Calibration in quantitative analysis: Part 2. Confidence Regions for the Sample Content in the Case of Linear Calibration Relations

The precision of analytical results determined from linear calibration graphs is discussed. It is assumed that the standard deviation of the analytical procedure has the same value sx for all calibration levels and for the sample measurements. The collective and individual influence exerted by the number of both calibration and sample measurements (N and n, respectively) on the precision of the results is calculated on the basis of widths of confidence intervals. In addition, the effects on precision caused by the distribution of the calibration measurements over the calibration range and by the location of the sample in the analytical range are calculated. A quantity, comprising an “eccentricity term”, is developed, which contributes to better insight into the problems considered. For the special case that the k calibration levels are equidistant and the same number of replicate measurements m is made at each of these levels (km = N), the length of the confidence region is given by δx = 4s_x (n^-1 + z/N)^{$\text{1}\text{2}$} for α = 0.05. The value of z depends on the “eccentricity” of the sample value compared to the centre of gravity of the calibration values, and on the value of k. If the sample value coincides with the centre of gravity of the calibration values, z = 1. If the sample value coincides with one of the extreme values of the calibration experiments, the value of z increases to z = 2.00 for k = 2 and to z = 3.81 for k = 32.     

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).     

The effects of errors in measurements of absorbance and time on the calculated value of a pseudo-first-order rate constant

When data on the variation with time of the absorbance of a reactant or product are used to evaluate the rate constant of a first- or pseudo-first-order reaction, the precision of the result depends on the precisions with which both the time and the absorbance are measured. The natures of the dependences, and the ways in which they are affected by both constant and linearly varying background absorbances, are examined. If the standard error σ_t of a measurement of time is below about 0.005 t_{$\text{1}\text{2}$}, the standard error of the rate constant is virtually identical for an experiment in which the concentration of the reactant is followed as for one in which the concentration of the product is followed, but for larger values of σ_t it is better to follow the concentration of the reactant. In any event, errors in the measurements of time are much more likely to be significant than they are usually assumed to be. Other sources of error in such experiments, including constant and time-dependent background absorbances, are examined more briefly, with emphasis on the requirements that should be satisfied in work of the highest quality.     

Anomalies in the properties of Hf(S2−xTex)1−y and Hf(Se2−xTex)1−y near the metal-insulator transition

The solid solutions Hf(S_2?xTe_x)_1?y and Hf(Se_2?xTe_x)_1?y were prepared to examine changes in the electronic structure in a narrow composition range near the metal-insulator transition. Powder X-ray diffraction analysis and resistivity measurements are presented. The x³ dependence of the hexagonal c parameter for the sulfur solutions is due to large packing mismatches in the layers. In the selenium solutions, anomalous behavior is observed in the composition dependence of the a and c hexagonal lattice parameters in the range 0.05 < x < 0.10, y ～ 0. Between these concentration limits, both lattice parameters show positive deviations from smooth behavior and the diffraction linewidths broaden. The temperature dependence of the resistivity suggests that the Hf(S_2?xTe_x)_1?y solutions have an energy of activation for conduction at room temperature for $\text{0 < x <}\text{7}\text{8}$; the sample of composition $\text{x =}\text{5}\text{8}$ has a metal-insulator transition as a function of temperature below room temperature. The band gap goes to zero with composition for Hf(Se_2?xTe_x)_1?y in the range x = 0.080 to x = 0.095. Nonstoichiometry, phase separation, and changes from covalent to metallic bonding explain the structural and electronic changes observed in the seleno-telluride system near the metal-insulator transition.     

Neutron powder diffraction investigations of hydrogenated Pd3P0.80 and Pd6P

The solid solutions Sr_3?xLa_xMn₂O₇ (0 ≤ x ≤ 1.50) and Sr_3?xLn_xMn₂O₇ (Ln = Nd, Sm, Gd; 0 ≤ x ≤ 1.40) with Sr₃Ti₂O₇-type structure have been prepared. Their cell parameters and $\text{c}\text{a}$ ratios are related to the size of the rare earths and to the Mn³⁺ ion concentration.     

Excess molar enthalpies of {(1 − x2 − x3)Al + x2Bi + x3Ga}(I) alloys

The excess molar enthalpies H_m^E{(1 ? x₂ ? x₃)Al + x₂Bi + x₃Ga}(I) have been measured between 725 and 1170 K along the sections $\text{(1 ? x}{}_2\text{? x}{}_3\text{)}\text{x}{}_3\text{=}\text{1}\text{3}$, 1, and 3, and $\text{x}{}_2\text{x}{}_3\text{=}\text{1}\text{3}$, 1, and 3, with a high-temperature Calvet calorimeter using both the direct- and indirect-drop methods of mixing; experimental uncertainty is quoted respectively at 6.7 per cent and 9.9 per cent. The equilibrium temperatures confirmed phase boundaries previously determined by potentiometry, d.t.a., and calculation. Extrapolation of the experimental excess molar enthalpies to the limiting binary alloys {(1 ? x₂)Al + x₂Bi} allows new values for the excess molar enthalpies of these alloys to be proposed. The excess molar enthalpies of the ternary liquid mixtures can be represented correctly using these new values and Bonnier's equation.     

