Polynomial products modulo primes and applications

Monatshefte für Mathematik - For any polynomial $$P(x)\in {\mathbb {Z}}[x],$$ we study arithmetic dynamical systems generated by $$\displaystyle {F_P(n)=\mathop {\prod \nolimits _{k\le...     

Soluble {[Rh2(μ-OOCCH3)2(dbbpy)2][BF4]}n molecular wire and [Rh2(μ-OOCCH3)2(dbbpy)2L2] complexes; dbbpy = 4,4′-di-tert-butyl-2,2′-bipyridine: Synthesis and physicochemical characterization

Binuclear Rh(II) compounds [Rh₂(μ-OOCCH₃)₂(dbbpy)₂(H₂O)₂](CH₃COO)₂ (1) (dbbpy = 4,4′-di-tert-butyl-2,2′-bipyridine), [Rh₂(μ-OOCCH₃)₂(dbbpy)₂(H₂O)₂](BF₄)₂·H₂O·CH₃CN (2), [Rh₂(CH₃COO)₂(C₁₈H₂₄N₂)₂(CH₃CN)₂](BF₄)₂·4CH₃CN (3) and {[Rh₂(μ-OOCCH₃)₂(dbbpy)₂][BF₄]}_n (4) have been synthesized and characterized with spectroscopic methods. Structure of complex 3 has been determined using X-ray crystallography. Rhodium atoms in compound 3 have distorted octahedral coordination with O and N atoms in equatorial positions and Rh atom and CH₃CN molecule in axial coordination sites. Reduction of rhodium(II) compounds with aqueous 2-propanol leads to the formation of polymetallic compound {[Rh₂(μ-OOCCH₃)₂(dbbpy)₂][BF₄]}_n (4) containing [Rh₂]³⁺ core. Compound 4 shows strong antiferromagnetic properties, μ = 0.18–1.73 M.B. in the range 1.8–300 K, J = −597 cm⁻¹. Electrochemistry of compounds 3 and 4 in CH₃CN has been investigated. Compound 4 exhibits a poorly reversible oxidation system at E_1/2 = −0.92 V (ΔE_p = 0.19 V) and in solution in DMF is slowly oxidized to 3 even in total absence of oxygen. Complex 3 is irreversibly oxidized to Rh(III) compound at E_pa = 1.48 V and irreversibly reduced at E_pc = −1.02 V to lead to the unstable polynuclear complex 4 in CH₃CN.     

Flow ultramicroscopic study of interparticle interaction in suspensions and kinetics of reversible aggregation

The kinetics of coagulation leading, in the long run, to the establishment of the aggregation equilibrium is studied by the flow ultramicroscopy method with allowance for the probability of aggregate formation and disintegration. The case of a slight aggregation is considered when the doublet-to-singlet concentration ratio in a disperse system is low. An equation characterizing the time dependence of the average sizes of aggregates is derived. The equation is analyzed and methods are proposed for determining the repulsive barrier and the depth of the energy minimum characterizing the potential of interparticle pair interaction from experimental data on coagulation kinetics. The case of long-range coagulation is investigated. The effects of particle size, Hamaker’s constant, and electrolyte concentration in a dispersion medium on the probability of disaggregation are estimated in terms of the theory of surface forces. Limits of the flow microscopy method in the determination of the secondary energy minimum value are considered.     

On the stability in periodic and almost periodic difference systems

In this paper we consider the system of difference equations

     

Experimental and theoretical study of the surface-controlled dissolution of cylindrical particles. Application to solubilization of potassium hydrogen carbonate in hot dimethylformamide

In this paper we present a mathematical model for the surface-controlled dissolution of cylindrical solid particles. This is employed to interpret experimental data published previously for the dissolution of potassium bicarbonate in dimethylformamide at elevated temperatures. Significant kinetic differences in assuming cylindrical rather than spherical shapes are reported with the former representing a closer approximation to the true shape of the particles as revealed by scanning electron microscopy. From the fits of experimental data to the cylindrical model for the surface-controlled dissolution, the dissolution rate constant, k, for the dissolution of KHCO(3) in DMF was found to be (9.6 +/- 1.6) x 10(-9) mol cm(-2) s(-1) at 100 degrees C, and the activation energy for the dissolution was 34.5 kJ mol(-1) over the temperature range of 60-100 degrees C. Comparison between cylindrical and spherical dissolution theory highlights the importance of considering the particle shapes for realistic modeling of surface-controlled dissolution kinetics.     

