Basic equations and uncertainties in isotope-dilution mass spectrometry for traceability to SI of values obtained by this primary method

A current interest in chemistry concerns traceability of analytical measurements to the International System of Units (SI) and the estimation of their uncertainties in accordance with principles of metrology, that is, measurement science. “Primary methods of measurement” achieve traceability to SI directly without intermediate reference standards or materials and without significant empirical correction factors. Isotope-dilution mass spectrometry should be regarded as such a method. It has the potential of smallest presently achievable uncertainties for analytical measurements directly or for the certification of reference materials including those with abnormal isotopic composition. A simple explanation of the method including its basic equations is given. Full uncertainty estimation is emphasized in terms of these equations. The wider use of concepts of metrology in chemistry is discussed.     

A group additivity approach for the estimation of heat capacities of organic liquids and solids at 298 K

A group additivity method is described which provides heat capacity estimates of the condensed phase. The data base consists of 810 liquids and 446 solids. Group values for carbon in various common substitution and hybridization states and for 47 functional groups are provided. The standard error of estimation using this approach on this data base is 19.5 (liquids) and 26.9 J/ (mole K) (solids). This can be compared to typical experimental uncertainties of 8.12 and 23,4 J/ (mole K) associated with these measurements, respectively. Experimental uncertainties were estimated from the numerical differences obtained for a given substance from multiple independent literature reports.     

Use of uncertainty information for estimation of certified values. Comparison of four approaches using data from a recent certification exercise

Four approaches for estimation of reference values and their respective uncertainties of characterisation were compared using data from the recently finalised certified reference materials ERM-EC680k and ERM-EC681k, elements in plastics. Reference values and uncertainties of characterisation were estimated as mean of laboratory means and their respective standard deviations, using equal weights and the weighting procedure of Mandel–Paule. In addition, two approaches taking into consideration uncertainty information reported by the participants, namely the consistency check and simulation procedure proposed by Cox for CCQM Key comparisons and an approach suggested by Pauwels et al. (Accred Qual Assur 3:180–184, 2000) were used. No difference between the equally-weighted and Mandel–Paule consensus means was observed and the reference value from the Cox approach was in all cases within ±2 u _char of each consensus mean. Uncertainties varied between the three approaches. Uncertainties derived from equally-weighted mean of means approach are on average 14% above uncertainties using the Mandel–Paule consensus mean, 36% above the uncertainties estimated by Pauwels et al., and 54% above the uncertainties from the Cox approach. Robustness of the uncertainty estimation against incorrect estimation of uncertainties was assessed. Assumption of a 50% uncertainty of the individual uncertainties resulted in an uncertainty of 30% of the uncertainty of characterisation. Differences between the four approaches are negligible for this dataset when combined with the uncertainty contribution from heterogeneity and stability as prescribed in ISO Guide 35. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Critical Eotvos numbers for buoyancy-induced oil drop detachment based on shape analysis

Critical values of the Eotvos number, which is half the Bond number, above which buoyancy induced drop detachment occurs, are estimated based on force balance equations available in the literature [Colloids Surf. A: Physicochem. Eng. Aspects 178 (2001) 249]. Since there are two significantly different expressions of the capillary retention force responsible for holding oil drops on a solid substrate in an aqueous phase, the critical dimensionless number is estimated with these two distinct equations. The differential equation defining the drop shape, with the constraints of the drop volume and the 'pinned' or 'receding' contact line, is numerically solved. The equilibrium drop shapes predicted are shown to match the experimentally observed variations in drop shape. From the numerical solution, it is observed that for interfacial tension (IFT) values lower than a certain limit for a given drop size, no numerically estimated drop shape can fulfil the drop volume constraint. Similarly, for the dimensionless number above a critical value, no shape can meet all the constraints. These critical Eotvos numbers are estimated, based on the above numerical approach, for initial contact angles measured in oil varying from 20 degrees to 90 degrees. It is found that the critical Eotvos numbers estimated from the numerical shape analysis are between the critical values estimated from the two force-balance equations. Near 90 degrees, the critical values estimated from the drop shape analysis matches the values from one of the force balance estimates, but merges with the critical values of the dimensionless number, estimated from the other force balance model near 10 degrees. From this analysis, it appears that a combination of the two equations for the capillary retention force is required, with one dominating when the contact angles are high, while the other applies for low values of the contact angle.     