Magnetic properties of compounds (Mn,Cr)1+xSb,V1+xSb and (Mn,V)1+xSb with B8-type structures

Investigations of the compounds (Mn,Cr)_1+xSb, V_1+xSb and (Mn,V)_1+xSb with B8-type structures are described. The homogeneity ranges of (Mn,Cr)_1+xSb and (Mn,V)_1+xSb shift to metal-richer compositions with increasing temperature. These compounds are ferrimagnetic. The magnetic-ordering temperature and the spontaneous magnetization decrease with increasing $\text{Cr}\text{Mn}$ and $\text{V}\text{Mn}$ ratios, respectively.The homogeneity range of the high-temperature phase V_1+xSb is situated around the composition x = 0.40. V_1.40Sb shows nearly temperature-independent (Pauli) paramagnetism (except at low temperature).     

Oxydes de type perovskite du systeme CaNbO

Three perovskite-like phases of the CaNbO system have been isolated, $\text{Ca}{}_{\mathrm{x}}\text{NbO}{}_3\text{(0.9 ? x ? 1),}\text{Ca(Ca}{}_{\mathrm{x}}\text{Nb}{}_{\mathrm{1?x}}\text{)}\text{O}{}_3\text{(0 ? x ?}\text{1}\text{3}\text{)}$, and Ca₃Nb₂O_8?x. X-ray powder diffraction and electron microscopy and diffraction were used to obtain crystallographic data of these phases. The high temperature form of Ca(Ca_xNb_1?x)O₃ perovskites, where calcium ions occupy partially the octahedral sites, presents an orthorhombic symmetry and quasi-periodic twin bands like Ca_xNbO₃ compounds. Two monoclinic forms (for $\text{x =}\text{1}\text{3}$ and $\text{x =}\text{1}\text{4}$) were found by heating Ca(Ca_xNb_1?x)O₃ compounds at lower temperature for long periods; ordering of calcium and niobium ions in octahedral sites is proposed for that structure. The Ca₃Nb₂O_8?x compound presents a quadratic cell: ordering of the cations derived from (NH₄)₃FeF₆ structure is proposed in agreement with the observed space group $\text{P4}\text{nnc}$. A new multiple-phase family, with general formula Ca_nNb_nO_3n+2, was obtained for n = 4.25, 4.5, and 5; electron microscopy shows intergrowth phenomena.     

In situ studies of electrode reactions: The mechanism of lithium intercalation in TiS2

The mechanism of lithium intercalation into TiS₂ is poorly understood even though the system Li_xTiS₂ (0 ? x ? 1) has been extensively studied. This mechanism is critical in the functioning of $\text{Li}\text{TiS}{}_2$ nonaqueous batteries. In this report we describe several techniques which are used to study this mechanism. These methods consist of ex situ and in situ optical and X-ray techniques which yield information regarding the mechanism of lithium intercalation and factors which effect this process during the operation of $\text{Li}\text{TiS}{}_2$ batteries.     

Studies on the solvent extraction of yttrium thenoyltrifluoroacetonate

Studies of extractions of yttrium from aqueous solution into chloroform, benzene, methyl isobutyl ketone, and mixtures of these solvents, containing thenoyltrifluoroacetone are described. The effects of reagent concentration, added n-butylamine, and added fluoride were investigated. Values of pH$\text{1}\text{2}$ and P_e were determined, and identifications of the involved species are suggested.     

Bismuth substituted calcium,strontium, and lead apatites

Bismuth substituted apatites of two general types have been prepared: M_10?2xBi_xNa_x(PO₄)₆Y₂ (M = Ca, Sr, Pb; Y = F, Cl) and lead apatites with the Y ion completely vacant Pb_10?(2x+2)BixNa_x+2(PO₄)₆. X-ray powder diffraction patterns of all the compounds show the $\text{P6}{}_3\text{m}$ hexagonal apatite type structure. The change of the lattice parameters and $\text{c}\text{a}$ values with compositions indicate the preference of the bismuth ions to occupy the 6h triangular positions. Bi³⁺ tends to incorporate in apatites with unoccupied halide positions.     

