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.     

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.     

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.     

Electrocatalytic oxidation of ascorbic acid and voltammetric determination with a ferrocene-modified platinum electrode

Ferrocene attached to the surface of a platinum electrode catalyses the electrochemical oxidation of ascorbic acid in acidic buffer solutions. The overpotential for ascorbic acid oxidation is decreased by 150 mV at pH 2.2 compared with reaction at bare platinum; and an increase in anodic current and decrease in cathodic current for the redox reaction of ferrocene occurs on addition of ascorbic acid to the solution. The ferrocene-modified electrode is useful for the voltammetric determination of ascorbic acid in natural fruit juices. The advantages result from the electrocatalytic effect and from the prevention of adsorption of inhibitory substances from solution.     

Aqueous cholesteric liquid crystals using uncharged rodlike polypeptides

The aqueous, lyotropic liquid-crystalline phase behavior of the alpha-helical polypeptide, poly(N(epsilon)-2-[2-(2-methoxyethoxy)ethoxy]acetyl-lysine) (1), has been studied using optical microscopy and X-ray scattering. Solutions of optically pure 1 were found to form cholesteric liquid crystals at volume fractions that decreased with increasing average chain length. At very high volume fractions, the formation of a hexagonal mesophase was observed. The pitch of the cholesteric phase could be varied by a mixture of enantiomeric samples L-1 and D-1, where the pitch increased as the mixture approached equimolar. The cholesteric phases could be untwisted, using either magnetic field or shear flow, into nematic phases, which relaxed into cholesterics upon removal of field or shear. We have found that the phase diagram of 1 in aqueous solution parallels that of poly(gamma-benzyl glutamate) in organic solvents, thus providing a useful system for liquid-crystal applications requiring water as solvent.     

Stereodynamics of 9,11-Diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecanes. Highly Restricted Nitrogen Inversion and Isolated Phenyl Rotation. X-ray Crystallographic, Dynamic NMR, and Molecular Mechanics Studies

The (1)H NMR spectra of 10-benzyl-9,11-diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecane (BnPh(2)()) and 10-methyl-9,11-diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecane (MePh(2)()) decoalesce due to slowing inversion at nitrogen and to slowing isolated bridgehead phenyl rotation. The high nitrogen inversion barriers in MePh(2)() (DeltaG() = 12.2 +/- 0.1 kcal/mol at 250 K) and BnPh(2)() (DeltaG() = 10.6 +/- 0.1 kcal/mol at 215 K) are typical of tertiary amines in which at least one C-N-C bond angle is constrained to a small value. Compared to the minuscule rotation barriers about sp(2)-sp(3) carbon-carbon bonds in simple molecular systems, the bridgehead phenyl rotation barriers in MePh(2)() (DeltaG() = 9.8 +/- 0.1 kcal/mol at 210 K) and BnPh(2)() (DeltaG() = 9.8 +/- 0.1 kcal/mol at 210 K) are unusually high. Molecular mechanics calculations (MMX force field) suggest that the origin of the high phenyl rotation barriers lies in the close passage of an o-phenyl proton and a methyl (or benzylmethylene) proton in the transition state. BnPh(2)() crystallized from hexane as white needles in the monoclinic system Pn. Unit cell dimensions are as follows: a = 12.198(1) ?, b = 6.1399(6) ?, c = 14.938(2) ?, beta = 107.470(4) degrees, V = 1067.1(2) ?(3), Z = 2. In the crystal molecular structure, the imine bridge CNC bond angle in BnPh(2)() is constrained to a small value (96 degrees ). The benzylic phenyl group is oriented gauche to the nitrogen lone pair.