Calculation of equilibrium constants from multiwavelength spectroscopic data-IV Model-free least-squares refinement by use of evolving factor analysis

The newly developed algorithm of evolving factor analysis has been supplemented by iterative refinement. It allows the completely model-free calculation of concentration profiles and spectra from spectrophotometric and other spectroscopic data. Not even implicit use is made of the law of mass action. The results are practically identical with those based on a specific chemical model and classical least-squares refinement. Iterative evolving factor analysis is based on applying factor analysis successively to the set of the first 1,2 M spectra of a spectrometric titration. The analysis is repeated from the opposite end and the eigenvalues thus calculated are combined into “concentration profiles” of completely abstract “species”. These “concentration profiles” are iteratively refined by normalization, calculation of the absorption spectra from the normalized concentrations and recalculation of the concentration profiles from the absorption spectra. Evolving factor analysis is not restricted to spectrometric titrations, and can also be applied to peak resolution in chromatography using a multiwavelength (diode array) photometric or mass-spectrometric detection system, or to any other ordered set of multichannel data.     

Calculation of equilibrium constants from multiwavelength spectroscopic data-I Mathematical considerations

Multiwavelength spectrophotometric and spectroscopic data in general contain considerably more information about complexation equilibria than potentiometric data do. With the construction of a fully automatic titration set-up built into a high-precision spectrophotometer, the problems related to the wider use of this method have shifted from the quality of the primary data to the complexity of their numerical treatment. Matrix algebra is used to show how these problems can be overcome. An algorithm is described for calculation of stability constants and absorption spectra, together with the associated standard errors, at a reasonable expense of computer time. Problems in finding the minimum in a multidimensional parameter space are reduced by elimination of the molar absorptivities from the algorithm for the iterative refinement. Numerical safety and speed of calculation are improved by use of analytical instead of numerical derivatives. The number of data to be fitted is decreased by principal-component analysis.     

Calculation of equilibrium constants from multiwavelength spectroscopic data--II: SPECFIT: two user-friendly programs in basic and standard FORTRAN 77

A new program (SPECFIT), written in HP BASIC or FORTRAN 77, for the calculation of stability constants from spectroscopic data, is presented. Stability constants have been successfully calculated from multiwavelength spectrophotometric and EPR data, but the program can be equally well applied to the numerical treatment of other spectroscopic measurements. The special features included in SPECFIT to improve convergence, increase numerical reliability, and minimize memory as well as computing time requirements, include (i) elimination of the linear parameters (i.e., molar absorptivities), (ii) the use of analytical instead of numerical derivatives and (iii) factor analysis. Calculation of stability constants from spectroscopic data is then as straightforward as from potentiometric titration curves and gives results of analogous reproducibility. The spectroscopic method has proved, however, to be superior in discrimination between chemical models.     

A program for evaluating equilibrium constants from spectrophotometric data by non-linear regression analysis

A new program, CFTSP, is described for computing stability constants and molar absorptivities from data on the absorbances of mixtures of a metal ion with a ligand. It is written in FORTRAN 77, can run on personal computers, and has facilities for interactive data analysis and presentation that ease the operator's task in searching for the equilibrium model that best fits the experimental data. A critical evaluation of the performance of the program is presented, and some general criteria for selecting equilibrium parameters, making use of both synthetic and experimental data, are described.     