Thermodynamic study of niobium oxides with ONb ratios from 2.47 to 2.50 using a high-temperature galvanic cell

The partial molar free energy, enthalpy, and entropy of oxygen in niobium oxides with $\text{O}\text{Nb}$ ratios from 2.47 to 2.50 were measured with a galvanic cell in the temperature range from 1084 to 1325 K. The partial molar enthalpies of oxygen of the Nb₂O_5?x and V phases were observed to be nearly independent of composition, indicating the presence of only weak interactions between defects. The value of the slope for the plots of log x in Nb₂O_5?x against log P_O2 was observed to be $\text{?1}\text{5.2}$ which is interpreted in terms of a defect structure involving both singly ionized and doubly ionized oxygen vacancies. The previously proposed phase diagram in the vicinity of Nb₂O_5?x was confirmed by the present emf measurements.     

Luminescent properties of Nd3+ in the NaxNdxM(1−2x)Ga2S4 thiogallates (M = Ca,Sr,Ba; x ≤ 0.5): A family of materials characterized by weak self-quenching and efficient band excitation

As a consequence of the weak phonon energies and the low crystal field, several excited states of Nd³⁺ are emitters in the Na_xNd_xM_1?2xGa₂G₄ thiogallates (x ≤ 0.5 for M = Ca or Sr and ≤0.2 for M = Ba). The infrared ${}^4\text{F}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ emission is little affected by concentration quenching. NaNdGa₄S₈ is the first efficient stoichiometric sulfide so far reported. Unlike other sulfides previously investigated, the neodymium thiogallates show an intense excitation band, ascribed to electron transfer from the valence band to states constituted essentially by neodymium orbitals.     

Structural intergrowth in the CaxLa1−xFeO3−x2 system (0 ≤ x ≤ 1): An electron microscopy study

An electron microscopy and diffraction study of several samples of the system $\text{Ca}{}_{\mathrm{x}}\text{La}{}_{\mathrm{1?x}}\text{FeO}{}_{\mathrm{<mtext>3?</mtext><mtext>x</mtext><mtext>2</mtext>}}$ shows these solids to be formed by the disordered intergrowth of two out of three of the ordered terms corresponding to $\text{x = 0, x =}\text{2}\text{3}$, and x = 1 (i.e., LaFeO₃, Ca₂LaFe₃O₈, and Ca₂Fe₂O₅). The relative amounts of the intergrowing individuals vary with x and may extend down to unit cell thickness; i.e., to the level of isolated extended defects.     

Comparative study on the precision of potentiometric techniques applied with ion-selective electrodes: Part 1. Direct techniques

The measurement techniques used with ion-selective electrodes, e.g. standard edition, double known addition, known subtraction, etc., are summarized. In each case, the error in the unknown concentration measured (c_x) can be characterized by the term [σ(C_x)/C_x]/σ(E) where σ(E) is the standard deviation of the potential measurement. The formulae developed make it possible to calculate the expected error for any strategy of standard addition or subtraction and to optimize precision by proper selection of the concentration and volumes of standard solutions.     

Thermodynamic study of lithium-vanadium bronzes by coulometric titration

Thermodynamic study has been made of lithium-vanadium bronzes β-Li_xV₂O_5?y in reversible cells with a lithium-conducting solid electrolyte using coulometric titration. The linear dependence of $\text{Δ}\text{G}{}_{\mathrm{<mtext>Li</mtext>}}$ in the bronze on parameters x and T was found. It is concluded that the dependence of $\text{Δ}\text{G}{}_{\mathrm{<mtext>Li</mtext>}}$ on the P_O2 value is weak. The limits of the homogeneity region of β-Li_xV₂O_5?y have been more clearly defined.     

Investigations on the influence of charge density on the mutual deactivation of polyion-macroradicals—II

Bimolecular rate constants (2 k_R) of the reaction R· + R· in aqueous solution were measured using the pulse radiolysis technique. R· designates radicals produced by H atom abstraction from polyvinyl alcohol partially acetalyzed with glyoxylic acid. Due to the expansion of the coils, 2 k_R increases with X_i the content of ionized base units in mole %. Above a critical value x_i (crit), 2 k_R decreases due to intermolecular repulsion. x_i(crit) decreases with increasing chain length $\text{n}{}_{\mathrm{<mtext>n</mtext>}}$ [x_i (crit) ≈ 0·5 mole % at $\text{n}{}_{\mathrm{<mtext>n</mtext>}}\text{= 3000}$ and x_i (crit) ≈ 1·5 mole % at $\text{n}{}_{\mathrm{n}}\text{= 270}$]. In the presence of NaClO₄, the following salt effects were observed: for x_i >x_i (crit), 2 k_R is increased because the intermolecular repulsion is partially depressed by screening. Below x_i (crit), 2 k_R is decreased since the coil expansion is diminished. At very low values of x_i where the coil expansion due to intramolecular repulsion is negligible, no salt effect was detectable.     

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.