Justification of the Shallow-Water Limit for a Rigid-Lid Flow with Bottom Topography

The so-called lake equations arise as the shallow-water limit of the rigid-lid equations—three-dimensional Euler equations with a rigid-lid upper boundary condition—in a horizontally periodic basin with bottom topography. We prove an a priori estimate in the Sobolev space H ^m for m≥ 3 which shows that a solution to the rigid-lid equations can be approximated by a solution of the lake equations for an interval of time which can be estimated in terms of the initial deviation from a columnar configuration and the magnitude of the initial data in H ^m , the gradient of the bottom topography in H ^m+1 , and the aspect ratio of the basin. In particular, any solution to the lake equations remains close to some solution of the rigid-lid equations for an interval of time that can be made arbitrarily large by choosing the aspect ratio of the basin small. Received 10 October 1996 and accepted 15 May 1997     

Consecutive reactions of diethoxydimethylsilane and triethoxymethylsilane with methoxide ions in methanol solution

The consecutive reactions of (CH₃)₂Si(OC₂H₅)₂ and CH₃Si(OC₂H₅)₃ with methoxide ions were investigated in methanol solutions. The reverse transesterification reactions with ethoxide ions could be neglected in both cases since the concentration of ethoxide in methanol solution was assumed to be low due to the fast equilibrium reaction C₂H₅O^? + CH₃OH ? C₂H₅OH + CH₃O^?. The progress of the reactions was followed by monitoring the formation of ethanol with a Fourier-transform infrared spectrometer. All rate constants were determined at 295 K. The reactions between the dialkoxydimethylsilanes and methoxide ions were assumed to consist of two consecutive steps that can be represented by the net reaction; (CH₃)₂Si(OC₂H₅)₂ + 2CH₃O^? → (CH₃)₂Si(OCH₃)₂ + 2C₂H₅O^?. The two consecutive rate constants were established as 1.93 ± 0.12M^?1s^?1 and 1.00 ± 0.12M^?1s^?1, respectively. The consecutive rate constants for the reactions between the trialkoxymethylsilanes and methoxide ions can be written according to the total reaction; CH₃Si(OC₂H₅)₃ + 3CH₃O^? → CH₃Si(OCH₃)₃ + 3C₂H₅O^?. The three rate constants corresponding to each consecutive step were established as 1.12 ± 0.09 M^?1s^?1, 0.82 ± 0.10 M^?1s^?1, and 0.51 ± 0.06 M^?1s^?1, respectively. © 1995 John Wiley & Sons, Inc.     

Rearrangements in cobaltic fluoride fluorinations of cycloalkanes with seven or more ring carbons

Cyclo-heptane, -octane, -decane, -dodecane and -octadecane have been fluorinated in the vapour phase with cobaltic fluoride. The perfluorocarbons obtained revealed varying degrees of skeletal rearrangements. Thus, cycloheptane gave perfluoro-cycloheptane and -methylcyclohexane; cyclooctane gave at least eight fluorocarbons including-bicyclo(3,3,0)octane and -cyclooctane; cyclodecane gave a complex mixture from which were isolated perfluoro cis- and trans-bicyclo(4,4,0)decanes but no perfluorocyclodecane; cyclododecane and cyclooctadecane gave complex fluorocarbon mixtures in which none of the fluorocarbons with the original cycloalkene skeleton could be detected.Tetradecafluorobicyclo(3,3,0)octane gave dodecafluorobicyclo (3,3,0)oct-1(5)-ene and decafluorobicyclo(3,3,0)octa-di-1(5), 2-ene when pyrolysed over iron gauze.The fluorination of cyclooctane also afforded 1$\text{H}$-pentadecafluorocyclooctane which was dehydrofluorinated to tetradecafluorocyclooctene. This was converted to a range of derivatives including dodecafluorosuberic acid, 1,2-dichlorotetradecafluorocyclooctane, 1-methoxy tridecafluorocyclooctene and tetradecafluorocyclooctanone.     