Sequential imputation for missing values

As missing values are often encountered in gene expression data, many imputation methods have been developed to substitute these unknown values with estimated values. Despite the presence of many imputation methods, these available techniques have some disadvantages. Some imputation techniques constrain the imputation of missing values to a limited set of genes, whereas other imputation methods optimise a more global criterion whereby the computation time of the method becomes infeasible. Others might be fast but inaccurate. Therefore in this paper a new, fast and accurate estimation procedure, called SEQimpute, is proposed. By introducing the idea of minimisation of a statistical distance rather than a Euclidean distance the method is intrinsically different from the thus far existing imputation methods. Moreover, this newly proposed method can be easily embedded in a multiple imputation technique which is better suited to highlight the uncertainties about the missing value estimates. A comparative study is performed to assess the estimation of the missing values by different imputation approaches. The proposed imputation method is shown to outperform some of the existing imputation methods in terms of accuracy and computation speed.     

Resolution and uniqueness of estimated parameters of a model of thin filament regulation in solution

The estimation of chemical kinetic rate constants for any non-trivial model is complex due to the nonlinear effects of second order chemical reactions. We developed an algorithm to accomplish this goal based on the Damped Least Squares (DLS) inversion method and then tested the effectiveness of this method on the McKillop–Geeves (MG) model of thin filament regulation. The kinetics of MG model is defined by a set of nonlinear ordinary differential equations (ODEs) that predict the evolution of troponin–tropomyosin–actin and actin–myosin states. The values of the rate constants are estimated by integrating these ODEs numerically and fitting them to a series of stopped-flow pyrene fluorescence transients of myosin-S1 fragment binding to regulated actin in solution. The accuracy and robustness of the estimated rate constants are evaluated for DLS and two other methods, namely quasi-Newton (QN) and simulated annealing (SA). The comparison of these methods revealed that SA provides the best estimates of the model parameters because of its global optimization scheme. However it converges slowly and does quantify the uniqueness of the estimated parameters. On the other hand the QN method converges rapidly but only if the initial guess of the parameters is close to the optimum values, otherwise it diverges. Overall, the DLS method proves to be the most convenient method. It converges fast and was able to provide excellent estimates of kinetic parameters. Furthermore, DLS provides the model resolution matrix, which quantifies the interdependence of model parameters thereby evaluating the uniqueness of their estimated values. This property is essential for estimating of the dependence of the model parameters on experimental conditions (e.g. Ca²⁺ concentration) when it is assessed from noisy experimental data such as pyrene fluorescence from stopped-flow transients. The advantages of the DLS method observed in this study should be further examined in other physicochemical systems to firmly establish the observed effectiveness of DSL vs. the other parameter estimation methods.     

Traceability of measurement results in chemistry: a case study of certification of three new pharmaceutical reference materials

This paper compares the results of the studies carried out with three new candidate certified reference materials (CRM) of captopril, metronidazole, and sodium diclofenac, which are the first CRMs of these active pharmaceutical ingredients (APIs) reported in the literature. The studies were carried out according to the ISO Guides 34: 2009 and 35: 2006 and included the determination of organic, inorganic, and volatile impurities mass fractions, evaluation of homogeneity and stability under transport and storage conditions, calculation of API mass fractions by mass balance, cross-checking of obtained results, and estimation of measurement uncertainties. The certified property values and corresponding expanded uncertainties, obtained from the combined standard uncertainties multiplied by the coverage factor (k?=?2), for a confidence level of 95?%, were (995.65?±?0.93) mg/g for captopril, (998.87?±?0.15) mg/g for metronidazole, and (999.76?±?0.10) mg/g for sodium diclofenac. These new CRMs are intended to be used in assay and tests methods, including equipment calibration, method validation, as well as assignment of traceable property values and corresponding uncertainties to non-certified reference materials, with the objective to ensure metrological traceability of measurement results to the International System of Units, as well as results accuracy and comparability.     

