Critical mathematical considerations involving the Abel integral equation for converting side-on experimental data in plasma spectroscopy—I: A study of basic analytical solutions

The particular form of the Abel integral equation used in plasma spectroscopy is investigated from the point of view of the computational error introduced in radial emission coefficient determinations from side-on experimental intensities. Since this emission coefficient is the basis of measurements of fundamental physical properties of the plasma, its values are to be obtained with minimal error.In this first part of this work, the mathematical properties of basic analytical solutions are investigated. Polynomial solutions resulting from positive powers of the variable in intensity profiles are described with reduced coordinates in terms of “rough parabolas” I(x) = 1 ?xⁿ (n = 2p, x = $\text{x}\text{?}$_exptl/$\text{x}\text{?}$_{max exptl}, I(x)=$\text{I}\text{?}$ ($\text{x}\text{?}$)_exptl/$\text{I}\text{?}$_{max exptl}) up to the 26th degree. For negative powers a new and original method is developed; it leads to ABEL inversion of intensity profites of the Lorcntzian kind. These profiles may be of interest when center-to-edgc intensity decay ends with a positive curvature.In following papers, all these results will then be applied to numerical approximation of theoretical standards and side-on experimental data in the spectroscopy of an inductivcly coupled plasma (ICP).     

Crystal structures of some niobium and tantalum oxides. II. The 4Rb2O:11Nb2O5 phase—A tunnel structure

Rb_12.20Nb_33.56O₉₀ crystallizes in the trigonal system with unit-cell dimensions (from single-crystal data) a = 7.527(6), c = 43.17(2)Å and space group $\text{R}\text{3}\text{m, z = 1}$. The structure was solved using three-dimensional Patterson and Fourier techniques. Of the 843 unique reflections measured by counter techniques, 675 with I ? 3σ(I) were used in the least-squares refinement of the model to a conventional R of 0.072 (wR = 0.055). The structure of Rb_12.20Nb_33.56O₉₀ consists of layers of corner-sharing groups of six edge-shared octahedra separated by pyrochlore-like layers of octahedra. The structure can be considered to be derived from that of pyrochlore by the ordered omission of layers of single octahedra.     

An improved absorption correction based on a Gaussian ionization distribution model for electron probe microanalysis

The electron range equation R, employed in a Gaussian absorption correction, is optimized making use of f(χ) values which were calculated from ionization distributions determined experimentally by Brown and Parobek. A new expression of R[R = $\text{(1 + 2}\text{h}\text{)}\text{(1 +}\text{h}$)2.59 × 10^?5(E^1.3₀ ?E^1.3_c)] g cm^?2 is proposed in this paper.The absorption correction using the new electron range equation R is applied to oxygen determination in several silicates and sulfates with greater accuracy than other methods. This equation also predicts one of the most accurate mean depths of ionization at an accelerating voltage of 25 kV for Zn and Cd tracers in Al, Cu, Ag and Au matrices.     

The chemistry of phenalenium systems XXXIII. The Dibenzo[de;thi]naphthacenyl dication and the dianon

The dibenzo[de;hi]naphthacenyl dication ($\text{4}$) and dianion ($\text{5}$) have been generated from a mixture of the corresponding precursor hydrocarbons. The pmr spectra of $\text{4}$ and $\text{5}$ indicated that the positive and the negative charges were found to be fully delocalized over the molecules, respectively, consistent with the C_2v-symmetry structures. The pmr chemical shifts of $\text{4}$ correlate well with the Hückel charge densities.     

