239 resultados para Legendre Formula


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A Schiff base metal complex, [Cu(II)(PLP-DL-tyrosinato)(H2O)].4H2O (PLP = pyridoxal phosphate), with the molecular formula CuC17O13N2H27P has been prepared and characterized by magnetic, spectral, and X-ray structural studies. The compound crystallizes in the triclinic space group P1BAR with a = 8.616 (2) angstrom, b = 11.843 (3) angstrom, c = 12.177 (3) angstrom, alpha = 103.40 (2)degrees, beta = 112.32 (2)degrees, gamma = 76.50 (1)degrees, and Z = 2. The structure was solved by the heavy-atom method and refined by least-squares techniques to a final R value of 0.057 for 3132 independent reflections. The coordination geometry around Cu(II) is distorted square pyramidal with phenolic oxygen, imino nitrogen, and carboxylate oxygen from the Schiff base ligand and water oxygen as basal donor atoms. The axial site is occupied by a phosphate oxygen from a neighboring molecule, thus resulting in a one-dimensional polymer. The structure reveals pi-pi interaction of the aromatic side chain of the amino acid with the pyridoxal pi system. A comparative study is made of this complex with similar Schiff base complexes. The variable-temperature magnetic behavior of this compound shows a weak antiferromagnetic interaction.

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We compute the entropy and transport properties of water in the hydration layer of dipalmitoylphosphatidylcholine bilayer by using a recently developed theoretical scheme two-phase thermodynamic model, termed as 2PT method; S.-T. Lin et al., J. Chem. Phys. 119, 11792 (2003)] based on the translational and rotational velocity autocorrelation functions and their power spectra. The weights of translational and rotational power spectra shift from higher to lower frequency as one goes from the bilayer interface to the bulk. Water molecules near the bilayer head groups have substantially lower entropy (48.36 J/mol/K) than water molecules in the intermediate region (51.36 J/mol/K), which have again lower entropy than the molecules (60.52 J/mol/K) in bulk. Thus, the entropic contribution to the free energy change (T Delta S) of transferring an interface water molecule to the bulk is 3.65 kJ/mol and of transferring intermediate water to the bulk is 2.75 kJ/mol at 300 K, which is to be compared with 6.03 kJ/mol for melting of ice at 273 K. The translational diffusion of water in the vicinity of the head groups is found to be in a subdiffusive regime and the rotational diffusion constant increases going away from the interface. This behavior is supported by the slower reorientational relaxation of the dipole vector and OH bond vector of interfacial water. The ratio of reorientational relaxation time for Legendre polynomials of order 1 and 2 is approximately 2 for interface, intermediate, and bulk water, indicating the presence of jump dynamics in these water molecules. (C) 2010 American Institute of Physics. doi:10.1063/1.3494115]

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The general time dependent source problem has been solved by the method of transforms (Laplace, Lebedev–Kontorovich in succession) and the solution is obtained in the form of an infinite series involving Legendre functions. The solutions in the case of harmonic time dependence and the incident plane wave have been derived from the above solution and are presented in the form of an infinite series. In the case of an incident plane wave, the series has been summed and the final solution involves an improper integral which behaves like a complementary error function for large values of the argument. Finally, the far field evaluation has been shown. The results are compared with those of Sommerfeld's half-plane diffraction problem with unmixed boundary conditions.

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The method of structured programming or program development using a top-down, stepwise refinement technique provides a systematic approach for the development of programs of considerable complexity. The aim of this paper is to present the philosophy of structured programming through a case study of a nonnumeric programming task. The problem of converting a well-formed formula in first-order logic into prenex normal form is considered. The program has been coded in the programming language PASCAL and implemented on a DEC-10 system. The program has about 500 lines of code and comprises 11 procedures.

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A series of anion-deficient pyrochlore oxides of the formula A2MoTiO7−x (xless-than-or-equals, slant0.5), where Atriple bond; length as m-dashSm, Gd, Tb, Dy, Ho, Er, Lu and Y, has been prepared by reduction of A2MoTiO8 scheelites. The scheelite-to-pyrochlore conversion is reversible, indicating that the reaction is likely to be topochemical. The oxidation states of molybdenum and titanium are most probably Mo(III) and Ti(IV) for the limiting composition of the pyrochlores A2MoTiO6.5. The new pyrochlores are non-metallic and paramagnetic as expected.

