19F NMR study of the equilibria and dynamics of the Al3+/F- system

A careful reinvestigation by high-field 19F NMR (470 MHz) spectroscopy has been made of the Al3+/F- system in aqueous solution under carefully controlled conditions of pH, concentration, ionic strength (I), and temperature. The 19F NMR spectra show five distinct signals at 278 K and I = 0.6 M (TMACl) which have been attributed to the complexes AlFi(3-i)+(aq) with i < or = 5. There was no need to invoke AlFi(OH)j(3-i-j)+ mixed complexes in the model under our experimental conditions (pH < or = 6.5), nor was any evidence obtained for the formation of AlF6(3-)(aq) at very high ratios of F-/Al3+. The stepwise equilibrium constants obtained for the complexes by integration of the 19F signals are in good agreement with literature data given the differences in medium and temperature. In I = 0.6 M TMACl at 278 K and in I = 3 M KCl at 298 K the log Ki values are 6.42, 5.41, 3.99, 2.50, and 0.84 (for species i = 1-5) and 6.35, 5.25, and 4.11 (for species i = 1-3), respectively. Disappearance of the 19F NMR signals under certain conditions was shown to be due to precipitation. Certain 19F NMR signals exhibit temperature- and concentration-dependent exchange broadening. Detailed line shape analysis of the spectra and magnetization transfer measurements indicate that the kinetics are dominated by F- exchange rather than complex formation. The detected reactions and their rate constants are AlF2(2+) + *F- reversible AlF*F2+ + F- (k02 = (1.8 +/- 0.3) x 10(6) M-1 s-1), AlF3(0) + *F- reversible AlF2*F0 + F- (k03 = (3.9 +/- 0.9) x 10(6) M-1 s-1), and AlF3(0) + H*F reversible AlF2*F0 + HF (kH03 = (6.6 +/- 0.5) x 10(4) M-1 s-1). The rates of these exchange reactions increase markedly with increasing F- substitution. Thus, the reactions of AlF2+(aq) were too inert to be detected even on the T1 NMR time scale, while some of the reactions of AlF3(0)(aq) were fast, causing large line broadening. The ligand exchange appears to follow an associative interchange mechanism. The cis-trans isomerization of AlF2+(aq), consistent with octahedral geometry for that complex, is slowed sufficiently to be observed at temperatures around 270 K. Difference between the Al3+/F- system and the much studied Al3+/OH- system are briefly commented on.     

Norm-graphs and bipartite turán numbers

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|c _t ·n ^2–1/t which do not contain a complete bipartite graghK _t,t !+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K _t,s ) for any fixedt and allst!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.Research supported in part by Hungarian National Foundation for Scientific Research Grant     

Slow water diffusion in micellar solutions

Slowly diffusing water molecules were found by quasi-elastic neutron scattering (QENS) in a sodium dodecyl sulfate (SDS) micellar solution, and both their diffusion coefficient (4.33 x 10(-6) cm2 x s(-1)) and mole fraction (0.057) were determined. After successfully checking the mean slowing down of solvent molecules by the gradient compensated stimulated spin-echo (GCSTE) pulse sequence NMR method, a similar effect was observed with this technique in the solvent phase of dodecyl trimethylammonium bromide (DTAB) and differing chain length (X = 12, 20, 30, and 40) ethoxylated nonyl phenol (9NX) micellar systems. Following the literature, the experimental results are qualitatively explained by assuming that, apart from ionic hydration, H-bonds may form between the solvent molecules and the O or N atoms present in the hydrophilic (head)groups of the micelle-forming monomers.     

Dynamic NMR properties of DOTA ligand: variable pH and temperature 1H NMR study on [K(HxDOTA)](3-x)- species

The (1)H NMR spectra of [H(x)DOTA]((4-x)-) species are reported as a function of pH and temperature in aqueous solution. The spectra show line broadening both in ligand proton signals and also in the water proton signal by titration with KOH solution. The formation of different [K(H(x)DOTA)]((3-x)-) complexes is found to be responsible for this behaviour. At high pH the usual fluxional motions, i.e. the ring inversion and the change in the acetate arms' helicity, which are also characteristic for other but inert metal-DOTA complexes, have been detected. However, because of the kinetic lability of K(+)-O and K(+)-N coordinative bonds a new type of rearrangement appears. This new motion requires breaking of coordinative bonds in the complex and can be described as a certain type of "ring slewing" around the ring C-C bonds. At low temperature (about 270 K) the ring slewing slows down and becomes negligible compared with the ring inversion and the change in the arms' helicity. These two latter processes have the same rate. When the temperature is higher (about 320 K) the ring slewing accelerates and its rate exceeds the rate of ring inversion. At this temperature the change in the acetate arms' helicity has the same rate as the ring slewing. Additionally, in the pH range 4-5 a slow intermolecular proton exchange process has been observed between the water and the dissociable protons of [K(H(x)DOTA)]((3-x)-). A water-assisted proton exchange mechanism is proposed on the basis of the activation parameters. This finding supports the previously suggested slow proton motion hypothesis for the formation of DOTA complexes.     

Surface Complexation Modeling of K(+), NO(3)(-), SO(4)(2-), Ca(2+), F(-), Co(2+), and Cr(3+) Ion Adsorption on Silica Gel

Surface complex formation of K(+), NO(3)(-), SO(4)(2-), Ca(2+), F(-), Co(2+), and Cr(3+) ions was determined on the surface of silica gel. Experimental data obtained by acid-base titration of suspensions were interpreted in terms of the triple-layer model. The value of the deprotonation constant of surface OH could be determined precisely but the protonation constant was rather uncertain. The logarithms of ion pair formation constants for K(+), NO(3)(-), Ca(2+), and SO(4)(2-) adsorbed in the beta-plane are log K(ipM,X) approximately 0, therefore these species can be considered inert ions in the investigated pH range. F(-), Co(2+), and Cr(3+) ions were found to be strongly sorbed in the o-plane. In order to provide a good fit and to obtain parameters independent of their initial values, all possible equilibrium must be accounted for in the models. Copyright 2001 Academic Press.     

Zero divisors in quaternion algebras

Let F be a finite field or an algebraic number field. In previous papers we have shown how to find the basic building blocks (the radical and the simple components) of a finite dimensional algebra over F in polynomial time (deterministically in characteristic zero and Las Vegas in the finite case). A finite-dimensional simple algebra A is a full matrix algebra over some not necessarily commutative extension field G of F. The problem remains to find G and an isomorphism between A and a matrix algebra over G. This, too, can be done in polynomial time for finite F. We indicate in the present paper that the problem for F = Q might be substantially more difficult. We link the problem to hard number theoretic problems such as quadratic residuosity modulo a composite number. We show that assuming the generalized Riemann hypothesis, there exists a randomized polynomial time reduction from quadratic residuosity to determining whether or not a given 4-dimensional algebra over Q has zero divisors. It will follow that finding a pair of zero divisors is at least as hard as factoring squarefree integers.     

Irreducible tensors for the groupSU 3

The explicit determination of the matrix elements of theSU ₃ irreducible tensors is carried out by a purely algebraic method. These expressions may be used to compute the Clebsch-Gordan coefficients by orthogonalisation. For the special case of (0,q) tensors simple formulas are derived.In partial fulfillment for the requierements of the doctoral degree at the Institute of Physics of the Academy.     

On algebraic properties of monotone clones

Let R ₈ denote the 8-element bounded tower. G. Tardos has shown that C(R ₈), the clone of all monotone functions on R ₈, is not finitely generated. In this paper we show that the clone of all nonsurjective functions is finitely generated.