The thermodynamic dissociation constants of methotrexate by the nonlinear regression and factor analysis of multiwavelength spectrophotometric pH-titration data

The mixed dissociation constants of methotrexate — chemically (2S)-2-[(4-{[(2,4-diamino-7,8-dihydropteridin-6-yl)methyl] (methyl)amino}phenyl)formamido]pentanedioic acid (the cas number 59-05-2) at various ionic strengths I of range 0.01–0.4, and at temperatures of 25°C and 37°C, were determined with the use of two different multiwavelength and multivariate treatments of spectral data, SPECFIT32 and SQUAD(84) nonlinear regression analyses and INDICES factor analysis according to a general rule of first, determining the number of components, and then calculating the spectral responses and concentrations of the components. Concurrently, the experimental determination of the thermodynamic dissociation constants was in agreement with its computational prediction of the PALLAS programme based on knowledge of the chemical structures of the drug. The factor analysis in the INDICES programme predicts the correct number of light-absorbing components when the data quality is high and the instrumental error is known. Three thermodynamic dissociation constants were estimated by nonlinear regression of {pK _a , I} data: for methotrexate pK_a1^T= 2.895(13), pK_a2^T= 4.410(14), pK_a3^T= 5.726(15) at 25°C and pK_a1^T= 3.089(15), pK_a2^T= 4.392(12), pK_a3^T= 5.585(11) at 37°C, where the figure in brackets is the standard deviation in last significant digits. The reliability of the dissociation constants of the drug were proven by conducting goodness-of-fit tests of the multiwavelength spectrophotometric pH-titration data.      

Handling of electronic absorption spectra with a desk-top computer-II: calculation of stability constants from spectrophotometric titrations

A fully automatic system for combined spectrophotometric and pH titrations was described in Part I. Its performance in the titration of weak acids and metal complexes is discussed, along with a computer program for numerical treatment of the data, based on Marquardt's modification of the Newton—Gauss non-linear least-squares method. The deprotonation of p-nitrophenol at concentrations of 4 × 10⁻⁵ and 4 × 10⁻⁶M was studied in order to test the sensitivity. Results identical within the reproducibility of the pH-meter were obtained: pK^H = 7.00 ± 0.01 and 7.02 ± 0.01, respectively. Three complexation reactions were studied: (1) the interaction of SCN⁻ with the Co²⁺ complex of 1,4,8,11-tetramethyl-1,4,8,11-tetra-azacyclotetradecane (TMC); five independent experiments gave pK [CoTMC (SCN)⁺ CoTMC²⁺ + SCN⁻] = 3.099 ± 0.003: (2) the deprotonation of the Cu²⁺ complex of 3,7-diazanonanediamide (DANA); five experiments gave pK (CuDANA²⁺ CuDANAH⁺₋₁ + H⁺) = 7.14 ± 0.01 and pK (CuDANAH⁺₋₁ CuDANAH₋₂ + H⁺) = 8.38 ± 0.01: (3) for the reaction of Cu²⁺ with 1,3,7-triazacyclodecane (L), data from different ligand: metal ratios had to be combined to obtain pK (CuL²⁺ Cu²⁺ + L) = 16.19 ± 0.01, pK (CuL²⁺₂ CuL²⁺ + L) = 10.30 ± 0.01, and pK (Cu₂L₂ (OH)²⁺₂ 2 CuL²⁺ + 2 OH⁻) = 14.58 ± 0.03. Titration curves with a total change in absorbance of as little as 0.03 units could be analysed satisfactorily, extending considerably the useful range of concentrations for spectrophotometric titrations. In combined spectrophotometric/pH titrations the accuracy of the glass electrode is normally the limiting factor. Other equilibrium constants can easily be reproduced with standard errors of less than 0.01 log unit.     

Determination of equilibrium constants of strong acidic ion exchange resins by potentiometric titrations

A new method for the determination of ion exchange equilibrium constants is developed. A technique based in the application of potentiometric titrations is used in the study of non selective strong acidic resins. Potentiometric titrations are carried out without constant ionic strength, so a mathematical treatment is developed in order to take into account the estimation of the activity coefficients and the liquid junction potentials. The ion exchange thermodynamic equilibrium constants for two resins, Dowex CM-15 and Dowex C650, between the proton form and the alkaline metals at 25 degrees C are given.     