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We present a biquadratic Lagrangian plate bending element with consistent fields for the constrained transverse shear strain functions. A technique involving expansion of the strain interpolations in terms of Legendre polynomials is used to redistribute the kinematically derived shear strain fields so that the field-consistent forms (i.e. avoiding locking) are also variationally correct (i.e. do not violate the variational norms). Also, a rational method of isoparametric Jacobian transformation is incorporated so that the constrained covariant shear strain fields are always consistent in whatever general quadrilateral form the element may take. Finally the element is compared with another formulation which was recently published. The element is subjected to several robust bench mark tests and is found to pass all the tests efficiently.

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A series of isomeric cationic surfactants (S1-S5) bearing a long alkyl chain that carries a 1,4-phenylene unit and a trimethyl ammonium headgroup was synthesized; the location of the phenyl ring within the alkyl tail was varied in an effort to understand its influence on the amphiphilic properties of the surfactants. The cmc's of the surfactants were estimated using ionic conductivity measurements and isothermal calorimetric titrations (ITC); the values obtained by the two methods were found to be in excellent agreement. The ITC measurements provided additional insight into the various thermodynamic parameters associated with the micellization process. Although all five surfactants have exactly the same molecular formula, their micellar properties were seen to vary dramatically depending on the location of the phenyl ring; the cmc was seen to decrease by almost an order of magnitude when the phenyl ring was moved from the tail end (cmc of S1 is 23 mM) to the headgroup region (cmc of S5 is 3 mM). In all cases, the enthalpy of micellization was negative but the entropy of micellization was positive, suggesting that in all of these systems the formation of micelles is both enthalpically and entropically favored. As expected, the decrease in cmc values upon moving the phenyl ring from the tail end to he headgroup region is accompanied by an increase in the thermodynamic driving force (Delta G) for micellization. To understand further the differences in the micellar structure of these surfactants, small-angle neutron scattering (SANS) measurements were carried out; these measurements reveal that the aggregation number of the micelles increases as the cmc decreases. This increase in the aggregation number is also accompanied by an increase in the asphericity of the micellar aggregate and a decrease in the fractional charge. Geometric packing arguments are presented to account for these changes in aggregation behavior as a function of phenyl ring location.

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A series of mixed ligand cobalt(III) complexes having the general formula Co(EA)X [where EA = dianion of N,N′-ethylenebis(acetylacetonimine) and X = anion of isonitroso-acetylacetone, IAA; isonitrosobenzoylacetone, IBA; isonitrosodibenzoylmethane, IDBM; isonitrosoethylacetoacetate, IEA; isonitrosoacetoacetanillide, IAN; isonitrosoethylmethylketone, IEMK; isonitrosobenzylmethylketone, IBMK and isonitrosopropiophenone, IPP] have been synthesised and characterised. A facial-cis-β structure (cis with respect to the coordinated two oxygen atoms of EA) with N,N,N,O,O,O ligational environment has been assigned for the complexes. The characterisation of the complexes has been based upon chemical analysis, electrical conductivity, magnetic moment, IR, PMR and electronic spectra.

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Five new complexes of lanthanide perchlorates with a new ligand O,O' diisopropyl N(-4-antipyryl) phosphoramidate (DIAP) of the general formula Ln(DIAP)4(ClO4)3 where Ln = La, Pr, Nd, Sm and Gd, have been synthesised and characterized by chemical analysis, IR(200–4000cm−1) and electronic spectra and electrical conductance data. Infrared spectral data indicate the coordination of the ligand to the metal ions in a bidentate fashion, through the C=O oxygen of the antipyrine group and the P=O group. IR and conductance values show that the three perchlorate groups are ionic. Electronic spectrum of the Nd3+ complex in the visible region, indicates reasonable covalency in the metal-ligand bond. The available data point to an eight coordinate geometry around the metal ions, with each ligand behaving in a bidentate ‘00’ fashion.