Dry mass determination: what role does it play in combined measurement uncertainty?

Elemental mass fractions are normally reported on a dry-mass basis. In addition, producers of reference material state their property values for dry masses and relate them to a defined dry mass determination method. This paper describes how biases and uncertainties in dry mass correction factors may affect values reported as ‘concentration per dry mass’. The influence of the use of different drying methods, accuracy of drying temperature, length of drying, environmental humidity, and the sample mass used for dry mass determination have been evaluated using two candidate International Atomic Energy Agency reference materials. Deviation from proposed drying methods was observed to lead to differences of up to 5% in dry mass correction factors. In addition, the heterogeneity of sample moisture as well as hygroscopic behaviour during weighing play an important role in the uncertainty estimation. Proposals are made regarding which sources of uncertainty arising from dry mass determination should be considered and included in the combined uncertainty of analytical results.     

Limiting conditions for applying the spherical section assumption in contact angle estimation

The shape of liquid drops on solid surfaces deviates from the spherical as tension decreases and gravity effects start affecting the drop shape. This paper attempts to define this deviation and estimates the dimensionless Eotvos number limits above which the deviation becomes "significant." The use of these limiting values can facilitate estimation of contact angle in the following manner. It is well known that the equilibrium contact angle made by a liquid drop on a solid surface can be estimated from measurements of two drop parameters. These parameters can be any two chosen from the drop volume, height, and wetted radius. In case the effect of gravity on the drop shape is negligible, simple algebraic relations derived from the spherical section assumption exist, from which the contact angle can be estimated. In systems where the "spherical section" assumption is invalid, the Laplace equation for the drop shape has been solved numerically with any two of the above parameters as the constraints, to obtain the contact angle. In this paper, Eotvos numbers at which the deviation of the drop profile from the spherical is significant enough to result in contact angle deviation of 1 degrees are estimated. The limiting values of Eotvos number, expressed as a function of the original contact angle made by the spherical profile, are obtained by solving the Laplace equation for the drop shape with the drop volume and wetted radius constraints for decreasing values of Interfacial tension. These limiting values are also estimated for different drop sizes and for cases where the drop phase is heavier (sessile) and lighter (buoyant) than the surrounding fluid. The independence of the Eotvos number estimates from the sign of the density difference as well as the drop size is shown. These Eotvos number limits can be used to check if the spherical section assumption, with the resulting simple algebraic relations, can be used for contact angle estimation and other shape-related analysis for a system.     

Probability distributions of molecular observables computed from Markov models

Molecular dynamics (MD) simulations can be used to estimate transition rates between conformational substates of the simulated molecule. Such an estimation is associated with statistical uncertainty, which depends on the number of observed transitions. In turn, it induces uncertainties in any property computed from the simulation, such as free energy differences or the time scales involved in the system's kinetics. Assessing these uncertainties is essential for testing the reliability of a given observation and also to plan further simulations in such a way that the most serious uncertainties will be reduced with minimal effort. Here, a rigorous statistical method is proposed to approximate the complete statistical distribution of any observable of an MD simulation provided that one can identify conformational substates such that the transition process between them may be modeled with a memoryless jump process, i.e., Markov or Master equation dynamics. The method is based on sampling the statistical distribution of Markov transition matrices that is induced by the observed transition events. It allows physically meaningful constraints to be included, such as sampling only matrices that fulfill detailed balance, or matrices that produce a predefined equilibrium distribution of states. The method is illustrated on mus MD simulations of a hexapeptide for which the distributions and uncertainties of the free energy differences between conformations, the transition matrix elements, and the transition matrix eigenvalues are estimated. It is found that both constraints, detailed balance and predefined equilibrium distribution, can significantly reduce the uncertainty of some observables.     