Thermal decomposition studies. Part XII. Kinetics of dehydration of calcium oxalate monohydrate. Multiple correlation with heating rate and sample mass

The kinetic parameters (energy of activation, E, and pre-exponential factor, A) from non-isothermal TG data have been correlated, for the first time, with simultaneous variations of both the procedural factors (heating rate and sample mass) by multiple regression analysis. The unique equation based on the mechanism of the reaction as well as three general mechanism-non-invoking integral equations were used to calculate E and A from the TG data for the dehydration of CaC₂O₄ · H₂O. The kinetic parameters calculated using all four equations showed a systematic trend and the results can be expressed as E(or log A) = $\text{constant}\text{heating rate}$ + $\text{constant}\text{mass}$ + constant     

Contribution à l'étude structurale de la wüstite solide de haute température

The structure of wüstite Fe_1?zO is studied by neutron diffraction on one polycrystalline sample under equilibrium conditions at high temperature. The mean isotrope temperature factor B is expressed as a sum of two parts, B_Th and B_St, which vary linearly with a single parameter, respectively temperature and z. A classification is established for clusters ($\text{m}\text{n}$) settled from m vacancies in octahedral sites and n Fe^III ions in interstitial sites. Sixteen values have been experimentally determined for the vacancies to interstitials ratio $\text{? =}\text{(z + t)}\text{t}\text{=}\text{m}\text{n}$. A constant value of ?, which is lower than 3, is observed. This result characterizes the short-range order. It eliminates several possibilities of clusters like those obeying the relation $\text{? =}\text{(1 + 3n)}\text{n}$. Other clusters, namely ($\text{16}\text{6}$) or ($\text{40}\text{14}$), might agree. The ($\text{8}\text{3}$) and ($\text{9}\text{4}$) clusters obtained from ($\text{4}\text{1}$) clusters joined by an edge would be the more likely. An analysis of diffuse scattering eliminates the hypothesis of large domains with inverse spinel structure. The structural differences between the three varieties W₁, W₂ and W₃ would not be found in a structural change of clusters.     

Fluorimetric determination of lactate with immobilized lactate dehydrogenase

In measuring inter-phase distribution coefficients (k_{$\text{1}\text{11}$})it is often convenient, and sometimes necessary, to obtain all of the analytical data from one phase. A rigorous formula, applicable to both low and high k_{$\text{1}\text{11}$} values has been derived; measurements of the distribution of mercury, methylmercury and ⁶⁷Cu²⁺ between various pairs of phases with this formula gave k_{$\text{1}\text{11}$} values similar to those obtained from other approaches.     

The polarography of oximes: II. 1,2-Acenaphthaquinone monoxime

Polarography of 1,2-acenaphthaquinone monoxime has been carried out in buffers (pH 3.5–13.0) of constant ionic strength 0.5 M and 40% alcohol $\text{v}\text{v}$ at 25 ± 0.5 °C. The oxime group underwent diffusion-controlled reduction (4e) over the whole pH range. The number of electrons involved in the reduction was found by coulometric method as well as by incorporating the values of diffusion coefficients, obtained by using a McBain-Dawson cell, into the Ilkovi? equation. Koutecky's method has been used to compute the kinetic parameters (αn_aand ?log K_f.h) for the reduction of the oxime group and a reduction mechanism is proposed.     

Polarographic determination of the stability constants of metal complexes based on the shifts in half-wave or peak potentials of the anodic oxidation of mercury: methylthioacetato- and 2,2′-thiobisacetato complexes

The stability constants of metal-ion complexes, other than those of mercury ions, can be determined from the shifts, caused by an excess of that metal ion, in the anodic waves of mercury oxidation in the presence of the ligand.Let a solution contain L, a ligand that can react with mercury ion from the anodic oxidation of mercury to form the complexes HgL_p. The half-wave potential of the anodic wave will be shifted by the presence of an excess of a metal ion M, which can also react with L forming the complexes M_jL (where j = 1, $\text{1}\text{2}$,…1/N). A general expression is derived relating the half-wave potential shift ΔE_{$\text{1}\text{2}$} to the overall stability constants. If the excess of metal ion M is great enough for the 1:1 complex to predominate in solution (and if HgL₂ is the predominating mercury complex), for reversible, diffusion-controlled processes the general equation can be simplified to:

where c_M is the total concentration of metal ion M.This equation provides an easy and fast method for β₁ determination. The simplified equation is best suited for experimental data obtained from techniques such as DP, AC₁ and AC₂ polarography, whose increased precision in ΔE measurement enables poorly-developed dc-polarographic anodic waves to be used for β₁ determination. Since the metal-to-ligand concentration ratio must be large in order to apply the simplified equation, the determination can be carried out at very small ligand concentrations. This fact renders the new method especially useful when the ligand solubility does not allow the high ligand concentrations needed in the DeFord—Hume method to be reached. The adequacy of the deduced equations when applied to polarographic processes which are irreversible or not controlled by diffusion is discussed.The simplified equation is tested using several metal complexes of methylthioacetate and 2,2′-thiobisacetate ligands and comparing some of the values obtained from experimentally measured ΔE's with known β₁ values taken from the literature. Good agreement is found. For the U(VI)—methylthioacetate complex, which has not been previously reported in the literature, log β₁ = 1.75 ± 0.1.     

Photometric titrations with indicators

Various types of photometric titration curves are discussed. If a metal M is titrated conipleximetrically using a metal indicator and the absorbance is plotted vs. the titrant consumed, the inflection point appears at a pM value defined by the equation 3 pM_infl = pM_trans + 2 pM_eqThis expression is valid when M combines in a 1 : 1 ratio with the complexing agent and the indicator and when the indicator concentration is small compared to the total metal concentration.The difference between the pM values at the inflection and equivalence points can be calculated from the equation ΔpM = pM_infl — pM_eq = $\text{1}\text{3}$(pM_trans — pM_eq) = $\text{1}\text{6}$log(C_MK²_MI/K_MY)If the inflection point is taken as the equivalence point, the error arising can be calculated from ΔpM, or more simply, read from a diagram.If transmittance, instead of absorbancc, is plotted as a function of the titrant volume, the inflection point depends on the added amount of indicator. However, at high transmittance values, i.e., at low indicator concentrations, the inflection point of a transmittance curve occurs practically at the same volume of added titrant as the inflection point of an absorbance curve. Rules are given for applying an indicator correction for the amount of metal bound to the indicator at the end-point.The derived equations and discussions can also be applied to acid-base titrations.     

Deformation of the valence electron distributions in the icosahedral boron structure of α-AlB12

X-ray difference electron densities in α-AlB₁₂ were examined. Broad peaks presumably characteristic of highly delocalized three-center bonds were observed at the centers of the triangular faces of the B₁₂ icosahedron. Another kind of prominent peak on the external BB bonds of the icosahedron indicated that the inter-icosahedral BB bonds are two-center bonds. An averaging over the icosahedral symmetry significantly reduced the random noises in the difference densities. Theoretical difference electron densities for a B₁₂H^?2₁₂ molecular ion, calculated by the $\text{CNDO}\text{2}$ molecular orbital method, were qualitatively in good agreement with the observed densities mentioned above. Residual peaks were observed on all the five edges of the open pentagonal face of the B₁₉ unit; they had broad tails extending toward the aluminum triplet sites. The observed difference densities around the “single” B(23) atom suggested that the atom belongs to the B₁₉ unit rather than being single.     

The crystal structure of a complex cerium(III) molybdate; Ce6(MoO4)8(Mo2O7)

Ce₆Mo₁₀O₃₉ crystallizes in the triclinic system with unit-cell dimensions (from single-crystal data) a = 10.148(5), Å, b = 18.764(6), Å, c = 9.566(5), Å, α = 103.12(7)°, β = 78.07(7)°, γ = 107.69(7)°, and space group $\text{P}\text{1}\text{, z = 2}$. The structure was solved using direct methods with 3113 countermeasured reflections (MoKα radiation), and refined using Fourier and least-squares techniques to a conventional R of 0.039 (ωR = 0.047). Ce₆Mo₁₀O₃₉ has a structure that consists of isolated MoO₄ tetrahedra together with one corner-shared pair of tetrahedra, linked to irregular eight-coordinate Ce(III) polyhedra. The average MoO distance of 1.77 Å, and average CeO distance of 2.52 Å are in good agreement with previously reported values.     

