the analogous compounds of carbon. However, surprises can still occur even with “small” phosphorus-containing molecules, as shown by the likewise recently discovered field of phosphorus three-membered ring compounds.     

Beiträge zur Chemie des Phosphors. 111. 1,2-Dichlor-1,2-di-tert-butyldiphosphan

Contributions to the Chemistry of Phosphorus. 111. 1,2-Dichloro-1,2-di-tert-butyl-diphosphane The reaction of tri-tert-butyl-cyclotriphosphane, (t-BuP)₃, with phosphorus(V) chloride (molar ratio 1:2) leads to the title compound Cl(t-Bu)P? P(t-Bu)Cl ( 1 ), which is remarkably stable against disproportionation reactions. As the first 1,2-dichloro-1,2-diorganyldiphosphane, 1 has been isolated in a pure state and was thoroughly characterized. At room temperature, 1 exists in a mixture of the d,l and meso form (about 20:80). The mutual repulsion of the negative polarized Cl atoms and their preferred gauche arrangement to the free electron pairs of neighboured P atoms leads to a gauche-conformation of the lone electron pairs in the d,l- and to a trans-conformation in the meso-configuration.     

Physicochemical Investigations on the Interaction of Surfactants and Salts with Polyvinylpyrrolidone in Aqueous Medium

Microcalorimetric investigations have been carried out onthe interaction of the surfactants sodium cholate, sodium deoxycholate, tetradecyltrimethylammonium bromide, cetyl(hexadecyl)trimethylammonium bromide, and p-tert-octylphenoxy polyoxy-ethylene ether (Triton X-100) and the salts potassium iodide, sodium benzoate, sodium bromide, and sodium salicylate with the neutral polymer polyvinylpyrrolidone (PVP). The enthalpy of dilution of the surfactants has been measured in the absence and presence of the polymer and the results are compared to determine the effect of PVP on the micellization of the surfactants and the energetics of the process. As well, the micellization activity of the surfactants in the presence of the polymer has been studied by conductometric and fluorimetric methods. The enthalpy of dilution of the salts has been measured to provide an understanding of the nature and magnitude of their interaction with PVP. Copyright 2001 Academic Press.