Synthesis and 197Au-Mössbauer spectroscopic studies of dihalo(pentafluorophenyl)(bidentate ligand)gold(III) complexes

Addition of a bidentate ligand (LL = 1,10-phenanthroline, o-phenylenebis(dimethylarsine)) to solutions of Au(C₆F₅)X₂(tht) (X = Cl, Br; tht = tetrahydrothiophene) leads to potentially five-coordinate gold(III) derivatives. ¹⁹⁷Au Mössbauer spectroscopy points, however, to four-coordinate square-planar complexes with a weak penta-coordination in the phen-containing derivatives. The complexes react with AgClO₄ to give four-coordinate cationic complexes of the types [Au(C₅F₅)X(LL)]ClO₄ or [Au(C₆F₅)(PPh₃)(LL)](ClO₄)₂.     

[RhCl(CO)(PPh3)]2. A precursor for the synthesis of cationic rhodium complexes with nitrogen donor ligands

Summary The use of [RhCl(CO)(PPh₃)]₂ as a precursor for the synthesis of complexes of the types [Rh(CO)L₂(PPh₃)]A (A = [ClO₄]^– or [BPh₄]^–; L = pyridine type ligand) and [Rh(CO)(L-L)(PPh₃)]A (A = [ClO₄]^– or [BPh₄]^–; L-L = bidentate nitrogen donor) and the preparation of several complexes of the types [Rh(CO)L(PPh₃){P(p-RC₆H₄)₃}]BPh₄ and [Rh(CO)(phen)(PPh₃){P(p-RC₆H₄)₃}]A (A = [ClO₄]^– or [BPh₄]^–; R = H or Me) is described.Author to whom all correspondence should be directed.     

Biimidazolate- and bibenzimidazolate-bridged binuclear substituted manganese(I) carbonyl complexes

Novel binuclear substituted manganese(I) carbonyls [Mn(CO)_4?nL_n]₂(μ-N-N)₂ (n = 1, (N-N)₂ = biimidazolate, L = PBuⁿ₃; (N-N)₂ = bibenzimidazolate, L = P(OMe)₃, P(OPh)₃, PPh₃, PEt₃ or PBuⁿ₃, as well as n = 2; (N-N)₂ = biimidazolate, or bibenzimidazolate, L = PBuⁿ₃, PEt₃ or P(OMe)₃) are described, in which the anions (N-N)₂ act as tetradentate bridging-groups. They were prepared by treating [Mn(CO)₄(μ-Br)]₂ with thallium or potassium salts of 2,2′-biimidazole or 2,2′-bibenzimidazole and subsequent displacement of CO by L. The structures of the complexes are discussed.     

The electrical double layer of micelles in ionic surfactant solutions in the presence of a background electrolyte: 1. Diluted micellar solutions of sodium dodecyl sulfate

The structure of the electrical double layer (EDL) of micelles in dilute micellar solutions in the presence of a background electrolyte is studied within the framework of the Gouy-Chapman-Stern theory. On the basis of the Stern isotherm for counterion adsorption, conditions of electroneutrality, and the Gauss condition at the interface between the diffuse and dense parts of EDL, three equations are derived for the electrostatic potentials of the surface of micelle cores and the diffuse part of EDL as well as for the potential of the specific adsorption of counterions. Model parameters are verified by the example of sodium dodecyl sulfate (SDS). Potentials of the diffuse part of EDL, the degree of binding of counterions with micelles, and the specific adsorption potential are calculated from the experimental data on the potential of the surface of SDS micelle cores and their sizes, critical micellization concentration, aggregation numbers, and the constants of premicellar association. The specific adsorption potential of SDS is found to be ?(4.6 ± 0.1)?, where ? is the product of Boltzmann’s constant and absolute temperature. The specific adsorption potential is independent of the background electrolyte concentration, remains constant within the determination error of the parameters, and substantially contributes to the formation of EDL of micelles.     

2,2′-biimidazole, 2,2′-bibenzimidazole and imidazoles as ligands in cationic rhodium(I) complexes

Summary Cationic rhodium(I) complexes of the type [Rh(diolefin)(L-L)]ClO₄ and [Rh(diolefin)L₂]ClO₄, (diolefin = 1,5-cyclooctadiene, 2,5-norbornadiene and tetrafluorobenzobarrelene; L-L = 2,2-biimidazole, 2,2-bibenzimidazole; L = pyrazole or imidazoles) are described. [Rh(CO)₂(L-L)]-C1O₄ complexes, which can be obtained by reaction of cyclooctadiene derivatives with CO, react with P-donor ligands in equimolar ratios to yield [Rh(CO)(P-donor)(L-L)]ClO₄ monocarbonyl derivatives. The catalytic activity of some of these complexes is considered.