The crystal structure of the 42-Å subcell of a layer structure with approximate composition Ba4Nb2S9

Platy crystals from the products of a mixture 4 Bas : 2 Nb : 5 S reacted at 1000°C have cell constants a = 13.754(3) Å, c = 83.73(2) Å, $\text{R}\text{3}\text{m}$. The reciprocal lattice had a pronounced subcell with dimensions a = 6.877(1) Å, c = 41.84(1) Å, same space group. Three dimensional X-ray diffraction data were collected using monochromatized Mo Kα radiation and of 5051 measured intensities 1892 were considered observed. From the set of observed intensities 611 reflections having all even indices were used to refine the crystal structure of the 42 × 7-Å subcell. The final R = 0.036 and ωR = 0.052 for the 611 observed amplitudes and R = 0.046, ωR = 0.052 for all 711 amplitudes of the subcell. The structure is based on the stacking of hexagonal BaS₃ layers with the sequence DABABDBCBCDCACAD. The D layer denotes a disordered level and occurs at $\text{z = 0,}\text{1}\text{3}$ and $\text{2}\text{3}$. The different letters for the ordered layers are based on the Ba positions in that layer. The Nb ions occupy octahedral interstices and form a unit of three face sharing octahedra parallel to c. The column is terminated above and below by disordered levels. The NbNb distances are 3.22 Å, causing displacement of Nb from the centers of the two outside octahedra. One Ba is in the center of a triangular orthobicupola formed by 12 S atoms. The other Ba is in the center of a hexagon of 6 S with 3 additional S above this layer forming $\text{1}\text{2}$ of a cuboctahedron. The lower half consists of a disordered layer of atoms. The NbS distances are 2.279, 2.433, and 2.683 Å; BaS distances vary between 3.1 and 3.5 Å. The subcell content based on the ordered structure only is Ba₁₂Nb₉S₃₆. The placement of disordered Ba and S at $\text{z = 0,}\text{1}\text{3}$, and $\text{2}\text{3}$ levels of the subcell leads to the unlikely composition Ba_16.5Nb₉S₄₂. The ordered structure most likely has a composition Ba₄Nb₂S₉, z = 36, so that the subcell composition should be Ba₁₈Nb₉S_40.5. The completely ordered structure has not been solved.     

ESR studies on the reactions of sulphite radical anion,SO3.̄−, with nitroalkane compounds in aqueous solutions

The reactions of the sulphite radical anion, SO₃^{$\text{.}\text{?}$?} (generated from the Ti³⁺-H₂O₂-Na₂SO₃ system at pH 9), in aqueous solutions with some nitroalkane compounds were investigated by using a rapid-mixing flow technique coupled with electron spin resonance (ESR) which can detect the radicals having a lifetime of 5–100 ms. The SO₃^{$\text{.}\text{?}$?} radical added to the double bond of CN in the nitroalkane aci-anions which are the main form of nitroalkanes in aqueous alkaline solutions. From the observed hyperfine splitting constants of the SO₃^{$\text{.}\text{?}$?} adducts of nitroalkane aci-anions, the preferred conformation of the adducts was deduced.     

A study of polytypism in AgIn5Se8 by combined electron microscopy techniques

Three different polytypes are found to exist in AgIn₅Se₈. They have been studied by means of conventional electron microscopy and diffraction as well as by high-resolution microscopy. The three different structures have tetragonal symmetry and can be derived from each other by means of periodic antiphase boundaries; they correspond to $\text{c}\text{a}\text{= 4,}\text{c}\text{a}\text{= 2}$, and $\text{c}\text{a}\text{= 1}$, respectively. Regions of $\text{c}\text{a}\text{= 4}$ often occur as microsyntactic intergrowths into $\text{c}\text{a}\text{= 2}$ areas, and a large number of nonperiodic antiphase boundaries also occur. After fast cooling from above T_c a transition state characterized by the presence of diffuse intensity has been observed and analyzed.     

