Determination of equivalence points of sigmoidal potentiometric titration curves

The equivalence point of potentiometric titration curves coinciding with the point of inflection is calculated from the maximum of the first derivative or the zero of the second derivative. The approximation of the curve is carried out by means of cubicB-splines. The quality of the fitted curve depends on the number and position (proportionally spaced or equidistant) of knots. Series of simulated and measured titration curves were evaluated. Optimum fits were obtained with knot numbers about 20–35% of the number of data points. Only proportionally spaced splines are applicable for all types of curves and produce results of high accuracy.     

A note on the calculation of acid-base titration curves and their equivalence points

An equation has been developed for the calculation of the titration curve of any mixture of acids and bases, and an alternative way of defining an equivalence point, suitable for computer calculations, is presented.     

Approximation of titration curves and other sigmoidal functions by proportionally spaced cubic B-splines

Proportionally spaced cubicB-splines are an appropriate function system for the approximation of titration curves and other sigmoidal functions. They give a smooth weighted leastsquares fit without instability problems. The degree of smoothing depends only on the number of spline function which can be chosen by the user. The equivalence points of titration curves can be estimated with high accuracy from the zeros of the second derivative.The method gives good approximation curves even in the case of empty data regions, i. e. there are no artefacts in subranges where no data points exist. The routine has been tested successfully with large series of simulated and experimental data.     

Theory of titration curves : The locations of inflection points on acid-base and related titration curves

Consideration of the effect of dilution on the locations of the inflection points of potentiometric titration curves for titrations of monobasic acids or bases and for precipitation titrations in which the ions of the precipitate have numerically equal valences shows that;(I) In a strong acid-strong base or isovalent precipitation titration, the inflection point always precedes the equivalence point. No physically meaningful inflection point exists, regardless of the concentration of the substance titrated, if the concentration of the reagent is smaller than a certain value or if, when the concentration of reagent exceeds this limit, the concentration of the substance titrated is smaller than another limiting value.(2) In a weak acid-strong base or weak base-strong acid titration, the inflection point at which the slope is greatest also precedes the equivalence point, and vanishes under certain conditions. Earlier calculations are shown to have given incorrect information regarding the location and existence of this inflection point. The location of the inflection point at which the slope is smallest— the “point of maximum buffer capacity”—is shown to depend on the concentrations of the reagents employed.     

Four points function fitted and first derivative procedure for determining the end points in potentiometric titration curves: statistical analysis and method comparison.

A new method that belongs to the differential category for determining the end points from potentiometric titration curves is presented. It uses a preprocess to find first derivative values by fitting four data points in and around the region of inflection to a non-linear function, and then locate the end point, usually as a maximum or minimum, using an inverse parabolic interpolation procedure that has an analytical solution. The behavior and accuracy of the sigmoid and cumulative non-linear functions used are investigated against three factors. A statistical evaluation of the new method using linear least-squares method validation and multifactor data analysis are covered. The new method is generally applied to symmetrical and unsymmetrical potentiometric titration curves, and the end point is calculated using numerical procedures only. It outperforms the "parent" regular differential method in almost all factors levels and gives accurate results comparable to the true or estimated true end points. Calculated end points from selected experimental titration curves compatible with the equivalence point category of methods, such as Gran or Fortuin, are also compared with the new method.     

Redox Titrations-II Location of inflection points of titration curves for homogeneous redox reactions

The relative positions of the inflection points and equivalence point of a homogeneous redox reaction have been studied by using the redox buffer capacity to derive an equation for the titration curve. The position of the inflection point corresponding to the maximum slope of the titration curve relative to the equivalence point depends on the electron transfers of the analyte and the titrant (the stoichiometric coefficients of the reaction equation) and on the difference between the formal potentials of the redox couples in a more complicated way than has been described previously.     

On potentiometric titration curves in complexometry

An equation is given which describes the potential change during complexometric titration of metals with potentiometric end-point detection. The potential of the indicator electrode (e.g., silver or mercury electrode) depends on the equilibrium of the reaction MY + W right harpoon over left harpoon WY + M, where W represents the indicator metal ion, and M the metal ion determined by titration with the ligand Y, forming 1:1 chelates. Introduction of the conditional stability constants and corresponding side-reaction coefficients facilitates the prediction of the limits of application of this type of titration as well as the optimum conditions for the determination.     

