A note on the solution of a class of two-point boundary value problems involving a mixed boundary condition

In this note, we consider a class of two-point boundary value problems involving a pa- rameter in one of the boundary conditions. We shall show that if we know the solution corresponding to a particular value of the parameter, then the solution for any other value of the parameter can be obtained by a simple algebraic method.     

Determination of rhodium in water samples using cloud point extraction (CPE) coupled with flame atomic absorption spectrometry (FAAS)

Cloud point extraction was applied as a method for preconcentration of rhodium after formation of a complex with 2-propylpiperidine-1-carbodithioate (2-PPC), and later determined by flame atomic absorption spectrometry using TritonX-114 as surfactant. Rhodium was complexed with 2-PPC in an aqueous phase and kept for 15 min in a thermostatted bath at 40 °C. Separation of the two phases was accomplished by centrifugation for 15 min at 4000 rpm. The chemical variables affecting the cloud point extraction were optimized and successfully applied to rhodium determination in various water samples. Under optimized conditions, the preconcentration system (100 mL sample) permitted an enhancement factor of 50. The detection limits obtained under optimal conditions was 0.052 ng mL⁻¹. The extraction efficiency was investigated at different rhodium concentrations (7.0–42.0 μg mL⁻¹), and good recoveries (96.42–99.14%) were obtained using this method. It has been applied to the determination of rhodium in water and was compared with reported methods in terms of Student’s ‘t’-test and variance ratio ‘f’-test.     

Unified treatment of free convection adjacent to a vertical plate with three thermal boundary conditions

Free convection adjacent to a vertical plate is considered with three different boundary conditions, namely, the plate is subjected to a prescribed temperature, a prescribed heat flux or a prescribed heat transfer coefficient. By a unified treatment of similarity analysis, the governing equations of free convection are reduced to identical system of ordinary differential equations for all the cases. It is shown that these equations are invariant under a certain transformation group and solution for one case can be used to obtain the solutions for the other two cases by a simple method. The critical cases are found for which the solutions for all the three cases are identical.

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Einheitliche Behandlung der freien Konvektion längs einer vertikalen Platte für drei verschiedene Randbedingungen

Zusammenfassung Die freie Konvektion längs einer vertikalen Platte wird für drei verschiedene Randbedingungen betrachtet: einmal wird die Platte auf einer bestimmten Temperatur gehalten, dann einem bestimmten Wärmestrom ausgesetzt und letztens mit einem bestimmten Wärmeübergangskoeffizienten untersucht. Bei einer einheitlichen Behandlung der Ähnlichkeitsanalyse werden für alle Fälle die bestimmenden Gleichungen für freie Konvektion zu einem identischen System von gewöhnlichen Differentialgleichungen reduziert. Es wird beweisen, daß diese Gleichungen unter bestimmten Transformationsgruppen invariant sind und Lösungen von einem Fall dazu benützt werden können, um die Lösungen der restlichen Fälle zu erhalten. Es werden die kritischen Fälle gefunden, für die alle Lösungen der drei entsprechenden Randbedingungen identisch sind.

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Nomenclature c _o positive constant - c _p specific heat of the convective fluid - C function defined by Eq. (6a) - E energy convection by the boundary layer - f dimensionless stream function - g gravitational acceleration - G function defined by Eq. (6b) - k thermal conductivity - N heat transfer coefficient - PHF prescribed heat flux - PHTC prescribed heat transfer coefficient - Pr Prandtl number - PT prescribed temperature - q _w surface heat flux - T temperature - T _e ambient temperature - u velocity component inx direction - v velocity component iny direction - x coordinate in the vertical direction - y coordinate in the horizontal direction - coefficient of thermal expansion - dimensionless similarity variable - kinematic viscosity - dimensionless temperature - density of the convective fluid - stream function     

Hansen Solubility Parameters in the Analysis of Solvent–Solvent Interactions by Inverse Gas Chromatography

The Flory–Huggins interaction parameters, for a solute with pure nonvolatile solvents and with binary mixtures of nonvolatile solvents were evaluated using Inverse Gas Chromatography (IGC) data at 403.15 K. The five solvents used have similar structure and nearly the same molar volume and their mixtures form regular solutions. The values were used to evaluate the Hansen solubility parameters (HSP) for the pure solvents and mixed solvent systems. The HSP are plotted as a function of composition of the polar solvent. The dispersion component, , and the polar component, , exhibited negative deviation whereas the hydrogen bonding component, , showed positive deviation in the four mixtures. However, the total solubility parameter was found to be almost linear with composition in the four binary systems. Further the total solubility parameter, , was used to calculate the molar excess heat of vaporization for the solvents (C78 + POH, C78 + PCN, C78 + TTF, and C78 + TMO) that was found to be negative for C78 (19,24-dioctadecyldotetracontane) + POH(18,23-dioctadecylhentetracontan-1-ol) and C78 + PCN(1-cyano-18,23-dioctadecylhentetracontane) systems and positive for C78 + TTF [19,24-bis(18,18,18-trifluorooctadecyl)-1,1,1,42,42,42-hexafluorodotetracontane), and C78 + TMO(17,22-bis-(16-methoxyhexadecyl)-1,38-dimethoxyoctatriacontane] and the other systems.     

An efficient and simple method for synthesis of 2,2-disubstituted-2H-chromenes by condensation of a phenol with a 1,1-disubstituted propargyl alcohol using BF3·Et2O as the catalyst

An efficient and simple method for the synthesis of 2,2-disubstituted-2H-chromenes by one-step cyclocondensation of a phenol with a variety of 1,1-disubstituted propargyl alcohols using BF₃·Et₂O as the catalyst is described.     

Regio- and diastereoselective synthesis of bifunctionalized limonenes

High regio- and diastereoselective ring opening of limonene aziridines with a variety of nucleophiles is described. The resulting novel chiral derivatives are readily accessible from limonene aziridine in either enantiomeric form in 35-94% yield.     

Photonic nanojet‐mediated SERS technique for enhancing the Raman scattering of a few molecules

We report a two‐step enhancement of Raman scattering signal (η) of a few dye molecules. In the first step, high‐quality surface‐enhanced Raman scattering (SERS) substrates have been used. The SERS substrates were fabricated by direct current sputtering of Au followed by thermal annealing. The role of thermal annealing of the SERS substrates and numerical aperture of Raman microscopic objective lens on the enhancement has been studied for optimizing the enhancement in the SERS technique. In the second step, the value of η obtained with conventional SERS technique has been improved significantly with the help of photonic nanojet (PNJ) of an optical microsphere (PNJ‐mediated SERS technique). The signal to noise ratio and reproducibility of the experimental results have been found to be very high. Based on our theoretical simulations on PNJ, a few suitable parameters have been proposed for obtaining better enhancement using this technique. To the best of our belief, this report will enable the SERS community to improve η value with ease. Copyright © 2016 John Wiley & Sons, Ltd.     

Optical and EPR studies of iron bearing phosphate minerals: satterlyite and gormanite from Yukon Territory,Canada

The iron phosphate minerals satterlyite and gormanite have been investigated by EPR and optical absorption studies. The optical results indicate the presence of ferrous and ferric ions in both minerals. In gormanite the site symmetry of Fe(III) is near octahedral whereas in satterlyite it is tetragonally distorted. On the other hand, the Fe(II) ions are in tetragonally distorted octahedral site in both minerals. In satterlyite the EPR results indicate the presence of the ferric ion in a tetragonally distorted state together with a small percentage of Mn(II). Crystal field (Dq) and interelectronic parameters (B and C) are evaluated.