Crystal structures of some niobium and tantalum oxides. VIII. The 5Rb2O:14.6Ta2O5 phase—A tunnel structure

Rb₁₀Ta_29.20O₇₈ crystallizes in the hexagonal system with unit-cell dimensions (from single-crystal data) a = 7.503(4)Å, c = 36.348(4)Å, and space group $\text{P6}{}_3\text{mmc}\text{, z = 1}$. The structure was solved using three-dimensional Patterson and Fourier techniques. Of the 666 unique reflections measured by counter techniques, 515 with I ? 3σ(I) were used in the least-squares refinement of the model to a conventional R of 0.057 (R_ω = 0.039). The structure of Rb₁₀Ta_29.20O₇₈ consists of layers of corner-sharing groups of six edge-shared octahedra separated by layers of single octahedra and double hexagonal tungsten bronze-like layers, these layers being perpendicular to the hexagonal c-axis. Nine-coordinate tricapped trigonal prismatic sites between the hexagonal tungsten bronze-like layers are partially occupied by Ta(V) ions.     

Characterization of noise in inductively-coupled plasma emission spectrometry

The nebulizing process in inductively-coupled plasma emission spectrometry is a major source of the instrumental variability; an attempt has been made to isolate this source so that its influence on the final result could be evaluated. Direct measurements of the time-dependent variations of the sample supply to the plasma were achieved by using a device based on the light-scattering property of nebulized test solutions. The noise from the nebulizer and of the emission signal was characterized by autocovariance functions and power spectral densities. The types of noise observed showed f^{-$\text{1}\text{2}$} and f^{-$\text{1}\text{4}$} character, depending on the concentration of the test elements employed. Cross-covariance revealed a strong correlation between the noise sources considered. The applicability and limitations of possible correction and electronic filtering procedures are indicated.     

Electroanalytical simulations. Orthogonal collocation simulation of fast second-order chemical reactions coupled to an electron transfer with a heterogeneous equivalent formulation

The combination of orthogonal collocation and the heterogeneous equivalent technique is extended to simulate cyclic voltammograms of fast second-order follow-up reactions coupled to an electron transfer at an electrode surface. The $\text{ED}$ (reversible electron transfer with irreversible follow-up dimerization) and $\text{EC}$₂ (reversible electron transfer with irreversible second-order follow-up reaction) models are considered. The non-linear boundary equations are solved numerically. No linear approximation of the concentration profiles is required. The use of non-linear space coordinate transformations is described. Peak potential and peak current function results are compared with literature values and agreement is found. The transition between the second-order $\text{EC}$₂ and the corresponding first-order mechanisms is discussed.     

Dilute solutions of nylon 12—II. Relationship between intrinsic viscosity and molecular weight

Samples of poly(12-dodecanelactam) (polylaurolactam, nylon 12) with $\text{M}$_n 1 × 10³?33 × 10³ were prepared. Polymerizations initiated with water or with lauric acid proceeded under conditions for minimum changes in end-group concentration. Values of $\text{M}$_n were calculated from the end-group content and $\text{M}$_n from light scattering in the mixture m-cresol/60 vol% of 2.2,3,3-tetrafluoropropanol. From measurements of intrinsic viscosity in m-cresol, the relationship [n] ? $\text{M}$_n was established in the given range of $\text{M}$_n. The relationship [n] ? $\text{M}$_w for $\text{M}$_w from 3.3 × 10³ to 125 × 10³ has been established.     

Identification of Cu(II)/Cu(III) couples in binuclear copper(II) complexes

A new series of binuclear copper(II) complexes were synthesised and studied by magnetic, spectral, ESR and cyclic voltammetry methods. The μ_eff values per copper atom correspond to the values observed for mononuclear copper(II) complexes. ESR spectral data in solution indicate weak interactions resulting from the electron delocalisation through the ligand system. Two nearly reversible red-ox couples are identified at $\text{+}\text{?}$0.50 V and $\text{+}\text{?}$0.75 V vs SCE. They correspond to Cu(II)αCu(III) red-ox processes, successively occurring at the two copper sites in the binuclear complexes.