SPECA: a program for the calculation of thermodynamic equilibrium constants from spectrophotometric data

A computer program called SPECA and written in compiled BASIC (Microsoft QuickBasic Ver. 4.5) has been written for the calculation of thermodynamic constants in solution equilibria studies from spectrophotometric measurements. In order to calculate the thermodynamic constants it is necessary to introduce different theories to estimate the activity coefficients of the system involved under investigation. In this version of the program the equations proposed in the models of Debye-Hückel. Specific Interaction Theory (SIT) or the Modified's Bromley Theory can be chosen. To achieve its purpose, the program uses absorbance and activity or free concentration data of one of the components, normally H(+), and it is not limited with respect to the number of components that can be studied. Minimization of the error square sum in the absorbance, which can be absolute or relative, is carried out using the Levenberg-Marquardt-Nash algorithm at three different levels: equilibrium constants in the first level, molar absorptivities in the second and interaction coefficients in the third for those systems using the SIT or the Modified Bromley's Theories.     

New spectrophotometric method for determining the stoichiometry and equilibrium constants of some redox reactions

A new spectrophotometric method is proposed for determining the stoichiometries of redox reactions; it can be used in many cases for determining the equilibrium constant of the reaction. The method is based on that of Holme and Langmyhr. Good results were obtained for several redox reactions involving iodate, bromate or periodate, and manganese(II) or some cyclohexanedione bisthiosemicarbazones.     

Development of algorithms for automated spectrophotometric titrations

A versatile method for evaluation of equilibrium constants from automated spectrophotometric titrations is described. General algorithms were derived for normalized titration curves of reactions having various stoichiometric ratios. Normalization of data involve converting it from absorption versus time to product formed (0 to 1) versus equivalents of titrant added. Relevance of derived algorithms was assessed by the quality of their fit to real data. The demonstration system chosen for this investigation was the 1:3 complexation reaction of Fe(II) and 1,10-phenantroline (o-Phen). Titrations were performed at three distinct infusion rates under controlled pH of 5.0 and at 25.0 °C, with sets produced using both Fe(II) and o-Phen as the titrant. Direct fitting of titration curves yielded values for overall equilibrium constants. Furthermore, resulting curves revealed reproducible initial deviation, consistent with step-wise reaction, when titrating Fe(II) with o-Phen. No unusual deviations were evident anywhere in the titration curves when Fe(II) was the titrant. Overall, our approach facilitates adoption of automated titrimetry and data analysis methodologies for the determination of stoichiometries and equilibrium constants of many UV/Vis active reactions.     

Nuclear magnetic resonance spectroscopic method for the estimation of competitive equilibrium constants

A nuclear magnetic resonance method is described which yields precise estimates of competitive equilibrium constants for systems in which two or more complexing agents compete for common species. Chemical-shift data under rapid exchange conditions are required, but neither analytical not equilibrium concentrations are needed. The method is demonstrated by estimating equilibria of proton exchange between aniline and pyridinium ion, proton exchange between triethylamine and diisopropylammonium ion, and cyclohexaamylose exchange between m- and p-hydroxybenzoic acids. The precisions of the equilibrium constants are similar to or better than those obtained from classical methods and are relatively free from interferences.     

Calculation of equilibrium formation constants of complexes with a polydentate oligomer

A methodology of the calculation of ligand-oligomer complex formation and cooperative binding constants by the matrix method is described. Theoretical analysis of the equilibrium binding between a ligand and a polydentate oligomer having a system of sites with different ligand affinities is performed. The simulated equilibrium complex formation between the oligonucleotide G4-DNA and porphyrin TMPyP4 was used to construct and analyze diagrams of the relative fraction of DNA-ligand complexes by the matrix method. Cooperative ligand binding to a system of nonequivalent sites can be established by chemometric analysis and rigid modeling of spectrophotometric titration data (in terms of the chemical equilibrium model).     