Uncertainty calculations in the certification of reference materials. 1. Principles of analysis of variance

 The preparation and certification of reference materials is a rapidly developing area. Many innovative reference materials have limited homogeneity and stability, and, additionally, the uncertainty estimation of the property values must be brought in agreement with the principles of the “Guide to the expression of uncertainty in measurement” (GUM). The results of the homogeneity and stability studies must be included to a certain extent in the uncertainty of the property values of the reference material, in order to comply with these requirements. The basic theory needed to accomplish this is essentially the theory of analysis of variance (ANOVA). As GUM also allows alternative evaluations other than Type A evaluations, a reinterpretation of the theory of ANOVA is necessary to establish a model for the certification of reference materials that is widely applicable. For this, analysis of variance can be used as a statistical technique to derive standard uncertainties from homogeneity, stability and characterisation data. Received: 10 May 2000 / Accepted: 29 July 2000     

Development of a certified reference material (NMIJ CRM 7505-a) for the determination of trace elements in tea leaves

A certified reference material (CRM) for trace elements in tea leaves has been developed in National Metrology Institute of Japan (NMIJ). The CRM was provided as a dry powder (<90 μm) after frozen pulverization of washed and dried fresh tea leaves from a tea plant farm in Shizuoka Prefecture, Japan. Characterization of the property value for each element was carried out exclusively by NMIJ with at least two independent analytical methods, including inductively coupled plasma mass spectrometry (ICP-MS), high-resolution (HR-) ICP-MS, isotope-dilution (ID-) ICP-MS, inductively coupled plasma optical emission spectrometry (ICP-OES), graphite-furnace atomic-absorption spectrometry (GF-AAS) and flame atomic-absorption spectrometry (FAAS). Property values were provided for 19 elements (Ca, K, Mg, P, Al, B, Ba, Cd, Cu, Fe, Li, Mn, Na, Ni, Pb, Rb, Sr, Zn and Co) and informative values for 18 elements (Ti, V, Cr, Y, and all of the lanthanides, except for Pm whose isotopes are exclusively radioactive). The concentration ranges of property values and informative values were from 1.59% (mass) of K to 0.0139 mg kg(-1) of Cd and from 0.6 mg kg(-1) of Ti to 0.0014 mg kg(-1) of Lu, respectively. Combined relatively standard uncertainties of the property values were estimated by considering the uncertainties of the homogeneity, analytical methods, characterization, calibration standard, and dry-mass correction factor. The range of the relative combined standard uncertainties was from 1.5% of Mg and K to 4.1% of Cd.     

Experimental studies of uncertainties associated with chromatographic techniques.

The paper describes experiments for the evaluation of uncertainties associated with a number of chromatographic parameters. Studies of the analysis of vitamins by HPLC illustrate the estimation of the uncertainties associated with experimental "input" parameters such as the detector wavelength, column temperature and mobile phase flow-rate. Experimental design techniques, which allow the efficient study a number of parameters simultaneously, are described. Multiple linear regression was used to fit response surfaces to the data. The resulting equations were used in the estimation of the uncertainties. Three approaches to uncertainty calculation were compared--Kragten's spreadsheet, symmetric spreadsheet and algebraic differentiation. In cases where non-linearity in the model was significant, agreement between the uncertainty estimates was poor as the spreadsheet approaches do not include second-order uncertainty terms.     

Estimation of reactivity ratios using emulsion copolymerization data

An approach for the estimation of reactivity ratios using cumulative copolymer composition–overall conversion data obtained in batch emulsion copolymerization is presented. The approach is based on an algorithm for parameter estimation in ordinary differential equations and takes into account the partition of the comonomers between the different phases present in the system. Both copolymer composition and conversion were considered to be affected by experimental errors. The method was first checked by using simulated data that included random errors. The effect of the initial guess and level of errors on the values of the estimated reactivity ratios was investigated. Once the approach was checked, it was used to estimate the reactivity ratios of the styrene–acrylonitrile system based on data obtained in unseeded batch emulsion copolymerizations of these monomers.     