Über die Dispersität von Metallen in Trägerkatalysatoren. II. Röntgenkleinwinkeluntersuchungen an Platin- und Palladium-Tonerde-Katalysatoren

On the Dispersion of Metals in Supported Catalysts. II. Small Angle X-ray Investigation. on Platinum/Alumina and Palladium/Alumina Catalysts The metal dispersion in 0.5% Pd/A1₂O₃. 0.5°% Pt/Al₂O₃ abd 1.5% Pt/Al₂O₃ catalysts was investigated by small angle X-ray scattering, Using a Kratky camera the difference scattering curves catalyst-carrier were evaluated. The assumption was made that the scattering centres are globular and that there is no interparticular scattering. From the Guinier approximation the Guinier radii R_G of the metal particles were determined. With the Pd catalyst always positive difference scattering curves were found, whereas negative values were found with the Pt catalysts at a scattering angle greater than 0.022 rad. Therefore only for the sample with 0.5% Pd the Porod invariant \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \tilde Q}$\end{document} and the Porod radius R_P could be determined. From R_G and R_P the parameters of a two- and of a one-parametric distribution were calculated. The results are in good agreement with data from wide angle X-ray diffraction measurements.     

Sorptionseffekte an Eisen(III)-hydroxid-F?llungen

Zusammenfassung Um Material zur Deutung des Mechanismus der Sorption von Sr²⁺-Ionen an Eisen(III)-hydroxid zu gewinnen, werden Meßreihen mitgeteilt, in denen die pH-Abhängigkeit der Sorption sowie deren Beeinflussung bei Fällung mit verschiedenen Basen (NaOH, Ammoniak, Trimethylamin, Benzylamin) untersucht wird. Aus den charakteristischen Unterschieden der Sorptionskurven muß gefolgert werden, daß die Art der Base für die Sorption eine wichtige Rolle spielt.

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Summary The sorption of Sr²⁺ ions on iron(III) hydroxide was studied as a function of the pH, of the precipitating reagent (NaOH, ammonium hydroxide, benzylamine, trimethylamine), of the temperature and of the total concentration.The curves of the relative sorption vs. pH are found to show characteristic differences which have to be explained in a detailed theory of the mechanism.

     

Intramolecular metal—double bond interactions : VII. Intramolecular cyclization of alkenyl derivatives of lithium,aluminum, gallium and indium

The intramolecular cyclization reactions of tri-1-hex-5-enyl Group III derivatives and of 1-hex-5-enyllithium have been investigated and it is proposed that the reactions may be depicted as internal addition across the terminal double bonds of the alkenyl species. The cyclization reaction proceeds more readily for the 1-hex-5-enyl derivatives than for any other chain length and is dependent on temperature, concentration and solvent. ¹H NMR data and cryoscopic molecular weight data are presented for several diisobutyialkenyl- and diisobutylalkyl-aluminum derivatives. These data have been interpreted in terms of a direct intramolecular interaction between the metal center and the π-electron system of the olefinic site. The intramolecular cyclization reactions for the 1-hept-6-enyl, 1-oct-7-enyl- and 1-undec-10-enylaluminum derivatives also were investigated and it was found that cyclization occurred only for the 1-hept-6-enyl derivative.     

Polynuclear manganese grids and clusters—A magnetic perspective

High nuclearity paramagnetic, spin-coupled transition metal clusters and grids are fascinating chemists and physicists partly because of their structural beauty, and the challenge of creating them, but also because of their novel physical properties. Magnetic interactions between the spin centers are a primary focus. This review will examine a selection of Mn(II) polynuclear grids and clusters, with nuclearities in the range Mn₄ to Mn₉. Theoretical treatments of the magnetic properties are discussed, and approaches to solving the exchange problem for ‘large’ spin systems related to computational difficulties. A freely available software package (MAGMUN4.1) is presented as a means of dealing simply with spin-coupled clusters in general, and symmetry reduction methods are discussed briefly as a means of dealing with ‘large’ spin systems.     

Zur Verbindungsbildung von MeO: M2O3. IV. Zur Struktur von monoklinem SrAl2O4

Einkristalle von SrAl₂O₄ wurden durch Hochtemperaturreaktionen von SrO mit Al₂O₃ dargestellt und röntgenographisch untersucht (a = 844,7; b = 881,6; c = 516,3 pm;β = 93,4₂°; Raumgruppe C–;P2₁, Z = 4). Die Tetraederabfolge in den Sechsringen dieser aufgefüllten Tridymit-Struktur wird mit der ähnlicher Oxometallate verglichen. Compound Formation MeO:M₂O₃. IV. Structure of Monoclinic SrAl₂O₄ Single crystals of SrAl₂O₄ were prepared by high temperature CO₂-LASER technique and investigated by X-ray methods. (a = 844.7; b = 881.6; c = 516.3 pm; β = 93.4₂°; space group C–P2₁, Z = 4). The orientation of the tetrahedra in this stuffed tridymite structure is discussed in comparison with related oxometallates.