Talanta 79 (2009) 1398–1405 Contents lists available at ScienceDirect Talanta journal homepage: www.elsevier.com/locate/talanta End-point detection in potentiometric titration by continuous wavelet transform Małgorzata Jakubowska ∗ , Bogusław Ba´ s, Władysław W. Kubiak Faculty of Materials Science and Ceramics, AGH University of Science and Technology, 30-059 Kraków, al. Mickiewicza 30, Poland article info Article history: Received 24 March 2009 Received in revised form 29 May 2009 Accepted 2 June 2009 Available online 12 June 2009 Keywords: Wavelet theory Dedicated mother wavelet Potentiometric titration End-point detection abstract The aim of this work was construction of the new wavelet function and verification that a continuous wavelet transform with a specially defined dedicated mother wavelet is a useful tool for precise detection of end-point in a potentiometric titration. The proposed algorithm does not require any initial information about the nature or the type of analyte and/or the shape of the titration curve. The signal imperfection, as well as random noise or spikes has no influence on the operation of the procedure. The optimization of the new algorithm was done using simulated curves and next experimental data were considered. In the case of well-shaped and noise-free titration data, the proposed method gives the same accuracy and precision as commonly used algorithms. But, in the case of noisy or badly shaped curves, the presented approach works good (relative error mainly below 2% and coefficients of variability below 5%) while traditional procedures fail. Therefore, the proposed algorithm may be useful in interpre- tation of the experimental data and also in automation of the typical titration analysis, specially in the case when random noise interfere with analytical signal. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Titration is a very useful and reliable technique, which is widely used in different fields such as the food industry, scientific research, and also chemical, clinical and pharmaceutical laboratories. Titri- metric procedures based on a detection of the end-point, i.e., the point at which volumetric titration is completed, are successfully employed over a wide range of concentrations and are popular because of their simplicity, speed, accuracy and good reproducibil- ity. The importance of titrimetric analysis has increased with the advance of instrumental methods (potentiometric, amperometric, conductimetric, photometric) of end-point detection. The accuracy and precision of the results of a titrimetric determination depend on the nature of the titration reaction, but they are also influenced by the technique of the end-point location. The methods for the determination of the end-points of titration can be graphical or numerical [1,2]. Graphical methods are simpler and are based on the graphical determination of the end-point by searching for the inflection point. The mathematical methods were developed in two ways: as numerical approximate and numerical modeling methods. The approximate methods are based on the pre- sumption that the end-point of a titration is the inflection point of the titration curve. In these methods, the chemical reaction is not important. Only a small number of points in the vicinity of the inflection point are used for the calculation. Common meth- ∗ Corresponding author. Fax: +48 12 6341201. E-mail address: jakubows@agh.edu.pl (M. Jakubowska). ods of automated end-point detection rely on using the first and second derivatives of the classic sigmoid titration curve. The end- point usually occurs at the point of maximum deflection, where the absolute value of the first derivative reaches a maximum and the second derivative changes sign. Unfortunately, they lack of accu- racy in the end-point calculation. In Ref. [3] a new method that belongs to the differential category is presented. It uses a prepro- cess to find first derivative values, by fitting four data points in and around the region of inflection to a non-linear function Then the end-point is usually calculated as maximum or minimum. In Ref. [4], the end-point is found by means of the Fibonacci method, adapted to act as a one-dimensional optimization algorithm for fast titration. The category of differential methods depends on the pres- ence of an inflection point of function, but does not require any prior information about the nature of the analyte. The imperfection of approximation methods for the determination of the inflection point relies in the fact, that it considers only measurements in the vicinity of the equivalence point, what may result in increasing of errors. In contrast to approximate methods, the numerical model meth- ods are used for the determination of the end-point on the basis of the mathematical model of the titration curve, which presents the interdependence of the volume of the titrant added and the poten- tial of the indicator electrode. The precision of the method depends on the accuracy of model parameters. In most cases, they involve a derived equation for each analyte and titration type considered without the necessity of the presence of an inflection point in the titration curve. Gran’s linearization method is one of the examples known since 1950 [5–11]. Each type of analyte and titration requires 0039-9140/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.talanta.2009.06.014