Stochastic ensembles of thermodynamic potentials

The provision of uncertainty estimates along with measurement results or values computed thereof is metrologically mandatory. This is in particular true for observational data related to climate change, and thermodynamic properties of geophysical substances derived thereof, such as of air, seawater or ice. The recent International Thermodynamic Equation of Seawater 2010 (TEOS-10) provides such properties in a comprehensive and highly accurate way, derived from empirical thermodynamic potentials released by the International Association for the Properties of Water and Steam (IAPWS). Currently, there are no generally recognised algorithms available for a systematic and comprehensive estimation of uncertainties for arbitrary properties derived from those potentials at arbitrary input values, based on the experimental uncertainties of the laboratory data that were used originally for the correlations during the construction process. In particular, standard formulas for the uncertainty propagation which do not account for mutual uncertainty correlations between different coefficients tend to systematically and significantly overestimate the uncertainties of derived quantities, which may lead to practically useless results. In this paper, stochastic ensembles of thermodynamic potentials, derived from randomly modified input data, are considered statistically to provide analytical formulas for the computation of the covariance matrix of the related regression coefficients, from which in turn uncertainty estimates for any derived property can be computed a posteriori. For illustration purposes, simple analytical application examples of the general formalism are briefly discussed in greater detail.     

Zone broadening in electrophoresis with special reference to high-performance electrophoresis in capillaries: an interplay between theory and practice

Approximate equations have been derived for the total (final) zone width (plate height, plate number and resolution) as a function of the width of the starting zone and of the zone broadening caused by diffusion. Joule heat, adsorption and the difference in conductivity (delta kappa) between a solute zone and the surrounding buffer. Two cases are treated: (A) the conductivity differences eliminate entirely or (B) partially, the diffusional broadening at one boundary of a zone. When adsorption is negligible one can derive from these equations the field strength - and for case A also the electrical conductivity - that gives the minimum zone broadening (plate height). Interestingly, at this minimum, contributions to the zone broadening from diffusion. Joule heat and conductivity differences have the ratio 4:1:1 in case A. In case B the ratio between the broadening caused by diffusion (including that caused by conductivity and pH differences) and broadening due to Joule heat is 4:1. The total zone width, plate height and optimal field strength calculated from the derived equations agree satisfactorily with experimental values. A simple method to estimate the variance of the zone broadening caused by the Joule heat led to a formula similar to that calculated mathematically. An appropriate width of the starting zone can be calculated rapidly by means of a simple formula. Following a run the true width can be estimated graphically from measurements of plate heights or zone widths at low field strengths. For high resolution the width of the starting zone usually should not exceed 0.5 mm. A new principle for the design of multi-buffer systems which generate sufficiently narrow starting zones has been developed for carrier-free zone electrophoresis. This zone sharpening method permits application of wide zones of concentrations below the detection limit of the monitor. The diffusion coefficient (D) and the universal parameter D/mu (mu = mobility) appear in many of the equations derived and are often the only variables which are not easily accessible. Simple methods have therefore been developed by which they can be determined with sufficient accuracy. Fortunately, they are raised to the power of 1/5 in many formulas and therefore only a rough estimation is required. True plate numbers (calculated in the absence of electroendosmosis) often differ considerably from apparent plate numbers (calculated in the presence of electroendosmosis). A mathematical relationship between the true and apparent plate numbers has been derived.(ABSTRACT TRUNCATED AT 400 WORDS)     

Paradoxes and paradigms: observations on pyrohydrolysis,oxygen bomb combustion,and alkaline carbonate fusion,most frequently used decomposition methods for subsequent determination of fluorine and accompanying thermochemistry

Pyrohydrolysis, oxygen bomb combustion, and alkaline carbonate fusion are the most frequently used methods for decomposition of fluorine containing materials. The efficiency of these methods was proven by the determination of fluorine content in certified reference materials of clay and vegetation. Possible reactions proceeding during decomposition were suggested and accompanying thermochemistry discussed. The Gibbs energies were estimated to establish if suggested reactions are thermodynamically favorable or not. In addition, linear relationships between the enthalpies of formation of metal fluorides and the balanced values of the enthalpies of formation of the plausible reaction products (metal tungstates, metal oxides, or metal carbonates), electronegativity of metals, and number of fluorine atoms in metal fluorides were established. These equations were suggested for the estimation of the enthalpies of formation of metal tungstates, metal oxides, or metal carbonates, for which experimental data are not available.     