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Tris(dicyclohexylamino)silane. (DCA)3SiH. is prepared by the reaction of trichlorosilane with dicyclohexylamine. This is found to undergo transamination reactions with other secondary amines (R2NH). such as pyrrolidine, piperidine, hexamethyleneimine. morpholine. N-methylpiperazine and diethylamine to yield mixed tri(amino)silanes of the formula (DCA)(R2N)2SiH in quantitative yields. These new derivatives are found to be moisture sensitive and hydrolyze to yield their respective amines, hydrogen and silica. They are found to be stable in an inert atmosphere. They have been characterized by IR, NMR (H-1, Si-29), mass spectroscopy and CHN analysis. N-15 NMR for one of the compounds has been done.

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Solid solutions of the formula, A2–xLa2Ti3–xNbxO10(A = K, Rb), exist for the range 0[less-than-or-eq]x[less-than-or-eq]1.0, bridging n= 3 members of the Ruddlesden–Popper series (A2La2Ti3O10) and the Dion–Jacobson series (ALa2Ti2NbO10). For 0[less-than-or-eq]x[less-than-or-eq]0.75, the phases possess body-centred structures characteristic of the Ruddlesden–Popper phases, while the x= 1 members are isostructural with KCa2Nb3O10(A = K) and CsCa2Nb3O10(A = Rb). Protonated derivatives, H2–xLa2Ti3–xNbxO10, which are prepared by ion exchange, retain the structural difference of the parent phases. A difference in the Brønsted acidity of the protonated derivatives revealed by intercalation experiments with organic bases seems to be related to this structural difference.

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Ferrites of the formula MoxFe3-xO4, prepared by a soft-chemistry route, show mixed valence states of both iron and molybdenum cations. Mössbauer studies show that Fe2+ and Fe3+ ions are present on both the A and B sites, giving Fe an average oxidation state between 2+ and 3+. Molybdenum is present in the 3+ and the 4+ states on the B sites. The presence of Mo in the 3+ state has been established by determining the Mo3+-O distance (2.2 Å), for the first time, by Mo K-EXAFS. The mixed valence of Fe on both the A and B sites and of Mo on the B sites is responsible for the fast electron transfer between the cations. All the Mössbauer parameters including the line width show a marked change at a composition (x ? 0.3) above which the concentration of Fe2+A increases rapidly.

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A series of layered perovskite oxides of the formula K1-xLaxCa2-xNb3O10 for 0 < x ≤ 1.0 have been prepared. All the members are isostructural, possessing the structure of KCa2Nb3O10. The interlayer potassium ions in the new series can be ion-exchanged with protons to give H1-xLaxCa2-xNb3O10. The latter readily forms intercalation compounds of the formula (CnH2n+1NH3)1-x LaxCa2-xNb3O10, just as the parent solid acid HCa2Nb3O10. The end member LaCaNb3O10 containing no interlayer cations is a novel layered perovskite oxide, being a n = 3 member of the series An-1BnX3n+1.

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Let O be a monomial curve in the affine algebraic e-space over a field K and P be the relation ideal of O. If O is defined by a sequence of e positive integers some e - 1 of which form an arithmetic sequence then we construct a minimal set of generators for P and write an explicit formula for mu(P).

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Cuprates of the formula TlSr3−xLnxCu2O7 (Ln=Pr, NdorY) derived from the hypothetical TlSr3Cu2O7 show superconductivity with Tcs up to 95 K when 0.5less, approximatex≤0.75, the x=1.0 compositions being insulators. Rietveld analysis of X-ray diffraction profiles has been carried out for two superconducting members of this family. The unit cell a-parameter, and hence the in-plane Cu-O distance, increases with increase in x. The Tc value decreases with increase in x or the in-plane Cu-O distance in all the series of cuprates. Superconductivity in the Tl1−yPbySr3−xNdxCu2O7 systems is found with the highest Tc of 95 K when y=0.2 and x=0.5. The in-plane Cu-O distances in all the cuprates studied fall in the range found in the Sr-class of cuprate superconductors.