Species formed in the potentiometric titration of fluoride with thorium or lanthanum nitrate and functions suitable for the evaluation of the equivalence points

Calculations by means of SILLÉN's HALTAFALL program have been performed on experimental curves by LINGANE for the potentiometric titration of fluoride with thorium and lanthanum in different media. After adjusting the stability constants for some of the main species formed during the titration a good fit was obtained between the experimental data and the calculated titration curves. In the cases where the solution was buffered with acetate it was necessary to consider the formation of mixed fluoride-acetate complexes. Suitable functions for the evaluation of the equivalence point could be derived from the calculations. If the functions: F₁=(v₀+v)[F^-] and F₃=(v₀+v)[F^-]^-3 are used before and after the equivalence point, respectively, it ought to be possible to determine fluoride very accurately by titration with lanthanum nitrate in an unbuffered solution. A value proportional to [F^-] is obtained from 10 exp (—EF/RT In 10), the Nernst e.m.f. equation.     

Theory of titration curves : Points of minimum slope on potentiometric strong acid-strong base and precipitation titration curves

Strong acid-strong base and precipitation titration curves, like other kinds of potentiometric titration curves, inherently possess an inflection point where the slope is a minimum as well as one where it is a maximum. In any kind of titration the first of the inflection points can be caused to occur earlier, and can eventually be made to disappear altogether, by adopting certain expedients. For a weak acid-strong base titration, for example, these include decreasing the concentration of the acid titrated and titrating in the presence of an excess of its conjugate base. For strong acid-strong base and precipitation titrations they include decreasing the concentration of the substance titrated and increasing the concentration of the titrant. The conditions under which a physically significant inflection point of minimum slope can exist are defined, and explicit equations are given from which its position can be calculated under various experimental conditions, for strong acid-strong base and for both isovalent and heterovalent precipitation titrations.     

Location of saddle points and minimum energy paths by a constrained simplex optimization procedure

Two methods are proposed, one for the location of saddle points and one for the calculation of steepest-descent paths on multidimensional surfaces. Both methods are based on a constrained simplex optimization technique that avoids the evaluation of gradients or second derivative matrices. Three chemical reactions of increasing structural complexity are studied within the PRDDO SCF approximation. Predicted properties of reaction hypersurfaces are in good overall agreement with those determined by gradient minimization and gradient following algorithms in connection with various ab initio SCF methods. Computational efforts required by the new procedures are discussed.     

Simplex optimization of acoustic assay for plasminogen activators

This article discusses the optimization of a newly developed method for measuring the activity of plasminogen activators using a thickness-shear-mode acoustic sensor. A variable-size simplex algorithm was used for optimization. Preliminary tests were performed to design the first simplex. A desirability function was defined to translate each performance value to a membership value of 0 to 1. If there was more than one performance variable, their membership values were translated to an aggregated membership value using another function that considers their individual influence on sensor performance. Two rounds of optimization were carried out for streptokinase followed by a single optimization for tissue-type plasminogen activator. In the last optimization, ratios of control variables were used in order to reduce the number of parameters and to formulate easily adjustable assay conditions. The results showed the usefulness of the simplex method for optimizing this type of assay, and the importance of preliminary tests and prior knowledge in providing rapid convergence using fewer experiments. The optimized plasminogen activator assay can be considered a reference method for measurement of all members of this drug class.     

Evaluation of acid-base titration curves obtained by the triangle-programmed titration technique in flowing solutions

Different evaluation techniques for triangle-programmed coulometric acid—base titration curves obtained with various detector systems are compared. In the case of potentiometric detection, hardware differentiation and linearization of the titration curves was investigated. Photometric end-point detection with single or mixed indicators was also studied. It was concluded that among the techniques studied the hardware potentiometric differentiation and the photometric detection with an indicator mixture are the most advantageous.     

A graphical method for locating the end point in potentiometric titration curves

Summary A new graphical method for locating the end point in potentiometric titration curves is described. By moving two glass rods to scan the entire curve, the end point is determined when the difference between the two refracted lines is maximum. The method is particularly useful when the end point is ill-defined.

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Graphisches Verjahren zur Bestimmung des Endpunktes potentiometrischer Titrationskurven

Zusammenfassung Zwei gleich starke Glasstäbe werden senkrecht zur Volumenachse über die Titrationskurve geführt. Der Endpunkt ist durch den maximalen Abstand der durch die Refraktion verzerrten Titrationskurve gekennzeichnet. Das Verfahren eignet sich auch noch bei Titrationskurven mit sonst schlecht definiertem Endpunkt.

     

A procedure for determining reaction paths and saddle points

To determine reaction paths, a method that does not require the previous location of extrema is presented and illustrated by an example. The procedure is based on a local symmetry property of the potential surface.