The determination of an accepted reference value from proficiency data with stated uncertainties

Proficiency data with stated uncertainties represent a unique opportunity for testing that the reported uncertainties are consistent with the Guide to the expression of uncertainty in measurement (GUM). In most proficiency tests, however, this opportunity is forfeited, because proficiency data are processed without regard to their uncertainties. In this paper we present alternative approaches for determining a reference value as the weighted mean of all mutually consistent results and their stated uncertainties. Using an accepted reference value each reported uncertainty estimate can be expressed as an E _n number, but a value of confirms its validity only if the uncertainty of the reference value is negligible in comparison.Reference values calculated for results from an International Measurement Evaluation Programme (IMEP-9) by “bottom up” as well as “top down” methods were practically identical, although the first strategy yielded the lowest uncertainty. A plot of individual coefficients of variation (CV) versus E _n numbers helps interpretation of the proficiency data, which could be used to validate relative uncertainties down to <1%.     

Determination of the Kinetic Parameters of Chemical Reactions on the Basis of Non-Isothermal Measurements A critical review

A detailed analysis is presented of the applicability of several dependences commonly used for the determination of activation energies from non-isothermal measurements. Reactions proceeding according to different kinetic equations are simulated and the validity of the activation energy values obtained is discussed. The general conclusion is drawn that none of the examined dependences should be used to determine the activation energy. For a rough estimation of activation energy, the Kissinger equation can be applied according to Ockham's razor. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the computation of the fundamental derivative of gas dynamics using equations of state

The value of the fundamental derivative of gas dynamics, Γ$\mathrm{\Gamma }$, is a quantitative measure of the variation of the speed of sound with respect to density in isentropic transformations, such as those occurring, for example, in gas-dynamic nozzles. The accurate computation of its value, which is a constant for a perfect gas, is key to the understanding of real-gas flows occurring in a thermodynamic region where the polytropic ideal gas law does not hold. The fundamental derivative of gas dynamics is a secondary thermodynamic property and so far, no experiments have been conducted with the aim of measuring its value. Several studies document the estimation of Γ$\mathrm{\Gamma }$ for fluids composed of complex molecules using mainly simple thermodynamic equations of state, e.g., that of Van der Waals. A review of these studies has revealed that the calculated values of Γ$\mathrm{\Gamma }$ are affected by large uncertainties; these uncertainties are due to the functional form of the adopted equations and because of uncertainties in the available fluid property data on which these equations were fitted. In this work, the fundamental derivative of gas dynamics of molecularly simple fluids is computed with the aid of, among other models, modern reference equations of state. The accuracy of these computations has been assessed. Reference thermodynamic models however, are not available for molecularly complex fluids; some of these molecularly complex fluids are the substances of interest in studies on the so-called nonclassical gas dynamics. Therefore, results of the computation of Γ$\mathrm{\Gamma }$ for few, molecularly simple hydrocarbons, like methane, ethane, etc., are used as a benchmark against which the performance of simpler equations of state, can be assessed. For the selected substances, the Peng–Robinson, Stryjek–Vera modified, cubic equation of state yields good results for Γ$\mathrm{\Gamma }$-predictions, while the modern multiparameter technical equations of state, e.g., the one in the Span–Wagner functional form, are preferable, provided that enough accurate thermodynamic data are available. Another notable result of this study, is that Γ$\mathrm{\Gamma }$ for a fluid composed of complex molecules is less affected by the inaccuracy of _Cv$C_v$-information (_Cv$C_v$ is the isochoric heat capacity), if compared to the estimation of Γ$\mathrm{\Gamma }$ for simple molecules. Inspection of the results of the calculation of Γ$\mathrm{\Gamma }$ in the proximity of the critical point confirms that analytical equations of state fail to predict the correct physical behavior, even if they include terms which allow for the correct estimation of thermodynamic properties.