Raman spectroscopic determination of equilibrium constants of uranyl sulphate complexes in aqueous solutions

Raman spectra are used to determine the formation constants of uranyl sulphate complexes in aqueous solutions at 20 degrees and remedy the confusion existing in this area in the available literature. Solutions with a varying total sulphate concentration and an ionic strength lower than 0.4M are analysed. The species UO(2)SO(4) and UO(2)(SO(4))(2-)(2) are characterized by a resolved Raman band at 861 cm(-1) and an unresolved one at 852 cm(-1), corresponding to the uranyl symmetrical stretching vibration. The equilibrium constants, in term of activity (standard state 1M), are found to be about 1400 and 11, respectively, for the consecutive reactions: UO(2+)(2)(aq)+SO(2-)(4)(aq)=UO(2)SO(4)(aq) and UO(2)SO(4)(aq)+SO(2-)(4)(aq)=UO(2)(SO(4))(2-)(2)(aq).     

Determination of equilibrium constants from chromatographic and electrophoretic measurements

Chemical interactions, such as acid-base, complex-forming, ion association and other equilibria, are widely exploited to improve the separation efficiency in liquid chromatography as well as in electrophoresis. On the other hand, these techniques can be advantageously used to study the chemical equilibria affecting the separations. If the equilibium is sufficiently fast in comparison with the separation process, then the retention characteristics in chromatography (retention factors) or the migration characteristics in electrophoresis (effective mobilities) may be expressed as functions of the composition of mobile phase or background electrolyte (BGE), respectively. Using a proper experimental arrangement, the dependencies of retention (migration) characteristics on the mobile phase (background electrolyte) composition can be measured and utilized to calculate the equilibrium constants for equlibria taking place in the mobile phase (background electrolyte). Although principles of these measurements have been known for a long time, only more recent studies utilizing HPLC and capillary electrophoretic techniques are reviewed in this paper.     

Speciation of the uranyl nitrate system via spectrophotometric titrations

The speciation of the uranyl nitrate system has been studied previously with limited success producing a wide range of stability constant values. The current literature has values for the mononitrate species, with scattered data for higher nitrate species. Furthermore, the reported values vary with experimental method. The work presented here examines the stability of the uranyl/nitrate/perchlorate/water system via spectrophotometric titrations with a focus on the predominance of the uranyl dinitrate species at low nitrate concentrations. Stability constants were determined at ionic strengths ranging from 1 to 6 molal followed by refinement with the specific ion interaction theory. The zero ionic strength stability constant of the uranyl dinitrate species was refined as log β (2,1)º = 3.37 ± 0.02 when including the stability constant for the uranyl mononitrate from Ahrland and 2.66 ± 0.02 without. These values are considerably higher than previous studies, which is attributed to the alternate speciation model used. The values generated in this work produce speciation diagrams that are consistent with published solvent extraction data of the uranyl nitrate system.     

Calculation of stability constants from photometric data

A method originally designed for the determination of stability constants by using a wedge calorimeter has been adapted to normal photometric measuring techniques. The calculations are simple and the method is especially suitable for use when the absorptivities of the absorbing species are unavailable because these species cannot be obtained alone in solution. Results of some determinations are given.     

Determination of ion exchange equilibrium constants of strongly acidic resins with alkaline-earth metals by means of the potentiometric titrations technique

A recently developed methodology for the determination of ion exchange equilibrium constants has been applied to ion exchange systems of 1:2 stoichiometry. Potentiometric titrations with variable ionic strength were carried out. Ionic medium titrations were performed for the estimation of the liquid junction potential. The modified Bromley's methodology and the Wilson model were used for the estimation of the activity coefficients of the species in the aqueous and resin phase, respectively. A modification of the Henderson equation is used for the estimation of liquid junction potentials in the mixtures including 1:2 electrolytes. Equilibrium constants for the H(+)/M(2+) (M=Mg, Ca, Sr and Ba) exchange systems in the strongly acidic resins Dowex CM-15 and Dowex C650 were studied.